At some station in a laminar boundary layer developing in air ( )

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

At some station in a laminar boundary layer developing in air (ρ=1.20 kg m−3\rho=1.20\,\mathrm{kg\,m^{-3}}), the outer speed is U=5.0 m s−1U=5.0\,\mathrm{m\,s^{-1}}, the momentum thickness is δ2=2.0 mm\delta_2=2.0\,\mathrm{mm}, the shape factor is Hsh=3H_{\rm sh}=3 (as for a linear profile) and the wall shear stress is τw=0.15 Pa\tau_w=0.15\,\mathrm{Pa}. The general momentum integral equation, obtained by integrating the boundary-layer equation across the layer, reads τwρU2=dδ2dx+(2+Hsh)δ2UdUdx.\frac{\tau_w}{\rho U^2}=\frac{\mathrm{d}\delta_2}{\mathrm{d}x}+(2+H_{\rm sh})\frac{\delta_2}{U}\frac{\mathrm{d}U}{\mathrm{d}x}. ((a)) Find dδ2/dx\mathrm{d}\delta_2/\mathrm{d}x at this station for an outer flow that is locally decelerating with dU/dx=−2.0 s−1\mathrm{d}U/\mathrm{d}x=-2.0\,\mathrm{s^{-1}} (adverse gradient). ((b)) Find dδ2/dx\mathrm{d}\delta_2/\mathrm{d}x for a flat plate at the same station, dU/dx=0\mathrm{d}U/\mathrm{d}x=0. ((c)) Find dδ2/dx\mathrm{d}\delta_2/\mathrm{d}x for an outer flow that is locally accelerating with dU/dx=+2.0 s−1\mathrm{d}U/\mathrm{d}x=+2.0\,\mathrm{s^{-1}} (favourable gradient). ((d)) Using the wall relation μ ∂2u/∂y2∣y=0=dp/dx\mu\,\partial^2u/\partial y^2|_{y=0}=\mathrm{d}p/\mathrm{d}x, identify which of the three cases is forced to have an inflection point in its profile at this station, and explain why the wall relation alone does not settle the question for the other two.

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