Low Frequency Axial Fluid Acoustic Modes in a Piping System that
Forms a Continuous Loop
by
Eric R. Marderness
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF SCIENCE in MECHANICAL ENGINEERING
Approved:
_________________________________________
Dr. Ernesto Gutierrez-Miravete, Thesis Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
April 2012
(For Graduation May 2012)
© Copyright 2012
by
Eric R. Marderness
All Rights Reserved
ii
CONTENTS
Low Frequency Axial Fluid Acoustic Modes in a Piping System that Forms a
Continuous Loop ......................................................................................................... 1
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
GLOSSARY OF KEY WORDS ...................................................................................... ix
NOMENCLATURE .......................................................................................................... x
ACKNOWLEDGMENT ................................................................................................ xiii
ABSTRACT ................................................................................................................... xiv
1. Introduction.................................................................................................................. 1
2. Problem Formulation and Description......................................................................... 5
2.1
Fluid Filled Elastic Cylinders ............................................................................ 6
2.2
System Description, Properties and Assumptions ........................................... 14
3. Theory and Methodology .......................................................................................... 20
3.1
Waves and Wave Propagation ......................................................................... 20
3.2
Helmholtz Equation ......................................................................................... 25
3.3
Uniform Loop: Theory ..................................................................................... 32
3.4
Uniform Loop: Transfer Matrix Method.......................................................... 36
3.5
Uniform Loop: COMSOL FEA Models .......................................................... 45
4. Discussion .................................................................................................................. 48
4.1
Uniform Loop: Elasticity Effects ..................................................................... 48
4.2
Non-Uniform Loop .......................................................................................... 52
4.3
4.2.1
Net Change in Phase around the Loop ................................................. 53
4.2.2
Net Change in Volume ......................................................................... 54
4.2.3
Continuity............................................................................................. 56
4.2.4
Acoustic Impedance Changes .............................................................. 57
Non-Uniform Loop: Transfer Matrix Method ................................................. 61
iii
4.4
Non-Uniform Loop: Loop with a Single Cavity .............................................. 64
4.4.1
Phase Changes at Impedance Discontinuities ...................................... 82
4.5
Non-Uniform Loop: Elastic Discontinuity....................................................... 95
4.6
Non-Uniform Loop: Elbows and Pipe Bends .................................................. 98
4.7
Full System .................................................................................................... 100
4.7.1
Summary of Full System Analysis ..................................................... 116
4.7.2
Implications and Physical Interpretation of Axial Modes.................. 117
5. Results and Conclusions .......................................................................................... 119
6. Areas for Future Work ............................................................................................. 122
7. References................................................................................................................ 124
Appendix A – Derivation of Linear Acoustic Wave Equation ....................................... A1
Appendix B – Derivation of Korteweg Lamb Correction ............................................... B1
Appendix C – Transmission Matrix Method ................................................................... C1
Appendix D – TMM Results Figures ............................................................................. D1
Appendix E – COMSOL FE Model Results.................................................................... E1
Appendix F – Additional Related References ................................................................. F1
iv
LIST OF TABLES
Table 1 – Example System Material Properties .............................................................. 16
Table 2 – Example System Component Dimensions ...................................................... 18
Table 3 – Axial Modes; Theoretical, Baseline “Loop” ................................................... 35
Table 4 – Axial Modes; TMM, Baseline “Loop” ............................................................ 40
Table 5 – Axial Modes; COMSOL, Baseline “Loop” ..................................................... 46
Table 6 – Axial Modes; Theoretical, Elastic Baseline “Loop” ....................................... 50
Table 7 – Axial Modes; TMM & COMSOL, Elastic Uniform Loop .............................. 51
Table 8 – Axial Modes; TMM & COMSOL, “Loop” with 8A Cavity ........................... 65
Table 9 – Axial Mode Frequencies (Hz); TMM, Effect of Cavity Cross Sectional Area 92
Table 10 – Axial Modes; TMM & COMSOL, Variation in Cavity Elasticity ................ 97
Table 11 – Axial Modes; TMM & COMSOL, Example System .................................. 101
Table 12 – Changes in Phase; TMM Example Systems, 1 Mode A and B ................. 112
Table 13 – Changes in Phase; TMM Example System RW, 4 Mode A and B ........... 115
v
LIST OF FIGURES
Figure 1 – Schematic of an Example Fluid Filled Piping System with Components that
form a Continuous Loop of Fluid ...................................................................................... 1
Figure 2 – Example System Piping Dimensions ............................................................. 17
Figure 3 – Example System Elbow Dimensions ............................................................. 18
Figure 4 – Example System Arrangement and Dimensions ............................................ 19
Figure 5 – Real Component of Radial Pressure Modes: [n,0,0] ...................................... 28
Figure 6 – Radial Pressure Modes: [n,0,0] ...................................................................... 28
Figure 7 – Real Component of Circumferential Pressure Modes: [0,m,0] ...................... 29
Figure 8 – Circumferential Pressure Modes: [0,m,0] ...................................................... 29
Figure 9 – Real Component of Axial Pressure Modes: [0,0,l] ........................................ 30
Figure 10 – Axial Pressure Modes: [0,0,l] ....................................................................... 30
Figure 11 – Baseline Loop and an “Unwrapped” Baseline Loop .................................... 32
Figure 12 – Baseline Loop, Pressure Modes Shape for 1 Mode A and Mode B........... 35
Figure 13 – Baseline Loop, Displacement Modes Shape for 1 Mode A and Mode B .. 36
Figure 14 – TMM Uniform Pipe Element ....................................................................... 37
Figure 15 – Baseline Loop, TMM, Characteristic Equation and Roots .......................... 41
Figure 16 – Baseline Loop, TMM Pressure Modes Shape for 1 Mode A ..................... 43
Figure 17 – Baseline Loop, TMM Pressure Modes Shape for 1 Mode B ..................... 44
Figure 18 – Baseline Loop, COMSOL FE Models ......................................................... 45
Figure 19 – Baseline Loop, COMSOL Pressure Mode Shapes, Actual Loop ................. 47
Figure 20 – Characteristic Equation and Roots; TMM, Elastic Uniform Loop .............. 50
Figure 21 – Calculated Change in Volume; 1 Mode A, Rigid Wall Uniform Loop ..... 55
Figure 22 – Non-Dimensional Impedance; 1 Mode A, Rigid Wall Uniform Loop ...... 60
Figure 23 – Schematic of Loop System with a Single Cavity ......................................... 64
Figure 24 – Characteristic Equation and Roots; TMM, Loop with 1 Cavity .................. 65
Figure 25 – Characteristic Eq.; TMM, Loop w/ 1 Cavity, Larger Frequency Range ...... 66
Figure 26 – Pressure Mode Shape, Loop with 1 Cavity, 1 Mode A, Surface Plot ........ 68
Figure 27 – Pressure Mode Shape, Loop with 1 Cavity, 1 Mode B, Surface Plot ........ 68
Figure 28 – Pressure Mode Shape, Loop with 1 Cavity, 1 Mode A, COMSOL FE ..... 69
vi
Figure 29 – Pressure Mode Shape, Loop with 1 Cavity, 1 Mode B, COMSOL FE ..... 69
Figure 30 – Pressure Mode Shape, Loop with 1 Cavity, 1 Mode A.............................. 71
Figure 31 – Pressure Mode Shape, Loop with 1 Cavity, 1 Mode B .............................. 71
Figure 32 –Axial Displacement, Loop with 1 Cavity, 1 Mode A ................................. 74
Figure 33 – Axial Displacement,, Loop with 1 Cavity, 1 Mode B................................ 74
Figure 34 – Axial Volume Displacement, Loop with 1 Cavity, 1 Mode A .................. 75
Figure 35 – Axial Volume Displacement,, Loop with 1 Cavity, 1 Mode B .................. 75
Figure 36 – Change in Volume, Loop with 1 Cavity, 1 Mode A .................................. 77
Figure 37 – Change in Volume, Loop with 1 Cavity, 1 Mode B .................................. 77
Figure 38 – Non-Dimensional Impedance, Loop with 1 Cavity, 1 Mode A ................. 79
Figure 39 – Non-Dimensional Impedance, Loop with 1 Cavity, 1 Mode B ................. 79
Figure 40 – Unit Circles for Two Pressure Waves in the Complex Plane, “Mode A”, on
opposite sides of the Impedance Discontinuity ............................................................... 82
Figure 41 – Mode A 1, Pressure and Non-Dimensional Impedance ............................. 87
Figure 42 – Mode A 1, Effective Unit Circle ................................................................ 89
Figure 43 – Unit Circles for Two Pressure Waves in the Complex Plane, “Mode B”, on
opposite sides of the Impedance Discontinuity ............................................................... 90
Figure 44 – Mode B 1, Pressure and Non-Dimensional Impedance ............................. 90
Figure 45 – Mode B 1, Effective Unit Circle ................................................................ 91
Figure 46 – Characteristic Eq.; TMM Variation in Cavity Cross Sectional Area ........... 93
Figure 47 – Roots of Char. Eq.; TMM Variation in Cavity Cross Sectional Area .......... 94
Figure 48 – Characteristic Eq.; TMM, Variation in Cavity Elasticity............................. 96
Figure 49 – Pressure Mode Shape, 1 Mode A; TMM, Variation in Cavity Elasticity .. 98
Figure 50 – Example System with Three Cavities; TMM and COMSOL FEA models 100
Figure 51 – Characteristic Eq.; TMM, Example System, RW, Steel & Aluminum ...... 103
Figure 52 – Characteristic Eq.; TMM, Example System, RW, Steel & Aluminum ...... 103
Figure 53 – Pressure Mode Shape, Example System RW, 1 Mode A, COMSOL FE 105
Figure 54 – Pressure Mode Shape, Example System RW, 1 Mode B, COMSOL FE 105
Figure 55 – Pressure Mode Shape, 1 Mode A TMM, Example System ..................... 106
Figure 56 – Pressure Mode Shape, 1 Mode B; TMM, Example System .................... 106
vii
Figure 57 – Axial Displacement, TMM, Example System RW, 1 Mode A................ 108
Figure 58 – Axial Displacement, TMM, Example System RW, 1 Mode B ................ 108
Figure 59 – Change in Volume, TMM, Example System, RW, 1 Mode A ................ 109
Figure 60 – Change in Volume, TMM, Example System, RW, 1 Mode B ................ 109
Figure 61 – Non-Dimensional Impedance, TMM, Example System, RW, 1 Mode A 110
Figure 62 – Non-Dimensional Impedance, TMM, Example System, RW, 1 Mode B 110
Figure 63 – Pressure Mode Shape; TMM, RW Example System, 4 Mode B ............. 114
Figure 64 – Non-Dimensional Impedance; TMM, RW Example System, 4 Mode B 114
Figure A.1 - Differential Volume Element in Cylindrical Coordinates ......................... A1
Figure B.1 - Fluid Filled Elastic Cylinder ....................................................................... B1
viii
GLOSSARY OF KEY WORDS
Acoustics
Study of sound and vibration in structures and fluids typically
characterized by small displacements
Acoustic Impedance
Relationship between pressure and volume velocity that can be
defined for plane waves in a given fluid based on the density speed
of sound and cross sectional area
Axial Mode
Mode characterized by changes in the phase in the axial direction
Circumfrential Mode
Mode characterized by changes in the phase in the circumferential
direction
COMSOL
Multi-Physics modeling software; www.comsol.com
Dispersion
Frequency and Mode shape dependent speed of sound
Loop Mode
Mode within a continuous fluid loop that follows the centerline of
the loop, also called axial loop mode
MATLAB
Mathematical software; www.mathworks.com
Modal Index
(n, m, l) Subscripts used to denote the component modes in each of
the coordinate system directions that make up a three dimension
mode shape
Mode
System specific pattern of sinusoidal vibration at a single
frequency
Mode Shape
Phase relationship between all points in system for a given mode
that describes the “shape” of the mode
Phase Velocity
Velocity at which a plane or regions of constant phase travels in a
given direction through a system
Radial Mode
Mode characterized by changes in the phase in the radial direction
Speed of Sound
Speed at which sound travels through a given material
System Impedance
Complex relationship between the complex pressure and complex
volume velocity waves in a fluid system
Wave Number
Quantity that gives the change in the phase angle of a wave in
radians per unit distance
ix
NOMENCLATURE
Symbol
Quantity
Units
A, B
Amplitude Coefficients
Pa
Anml
Amplitude Coefficient for mode n,m,l
Pa
Cross Sectional Area
m2
radius
m
Beff
Effective Bulk Modulus
Pa
Bf
Fluid Bulk Modulus
Pa
Bs
Shell Bulk Modulus
Pa
c
Phase Velocity
m/s
cc
Compressional Phase Velocity
m/s
ceff
Effective Phase Velocity
m/s
co
Characteristic Fluid Speed of Sound
m/s
E
Young’s Modulus
Pa
FE
Finite Element
Ax
a
-
f
Frequency
Hz
G
Shear Modulus
PA
g
Gravity
m/s2
h
Thickness
m
ID
Inner Diameter
m

-
Jm
mth order Bessel function of the first kind
-
knml
Acoustic wave number
rad/m
kzl
Axial wave number
rad/m
i
l
Cylindrical modal index, axial direction
-
Lc
Length of cylinder
m
LCL
Center Line Length
m
LSEG
Length of segment
m
x
NOMENCLATURE (Continued)
Symbol
M
Quantity
Units
Pa
NPS
Magnitude
Cylindrical modal index, circumferential
direction
Nominal Pipe Size
n
Cylindrical modal index, radial direction
-
Outer diameter
m
Pressure
Pa
pnml
Pressure of cylindrical mode n, m, l
Pa
PR
Real Pressure
Pa
q
Volume Velocity
m3/s
r
Radial coordinate
m
RE
Elbow bend radius
m
[S]n
TMM Matrix for Segment n
-
Element i,j of TMM Segment Matrix
-
Transfer Matrix Method
-
Time
s
TMM System Matrix
-
Tij
Element i,j of TMM System Matrix
-

u

v
Displacement Vector
m
Velocity Vector
m/s
Vf
Fluid Volume
m3
vr
Radial velocity
m/s
v
Circumferential velocity
m/s
vz
Axial velocity
m/s
z
Axial coordinate
Zp
Acoustic impedance
kg/(m4s)
Zsys
System impedance
kg/(m4s)
m
OD
p
Sij
TMM
t
[T]sys
-
m
xi
NOMENCLATURE (Continued)
Symbol
n
sys
Quantity
Phase angle on the complex pressure plane
Non-Dimensional System Impedance
Units
radians
-
E
Elbow bend angle

Hoop Strain
m

Circumferential coordinate
m

Wavelength
m

Poison’s Ratio
-

Density
kg/m3
o
Nominal Density
kg/m3

Hoop Stress
Pa

Shear stress
Pa

Phase Angle
radians
radians
(P)
Pressure Mode Shape
-
(X)
Displacement Mode Shape
-

