8th AIAA/CEAS Aeroacoustics Conference June 17 19, 2002 Breckenridge, CO

Transcription

1 AIAA 598 Recent Progress Towards a Large Eddy Simulation Code for Jet Aeroacoustics A. Uzun, G. A. Blaisdell, and A. S. Lyrintzis School of Aeronautics and Astronautics Purdue University West Lafayette, IN th AIAA/CEAS Aeroacoustics Conference June 7 9, Breckenridge, CO For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 8 Alexander Bell Drive, Suite 5, Reston, VA

2 Recent Progress Towards a Large Eddy Simulation Code for Jet Aeroacoustics A. Uzun, G. A. Blaisdell, and A. S. Lyrintzis School of Aeronautics and Astronautics Purdue University West Lafayette, IN 4797 In this paper, we present 3-D Large Eddy Simulation(LES) results for turbulent round jets. Our recently developed LES code is part of a Computational Aeroacoustics(CAA) methodology that will eventually couple the near field unsteady LES data with an integral acoustic formulation for the far field noise prediction of turbulent jets. The code employs high-order accurate compact differencing together with implicit spatial filtering and state-of-the-art non-reflecting boundary conditions. The classical Smagorinsky subgrid-scale(sgs) model is used for representing the effect of the unresolved scales on the resolved scales. Preliminary results obtained in our simulations seem encouraging when compared with experimental data. Introduction THE turbulent jet noise problem still remains one of the most complicated and difficult problems in aeroacoustics. There is a need for substantial amount of research in this area that will lead to improved jet noise prediction methodologies and aid in the design process of aircraft engines with low jet noise emissions. With the recent improvements in the processing speed of computers, the application of Direct Numerical Simulation(DNS) and Large Eddy Simulation(LES) to jet noise prediction methodologies is becoming more feasible. The first DNS of a turbulent jet was done for a Reynolds number, supersonic jet at Mach.9 by Freund et al. The computed sound pressure levels were compared with experimental data and found to be in good agreement with jets at similar convective Mach numbers. Freund also simulated a Reynolds number 36, Mach.9 turbulent jet using 5 million grid points, matching the parameters of the experimental jet studied by Stromberg et al. 3 Excellent agreement with the experimental data was obtained for both the mean flow field and the radiated sound. Such results clearly show the attractiveness of DNS to the jet noise problem. However, due to a wide range of length and time scales present in turbulent flows, DNS is still restricted to low-reynolds-number flows in relatively simple geometries at present. DNS of high-reynolds-number jet flows of practical interest would necessitate tremendous resolution requirements that are far beyond the reach of the capability of even the fastest supercomputers available today. Therefore, as frustrating as it might seem, turbu- Graduate Research Assistant, Student Member AIAA. Associate Professor, Senior Member AIAA. Professor, AIAA Associate Fellow. Copyright c by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. lence still has to be modelled in some way to do simulations for problems of practical interest. LES, with lower computational cost, is an attractive alternative to DNS. In an LES, the large scales are directly solved and the effect of the small scales or the subgrid scales on the large scales are modelled. The large scales are generally much more energetic than the small ones and are directly affected by the boundary conditions. The small scales, however, are usually much weaker and they tend to have more or less a universal character. Hence it makes sense to directly simulate the more energetic large scales and model the effect of the small scales. LES methods are capable of simulating flows at higher Reynolds numbers and many successful LES computations for different types of flows have been performed to date. Since noise generation is an unsteady process, LES will probably be the most powerful computational tool to be used in jet noise research in the foreseeable future since it is the only way, other than DNS, to obtain time-accurate unsteady data. One of the first attempts in using LES as a tool for jet noise prediction was carried out by Mankbadi et al. 4 They employed a high-order numerical scheme to perform LES of a supersonic jet flow to capture the time-dependent flow structure and applied Lighthill s theory 5 to calculate the far field noise. The first application of Kirchhoff s method in combination with LES to far-field jet noise was conducted by Lyrintzis and Mankbadi. 6 LES has also been used together with Kirchhoff s method 7 for the noise prediction of a Mach. jet by Gamet and Estivalezes 8 as well as for a Mach. jet with Mach. coflow by Choi et al. 9 with encouraging results obtained in both studies. Zhao et al. did LES for a Mach.9, Reynolds number 36 jet obtaining mean flow results that compared well with Freund s DNS and experi- of American Institute of Aeronautics and Astronautics Paper 598

