Ch9
9.1 Introduction
Linearized Flow
up to the middle 1950s, before CFD comes
A uniform flow is changed, or perturbed, only slightly

Small-perturbation theories
1. Frequently (but not always) linear theory, e.g. acoustic
theory in Sec. 7.5.
2. Highlighting some important physical aspects of the flow,
explicitly identifying trends and governing parameters,
providing practical formulas for the rapid estimation of
aerodynamic forces and pressure distributions.
-A slender body immersed in a uniform flow
9.2 Linearized Velocity Potential Equation
V  Vx i  Vy j  Vz k
Vx  V  u '
Vy  v '
where u ' , v ' , w ' denotes velocity
perturbations from the uniform flow
Vz  w '
  V  (V  u ')i  v ' j  w ' k
total velocity potential
Define a new velocity potential － perturbation velocity potential 



 v'
 u '，
 w'
，
y
x
z


Vx 
 V x  u '  V x 
then
x
x
( x, y, z)  V x   ( x, y, z) ，where
 

 v'
y y
 
Vz 

 w'
z z
Vy 
 2
 2
 xx  2  xx  yy  2
，
y
x
substitute into
  2y
  2x 
1  2   xx  1  2
a 
a


Also，
 2
  yy
 zz  2  zz
，
z

  2z 
  yy  1  2   zz
a 


 x y
 y z
 x z
2 2  xy  2 2  xz  2 2  yz  0
a
a
a

2
2
2
2
2
2


 2 





   










2
2
a

V


a


a


   2 
 
  2 
   2
x   x 

 z   z
 y   y 

    2
    2
   2


2  V  
 2  V  
2
0
x  y xy
x  z xz
y z yz


－ perturbation-velocity potential equation
or
 a 2  V  u '2  u '   a 2  v '2  v '   a 2  w '2  w '
 y 
 z

 x 
v '
u '
v '
2 V  u ' v '
 2 V  u ' w '
 2v ' w '
0
y
z
z
h0  const. throughout the flow
2
2
2
V  u '  v '2  w '2

V
V
h0  h 
 h
 h
2
2
2
or
2
2
2
2
2
2
V

u
'

v
'

w
'


a V
a


 
 1 2  1
2
a a 
2
2

 1
2
2
2
2
2
u
'
V

u
'

v
'

w
'
 

－ (**)
Substitute (**) into (*), and algebraically rearranging
－ (*)
1  M 2 
u ' v ' w '


x y z
linear

u '    1  u '2    1   v '2  w '2   u '
 M    1  

 2 

2
V
2
V
2
V

  



  x


u '    1  v '2    1   w '2  u '2   v '
2
 M     1  

 2 

2
V
2
V
2
V








  x


u '    1  w '2    1   u '2  v '2   w '
2
 M     1  

 2 

2
V
2
V
2
V








  x

2

nonlinear
 v '  u '   u ' v '  w '  u '   u ' w '  u ' w '  w ' v '   v '
 M 2  1   



  1   



V
V

y

x
V
V

z

x
V

y

z


 
  x
 
 
 
 
－ an exact equation for irrotational, isentropic flow
Now specialize to the case of small perturbation, i.e. , u ' , v ' , w ' are
small compared to V
u'
V
v'
， V
w'
 1
， V
2
2
2
 u '   v'   w' 
       1
 V ，  V  ， V 

1. 0≦M∞≦0.8 and M∞≧1.2 （ transonic flow (0.8≦M∞≦1.2) is excluded ）
u'
2 
M


1
 
the magnitude of
 
V

2. M∞≦5 (approximately)

u'
M 2    1 
V

 u '
2 u '

1

M
   x


x

（ hypersonic flow (M∞≧5 ) is excluded ）
 u '
2 u '

1

M
   x


x


 v '
u'
u'
v '
2 
M    1 
M    1  

V
V
y ，


 y


v' 
u '   u ' v ' 
M 2    1 1   


  1 (  0)

V  V   y x 


2


1  M  ux'  vy'  wz '  0
or
 2  2  2
1  M  x2  y2  z 2  0
 w '
w '


z
 z
2

2

approximate equations are valid for subsonic
＆ supersonic flow only
Note： The real physical problems associated with
1. transonic flow：mixed subsonic –supersonic regions
with possible shocks, and extreme sensitivity to
geometry changes at sonic conditions.
2. hypersonic flow：strong shock waves closed to the
geometric boundaries, i.e., thin shock layers, as well
as high enthalpy, and hence high-temperature
conditions in the flow.
9.3 Linearized Pressure Coefficient
p  p
Pressure coefficient： C p 
1
V2
2
2

p
V
1
1



V2 
V2  p 2  p M 2
2
2  p
2
a 2

 p

p 
 1
p


Cp 

p M 2
2
V2
V2
h
 h 
2
2


2  p
Cp 
 1
2 
 M   p 

V2
V2
T
 T 
2c p
2c p
V2  V 2
V2  V 2
T  T 

2c p
2 R /   1
2
2
2

V

V

u
'

v
'

w
'