Angular frequency

Laplacian
rad/s
-
xii
ACKNOWLEDGMENT
Dr. Ernesto Gutierrez-Miravete
Mr. Nathan Henry
Mr. Chris Hoddinott
Ms. Shari Sanger
xiii
ABSTRACT
This thesis discusses the frequency and mode shapes of axial fluid resonances
within a system of piping and components that form a continuous loop. The analysis
focuses on the low frequency axial modes within the fluid loop. The fluid loop provides
a continuous and finite axial dimension that allows for modes to occur in the axial
direction. The frequency at which the axial modes occur is a function of the phase
velocities and changes in impedance within the fluid loop. The analysis is restricted to
axial modes of the lowest order axially symmetric radial mode or “plane waves”.
Acoustic equations are developed to treat the fluid loop as a special case of a cylindrical
Helmholtz resonator. The ends of a straight finite circular cylindrical fluid column are
set equal to each other to model a fluid “loop” with an infinite radius of curvature. The
Transfer Matrix Method (TMM) and finite element models in COMSOL are
implemented to investigate how local differences in elasticity and impedance influence
the frequencies and mode shapes of the axial fluid loop modes. It is shown that for a
fluid filled loop of uniform cross section the axial modes occur in pairs at frequencies
corresponding to integer multiples of whole wave lengths. The pairs of axial modes in
non-uniform loops are shown to shift independently resulting in two axial modes of the
same “wavelength” occurring at different frequencies within the same physical system.
The modes shapes are shown to have an uneven axial distribution in phase resulting in
an apparent “kink” in the mode shape. The independent shifting of the modes is shown
to be modally specific and related to the local phase velocities and impedances that
result from the dimensions and materials of the system. The “kinks” in the modes shapes
are explained and it is shown that the pressure and displacement mode shapes maintain
the necessary continuity requirements. The results from theory, previous works on radial
and circumferential modes in fluid filled elastic cylinders, and the TMM models are
compared to the eigenvalue analysis results from finite element models developed in
COMSOL. The finite element model results for the axial loop modes are verified to be
reasonable. Some physical interpretations of the results are provided.
xiv
1. Introduction
This thesis presents an analysis of the low order axial fluid acoustic modes within a
system of piping and components that form a continuous loop. The “loop” arrangement
of the system allows for axial modes to occur in the fluid column that follow the
centerline of the loop and wrap around the entire system form a continuous loop or loop
mode. This analysis focuses on predicting the frequency and mode shape of the system
wide axial modes and how changes to the system parameters affect the frequencies and
modes shapes of those modes. Figure 1 shows an Example piping system arranged in a
“loop” that includes three cylindrical cavities representing generic tanks or components.
Component #1
Component #2
Continuous
loop of fluid
Component #3
Figure 1 – Schematic of an Example Fluid Filled Piping System with Components
that form a Continuous Loop of Fluid
In its simplest form a piping system is a series of inter-connected cylindrical shells
intended to transport a fluid from one location to another. The fluid is used to transfer
heat, mass, pressure or mechanical energy between the connected locations. In some
cases the fluid is the quantity that is being transferred between locations such as in oil
pipelines or water distribution systems. In other cases the fluid is only the transport
medium being used to transfer thermal or mechanical energy between two points such as
the fluids in cooling or hydraulics systems.
1
Two of the most common examples of an industrial fluid system that involves a
loop are hydraulic and heating/coolant systems. These systems consist of varying lengths
of pipes and hoses connecting multiple components together. The fluid passes from a
starting point along a supply path to a location where the quantity within the fluid is
transferred into another process through a component such as a heat exchanger, separator
or hydraulic actuator. Once the quantity of interest has been transferred, the fluid is
recirculated through a return leg to the original location forming a “continuous” loop of
fluid. The original location may also be a component that adds a desired quantity to the
fluid such as a boiler, pump or reservoir. These types of industrial systems exist in a
wide range of sizes from commercial oil and power plants to cars and hand tools.
Another system that is very similar to the fluid system schematic shown in Figure 1
is a forced flow test loop, similar to the example provided by Axisa and Antunes in their
book on Fluid Structure Interaction [1]. A forced flow test loop consists of a section of
pipe located between two large cavities to isolate that section of the loop from the noise
generated elsewhere in the loop by the pump that generates the flow.
This type of system can be used to test components and component flow related
noise. However if the test loop has resonances based on the loop arrangement, those
resonances can interfere with the desired test results. One type of resonance that can
occur within the test loop will occur at frequencies coincident with the frequencies of the
axial loop modes
The analysis of axial fluid resonances within a system loop of piping is an acoustic
problem which includes information from a variety of subjects. These resonances are
sometimes simply referred to as “acoustic” resonances as opposed to structural
resonances, even though the fluid and structural resonances are not actually independent
of each other. Several acoustic specific concepts are required to provide a detailed
discussion of the frequency and mode shapes of the axial resonances within the fluid and
put the analysis into context relative to existing literature on fluid filled elastic cylinders.
These topics will be introduced in the following sections with several examples that will
help to conceptualize the resonances within the fluid loop. More detailed derivations
related to the key pieces of acoustic theory are provided in Appendix A and B.
2
Fluid resonances can detrimentally impact the operation of fluid systems and
components. The unwanted impacts of the fluid resonances include increased system
noise, excessive component fatigue, interference with test measurements and monitoring
instrumentation, improper system and potentially system or component failure. When
the piping system is elastic and contains a dense fluid (such as commonly used steel or
aluminum pipes containing water) the fluid-structure interaction (FSI) between the
piping structure and the enclosed fluid can significantly alter the acoustic response of
both the fluid and structural portions of the system making it necessary to consider FSI
during the system design as discussed in [2], [3], [4], [5] and [6]. How the structure of
the piping (or components) and the internal fluid interact depends on many system
specific factors including the properties of the structural materials, the properties of the
internal fluid, the geometry of the system, and the frequency range of interest.
For as common as fluid filled piping systems are in industry, with applications
ranging from extremely large and complex to very small and simple, calculating the
fluid structure interaction of the piping system is anything but simple. This can be
observed from the large body of literature available on cylindrical shells, fluid filled
elastic shells, fluid structure coupling, transient analysis of piping systems, and the
numerous analysis and modeling techniques devised to investigate the FSI of specific
arrangements and problems. References [7], [8], [9], [10] and [11] provide a sample of
articles focused on the study of FSI in piping systems. A more diverse list of FSI related
references, consulted during the research for this paper, is provided in Appendix F.
Modern finite element (FE) software makes it easy to develop models of very
complicated systems and structures. The analysis of FE models however is only as good
as the assumptions used to generate and analyze the model. As computational
capabilities have evolved, FE models have also evolved to include more detail and
capabilities such as calculating the fluid structure interactions of piping systems. These
capabilities can provide insight into many problems that were previously impractical to
evaluate before actually building and testing a prototype of the system. However when a
complicated FE model produces results that do not intuitive make sense it becomes
necessary to simplify the model and establish a realistic set of physics based
expectations to compare with the model results.
3
This analysis was initiated to better understand the results from a finite element
model of the example fluid filled piping system with a continuous loop of fluid shown in
Figure 1. The example system used in this paper has been modified to simplify the
problem and was arbitrarily sized to avoid resemblance to any specific system.
A finite element model was generated for the example system shown in Figure 1.
The frequencies and mode shapes of the pressure modes were calculated with an
eigenvalue analysis of the model. The results showed a series of axial fluid loop modes
that appeared to occur primarily within the fluid loop. The observed axial fluid loop
modes did not agree with the brief interpretation of the low frequency modes of the
similar loop system found in [1].
The loop resonances were observed to occur in multiples of whole wave lengths and
were not spaced consistently or evenly in frequency. The axial fluid modes were
calculated to exist much lower in frequency than anticipated based on the centerline
length of the system and the fluid material properties. The pressure modes shapes
revealed that there were two modes for each full wavelength multiple. Each full wave
length mode of the mode pair occurred at different individual frequencies, even though
they were calculated from the same model with the same material properties. The mode
pairs were also not consistently or harmonically spaced in frequency. The pressure
modes in general also appeared to be non-symmetric with “kinks” in the modes shapes
and different distances between nodes and anti nodes within the same modes shape.
The purpose of this analysis is to use acoustic theory, published results from similar
studies and simple numerical models to establish a set of realistic expectations for the
frequencies and mode shapes of the axial fluid loop resonances. The system loop
provides a finite length of pipe that can be used to study the behavior of axial modes
within the fluid column. The frequency and shape of the axial fluid acoustic resonances
for the lowest order axially symmetric radial mode, also called “plane waves”, will be
determined using analytical and 1D numerical formulation. These physics based
expectations will also be compared to the results from other related studies on fluid filled
elastic shells and to results from a representative finite element model to show that the
FE model results are reasonable.
4
2. Problem Formulation and Description
In general there were very few fluid loop examples found in literature. The force
flow fluid test loop example in [1] provided an example. However, the fluid loop
example was used to demonstrate how a system could be arranged with two large
cavities to reduce the noise over the small test length between the cavities. This example
was focused exclusively on the attenuation of the sound amplitude over a given
frequency range for a forced frequency analysis and did not provide a modal analysis of
the fluid loop modes. The example noted that the isolation tanks added low frequency
resonances to the system and that these resonances were interpreted as Helmholtz
resonances between the two tanks. This is not consistent with the initial observations for
the pressure modes calculated for the example system shown in Figure 1.
The forced frequency example is similar to many of the examples in the large body
of literature on mufflers such as [12] and [13]. Much of the muffler literature however is
focused on air or gas filled piping and the attenuation provided across a specific muffler
design. No relevant muffler examples were found with a loop system arrangement. The
example provided in [1] and other examples for mufflers in straight pipes in [12] and
[13] show that the attenuation provided by a muffler like device is frequency dependent.
These examples provide some insight into the potential for the axial loop modes in a
system with components to have regions within the system that have different
amplitudes. These differences in amplitude may support some of the non-symmetries
observed in the initial model results.
Mufflers provide a change in the acoustic impedance to the sound traveling along a
pipe. Impedance, as an acoustic quantity, relates the pressure to the volume velocity at
given location within the fluid column. The change in impedance causes part of the
acoustic pressure wave to be reflected back towards the source of the sound, while the
remainder of the wave is transmitted through the muffler away from the source [1] [12]
[13]. The muffler essentially establishes an elastic boundary condition on the end of the
fluid column where the pipe meets the muffler. The fluid cavities in the example system
are similar in that they will also result in elastic boundary conditions in the axial
direction of the fluid loop.
5
Research into propagation of radial and circumferential modes along an infinite
fluid filled elastic shell provides several similarities and insights into the study of the
axial fluid resonances within a piping loop.
The speeds at which radial and
circumferential modes propagate in the axial direction, called the phase velocity, become
a function of frequency and mode shape causing different modes to travel along the
same system at different rates. This frequency dependent “speed of sound” for specific
modes is referred to as “dispersion” in the field of acoustics.
The changes in phase velocity as a function of frequency are unique for each mode
and occur due to the interaction of the fluid column and the solid elastic shell. Basically
the shell provides complex elastic boundary conditions at the cylindrical wall which in
turn alters the speed at which the sound propagates in the axial direction. A frequency
dependent speed of sound could result in standing waves within the same physical
system occurring at non-harmonically spaced frequencies.
There is a large body of literature on fluid filled elastic shells. Studying how a wave
travels along an infinite circular cylindrical elastic shell filled with a fluid provides
useful insight into how specific modes will propagate along a finite length piping system
and how those modes may form standing waves in the axial direction. If the elastic
boundary conditions on the cylindrical wall can affect the axial phase velocity of the
radial and circumferential modes then it may be possible that under certain conditions
the axial modes may also show “dispersive like” behavior. Modally specific behavior
due to elastic boundary conditions on the fluid column may help to explain, in part, the
axial fluid resonances within the fluid occurring at frequencies much lower than
expected and not occurring in any obvious pattern. The following section summarizes
some of the investigations into fluid filled elastic cylinders.
2.1 Fluid Filled Elastic Cylinders
Waves traveling along a fluid filled elastic shell have been studied extensively over
the last century. Investigations and research into fluid filled elastic shells began more
than a century ago with works by Lord Rayleigh [14] and Horace Lamb [15] originally
published in the late 1800’s and early 1900’s. Much of the early acoustic work was
6
focused on airborne noise or gases contained within a tube, such as an organ pipe and
woodwind instruments as discussed by P. Morse [16]. There was an increase in the
studies of fluid filled elastic pipes filled with water in the early to mid twentieth century,
when research into underwater acoustics was accelerated by the increased use of
submarines during World War I and II.
The study of sound propagation in a fluid filled elastic pipe received attention from
Fay et al. [17] as part of the development of fluid filled impedance tubes. An impedance
tube is used to measure the acoustic properties of a submerged material. A sample of the
material is placed at one end of the impedance tube. A known sound source is placed at
the other end of the tube. The impedance of the submerged material sample is
determined by measuring the amount of sound that is reflected from the sample. Initially
impedance tubes were constructed from metal tubes and filled with air. The large
differences in stiffness and density between the tube walls and the internal air, allowed
the air to be treated like it was enclosed within a perfectly rigid tube. Assuming a “rigid
wall” condition greatly simplified the mathematics and measurements of the material
being tested. In [17], Fay et al. investigated and tested water filled impedance tubes to
measure the acoustic performance of submerged materials. The reflected sound from a
material sample was studied by measuring the interference patterns between the incident
wave and the reflected wave inside of the tube. It was found that, while the metal tube
could be considered rigid (and neglected) for an air filled impedance tube, it could not be
considered perfectly rigid for a fluid filled tube. The elasticity of the tube walls had to be
included in the analysis.
The metal impedance tube acted as an elastic boundary for the column of water. The
non-rigid boundary condition of the cylindrical wall resulted in changes to the velocity at
which sound propagated along the fluid in the axial direction. The axial phase velocity
along the tube was found to be less than the characteristic speed of sound within a free
field of the same fluid. The test results presented in [17] were in good agreement with
predicted phase velocities over the tested range of 1 kHz to 3.5 kHz.
In [18] Jacobi discussed the propagation of axisymmetric waves along a fluid
column with various boundary conditions. Some of the boundary conditions investigated
were perfectly rigid cylindrical walls, pressure release walls, and thin solid “elastic”
7
walls. It was also recognized in [18] that the compliance of the elastic cylindrical walls
must be included when the internal fluid is a liquid. Jacobi studied the lowest order
axisymmetric radial fluid modes over a wide frequency range and identified that, with
the exception of the 0th order mode, all of the higher order modes have cut off
frequencies. The higher order modes were shown to only exist above their cut off
frequencies. Jacobi included the interaction of the surrounding elastic shell as an applied
boundary condition to the linear lossless wave equation representing the fluid column.
Only the approximate “normal” influence of the cylindrical shell was included in
Jacobi’s study [18]. Poisson’s ratio and the effects of flexural and axial waves in the
solid shell were also neglected.
Thompson provided a discussion of sound propagating along infinite liquid filled
elastic tubes in [19] which also focused on low order axially symmetric radial fluid
modes. Thompson included a derivation of cylindrical thin shell equations for use with
calculating the phase velocities within the fluid. The shell equations that were used by
Thompson included Poisson’s ratio but ignored the rotary inertia and transverse shear
terms. It was found in [19] that for a given frequency only a finite number of axially
symmetric modes exist within the fluid. The 0th order or “plane wave” mode was shown
to exist at all frequencies and propagate at a phase velocity less than the fluid’s
characteristic speed of sound. The higher order modes where shown to exist only above
their cut off frequencies and propagate at phase velocities above characteristic fluid
speed of sound. (A review of the cylindrical modes and mode shapes is provided in
Section 3.2 and Figure 5 - Figure 10.)
Thompson provided results for the phase velocities of the first three axially
symmetric radial modes in a non-dimensional form plotted against normalized wave
numbers. The phase velocities were normalized to the fluid’s characteristic speed of
sound. Phase speeds less than unity represented modes traveling slower than the
characteristic speed of sound in the fluid while phase speeds above unity were traveling
faster than the characteristic fluid speed of sound. This type of curve is referred to as a
“dispersion curve” in the field of acoustics and is used to present the phase velocity for a
specific mode shape over a range of wave numbers. The dispersion curve presented in
[19] shows that the phase velocity of each mode shape is frequency dependent at lower
8
to mid wave numbers and then converges to a single value at higher wave numbers. The
phase velocity of the lowest order mode was shown to be less than the characteristic
speed of sound, in agreement with the results reported in [17] and [18], and only
exhibits very small changes in phase velocity at very low frequencies or wave numbers.
In the same work Thompson also studied the effects of including viscosity in the
fluid formulation to investigate attenuation as well as phase speed. He found that the
attenuation from viscosity had less of an effect on the 1st axially symmetric radial fluid
mode than the 0th and 2nd modes indicating that attenuation and phase velocity are
frequency dependent and unique to each mode.
Lin and Morgan [20] presented an analysis of axisymmetric waves propagating
through an inviscid compressible fluid in an infinite elastic shell. The material properties
and dimensions used were the same as those used by Thompson in [19] to allow a direct
comparison of the results. The effects of four non-dimensional parameters on the phase
velocities of the axisymmetric waves were studied including; the ratio of the fluid speed
of sound to the compressional speed of sound in the shell (cf/cp), the ratio of the fluid
density times the fluid radius to the density of the shell times thickness of the shell
(fa/ph), the ratios of the shell thickness to the radius of the fluid (h/a) and Poisson’s
Ratio (. Unlike the previous references the elastic shell formulation in [20] included
the effects of rotary inertia and transverse shear. Several limiting cases were also studied
including a perfectly rigid tube and infinitely flexible tube.
Contrary to results in [18] and [19], Lin and Morgan showed that there are actually
two axisymmetric radial modes which exist at all frequencies, the 0th and 1st modes. The
0th mode was shown to have a phase velocity that is less than the characteristic fluid
speed of sound, in agreement with the results in [18] and [19]. The 1st mode has a phase
velocity greater than characteristic fluid speed of sound at low frequencies and then
converges to the characteristic speed of sound at high frequencies. The higher order
modes were shown to exhibit a cut off frequency behavior as observed by previous
authors however the cut off frequencies were shown to be considerably lower in
frequency than previously reported by Thompson in [19]. The cut off frequencies were
found to be calculated correctly however because only a limited range of phase
velocities was included in [19] the cut off frequencies were associated the wrong modes.
9
Lin and Morgan presented dispersion curves similar to the one provided in [19] however
they included a much larger range of phase velocities to demonstrate the character in the
dispersion behavior of the 1st mode not captured in [19] and to illustrate the differences
in cut of frequencies. It also should be noted that the phase velocities presented in [20]
were normalized to the compressional sound speed in a solid bar of the elastic shell
material instead of the characteristic speed of sound in the fluid which was used for
normalization in [19]. While the dimensions and material properties in [20] where
chosen to match those in [19] the difference in the normalization value makes visual
comparison of the plotted results in [19] and [20] more difficult.
The influence of rotary inertia term for the cylindrical shell was found to have very
little effect on the modal phase velocities even at higher frequencies. The transverse
shear term was found to only affect the 0th mode by further reducing the phase speed
most noticeably for wave numbers above 1. The magnitudes of the four non-dimensional
parameters mentioned above were shown to affect the shape of the dispersion curves
changing the phase speed at which specific modes travel along the fluid filled elastic
cylinder.
Fuller and Fahy in [21] provided an analysis of the dispersion behavior of waves
and the energy distribution in an infinite fluid filled cylindrical elastic shell. In contrast
to the work described in [17] - [20] which focused on axially symmetric radial modes,
Fuller and Fahy focused on the circumferential modes propagating in the cylindrical
elastic shell and did not restrict their analysis to axisymmetric modes. The DonnellMushtari thin shell equations were used to describe the cylindrical elastic shell similar to
the discussions of harmonic motion for an infinite cylinder found in Junger and Fiet’s
Sound, Structures and Interactions [22]. The pressure field within the fluid was assumed
to only influence the radial motions of the shell. The influence of the internal fluid was
reduced to a fluid loading term which was combined with the shell equation that
describes the radial motion of cylinder.
Fuller and Fahy discussed real, imaginary and complex branches to the dispersion
curves of the modes propagating in the shell. It was shown in [21] that there are eight
propagating waves in the fluid loaded shell, representing four unique waves traveling in
opposite directions along the infinite cylinder. Dispersion curves were presented for the
10
shell waves; however [21] plots radial wave numbers versus non-dimensional frequency
instead of phase speed. The waves described in [21] include both propagating waves as
well as evanescent waves, which represent near field waves that decay quickly instead of
propagating. Similar to [20] the behavior of the waves in the shell was found to be
strongly influenced by the ratio of fluid to structure density and the ratio of the fluid
radius to the shell wall thickness. Fuller and Fahy provide a detailed discussion of the
wave types for each branch of the dispersion curve; however this approach can be
somewhat difficult to intuitively apply to the evaluation of a physical piping system.
While this analysis did identify branches that behaved as “fluid like” waves, treating the
fluid as loading term on the radial displacement of the shell wall focuses the analysis
heavily on the waves in the shell while ignoring the modes that are predominantly within
the fluid column, such as the axial fluid loop modes that are the focus of this study.
DeJong noted in [23] that the Donnell-Mushtari thin shell equations used in [21] and
[22] result in cut off frequencies that are 30% too low for the first and second
circumferential order modes of the shell. This result is similar to the results presented in
[18] – [20] which showed that the dispersion curves showed slight variations as
additional shell equation terms were included in the formulation by successive authors.
DelGrosso [24] provided a formulation of the fluid and shell interactions using the
exact equations for the longitudinal and shear waves. The equations for the radial and
axial displacement of the fluid and the cylinder explicitly include the inner and outer
radius of the elastic cylinder and do not use a thin shell approximation as was used in
[17] through [21]. These equations are significantly more complicated than the
formulations provided in the previous references; however DelGrosso’s formulation
allows the dispersion of propagating waves within both the shell and the fluid to be
studied. DelGrosso’ analysis was focused on axisymmetric waves in a lossless fluid and
structure similar to studies presented in [17] - [21]. DelGrosso found that a summation of
the fluid modes, at the phase speeds calculated using this method, was adequate to
capture the sound field from a piston drive. This result was motivated by a desire to
correct the observed discrepancies between predictions and measurements of the sound
field within a water filled impedance tube, particularly in the ultrasonic frequency range.
11
Lafleur and Shields used DelGrosso’s formulation to study low frequency
propagating modes in liquid filled elastic tubes in [25]. They evaluated the phase
velocities and radial particle displacements within the fluid and elastic wall for the 0th
and 1st axisymmetric modes. They showed that for certain combinations of fluid and
solid material properties that the displacement profile of the 1st mode could be more
“plane wave like” than the 0th mode. The modes were found to become plane waves,
with most of the fluid displacement in the axial direction, as the phase velocity of the
mode approached the characteristic speed of sound within the fluid regardless of the
radial mode number. Results for an aluminum/water and PVC/water wave guide are
provided along with experimental phase velocity measurements for each wave guide.
The measured phase velocities were in good agreement with the predicted phase speeds
over the tested frequency range of approximately 8 kHz to 20 kHz.
In a more recent paper [26] Baik et al. adapted DelGrosso’s formulation into a
complex formulation to predict attenuation as well as phase speed. The real portion of
the solution was used to predict phase speed while the imaginary portion of the solution
gave the attenuation of the propagating waves. The focus of the study was on mercury
filled pipes. However due to safety issues a mercury filled piping system could not be
tested so the authors chose to validate their prediction using a different system with
similar ratios between the fluid and structural material properties. Similarly dispersive
phase speed results were calculated for the first several axisymmetric modes as in [20] [25]. It was demonstrated in [26] that in the inviscid case the extension of DelGrosso’s
formulations by Baik et al. reduces to the original equations given in [24] and [25]. It
must also be noted that in a previous work where the authors re-derived DelGrosso’s
original formulation [27], a detailed discussion was provided of several typographical
errors that occurred in the equations given in [25] and in [9].
Similar to the works cited in [17] and [25] the testing performed by Baik et al. was
concentrated in the high frequency range, where the chosen system dimensions resulted
in considerable changes in the model phase velocity. It was shown in [26] that the
measured phase velocities between approximately 50 kHz and 400 kHz showed good
agreement with the predicted values. It was also found that the measured attenuation was
slightly larger than predicted by the complex extension of Delgrosso’s equations. The 0 th
12
mode, or plane wave, was not observed during testing as the sound source used was not
capable of exciting this particular mode shape. The investigations provided in [26] and
[27] provide a good example of applying the existing studies of fluid filled elastic
cylinders to investigate an actual system arrangement too difficult to prototype and test.
Summarizing, references [17] through [27] use a variety of approximations and
exact approaches to study the phase speed of radial and circumferential modes in fluid
filled cylinders. References [17] through [20] approximated the elastic shell as boundary
conditions on the fluid equations while [21] and [22] approximated the fluid loading as
boundary conditions on the structural equations. References [24] through [27] use a
more exact method to calculate response of both the fluid and structure.
The dispersion results presented in [17] through [20] and [24] through [27] agree
that at low frequencies there exists a 0th order axially symmetric radial mode with a
phase velocity that is less than the characteristic fluid speed of sound. This 0th order
mode, or “plane wave” mode, was shown to vary in phase velocity over a large range of
wave numbers or frequencies and then converge to a single velocity at high frequencies.
As discussed in [19], [20] and [24] - [27] both the 0th order and 1st order axially
symmetric radial modes exist at all frequencies while the higher order modes only exist
above a mode specific cutoff frequency. The 1st order and higher order modes showed
more extreme changes in phase velocity than the 0th order mode over the range of
frequencies where the higher order modes exist.
The dispersion results indicate that the radial and circumferential modes travel along
a fluid filled elastic cylinder at different velocities. These phase velocities are unique to
each type of mode, a function of frequency and dependent on the geometry and materials
of the fluid filled elastic cylinder.
As previously noted the analyses in [17]-[22] and [24]-[27] assume that the fluid
filled cylinders are infinite and utilize modal solutions to the linear wave equation that
include traveling waves in the axial direction. Because only axially traveling waves were
assumed there were no axial modes or resonances in these analyses.
If the length of the fluid filled cylinder is finite then the boundary conditions at the
ends of the elastic cylinder and the fluid column will result in axial modes. The axial
modes or standing waves will be made up of traveling waves moving in opposite
13
directions around the loop and traveling at frequency and modally dependent phase
velocities. The axial loop resonances will be related to all of the local phase speeds and
boundary conditions in the system. Additional system parameters, such as sudden
impedance changes, may also result in changes to the resonant frequencies.
The loop of fluid filled elastic pipe studied in this analysis establishes a finite length
of pipe so modal solutions to the linear acoustic wave equation with standing waves in
the axial direction will be used. The axial modes will have radial and circumferential
components that are similar to the radial and circumferential modes discussed in [17][27]. The analysis presented in this paper focuses on the axial fluid resonances with 0th
order radial and circumferential modal components. The frequencies and mode shapes of
the axial fluid loop resonances will be investigate using acoustic theory, the Transfer
Matrix Method implemented in MATLAB and simple finite element models made in
COMSOL.
2.2 System Description, Properties and Assumptions
The example system shown in Figure 1 was specifically developed to support the
present study of axial loop resonances. As previously mentioned, it was designed to be
similar to the acoustic forced flow test loop discussed briefly as an example in [1]. The
system is assumed to be a closed system, in-vacuo and will be analyzed in a free-free
boundary condition. Assuming the loop is in a free-free boundary condition allows the
analysis to neglect body forces, neglect the effects of pipe or component foundations and
assume uniform conditions in the radial direction of the pipe. The structures of the
piping and component walls are assumed to be either rigid or linearly elastic. The
internal fluid is a liquid and assumed to be compressible, lossless, inviscid, irrotational
and free from bubbles or dissolved particulates. The fluid will also be assumed to be at
rest and at a uniform temperature and pressure.
These assumptions are generally in agreement with the analyses discussed in
references [17] through [27], except where the previous authors specifically evaluated
changing one of these assumptions such as Thompson did by adding viscosity in [19].
Unlike the analyses discussed in [17] through [27] the “loop” system does not have an
14
infinite axial length. The finite axial length of the system loop is what allows the
existence of fluid resonances in the axial direction.
The analysis will be restricted to low frequencies where the wave lengths are very
long relative to radial dimensions of the system. Limiting the analysis to low frequencies
also restricts the analysis to very low wave numbers where at most only 2 axially
symmetric radial modes will exist in an elastic fluid filled cylinder and only the 0th order
mode exists in a rigid wall fluid filled cylinder as discussed in [20]. The axial loop
resonances investigated are restricted to resonances of the lowest order axially
symmetric radial mode or plane waves.
These restrictions are different from the analyses presented in [17] through [27]
where previous authors have been more focused on ultrasonic frequencies. The low
frequency restriction is based on prior knowledge of the example system and the
previously stated objective of this analysis to establish and understanding of the results
generated in the FE model. In general, low frequency analysis is also more conducive to
comparisons with FE models.
The materials of the example system are steel and water. The material properties for
steel and water are from [28] and are consistent with the material properties used in [25].
Several other material combinations were used in the analyses presented in [17] through
[27] including brass and water, aluminum and water, and mercury and steel. Steel and
water were chosen for this example to provide a more direct comparison to numerous
steel and water piping systems found in industry. The properties for Aluminum are also
included for an elasticity comparison later in this document. The material properties of
the example system are given in Table 1.
15
Metric
English
Material
Property
Symbol
Value
Units
Value
Units
Steel
Young’s Modulus
E
1.95E+11
Pa
2.8282E+7
psi
Shear Modulus
G
8.30E+10
Pa
1.2038E+7
psi
Poisson’s Ratio

0.28
-
0.28
-
Density

7700
kg/m3
2.7818E-1
lb/in3
Speed of sound
cc
6100
m/s
2.4016E+5
in/s
Young’s Modulus
E
7.1E+10
Pa
1.0298E+7
psi
Shear Modulus
G
2.4E+10
Pa
3.4809E+6
psi
Poisson’s Ratio

0.33
-
0.33
-
Density

2700
kg/m3
9.7543E-2
lb/in3
Speed of sound
cc
6300
m/s
2.4803E+5
in/s
Density

998
kg/m3
3.6055E-2
lb/in3
Speed of Sound
co
1481
m/s
58307.1
in/s
Bulk Modulus
Bf
2.18E+9
Pa
3.1618E+5
psi
Aluminu
m
Water*
* Water is at 20 oC and 1 atm
Table 1 – Example System Material Properties
The system was sized to have a nominal centerline length equivalent to the length of
1 full wave at 40 Hz using the characteristic speed of sound for water listed in Table 1.
As shown below the system length was calculated to be 37.025 m.
L sys  n    n
co
1481m / s
 1 
 37.025m
f
40Hz
16
(2.4)
The piping of the example system is constructed from Schedule 80 pipe of Nominal
Pipe Size (NPS) 10. The elbows are assumed to be long radius elbows of the same inner
diameter, outer diameter and thickness as the straight piping. The cross sectional
dimensions of the straight sections of pipe are shown in Figure 2 and the dimensions of
the elbows are shown in Figure 3. The piping dimensions for schedule 80 NPS 10 piping
were obtained from [29]. The schedule 80, NPS 10 piping was chosen because it
provides a thickness to radius ratio of 0.124 which is approximately the same as the 1/8
ratio used in [19] and [20].
h
OD
ID
a
Pipe
Metric
units
English
units
-
80
-
80
-
Nominal Pipe Size
NPS
10
-
10
-
Outer Diameter
OD
27.305
cm
10.75
in
Inner Diameter
ID
24.29256
cm
9.564
in
Thickness
h
1.50622
cm
0.593
in
Radius
a
12.14628
cm
4.782
in
Schedule
Figure 2 – Example System Piping Dimensions
17
OD
ID
E
h
RE
Long Radius Elbow
Metric
units
English
units
Outer Diameter
OD
27.305
cm
10.75
in
Inner Diameter
ID
24.29256
cm
9.564
in
Thickness
h
1.50622
cm
0.593
in
Bend Radius
RE
38.1
cm
(1.5*NPS) = 15
in
Bend Angle
E
/2
rad
90
deg
Center Line Length
LCL
59.84723
cm
23.5619
in
Figure 3 – Example System Elbow Dimensions
The components are assumed to be cylindrical and have the same wall thickness as
the Schedule 80 NPS 10 piping. Table 2 provides the inner diameter for the three
components shown in Figure 4.
-
Metric
Component 1
ID
Component 2
Component 3
Inner Diameter
units
English
units
68.70974
cm
27.05108
in
ID
68.70974
cm
27.05108
in
ID
34.35486
cm
13.52554
in
Table 2 – Example System Component Dimensions
Figure 4 provides a dimensioned sketch of the example system developed for this
analysis. Some of the system dimensions may be modified later to illustrate specific
findings or observations. The total system length will however remain constant
throughout this study to allow for a consistent point of comparison.
18
L1
EL4
EL1
L7
L2
C1
C2
L6
L3
EL3
EL2
L5
C3
L4
Segment
#
Metric
Units
English
Units
Length 1
L1
865.7777
cm
2199.075
in
Elbow 1
EL1
59.84723
cm
23.5619
in
Length 2
L2
166.44
cm
422.7576
in
Component 2
C2
100
cm
254
in
Length 3
L3
599.3377
cm
1522.318
in
Elbow 2
EL2
59.84723
cm
23.5619
in
Length 4
L4
416.8885
cm
599.3377
in
Component 3
C3
33.0
cm
0.8382
in
Length 5
L5
416.8885
cm
599.3377
in
Elbow 3
EL3
59.84723
cm
23.5619
in
Length 6
L6
599.3377
cm
1522.318
in
Component 1
C1
100
cm
254
in
Length 7
L7
166.44
cm
422.7576
in
Elbow 4
EL4
59.84723
cm
23.5619
in
Figure 4 – Example System Arrangement and Dimensions
19
3. Theory and Methodology
This section provides a review of acoustic theory as it relates to resonances within a
cylindrical fluid column. Modal solutions of the cylindrical Helmholtz equation, various
boundary conditions, changes in impedance and phase velocity are also discussed. The
theory and modal solutions are briefly compared to the previously cited references on
elastic fluid filled cylinders. The Transfer Matrix Method (TMM) is introduced to
provide a numeric means of evaluating the axial fluid resonances. Acoustic theory and
the TMM are then used to evaluate the axial loop resonances using the dimensions and
material properties defined in the previous section. The results from theoretical and
numerical examples are compared to results from representative FE models built using
COMSOL.
3.1 Waves and Wave Propagation
Waves are found in many fields of science and engineering. The field of study that
focuses on how vibrational waves in fluid structure systems propagate and interact is
known as Structural Acoustics, Vibroacoustics or simply Fluid Structure Interaction
(FSI). The study of FSI spans a wide range of practical engineering problems including,
but not limited to; sound radiation of submerged objects, architectural and room
acoustics, environmental acoustics, ultrasonics, and fluid filled piping systems. All of
these FSI applications study how mechanical waves are transmitted through various
systems which include both fluid and structural components.
This section provides a brief discussion of acoustic waves as well as several
simplistic examples to relate the axial fluid loop resonances to the cylindrical waves
discussed in the references cited in Section 2.1.1. More thorough and detailed
introductions to wave mechanics and the propagation of waves in fluids and structures
can be found in [1], [16], [22], [28], [30], and [31].
The displacements due to acoustic vibrations are typically assumed to be small
relative to the dimensions of the system. The wave motion of both linear elastic
structures and compressible fluids can be described using the linear wave equation. The
linear lossless wave equation for longitudinal or compressional waves in linear elastic
20
solids is shown in (3.1), where 2 is the Laplacian operator, ū is a displacement vector,
cc is the compressional phase velocity and t is time.

