Time-Varying Incident Mach Number
As previously discussed in Section 8.13, for a rotor in forward flight a blade element will encounter a time-varying incident velocity and Mach number. Under these conditions, there are additional unsteady aerodynamic effects to be considered, including the effects of the nonuniform convection of the shed wake (see Fig. 8.25) as well as noncir – culatory effects. If compressibility effects are to be included, the mathematical modeling of the problem becomes relatively difficult because, unlike the incompressible flow case, no exact solution can be found. However, extending the compressible indicial theory from the previous sections it can be shown that for nonsteady variations in Mach number at a constant AoA a that for the circulatory part of the unsteady lift
and for the noncirculatory part of the unsteady lift
where Mo is the initial Mach number and all the symbols have their usual meanings as defined in the preceding sections. See Jose & Leishman (2005) for details.
Representative calculated results are shown in Fig. 8.40 for time-dependent free-stream Mach number variations with the airfoil at a constant angle of attack, and are compared with direct CFD calculations based on an Euler method. Notice again the applicability of the indicial method for this type of calculation, which is several orders of magnitude more computationally efficient than the CFD solution, and so is ideal for various forms of helicopter rotor analyses where the treatment of these unsteady effects is necessary. It will be apparent that the unsteady incompressible flow theory (see Section 8.13) fails for these conditions, again emphasizing that the proper treatment of compressibility effects will be necessary if accurate unsteady airloads predictions are to be obtained on the rotor.