Numerical Simulation of Transient Supersonic Nozzle Flows

Transcription

1 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition January 2012, Nashville, Tennessee AIAA Numerical Simulation of Transient Supersonic Nozzle Flows A. Chaudhuri and A. Hadjadj National Institute of Applied Sciences, CORIA - UMR, CNRS 6614, Rouen, France The starting process of supersonic nozzles has been investigated numerically. The objective of this study is to identify the origin of side-loads during start-up and shutdown phases of rocket nozzles. In such cases, a complex flow structure with significant separated boundary layer and strong moving shocks have been observed. Special attention has been paid to the early phase of the starting process and to the appearance of a secondary shock wave and its interaction with the wall. The computational results are compared with the experiments in terms of shock velocities. When turbulence is included in the simulation, the flow evolution is proved to be more complex due to the strong coupling between shock and turbulence, in particular, when large-scale turbulence are created within the mixing layer at the separation point. This phenomenon is considered to be one of the basic characteristics of shock unsteadiness in transient nozzle flows. A c c p c C w δ n Nomenclature Pre-multiplier in energy spectrum Speed of sound Specific heat at constant pressure Speed of sound at reference state Model constant Filter width Grid size Wall-normal non-dimensional distance of the near wall grid point E Energy Ĕ Resolved energy f Filtered variable f Favre averaged filtered variable g ij Resolved velocity gradient tensor G δ Filter function γ Heat capacity ratio at reference state h t Throat length ke Wave number associated with most energetic length scale le Most energetic length scale L Reference-length scale M s Shock Mach number µ t Eddy-dynamic viscosity ν t Eddy-kinematic viscosity p Static Pressure P a Atmospheric pressure P c Reservoir pressure P r t Turbulent Prandtl number SGS heat flux q sgs j Research Associate, National Institute of Applied Sciences, Rouen, France. AIAA Member, Professor - National Institute of Applied Sciences, Rouen, France. 1 of 13 Copyright 2012 by the, Inc. All rights reserved.

2 Re ac R ij ρ s d ij s ij σ ij t t T a T c τ ij u u p u τ x Subscript Acoustic Reynolds number Resolved Reynolds stresses Density Traceless symmetric part of the square of the resolved velocity gradient tensor Symmetric part of the resolved velocity gradient tensor Resolved stress tensor Time Non dimensional time Atmospheric temperature Reservoir temperature SGS stress tensor Velocity Turbulent velocity Friction velocity Non dimensional length 1 Stagnant state 2 Shocked state x, y, z Coordinate directions I. Introduction Flow separation control in rocket nozzles is a challenging problem in aerospace science, not only for current engines confronted with problems of thermo-mechanical loads, but also for future engines which could work with very wide separation zones. This phenomenon is related to an usually unstable type of flow occurring in large expansion nozzle and producing unstable forces known as side-loads. Theses forces are prejudicial to the mechanical structure of the nozzle and can cause damages. Basically, the physical problem met in those configurations is essentially due to the boundary layer separation during nozzle startup process, caused by the ambient high pressure gradient, resulting in a complex phenomenon with shock/shock and shock/boundary layer interactions (see Fig.1). Several complex phenomena, such as boundary layers with adverse pressure gradients, shocks, induced separation, recirculation bubbles, shear layers can occur in nozzles and may strongly affect the rocket engine s performance. Transient nozzle flow which exists during engine start-up has been investigated by only a few researchers. Some experimental work has been performed by Smith, 1 Amann, 2 Saito et al. 3 Amann, 2 for instance, studied the influence of several parameters (nozzle half-angle, throat width and nozzle inlet radius) on the starting process in supersonic nozzles driven by a shock tube. Besides, special interest has been paid to the duration of the starting process, since it decreases the useful testing time of short-duration facilities. However, the evolution of a complex wave structure has also been shown. From a numerical point of view, some studies were undertaken to simulate nozzle flow transients. Most of the simulations performed were two-dimensional plane or axisymmetric owing to the large amount of computational time required for three-dimensional ones. From previous studies, 4, 5 it may be concluded from these studies that inviscid computations satisfactorily predict the main flow features (namely the primary and secondary shock waves, multiple shock wave reflections and slip surfaces). Concerning rocket nozzles, Chen and Chakravarty 6 examined the flow structures of the start-up and shut-down processes using a Navier-Stokes solver. The configuration they studied was a sub-scale nozzle of a J-2S rocket engine (i.e. a precursor of the American Space Shuttle main engine). In a recent work, Mouronval et al. 7, 8 studied numerically the early transient flow induced in an expanding nozzle by an inflow preceded by an incident planar shock wave using high-order numerical methods. Special attention has been paid to the early phase of the starting process and to the appearance of a strong secondary shock wave. A detailed analysis of the wave structure was given and the mechanism of formation of vortices on the contact surface has been clearly shown as a result of the high accuracy of the simulations. In this paper, we report recent progress on computations of transient supersonic nozzle flows using both 2 of 13

