# Axial Compressor Design

```Axial Compressor Design
Onur Tuncer
Istanbul Technical University
Faculty of Aeronautics and Astronautics
Department of Aeronautical Engineering
Maslak, Istanbul 34469
[email protected]
November 23, 2011
Outline
Introduction
Thermodynamics
Fluid Mechanics
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Introduction
Axial Flow Compressor Basics
Axial vs. Centrifugal Compressors
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Introduction
Axial Flow Compressor Basics
Construction of an Axial Compressor
A stationary row of blades (stator) is followed by a rotating row of blades
(rotor).
A compressor stage is made up of a rotor and a stator.
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Introduction
Basic Velocity Diagrams for a Stage
Polar Surface View of a Stage
The rotor row is rotating with a velocity U = ωr
Viewed in a reference frame rotating with the rotor, the upstream velocity
W is called the relative velovity.
The rotor deflects the flow such that the velocity in the stationary frame
of reference of the stator (the absolute velocity), C is properly aligned to
enter the stator row.
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Introduction
Basic Velocity Diagrams for a Stage
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Introduction
Basic Velocity Diagrams for a Stage
Guide Vane Velocity Triangles
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Introduction
Basic Velocity Diagrams for a Stage
Velocity Triangle for a Rotor
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Introduction
Basic Velocity Diagrams for a Stage
Velocity Triangle Calculations
Wθ = Cθ − ωr
The axial components of velocity are identical in both reference frames.
W z = Cz
The absolute
and relative velocities are,
q
2
C = Cz + Cθ2
q
W = Cz2 + Wθ2
The absolute β and relative β 0 flow angles are,
tan β = Cθ /Cz
tan β 0 = Wθ /Cz
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Introduction
Similitude and Performance Characteristics
Similarity
Two turbomachines are completely similiar if the ratios of all corresponding
length dimensions, velocity components and forces are equal.
Equivalent Flow Rate Parameter
Q = ṁ/ρt
Local Axial Flow Velocity
Q = ṁ/ρt
Volume Flow Machines
ρ0 /ρt0 is a function of Cz 0. Therefore, unique velocity diagrams are
associated with a unique Q0 /A0 , yet can correspond to many values of
ṁ/A0 .
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Introduction
Similitude and Performance Characteristics
An Equivalent Performance Map
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Introduction
Similitude and Performance Characteristics
More on Similarity
I
True equivalent performance is obtained if working fluids obey the
perfect gas equation.
I
Similarity is also compromised if the compressors operate at
substantially different Reynolds numbers.
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Introduction
Similitude and Performance Characteristics
More on Similarity
Alternative Equivalent Flow Parameters
√
√
Q/at0 ∝ ṁ Tt0 /Pt0 ∝ ṁ θ/δ
Sound
√ of speed is calculated by,
a = kRT
θ and δ relate inlet total conditions to some reference condition (most
often to standart atmospheric conditions).
θ = Tt0 /Tref
δ = Pt0 /Pref
The equivalent speed can be replaced by,
N/at0 ∝ N/
O. Tuncer (ITU)
p
√
Tt0 ∝ N/ θ
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Introduction
Similitude and Performance Characteristics
Efficiency
Previous figure only shows part of the information. However, it is
important to know how much work is necessary to drive the compressor.
η = ∆Hrev /∆H
Z
∆Hrev =
rev
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dP
ρ
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Introduction
Similitude and Performance Characteristics
An Equivalent Efficiency Map
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Introduction
Stage Matching and Stability
Stage Matching and Stability
I
Each blade row achieves best performance for a specific inlet flow
angle, where losses are minimum.
I
The designer seeks to ”match” succeeding blade rows such that all
operate close to their optimum inlet flow angles, at a specific
operating condition (i.e. design point or match point)
I
At lower flow rates the characteristic has a positive slope which is
theoretically unstable. This severe unstable operation is commonly
called as surge.
I
In other cases abrupt stall might occur.
I
For higher flow rates no rise in pressure might occur. This is called
choke.
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Introduction
Dimensionless Parameters
Dimensionless Parameters
Euler Turbine Equation
∆H = U(Cθ2 − Ctheta1 )
Total Enthalpy
1
H = h + C2
2
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Introduction
Dimensionless Parameters
Dimensionless Parameters
Stage Work Coefficient
Ψ = ∆H/U 2 = (Cθ2 − Ctheta1 )/U
Stage Flow Coefficient
φ = Cz 1/U
Stage Reaction
R = (h2 − h1 )/(h3 − h1 )
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Introduction
Dimensionless Parameters
Relationship Between φ, Ψ, R and Velocity Diagrams
50% Reaction Stages
tan β10 = −(Ψ/2 + R)/φ
tan β20 = (Ψ/2 − R)/φ
tan β1 = (1 − R − Ψ/2)/φ
tan β2 = (1 − R + Ψ/2)φ
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Thermodynamics
First and Second Laws of Thermodynamics
First Law of Thermodynamics
I
I
I
Open system
1 2
q̇ + ẇ = ṁ∆ u + C + P/ρ
2
Note that,
h ≡ u + P/ρ
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Thermodynamics
First and Second Laws of Thermodynamics
Second Law of Thermodynamcis
Specific Entropy
ds =
dqrev
T
Second Law
∆s ≥ 0
Fundamental Thermodynamic Equation for Entropy
Tds = dh − VdP
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Thermodynamics
Efficiency
An Enthalpy Entropy Diagram
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Thermodynamics
Efficiency
Polytropic Efficiency
I
Polytropic efficiency is also known as small stage efficiency or true
aerodynamic efficiency.
I
Instead of using a path of constant entropy as the reversible path,
polytropic efficiency uses as path of constant efficiency defined by
ηP = ρ1 dP
dh .
∆Hp = ∆H − (sd − si )(Ttd − Tti )/ ln(Ttd /Tti )
Total to Polytropic Efficiency
ηP =
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∆Hp
∆H
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Thermodynamics
Fluid Equation of State Fundamentals
Fundamental Relations
Thermal Equation of State
P = P(ρ, T )
Calorific Equation of State
h = h(T , P)
u = u(T , P)
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Thermodynamics
Fluid Equation of State Fundamentals
Ideal Gas Law
P = ρRT
R = Ru /M
Ru = 8314 Pa.m3 /(kmol.K )
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Thermodynamics
Fluid Equation of State Fundamentals
A Pressure Enthalpy Diagram Schematic
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Thermodynamics
The Calorific Equation of State
The Calorific Equation of State
∂h◦
=
∂T
◦ P
∂h
cv◦ (T ) =
∂T V
cp◦ (T )
For a thermally perfect gas,
cp◦ (T ) − cv◦ (T ) = R
◦
◦
T
Z
h (T ) = h (Tref ) +
u ◦ (T ) = u ◦ (Tref ) +
Tref
T
Z
cp◦ (T )dT
cv◦ (T )dT
Tref
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Thermodynamics
Entropy and the Speed of Sound
Entropy and the Speed of Sound
Specific Entropy
◦
◦
Z
T
s (T , P) = s (Tref , Pref ) +
Tr ef
cp◦ (T )
− R ln(P/Pref )
T
Specific Entropy for a Calorifically Perfect Gas
s ◦ (T , P) = s ◦ (Tref , Pref ) + cp◦ ln(T /Tref ) − R ln(P/Pref )
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Thermodynamics
Entropy and the Speed of Sound
Speed of Sound
Thermodynamic Relation for the Speed of Sound
◦
a =
∂P
∂ρ
=k
s
∂P
∂ρ
T
Ratio of Specific Heats
k = cp /cv
Speed of Sound for a Thermally Perfect Gas
a◦ =
O. Tuncer (ITU)
√
kRT
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Thermodynamics
The Thermal Equation of State for Real Gases
The Thermal Equation of State for Real Gases
General Thermal Equation of State for a Real Gas
P/(ρRT ) = z(T , P)
where z is the compressibility factor.
