\jAST. oFcf
yivl
AUG 16 1962
LIBRADRy9
ON PERTURBED BOUNDARY IAYER FLOWS
by
James Doyle McClure
S.B., University of Washington, 1956
S.M., University of Washington, 1957
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1962
Signatur
of Author
Signature redacted
partment of Aeronautics
and Astronauties, June 1962
Certified by
Signature redacted
Thesis Supervisor
Accepted by
Signature redacted
Chairman, DertmentaI
Graduate Comttee
ii
ON PERTURBED BOUNDARY LAYER FLOWS
by
James Doyle McClure
Submitted to the Department of Aeronautics and
Astronautics on May 11, 1962 in partial fulfillment
of the requirements for the degree of Doctor of
Philosophy.
ABSTRACT
The object of this investigation is to develop the
concepts necessary for understanding the character of
perturbed boundary layer flows.
While the study is
relevant to disturbances in the boundary layer adjoining
a flat
rigid wall, attention is imrkarily directed toward
the role of the boundary layer in flows over perturbed
The study ranges from the fundamental consurfaces.
siderations regarding the qualitative and quantitative
effects of the boundary layer to the explicit evaluation
of these effects in problems of current interest.
The theory for the laminar boundary layer with freestream Mach number ranging from O-e-2 is first considered.
This part of the investigation is largely of an introductory
The latter
nature, although several new points are made.
concern waves which travel with supersonic phase velocity
relative to the free stream as well as the development and
comparison of alternate analytical techniques associated
with the "inviscid" solutions of the equations of motion.
Some implications and limitations of the theory are disThe effects of the boundary layer are further
cussed.
illustrated by the examination of the stability of traveling
wave disturbances in the fluid and an adjoining flexible
Significant boundary layer effects are found to
surface.
derive from one of three more-or-less distinct causes:
1) the boundary layer thickness is comparable to the scale
ABSTRACT, contId
of the disturbance, 2) the disturbances are similar in
nature to those arising in boundary layer instability
and can cause "resonance" phenomena in the fluid, and
3) the disturbances exhibit "transonic" behavior.
The investigations of the laminar flow are relevant
to the studies of the perturbed turbulent boundary layer
since the latter are based on an analogy with the laminar
The analogy is established by ensemble average
model.
However, certain conjectures regarding the
techniques.
The associated
nature of the mean flow are required.
difficulties are largely removed by a detailed examination
of the turbulent boundary layer over a rigid "wavy" wall
and by correlation of the measured and predicted effects
of the boundary layer on the surface pressure. These
effects are found to be quite significant for certain
geometries. The resulting clarification of the theory
makes possible a fairly successful prediction of an
observed "panel flutter" result under conditions for which
an analysis based on the inviscid, irrotational fluid model
is in serious error.
Thesis Supervisor: Erik Mollo-Christensen
Title:
Associate Professor of
Aeronautics and Astronautics
iv
ACKNOWLEDGEMENTS
The author wishes to express his gratitude to
Professor Erik Mollo-Christensen for his encouragement,
advice, and perhaps most of all for his broadening
influence as a teacher; and to Professor Marten Landahl
for his helpful criticism and advice on many aspects
of this study.
Appreciation is
also expressed to Miss Theo
Coughlin for her unstinting efforts in preparing the
final manuscript and to the author's wife, Moyra, for
the impossible task of translating the hand-written
copy to draft form.
The work was supported under the Air Force Office
of Scientific Research Grant AF-AFOSR-62-187.
If
V
TABLE OF CONTENTS
Chapter No.
Page
1
Introduction
2
The Boundary Value Problem for
Laminar Viscous Flow Over a Perturbed
Surface
2-1
2-2
2-3
2-4
2-5
2-6
3
7
9
17
28
33
37
The Perturbed Surface with a Turbulent
Boundary Layer
3-1
3-2
3-3
3-4
4
Nature of the Investigation
Formulation of the Problem
Asymptotic Solutions
Boundary Conditions
The Surface Pressure
Discussion
1
Introduction
Formulation of the Problem
Relation to the Laminar Model
Application of the Theory
48
50
63
69
The "Supersonic Wavy Wall Problem"'
With a Turbulent Boundary Layer - An
Experimental Investigation
4-1
4-2
4-3
4-4
5
75
80
84
88
Stability of Finite Chord Panels
Exposed to Low Supersonic Flows with
a Turbulent Boundary Layer
5-1
5-2
5-3
6
Nature of the Investigation
Experimental Apparatus
Experimental Procedure and
Results
Discussion and Conclusions
Introduction
Analysis
Results
Conclusion
92
99
106
ill
WWT
vi
Page
Appendix
Stability of a Flexible Surface
A
Exposed to Potential and Laminar
Vis cous Flows
A-2
A-3
A-4
Introduction and Qualitative
Results
The Characteristic Equations
and Techniques for Determining
Eigenvalues
Potential Flow over a Surface
with Dissipation
Laminar Viscous Flow
Hydrodynamic and Aeroelastic
Instability
117
126
134
-
A-1
140
Figures
1
Predicted Effect of the Turbulent
Boundary Layer on the Pressure
Distribution over a Rigid "?Wavf' Wall
2
3
5
6
7
8
Mean Velocity Profiles for the
Turbulent Boundary Layer Adjoining
a Flat Rigid Wall
Wavy Wall Model
Mean Velocity Profiles for the
"Unnerturbed" Turbulent Boundary
Layer - M.= 1.405
"Average" Mean Velocity Profiles,
over the Second Wave - M, = 1.405
Detailed Mach Number Profiles over
the Second Wave
Pressure Distributions over a TwoDimensional Wavy Wall - M = 1.405
Variation of Complex Pressure
Amplitude with Reynolds Number and
Surface Wave Amplitude M, = 1.405
9
10
148
149
150
151
152
153
154
155
Thickness Required to Prevent
Instability of a Flat Simply Supported
Panel - Theoretical and Experimental 156
The Single Degree-of-Freedom AnalysisEffects of the Approximation (5.4),
Viscosity, and Structural Distipation157
U
vii
Page
Figures
11
12
Stability Boundaries
158
cPrzY
The Aeroelastic Mode-shape,
= .0785 - Two Mode Calerkin
for *
Analysis with Viscous Effects included
158
A-1
The Function
A-2
(aoicr)
A-3
Stability Boundaries for Laminar
Viscous Flow over a Rigid "Adiabatic"
160
Wall - MO = 0, 1.3
Stability Boundaries for Laminar
Viscous Flow over a Flexible
161
"Adiabatic" Wall - M. = 1.3
Pro for Laminar Viscous Flow over
a Flexible "Adi batic" Wall = 1,= 1.3
A-4
A-5
R q=
A-6
for M,.
159
159
1.3
162
1.024 x 10
Effects of the Laminar Boundary Layer
and Structural Dissipation on the
Growth Rate of an Unstable Disturbance
163
Neutral "Surf&ces" for.Laminar
Viscous Flow over a Flexible Wall
164
Mw = 0, R, = 1.04 x 10
Neutral "Surfaces" for Laminar Viscous
Flow over a Fle ible Wall - 1, = 1.3,
165
Re = 1.04 x l0&
Qualitative Neutral "Surface" for
Laminar Viscous Flow over a Flexible
166
Wall
-
A-7
vrea
A-P
A-9
References
167
Dimensionial
Non-Dimensional
Description
Dimensionial
Non-Dimensional
Description
I
Quantities Associated with the General Equations of Gas Flow
*
*0
z= C4."e"
/
'W.
e*/
,r
Z.
7-'
--
Cartesian coordinates, scaled on total boundary
layer thickness for laminar flows and on momentum
thickness for turbulent flows
Time coordinate, scaled on free stream velocity
in i=1 direction and boundary layer thickness as
above
Fluid velocities scaled on free stream velocity as
above - when (
is deleted, x1 direction is implied
Fluid mass density, scaled on free stream density
Thermodynamic pressure, scaled on free stream
astatic
Stagnation pressure associated with isentropic
7 o
expansion to Al
Thermodynamic temperature, scaled on free stream
static
- ~Coefficient
static
A7
<r"
4- 7-
. .Internal
of viscosity, scaled on free stream
Kinematic coefficient of viscosity, scaled on free
stream static
Second coefficient of viscosity, scaled on free
stream value of ordinary coefficient of viscosity
energy of the gas
0
tlj
Quantities Associated with the General Equations of Gas Flow.: (conttd)
I
r- =
*C.
~
,
Coefficient of heat conduction
Prandtl number
Local propagation -peed
of sound waves, based
on unperturbed (mean) quantities
S
a3
Description
Non-Dimensional
Dimensional
Ratio of specific heats - assumed constant
Local Mach number based on unperturbed velocity
in i=l direction, 4'.
.etj
I-i.
e'-= f(f,
= (,b
ts
t~~d
a?d'
r
Rate of strain tensor
2:
Stress tensor,
signifies 4
X
/
,
repeated index
4C
dipari,~.
0
fr~
"Total" local boundary layer thickness
0)v
Local momentum thickness (=-C7/VL,'A
laminar boundary layer, see Lees (6)
---()R
RIX L'E'L- 7.;6
for the
Local displacement thickness
Reynolds numbers based on "total" momentum
thickness
Denotes transfer coefficients associated with
"eddy" transport
Dimensional
Description
Non-Dimensional
Quantities Associated with the Small Disturbance Equations of Gas Flow over a
Perturbed Surface
Jr
A'
.A
ITg
Wave number corresponding to a periodic
disturbance
U
c=
C.
W
c
--
Phase velocity of a traveling wave disturbance
0
Circular frequency for harmonic motion
SrxVJ
C it
!
e7s
Wavelength of a periodic disturbance
Agj-
w4 er
O*-t
a Swe'r'o-t)
4,
2
=
7-'*
r(~)
e(~)
;o
84F
to
'I
/
7
A
'a
Small disturbance quantities for traveling wave
disturbances in a frame with the x-axis aligned
y with the phase velocity,; steady (mea4 quantities
"
g,
a
,%
,
ae
Coordinate describing surface deviation from
x , f plane
Small perturbation quantity in partial differential
in ordinary differential equations
equations, 4/,d
/J'pr,,
,
II
,,
associated with the (parallel) shear profile
x
Mom
Dimensional,,
Quantities Associated with the Small Disturbance Equations of Gas Flow
Perturoect 3urrace (conl'd)
-Exponent
e
2 -U
_
11
-o
cit-c,
'A)
F-If
for y dependence of "Inviscid" Solutions
Quantity in (2.20) deriving from the "Inviscid"
Solutions
Quantity representing viscous effects in (2.20)
; SWA$=ssc
I
j,/3$
over a
Arbitrary small number associated with approximate
evaluation of integrals in (6"-'d
.gF
-
II
Description
Non-Dimensional
Non-imesioalDesripio
/f4j(Ii
Quantity representing viscous effects in (2.20)
see Miles (11) for complete tables
'I Tieti enst function
-
Dimensional
I-.
Dimensional
II
Description
Non-Dimensional
Quantities Associated with the Small Disturbance Equations of Gas Flow over a
Perturpea Surrace (cont'd)
J,
f.. .aw
,q
= re'-c
e-aj,
F-e
C
'ir. C ~ g =-
f
(Y:.
Js
-.
Independent variable for "viscous" solutions
a
=e
3/z
Yee
I-I.
Vf Argument of the Hankel function, /l,
in the "Viscous" solutions
jrj
Quantities Associated with the Unbounded Flexible Surface and the Finite Elastic
Panel
S
*j/
P*
'
III
Argument of
Mass ratio for flexible surface - based on
boundary layer thickness
Inverse mass ratio for flexible surface - based
on wavelength
III
Description
Non-Dimensional
Dimensional
Quantities Associated with the Unbounded Flexible Surface and the Finite Elastic
Panel, (cont'd)
Mass ratio for elastic panel - based on panel chord
C.
+
(C.VWOv)
I
ce is the flexure speed (phase velocity for a
) in the
traveling wave with wavelength 2*r/W
unbounded surface "in vacuum"
I
4A*
(/)=.
.
Oa4
is the natural frequency of the fundamental
mode of vibration
a'tr
4z~Z
Viscous damping coefficient,
-6
dimensions
F'r/.
Structural damping coefficient
9
Kinetic energy for a two-dimensional panel with
chord Z4*
P0at"
0o
V-
;-
4,
X &I
(as)
Potential energy for a two-dimensional panel with
and characterized by flexural stiffness
chord 2'
only
Deflection distribution for vibration in the nth
mode
8.
I
Dimensional
Quantities Associated with the Unbounded Flexible Surface and the Finite Elastic
Panel, (Cont'd)
-A
P,
~Ds*
44Gira= 114,
fa SD
3
The generalized coordinate for the nth vibration
mode (specialized to harmonic vibration)
Unbounded-surface/panel thickness and panel
thickness ratio
Unbounded-surface/panel mass density
Effective flexural stiffness, dimensions /I4
Coefficient of membrane tension, dimensions
7- Ow
/
- 39
*e'0
,
III
Description
Non-Dimensional
Spring constant for elastic foundation, dimensions
Young's modulus, dimensions
he F
7
(Streamwise) spatial and time coordinate for panel
X0.*
c *=z' 4
Poisson's ratio
Pc
Panel chord
CHAPTER 1
INTRODUCTION
This study is primarily concerned with the role
of the boundary layer and viscosity in modifying the
pressure on a perturbed surface that bounds the flow.
One of the principal factors that motivated the study
is the lack of correlation between theoretical and
experimental studies of "panel flutter" (i.e., the
"self-excited" vibration of an elastic panel exposed
to a gas flow) at low supersonic Mach numbers.
For the
most part, the analytical studies of panel flutter have
been based on the inviscid, irrotational
(potential)
flow model, for example, see the literature survey by
Fung (32) which encompasses some seventy papers.
For
low supersonic Mach numbers, Lock and Fung (21) have
recently shown that the results of an analysis based on
this model disagree sharply with measurements, the two
sets of results converging with increasing Mach number
and reaching agreement at M 7
2.
In this case, the
observed boundary layer was turbulent and had a "total"
2
thickness of roughly 10' of the panel chord.
One may
note that panels with small dimensions relative to the
wing chord or body length are common in aircraft and,
therefore, the above ratio of boundary layer thickness
to panel chord may not be exceptionally large, even for
large Reynolds numbers.
Miles (14) has examined the effect of a shear
profile on the stability of traveling wave disturbances
in an unbounded flexible surface and the adjoining
inviscid flow.
Although Miles' analysis indicated that
the shear profile could have significant effects on
stability, his results are rather qualitative and, in
any case, the study of the infinitely long traveling
wave is not directly applicable to realizable configurations.
As is pointed out in Chapter 2, Miles' fluid
model is of doubtful validity.
Miles (14) also used
this fluid model to examine waves generated by wind
over water and Benjamin (1) extended the theory to the
laminar viscous fluid.
Both authors recognized the
importance of the turbulent boundary layer (for high
Reynolds numbers, the thickness of the turbulent layer
can be a good deal larger than that of the laminar and,
for this reason, the turbulent layer can be important
for disturbances with larger spatial scales - e.g.,
wavelengths) and they speculated that the laminar flow
3
theory could be used in this case, provided that the
laminar velocity profile was replaced by the mean
turbulent profile.
The investigations regarding the stability of the
laminar boundary layer and an adjoining flexible surface,
for examples see the papers by Benjamin (2), Landahl (3),
and Linebarger (3a), have given additional impetus to
this study.
This physical system exhibits instabilities
that derive from the laminar boundary layer over a rigid
wall as well as those from a flexible surface exposed to
potential flow, the former adding a new facet to panel
flutter.
This study corresponds to flow over an unbounded
surface that is defined by small deviations from a plane,
the plane being parallel to the undisturbed free stream
and the material "particles" of the surface being
constrained to move in a direction perpendicular to
this plane.
The study is then applicable to gas flows
over rigid or elastic solids that have negligible motion
in the streamwise direction.
The unbounded nature of
the surface will present no conceptual difficulty since
the unperturbed boundary layer is characterized by a
parallel shear flow.
layers are considered.
Both laminar and turbulent boundary
The Mach number is taken to be
in the range zero to approximately two, and the Reynolds
4
number is taken to be large, although finite, of
course.
The boundary value problem corresponding to a
surface perturbation described by a planar traveling
wave is considered in Chapters 2-4.
The main object
is to determine the expression for the surface pressure.
The theory for the laminar flow, given in Chapter 2, is
based on the earlier studies concerning stability of
the laminar boundary layer.
Beyond serving as back-
ground material, the study of the laminar flow deals
mainly with two points that have not been clarified by
previous authors.
The first concerns disturbances that
propagate with supersonic phase velocity relative to
the undisturbed free stream (hereafter these will be
designated "supersonic disturbances").
The theoretical
capability for handling these disturbances is essential
for calculating the pressure distribution on standing
wave disturbances with arbitrary frequency of oscillation
if M)>l and for those with a frequency above a certain
value,
dacm)
if
M<l.
This capability is also
necessary for a thorough examination of the stability
of the boundary layer over rigid and compliant walls.
The second point is the consideration of alternate
analytical techniques developed by Lighthill (10) and
5
Heisenberg for determining the "inviscid" solutions to
the equations of motion.
The Heisenberg technique has
been extended to include compressible flows (Lin (5),
p. 85) and has received more general usage, exceptions
being Benjamin's papers (1) and (2).
The Lighthill
technique is here extended to include compressible
flow and the resulting solution is compared with that
due to Heisenberg.
The boundary value problem for the turbulent flow,
considered in Chapters 3 and 4, is based on an analogy
with the laminar flows, the analogy being established
by ensemble average techniques.
As anticipated by
Miles (14) and Benjamin (1), the results do take on a
"pseudo laminar" form, i.e., they are of the same form
as the laminar results with the mean turbulent velocity
profile replacing the laminar profile.
The more
rational approach considered here gives rise to several
interpretations of the theory that were previously overlooked.
It is found necessary to resort to empirical
observations to gain more insight into the factors governing
the choice of the proper interpretation.
However, the
resulting semi-empirical theory is more securely based
and of more general validity than that proposed by Miles
and Benjamin.
The results of the boundary value analyses
display the viscous effects explicitly, thus yielding.
6
qualitative criteria for estimating the significance
of these effects in a given circumstance as well as
the expressions required for a quantitative analysis.
The remainder of the study concerns the application of these results to the stability of flexible
surfaces in the presence of boundary layer flows.
The
analysis corresponding to the Lock-Fung measurements,
i.e., the stability of a two-dimensional, simplysupported panel exposed to a low supersonic gas flow
with a turbulent boundary layer, is given in Chapter 5.
The analysis is based on the conventional Galerkin
technique with the assumed mode shapes corresponding to
the natural vibration mode shapes of the panel "in
vacuum."
In this instance, the Galerkin procedure
permits a convenient approximation for the surface
pressure, the approximation facilitating a better understanding of the mechanism underlying the effect of the
boundary layer on panel flutter and simplifying the
calculations as well.
The roles of structural dissipation and viscous
effects in the stability of laminar flow over an unbounded
flexible surface are examined in Appendix A.
As was
indicated above, this system exhibits instabilities that
derive from the laminar boundary layer over a rigid wall
as well as those from the flexible surfaces exposed to
-I
6a
potential flow.
Indeed, one of the primary objects
of this study is to view the general problem alternately
from the points of view previously taken in studies of
these two limiting cases.
The supersonic disturbance in
the laminar boundary layer adjoining a rigid wall is also
examined.
Further introductory remarks more relevant to
particular facets of the
section of each chapter.
study are given in the first
7
CHAPTER 2
THE BOUNDARY VALUE PROBLEM FOR LAMINAR VISCOUS FLOW
OVER A PERTURBED SURFACE
2-1
Nature of the Investigation
In the following, a laminar viscous gas flow over
a surface undergoing prescribed motion is examined.
The ostensible purpose is to determine the normal surface
stress that is induced by the motion.
The amplitude of
the motion is restricted to be small, and the resulting
linear analysis permits the usual superposition techniques.
Therefore, only traveling wave disturbances are considered.
The generalization to more arbitrary motion via Fourier
transforms and a discussion of the limitations and
implications of the theory are given in 2-6.
In Appendix
A, further insight into the latter is gained through an
application to the stability of traveling wave disturbances
in the fluid and an adjoining flexible surface.
The foundations of the development rest on the theory
of hydrodynamic stability which was developed by Tollmien,
Lin, Lees, Dunn and others; see (4)-(7).
Benjamin
(1)
(2) and Landahl (3) have considered the problem of finding
the pressure on a perturbed surface for low speed flows.
8
Linebarger (3a) has extended Landahl's work to compressible
flows under the restriction that the waves travel at
subsonic speeds relative to the free stream.
The
investigations (2)-(3a) are concerned with the stability
of a laminar boundary layer over a flexible surface and
portions of Appendix A are complementary to these.
