Navier-Stokes simulation of self-excited shock induced oscillations

Abstract

A numerical study investigating the ability of the full massaveraged Navier-Stokes equations to accurately predict selfexcited shock induced oscillations on an eighteen per cent thick circular-arc airfoil is presented. An explicit cell-vertex-centred Navier-Stokes code is employed in conjunction with a hyperbolic C-grid generator. Turbulence closure is accomplished using the zero equation algebraic Baldwin-Lomax model. The code accurately predicts the shock induced oscillation onset boundary, reduced frequencies and hysteresis region. Comparing these results with those obtained from a thin-layer version of the code highlights the limitations of the latter technique. Only by employing the full mass averaged Navier-Stokes equations, operating on a suitably fine grid, can the dynamic shear layers be adequately resolved. Introduction The need to accurately model the evolution of unsteady aerodynamic phenomena, such as buffet, flutter, limit cycle oscillations and buzz has thrust unsteady computational fluid dynamics to the forefront of modern research. The classical experiments performed by Tijdeman' on a NACA64A006 airfoil with a trailing-edge flap detected three types of shock motion. Tijdeman's results have proven equally valid for rigid airfoil studies and form the basis for the classification of self-excited shock induced oscillations, or SIO. Tijdeman type A SIO is were the shock wave remains distinct throughout the oscillation with a cyclic change in both the shock wave strength and location. In Tijdeman type B SIO the shock wave vanishes for part of the cycle, normally whilst the shock is propagating upstream. Finally, Tijdeman type C SIO describes the shock wave motion whereby the shock remains distinct as it propagates upstream past the leading-edge and into the on coming flow. Throughout the past two decades extensive experimental investigations into periodic flow over varying thickness circulararc airfoils have been p e r f ~ r m e d ~ ~ . These experiments, which were conducted over a wide range of free stream Mach and Reynolds numbers, have detected all three types of SIO. References 2 and 3 identified Tijdeman type C SIO, with a reduced frequency of, k = 0.49, for an 18 % thick circular-arc airfoil at zero incidence over a narrow range of Mach numbers, namely: 0.73 < M, < 0.78. Small regions of Tijdeman type A SIO were also detected in the extremities of the periodic flow band. Figure 1 indicates that the extent of this periodic band decreases considerably as the Reynolds number approaches 'ResearchTechnicalEngineer, Nacelle Systems, Member AIAA. Copyright 1995 by Mark Gillan. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. 3 x 10'. A similar phenomenon was detected by Mabey et a1.' whereby the SIO vanished within the Reynolds number range of: 3 x lo6 < Re, < 5 x 10'. Furthermore, the region of flow hysteresis which McDevitt et al.' discovered was later verified by Seegmiller et aL4 and LevyS (fig.1). Finally, Gibbg, who conducted a comprehensive study of flow over a 14% thick circular-arc airfoil, detected strong Tijdeman type B SIO, with small regions of Tijdeman type A SIO, occurring within a narrow Mach number band, namely: 0.84 < M, < 0.88. Following various steady-state c~mputations~.'~-, Levy5 eventually produced the first unsteady Navier-Stokes computation over an 18 7% thick circular-arc airfoil that was capable of reproducing the intrinsically asymmetric periodic flow. Levy used a modified version of the code employed by DeiwertloaL' which included solid-wall test section inviscid boundary conditions. Despite employing an extremely coarse 78 X 35 grid and a 1 % chord nose radius (to alleviate numerical difficulties), the impulsively started computation produced SIO with a reduced frequency of, k=0.4, comparable to that produced experimentally by McDevitt et a1.I of, k=0.49. Furthermore, in an attempt to show that the periodic flow solution was the result of viscous effects alone and not due to spurious numerical inaccuracies, Levy ran the periodic test case for various different configurations. Both an inviscid test case and a viscous computation, with a 114 chord trailing-edge splitter plate, which effectively prevents pressure wave communication across the wake, produced steady flow solutions, thereby corroborating Levy's hypothesis. Seegmiller et aL4 developed Levy's research further by presenting an in-depth explanation regarding the exact nature of the periodic mechanism. Furthermore, they4 