3 inviscid and viscous computations. An immersed boundary (IB) method based on a direct forcing is coupled with a high-order weighted-essentially non-oscillatory (WENO) scheme to simulate complex geometries. TW Separated Flow CD SL SS MS PS SWBLI Figure 1. Starting process in a planar supersonic nozzle, PS: primary shock, MS: Mushroom structure, CD: contact discontinuity, SS: secondary shock, SWBLI: shock wave boundary layer interaction, SL: slip line and TW: transverse wave. II. Inviscid computations The first part of this paper concerns the transient flow developing in the Vulcain nozzle (designed for the European rocket Ariane V) during the engine start-up. After burning in the combustion chamber, the exhaust gases are expanded in the nozzle to deliver an optimal thrust to the launcher. The starting process begins with the primary shock (blast wave) entering the nozzle and ends when a quasi-steady flow has been achieved. In the simulation, the flow starts at t = 0 after the rupture of a diaphragm located at the nozzle inlet that separates two regions of quiescent air. The high-pressure region is at reservoir conditions, pressure P c =110 bar and temperature T c =2500 K, whereas the low pressure is at atmospheric conditions, P a =1 bar and T a =298 K. As a first step, viscous effects were neglected during the transient phase in order to focus mainly on the evolution of the rapidly moving shock wave structures which might later interact with viscous effects. Initially, a strong incident shock moves rapidly through the stagnant low-pressure medium. When the shock enters the divergent part of the nozzle, because the area increases, the flow undergoes an expansion and a contact discontinuity is formed. This discontinuity quickly becomes distorted owing to a Richtmyer-Meshkov type instability. In addition, a left-running (with respect to the fluid) secondary shock appears and is carried to the right because of the supersonic carrier flow. This shock wave links the high Mach number, low pressure flow, downstream of the throat, with the lower velocity high pressure gas behind the primary shock. The computed flow-field at time 2ms is shown by schlieren picture (see the inset of Fig. 2). The mean flow features (namely the primary and the secondary shock waves, multiple shock wave reflections and slip surfaces) are well reproduced by the simulation. Also, Fig. 2 displays additional details on the multiple wave reflection system (triple points, recompression shocks) formed by the bifurcation of the shocks near the wall as well as the important role played by the internal shock occurring at the nozzle throat. III. URANS computation of a 2D axisymmetric nozzle To complete this study a transient flow simulation at a very low regime ( Nozzle Pressure Ratio, NPR=5) is investigated. The objective of such investigation is to understand the origin of side-loads amplification observed during transients in the experiment of a TIC nozzle done by Kwan and Stark. 9 Fig. 3 shows a sequence of numerical pictures, where it is possible to identify different flow structures. In particular, the formation of the secondary shock is clearly visible at the early stage of the start-up process. This strong shock interacts with the boundary layer and large separation occurs. Since the jet plume is not sufficiently formed, the supersonic flow reattached further downstream, leading to a formation of a high-pressure region with a recirculation bubble. A restricted-shock separation (RSS) configuration is then formed, while the 3 of 13

