Variations in flow field physics caused by algebraic turbulence model modifications for a supersonic 2-D open cavity

This paper explores the effects of turbulent viscosity on the computation of supersonic open cavity flow physics. Several modifications of the Baldwin-Lomax turbulence model are used, including the relaxation, first peak, multiple-wall, Baysal's, Subs', inverse Subs', and laminar in and above the cavity approaches as well as laminar flow. Results from these methods are presented in terms of surface and field data properties. It was found that the majority of the models overpredict the time-averaged sound pressure level as compared to experimental data, with the exception of the standard Baldwin-Lomax model. The flow field results depicted as simulated Schlieren and vorticity contours show similar flow features, with the exceptions being the standard Baldwin-Lomax and laminar in and above the cavity models. In addition, a distinct link between the eddy viscosity and the quantitative results for sound pressure level and dominant frequency was observed. An investigation was also conducted into the number of characteristic times needed to purge transients. Limit cycle convergence was observed to occur between 42 and 114 characteristic times, much longer than previously reported by other researchers. INTRODUCTION Cavities can be found on high speed military aircraft in many places including weapons bays and wheel wells. Flow over a cavity can be classified in several distinct ways. The first distinct flow is the closed cavity, which typically has a length-to-depth ratio (L/D) larger than 13. These flows are known to have high drag coefficients and heat transfer characteristics which make them less desirable for design purposes. The second type of flow is the open cavity, which occurs when L/D is less than 10. In this flow the shear layer bridges the cavity and periodically impinges * Student Member AIAA, Research Assistant. t Senior Member AIAA, Assistant Professor of Aerospace Engineering & Engineering Mechanics. t Member AIAA, Associate Professor of Aerospace Engineering & Engineering Mechanics. Copyright © 1997 by Chung-Jen Tam, Paul D. Orkwis and Peter J. Disimile. Published by the American Institute of Aeronautics & Astronautics, Inc. with permission. on the aft wall. Supersonic open cavities are known to have unsteady flow fields caused by pressure oscillations within the cavities and to shed vortices across the mouth of the cavities. The design of an open cavity can result in potential damage to the aircraft structure and lead to possible missile fly back after release. The last category of cavities, with 10 < L/D < 13, are classified as transitional and have behaviors which combine the open and closed descriptions, as discussed in detail by Tracy and Plentovich [1]The details and features of open cavity flow fields have been reported by many experimental researchers, including Rossiter [2], Heller and Bliss [3] and Rockwell and Naudascher [4]. Rossiter used an edge-tone analysis to derive an empirical formula that predicts the resonant frequencies inside the open cavities. Heller and Bliss used water table visualizations to conclude that the unsteady flow field inside open cavities is caused by pressure wave oscillations that are initiated by the impingement of the shear layer on the aft wall. Rockwell and Naudascher's experimental results stipulated that shed vortices impinge on the aft wall of the cavity and create an upstream moving pressure wave. This pressure wave eventually hits the forward wall and causes the shear layer to initiate another shed vortex, thus closing the feedback loop. Recently, the authors [5] used numerical simulations to illustrate the physical mechanism and describe the driving phenomena behind the pressure oscillations which occur in a supersonic open cavity. Their mechanism combines both the Heller and Bliss, and Rockwell and Naudascher descriptions and illustrated the simultaneous contribution of the pressure waves and vortex interactions to the oscillation cycle. Besides the efforts described above, considerable computational research has been performed in the study of supersonic open cavity resonance cycles. Unlike the experimental efforts, the primary issue that computational researchers face for this problem is the choice of a turbulence model. Algebraic turbulence models [6,7,8,9,10,11], two-equation turbulence models [12,13, 14], and recently, VLES [15] have been used in supersonic open cavity simulations. Rizzetta [8] applied the BaldwinLomax model with the relaxation modification and obtained two-dimensional results which overpredicted the time-averaged sound pressure levels along the cavity floor by about 8 dB. Time series of velocity vectors were presented that indicated the generation of a vortex which formed at the forward cavity lip and was then convected downstream. Instantaneous pressure contours were presented to illustrate the upstream and