(Solved): Turbulent boundary layer skin friction is one of the fluid phenomena that limit the travel speed o ...

Turbulent boundary layer skin friction is one of the fluid phenomena that limit the travel speed of aircraft and ships. One means for reducing the skin friction of liquid boundary layers is to inject a gas (typically air) from the surface on which the boundary layer forms. The shear stress, \( \tau_{\mathrm{w}} \), that is felt a distance \( \mathrm{L} \) downstream of such an air injector depends on: the volumetric gas flux per unit span \( \mathrm{q}\left(\right. \) in \( \left.\mathrm{m}^{2} / \mathrm{s}\right) \), the free stream flow speed \( \mathrm{U} \), the liquid density \( \rho \), the liquid dynamic viscositv u. the surface tension \( \sigma \) (in \( \mathrm{N} / \mathrm{m} \) ). and the gravitational acceleration \( g \). 1. Formulate a dimensionless law for \( \tau_{\mathrm{w}} \) in terms of the other parameters. 2. Experimental studies of air injection into liquid turbulent boundary layers on flat plates has found that the bubbles may coalesce to form an air film that provides near perfect lubrication, \( \tau_{\mathrm{w}} \rightarrow 0 \) for \( \mathrm{L}>0 \), when \( \mathrm{q} \) is high enough and gravity tends to push the injected gas toward the plate surface. Reformulate your answer to 1) by dropping \( \tau_{\mathrm{w}} \) and \( \mathrm{L} \) to determine a dimensionless law for the minimum air injection rate, \( \mathrm{qc} \), necessary to form an air layer. 3. Simplify the result of 2) when surface tension can be neglected. 4. Experimental studies (Elbing et al., 2008) find that \( \mathrm{q}_{\mathrm{c}} \) is proportional to \( \mathrm{U}^{2} \). Using this information, determine a scaling law for qc involving the other parameters. Would an increase in \( \mathrm{g} \) cause \( \mathrm{qc} \) to increase or decrease?