4 20 Computation Experiment 15 SS Normalized time, t* 10 IS CD PS SS PS Normalized distance from the throat, x* Figure 2. Comparison of shock-waves trajectories on the axis. All length scales are non dimensionalized by the throat height h t = 6.15 mm and the elapsed dimensionless time is defined as t = tc 1 /h t where c 1 is the speed of sound of the gas initially at rest (c m.s 1 ).The origin of time is chosen in such a way that the primary shock reaches the nozzle throat at t = 0. Inset showing numerical schlieren picture, IS: Internal Shock. major structures of the internal jet remained similar during the transient process. The separated-reattached flow structure moves downstream and gradually discharges from the nozzle exit until a sudden transition from restricted-shock separation (RSS) to free-shock separation (FSS) configuration appears. During a very short time, a high pressure peak was found. The occurrence of the side-loads has been reported when the transition from FSS to RSS occurred during the start-up transient of TOC (Thrust Optimized Contour) nozzles. 10 IV. Large-eddy simulations of a 3D planar nozzle For further analysis, we have computed a small-scale laboratory nozzle similar to the one studied experimentally by Amann 2 and the existing shock-tube facility of the Ben-Gurion University, Beer-Sheva, Israel. For this purpose, large-eddy simulations are carried out to investigate the propagation of shock waves, flow separation and complex shock wave / boundary layer interaction associated with shock induced transient flows through planar nozzles. In the following subsections, we will briefly describe the numerical method as well as the flow initialization procedure. The obtained results will be presented and discussed in the remaining part of the paper. A. Numerical method 1. Filtered Navier-Stokes equations An implication of Kolmogorov s (1941) theory of self-similarity is that the large eddies of the flow are dependent on the geometry while smaller scales are more universal in nature. This feature allows one to explicitly solve for the large eddies in a numerical simulation and implicitly account for the smaller eddies by using a subgrid scale (SGS) model. The triumphant journey of LES started with the pioneering work of Smagorinsky, 12 Lilly, 13 Deardoff, 14 Germano 15 and others. Comprehensive accounts on LES are provided by Sagaut 16 and Pope 17 and reviews at different stages of the development are provided in The definition of any filtered quantity with a filter function G δ and associated filter width δ = ( x y z ) 1/3 can be 4 of 13

5 Figure 3. Formation of shock waves, contact discontinuities and boundary layer separation during the start-up of a Truncated Ideal Contoured (TIC) nozzle (numerical schlieren pictures). TIC nozzle at NPR= of 13