3 mental data. They also studied the far field noise from a Mach.4, Reynolds number 5 jet. They compared Kirchhoff s method results with the directly computed sound and observed good agreement. Morris et al. simulated high speed round jet flows using the Nonlinear Disturbance Equations(NLDE). 5 In their NLDE method, the instantaneous quantities are decomposed into a time-independent mean component, a large-scale perturbation and a small-scale perturbation. The mean quantities are obtained using a traditional Reynolds Averaged Navier-Stokes(RANS) method. As in LES, they resolved the large-scale fluctuations directly and used a subgrid-scale model for the small-scale fluctuations in their unsteady calculations. In a recent paper, 6 they also report the noise from a supersonic elliptic jet. Chyczewski and Long 7 conducted a supersonic rectangular jet flow simulation and also did far field noise predictions with a Kirchhoff method. Boersma and Lele 8 did LES for a Mach.9 jet at Reynolds numbers of 36 and 36 without any noise predictions. More recently, Bogey et al. 9 simulated a Reynolds number 65, Mach.9 jet using LES and obtained very good mean flow results, turbulent intensities as well as sound levels and directivity. Constantinescu and Lele did simulations for a Mach.9 jet at Reynolds numbers of 36 and 7. They directly calculated the near-field noise using LES. Their mean flow parameters and turbulence statistics were in good agreement with experimental data and results from other simulations. The peak of the power spectra of the sound waves was also captured accurately in their calculations. In general, the LES results in the literature are encouraging and show the potential promise of LES application to jet noise prediction. Except for the study of Choi et al. 9 which is not a well resolved LES, the highest Reynolds numbers reached in the LES simulations so far are still below those of practical interest. Simulations of jets at higher Reynolds numbers would be very helpful for analyzing the broad-banded noise spectrum at such high Reynolds numbers. On the other hand, none of the LES studies done so far have predicted the high-frequency noise associated with the unresolved scales. Obviously, a subgrid-scale model for noise is highly desirable. Piomelli et al., Rubinstein and Zhou, Seror et al., 3 and Zhao et al. 4 did some initial work on the investigation of the contribution of small-scales to the noise spectrum. More research in this area that will lead to development of subgrid-scale acoustic models is needed. With this motivation behind the current study given, we now present the two main objectives of this research:. Development and validation of a versatile 3-D LES code for turbulent jet simulations. A highorder accurate 3-D LES code utilizing a robust of dynamic subgrid scale model is being developed. Generalized curvilinear coordinates are used, so the code can be easily adapted for calculations in several applications with complicated geometries. Since the sound field is several orders of magnitude smaller than the aerodynamic field, we make use of high-order accurate non-dissipative compact schemes which satisfy the strict requirements of CAA. Implicit spatial filtering is employed to get rid of the high-frequency oscillations resulting from unresolved scales and boundary conditions. Non-reflecting boundary conditions are imposed on the boundaries of the domain to let outgoing disturbances exit domain without spurious reflection. A sponge zone that is attached downstream of the physical domain damps out the disturbances before they reach the outflow boundary. Significant progress towards this first goal has already been made. The current version of the LES code has the standard Smagorinsky model implemented. As will be shown in the results section, promising results were obtained with our current code.. Accurate prediction of the far field noise. Even though the sound is generated by a nonlinear process, the sound field itself is known to be linear and irrotational. This implies that instead of solving the full nonlinear flow equations out in the far field for sound propagation, one can use a cheaper method instead, such as Lighthill s acoustic analogy 5 or surface integral acoustic methods such as Kirchhoff s method 7 and the porous Ffowcs- Williams Hawkings (FW-H) equation. 5, 6 In our future studies, the near field provided by the LES code will be supplied to a surface integral acoustic method for computing noise propagation to the far field. Results will be compared with experimental as well as computational data available in the literature. Governing Equations Large Eddy Simulation(LES) can be thought of as a compromise between Direct Numerical Simulation(DNS) and the Reynolds Averaged Navier- Stokes(RANS) equations. In a DNS, all the relevant scales of turbulence have to be directly computed, whereas in a RANS calculation, all the relevant scales of turbulence are modelled. In an LES, the flowfield is decomposed into a large-scale or resolved-scale component ( f) and a small-scale or subgrid-scale component (f sg ), f = f + f sg () The large-scale component is obtained by filtering the entire domain using a grid filter function, G, as follows f( x) = G( x, x )f( x ) d x () American Institute of Aeronautics and Astronautics Paper 598 V

4 The filtering operation removes the small-scale or the subgrid-scale turbulence from the Navier-Stokes equations. The resulting governing equations are then solved directly for the large-scale turbulent motions while the effect of the subgrid-scales are computed using a subgrid-scale model, such as the classical Smagorinsky model 7 or the more sophisticated dynamic model proposed by Germano et al. 8 For compressible flows, the large-scale component is written in terms of a Favre-filtered variable f = ρf ρ (3) The Favre-filtered unsteady, compressible, nondimensionalized Navier-Stokes equations formulated in conservative form are solved in this study. The continuity equation is given by The momentum equation is ρũ i t + ρũ iũ j x j ρ t + ρũ i x i = (4) + p ( σ ij τ ij ) = (5) x i x j where the resolved shear stress tensor is given by σ ij = µ ( Sij Re 3 S ) kk δ ij (6) with the Favre-filtered strain rate tensor defined as S ij = ũj + ( ũ ) i (7) x i x j The subgrid-scale stress tensor is modelled as τ ij = C ρ SM ( Sij 3 S kk δ ij ) + 3 C I ρ S M