  1   2u
 u   2  2  0
 c c  t
2
(3.1)
The linear wave equation for compressible fluids, also known as the linear
“acoustic” wave equation, is shown in (3.2), where p is pressure, co is the phase velocity
and t is time.
 1  2 p
 p   2  2  0
 co  t
2
(3.2)
The linear acoustic wave equation is derived from the equation of continuity and the
equation of motion [32]. A detailed derivation of the linear lossless acoustic wave
equation in cylindrical coordinates is provided in Appendix A.
The wave equation describes changes within the solid or fluid as a function of both
time and space [28]. Note that while Equations (3.1) and (3.2) have the same form, that
the wave equation for linear elastic solids is in terms of strains or displacements and the
linear acoustic wave equation is in terms of pressure. This analysis is focused on the
fluid within a cylindrical loop or pipe and will use the linear lossless acoustic wave (3.2).
Assuming a harmonic solution for a one dimensional “plane wave” in a
compressible fluid propagating in the z (axial) direction, the solution of (3.2) can be
written as
p z , t ,    Ae i k zl z t   Be i k zl z t 
(3.3)
Where p(z,t,) is the complex pressure at some location z, for time t and a given
angular frequency . The real part of the complex pressure p(z,t,) in (3.3) is the
amplitude of the pressure wave while the imaginary part is the phase angle. The term i is
21
√ (-1). The first term on the right hand side represents a traveling wave moving in the
positive z-direction while the second term represents a traveling wave in the negative zdirection. The terms A and B represent amplitude constants determined by the system
boundary conditions. The wave number, k, is related to wave length , angular
frequency and phase velocity of the wave c as shown below.
k 
2



c
(3.4)
Assuming a frequency dependent solution is typical for most solutions of the
acoustic wave equation. The complete pressure at a given point in time and space is the
linear combination of the pressure at all frequencies.
For two and three dimensions, separation of variables is used to develop a solution
to the linear acoustic wave equation based on the coordinate system of the problem.
Examples of separable solutions to the acoustic wave equation in Cartesian, cylindrical
and spherical coordinates are given in [28]. The separable differential equations for
cylindrical coordinates are provided in Appendix A. Similar solution methods are
employed to solve the wave equation (3.1) for plates and shells as discussed in
References [28], [31], [33], and [34]. The separable portions of the solution are related
to each other through a separation constant that relates the wave numbers in each of the
coordinate system directions. The multi-dimensional solution to the wave equation
establishes a solution that is a function of frequency, time, location and mode shape. A
fluid filled pipe described in cylindrical coordinates for example will have radial,
circumferential and axial modes that are functions of frequency, time and location.
As previously mentioned, this paper is concerned with the axial fluid modes within
a fluid filled elastic piping system that forms a continuous loop. The cross section of the
piping and fluid column are assumed to be circular, so this analysis will focus on
solutions to the wave equation in cylindrical coordinates. The solution to the threedimensional wave equation (2.2) in cylindrical coordinates is given in (3.5) as developed
in Appendix A.
22
p nml r , , z ,t   A nml J m k nm r cosm e ik zl z e it
(3.5)
Where Pnml is an amplitude coefficient determined from boundary conditions, knm is
the radial wave number and kzl is the axial wave number. The radial and axial wave
numbers are related to the angular frequency and phase velocity as shown in (3.6)
 
2
2
2
   k nml  k nm  k zl
c 
2
(3.6)
In order to solve (3.5) a radial, circumferential and axial component to the mode
shape must be specified using the appropriate values for the modal indices “n”, “m” and
“l”. The subscripts “n”, “m” and “l” are used to denote the normal mode shapes in the
radial, circumferential and axial directions respectively. Each of the modal indices
begins counting at 0. The lowest order mode is referred to as the 0th mode.
The pressure in (3.5) is a function of position in cylindrical coordinates (r, , z) at
time t for angular frequency  and mode shape “n,m,l”. Each of the first three terms on
the right hand side of the equation represents a solution in terms of one of the coordinate
system directions. These solutions can be in the form of traveling or standing waves
depending on the boundary conditions of the specific problem. For cylindrical problems
such as fluid filled pipes, the radius and circumference are finite dimensions with known
boundary conditions so the solutions in the radial and circumferential directions are
typically standing wave solutions. The solution in the axial direction (along the pipe) can
be given as a traveling wave or as a standing wave. References [17] through [21] and
[24] through [27] assumed infinite pipes to study the propagation of radial and
cylindrical modes. By assuming an infinite pipe there are no reflected waves from the
end of the pipe which allows the authors to study traveling waves. If the axial dimension
of the pipe or fluid column is finite and has known boundary conditions a standing wave
solution can be used in the axial direction. The fifth term on the right hand side of (3.5)
denotes the time dependence of the entire solution.
23
Not all modes exist at all frequencies as discussed in References [20], [25] and [27].
For example, higher order axially symmetric radial modes exhibit a cut off behavior
where certain modes only exist above a cut off frequency specific to each mode. Below
the cut off frequency for the second order radial mode (n=2) only the 0th and 1st order
axially symmetric radial modes exist in a fluid filled elastic shell. Other, non-axially
symmetric modes do exist in the fluid loaded cylindrical shell at low frequencies such as
bending modes.
All of the systems studied in the references cited in Section 2.1.1 are fluid structure
coupled systems. The modes within the fluid filled elastic shells are technically coupled
modes not separate modes of the fluid or structure. Due to how the field of acoustics has
developed it is easier to conceptualize many of the modes as either “fluid modes”
affected by the structure or “structural modes” influenced by the fluid. The fluid will
have radial, circumferential, and axial modes as discussed [28]. The elastic shell will
have circumferential, torsional, bending and axial modes as discussed in [35] and [22].
For an inviscid fluid within a cylinder, the primary interaction between the fluid and
structure is through the interaction of the fluid radial velocity and the radial velocity of
the cylindrical wall. This directionally specific interaction allows for various
simplifications to be used depending on the focus of the analysis.
The formulations used to account for the fluid structure interaction in [17] - [27] are
tailored to the specific problem and the modes of interest. References [17] - [20] and
[24] – [27] used the axially symmetric restriction on the fluid modes to focused on how
the elasticity of the shell affects the phase velocity of the axially symmetric radial modes
while ignoring the non-axially symmetric modes such as bending and torsion. In [21]
and [22] the focus was predominantly on modes of the elastic cylindrical shell. The fluid
was therefore only included as a radial loading term on the elastic shell. Modes that are
predominantly within the fluid column, such as the axial fluid resonances, are not
captured by the analysis in [21] and [22].
Similar to References [17] - [20] and [24] – [27], this analysis is focused on system
modes that primarily occur within the fluid column. The axial loop modes of interest
within the fluid column are restricted to low frequencies where at most only two axially
symmetric radial modes exist; the 0th order plane wave mode and the 1st order mode. For
24
a rigid wall cylinder only the 0th order mode exists at very low frequencies, [20], which
further simplifies the analysis and comparison of TMM and FE models. Assuming
axially symmetric waves of the lowest order radial mode only, the subscripts in (3.5)
become n=0 and m=0 resulting in plane wave solutions of the form
p 00l r , z , t   P00l e i  k zl z t 
(3.7)
3.2 Helmholtz Equation
The Helmholtz equation is a special case of the wave equation which is only a
function of position, as opposed to a function of position and time. The Helmholtz
equation describes standing waves within a fluid volume. A fluid column within a
cylindrical elastic shell that forms a complete loop is an example of an enclosed fluid.
The Helmholtz equation is obtained from the wave equation by assuming a solution of
the form in (3.8) where p is function of position and time and P is a function of position
p r , , z ,t   P r , , z e it
(3.8)
Differentiating Equation (3.8) with respect to time twice gives
2 p
  i  i Pe  it   2 Pe  it
2
t
(2.9)
Substituting Equations (3.8) and (3.9) into Equation (3.2)
 1 
 2 Pe it   2   2 Pe it  0
c 
(3.10)
Eliminating the time dependent term and using the definition of wave number in
(3.4) gives the Helmholtz equation as discussed in [22] and [28]
25
(3.11)
2P  k 2P  0
Following the same procedure used to solve the linear acoustic wave equation, a
solution is assumed for the Helmholtz equation. This solution is also separable and
results in the same three differential equations as the solution to the linear acoustic wave
equation in cylindrical coordinates. If a standing wave solution is assumed for the axial
direction, the axial component to the solution becomes (3.12).
Z z  
The
cos
k zl z 
sin
cos
k zl z 
sin
(3.12)
term in (3.12) represents the use of either cos(kzlz) or sin(kzlz) as
determined by the boundary conditions on the ends of the cylinder, not a cotangent
(cosine/sine) function. The three dimensional solution to (3.11) for a given pressure
mode becomes
p nml r , , z ,t   A nml J m k nm r cosm  
cos
k zl z e it
sin
(3.15)
By inspection it can be seen that (2.15) is very similar to (2.5). The term Anml is an
amplitude coefficient determined by boundary conditions and the term Jm(knmr) is a
Bessel function of the first kind and order m [36] which represents standing waves in the
radial direction. The term cos(m) resents standing waves in the circumferential
direction and the cos(kzlz) or sin(kzlz) term represents standing waves in the axial
direction. The choice of either the cosine or sine function in the axial direction is
typically based on the boundary conditions in the axial direction. While the time
dependent term was eliminated in the derivation of the Helmholtz equation, the solution
can be written with a time dependent term. The solution to the Helmholtz Equation in
(3.15) is shown with a time dependent term eit. This term represents the time
dependence of the amplitude of the standing wave.
26
Figure 5 through Figure 10 present the real portion of the first four pressure modes
in each of the cylindrical directions within a finite cylinder of compressible inviscid fluid
with rigid wall boundary conditions. Each figure shows the mode shapes obtained by
incrementing one of the modal subscripts from 0 to 3 while the other two subscripts are
held constant at 0. Figure 5 through Figure 10 were created by plotting solutions to the
Helmholtz equation in (3.15) using MATLAB. The term Anml was set to unity and the
cosine term was used for the axial solution assuming that the circular ends of the
cylinder are rigid. The “n”, “m” and “l” subscripts are annotated for each of the modes in
Figure 5 through Figure 10.
In Figure 5, Figure 7, and Figure 9 the blue and green dashed lines represent the
same mode shape but are out of phase by a factor of . The solid black line in each
figure represents the nominal or undisturbed value of pressure. The nominal value in
Figure 7 was set to 1 instead of 0 for presentation purposes. (The x axis and y axis in
Figure 7 represents location while the radial distance represents the pressure magnitude.)
Figure 6, Figure 8, and Figure 10 present the real portion of the pressure mode shapes
using a color map presented on a three dimensional cylinder. The color maps mode
shapes have been normalized to present the maximum real component of each pressure
mode on a scale of -1 to +1. Positive pressures are represented in red, nulls or zeros are
green, and negative pressures are represented in blue on the color map. The pink line
overlaid on each of the 0th mode subfigures indicates a line along the cylinder that
corresponds to the real portions of the pressure presented in Figure 5, Figure 7, and
Figure 9. The pressure presented in Figure 6, Figure 8, and Figure 10 along the pink line
corresponds with the blue dashed lines in Figure 5, Figure 7, and Figure 9 respectively.
Previously cited works [17] - [20] and [24] - [27] investigated the phase velocity of
axially symmetric radial modes propagating along fluid filled cylinder such as those
shown in Figure 5 and Figure 6. The mode shapes plotted in Figure 5 are similar to the
mode shapes presented for velocity potential in [18].
27
[1,0,0]
Pressure
[n,m,l]=[0,0,0]
[0,0,0]
[1,0,0]
Radius
Radius
[3,0,0]
Pressure
[2,0,0]
[2,0,0]
[3,0,0]
Figure 5 – Real Component of Radial Pressure Modes: [n,0,0]
[1,0,0]
[2,0,0]
[3,0,0]
Pressure
Pressure
[0,0,0]
Figure 6 – Radial Pressure Modes: [n,0,0]
28
[0,1,0]
Pressure
[0,0,0]
[0,0,0]
[0,1,0]
Pressure
Pressure
[0,2,0]
Pressure
[0,3,0]
[0,2,0]
[0,3,0]
Figure 7 – Real Component of Circumferential Pressure Modes: [0,m,0]
[0,1,0]
[0,2,0]
[0,3,0]
Pressure
Pressure
[0,0,0]
Figure 8 – Circumferential Pressure Modes: [0,m,0]
29
[0,0,1]
[0,0,0]
[0,0,1]
Pressure
[0,0,0]
Length
Length
[0,0,3]
[0,0,2]
[0,0,3]
Pressure
[0,0,2]
Figure 9 – Real Component of Axial Pressure Modes: [0,0,l]
[0,0,1]
[0,0,2]
[0,0,3]
Pressure
Pressure
[0,0,0]
Figure 10 – Axial Pressure Modes: [0,0,l]
30
As previously mentioned Fuller and Fahy studied the first two circumferential
modes of the elastic cylindrical shell as shown in Figure 1 of Reference [21]. These
modes are similar to the mode shapes shown in Figure 7 if the figure represented
displacement instead of pressure. The waves studied in [21] and [22] are “displacement”
waves of the cylindrical shell traveling along the elastic cylinder. The internal fluid was
treated as a loading term on the radial motion of the elastic cylinder walls. The mode
shapes and vibrations of cylindrical shells are described in more detail in [31], [33] and
[34]. The axisymmetric condition imposed by the analyses in [17] - [20] and [24] - [27] ,
excludes all of the circumferential modes except the 0th mode.
The modes shown in Figure 9 and Figure 10 are the modes referred to as “plane
wave modes”. The cylindrical cross section of the axial mode shapes move as a plane of
constant phase much like a piston moving back and forth in the axial direction. The
analyses in [17] - [21] and [24] - [27] assumed a traveling wave in the axial direction and
only considered axial modes of the 0th order.
The present analysis is focused on the “plane wave” axial modes within a
cylindrical fluid column with 0th order radial and circumferential components, similar to
the modes shown in Figure 9 and Figure 10. If the time dependence term for the standing
wave is included and the solution to the Helmholtz equation in (3.15) is restricted to
plane wave modes (axisymmetric modes of the 0th order radial modes) then the radial
and circumferential components of the solution reduce to 1. The solution (3.15) takes the
form
p 00l r , , z   A 00l
cos
k zl z e it
sin
31
(3.16)
3.3 Uniform Loop: Theory
To investigate the axial loop resonances within the example system shown
previously in Figure 4, the resonances within a uniform fluid loop will first be
considered. This uniform fluid loop provides a baseline set of results that make intuitive
sense. The baseline results will be used as a point of comparison for the studies on
component cavities and elasticity presented in later sections. Only the axial resonances
of plane waves, the lowest order axisymmetric radial mode (n,m,l = [0,0,n]) will be
considered in this analysis.
The uniform loop is assumed to be perfectly rigid, filled with water and in a freefree boundary condition. The water is a liquid and assumed to be compressible, lossless,
inviscid, irrotational, free from bubbles or dissolved particulates, and have the properties
specified in Table 1.
The loop will be “unwrapped” into an equivalent straight fluid column, as shown in
Figure 11. Continuity boundary conditions will be applied to either end of the straight
pipe such that the magnitude and phase of both the dynamic pressure and volume
velocity are equal at Section A-A in Figure 11. By applying the continuity boundary
conditions a continuous loop of fluid will be simulated which ignores any potential
effects that the elbows or a continuous curvature might have on the frequency and mode
shape of the axial fluid resonances.
A
A
A
A
A
A
Figure 11 – Baseline Loop and an “Unwrapped” Baseline Loop
32
By “unwrapping” the system into an equivalent straight system and applying
appropriate boundary conditions the Helmholtz equation in cylindrical coordinates
(3.16) can be used to calculate the axial resonances within the fluid. The well known
solutions to the Helmholtz equation discussed in Reference [22] and Reference [28]
provide a theoretical solution that is relatively intuitive to understand. The result of the
unwrapped baseline loop are also more easily compared to the dispersion results for the
plane wave modes discussed in References [17] through [21] and References [24]
through [27] .
At low frequencies, the wave lengths of the axially fluid resonances are much longer
than the cross sectional dimensions of the piping or elbows. The pipe will act like a wave
guide as discussed in Reference [28]. Ignoring the curvature or elbows of the system is
not uncommon when evaluating guided waves. This assumption is reasonable to
establish some initial expectations for the axial loop resonances based on acoustic
theory. The assumption will be shown to be reasonable using results from FE models in
Section 3.5 and revisited briefly in Section 4.6
The solution to the Helmholtz equation in cylindrical coordinates for the plane wave
modes was given in (3.16) and is repeated in (3.17). Since the analysis is restricted to
only plane wave axial modes the modal notation used throughout the remainder of the
paper will drop the radial and circumferential notation for simplicity. The time
dependent term is also common to all of the subsequent solutions. This term will also be
omitted to simplify the notation as is a typical convention in acoustic literature.
p l z   A l
cos
k zl z 
sin
(3.17)
For pure plane wave modes the radial wave number becomes 0, so the axial wave
number simplifies to
 
   k zl
c 
33
(3.18)
The sine or the cosine term is chosen for the solution based on the boundary
conditions at the end of the cylinder. For free ends or a pressure release boundary, the
pressure at each end must go to 0, while the volume velocity must be at an extrema. The
sine term is therefore used for the pressure solution. If the end conditions are fixed, the
pressure must go to an anti-node or extrema while the volume velocity goes to 0,
dictating the use of the cosine term for the pressure solution. In each of these boundary
condition cases, axial resonances will occur in half wave length multiples or at (n. For
the system dimensions and material properties given in Section 2.2 the half wave length
resonances will occur at 20 Hz, 40 Hz, 60 Hz, etc.
The continuity boundary condition only requires that the pressure and volume
velocity be continuous at the end. For both pressure and volume velocity to be
continuous the axial resonances can only occur in whole wave length multiples
(n2within the “loop”. At half wavelength multiples only the pressure or the volume
velocity can be continuous, while the other quantity would have a significant
discontinuity.
The continuity boundary condition results in both the sine and cosine terms being
valid solutions to the Helmholtz equation (3.17). Unlike the free and fixed boundary
conditions, the continuity boundary condition does not specify an axial location for a
particular point in the pressure or volume velocity mode shape. For a “uniform” loop the
axial mode shapes could occur in any rotational orientation along the axis of the loop. In
an actual system some perturbation in the loop would exist and lead to a specific
orientation of the standing wave nodes and extrema.
The continuity boundary condition for the “baseline” loop results in pairs of whole
wave length modes which satisfy both the sine and cosine solutions to (3.17). The modes
that make up each pair occur at the same frequency for a uniform loop. Table 3 provides
the frequency of the first four pairs of axial loop resonances calculated for the baseline
loop using (3.17). The 0 Hz rigid body mode has been omitted. The modes within each
pair will be designated as Mode A and Mode B. This notation will be used in later
sections to make a distinction between the two modes that make up the mode pair for
each whole wave length multiple.
34
Mode
Frequency
A
B
A
B
A
B
A
B
[Hz]
40
40
80
80
120
120
160
160
Wave Length
Multiple
[n
1
1
2
2
3
3
4
4
Table 3 – Axial Modes; Theoretical, Baseline “Loop”
Figure 12 shows the normalized pressure mode shapes for the 40 Hz 1 Mode A
and Mode B as determined from (3.17). Figure 13 shows the normalized axial
displacement modes shapes obtained by combining (3.17) with the linear Euler equation
in the z direction given in equation (A.8c) of Appendix A. The equation is then
integrated with respect to time and solved for displacement.
Pressure Mode Shapes; 40 Hz
1
Normalized Pressure
0.5
0
-0.5
Mode A
Mode B
-1
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 12 – Baseline Loop, Pressure Modes Shape for 1 Mode A and Mode B
35
Displacement Modes Shapes; 40 Hz
Normalized Displacement
1
0.5
0
-0.5
Mode A
Mode B
-1
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 13 – Baseline Loop, Displacement Modes Shape for 1 Mode A and Mode B
The pairs of modes for whole wave length multiples are consistent with the original
observations that there is more than one mode for each full wave length in the Example
loop system. Based on the solution to the Helmholtz equation with continuity boundary
conditions in (3.17) the mode pairs are an expected result for a continuous loop of fluid.
However, unlike the results to the example system, the mode pairs in the baseline loop
occur in a harmonic series with each mode of a given pair occurring at the same
frequency.
3.4 Uniform Loop: Transfer Matrix Method
To investigate changes to material properties, elasticity and cross sectional
dimensions over finite lengths of the loop the Transfer Matrix Method (TMM) was
implemented using MATLAB. TMM is a numerical method that was originally
developed for analyzing electrical networks [37]. There are many parallels between the
differential equations that govern electrical networks and acoustics as discussed in [37]
and [38] .
36
TMM use a matrix formulation to represent each element or section of the “pipe”.
Volume velocity and pressure are used as the state variables. The element matrix relates
the inlet volume velocity (qin) and inlet pressure (pin) to the outlet volume velocity (qout)
and outlet pressure (pout). This numerical method focuses on the axial wave propagation
along the fluid column and uses sine and cosine functions to relate the inlet and outlet
variables. This approach is ideally suited for investigating axial waves within a fluid
loop. TMM provides the flexibility to change many of the system parameters such as the
axial phase velocity and cross sectional area over finite portions of the system to
investigate the impact these changes have on the axial loop resonances. The TMM
formulation for standing waves in a cylindrical fluid filled pipe provided in [1] was
implemented using MATLAB. All of the associated MATLAB functions and scripts
developed to support this work can be found in Appendix C.
The acoustic impedance of the fluid in the pipe element, Zp, is given for a plane
wave by
Zp 
c
(3.19)
Ax
Figure 14 shows variables for a uniform tube element, where q is volume velocity, p
is pressure,  is the fluid density, c is the axial phase velocity, Ax is the cross sectional
area of the fluid, Lseg is the length of the element and Zp is the acoustic impedance of the
tube element.
z
c
Zp 
q in 
p 
 in 
q out 
p 
 out 
Ax
z=0
z = Lseg
Figure 14 – TMM Uniform Pipe Element
37
The matrix representing a single uniform pipe element [S] n is given in (3.20) where
the “n” represents the element number in pipe network, Sij, represent the “ith” by “jth”
element of [S] n, kzl is the axial wave number and z is the coordinate direction along the
axis of the element. Note that unlike many finite element matrices the TMM element
matrix is not symmetric about the diagonal of the matrix.
S n

S 11 S 12 
 cos( k zl z )



S 21 S 22  n  iZ sin( k z )
p
zl

i

sin( k zl z )
Zp

cos( k zl z ) 
(3.20)
n
Other elements are available to represent different types of connections or piping
system features such as side branches, nozzles, and tanks. Only the pipe element shown
in (3.20) and (3.21) will be used in this analysis. The element matrix, input vector and
output vector for a single pipe are shown in (3.21), where qin , pin, qout , pout are complex
volume velocity and pressure.

 cos( k zl z )

 iZ p sin( k zl z )
i

sin( k zl z ) q in  qout 
Zp


  p in   pout 
cos( k zl z ) 
(3.21)
n
Since the matrix for each element relates input qin and pin to the output qout and pout,
the system matrix [T] sys can be assembled by multiplying successive element matrices
together to form a single matrix as shown in (3.22)
T sys  S 1  S 2  S 3 S n
(3.22)
The system matrix [T] sys relates the input qin and pin to the output qout and pout across
the entire pipe system made up multiple elements. The individual elements of [T] sys are
denoted as Tij , which gives the “ith” by “jth” element of T as shown in (3.23).
T 11 T 12 
T

 21 T 22  sys
 q  q 
  in    out 
 p in   pout 
38
(3.23)
Expanding the matrices in (3.23) gives the equations
T11q in  T 21 p in  qout
T 21q in  T 22 p in  pout
(3.24)
A common input file structure was implemented in MATLAB to assemble the
element matrix for each individual segment. A separate routine assembles the elements
into a system matrix based on a sequence provided in the input file. In order to evaluate
a “loop” of piping the continuity boundary condition (3.25) was applied to the equations
in (3.23) and (3.24) as shown in (3.26)
q in  qout
pin  pout
(3.25)
T11q in  T 21 p in  q in  0
T 21q in  T 22 p in  p in  0
(3.26)
,
To find the frequencies of the axial loop modes a characteristic equation was
developed from the equations in (3.26). Collecting like terms from (3.26)
T11  1q in  T 21 pin
T 21q in  T 22  1p in
(3.27)
Dividing the two equations in (3.27) to eliminate qin and pin
T11  1q in
T 21q in

