91Ó°ÊÓ

Standard air flows over a flat surface at a speed of \(25 \mathrm{~m} / \mathrm{s}\). The boundary layer is turbulent from the leading edge of the surface, and the velocity distribution within the boundary layer can be approximated by the one-seventh law distribution. The thickness of the boundary layer at two locations along the surface are measured as \(5 \mathrm{~mm}\) and \(15 \mathrm{~mm}\). Estimate the distance between the locations where the boundarylayer thicknesses were measured.

The momentum integral equation is derived in the text for the case in which the free-stream velocity, \(U,\) remains constant (see Equation 11.76 ). In cases where the free-stream velocity varies as a function of the distance, \(x\), along a flat surface, a force balance between two sections along the boundary layer yields the following momentum equation: $$ -\delta \frac{\mathrm{d} p}{\mathrm{~d} x}-\tau_{\mathrm{w}}=\frac{\partial}{\partial x} \int_{0}^{\delta} u \rho u \mathrm{~d} y-U \frac{\partial}{\partial x} \int_{0}^{\delta} \rho u \mathrm{~d} y $$ where \(\mathrm{d} p / \mathrm{d} x\) is the pressure gradient along the surface in the streamwise direction. Show that Equation 11.118 can be expressed without the pressure-gradient term as $$ \tau_{\mathrm{w}}=-\frac{\partial}{\partial x} \int_{0}^{\delta} u \rho u \mathrm{~d} y+U \frac{\partial}{\partial x} \int_{0}^{\delta} \rho u \mathrm{~d} y+\frac{\mathrm{d} U}{\mathrm{~d} x} \int_{0}^{\delta} \rho U \mathrm{~d} y $$