Laminar Boundary Layer

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Laminar Boundary layer

Unit#2

(Ref 8.6.2, 8.6.3 Potter)



Boundary layer

• Types of boundary layer - At Rex < Recr Laminar layer

- At Rex > Recr Turbulent layer

Critical Reynolds number, Recr = 500,000

Boundary layer properties is found by either

1. Rigorous solution of Navier Stokes equation or (Blasius)

or 2. Simple Von Karman momentum integral method



Von Karman Integral Method .

It is valid for both laminar and turbulent flow



Von Karman Integral Equation for

• Force balance on fluid element from previous pagegives the Momentum eqn. in flow direction when dP/dx

=0

• Mass balance gives

• Neglecting pressure gradient

where θ is the momentum thickness

[ ]

∫ ∫

∂

∂−

∂

∂=−−

−−=+++−++

δ δ

ρ ρ τ δ

τ δ δ δ δ

0 0

2

0

0

)(

))(()

2

(

xU udydx x

dydxu x

dxdp

M M M dxd dp pd dp

p p topinout

δ ρ

δ

Ud dydxu x

mmm inout top =∂

∂=−= ∫

0

∫∫∫ −=−=

δ δ δ

ρ ρ ρ τ dyuU ud

dyud

dyU d

)(2

∞∞dxdxdx

00

0



Laminar velocity profile• Assume laminar profile

u = ay + b y2+ cy3

• Use boundary conditions y=0, u=0; u=U,

(du/ d y)= 0 at y =δ (d2u/ d y2)= 0 at y = 0

• It gives laminar velocity

profile

• (2)3

3

2

1

2

3

δ δ

y y

U

u−=



Find Boundary layer thickness of

Laminar BL using cubic velocity profile

• Substitute velocity profile (2) in Von Karman eqn (1)

• From the definition of shear stress

• Equating above two one gets

.

dx

d U dy

y y y y

dx

d U dyuU u

dx

d δ ρ

δ δ δ δ ρ ρ τ

δ δ 2

0 3

3

3

32

0

0 139.022

31

22

3)( =

+−

−=−= ∫∫

δ

µ µ τ

2

3

0

0

U

dy

du

y

=

=

=

x x Re

65.4

=

δ



Displacement thickness & skin friction (Appx)

• Displacement thickness is defined as

• Skin friction on the wall is

found by using value of δ

• From the definition of local skin friction coefficient, C fx

• Average skin friction coefficient

∫

−=

δ

δ 0

* 1 dyU

u

x

fxU

C Re

646.022

0 == ρ

τ

L

L L

x

fx f

L

dxC

L

C

Re

29.1

Re

646.011

0 0

=== ∫ ∫

x

U U

dy

du

yRe

323.0

2

3 2

0

0

ρ

δ µ µ τ ==

= =



Exact solution of Blasius• Blasius through more rigorous solution of BL eqns.

found values of BL thickness, displacement thickness

& local friction coefficient.

•He found average skin friction coefficient for the entire

length L

• Use these equations for solving problems. Shear stress

;Re

664.0;

Re

72.1;

Re

5 *

5.0

x

x

x x

C x x

=== δ δ

L

f C Re

33.1

=



Laminar boundary layer

[Douglas p-393]

• Oil with a free stream velocity of 3 m/sflows over a thin plate of 1.25 m wide and 2

m long. Determine the boundary layer

thickness and the shear stress at mid-lengthand calculate the total, double sided

resistance of the plate (density 860 kg/m3,kinematic viscosity 10-5 m2/s)



Problem

• The floor of the wind tunnel of a model of FrenchTGV moves with the speed corresponding to that

of the main flow. This prevents a boundary layer

from building up on the floor as the fluid reachesthe turbo-train and more closely resembles the

actual flow relative to a moving train. Find what

would be boundary layer if floor was not moving.Air velocity is 6 m/s and train is 2.5 m from the

leading edge. Take viscosity 1.55x10-5 m2/s



Model testing of world’s fastest train

Laminar Boundary Layer

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