Subgrid-scale Model Research Articles

Joint subgrid velocity-scalar filtered mass density function method for compressible turbulent flows

In large eddy simulations (LESs) and scalar filtered mass density functions, a gradient diffusion model is generally used for subgrid-scale (SGS) scalar flux modeling. However, this hypothesis is known to generate evident errors under certain conditions, particularly when the counter-gradient scalar transport effect dominates. Herein, a joint subgrid velocity-scalar filtered mass density function (VSGSSFMDF) method is developed to address this problem. The exact FMDF transport equation is derived in detail. Based on the order of the magnitude analysis and simply Langevin model, the FMDF transport equation is modeled, and a system of stochastic differential equations is, thus, proposed. Theoretical derivation and analysis suggest that both the first- and second-order moments are consistent. A compressible mixing layer and hydrogen/air-reactive mixing layer are simulated to verify the proposed method. Based on the diffusion model, a direct numerical simulation and an LES are performed for comparative verification. A generally reasonable SGS velocity distribution is obtained using the proposed VSGSSFMDF method. Consequently, the counter-gradient scalar transport effect is effectively simulated using this method. Overall, the VSGSSFMDF produces better results than the SFMDF and LES in both cases, particularly in the reactive case.

Data-driven stochastic spectral modeling for coarsening of the two-dimensional Euler equations on the sphere

A resolution-independent data-driven subgrid-scale model in coarsened fluid descriptions is proposed. The method enables the inclusion of high-fidelity data into the coarsened flow model, thereby enabling accurate simulations also with the coarser representation. The small-scale model is introduced at the level of the Fourier coefficients of the coarsened numerical solution. It is designed to reproduce the kinetic energy spectra observed in high-fidelity data of the same system. The approach is based on a control feedback term reminiscent of continuous data assimilation implemented using nudging (Newtonian relaxation). The method relies solely on the availability of high-fidelity data from a statistically steady state. No assumptions are made regarding the adopted discretization method or the selected coarser resolution. The performance of the method is assessed for the two-dimensional Euler equations on the sphere for coarsening factors of 8 and 16 times. Applying the method at these significantly coarser resolutions yields good results for the mean and variance of the Fourier coefficients and leads to improvements in the empirical probability density functions of the attained vorticity values. Stable and accurate large-scale dynamics can be simulated over long integration times and are illustrated by capturing long-time vortex trajectories.

Momentum boundary layers in transcritical channel flows

We present well-resolved large-eddy simulations (LES) of a channel flow solving the fully compressible Navier–Stokes equations in conservative form. An adaptive look-up table method is used for thermodynamic and transport properties. We apply a physically consistent subgrid-scale turbulence model, that is based on the Adaptive Local Deconvolution Method (ALDM) for implicit LES. The wall temperatures are set to enclose the pseudo-boiling temperature at a supercritical pressure, leading to strong property variations within the channel geometry. In total seven cases are computed covering different Reynolds numbers, pseudo-boiling positions and pressure values. The hot wall at the top and the cold wall at the bottom produce asymmetric mean velocity and temperature profiles which result in different momentum layer thicknesses. All cases feature a turbulent mass flux which is essential in turbulence modelling. Furthermore, we analyse the turbulent kinetic energy budgets, perform a quadrant and octant analysis of the Reynolds shear stress and employ an invariant map to study the anisotropy in transcritical turbulent channel flows.

Large eddy simulation of plunging solitary wave: Understanding the breaking and turbulent mechanisms along shoaling region

A large eddy simulation (LES) is conducted to investigate the distribution of turbulence kinetic energy (TKE) under a plunging solitary wave over a 1:15 slope. This study provides a novel contribution to the field by examining the roles of resolved and sub-grid scale TKE in plunging solitary waves at the different stages of wave breaking. Furthermore, comparing the performances of two sub-grid scale (SGS) models in simulating the distribution of TKE was carried out to identify their performances. The separate investigation of these components in the context of wave breaking and recognizing the importance of an appropriate sub-grid scale model to consider the effects of small-scale eddies provide a significant advancement in understanding coastal morphological changes and nearshore sediment transport. Both the zero-equation and one-equation SGS models demonstrated acceptable performance in simulating water surface and kinematic properties. The one-equation SGS model, however, provided more accurate results on TKE transport during the breaking process and as the wave approaches its collapsing point. The study’s results reveal that an SGS model’s inability to simulate TKE transport (such as in the zero equation model) leads to inaccurate simulations of the TKE level and breaking location in the breaking zone. Additionally, the results of the one-equation model demonstrated that the maximum horizontal fluid velocity around the wavefront surface is a better predictor of breaking wave onset than the horizontal fluid velocity at the wave crest.