For ideal gases z = 1.
Parametric Equations
Simple two-parameter equations of state are a good choice for general
aerothermodynamic design and analysis.
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Thermodynamics
The Thermal Equation of State for Real Gases
Redlich-Kwong Equation of State
Redlich-Kwong Equation
P=
a
RT
√
−
V − b V (V + b) Tr
where Tr = T /Tc is the reduced temperature and,
a = 0.42747R 2 Tc /Pc
b = 0.08664RTc /Pc
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Thermodynamics
Thermodynamic Properties of Real Gases
Thermodynamic Properties of Real Gases
Departure Functions
I
Specification of the calorific equation of state, h◦ , u ◦ are limited to
state points where the fluid is thermally perfect.
I
For non-ideal fluids h and u are functions of P and T .
I
Thermodynamics properties of a non-ideal fluid are best accomplished
utilizing departure functions.
I
Departure functions are defined as the difference between the actual
value of a parameter and its value under conditions where the fluid is
thermally perfect.
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Thermodynamics
Thermodynamic Properties of Real Gases
Departure Functions
Corresponding Specific Volume
V ◦ = RT /P ◦
If A=Helmholtz Energy
A − A◦ = −
Z
V
(P − RT /V )dV − RT ln(V /V ◦ )
∞
◦
s −s =−
∂(A − A◦ )
∂T
V
h − h◦ = (A − A◦ ) + T (s − s ◦ ) + RT (z − 1)
u − u ◦ = (A − A◦ ) + T (s − s ◦ )
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Thermodynamics
Thermodynamic Properties of Real Gases
Redlich-Kwong Departure Functions
h−h
◦
s − s◦
a
V +b
−n
= PV − RT − (n + 1)Tr ln
b
b
V V −b+c
na −n
V +b
= −R ln
−
T ln
V◦
V
bT r
b
where, c = 0 and n = 0.5 for the original Redlich-Kwong equation of state.
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Thermodynamics
Thermally and Calorifically Perfect Gases
Thermally and Calorifically Perfect Gases
When the fluid can be considered thermallyperfect (z = 1) and calorifically
perfect (cp , cv and k are constants), equation of state calculations are
greatly simplified.
Calculations for Enthalpy and Entropy
h = href + cp (T − Tref )
s = sref + cp ln(T /Tref ) − R ln(P/Pref )
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Thermodynamics
Thermally and Calorifically Perfect Gases
Thermally and Calorifically Perfect Gases
Relation Between Total and Static Conditions
1
cp (Tt − T ) = C 2
2
T /Tref = (P/Pref )
k−1
k
= (ρ/ρref )k−1
Efficiency Calculations
k−1
=
ηp =
O. Tuncer (ITU)
(Ptd /Pti ) k − 1
Ttd /Tti − 1
k − 1 ln(Ptd /Pti )
k ln(Ttd /Tti )
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Thermodynamics
The Pseudo-Perfect Gas Model
The Pseudo-Perfect Gas Model
The concept is to use fictitious values of cp , cv and k in an otherwise
standart calorifically and thermally perfect gas model.
√
R̄ = R z1 z2
c¯p = (h2 − h1 )/(T2 − T1 )
c¯v
= (u2 − u1 )/(T2 − T1 )
k̄ = ln(P2 /P1 )/ ln(ρ2 /ρ1 )
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Thermodynamics
Component Performance Parameters
Diffuser
Diffuser Efficiency
ηdiff =
∆h
Pressure Recovery Coefficient
cp =
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Pd − Pi
Pti − Pi
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Thermodynamics
Component Performance Parameters
Nozzle
Pressure Loss Coefficient
Nozzle Efficiency
ηnoz =
ω̄ =
Cd2 − Ci2
2 − C2
i
ηnoz = 1 −
∆Pt
1
2
2 ρC
Pti − Ptd
Pi − Pd
Pressure Loss
∆Pt
Pti − Ptd
=
= T ∆s
ρ
ρ
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Thermodynamics
Gas Viscosity
Gas Viscosity
Dean and Stiel Model
1/6
ξ = Tc /
√
2/3
MPc
Low pressure fluid viscosity,
8/9
µ0 ξ = (3.4.10−4 Tr
, Tr ≤ 1.5
µ0 ξ = 0.001668(0.1338Tr − 0.0932)5/9 , Tr > 1.5
Then, the viscosity at any pressure can be defined by the following
departure function.
(µ − µ◦ )ξ = (1.08.10−4 )[exp(1.439ρr ) − exp(−1.111ρr1.858 )]
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Fluid Mechanics
Flow in a Rotating Coordinate System
Flow in a Rotating Coordinate System
The analysis of the flow in the rotor blade rows is accomplished in a
coordinate system that rotates with the blade.
Wθ = Cθ − ωr
The axial and radial velocity components are independent of rotation.
Wz
= Cz
Wr
= Cr
Meridional Velocity Component
Wm =
O. Tuncer (ITU)
q
Wz2 + Wr2 = Cm
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Fluid Mechanics
Flow in a Rotating Coordinate System
Stream Surface and Natural Coordinate System
A stream surface is defined as a surface having no fluid velocity
component normal to it.
Schematic of a Stream Surface
Natural Coordinate System
(dm)2 = (dr )2 + (dz 2 )
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Fluid Mechanics
Flow in a Rotating Coordinate System
Euler Turbine Equation
Consider the flow through a thin stream sheet (i.e. a thin annular passage
bounded by two stream surfaces). The torque τ acting on the fluid
between merional stations 1 and 2 is provided by the conservation of
angular momentum.
τ = ṁ(r2 Cθ2 − r1 Cθ1 )
The torque must balance the power input.
ẇ = ωτ = ω ṁ(r2 Cθ2 − r1 Cθ1 )
Euler Turbine Equation
H2 − H1 = ω(r2 Cθ2 − r1 Cθ1 )
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Fluid Mechanics
Flow in a Rotating Coordinate System
Rothalpy
Total enthalpy change is produced by a transfer of mechanical energy
between the fluid and the rotating blade row.
I = H − ωrCθ
For a stationary blade row, I = H.
Aerodynamic analysis of axial compressors involve the solution of
conservation equations in both rotating (rotors) and stationary (stators)
coordinates.
The relationship between relative total enthalpy H 0 in a rotating, and
absolute total entalhalpy in a stationary coordinate system.
