Fluid Mechanics · Dimensional analysis and similitude
Apply the von Kármán momentum-integral relation dθ/dx = τ_w/(ρU^2) to a…
Problem
Apply the von Kármán momentum-integral relation dθ/dx = τ_w/(ρU^2) to a zero-pressure-gradient flat-plate boundary layer. Assume a linear velocity profile u/U = y/δ for 0 ≤ y ≤ δ, evaluate the momentum thickness θ in terms of δ, estimate τ_w from a linear gradient with an effective viscosity equal to the molecular viscosity, and obtain a differential equation for δ(x). Integrate from a virtual origin at x = 0 and compare your δ/x and C_{f,x} with the Blasius values δ/x = 5.0 Re_x^{-1/2} and C_{f,x} = 0.332 Re_x^{-1/2} for a plate 0.40 m long in 20 °C air at 3.0 m/s.
Hint
For a linear profile, evaluate θ/δ and estimate τ_w from the molecular viscosity times the wall gradient U/δ.
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