CE F312 Hydraulics Engineering 1

BOUNDARY LAYER

THEORY

Dr. Sanghamitra Kundu

Civil Engineering Department

BITS Pilani

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INTRODUCTION

� When a real fluid past a solid boundary, a layer of fluidwhich comes in contact with the boundary surfaceadheres to it on account of viscosity. Since this layer offluid can not slip away from the boundary surface it attainsthe same velocity as that of the boundary.

� This is called no slip condition.

� In the boundary layer region since there is a largervariation of velocity in a relatively small distance, thereexists a fairly large velocity gradient (dv/dy) normal to theboundary surface� Even if the fluid has small viscosity, the corresponding shear

stress is of appreciable magnitude

v

y

∂τ = µ

∂2

CE F312 Hydraulics Engineering 2

BOUNDARY LAYER

The idea of the boundary layer dates back at least to the time

of Prandtl (1904, see the article: Ludwig Prandtl’s boundary

layer, Physics Today, 2005, 58, no.12, 42-48).

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BOUNDARY LAYER

� As a fluid flows over a body, the no-slip condition ensuresthat the fluid next to the boundary is subject to largeshear.

� If the boundary is stationary, the fluid velocity at theboundary surface will be zero. Thus at the boundarysurface the layer of the fluid undergoes retardation.

� Therefore in the immediate vicinity of the boundarysurface, the velocity of the fluid increases gradually fromzero at boundary surface to the velocity of themainstream. This region is known as BOUNDARY LAYER.

For example, boundary layers form on the surfaces of cars, in

the water running down the gutter of the street, and in the

atmosphere as the wind blows across the surface of the earth

(land or water). 4

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CAUSES OF ITS FORMATION

� Large velocity gradient (Larger variation of velocity in

relatively smaller distance) leading to appreciable

shear stress.

� Consists of two layers:

� INSIDE THE BOUNDARY : the friction is significant and across

the width of which the velocity increases rapidly from zero

(at the surface) to the value inviscid flow theory predicts.

� OUTSIDE BOUNDARY LAYER: the velocity gradients normal to

the flow are relatively small, and the fluids acts as if it were

inviscid, even though the viscosity is not zero.

BOUNDARY LAYER ON SOLID SURFACE

� Consider the flow over a flat plate as shown, the boundary layer is laminar for

a short distance downstream from the leading edge; transition occurs over a

region of the plate rather than at a single line across the plate.

Distortion of a fluid particle as it flows within the boundary layer. 6

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FACTORS AFFECTING THE THICKNESS OF

BOUNDARY LAYER

� BLT increases as the distance from the leading edge increases

� BLT decreases with the increase in the velocity of flow ofapproaching stream fluid

� Greater is the kinematic viscosity, greater is the BLT

� Considerably affected by the pressure gradient in the direction offlow

� If pressure gradient is –ve in the case of converging flow, boundary layergrowth is retarded because the resulting pressure force acts in thedirection of flow and it accelerates the retarded flow in the BL.

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BOUNDARY LAYER CONCEPTS

� Two flow regimes

� Laminar boundary layer

� Turbulent boundary layer

� with laminar sub-layer

� Velocity distribution

� Parabolic (laminar region)

� Logarithmic (turbulent region)

� The change of boundary layer from laminar to turbulent depends on:

� Velocity of flow U of the approaching stream of fluid

� Length x measured along the plate from the leading edge

� The mass density ρ of fluid and dynamic viscosity µ

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CE F312 Hydraulics Engineering 5

FLAT PLATE: PARALLEL TO FLOW

Ux

y

U U U

δ

τ

Why is shear maximum at the leading edge of

the plate?

boundary

layer

thickness

shear

du

dyis maximum

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BOUNDARY LAYER CONCEPTS

µ

ρVxx =Re

The value of Rex at which the boundary layer may change

from laminar to turbulent varies from 2 × 105 to 3× 106

Unless otherwise stated, we will use ReReReRexcrxcrxcrxcr = 5 = 5 = 5 = 5 ×××× 101010105555

in our calculations

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THICKNESS OF BOUNDARY LAYER

� Velocity within the boundary layer increases from

zero at the boundary surface to the velocity of the

main stream asymptotically

A pipe is enclosed, so the fluid is fully bounded, but in an

open flow at what distance away from the boundary can we

begin to ignore this shear?

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THICKNESS OF BOUNDARY LAYER

� Standard Boundary Layer Thickness

� Displacement Thickness

� Momentum thickness

� Energy thickness

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