The principal results of the present investigation,
and those of previous researchers as well, depend on
certain asymptotic solutions of the governing differential
equations.
These solutions require that the waves travel
upstream with respect to the fluid at rest and they are
expected to fail for M > 2, see Dunn(7).
Also, the
Reynolds number and wave number are required to be large
and small respectively.
Kurtz (8) has developed a numerical
technique for incompressible flow that obviates the need
of these asymptotic solutions and thus relieves the
corresponding restrictions.
The counterpart for com-
pressible flow involves considerably more labor, but
there appear to be no fundamental obstructions.
The primary features of the present chapter are the
consideration of disturbances traveling with supersonic
velocity relative to the free stream as well as the
development and comparison of alternate forms of the
asymptotic solutions.
The latter is useful since the
various forms are in current use for low speed flow.
9
The general outlines of the theory are presented for
the sake of continuity, even though some aspects are
covered in one or other of the references above.
Emphasis is placed on questions not previously resolved,
as well as those that have particular relevance to the
turbulent boundary layer discussion of Chapter III.
2-2
Formulation of the Problem
Consider the flow of a viscous fluid over a perturbed
planar surface of unbounded extent.
The lack of a leading
edge will present no conceptual difficulty since the
boundary layer will be characterized by a parallel flow.
The surface is defined by prescribing its transverse
deviations,
state.
=
The "particle
xI
, e*
,
from the planar
points" of the surface are assumed
to have motion only in the direction normal to the initial
plane.
The governing equations are the full Navier-Stokes
equations and those expressing conservation of mass and
energy, as well as the thermodynamic equation of state.
The boundary conditions are (1) that the fluid disturbances
are everywhere finite and that no energy is being radiated
toward the surface and (2) that the fluid 'barticle; in
contact with the surface move with the surface "particles.
10
The investigation is closely connected to the
The latter concerns
studies of hydrodynamic stability.
the growth or decay of small disturbances in laminar
The flow quantities are described
boundary layer flows.
by small deviations from the steady-state laminar
solutions, i.e.,
Dunn (7)
justifies the use of a parallel flow model
for the steady two-dimensional boundary layer(Or)PQ$
and derives the self-consistent perturbation equations.
These are listed by Lin (5) pp. 76.
It is then sufficient
to consider only periodic disturbances, i.e.,
ac
Ce*
=,4<'))
(jr
e
In the present circumstance, the surface deflection
is given by
*/
rA
ReIT
C
(~~c4~dZ
-/77 VJ
- (*
Ie
11
In a linearized theory, it is consistent to express the
flow quantities in the same form given above, where the
steady laminar solutions are to be continued analytically
for y <
Thus, the mathematical statement of the
0.
problem takes exactly the same form as that for hydrdynamic stability, except for the surface boundary conditions.
A detailed discussion of the latter is given at the end of
this section.
For purposes of calculation, it is convenient to
reorient the coordinate frame so that the
aligned with the phase velocity.
X
axis is
In such case,
C
_0%'C'
j1
The linearized flow equations for the disturbances of
this form are listed by Lin (5) and Dunn (7).
Beyond
the approximations involved in the linearization process,
certain linear terms which are expected to be small from
order of magnitude'have been deleted.
In terms of the
non-dimensional variables defined in the nomenclature,
these equations are:
Navier-Stokes:
Ck/
Fj i
(Nei- c)
' J
cl;7
(2.1)
C)'-1
12
Conservation of Mass:
j(Ug>)- C)+I,+e|,
1,, +9(,(i<9 +1(3>)
0
Conservation of Energy:
State:
et)
, e(,) -A7/f-:
ry)
where
in the reoriented frame.
The unperturbed static
pressure is taken as constant,
i.e.,
A';
=I
and
the equation of state for the unperturbed variables is
simply
.1.
9Pcj
Tr'W
13
The simplicity of (2.1) in the new frame is
evident, z does not appear and 9c5
appears only as
an auxiliary function irrelevant to the determination
of the other variables.
One should be cautioned,
however, that this system is only pseudo two-dimensional
since the Reynolds number and Mach number are based on
the free stream velocity component in the x-direction,
while the velocity, density and temperature profiles
depend on Mach number based on the total free stream
velocity.
The order of magnitude estimates of the derivatives
of the unperturbed quantities and of the perturbation
quantities will be listed here, see Dunn (7).
These
estimates are determined by heuristic reasoning and
requiring that the reduced form (2.1) is self-consistent.
Their use is required in the solution of (2.1) as well as
for intelligent use of the resulting theory.
The solutions
for the perturbations will be divided into two classes,
the so-called "Inviscid Solutions" and the"Viscous
Solutions" to be defined later.
The order of magnitude
estimates are:
Undisturbed Flow:
d "',
P 7(2
.,
= o()
or /22ss
r=/j2,-)
14
(The scale used for the non-dimensional independent
variable is chosen so that this is so.)
"Inviscid Solutions":
''
r -f
&-
49~ 7r,
7T')~'4,'e, rf.
Tr)
"Viscous Solutions":
4
(P 40r-'Y
'f
e
~Z
r 1i0 )e 1,-,
-I
I
TI- << 6 iT
The remaining task of this section is to state
the boundary conditions appropriate to (2.1).
Lin (5)
has shown that (2.1) (excepting the third Navier-Stokes
equation) corresponds to a 6th order system.
expect six boundary conditions.
Thus, we
In dimensional form,
these are:
/,
4
74
=
bounded and the radiation
(if any) is in the positive
y*
direction
1c,7 s.) "/ , ,J,=0
L~
~~~,
i]6x~IJ~t*
15
Fixed Wall
Temperature
*
[-7-/-)
or
"Adiabati c
3(T y w) -/- 7-
,r
e~t.4
In
=
Wall"
order to obtain linear wall conditions, the various
quantities are expanded in Taylor series about y *=
0
and only the first order terms in the perturbations are
retained.
The conditions appropriate to (2.1) are then:
bounded and the radiation
(if any) is in the positive
J direction.
9]
jf-,
(2.3)
A
.*
C, 7 ,
4iA
&J7 J~=A
LJ4
=.J34
0 Fixed Wall Temperature
. "Adiabatic Wall"
Some comments are in order regarding (2.3).
Various
investigators (see Benjamin (1), Miles (9)) have been
16
concerned with restrictions imposed by the conditions
at the wall.
if
The problems are of two kinds.
~V- 1=0
is large, then
2
First,
may have to be
required unrealistically small in order that the
velocity perturbations be reasonably small.
Secondly,
the linearized form (2.3) may be in serious error if
'1
is larger than the distance over which the
velocity profile is linear.
The former difficulty is
probably the more serious for the velocity profiles
usually encountered.
For incompressible flow, Benjamin
has replaced the above restriction with a less serious
one, namely that
i
be small.
This was done by
choosing a curvilinear coordinate system in which the
wavy surface was coincident with a coordinate surface.
These questions are certainly present in principle.
However, there are reasons to believe that they are less
serious than might be expected.
to be derived for the pressure,
Benjamin's for M = 0.
First, the expression
(2.20), is identical with
One should also note that (2.20)
is not a direct function of
f
,but
rather of
and that it merges into the potential flow result in the
limit as the boundary layer becomes infinitesimally thin.
17
The latter profile is the worst extreme as regards
the limitations above.
Lastly, the experimental
results of Chapter 4 indicate that good results can
be obtained even though the actual measured perturbations
are so large as to indicate that any linear theory is
questionable.
Some researchers have proposed the use of the
inviscid fluid model with a shear profile, see Miles
It will be demonstrated that the analysis as
(9).
developed here is questionable for this fluid model,
at least for small c.
The primary reasons are that the
condition (2.3) on the x velocity at the wall is irrelevant
and the condition on the y velocity becomes of second order
as c-+ 0.
Miles proposes an alternative in attempting
to apply the boundary conditions away from the wall.
However, this induces some uncertainties and the resulting
theory is no more attractive in form than that for the
full viscous model.
2-3
Asymptotic Solutions
The concern here is with finding the six linearly
independent sets of solutions to the simultaneous system
(2.1).
Considerable attention has been focused on
18
finding approximations to these solutions for large R
and small
W
real positive
; see Lin (5) and Lighthill (10).
o&
will be considered.
Only
For negative o,
the complex amplitude function defining the pressure is
the complex conjugate of that for positive
oe
with the
other parameters fixed.
"Inviscid Solutions"
For large values of
R, useful approximations to
two independent solutions of (2.1) have been obtained in
the following form:
The determining equations for the
C',
are obtained
by putting (2.4) into (2.1), interchanging the operations
of differentiation and summation, and equating the
coefficient of each power of
The
0)
to zero individually.
formulation is identical to (2.1) with
set equal to zero, hence the designation "Inviscid
Solutions."
19
Performing the operations indicated above yields
4(r-c) n4
4,'
444~tf
(2.5)
The zeroth order solution of (2.5) is the only one
to be retained in (2.4).
The corresponding formulation
(2.5) can be reduced to
C(Ac) 41
(-c)
117.'-
41
0
7'19
-49
SZ$-(7r
r=
7T
where the
4.ej(
e"
7e54r
(Jo has been deleted.
gives the two independent
The first of (2.5')
<f solutions and the remaining
auxiliary relations serve to fill out each independent
set.
It is not possible to obtain closed form solutions
to (2.5'), except for certain special
velocity profiles,
20
U)
.
The techniques previously used to
find approximate solutions have utilized the assumption
that the various quantities
functions of
ce
near
()(!dCCjA1)
0(= O
.
quantities in a Taylor series in
similar fashion as with (2.4),
were analytic
Thus, one expands these
Oe
and proceeds in
(2.5).
Two such solutions have been put forward for M = 0.
The first, due to Heisenberg, involves a direct expansion
of
4V
in a Taylor series in
oC(z
.
This has been extended
to include compressible flow, see Lin (5).
The principle
disadvantage, if it can be called such, is that neither
solution so determined satisfies the boundary condition
at y-+- co
.
Thus, a linear combination of the two
resulting solutions is required.
In order to find a solution satisfying the proper
boundary condition above, Lighthill rewrote
47
as an
exponential satisfying this boundary condition, times a
new dependent variable.
Putting this into (2.5') gives
the formulation for the new dependent variable.
Cle
now enters to the first power
series expansion in oel is required.
However,
so that a Taylor
Thus, the solution
converges less rapidly than the previous one.
The
compressible flow equivalents of both solutions will be
presented here and some comments regarding their
21
application are made in section 2-6.
It is sufficient
to say here that the Heisenberg solution appears to
have more general validity.
It is also possible to obtain an expansion for
large
o
by proceeding in a fashion similar to the
Lighthill technique, with the difference that the new
-g
dependent variable is expanded in powers of
.
The
author has shown that the result is attractive in form,
but its rapidity of convergence etc.has not been investigated.
The Heisenberg Expansion
Writing
71=0
(2.6)
yields upon substitution into the first of (2.5') and
/ /
-/1
W
to zero,
/
equating the coefficients of each power of
0
---
(2.7)
Making a compressible flow transformation
(2.7')
22
reduces
(2.7) to a more attractive form.
The
4
are obtained by inspection and the higher order terms
by the method of variation of parameters.
The two
fre
a~I
J-and
y
(q',-
Dil(OC0
)
i
lo (2.8)
I
1
The integrals are defined by generalizing the problem into
the complex y plane and interpreting them as contour
The contour path will be chosen such that
(real c) correspond to the limit
of growing disturbances (i.e., for
Gt'>,
Czr
ilr
For monotonicfy increasing velocity profiles ( C4sg4
)
>
0
)
the neutral disturbances
.
integrals.
this implies that the contour in the complex y plane
passes below the point yc'
th nutalditubace
(ea
c
crrspndtoth6lmi
The Lighthill Expansion
In section (2-4) the boundary condition appropriate
to the "Inviscid Solution" will be shown to be (see 2.16)
)
independent solutions to (2.5) are
23
where
is defined in the nomenclature.
Again, the compressible flow transformation (2.7')
is convenient in the solution of (2.5') for M > 0.
It
is thus convenient to parallel the Lighthill approach by
writing
where
is the new dependent variable and where it is
evident that the exponential behavior is proper for yPy.
Putting this into the first of (2.5') yields
(2.10)
Writing
477
(Y.; C(A2)
(2.11)
24
a nd duplicating the procedure for the Heisenberg
expansion yields
0/
r
(Cr-c)
z
T-A~. (Or-c)
Z
/
I
jz
1,z
IZ
-1
e,
-
-
z
4
(2.12)
With the lower limit taken as infinity, it is clear that
i = 1 is the proper solution.
[h(Crc
z
)e
Thus
--L
We
fF I- CZ-I Ijk
I
%
1J5#Od]4
0(r
[T
(2.13)
The singular integrals are to be interpreted as in (2.8)
"Viscous Solutions"
It remains to find the four remaining sets of solutions
corresponding to (2.1),.
The characteristic feature of the
use of (2.4) in deriving (2.5) is that the viscous stresses
do not enter the first order approximations.
Thus, the
25
order of the system of differential equations was
reduced and only two independent solutions resulted.
It is evident that the viscous stresses must play an
important role in the remaining solutions and hence,
the designation "Viscous Solutions."
The condition that the viscous stresses should
be of the order of the other terms in (2.1) is precisely
the determining factor in the order of magnitude
estimates of (2.2).
In particular, this requirement
forms the basis for the concept that differentiation
of a perturbation quantity with respect to y increases
the order.
The development is strongly analogous to
classical boundary layer theories except that in the
present case, the viscous effects are predominant in
two sublayers that are rather thin as compared to the
total boundary layer thickness.
One of these layers
adjoins the boundary ("wall friction layer") and the
other surrounds the point yc ("critical layer").
More
Complete discussions of these points are given by Lin
(5) and Dunn (7).
Eliminating W between the first two of (2.1) and
applying the "Viscous Solution" order of magnitude
estimates to this result and the remainder of (2.1)
gives, upon deleting terms of order
E
as compared
26
to 1 (the terms deleted may actually be large near y
so the resulting solutions are suspect there)
09
--
W
(2.14)
with the auxiliary relations:
The solution,
9
=
constant is rejected as not having
the behavior of a viscous solution (i.e., the behavior
).
assumed in deriving (2.14) was
solutions with
remaining i
The two
9 "0 along with the auxiliary
relations give two of the desired sets of solutions and
vice versa with the
solutions.
6
Transforming to the new independent variable,
yeilds for the first
d2
'Ii
I.
Ir,-C-/
two of (2.14)
.
-
1/3
27
where again terms of order
were deleted.
It
should be noted that the coordinate transformation
introduces the restriction c < 1.
With the further
transformations
.33/z
the above is reduced to a Bessel equation.
The Hankel
function solutions are appropriate since there are
boundary conditions to be satisfied for infinite values
Thus
(
of the argument.
.4==35
4
/2)
/4///
With Dunn we note that, for large
W'R
,
.3
is a large
constant times a function that varies slowly compared to
the Hankel function.
F~?f (4)
Thus one can make the approximation
-I
-. %.P
Sif
which satisfies the original differential equation with
errors of the order already made.
The homogeneous solutions
28
of (2.14) are then given by
-f
I/a
~j-+f
3 1 ~
,~
4 3 ~ ~
3k
~
6-Y, 4
(2.15)
3
jZ)
.,/z
4
are obtained from (2.14).
2-4
Boundary Conditions
The six sets of homogeneous solutions are given
by (2.8) and (2.15) and the corresponding boundary
conditions by (2.3).
One could immediately write for
all perturbation quantities
n
=z6
fi/
L
The boundary conditions then give six simultaneous
equations in the Kd .
However, it is more convenient
to apply the conditions at infinity first.
that
P
and
gi, of (2.8) as well as
;F
It is clear
,,
of (2.15)
(see the phase definitions in the nomenclature and the
29
asymptotic expansions for Hank'el functions) are not
bounded as y -lo
.
a
Only two solutions remain if
all of these are rejected and thus the three remaining
boundary conditions could not be satisfied.
The
alternative is that two of the former could be combined
to give another solution
which is everywhere finite.
This is of course not the case with the solutions of
(2.15) so we are left with those of (2.8).
Although
it can be formally shown that the latter correspond to
series expansions of functions that can be so combined;
the procedure given by Lin (5) is followed here.
Consider the "Inviscid Equation" (2.5') for
where
L.,)-+I
,
>
i.e.,
Thus two linearly independent solutions to (2.5') exist
which have the asymptotic behavior:
For disturbances traveling with subsonic speed relative
to the free stream, i.e.,
/-,
clear that the proper solution is
O6/2=e
c< /-e
;<
, it is
30
where
P
is real positive.
Both solutions are every-
where finite for disturbances traveling at supersonic
speeds relative to the free stream, i.e.,
C>/*
y.Thus,
c<<~ 1
the finiteness condition no longer
serves to eliminate one solution and, unless some other
condition is imposed,
well set.
the boundary value problem is not
The difficulty posed is precisely the one which
occurs in choosing the proper solution for supersonic
potential flow over a rigid wavy wall.
The difficulty
is easily resolved by use of the Sommerfeld radiation
condition, that is the requirement that the disturbances
radiate away from the perturbed wall (in a frame fixed
with respect to the free stream) not vice versa.
Lees
and Lin (33) (pp. 37, 49) have discussed this point in
connection with boundary layer stability, but they did
not appear to recognize that the solution with no incoming
radiation was of particular interest.
In a frame fixed
relative to the free stream, the complete solutions have
'
'
-
the form
Examination of the above will show that the radiation is
in the positive y direction for both
if the solution is taken as
C.f
with #
jp2(2.16)
defined as in the nomenclature.
C<
and C
I
,
31
The Lighthill procedure in the derivation of (2.13)
is now clear.
To find the proper linear combination of
the Heisenberg solutions of (2.8), one notes from (2.16)
that
or denoting the proper solution
VIIJ) := ?V
1 _ _Pwhere
r
.
9P
1
2
(2.17)
4qqf~)
The general solutions for the perturbation quantities
may now be written in the form
gives three inhomogeneous algebraic equations in the
The inversion of these equations yields
A
.
Utilization of the three boundary conditions at the wall
32
I
0
Kz
I
--9
M*
d)
k441;rAT
('M
~5
fo
(*)
h30
From (2.2), the terms denoted
times the others.
(0)
e~- 0
~~JJ
were used.
are of order
is given by
0
Kz
Al
-r
- 3o
0
0
ulo
0
1.
9r o f3
Ks)j
C
Thus, the first approximation for
-
K.
0
I
where the relations
-
I(TO
(.1)
4'6-o
JO3
the
1c
(4)
(
I
J3 0
c0i
(1
_fro 0G(A
1c
0
7;ww
i 30 (' JO
47)r
fir
(2.18)
33
The reduction of the denominator is not entirely consistent
with the order of magnitude estimates.
The approximation
of small velocity fluctuations in the temperature mode
I),
is made as well.
The denominator of (2.18) is
precisely the characteristic equation encountered in
the classical theory of hydrodynamic stability.
2-5
The Surface Pressure
The quantity of primary interest is the normal
stress at the surface.
viscuus
stress is of the order E times the thermodynamic
pressure.
here.
It can be shown tiiau Tiie normiai
Thus, only the latter will be investigated
The pressure could be obtained from the expression
//o
=ZI
h7
777(0)
Benjamin (1) discusses more convenient procedures of
obtaining the pressure directly from the x or y momentum
equations.
Although the use of the y equation is probably
more accurate, the x equation will be used here due to
its convenience.
Landahl has shown that they give the
same result for small c.
34
Elimination of
between the fourth and fifth
of (2.3) yields
- tC
Thus,
-
=
of (2.1),
from the first
J*
:Fsero,
-7-oi
terms of order
E
,
or from (2.18) and (2.15) one obtains, upon deleting
2
r'A4.
~(2.19)
Integrating the fourth of (2.14) under the same approximation
used in deriving the first of (2.15) yields, with use of
the latter,
Cor,
C
cIe
,'
Yr,9V
(2.19')
where
N and
%,,
Z
are defined in the nomenclature.
Integrating the first of (2.14) under the above approximation gives
f
+
in terms of
and with the first of
(2.15)
//
h3o
30C
JwR
C
-/
(2.19'')
35
Expressing the quantity
40 fro in terms of the quantity
as defined in the nomenclature gives the identity
(2.19''')
-
Combining the equations
(2.19) gives finally
-1
r/- C) 'o/'+
-
(M a
00 <
c<
iAj (
-
S -
ir)](1+ AccSi
(2.20)
(7
s/;O
4 is written in terms of 4Pr with the aid of (2.5').
Thus, with
0,
given by (2.17) and (2.8), i.e., corres-
ponding to the Heisenberg expansion,
U(-)(+g:-/~MVZ(/--)2#W(/-c)ZJZ
+ir
o
+r~~
-z
___-_
__
(W2)
(2.21)
Equation (2.20) with this result is equivalent to the
results given by Linebarger (3a) except that the
limitation
c 7/- -1
Alternatively, with
has been removed in (2.20).