1
1
h = H0 − W 2 = H − C 2
2
2
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Fluid Mechanics
Flow in a Rotating Coordinate System
Rothalpy (Continued)
The relative velocity W ,
W =
q
Wm2 + Wθ2
Total enthalpies,
1
1
H 0 = H − ωrCθ + (ωr )2 = I + (ωr )2
2
2
Since I is constant on the stream surface, above equation allows the
calculation of H 0 at all points on a stream surface when one value is
known, e.g. at the inlet.
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Fluid Mechanics
Vector Form of Momentum Equation in a Rotating Coordinate Frame
~
~
1~
DC
DW
~ )+ω
= − ∇P
+ 2(~
ωx W
~ x(~
ω x~r )
=
Dt
ρ
Dt
where,
~
~
DW
∂W
~ .∇)
~
~ W
=
+ (W
Dt
∂t
Hence the momentum equation in rotating coordinates is,
~
~
∂W
∇P
~ .∇)
~ + 2(~
~ )+ω
~ W
+ (W
ωx W
~ x(~
ω x~r ) =
∂t
ρ
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Fluid Mechanics
Vector Form of Continuity and Energy Equations
∂ρ ~
~ )=0
+ ∇.(ρW
∂t
1 ∂P
∂I
~ .∇)I
~ =0
−
+ (W
∂t
ρ ∂t
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Fluid Mechanics
Governing Equations in Natural Coordinates
∂ρ 1 ∂r ρWm ∂ρWθ
+
+
+ κn ρWm = 0
∂t
r
∂m
∂θ
Wm Wθ ∂Wm sin φ
1 ∂P
∂Wm
+ Wm
+
−
[Wθ + ωr ]2 = −
∂t
∂m
r ∂θ
r
ρ ∂m
∂Wθ
∂Wθ
Wθ ∂Wθ
Wm sin φ
1 ∂P
+ Wm
+
+
[Wθ + 2ωr ] = −
∂t
∂m
r ∂θ
r
r ρ ∂θ
κm Wm2 +
cos φ
1 ∂P
[Wθ + ωr ]2 =
r
ρ ∂n
∂I
1 ∂P
∂I
Wθ ∂I
−
+ Wm
+
=0
∂t
ρ ∂t
∂m
r ∂θ
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Fluid Mechanics
Governing Equations in Natural Coordinates
The curvature of the stream sheet κm and the normal of the surface κn
are related to the angle φ as follows.
∂φ
∂m
∂φ
1 ∂b
=
∂n
b ∂m
κm = −
κn =
Parameter b is the thickness of the stream sheet bounded by two stream
surfaces.
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Fluid Mechanics
I
To determine the flow in the meridional plane
I
I
Hub to shroud flow solutions
I
Quasi three dimensional analysis
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Fluid Mechanics
Boundary Layer Analysis
Boundary Layer Analysis
The basic premise of boundary layer theory is that the viscous effects are
confined to a thin layer close to the physical surfaces bounding the flow
passages.
I
I
Blade surface boundary layers play an important role in viscous losses
and stall or boundary layer seperation.
Endwall boundary layers can produce substantial viscous blockage
effects that significantly affect compressor performance.
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Fluid Mechanics
Two-Dimensional Boundary Layer Analysis
Two-Dimensional Boundary Layer Analysis
I
Two-dimensional boundary layer analysis is a useful approximation in
I
Two dimensional blade sections designed between the hub and tip are
stacked together to create the actual three dimensional compressor
Boundary Layer Equations
Basic conservation of mass and momentum provide the governin equations
for two dimensional boundary layer flow over an adiabatic wall.
∂ρbu ∂ρbv
+
=0
∂x
∂y
u
O. Tuncer (ITU)
∂u
∂u 1 ∂P
1 ∂τ
+v
+
=
∂x
∂y
ρ ∂x
ρ ∂y
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Fluid Mechanics
Two-Dimensional Boundary Layer Analysis
Integral Form of Boundary Layer Analysis
Integrating the conservation of mass across the boundary layer and
applying the Liebnitz rule to interchange the order of integration and
differentiation yields.
∂
∂x
Z
δ
bρudy = bρe ue
0
∂δ
∂
− bρe ve =
[bρe ue (δ − δ ∗ )]
∂x
∂x
Displacement Thickness
∗
Z
δ
[ρe ue − ρu]dy
ρe ue δ =
0
It is a fictitious thickness used to correct the mass balance relative to the
inviscid flow solution.
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Fluid Mechanics
Two-Dimensional Boundary Layer Analysis
Integral Form of Boundary Layer Analysis (Continued)
Momentum Thickness
ρe ue2 θ =
Z
ρu[ue − u]dy
Combining displacement and momentum thicknesses yields,
Z
δ
ρu 2 dy = ρe ue2 [δ − δ ∗ − θ]
0
If the free stream conditions are applied within the boundary layer with no
flow in the thickness δ ∗ , and no momentum in the thickness θ, momentum
conservation will be corrected for viscous effects. This is the basis of
integral boundary layer analysis method.
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Fluid Mechanics
Two-Dimensional Boundary Layer Analysis
Integral Form of Boundary Layer Analysis (Continued)
Integrating the momentum equation across the boundary layer. Further
noting that P = Pe across the boundary layer,
∂
∂x
Z
0
δ
bρu 2 dy − ρe ue2
∂δ
∂Pe
+ ρe ue ve + δ
= −τw
∂x
∂x
Arranging and manipulating this statement one arrives at the well-known
momentum integral equation.
1 ∂bρe ue2 θ
∂ue
+ δ ∗ ρe ue
= τw
b ∂x
∂x
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Fluid Mechanics
Two-Dimensional Boundary Layer Analysis
Flow Entrainment into the Boundary Layer
The momentum integral equation is valid for both laminar and turbulent
boundary layers.
Laminar boundary layer analysis usually employs specific boundary layer
flow profile assumptions to permit direct integration of the momentum
integral equation.
Turbulent boundary layer analysis usually employs several empirical models
for solution, which may include specific boundary layer flow profile
assumptions.
Usually turbulent boundary layer analysis employs a second conservation
equation (i.e mass, energy, moment of momentum).