T
  1 V  V
  1 V  V
 1   

1 


T
2  RT
2
a2
2
a2
2

2
2
2
2
T
 1
 1  2  2u 'V  u '2  v '2  w '2 
T
2a

p  T   1
isentropic,
 
p  T 


 1

p   1
 1  2  2u 'V  u '2  v '2  w '2  
p  2a 


   1 2  2u ' u '2  v '2  w '2    1
 1 
M 


2
2
V
V
 



－exact
2
2
2
v
'
w
'
u
'
u'
 1
Consider small perturbations：
 1
2
2
2
， V  ， V  ， V
V

p



1
 1     1 

p
 1
p
 2  2u ' u '2  v '2  w '2 
 1 M 


2
p
2
V
 V


2   2  2u ' u '2  v '2  w '2 
Cp  
1 M 

   1
2 
2
 M  2
V
 V





2u ' u '2  v '2  w '2



2
V
V
2u '
Cp  
－linearized pressure coefficient
V
valid for small perturbations
depends only on x-component of the
perturbation velocity
9.4 Linearized Subsonic Flow
-Take incompressible results (theory or experiment) and modify
them to take compressibility into account.
2-D flow over an airfoil
-Applied for any 2-D shape, including
Flow over a bumpy or wavy body
which satisfies the assumptions of small perturbations.
Consider the compressible subsonic flow over a thin airfoil at
small angle of attack (i.e. small perturbations)
－inviscid flow boundary condition holds at the surface ,
i.e. V at surface // to the surface

if


df
v'

 tan 
dx V  u '
u '  V
tan   
df
v'


dx V

df
 V
y
dx
for subsonic compressible flow over a 2-D airfoil
 2xx  yy  0 ；   1  M 2
transformed to a familiar incompressible form by
considering a transformed coordinate system
x
 ,  ，
 y
  ,     x, y 
－transformed perturbation
velocity potential





0

1
0
， y
， x
， y
x
 1  1       1  
x 

 



x  x    x  x    
1
xx  

 1  1       
y 

 

 

y  y    y  y  
 yy   

2
2 1
 xx  yy        


    0 Laplace’s equation for incompressible flow




 represents an incompressible flow in  , 
space, which is related to compressible flow 
in (x, y) space
The shape of airfoil is given by y=f(x) and   q   in (x, y)
and  ,  space, respectively.
df  1  
V



dx y  y 
dq 
df dq
V


d 
dx d 



The shape of the airfoil in (x, y) ＆ ,  space is
the same.

The compressible flow over an airfoil in (x, y)
space transforms to the incompressible flow over
the same airfoil in  ,  space.
2u '
2 
2 1 
2 1 
Cp  



V
V x
V  x
V  
Denoting the incompressible perturbation velocity in the  direction
by u , where u   / 
1  2u 
Cp    
  V 
Cp 
CL 
CM 
C p0
1  M 2
L
1
V2 S
2
M
1
V2 Sl
2
2u －incompressible pressure
Cp0  
V coefficient in  ,  space
Prandtl-Glauert rule
L：Lift force is perpendicular to the V∞
S：a reference area, for a wing, usually
the platform area of the wing
l：a reference length, for an airfoil,
usually the chord length
CL 
CL 0
1  M 2 ，
CM 
CM 0
1  M 2
－Prandtl-Glauert rule
valid up to M∞≒0.7
An important effect of compressibility on subsonic flowfields
 1  1  u
u
u'


 
x  x   
1  M 2
 M   ， u '  －Compressibility strengthens the
disturbance to the flow introduced by a solid
body！
c.f.
incompressible flow
－A perturbation of given strength reaches further away
from the surface in compressible flow.
－The spatial extent of the disturbed flow region is
increased by compressibility.
－The disturbance reaches out in all directions, both
upstream and downstream.
In classical inviscid incompressible flow theory
d’Alembert’s paradox：a 2-D closed body experiences no
aerodynamic drag.
∵ No friction and its associated separated flow.
∴the pressure distributions over the forward and rearward portions
of the body exactly cancel in the flow direction.
Cp 
C p0
1  M 2
∴d’Alembert’s paradox can be generalized to
include subsonic compressible flow as well as
incompressible flow.
Similar results are obtained from nonlinear subsonic
calculations (thick bodies at large angle of attack)
Ex 9.1
An uniform upstream M∞ flow over a wavy wall yw  h cos  2 x / l 
Using the small perturbation theory, derive an equation for  ＆ C p
sol：
Assume h / l  1
2
2
d
F
d
G
2
1  M   G dx2  F dy 2  0
1 d 2F
1
1 d 2G