T 21 p in
T 22  1pin
(3.28)
Gives the following relationship between the elements of the system matrix
T11  1 
 T 21
 T 21
T 22  1
(3.29)
The equation in (3.29) is rearranged to generate the Characteristic Equation for a
“loop” of pipe using TMM (3.30).
39
T11 1T 22 1 T12T12  0
(3.30)
Since the element matrices are functions of kzl (3.20), and kzl is a function of 
(3.18), the system matrix [T] sys and the Characteristic Equation (3.30) are also functions
of angular frequency. Using MATLAB the characteristic equation was calculated by
varying  over a specified range, calculating the element and system matrices for each
value of  and then solving (3.30). The roots of the characteristic equation (3.30)
indicate the frequencies where axial modes exist within the loop of piping.
Since (3.30) was evaluated numerically with MATLAB a routine was developed to
bracket potential root locations in frequency from an arbitrarily chosen frequency vector.
The brackets and the characteristic equation were then input into a root finding algorithm
that used Muller’s Method [39] to iteratively find the frequency of each root. Table 4
shows the first four roots calculated using TMM in MATLAB for the baseline loop
analyzed in previously Section 3.3.
[Hz]
Wave Length
Multiple
[n
A
40
1
A
80
2
A
120
3
A
160
4
Mode
Frequency
Table 4 – Axial Modes; TMM, Baseline “Loop”
The implementation of the root finder developed for MATLAB does not
discriminate between single or double roots, hence only a single root is returned at the
frequency for each of the full wave length axial mode. Figure 15 provides a plot of the
characteristic equation and the roots calculated using MATLAB.
40
Characteristic Equation
4
Ch EQ Uniform Loop
Roots Uniform Loop
3.5
3
2.5
2
1.5
1
0.5
0
0
20
40
60
80
100
120
Frequency, Hz
140
160
180
200
Figure 15 – Baseline Loop, TMM, Characteristic Equation and Roots
The TMM results are in good agreement with the theoretical values calculated from
(3.17) in Section 3.3. The characteristic equation will be plotted in later sections when
changes are added to the system to identify how the modes are shifting in frequency.
MATLAB scripts were also developed to calculate the pressure mode shape,
displacement mode shape, change in volume per unit length, impedance as a function of
axial location, Zsys(z), and the non-dimensional impedance as a function of axial location,
sys(z) , for the system at the frequency of each root. These quantities will be used in
later analyses to calculate the axial pressure and displacement modes shapes for nonuniform loops of pipe. The change in volume per unit length will be plotted to help
characterize some of the results for the pressure modes shapes.
Since there are no impedance discontinuities within the uniform baseline loop, the
axial orientation of the pressure and displacement modes shapes is arbitrary as
mentioned in the previous section. In order to plot the A and B modes shapes a phase
must be specified for a single axial location.
41
Figure 16 and Figure 17 present the plots of the 40 Hz pressure mode shape for
Mode A and Mode B respectively. The upper portion of each figure presents the pressure
as a normalized sine or cosine function with an arbitrary orientation relative to the pipe
axis. The lower portion of each figure also presents the pressure mode shape displayed
as a colored surface plot, similar to the presentation provided for pressure modes in most
FE software. The color plot shown corresponds to the blue line on each plot above it.
Note that the transverse dimensions of the surface plot are presented on a different scale
to improve the visibility of the color distribution.
As previously noted the root finder that was implemented cannot distinguish if
double roots exist at the calculated root to the characteristic function. Plotting the A and
B mode in Figure 16 and Figure 17 was based on specifying an orientation for the modes
and the results discussed in Section 3.3 that indicated each of the roots shown in Figure
15 are double roots.
As shown in Figure 16 and Figure 17 the pressure mode shapes are continuous and
in good agreement with the pressure modes previously present in Figure 12. The axial
displacement mode shapes are also in good agreements with the mode shapes shown
previously in Figure 13. Plots of the non-dimensional system impedance, pressure mode
shape, axial displacement mode shape, and change in volume per unit length for the first
four mode pairs are provided in Appendix D.
42
Pressure Mode Shape; Mode A, Root 2, 40 Hz
1
Normalized Pressure
0.5
0
-0.5
-1
0
5
10
15
20
25
Axial Location, m
30
35
40
Projection of 1D Pressure Mode Shape; Mode A, Root 2, 40 Hz
2
1
0.8
1.5
0.6
1
0.4
Z, m
0.5
0.2
0
0
-0.2
-0.5
-0.4
-1
-0.6
-1.5
-2
-0.8
0
5
10
15
20
X, m
25
30
35
40
Figure 16 – Baseline Loop, TMM Pressure Modes Shape for 1 Mode A
43
-1
Pressure Mode Shape; Mode B, Root 2, 40 Hz
1
Normalized Pressure
0.5
0
-0.5
-1
0
5
10
15
20
25
Axial Location, m
30
35
40
Projection of 1D Pressure Mode Shape; Mode B, Root 2, 40 Hz
2
1
0.8
1.5
0.6
1
0.4
Z, m
0.5
0.2
0
0
-0.2
-0.5
-0.4
-1
-0.6
-1.5
-2
-0.8
0
5
10
15
20
X, m
25
30
35
40
Figure 17 – Baseline Loop, TMM Pressure Modes Shape for 1 Mode B
44
-1
The results in Figure 16 and Figure 17 are in good agreement with the theoretical
results for axial fluid modes in a uniform loop of rigid wall pipe. The axial fluid modes
are standing waves made up of sine and cosine functions. For a uniform loop the axial
resonances will occur in pairs, as discussed in Section 3.3, and occur at multiples of
whole wave lengths forming a harmonic series.
3.5 Uniform Loop: COMSOL FEA Models
Two finite element models of the baseline loop were constructed using the acoustics
module of the COMSOL multi-physics software. The first model was a straight fluid
column, with rigid walls, filled with water using the dimensions and material properties
used in Section 3.3 and Section 3.4. The second model was an actual fluid loop with
rigid walls, filled with water, and using t the dimensions and material properties given in
Table 1 and Table 2. The actual loop model uses the long radius elbow dimensions given
in Figure 3 and has the same total centerline length as the simulated baseline loop
models. The COMSOL models are shown in Figure 18.
Simulated Loop
Actual Loop
Figure 18 – Baseline Loop, COMSOL FE Models
45
An Eigen value analysis was performed on each of the models shown in Figure 18.
Detailed discussions regarding eigenvalues and the calculation of eigenvalues from finite
element models can be found in [40], [41] and [42]. Table 5 shows the first eight
eigenvalues for each of the FE models, neglecting the single rigid body mode calculated
near 0 Hz. The model reports generated from COMSOL can be found in Appendix E.
“Simulated Loop”
Mode
Frequency
“Actual Loop”
Mode
[Hz]
[Hz]
Wave
Length
Multiple
[n
Frequency
A
39.9998
A
40.0242
1
B
40.0002
B
40.0323
1
A
79.9996
A
79.9992
2
B
80.0004
B
80.1138
2
A
119.9995
A
120.0823
3
B
120.0005
B
120.0866
3
A
159.9994
A
160.0056
4
B
160.0006
B
160.2191
4
Table 5 – Axial Modes; COMSOL, Baseline “Loop”
The frequencies of the eigenvalue results calculated by COMSOL are in reasonably
good agreement with the resonant frequencies calculated using acoustic theory in
Section 3.3 and using the TMM in Section 3.4. The COMSOL eigenvalue solver
calculated Eigen frequencies that are slightly shifted apart in frequency. This may be due
to discretization of the FE model or the solver may have had convergence problems due
to the existence of double roots. The calculated frequencies for the simulated loop are
less than 0.001% different than the theoretical and TMM calculated roots.
The calculated Eigen frequencies for the actual loop model are less than 0.2%
different than the roots or Eigen frequencies calculated using theory and TMM. Note
that for the actual loop the each mode of a pair is slightly shifted and occurs at a
frequency that is different that the frequency of the associated mode pair. This
observation will be revisited later in this analysis. As shown in Table 5, approximating
46
the loop as a straight fluid column with continuity boundary conditions results in less
than a 0.2% difference in the calculated eigenvalue frequencies as compared to the
frequencies calculated from the model with explicit elbows. This supports the initial
assumption that the rigid wall fluid column acts as a wave guide at low frequencies.
Figure 19 presents the pressure mode shapes for Mode A and Mode B of the 1
wavelength axial loop resonance at nominally 40 Hz from the actual loop model.
Figure 19 – Baseline Loop, COMSOL Pressure Mode Shapes, Actual Loop
47
4. Discussion
4.1 Uniform Loop: Elasticity Effects
Previous authors have shown that the phase velocities of circumferential and radial
fluid modes within a fluid filled cylinder are affected by the elasticity of the cylindrical
wall [17] - [21] and [24] - [27]. The resulting phase speeds are modally and frequency
dependent. This means that each mode travels along the fluid column at a different phase
velocity and that the phase velocity for a given mode changes as a function of frequency.
The present analysis is only concerned with the axial modes with 0th order radial and
circumferential components or plane waves. If the 0th mode propagates at a velocity
other than the fluid characteristic speed of sound, then the axial modes within the fluid
loop will occur at frequencies that are different than the frequencies calculated for the
rigid wall loop shown in Table 3 and Table 4.
The results for the phase velocity of the 0th mode reported in [17] - [21] and [24] [27] show similar characteristics for most of the material and dimension combinations
reported by the respective authors. The reported phase velocity of the 0th mode is less
than the characteristic fluid speed of sound and there is almost no perceptible variation
with frequency at the very low wave numbers which correspond to the low frequency
range of this analysis.
The exact equations used by DelGrosso [24], Lafleur and Shields [25] and Baik
et.al. [26] – [27] require a complex numerical solution to determine the phase velocities
of each mode that exist for a given frequency. These authors were primarily focused on
the high frequency responses of fluid filled elastic cylinders. At high frequencies
multiple radial modes exist at each frequency and there are rapid changes in the phase
velocity over relatively small frequency bands.
The equations used by Lin et. al. [20] and Thompson [19] included the elastic
cylinder as a shell approximation. As mentioned in Section 2.1, it was shown in [20] that
including the transverse shear term in the approximation of the cylindrical shell resulted
in a lower axial phase velocity for the 0th mode than the phase velocity calculated in
[19]. However the phase velocities calculated for 0th order mode with or without the
transverse shear term included are not perceptibly different for low wave numbers, as
48
shown in Figure 4 in [20]. In fact, at very low wave number the results for the 0th mode
phase velocities in [20] are in reasonable good agreement with the results calculated in
[19].
For the purposes of this analysis which is restricted to plane waves at low
frequencies (or very low radial waves numbers) the Korteweg-Lamb correction factor
will be used. The Korteweg-Lamb correction, shown in Equation (4.1), is used to
calculate an effective axial phase velocity ceff for a fluid of density  and characteristic
speed of sound co within an elastic cylinder. The elastic cylinder has a thickness h, radius
a, and is made of a material with a Young’s Modulus E and Poisson’s ratio .
The correction factor modifies the characteristic fluid speed of sound in the axial
direction to account for the combined radial expansion of the fluid column and elastic
shell. The correction factor was developed by equating the change in volume of the
cylindrical shell and cylindrical fluid column due to an applied internal pressure.
c eff
 2ac o2 1    

 c o 1 
Eh


1 / 2
(4.1)
The derivation of the Korteweg-Lamb correction factor is provided in Appendix B
following the derivations in [22] and [37] and using cylindrical shell equations from
[43]. The correction factor is determined from the material properties and dimensions of
the fluid column and cylindrical shell and it is not frequency dependent. As noted in [22]
and [23] the Korteweg-Lamb correction factor is a reasonable approximation for the
very low frequency effect of a cylindrical elastic shell on the phase speed of the 0 th
mode. Thompson also noted that his equation developed for the phase velocity of the 0th
mode in [19] reduces to the Korteweg-Lamb correction factor at 0 Hz.
Using the dimensions and material properties given in Section 2.2 the KortewegLamb correction factor is calculated to be approximately 0.9406 for a water filled steel
pipe resulting in an effective axial phase velocity of ceff = 1393 m/s. Table 6 provides
the frequencies of the axial resonances from the uniform loop calculated by substituting
ceff into (3.17). The frequencies of the axial resonances shifted lower in frequency, by
49
approximately 6%. The pairs of modes remain pairs and harmonically spaced as
expected for a uniform change to the entire loop.
Mode
Frequency
(A or B)
A
B
A
B
A
B
A
B
[Hz]
37.6
37.6
75.2
75.2
112.8
112.8
150.4
150.4
Wave Length
Multiple
[n
1
1
2
2
3
3
4
4
Table 6 – Axial Modes; Theoretical, Elastic Baseline “Loop”
Figure 20 shows a plot of the characteristic equation for the uniform elastic loop as
a function of frequency which was calculated using TMM. The characteristic equation
and roots for the baseline “rigid wall” loop are shown for comparison.
Characteristic Equation
4
Ch EQ Elastic Loop
Roots Elastic Loop
Ch EQ Rigid Wall Loop
Roots Rigid Wall Loop
3.5
3
2.5
2
1.5
1
0.5
0
0
20
40
60
80
100
120
Frequency, Hz
140
160
180
200
Figure 20 – Characteristic Equation and Roots; TMM, Elastic Uniform Loop
50
Similar results for the frequencies of the axial loop modes were obtained from the
TMM and COMSOL models. The results from the TMM and COMSOL simulated loop
models with Korteweg-Lamb corrected phase velocity are shown in Table 7. These
results are also in good agreement with the results in Table 6.
Elastic Uniform Loop TMM
Mode
Frequency
Elastic Uniform Loop COMSOL
Mode
[Hz]
[Hz]
Wave
Length
Multiple
[n
Frequency
A
37.6
A
37.6231
1
B
37.6
B
37.6234
1
A
75.2
A
75.2461
2
B
75.2
B
75.2468
2
A
112.8
A
112.8692
3
B
112.8
B
112.8702
3
A
150.4
A
150.4923
4
B
150.4
B
150.4935
4
Table 7 – Axial Modes; TMM & COMSOL, Elastic Uniform Loop
The frequencies of the axial loop modes occurred at lower values than those
calculated for the axial modes within a rigid wall loop of the same fluid and dimensions.
The pressure and displacement mode shapes remained sinusoidal and did not show any
significant differences from the mode shapes previously presented in Figure 12 and
Figure 13.
Unlike the high frequency results in [17] - [21] and [24] - [27], the elasticity alone
does not result in a unique shifting or spacing of the low frequency axial modes when
implemented uniformly for the entire loop. Even if a slightly frequency dependent phase
velocity had been calculated over this range of frequencies, based on the minimal
changes in phase velocity reported in previous works, the modes would still have shifted
as pairs and resulted in a spacing that was only slightly off from a harmonic spacing. So
while the elasticity of the cylindrical wall provides one explanation for how the axial
51
loop modes can occur at frequencies lower than the frequencies of the equivalent rigid
wall system, it does not completely explain the larger shifts in frequency or unique nonharmonic frequency spacing of the axial modes observed in the example system results
in discussed in more detail in Section 4.7.
The effects of elasticity will be revisited in a later section when local differences in
phase velocity will be considered over finite regions of the loop. Since elasticity makes
the phase velocity dependent on the materials and cross sectional dimensions of the
system, the phase velocity will vary with position around the loop if the materials or
dimensions are a function of axial position.
4.2 Non-Uniform Loop
The uniform loop discussed in the previous sections provided a reasonable case
to establish some baseline for analyzing the axial modes in an arbitrary loop of fluid.
The axial modes will occur in pairs based on the solution to (3.17). Including the
elasticity of the cylindrical wall will result in a lower axial phase velocity, [17] - [21],
[24] - [27]. The lower axial phase velocity causes the pairs of axial loop modes to occur
at a lower frequency than the axial modes in an equivalent loop with rigid walls.
The frequencies of the mode pairs calculated from the TMM and FE models are
similar to the theoretical solutions to within a fraction of a percent. The theoretical,
TMM, and FE results for the frequencies and modes shapes of the axial loop modes in a
simulated loop (a straight pipe with continuity boundary conditions) are similar to the
results calculated from a FE model of an actual loop that included the elbows. This result
indicates that the simulated loop models are reasonable engineering approximations of
the axial loop modes at low frequencies.
A more representative or practical loop arrangement will not be uniform around
the entire loop. A system that includes components will have variations in the material
properties, dimensions, radial elasticity and the shape of the internal fluid column. These
location dependent properties will result in changes to the phase velocity, the impedance
of the fluid column, the frequency and the shapes of the axial loop modes. The following
sections establish several additional physics based expectations for the axial loop modes.
52
These expectations will be compared with the results for the axial loop modes calculated
from the TMM and FE models of non-uniform loops
4.2.1
Net Change in Phase around the Loop
As discussed in Sections 3.3 to 3.5, the axial fluid loop modes occur in integer
multiples of whole wave lengths. An integer number, l, of whole wave lengths represents
a total change in phase of l2 around the loop. The summation of phase for each axial
fluid loop mode is expected to sum to l2 where l is equal to the number of whole
wavelengths and the axial modal subscript “l” in the solution to the Helmholtz equation
(3.15) discussed in Section 3.2.
For a non-uniform loop, different phase velocities may exist at various locations
within the loop. The angular frequency will be constant in all portions of the system for a
given mode. If the angular frequency remains constant and there are different phase
velocities along the axis of the loop then there will be different wave numbers kzl within
the loop as defined in (3.18). The wave number kzl describes the change in phase of the
wave with respect to the change in axial location. The change in phase of an axial wave,
seg, across a cylindrical segment of length Lseg, is given by
seg  k zl L seg
(4.2)
The net change in phase around a non-uniform loop at a given frequency is given by
net   seg
(4.3)
where the change in phase seg is specific to the phase velocity of the fluid and the
length of each segment. Whole wave lengths will occur when the net change in phase
around the loop is equal to an integer multiple of l2.
net  l 2
(4.4)
The frequencies where the axial loop modes occur will be a function of all of the
axial phase velocities present in the system and the length of the system where these
axial phase velocities exist. This indicates that the axial modes will occur at different
53
frequencies for non-uniform loops with multiple phase velocities. Similar to the
elasticity results discussed in Section 4.1, this also indicates that the pairs of modes will
shift together in frequency as a pair.
4.2.2
Net Change in Volume
All of the loop systems studied in this thesis are closed systems. For a closed
system, uniform or non-uniform loop the summation of the change in volume per unit
length must be equal to 0, i.e.
 V z   0
f
(4.5)
For a differential element of a compressible or “acoustic” fluid a change in volume
(Vf ) can be define from (A.12) in Appendix A for an applied pressure as
V f 
pV f
Bf
(4.6)
Where p is an applied internal pressure, Vf is the original volume of the differential
fluid element and Bf is the fluid bulk modulus. The fluid bulk modulus is a function of
the fluid density f and characteristic speed of sound co of the fluid as shown in (4.7).
B f   f c o2
(4.7)
As shown in (4.6), a positive internal pressure causes an increase in volume. The
change in volume as a function of axial position (4.8) can be calculated for the axial
modes in a fluid loop by combing (4.6) and (4.7) and using the pressure p(z) for a
specific mode shape such as the pressure plotted in Figure 12 for Mode A of the 1
mode pair of the uniform loop.
 V 
V f z   p z  f 2 
 f c 
54
(4.8)
The change in volume per unit length for Mode A of the 1 mode pair from the
uniform loop is shown in Figure 21. The blue line and the green line represent the
normalized change in volume per unit length for the same mode shape but are out of
phase by  or 180 degrees. The positive values in Figure 21 represent locations where
the pressure mode shape results in an increase in the volume of the differential fluid
elements and the negative values represent a decrease in volume. The shaded regions
Volume / Length [m3/m]
represent the total positive and negative changes in volume (Vf ).
+ Vf
- Vf
Figure 21 – Calculated Change in Volume; 1 Mode A, Rigid Wall Uniform Loop
The summation of the shaded regions in Figure 21 represents total change in volume
for the 1Mode A pressure mode. Integrating the shaded regions in Figure 21 separately
and summing the total positive and negative changes in volume, results in a total system
change in volume of 0. This is consistent with the assumption that the system is a closed
system where fluid cannot enter or leave the system.
55
The change in volume in (4.8) is a function of the original volume. The larger the
original volume of the differential element is the larger the change in volume for a given
value of pressure. For a non-uniform loop which may have several different cross
sectional areas and shapes, the change in volume per unit length will increase in regions
of the system that have larger cross sectional areas.
Note that the change in volume is also a function of the phase velocity. Changes in
the radial elasticity which result in different axial phase velocities, such as those
discussed in Section 2.1 and Section 4.1, will also impact the change in volume per unit
length. Essentially if an internal pressure is applied to a differential cylindrical element
of fluid, the element will change volume in the radial and axial directions.
The total change in volume of the cylindrical element for a given applied pressure
is given by (4.8) and remains the same independent of the boundary conditions. If the
element has a rigid boundary in the radial direction and elastic boundaries in the axial
direction, like the uniform loop discussed in Section 3.3 to 3.5, the change in volume
will occur only in the axial direction. If the element has elastic boundary conditions in
both the radial and axial directions the change in volume will occur both radially and
axially. The greater the change in volume in the radial direction the lower the change in
volume expansion will be in the axial direction as the total change in volume must
remain constant for a given applied pressure (4.8).
Regardless of the axial distribution of the change in volume, the summation of the
volume changes over the loop for any given axial mode of a closed system of continuous
fluid must equal zero.
4.2.3
Continuity
The continuity conditions for a fluid system require that the pressure and the volume
velocity within the fluid loop must remain continuous. The fluid must also remain in
contact with the system walls and the adjacent fluid in all directions.
For a non-uniform loop the continuity requirement can result in sudden local
changes and discontinuities in the axial displacement mode shape. A sudden change in
cross sectional area provides a simple example of this occurrence. To maintain the
56
continuity of volume velocity across a sudden change in cross sectional area the axial
particle velocity must change in inverse proportion to the change in cross sectional area.
The continuity in volume velocity is due to the conservation of mass in a closed system.
This also results in discontinuities in the axial displacement mode shapes.
The continuity condition for the displacement mode shapes can be checked by
multiplying the axial displacement per unit length times the cross sectional area to
calculate a volume displacement. The axial volume velocity is the time derivative of the
axial volume displacement. Since the volume velocity must be continuous in the axial
direction the volume displacement will also be continuous in the axial direction. The
axial displacement mode shapes will be discussed further during the analysis of the nonuniform loop systems.
4.2.4
Acoustic Impedance Changes
Several definitions of impedance are used in acoustics. As discussed in [37] the
choice of impedance formulation is based on the particular problem considered. This
analysis primarily uses the acoustic impedance based on the acoustic volume flow. The
acoustic impedance for a plane wave in a uniform cylindrical tube or pipe element, Zp,
was defined in (3.19). The acoustic impedance for each section of the loop is a function
of the density, , the cross sectional area, A, and the phase velocity, c, in that section of
the loop.
Differences in fluid density, phase velocity or cross sectional area result in
different acoustic impedances at various locations around a non-uniform loop. In
addition, any system properties that affect the fluid density, phase velocity, or cross
sectional area will also cause changes to the acoustic impedance. These changes can be
gradual such as the one due to a temperature gradient over the length of a heat
exchanger, which changes the density and characteristic fluid speed of sound, or sudden
such as the change in cross sectional area where a pipe connects to a large tank. System
properties that can cause either gradual or sudden changes to the acoustic impedance
include, but are not limited to; temperature, pressure, the amount of entrained gases,
57
structural materials, the thickness of piping and component walls, foundations and
piping supports, and intricate component internal arrangements.
At a location where a change in impedance occurs, a portion of the acoustic wave
can be reflected back from the location while the remainder of the acoustic wave is
transmitted through the impedance change. The amount of the acoustic wave that is
reflected or transmitted depends on how severe the mismatch in impedance is and how
suddenly the change occurs. The analysis of transmitted and reflected waves in a piping
system is typically focused on traveling waves and determining how much transmission
loss occurs for an incident signal through a system of impedance changes. One of the
most common examples is a muffler in the exhaust system of a car. In depth discussions
on the reflections and transmission of acoustic waves can be found in [1], [13], [28] and
[37].
The analysis of the reflected and transmitted acoustic waves within a fluid loop is
complicated by the loop arrangement. An acoustic wave traveling in one direction
around the loop from a source will eventually traverse the entire loop and interact with
its own reflections as well as the reflected and transmitted portions of the acoustic wave
that initially traveled in the opposite direction around the loop. On a traveling wave basis
a series summation would be required to calculate the steady state pressures in a
continuous loop due to a known source at a specific location in the loop. This could be
particularly interesting if a non-zero fluid velocity is considered, which would travel in
only one direction around the loop while the acoustic waves would propagate in both
directions. Studying the combinations of traveling waves around a fluid loop system due
to various types and locations of acoustic sources is beyond the scope of the present
analysis and will require further study.
This analysis focuses only on the frequency and modes shapes of axial modes in
the fluid loop. The axial modes are a function of the system dimensions, materials and
boundary conditions and can be calculated without applying a known source as shown in
Sections 3.3 - 4.1.
The reflected and transmitted acoustic waves at changes in local impedance do
provide a potential explanation for the initial observations that modes of the example
system had amplitude variations within each mode shape. The “boundary” between two
58
fluid columns that have different acoustic impedances causes an elastic boundary
condition in the axial direction. The variation in local impedance, around the loop, will
result in pressure mode shapes or standing waves that have different magnitudes in
different regions of the system and may explain the “kinked” appearance of the axial
modes calculated by the models of the example system.
The other forms of impedance used in this analysis are the system impedance
Zsys(z) and the non-dimensional system impedance sys(z). The system impedance,
Zsys(z), is the ratio of complex pressure as a function of axial location p(z) over the
complex volume velocity q(z) as a function of axial location as shown in (4.9).
Z sys (z ) 
p (z )
q (z )
(4.9)
A non-dimensional form of the system impedance sys(z) can be calculated as
shown in (4.10)
 A

