Compressible Frictional Flow Past Wings

P M V Subbarao Professor

Mechanical Engineering DepartmentI 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

17521904

Ideal to Real1822

1860

Concept of Solid Fluid Interaction

Diffuse reflection

U2

U U

Φ

U2

Φ

U1

U1

Φ

U2

Specular reflection

• Perfectly smooth surface (ideal surface) Real surface

• 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

eyVu 99.0

The governing equations for steady two dimensional incompressible fluid flow with negligible viscous dissipation:

Boundary Conditions

0

0

Twall

Ve

0

T

Define dimensionless variables:

L

xx *

L

yy *

u

uu*

u

vv*

s

s

TT

TT

2*

u

pp

Lu

Re

Similarity Parameters:

Pr PrRePe

0*

*

*

*

y

v

x

u

22 *

*2

*

*2

*

*

*

**

*

**

Re

1

y

u

x

u

x

p

y

uv

x

uu

L

2*

2

**

**

PrRe

1

yyv

xu

L

Similarity Solution for Flat Plate Boundary Layer

2*

*2

*

**

*

**

Re

1

y

u

y

uv

x

uu

L

**

** &

xv

yu

Similarity variables :

**

& x

uy

ux

u

f

3*

3

2*

2

***

2

* Re

1

yyxyxy L

022

2

3

3

d

fdf

d

fd

Substitute similarity variables:

1 and 000

d

dff

d

df

3*

3

2*

2

***

2

* Re

1

yyxyxy L

This is called as Blasius Equation. An ordinary differential equation with following boundary conditions.

Numerical Solution of Blasius Equation

022

2

3

3

d

fdf

d

fd

)()( 1 ffLet

''1

12

)()( ff

d

dffLet

''''12

12

23

)()()( ff

d

fd

d

dffLet

Substitute in Blasius Equation

02 313 ff

d

df

32 f

d

df

21 f

d

df

Fourth-order Runge-Kutta method

Blasius Solution

f f’ f’’

0 0 0 0.3321

0.2 0.00664 0.06641 0.3320

0.4 0.02656 0.13277 0.3315

0.6 0.05974 0.198994

0.8 0.10611 0.26471

1.0 0.16557 0.32979

2.0 0.65003 0.62977

3.0 1.39682 0.84605

4.0 2.30576 0.95552

5.0 3.28329 0.99155

022

2

3

3

d

fdf

d

fd

Blasius Similarity Solution

• Blasius equation was first solved numerically (undoubtedly by hand 1908).

u

1 and , x

•Conclusions from the Blasius solution:

Variation of Reynolds numbers

All Engineering Applications

What Sort of Reynolds Numbers do We Encounter in Supersonic Flight?

“Transition Line”SpaceShuttle

Velocity Profile in Boundary Layer

u

V

y

V

u(y)

y

dy

• Simple Velocity Profile Models

Laminar Turbulent

u(y)

Ve

y

1

n

u

V

y

u(y)

Ve

2y

y

2

• Laminar • Turbulent

CD fric

4

15

c

CD fric

c

2n

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

Re 1

n

Compare Laminar and Turbulent Skin Friction

CD fric

32

15

1

Re 1

2laminar

CD fric

0.32n

n 1 n 2

Re 1

nturbulent

Plot these Formulae

Versus Re

Compare Laminar and Turbulent Skin Friction

Comparison of Reynolds Number and Mach Number

V 2

2e

1

2M 2

Reynold’s number is a measure of the ratio of the Inertial Forces acting on the fluid -- to -- the Viscous Forces Acting on the fluid -- Fundamental Parameter of Viscous Flow

w 2

Ve

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

Ve2

2

Ve

• Re cVe

cVe

2

Ve

c

2c

Ve2

w

Inertial Forces

Viscous Forces

Empirical Skin Friction Correlations

Re~500,000

M=0

“Smooth Flat Plate with No Pressure Gradient”

Empirical Skin Friction Correlations

“Smooth Flat Plate with No Pressure Gradient”

CD fric

32

15

1

Re 1

2

CD fric

0.32(7)

(7) 1 (7) 2

Re 1

7

Plot the laws

“exact solution for Laminar Flo”

1.3281

Re 1

2

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

CD fric

0.32n

n 1 n 2

Re 1

n

7

225 Re 1

7turbulent

CD fric1.328

1

Re 1

2laminar

Exact Blasius Solution

Re < 500,000

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 ) (Ts )T

Ts

3/2T

s Cs

T Cs

CD fric

compressible

7

225

T

Tavg

cV

(T )Tavg

T

3/2T

Cs

Tavg Cs

1

7

7

225cV

(T )T

Tavg

5 /2T

avg Cs

T Cs

1

7

7

225cV

(T )

17 T

Tavg

5 /2T

avg Cs

T Cs

1

7

CDfric

compressible

CDfric

incompressible

T

Tavg

5/2T

avgCs

T Cs

1

7

7/1,, Re225

7turbulentfirctionDC

• What is “Tavg” in the boundary layer?

CD fric

compressible

CD fric

incompressible

T

Tavg

5 /2Tavg Cs

T

Cs

1

7

• Look at small segment of boundary layer, dy

V

u(y)

y

dy

• Enthalpy Balance

T V

2

2cp

T (y) u(y)2

2cp

T (y) T V

2

2cp

1 u(y)

Ve

2

• Taking the average (Integrating Across Boundary layer)

V

u(y)

y

dy

yTavg T 1

V2

2cp

1 u(y)

Ve

2

d

0

T V2

2cp

1 2

7

dy

0

1

1

T V2

2cp

1

9

7

0

9 / 7

T 2

9

V2

2cp

T 1 2

9

1

2M

2

• Valid for Turbulent Flow

Collected Algorithm

• Valid for Turbulent Flow

CD fric

incompressible

7

225 Re

1

7

Re

Vc

Re

Vc

CD fric

compressible

CD fric

incompressible

T

Tavg

5 /2Tavg Cs

T

Cs

1

7

Cs 1200 K for air

Tavg T 12

9

1

2M

2

Skin Friction Versus Mach Number

Symmetric Double-wedge Airfoil … L/D (revisited)

+t/c = 0.035

• Inviscid Analysis

• Mach 3

t/c

Symmetric Double-wedgeAirfoil … L/D (revisited)

=

• Analysis Including skin Friction Model

• Mach 3

L/Dmax =7.4

• Blow up of Previous page

=

• Mach 25

• 60 km Altitude

L/Dmax =3.18