6 given by, f( x, t) = f( y, t)g δ ( x y)d y (1) R 3 To reduce the SGS terms, the Favre averaged definition is generally used in compressible flow simulations, defined as, f = ρf/ ρ. Applying the above definitions and neglecting the SGS terms having negligible contributions, 22 the filtered compressible Navier-Stokes system of equations can be written as, ρ t + ρũ i x i = 0 (2) ρũ i t + ρũ iũ j x j p + δ ij = σ ij τ ij (3) x j x j x j ρĕ t + ( ρĕ + p)ũ j = (ũ i σ ij ) q j qsgs j x j x j x j x j ũ i τ ij x j (4) where τ ij = ρ(ũ i u j ũ i ũ j ) and q sgs j have to be modeled in order to close the system of equations. In the present work, the Wall-Adapting Local Eddy-viscosity (WALE) model is used to close these SGS terms. 2. WALE model The Wall-Adapting Local Eddy-viscosity model proposed by Nicoud and Ducros 23 is basically designed to reproduce more accurate scaling for simulations containing wall boundary conditions. It includes the effect of both the strain and the rotation and thereby gives a better prediction in the region where vorticity dominates irrotational strain. The WALE model reproduces a proper near wall scaling so that the eddy viscosity is ν t = O(y 3 ). It is estimated from the velocity gradient tensor s invariant as follows, ν t = Cw 2 2 ( s d ij sd ij )3/2 ( s ij s ij ) 5/2 + ( s d (5) ij sd ij )5/4 ( ) where s ij = 1 ũ i 2 x j + ũj x i and s d ij is the traceless symmetric part of the square of the resolved velocity gradient tensor ( g ij = ũ i / x j ), namely, s d ij = ) ( g ij + g ji δ ij g kk 2 with g2 ij = g ik g kj. The model constant C w = 0.5 is recommended to be optimal from priori tests of freely decaying isotropic homogeneous turbulence. It can be emphasized that, the LES model based on s d ij sd ij detects turbulence structures with either (large) strain rate, rotation rate or both. Moreover, it avoids any dynamic procedure while maintaining the desired near wall scaling. No eddy-viscosity is being produced in case of wall bounded laminar flow (Poiseuille flow). This is distinctively advantageous over the Smagorinsky model (based on s ij s ij, but not on rotation rate) which is unable to reproduce the laminar to turbulent flow transition. The WALE model based on the s d ij sd ij invariant is known to be capable of handling the transitional pipe flow. 23 The SGS heat flux, q sgs j is modeled using the eddy-diffusivity hypothesis assuming constant P r t = 0.72 and is given by, q sgs j = µ tc p P r t T x j (6) An in-house 3D compressible Navier-Stokes solver equipped with a fifth-order WENO scheme, 24 LES models and an immersed boundary method is used for the present simulations. The use of low-dissipation, high-order shock-capturing schemes is an essential ingredient for computing complex compressible flows with shock waves. The objective is to avoid excessive numerical damping of the flow features over a wide range of length scales as well as to prevent spurious numerical oscillations near shock waves and discontinuities. For instance, the family of WENO schemes is a good choice to achieve this goal. The diffusion terms are determined by means of fourth-order compact central difference formulas. An immersed boundary technique is adopted to handle the complex geometries while making use of the cartesian grid arrangement. The discretized equations are integrated in time by means of the explicit third-order total variation diminishing Runge-Kutta algorithm (RK3-TVD). The CFL number is set to 0.7 for all simulations. Detailed description 31, 32 of the applied methodology is reported in our previous work. The simulations are performed on a SGI Altix ICE 8200EX and an IBM Power6 parallel computer using up to 512 processors consuming about CPU hours for each test case. 6 of 13

7 Figure 4. Computational domain, IS: Incident Shock wave, Rn = 10 mm, Ln = mm, nozzle angle = 15, throat length, h t = 9.5 mm, p 1 = Pa, T 1 = 291.5K, Re (based on h t and flow properties at the left state). 3. Problem setup Figure 4 shows the computational domaine used in the LES simulations. Although the experimental setup is having a cross-section of 80 mm 80 mm, a computational domain having (80 2π) mm 2 cross section is chosen. An incident shock wave with a Mach number M s = 1.86 is allowed to pass through the nozzle situated at the end test section. The Rankine-Hugoniot relations for a moving shock in air is used to fix both the left (shocked gas, subscripted as 2 ) and right (stagnant gas, subscripted as 1 ) states. In order to make direct comparison with the experiments, the shock wave is initially located at the leading edge of the nozzle. A cartesian grid is used along with an immersed boundary technique to handle the nozzle geometry. The top and the bottom boundaries of the nozzle are set to no-slip and adiabatic conditions while the span-wise direction (z-axis) is considered as homogeneous. However, for the most refined simulation, only half of the domain is considered exploiting the symmetry of the problem, which allow us to reduce substantially the computational cost. It is worth mentioning that all the computations are stopped before the shock wave reaches the boundaries. As summarized in Table 1, three test cases, namely C 1, C 2 and C 3 are considered on different grid sizes. To initialize the flow field, a homogeneous isotropic turbulence is superimposed to the mean flow in the shocked gas. The initial velocity fluctuations are by a prescribed energy spectrum of Passot-Pouquet, ( ) k E(k 4 ) = A ke e 2 k 2 k 2 u 2 e p (where A = 16 π ke ). An open source code 33 for turbulent flow-field generation is used first to get an initial box of turbulent flow-field and this periodic data is repeatedly assigned to fit in the computational domain. To generate a (2π) 3 box of turbulence, the following inputs are assumed i) Acoustic Reynolds number Re ac, ii) non-dimensional turbulent velocity u p and iii) most energetic length scale le. The different parameters used to generate this flow-field are summarized in Table 2. To compare with the experimental results, the centerline density is recorded to construct the speed of the two shock waves, namely the PS and SS. Flow field data at numerical probes on the walls of the nozzle are also registered to get the wall properties and their unsteady evolution. Case C 1 C 2 C 3 x, y, z 4, 2, 8 4, 2, 4 2,, 4 N total N N/2 N/8 Table 1. Parameters for different test cases, 24µm. 7 of 13