∂
[bρe ue (δ − δ ∗ )] = bρe ue E
∂x
Entrainment Function
E=
O. Tuncer (ITU)
∂δ
ve
−
∂x
ue
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Fluid Mechanics
Two-Dimensional Boundary Layer Analysis
Axisymmetric Three-Dimensional Boundary Layer Analysis
The governing equations for axisymmetric three-dimensional boundary
layer flow in a rotating coordinate system in natural coordinates are,
1 ∂ρWm ∂ρWy
+
=0
r ∂m
∂y
∂Wm
∂Wm sin φ
1
∂Pe
∂τm
2
Wm
+ Wy
−
(Wθ + ωr ) =
fm −
−
∂m
∂y
r
ρ
∂m
∂y
∂Wθ
∂Wθ
sin φ
1
∂τθ
Wm
+ Wy
+
Wm (Wθ + 2ωr ) =
fθ −
∂m
∂y
r
ρ
∂y
Body force terms,
fme = ρe Wme
fθe = ρe Wme
O. Tuncer (ITU)
∂Wme
∂Pe
sin φ
+
−
ρe (Wθe + ωr )2
∂m
∂m
r
∂Wθe
sin φ
ρe Wme ∂rCthetae
+
ρe Wme (Wθe + 2ωr ) =
∂m
r
r
∂m
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Fluid Mechanics
Two-Dimensional Boundary Layer Analysis
Integral Form of Axisymmetric Three Dimensional
Boundary Layer Equations
Boundary layer equations are converted into integral form in the same
manner as described earlier. The resulting integral equations are,
∂
[r ρe Wme (δ − δ1∗ )] = r ρe We E
∂m
∂
∂Wme
2
[r ρe Wme
θ11 ] + δ1∗ r ρe Wme
∂m
∂m
− ρe Wθe sin φ[Wθe (δ2∗ + θ22 ) + 2ωr δ2∗
= r [τmw + fme vm ]
∂ 2
∂Wθe
∗
[r ρe Wme Wθe θ12 ] + r δ1 ρe Wme r
+ sin φ(Wθe + 2r ω)
∂m
∂m
= r 2 [τθw + fθe vθ ]
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Fluid Mechanics
Two-Dimensional Boundary Layer Analysis
Mass, Momentum and Force Defects
ρe Wme δ1∗
Z
δ
(ρe Wme − ρWm )dy
=
0
2
ρe Wme
θ11 =
Z
δ
ρWm (Wme − Wm )dy
0
Z
δ
ρWm (Wθe − Wθ )dy
ρe Wme Wθe θ12 =
0
ρe Wθe δ2∗
Z
δ
=
(ρe Wθe )dy
0
2
ρe Wθe
θ22
Z
δ
ρWθ (Wθe − Wθ )dy
=
0
Z
vm fme
δ
(fme − fm )dy
=
0
Z
vθ fθe
O. Tuncer (ITU)
δ
(fθe − fθ )dy
=
0
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Basic Airfoil Geometry
θ = χ1 + χ2
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σ = c/s
Stagger Angle γ
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κ1 and kappa2
The angles between slopes to the camberline and the axial direction, at
the leading and trailing edges respectively.
Incidence Angle
i = β 1 − κ1
Deviation Angle
δ = β2 − κ2
Angle of Attack
α = β1 − γ
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I
well-defined camberlines such as the circular arc and parabolic arc
camberlines (typical of British practice).
I
American practice is based on NACA airfoils, which typically have
infinite camberline slopes at the leading and trailing edges. A suitable
approximate reference is needed to define κ, χ, θ, i and δ. Common
practice is to use an equivalent circular arc camberline as a reference.
I
Construction of blades from the base camberline and profile is
occasionally a source of confusion. When imposing a profile on a
blade with camber, the thickness distribution data should be
interpreted in terms of dimensionless distance along the camberline
rather than along the chord line.
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November 2011
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Dimensionless Data for Axial Flow Compressor Blades
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NACA 65-Series Profile
NACA 65-Series Profile
I
The NACA 65-series blades are derived from NACA aircraft wing
I
NACA 65-series airfoils are designated by their lift coefficients and
maximum thickness to chord ratio. The lift coefficient in tenths first
appear in parentheses followed by the thickness to chord ratio.
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NACA 65-Series Profile
NACA 65-12 and Equivalent Circular Arc Camberline
Profiles
Relation between the effective camber angle and the lift coefficient.
tan(θ/4) = 0.1103Cl0
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Circular Arc Camberline
Circular Arc Camberline
Commonly used in conjunction with the British C.4 profile. Also the
camberline used for double circular arc profile.
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Circular Arc Camberline
Circular Arc Camberline (Continued)
c/2 = Rc (sin(θ/2)
yc = −Rc cos(θ/2)
y = yc +
p
Rc − x 2
2y (0)/c = [1 − cos(θ/2)]/ sin(θ/2) = tan(θ/4)
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Parabolic Arc Camberline
Parabolic Arc Camberline
Parabolic arc camberline is also used with British C.4 profile and with
others as well.
circular arc. Front, mid and rear loaded blades are all possible. This
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Parabolic Arc Camberline
Parabolic Arc Camberline (Continued)
The point of maximum camber is located at x = a, y = b.
The basic constraints are,
y (0) = 0
y (c) = 0
y (a) = 0
y 0 (a) = 0
The camberline is generated using the general second-order equation.
√
Ax 2 + 2 AE xy + By 2 + Cx + Dy + E = 0
Note that one of the coefficients is arbitrary.
x2 +
O. Tuncer (ITU)
c − 2a
(c − 2a)2 2
c 2 − 4ac
xy +
y
−
cx
−
y =0
b
4b 2
4b
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November 2011
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Parabolic Arc Camberline
Parabolic Arc Camberline (Continued)
To evaluate the blade angles the derivative of the previous expression at
x = 0 and x = c can be used.
tan χ1 = 4b/(4a − c)
tan χ2 = 4b/(3c − 4a)
Parabolic arc camberline in terms of camber and the ratio a/c
q
b/c = [ 1 + (4 tan θ)2 [a/c − (a/c)2 − 3/16] − 1]/(4 tan θ)
Note that,
0.25 < a/c < 0.75s
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British C.4 Profile
British C.4 Profile
I
One of the several profiles in the British C series.
I
With respect to NACA-65 series, C.4 is thicker towards the leading
edge.
I
Maximum thickness at 30% chord.
I
Less effective at higher Mach numbers but higher structural integrity.
I
C.7 profile has more use in compressors and quite similiar to
NACA-65 series.
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British C.4 Profile
Designation of C Series Profiles
I
C-series profiles are designated by a code giving tb , profile, θ,
camberline and a/c.
I
10C4/20P40 is a 10% thick C.4 profile with 20◦ camber angle using a
parabolic arc camberline with a/c = 0.4.
I
10C4/20C50 similiar but with a circular arc camberline.
Note: Well established empirical performance prediction models exist
for C.4 profiles.
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Double Circular Arc Profile
Double Circular Arc Profile
Double circular arc profiles are constructed with both surfaces formed by
circular arcs, that blend with a nose radius r0 applied both at the leading
and trailing edgrs.
∆xU = (RU − r0 ) sin(θU /2) = c/2 − r0 cos(θ/2)
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Double Circular Arc Profile
Double Circular Arc Profile (Continued)
∆yU = Ru − y (0) − tb /2 + r0 sin(θ/2) = RU − d
d = y (0) + tb /2 − r0 sin(θ/2)
The Phythagorean theorem applied to the right triangle requires,
[RU − r0 ]2 = [RU − d]2 + [c/2 − r0 cos(θ/2)]2
Ru =
d 2 − r02 + [c/2 − r0 cos(θ/2)]2
2(d − r0 )
y = r0 sin(θ/2) and x = ±[c/2 − r0 cos(θ/2)] to blend with the circular arc.
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NACA A4 K6 63-Series Guide Vane Profile
NACA A4 K6 63-Series Guide Vane Profile
This vane has excellent flow guidance and a wide incidence operating
range.
The camberline is developed by combining a front-loaded A profile with
Cl0 = 0.4 and a uniform loaded K profile with Cl0 = 0.6, which is
designated as the A4 K6 camberline corresponding to Cl0 = 1. This
geometry is combined with the 6% thick NACA-63 series profile as the
base guide vane geometry. Similiar to 65 series blades, the camberline
coordinates can be scaled directly by lift cooefficient to alternate
camberlines. The thickness distribution can be scaled as well.