Z sys z 
 f co

 sys z   tan 1 
(4.10)
As discussed in [1] and mentioned in Section 3.4, the non-dimensional system
impedance can be used to generate the pressure and the axial displacement mode shapes.
Figure 22 shows the plot of the non-dimensional impedance and the inverse of the nondimensional impedance calculated for Mode A of the 1 mode pair of the uniform loop.
The system impedance Zsys(z), has the form of a tangent function. The non-dimensional
system impedance sys(z) becomes a linear function with discontinuities at the locations
where Zsys(z) approaches +/- infinity. Figure 22 shows that sys(z) cycles in amplitude
between –/2 and +/2 which is consistent with Zsys(z) have the form of a tangent
function. The slope of the line indicates the change in phase per change in axial location
of a standing wave, (z), similar to the discussion in Section 4.2.1.
59
Non-Dimensional Impedance; Normalized Pressure
Non-Dimensional Impedance, Root 2, 40 Hz
Non-Dim Imp
inv Non-Dim Imp
Mode A Pressure
Pressure Node
Pressure Anti-Node
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 22 – Non-Dimensional Impedance; 1 Mode A, Rigid Wall Uniform Loop
Figure 22 also shows the normalized Mode A pressure for the 1 mode pair of
the uniform loop. By inspection it can be seen in Figure 22 that the roots of sys(z)
correspond to the nodes or zeros of the pressure mode shape (o) and the roots of 1/sys(z)
correspond to the anti-nodes (x) of the pressure mode shape. This relationship is used by
the TMM MATLAB scripts in Appendix C as a part of the process to calculate the
pressure and displacement mode shapes for non-uniform loops.
The non-dimensional system impedance sys(z) corresponds to the phase angle
(z)=kzl z. of the pressure mode shapes. Summing the values of sys(z) shown in Figure
22 results in a value of 2 for a 1 mode which is consistent with the expectation for
whole wave length mode shapes established in Section 4.2.1. The non-dimensional
system impedance can therefore be used to evaluate the changes in phase across various
locations in the fluid loop.
60
4.3 Non-Uniform Loop: Transfer Matrix Method
The Transfer Matrix Method that was implemented in MATLAB (Section 3.4)
allows for cylinders with different dimensions and materials to be combined to form a
continuous loop. The different segments that make up the non-uniform loop are assumed
to be cylindrical and have constant cross sectional dimensions and material properties
over the length of the element.
As shown in (3.23) the system matrix [T] sys relates the volume velocity and pressure
at the inlet of the system to the volume velocity and pressure at the outlet of the system.
The continuity boundary condition is applied at the inlet and the outlet to represent the
same axial location on the loop (3.26). The axial location of the inlet/outlet on a loop of
pipe is essential arbitrary. The same characteristic equations and roots are calculated
regardless of where the TMM matrices start and stop as long as they represent the same
loop.
In order to calculate the pressure, volume velocity and impedance of a non-uniform
loop as function of axial position an additional set of MATLAB scripts was developed.
This set of scripts was written to specifically exploit the fact that the system of interest is
a continuous loop. As discussed in Section 3.4 the TMM models simulate a loop by
assembling a set of straight finite segments into a system with continuity boundary
conditions on the ends. The straight finite length of the system establishes a frame of
reference where there is a starting and ending location of the simulated loop. The
MATLAB scripts were written to take a given loop system and “rotate” the TMM frame
of reference around the piping system loop.
The frame of reference is moved along the axis of the loop using a specified set of
axial locations. This is achieved by removing a short length from the starting element
and attaching it to the end of the last element. Since the system is a continuous loop,
moving a segment to the end of the loop results in the same system with a different
starting location for the frame of reference used for the analysis of the TMM models. By
rotating the frame of reference the pressure and volume velocity can be calculated at a
new inlet/outlet location. This method also allows the system impedance Zsys(z) and
non-dimensional impedance sys(z) to be calculated at the new starting location.
61
The pressure, volume velocity, system impedance and non-dimensional system
impedance can be calculated for any location around the loop by rotating the frame of
reference for the TMM model to “start” at that location. First the system impedance,
Zsys(z), is calculated at the frequency of a specific axial mode determined from the roots
of (3.30). Zsys(z) is the complex relationship between the complex pressure and volume
velocity at each axial location within the piping loop (4.10).
Z sys (z ) 
p (z )
q (z )
(4.10)
Zsys(z) is calculated from the TMM system matrix [T] sys using one of two methods
shown below in (4.11) which are obtained by solving (3.26) for pin / qin .
Z sys
x start

T 11  1 
T 12
T 21
T 22  1
(4.11)
Zsys(z) has the form of a tangent function for a non-uniform loop with axial locations
where the function approaches (+/-) infinity and becomes undefined. Based on the
numeric conditioning of the system matrix [T] sys one of the two equations in (4.11) is
selected to calculate Zsys(z) from the system matrix.
After calculating Zsys(z) the non-dimensional impedance sys(z) is calculated from
Zsys(z) as shown in (4.10). The non-dimensional impedance sys(z) is then used to
generate the normalized pressure mode shapes (P)(z) and displacement mode shapes
(X)(z) for the TMM model by taking the sine and cosine of sys(z) as shown below in
(4.12) and (4.13) respectively and discussed in [1]. Because sys(z) is a tangent function,
several steps are utilized in the MATLAB scripts for continuous normalized mode
shapes. The MATLAB scripts developed to support this analysis are documented in
Appendix C.
 P  z   sin  z 
(4.12)
 X  z   cos z 
(4.13)
62
The impedance and non-dimensional impedance functions can be used to identify
where the nodes and anti nodes of the mode shapes occur as previously mentioned in
Section 4.2.4. From the definition of Zsys(z) in (4.10), it can be seen that Zsys(z) goes to 0
when p(z) goes to 0. Similarly Zsys(z) goes to +/- infinity and becomes undefined when
q(z) goes to 0. The roots of the inverse of impedance or non-dimensional impedance
(1/Zsys(z) or 1/sys(z) ) identify the axial location of the anti-nodes of the pressure mode
shape.
Because the elastic boundary conditions in the axial direction will result in reflected
and transmitted waves, the resulting standing waves are expected to occur in a specific
orientation for each mode shape within the loop. The orientation of the mode shapes will
be a function of Zsys(z) or sys(z) which will be shown to be different for each of the
axial resonances within the non-uniform loop. The location specific system impedance
is different from the uniform loop case where, Zsys(z) was a constant at all locations
around the loop and had an infinite number of orientations within the loop for each
mode.
The change in volume as function of axial location Vf(z) is also calculated for each
mode. Using the pressure mode shape calculated by MATLAB (4.12) and the change in
volume of a differential fluid cylinder for an applied pressure (4.9) gives
V f z  
  P  z   V f z 
B f z 
(4.14)
All of the loops considered are closed systems, meaning no fluid either enters of
leaves the system. Therefore the summation of the change in volume around the entire
system must equal 0. The change in volume will be used in several of the following
sections as part of the analysis for a non-uniform loop. A complete set of results for each
TMM system analyzed can be found in Appendix D.
63
4.4 Non-Uniform Loop: Loop with a Single Cavity
To evaluate the effects of component cavities and how changes in the local axial
impedance affects the frequency and modes shapes of the axial fluid loop modes a single
cylindrical cavity was inserted into the previously analyzed uniform loop with rigid
walls. Figure 23 shows a representative diagram of a loop with a single cavity.
A2
L2
Lsys
A1
*Not to Scale
Figure 23 – Schematic of Loop System with a Single Cavity
The fluid cavity was assumed to be 2 meters long with a diameter of approximately
34.4 cm. The diameter of the cavity was chosen such that the cross sectional area within
the cavity would be 8 times the cross sectional area of the piping. The cavity dimensions
are arbitrary and were chosen to cause a noticeable change in acoustic impedance while
the length of the cavity remains small relative to the wave length of the axial modes. The
simulated piping loop and cavity were assumed to have rigid cylindrical walls. The
annular ends of the cavity were also assumed to be rigid. The axial fluid loop resonances
were then calculated for the system using TMM and FE models. Table 8 compares the
frequency of the first eight non-zero axial modes for the rigid wall loop with a single
cavity as calculated by TMM and COMSOL to the frequencies of the axial resonances
from the uniform loop. The results of two COMSOL models are included; a straight pipe
with continuity boundary conditions on the ends to simulate a loop and an actual loop.
Figure 24 shows the characteristic equation and roots for the loop with a single
cavity as calculated by TMM compared to the characteristic equations and roots for the
baseline uniform loop previously shown in Figure 15.
64
Mod
e
A
B
A
B
A
B
A
B
Loop w/ 8A
Cavity
TMM
Simulated Loop
w/ 8A Cavity
COMSOL FE
Actual Loop w/
8A Cavity
COMSOL FE
Uniform
Loop
TMM
Wave
Length
Frequency
Frequency
Frequency
Frequency
Multiple
[Hz]
31.2727
41.9813
68.7370
83.9428
109.0165
125.8593
150.2450
167.6916
[Hz]
31.1940
41.8628
68.4783
83.5565
108.6846
125.5890
149.6399
167.2518
[Hz]
30.7127
41.8612
68.1512
83.6459
108.3634
125.4856
149.5034
167.2227
[Hz]
40
40
80
80
120
120
160
160
[n
1
1
2
2
3
3
4
4
Table 8 – Axial Modes; TMM & COMSOL, “Loop” with 8A Cavity
Characteristic Equation
12
Ch EQ; 8A Cavity
Roots; 8A Cavity
Ch EQ; Uniform Loop
Roots; Uniform Loop
10
8
6
4
2
0
-2
-4
-6
-8
0
20
40
60
80
100
120
Frequency, Hz
140
160
180
200
Figure 24 – Characteristic Equation and Roots; TMM, Loop with 1 Cavity
As shown in Table 8 and Figure 24 the axial loop modes have split and shifted to
different frequencies. The pairs of modes for each whole wavelength multiple are no
65
longer located at the same frequencies. For each of the axial mode pairs one of the
modes has shifted lower in frequency while the other mode has shifted higher in
frequency. This result is similar to the original observations for the example system that
the axial modes where not spaced as a harmonic series and that two modes for each
whole wavelength multiple existed at different frequencies.
The characteristic equation shown in Figure 22 has changed significantly relative to
the characteristic equation for a uniform loop. The characteristic equation for the loop
with a single cavity is no longer a pure sinusoidal function and includes changes in
amplitude such that the zero crossings have spread out in frequency. The characteristic
equation for the uniform loop was completely positive while the characteristic equation
for the loop with a single cavity has negative values. Both of the characteristic equations
appear to oscillate about a value of 2 on the y-axis. Figure 25 provides the same
comparison given in Figure 22 but with a much higher frequency range. As shown in
Figure 25 the characteristic equation of the loop with a cavity displays a “beating” effect
over the range of frequencies shown.
Characteristic Equation
12
Ch EQ Elastic Loop
Roots Elastic Loop
Ch EQ Uniform Loop
Roots Uniform Loop
10
8
6
4
2
0
-2
-4
-6
-8
0
50
100
150
200
250
300
Frequency, Hz
350
400
450
500
Figure 25 – Characteristic Eq.; TMM, Loop w/ 1 Cavity, Larger Frequency Range
66
Table 8, Figure 24, and Figure 25 show that the roots of the characteristic equation
for the loop with a cavity shift away from the frequencies of the roots for the uniform
loop by different amounts for each root pair. The roots that make up each pair shift away
from each other by increasing amounts as the absolute magnitude of the extrema for
each cycle of the characteristic equation increases. The frequencies of the axial modes
for the rigid wall loop with a cavity shown in Table 8 are not harmonically spaced and
have shifted further in frequency than the modes for the simulated elastic loop shown
Table 7.
While the TMM code does not have a bound on the frequency span that can be
calculated, the low wave number assumption does limit the applicability of the analysis
to low frequencies. Figure 25 is provided to show a more representative portion of the
characteristic equation for the loop with a single cavity. The shifting of the frequencies
for the axial modes due to the addition of the cavity appears to be more complicated than
the shifting observed for a uniform change in elasticity discussed in Section 4.1.
The complexity of the changes observed for the frequencies of the axial modes due
to the insertion of a single cavity is more similar to the changes of the higher frequency
modes discussed in Section 2.1. As noted by previous authors studying the propagation
of radial and circumferential modes along an elastic fluid filled cylinder, the changes in
the phase were unique for each of the higher order modes. As shown in Table 8 the
changes to the frequency of the axial loop modes are also unique to each mode including
unique changes for each of the modes that make up a whole wave length pair.
The TMM surface plots of the pressure mode shapes for the 1 modes are shown in
Figure 26 and Figure 27 respectively. Mode “A” will be used to designate the mode that
has a pressure anti-node within the cavity, while Mode “B” will denote the mode with a
node located within the cavity. The corresponding pressure plots from the COMSOL FE
model of an actual loop with a single cavity are shown in Figure 28 and Figure 29.
Figure 26 through Figure 29 show the pressure modes using a color scale relative to
the layout of the loop models. In each of these figures it can be seen that the pressure
mode displayed is a 1 mode. A set of red arrows in Figure 28 and Figure 29 indicate the
approximate start location of the “loop” modeled using TMM.
67
Figure 26 – Pressure Mode Shape, Loop with 1 Cavity, 1 Mode A, Surface Plot
Figure 27 – Pressure Mode Shape, Loop with 1 Cavity, 1 Mode B, Surface Plot
68
Approx. TMM Start
Figure 28 – Pressure Mode Shape, Loop with 1 Cavity, 1 Mode A, COMSOL FE
Approx. TMM Start
Figure 29 – Pressure Mode Shape, Loop with 1 Cavity, 1 Mode B, COMSOL FE
69
A complete set of plots for the first four mode pairs from the TMM model is
included in Appendix D and the eight corresponding pressure mode figures from the
COMSOL models can be found in Appendix E.
A closer inspection of the pressure mode shapes for the 1 Mode A from the TMM
and COMSOL models identified several subtleties in the pressure mode shapes. In
Figure 26 and Figure 28 the color of the anti-node within the cavity does not reach the
maximum color on the scale. This indicates that the maximum pressure at the anti-node
within the cavity is less than the maximum pressure of the anti node within the piping.
Figure 26 and Figure 27 were generated to present the pressure modes calculated by
the TMM models in a manner that is similar to the presentation of the mode shapes from
the COMSOL FEA models. The results shown in Figure 26 and Figure 27 are actually a
projection of the 1D pressure mode shape calculated using the non-dimensional system
impedance. Figure 30 and Figure 31 present the 1 pressure mode shapes as 2dimensional linear plots with the pressure amplitude on the y-axis versus the axial
location on the x-axis. The pressure mode shapes have been normalized to such that an
anti-node in the first section of the TMM loop will have an absolute value of 1. The
green lines in Figure 30 and Figure 31 correspond to the pressures plotted as a color
scale in Figure 26 and Figure 27 respectively. The blue lines in Figure 30 and Figure 31
are the pressure mode shape plotted in green but shifted by a factor of  to show the
extremes of the mode shapes as they cycle through 0 to 2. Plotting the pressure modes
shapes twice out of phase by a factor of p helps to illustrate the “standing wave” nature
of the axial loop modes. A simple schematic of the loop and cavity are plotted in black
near the bottom of Figure 30 and Figure 31. The schematic is provided to put the
pressure mode results into context with the location of the cavity in the loop.
As shown in Figure 30 and Figure 31 both of the mode shapes maintain the required
continuity of pressure, however there are clear changes in the slope and amplitude of the
pressure modes shapes where the cavity meets the loop piping. There are discontinuities
in the slope of the pressure at the interfaces between the cavity and the piping that cause
“kinks” in the pressure mode shape.
70
Pressure Mode Shape, Root 2, 31.2727 Hz
1
Normalized Pressure
0.5
0
-0.5
-1
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 30 – Pressure Mode Shape, Loop with 1 Cavity, 1 Mode A
Pressure Mode Shape, Root 3, 41.9813 Hz
Normalized Pressure
1
0.5
0
-0.5
-1
0
5
10
15
20
25
Axial Location, m
30
35
Figure 31 – Pressure Mode Shape, Loop with 1 Cavity, 1 Mode B
71
40
The nodes of Mode A and the anti-nodes of Mode B are not evenly spaced around
the loop. For the pressure mode shapes in the uniform pipe, shown previously in Figure
12, the nodes of Mode A were located at the same locations as the anti nodes for Mode
B, and vice versa. As shown in Figure 30 and Figure 31 the anti-nodes of Mode A still
correspond to the nodes of Mode B however the reverse is no longer true.
The nodes for Mode A are located at 2.5832 m and 15.9293 m in a loop with a total
length of 37.025 m. The distance between the nodes as measured through the cavity is
13.3462 m while the distance between the nodes without passing through the cavity is
23.6788 m. For a single wavelength in the same liquid at the frequency of Mode A,
31.2727 Hz, the nominal distance between the nodes for a uniform wave would be
23.6788 m, the same as the distance between the nodes as measured without passing
through the cavity. Note that the distance between the nodes, 23.6788m, is not half of the
total system length, 37.025m as a uniform loop of the same material does not have an
integer whole wave length at 31.2727 Hz. The anti-nodes of Mode A are spaced evenly
are the loop with 18.5125 m between them.
For mode B, the nodes are equally spaced at 18.5125 m however the anti-nodes are
not centered between the nodes. The anti-nodes for Mode B are located 9.6931 m from
one node and 8.8194 m from the other node. As mentioned preciously, the nodes of
Mode B correspond to the anti nodes of Mode A but the anti-nodes of Mode B do not
correspond to the nodes of Mode A. At the frequency of Mode B in a uniform loop of
the same fluid the nodes of the pressure waves are located 17.6388 m apart, which is less
than half of the actual centerline length of the system.
In the uniform loop the nodes and anti-nodes of the mode pairs are spaced uniformly
and could occur in any orientation within the loop. For the loop with a single cavity the
nodes and anti-nodes are not spaced uniformly around the loop and the mode shapes
specifically align to center an anti-node or node on the cavity.
A review of the higher order mode pairs revealed that there are similar nonsymmetries in the spacing of the nodes and anti-nodes for all of the axial modes. The
non-uniformity in each mode is centered on the cavity. It was also noted that for each of
the mode pairs, Mode A was observed to always shift to a lower frequency while the
Mode B always shifted higher in frequency. Mode A and Mode B of each pair shifted in
72
frequency by different amounts and the changes in frequency are not symmetric about
the frequencies of the uniform loop.
For a loop with a single cavity it makes intuitive sense that a node and an anti-node
occur centered in the cavity. The system is essentially made up of two cylinders. Each
cylinder will have matching elastic boundary conditions in the axial direction which will
result in the pressure mode being symmetric within each cylinder.
Figure 32 and Figure 33 present the axial displacement mode shapes for Mode A
and Mode B of the 1 mode. The displacement mode shapes are normalized to the
maximum anti-node of each shape. The displacement mode shapes were also plotted
using the non-dimensional system impedance sys(z). As shown in Figure 32 and Figure
33 the axial displacement mode shapes have significant discontinuities where the cavity
connects to the piping.
The continuity conditions require that the volume velocity be continuous. By
extension the volume displacement must be also be continuous. Multiplying each axial
location in the displacement modes shapes in Figure 32 and Figure 33 by the cross
sectional area for each axial location produces the volume displacement mode shapes
shown in Figure 34 and Figure 35. The volume displacement curves in Figure 34 and
Figure 35, and the volume displacement curves for the other mode pairs given in
Appendix D show that the displacement modes satisfy the continuity requirements.
The discontinuity in the displacement mode shapes makes intuitive sense as well. A
fixed volume of fluid in two cylinders with different cross sectional areas will have
different axial lengths. As the fluid passes from the smaller cross sectional area to the
larger cross sectional area the volume must remain continuous and in contact with the
sides of the cylinders. If the end of the differential fluid volume is assumed to remain
plane, like a plane wave, the axial progression of the differential volume will slow down
as the cross sectional area increases. The discontinuity in axial displacement for the
sudden change in cross sectional area is consistent with expectations for the continuity of
volume velocity. The continuous function for the axial volume displacement verifies that
the conditions of continuity are satisfied for the calculated mode shapes. The continuous
displacement mode shapes have a somewhat different character than the continuous
pressure mode shapes.
73
Axial Displacement Mode Shape, Root 2, 31.2727 Hz
Normalized Axial Displacement
1
0.5
0
-0.5
-1
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 32 –Axial Displacement, Loop with 1 Cavity, 1 Mode A
Axial Displacement Mode Shape, Root 3, 41.9813 Hz
Normalized Axial Displacement
1
0.5
0
-0.5
-1
0
5
10
15
20
25
Axial Location, m
30
35
Figure 33 – Axial Displacement,, Loop with 1 Cavity, 1 Mode B
74
40
Axial Volume Displacement Mode Shape, Root 2, 31.2727 Hz
Normalized Axial Volume Displacement
1
0.5
0
-0.5
-1
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 34 – Axial Volume Displacement, Loop with 1 Cavity, 1 Mode A
Axial Volume Displacement Mode Shape, Root 3, 41.9813 Hz
Normalized Axial Volume Displacement
1.5
1
0.5
0
-0.5
-1
-1.5
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 35 – Axial Volume Displacement,, Loop with 1 Cavity, 1 Mode B
75
The change in volume as function of axial location Vf(z) was calculated for the first
four mode pairs as discussed in Section 4.2.2 using (4.8). Figure 36 and Figure 37
present the change in volume per unit length for Mode A and Mode B of the 1 mode.
The plots of the change in volume per unit length also show discontinuities at the
location of the cavity. The magnitude of the change in volume increases inside of the
cavity relative to the magnitude of the change in volume within the adjacent loop. As
shown in (4.8) the change in volume is a function of the applied pressure, the original
volume, the fluid density and the fluid’s phase velocity. The change in volume for an
applied pressure is higher for a larger original volume. The results in Figure 36 and
Figure 37 are consistent with the expectations from equation (4.8).
The loop with a cavity is a closed system. The net change in volume per unit length
over the length of the loop must be equal to 0 as discussed in Section 4.2.2. For Mode B
shown in Figure 37 the magnitude of the change in volume is symmetric about the node
located within the cavity. It is easy to see by inspection that the area above and below
the Z axis are equal. For Mode A, shown in Figure 36, the shape of the positive and
negative changes in volume are not similar. These regions have been shaded to highlight
the areas that are expected to be equal. Summing the numeric values plotted in Figure 36
using MATLAB it was confirmed that the region shaded in blue is equal to the region
shaded in grey indicating that the net change in volume is zero. The net change in
volume for the three higher mode pairs was also verified to be equal to 0, confirming
that the results are consistent with a closed system.
The larger volume of the cavity essentially acts like a soft volumetric spring in the
axial direction. This “softer” volume has a greater change in volume per unit pressure
than the remainder of the loop system. Hence the volumetric displacement is greatest at
the location of the softest volumetric spring.
76
Volume / Length [m3/m]
+ Vf
- Vf
Volume / Length [m3/m]
Figure 36 – Change in Volume, Loop with 1 Cavity, 1 Mode A
+ Vf
+ Vf
- Vf
Figure 37 – Change in Volume, Loop with 1 Cavity, 1 Mode B
77
As discussed in Section 3.3, the axial loop modes occur in pairs at multiples of
whole wave lengths. The most interesting feature about the results in Figure 30 through
Figure 37 is that for the same dimensions and material properties, two different modes
for each whole wave length occur within the loop at different frequencies.
If the loop modes are full wave lengths then the distance between the nodes
represents half a wave length or a change in the phase angle  of . The distances
between the nodes in the pressure modes of the loop with a single cavity are not equal.
This observation implies that either the phase velocity between the nodes is different or
there is some other feature causing there to be a rapid change or discontinuity in the
phase.
The system impedance, Zsys(z), and non-dimensional impedance, sys(z), were
calculated for the first four mode pairs using the “rotating” frame of reference discussed
in Section 4.3. The non-dimensional impedance and the inverse of the non-dimensional
impedance for Mode A and Mode B of the 1 mode are shown in Figure 38 and Figure
39 respectively. The roots to the non-dimensional impedance and its inverse are also
plotted in Figure 38 and Figure 39. The roots of the non-dimensional impedance
correspond to the nodes of the pressure mode and the roots of the inverse of the nondimensional impedance correspond to the anti-nodes of the pressure mode shapes.
As mentioned in Section 4.3 the system impedance, Zsys(z), has the form of a tangent
function. The non-dimensional impedance takes the form of a straight line that extends
from –/2 to /2 in amplitude. The non-dimensional system impedance sys(z) is
equivalent to the phase angle of the pressure mode shape (z)at a given axial location.
Figure 38 and Figure 39 show that there are discontinuities in sys(z) at the inlet and
outlet of the cavity. These discontinuities correspond with the discontinuities in the slope
of the pressure mode shapes, shown in Figure 30 and Figure 31 and the axial
displacement mode shapes in Figure 32 and Figure 33. The discontinuities at the cavity
inlet and outlet are the locations where there are sudden changes in the local acoustic
impedance, Zp.
78
Non-Dimensional Impedance, Root 2, 31.2727 Hz
Non-Dim Imp
inv Non-Dim Imp
Non-Dim Imp Roots
inv Non-Dim Imp Roots
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 38 – Non-Dimensional Impedance, Loop with 1 Cavity, 1 Mode A
Non-Dimensional Impedance, Root 3, 41.9813 Hz
Non-Dim Imp
inv Non-Dim Imp
Non-Dim Imp Roots
inv Non-Dim Imp Roots
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
5
10
15
20
Axial Location, m
25
30
35
40
Figure 39 – Non-Dimensional Impedance, Loop with 1 Cavity, 1 Mode B
79
The slope of sys(z) represents the change in the phase angle of the pressure mode
with respect to the change in axial location which is the axial wave number kzl as
discussed in Section 4.2.1. The slope of sys(z) is related to the angular frequency and the
phase velocity of the fluid in each segment of the system. The angular frequency must be
constant throughout the entire system. Since the entire system was assumed to have rigid
cylindrical walls, the phase velocity in each segment (the loop and the cavity) is equal to
the characteristic speed of sound in water. Therefore the slope of the non-dimensional
impedance is expected to be the same throughout the system for each frequency.
The slope of the non-dimensional impedance within the cavity was confirmed to be
the same as the slope in the remainder of the system. This can be seen in the nondimensional impedance plots for the uniform system shown in Figure 22. The nondimensional system impedance shows that there are no changes in the phase velocity
within the loop system containing a single cavity. While this result is expected based on
acoustic theory it is not immediately clear by inspection of only the pressure mode
shapes shown in Figure 26 through Figure 30.
Since the phase velocity is the same throughout the system and the center line length
of the system with a cavity is the same as the centerline length of the uniform system,
the net change in phase angle can only be equal to multiples of 2 at frequencies which
correspond to the roots of the uniform system. The summation of the change in phase
angle for the segments around the loop for the axial modes of the system with a single
cavity do not equal whole number multiple of 2.
The summation of the change phase angle for Mode A, shown in Figure 38, is
4.9123 radians and 6.5944 radians for Mode B shown in Figure 39. Mode A is
equivalent to (0.7818)*2 and Mode B is (1.0495)*2 These non-integer multiples of
2 are consistent with Mode A shifting down in frequency and Mode B shifting up in
frequency.
The discontinuity in the non-dimensional impedance at the inlet and the outlet of the
cavity for Mode A each account for a positive “jump” in the phase angle of (0.1091)*2
radians for a total of (0.2182)*2 radians. The discontinuities in Mode B account for a
negative “jump” in phase of (-0.02475)*2 radians each for a total of (-0.0495)*2
radians. When the summation of phase is calculated for each mode, including the change
80
in phase at the discontinuities, the net change in phase for Mode A and Mode B is equal
to 2. Although it is not provided in detail, the summation of the change in phase for all
of the higher order modes were also confirmed to be equal to the expected whole number
multiples of 2 when the change in phase at the impedance discontinuities was included.
The change in phase angle at the impedance discontinuities essentially changes the
effective length of the system. The positive change in phase angle in Mode A essentially
allows the phase to “jump ahead” and results in a longer effective system length of
47.3576 m. The negative change in phase angle in Mode B causes the phase angle to be
“set back” resulting in a shorter effective system length of 35.2776 m. The different
effective lengths result in full wave length axial modes occurring at frequencies that are
different from the frequencies of the axial loop modes in the uniform loop.
What still remains interesting about the change in phase at the impedance
discontinuities is that the change is different for Mode A and Mode B of the same whole
wave length pair. The change in phase at the impedance discontinuity is also different
for each of the higher order mode pairs and different between the Mode A modes and
Mode B modes. Each of the axial modes experiences a unique change in phase at the
impedance discontinuities even though the modes exist simultaneously in exactly the
same system. These modally specific shifts in frequency result in a non-harmonic
spacing of the axial mode which is consistent with the results observed from the initial
models of the example system.
What is also unique to each of the eight modes that make up the first four whole
wave length pairs is the amplitude and slope of the pressure mode shape where it
intersects with the impedance discontinuities. The amplitude and slope of the mode
shape represents specific magnitude and phase angle within the loop “piping” at the
location of the impedance discontinuity. If the sudden change in phase angle at the
impedance discontinuity is a function of the spatial orientation of each pressure mode
within the loop, then each pressure mode would experience a unique change in phase
and shift in frequency.
81
4.4.1
Phase Changes at Impedance Discontinuities
The change in phase angle of the pressure mode shape at the impedance
discontinuity was investigated further. A relationship was established to calculate the
change in phase angle using the system properties and dimensions, a known phase angle
for the pressure on one side of the discontinuity, the pressure cycle in the complex plane
and the linear Euler equation in the axial direction.
Figure 40 shows the unit circles for two pressure waves on the real and imaginary
plane. The two circles generically represent the pressure mode on either side of the
impedance discontinuity. The circle on the left represents the pressure within the piping
loop and has a magnitude, M1. The unit circle on the right represents the pressure in the
cavity, with a magnitude of M2. The axes of each unit circle have the same scale. The
dashed red line represents the level of the real pressure, P, which must remain
continuous across the interface. The phase angle shown for each unit circle represents
the phase of the pressure mode in the piping and the cavity on either side of the interface
where the sudden change in the impedance occurs. The phase angle, , also corresponds
to the non-dimensional system impedance sys(z) . As shown in Figure 38 and Figure 39,
there is a discontinuity in sys(z) at the interface, therefore 12..
Real (P1)
Real (P2)
P1 / 
P2 / 
M1
1
M2
PR
2
Imag (P1)
PR
Imag (P2)
Figure 40 – Unit Circles for Two Pressure Waves in the Complex Plane, “Mode A”,
on opposite sides of the Impedance Discontinuity
82
The real pressure PR that must remain continuous at the interface is related to the
absolute magnitude Mn, and the phase angle n of each pressure wave as shown below
PR 1   M 1 sin 1 
PR  2   M 2 sin  2 
(4.15)
The change in the real pressure with respect to the change in phase angle, or the
slope of the pressure on the unit circle, is annotated in Figure 40 and given by (4.16)
 PR 

