Someone argues: “Within a boundary layer the fluid slows from the free-stream speed…

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

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

Problem

Someone argues: “Within a boundary layer the fluid slows from the free-stream speed

U

down to zero at the wall. Applying Bernoulli's theorem along the vertical line from a point at the outer edge of the layer down to the wall beneath it therefore gives

p_{\rm wall}=p_{\rm edge}+\tfrac12\rho U^2

: the wall pressure exceeds the pressure at the edge of the layer by the full free-stream dynamic pressure. For air at

U=12\,\mathrm{m\,s^{-1}}

(

\rho=1.20\,\mathrm{kg\,m^{-3}}

) that is a difference of

86.4\,\mathrm{Pa}

.” ((a)) Find the error in this argument. ((b)) Give an order-of-magnitude estimate of the actual pressure difference across a layer of thickness

\delta=5.0\,\mathrm{mm}

on a wall of radius of curvature

R=2.0\,\mathrm{m}

, at the same speed, and compare it with the

86.4\,\mathrm{Pa}

claimed above.

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