Transcript Compressible Frictional Flow Past Wings

Compressible Frictional Flow Past Wings
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
A Small and Significant Region of Curse ……..
A continuously Growing Solid affected Region.
The Boundary Layer
An explicit Negligence by Potential Flow Theory.
Great Disadvantage for Simple fluid Systems
De Alembert to Prandtl
1822
Ideal to Real
1752
1904
1860
Concept of Solid Fluid Interaction
• Perfectly
smooth surface (ideal surface)
Real surface
U2
U1
U1
U2
U2
U
U
Φ
Φ
Φ
Specular reflection
Diffuse reflection
• The Momentum & convective heat transfer is defined for a
combined solid and fluid system.
• The fluid packets close to a solid wall attain a zero relative
velocity close to the solid wall : Momentum Boundary Layer.
• The fluid packets close to a solid wall come to mechanical
equilibrium with the wall.
• The fluid particles will exchange maximum possible
momentum flux with the solid wall.
• A Zero velocity difference exists between wall and fluid
packets at the wall.
• A small layer of fluid particles close the the wall come to
Mechanical, Thermal and Chemical Equilibrium With
solid wall.
• Fundamentally this fluid layer is in Thermodynamic
Equilibrium with the solid wall.
Introduction
• A boundary layer is a thin region in the fluid adjacent to a
surface where velocity, temperature and/or concentration
gradients normal to the surface are significant.
• Typically, the flow is predominantly in one direction.
• As the fluid moves over a surface, a velocity gradient is
present in a region known as the velocity boundary layer,
δ(x).
• Likewise, a temperature gradient forms (T ∞ ≠ Ts) in the
thermal boundary layer, δt(x),
• Therefore, examine the boundary layer at the surface (y
= 0).
• Flat Plate Boundary Layer is an hypothetical standard for
initiation of basic analysis.
Boundary Layer Thickness
u y   0.99Ve
The governing equations for steady two dimensional
incompressible fluid flow with negligible viscous dissipation:
Boundary Conditions
0
Ve
0
0
Twall
T
Define dimensionless variables:
x
x 
L
*
v
v 
u
*
y
y 
L
*
p
p  2
u 
*
u
u 
u
*
T  Ts

T  Ts
Similarity Parameters:
Re 
u L


Pr 

Pe  Re Pr
u v
 * 0
*
x y
*
*
2 *
2 *

p
1   u  u 
* u
* u
u
v
 * 
2 
*
*
*
*2 

x
y
x Re L  x
y 
*
*
*
2




1


*
*
u
v

*
*
*2
x
y
Re L Pr y
Similarity Solution for Flat Plate Boundary Layer
u
1 u
* u
u
v

*
*
*2
x
y
Re L y
*
*
2 *
*


*
u  *
&
v  *
y
x
2
2
3
  
  
1 
 * *2 
*
*
*
*3
y x y x y
Re L y
*
Similarity variables :

f   
u
x *
u
&
u
y
x *
  
  
1 
 * *2 
*
*
*
*3
y x y x y
Re L y
2
2
3
Substitute similarity variables:
d3 f
d2 f
2 3f
0
2
d
d
This is called as Blasius Equation. An ordinary differential
equation with following boundary conditions.
df
 f 0  0
d  0
and
df
1
d  
Numerical Solution of Blasius Equation
d3 f
d2 f
2 3f
0
2
d
d
Let f ( )  f1 ( )
df1 ( )
'
'
Let f 2 ( ) 
 f1  f
d
df 2 ( ) d 2 f1 ( )
''
''
Let f 3 ( ) 

 f1  f
2
d
d
Substitute in Blasius Equation
df 3
2
 f1 f 3  0
d
df 2
 f3
d
df1
 f2
d
Fourth-order Runge-Kutta method
Blasius Solution
d3 f
d2 f
2 3f
0
2
d
d

0
0.2
0.4
0.6
f
f’
f’’
0
0.00664
0.02656
0.05974
0
0.06641
0.13277
0.198994
0.3321
0.3320
0.3315
0.8
1.0
2.0
0.10611
0.16557
0.65003
0.26471
0.32979
0.62977
3.0
4.0
5.0
1.39682
2.30576
3.28329
0.84605
0.95552
0.99155
Blasius Similarity Solution
• Blasius equation was first solved numerically (undoubtedly
by hand 1908).
•Conclusions from the Blasius solution:
 
x
,
  
and
 
1
u
Variation of Reynolds numbers
All Engineering Applications
What Sort of Reynolds Numbers do We Encounter in
Supersonic Flight?
“Transition Line”
Space
Shuttle
Velocity Profile in Boundary Layer
V
u(y)
u
V
dy
y
y

• Simple Velocity Profile Models
Laminar
 2y  y 
   
Ve     
u( y)
u
V
y

Turbulent
2



u( y)
 y
 
Ve   
1
n
• Laminar
C D fric
4 

15 c
• Turbulent
CD fric


2n
 

c  n  1n  2 
Turbulent Skin Friction
• Turbulent Boundary Layer
• No Theoretical Prediction for Boundary Layer
Thickness for Turbulent Boundary layer
• Statistical Empirical Correlation
“Time averaged”