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NACA A4 K6 63-Series Guide Vane Profile
NACA A4 K6 63-Series Guide Vane Profile (Continued)
The general vane designation is 63 − (Cl0 A4 K6 )nn , where nn is the
maximum thickness as percent of chord.
The leading and trailing edge camberline slopes are infinite. An equivalent
parabolic arc camberline can be used to provide viable definitions of
tan θ =
O. Tuncer (ITU)
291.5cl0
468.75 − (5.83Cl0 )2
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Controlled Diffusion Airfoils
Controlled Diffusion Airfoils
I
I
I
I
I
I
I
Standart blade profiles are used extensively for axial compressors.
These are well-understood, reliable and can yield excellent
performance if properly applied.
However, many investigators have explored alternatives offering better
Mach number range and efficiency.
This is possible by specifying Mach number distributions on the
surface.
A continuous acceleration along the suction surface and near the
leading edge to avoid boundary layer separation or premature
separation.
The peak Mach number should not exceed 1.3 in order to avoid
shock-wave induced separation.
Carefully controlled deceleration along the suction surface from the
peak Mach number to avoid turbulent boundary layer separation
A nearly constant Mach number distribution on the pressure surface.
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Controlled Diffusion Airfoils
Controlled Diffusion Airfoils (Continued)
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Governs the onset of local flow choking within the blade passage.
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Stagger Angle Parameter
1.5
1.5
φ = γ(1 − 0.05Cl0
) + 5Cl0
−2
Throat Opening to Pitch Ratio
√
√
o/s = [1 − tb σ/c) cos φ] σ
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Extended Throat Opening Correlation
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Once the camberline and profile coordinates are generated along the
chord, the geometry of the staggered blade in the cascade is obtained by a
simple rotation of coordinates to the stagger angle γ.
The staggered blade inlet and discharge angles are given by,
κ1 = χ1 + γ
κ2 = γ − χ2
θ = κ1 − κ2
For circular arc or NACA 65 series equivalent circular arc approximation.
χ1 = χ2 = θ 2
γ = (κ1 + κ2 )/2
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I
Offers a very natural view of cascade fluid dynamics to make it easier
for designers to develop an understanding of the basic flow processes
involved.
I
two dimensional flow on a stream surface in an annular cascade.
I
Two dimensional boundary layer analysis can be used to approximate
viscous effects.
I
Ignores secondary flows.
I
Loses accuracy when significant flow separation is present.
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Coordinate System and Velocity Components
Coordinate System and Velocity Components
Convenient to use a coordinate transformation into new coordinates (ξ, η)
such that blade surfaces correspond to η = constant.
Z
m
dm
0 cos β
η = [θ − θ0 ]/[θ1 − θ0 ]
ξ =
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Coordinate System and Velocity Components
Coordinate System and Velocity Components
r ∂θ
tan β =
∂m
O. Tuncer (ITU)
= tan β0 + [tan β1 − tan β0 ]η
η
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November 2011
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Coordinate System and Velocity Components
Coordinate System and Velocity Components
The following equations relate velocity components to the more usual Wm
and θ components.
Wξ = Wm cos β + Wθ sin β
Wq = Wθ cos β − Wm sin β
Wm = Wξ cos β − Wq sin β
Wθ = Wq cos β + Wz sin β
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Assumptions,
I
I
Inviscid
I
I
Rothaply and entropy constant on the flow plane.
Conservation of Mass
"
2∆m
ρbWq
cos β
−
m,η−∆η
ρbwq
cos β
#
+2∆η[(SρbWm )m−∆m,η −(SρbW
m,η+∆η
where,
S = r (θ1 − θ0 )
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Conservation of Mass (Continued)
Taking the limit,
Continuity
∂ ρbWq
∂(SρbWm
+
=0
∂η cos β
∂m
Irrotational flow in the stream surface requires that the component of
absolute vorticity normal to the stream sheet to be zero.
~ ) = ~en .[∇x(
~ + r ω~eθ )] = 0
~ C
~ W
~en .(∇x
Stokes Theorem
I
~ .d~r =
W
C
O. Tuncer (ITU)
Z
~ )]da = −
~ W
[~en .(∇x
A
Z
~ ω~eθ )]da
~en .(∇xr
A
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Irrotationality (Continued)
Applying to control volume,
"
2 ∆m
Wξ
cos β
−
m,η−∆η
Wξ
cos β
#
m,η+∆η
+ 2 ∆η[(SWθ )m−∆m,η − (SWθ )m+∆m,η ] = 4∆η∆m
S ∂r 2 ω
r ∂r
Taking the limit,
Wξ
∂
∂(SWθ )
+ 2Sω sin φ
=
∂η cos β
∂m
where φ is the stream sheet angle w.r.t the axial direction.
sin φ =
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∂r
∂m
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Stream Function
A strem function Ψ is defined by,
ṁ
∂Ψ
0 − ρb(Wθ − Wm tan β)
∂m
ṁ
Ψ
= SρbWm
∂η
ṁ is the stream sheet mass flow rate.
Velocity components are given by,
Wm =
Wθ =
O. Tuncer (ITU)
ṁ ∂Ψ
Sbρ ∂η
ṁ tan β ∂Ψ ∂Ψ
−
bρ
S ∂η
∂m
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Irrotationality in Terms of Stream Function
Introducing stream function into the vorticity equation
=
∂ ṁ(1 + tan2 β) ∂Ψ ṁ tan β ∂Ψ
−
∂η
Sbρ
∂η
bρ ∂m
∂ ṁ tan β ∂Ψ ṁS ∂Ψ
−
+ 2Sω sin φ
∂m
bρ ∂η
bρ ∂m
This equation can be simplified into,
A
∂2Ψ
∂Ψ
∂2Ψ
∂Ψ
∂2Ψ
+
C
+E
= 2Sω sin φ
−
2B
+D
∂η 2
∂η∂m
∂m2
∂η
∂m
where,
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Irrotationality in Terms of Stream Function (Continued)
Boundary Conditions
Ψ(m, 0) = 0
Ψ(m, 1) = 1
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Irrotationality in Terms of Stream Function (Continued)
Boundary Conditions (Continued)
Periodic boundary conditions on the sides,
Ψ(m, η + 1) = Ψ(m, η) + 1
ρ(m, η + 1) = ρ(m, η)
Wm (m, η + 1) = Wm (m, η)
Wθ (m, η + 1) = Wθ (m, η)
Uniform flow at the upstream and downstream boundaries (This requires
Ψ vary linearly in the tangential direction.
Another choice is to require constant flow angle. If the geometry of the
side boundaries coincide with the local flow angles.
∂Ψ
∂Ψ
= cos β
=0
∂ξ
∂m
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Irrotationality in Terms of Stream Function (Continued)
Boundary Conditions (Continued)
Periodic boundary conditions on the sides,
Ψ(m, η + 1) = Ψ(m, η) + 1
ρ(m, η + 1) = ρ(m, η)
Wm (m, η + 1) = Wm (m, η)
Wθ (m, η + 1) = Wθ (m, η)
Uniform flow at the upstream and downstream boundaries (This requires
Ψ vary linearly in the tangential direction.
