For flow over a flat plate at zero pressure gradient

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity

Fluid Dynamics, Ch The boundary layer: Prandtl's equations and the Blasius solution

Problem

For flow over a flat plate at zero pressure gradient, the momentum integral method uses an assumed velocity profile

u/U=f(\eta)

\eta=y/\delta

, with

f(0)=0

f(1)=1

f'(1)=0

. Suppose the quadratic profile

f(\eta)=2\eta-\eta^2

is used instead of the linear profile (

\delta=3.46x/\sqrt{Re_x}

C_f=0.577/\sqrt{Re_x}

) or the Pohlhausen cubic profile (

\delta=4.64x/\sqrt{Re_x}

C_f=0.646/\sqrt{Re_x}

). ((a)) Using the quadratic profile, derive

\delta_1

and

\delta_2

in terms of

\delta

, then use the momentum integral equation to derive

\delta(x)=A\,x/\sqrt{Re_x}

and

C_f(x)=B/\sqrt{Re_x}

, giving

A

and

B

. ((b)) State the percentage by which this

C_f

differs from the Blasius value

0.664/\sqrt{Re_x}

, and say whether the quadratic profile is more or less accurate than the linear and cubic profiles quoted above. ((c)) The wall relation

\mu\,\partial^2u/\partial y^2|_{y=0}=\mathrm{d}p/\mathrm{d}x

must hold. Check whether the quadratic profile satisfies it for a flat plate (

\mathrm{d}p/\mathrm{d}x=0

), and explain why none of the linear, quadratic or cubic profiles used here could represent a layer with an inflection point, of the kind an adverse pressure gradient produces.

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