δ ij (8) where S M = ( S ij Sij ) / (9) The resolved heat flux is [ ] µ T q i = (γ )M () r ReP r x i and the subgrid-scale heat flux is modelled as The ideal gas relation Q i = C ρ SM P r t p = T x i (3) ρ T (4) γm r and Sutherland s law for the molecular viscosity µ = µ( T ) are also used in these equations with the Sutherland constant and the reference temperature chosen as K and 88K, respectively. The three coefficients for the subgrid-scale models are C, C I and the turbulent Prandtl number, P r t. In the standard Smagorinsky model with compressibility corrections, these coefficients are set to constant values based on previous studies. It is also possible to represent these coefficients as functions of space and time, and compute them dynamically as part of the 8, 3 flowfield simulation. In order to do direct numerical simulations using these Favre-filtered equations, all spatially filtered variables can be replaced by their unfiltered forms and the subgrid-scale stress tensor and the subgrid-scale heat flux terms are simply set to zero. The equations given so far are formulated in Cartesian coordinates. For problems involving complex geometries, the form of the governing equations in generalized curvilinear coordinates should be used. Extension of the equations from Cartesian coordinates to generalized curvilinear coordinates is straightforward and will not be presented here. In this work, we use the governing equations formulated in curvilinear coordinates and solve the discretized equations using the methods described in the next section. and C, C I are the model coefficients, and is the filter width or the eddy viscosity length scale. In the subgrid-scale stress equation, the first term on the right hand side is the original incompressible term in Smagorinsky s model, 7 and the second term is the compressible correction proposed by Yoshizawa 9 and implemented by Moin et al. 3 Finally, the energy equation is ē t t + ũ i(ē t + p) ũ j ( σ ij τ ij )+ ( q i +Q i ) = x i x i x i () where the total energy is defined as ē t = ρũ iũ i + p γ () 3 of Numerical Methods We first transform a given non-uniformly spaced curvilinear computational grid in physical space to a uniform grid in computational space and solve the discretized governing equations on the uniform grid. To compute the spatial derivatives at interior grid points away from the boundaries, we employ the nondissipative sixth-order compact scheme of Lele. 3 For boundary points, we apply the third-order one-sided compact scheme and for the points next to the boundaries, we use the fourth-order central compact scheme formulation. Spatial filtering is sometimes used as a means of suppressing unwanted numerical instabilities that can arise from the boundary conditions, unresolved scales American Institute of Aeronautics and Astronautics Paper 598

5 and mesh non-uniformities. In our study, we considered two different filters. We initially used the following fourth-order compact filter developed by Lele 3 β ˆφ i + α ˆφ i + ˆφ i + α ˆφ i+ + β ˆφ i+ = aφ i +b(φ i+ + φ i ) + c(φ i+ + φ i ) +d(φ i+3 + φ i 3 ) (5) Characteristic inflow boundary conditions Radiation boundary conditions Flow direction Outflow boundary conditions where φ i and ˆφ i represent the solution variable and the spatially filtered solution variable at point i, respectively and the coefficients are given by α =.6547, β =.793 a = + 3α 9 + 6α + β, b = (6) 4 3 c = α + 4β, d = 6β 8 3 The application of this filter results in a penta-diagonal system of equations. It was determined by Zhao 3 that filtering at and near the boundaries is not necessary and therefore, this filter is used on grid points i = 5 through i = N 4 where N is the total number of grid points along the grid line. The second filter we considered is the following sixth-order tri-diagonal filter used by Visbal and Gaitonde 33 ˆφi + ˆφ i + ˆφi+ = where 3 a n (φ i+n + φ i n ) (7) n= a = , a = , a = , a 3 = 3 (8) 6 The parameter must satisfy the inequality.5 < <.5. A less dissipative filter is obtained with higher values of within the given range. With =.5, there is no filtering effect. Since this filter has a 7-point right-hand side stencil, it obviously cannot be used at near-boundary points. Instead, one-sided, sixth-order equations given by Visbal and Gaitonde 33 are used for near-boundary points. The boundary points, i = and i = N are left unfiltered. The solution is normally filtered in every spatial direction at the end of every time step. In DNS calculations, filtering is typically used to maintain numerical stability. The filter eliminates all the scales that cannot be resolved by the finite difference scheme. Hence, in LES calculations, the filter is treated as the grid filter. The standard fourth-order explicit Runge-Kutta scheme is used for time advancement. For boundary conditions, we initially employed Thompson s nonreflecting boundary conditions 34 on all boundaries of Fig. Radiation boundary conditions Sponge zone Schematic of the boundary conditions. the domain except for the inflow boundary. However, during our 3-D turbulent round jet simulations at a Reynolds number of 36(based on jet diameter), we discovered that Thompson s boundary conditions actually did not provide the correct entrainment flow physics on the boundaries of a Cartesian grid. Hence, we decided to switch to the boundary conditions of Tam and Dong. 35 The original twodimensional boundary conditions of Tam and Dong were recently extended to 3-D by Bogey and Bailly. 