Another choice is to require constant flow angle. If the geometry of the
side boundaries coincide with the local flow angles.
∂Ψ
∂Ψ
= cos β
=0
∂ξ
∂m
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Irrotationality in Terms of Stream Function (Continued)
Kutta Condition
Typically β will be assigned to vary uniformly along the side boundary
from the upstream boundary flow angle to the blade leading edge angle,
and analogously for the downstream boundary.
Upstream and downstream flow angles depend upon each other. A
prediction of the downstream flow angle for any upstream flow condition is
required. Therefore an additional constraint is needed.
[W (m, 0)]te = [W (m, 1)]te
Iterative adjustment of the discharge flow angle is needed until the above
condition is met.
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Irrotationality in Terms of Stream Function (Continued)
Finite Difference Approximations
For interior points,
∂Ψ
∂m
∂Ψ
∂η
∂2Ψ
∂m2
∂2Ψ
∂m∂η
O. Tuncer (ITU)
=
=
=
Ψi+1,j − Ψi−1,j
2∆m
Ψi,j+1 − Ψi,j−1
2∆η
Ψi+1,j − 2Ψi,j + Ψi−1,j
(∆m)2
=
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November 2011
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Irrotationality in Terms of Stream Function (Continued)
Finite Difference Approximations
For points at the boundary,
∂Ψ
∂m
∂Ψ
∂η
∂Ψ
∂m
∂Ψ
∂η
O. Tuncer (ITU)
=
=
=
=
4Ψi+1,j − 3Ψi,j − Ψi+2,j
2∆m
4Ψi,j+1 − 3Ψi,j − Ψi,j+2
2∆η
3Ψi,j − 4Ψi−1,j + Ψi−2,j
2∆m
3Ψi,j − 4Ψi,j−1 + Ψi,j−2
2∆η
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Irrotationality in Terms of Stream Function (Continued)
Algebraic Form of the PDE for Interior Points
Ψi,j
+ Ãi,j Ψi−1,j + B̃i,j Ψi+1,j + C̃i,j Ψi,j−1 + D̃i,j Ψi,j+1
+ Ẽi,j [Ψi+1,j+1 − Ψi+1,j−1 − Ψi−1,j+1 + Ψi−1,j−1 ] = Q̃i,j
where,
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Irrotationality in Terms of Stream Function (Continued)
Coefficients
Ãi,j
=
B̃i,j
=
C̃i,j
=
Ei,j
Ci,j
− 2∆m
(∆m)2
− 2A
2Ci,j
i,j
+ (∆m)
2
(∆η)2
Ci,j
Ei,j
+ 2∆m
(∆m)2
− 2A
2Ci,j
i,j
+ (∆m)
2
(∆η)2
Ai,j
Di,j
− 2∆η
(∆η)2
− 2A
2Ci,j
i,j
+ (∆m)
2
(∆η)2
O. Tuncer (ITU)
Axial Compressor
Di,j
Ai,j
+ 2∆η
(∆η)2
− 2A
2Ci,j
i,j
+ (∆m)
2
(∆η)2
Bi,j
2∆m∆η
2Ai,j
2Ci,j
+ (∆m)
2
(∆η)2
D̃i,j
=
Ẽi,j
=
Q̃i,j
= −
2sω sin φ
2Ai,j
(∆η)2
+
2Ci,j
(∆m)2
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Solution Procedure
I
A relaxation technique or a matrix method can be used for solution.
I
After each solution for the stream function density field must be
updated. Rothalpy is constant on the stream sheet,
1
1
1
h = H 0 − W 2 = I + (r ω)2 − W 2
(3)
2
2
2
Since entropy is also constant, all thermodynamic properties can be
calculated from (h,s). Density can be calculated using an appropriate
equation of state.
I
As long as the flow is subsonic this lagging density solution offers very
good numerical stability and rapid convergence.
I
If the flow is transonic or supersonic other solution procedures are
needed as the PDE is no longer elliptic in nature.
I
Stability of numerical solution is significantly influenced by the grid
structure.
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Potential Flow Results
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Transonic Potential Flow Results
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Linearized Potential Flow Analysis
Linearized Potential Flow Analysis
Motivation
The real purpose of the linearized method is its use in a
are conducted on several stream sheets and must be repeated many times.
Linearization of the Stream Function
Ψ(m, n) = a(m)[η − η 2 ] + η 2
This is tantamount to assuming ρbWm vary linearly with η.
Note that W = Wξ on the blade surface.
W1
W0
−
=
β1
cos β0
O. Tuncer (ITU)
Z
0
1
∂SWθ
+ 2Sω sin φ
∂m
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Linearized Potential Flow Analysis
Linearized Potential Flow Analysis (Continued)
The velocity normal to the blade surfaces must be zero,
W =
Wm
1 ṁ ∂Ψ
=
cos β
cos β Sbρ ∂η
Combining,
W1
cos β1
W0
cos β0
=
=
ṁ(2 − a)
Sbρ cos2 β1
ṁa
Sbρ cos2 β0
SWθ = ṁ[tan β(a − 2aη + 2η) − a0 S(η − η 2 )]/(bρ)
where the prime denotes total derivative w.r.t. m.
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Linearized Potential Flow Analysis
Linearized Potential Flow Analysis (Continued)
In order to simplify the analysis we define,
u(m, η) = ṁ tan β/(bρ)
v (m, η) = ṁS/(bρ)
Differentiating the previous eqaution and substituting new variables,
∂SWθ
∂u
∂v 0
=
(a − 2aη − 2η) + (1 − 2η)ua0 − (va00 +
a )(η − η 2 )
∂m
∂m
∂m
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Linearized Potential Flow Analysis
Linearized Potential Flow Analysis (Continued)
Using truncated Taylor series expansion for any function F (η), for values
at 0, 0.5, and 1, a three-point difference approximation to the integral is
obtained.
Z
1
F (η)dη = (F0 + 4F̄ + F1 )/6
0
where overbar denotes the function value at η = 0.5. Therefore,
Z
0
1
∂SWθ
dη = [au00 + u0 a0 + 4ū 0 − v̄ a00 − v̄ 0 a0 + u10 (2 − a) − u1 a0 ]/6
∂m
Combining,
a00 + Aa0 + Ba = C
where A,B, and C are functions of m only.
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Linearized Potential Flow Analysis
Linearized Potential Flow Analysis (Continued)
A(m) = [v̄ 0 − u0 + u1 ]/v̄
B(m) =
u10 − u00
v0
6
v1
+
−
v̄
v̄ S 2 cos2 β1 cos2 β0
C (m) =
2u10 + 4ū 0 + 12ω sin φ
12v1
−
2
v̄
v̄ S cos2 β1
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Linearized Potential Flow Analysis
Linearized Potential Flow Analysis (Continued)
The leading edge boundary condition follows from the known inlet angular
momentum supplied by the upstream flow, Wθ,in .
Integrating Wθ across the passage at the leading edge,
a0 + a[u1 − u0 ]/v̄ =
Kutta Condition
Kutta condition is used as the trailing edge boundary condition (i.e.