8 L Re ac u p u p k e l e 1 mm c L /γ u p /c 0.1U 2 6 2π/k e Table 2. Different parameters for the Passot-Pouquet spectrum, with the reference state ( ) taken in the shocked gas. Figure 5. Numerical schlieren at different stage, top left: at 16µs, top right: at 65.5µs, bottom left: at 115µs and bottom right: at 165µs. 8 of 13

9 V. Results and discussion The flow-field is averaged in the homogeneous direction (z-axis) to compare with the experimental data. The complex flow evolution of different stages are shown in Figs. 5 for the most refined case C 3. It can be seen that a part of the incident shock reflects from the end-wall and returns back upstream of the nozzle section as RS, while the part of the shock entering the nozzle evolves as PS front. This shock is usually followed by a typical mushroom shaped, noted CS, due to the Richtmyer-Meshkov instability. At the wall, the boundary layer strongly interacts with the secondary shock, SS, and gives rise to multiple transverse waves. The mean characteristics of the flow are very similar to the experimental schlieren showed in Fig. 1. These pictures show that the numerical simulation captured the most dominant wave structures of the flow-field. 20 Nondimensional distance PS : Experimental SS : Experimenral PS : Simulation SS : Simulation Non dimensional time Figure 6. X-T diagram: density contour on non-dimensional time (y-axis) and non-dimensional position (centerline) x-axis. In order to visualize the propagation of different waves, the X-T diagram is constructed from the instantaneous density field. Fig. 6 clearly shows the evolution of PS, CD and SS. It can be seen that a very good agreement of the shock positions with the experimental results is found. All three cases reproduce nearly identical outcome for speed of the principle shock waves. Deviation of the position of the secondary shock at the later stages is inevitable due to the inaccuracy of the determination of the exact location of the secondary shock from the first derivative of the density field. From the registered flow-field data, we estimated the near wall properties at t 165µs (see Fig. 7). The position of the flow separation can be deduced from the wall shear stress profile. It can be seen that + n 40 ( + n = n u τ /ν w, for C 3 and is in acceptable range of grid resolution for LES. It is worth mentioning that the mean wall pressure and density profiles show symmetrical behavior on top and bottom walls for coarser 9 of 13

10 meshes. The 3D numerical schlieren, Q criterion and vorticity field are shown in Fig. 8 to highlight the later stage flow features and different scales of turbulence. It is evident that the shear layer region is dominantly turbulent and a wide range of flow structures are captured by the present simulation u τ (m/s) n Nondimensional distance Figure 7. Near wall parameters A qualitative comparison of the three test cases is shown in fig.9. It can be seen from the experimental schlieren that the secondary shock is related to a Mach reflection and is affected by the turbulent separated zone. The experimental picture clearly indicates that the boundary layer, upstream of the flow separation zone, is turbulent in nature, resulting in a strong shock interaction with the boundary layer. It is clear that for C 1 and C 2, the oblique shock reflections are close to a regular reflection rather than a Mach reflection. The flow features depict the inability to reproduce the turbulent flow-field and coupled interactions. Moreover, the strong expansion fan downstream of the throat region is a clear indication of lacking of grid resolution. On the other hand, the existence of a turbulent separation zone and the improved of the shock boundary layer interactions is visible for C 3. Again, it can be noted that the deviation from the experimental flow structures are essentially related to the onset of turbulence in the region between throat and the separation point. The weak Mach reflections predicted by the simulations are related to the nature of the upstream boundary layer. Nevertheless, it is clearly visible that there exists a noticeable improvement in the simulations from case C 1 to case C 3. VI. Conclusions In this work, we made an initial attempt to simulate the complex flow features associated with shock induced supersonic flow inside a planar nozzle in a shock-tube arrangement. A comprehensive study have been made related to complex flow features of nozzle-startup process. Initial inviscid and URANS computations show interesting findings related to the principle shock-wave features and agrees with experimental data on the centerline of the nozzle. This depicts the governing inviscid characteristic speed of the PS and SS. However, boundary layer separation and shock/boundary layer interactions are much more important to predict the unsteady flow evolution associated with the shock induced flow inside a nozzle. Large-eddy simulations are carried out with a flow solver equipped with a high-order WENO scheme and an immersed boundary technique. The global flow features of primary, secondary shock waves and contact discontinuity are well captured and are in good agreement with the experimental data. Simulations show quantitative agreement with experiments for the speed of the primary as well as the secondary shocks. Homogeneous isotropic turbulent fluctuating flow-field has been assigned to have realistic initial flow-field. Nevertheless, 10 of 13