astutely recognised that the period of the SIO depended on the time taken for the flow to adjust to, and counteract, the airfoil's effective change in camber due to the asymmetry of the shock-induced separation. This reasoning is supported by the earlier experimentation of ~ i n k e ' ~ . Subsequent Navier-Stokes computations over circular-arc airfoils have been performed by Steger, Edwards and ThomasL4, Gerteisenl' and illa an^^^'^. Steger computed periodic flow with a reduced frequency of, k=0.41, at a free stream Mach number of, M, =0.783. Edwards and tho ma^'^ employed anupwind-biased Navier-Stokes scheme and computed Tijdeman type B SIO with a reduced frequency of, k=0.406, at a free-stream Mach number comparable to Steger's, of M, =0.78. GerteisenLs failed to reproduce either Tijdeman type B, or type C, SIO. Gertesin believed that this was due to a combination of insufficient shock resolution, inadequate turbulence modelling and the existence of transitionally dominated flow. Gillan16 computed periodic flow over an 18 % thick circular-arc airfoil at 0' angle of attack and a free stream Mach number of M, =0.771 (see fig.2). This Mach number lies within the periodic region which was detected both experimentally and computationally by Levy5 as depicted in fig. 1. No attempt was made during the computational analysis to detect the hysteresis region depicted in this figure. A 320x64 cells was employed, with 256 cells placed on the airfoil's surface. A far-field boundary of 20 chord lengths was enforced, with the initial near-wall normal grid spacing corresponding to a value of y+ < 2 for the assigned Reynolds number of Re, = 1 1 X 10.~. Moreover, a 1 % leading-edge radius was introduced in an attempt to avoid any unnecessary computational difficulties. The computational test case, which was impulsively started from free stream conditions, produced an unsteady periodic Tijdeman type B motion with a reduced frequency of, k = 0.396. Although this reduced frequency is approximately 21 % lower than that obtained experimentally by McDevitt et aL2 it compares favourably with the computed values of 0.4 and 0.406 obtained by Levy5 and Edwards and Thomasr4 respectively. In an attempt to introduce some non-equilibrium history effects into periodic flow computations various pre-eminent researchers have temporally 'abandoned' the use of NavierStokes solvers. Le Balleur and Girodroux-La~igne'~'~ used a small-disturbance potential method with a two-equation integral viscous solver to compute SIO with a reduced frequency of, k=0.34, at a free-stream Mach number of, M ~ 0 . 7 6 . Although this frequency is significantly lower than previous results, it indicates the ability of a non-Navier-Stokes solver to qualitatively reproduce unsteady periodic flows. Finally, Edwards2 has recently employed a new lag-entrainment integral boundary layer method coupled with a transonic small disturbance potential code to compute periodic flow over a range of configurations. Edwards accurately modelled the SIO period for an 18% thick circular-arc airfoil, detecting a seduced frequency of, k=0.47, at a free-stream Mach number of, M, =0.76. Furthermore, he reproduced, for the first time, the hysteresis region which was initially discovered experimentally by McDevitt et al.'. The primary objective of this paper is to investigate the ability of the full mass-averaged Navier-Stokes equations, operating on a suitably fine grid, to accurately predict SIO on an 18% thick circular-arc airfoil. In order to perform this analysis a hyperbolic grid generation code and an explicit finite volume cell vertex-centred Navier-Stokes code have been employed. The following section briefly describes both the numerical model and the results. Numerical Model A recently developed explicit Navier-Stokes finite volume code (MGENS2D) has been used in conjunction with an orthogonal boundary conforming hyperbolic C-grid. A detailed description of both the Navier-Stokes scheme and the grid generation code (MGHYPR) is given in ref. 16. If Q represents an arbitrary control volume, with a domain boundary, an, and a unit outward normal, n, then, neglecting body forces and external heat addition, the integral form of the 2-0 non-dimensional mass-averaged Navier-Stokes equations may be written as: where W is the vector of the conserved quantities: with p,u,v and E denoting the density, the cartesian velocities and the specific total internal energy respectively. The flux tensor P is then split into a convective part, Few, and a viscous part, F,,,: P = PCMV Fam