36 The 3-D radiation boundary conditions formulated in spherical coordinates are applied on the lateral boundaries of the computational domain illustrated in figure. It should be noted that Tam and Dong s radiation boundary conditions are formulated for boundaries to which only acoustic disturbances reach. The lateral boundaries of figure are therefore suitable for the application of these boundary conditions. However, radiation boundary conditions cannot be applied on the outflow boundary of figure since there are entropy and vorticity waves in addition to acoustic waves crossing this boundary. Instead, Tam and Dong s outflow boundary conditions are imposed on the outflow boundary. For the outflow boundary, we use the sponge zone method proposed by Colonius et al. 37 In this method, a sponge zone is attached downstream of the physical domain. We apply grid stretching and explicit filtering in this exit zone to dissipate the vortices present in the flowfield before they hit the outflow boundary. We then use Tam and Dong s outflow boundary conditions without any modification at the end of the sponge zone. This method has been shown to be quite effective in minimizing reflections from the outflow boundary. The inflow boundary is responsible for both allowing the outgoing acoustic waves to leave the domain as well as generating disturbances that trigger the growth of instabilities in the flowfield. For the inflow boundary conditions, we apply a procedure based on characteristics. 3 4 of American Institute of Aeronautics and Astronautics Paper 598

6 y / r o 5 z / r o y / r o Fig. The x y section of the computational grid used in the Re D = 36 simulation. (Every nd node is shown.) Results Before doing any 3-D simulations, we studied a -D planar mixing layer problem to test the various numerical algorithms implemented in our 3-D LES code. Our -D results compared well with the available experimental and computational data in the literature. The first 3-D LES test case we considered is a turbulent round jet at a Mach number of.9. The Reynolds number of the jet is Re D = ρ ju j D j µ j = 36 (9) where ρ j, U j, µ j are the jet centerline density, velocity and viscosity at the nozzle exit, respectively and D j is the jet diameter. This is the same test case studied by Freund in his DNS. We obtained promising results in this simulation. Due to space restrictions, we will skip the details of this test case here. For our second test case, we studied a turbulent round jet at a Reynolds number of 36. The stretched Cartesian grid used in this test case had points in the x, y, and z directions respectively. Figures and 3 illustrate the x y and y z sections of our grid. The domain extends to about 45r o in the streamwise direction and from 5r o to 5r o in the transverse y and z directions, where r o is the jet radius. In the x direction, the physical region ends at around x = 3r o. The grid spacing along the x direction in the physical portion of the grid is uniform and about.r o. The minimum grid spacing in the y and z directions is about.5r o around the jet centerline. Exponential grid stretching is applied along the transverse directions such that the maximum spacing around the external boundaries is about.66r o. For the inflow forcing in this test case, we applied the same kind of forcing used by Constantinescu and Lele in their LES calculations. In this forcing, Fig. 3 The y z section of the computational grid used in the Re D = 36 simulation. (Every nd node is shown.) time-harmonic fluctuations are imposed on the mean streamwise velocity on the inflow boundary as follows: v x (r) = [ [ ( r U o tanh b r )]] o ( + r o r α sin(πst t)) () where the Strouhal number, St = r o f/u o is.9, r o is the jet radius, U o is the mean jet centerline velocity on the inflow boundary, f is the frequency of the perturbations, b = 3.5 is the shear layer thickness parameter, and the amplitude of the sinusoidal oscillations, α is.5. Randomly generated perturbations are applied on the azimuthal velocity component using the following equation ( ) ) v θ (r, θ) =.5U o ɛ exp 3 ( rro () where ɛ is a random number between.5 and.5. The exponential function in the above equation localizes the perturbations within the shear layer region only. Both the mean azimuthal velocity and the mean radial velocity on the inflow boundary are zero. We did not impose any fluctuations on the radial velocity component. As will be explained below, we did several runs in this test case. For all the simulations presented in this section, we used 8 processors on Indiana University s IBM-SP research computer. Each run required about 3 days to complete. For every simulation, we ran a total of 5 time steps during which an acoustic wave travelling at the ambient sound speed moved a distance of about 37 times the domain length in the streamwise direction. The initial transients exited the domain over the first time steps. We then collected the flow statistics over the other 4 time steps. Although a value of C =. was used 5 of American Institute of Aeronautics and Astronautics Paper 598

7 for our Re D = 36 simulation, the Smagorinsky coefficient at this higher Reynolds number (36) was chosen as C =.8 in an ad hoc manner. As the Reynolds number is increased, the dissipative length scales are hardly resolved on relatively coarse grids, hence all the dissipation has to come from the SGS model. We did a Kolmogorov length scale analysis and determined that our coarsest grid resolution for this Reynolds number was between 5 and 6 times the local Kolmogorov length scale. Assuming that the upper limit of the dissipative length scales is 6 times the Kolmogorov length scale, 38, 39 we see that we hardly resolved any of the dissipative length scales in portions of the flow where the local length scales were very small. We believe that a larger coefficient should represent the effect of the dissipative length scales on the large scales more accurately at high Reynolds numbers on a relatively coarse grid. For