11 Figure 8. Flow visualization for C 2, top: 3D numerical schlieren, bottom: Q iso-surfaces flooded with streamwise velocity and mid-plane vorticity field. 11 of 13

12 Figure 9. Comparison of schlieren pictures at 165µs top left: experimental, top right: C 1, bottom left: C 2 and bottom right: C 3. the lack of information of the initial level of turbulence in the experiments leads to the difficulties involved in the proper choice of the initial flow-field and the assumption of initial turbulent parameters. Results of the finest mesh (C 3 ) shows better predictions of the separation point, Mach reflection associated with SS and shock/boundary layer interactions compared to C 1 and C 2. It can be realized that the present simulations are extremely demanding in terms of CPU usage. The reduced domain for C 3 utilizes 341 million mesh points run on 512 processors. It is worth noticing that grid resolution near the wall is essentially driving the computation requirement. Future work are planned to use a further refined mesh for LES and adoption of Detached Eddy Simulation (DES) approach in the present flow solver. Acknowledgments A part of this work has been done during the summer school at the TU München, Special thanks to Prof. Rainer Friedrich (Technische Universität München, Germany) and Prof. Tom Gatski (University of Poitiers, France) for their valuable discussion during the course of this research. The support of the first author by the German Research Foundation (Deutsche Forschungsgemeinschaft DFG), in the framework of the Sonderforschungsbereich Transregio 40 and the IGSSE (International Graduate School of Science and Engineering) during the summer program, is acknowledged. Authors are also thankful to the group of researchers working with Prof. Gabi Ben-Dor and Dr. Oren Sadot of Ben-Gurion University, Beer-Sheva, Israel, for providing some of their experimental results. Also, the authors wish to express their gratitude to Dr. Anne-Sophie Mouronval (Ecole Centrale Paris) and Dr. Guy Moebs (Uni. of Nantes, France) for their help with the optimization of the code. This work was performed using the computational facilities from GENCI [CCRT/CINES/IDRIS] (grant ). References 1 Smith C. E., The starting process in a hypersonic nozzle, J. Fluid Mechanics, Vol. 24, pp , Amann H. O., Experimental study of the starting process in a reflection nozzle, Phys. Fluids, Vol. 12, pp , Saito T., Timofeev E. V., Sun M. and Takayama K., Numerical and experimental study of 2-D nozzle starting process, In the proceedings of the 22nd ISSW, paper No. 4090, Imperial College, London, UK, July 18-23, of 13