  M 1 cos1 


 1
 PR 

  M 2 cos 2 


 2
(4.16)
The real portion of the pressure must remain continuous across the impedance
discontinuity as discussed in Section 4.2.3. The continuity of real pressure between the
two unit circles is easily observed in Figure 40 and is annotated with the red dashed line.
The continuity of volume velocity must also be maintained across the impedance
discontinuity. Using the linear Euler equation and the continuity of volume velocity it
can be shown that the slope of the real pressures on each of unit circles must be
proportional to the acoustic impedance on either side of the discontinuity. The linear
Euler equation in the axial direction, as given in (A.8c) of Appendix A is
P
 v z 

z
 t 

(4.17)
Defining the volume velocity in the axial direction, qn, where the subscript “n”
denotes a segment number, and the change in the axial fluid particle velocity vz / t as
q n  v z n A xn
 v z n 

  i v z n
 t 
(4.18)
The linear Euler equation in the axial direction becomes
 qn
 A xn
 n i
  P 
  
 0
  z  n
83
(4.19)
Rearranging (4.19), it can be seen in (4.20) that the linear Euler equation relates the
change in pressure with respect to the axial location along the loop to the volume
velocity. Equation (4.20) is consistent with the form of the liner Euler equation provided
in [1] as part of the derivation of the 1-D linear acoustic wave equation.
 A  P 
iq n   xn 
 0
  n  z  n
(4.20)
Using (4.20) the slope of the pressure with respect to axial location can be defined
for the fluid on either side of the interface as
 P    i1q1 


  
 z 1  A x 1 
 P    i 2q 2 


  
 z  2  A x 2 
(4.21)
Taking the ratio between the slope of the pressure within the cavity and the slope of
the pressure within the piping and eliminating  and q gives
 P 


 z  2 i 2q 2 A x 1  2 A x 1


i1q1A x 2 1A x 2
 P 


 z 1
(4.22)
Simplifying (4.22), the changes in pressure with respect to axial location on either
side of the interface are shown to be related. Not only must the pressure remain
continuous across the impedance discontinuity but the slope of the pressure on either
side of the impedance discontinuity must also satisfy the relationship in (4.23) which is
based on the material properties and dimensions of the system.
 P   P    2 A x 1 


 
 
 z  2  z 1  1A x 2 
(4.23)
The change in pressure with respect to the axial location in the loop is a function of
the change in pressure with respect to the phase angle and the change in phase angle
with respect to axial location
84
 P   P

  
 z  n   n
  n 


 z 
(4.24)
The change in phase angle relative to the axial location for each cylindrical section n
is the axial wave number kzl for each segment of the system (4.25).

  n 

  k zl 
cn
 z 
(4.25)
Substituting (4.16) and (4.25) into (4.24) relates the change in pressure along the
axis of the loop from the linear Euler equation to the change in pressure relative to the
phase angle derived from the unit circle in Figure 40.

 P 

  M n cos n 
 z  n
 cn



(4.26)
Substituting (4.15) into (4.26) for the magnitude of the pressure Mn gives
 P    

   P cot  n 
 z  n  c n 
(4.27)
Substituting (4.27) into the relationship between the slopes of the pressure (4.23)
provides a relationship between the phase angle on either side of the interface
 
 
 A 
 P cot  2    P cot 1  2 x 1 
 c2 
 c1 
 1A x 2 
(4.28)
c  A 
cot  2   cot 1  2 2 x 1 
 c1 1A x 2 
(4.29)
Substituting the acoustic impedance for a cylindrical element defined previously in
(3.19) and solving for the phase angle after the discontinuity gives
85
Z 
cot  2   cot 1  p 2 
Z 
 p1 



 Z p2 


Z
 p1  
 2  cot 1  cot 1 
(4.30)
(4.31)
Equations (4.30) and (4.31) relate the phase angles of the pressure mode on either
side of the impedance discontinuity. If the phase angle, which is also the nondimensional system impedance, is known on either side of the discontinuity then the
phase angle on the opposite side of the discontinuity can be calculated as well as the
change in phase 1-2 due to the discontinuity.
12   2  1
(4.32)
Since the real pressure must remain continuous across the impedance discontinuity
equation (4.30) can be rearranged using (4.15) and (4.16) to provide the slope
relationship in more intuitive form.
 P   P  Z 2 

  






Z
 2   1  1 
(4.33)
As shown in (4.33), the slope of the pressure modes on either side of the
discontinuity, annotated on the unit circle in Figure 40, are related through the ratio of
the acoustic impedance for the cylinders on either side of the discontinuity.
To maintain continuity in the real pressure and the slope relationship shown in
(4.33) the unit circle after the discontinuity must have a different magnitude and phase
angle as shown in Figure 40. The magnitude of the pressure after the discontinuity can
be found using either (4.34) or (4.35).
M2 
P
sin  2 
86
(4.34)
M 2 sin 1 

