. Modeling of Dynamic Stall

Mathematical models that attempt to predict the effects of dynamic stall currently range from relatively parsimonious empirical or semi-empirical models to sophisticated computational fluid dynamics (CFD) methods. Because dynamic stall is characterized by large recirculating, turbulent separated flow regimes, a proper CFD simulation can only be achieved by solving numerically the full Navier-Stokes equations with a suitable turbulence model. CFD methods have now begun to show some promise in predicting 2-D and 3-D dynamic stall events – see, for example, Srinivasan et al. (1993), Ekaterinaris et al. (1994),

and Spentzos et al. (2004) and also the discussion in Section 14.10.1. However, the quantita­tive predictions of the airloads are not yet satisfactory in the stalled regime and during flow reattachment and especially not at the Reynolds numbers and Mach numbers appropriate to helicopter rotors. In this regard, the accurate prediction of the transition from a laminar to turbulent boundary layer is a key issue. Furthermore, the computational resources for these CFD solutions are prohibitive other than for use as research tools, and for the foreseeable future more approximate models of dynamic stall will still have to be used in a variety of rotor airloads prediction problems, including rotor design work.

Some of the mathematical models of dynamic stall in current use are a form of resyn­thesis of the measured unsteady airloads, which are based on results from 2-D oscillating airfoils in wind tunnel experiments. Other so-called semi-empirical models of dynamic stall contain simplified representations of the essential physics using sets of linear and nonlinear equations for the lift, drag, and pitching moment. The nonlinear equations may have many empirical coefficients, which must be deduced (extracted) from unsteady airfoil measurements by parameter identification methods. However, the root of these models is usually based on classical unsteady thin-airfoil theory, as discussed in Chapter 8. The de­velopment of the nonlinear part of such models are more subjective and require skillful interpretation of experimental data. As a result, most of these models remain in a perpetual state of flux as the level of detail is refined and/or more experimental data become available for formulation and/or correlation purposes.

While semi-empirical models are usually adequate for most rotor design purposes, they often lack rigor and generality when applied to different airfoils and at different Mach numbers for which 2-D experimental measurements may not be available. Another major problem with some of these types of models is that a significant number of empirical co­efficients must be derived. Generally, a set of coefficients for the model must be derived for each and every airfoil, and also over the appropriate range of Mach numbers, assuming such measurements are available. In cases where experimental measurements are not avail­able, the models cannot be used with the same confidence levels to predict the nonlinear airloads. Other common limitations with these models include the accuracy with which the stall onset can be predicted, that is, the prediction of the combination of unsteady AoA and Mach number that produce the onset of dynamic leading edge flow separation. In these cases, computer coding of the model must be done with extreme care to ensure that logic or conditional branching in the algorithm does not cause nonphysical transients in the pre­dictions of the unsteady airloads, especially if large time (azimuth) steps are involved. This undesirable behavior may produce erroneous predictions of stall and aeroelastic behavior, which would be considered unacceptable for rotor design purposes.

Therefore, it is important for the analyst to build up a confidence level with any model selected for the design process. While a review of the literature will show a large number of experiments on dynamic stall that could be used for such purposes, the problem is usually that the full range of Reynolds numbers, Mach numbers, reduced frequencies, and, to some extent, airfoil shapes, cannot be studied in the same test facility or wind tunnel. Therefore, the same problems and uncertainties in data quality that were discussed in Section 7.9 in regard to comparing static airfoil characteristics also apply in the dynamic case.