0
2
2
2
F dx
1  M   G dy

f(x) only
f(y) only
1 d 2F
2



k
F dx 2
1
1 d 2G
2


k
2
2
G
dy
1

M
 

d 2G
2
2

k
1

M

 G  0
2
dy
Ae
 G( y)  A e
 F (x)  B sin kx  B cos kx
 k 1 M 2 y
k 1 M 2 y
1
d 2F
2

k
F 0
2
dx
1
2
2
A1, A2, B1, B2 are determined by BCs.
1. y→∞ , V (  ) remains finite. →A2=0
→   x, y    B1 sin kx  B2 cos kx  A1e
2.
dyw
v 'w
v 'w 1   


  
dx V  u 'w V V  y w
  
 2
   V h 
 l
 y w
  2 x    
 sin 
 
  l   y  y 0
 k 1 M 2 y
  
2 k

B
sin
kx

B
cos
kx
A

k
1

M
 1 
2
e
   1
 y 
1 M 2 y
  
2


A
k
1

M
1
  B1 sin kx  B2 cos kx 
 
 y  y 0
 2   2 x 
 V h 
 sin 

l
l

 

V h
 2 
2
2
A1B1k 1  M   V h 
 B2  0, k 
 , A1B1 
l
 l 
1  M 2
 2 1  M 2   2 x 
V h

  x, y  
exp 
y  sin 



l
l
1 M
＃

 
2



2

1

M
2

x



y

 cos 
 exp 

l

 l 


2



2

1

M
2u '
4  h 
 2 x 
y


Cp  

exp
cos


2  l 


V
l
l


1 M  


2u '
4  h 
 2 x 
C pw  

  cos 

2
V
l
l


1 M  
＃
V h

u' 

x
1  M 2
 2

 l
－the same cosine variation as
the shape of the wall, but is
180˚ out of phase.
－symmetrical distribution
∴ no net force in x-direction.
no drag.
2



2

1

M
y
1

0

exp 


l
1  M 2


as y  0
C pw 
C 
C 
pw M
1
pw M
2
1
1  M 2
1  M 2 2

1  M 21
if incompressible

C pw 
M∞≈0
C 
pw 0
1  M 2
9.5 Improved Compressibility Corrections
－linearized solutions are influenced predominantly by free-stream
conditions; they do not fully recognize change in local regions of
the flow

nonlinear phenomena

Improved compressibility correction
－Laitone
Cp 
local pressure coefficient

Cp 
Cp0
1 M 2
local Mach No., can be related to M∞
Cp0
 2   1 2 
2 
1  M   M  1 
M   / 2 1  M   C p0
2




2
Cp0<<1
Cp 
C p0
1  M 2
－Karman and Tsien：hodograph
solution of the nonlinear equations of
motion along with a simplified “tangent
gas” equation of state.
C p0
Cp 
C p0
2
2
2
1 M  M / H 1 M
2


－Karman-Tsien rule

9.6 Linearized Supersonic Flow
Linearized perturbation-velocity potential equation for 2-D flow
2
2
－elliptic P.D.E.


1

M
 xx  yy  0 for subsonic flow

 xx  yy  0 for supersonic flow
2

 
M 2

1－hyperbolic P.D.E.
Consider the supersonic flow over a body or surface which introduces
small changes in the flowfield, e.g. flow
over a thin airfoil,
over a mildly wavy wall,
over a small hump in a surface.
 2xx  yy  0 －wave equation
  f x  y   g x  y  if g=0 ,   f x  y  －lines of
 =const. correspond
to x-λy=const. (left-running Mach
lines)
dy 1
1
 
dx 
M 2  1
if f=0,   g x  y 

  arcsin1 / M    arctan1 / M 2  1


－lines of  =const. correspond
to x+λy=const. (right-running
Mach lines)
dy
1
1
 
－disturbances propagate along Mach lines. ∴the
2
dx

M   1 flowfield upstream of a disturbance does not feel
the presence of the disturbance.
c.f.
M∞<1, disturbances propagate everywhere in the
flowfield.( upstream ＆ downstream )
Letting g=0 ,   f x  y 


 f '
u' 
 f ' v' 
， y
x
 u'  
v'