4
given by the expression (2.13),
which corresponds to the Lighthill expansion
- m. (a3
W(/-C)/-,
(-
+
+-
(o1)
qc)l- c)
4 El
f
(2.22)
36
The function
in the literature.
rzi) has been extensively tabulated
Probably the most complete tabulation
is given by Miles (11) .
The quantities
CZy)
and
7e
for the laminar boundary layer are given by Lees (6) and
Dunn (5).
The same singular integral appears in (2.21) and
It is necessary to remove the singular part
(2.22).
since the velocity profiles are usually given in numerical
form only.
This is done by expanding the argument in
Taylor series about y c and integrating across a narrow
region including y .
For example, one obtains for small c
(Je
)r
(2.23)
where
E
is an arbitrary small number.
It should be noted that (2.21) has been carried to
higher order in
c(
than (2.22).
The order of both can
be extended by retaining more terms in (2.8) and (2.13).
However, examination will show that the particular orders
retained are unique, in that they represent the modification
due to viscous effects, yet give a non-trivial value for
the surface pressure in the limit as c-* 0 (i.e., the rigid
wavy wall).
w
37
2-6
Discussion
flow (inviscid, irrotational fluid model with
p m.f
)
One may note that (2.20) is simply the potential
result with the denominator modified by viscous effects.
r*vO'-)
contains the term arising with the potential
flow model and a term representing the influence of the
shear profile.
r"ez) from unity
The deviation of
represents the effect of the viscous stresses associated
with the perturbations.
Thus, as
aO
J
(i.e.,
c'-*4
),
(2.20) merges uniformly into the potential flow prediction.
co
as
for
Also,
- , (2.20) takes the form
R-e
,
the inviscid fluid model with a shear profile.
However,
the viscous stresses are generally rather important as
will be indicated shortly.
There are three more-or-less distinct instances where
the viscous effects can be of paramount importance for
the traveling wave disturbance.
The first
is the obvious
condition of the boundary layer thickness being rather
large relative to the spatial scale of the perturbation,
*l/l.* is large. This is perhaps of
i.e.,
lesser significance for the laminar layer, but will be
important in the next chapter concerning the turbulent
problem.
The second is the condition of transonic
disturbances, i.e.,
C ./-
L
.
In this case, (
of
(2.21), (2.22) becomes small relative to the other terms.
38
The last, which is not entirely exclusive of the above,
is associated with the classical hydrodynamic stability
problem.
The denominator of (2.20) is precisely the
characteristic equation for the latter problem.
Thus,
it is clear that the pressure is strongly modified by
viscous effects near such eigenvalues.
Certain of these
eigensolutions are associated with the conditions mentioned
above.
by small
However, there are other eigensolutions characterized
O
and large
R
see Lin (5) and Appendix A.
,
As indicated in 2-1, these comments are further illustrated
in Appendix A.
The role of the supersonic disturbance in
the stability of the laminar boundary layer over a rigid
wall is also considered there.
It is a curious, but regrettable, fact that the
inherent limitations of the theory are most severe in
precisely the instances where the viscous effects are
most important.
For example, the expression (2.20) is
singular for those values of the parameters corresponding
to eigensolutions of the hydrodynamic stability problem.
The physical explanation is quite simple.
This is a
resonance condition for the fluid, that is, finite (but
indeterminate) disturbances can exist for the rigid wall
(7
O
) and the linear theory predicts infinite
A
disturbances for finite 'Z
.
In order that the linear
theory be applicable, one must restrict to increasingly
39
smaller anplitudes in approaching the eigenvalue
condition.
However, it is expected that the singular
behavior of the theory does reflect a real physical
effect as is the usual case with resonance phenomena.
At leastto some degree, the usual difficulties
associated with the linearized theory for transonic
flow are present here.
That is, outside the boundary
layer the order of magnitude estimates (for the "inviscid
solutions") break down as
.
However,
examination of (2.20) shows that the viscous effects
do remove the singularity in the surface pressure and
that (2.20) evidences a smooth transition from flow
The
of supersonic nature to that of subsonic nature.
fact that the transonic singularity is mild for inviscid
flow is indicated by the fact that the latter model is
well-behaved for unsteady flow problems (excepting
often
the infinite traveling wave, of course).
Thus, there
seems little doubt that the full theory postulated
here is acceptable for such problems.
The inherent difficulties with
eo
being large
are, of course, associated with the small cf
of 2-3.
expansion
The asymptotic solution for large c
complicates the picture.
also
The alternate large ce
expansion mentioned in 2-3 can possibly be used to
40
alleviate the situation.
However, the latter procedure
is not without its difficulties since it implies expanding
about a condition where the initial hypothesis, that
the x scale of motion be much larger than the y scale,
is not met.
lies with
At any rate, the practical difficulty
= 011)
where both procedures are suspect.
The same difficulty is present in the experimentally
confirmed problem of hydrodynamic stability.
The only
real alternative is the use of numerical methods, in
which case the restrictions on a/, R , C
can be
,
largely removed, see 2-1.
A comparison of the quantities (2.21) and (2.22) is
made in the following.
It is to be noted that the essential
differences occur in the real parts of the terms of order
(
(and the higher order terms).
-Cllzis
real and of order unity.
For low subsonic flow,
Thus, the differences
are of lesser importance in this instance.
In fact, Lin
(5) postulated that the relevant term was of little
However, for
importance in the characteristic equation.
'~'
transonic and supersonic disturbances, /MA
becomes
d
small and pure imaginary respectively.
Uner these
conditions, the differences become of real importance.
One notes that (2.21) is accurate to higher order in
than is (2.22).
The rationale for retaining these
41
particular orders is given in 2-5.
The experimental
results given in Chapter 4 indicate that (2.21) is
far superior in an extreme condition with c = 0.
One
may also note that the higher order terms of (2.13)
are singular for sonic disturbances.
This is not true
of the seemingly less attractive form (2.17).
A word is in order regarding applications for c-0.
Equation (2.5') has a singular point at yc.
Also,
viscous solutions (2.15) are suspect near y
due to
approximation made in their derivation.
Thus, the use
of (2.20) would appear questionable as c-0.
with
o"= 0
and
7 */0
the
However,
("Adiabatic Wall") the
singular point in (2.5') is only apparent.
The relation
to the potential flow model holds with c = 0 and one may
expect that the judiciously chosen forms (2.21) and (2.22)
are representative.
The viscous effects are generally of
a simpler nature for c = 0 since the conditions associated
with the hydrodynamic stability problem are absent (except
at very high Reynolds numbers).
Preliminary calculations
indicate that the viscous effect at low subsonic Mach
numbers is primarily a reduction in pressure amplitude
and
that at transonic amd moderate supersonic Mach
numbers, a significant modification in phase shift is
indicated as well.
42
It should be noted that the results for c-O
are not
valid
this case,
for the inviscid fluid model.
In
the denominator contains a term /L
c-WO
which gives zero pressure.
c
V
The reasons for this failure
were given at the end of 2-2.
Preliminary calculations
have shown that the direct effect of the viscous stresses
are substantial for c rather greater than zero.
There is
no apparent reason for using the inviscid approximation
except that the limitation c<l can be removed.
The remaining task of this chapter is to generalize
the foregoing results to more arbitrary surface motion.
A full rigorous treatment cannot be given due to the
inherent limitations of the asymptotic theory.
However,
the results should be useful for certain classes of
motion.
Benjamin has considered a similar problem for
the two-dimensional rigid wall (with the surface elevation
varying in the streamwise direction only) with incompressible
flow (See .(1) and for the surface elevation varying in
two directions (see(12)).
The latter concerned fluid of
finite depth flowing down an incline.
The lateral
displacement of the surface considered here will vary
only with the streamwise coordinate, but extension to
more arbitrary motion is straightforward.
Consider the linearized Navier-Stokes equations
mentioned in 2-2 (see Lin (5), p. 76) after having
i
43
specialized to harmonic time dependence (or after having
taken Fourier transforms with respect to time).
One
obtains the set (2.1) (with the dependent variables
interpreted as transforms of the original physical
quantities) by Fourier transforming the above equations
and removing the x derivatives by partial integration,
providing that the Fourier transforms are defined by
C:>O
alWX' Y)-dA
(2.24)
The same process yields the boundary condition (2.3)
where
-j
deflection.
eve)
is the transform of the surface
For example, suppose
e
t're'c=
,d *-
(2
.25a)
is the reduced frequency
Positive
-
Real
is the specified spatial distribution
Then
~,
~
(2.25b)
T
44
In such case,
(2.20) corresponds to the Fourier transform
Ibq-
* I
4(-
-
of the pressure and the inversion integral (2.24) yields
4w,
~y----
____
<
(2. 2 c)
In general, it will be necessary to compute the inversion
integral (2.25c) by numerical means.
The extension to
a number of such modes of motion is easily accomplished
by superposition.
The principal difficulty encountered is that the
pressure transform is not available for all values of
That is, due to the expansion for small
c(
and the fact
that the requirement c <1 was imposed, the transform is
not available in the indicated regions
91-77
x
- eI
(ce i)
r'.
45
Thus, the class of functions
4'rz> for which this
I "'I
decays rapidly for
ice)
.
Of course,
*
approach is to be useful, must be restricted so that
must also be small with respect to the range of interest
in o(
(since one must estimate the transform through
the region
o<-4*'
).
A second difficulty arises in the numerical
The eigenvalues for the hydro-
evaluation of (2.25c).
dynamic stability problem imply that the argument has
first order poles.
are from one to three such values of
C=
corresponding values of
-;
(i.e.,
for
A
=
will vary rapidly near
critical.
,
see
and
Appendix A. This
e'; 4
Q14-
although the
),
for values of
M
integrand
near the
The resolution of the problem for these
critical values lies in
contour,
C.
/'
W'
will only be encountered for special values
difficulty
of
/R , there
In general, for fixed
to the complex
(13), p. 459.
If
4/g-xj
in the pressure only for
formulation is
generalizing the transform
see morse-Feshbach
ot
plane,
o
and if one is interested
A'O
,
then the present
appropriate and the
be passed under the critical point,
contour is to
D/
cI'
.
Otherwise,
the relevant spatial functions must be split into two
one-sided functions as in Morse-Feshbach.
In order to
wL
46
treat the function nontrivial for x <O, one must pass
the contour above the critical point
c'-
.
All of these
considerations are based on the assumption that
The portion of the integral (2.25c) on
4>
W
the segment of contour near
analytically when
value.
*
0'
should be evaluated
is near or equal to the critical
The latter is accomplished by expanding the
argument in a Laurent series about
4" and integrating
term by term.
A related
approximate technique that is appropriate
to certain special problems is utilized in Chapters 4 and
5.
If the surface deflection can be represented by a
finite number of traveling waves in the region of interest,
i.e.,
then the following approximation for the pressure is
suggested
where the
7T
are given by (2.20), i.e., they are the pressure
amplitude ratios for the infinite traveling wave disturbance.
mom
47
This approximation is not of universal validity, but
it is extremely convenient where appropriate.
Of
course, the pressure expression becomes exact if
V., Y, -- yPtd
and this
procedure
then represents
a trivial reduction of the general Fourier transform
application.
48
CHAPTER 3
THE PERTURBED SURFACE WITH A TURBULENT BOUNDARY LAYER
3-1
Introduction
The physical problem considered here is essentially
that of Chapter 2, except that the boundary layer is
taken to be fully turbulent.
The main object is to
determine the regular surface-pressure perturbation
induced by a regular
surface deflection.
prescribed perturbation in the
The surface deflection may be
fully described by the regular perturbation, or may
have an additional unprescribed random component.
The
relevance of the resulting theory to the examination of
regular "self-excited" oscillations in the flow over
rigid and flexible surfaces is considered in section 3-4.
Benjamin (1), and Miles (9),
(14) have postulated
the direct use of the laminar results for the turbulent
layer.
UeVr
was interpreted as the turbulent mean
profile, i.e., the problem was considered "pseudo laminar.
Although this is attractive for engineering purposes,
49
several conceptual difficulties arise.
For example,
the use of the turbulent profile accounts for the
macroscopic difference from the laminar flow, but
ignores the mechanism causing this difference.
Also,
the physical processes in the laminar layer are intimately
connected with the existence of a critical layer where
bef
- c = 0
.
Benjamin (private communication) later
became skeptical of the "pseudo laminar" formulation
due to the fact that a given turbulent flow exhibits
no such distinct layer.
Questions regarding the
interpretation of the mean flow quantities for the
flexible and/or perturbed surfaces also arise.
A rigorous analysis for the turbulent layer is
quite complicated.
The following investigation proceeds
largely along qualitative lines with the primary intent
being to set forth a rational analogy between the turbulent
and laminar problems.
A rather precise analogy is found
for the incompressible fluid if the surface deflection has
no random component.
The results do take on a "pseudo
laminar" form and the earlier objections are removed.
The more general problem is on a less secure theoretical
base.
An analogy with the laminar problem is established
and a formal iteration process - with the zeroth iteration
corresponding to the "pseudo laminar" result - is developed.
However, the higher order iterations are complicated and
50
depend on cross correlations of the turbulent
fluctuations which are difficult to determine.
No
direct evaluation of their significance is made.
The analogy to the laminar problem is based on
one of various possible conjectures regarding the
mean flow quantities.
The question of which, if any,
of these conjectures is appropriate to a given class
of physical situations can be established only by
experiment.
This question and the role of the "pseudo
laminar" results are clarified in 3-4 and Chapter 4
It is sufficient to say here that measurements indicate
large boundary layer effects under commonly encountered
situations and that these can be successfully predicted
by the theory.
3-2
Formulation of the Problem
The physical situation under consideration is the
same as described at the beginning of 2-2.
However, one
must keep in mind that all quantities, including the
surface deflection
(,?,t.,
,
are now random functions.
The special case with no random surface-deflection
component will be an obvious reduction of the more
general case.
The governing equations in their non-
I
dimensional form (see nomenclature) are, in tensor
notation,
2 (e',~~ 1'f4 *)
&
4?(PA
d6P ---+
a(iI)
=
-~--
=0
CP i~A
7
(3.1)
Note:
In order for the first of (2.2) to remain valid,
and thus Re,,
J-;
the length scale is taken to be
will appear throughout.
where
%
+
4.4- ,)
'=-~1:,
J.:-Z3
--
d
-
~ -4
~ 4VW.'
~ _~a'4 j
The boundary conditions are
=
W .0 , 1J T
lat
-
and the radiation
(if any) is in the
positive y direction
bounded
.0
11*.
(4e)
=
--
(3.1')
52
- Fixed Wall Temperature
or
-
- "Adiabatic Wall"
The decomposition of these quantities into a
mean
steady component and a superimposed random part
with zero mean is well known in the applications for
the rigid wall.
In the present circumstance, the
averages will be construed in the ensemble sense because
of the time-dependent nature of the boundary conditions.
That is, one consideres a large number of physical flow
realizations with the regular portion of the boundary
motion identical in each realization.
The mean quantities
to be discussed are then the numerical averages of the
total random functions.
this concept is
The mathematical statement of
(taking the velocities as an example)
r(3.2)
where
a.'.
53
The Oi signify the time dependent regular motion and
the
di
the random fluctuations.
It is assumed that the
M.'
can be further sub-
divided into a regular time-dependent perturbation
corresponding to the surface traveling wave and the
mean steady portion,
&l
e
The existence of the
former is quite clear for the boundary value problem
under consideration, providing that the amplitude of the
surface perturbation is not so large so as to induce
serious separation.
Although the obvious analogy to the laminar problem
is to interpret the
VWCA-2,z) as the mean turbulent
profiles over a flat rigid surface, some special difficulties
arise.
There is reason to believe that the turbulent
fluctuations, and likely the mean profiles as well, are
affected by a compliant wall.
A more serious consequence
arises from the fact that the mean profiles are strongly
influenced by the turbulent fluctuations.
Thus, any
modification of the latter, due to the regular perturbations,
could have significant effect on
Zo
r z,
.
The
various interpretations are discussed below and a partial
resolution of the question is given in Chapter 4.
The assumption is made that the
of
E
x,
variations
(and all other mean steady quantities) are
54
due to boundary layer growth.
The parallel flow model
will be used since only then do the fluid equations
admit solutions periodic in
X
Calculations
,
based on turbulent flow over a rigid wall show that
the assumption of the parallel flow implies errors in
the differential equation of order
to
7,
-/a
as compared
for the laminar layer.
In analogy with classical studies of the turbulent
boundary layer, it is convenient to consider the ensemble
mean of (3.1).
Summing Eqs. (3.1) over many realizations
and taking the numerical average of the results yields
the following expressions, where the products of the
regular perturbations have been
deleted as small with
respect to the terms which are linear in the regular
perturbation:
S-
)4-J
42
(3.3)
10
714
55
at
xax-
0-
e thaut
-(a
a7-
,;-J
-
f!J
n
piu
n Irute
ua
th
t
en
e
4X 'p
in
y
'Poll,'
Alf
4L
4,
temswoldb
boundaryX
zeo__erl
o;r
o
4rltpae) h em o
FAye
the
rth
ubln
-1ayer
The quantities marked off in the E r .brackets are
the counterparts of the ensemble average equations
appropriate to the turbulent boundary layer in the
absence of the regular perturbations.
(Many of these
terms would be zero or nearly so for the turbulent
boundary layer over a flat plate.)
The terms not
indicated by the I I or fIbrackets are the counterparts of the equations for the perturbed laminar boundary
56
layer; and the terms marked by the
f ,I
arise from
direct coupling between the random and regular fluctuations.
The latter are primarily associated with density
fluctuations in the convective derivatives of the
Navier-Stokes equations and would vanish for incompressible
flow.
In a strict sense, the introduction of (3.2)
triples the number of unknown functions.
From this
point of view, Eqs.(3.3) merely imply certain constraints
on these functions and are by no means sufficient to
determine
them.
However, the equations can be reduced
to a form which serves as a small perturbation theory
for the regular disturbances.
This reduction rests on
alternate arguments which are intimately related to the
various interpretations of
V
above.
The structure
of the resulting equations is the same in all cases, but
the interpretation of the quantities
serving as
coefficients depends on the particular argument.
For example, suppose that the mean flow quantities
and the turbulent correlations are associated with the
unperturbed boundary layer over a flat surface, be it
rigid or flexible.
This would imply that the 1) quantities
in
57
correspond to the equations of motion in the absence
of the regular perturbations and thus can be eliminated
from (3.3).
The condition of the turbulent boundary
layer is reflected by the difference in the macroscopic
environment as represented by the mean quantities
appearing as coefficients and by the
f I
terms which
reflect the average influence of the turbulent fluctuations.
These coefficients are presumed to be known from theoretical
and/or empirical studies of the unperturbed boundary layer.
This interpretation, with the
Ii
deleted,
corresponds
to the "pseudo laminar" model considered by Benjamin and
Miles.
For the incompressible fluid,
Yc
about
the critical layer
is seen to exist in an average sense.
More
will be said about the compressible flow in 3-3.
An alternate set of perturbation equations which
directly reflect the turbulent transport of momentum and
energy,
can be found by accounting for the influence of
the regular perturbations on the other quantities in a
The turbulent transport phenomena are
represented by the terms
and
the classical boundary layer equations.
ew
in
The use of the
"eddy" viscosity and conduction coefficients,
to account for these effects is
z.
well known.
4
(
,
particular way.
If one
proceeds in an analogous fashion for the unsteady problem
58
and replaces the corresponding terms in (3.3) by
and
C-4
n
respectively, a curious result is obtained.
O-
That is,
again linearizing in the regular perturbations and
assuming that the remaining mean flow and correlation
terms are unaffected yields a formulation identical to
the one above, except that, in the terms of interest,
the laminar coefficients of viscosity and heat conduction
are replaced by the sum of the laminar and "eddy" values.
It is presumed that the "eddy" coefficients are available
from studies of the unperturbed boundary layer.
The
physical difference between this and the previous
interpretation is easily explained.
Due to the large
magnitude of these particular correlation quantities,
small percentage changes can give sizeable contributions
in the perturbation equations.
It will be shown later that
the two procedures essentially coalesce for surface waves
with small c.
One can write still another set of perturbation
equations which account for a postulated effect of large
amplitudes.
It is expected that the mean flow quantities
are strongly affected if the surface perturbations become
59
The mean level of the correlations may change
and they may have wave-like variations in X ,
-,
-
large.
for the surface traveling waves to be considered.
However, it is postulated that the latter can be
largely removed by the "eddy"
transport concept and
that the remaining correlation quantities can be taken
as functions of y only.
The terms involving the regular
disturbances are periodic in x
,
for surface
traveling waves and due to the independent nature of
the functional forms, the
removed.
CI
terms can again be
The remaining equations have the same form
as before, but the mean flow and correlation quantities
which serve as coefficients are unknown.