M 1 sin  2 
(4.35)
The sudden changes in phase shown in Figure 38 and Figure 39, the discontinuities
in the slope of the pressure mode shapes and the differences in magnitude of the pressure
modes in the cavity versus the piping shown in Figure 30 and Figure 31 are consistent
with the expectations for continuity shown in (4.33) and (4.35). Essentially the
continuity conditions require that the slope of the pressure must change in proportion
with the change in impedance while the real portion of the pressure remains constant.
Mode A and Mode B of the 1 mode can used to illustrate the unit circle and
continuity concepts discussed above. Figure 41 shows the non-dimensional impedance
the pressure mode shape for Mode A of the 1 mode, as well as the pressure mode
shape.
Non-Dimensional Impedance; Normalized Pressure
Non-Dimensional Impedance, Root 2, 31.2727 Hz
Non-Dim Imp
inv Non-Dim Imp
Non-Dim Imp Roots
inv Non-Dim Imp Roots
Pressure Mode A
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
Inlet
-2
0
5
Outlet
10
15
20
25
Axial Location, m
30
35
Figure 41 – Mode A 1, Pressure and Non-Dimensional Impedance
87
40
The cavity has the same fluid density and speed of sound as the piping of the loop in
this example. The only difference between the cavity and the piping is that the cross
sectional area of the cavity is 8 times larger than the cross sectional area of the piping.
This results in an impedance ratio Z1/Z2 at the cavity inlet of 0.125 and an impedance
ratio of Z2/Z1 of 8 at the cavity outlet. Substituting the inlet impedance ratio into
Equation (4.33), the slope of the pressure on the cavity side of the inlet will be 1/8th of
the pressure slope on the pipe side of the inlet.
As shown in Figure 41, the slope of the pressure in the piping at the cavity inlet is
positive but decreasing towards a slope of zero. Since the slope of the pressure is
decreasing with increasing axial position reducing the slope of the pressure to 1/8 of the
original value the phase angle essentially “jumps ahead”. The change in phase angle at
the impedance discontinuity is 0.6854 radians or approximately 39.27 degrees. At the
outlet of the cavity, the slope of the pressure within the cavity is negative and is
becoming more negative. The impedance ratio at the outlet of the cavity to the piping is
8. With a ration of 8 the slope of the pressure becomes suddenly more negative at the
outlet discontinuity again causing the phase angle to “jump ahead”. The magnitude of
the change in phase angle at the outlet is the same as at the inlet, 0.6854 radians. These
two discontinuities in the phase angle plus the changes in phase angles over the
cylindrical segments of the system result in an effective system length of 47.3576 m,
which is more than 10 m longer than the centerline of the physical system. The longer
effective length results in the frequency of the 1 Mode A occurring at a frequency that
is significantly lower than the 1 Mode A of the uniform loop.
Figure 42 provides the effective unit circle for Mode A of the 1 mode in the loop
with a single cavity. The blue arrow approximately represents the phase angles that
occur within the piping while the green arrow approximately represents the phase angles
that occur within the cavity.
The discontinuities in the unit circle represent the instantaneous changes in phase
due to the impedance discontinuities. Note that the real portion of the pressure is
continuous at the inlet and outlet of the cavity and that the magnitude of the pressure
within the cavity is different that the magnitude of the pressure in the loop. The
88
summation of the change in phase due to the length of the piping, the length of the cavity
and due to the impedance discontinuities is equal to 2.
Real (P)
Cavity
M2
PR
M1
Imag (P)
Piping
Figure 42 – Mode A 1, Effective Unit Circle
The change in the slope of the pressure causes the phase angle to either increase
(jump ahead) or decrease (be set back). The changes in phase angle at the impedance
discontinuities combine with changes in the phase angle over the cylindrical lengths of
the system resulting in an effective system length that can be longer or shorter the actual
physical system length. The axial loop modes occur at the frequency where the effective
system lengths result in whole wave length multiples.
Figure 43 shows two unit circles representing the pressure of Mode B in the piping
and the cavity, similar to those shown previously in Figure 40 for Mode A. Figure 44
presents the non-dimensional impedance the pressure mode shape for Mode B of the 1
mode. Note that because Mode B pressure mode shape has a node or zero within the
cavity that the phase angle where the pressure mode encounters the cavity is in a
different location on the unit circles. The pressure at the inlet and the outlet of the cavity
for Mode B are of the same magnitude however the inlet pressure is positive while the
outlet pressure is negative.
89
Real (P1)
Real (P2)
P()
M1
-P()
-P()
1
P()
M2
2
Imag (P1)
Imag (P2)
P / 
P / 
Figure 43 – Unit Circles for Two Pressure Waves in the Complex Plane, “Mode B”,
on opposite sides of the Impedance Discontinuity
Non-Dimensional Impedance; Normalized Pressure
Non-Dimensional Impedance, Root 3, 41.9813 Hz
Non-Dim Imp
inv Non-Dim Imp
Non-Dim Imp Roots
inv Non-Dim Imp Roots
Pressure Mode B
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
Inlet
-2
0
5
Outlet
10
15
20
25
Axial Location, m
30
35
Figure 44 – Mode B 1, Pressure and Non-Dimensional Impedance
90
40
Mode B exists in the same loop with a single cavity as Mode A so the impedance
ratio at the inlet is 0.125 and 8 at the outlet. The slope of the pressure for Mode B in the
piping at the inlet to the cavity is negative and becoming more negative as it approaches
the maximum negative slope which occurs when the pressure is equal to 0. Multiplying
the negative slope by a factor of 1/8 essentially sets the phase angle back by a small
amount. At the outlet of the cavity the slope is negative and becoming less negative as
the slope approaches a value of 0 at the anti-node of the pressure mode shape.
Multiplying the pressure slope at the outlet by a factor of 8 also “sets back” the phase
angle by a small amount.
Figure 45 presents the effective unit circle for Mode B. The real value of the
pressure remains continuous at the inlet and outlet of the cavity. The magnitude of the
pressure within the cavity is less than that of the pressure magnitude in the piping.
Real (P2)
Cavity
M2
P()
-P()
M1
Imag (P2)
Piping
Figure 45 – Mode B 1, Effective Unit Circle
It can be seen more clearly in Figure 45 that by maintaining the real value of
pressure (on the vertical axis) and increasing the slope of the pressure causes a small “set
back” in the phase angle when transitioning from the piping to the cavity and vice versa.
The change in phase angle at the inlet and outlet is -0.1556 radians or approximately 91
8.92 degrees. The changes in phase angle are negative at the impedance discontinuities
for Mode B indicating a clockwise rotation on the unit circle in Figure 45. The negative
changes in phase angle result in an effective system length of 35.2776 m, a length which
is shorter than the centerline length of the physical loop and cavity. The shorter effective
length results in the frequency of Mode B shifting higher in frequency. The “set back” in
phase angle, is equivalent to the discontinuity in the non-dimensional impedance shown
in Figure 44.
Summing the change in phase angle within the piping and the cavity and then
including also the negative phase angle changes at the impedance discontinuities results
in a total of 2. This is consistent with the expectation set by the full uniform loop of
whole wave length axial modes
The analysis of the unit circle also implies that if instead of a cavity a constriction
were introduced into the loop, the loop modes would shift in the opposite direction in
frequency. If a constriction with 1/8th the cross sectional area of the piping with the same
length as the cavity were inserted into the system, the ratios of acoustic impedance at the
inlet and outlet would reverse. The reversal of the impedance ratios indicates that Mode
B would shift down in frequency while Mode A shifts up in frequency.
Using the TMM models the cross sectional area dimension of the cavity was varied
from 1/8th to 8 times the cross sectional area of the piping. The TMM results for the
frequencies of the first 8 axial loop modes are presented in Table 9.
1
1
2
2
3
3
4
4
Cav. Area
Mode A
Mode B
Mode A
Mode B
Mode A
Mode B
Mode A
Mode B
0.125
41.9813
31.2727
83.9428
68.737
125.8593
109.0165
167.6916
150.245
0.25
41.6832
35.0227
83.3299
72.4091
124.8952
111.8088
166.3121
152.3027
0.5
41.1016
38.0466
82.1493
76.5883
123.0814
115.8241
163.82
155.6908
1
40
40
80
80
120
120
160
160
2
38.0467
41.1016
76.5883
82.1493
115.8241
123.0814
155.6908
163.82
4
35.0227
41.6832
72.4091
83.3299
111.8088
124.8952
152.3027
166.3121
8
31.2727
41.9813
68.737
83.9428
109.0165
125.8593
150.245
167.6916
Table 9 – Axial Mode Frequencies (Hz); TMM, Effect of Cavity Cross Sectional
Area
92
The TMM results presented in Table 9 confirm that for a constriction, Mode B of
each pair shifts down in frequency while Mode A shifts higher in frequency. It is also
interesting to note the symmetry in the frequency shifts between the Mode A and Mode
B pairs where they obtain the same frequency shift due to the opposite changes to the
insert impedance.
Figure 46 presents the characteristic equations for the uniform loop and those of the
increased cavity sizes presented in Table 9. The axial modes occur at the frequencies
where the characteristic equation is equal to zero. In Figure 46 Mode A of each pair has
shifting lower in frequency while Mode B of each pair has shifting higher in frequency.
If the decreases in cavity area had been plotted instead, the exact same curves would be
obtain however Mode B of each pair would have shifted lower in frequency while Mode
A of each pair shifter higher in frequency. Figure 47 plots the frequency of the roots
provided in Table 9 and shown in Figure 46 versus the ratio of the cross sectional area of
the insert relative to the piping.
Characteristic Equation
10
5
0
-5
Ch EQ;
Ch EQ;
Ch EQ;
Ch EQ;
-10
0
20
Uniform Loop
2A Cavity
4A Cavity
8A Cavity
40
60
80
100
120
Freqeuncy, Hz
140
160
180
200
Figure 46 – Characteristic Eq.; TMM Variation in Cavity Cross Sectional Area
93
Roots of Characteristic Equations
8
ModeA
ModeB
7
Insert Area Multiple
6
5
4
3
2
1
0
0
20
40
60
80
100
120
Frequency, Hz
140
160
180
200
Figure 47 – Roots of Char. Eq.; TMM Variation in Cavity Cross Sectional Area
For this study only the cross sectional area of the insert (cavity or constriction) was
changed. The acoustic impedance however is defined by the fluid density, the axial
phase velocity and the cross sectional area as shown in (3.19). Any sudden change that
can affect these properties will cause a sudden change in the impedance. The sudden
change in impedance will cause discontinuities in the phase angle and shift the frequency
of the loop modes away from the frequencies of the uniform loop.
Cross sectional area can vary widely in piping systems and fluid components. The
fluid density within a single fluid system will not likely change much over a short
section but changes due to temperature and pressure are possible. Changes in the axial
phase velocity can occur for a variety of reasons as discussed previously. All of these
changes can impact the frequency of the axial loop modes and must be considered to
determine the frequencies of the axial loop modes.
94
4.5 Non-Uniform Loop: Elastic Discontinuity
The elasticity of the pipe and component walls reduces the axial phase velocity
within the fluid as previously discussed in Section 4.1. Reducing the axial phase velocity
lowers the frequencies of the axial loop resonances. The effect of elasticity on the axial
loop modes can occur in addition to the effects caused by the impedance discontinuities.
As defined in (3.19) the axial phase velocity along with the density and cross sectional
area defines the local acoustic impedance of each section of the loop. Reducing the
phase velocity also reduces the value of the local acoustic impedance which in turn will
affect the ratios of impedances in (4.31) and influence the phase shift at each impedance
discontinuity.
To demonstrate the influence of elasticity the phase velocity of the single cavity was
adjusted to represent an elastic shell around cavity while the remainder of the loop was
assumed to have a rigid wall. The single cavity in the loop has a cross sectional area that
is 8 times the cross section area of the pipe, the thickness of the cylindrical shell around
the cavity was assumed to be the same as the piping thickness shown in Figure 2. The
annular surfaces that connect the cylindrical pipe walls to the cylindrical cavity wall are
assumed to be rigid. Two cases were evaluated; a steel cavity shell and an aluminum
cavity shell using the material properties in Table 1.
Using the Korteweg-Lamb correction factor (4.1) the modified axial phase velocity
of the steel cavity was calculated to be 1265.906 m/s or 85.48 % of the rigid wall phase
velocity of water. The modified phase velocity of the aluminum cavity was calculated to
be 1062.67 m/s or 71.75 % of the rigid wall axial phase velocity of water. Figure 48
presents the characteristic equations of the loop with a single cavity where the
cylindrical cavity wall is rigid, steel and aluminum.
95
Characteristic Equation
15
Ch EQ. RW Cavity
Ch EQ. Steel Cavity
Ch EQ. Aluminum Cavity
10
5
0
-5
-10
0
20
40
60
80
100
120
Freqeuncy, Hz
140
160
180
200
Figure 48 – Characteristic Eq.; TMM, Variation in Cavity Elasticity
The roots of the characteristic equations in Figure 48 show that the elasticity of the
cavity wall causes the axial loop modes to shift in frequency by different amounts for
each mode. Table 10 gives the frequency of the first 8 modes for the loop with a single
cavity as calculated using the TMM and FEA models. The results from the TMM and
FEA models are again in reasonable agreement. The frequencies of the rigid wall
uniform loop and the rigid wall loop with a single cavity are also provided for
comparison. As shown in Table 10 the axial loop modes shift to lower frequencies with
Mode A of each pair shifting further than the associated Mode B of the pair. The
magnitude of the frequency shift is also more significant at the lower wave length
modes. Mode A of the 1 pair has shifted approximately 22% and 30 % lower in
frequency than the 1 Mode A of the uniform loop for the steel and aluminum cases
respectively.
96
Frequency
[Hz]
Actual
Loop w/ 8A
Cavity
Steel
COMSOL
FE
Frequency
[Hz]
Actual
Loop w/ 8A
Cavity
Aluminum
COMSOL
FE
Frequency
[Hz]
Multiple
[n]
29.5663
27.8116
29.5058
27.74077
1
41.9813
41.9801
41.9782
41.80797
41.80612
1
80
68.737
67.4377
66.3016
67.2129
66.06399
2
B
80
83.9428
83.9323
83.9152
83.55364
83.53685
2
A
120
109.0165
108.1026
107.3417
107.6736
106.9068
3
B
120
125.8593
125.8179
125.7447
125.3115
125.2404
3
A
160
150.245
149.555
148.9886
148.9442
148.3744
4
B
160
167.6916
167.5657
167.3112
166.9011
166.6569
4
Uniform
Loop
Loop w/ 8A
Cavity
Loop w/ 8A
Cavity
Loop w/ 8A
Cavity
Rigid Wall
Rigid Wall
Steel
Aluminum
TMM
TMM
TMM
TMM
Mode
Frequency
[Hz]
Frequency
[Hz]
Frequency
[Hz]
A
40
31.2727
B
40
A
Wave
Length
Table 10 – Axial Modes; TMM & COMSOL, Variation in Cavity Elasticity
The pressure modes shapes are similar in all three cases for the loop with a single
cylindrical cavity. For the 1 Mode A pressure mode shapes the anti-nodes were
observed to stay in the same location. The nodes of Mode A moved closer to the cavity
as the cavity became more elastic and the frequency shifted to a lower value. The Mode
B pressure mode shapes are nearly indistinguishable between the three single cavity
cases.
Figure 49 compares the 1 Mode A pressure mode shapes from the loop with a
single rigid wall, steel and aluminum cavity. The solid line and the dotted line of the
same color represent a single mode shape and are out of phase by a factor of . The antinode within the cavity becomes lower in amplitude as the elasticity of the cavity wall
increases. This is consistent with the conditions of continuity for pressure and volume
velocity as discussed in Section 4.4.1. The full sets of figures for each of the single
cavity cases can be found in Appendix D.
97
Pressure Mode Shape, 1 Wave Length, Mode A
1L ModeA 31.2727 Hz Rigid Wall
1L ModeA 29.5663 Hz Steel
1L ModeA 27.8116 Hz Aluminum
Normalized Pressure
1
0.5
0
-0.5
-1
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 49 – Pressure Mode Shape, 1 Mode A; TMM, Variation in Cavity
Elasticity
The elasticity and the changes in impedance affect the frequencies of the axial loop
modes. When both occur in the same system the modally specific shifts in frequency will
result in a unique frequency spacing. These results support the original observations of
the frequency spacing of the axial loop modes within the original example system. The
elastic cavity examples demonstrate how the choice of structural materials for the
components and piping will affect the frequencies and shapes of the loop modes.
4.6 Non-Uniform Loop: Elbows and Pipe Bends
The impact that the elbows have on the loop modes was discussed briefly in Section
3.5. The frequencies of the axial loop modes calculated from an FEA model of a
simulated loop and an actual loop were presented in Table 5. In general all of the loop
modes in the actual loop model occur at slightly higher frequencies than the same mode
in the simulated loop with the exception of the 2 Mode A.
98
Rostafinski studied the propagation of waves in curved ducts in [44]. He found that
the long wave length waves traveled through a curved duct at a phase velocity that was
higher than the phase velocity in a straight section of duct with an equivalent center line
length. It was demonstrated that the increase in the phase velocity was proportional to
the sharpness of the bend in the duct.
In [45] El-Raheb also found that the phase velocity of long wave lengths traveled
through bends in rectangular ducts faster than equivalent length straight ducts. This
resulted in the positive shift in frequency for the low frequency resonances of the bend.
El-Raheb concluded that the frequency of the axial resonances within the duct shifted up
in some frequency ranges and down in other ranges as part of “a cyclic and involved
mechanism”.
The volume velocity from the inlet to the outlet of the duct or pipe bend would have
to remain equal to maintain the continuity in volume velocity across the bend. The
pressure at the inlet and outlet of the elbow must remain continuous with the adjacent
ducting or piping. If the bend or elbow results in an increased phase velocity for low
frequency axial waves then the elbow will cause a slight impedance discontinuity in the
loop. As discussed in Section 4.4.1 changes to the impedance also cause changes to the
frequency of the axial loop modes.
An increase in the phase velocity would result in a higher acoustic impedance in the
elbow than the adjacent piping assuming that the density of the fluid and the cross
sectional areas in the piping and elbow are constant. The impact that the change in
impedance will have on each mode will be based on the magnitude and the slope of the
pressure mode where the mode shape encounters each elbow. These changes will be
small based on the results in [44] and [45].
In general the effect of the elbows on the low frequency axial modes is very small as
previously shown in Table 5. Approximating the loop in the TMM models as a straight
pipe with continuity boundary conditions on the ends is still a reasonable engineering
approximation of the loop introducing less than a 1 % error in the frequency of the axial
loop modes. Additional analyses of acoustic waves in elbows and bends can be found in
[46], [47], and [48].
99
4.7 Full System
The axial fluid loop modes for the example system shown in Figure 1 were
evaluated using the modeling techniques and assumptions developed in Section 3
through Section 4.6. The frequencies and mode shapes of the axial loop modes for the
example system were calculated using TMM and FEA models. The COMSOL FEA
model and a schematic of the TMM model of the example system are shown in Figure
50. The dimensions of the example system are in accordance with the dimensions
provided in Table 2 and Figure 2 - Figure 4. The centerline length of the example system
is 37.025m which is the same as the centerline length of the uniform and non-uniform
loops evaluated in Sections 3.3 through 4.4. The cylindrical cavity walls have the same
thickness as the piping given in Figure 2 and the annular cavity walls are assumed to be
rigid in all cases.
TMM Start
L3, A3
L2, A2
L3, A3
Lsys, A1
TMM Example System Model
Approx. TMM Start
L3, A3
L3, A3
L2, A2
Lsys, A1
FEA Example System Model
Figure 50 – Example System with Three Cavities; TMM and COMSOL FEA
models
100
The example system was analyzed using the material properties, dimensions and
assumptions discussed in Section 2.2. The example system was evaluated with rigid
walls, elastic steel walls and elastic aluminum walls. The elasticity of the steel and
aluminum walls was incorporated into the TMM and FEA models by modifying the
speed of sound for the water using the Korteweg-Lamb correction factor discussed in
Section 4.1, Section 4.5 and Appendix B.
Table 11 provides the frequency of the first four, non-zero mode pairs for the three
cylindrical wall boundary conditions as calculated by the TMM and COMSOL FEA
models. The whole wave length multiple and mode designation (A or B) for each of the
axial modes are also listed in Table 11.
Wave
Length
Multiple
[n]
TMM
TMM
TMM
Frequency
[Hz]
Frequency
[Hz]
Frequency
[Hz]
Example
System
Rigid Wall
COMSOL
FE
Frequency
[Hz]
A
1
30.5352
27.2539
23.7382
30.4836
27.2028
23.6909
B
1
39.3763
36.6634
33.3931
39.2190
36.5185
33.2641
A
2
69.4573
63.8355
57.4958
69.1463
63.5194
57.1889
B
2
74.7659
69.5400
63.3773
74.3822
69.1724
63.0371
A
3
105.9187
98.5764
89.9189
105.5981
98.2619
89.6254
B
3
118.5776
110.7070
101.2437
117.6924
109.8507
100.4425
A
4
141.7038
132.3144
121.1174
141.0630
131.6906
120.5359
B
4
166.3593
156.3237
143.7695
165.3277
155.3608
142.9017
Mode
Example
System
Rigid Wall
Example
System
Steel
Example
System
Aluminum
Example
System
Steel
COMSOL
FE
Frequency
[Hz]
Example
System
Aluminum
COMSOL
FE
Frequency
[Hz]
Table 11 – Axial Modes; TMM & COMSOL, Example System
The axial loop modes occur in pairs in agreement with the results discussed in
Section 3.3. The results from the TMM models are in good agreement with the results
from the COMSOL FEA models. The calculated frequencies of the axial loop modes
show less than a 0.8 % difference between the TMM and FEA models.
101
The frequencies of the axial loop modes in the Example systems have shifted away
from the frequencies of the axial loop modes of the uniform rigid wall loop presented in
Table 3 - Table 5 . In general the frequencies of both the A and B modes for the three
example systems have shifted lower in frequency. The maximum difference in frequency
occurs at the 1 Mode A for the Aluminum system which has shifted approximately 40
% lower in frequency. Only the 4 Mode B for the rigid wall case has shifted higher
than the frequency of the associated mode in the rigid wall uniform loop.
The characteristic equations for the Example systems with rigid walls, elastic steel
walls and elastic aluminum walls are shown in Figure 51 and Figure 52. The
characteristic equation for the uniform rigid wall loop is also shown in both figures for
reference. Figure 52 presents the characteristic equations over a smaller frequency range
and more clearly shows 1 and 2 mode pairs.
The shifting of the axial loop modes to lower frequencies is in agreement with the
results and expectations established in Section 3.3 through Section 4.4. The frequencies
of the axial loop modes are affected by the reduction in the phase velocity due to the
elasticity of the cylindrical walls and by the impedance discontinuities where the
components meet the system piping.
Note that there are more roots shown in Figure 51 than there are reported in Table
11. This is due to the roots of the 5 modes shifting down in frequency and occurring
below 200 Hz where the 5 modes occur for the rigid wall uniform loop.
102
Characteristic Equation
40
20
0
-20
Ch EQ, Uniform Loop, RW
Roots, Uniform Loop, RW
Ch EQ, 3 Cav Sys, RW
Roots, 3 Cav Sys, RW
Ch EQ, 3 Cav Sys, Steel
Roots, 3 Cav Sys, Steel
Ch EQ, 3 Cav Sys, Aluminum
Roots, 3 Cav Sys, Aluminum
-40
-60
-80
0
20
40
60
80
100
120
Freqeuncy, Hz
140
160
180
200
Figure 51 – Characteristic Eq.; TMM, Example System, RW, Steel & Aluminum
Characteristic Equation
30
Ch EQ, Uniform Loop, RW
Roots, Uniform Loop, RW
Ch EQ, 3 Cav Sys, RW
Roots, 3 Cav Sys, RW
Ch EQ, 3 Cav Sys, Steel
Roots, 3 Cav Sys, Steel
Ch EQ, 3 Cav Sys, Aluminum
Roots, 3 Cav Sys, Aluminum
25
20
15
10
5
0
-5
-10
0
10
20
30
40
50
60
Freqeuncy, Hz
70
80
90
100
Figure 52 – Characteristic Eq.; TMM, Example System, RW, Steel & Aluminum
103
As shown in Figure 51 and Figure 52, the characteristic equations for the example
systems are significantly more complex than the previously shown characteristic
equations in Figure 24 and Figure 46. The characteristic equation for the rigid wall
system shows the changes caused by the impedance discontinuities at the sudden
changes in cross sectional area. The characteristic equations for the elastic steel and
aluminum walls include the effects of the reduced phase velocities and the impedance
discontinuities.
The 1 Mode A and Mode B pressure mode shapes calculated by the COMSOL
FEA model of the rigid wall example system are shown in Figure 53 and Figure 54
respectively. Figure 55 and Figure 56 show the pressure mode shapes for the 1 Mode A
and Mode B for the rigid wall, elastic steel wall and elastic aluminum wall example
system as calculated by the TMM model. It can be seen in Figure 53 through Figure 56
that the modes resemble the full wave length modes shown in previous analyses.
However the pressure mode shapes in Figure 55 and Figure 56 show that the 1 modes
in the example system have some relatively significant differences from the purely
sinusoidal pressure modes shapes of the uniform loop. The non-sinusoidal shapes of the
pressure modes are more obvious in Figure 55 and Figure 56 than in Figure 53 and
Figure 54. A close inspection of Figure 53 and Figure 54 show that the color
distributions from the FEA model results are in agreement with the pressure mode
shapes Figure 55 and Figure 56.
The pressure mode shapes for the example system have several sudden changes in
the slope of the pressure. The changes in the slope of the pressure occur at the locations
where impedance differences occur between the components and the piping, which is
consistent with the results presented in Section 4.4.1.
104
Approx. TMM Start
Figure 53 – Pressure Mode Shape, Example System RW, 1 Mode A, COMSOL FE
Approx. TMM Start
Figure 54 – Pressure Mode Shape, Example System RW, 1 Mode B, COMSOL FE
105
Pressure Mode Shape, 1L Mode A
1.2
RW, 30.5352 Hz
Steel, 27.2539 Hz
Aluminum, 23.7382 Hz
1
0.8
Normalized Pressure
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 55 – Pressure Mode Shape, 1 Mode A TMM, Example System
Pressure Mode Shape, 1L Mode B
RW, 39.3763 Hz
Steel, 36.6634 Hz
Aluminum, 33.3931 Hz
2
1.5
Normalized Pressure
1
0.5
0
-0.5
-1
-1.5
-2
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 56 – Pressure Mode Shape, 1 Mode B; TMM, Example System
106
In general the pressure modes of the three example systems are similar, occurring at
slightly different frequencies and showing small differences in the amplitudes of the
mode shape. The pressure mode shapes shown in Figure 55 and Figure 56 occur at
different frequencies as noted in the legend and Table 11. A complete set of plots for the
first 8 pressure mode shapes calculated with the TMM and FEA models can be found in
Appendix D and Appendix E.
The 1 Mode A and 1 Mode B displacement mode shapes for the example system
with rigid walls are shown in Figure 57 and Figure 58. The displacement modes shapes
are similar to the displacement mode shapes for the 1 Mode A and Mode B in the
example system with elastic steel and aluminum walls. Similar to the results presented in
Section 4.4 for a non-uniform loop with a single cavity, the displacement mode shapes
are not continuous. The axial displacement within the component cavities is reduced due
to the increase in cross sectional area and changes in the phase velocity. The volume
displacement was confirmed to be continuous for each of the mode shapes calculated
indicating that the TMM and FEA modes satisfy the continuity of volume velocity. (See
Appendix D for plots of volume displacement).
Figure 59 and Figure 60 present the change in volume per unit length of the 1
Mode A and Mode B for the example system with rigid walls. Consistent with
previously presented results the magnitude of the change in volume is larger within the
cavities which have a larger initial volume. The example systems are closed systems. As
discussed in Section 4.2.2 the net change in volume for a closed system must be equal to
zero. The summation of the change in volume was verified to be equal to zero for all of
the example system modes in each of the three boundary conditions.
Figure 61 and Figure 62 provide the non-dimensional impedance for the 1 Mode A
and Mode B for the example system with rigid walls. Figure 61 and Figure 62 show
discontinuities in the non-dimensional impedance for each mode. These discontinuities
occur at the sudden changes in impedance where the piping connects to the components.
As discussed in Section 4.4.1 the non-dimensional impedance also represents the phase
angle of the pressure mode shape in the complex plane. The discontinuities in the nondimensional system impedance represent sudden jumps in the phase angle of the
pressure modes which also affect the frequency of the axial loop modes.
107
Axial Displacement Mode Shape, Root 2, 30.5352 Hz
Normalized Axial Displacement
1
0.5
0
-0.5
-1
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 57 – Axial Displacement, TMM, Example System RW, 1 Mode A
Axial Displacement Mode Shape, Root 3, 39.3763 Hz
2
Normalized Axial Displacement
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 58 – Axial Displacement, TMM, Example System RW, 1 Mode B
108
Volume / Length [m3/m]
+ Vf
- Vf
Volume / Length [m3/m]
Figure 59 – Change in Volume, TMM, Example System, RW, 1 Mode A
+ Vf
- Vf
- Vf
Figure 60 – Change in Volume, TMM, Example System, RW, 1 Mode B
109
Non-Dimensional Impedance, Root 2, 30.5352 Hz
Non-Dim Imp
inv Non-Dim Imp
Non-Dim Imp Roots
inv Non-Dim Imp Roots
2
1.5
1
0.5
0
-0.5
-1
-1.5
Lm1
-2
Lm2
Lm3
Lm4
Lm5
Lm
Lm7
6
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 61 – Non-Dimensional Impedance, TMM, Example System, RW, 1 Mode A
Non-Dimensional Impedance, Root 3, 39.3763 Hz
Non-Dim Imp
inv Non-Dim Imp
Non-Dim Imp Roots
inv Non-Dim Imp Roots
2
1.5
1
0.5
0
-0.5
-1
-1.5
Lm1
-2
0
Lm2
5
Lm3
10
Lm4
Lm5
15
20
25
Axial Location, m
Lm6
30
Lm7
35
40
Figure 62 – Non-Dimensional Impedance, TMM, Example System, RW, 1 Mode B
110
The segments of the TMM model for the example system have been labeled on the
schematics in Figure 61 and Figure 62. The TMM model is made up of seven cylindrical
lengths; three cavities (Lm2, Lm4, and Lm6) connected by four lengths of piping (Lm1,
Lm3, Lm5, and Lm7). The segments of the TMM model have been labeled to help
describe where specific results are observed in the model.
The non-dimensional impedance for the 1 Mode A of the RW system in Figure 61
is similar to the results for the 1 Mode A of the non-uniform loop with a single cavity
previously shown in Figure 38. The discontinuities in impedance at the inlet and the
outlet of the two larger cavities cause the phase angle of the pressure mode to jump
ahead. However unlike the non-uniform loop with a single cavity the pressure mode is
not centered symmetrically on either of the larger cavities. The asymmetry causes there
to be different changes in phase at the inlet and outlet of the large cavities. The change in
phase at the interfaces between Lm1-Lm2 and Lm6-Lm7 is 0.6083 radians and the
change in phase at the opposite end of each cavity, Lm2-Lm3 and Lm5-Lm6, is 0.1457
radians. The net change in phase due to the impedance discontinuities for each of the
larger cavities is 0.7540 radians. The smaller cavity causes symmetric changes in phase
for a net change of -0.0214 radians. Overall the impedance discontinuities in the rigid
wall system for 1 Mode A account for a total of 1.4867 radians which is approximately
23.5 % of the total change in phase for a 1 wavelength long mode.
Table 12 provides the change in phase over each segment of the Example systems,
the change in phase due to the impedance discontinuity between each of the segments
and the total phase for the 1 Mode A and 1 Mode B. The change in phase for each
segment of the example system was calculated by multiplying the wave number in each
segment by the length of the segment. The change in phase at the interface between each
segment was calculated from the discontinuities in the non-dimensional system
impedance calculated using the TMM models. Note that the total change in phase around
the loop for the 1 modes is equal to 1 x (2) = 6.2832 radians which is consistent with
expectation for whole wave length modes and the results discussed in Section 4.4.1.
111
Segment
Length
Lm1
Lm2
Lm3
Lm4
Lm5
Lm6
Lm7
Total
Segment
Interface
Lm1-Lm2
Lm2-Lm3
Lm3-Lm4
Lm4-Lm5
Lm5-Lm6
Lm6-Lm7
Lm7-Lm1
Total
Change in Phase over Each Segment of Piping or Cavity
RW
Steel
Aluminum
RW
Steel
1
1
1
1
1
Mode A
Mode A
Mode A
Mode B
Mode B
[radians] [radians] [radians] [radians] [radians]
0.6633
0.6294
0.5954
0.8553
0.8467
0.1295
0.1353
0.1404
0.1671
0.1820
1.3934
1.3222
1.2507
1.7968
1.7787
0.0428
0.0415
0.0403
0.0551
0.0559
1.3934
1.3222
1.2507
1.7968
1.7787
0.1295
0.1353
0.1404
0.1671
0.1820
1.0446
0.9913
0.9377
1.3471
1.3335
4.7964
4.5771
4.3554
6.1852
6.1574
Aluminum
1
Mode B
[radians]
0.8375
0.1974
1.7594
0.0567
1.7594
0.1974
1.3190
6.1269
Change in Phase at Interfaces between Piping and Cavities
RW
Steel
Aluminum
RW
Steel
Aluminum
1
1
1
1
1
1
Mode A
Mode A
Mode A
Mode B
Mode B
Mode B
[radians] [radians] [radians] [radians] [radians] [radians]
0.6083
0.6528
0.6972
-0.8597
-0.8756
-0.8876
0.1457
0.2109
0.2773
0.8811
0.9094
0.9346
-0.0107
-0.0106
-0.0106
0.0275
0.0292
0.0312
-0.0107
-0.0106
-0.0106
0.0275
0.0292
0.0312
0.1457
0.2109
0.2773
0.8811
0.9094
0.9346
0.6083
0.6528
0.6972
-0.8597
-0.8756
-0.8876
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
1.4867
1.7062
1.9278
0.0980
0.1259
0.1563
6.2833
6.2832
Total Change in Phase Around Loop
Total
6.2832
6.2833
6.2832
6.2832
Table 12 – Changes in Phase; TMM Example Systems, 1 Mode A and B
Figure 62 shows that the non-dimensional impedance is somewhat similar to the 1
Mode B for the loop with a single cavity, where the impedance discontinues at the large
cavities causes a setback in the phase angle. However the 1 Mode B for the example
system shows that instead of a setback in phase angle at both ends of the large cavities
the opposite end of the each large cavity causes the phase angle to jump ahead. The
changes in phase angle shown in Figure 62 are also significantly larger in than the
changes in phase angle observed in Figure 39 for the non-uniform loop with a single
112
cavity even though the two large cavities in the example system have the same cross
sectional area as the single cavity previously evaluated.
As discussed in Section 4.4.1 the magnitude of the pressure and the slope of the
pressure on either side of the impedance discontinuity are related through the linear
Euler equation and conditions of continuity for pressure and volume velocity. The 1
Mode B pressure mode shape for the loop with a single cavity has a node within the
cavity. The single cavity evaluated in Section 4.4.1 caused two setbacks or reductions in
the phase angle. The reductions in phase angle resulted in an effective length that was
shorter than the physical length of the system. The shorter effective system length
shifted the frequency of the 1 Mode B higher in frequency than the associated mode for
a uniform rigid wall loop.
The 1 Mode B pressure mode shown in Figure 56 for the example system does not
contain a node in either of the larger component cavities. As shown in Figure 62 and
Table 12 the change in phase angle at interfaces Lm1-Lm2 and Lm6-Lm7 is -0.8597
radians. The change in phase at the opposite ends of the large cavities, Lm2-Lm3 and
Lm5-Lm6, is 0.8811 radians. The net change in phase for impedance discontinuities for
each of the larger cavities is +0.0214 radians. The smaller component cavity also results
in a net positive change in phase angle of 0.0550 radians. The positive changes in phase
angle are due to the position of the cavities within the pressure mode shape. The net
change in phase angle for all of the impedance discontinuities is 0.0980 radians which
results in an increased effective system length and a reduction in the frequency of the
mode.
It was noted previously that the 4 Mode B for the rigid wall example system was
the only Mode from the first 4 mode pairs that shifted higher in frequency than the
associated rigid wall uniform loop mode. Figure 63 and Figure 64 present the 4 Mode
B pressure mode shape and the 4 Mode B non-dimensional impedance for the rigid
wall example system.
113
Pressure Mode Shape, Root 9, 166.3593 Hz
1.5
Normalized Pressure
1
0.5
0
-0.5
-1
-1.5
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 63 – Pressure Mode Shape; TMM, RW Example System, 4 Mode B
Non-Dimensional Impedance, Root 9, 166.3593 Hz
Non-Dim Imp
inv Non-Dim Imp
Non-Dim Imp Roots
inv Non-Dim Imp Roots
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
5
10
15
20
25
Axial Location, m
30
35
40
Figure 64 – Non-Dimensional Impedance; TMM, RW Example System, 4 Mode B
114
As shown in Figure 63 and Figure 64 the 4 Mode B for the rigid wall example
system has a node within both of the larger component cavities. Similar to the results in
Section 4.4.1 the impedance changes at both ends of the larger cavities result in a
negative change in the phase angle. Table 13 gives the changes in phase angle for each
of the example system segments and the change in phase angle due to each of the
interfaces between the segments of the example system for the 4 Mode A and 4 Mode
B. it can be seen in Table 13 that the net change in phase due to the all of the impedance
discontinuities for 4 Mode B is negative. The negative change in phase angle results in
a shorter effective system length and increases the frequency of the 4 Mode B above
the frequency of the 4 Mode B in the uniform rigid wall loop.
Change in Phase over Each
Segment of Piping or Cavity
RW
RW
Segment
4
4
Length
Mode A
Mode B
[radians] [radians]
Lm1
3.0780
3.6135
Lm2
0.6012
0.7058
Lm3
6.4661
7.5912
Lm4
0.1984
0.2329
Lm5
6.4661
7.5912
Lm6
0.6012
0.7058
Lm7
4.8477
5.6912
Total
22.2587
26.1316
Change in Phase at Interfaces
between Piping and Cavities
RW
RW
Segment
4
4
Length
Mode A
Mode B
[radians] [radians]
Lm1-Lm2
0.6337
-0.3882
Lm2-Lm3
0.8527
-0.2247
Lm3-Lm4
-0.0495
0.1134
Lm4-Lm5
-0.0495
0.1134
Lm5-Lm6
0.8527
-0.2247
Lm6-Lm7
0.6337
-0.3882
Lm7-Lm1
0.0000
0.0000
Total
2.8740
-0.9989
Total Change in Phase
Total
25.1327
25.1327
Table 13 – Changes in Phase; TMM Example System RW, 4 Mode A and B
The reduction in the phase velocity, due to the elasticity of the steel and aluminum
cylindrical walls, also lowers the frequencies of the axial loop modes as shown in Table
11 and Figure 51. The reduction in frequency of the axial loop modes due to a lower
phase velocities within each of the segments that make up the system makes intuitive
sense and is consistent with the results presented in Section 4.1 and Section 4.5. The
change in the phase velocities reduced the amount of the total phase change that
115
occurred in the cylindrical segments of the full system as shown in Table 12. The phase
velocity also impacts the acoustic impedance of each section of the example system as
shown in (3.19). The different local acoustic impedances result in changes to the
magnitude of the discontinuity in the non-dimensional impedance at the interfaces
between the piping and components. In general it was found that as the elasticity
increased and the phase velocity decreased, the magnitude of the impedance
discontinuity became larger at each of component and piping interfaces. The increased
impedance between the adjacent segments resulted in larger changes to the phase angle
at the impedance discontinuities. As the phase velocity within the example system
decreased the total phase change in phase angle due to the impedance discontinuities
increased while the change in phase angle within the cylindrical lengths of the system
decreased. For the 1 Mode A in the aluminum example system the impedance
discontinuities make up more than 30 % of the total change in phase angle
The changes in the phase angle due to the impedance discontinuities shown in Table
12 are consistent with the results and expectations discussed in Section 4.4.1. It can be
seen in Figure 55 and Figure 56 that the pressure modes remain continuous around the
entire loop. The changes in the slope of the pressure relative to the ratio of impedance on
either side of an interface between segments also agree with the expectations discussed
in Section 4.4.1 which were developed from the linear Euler equation and the continuity
of volume velocity.
4.7.1
Summary of Full System Analysis
In summary, the frequencies and modes shapes of the axial loop modes calculated
by the FEA models for the example systems are reasonable relative expectations based
on acoustic theory and the numeric models studies presented in Section 3 through
Section 4.6. The axial loop modes satisfy the conditions of continuity for pressure and
volume velocity. The net change in volume for each mode sums to zero in agreement
with the requirements for a closed system and the net change in phase for each mode is
equal to integer multiples of 2 consistent with whole wave length axial modes.
116
The frequencies and mode shapes of the axial loop modes are influenced by the
phase velocity and impedance discontinuities within the system. The elasticity of the
cylindrical wall reduces the axial phase velocity of the fluid and reduces the frequency
of the loop modes. Impedance discontinuities within the system can cause either an
increase or a decrease in the frequency of the loop modes as well as cause “kinks” in the
pressure mode shapes.
The change in phase angle and magnitude of the pressure results in pressure mode
shapes that are not purely sinusoidal. The jumps in phase angle also cause each of the
full wave length modes to see a different effective system length and shift to a frequency
that is different from the frequency of the same mode in a uniform rigid wall loop of the
same physical length. The amount and the direction in frequency that each axial mode
shifts due to an impedance discontinuity are based on the pressure and slope of the
pressure at the location where the mode encounters the change in impedance.
The shifted frequencies of the example system with rigid walls, elastic steel walls
and elastic aluminum walls in Table 11 are in agreement with acoustic theory and the
numerical models discussed in Section 3 and Section 4.
4.7.2
Implications and Physical Interpretation of Axial Modes
It is important to be able to predict the frequency and mode shape of resonances
within a piping system during system design. The frequency of a resonance is necessary
to determine potential sources of excitation within the system. The mode shape provides
locations where a specific system mode might cause excessive or damaging responses as
well as locations that may efficiently excite the mode. Understanding how specific
system properties affect the frequencies and mode shapes of the system resonances
provides additional information into how design choices will affect the noise
performance of the system.
It is not always practical or cost efficient to design a system around the noise
performance. However when an existing system has a noise or vibration problem,
understanding which portions of the system strongly influence the problematic
117
frequencies and mode shapes will help identify potential system modification that could
resolve the problem.
The axial modes within a continuous loop of fluid are a specific example of one
type of mode that can occur in a piping system. Several other types of modes within
fluid filled elastic cylinders were discussed in Section 2.1Error! Reference source not
found.. The fluid structure interactions of piping systems can be very complex with
multiple types of modes existing in similar frequency ranges. If an axial mode in a fluid
loop were identified as a potential vibration problem, the results of this study suggest
that by altering the frequency of that mode might be able to be shifted to a different
frequency where the mode may be less of an issue. Tuning the axial loop modes might
be achieved by altering the impedance discontinuities at the inlet and or outlet of the
existing components or by adding additional impedance discontinuities at a specific
location within the system. This concept of tuning the frequency of the axial loop
resonances using impedance discontinuities will require further study.
118
5. Results and Conclusions
The purpose of this analysis was to establish a set of physics based expectations for
the frequencies and modes shapes of the axial fluid modes within a system of piping and
components that form a continuous loop of fluid. The axial modes in a fluid loop were
studied using acoustic theory, the transfer matrix method (TMM) and finite element
models. The frequency and the pressure mode shapes were calculated for a fluid loop
sized to have a 1 long axial mode at 40 Hz with rigid boundary conditions. The analysis
was restricted to low frequency axial modes with 0th order circumferential and radial
components and wavelengths that were greater than or equal to 1/4th of the centerline
length of the fluid loop.
The Helmholtz equation in cylindrical coordinates was solved with a rigid boundary
condition on the cylindrical wall and a continuity boundary condition on the circular
ends to simulate uniform “loop”. It was shown that the there are pairs of whole wave
length modes that satisfy the Helmholtz equation with these boundary conditions. Each
mode of a pair occurs at the same frequency and the pairs of modes occur at frequencies
that form a harmonic series. The theoretical, TMM and FE results were shown to be in
good agreement for the uniform rigid wall simulated loop. The FE results from an actual
loop model with four elbows, uniform material properties and cross sectional areas were
also shown to be in good agreement with the simulated loop results.
The effects that changes in the phase velocity and the local impedance have on the
frequency and modes shapes in a non-uniform loop where studied using the TMM and
FE models. It was found that changes in the acoustic impedance can have a larger effect
on the frequencies and pressure mode shapes of the axial loop modes than change in the
phase velocity of the system.
The Korteweg-Lamb correction was used to adjust the axial phase velocity of the
fluid to account for the effects of the radial elasticity of the piping and the components.
It was shown that the lower phase velocity causes the pairs of axial modes to occur at
lower frequencies. This result is consistent with previously published results for the
phase velocity of plane waves in a fluid filled elastic cylinder.
Short cylindrical sections with larger cross sectional areas than uniform loop were
inserted into the loop to investigate changes due to differences in local impedances. It
119
was shown that at the location of sudden changes in impedance, the conditions of
continuity can cause a discontinuity in the phase angle or imaginary portion of the axial
mode. The discontinuities in phase angle can cause each of the axial loop modes to shift
lower or higher in frequency, depending on the net change in phase angle. It was shown
that the summation of the phase angle within the cylindrical components and piping plus
the summation of the phase angles at the impedance discontinuities resulted in the net
change in phase angle equivalent to integer multiples of whole wave lengths.
The net change in phase angle was found to be specific to each axial mode resulting
in a unique change in frequency for all of the axial loop modes. The unique shifting in
frequency resulted in a non-harmonic spacing of the axial modes. The modes of each
whole wavelength axial mode pair were also shown to shift uniquely. This resulted in
two modes for each integer multiple of whole wavelength occurring at different
frequencies in the same physical system.
The pressure mode shapes were found to have a “kink” at the locations of the
impedance and phase angle discontinuities. While the real portion of the pressure
remained continuous, the slope of the pressure mode shape was not continuous. The
discontinuities in the slope of the pressure resulted in mode shapes that were not purely
sinusoidal and had amplitude variations within a single mode.
The variation in
magnitude and the discontinuity in the slope of the pressure modes were shown to be
related to the phase angle discontinuity using the linear Euler equation and the continuity
of volume velocity.
The shifted frequencies and the “kinked” pressure mode shapes where both shown
to be related to the discontinuity in phase angle that occur due at impedance
discontinuities. Using a unit circle in the complex pressure plane the phase angle
discontinuity was demonstrated to be a function of the pressure and the slope of the
pressure at the axial location of the impedance discontinuity. Because each pressure
mode in the fluid loop intersects the impedance discontinuities at a unique pressure and
phase angle the direction and magnitude of the frequency shift is unique for each axial
mode. This result explains the non-harmonic frequency spacing of the axial modes, the
pairs of whole wave length modes existing at different frequencies in the same system
and the kinked appearance of the pressure mode shapes.
120
The changes in the frequency and mode shapes of the axial loop modes were much
larger due to the impedance discontinuities than the changes in phase velocity due to the
elasticity of the cylindrical components and piping. The combined effects of the reduced
phase velocity and the impedance changes for the example system made from aluminum
with three component cavities resulted in the axial loop modes being shifted up to 40%
lower in frequency than the associated modes in the rigid wall uniform loop.
Overall the frequencies and modes shapes of the axial loop modes calculated by the
COMSOL FE models were in good agreement with acoustic theory and the results from
the TMM models. The whole wavelength modes, mode pairs at different frequencies,
non-harmonic frequency spacing of high order axial modes and the kinked pressure
mode shapes were all shown to be reasonable and make physical sense. The results
presented in this analysis provide a set of physics based expectations for the frequencies
and modes shapes of the axial modes in a system that forms a continuous loop of fluid.
121
6. Areas for Future Work
There is potential for future work on the axial loop modes including numerical
model studies with different materials, geometries, and boundary conditions, expansion
and refinement of the TMM scripts to include more types of components and piping
segments and testing of a fluid loop built to the dimensions specified in this study.
Future model studies could leverage the extensive body of FSI literature to include
additional fluid effects within the system such as viscosity, temperature and pressure
variations, entrained or dissolved gasses and a recirculation flow. The FE models could
be expanded to include the elasticity of the piping and components more explicitly as
well as more representative elbows and components.
Additional modeling studies could also add more realistic boundary conditions to
the structural portions of the system, add various material combinations such as
composite or non-linearly elastic materials to the system, and leverage existing literature
to study the effects weld joints, flanges and nozzles have on the fluid loop resonances.
The flow effects within a loop system may pose several interesting challenges
particularly if the fluid velocity were increased significantly relative to the phase
velocities within the system. The flow velocity would only exist in one direction around
the loop while the phase velocity will exist in both directions around the loop. The flow
velocity would increase the phase velocity of the fluid in one direction and decrease the
phase velocity the opposite direction. This would result in two different directionally
deponent values for the speed of sound within the same fluid column and would likely
impact the frequency, mode shape and stability of the axial modes and resonances within
the loop. In addition to the physical implications of the fluid velocity to the axial modes,
implementing the directionally dependent phase velocity in the numerical models would
be an interesting challenge.
The analysis of the fluid loop could be expanded to include axial modes of higher
order radial and circumferential modes, force frequency analysis of the fluid loop and
damping in the fluid and structural portions of the model. Adding acoustic sources and
forcing functions to the FEA and TMM models would also provide additional insight
into how a loop system may function over the entire frequency range, the attenuation and
amplification effects of various system arrangements, and allow for studies on power
122
flow from the acoustic sources along the fluid loop and into the surrounding structures
and foundations.
The TMM models developed for this analysis provided useful insights into the axial
fluid resonances that were not immediately obvious or even available from the FE
models. The understanding of the impacts that the impedance discontinuities have on the
frequencies and modes shapes of the axial modes was developed based on the additional
information available in the TMM models. Expanding the TMM MATLAB models to
include matrices representing additional component types, forced frequency analysis and
other system configurations would result in a useful tool for efficiently scoping and
design new fluid piping systems and test loops.
As discussed in detail above, the sudden changes of impedance in the example
system resulted in the large changes to the center frequencies of the axial fluid loop
resonances. This result along with studies on nozzles, horns and musical instruments
suggest that combinations of gradual and sudden changes to the impedance of the fluid
could be used to tune the frequency of the axial fluid loop modes. Tuning the axial
modes could provide a significant design advantage. For a component test loop, axial
modes within the test loop may be excited and “contaminate” some of the frequency
regions of interest. It may be possible to design a removable portion of the test loop such
that various insert configurations could be added or removed from the system which
would result in different axial resonances. This interchangeable portion of the loop
would allow for accurate measurements to be acquire in a frequency regions previously
dominated by the test loop axial resonance by shifting the axial resonances to different
frequency regions and re-testing.
The system sizing used in this analysis was chosen to reflect piping sizes that are
commercially available. It would be possible to build a representative loop of piping that
could be tested to verify the results of this analysis. This test loop could also be
constructed to support investigations into fluid and structural material combinations,
recirculation flow and tunable inserts.
123
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127
Appendix A – Derivation of Linear Acoustic Wave Equation
A.1
Equation of Continuity
The Equation of Continuity relates the flow of mass into and out of a fixed volume
to the change in density within the volume. Since mass must be conserved the time rate
of change for the density within the element must equal the difference between the rate
of mass entering the volume and the rate of mass exiting the volume. The Equation of
Continuity is developed by performing a mass balance over a differential volume
element as described in [32]. A fixed volume differential element in cylindrical
coordinates is shown in Figure 1.
1
v  
r
r
1
rv r  r  r
r
vz  z
vz  zz