comparison, Bogey et al. 9 used a larger constant of C =.8 =.34 in their LES calculation for a jet at a Reynolds number of 65. We kept the compressibility correction constant at C I =.66, and the turbulent Prandtl number at P r t =.7, same as the values used in the Re D = 36 simulation. As mentioned previously, we discovered some entrainment flow problems with Thompson s boundary conditions during the runs done in this test case. Since we are dealing with a round jet problem, the ambient fluid that is entrained by the jet should be coming in radially towards the jet centerline along the boundaries of the y z section of the grid that was previously shown in figure 3. However, it was found that with Thompson s boundary conditions, the entrained fluid came in at almost normal angles pretty much everywhere along the boundaries except at the corners. Clearly, this is not physical. Hence, starting from the current test case, we decided to employ the state-ofthe-art Tam and Dong s boundary conditions which solved the entrainment problem. Figure 4 shows the instantaneous contours of vorticity magnitude on the x z plane. The jet is initially laminar, then the transition process starts and the jet becomes fully turbulent after its potential core breaks up. In order to study the effect of spatial filtering on our results, we did a run using the 4 th -order accurate implicit penta-diagonal filter as well as another run using the 6 th -order accurate implicit tri-diagonal filter with =.49. The solution was filtered once in each spatial dimension at the end of every time step in both runs. Figures 5 through 9 summarize the mean flow results obtained with the 4 th -order accurate penta-diagonal filter. Our results are compared with the experimental data of Hussein et al. 4 for an incompressible jet at Re D = 955 as well as that of Panchapakesan et al. 4 for an incompressible jet at Re D =. In figure 5, we plot the streamwise z / r o Fig. 4 Instantaneous contours of vorticity magnitude on the x z plane at y =, Re D = levels shown. U o / U c th-order penta-diagonal filter =.8 slope =.974 B =.5/slope = 5.3 experimental values of B: Fig. 5 Inverse of the mean streamwise velocity on the centerline normalized by the jet inflow velocity, 4th-order accurate filter, =.8. variation of the inverse of the mean centerline velocity normalized by the jet inflow velocity. After transition to turbulence, we get a linear growth. The slope of the line can be used to compute the jet decay coefficient and our value is around 5.3. Experimental values of the jet decay coefficient are in between 5.4 and 6.. The reason for our low value will be explained shortly. The growth of the half-velocity radius, illustrated in figure 6, also shows a linear growth in the turbulent region of the jet, consistent with experimental observations. The slope of the linear line which is defined as the jet spreading rate, is about.98. Experimental values for the jet spreading rate range from.86 to.96. Mean streamwise velocity profiles at two downstream locations are plotted in figure 7. From this figure, it can be observed that the profiles of our compressible jet at the two downstream locations collapse onto each other fairly well and exhibit 6 of American Institute of Aeronautics and Astronautics Paper 598

8 3 4th-order penta-diagonal filter =.8. x = 5r o x = 8r o th-order penta-diagonal filter =.8 r / / r o.5 σ xx.5.5 slope = A =.98 experimental values of A: Fig. 6 Half-velocity radius normalized by the jet radius, 4th-order accurate filter, =.8. where v x, v r, v θ are the axial, radial and azimuthal components of the fluctuating velocity, respectively, U c is the mean jet centerline velocity at a given axial location, and the overbar denotes time-averaging. In order to compare our Reynolds stresses with the ex r / r / Fig. 8 Normalized Reynolds stress σ xx profiles and comparison with experimental data, 4th-order accurate filter, = x = 5r o x = 8r o 4th-order penta-diagonal filter =.8.5. x = 5r o x = 8r o 4th-order penta-diagonal filter =.8 u / U c.5.4 σ rx r / r / Fig. 7 Normalized mean streamwise velocity profiles and comparison with experimental data, 4thorder accurate filter, =.8. self-similarity which is consistent with experimental observations. Our results are also in very good agreement with the experimental profiles of incompressible jets. We also compared our Reynolds stresses with the available experimental data of Hussein et al. 4 and Panchapakesan et al. 4 The normalized Reynolds stresses in cylindrical coordinates are defined as follows: σ xx = v x v x σ rr = v r v r U c U c σ θθ = v θ v θ U c σ rx = v r v x () U c r / r / Fig. 9 Normalized Reynolds stress σ rx profiles and comparison with experimental data, 4th-order accurate filter, =.8. perimental data, we transformed our Reynolds stresses from the Cartesian coordinate system to the cylindrical coordinate system. The transformation process was trivial and the details of it will be skipped here. Figures 8 and 9 plot our computed Reynolds stresses and compare them with the experimental data. From these plots, we see that our Reynolds stress profiles also exhibit self-similarity, consistent with experimental evidence, and our data are again in fairly good agreement with experiment. Figures through 4 summarize the mean flow results obtained with the 6 th -order accurate tri-diagonal filter. In general, the two sets of results obtained with different filters do not seem to differ much although the Reynolds stress profiles computed with the 6 th -order filter match the experimental profiles of Hussein et al. 4 slightly better. The jet decay coeffi- 7 of American Institute of Aeronautics and Astronautics Paper 598