Ensemble data assimilation-based mixed subgrid-scale model for large-eddy simulations

An ensemble Kalman filter (EnKF)-based mixed model (EnKF-MM) is proposed for the subgrid-scale (SGS) closure in the large-eddy simulation (LES) of turbulence. The model coefficients are determined through the EnKF-based data assimilation technique. The direct numerical simulation (DNS) results are filtered to obtain the benchmark data for the LES. Reconstructing the correct kinetic energy spectrum of the filtered DNS (fDNS) data has been adopted as the target for the EnKF to optimize the coefficient of the functional part in the mixed model. The proposed EnKF-MM framework is subsequently tested in the LES of both the incompressible homogeneous isotropic turbulence and turbulent mixing layer. The performance of the LES is comprehensively examined through the predictions of the flow statistics including the velocity spectrum, the probability density functions (PDFs) of the SGS stress, the PDF of the strain rate, and the PDF of the SGS energy flux. The structure functions, the evolution of turbulent kinetic energy, the mean flow, the Reynolds stress profile, and the iso-surface of the Q-criterion are also examined to evaluate the spatial–temporal predictions by different SGS models. The results of the EnKF-MM framework are consistently more satisfying compared to the traditional SGS models, including the dynamic Smagorinsky model, the dynamic mixed model, and the velocity gradient model, demonstrating its great potential in the optimization of SGS models for the LES of turbulence.

On the robustness and accuracy of large-eddy simulation in predicting complex internal flow of a gas-turbine combustor

The objective of this study is to evaluate the effects of numerical and model setups on the large-eddy simulation (LES) predictive capability for the internal flow of a propulsion-relevant configuration. The specific focus is placed on assessing the LES technique with lower mesh resolutions, which is of technological relevance to practical industrial design. A set of Riemann flux formulations and commonly used subgrid-scale models are considered in this work to produce a hierarchy of LES setups with different dissipation effects (both numerically and physically). The LES results obtained from different setups are compared qualitatively in terms of the key flow characteristics and evaluated quantitatively against the experimental measurements. The error landscape is generated to reveal the predictive qualities of different LES setups. The study shows that the choice of numerical flux formulation plays a prominent role in governing the general flow patterns, while the effect of subgrid-scale model is mainly manifested in transient flow characteristics, such as vortex breakdown and swirl-induced vortical structures. Based on the error analysis, it is found that lower dissipative LES setup is not always beneficial to the LES accuracy. This is in contrast to the commonly accepted understanding in literature for the LES, which was established solely with canonical flow configurations.

Dynamic Mixed Modeling in Large Eddy Simulation Using the Concept of a Subgrid Activity Sensor

Following the relative success of mixed models in the Large Eddy Simulation of complex turbulent flow configurations, an alternative formulation is suggested here which incorporates the concept of a local subgrid activity sensor. The general idea of mixed models is to combine the advantages of structural models (superior alignment properties), usually of the scale similarity type, and functional models (superior stability), usually of the eddy viscosity type, while avoiding their disadvantages. However, the key question is the mathematical realization of this combination, and the formulation in this work accounts for the local level of underresolution of the flow. The justification and evaluation of the newly proposed mixed model is based on a priori and a posteriori analysis of homogeneous isotropic turbulence and laminar–turbulent transition in the Taylor–Green vortex, respectively. The suggested model shows a robust and accurate behavior for the cases investigated. In particular, it outperforms the separate structural and functional base models as well as the simulation without an explicit subgrid-scale model.

Application of a Local Dynamic Model of Large Eddy Simulation to a Marine Propeller Wake

With recent development of computer technology, the use of large eddy simulation method to solve industrial problems is gaining acceptance. From a theoretical and applied perspective, a local dynamic k-equation subgrid-scale model is applied to study the flow over a marine propeller. This local dynamic SGS model of LES have already been used in simple flows such as classical Taylor-Green vortex flow to investigate its robustness and superior than other dynamic SGS models. In this paper it will be applied to a more complex flows, i.e., simulation of a marine propeller wake, to further evaluate its ability range. 42 intentionally selected numerical experiments were conducted. The results of this local dynamic model of LES shows some superior than the dynamic Smagorinsky model, and well captures the wake evolution mechanism of a propeller, although which actually depends on its geometry.

