Air ( , , ) streams at over a flat plate aligned with the flow

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

Air (ρ=1.20 kg m−3\rho = 1.20\,\text{kg m}^{-3}, μ=1.81×10−5 Pa s\mu = 1.81\times10^{-5}\,\text{Pa s}, ν=1.51×10−5 m2 s−1\nu = 1.51\times10^{-5}\,\text{m}^2\,\text{s}^{-1}) streams at U=25 m s−1U = 25\,\text{m s}^{-1} over a flat plate aligned with the flow. A hot-wire anemometer is to be calibrated at the station x=0.15 mx = 0.15\,\text{m} from the leading edge. ((a)) Starting from the Blasius similarity form u=UF′(η)u = U F'(\eta) with η=yU/(νx)\eta = y\sqrt{U/(\nu x)}, obtained by reducing the boundary-layer equations for a flat plate (dU/dx=0dU/dx = 0) to a single ordinary differential equation, derive an expression for the wall shear stress τw(x)\tau_w(x) in terms of F′′(0)F''(0), and show that Cf(x)≡τw/(12ρU2)=2F′′(0)/RexC_f(x) \equiv \tau_w/(\tfrac12\rho U^2) = 2F''(0)/\sqrt{Re_x}. Using the shooting result F′′(0)=0.332F''(0) = 0.332, confirm Cf=0.664/RexC_f = 0.664/\sqrt{Re_x}. ((b)) The momentum integral equation for zero pressure gradient, obtained by integrating the boundary-layer xx-momentum equation across the layer, reads τw=ρU2 dδ2/dx\tau_w = \rho U^2\,d\delta_2/dx. Combine this with part (a) and integrate from the leading edge, where δ2=0\delta_2 = 0, to derive δ2(x)=0.664νx/U\delta_2(x) = 0.664\sqrt{\nu x/U}. ((c)) Evaluate RexRe_x, τw\tau_w and δ2\delta_2 at x=0.15 mx = 0.15\,\text{m}, and confirm that RexRe_x lies below the value at which transition is first observed (Rex≈3×105Re_x \approx 3\times10^{5}), so that treating the layer as laminar there is justified.

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