9 3.5 =.8 = x = 5r o x = 8r o =.8 =.49 U o / U c.5.5 slope =. B =.5/slope = 5. experimental values of B: u / U c Fig. Inverse of the mean streamwise velocity on the centerline normalized by the jet inflow velocity, 6th-order accurate filter, =.8, =.49. cients obtained in both runs are a bit low relative to the experimental values. The jet spreading coefficient obtained with the 4 th -order accurate filter is just below the upper limit of the experimental values while the one obtained with the 6 th -order accurate filter is slightly larger. The mean streamwise velocity profiles as well as the Reynolds stress profiles in both cases compare fairly well with experiment. We also investigated the effect of the Smagorinsky coefficient on the mean flow properties. We did a new run using C =.9 and the 6 th -order filter with =.49. The mean flow results for this simulation are plotted in figures 5 through 9. As can be seen from these results, a slightly larger Smagorinsky coefficient results in a jet decay coefficient of 5.76 which is within the range of experimental observations. The jet decay coefficient obtained with C =.8 was 5., a bit smaller than the lower limit of experimen r / r / Fig. Normalized mean streamwise velocity profiles and comparison with experimental data, 6thorder accurate filter, =.8, = =.8 =.49. x = 5r o x = 8r o.75 =.8 =.49 r / / r o.5 slope = A =.6 σ xx.5.5 experimental values of A: Fig. Half-velocity radius normalized by the jet radius, 6th-order accurate filter, =.8, = r / r / Fig. 3 Normalized Reynolds stress σ xx profiles and comparison with experimental data, 6th-order accurate filter, =.8, =.49. tal data. The reason for this difference is believed to be the fact that with a higher SGS model coefficient, the jet loses more of its energy into dissipation and does not have as much energy left to spread out. Furthermore, with increasing jet decay coefficient, the jet spreading rate gets smaller. This is justified by the fact that a jet spreading rate of.6 is obtained with C =.8 while a value of.86 is obtained with C =.9. Hence, the jet decay coefficient as well as the jet spreading rate are directly affected by the SGS model constant. Another interesting fact is that even though the simulation with C =.9 produced a jet decay coefficient and a jet spreading rate in good agreement with experiment, we see that the Reynolds stress profiles in this simulation have significantly changed relative to the ones obtained in the simulation done with C =.8. This must be due to the too dissipa- 8 of American Institute of Aeronautics and Astronautics Paper 598

10 .5. x = 5r o x = 8r o =.8 = =.9 =.49 σ rx.5..5 r / / r o.5.5 slope = A =.86 experimental values of A: r / r / Fig. 4 Normalized Reynolds stress σ rx profiles and comparison with experimental data, 6th-order accurate filter, =.8, = Fig. 6 Half-velocity radius normalized by the jet radius, 6th-order accurate filter, =.9, = =.9 = x = 5r o x = 8r o =.9 =.49 U o / U c.5.5 slope =.868 B =.5/slope = 5.76 experimental values of B: u / U c Fig. 5 Inverse of the mean streamwise velocity on the centerline normalized by the jet inflow velocity, 6th-order accurate filter, =.9, =.49. tive nature of the Smagorinsky SGS model. A slight increase in the model coefficient provides more dissipation but at the same time it suppresses more of the turbulence and subsequently causes a significant change in the Reynolds stresses. Finally, in order to see how much the filtering parameter affects the mean flow results, we did a simulation using =.45 in our 6 th -order tridiagonal filter with the Smagorinsky coefficient chosen as C =.8. Shown in figures through 4 are the results obtained in this run. It is easily observed that the lower value of the filtering parameter qualitatively has the same effect as increasing the Smagorinsky coefficient. This makes sense since the lower values of correspond to a more dissipative filter. The dissipation coming from the filter at this value of seems to be causing noticeable changes in the Reynolds stress r / r / Fig. 7 Normalized mean streamwise velocity profiles and comparison with experimental data, 6thorder accurate filter, =.9, =.49. profiles relative to the ones obtained using =.49 with C =.8. Concluding Remarks - Future Work As part of a jet noise prediction methodology, we have developed and tested a 3-D Large Eddy Simulation code. So far, our main concern was the validation of the LES code, therefore studies related to the aeroacoustics of turbulent jets have not been performed yet. In our initial 3-D test cases, we performed turbulence simulations on grids consisting of about 6 million grid points in reasonable amounts of run time of less than 4 days using up to 8 processors on SGI Origin and IBM SP parallel computers. Calculations were done for turbulent jets at Reynolds numbers of 36 and 36. Acceptable mean flow results were obtained in our simulations and we believe we gathered enough evidence towards the validation of our LES 9 of American Institute of Aeronautics and Astronautics Paper 598