Indeed, the
theory is non-linear since it implies linearization
about a condition that depends on the perturbation.
The problem of predicting the mean quantities as
functions of the perturbations must be solved before
the theory could be used to predict the flow for a given
perturbation.
This question is not considered in the
present work, but the value of the general procedure
is demonstrated in Chapter )4.
It can be shown that the order of magnitude estimates
given in (2.2) are again self-consistent with the reduced
form of (3.3) provided one modification is made, namely,
that for the viscous solutions r-e- -
-f
.0
60
4a. and
Ca
are numbers characterizing the amplitudes
of
and
/4 4
/4s
This modification
respectively.
implies no new restrictions on the theory for typical
values of the correlation coefficients over a rigid
wall, see Hinze
(15),
ilorkovin (16).
Adopting the parallel flow model and re-orienting
the coordinate system so that the x axis points in the
cirection of wave propagation yields for (3.2) and the
surfacc deflection under consideration
L3
e
.0
-A(X-CR/
(3.4
xV ,e,-vx
The use of (2.2) to delete the nonessential terms in
the reduced form o2 (3.3) yields for the special motion
of (3.4) (see nomenclature)
"4P 7- g!E- f
~2 e[uo~4J
I
[,
Pa
,1,Z
r+(a A6
I1,
(3.5)
4t
61
7-
7F.= Pg 7-IP
I
, the mean equation of state is
where again
=
#7-
including
7'
.. ej (.fr)
,
and where the mean flow quantities
can be interpreted in the various
ways mentioned above.
The boundary conditions appropriate to (3.5) are
found from (3.l') by considerations similar to those
The process will be demonstrated
used in deriving (3.5).
for the first
of the surface conditions of (3.1').
Expansion of the left hand side in a Taylor series from
the wall yields with (3.2)
Deleting non-linear terms in the perturbations and taking
the ensemble average of the result gives (with
tr
O)
P
/-*--
*
12
I
L
-I
~
... =0
62
Any one of the alternate assumptions that permitted
elimination of the
Fl
eliminated here as well.
in (3.5) allows them to be
One does not expect the
higher velocity derivatives to be strongly correlated
with the surface displacement and thus it is reasonable
t o assume
-;t
.0,2
The further restriction that
An estimated equivalent of this condition is
11
e<6
12
a
*
is made.
The previous expression then reduces to
2
(i~)(44~~J
Similar expressions can be derived for the other boundary
For the particular motion of (3.4)
conditions.
these
become
.64.
zoJa.
(3.6)
63
0
Fixed wall temperature
"Adiabatic Wall"
=
4,
P. &
3-3
bounded and any radiation is in
the positive y direction.
Relation to the Laminar Model
The similarity of (3.5) and (3.6) with (2.1) and
(2.3) suggests that the techniques of Chapter 2 will
be useful here.
In fact, the mathematical formulations,
including order of magnitude estimates, are identical
for the incompressible fluid if the random surfacedeflection component is absent.
The results of Chapter
2 are directly noolication to this case.
Although the general case is more complex,
one may
note that the differences from the simple case above
arise from two sources: 1) density and temperature
variations which couple the random and regular fluctuations and 2) the expansion of the non-linear wall condition.
With this in mind, an iteration scheme is established for
which these differences enter only in the higher order
iterations.
As was indicated in 3-1, the latter involve
considerable labor in applications.
Chapter I4 show that their use
refined calculations.
The results of
would be warranted only in
64
The dependent variables of (3.5),
(3.6)
are now
functions of the correlations of the turbulent
With the largest of these characterized
fluctuations.
, a perturbation expansion can
by the amplitude,
be written in the form
()%
(3.7)
theories,
(i
i.e.,
is taken to be independent of
7
.
The subsequent development will be limited to the linear
The mean flow quantities will be taken to correspond to
a given value of 4' so that the introduction of
into (3.5} and (3.6) yields
+~~
CV
/(c7-c)
24,
(3.7)
(3.8)
7Tz
-
a-
7
+<
4:I(ZXcJ&,r
-7.rv =-~r~me
4f~ -/11jjC~~~4A,)
+(,
e
vs
-
4 (4-c
}
65
4M3&)
frai, 4/r,
+
Jb
, Gfa
)
bounded and any radiation is
in the positive y direction
are known functions.
where 7/,87
,
The estimates of (2.2) are independent of
hence they are expected to hold for each individual
iteration.
Thus, the same concepts leading to the
asymptotic solutiors ol Chapter 2 apply here.
open to question whether the set
It
is
(3.5) has a critical
layer - i.e., note that the coefficient TC
in a different fashion in each of (3.5)
-
is modified
and thus the
fact that (3.8) implies an iteration on a set which does
have a critical layer may be disturbing.
However, the
restrictions on the solutions are no more severe than
on the laminar solutions since the latter fail near
Ic
as well.
Expanding the quantities of (3.8) in the power
series (2.4) yields the formulation for the "Inviscid
Solutions."
The set corresponding to
put into the form
(g)
can be
T
66
/ra
___
_
419__
A1047c
~Ca
a
,
40
jffar'/~*sg(LC&
s~>~ ~p~Z -
r
~
~
C
(-c)
T
.
___
0
V.,&(H.></..)
'
.
-'
.+
,,
+
___
/ ,,em..) I
-, (0
q
4.
7-Praj
~~4;
4
(4j
(7)
4Z
'2~
I A(4
Z-V
2-::
r
-
' *
I
-F'ZAlf
(
63 fAI
I eew)
-
f417
P
eAM-0)
4
4
* g~'
J15/40
4
-'C-0:iAP
The zeroth order (homogeneous) solutions of
are found in the same manner as those of
.)
.
(3.9)
only
the particular solutions of the higher iterations are
67
needed and these are found in terms of the homogeneous
solutions by the method of variation of parameters.
The '"viscous solutions" are obtained by the same
considerations that led to (2.14).
The corresponding
formulation is
w-(3.10)
4,9
/C31
7*J
Again the zeroth order formulation is identical with (2.14)
and the higher order particular solutions are given by
variation of parameters.
Note that the streamwise
velocity fluctuation corresponding to the thermal mode
is not zero in the higher order iterations.
Application of the boundary conditions away from
()
h
e
sam
d
rUa
)
the wall again shows that the solutions are of the form
where, to the same order as (2.18), applicatlion of the
wall conditions in (3.8) gives
68
0
0
Lf
.,
4X0
zj
-fro
;~-
e-P~rL
13r
~ffa~
$"a
Vyd)
Jjrdl~b~,)ki
071a
7.-i
eb
AM~w
f3OtA~j
-ro
(3.11)
Eliminating
-iC
ro>
A
between the first two of (3.6) yields
+y,',,=
-
AECf,rzC'r>
Keeping terms of the same order as in (2.19) one has
(3.12)
The first term is identical in form with (2.19) and (2.20).
Equations (3.9)-(3.11) provide the necessary information to
compute the higher order terms.
7
69
3.4
Application of the Theory
The assumptions and restrictions implied in the
use of the "pseudo-laminar" result for the turbulent
layer depend on the particular application.
The results
of this investigation show that the earlier objections
and restrictions (see 3-1 and Miles (9),
relieved or removed.
(14)) can be
For example, the application to
the incompressible fluid over a surface with a negligible
random deflection component requires only that the time
and spatial variations of the turbulent correlations be
small.
Even this requirement can be made less severe
by the use of the "eddy" transport coefficients.
The
more general case requires that terms in the equations
of motion of the order of the largest correlation
coefficient be deleted, although this restriction can
be relieved under the assumption that the regular
perturbations do not affect the mean flow quantities.
The limitations of the laminar theory apply here as
well.
Indeed, the magnitude of the viscous effects
is increased and the objections to the asymptotic
solutions are accentuated due to the diffuse character
of the turbulent profile.
Two aspects of the theory are still to be resolved.
These are connected with the conjectures made in the
reduction of the system (3.3) and no way has been found
to resolve them on a deductive basis.
The first concerns
70
the alternatives of using the mean quantities for
flow over an unperturbed surface, be it rigid or
flexible, or those in the presence of the perturbation.
One expects the proper choice to hinge on disturbance
amplitude.
Indeed, for the experimental configuration
of Chapter 4, the first alternative is found adequate
for disturbances of moderate amplitude and the second
is successfully used for extreme amplitudes.
No attempt
is made to predict the effect of the large amplitude
disturbances on the mean flow quantities, but
Uf,'
over a rigid wavy surface is shown to be qualitatively
similar to the profile for a rough wall.
The remaining question is whether the coefficient
of viscosity corresponds to the laminar, or the "laminar
plus eddy" value.
If
C
is small, the question is
shown to be of little significance in calculations for
the surface pressure.
The fact that variations in 92
of order 1-2 change the viscosity dependent term in
(2.20) by only 25-300, is evident from the definitions
of
,
g'.)
.
Preliminary calculations for the
turbulent layer show that the latter term is reasonably
small with respect to the
tA*407) term.
Hence, the
surface pressure is insensitive to such changes in
It is also important to note that the values of
Z
entering the calculation are associated with the layer
-1
71
-
c
The following sketch shows the qualitative
.
distribution of the "eddy" viscosity which was determined
by Clauser (17) for incompressible flow over a flat
rigid wall.
i,;0
I
S-~
-
-9
4~J
T
p
4I
P
-t
.) ~
0
,I
I
l.a
-z
From these considerations, one can conclude that the
inclusion of the "eddy" coefficient is immaterial for
c~o
C
0}
or less.
, rr
Preliminary calculations for large
C
indicate
that the addibion of the "eddy" value brings the result
much closer to that for potential flow, but the basic
question is still not resolved.
72
It is evident that this study is basic to the
examination of regular"self-excited" disturbances in
the turbulent flow over rigid and flexible surfaces.
The expression for the surface pressure is required
in the case with the flexible surface and the denominator
of (2.20) is the approximate characteristic equation for
the disturbances over the rigid surface, i.e., the
"equivalent" Tollmien-Schlichting waves in the turbulent
boundary layer.
These applications have special features that are
not associated with similar investigations for the laminar
viscous flow or for the boundary value problem with the
turbulent flow.
In particular, the existence of the
regular time-dependent perturbation in each flow
realization is only postulated (see Eq.
(3.2)) and the
equations (3.5), (3.6) are not the governing equations
for one physical system, but rather the (approximate)
average equations for an ensemble of such systems.
Aside from the approximations involved in their
derivation,
(3.5),
(3.6) must describe any such perturbations
that actually exist.
However, it is conceivable that
the inverse is not true, i.e.,
solutions to the set
(3.5), (3.6) may not imply that the regular perturbations
exist for each realization.
In mathematical terms this
would mean that these solutions imply sufficiency, but
73
not necessary, conditions for the existence of the
Further theoretical and
regular perturbations.
experimental studies will be required to determine
if these comments are of more than academib interest.
One certainly expects that the theory will be
useful for examining the effects of the gas flow on
disturbances deriving from the natural vibrations of
an elastic surface.
This is verified for a particular
example in Chapter 5.
The existence of the "equivalent"
Tollmien-Schlichting waves remains to be resolved since
they have neither been predicted theoretically nor
observed in the laboratory.
As was indicated in
Chapter 2, the theoretical prediction of these waves
would imply that the expression for the surface pressure
has singularities.
The fact that the mean profiles are not generally
available in analytic form causes difficulties in the
For example, one needs to know
pressure calculations.
such quantities as
,
,
.
These quantities
can be estimated from measured profiles, but perhaps a
better procedure is to fit universal profiles - which
have been developed in classical studies of the turbulent
boundary layer - to measured data.
Jo
can be determined
from knowledge of the coefficient of viscosity and the
wall stress in cases where the unperturbed profile can
be used, see (18).
724
The question of determining the temperature
profile also arises.
It is proposed that the usual
assumption of constant total temperature be made for
applications with the "Adiabatic Wall."
However, heat
transfer is not expected to be very important for
pressure calculations in the Mach number range under
consideration.
75
CHAPTER 4
THE'"SUPERSONIC WAVY WALL PROBLEM"
WITH A TURBULENT BOUNDARY LAYER - AN EXPERIMENTAL
INVESTIGATION
4-1
Nature of the Investigation
Preliminary calculations based on (2.20),
(2.21)
indicate that the presence of a typical turbulent
boundary layer can significantly influence the pressure
on a perturbed surface.
These effects are most dramatic
in a broad transonic and moderate supersonic Mach
number range.
on a
For example, the pressure distributions
rigid wavy wall witu
u.uulent
boundary layers
characterized by the velocity profiles measured by
Lock and Fung
(21) and Morkovin (16) are shown in Fig. 1.
These profiles are exhibited in Fig. 2.
The purpose of
this experimental investigation is to examine the interaction between the boundary layer and the surface
perturbation.
In particular,
the orders of the
aforementioned effects are to be firmly established and
the character of the boundary layer is to be investigated,
76
both for its intrinsic interest and to clarify the
The physical
configuration for the study is a fully turbulent
-
intuitive concepts of Chapter 3.
although not necessarily equilibrium - boundary layer
adjoining a rigid wall whose surface elevation varies
in a sinusoidal fashion.
Mach number is 1.4.
The undisturbed free stream
This physical configuration permits
simple experimental techniques, yet gives a severe test
of the theory (e.g., c = 0 and ce is large) and gives a
large and interesting boundary layer effect.
Similar experiments have been carried out for low
speed flows, see Stanton, Marshall and Houghton (15),
and Motzfeld (20).
Benjamin (1) gives a rather complete
discussion of these results.
No regular surface pressure
component was discovered in (15) and this implies that
the boundary layer was badly separated.
Benjamin uses
his form of the "pseudo laminar" theory to predict the
Motzfeld result and finds good correlation between the
theory and experiment.
The boundary layer effect for
this condition is largely modification of the pressure
amplitude.
Benjamin predicts the amplitude within 10%
while potential flow theory is in error by 50%.
The use
of the sligntly different rormulation (2.20) gives
essentially the same result, the amplitude is predicted
within 15% with (2.21) and within 10% with (2.22).
77
There are several features in the Motzfeld experiment
which can be adversely criticized.
Motzfeld's observed
velocity profile more nearly resembles fully turbulent
pipe flow than a flat plate boundary layer, e.g., the
"boundary layer thickness" was half the tunnel height.
The fact that the wavelength was larger than the tunnel
Thus, it is felt that the
height is also unsettling.
correlation between the theories and experiment could
at best be taken only as a tentative confirmation of
the former.
The present measurements show that the mean profile,
is not strongly affected for moderate disturbance
amplitudes.
Measures of the upper limit for which this
is found to hold true are
0 (/,/a).
-p
on
UZry)
roughened.
E
=.>4
= O('4)or
For larger amplitudes,
the effect
is qualitatively the same as if the wall were
It is curious to note that this implies an
increased momentum loss through the boundary layer and
hence increased friction drag.
However, the boundary
layer effect decreases the supersonic wave drag.
Although
no total drag is estimated, it would appear that the
latter effect is dominant for the conditions of the
experiment.
The existence of mild separation is not.
78
necessarily of great importance for perturbed boundary
layer flows.
That is, it is possible to predict flow
reversal near the wall with linear theory.
However,
even at the maximum disturbance amplitude,
7=.oZA,
no indication of the flow reversal is found in the profile
measurements.
This indication that no serious separation
occurred is reinforced by the nearly sinusoidal pressure
distribution on the surface.
The detailed variation of
the velocity profile along the wave is found to correspond
directly to the usual considerations of favorable and
adverse pressure gradients.
The measurements confirm the boundary layer effects
indicated in the preliminary calculations for the surface
pressure.
The use of the "pseudo laminar" expression
(2.20) - with the auxiliary relation (2.21), which is
derived from the asymptotic Heisenberg expansion - is
shown to give results which are well correlated with
the measurements.
On the other hand, the auxiliary
result (2.22), which is derived from the asymptotic
Lighthill expansion, is found to underestimate the
boundary layer effect considerably, although it does
predict the correct trend.
Recall that the latter has
some inherent inadequacies, particularly for high speed
flows.
It is remarkable that (2.20) gives such good
results since the quantity
a.
zr
is of
79
the order 1.5.
(Note that (2.20) is independent of
the scale factor and that the upper limit of the
integrals of (2.21), (2.22) are of order unity if
is used.
Thus, the quantity
O(
is the proper
measure of the boundary layer effects even though c'e
is the quantity appearing in the analysis for the
turbulent boundary layer.)
However, the success in
predicting the effects of variation in Reynolds number
and amplitude - and Mach number as well if the tentative
correlation with the Motzfeld result is included - is
too complete to be explained by coincidence.
One may
also note that the asymptotic theory of hydrodynamic
stability has been experimentally confirmed for large
values of
Oe
.
Of course, the surface pressure
calculations depend on the interpretation of
7
(jIJ
The use of the mean profile in the perturbed flow or
-
the profile corresponding to the unperturbed flow
which for the first few wavelengths is taken to be
identical to the profile measured immediately upstream
from the wall perturbation - corresponds to the alternate
conjectures made in 3-2.
Thus, separate calculations are
made with each interpretation, and comparison with
the measured values enables one to infer which, if any,
of the procedures of 3-2 is proper.
Both profiles
80
give good results for the moderate amplitude disturbance,
although it is found that the use of the Pnperturbed
profile is preferable.
However,
the use of the
modified profile is found to largely explain the
non-linear variation of the pressure amplitude with
surface-wave amplitude when the latter is large.
The
sinusoidal variation in surface elevation has its
starting point in the test section and it is found
that the pressure distribution settles down to the
"infinitiwe wave train" result (recall the approximation
discussed in the final paragraph of 2-6) in half a wave
length or less.
The question regarding the use of
"eddy" transport coefficients cannot be answered with
a rigid wall experiment.
4-2
Experimental Apparatus
The measurements were made in the 18" x 23" x 30"
test section of the Aerophysics Research Laboratory.
This is a continuous flow, closed return wind tunnel
and permits relatively large variation in stagnation
pressure (1 psi-20 psi), but is restricted to a small
variation in stagnation temperature (600 F-120 0 F), see
(22).
The lowest design supersonic Mach number is 1.5.
81
Although a set of slotted transonic nozzle blocks were
available, the corresponding flow conditions were
considered too rough for precise measurements.
Since
a supersonic free stream Mach number below 1.5 was
desired, a "bastard" configuration consisting of an
upper nozzle block for M = 1.5 and a lower nozzle block
for subsonic flow was used.
This configuration had
been previously found to give good flow conditions with
a Mach number of 1.35, although the latter was raised
to 1.405 as a result of fairings required to accommodate
the wavy wall model.
The latter was mounted on the floor,
i.e., on the subsonic block.
The test section is equipped with a traversing
mechanism giving arbitrary streamwise and vertical
motion in the vertical center plane.
A two probe rake,
which consisted of an ordinary 1/16" O.D., 1/32" I.D.
total head tube and a small boundary layer probe mounted
1/2" below the former, was mounted on the traversing
mechanism for free stream calibration and boundary layer
survey measurements.
The centerline of the boundary
layer probe was inclined downward 70 from the horizontal
in order to obtain complete velocity profiles on the
positively inclined portions of the waves.
The tip
was tapered to .011" x .038" O.D. and .003" x .030" I.D.
The upper probe was used to check the accuracy of the
82
lower and to decrease the running time required for
the boundary layer surveys.
The total pressures were
measured with a 60" differential manometer containing
silicone with a pressure-head relation,
in the case of low stagnation pressure and with Wallace
and Tiernan absolute pressure gauges for high stagnation
pressures.
The reference pressure was essentially
vacuum in all cases.
The vertical positioning of the
probe was determined by touching the surface and using
the traversing counter system to measure vertical distance
from this location.
Although the "touch point" was
judged by eye, extensive initial calibration indicated
that an accuracy of .002" could be obtained.
Streamwise
positioning was determined by the traverse counter using
a fixed forward reference.
Serious lateral probe flutter
was initially encountered near the surface.
The vibration
amplitude was reduced to 1/16" by stiffening the probe.
Comparison of profiles before and after stiffening
indicated that this final vibration caused no measurable
error in total pressure.
No vibration in the vertical
plane was observed.
The wavy wall models were constructed from 1" thick
mahogany blocks and were firmly attached to a 3/8" thick
83
aluminum base plate, see Fig. 2.
Eight wavelengths
of sinusoidally varying surface deflection were cut
with milling cutters made especially for this purpose
and the surface was hand-rubbed and filled.
The two-
dimensional waves spanned the 18" test section.
Two
such models were constructed with waves of .04" and
.08" double amplitude respectively, both having 2"
wavelengths.
Careful measurements after installation
showed that the surface elevation of the small amplitude
model had a tolerance of
-
0.0005".
The same tolerance
applies for the large amplitude model, except that the
elevations were measured relative to a mean surface which
was characterized by a smooth bow with 0.002" amplitude
over a 6" chord.
Thermocouples were mounted on the upper
and lower surfaces of the mahogany block upstream from
the surface perturbation.