1
rv r  r
r
1
v    
r
z
Figure A.1 - Differential Volume Element in Cylindrical Coordinates
The mass fluxes are annotated for each side of the element shown in Figure A.1
where r, , and z are the cylindrical coordinates,  is the fluid density and v is velocity in
the cylindrical directions.
Equation (A.1) is the exact Equation of Continuity developed from the mass balance
in vector notation and expanded in cylindrical coordinates in (A.2) as given in [32]. The
Equation of Continuity will be used with the linear Euler equation to develop the linear
lossless wave equation.


   v 
t
A-1
(A.1)
 1 
rv r   1  v    v z   0

t r r
r 
z
A.2
(A.2)
Equation of Motion
The equation of motion for an element of fluid is obtained by performing a
momentum balance on a fixed volume differential fluid element. The Equation of
Motion in vector notation as developed in [32] is shown in Equation (A.3). The terms
from left to right represent the rate of momentum increase per unit volume, rate of
momentum addition by convection per unit volume, rate of momentum by molecular
transport (pressure and viscosity) per unit volume and the external body forces acting on
the fluid per unit volume.

 

v    v v   p      g
t
(A.3)
Neglecting viscous term and body forces and collecting density terms, (A.3)
becomes
 

v    v v   p
t
(A.4)
Expanding (A.4) into cylindrical coordinates gives
 v r
v r v v r
v r v2 
p


 vr

 vz
   
r
r 
z
r 
r
 t
(A.5a)
v
v v
v
vv 
1 p
 v
 vr      v z   r    
r
r 
z
r 
r 
 t
(A.5b)
v v z
v
v 
p
 v z
 vr z  
 vz z   
r
r 
z 
z
 t
(A.5c)


A-2
For acoustic processes that are assumed to have small displacements and velocities
it is assumed that
 

  v v 
v
t
(A.6)
This assumption results in the linear Euler’s equation, shown in vector notation in
(A.7) and in cylindrical coordinates in (A.8)
 
v  p
t
(A.7)

p
 vr 

r
 t 
(A.8a)
1 p
 v 

r 
 t 
(A.8b)
p
 v z 

z
 t 
(A.8c)


The linear Euler’s equation relates the change in fluid pressure with respect to space
to the change in fluid particle velocity with respect to time and is valid for acoustic
processes of small amplitude.
A.3
Acoustic Wave Equation
Detailed derivations of the linear lossless wave equation can be found in [22] and
[28]. It is worth noting that there are slight notational differences between the equations
of continuity derived in [22] and [28]. Reference [22] derives the continuity condition in
terms of a displacement vector while [28] derives it in terms of a velocity vector. The
equations of continuity and motion discussed in the previous sections are based on [32]
and closely follow [28]. The derivation of the wave equation presented below uses the
velocity vector notation along with the development of the fluid bulk modulus presented
A-3
in [22]. The bulk modulus relates the change in volume of a compressible fluid to an
applied pressure as shown in (A.9).
Bf 
p
 d 
 
  
(A.9)
If the mass of a fluid element of volume Vf is held constant then
d V f   dV f  dV f  0
(A.10)
Rearranging gives
 dV f
d

Vf

(A.11)
Substituting (A.11) into (A.9) and rearranging
 d 
p  B f    B f
  
 dV f
 
 Vf



(A.12)
It can be seen by inspection that the first portion of Equation (A.12) represents an
extension of Hooke’s law to a fluid media. The fluid element will essentially act as a
volumetric spring changing in volume base on an applied pressure.
The Bulk modulus of a fluid for a given temperature and pressure can also be
defined as shown in (A.13) where co is the characteristic speed of sound within the fluid.
Both definitions of the fluid bulk modulus will be used.
B f  c o2
(A.13)
Defining density  as a function of a mean value o plus a change in density d
gives (A.14)
  o  d
A-4
(A.14)
Replacing  with o in (A.11), substituting (A.11) into (A.14) and then taking the
derivative with respect to time gives
 o o   dV f



t
t
t  V f

   dV f
  o 
t  V f




(A.15)
Since o is constant the terms containing derivatives of o with respect to time go to
0. Substituting (A.12) into (A.15) and equating the right hand side of (A.15) to the
equation of continuity (A.1) gives

 p 

    v 
 o 
t
t  B f 
(A.16)
Substituting (A.11) and (A.14) into the RHS of (A.16)
 
  dV f

   v       o  o 

 Vf
 
   
 p 
  v  o 

 

t
B
 
 f 
Assuming that changes in volume are very small    dV f
o
 Vf

  o

(A.17)
gives
p o

  o  v 
t B f
(A.18)
p

 B f  v 
t
(A.19)
Which reduces to
Taking the partial derivative of (A.19) with respect to time
2 p






B


v
f
t 2
t
(A.20)
Taking the divergence of (A.7)
  
 v     2 p
 t 
A-5
(A.21)
Since differentiation in this case is commutative (A.21) becomes

   
 v    v 
 t  t
(A.22)
Setting Equations (A.20) and (A.21) equal to each other and rearranging gives the
linear lossless acoustic wave equation which provides a relationship between the
variation of pressure in time and space.
 
 2 p  
 Bf
 2 p
 2  0
 t
or
 1  2 p
 2 p   2  2  0
 co  t
(A.23)
In cylindrical coordinates (A.23) becomes
 1  2 p
 2 1 
1 2
2 
 2 2
 2 

 2

p

2
2 
 c  t
r

r

r
r



z


 o 
(A.24)
If a solution of the form shown in (A.25) is assumed and substituted into (A.24), the
acoustic wave equation in cylindrical coordinates can be separated into three differential
equations as discussed in [22] and [28]. The three differential equations that satisfy the
assumed form of the assumed solutions are shown in Equations (A.26) through (A.28)
p r , , z ,t   R (r )( )Z (z )e it
(A.25)
d 2R 1 dR  2 m 2 

  k nm  2 R  0
dr 2 r dr 
r 
(A.26)
d 2
 m 2  0
2
d
(A.27)
A-6
d 2Z
 k zl2 Z  0
2
dz
(A.28)
Where the subscripts n, m, and l are used to identify specific normal modes, knm is
the radial wave number and kzl is the axial wave number for mode nml. The radial and
axial wave numbers are related to the acoustic wave number knml , angular frequency, ,
and the phase velocity, c, as shown in (A.29)
 
2
2
2
   k nml  k nm  k zl
c 
2
(A.29)
Solutions to (A.26) are combinations of Bessel functions of the first and second kind
as detailed in [22] and [28]. Cylindrical acoustic waves that include the z axis, therefore
solutions to (A.26) must have real values at the center line of the cylinder when r=0.
Bessel functions of the second kind diverge as the argument approaches zero so the
solution to (A.26) for problems which include the z axis will include only Bessel
functions of the first kind which have finite values when r=0, as shown in Equation
(A.30). The index m=0,1,2,… and Jm is the mth Bessel function of the first kind.
R r   J m k nm r 
(A.30)
The value of (knma), where a is the radius of the fluid column, is equal to the nth
extrema of the mth Bessel function as discussed in [28].
Solutions to (A.27) and (A.28) can be expressed as either complex exponential
functions representing traveling waves or sinusoidal indicating standing waves. Equation
(A.31) provides the solution to (A.27) assuming a circumferential standing wave and
(A.32) gives the solution to (A.28) assuming an axially traveling in the + or – z
direction.
   cosm 
A-7
(A.31)
Z z   e ik zl z
(A.32)
The pressure of a given mode, nml, within a cylindrical column of fluid can be
expressed using Equation (A.25) and Equations (A.29) through (A.32) as
p nml r , , z ,t   A nml J m k nm r cosm e ik zl z e it
(A.33)
Where Anml is an “amplitude” coefficient determined from the boundary conditions.
If the analysis is restricted to axially symmetric waves of the lowest order radial mode,
i.e. plane waves, where the n=0 and m=0 then (A.33) reduces to solutions of the form
p 00l r , z , t   A 00l e  ik zl z e  it
(A.34)
The cosine term and Bessel function term in (A.31) each reduce to a value of 1
when the subscripts n and m are set to 0. Equation (A.34) represents a traveling plane
wave in a cylindrical fluid column with rigid walls.
A-8
Appendix B – Derivation of Korteweg Lamb Correction
The development of the Korteweg-Lamb correction provides a simple example of
the effects that the elastic walls have on the phase velocity or speed of sound for a
pressure wave within a fluid filled elastic cylinder. Originally discussed separately by D.
Korteweg (Appendix D [1]) and H. Lamb (Appendix D [2]) more than a century ago, it
was observed that wave propagation within a fluid surrounded by elastic walls was
slower than the wave propagation in a free field of the same fluid. The Korteweg-Lamb
correction is a correction factor that can be applied to the characteristic fluid speed of
sound to account for the elasticity of the cylindrical tube or pipe wall. The correction
factor is determined using the geometry and material properties of the cylinder and
internal fluid. Figure B.1 shows a basic fluid filled cylinder with the relevant geometric
dimensions noted.
h
E, 
r
, co
Lc
Figure B.1 - Fluid Filled Elastic Cylinder
Reference [37] and Reference [22] provide similar derivations of this correction
factor as originally provided in Reference [15]. However unlike the references cited in
Section 2.1, the Korteweg-Lamb correction is essentially a static correction and does not
provide the frequency dependence to the speed of sound or phase velocity calculated in
References [17] through [27]. Jacobi noted in [18] that his formulation of phase velocity
in a fluid filled elastic tube converged to the Korteweg-Lamb correction factor at 0 Hz.
If a pressure is applied to the internal fluid column, Pascal’s principle states that the
pressure will act in the axial and radial directions. A positive internal pressure causes
B-1
both the fluid volume and the cylindrical shell to expand “outward” in the positive radial
direction. If the fluid is assumed to be inviscid, only the radial component of the fluid
and shell displacement are coupled. The Korteweg-Lamb correction factor is calculated
by determining the volumetric expansion of the elastic cylinder and the cylindrical fluid
volume and setting the volumetric expansions equal to each other. Essentially if the fluid
column is allowed to expand radially due to the elasticity of the cylindrical wall the fluid
column will not expand as far in the axial direction as it would if the cylindrical wall
were perfectly rigid.
The change in the volume V of a fluid column of known volume V is related to the
applied pressure through the fluid bulk modulus Bf as shown in (B.1). The fluid bulk
modulus can be defined using the characteristic density and speed of sound for the fluid
as also shown in (B.1)
Bf 
p
 V 


V 
B f  co2
(B.1)
Roark’s Formulas for Stress and Strain [43] provide equations for the stress, strain
and change in volume of an elastic cylindrical shell due to an internally applied pressure.
The hoop stress of an elastic cylinder with a thickness of h and a radius of a due to an
applied internal pressure p is given by (B.2)

pa
h
(B.2)
The hoop strain of an elastic cylinder with a Young’s Modulus E is determined
using Hooke’s law (B.3)


E

pa
Eh
(B.3)
Using the small angle approximation, the change in radius for an elastic cylinder is
B-2
r  r 
pa 2
Eh
(B.4)
The initial volume of the cylinder is given by (B.5) if the length of the cylinder is Lc
(B.5)
V  a 2 Lc
The change in volume V due to a change in the radius a of the cylinder is
V   (a  a ) 2 Lc  a 2 Lc
(B.6)
V  a Lc   2aaLc  a Lc  a Lc
2
2
2
Assuming that the change in radius is small compared to the initial radius and that
the square of the change is much smaller than the original radius
a  a, a 2  a
(B.7)
The change in volume due to a change in radius can be reduced to only the second
term in (B.6). Substituting (B.4) into (B.6) gives
V   2aaLc 
2pa 3Lc
Eh
(B.8)
Dividing (B.8) by the original formulation of the cylindrical volume in (B.5) results
in an expression for the fractional change in cylindrical volume due to an applied
internal pressure as shown in (B.9)
2pa 3 Lc
V
2 pa
 Eh

2
V
a Lc
Eh
B-3
(B.9)
Using the expression for the bulk modulus of fluid given in (B.1) an equivalent bulk
modulus for the elastic shell can be determined as shown in (B.10)
Bs 
p
p
Eh


 V   2 pa  2a

 

 V   Eh 
(B.10)
The total change in volume of the elastic cylinder and the fluid column is given by
V
V

total
V
V

fluid
V
V
(B.11)
shell
Using (B.1) and (B.10) expressions for fluid bulk modulus and the effective shell
bulk modulus are substituted into (B.11) as shown below to calculate an effective bulk
modulus of the compressible fluid / elastic shell system
V
V

total
1
p
p
1

 p 

Bf
Bs
 B f Bs

p
 
 B eff
(B.12)
The effective bulk modulus Beff is related to an effective axial phase velocity ceff or
effective fluid speed of sound as shown in (B.13)
B eff  ceff2
(B.13)
Solving for the effective axial phase velocity in terms of the fluid bulk modulus and
characteristic speed of sound, gives (B.14)
c eff
B
  eff
 



1 / 2
 1
1  Bf
 2
  

  B f B s  co




1 / 2
 B
 c o 1  f
 Bs



1 / 2
(B.14)
Dividing (B.14) by characteristic speed of sound and substituting (B.1) and (B.10)
into (B.14) gives
B-4
c eff
co

 c 2
 1  o
Eh


2a







1 / 2
 2ac o2
 1 
Eh





1 / 2
(B.15)
The ratio of the effective phase velocity to the characteristic fluid speed of sound
shown in (B.15) is the Korteweg-Lamb correction. This correction was derived on a
static basis and only considers volumetric changes in the radial direction due to a static
or quasi static internal pressure. In Reference [22] Poisson’s ratio for the elastic shell
material, , was included in the formulation of the Korteweg-Lamb correction factor as
shown in (B.16).
 2ac o2 1    
c eff

 1 
co
Eh


B-5
1 / 2
(B.16)
Appendix C – Transmission Matrix Method
Available in Electronic Format Only
C-1
Appendix D – TMM Results Figures
Available in Electronic Format Only
D-1
Appendix E – COMSOL FE Model Results
Available in Electronic Format Only
E-1
Appendix F – Additional Related References
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F-1
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F-3
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