11 ..75 x = 5r o x = 8r o =.9 = =.8 =.45 σ xx.5.5 U o / U c.5.5 slope =.9 B =.5/slope = 5.48 experimental values of B: r / r / Fig. 8 Normalized Reynolds stress σ xx profiles and comparison with experimental data, 6th-order accurate filter, =.9, = Fig. Inverse of the mean streamwise velocity on the centerline normalized by the jet inflow velocity, 6th-order accurate filter, =.8, = x = 5r o x = 8r o =.9 = =.8 =.45 σ rx.5 r / / r o slope = A =.93 experimental values of A: r / r / Fig. 9 Normalized Reynolds stress σ rx profiles and comparison with experimental data, 6th-order accurate filter, =.9, =.49. code. However, we found that the Smagorinsky SGS model is too dissipative and the mean flow results are sensitive to the particular choice of the model constant. The effect of different grid filters on the results was also studied. In general, the 4 th -order penta-diagonal filter results and the 6 th -order tri-diagonal filter with =.49 results do not differ much. Yet, the more dissipative tri-diagonal filter with =.45 has similar effects as increasing the SGS model constant. In our future work, we will implement a dynamic SGS model that is applicable to inhomogeneous turbulent flows in complex geometries. Currently, one model under consideration for implementation into our code is the localized dynamic SGS model by Kim and Menon This model is based on the subgrid kinetic energy which is computed by solving a transport equation. The model coefficients are determined using 3 Fig. Half-velocity radius normalized by the jet radius, 6th-order accurate filter, =.8, =.45. a localized dynamic procedure. This is actually a oneequation SGS model. Unlike the algebraic Smagorinsky model, it does not require the assumption of local equilibrium between the SGS energy production and dissipation rate. As a result, the model allows nonnegligible kinetic energy to remain unresolved in the small scales. This implies that the computational grids to be used with this model can be relatively coarser than those that would be needed with an algebraic SGS model. Our eventual goal in this study is to accurately capture the noise field generated by the turbulent mixing of the jet and compute the sound propagation to the far field. For this purpose, the near-field data computed by our LES code will be coupled with a surface integral acoustic method. Surface integral acoustic methods based on the Kirchhoff and the porous Ffowcs Williams-Hawkings (FW-H) formulations have been of American Institute of Aeronautics and Astronautics Paper 598

12 .9 x = 5r o x = 8r o.5 x = 5r o x = 8r o.8.7 =.8 =.45. =.8 = u / U c.5.4 σ rx r / r / r / r / Fig. Normalized mean streamwise velocity profiles and comparison with experimental data, 6thorder accurate filter, =.8, =.45. σ xx x = 5r o x = 8r o =.8 = r / r / Fig. 3 Normalized Reynolds stress σ xx profiles and comparison with experimental data, 6th-order accurate filter, =.8, =.45. widely used in Computational Aeroacoustics (CAA) to date. Although Kirchhoff s method has been used in jet noise problems before, the porous FW-H equation has not been widely applied to this problem yet. One disadvantage of the Kirchhoff method is that the control surface must be chosen to be in the linear flow region since Kirchhoff s formulation is based on the linear wave equation and does not include nonlinear flow effects. The location of the linear region is not well defined and is problem dependent. However, the FW-H formulation is based on the conservation laws of fluid mechanics rather than on the wave equation. The FW- H formulation includes nonlinearities and hence the integration surface can be placed in the nonlinear regions of the flow field if the quadrupole source is included. The quadrupole source term accounts for nonlinear effects such as nonlinear wave propagation, variations on Fig. 4 Normalized Reynolds stress σ rx profiles and comparison with experimental data, 6th-order accurate filter, =.8, =.45. the local sound speed and noise generated by shocks, vorticity and turbulence in the flow field. Brentner and Farassat 45 showed that if the quadrupole term is neglected outside the control surface, the FW-H solution is less sensitive than the Kirchhoff method to the placement of integration surface. They also showed that the FW-H formulation is equivalent to the Kirchhoff formulation when the integration surface is located in the linear flow region. Hence, our efforts will be concentrated on the application of the FW-H method to jet noise prediction. In order to avoid boundary effects, the control surface needed for the FW-H method will be kept outside the jet flow, but inside the boundaries of the computational domain used in LES calculations. The quantities needed on the control surface will be provided by the LES code. Acknowledgments This work is sponsored by the Indiana st Century Research & Technology Fund and was also partially supported by National Computational Science Alliance under the grant CTS3N, and utilized the SGI Origin computer systems at the University of Illinois at Urbana-Champaign. Some of the computations were performed on the IBM-SP research computers of Indiana University and Purdue University. References Freund, J. B., Lele, S. K., and Moin, P., Direct Numerical Simulation of a Mach.9 turbulent jet and its sound field, AIAA Journal, Vol. 38, No., November, pp Freund, J. B., Noise Sources in a low-reynolds-number turbulent jet at Mach.9, Journal of Fluid Mechanics, Vol. 438,, pp Stromberg, J. L., McLaughlin, D. K., and Troutt, T. R., Flow Field and Acoustic properties of a Mach Number.9 Jet at a low Reynolds Number, Journal of Sound and Vibration, Vol. 7, No., 98, pp of American Institute of Aeronautics and Astronautics Paper 598