downstream pressure waves. Suhs et al. [14] used a two-equation turbulence model with wall functions and reported that the time-averaged sound pressure levels along the cavity floor differed by less than 2 dB compared to experimental data, however, no field data were presented. Although many different turbulence models were used in the above studies, few reasons were given for their choice and little discussion was offered regarding how the eddy viscosity influences the predictions of the surface and field properties. It was therefore necessary for a comparison to be made between the models so that a reasoned choice could be made. Another issue which needed to be explained is the computational run time required to purge transients. Only a few researchers addressed this question for open cavity computations. Suhs [7] suggested that 6 characteristic times are sufficient to wash out the transients. In an earlier work, the authors [16] used this idea and sampled data from 6 to 18 characteristic times, however, they later realized [5] that the solution had not attained a limit cycle oscillatory state. It was found that a longer computational time was needed to wash out the transients. This concept was expanded in the current work where a convergence test was performed to ensure that the results had in fact reached the limit cycle. The current analysis is similar to that done by Zhang [13]. Comparisons between the short duration and limit cycle converged test results are included so that the reader might judge the benefits of employing the longer run time. As stated above, the main goal of this research was to investigate the changes in flow physics caused by using the Baldwin-Lomax turbulence model [17] and several of its modifications for supersonic 2-D open cavity simulations. The computational results were compared with the experimental data of Disimile and Orkwis [18] in terms of time-averaged sound pressure levels and dominant frequency. The following sections discuss the governing equations; the numerical scheme; the tested algebraic turbulence models; the results, including the limit cycle convergence test; and finally the conclusions drawn from the study. NUMERICAL APPROACH The equations solved discretely in the current work are the thin layer Navier-Stokes (TLNS) equations in which the viscous terms in both the x and y directions are retained. The TLNS equations are solved using an implicit approximate factorization, multi-block finite volume scheme developed by Simpson and Whitfield [19] and modified by Tam et al. [16]. The current procedure was detailed in earlier works by the authors [5,16] and will not be included in the present discussion. The scheme utilizes Roe flux difference splitting for the residual and an approximate Steger-Wanning splitting for the Jacobian matrix. It is third order accurate in space and second order accurate in time when a Newton-like subiteration procedure is employed [19]. Four subiterations per global iteration were used, as per Simpson and Whitfield. Boundary and Initial Conditions Boundary conditions for the computation were prescribed variables on the inflow plane, no-slip adiabatic walls on solid surfaces, and extrapolation on the outflow boundaries. The upstream profile was determined by using a flat plate computation from which the proper profile was chosen by finding the x-location with the same momentum thickness as that found experimentally [18]. Initial conditions for the flow field external to the cavity were obtained by extrapolating the inflow profile, i.e., a constant boundary layer profile was set along the flat plates and above the cavity. Flow inside the cavity was set by using the wall pressure and density of the flat plate. The nondimensional velocity components were prescribed via the equations pu=0.5, pv=pw=0.0, which represented a slow uniform flow in the x-direction inside the cavity. The total energy was then defined from these quantities. Note that these are somewhat arbitrary settings for the flow conditions inside the cavity, as other conditions are possible. Grids The block grid used in this study is the same as that used previously by the authors [5,16]. The dimensions were 66x55, 66x120 and 66x55 for the upstream, cavity and downstream blocks, respectively, with points clustered along all walls. This choice was based on a previous grid resolution study, in which half and double x and y direction grid point densities were tested with only marginally different results. Data Sampling A sampling rate 4.292x10" sec corresponding to a nondimensional time step of 1x10" was employed for surface pressure, while 8.584xlO~sec was used to sample the field properties for the entire flow. The former corresponds to taking data each iteration while the latter to data taken every 200 iterations. The time histories of the surface pressure were used to compute the time-averaged sound pressure levels ( SPL ) at every grid point along the cavity walls via the equation