W0 = W1 ).
a = 2cosβ0 /[cos β1 + cos β0 ]
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Linearized Potential Flow Analysis
Solution Method
I
The governing equations are cast into matrix form, Ax = B.
I
Matrix A is in tri-diagonal form. Except for the equation at the
I
Gas density field update is lagging the velocity field. Same procedure
as before.
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Linearized Potential Flow Analysis
Tri-Diagonal Systems
Tri-diagonal systems commonly arise in the solution of many engineering
problems. Ordinary matrix inversion for a tri-diagonal system with n
unknowns requires O(n3 ) operations. Such an approach is unnecessarily
computationally intensive. “Thomas Algorithm”only requires O(n)
operations. A tri-diagonal system can be written as,
ai xi−1 + bi xi + ci xi+1 = di , with, a1 = 0, cn = 0
This system can be cast into matrix Ax = b form with,





b1 c1 0 · · ·
0 0
x1
 a2 b2 c2 · · · 0 0 



 x2 




b3
c3 0 
A =  0 a3
 , x =  ..  , b = 



 . 
..
..
 0 ...
.
. 0 
xn
0 0
···
an bn
d1
d2
..
.





dn
Note that in matrix A all entries are zero except the ones in the diagonal,
the super-diagonal and the sub-diagonal.
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Linearized Potential Flow Analysis
Thomas Algorithm
The solution of this system is performed in two steps as per the Thomas
Algorithm. First step involves modifying the coefficient vectors.
(c
1
i =1
ci0 = b1 ci
i = 2, 3, · · · , n − 1
0 a
bi −ci−1
i

 d1
i =1
b
0 a
di0 = d1i −di−1
i
 b −c 0 a i = 2, 3, · · · , n − 1
i
i
i−1
After the new coefficients are obtained solution is reached through back
substitution.
(
i =n
dn0
xi =
0
0
di − ci xi+1 i = n − 1, n − 2, · · · , 1
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Linearized Potential Flow Analysis
Listing 1: Source Code to Implement the Thomas Algorithm
i n t Thomas ( const double ∗a , const double ∗b , double ∗ c
double ∗d , double ∗x , i n t n ) {
int i ;
/∗ Mo di fy t h e c o e f f i c i e n t s . ∗/
c [ 0 ] /= b [ 0 ] ;
d [ 0 ] /= b [ 0 ] ;
f o r ( i = 1 ; i < n ; i ++){
double i d = 1 . 0 / ( b [ i ] − c [ i − 1 ] ∗ a [ i ] )
c [ i ] ∗= i d ; /∗ L a s t v a l u e i s r e d u n d a n t
d [ i ] = (d [ i ] − d [ i − 1]∗ a [ i ])∗ id ;
}
/∗ Back s u b s t i t u t i o n ∗/
x [ n − 1] = d [ n − 1 ] ;
f o r ( i = n − 2 ; i >= 0 ; i −−)
x [ i ] = d [ i ] − c [ i ]∗ x [ i + 1 ] ;
return 0;
}
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Linearized Potential Flow Analysis
Linearized Potential Flow Results
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Time Marching Method
Time Marching Method
I
Potential flow method has limitations in terms of Mach number.
I
Time marching method provides a more general solution capability.
I
It is applicable to subsonic, transonic and supersonic flows.
I
Suggested first by von Neumann and Richtmayer (1950).
I
Solution is accomplished using the same velocity components and
coordinate system.
I
The governing equations are solved in their full unsteady form.
I
Solution is advanced in time until variations is time become negligible.
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Time Marching Method
Time Marching Method (Integral form of Conservation
Equations)
Mass
Z
V
∂ρ
dV +
∂t
Z
~ .~n)dA = 0
ρ(W
A
Momentum
Z
V
~
∂ρW
dV +
∂t
Z
~ (W
~ .~n)dA +
ρW
Z
A
Z
P~e (~e .~n)dA =
A
~f dV
V
Energy
Z Z
Z
∂H 0 ∂P
~ .~n)dA =
~ )dV
ρ
−
dV +
ρH 0 (W
ρ(~f .W
∂t
∂t
V
A
V
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Time Marching Method
Time Marching Method (Integral form of Conservation
Equations)
~e is the unit vector along w
~.
~f is a body force used to account for the Coriolis and centrifugal
acceleration terms in the rotating curvilinear system.
Equations in conservative form,
Sb
Sb
∂ρWm
∂t
+
=
O. Tuncer (ITU)
∂ρ
∂
∂
+
[SbρWm ] +
[bρQ] = 0
∂t
∂m
∂η
∂
∂bP
∂
[Sb(ρWm2 + P)] − tan β
+
[bρQWm ]
∂m
∂η
∂η
1
∂Sb
SBρ sin φ(Wθ + r ω)2 + P
r
∂m
Axial Compressor
November 2011
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Time Marching Method
Time Marching Method (Integral form of Conservation
Equations)
Sb
∂ρWθ 1 ∂
∂
∂
+
[Sb(ρWm (Wθ +r ω)]+ [b(QWθ +P)] = r ω
[SbρWm ]
∂t
r ∂m
∂η
∂m
Sb
∂(ρI − P)
∂
∂
+
[SbρWm I ] +
[bρQI = 0
∂t
∂m
∂η
Q is a special velocity component to conserve properties at the constant η
boundaries of the control cell.
Q = Wq / cos β = Wθ − Wm tan β
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Time Marching Method
Time Marching Method (Integral form of Conservation
Equations)
Note that Q = Wq = 0 on the blade surface. For these points there is only
Wξ component of velocity.
Applying the integral momentum equation in the ξ-direction.
Sb
∂ρWξ
∂t
∂
∂
[Sb(ρWm Wξ + P cos β)] +
[bρQWξ ]
∂m
∂η
∂
[Sb cos β] + Sbρ sin φ cos βr ω 2
= P
∂m
+
Wm = Wξ cos β
Wθ = Wξ sin β
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Time Marching Method
Time Marching Method (Boundary Conditions)
I
On the blades velocity normal to the surface is zero.
I
For side boundaries outside the blade passage periodic boundary
conditions are used (as before)
I
Upstream and downstream boundary conditions are rather tricky.
I
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Time Marching Method
dm
dt
dm
dt
dm
dt
O. Tuncer (ITU)
= Wm + a
= Wm − a
= Wm
Axial Compressor
November 2011
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Time Marching Method
Subsonic Upstream Boundary
At t + ∆t one of the characteristics is inside the domain.
A logical choice for the computed dependant variable is density.
P, Wm from equation of state and rothalpy (if entropy is known).
At the upstream boundary specify Wθ , Pt , Tt .
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Time Marching Method
Supersonic Upstream Boundary
All characteristics outside the solution domain.
All dependant variables must be assigned when Wm > a on an upstream
boundary.
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Time Marching Method
Subsonic Downstream Boundary
At t + ∆t one of the characteristics is inside the domain.
Specify discharge static static pressure as the boundary condition.
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Time Marching Method
Supersonic Downstream Boundary
All characteristics inside the solution domain.
All dependent variables can be computed from the solution without any
boundary condition specification.
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Time Marching Method
Numerical Stability
I
No Kutta condition is needed!
