Flow and Turbulence Modeling and Computation of Shock Buffet Onset for Conventional and Supercritical Airfoils

Flow and turbulence models applied to the problem of shock bu et onset are studied. The accuracy of the interactive boundary layer and the thin-layer Navier-Stokes equations solved with recent upwind techniques using similar transport eld equation turbulence models is assessed for standard steady test cases, including conditions having signi cant shock separation. The two methods are found to compare well in the shock bu et onset region of a supercritical airfoil that involves strong trailing-edge separation. A computational analysis using the interactive boundary layer has revealed a Reynolds scaling e ect in the shock bu et onset of the supercritical airfoil, which compares well with experiment. The methods are next applied to a conventional airfoil. Steady shock-separatedcomputations of theconventional airfoil with the two methods compare well with experiment. Although the interactive boundary layer computations in the shock bu et region compare well with experiment for the conventional airfoil, the thin-layer Navier-Stokes computations do not. These ndings are discussed in connection with possible mechanisms important in the onset of shock bu et and the constraints imposed by current numerical modeling techniques. Introduction Shock bu et or shock-induced oscil lation (SIO) is large-scale ow-induced shock motion that involves alternating separation and reattachment of a boundary layer. In several recent computational studies, prominent features of the shock bu et of the 18-percent-thick circulararc airfoil have been computed with Navier-Stokes and thin-layer Navier-Stokes codes (refs. 1 and 2). Those studies highlighted the sensitivity of this problem to the type of turbulence and ow model and the importance of shock and trailing-edge separation in the onset of shock bu et. Although details of the shock bu et are sensitive to these factors, all computations have computed the onset Mach number for the circular-arc airfoil quite accurately. After the comprehensive time accurate calculations made for the shock bu et of the 18-percent circular-arc airfoil in reference 2, an assessment using current methods and turbulence models of predictive capabilities for several more widely used airfoils was undertaken. The present report shows the results of a computational study of this problem with both the interactive boundary layer (IBL) method and a thin-layer Navier-Stokes (TLNS) code. The physical mechanisms important in this problem can be investigated from a variety of viewpoints. For instance, shock strength is implicated in the identi cation of a Mach number range ahead of the shock for the 14-percent circular-arc airfoil in which shock bu et occurs (ref. 3). Geometry and trailing-edge viscous-inviscid interaction play a role as well. The 18-percent circular-arc airfoil has trailing-edge separation prior to shock separation and shock bu et onset (ref. 4). Trailing-edge separation has long been associated with the onset of shock bu et. (See refs. 5 and 6.) Shock bu et for this airfoil is antisymmetric and displays hysteresis in the onset Mach number range, the latter of which is discussed in reference 7 in connection with the coalescing of a shock and trail ing-edge separation. Questions remain, however, as to the important mechanisms involved for other airfoils. For instance, the NACA 0012 airfoil has a much weaker trailing-edge separation in the onset region and experiences one-sided shock bu et (ref. 8). Nor does onset for this airfoil have a hysteresis in Mach number, and it does not apparently display the sensitivity of the shock bu et range on Reynolds number that is evident for the 18-percent circular-arc airfoil (ref. 8). Experimental measurement of shock bu et onset is of course complicated by external e ects such as wind tunnel noise, Reynolds scaling, and walls. However, experiment and several computations show transonic Mach numbers within an angle-of-attack envelope for the NACA 0012 airfoil where shock motion intensity and chordwise extent change from a localized shock oscillatory (or steady in the case of the computations) to a large-scale motion displaying limit cycle behavior. This has been studied experimentally for the NACA 0012 airfoil in reference 8, which represents an e ort to provide quality steady and unsteady lifting surface results with minimal interference e ects. In the test of reference 8, tunnel walls were contoured to match free air streamlines for nominal test conditions derived from Navier-Stokes computations. A much less extensive study of the NACA 0012 airfoil has been presented in reference 9, which also reveals shock bu et behavior. Reference 10, in contrast, presents experimental data from a slotted-wall wind tunnel for the same airfoil and range of conditions that are steady. Experimental studies through the onset Mach number range for several supercritical airfoils con rm that these airfoils can also experience shock bu et (refs. 9, 11, 12, and 13). In summary, although di culties remain in verifying onset and sorting out the various extraneous e ects, it is clear that under the right conditions some conventional and supercritical airfoils experience shock bu et. The computations vary somewhat for airfoils other than the 18-percent circular arc. Steady interactive boundary layer and Navier-Stokes solutions of the NACA 0012 airfoil have been previously published (e.g., refs. 14 and 15) and compared with the steady data of reference 10. Shock bu et interactive boundary layer computations for the same airfoil have been shown in previous publications with a time accurate integral boundary layer and the classical transonic small disturbance (TSD) equation (ref. 16) and a TSD using an Euler-like streamwise ux and a steady integral boundary layer (ref. 7). This last reference has identi ed the onset behavior for the NACA 0012 airfoil as a Hopf bifurcation point where the solution changes from an equilibrium point to a limit cycle solution. The critical point or onset location is the point having a zero amplitude limit cycle solution. Whether a supercritical airfoil behaves like a conventional airfoil or like the 18-percent circular-arc airfoil in this and other respects, however, remains to be ascertained. But the fact that the interactive boundary layer approach has given shock bu et onset for the NACA 0012 airfoil that compares well with the onset of reference 8 (e.g., refs. 7 and 17) encourages one to pursue further investigation with this method. Although it is generally accepted that the boundary layer assumption is violated in many problems of this type, the interactive boundary layer method does make possible a broader study of the problem due to its e ciency. That is done here with a recently developed interactive boundary layer method using the CAP-TSD (Computational Aeroelasticity Program) potential code with a modi ed streamwise ux (ref. 18) and an unsteady compressible boundary layer solved in nite di erence form. This method is shown to give very accurate results for many widely used attached and shock-separated steady test conditions. Comparisons of wall shear, boundary layer velocity pro les, and pressure distributions are shown to match well with experiment and Navier-Stokes results. In view of the sensitivity of the 18-percent circular-arc airfoil shock bu et to turbulence and ow model, comparisons of shock bu et onset for the NACA 0012 airfoil using several turbulence models are shown. Results are presented for several variations of the k-! turbulence model. The k-! turbulence model embodies more ow physics than oneor zero-equation turbulence models and is applicable to boundary layer dominated ows. It allows solution of the turbulence equations to the wall including the viscous sublayer and also allows modeling of free-stream turbulence and the e ect of varying surface roughnesses. This allows the e ect of these modeling parameters on shock bu et onset to be investigated. The shear stress transport form of the model is used to compute details of the shock bu et of the NASA SC(2)-0714 airfoil (ref. 13). Comparisons with the experimental shock bu et data at high Reynolds numbers of reference 13 are shown at several Reynolds numbers; this represents the rst numerical study of the e ect of turbulent boundary layer Reynolds number scaling on shock