B.C. on the surface
dy
v'
tan 


dx V  u '
 u'  
 v'  V
u' V 
V

2u ' 2
Cp  

V

 Cp 
2
M 2  1
Cp   (local surface inclination
w.r.t. V∞)
1
Cp 
 M  , C p 
2
M  1 ，
C pA 
2 A
M 1
2

( )
，
C pB 
2 B
M 1
2

()
consistent with
 a net pressure imbalance  a drag (wave drag)
For the upper surface,   g x  y 
 2
Cp 
M 2  1
Note：
Although shock waves do not appear explicitly within the
framework of linearized theory, their consequence in terms of
wave drag are reflected in the linearized results.
d’Alembert’s paradox does not apply to supersonic flows！！
For M∞=2, linearized theory
yields reasonably accurate
results for Cp whenθ<4°
Note：CL ＆ CD are more
accurate at large angle of
attack then one would
initially expect.
Ex. 9.2
An uniform supersonic flow over the same wavy wall in ex. 9.1
 ？ Cp？
sol：
 2
1  2
 2
0
2
2
x
M   1 y
 

  f x  M 2  1 y  g x  M 2  1 y
Let g=0
  f x  M 2  1 y

B.C.



 
 
 f ' x  M 2  1y   M 2  1

y 
dy

 2   2x 
 V w  V h
 sin

y
dx
 l   l 
dyw v'w 1   
 2x 

  
yw  h cos

dx V V  y  w
 l 

 2   2x 
M  1 f ' x   V h
 sin 
 y0
 l   l 
1
 2   2x 
f ' x  
V h  sin

2
 l   l 
M  1
2

 2x 
f x  
cos
  const.
2
M  1  l 
 V h




 2

2
 x, y   f x  M  1y 
cos
x  M   1y   const.
2
 l

M  1
＃
Cp  
2

 V h


2u '
2 
4  h   2

2


sin
x

M

1
y
  


2
V
V x
l
l

M  1   
4
 h   2x 
Cp  
  sin

2
M   1  l   l ＃
Note：
1. The perturbations do not disappear
c.f.
at y→∞ ↔ subsonic
2
x

M
 1y  const .  C p  const . ,

2.
  const. ← characteristic lines
dy

dx
 1 

 tan     sin 1 
2
M  1
 M 
1
3. unsymmetrical streamlines
4. Cpw is a sine variation, whereas the
wall is a cosine shape →a net force
in the x-direction.(wave drag)
9.7 Critical Mach Number
Mc ≡M∞ at which sonic flow is first
encountered in the airfoil
Assuming isentropic flow,

   1 2   1
1
M 
PA 
2


P  1    1 M 2 
A

2

2
Cp 
 M 2
C pA





1


1
2 
 1 

M



2 

2


1



 M 2  1    1 M 2 
A


2



 P

  1
 P

MA=1
M∞=Mcr
 
 C pA  C pA
 C pcr
min

 1    1 M cr2
2 
2


 M cr2  1    1

2

+ Prandtl-Glauert rule
(or Laitone rule
or Karman-Tsien rule)
Mcr for a given airfoil






 1


 1



－derived from fundamental
gas dynamics, independent of
the size or shape of the airfoil.
Cp0
(measured or
calculated)
different for
different
airfoil
Thin airfoil
c.f.
Thick airfoil
mild expansion over
the top surface
stronger expansion
Cp0 small ( curve B
is low )
Cp0 larger ( curve B
is higher )
Mcr large
Mcr lower
∴An airfoil designed for a high Mcr must have a thin
airfoil.
1 > Drag-divergence Mach Number, MDD > Mcr
At MDD, the drag (CD) is
massively increased.
when M∞ > Mcr
“sonic barrier”
before 1947
The total pressure loss associated with the weak λ-shock will
be small, however, the adverse pressure gradient induced by
the shock tends to separate the boundary layer on the top
surface, causing a large pressure drag.  CD ↑ dramatically
Mcr ↑  MDD ↑ ( MDD ↑ more by “supercritical” airfoil)
ch 14
Two ways to increase Mcr：
(1)
t
↓
c
( thinner airfoil )
t
 0.09, M DD  0.88
c
t
 0.04  0.06, M DD  1
﹟
c
(2) sweep the wing
The flow behaves as if the airfoil section is thinner.
 Mcr , M DD 