1 Boundary Layer Theory

7/29/2019 1 Boundary Layer Theory

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BOUNDARY LAYER THEORY

(Re>1000)

Unit # 1

Sec 8.6.1 Potter


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Why study Boundary layer

It would help you understand how aircrafts fly It would help you calculate

drag on surfaces

skin friction on surfaces

boundary layer thickness


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Free-stream vs Boundary LayerA high Reynolds number fluid stream over a solid

surface may be divided into two parts:

a) A thin layer adjacent to the surface, whereonly viscosity dominates calledBOUNDARY LAYER

(It is a viscous flow. It is very thin. Thepressure within the boundary layer may betaken as that at the wall)

b) An inviscid flow outside this boundary layerthroughout calledFREE-STREAM

(It is a potential flow)


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Boundary layer on flat plate

Boundary layer begins as a laminar flow with zero thicknessat the leading edge. After some distance downstream the

laminar flow undergoes transition to turbulent flow.

Pressure in the boundary layer does not vary withy. It issame as that outside of it


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DEFINITIONS

BOUNDARY LAYER () thickness is thedistance from the wall up to a height wherethe fluid velocity is 99% of the free-stream

velocity

DISPLACEMENT () thickness is thedistance by which the boundary layer would

have to be displaced if the entire flow were

frictionless and the same mass flow ismaintained

MOMENTUM thickness ( ) is a measure ofthe deficit of momentum flux in the boundary

layer as compared to potential flow.

It is also a measure of total plate drag

)99.0( == Uuy

dyUu

=

0

* 1

dy

U

u

U

u

=

0

1


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Drag or Shear stress

dy

du

Fluid drag on a bodyis governed by shear

stress on the wall

It is known from thevelocity gradient on

the wall


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Transition of Laminar to Turbulent BL

Boundary layer is initially laminar, goes through a transitionregion when large-scale eddies are formed and then

develops into turbulentflow.

Transition from laminar to turbulent occurs at Rex >500,000. where Rex is local Reynolds number, U x/=U x/


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Turbulentboundary

layer

Turbulent BL has a steeper velocity gradient at the surface


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Problem (Fox-415) A laboratory wind tunnel has a test section, 305 mm

square. Boundary-layer velocity profiles aremeasured at two cross sections and displacementthicknesses are evaluated from the measuredprofiles.

At section(1), where the free-stream speed isU1=26 m/s, the displacement thickness is 1.5 mm.

At section (2), located downstream from section

(1) displacement thickness is 2.1 mm. Calculate the change in static pressure between

sections (1) and (2). Express the result as a fraction

of the free-stream dynamic pressure at section (1).


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Home work (Douglas 11.2)

Determine the ratio of momentum and

displacement thickness to the boundary layer

thickness, when the layer velocity profile is

given by

where u is the velocity at a height y above the

surface and the flow free stream velocity is U.

[ Ans: 0.166 m, 0.33m]

5.0

=

y

U

u


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Summary

Boundary layer applies to the region close to thesurface where viscosity is dominant.

Boundary layer thickness

Displacement thickness

Momentum thickness

)99.0( == Uuy

dyU

u

=

0

* 1

dy

U

u

U

u

=

0

1


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Wind Tunnel


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Boundary layer

Figure 8.24 Boundary layer in air withRecrit

= 3 x 105.

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1 Boundary Layer Theory