13 4 Mankbadi, R. R., Hayder, M. E., and Povinelli, L. A., Structure of supersonic jet flow and its radiated sound, AIAA Journal, Vol. 3, No. 5, May 994, pp Lighthill, M. J., On the Sound Generated Aerodynamically: I, General Theory, Proc. Royal Soc. London A, Vol., 95, pp Lyrintzis, A. S. and Mankbadi, R. R., Prediction of the far-field jet noise using Kirchhoff s formulation, AIAA Journal, Vol., No., 996, pp Kirchhoff, G. R., Zur Theorie der Lichtstrahlen, Annalen der Physik und Chemie, Vol. 8, 883, pp Gamet, L. and Estivalezes, J. L., Application of Large- Eddy Simulations and Kirchhoff method to jet noise prediction, AIAA Journal, Vol. 36, No., December 998, pp Choi, D., Barber, T. J., Chiappetta, L. M., and Nishimura, M., Large Eddy Simulation of high-reynolds number jet flows, AIAA Paper No. 99-3, January 999. Zhao, W., Frankel, S. H., and Mongeau, L., Large Eddy Simulations of Sound Radiation from Subsonic Turbulent Jets, AIAA Journal, Vol. 39, No. 8, August, pp Morris, P. J., Wang, Q., Long, L. N., and Lockhard, D. P., Numerical Predictions of High Speed Jet Noise, AIAA Paper No , May 997. Morris, P. J., Long, L. N., Bangalore, A., and Wang, Q., A Parallel Three-Dimensional Computational Aeroacoustics Method Using Nonlinear Disturbance Equations, Journal of Computational Physics, Vol. 33, 997, pp Morris, P. J., Long, L. N., Scheidegger, T. E., Wang, Q., and Pilon, A. R., High Speed Jet Noise Simulations, AIAA Paper No. 98-9, Morris, P. J., Long, L. N., and Scheidegger, T., Parallel computations of high speed jet noise, AIAA Paper No , Morris, P. J., Scheidegger, T. E., and Long, L. N., Jet noise simulations for circular nozzles, AIAA Paper No. - 8, June. 6 Boluriaan, S., Morris, P. J., Long, L. N., and Scheidegger, T., High speed jet noise simulations for noncircular nozzles, AIAA Paper No. -7, May. 7 Chyczewski, T. S. and Long, L. N., Numerical prediction of the noise produced by a perfectly expanded rectangular jet, AIAA Paper No , May Boersma, B. J. and Lele, S. K., Large Eddy Simulation of a Mach.9 Turbulent Jet, AIAA Paper No , May Bogey, C., Bailly, C., and Juvé, D., Computation of the sound radiated by a 3-D jet using Large Eddy Simulation, AIAA Paper No. -9, June. Constantinescu, G. S. and Lele, S. K., Large Eddy Simulation of a Near Sonic Turbulent Jet and its Radiated Noise, AIAA Paper No. -376, January. Piomelli, U., Streett, C. L., and Sarkar, S., On the computation of sound by large-eddy simulations, Journal of Engineering Mathematics, Vol. 3, 997, pp Rubinstein, R. and Zhou, Y., Characterization of Sound Radiation by Unresolved Scales of Motion in Computational Aeroacoustics, Tech. Rep. NASA/CR , ICASE Report No , ICASE, NASA Langley Research Center, October Seror, C., Sagaut, P., Bailly, C., and Juvé, D., On the radiated noise computed by large-eddy simulation, Physics of Fluids, Vol. 3, No., February, pp Zhao, W., Frankel, S. H., and Mongeau, L., Effects of Spatial Filtering on Sound Radiation from a Subsonic Axisymmetric Jet, AIAA Journal, Vol. 38, No., November, pp Williams, J. E. F. and Hawkings, D. L., Sound Generated by Turbulence and Surfaces in Arbitrary Motion, Philosophical Transactions of the Royal Society, Vol. A64, 969, pp Crighton, D. G., Dowling, A. P., Williams, J. E. F., Heckl, M., and Leppington, F. G., Modern Methods in Analytical Acoustics: Lecture Notes, Springer Verlag, London, Smagorinsky, J. S., General Circulation Experiments with the Primitive Equations, Monthly Weather Review, Vol. 9, No. 3, March 963, pp Germano, M., Piomelli, U., Moin, P., and Cabot, W., A Dynamic Subgrid-Scale Eddy Viscosity Model, Physics of Fluids A, Vol. 3, No. 7, July 99, pp Yoshizawa, A., Statistical Theory for Compressible Turbulent Shear Flows, with Application to Subgrid Modeling, Physics of Fluids, Vol. 9, No. 7, July 986, pp Moin, P., Squires, K., Cabot, W., and Lee, S., A Dynamic Subgrid-Scale Model for Compressible Turbulence and Scalar Transport, Physics of Fluids A, Vol. 3, No., November 99, pp Lele, S. K., Compact Finite Difference Schemes with Spectral-like Resolution, Journal of Computational Physics, Vol. 3, No., November 99, pp Zhao, W., A Numerical Investigation of Sound Radiated from Subsonic Jets with Application to Human Phonation, Ph.D. thesis, School of Mechanical Engineering, Purdue University, West Lafayette, IN, June. 33 Visbal, M. R. and Gaitonde, D. V., Very high-order spatially implicit schemes for computational acoustics on curvilinear meshes, Journal of Computational Acoustics, Vol. 9, No. 4,, pp Thompson, W. K., Time-Dependent Boundary Conditions for Hyperbolic Systems, II, Journal of Computational Physics, Vol. 89, No., August 99, pp Tam, C. K. W. and Dong, Z., Radiation and Outflow Boundary Conditions for Direct Computation of Acoustic and Flow Disturbances in a Nonuniform Mean Flow, Journal of Computational Acoustics, Vol. 4, No., 996, pp Bogey, C. and Bailly, C., Three-dimensional non-reflective boundary conditions for acoustic simulations: far field formulation and validation test cases, Acta Acustica, soon to appear. 37 Colonius, T., Lele, S. K., and Moin, P., Boundary Conditions for Direct Computation of Aerodynamic Sound Generation, AIAA Journal, Vol. 3, No. 9, September 993, pp Pope, S. B., Turbulent Flows, Cambridge University Press,. 39 Piomelli, U., Scotti, A., and Balaras, E., Large Eddy Simulations of Turbulent Flows, from Desktop to Supercomputer (Invited Talk), Vector and Parallel Processing - VECPAR, 4th International Conference, Porto, Portugal, Selected Papers and Invited Talks, June. 4 Hussein, H. J., Capp, S. C., and George, W. K., Velocity measurements in a high-reynolds-number, momentumconserving, axisymmetric, turbulent jet, Journal of Fluid Mechanics, Vol. 58, 994, pp Panchapakesan, N. R. and Lumley, J. L., Turbulence measurements in axisymmetric jets of air and helium. Part. Air jets. Journal of Fluid Mechanics, Vol. 46, 993, pp Kim, W.-W. and Menon, S., A new dynamic one-equation subgrid-scale model for large eddy simulations, AIAA Paper No , January Kim, W.-W. and Menon, S., An Unsteady Incompressible Navier-Stokes solver for Large Eddy Simulation of Turbulent Flows, International Journal for Numerical Methods in Fluids, Vol. 3, 999, pp Kim, W.-W. and Menon, S., LES of Turbulent Fuel/Air Mixing in a Swirling Combustor, AIAA Paper No. 99-, January Brentner, K. S. and Farassat, F., Analytical Comparison of the Acoustic Analogy and Kirchhoff Formulation for Moving Surfaces, AIAA Journal, Vol. 36, No. 8, August 998, pp of American Institute of Aeronautics and Astronautics Paper 598