EFFECT OF REYNOLDS NUMBER ON THE UNSTEADY FLOW AND ACOUSTIC FIELDS OF SUPERSONIC CAVITY

Proceedings of FEDSM 03 4TH ASME_JSME Joint Fluids Engineering Conference Honolulu, Hawaii, USA, July 6 11, 2003 FEDSM2003-45473 EFFECT OF REYNOLDS NUMBER ON THE UNSTEADY FLOW AND ACOUSTIC FIELDS OF SUPERSONIC CAVITY A. Hamed, D. Basu and K. Das Department of Aerospace Engineering and Engineering Mechanics University of Cincinnati, Cincinnati, OH-45219 ABSTRACT Numerical simulations are conducted to study the flow and acoustic fields for unsteady supersonic turbulent flow over an open cavity using Detached Eddy Simulations for two different Reynolds numbers. Results are presented for pressure fluctuations history, vorticity iso-surfaces and turbulent kinetic energy and sound pressure levels spectra. The results reveal higher sound pressure levels (SPL), and finer scale structures within the cavity at the higher Reynolds number INTRODUCTION High speed flows over open cavities produce complex unsteady interactions, which are characterized by a severe acoustic environment. Such flow fields are comprised of both broadband small-scale fluctuations typical of turbulent shear layers, and discrete resonance whose frequency and amplitude depend upon the cavity geometry and external flow conditions. Recent research on cavity flow control [1,2,3,4,5] underlines the need for understanding the unsteady flow mechanisms involved in acoustic emissions and their control-. Simulation of cavity flow field is challenging because of the massive separation regions and the complex coupling between turbulence and acoustics through a feedback process that leads to large amplitude; self sustained flow oscillations [1]. Early correlations of cavity acoustic resonance frequency were developed by Rossiter [6] based on the linear acoustic theory. However, the amplitude of pressure oscillations and the SPL is more challenging to predict. Numerical investigations of flow over open cavities based on time dependent Reynolds Averaged Navier- Stokes (URANS) simulations have been carried out for both two-dimensional [7-10] and three-dimensional [11,12] configurations. In general, these RANS simulations did not capture the cavity acoustic field. This was attributed to excessive turbulent dissipation in the used turbulence models. In addition, some studies reported that some two-equation turbulence models damped the cavity flow-field unsteadiness [13,14]. Direct Numerical Simulations (DNS), which do not recourse to conventional turbulence models have been limited by computational resources requirements to laminar flow over 2- D cavity. Large Eddy Simulations (LES) reduce the grid resolution requirements by resolving the larger, energy containing scales and modeling the smaller, more homogeneous scales. Sinha et al. [13,17] developed a hybrid LES/RANS code and to perform 2-D cavity flow simulations and compared the VLES results with the URANS results. Rizzetta et al.[18] performed LES to study L/D = 5 cavity flow control on a parallel computing platform of IBM SP3 with 254 processors. The simulations which were conducted at Re = 0.12 10 6 /ft required perturbation of the incoming flow variables to produce transition, and sufficiently long runs to generate statistically meaningful data for the LES simulations at the upstream boundary. Detached Eddy Simulations (DES) were initiated by Spalart et al. [19], who proposed that a single turbulence model can be used to function as a sub-grid scale model in the LES regions and as turbulence model in the RANS regions [20]. Hence, DES combined the fine-tuned RANS technology in the attached boundary layers and the power of LES in the separated region to enable three-dimensional unsteady Navier-Stokes simulations at high Reynolds number with realistic computational resources. Strelets [21] and Bush et al. [22] developed a Mentor s SST [23] based DES model through the introduction of an equivalent length scale, and implemented it into the WIND code [24]. Hamed et al. [14] used the SST based DES to investigate 3-D unsteady flow and acoustic fields in supersonic cavity. Our previous DNS investigations [16] demonstrated a dramatic increase in the pressure oscillations amplitude and the associated sound pressure level with free stream Mach number. However these DNS simulations were limited to laminar 2-D cavity. Subsequent DES investigations [14] revealed both broadband as well as tonal frequencies in the computed SPL spectra. In the present paper, the unsteady supersonic 3D cavity flow is investigated using SST [23] based DES model to 1 Copyright 2003 by ASME