These were used to set the
stagnation temperature at a value which corresponded to
the "Adiabatic Wall" condition.
The models were equipped
with 39 static pressure orifices which consisted of .02"
I.D. steel tubes mounted vertically and filed flush with
the surface.
The majority of the orifices were on the
lateral centerline, giving rough pressure distributions
over the first and seventh waves and detailed distributions
on the second and fourth, see Fig. 3.
Static pressures
were also measured before and after the waves and limited
lateral surveys were taken on the second and fourth
waves at streamwise locations where separation would
I
84
be expected to first set in.
The static pressures
were measured with 110" differential manometers
containing the fluid mentioned above.
A vacuum
reference was used for the low stagnation pressures
and the balance-house-chamber static pressure was
measured by gauge and used for reference at the high
stagnation pressures.
4-3
Experimental Procedure and Results
Figure 2 shows a boundary layer profile measured
at the Aerophysics Research Laboratory by Baron (23)
with approximately the same test section configuration
as that for the present experiment.
Profiles measured
elsewhere by Lock and Fung (21) and Morkovin (16) are
also shown.
The profile measured by Baron would give
less significant effects on the surface pressure at
the same Mach number and free stream Reynolds number.
Therefore,
the Reynolds number for the present experiment
was varied from roughly that of the above measurements
to an order of magnitude smaller in order to obtain
velocity profiles that more nearly resemble the LockFung-Morkovin profiles.
The variation in Reynolds number
was achieved by changing stagnation pressure.
85
Figure 4 gives the unperturbed velocity profiles
measured above the forward static pressure orifice.
The stagnation temperature was adjusted to correspond
to the adiabatic wall condition and the usual assumptions
of constant static pressure and total temperature through
the boundary layer, as well as normal shock and inviscid
pitot compression were made.
Measurements gave actual
variations in total temperature and static pressure
through the boundary layer of 312%.
The stagnation
temperature varied from 81 to 85 0 F depending on room
temperature.
Free stream surveys were made and they
indicated Mach number variations of less than .015.
The differences in the total pressures measured by the
two probes were less than those due to non-uniformities
in the stream.
Total pressure profiles were measured at various
streamwise locations along the second wave.
The
measurements were made at both the low and high Reynolds
number conditions for the low amplitude wave model and
at the low Reynolds number condition for the large
amplitude model.
The static pressure is, of coursenot
constant through the perturbed boundary layer.
However,
assuming that the static pressure and the velocity are
of the form
-fry) +
ax, ;arm
ki3, one can show
that the pitot pressure is of the same form including
86
all higher harmonics.
profiles with
Averaging the total pressure
4
streamwise separation distances of
removes the first and second harmonics and thus yields
the non-periodic portion with errors of third order in
the perturbation.
The non-periodic portion of the
pitot pressure is related to the non-periodic portions
of the .Mach
number,, ,
'
,
and static pressure
(this is equal to the undisturbed freestream static)
by the usual normal shock-isentropic compression
relations with error of second order in the perturbations.
Hence, the average velocity profile can be obtained
under the constant total temperature assumption.
The
errors incurred above were found to induce errors in
the velocity profiles of the order (1/3f%)
conditions of the experiment.
for the
In the averaging process,
y is interpreted as the vertical distance measured from
the surface.
The resulting profiles,
eY
,
are given
in Fig. 5.
A slightly different technique follows from the
"assumption" that the static pressure through the
boundary layer is constant (recall that this is not
the case) and equal to the surface value at that
particular streamwise location.
This enables one to
calculate Mach numbers and velocity profiles directly
from the total pressure measurements.
These profiles
87
are accurate to an undetermined distance (of order
A
<
'f.vp-~.'r)
from the wall.
The Mach number
profiles are shown in Figs. 6a-6b along with the average
profile calculated by the previous method.
It is to
the average of the other profiles even for
1>>4f
,
be noted that the latter corresponds very closely to
since the actual variations in pressure are periodic
in x and the errors average out.
check of the
Thus, a semi-independent
average profile is obtained.
The latter
technique is not very practical in the outer portion of
the boundary layer due to the fact that the computed Mach
number is very sensitive to small errors in total pressure
if M = 0 (1).
The first technique avoids this difficulty
since the pressure profiles are averaged before calculating
Schlieren photographs of the boundary layer and
external flow were taken.
These are not included in the
text since they were unclear in the area of most interest,
i.e., near the surface.
The most valuable contributions
of these photographs were to assure that no undesirable
disturbances were present upstream and to define the
character of the
model.
shock pattern which formed above the
An oblique shock wave formed in the outer
boundary layer at each streamwise point of maximum
compression.
These shocks were shown to be weak by
88
the fact that the shock angles were all identical and
equal to the Mach angle for M
=
1.4
The measured streamwise pressure distributions
are shown in Figs. 7a-7c.
These were recorded at the
beginning of a run (after the stagnation pressure and
temperature had stabilized) and at the end with the
elapsed time of the order 30-60 minutes.
The pressures
did not vary appreciably with time except where noted.
Small variations in stagnation temperature did not
appear to have any appreciable influence on the surface
pressure although no specific data was taken.
The pressures
measured in the lateral surveys did not differ appreciably
from that on the centerline.
4-4
Discussion and Conclusions
The "unperturbed" velocity profiles of Fig. 4 show
the anticipated effect of increasing the Reynolds number,
i.e., the profile becomes more full and the momentum
thickness decreases.
Figure 5 indicates that the surface
wave influences the average profile in the same fashion
as a rough wall, the inner portion of the profile being
less full and the outer part almost unaffected.
The
intercepts of the individual Mach number profiles in
Figs. 6a-6b are directly correlated with the compressions
89
and expansions indicated by the pressure distributions
of 7a-7b.
The Mach number profiles also indicate the
thickening and thinning of the boundary layer in adverse
and favorable pressure gradients, respectively, and the
results of 6b indicate that separation would probably
set in for
of the order .03-.04.
7
The pressure distributions given in Figs. 7a-c
are so nearly sinusoidal over the
wavelengths
first
one or' two
that they can be usefully characterized
by a complex amplitude factor, i.e., the ratio of pressure
amplitude to surface amplitude and a phase shift.
The
variation of boundary layer thickness over the wave
causes a slight distortion in this sinusoidal character.
That is, the effective forward phase shift varies + 50
in the regions of adverse and favorable pressure gradient,
respectively.
The upward drift in the mean static pressure
may be due to the losses in the weak shocks, although it
is not evident over the first one or two wavelengths
and could result from an inherent pressure gradient in
the test section.
Tne continuous aeuiease of pressure
amplitude and increase of phase shift with x is likely
caused by the changes in the boundary layer profile
which, in turn, result from the adverse gradient in
the mean static pressure.
~1
90
The results regarding surface pressure are
summarized in Figs. 8a-8b which exhibit the variation
of the complex pressure amplitude witn Reynolds numoer
ana suriace ampiitude.
The experimental results correspond
to averages over the first and second waves.
It is
seen that the deviations of these averages from the
potential flow calculations are an order of magnitude
larger than the distortion over a given wave.
The
pressure amplitude ratio is seen to rapidly approach
that given by potential flow with increasing Reynolds
number, but a strong phase shift remains.
Increasing
the wave amplitude at the low Reynolds number causes
even more serious deviations from potential flow and a
non-linearity is indicated, i.e., the phase shift varies
witn wave amplitude and the pressure amplitude varies in
a nun-linear fashion.
The theoretical predictions of the pressures are
also given by Figs. 8a-8b.
These results correspond
to the expressions (2.20), k2.21) with the profile data
taken from Figs. 4 and 5.
not shown.
The results using (2.22) are
The latter gives the proper trend from
potential flow, but the indicated boundary layer effect
is an order of magnitude too small.
the linear theory - i.e.,
It is seen that
(2.21) with the unperturbed
91
profiles of Fig.
4
-
gives the best results for the
variation of Reynolds number in the case of the small
amplitude model.
The non-linear theory - i.e.,
(2.20)
with the perturbed profiles of Fig. 5 - gives much
better results for the large amplitude model at the
lower Reynolds number.
One must exercise particular care not tovassociate
large boundary layer effects with uncommonly low Reynolds
numbers.
Comparison of the data given in Fig. 1 with
that in Fig. 8a indicates that the boundary layer effects
at high Reynolds number would be expected to be much
larger in the wind tunnels used by Lock-Fung-Morkovin
than at the Aerophysics Research Laboratory.
from
the
This results
differences between the Lock-Fung-Morkovin
velocity profiles given in Fig. 2 and those given in
Fig. 4.
As was indicated in section 4-3, it was precisely
these differences that motivated
the use of the lower
Reynolds numbers for this experiment.
Incidentally, it
is worthwhile to note that the direct effect of increasing
the Reynolds number in (2.20) (without accounting for the
change in the velocity profile) is to increase the
deviation from the results
model, see Fig. 1.
for
the potential flow
As is expected from order of
magnitude considerations, both the theory and experiment
show that the overall effect of increasing the Reynolds
number is to decrease these deviations.
92
CHAPTER 5
STABILITY OF FINITE CHORD PANELS
EXPOSED TO LOW SUPERSONIC FLOWS
WITH A TURBULENT BOUNDARY LAYER
5-1
Introduction
The "self-excited" vibration of an elastic panel
exposed to an airstream -
i.e.,
"panel flutter" - is
examined with particular emphasis on the effects of
a turbulent boundary layer.
The Mach number range is
taken to be low supersonic since the phenomenon can be
highly dependent on boundary layer effects in this
regime.
The structural model is a flat, two-dimensional
panel with "pinned" end-conditions and is characterized
by
flexural stiffness and structural dissipation.
The inadequacy of the potential fluid model under
these conditions has been demonstrated by Lock and
Fung (21).
In their analytical work, they used a two-
mode Galerkin technique and the "exact" linearized
potential flow aerodynamics.
Their primary results
are summarized in the "flutter boundaries" of Fig. 9
93
which show that the theoretical and measured results
converge at
/APYZ', but
disagree sharply
lower supersonic Mach numbers.
for the
Nelson and Cunningham
(28) have found that a similar four-mode analysis gives
essentially the same results for clamped-end panels
and that the additional degrees of freedom were of no
qualitative importance.
The latter results were compared
with experimental points taken by Sylvester and Baker
(29).
Although the Lock-Fung measurements were designed
to remove undesirable features of various earlier
measurements, they can still be adversely criticized on
several counts.
The wind tunnel in which the experiments
were made was only 4" wide and from their measured mean
velocity profiles over the panel - see Fig. 2 - one
anticipates that the side-wall boundary layers covered
10-20% of the span.
The authors realized this difficulty
and terminated the panel 0.2" short of the side walls.
Unfortunately, this procedure introduces the possibility
of more serious three-dimensional disturbances for the
low Mach numbers and small span-to-chord ratios - of
order (1) or less - of their experiment.
However, the
general agreement with the Nelson-Cunningham and SylvesterBaker stability boundaries indicates that the results are
qualitatively correct.
Attempts were also made to induce
static deformations in the panel by slight accelerations
94
in the free stream, but this may well lose the
The corresponding
character of a controlled experiment.
variations in the flutter boundary are indicated by the
shaded portion of Fig. 9 and these can perhaps best be
interpreted as uncertainties in the measured results.
Lock and Fung felt that the disagreement between the
analytical and experimental results was
due to failures
in the aerodynamic model used since this difference
arose as a function of Mach number.
They suggested
that the difficulty might be due to the boundary layer
because other obvious sources of error had been eliminated.
They also suggested that the effect of the slight flow
acceleration might be associated with the measured change
in the boundary layer profile.
To the author's knowledge, the only previous
theoretical attempt to account for the effect of a
turbulent boundary layer in the present context is the
qualitative analysis given by Miles (9).
In this,
Miles used an inviscid shear profile which roughly
corresponds to a turbulent mean velocity profile and
examined traveling waves in a'flexible surface of
infinite extent.
As is mentioned in Appendix A, the
applicability of this idealized model to realizable
systems is in doubt at supersonic Mach numbers.
It is
relevant to note that the Lock-Fung measurements were
definitely not associated with traveling wave motion.
95
The present study is not intended as a broad
investigation into the particulars of panel flutter,
but rather as a preliminary effort to ascertain the
physical role of the boundary layer in the instance
of the Lock-Fung experiment.
The measured results are
predicted reasonably wellwithe inclusion of boundary
layer effects, and the failure of the potential fluid
model is clarified.
While the results of Chapter 4
might lead one to expect that the boundary layer effects
would be important throughout the Mach number range
1.,
M < 1.7, the primary failure of the potential flow
model is found to derive from a special feature
dynamical system.
of the
The spurious portion of the predicted
flutter boundary in Fig. 9 is directly associated with
the single-degree-of-freedom instability noted by Miles
and Rodden (30),,
This is demonstrated by the fact that
the instability is predicted by a single assumed mode
analysis with potential flow aerodynamics.
This
instability is suppressed entirely by the inclusion
of boundary layer effects and the detailed role of these
effects is clearly exhibited.
Thus, two assumed modes
are required to predict a meaningful stability curve
with boundary layer effects included.
Again the neutral
solutions are characterized by a single-degree-of-freedom
96
oscillation (i.e., the ratio of the generalized
coordinate amplitudes is essentially real), but it
appears that the new instability does not derive from
the previous instability.
Although the boundary layer
effects are apparently significant for the new neutral
solutions, it is possible that a potential flow analysis
would give reasonable results if the spurious instability
is ignored.
This point is not investigated in detail
since it is not felt to be basic to the present study.
Of course, such a conclusion would be of real practical
importance to the practicing aeroelastician.
It may be relevant to note that the results of
Chapters 2-4 give, perhaps for the first time, tne
underlying information requisite to the rational study
of a perturbed surface with a turbulent boundary layer.
In view of the results of Chapter 4, the Lock-Fung
measurements strongly suggest that the assumption
regarding the compositionvof the field variables, i.e.,
(3.2), is appropriate and that the "linear" interpretation
of the mean profile as the one measured over the flexible
surface in the absence of the harmonic vibration is
reasonable.
While the question of the interpretation
of the coefficient of viscosity has not been resolved,
the particular wave speeds of interest are small and the
laminar viscosity is used.
The linear analysis - in which
97
the Galerkin technique is used - admits an especially
convenient approximation of the aerodynamic pressure
for the particular structural model considered; namely,
the superposition of pressures corresponding to a finite
number of traveling waves (recall the discussion at the
end of section 2-6).
This is the analogy of the direct
use of (2.20) in predicting the measurements given in
Chapter 4.
While this approximation is not of universal
validity, it is shown to be adequate for the present
purposes and its peculiarities are clearly isolated.
The approximation not only eases the computational
problem posed by the Fourier superposition techniques
of 2-6, 3-4, but facilitates understanding of the underlying mechanisms as well.
In
regard to more comprehensive
studies of panel flutter, it may be relevant to point out
that the analysis at hand is similar to the investigation
based on (A-2) of Appendix A (in contrast to the study
based on (A-2")).
This similarity appears to have no
importance for the present analysis, but it is conceivable
that this approach could overlook instability phenomena
associated with boundary layer instability over a rigid
wall (see Appendix A).
In this instance, the ratio of "total" boundary
layer thickness to chord length is of the order 0.3"/6.0"=0.05.
98
(Again recall that the appropriate measure of the
boundary layer effect for the turbulent layer is
a/..
r
a/
,
even though
that appears in the analysis).
0/e
is the quantity
Thus, one does not
expect this to be an exceptional case in which the
boundary layer measure is unduly large, although it
is evident that it can be of some significance.
However,
these order of magnitude considerations may be somewhat
misleading in applications regarding stability boundaries
and of course (2.20) does not correspond to the mode of
motion to be studied.
As is evidenced by the study of
structural dissipation in Appendix A, small changes in
certain parameters can have large effects on the stability
boundaries.
A more detailed examination of the eigenvalues
is expected to show that these areas correspond to small
growth rates, but the implication of the latter is another
question.
The boundary layer effects encountered here
appear to admit a similar interpretation, i.e., the
spurious single-degree-of-freedom instability mentioned
above appears to be only mildly unstable.
The latter is
demonstrated by the radical effects of structural
dissipation as shown by Lock and Fung (21).
99
5-2
Analysis
The analysis is based on the Galerkin technique
and it is assumed that the panel deflection can be
expressed in a truncated modal expansion, i.e., with
the same coordinate system as in previous chapters
are taken as the natural modes of
where the
the panel in the absence of the air.
With the panel
characterized by mass and flexural stiffness (see the
nomenclature for the kinetic and potential energy
expressions) and the forcing term arising from the
induced normal pressure, the application of Lagrange's
equation yields the equations of motion in the generalized
coordinates,
0 eej
.
.
The subsequent assumption of
the harmonic time dependence with
a set of homogeneous algebraic equations for the
.
and the introduction of the structural damping yields
For the panel under consideration with "pinned" endconditions
are for
/'=2
,
y
,
... ),
these equations
100
~~z
L
ITS
14q2
if 01 Jra
0IV0
aax"j
2. -kj,
z
I
rz
0
(5.2)
where the non-dimensional quantities are defined in the
nomenclature.
The
7r7,
are
('!'
the non-dimensional
complex pressure amplitudes due to oscillation in the
The equation for AVI
7it4 mode.
(i.e., the fundamental
mode) is simply that the first term of (5.2) vanish.
In general, Eq.
(2.20) gives the Fourier transform
A
and
(a'
AS
"
of the pressure where
is to be interpreted as the Fourier
transform of the mode shape,
OPM erP
.
The inversion
transform can be found in closed form for the potential
flow (i.e.,
'-+ 0
) and gives a convolution integral,
see for example Bisplinghoff, Ashley and Halfman (31),
pp. 364.
means.
Otherwise, it must be obtained by numerical
101
However, for the particular mode shapes of the
present example one can write
0
z4<X
i.e., in the range of interest the total deflection is
+
composed of the traveling waves with wave speeds
This fact suggests that the pressure due to deformation
in a given mode be approximated by the superposition of
waves.
The measurements of Chapter 4 - see Fig. 7
-
the pressures corresponding to the infinite traveling
indicate that this is good approximation for the
turbulent boundary layer over a rigid wall (i.e.,C = 0).
Unfortunately, the measurements do not include the first
half wavelength which is of paramount interest here.
Comparative studies made wi tishe potential flow model
give further insight into this approximation.
That is,
)
the power series expansions (in the reduced frequency,*
of the "exact" linearized expression for the pressure and
of (5.4) with (5.5) show that the approximation is in
102
error by terms of order
-
.
Further, in section 5-3,
the corresponding errors in the stability boundaries
for the single-degree-of-ireedom analysis are shown
to become of real significance for
i.e., as the downstream
C.= 1 * -
/-
wave becomes transonic.
The
reduced frequencies of interest here are rather less
than this number so the approximation is quite acceptable.
Therefore, the pressure is taken to be
where the
are dimensionless complex pressure
pl
amplitudes associated with the traveling waves of unit
dimensional amplitude, the latter having wave numbers
~and wave speeds
Assuming pressurized air behind the panel (with negligible
dynamics), the
)%_
are found from the "pseudo-laminar"
expression (2.20) with the Heisenberg form (2.21) and
are
103
-2 's'
(/
-
C-,) l
mma,'
OF-
04V
~c
--
/z
tZ.VC)l
2
9--0 (, -)
+J[-
O) w
$1
-
,
r;
~
(n
~-~:- /--.2) (/
e--.>4
-r..,
751 1*
7;.,
1
e
ir)]
0 6K,
aj
7
U-IGOda
4f
+;-
---e
440 -
KI -C44
-
'L
"
a
I-A&
0 1- C.N ) 4
7-
//.
if ft16
-;
jr
C-?C',
=-
7,7r
7 C Iz4
14
*
The computations for the
JJ~7 'O
~
RW,::
R ) 11/3
C
7.;&
are based on the
assumptions of the "Adiabatic Wall"
(the Lock-Fung
measurements were made with a cavity of still
air
behind the panel) and constant total temperature
across the boundary layer.
The profile derivatives
~~ar)
104
are found by the "Rapid Computing Technique" of Lees
(6) and were based on the laminar profiles given by
Lees with a scale change in the normal coordinate to
adjust
to the value taken from the measured
0
profiles of Fig. 2.
This point is not essential
since both the laminar and turbulent profiles are
essentially linear for the pertinent (small) values
of
Z~c
E CV
and therefore it is only the value O,
that is of importance.
The possibility of non-trivial solutions to (5.2)
requires that the (complex) characteristic determinant
vanish.
Utilizing (5.4), the resulting characteristic
equations for the single-mode analysis become
z
(5.6)
Similarly, for the two mode analysis,
105
c7
)(e)+
(A
(
C)- Jri -4j0
)(j#,w.
A (F*>()(
*.-
For the one-mode analysis,
and
*
=
= 0
+
(c
(.$)(.
