CE F312 Hydraulics Engineering 1

� Integrating

BOUNDARY LAYER EQUATIONS

10

p

y

∂− =ρ ∂

( )( )

for steady flow

, for unsteady flow

p f x

p f x t

=

=

Pressure remains constant across the boundary layer

2

2

1u u dp uu vx y dx y

∂ ∂ ∂+ = − + υ

∂ ∂ ρ ∂ The details of viscous

incompressible flow past any object

can also be obtained by solving the

governing Navier-Stokes equation.29

� In the region outside the boundary layer since

v=0 and u=U the free stream or ambient velocity

of the approaching stream,

BOUNDARY LAYER EQUATIONS

2

2

1u u dp uu vx y dx y

∂ ∂ ∂+ = − + υ

∂ ∂ ρ ∂

0 01U dp

Ux dx

∂= −

∂ ρ

1dU dpUdx dx

= −ρ

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CE F312 Hydraulics Engineering 2

HOW TO SOLVE BOUNDARY LAYER

� Momentum integral equation

� Used to obtain approximate information on boundary

layer growth for the general case ( laminar or

turbulent boundary layers, with or without a pressure

gradient).

� By Blasius (called Blasius solution)

� Limited to laminar boundary layer only, and for a flat

plate only ( no pressure variations).

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Used to obtain approximate information on boundary layer growth

MOMENTUM INTEGRAL EQUATION

32

CE F312 Hydraulics Engineering 3

33

MOMENTUM INTEGRAL EQUATION

� A complete description of the boundary layer with the aid of non-lineardifferential equation is very cumbersome.

� Therefore approximate solutions have been developed for predictingcharacteristics of boundary layer – e.g. momentum thickness,displacement thickness, skin friction.

� One such approximate method developed by Theodore von Karman isbased on the application of the momentum equation to the boundarylayer and the corresponding equation derived by him is known asmomentum integral equation of the boundary layer or von Karman’sintegral equation

� The momentum integral equation of the boundary layer expresses therelation that must exist between the overall rate of flux of momentum(dƟ/dx) across a section of the boundary layer, the shear stress at theboundary surface (τ0) and the pressure gradient in the direction of flow(dp/dx)

� Using a control volume analysis

�Gives a better physical understanding: i.e. balancing rate of

momentum flux with frictional forces at the surface and

pressure gradient forces.

BOUNDARY LAYER EQUATIONS

34

CE F312 Hydraulics Engineering 4

� Consider the control volume (CV) AEFD

� Boundary layer thickness δ, outer edge represented by BC

� AB and CD extended upto E and F where AE=DF=h, such that his slightly greater than the local boundary layer thickness δ.

MOMENTUM INTEGRAL EQUATION

35

� Mass flow and Conservation of mass:

� Rate of mass flow across AE into CV =

� Rate of mass flow across DF out of CV =

� Net rate of mass flow across DF and AE, out of AEFD =

MOMENTUM INTEGRAL EQUATION

0

h

udyρ∫

0 0

h hd

udy udy xdx

ρ + ρ δ

∫ ∫

0

hd

udy xdx

ρ δ

∫

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