13 4 Prodromou P. and Hillier R., Computation of unsteady nozzle flows, In the proceedings of the 18th ISSW, Sendai, Japan, Vol. II, pp , Igra O., Wang L., Falcovitz J., Amann O., Simulation of the starting flow in a wedge-like nozzle, Int. J. Shock Waves, Vol. 8, pp , Chen C. L., Chakravarty S. L. and Hung C. M., Numerical Investigation of Separated Nozzle Flows, AIAA J., Vol. 32, pp , Mouronval A.-S., Hadjadj A., Kudryavtsev A. and Vandromme D., Numerical investigation of transient nozzle flow, Int. J. Shock Waves, Vol. 12, pp , Mouronval A.-S. and Hadjadj A., Numerical study of the starting process in a supersonic nozzle, J. Propulsion & Power, Vol. 21, pp , Kwan W. and Stark R., Flow separation phenomena in subscale rocket nozzles, 38th AIAA/ASME/SEA/ASEE Joint Propulsion Conference & Exhibit, AIAA paper , Ostlund J., Damgaard T. and Frey M., Side-load phenomena in highly overexpanded rocket nozzles, Journal of Propulsion and Power, Vol. 20, No.4, Hadjadj A. and Perrot Y., Numerical simulation of transient supersonic nozzle flows, In the Proceedings of the 26th International Symposium on Shock Waves, Vol. 2, K. Hannemann, F. Seiler (Ed.) Springer, ISBN , Smagorinsky, J., General circulation experiments with the primitive equations. I. the basic experiment, Mon. Weather Rev, 91, 99, (1963). 13 Lilly, D. K., A proposed modification of the Germano subgrid-scale closure method, Phys. Fluids A, 4, 633, (1992). 14 Deardorff, J. W., Three-dimensional numerical study of the height and mean structure of a heated planetary boundary layer, Boundary Layer Meteorol., 7, 81, (1974). 15 Germano, M., Piomelli, U., Moin, P. and Cabot, W. H., A dynamic subgrid-scale eddy viscosity model, Phys. Fluids, A 3, 1760, (1991). 16 Sagaut, P., Large eddy simulation of turbulent flows, Springer, Berlin, (2001). 17 Pope, S. B., Turbulent flows, Cambridge university press, Cambridge, (2000). 18 Rogallo, R. S. and Moin, P., Numerical simulation of turbulent flows, Ann. Rev. Fluid Mech., 16, , (1984). 19 Galperin, B. and Orszag, S. A., Large eddy simulation of complex engineering and geophysical flows, Cambridge university press, Cambridge, (1993). 20 Lesieur, M. and Métais, O., New trends in Large-Eddy Simulations of turbulence, Ann. Rev. Fluid Mech., 28, 45-82, (1996). 21 Meneveau, C. and Katz, J., Scale-invariance and turbulence models for Large-Eddy Simulation, Ann. Rev. Fluid Mech., 32, 1-32, (2000). 22 Dubos, S. PhD Thesis, Simulation des grandes échelles d écoulements turbulents supersoniques, LMFN-CORIA, UMR 6614, France, (2005). 23 Nicoud, F. and Ducros, F., Subgrid-scale stress modelling based on the square of the velocity gradient tensor, Flow, Turbulence and Combustion, 62, , (1999). 24 Jiang, G. and Shu, C. W., Efficient implementation of weighted ENO schemes, J. Comp. Phys., 126, , (1996). 25 Peskin, C. S., Flow patterns around heart valves: a digital computer method for solving the equations of motion, PhD thesis, Albert Einstein Coll. Med., (1972). 26 Mittal, R. and Iaccarino, G., Immersed boundary methods, Ann. Rev. Fluid Mech., 37, , (2005). 27 Iaccarino, G. and Verzicco, R., Immersed boundary technique for turbulent flow simulations, Appl. Mech. Rev., 56, , (2003). 28 Tseng, Y. and Ferziger, J. H., A ghost-cell immersed boundary method for flow in complex geometry, J. Comp. Phys., 192, , (2003). 29 Gao, T., Tseng, Y. and Lu, X., An improved hybrid cartesian/immersed boundary method for fluid-solid flows, Int. J. Numer. Meth. Fluids, 55, , (2007). 30 Dadone, A. and Grossman, B., Ghost-cell method for inviscid two-dimensional flows on cartesian grids, AIAA J., 42, 12, , (2004). 31 Chaudhuri, A., Hadjadj, A. and Chinnayya, A., On the use of immersed boundary methods for shock/obstacle interactions, J. Comp. Phys., 230, , (2011). 32 Chaudhuri, A., Hadjadj, A., Chinnayya, A. and Palerm, S., Numerical study of compressible mixing layers using highorder WENO schemes. J. Sci. Computing, 47 (2), , (2011). 33 Thi3Dr8 (source code), Generation of incompressible 3D homogeneous isotropic turbulence, available at 13 of 13