')6
(5.6) gives the eigenvalues
, directly for particular values of
I/
I
real and
.
The two-mode analysis proceeds in
a similar fashion except that the simultaneous solutions
of (5.6') are determined graphically for given values
of
*'
and
9
.
Thus, stability boundaries in
space can be found with the parameter,
each boundary.
, varying along
The stable and unstable regions can easily
be determined by the technique associated with Eq.
(A.6).
For a given atmosphere, Mach number and panel material
and
1
take the forms
z2
106
where
2
is the panel thickness ratio
a second curve with the parameter
i
&,
in
space.
I
.
Thus,
can be constructed
The intersection of this curve
with the stability boundary gives the value of
'e
required to prevent instability at a given Mach number.
5-3
Results
The failure of the potential flow model and the
relevance of the approximation (5.4) can best be understood
by reference to Figs. 10 and 11.
Figure 10 shows that
the spurious portion of the two-mode boundary with "exact
linearized" potential flow aerodynamics
(derived by Lock
and Fung) is predicted by a single-degree-of-freedom
analysis with the same aerodynamics (discrete points)
or with the approximation (5.4) specialized to potential
flow, i.e.,
4-7140
in (5.5).
are erroneous for large values of
These latter boundaries
2
due to the fact
that the reduced frequency becomes large, which in turn
implies that the downstream wave becomes transonic relative
to the free stream, i.e., the approximation (5.4) fails.
Figure 11 shows the character of this failure in the A
domain.
The failure for small values of
'r
'Z
is due to
the fact that only one degree of freedom is allowed.
instability is entirely suppressed by the inclusion of
This
107
boundary layer effects.
The character of the instability
and the role of the boundary layer are illuminated by
using (5.4) to compute the energy flux to the panel
This is
over a cycle of harmonic oscillation.
Ir
,w'
A
fir
-
f
AZ~
zr
__________
Wr f,
(5.7)
It is evident that the out-of-phase components of the
pressure act like "damping" terms.
In the potential
flow limit, the pressure associated witn the downstream
wave corresponds to energy addition and that with the
wave to energy removal.
upstream
One finds that the
energy addition term dominates in a Mach number range
bounded by
that
C
Af'
.
Detailed calculations also indicate
is considerably smaller than each individual
contribution.
Owing to this, the boundary layer effects
are rather more important than for a simple traveling wave.
Indeed, for the boundary layer under consideration
for all
M > 1.
C<o
108
These effects are largely associated with
Ae 7
and
.
(rerlected by
Re
ii:t
The effects of the shear profile
/'i
)
are somewhat larger than the
direct viscous effects (reflected by the deviation
of
from unity), although the latter are certainly
'
appreciable.
The lesser value of
6
for the super-
imposed waves also leads one to expect that the indicated
instability for the finite panel corresponds to a smaller
growth rate than that for a wave traveling downstream
under otherwise similar conditions.
The increased flexibility of an analysis with two
assumed modes is then required for the new predicted
stability boundary of Fig. 9.
are again associated with an
of-freedom oscillation.
The neutral solutions
essentially single-degree-
That is, the ratio
is real and fairly large, see Fig. 12.
./'
However, tne
previous neutral curve was traced from the one-mode to
the two-mode boundary layer analysis (it lies in the
domain
-
<0
and so is meaningless) and hence the
new meaningful instability does not appear to derive
from the old.
The single-degree-of-freedom character
of the neutral solution agrees with the Lock-Fung observations,
but the reduced frequencies are rather too low.
change in the boundary layer profile due to "panel
The
109
curvature" would give the same trend as the LockFung observations, but this refinement was not felt
to be warranted here.
The corresponding two-mode analysis with the
approximate potential flow aerodynamics is not very
informative.
It predicts the same spurious single-
degree-of-freedom instability and two other less
critical modes.
One of these appears to arise directly
from the approximate aerodynamics (it is suppressed
both by boundary layer effects and apparently by the
texact" linearized potential flow) and has pathological
stability characteristics.
The second appears to
correspond to a second mode predicted by Lock and Fung,
but it is associated with rather large values of
for moderate values of
'/ZZ
.
A-
V*
and thus cannot be
adequately studied with the approximation (5.4).
The relation of the stability boundary predicted
in Fig. 9 (whichincidentally, is the only meaningful
instability with the effects of the boundary layer
included) to the second aeroelastic mode of the twomode analysis with "exact" linearized potential flow is
not examined.
However, the character of the predicted
and measured instabilities for
7
is well-known
and is associated with coupling between the two assumed
modes.
The smooth transition of the measured thickness
110
to prevent instability from
to
suggests that the potential flow analysis might
predict the observations rather well if the spurious
boundary is ignored.
ill
CHAPTER 6
CONCLUSION
In closing, the principal features of the preceding
The
chapters are summarized and criticized.
interrelated natures of various facets of the study
are indicated and topics needing further clarification are emphasized.
In Chapter 1, the limitations and qualitative
implications of the theory for a traveling wave disturbance in a laminar viscous flow and the adjoining
boundary is examined.
These implications are illus-
trated by the study of the stability of the laminar
flow over a flexible surface, see Appendix A.
The
theory for the perturbed flow with a turbulent boundary layer, given in Chapter j, is intimately related
to that for the laminar viscous flow.
Thus, the study
of the laminar flow has direct bearing on that for
the turbulent flow and vice versa.
As was pointed out in section 2-6, the boundary
layer significantly influences the pressure on a
surface whose deformation is described by a planar
112
C-/+
(and presumably
d/ C
),
c</
limited to
A R.,
is
ol= 27
traveling wave if a)
%w
,
large, b) C-+-/~
although the theory is
and c) the values of the parameters
M approach those values associated with
hydrodynamic instability over a rigid wall, i.e., the
surface disturbance excites natural oscillations in the
boundary layer.
The theoretical limitations on the expression for
the surface pressure (2.20), are most severe under these
same conditions.
and
imposed on
The restricitions
c/5 dA', C
by the use of the asymptotic expansions can be
Me
relieved by resorting to numerical techniques.
However,
the theory of hydrodynamic instability has been confirmed
for
=W
.
Also, the correlation between the theo-
retical and observed
pressure distributions on a rigid
wavy wall with 0( = 0(1.5)
(Fig. 8a) is good.
can conclude that the limit.tion on oe
restrictive.
Thus, one
is not overly
One should recall that this conclusion is
valid for the studies with
expansion, see Eqs.
l,
O
only if the Heisenberg
(2.8) and (2.21), is used.
The com-
parisons of the postulated Fourier transform applications
of (2.20) with and without viscous effects
(section 2-6)
leads one to expect that the "transonic difficulty"
(c-v/- -J-
) is not serious for most surface disturbances.
One may also note that the results for the viscous
- -9
fluid are well behaved in the worst extreme, i.e. a
traveling wave with C=-,.
The validity of the theory
for surface disturbances that excite the natural
oscillations in the boundary layer remains to be resolved.
Possible analogies with simpl,
mechanical
systems indicate that this point should be investigated further.
For example,
the effect of a mass-spring
oscillator attached to a massive body may be considerably overestimated
by a linear analypis
body vibrates with a frequency
to the oscillator
that is
if the large
nearly equal
frequency.
The results of Chapters 4 and 5 show that the
turbulent boundary layer can,
indeed, have significant
effects on the surface pressure and on phenomena which
depend on the surface pressure.
The observed effects
on the pressure distribution over a rigid wavy wall
(Chapter 4 and Fig.b) are directly associated with
large values of the measure,
equation
(2.20).
9
,
in
However, the panel-flutter analysis
of Chapter 5 (Fig. 9)
shows that the boundary layer
effects can be more signigicant than the value of
(sr
o0-04) might inaicate.
This results from the
fact that the out-of-phase componenti
distrioution for a standing wave is
of the pressure
sensitive to Dound-
ary layer effects.
The results of Chapters 4 and 5 also snow tnat
the analysis based on
Lhe
'linear'
conjecture made in
114
Chapter 3; namely, that the mean quantities describing
the turbulent boundary are unaffected by the regular
perturbation; is valid for small amplitude disturbances.
This conclusion can be questioned for values or
less than those considered
( A*I'/r= 0 ()
W
) and for
values of the phase velocity greater than those considered
(c %oC44)}.
The use of the "eddy" viscosity to account for
exceptions to the above conjecture is suggested, but
is still unresolved (except that it is shown to be
unimportant for small values of
C
, see pp. 70, 71).
An extension of the experimental investigation of
Chapter 4 is suggested for direct examination of this
point.
If one passes an endless moving belt over tne
rigid wavy wall, the condition of a traveling wave
with
c O O can be generated (as seen by an observer
in a frame translating with the belt).
An alternate
indirect procedure is the study of natural or forced
In order to generate sufficiently large values of
C
,
vibrations in a panel exposed to the boundary layer flow.
one would have to design the system so that the frequency
of oscillation is higher and/or the panel length is less
than those values considered in Chapter 5.
The correlation between the observed pressure
distribution for the large amplitude wavy wall model
115
(Fig. 8b) and that predicted by the analysis based on
the "non-linear" conjecture of Chapter 3; namely, that
it
is consistent to consider a first-order perturbation
on the mean quantities which are modified by the
perturbation; demonstrates that this concept does
describe the non-linear mechanism.
The theory is semi-
empirical in the context of Chapter 4, i.e., the mean
quantities are determined by measurement.
However, if
a first order analysis could be put forward to describe
the inverse problem (the determination of the perturbation
in the mean profile,
4ur'
=
for a specified
periodic disturbance), then the two linear theories could
be used in counterpoint to predict the basically nonlinear phenomena.
A final comment regarding the relevance of the
examination of panel flutter (Chapter 5) to aerospace
craft may be of interest.
The intent of Chapter 5 is to
demonstrate the significance of boundary layer effects
for a particular example and the studies are limited in
scope insofar as possible parameter variations are
concerned.
These studies roughly correspond to a
constant dynamic pressure trajectory with varying Mach
number and altitude (in the range 30,000-50,000 feet)9
The boundary layer effects are expected to increase
with altitude at constant Mach number.
Figure 9 shows
that the maximum required panel thickness corresponds to
,I
1.2, a condition for which the boundary layer effects
-I
116
are extreme.
The importance of these effects in
three-dimensional flows remains to be examined.
117
APPENDIX A
STABILITY OF A FLEXIBLE SURFACE
EXPOSED TO POTENTIAL AND LAMINAR VISCOUS FLOWS
A-1
Introduction and Qualitative Results
The roles of fluid viscosity and structural
dissipation in the stability of flows over a flexible
surface are examined.
The study is confined to the
unbounded, initially flat surface and the associated
traveling wave disturbances - which are taken to be
two-dimensional with their phase velocity in the stream
direction - because of their analytical convenience.
As may be anticipated, this model affords a convenient
vehicle for the illustration of the viscous effects
discussed under 2-6.
With viscous effects included,
the model exhibits instabilities that derive from two
limiting cases: 1) potential flow over a flexible
surface and 2) the boundary layer instability over a
rigid wall.
Emphasis is given to considerations of
the phenomenon from these alternate points of view.
A cursory examination of the "supersonic disturbance"
118
(i.e.,
C <
in the classical hydrodynamic
stability problem is also made.
The stability of a non-dissipative surface with
a uniform potential flow has been previously investigated
by Miles (24) and serves as a point of reference for
the present investigation.
Similar investigations,
which take structural dissipation into account, have
been made for the infinite cylinder, see Miles (25),
Leonard and Hedgepath (26).
The latter authors noted
that the neutral curves for the model including structural
damping did not converge to those without damping as
the damping coefficient was taken uniformly to zero.
The results for the non-dissipative model were rejected
as meaningless for real systems.
This phenomenon is
also present for the flat plate and a study of amplified,
as well as neutral, disturbances removes the conceptual
difficulty.
The results of the non-dissipative model
are indeed meaningless insofar as absolute stability
is concerned, but they indicate the onset of severe
instability for lightly damped surface.
AI/
The case for
with the flat, dissipative surface exhibits an
especially disturbing feature in that the mild instability
occurs for almost all values of the structural parameters,
excepting a small region corresponding to large dissipation.
This vast region of instability is sharply curtailed by
119
the presence of the boundary layer, even if the
boundary layer is infinitesimally thin as compared
to the wavelength of the disturbance.
The new portion
of the neutral surface is intimately associated with
the effect of the boundary layer on "transonic"
disturbances.
The previous investigators found that
structural dissipation can have a destabilizing effect.
This is reaffirmed and a qualitative explanation based
on recent work by Landahl is given.
Miles (9) has extended his investigations to include
an inviscid shear profile.
Adverse criticism of the
inviscid fluid model is given in Chapter 2.
Miles'
results were highly qualitative and were based on the
first iteration of an approximate solution to the
characteristic equation.
The potential flow studies
are here extended to include the full laminar viscous
model of Chapter 2 and more quantitative results are
found through the use of graphical techniques, due to
Landahl (3).
M = 0, 1.3.
Detailed calculations are carried out for
The latter is within a Mach number range
of considerable interest in connection with panel flutter.
The effects of the boundary layer and viscosity are
shown to be highly important for conditions approaching
those of hydrodynamic instability over a rigid wall,
120
although the amplitude limitations mentioned in Chapter 2
should be noted.
It is found that no simple rule can
be given concerning the stabilizing or destabilizing
effects of the boundary layer or structural dissipation,
although sufficiently thick boundary layers tend to
suppress those instabilities deriving from the elastic
surface exposed to the inviscid potential flow.
Certain
pathological neutral surfaces are found when the viscous
effects are included.
That is, these surfaces do not
demarcate regions, in parameter-space, for which an
eigenvalue changes its stability characteristic.
This
fact is of course disturbing in light of the usual
implications of neutral surfaces.
Even in the latter
case, the possibility of other eigensolutions which
have the same stability characteristic on either side
of a particular neutral surface exists.
Indeed, amplified
disturbances that are not indicated by the above
examination are found for certain boundary layer
configurations.
These disturbances are found to be
intimately related to the classical hydrodynamic
instability over a rigid wall.
The investigations mentioned above proceed along
lines conventionally used in aeroelastic investigations.
That is, the equation of motion for the panel is taken
directly as the characteristic equation and neutral
121
surfaces are defined in the structural parameterspace for discrete values of the aerodynamic parameters.
One may note that defining such
surfaces
as functions
of all parameters would require a hyperspace of six
dimensions.
conclusion
Perhaps the most important practical
of this analysis is that the difficulty
noted above points up the danger of dealing with a
parameter-space of reduced dimensions.
Some practical
difficulties occur near the eigenvalues of the hydrodynamic stability problem since the term arising from
the aerodynamic pressure is nearly singular.
An
alternate technique, which has been utilized by
Benjamin (2) and Landahl (3), proceeds along lines
similar to those used in the hydrodynamic stability
investigations.
In essence, this consists of considering
the inverse of the equation of motion of the panel and
determining neutral curves in the dj,R
discrete values of the other parameters.
space for
Of course,
the solutions must be identical with those found by
the method above, the differences being only in the
practical difficulties just mentioned and in that different
areas of the parameter space are investigated.
A brief study is made with the latter technique,
the ostensible surpose being to investigate the effect
of variable wall flexibility on boundary layer stability.
PO
122
The wall flexibility (or more properly wall rigidity)
CO
is characterized by the wave flexure speed,
defined in the nomenclature.
Co
, as
is a function of
the wavelength for the dispersive surface and the
Co
variation of
is of fundamental importance since
any one such surface admits disturbances with different
flexure speeds.
This analysis is applicable to both
dispersive and nondispersive surfaces, but only the
latter is considered in detail.
An obvious example
of The non-dispersive surface is the simple membrane
under tension.
The study, which corresponds to M = 1.3,
indicates that the wall flexibility is unimportant for
Co >/
(at least for the rather high values of panel
mass and damping coefficients considered, i.e.,
-5 = 2.5).
"
= 30,
However, the flexible surface begins to
react unfavorably with the flow for
first occurs at
Co <I
.
This
C,1.4 and further decrease in Co
causes amplified disturbances that are relatively
unimportant for the rigid wall to dominate the complete
A'R
domain.
These disturbances travel very slowly
and correspond to a visco-elastic interaction.
They
have "supersonic" phase velocities for 7,? 1, but are
approximately independent of the "inviscid solutions"
and hence M.. Landahl (3) has carried out more extensive
investigations regarding the variations of -4
and 7w
123
for
CO>,4, M=o
.
The comparison of the results
gained by this technique with those mentioned above
provides the key to the discovery of those amplified
disturbances for which no neutral surfaces were
previously found.
Detailed examination of the characteristic equation
for the rigid wall or the flexible surface with positive
dissipation reveals that no neutral disturbances exist
for
C<O
at any M.
This rules out the possibility of
neutral supersonic disturbances in subsonic flow (except
possibly for C>/
which cannot be investigated with the
It also rules out the possibility
asymptotic solutions).
of standing waves in the unbounded flexible surface.
The remaining possibility for supersonic disturbances
is
0
<C
<1---L
.
The calculations for Mr= 1.3
show that no such neutral disturbances exist for the
rigid wall.
Qualitative considerations indicate that
this conclusion should hold for all M and that it holds
for moderately unstable disturbances as well.
The
definition of moderate instability was taken as
/
0(Z
.
The possible exception to
these conclusions is for low
is untrustworthy in this case.
Lin (5), p.
69,
a'sR
,
but the theory
Thus, it appears that
lost no important information in
disregarding the supersonic disturbance, although the
124
work of Landahl (3) introduces the possibility that
Lin's energy arguments could be in error.
In conclusion,
some qualitative remarks are
made regarding the relevance of the study to surfaces
of bounded extent.
Quantitative criterion with which
to make such a judgment for a particular configuration
is difficult to obtain because the effect of the overall
geometry enters the solutions in an inexplicit fashion.
It is useful to note that the results of the studies
on hydrodynamic stability have been experimentally
confirmed near the leading edge of a plate.
This
implies that the hydrodynamic phenomena is local in
nature, except for the variation in the boundary layer
profile with distance from the leading edge.
The
requirements are more stringent for the flexible wall
because the structural end-conditions also enter.
The
effects of the end-conditions are confined to narrow
regions near the panel edges provided that the spatial
derivative terms in the panel equation of motion are
small with respect to the other terms.
Thus, one would
expect the unbounded panel result to be applicable in
the intermediate region.
Another way to characterize
the existence of this region is by requiring the energy
flux along the panel to be small with respect to the
125
local energy density.
Dugundji (27) has used a
potential flow analysis to show that the traveling
wave results correspond closely to a two-mode result
for a "long" panel with pinned ends if Mw< 1.
(The
panel was long in the sense that the modes of interest
contained 8-9 wavelengths.)
This was not the case for
the same configuration with M.> 1.
In the latter
condition, the two-mode analysis indicated standing
waves and, as was indicated above, such solutions do
not exist for the unbounded surface.
Thus, it is
doubtful that the analysis for the unbounded panel
is directly applicable to the study of bounded panels
in the case My,1> 1, unless further experimental or
analytical studies show that the bounded panels can
admit disturbances that resemble traveling waves.
4
r
126
The Characteristic Equations and Techniques for
Determining Eigenvalues
A-2
The Characteristic Equation
The characteristic equation considered here
corresponds to an initially flat, unbounded surface
that is characterized by inertia, viscous damping, and
generalized stiffness forces.
For purposes of definiteness,
the characteristic equation will be derived from the
equation of motion for a thin elastic panel with the
generalized stiffness forces corresponding to flexural
stiffness, membrane tension, and a spring foundation.
However, it is quite possible that more general surfaces
will lead to a characteristic equation of similar form.
The equation of motion for small deformations
perpendicular to the plane of the undeformed state is
in
dimensional form
/
-3Fr
* a'
(A.1)
'
is the total compressive surface stress arising
from the perturbed fluid on one side of the surface.
order of magnitude estimates of Chapter 2 show that
can be approximated by
order
(e
).
-'
within
The inclusion of
Ar
errors of
(i.e., the
undisturbed static pressure) can be regarded as a
The
'2'
127
pressurized fluid (with its dynamics ignored) on the
underside of the surface or as a linearization from
the statically deformed state with that deformation
disregarded.
As in Chapter 2, we consider traveling wave motion
and orient the X
axis to point in the direction of
the phase velocity.
Reverting to the non-dimensional
quantities of the nomenclature and utilizing (2.20)
(with Acc)I O ) yields the characteristic equation in
a form conventionally used by aeroelasticians, namely,
/
(-C)
.~
2
where the surface admittance, A (or the impedance,
.
)
(A.2)
is defined by
4'
(A.21)
C 2Zjc- Z4WC
It is to be noted that the parameters characterizing
the structure are defined so as to be independent of
Thus, one can consider
the potential flow model by simply letting
Q
.
the boundary layer thickness.
128
The Heisenberg result,
(2.21), and (2.23) are used
rka',W
in calculations for
.
