A two-dimensional laminar boundary layer grows along a wall over which the outer…

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

A two-dimensional laminar boundary layer grows along a wall over which the outer inviscid flow has speed U(x)U(x), so that the pressure gradient imposed on the layer is dp/dx=−ρU dU/dxdp/dx = -\rho U\,dU/dx, a result of the boundary-layer scaling applied to Euler's equation at the edge of the layer. The boundary-layer xx-momentum equation is u∂u∂x+v∂u∂y=UdUdx+ν∂2u∂y2,u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = U\frac{dU}{dx} + \nu\frac{\partial^2 u}{\partial y^2}, with continuity ∂u/∂x+∂v/∂y=0\partial u/\partial x + \partial v/\partial y = 0. ((a)) Integrate the momentum equation with respect to yy from y=0y=0 to a fixed height hh large enough that u→Uu \to U for y>δ99(x)y > \delta_{99}(x), using continuity to eliminate vv and the definitions δ1(x)=∫0h(1−uU)dy,δ2(x)=∫0huU(1−uU)dy,\delta_1(x) = \int_0^h\left(1-\frac{u}{U}\right)dy,\qquad \delta_2(x) = \int_0^h\frac{u}{U}\left(1-\frac{u}{U}\right)dy, to derive the general momentum integral equation τwρU2=dδ2dx+(δ1+2δ2)1UdUdx.\frac{\tau_w}{\rho U^2} = \frac{d\delta_2}{dx} + (\delta_1+2\delta_2)\frac{1}{U}\frac{dU}{dx}. ((b)) Show that for a flat plate (dU/dx=0dU/dx = 0) this reduces to τw/(ρU2)=dδ2/dx\tau_w/(\rho U^2) = d\delta_2/dx. ((c)) Take air (ν=1.51×10−5 m2 s−1\nu = 1.51\times10^{-5}\,\text{m}^2\,\text{s}^{-1}) at U=8.0 m s−1U = 8.0\,\text{m s}^{-1} over a flat plate and, at x=0.20 mx = 0.20\,\text{m}, evaluate both τw/(ρU2)\tau_w/(\rho U^2) (using Cf=0.664/RexC_f = 0.664/\sqrt{Re_x}) and dδ2/dxd\delta_2/dx (using δ2=0.664νx/U\delta_2 = 0.664\sqrt{\nu x/U}) to confirm they agree.

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