An Ultimate Combination of Physical Intuition with

Experiments…

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Boundary Layer Theory for Viscous Fluid Flows

Introduction of Boundary Layer Concept

Based on his experimental observations, Prandtl found that

effect of the viscosityis confined to a thin viscous layer that he called, the boundary layer.

Analytical Proof of Prandtls Intuition & Experiments

i

j

j

i

jij

ij

i

x

v

x

v

xx

p

Frk

x

vv

t

v*

*

*

*

**

*

*

**

*

*

Re

11ˆ

Consider non-dimensional of NS Equations

Steady State non-dimensional of NS Equations

i

j

j

i

jij

ij

x

v

x

v

xx

p

Frk

x

vv

*

*

*

*

**

*

*

**

Re

11ˆ

Steady State Incompressible non-dimensional of NS Equations

*2*

*

*

** *

Re

11ˆi

ij

ij v

x

p

Frk

x

vv

Equivalent ODE to NS

021

2

2

2

dy

d

dy

d

nn

yeyey ny

ny

flnn 1sinh

11cosh1)( 2

2

2

01ˆ*

Re

1*

*

*

***2

ij

iji x

p

Frk

x

vvv

A selected property of any fluid flow field can be approximated as:

General Response of A Second Order System

)(y

2n

y

)(y

y

2n

Toward Creeping

2n

)(y

y2n

)(y

y

Response of Flow Field towards Boundary Effects*2

*

*

*

** *

Re

11ˆi

ij

ij v

x

p

Frk

x

vv

y

)(y

1Re

1Re

1Re

The limit of Very large Re

Flow over a Wedge

Prandtls Large Reynolds Number 2-D Incompressible Flow

The free-stream velocity will accelerate for non-zero values of β:

2

1

m

edge L

xUxU

where L is a characteristic length and m is a dimensionless constant that depends on β:

1

2

m

m

The condition m = 0 gives zero flow acceleration corresponding to the Blausius solution for flat-plate flow.

The Measure of Wedge Angle

The boundary layer is seen to grow in thickness as x moves from 0 to L.

Two-dimensional Boundary Layer Flows

In dimensionless variables the steady Navier-Stokes equations in two dimensions may be written:

0Re

1 2

ux

puv

0Re

1 2

vy

pvv

0

y

v

x

u

The boundary layer is seen to grow in thickness as x moves from 0 to L.