The characteristics
of the laminar boundary layer for compressible flow
and a"rapid computing technique" for finding
It is to be noted that the computations
throughout.
(N4r/
are
The "Adiabatic Wall" is considered
taken from Lees (6).
for
P
are rather more lengthy for transonic and
supersonic disturbances than the simple low speed
approximation given by Lin (5), pp. 37 and 43.
As was mentioned previously, the denominator of
the right hand side of (A.2) is precisely the characteristic
equation for the problem of hydrodynamic stability over
a rigid wall.
Thus, to avoid practical difficulties
for combinations of parameters near the eigenvalues of
the latter problem and to investigate the modification
of those neutral curves due to flexibility, it is
convenient to consider the inverse of (A.2), i.e.,
(A.2")
The characteristic equation (A.2), or its alternate
(A.2"), is a complex equation and thus corresponds to
two constraints on the parameters.
the complex number
C
One can think of
as the double eigenvalue which
129
can in principle be determined if all other parameters
However, exact solutions can be found only
in special cases.
generally been restricted to real values of
Thus, one can take
have
The tabulated values of'
Ci
.
are fixed.
and one other parameter as the
eigenvalues corresponding to neutral eigensolutions.
In fact, it is usually more convenient to choose values
for
Cv
as well and determine two other parameters as
eigenvalues.
C
,
However, there are multiple eigenvalues,
for fixed values of the other parameters and one
can never be assured that all those corresponding to
unstable motion have been found.
Some specific techniques
for determining information about the eigenvalues are
explored in the following paragraphs.
The Nyquist Criteria
The Nyquist technique is well-known for its value
in
determining the areas in parameter space fcrwhich
unstable eigenvalues occur and the number of such
eigenvalues.
It is limited by the facts that it does
not give complete information regarding the eigenvalue
and that the behavior of the characteristic equation
must be known for complex
C
.
To illustrate the value
of this technique, let us specialize (A.2) to potential
flow by letting
W-50
,
see (2.21).
130
)
(/-c
(4-c Cc
-
/
;//
(A .3)
(See nomenclature for the definition of
Now E
has branch points at
C
C
analytic in the half plane
but it
,
/
0
.)
/-M('C-'
if
is
the branch
lines are chosen to fall into the region
can then be written in the form
In principle,
CZ:-Vo
(C)
.1:
one fixes the parameters -*
C4,
c&-
) and lets
in the half plane,
C
'
,
ea,
d
, 40-
traverse an infinite semicircle
S7'
,
will undergo a phase change of
is the number of zeros of 16:
then the function
(rC)
where
2/- (A-Al)
/-
in the given domain, and
the number of poles (in the present case M = 0).
A
Al,
Thus, the number of unstable eigenvalues,
can be found.
and
(and thus
AMw
Systematic variation of
,
,
-
If
enables one to determine the areas in the
parameter hyperspace for which unstable motion occurs.
Miles
of
(24) has exploited this technique in the examination
(A.3)
with
-4
=do .
The extension to
considered in the next section.
-t
4 -0
is
131
Other Graphical Techniques
Landahl (3) has usefully employed direct graphical
techniques for finding neutral solutions of (A.2").
Wrx
has been tabulated for
Cr- 0., see Fig. A.l.
If the structural parameters and the Mach number are
fixed, families of curves corresponaing to the right
hand side of (A.2") can be drawn; for example, see
Fig. A.2 which corresponds to the rigid wall ( A
0
The intersection of one of these curves with the
CezJ
curve gives a set of values W, , C, , 0, , which
satisfy (A.2") .
The corresponding value of As
then be found from the definition of Z
.
can
A simple
analytic equivalent of this technique has been given
for tne rigid wall, see Lin (5), p. 37.
Typical results
of these techniques for both rigid and flexible walls
are given in Figs. A.3 and A.4.
The same technique
can, of course, be applied to (A.2).
values for c/ , P , A& and graphs
of
Cs- cy1
as a function
, equating the imaginary parts of (A.2) for
a given value of -6
CA
7-
If one chooses
.Equating
implies one or more possible solutions
the real parts for a particular
yields
z
(
c ).
4, =
(A.4P
-9
132
C10 2
This gives a simple linear relation between
and
4-
the neutral surfaces in the
Thus,
.
domain can be easily found.
C
with
The variation of
is given in A.5 for several boundary layer
configurations (see Fig. A.3) with M,= 1.3
is
It
has large fluctuations in
to be noted that
the region near the eigenvalues for hydrodynamic
stability, and that this will be reflected in the
neutral surfaces.
Landahl has also utilized a convenient technique
for determining mildly unstable eigenvalues.
that one chooses all parameters except
(rc)
for
assures that
Z'
d'c)
C > 0
=
Now Liie aeivation
of Chapter 2
for any given value of
fCrweicCa-J
r
Oca
CGM)
)
and grapns
can be continued analytically to
Thus,
.
one can compute
E
.
ces c-
C
Suppose
J-I
4
1
(cCe) -'CZ*
Cr,
by
oc-
C'>o
(A.5)
This process is demonstrated in the accompanying sketch.
-TAW E
133
To first order,
r
to the tangent at
CR
on a ray perpendicular
Therefore, some particular
will be an eigenvalue -..
Cr
value of
lies
Cz)
C=
This value is, again to first order,
(A.6
c'(
=6
Thus, one can investigate the amplitude rates in the
vicinity of a known neutral curve by perturbing one
parameter from its neutral eigenvalue.
However, the
technique is not limited by the existence of a neutral
curve, but rather only by the accuracy of the first order
form of (A.5).
The equation (A.6) can easily be used to decide which
side of a neutral curve is unstable without calculating
the magnitude of
Cf*0
.I/c=That is, one needs
c
only to know how the trace of
E~
passes the origin
with a particular parameter perturbed from the neutral
eigenvalue.
However, implicit differentiation of the
characteristic equation yields this information more
directly for variations of certain parameters.
example,
Cal
all
For
if one differentiates (A.2) with respect to
under the constraint that (A.2) be satisfied for
Co
,
i.e.,
CCrz),
one finds
134
-~2
C
Evaluating this for real C
(i.e., at a neutral eigenvalue)
gives
dc)
ez
In this way, inspection of Fig. A.5 which is to be used
with (A.4) indicates whether increase or decrease of
Co
above the neutral eigenvalue corresponds to instability.
A-3
Potential Flow over a Surface with Dissipation
The object of this section is, as was indicated in
section A-1, to examine the effects of structural
dissipation in a surface exposed to an irrotational,
It is intended to be complementary to
inviscid flow.
the works of Miles (24) and Leonard, Hedgepath (26).
It is evident that one can obtain exact solutions
to the characteristic equations (A.3) for the special
case M = 0.
That is,
-Ac~
C
-2
(A-7)
135
The relation defining the parameter region for which
unstable solutions can exist
the condition -6w
O
,
is easily obtained under
i.e., one unstable solution
exists for
6~o
(A. 7')
and two neutral solutions exist otherwise.
(A.7) is less convenient for -- C'
,
The use of
but it is evident
from (A.3) that the only possible neutral solutions are
with
C
o
i.e.,
,
C..a
.
Examination of
(A.7)
shows that one unstable solution exists for
(A.7")
and that two stable solutions exist otherwise.
The fact
that the requirement (A.7") is independent of
-6
leads
to the apparent contradiction between (A.7') and (A.7")
in the limit
-
.
resolved through (A.7).
However,
the difficulty is
It is evident that there is
no discontinuity in the eigenvalue for given
as
%-+.
Therefore,
,C
in the area
1Co14
<
the one analysis indicates the existence of two neutral
solutions, while the other indicates one stable and one
136
that
approaches zero as --
"PO
.
unstable solution, each having an amplification rate
The exact solutions of (A.3) cannot be obtained for
MgY> 0, but Miles (24) has used the Nyquist criterion to
are similar in nature
show that the results for -d=O
The relation bounding the region for which
to (A.7').
unstable disturbances
(one actually) can occur is given
in parametric form by
3
2
CO
< c
C/- c)1
~2c.u/vZe(/-c;)"
It
(A.8)
2_-_________)
is convenient to examine the conditions
, <4/
0 and (A.3)
(I-c)e )
and
w40
,
/V.;>/
-4*
In the former,
separately.
the only neutral solutions are with
C
,
4f/~)
0
(with use of the Nyquist technique) shows the equivalent
of (A.7") to be
(A.8')
to#
137
(A.8 ) and (A.8') are summarized
The results
in the following sketch:
,t4a4* / o~
CO
CO
It
J4
i#Alv
A~o
04
-7 's)
0
i -as
(Note: The Nyquist technique shows that one unstable
o <c,, </ ) exists in the indicated
root (with
areas.
The amplification rate of the unstable mode is indicated
for small values of
--
.
The use of (A.7)
or of (A.6)
in conjunction with (A.3) indicates that the area between
the two curves is mildly unstable in these cases, but
Thus,
the conceptual difficulty
fully resolved.
at the lower curve.
for -/--o
,
that severe instability sets in
is
Of course, the practical implication of
of -
.
the region between the two curves depends on the magnitude
138
Considerations of the conservation of energy
immediately imply that, for the potential flow model,
C =O
neutral disturbance must correspond to
and /-
<<
.
if
That is, the total
energy of the panel and fluid in a volume one wavelength
long and infinitely high, is the same at
as it was at
.
;&
this element at
A
&
D
Further, the energy flux into
+7
is minus that at
5-+ e
there is no energy flux at
-
and
However, the
.
structural dissipation is positive definite.
Thus,
the only possible energy balance is with zero dissipation,
i.e.,
C wO .
One may note that this argument does
-
since the solutions do not decay as
.
not hold for the disturbance with supersonic character
One may further inquire into the mechanism of the
destabilizing effect of structural damping (which of
course corresponds to energy removal).
Landahl (3) has
investigated this point rather carefully.
The key concept
is that the total energy of the fluid can be decreasing
as the disturbance grows.
(An obvious example is
.)
Thus,
the increased
total energy of the panel, plus that dissipated, is equal
to the decrease in the total fluid energy.
termed such waves "energy deficient."
Landahl has
However, it is
important to note that this characterization depends on
139
In the next section, it is shown that
a wave exists which is energy deficient for
-6aC
,
the eigenvalue.
-
but is stabilized again by further increase of
,
i.e., it is destabilized by a slight increase in -
This implies that the wave eventually changes from the
The behavior of neutral solutions with .-6 e2,
/f
/
energy deficient character due to change in the eigenvalue.
are rather different than in the subsonic cases, although
(A.8) still
with
applies for
--ef4
,
(A.3)
2
For neutral solutions
.
requires that
- //-mJ
4~Cc
J6-O
(/- cejt
(A. 9)
*4c arbitrary
which can be easily solved by graphical means.
The
qualitative results are most conveniently presented in
a three-dimensional sketch.
A) 2
(4 - p)
(Note: The Nyquist criterion shows that one unstable
) exists in the indicated areas.
root (with Oc cI
140
These results are rather disturbing since an
4
unstable root exists for all
-
if
,
,
less than the generally large value,
is
The
.
variation of the amplification rate corresponding to
the unstable root is shown in Fig. A.6.
Its relation
A- =o is essentially the same as for
to the result for
Md< 1 except that the region of mild instability is no
longer bounded.
This difficulty is intimately related
to the fact that the potential flow model admits no
C< /-
neutral solutions for
"
,>
,
.
The
region of instability is sharply reduced by boundary
layer effects as will be shown in the following section.
A-4
Laminar Viscous Flow Hydrodynamic and Aeroelastic
Instability
With viscous effects included, the characteristic
equations (A.2) or (A.2") have six independent parameters,
(
C,
,.
, c
, R,
l,
(Recall that the relationship.,
panels is not considered.)
besides the eigenvalue
I =
CO z
C
for dispersive
In order to make the problem
tractable, certain of these parameters will be kept
fixed, or at most discrete variations will be considered.
With this in mind, the problem can be usefully framed in
two alternate ways.
First, one can inquire into the
141
modifications of the neutral surfaces for potential
,
flow, (in 4d ,
space) due to variations in
For a given Mach number, these variations
can be attributed to changes in the boundary layer
configuration.
Alternately, one can consider the
neutral curve in a, R
space and find the modifications
_,
from the condition
"
,
i.e.,
the influence
of wall flexibility on the classical problem of boundary
The first technique (which utilizes
layer stability.
A.2) is the natural extension of the previous section.
However, the second (which utilizes A.2") will be
explored next because it gives valuable perspective for
the first.
The neutral curves and corresponding domains of
4$'*
instability in
space are given in Fig. A.3 for
the rigid wall with Mr= 0, 1.3.
These are taken from
Lin (5), p. 73 and the curve for M = 1.3 is recalculated
with (A.2") (with X =0 ) for purposes of comparison.
It is to be noted that a second branch of the neutral
curve is found, at least for M = 1.3.
es
asymptotic to the
to small
aIR
.
and
'
This branch is
axes and always corresponds
Therefore, it is neither important nor
trustworthy for the rigid wall - which may be the reason
why it was omitted by previous authors.
However, it
can become of paramount importance when the flexible
wall is introduced.
The possibility of neutral solutions
142
to
(A.2") for
C<O
is entirely eliminated by noting
>O
that Fig. A.1 shows
shows
while Eq.
(2.21)
C)'-
and
)< 0
Polar plots of the right and left hand sides of (A.2")
are presented in Figs. A.1 and A.2 for the rigid wall
with M = 1.3.
It is evident tnat no neutral supersonic
disturbances exist in this case.
of
N
and
The possible values
are also indicated for small
(ak'r-)
cc
and these show the improbability of amplified supersonic
disturbances with
at low
o
WI?
cc /c.,
0 (2)
,
except possibly
where the theory is untrustworthy.
Comparison
tne data in Fig. A.2 with extensive calculations given
by Lees
(6)
gives a qualitative indication that these
conclusions hold for all
(supersonic) Mach numbers.
The modifications of the areas of instability due
to surface flexibility are given in Figs. A.4 for MNI= 1.3.
The value of
-w=
50
is taken as a compromise
between values typical for panel flutter and for
stabilization of the boundary layer.
-=-/5~
The value
is taken to be typical of lightly damped surfaces.
effect of decreasing the flexure speed,
Co
,
The
can best
be discussed in terms of the two branches of neutral
arve exhibited in Fig. A.3.
The introduction of wall
flexibility (in the form (A.2')) immediately causes
these two branches to join.
The upper branch is
143
largely independent of flexibility except in the
range
Co W.4
(see Fig. A.4b) in which case
resonance phenomena with
C ~ CO
is felt everywhere.
This phenomenon is seen to be largely destabilizing.
R
The lower branch moves rapidly in the increasing
direction with decreasing
C0
.
This of course
means that unstable disturbances exist in large portions
of the
Md
domain and that the concept of a minimum
critical Reynolds number ceases to have meaning.
o
i.e.,
C- /
C</-
For
these disturbances are supersonic,
.
However, this fact is not particularly
significant (except locally where the two branches join)
since the wave speed becomes extremely low with decreasing
Co
and to very good approximation (A.2") takes the
form
This means that the "inviscid" solutions play no role
and that the results are nearly independent of M,,.
Inverting the above to retrieve the panel equations of
motion shows that the panel forces are being balanced
by the viscous stresses in the fluid.
Since the wave
speed is extremely low, the panel inertia forces are
small and panel forces are themselves viscous and
elastic in origin.
In the final Fig. A.4d, this branch
i
144
d
has moved off to
.
I-+ appears to always
it
connect in some way to a third branch at low
No detailed calculations were made for the connecting
curve since it was associated with large values of O.
The results of the alternate procedure, i.e.,
, A
determining neutral surfaces in A
for fixed values of
(A.7)-(A.9).
Figs.
,
and
/R
/Wl,
,
are shown in
of
for M= 1.3.
&,,AP,
Pcc)
For M = 0,
found from quantities tabulated by Lees
Pe
space
These neutral surfaces correspond
to the solutions of (A.2), (A.4).
gives
C'
(6).
is
Figure (A.5)
In Fig. (A.3), the values
that are considered are shown relative to
the neutral curve for hydrodynamic stability.
It is
important to keep this relationship in mind throughout
the following discussion.,
For the most part, the results for M = 0 merge
uniformly with the potential flow results in the limit
Q'-+O
(see Fig.
is for
+
(A.7a) and Eq.
in which case
(A.8')).
C--/
The exception
and the results
with viscous effects included become ambiguous.
The
instabilities deriving from the surface exposed to a
potential
v'&
low are generally suppressed with increasing
, see Figs.
(A.7a-d), except for those disturbances
which are strongly affected by hydrodynamic instability
phenomenon.
The latter is evident in Fig.
(A.7c) and
14
145
is associated with the lower branch of the A'eo neutral
curve shown in Fig.
(A.3).
For a given atmosphere and
free stream velocity, the various values of
Re
de'
with
= constant correspond to disturbances with different
wavelengths with a boundary layer of given (constant)
However, the surface pressure is much less
thickness.
sensitive to
Pa
than to
W
except when hydrodynamic
stability phenomena play a dominant role.
the overall trends due to increasing
oa'
Therefore,
can be roughly
associated with the effects of increasing the boundary
layer thickness on a disturbance of given wavelength.
Another branch of the neutral surface that is not shown
in Fig. (A.7) is discussed below.
Figure (A.8a) shows that the potential flow results
for M = 1.3 (recall Eq.
(A.9) and the associated sketch)
are strongly modified by viscous effects, even if the
boundary
layer is extremely thin relative to the wave-
length of the disturbance.
The vast area of mild instabiii'cy
that was indicated by the potential flow analysis is
sharply reduced.
The portion of the neutral surfaces
in Figs. (A.8a,b) that extends to high values of
and .6
CO
(excepting that for c+o ) is strongly affected
by hydrodynamic instability phenomena.
The latter are
associated with the lowest branch of the neutral curve
of Fig.
(A.3) (and the disturbances are transonic in
146
nature) for Fig.
(A.8a) and with the lower branch of
the "loop" of Fig. (A.3) for Fig.
effects of increasing
(A.8b).
The overall
are rougnly similar for
W'
M = 1.3 and M = 0.
A brancii oi tiie neutral surfaces in Figs.
(A.7c,d)
The character of this
and (A.8b,c) has not been shown.
branch is similar in all cases and it is described in
(A.9).
Fig.
and those of
The associated values of -6
C
are
Cs< c41
are -d <
C -.
where
.25
This
4.
branch has rather pathological behavior in that its
intersection with certain planes of
-6 = constant
consists of two straight lines that intersect at some
A---
value of
where
is
-
(A.9).
shown in Fig.
Thus, this surface does not demarcate regions of
stability
and instability
-e
.
;-e,
in -#e ,
,
2 space for
Comparison of the results of Fig.
and (A.4) with Figs.
(A.3)
(A.7) and (A.8) brings another
interesting feature to light.
That is, the boundary
layer investigations show that unstable disturbances
exist for certain regions in
of
C
.
'
space for all values
Figures (A.7) and (A.8) give no information
regarding this characteristic value.
However, using
(A.6) in conjunction with (A.2) enauies one to investigate
the moderately unstable eigenvalues for the various cases.
The amplification rates for the various boundary layer
147
configurations and M = 1.3 are shown in Fig. A.6 and,
indeed, a root that is unstable for all
the case
/
=
cZ
exists for
.0416.
One may finally conclude that, while the study of
(A.2) gives considerable information regarding the
modifications of the potential
flow analysis due to
viscous effects, any comprehensive study should be based
on (A.2").
Further, stability boundaries corresponding to
the inviscid rluid moctel without structural dissipation
g-nerally correspond to tne onset or severe instability.
148
(~~G.)
FIG.I
'GO
P0 rEN/AL
/41<I
I
PO
EA
(oC
e
.4>/
= 0)
0
/
w1
_____
t
e
?*Xd= Re 9*e19
'5-
(Oe
0)
=Ref/i
A
P,
_
,
ii
A* =R"
-
=
iij
I,
0
00
TlE
NOTE.
BOolNOARY L AYER
/rROFILE
DATA ARE GIVEN
IN .=1& 2. T7hE CURVE
19S ]-
qOR
g,
=.R,0/2 DOE S NOT
THE EFFECT O R
VAR/A T/ON ON rlE PROPILE
/NCLUOE
/
/
o
/.
-
I
PRED/CTED EPP4 ECT OP T/E
>/3
1.6
A4,.
TURULENr BOUNDARY LAYER ON T4E.
7-RZ/4a/7r/OAf OV&R
A
1GID VV WAVY
WAL.
149
FIG.2
I
0.8
M, =/. /,
R'
/106
.0+-V5"
.04#8",
6|=
0.6
37 R//A/ = 3. 7 X
I
64/ =-/7;300,/5,800
LOal crz4l/Ne(2/
0
6*
6
=.
