Analysis of Physical Intuition …

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Two-dimensional Boundary Layer Flows

Division of Flow at Higher Reynolds Numbers

Prandtls Large Reynolds Number 2-D Incompressible Flow

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

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 Incompressible 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.

The Art of Asymptotic Thinking

This suggests that the term in x-momentum equation can be properly estimated as of order U2/L

uv .

In the dimensionless formulation, should be taken as O(1) at large Re.

uv .

If this term is to balance the viscous stress term, then the natural choiceis to assume that the y-derivatives of u are so large that the balance is with .

2

2

Re

1

y

u

This is due to the fact that the boundary layer on the plate is observed to be so thin.Thus it makes sense to define

A stretched variable yy Re

Local Reynolds Number

xURe

Shape of Boundary Layer In Stretched Coordinates

The stretched N-S Equations

2-D incompressible continuity equations

0

y

v

x

u

yy Re

0

Re

yv

x

u

In order to keep this essential equation intact and as of order unity: x

u

The stretched variable must be compensated by a stretched form of the y-velocity component:

y

yv Re

Stretched coordinate:

0

y

v

x

u

2-D incompressible continuity equations in stretched coordinates:

Prandtls Intuition

Prandtl would have been comfortably guessed this definition.

The boundary layer on the plate was so thin that there could have been only a small velocity component normal to its surface.

Thus the continuity equation will survive our limit Re .

0

y

v

x

u

X - Momentum Equation in Stretched Coordinates

0Re

12

2

2

2

y

u

x

u

x

p

y

uv

x

uu

Returning now to consideration of x-momentum equation, retain the pressure term as O(1).

x

p

0Re

12

2

2

2

y

u

x

u

x

puv

x-momentum equation in stretched coordinates:

In the limit Re , with stretched variables, this amounts to dropping the term

2

2

Re

1

x

u

02

2

y

u

x

p

y

uv

x

uu

y-Momentum Equation in 2-D Boundary Layer Flows

Use these stretched variables in y-momentum equation

0y

p

0Re

1

Re

1Re

ReRe 2

2

2

2

23

y

v

x

v

y

p

y

vv

x

vu

2

2

22

2

Re

1

Re

1

x

v

y

v

y

vv

x

vu

y

p

0Re

1 2

vy

pvv

Thus in the limit Re the vertical momentum equation reduces to

The Conclusions from Intuitive Mathematics

• The pressure does not change as we move vertically through the thin boundary layer.

• That is, the pressure throughout the boundary layer at a station x must be the pressure outside the layer.

• At this point a crucial contact is made with inviscid fluid theory.

• The “pressure outside the boundary layer” should be determined by the inviscid theory.

• Since the boundary layer is thin and will presumably not disturb the inviscid flow very much.

• In particular for a flat plate the Euler flow is the uniform stream- the plate has no effect and so the pressure has its constant free-stream value.