determine the effects of Reynolds number on the flow and acoustic fields. METHODOLOGY Unsteady compressible viscous flow solutions for the Navier-Stokes equations in conservation law form were obtained using the WIND codes [24]. The SST based DES model was used in the present investigation. Based on prior RANS experience of the lead author [25] who assessed both S- A [20] and Menter s SST [23] models in the simulations of supersonic flow with larger separated flow regions in overexpanded supersonic nozzles. The solution domain for the 20 4 2 inch cavity, allowed the turbulent boundary layer to develop on the flat plate upstream of the cavity s forward bulkhead. In order to maintain the incoming boundary layer thickness at 10% of the cavity depth D, the upstream plate length was 3D for the Reynolds number of 0.12 10 6 /ft and 4.5D for the higher Reynolds number of 0.60 10 6 /ft. Supersonic free stream conditions were specified at the upstream boundary and first order extrapolation was applied at the upper boundary, 3D above the cavity opening and at the downstream boundary, 3D behind the rear bulkhead. Periodic boundary conditions were specified at the lateral boundaries in the span-wise direction. The aspect ratio in the computational grid varied between 1 and 5 in all three directions. The grid was packed near the walls to maintain y + < 3 for the first grid point. The parallel computations were performed on a cluster of Linux machines using 205 94 40 grid points in the stream-wise, normal and span-wise directions for Re = 0.12 10 6 /ft and 253 112 80 grid points for Re = 0.6 10 6 /ft. Additional details about the computational grid are given in table 1. The third order upwind-biased Roe scheme was used for spatial discretization with TVD operator to suppress the numerical instabilities in the shear layer and near the shock waves. Explicit time marching scheme with Newton like subiterations was used for temporal advancement. The DES simulations were initiated in the unsteady mode and continued over 200,000 constant time-steps. It took 100,000 time steps to purge out the transient flow and establish resonance and the remaining 100,000 time steps to compute 20 cycles. RESULTS AND DISCUSSIONS The DES simulations were performed for the L/D = 5 and W/D = 0.5 cavity at free stream Mach number of 1.19 and Reynolds numbers of 0.12 10 6 /ft and 0.60 10 6 /ft. Computational results are presented for the pressure fluctuations, sound pressure levels and turbulent kinetic energy spectra, Mach number contours and vorticity iso-surfaces. Figure 1 presents sample pressure fluctuations histories near the front and rear bulkheads at the cavity mid-span. One can see that, in general, the pressure fluctuations are chaotic and the amplitude is higher at the rear bulkhead. The SPL spectra were computed from the pressure fluctuations based on 2048 sample data points. Figure 2 compares the computed sound pressure level (SPL) spectra variation along the cavity opening mid-span for the two Reynolds numbers. One can see that the SPL spectra have a broadband content with a wide range of frequency scales and that the SPL level increases towards the rear bulkhead. According to figure 2, the maximum SPL increased from 145 db to 160 db with the increase in Reynolds number from 0.12 10 6 /ft. to 0.60 10 6 /ft. Figure 3 shows sample turbulent kinetic energy (TKE) spectra in the shear layer near the cavity s rear bulkhead. One can see that the spectra have a broadband content over a wide range of frequency. The classical 5/3 Kolmogorov slope is shown in the figure for reference. Sample results for the axial vorticity iso-surfaces at Re=0.60 10 6 /ft are presented in figure 4. This figure demonstrates eddies formation within the cavity since the axial vorticity is zero in the incoming flow upstream of the cavity. The span-wise vorticity iso-surfaces are compared for the two Reynolds numbers in figure 5. They show the roll up of the shed vortex and the impingement of the shear layer on the rear bulkhead. Figure 5 shows a tangible deeper range of fine scale structures within the cavity at the high Reynolds number and a reduction in the size of the shed vortex. Figure 6 presents Mach number carpet plots at mid-span for the two Reynolds numbers. The figures show that an oblique shock is formed upstream of the cavity with a subsequent increase in the incoming boundary layer thickness due to shock interactions. One can see that the thickening of the boundary layer and the subsequent deflection of the shear layer is greater at the low Reynolds number. In addition to the different eddy structures inside the cavity, figure 5 shows that the Reynolds number also affects the shock and expansion waves pattern and strength outside the cavity. CONCLUSIONS Detached Eddy Simulations (DES) were performed to study the unsteady three-dimensional flow and acoustic fields of an open L/D = 5 cavity at free-stream Mach number of 1.19 for two different Reynolds numbers. The presented results reveal the basic flow features, including the vortex shedding, shock waves, the small scale eddy formation within the cavity and the three-dimensional flow characteristics. The results showed an increase in the range of predicted fine scale structures and an increase in the predicted SPL with the increase in Reynolds number ACKNOWLEDGMENTS This research was supported by a DAGSI grant AFRL/VA- UC-00-01; Mr. M. Stanek and Dr. M Visbal, technical monitors. The authors would like to thank Mr. Robert Ogden of UC for providing technical support with the linux cluster. 2 Copyright 2003 by ASME

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Table 1: Summary of computational parameters for numerical simulations Reynolds Number Grid y minimum Grid Points Grids within BL ( ) Grid points within cavity 0.12 10 6 /ft 205 94 40 1 10-3 D 770,800 15 184,040 0.60 10 6 /ft 253 112 80 6 10-4 D 2,266,800 20 396,800 Near Front Bulkhead Near Rear Bulkhead Figure 1 Pressure fluctuation history (Re = 0.60 10 6 /ft) Re= 0.12 10 6 /ft Re=0.60 10 6 /ft Figure 2 Sound Pressure Level spectra 4 Copyright 2003 by ASME