=
-=
72
/
=
3.3
I
/0 s
I
500
,'
I
P4/NNEY (/6)
I
/
AfOCRKOV/N
0.4
R //v
06 53"
.
A-f
.I
/0
o ~.028"
R
8AROW
/,000
(23) (A AL)
(i-h
I
I
/
I
0
TL/18UL ENT
0.,
MEAN
0.4VELOClry
/
If/
0.G
PROFILES
I
_
0.8
OVER
/.0
A
FL AT
Zcz,
RIC71
WALL
A
STA77/VONS FOR BOQND
I
I
4RY 4 AVER SU/RVEYS
II
I
S= Ref'ej, 0-o 4=.0,2" f . 04"'
R~
Fti
meI
-
--
jj~jJJJJJJ~fiJJJAU[ISTA
T1.1ERMOCof
7,-
5- 71
-
--
7f1-
cSSrEM
7
I
IkLEl
3/
"
AL (/IAU/I
84S2 tpLA T
H
0
2,CONSTRUCTEO
/IROM-
MAHOGANY
AS
p p
p "'e~pp
TI-ICK
/8
~p
*:et~p:
/
BLOCK
p
p
0
0
0
WAVY
WALL MODEL.
(4
151
/.0
i
FG. 4-
0.8
NOTE.
0.6 i
ALL PROF1 LES EXCEPT
84 RON (2,3) COA 'RESPOND TO T7WE
"48/ABA/C WALL " AID TO TIE
/
9CAT/ON OF TAIE
ST/?EAMWISE
C PRE SSC/RE
FORWAR
SAT D54
OR//
CE
SE E
-
F/G.
3.
IN
.iX/
F
4. -?5
x/0I
04//
=Z2.08 x
= 053"
Re
=-., /00
R
/0
5
J
S=3.62 e/O
/
R
1 8 400
0.2
5x/O
35, R//r-
A4=/
w 028"
R
1/
000
(23)
SARoN
/01
I
0.2
0.4-
VLNPERTUSEDTURBUL.sNr MEAN
0.6
vaLOcry
0.8
0.8
PRoP/4i.ES
7.to-
A4
(-,
=
/.
Zw
4.05
152
FG. 6
T .0
0.8
PEIT/Rukef o PROF/LES A1RE
BASE D ON T4E AVERAGE OF r9E
PR ESSURE PROFILES SEA SWRED AT
TH E STT/ONVS /ND/CA TE D /N F/G 3.
NOTE.: JHE
0.6
y/04
R/II/I 4.95
Ro
= 4030
0.2
R/N =3.62x
c5*
=
.078"
1=.
/NPERUR8EO
M
/i/I =4-.9 5X1/
/0
8; =.060"
Re =2, 600
R1= =-*..95.I"
RI/N
95 x/e'
ZINERTRSD
02"
4
0.2
0
';4VERAGE"MEAN
ti-2
Cal.5L
0.G
VELOCITY PROFILES,
0.8
/.0
trcq), OVER SECOAID WAVE-M4,=f.405
r
153
p.
I
y
'?)
YS STW'
872*'
q*02"
F16 6
,Q/I N = 4.95
.(01-
-dTr'f7r
It
AVERAGE
--
ROFILE
E
y-y
F/y6
-
-. 2
c
0
/
si
6)17
4.95.xi0 4
0~R/A
/ /
/
I
I
/00
0000,
.0
0
.8
.2
DE TA-E
MACH
MUM 8 ER PROFILES
/.0
1.2
OVER MWE SECOND WAVE - /oo
= /.405
154
40
a) >=.
9I=4..e./04
A'
p0
=
Ff6.7
4
C,
I
.35
A VER A6E
SrA r/C
.30 I
or
RL
a x
r
I
a
I
I
I
wPsTr/EA4
Oj/i .(/C&
S/
It
L7
.20C
b) ;
=
=_g1,
.02 "R/i=/_4A. 95X/0
_
,0
.3 '-t
-o
V
T111
C
__s__1lwx
.25-
.dA
-. o2"R/IN =3.62 x1
CKp
f
=+
)
C) ,
Ir
C-
.25L2NL
-
7NROOSWOUT
?ESSURE
o/STr'eu771oN
OVSR A
-P C
t.77~/
TWO--D/AM-NS/ONALI.- WAVY
w4LL-~, O= 1.405
1'55
/80
z2
'.o2"A
>RI //
gOraFw
I
1i
-
I
'O02
5
.4.
ss.9X/.,
ii
z
F/6 8
/810
ALL 1#,<I
. .0
/A,
/2A
4A40>1
60
4
*
I
0
I/
-2
Int=o)(Ze<
p
*/'Re
2
z/
e
tx
POTENtIAL FLOW(8
A4EA SURC
-0-
SECONO
(VERAGE
=0)
OVE/
/Sr
WAVE S)
Y( vVir4 vicous
L.,NE AR' rle o
E /FECT5 - SEE I-ROPFL ES O CIG.
_.-;
osN-LINEAg TI4EORY wiTH VIsCoi
Nff CrS - SEE PRO ,/..E 5 O# IG.
4
S/
5I
g5/
-1*~
/
R/IN X
2
3
-?-
0
-02
VA R/A r/ONS oF COMPLE X PRESSURE AMPLITUVE WI/- I REYMOLDs'
AMUmER A Ai SURFACE WAVE AMPLITUDE- M, = /.405
156
Ff6.9
bLE
-----
UNSTABL E
MEASURED RESULT WITH
TlRIULEN T
SEE LOCK
*
---
TWO A4O0e
SOUNDARY
AN ADJoNiMq
LAYER
-
3 7-4
TWO MODE 6AL ERI(IN ANA LYSIS,
WI/TH TE SURFACE PRESSURE GIVEN
BY 'FYACT"L/AEARIZEo POTENTIAL
FLOW
-SEE L 0CK
FUNG (~2)
FUING (2/)
G.ALERKIN
ANA4L'/SIS ftWlr
BY
T1E SURFACE PRESSUR& GIVEN
TA/E APPROX i MA rION (-4) INCLUo ING
$
VISCOUS
26*
EFFECTS.
MOTE.: rE ANALYTICAL RESULTS CORRESPOND
To _9=.O/. TE
MEASUREO> VALUES
: .! 2 - SEE (Cz)
oF 9 ARE .0O1?5
x
.L1I~
1
/.0
4
/.-f
TA'ICK/ESS
REQU/RE
SuPPORTE D PAN EL
I
I.
I
ro
-
PREVENT
/.6
&.4
co
(NST8AILITY OF A FLAT, SI MPL.Y
AND
EXPERMENTAL.
THEORETiCAL
157
FIG. /0
. vZol-=AILu 1
A Pitox
e
ODE
To 7HE
/MArON
-
-
TWO MOE GALERKAMA LS/S
WITM T4E SURFACE PRESSURE
-
(DOWNSTREAM
WA VN
BUrCOA4ES 7YR NS'OA/IC
)
GIVE A 8Y oE 4C74/NE ARIZEo
POTENrIA L FLOW - SEE (Z)
O ONE MODE GALERKIN ANALVSIS
THE SURFACE PRESSUR
WIT
GIVEN BV EXACT"LINEARIZED
.06
POTENT/AL
FLOW
ONE AM0OE GALERkIN ANALYfS
WiT/ T14E SURFACE PRESSwRE
C/VEN BY LNEARIZED POTENTIAL
FLOW UNEER TUE APPR0YIDA4ATI.N
00=.O5
(6-4)
.0/
o
.0/2
.0/5
NOTE.* THE INCLUSION OF VISCOUS EFFECTS
IN
THE APPRO1VATION
C-}
ENTIRELY SUPP/?ESSES
TIHE
INSTASILITY FOR THE ONE MODE
GALERKIPI ANALYSIS.
.008
11"Im-
PA/LURE D UE ro
SINGLE DsGREE OP
FREEDOf ANALYSIS
01
1.0
THE
1
SINGLE
I.
1
/-2
VEGREE
O
A/4 RROYIA4A r.nO N(s-4),
FEEOO-f
V/sCOSi)Y
1I
A"ALY~ss-E'PFEcrS
ANC
sTRqucruRAL
lo-Ico
/.-r
OF
TIHE
Dlss/PA r7o
r
158
/i
T4
A4,v = /.Z 19
Ac / loe-I -/If
_____
UNSrABLE
ONE
STA
-
8 LE
-
MOE
ONE MODE
-
FLOW
wlrH PoTENAL FLOW
ISCOUS EFFECrs
-(5-4)
WO l40E-(s-4)i I r
T
k= .0785
R-ACT POTENT/AL
.03
. / 78
..
__ __ _
DOWN STREAAoM
-
=.O 785
WA VE BECOA-ES
rRANSoNIC
.0755
,02
. 20
M -/37
0
-
./57
-.01
A-/..3
Ali
=39Z
wo
3
.11/717
Ow
-
.2,
.
0
.j5
./o
.05
SrA/,1lTY
8OUNOARIES
F6.12
C
I
K
4
A4,0 2 /,jAPPRdo
1.37 -. /7
-
/
/.2/9
-
. /51
\
Irk
0
N
x
/
.~&L 21
*-
= 5/v 77kI
-/ L-
TH E
AEM04EL
45T1C MO D E .OR /Z =VISCOUS
A A Z YSiS W/ 7
0 7&5-
TWO MADE GALtER I./
EAFle Cr3
1NCLUcSP
159
/6 4-1
3-2
"0
2.9
-4.o
2
-
2-f
25
20
;0..,1.5
.2
/
/
.. 2.0
a
--
.. 4
0(\-2.'
CZz
.6
.1.5
-2.0
/-2.
fPE ,=cUNDCr#o AIz
rr
FG. A -2
/0+I
c = 4.0
.0~
e =. co
.7
. 55'
OO5
C 2.09
-~0.-5
U
2.0
1o .5
C = 1-
-
r<
11n o
23
C
.25
A-5
A.
PF'PROXIAM4ArE 0OAIN
w / rH
r 14 -t 46 Vr
C
OR
I-g
,
/.LE/
.,
c
ibi = /- ,i3
FOR (u t it)
,I C
-*'-
0
16o
F1 . A-3
.161
M0A.=
N
1.
/2k-
.
.
ng
'4
N
I
.04H-
o
/0'
&T7AUJL/TY
l
22
W
104
P9
43OUJNARIEST =O
fAA4.1VAR W. VISCOUS$ PLOW OVEM
A /GOA0413qaT/C
WAL L
---.
.--
-
RIGID WA LL
FLEXIBLE WALL
1" =,O
-.0/7
=.05
F/6A-4
----. FLE/LE WALL
M1-=50
-46/m=.0A
(UNS7ABLE ROOTS ARE DETECTED ON
3or4 SIOES OF TUE NEUTR4L CURVE)
----UAISrABLEN
b) co = . l
a) Co= 1.0
ace
-I0
10
-51 i
.05
m
O /0
,
0
/os
(0 z
loomm-
/oz
/08
/04
R
c)
C
=
.10
oe
. /0
.1/0
.05
.05
---
*, c
--
1__k
.0
- - -,
6>m-.
a
f0'
/02
/o'
103
Re
R9
STA8/4/rY BOUNDARIES FOR LAMINAR
A FLEXI,98L2 "AD/ABA TIC" WALL
/ov-
/02
VSCOUS
-
,
FLOW OVER
.3= 1-3
II
j1
102
F16.4-5
t
= 0 (Po TEN 7l4 i)
3
cz
.00/04
.0/0+
2
.
.04/6
0
I
~
'I
~ZIKN
--I
~
~
'0+
l
.6
-I--
z
I
U -----
I
I-
.8
f4
-2
-
-3
c < I--
c C?/--j
r2,P
3
2
.o+0f
n
Nib.
l
i
Elam
.
i
m.
/l
.6'
-2
-.
62
o.
000
-U
-27
P
= 0(
'cTENTIAL)
F01 L.A M/4R V/SCO/S FLOW OVER A FLEX/&8E
WAL
L-
A4,= /. 3,
Re = /. 0*
/O 3
AD/484r/C
163
Ff6. 4-6
-
e
cxc
Re =/04x
-Ta.
A-3
103
.20
-/5
'Il
=.0-0/6
./0
.00/04
o(PO rENrAL-)
0
5
OC - .104
EFFECTS
DISSIpArToN
=.9
OF
4AMINAR
OV TI4E
=
0
-145A#401 l-
4t-V1N
72
ST rA 811- 1 r y
)
or
CW.C
SrRUC7TuRAL.
804NOARY LAYER AANI
AI4cPLIFICA TION RATE OF AN UNST ABLE DISTURBANCE
r
164
b)q
a) %0= .00/04
f~
C
-~
0
6.10
/
-0
F/&-7
=0/041
c.~. o2~~
Lu
I/
SI
'"'b
~ b0~2
1=.+
-
UNSTA8LE
.-
C~
%o(/ooeNTIALLoW~~#C
-
-
2.o
.3
2.0
c= o)
STASL. E
]------~m-~
C) oa = . 0 4- / 6
5.0
s
.ozs
02.
x
V
|
/-o
2s'
.265
0
;2
NeurRL 'L//U'FACES
Ic
FOR L A MINA R VISCOUS FLOW 0VER A FLEXIBLE 5VRrACE
20
5
M4,=0, / 9 e.=/.o4( 0 A3
~iq
-m
- -
165
C --w0
23
F1G A-8
0=9+
-
a
c-
b)c~ =0/01-
=.o 0 /4
C--0
mx8
25.0
25.0
C=.Oo
C= .03
C.5
37
.u
.328
/22.
~s
.265
20
2
K.0
c .335
.245
.37.50
c-0)
.FOW,4-,
ox-o(porNrIAl
c
S
U
UNSrAOLE
T44BLE
d) cc
= ./04-
C -- p.O
.9,
25.0
I
I
I
I
'
C
C=.55
C=.dSo
;05
C
25.0
.55
.23
/ .oS
-
.10
-
-
I- 0
-I
C2
MVEUTRAL
"'U/FAcES"
POR
0
LAM/MAR
SURFA CE A0
/.
VIScO&S
,
Re = /.
OVR
* /03
PL=OW
0
A PLEXISLa
166
F16/-7
.
. ...... .
S
qc/AL/r.4r1v
NEuRAL "SURPACE " FOR LAM/MAR
S(uRFACE
FiOW OVER A PLEX/SLE
V/SCOVS
'167
REFERENCES
1.
Benjamin,
T. Brooke,
"Shearing Flow over a Wavy
Boundary, " Journal of Fluid Mechanics,
Vol.
6,
part 2, August 1959.
2.
Benjamin, T. Brooke,
"Effects of a Flexible Boundary
on Hydrodynamic Instability," Journal of Fluid
Mechanics, Vol. 9, part 4, December 1960.
3.
Landahl, M., "On the Stability of a Laminar
Incompressible Boundary Layer over a Flexible
Surface," to be published in the Journal of Fluid
Mechanics.
3a. Linebarger, J.,
On the Stability of a Laminar
Boundary Layer over a Flexible Surface in Compressible
Flow, S.M.
Thesis, MIT Department of Aeronautics and
Astronautics, 1961.
4.
Tollmien, W.,
"Laminare Grenzschicten, Hydro-and
Aerodynamics," FIAT Review of German Science,
1939-1946,
5.
Wiesbaden,
1948.
Lin, C. C., The Theory of Hydrodynamic Stability,
Cambridge Monographs on Mechanics and Applied
Mathematics,
6.
Lees, L.,
Cambridge University Press, 1955.
The Stability of the Laminar Boundary
Layer in Compressible Flow,
formerly NACA TN 1360.
NACA TR 876 (1947),
168
7.
Dunn, D. W., On the Stability of the Boundary Layer
in a Compressible Fluid, Doctoral Dissertation,
MIT,
U.
1953.
Kurtz, E. F.
(Private Communication), MIT Mechanical
Engineering Department.
9.
Miles, J. W., "Panel Flutter in the Preseice of a
Boundary Layer," Journal of the Aerospace Sciences,
Vol. 26, February 1959.
10.
"The Fundamental Solution for
Small Steady Three-Dimensional Disturbances in a
Two-Dimensional Parallel Shear Flow," Journal of
Lighthill,
M. J.,
Fluid Mechanics, Vol. 3, p. 113, 1957.
11.
Miles, J. W., "The Hydrodynamic Stability of a Thin
Film of Liquid in Uniform Shearing Motion," Journal
of Fluid Mechanics, Vol. 8, part 4, August 1960.
12. Benjamin, T. Brooke,
"The Development of Three-
Dimensional Disturbances in an Unstable Film of
Liquid Flowing Down an Inclined Plane," Journal of
of Fluid
echanics, Vol. 10, part 3, May 1961.
13. Morse, P. M. and Feshbach,
Physics,
H., Methods of Theoretical
McGraw-Hill Book Company,
14. Miles, J. W.,
1953.
?On the Generation of Surface Waves
by Shear Flows," Journal of Fluid Mechanics, Vol. 3,
p. 185, 1957.
15.
Hinze, J. 0., Turbulence, McGraw-Hill Book Company,
1959.
A
169
16. Morkovin, M. V. and Phinney, R. E., Extended
Applications of Hot-Wire Anemometry to High Speed
Turbulent Boundary Layers, AFOSR TN 58-469.
17. Clauzer, F. H., "The Turbulent Boundary Layer,"
Advances in Applied Mechanics, Vol. 4, 1-51,
Academic Press, 1956.
18. Schubauer, G. B., and Tchen, C. M., "Turbulent Flow
and Heat Transfer.," Vol. 5, High Speed Aeroaynamics
and Jet Propulsion, Princeton University Press, 1960.
19. Stanton, T. E., Marshall, D. and Houghton, R., "The
Growth of Waves on Water due to the Action of Wind_,"
Proc. Roy. Soc A., 137, p. 283, 1932.
"Die Turbulente Stromunganwelligen
Wanden," Z. angew. Math Mech. Vol. 17, p. 193, 1937
20. Motzfeld, H.,
21. Lock, M. H. and Fung, Y. C. B., Comparative Experimental and Theoretical Studies of the Flutter of
Flat Panels in a Low Supersonic Flow, AFOSR TN 670,
May 1961.
22. "Massachusetts Institute of Technology Aerophysics
Laboratory and Naval Supersonic Facility," General
Information Bulletin, 1961.
23. Baron, J. R., et al., Analytic Description of a
Supersonic Nozzle by the Friedrichs Method, Including
Computation Tables and a Summary of Calibration Data,
WADC TR 54-279, June 1954
170
24. Miles, J. W., "On the Aerodynamic Instability of
Thin Panels, " Journal of the Aeronautical Sciences,
Vol. 23, August 1956.
23. Miles, J. W.,
"SuDersonic Flutter of a Cylindrical
Shell, " Journal of the Aeronautical Sciences, Vol. 24,
February 1957.
26. Leonard, R. W. and Hedgepath, J. M., On Panel Flutter
and Divergence of Infinitely Long Unstiffened and
Ring Stiffened Thin Walled Circular Cylinders, NACA
TR 1302, 1957.
Doe//, e. 4uJ Per
27. Dugundji, J.
. S.
Subsonic Flutter of Panels on Continuous
Elastic Foundations - Experiment and Theory, Massachusetts Institute of Technology Aeroelastic and
Structures Research Laboratory Report 74-4 (to be
published as AFOSR TR).
28. Nelson, H. C. and Cunningham, H. J., Theoretical
Investigation of Flutter of Two-Dimensional Flat
Panels with One Surface Exposed to Supersonic
Potential Flow, NACA TR 1280, 1956.
29. Sylvester, M. A. and Baker, J. E., Some Experimental
Studies of Panel Flutter at Mach Number 1.3, NACA
RM L52116.
30. Miles, J. W. and Rodden, W. P. "On Supersonic Flutter
of Two-Dimensional Infinite Panels," Readers' Forum,
Journal of the Aerospace Sciences, Vol. 26, March 1959.
31. Bisplinghoff, R. L., Ashley, H. and Halfman, R. L.,
Aeroelasticity, Addison-Wesley Publishing Company, 55.
171
32.
C. B., A Summary of the Theories and
Experiments on Panel Flutter, AFOSR TN 60-226
uing, Y.
May 1960.
33.
Lees, L., and Lin,
C. C.,
Investigation of the
Laminar Boundary Layer in a Compressible Fluid,
NACA TN 1115, September 1946.
BIOGRAPHICAL NOTE
James D. McClure was born in Clayton, New
Mexico on June 23, 1935.
He received his secondary
education at Eureka, Montana, and completed the studies
leading to the B.S. and M.S. degrees in Aeronautical
Engineering at the University of Washington in Seattle.
He was associated with the Dynamics Staff Group
of the Boeing Airplane Company from June, 1957 to
August, 1958.
Subsequently, he came to MIT to do doctoral work
and has been associated with the Aeroelastic and Structures
Research Laboratory and the Fluid Dynamics Research
Laboratory, both through the Division of Sponsored
Research at MIT, during the summer periods.