Results for Turbulent Boundary Layers on a Flat Plate

In this section, we discuss a few results for the turbulent boundary layer on a flat plate, both incompressible and compressible, simply to provide a basis of comparison with the laminar results described in the previous section. For considerably more detail on the subject of turbulent boundary layers, consult References 42 to 44.

For incompressible flow over a flat plate, the boundary-layer thickness is given approximately by

Note from Equation (19.1) that the turbulent boundary-layer thickness varies approx­imately as RejTl/5 in contrast to Re“I/2 for a laminar boundary layer. Also, turbulent values of <5 grow more rapidly with distance along the surface; 8 <x x4/5 for a turbulent flow in contrast to 8 oc x1 /2 for a laminar flow. With regard to skin friction drag, for incompressible turbulent flow over a flat plate, we have

Note that for turbulent flow, C f varies as Re~1//5 in comparison with the Re’1/2 vari­ation for laminar flow. Hence, Equation (19.2) yields larger friction drag coefficients for turbulent flow in comparison with Equation (18.22) for laminar flow.

The effects of compressibility on Equation (19.2) are shown in Figure 19.1, where C f is plotted versus ReTO with Мто as a parameter. The turbulent flow results are shown toward the right of Figure 19.1, at the higher values of Reynolds numbers where turbulent conditions are expected to occur, and laminar flow results are shown toward the left of the figure, at lower values of Reynolds numbers. This type of figure—friction drag coefficient for both laminar and turbulent flow as a function of Re on a log-log plot—is a classic picture, and it allows a ready contrast of the two types of flow. From this figure, we can see that, for the same Re^, turbulent skin

friction is higher than laminar; also, the slopes of the turbulent curves are smaller than the slopes of the laminar curves—a graphic comparison of the Re,/? variation in contrast to the laminar Re 1/2 variation. Note that the effect of increasing Мж is to reduce Cf at constant Re and that this effect is stronger on the turbulent flow results. Indeed, C/ for the turbulent results decreases by almost an order of magnitude (at the higher values of Re,*,) when M0c is increased from 0 to 10. For the laminar flow, the decrease in Су as Мж is increased though the same Mach number range is far less pronounced.