0.16c
1
n
R 
e
Compare Laminar and Turbulent Skin Friction
 CD fric
 CD fric
32

15
1
Plot these Formulae
1
2
R e 


0.32n


n  1n  2 


1
R e n
laminar
turbulent
Versus Re
Compare Laminar and Turbulent Skin Friction
Comparison of Reynolds Number and Mach Number
V2
2     1 M 2
e
2
Mach number is a measure of the ratio of the
fluid Kinetic energy to the fluid internal energy
(direct motion To random thermal motion of gas
molecules)
-- Fundamental Parameter of Compressible
Flow
2
Ve 2 c
Ve c Ve c 
2c Ve 2 Inertial  Forces


• Re      2
 w
Ve
Viscous  Forces
Ve

Reynold’s number is a measure of the
ratio of the Inertial Forces acting on the
2
fluid -- to -- the Viscous Forces Acting on
 w  Ve
the fluid

-- Fundamental Parameter of Viscous
Flow
2
Empirical Skin Friction Correlations
“Smooth Flat Plate with No Pressure Gradient”
M=0
Re~500,000
Empirical Skin Friction Correlations
“Smooth Flat Plate with No Pressure Gradient”


0.32(7)


(7)

1
(7)

2






1
R e 7
CD fric
Plot the laws
1.328
CD fric
1
1
2
R e 
“exact solution for
Laminar Flo”
32

15
1
1
2
R e 
Simple High Speed Skin Friction Model
• So For Our Purposes … we’ll use the “1/7th power”
Boundary layer law … and the Exact Laminar Solution
Re < 500,000
 CD fric  1.328
Exact Blasius Solution
 CD fric
1
1
2
R e 
laminar


0.32n


n

1
n

2




7




1
1
225 R e 7
R e n
turbulent
Comparison of Velocity Distributions
Supersonic Boundary Layers
• When a vehicle travels at Mach numbers greater than one, a significant
temperature gradient develops across the boundary layer due to the high levels
of viscous dissipation near the wall.
• In fact, the static-temperature variation can be very large even in an adiabatic
flow, resulting in a low density, high-viscosity region near the wall.
• In turn, this leads to a skewed mass-flux profile, a thicker boundary layer, and
a region in which viscous effects are somewhat more important than at an
equivalent Reynolds number in subsonic flow.
• Intuitively, one would expect to see significant dynamical differences between
subsonic and supersonic boundary layers.
• However, many of these differences can be explained by simply accounting for
the fluid-property variations that accompany the temperature variation, as
would be the case in a heated incompressible boundary layer.
• This suggests a rather passive role for the density differences in these flows,
most clearly expressed by Morkovin’s hypothesis.
Effect of Mach Number
• The friction coefficient is affected by Mach number as well.
• This effect is small at subsonic speeds, but becomes appreciable
for supersonic aircraft.
• The idea is that aerodynamic heating modifies the fluid
properties.
• For a fully-turbulent flow, the wall temperature may be
estimated from:
• An effective incompressible temperature ratio is defined:
GAS Viscosity Models Sutherland’s Formula
Result from kinetic theory that expresses viscosity as a function of
temperature
T
 (T )   (Ts )  
 Ts 
3/2
 T s  Cs 
 T  C 
s
7
C D , firction,turbulent 
225 Re1/ 7
CD 
 fric  compressible 
7


T




cV
Tavg


225 
3/2






T
C
T
s

  (T ) avg

 




 T   Tavg  C s  

7

 T 
cV


225 


  (T )  Tavg 

5/2
 Tavg  C s  



T  C s  
  C D fric 

compressible
1
7

CD 
 fric  incompressible


  T 
  Tavg 

5 /2
 T avg  C s  



T  C s  

1
7
7
1
7
  cV    T 

225 


  (T )    Tavg 

1
7
5/2
 T avg  C s  



T  C s  

1
7
• What is “Tavg” in the boundary layer?
CD 

fric

 compressible
C D 
 fric incompressible
 T  5 /2  T  C  
s
    avg


 Tavg   T  Cs  


1
7
• Look at small segment of boundary layer, dy
• Enthalpy Balance
2

 u(y)  
V
u(y)
V
T 
 T (y) 
 T (y)  T 
1  
 
2c p
2c p
2c p   Ve  
2
2
2
V
u(y)
dy
y
• Taking the average (Integrating Across Boundary layer)
2
1
2

 u(y)  

V 2 
1 V
7


x


 T 

1
T

 dy 
1 

  dy  

 2c p 0   Ve  
2c p 0
2 
Tavg
9


71
x



V 2 
0
1 

T 
2c p 
9/7 




2 V 2


T
T 1 
9 2c p

2   1
2 
M  


9 2
V
• Valid for Turbulent Flow
u(y)
dy
y
Collected Algorithm


V c
7

V
c
C D 
 
 R e 

R

 fric incompressible  
e
1




7


225
R
e

 
CD 
fric 

incompressible
0
C D 


C

120
K for air
s
1
 fric  compressible
 T  5 /2  T  C   7
s
    avg


 Tavg   T  Cs  


 2   1
2 
Tavg  T 1  
M  

• Valid for Turbulent Flow
 9 2





Skin Friction Versus Mach Number
Symmetric Double-wedge Airfoil … L/D (revisited)
• Inviscid Analysis
t/c
• Mach 3
t/c = 0.035
+
Symmetric Double-wedgeAirfoil … L/D (revisited)
• Blow up of
Previous page
• Analysis
Including skin
Friction Model
• Mach 3
L/Dmax =7.4
=
• Mach 25
• 60 km Altitude
L/Dmax =3.18
=