I
Specifiying more boundary conditions WILL NOT PRODUCE A
VALID SOLUTION. WILL CAUSE THE SOLUTION TO DIVERGE!!!
I
Explicit solution scheme numerically unstable.
I
Stabilizing terms (like artificial viscosity) is needed in governing
equations if an explicit solution is sought.
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Time Marching Method
Stability Analysis
ut = v (ξ, η, t) + µ(ξ) uξξ + µ(η) uηη
For a stable solution the coefficients must satisfy the following conditions.
µ(ξ) ≥
µ(η) ≥
µ(ξ) ≥
µ(η) ≥
O. Tuncer (ITU)
1
(|Wm | + a)2 ∆t
2
1
(|Wθ | + a)2 ∆t
2
1
[(|Wξ | + a) cos β]2 ∆t
2
1
[(|Wq | + a)/ cos β]2 ∆t
2
Axial Compressor
November 2011
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Time Marching Method
Stability Analysis (Continued)
If the grid structure is highly skewed and node spacing in the tangential
direction is much finer than the meridional direction an additional
meridional stabilizing term can be added.
1
µ(ξ) → µ(ξ) [(|Wθ | + a) sin2 β(∆m)/(S∆η)]2 ∆t
2
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Time Marching Method
Stability Analysis (Continued)
CFL Criterion
∆tmax
≤
∆tmax
≤
∆m
|Wm | + a
∆S∆η
|Wθ | + a
Actual time step is,
∆t = µ0 ∆tmax
A rule-of-thumb is,
0.1 ≤ µ0 ≤ 0.9
O. Tuncer (ITU)
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Time Marching Method
Stability Analysis (Continued)
If the time derivative is approximated by a forward time derivative,
u(ξ, η, t + ∆t) = u(ξ, η, t) + [v (ξ, η, t) + µ(ξ) uξξ + µη uηη ]∆t
I
Stabilizing terms second order w.r.t ∆t
I
Dynamic terms of the original PDE are first order w.r.t ∆t
I
I
Reduce the time step as the solution converges
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Time Marching Method
Time Marching Solution Results
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Time Marching Method
Transonic Time Marching Solution Results
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Motivation
I
Quantify viscous effects
I
Predict flow seperation
I
Predict the level of total pressure loss
Momentum Integral Equation
1 ∂bρe ue2 θ
∂ue
+ δ ∗ ρe ue
= τw
b ∂x
∂x
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Boundary Layer Analysis (Continued)
Momentum Thickness
ρe We2 θ
Z
ρW (We − W )dy
=
Displacement Thickness
∗
Z
δ
(ρe We − ρW )dy
ρe We δ =
0
Skin Friction Coefficient
Cf =
O. Tuncer (ITU)
τw
1
2
2 ρe We
Axial Compressor
November 2011
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Boundary Layer Analysis of Grushwithz (1950)
Boundary layer initially laminar, soon transitions into turbulence.
Universal Boundary Layer Profile
W
= C1 η + C2 η 2 + C3 η 3 + C4 η 4
We
Z
1 y ρ
δ 0 0 ρe
Z δ
ρ
=
0 ρe
η =
δ0
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Boundary Layer Analysis of Grushwithz (1950)
Shape Factor
Λ=
ρ2e (δ 0 )2 dWe
ρw µ dx
Matching Edge Conditions
C1 = 2 + Λ/6
C2 = −Λ/2
C3 = Λ/2 − 2
C4 = 1 − Λ/6
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Boundary Layer Analysis of Grushwithz (1950)
Momentum Thickness
Λ
Λ2
θ
37
−
−
=
δ0
315 945 9072
Energy Thickness
δE
=
δ0
Z
0
δ
ρW
W2
798048 − 4656Λ − 758Λ2 − 7Λ3
1 − 2 dy =
ρe We
We
4324320
Velocity Thickness
δW
=
δ0
O. Tuncer (ITU)
Z
0
δ
ρW
W
3
Λ
FWe2
−
+
1−
dy =
ρ e We
We
10 120 2cp Te
Axial Compressor
November 2011
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Boundary Layer Analysis of Grushwithz (1950)
Velocity Thickness
Λ 3
Λ
Λ 2
+ 17.8063
F = 0.232912 − 0.831483
+ 0.650584
100
100
100
Enthalpy Thickness
For adiabatic walls with Pr = 1
δh
=
δ0
Z
0
δ
ρW
ρe We
h
− 1 dy =
he
Displacement Thickness
δ ∗ = δh + δw
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Boundary Layer Analysis of Grushwithz (1950)
Following parameters introduced for convenience,
b0 =
ρe
T0
= t
ρ
Te
2
θ
ρe θ2 dWe
K = Λ 0 = b0
δ
µ dx
Gruschweitz (1950) shows that,
1
µ
Λ
cf =
1
+
2
ρe We δ 0
6
37
Λ
Λ2
K=
−
−
315 945 9072
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2
Λ
November 2011
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Boundary Layer Analysis of Grushwithz (1950)
Transition Criteria
<θ =
ρ e We θ
> 250
µ
Once boundary layer transitions into turbulence different methods are
needed.
Originally developed for incompressible boundary layers.
∂
[bρe We (δ − δ ∗ )] = bρe We E
∂x
Empirical relations are needed for E and cf as a function of θ and (δ − δ ∗ ).
O. Tuncer (ITU)
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November 2011
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Boundary Layer Analysis of Grushwithz (1950)
Shape Factors
H1 ≡ (δ − δ ∗ )/θ
H = δ ∗ /θ
Kinematic Shape Factor
Recommended by Green (1968)
1
Hk =
θ
δ
Z
0
ρ
ρe
W
1−
We
dy
For adiabatic walls with Pr = 1
H = (Hk + 1)Tt0 /Te − 1
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November 2011
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Boundary Layer Analysis of Grushwithz (1950)
Aungier 2003 recommends,
Hk = 1 + [0.9/(H1 − 3.3)]0.75
E = 0.025(Hk − 1)
Skin friction coefficient correlation of Ludwieg and Tillmann (1950)
commonly used for incompressible turbulent boundary layer analysis.
cf ,inc = 0.246exp(−1.561Hk )Reθ−0.268
Green’s correction to it,
cf = cf ,inc
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Boundary Layer Analysis of Grushwithz (1950)
At the transition (laminar to turbulent) point,
(δ − δ ∗ )turb = (δ − δ ∗ )lam
θturb = θlam
From Gruschwitz (1950) profiles,
δ − δ∗ = δ0
7
Λ
+
10 120
Seperation Criterion for Turbulent Boundary Layers
Hk ≥ 2.4
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Boundary Layer Analysis of Grushwithz (1950)
Useful to predict total pressure loss coefficient for the cascade from the
B.L. data at the trailing edge.
Total pressure loss coefficient based on cascade inlet velocity (Lieblein and
Roudebush, 1956), where the summation is carried out for the boundary
This loss coefficient can be used to estimate the rotor efficiency.
cos βin 2 2Θ + (∆∗ )2
cos βout
(1 − ∆∗ )2
P
θ
Θ=
S cos βout
P ∗
δ
∗
∆ =
S cos βout
∆Pt
ω̄ =
=
(Pt − P)in
Loss coefficients are approximate, since the analysis method ignores
secondary flows.
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