Improved numerical predictions of inclined negatively buoyant jet behaviour

The results of numerical simulations of inclined negatively buoyant jets are presented. These simulations address previously highlighted difficulties in capturing sufficient detail of critical flow processes to effectively predict the detailed flow behaviour. In particular, the new simulations are able to accurately capture the details of the buoyancy-induced instabilities, which are clearly evident in associated experimental investigations and that have significant impacts on the flow behaviour. This new information is captured for inclined negatively buoyant jets discharged at 45° above a horizontal reference plane. A Large Eddy Simulation (LES) approach is implemented that makes use of a Lagrangian Dynamic Sub-grid scale (SGS) model and a novel criterion for the adaptive meshing system. Comparisons with previously published simulation results and experimental data demonstrate that these new Adaptive LES simulations provide improved predictions of flow path, concentration and velocity fields, and associated mean and turbulent statistics. In addition, this study provides a set of methods for generating high-quality LES data sets for free shear flows, which are well beyond the level of detail that can be captured by current experimental systems.

Investigation of thermal hydraulic behavior of the High Temperature Test Facility's lower plenum via large eddy simulation

A high-fidelity computational fluid dynamics (CFD) analysis was performed using the Large Eddy Simulation (LES) model for the lower plenum of the High–Temperature Test Facility (HTTF), a ¼ scale test facility of the modular high temperature gas-cooled reactor (MHTGR) managed by Oregon State University. In most next–generation nuclear reactors, thermal stress due to thermal striping is one of the risks to be curiously considered. This is also true for HTGRs, especially since the exhaust helium gas temperature is high. In order to evaluate these risks and performance, organizations in the United States led by the OECD NEA are conducting a thermal hydraulic code benchmark for HTGR, and the test facility used for this benchmark is HTTF. HTTF can perform experiments in both normal and accident situations and provide high-quality experimental data. However, it is difficult to provide sufficient data for benchmarking through experiments, and there is a problem with the reliability of CFD analysis results based on Reynolds–averaged Navier–Stokes to analyze thermal hydraulic behavior without verification. To solve this problem, high-fidelity 3-D CFD analysis was performed using the LES model for HTTF. It was also verified that the LES model can properly simulate this jet mixing phenomenon via a unit cell test that provides experimental information. As a result of CFD analysis, the lower the dependency of the sub-grid scale model, the closer to the actual analysis result. In the case of unit cell test CFD analysis and HTTF CFD analysis, the volume-averaged sub-grid scale model dependency was calculated to be 13.0% and 9.16%, respectively. As a result of HTTF analysis, quantitative data of the fluid inside the HTTF lower plenum was provided in this paper. As a result of qualitative analysis, the temperature was highest at the center of the lower plenum, while the temperature fluctuation was highest near the edge of the lower plenum wall. The power spectral density of temperature was analyzed via fast Fourier transform (FFT) for specific points on the center and side of the lower plenum. FFT results did not reveal specific frequency-dominant temperature fluctuations in the center part. It was confirmed that the temperature power spectral density (PSD) at the top increased from the center to the wake. The vortex was visualized using the well-known scalar Q-criterion, and as a result, the closer to the outlet duct, the greater the influence of the mainstream, so that the inflow jet vortex was dissipated and mixed at the top of the lower plenum. Additionally, FFT analysis was performed on the support structure near the corner of the lower plenum with large temperature fluctuations, and as a result, it was confirmed that the temperature fluctuation of the flow did not have a significant effect near the corner wall. In addition, the vortices generated from the lower plenum to the outlet duct were identified in this paper. It is considered that the quantitative and qualitative results presented in this paper will serve as reference data for the benchmark.

Numerical Investigation of the Inviscid Taylor-Green Vortex Using an Adaptive Filtering Method for a Modal Discontinuous Galerkin Method

Implicit Large Eddy Simulation and under-resolved Direct Numerical Simulation bypass the complexity and uncertainty of turbulence modelling by using the numerical dissipation of the scheme as a subgrid scale model. High-order methods allow for more accurate capturing of smaller scale structures but suffer from energy pile-up in the higher modes which leads to instability in under-resolved applications. This work presents a filtered modal Discontinuous Galerkin method which adaptively determines the filter strength, avoiding unnecessary degradation of accuracy while maintaining stability. The method is applied to the inviscid Taylor-Green Vortex, a challenging test case which exhibits under-resolved turbulence for which few published results exist. This goal of this work is to present the adaptive filtering method which achieves robustness and accuracy despite a low number of degrees of freedom, as well as to publish a quantity of relevant data for the inviscid TGV problem.

Adjoint-based variational optimal mixed models for large-eddy simulation of turbulence

An adjoint-based variational optimal mixed model (VOMM) is proposed for subgrid-scale (SGS) closure in large-eddy simulation (LES) of turbulence. The stabilized adjoint LES equations are formulated by introducing a minimal regularization to address the numerical instabilities of the long-term gradient evaluations in chaotic turbulent flows. The VOMM model parameters are optimized by minimizing the discrepancy of energy dissipation spectra between LES calculations and a priori knowledge of direct numerical simulation using the gradient-based optimization. The a posteriori performance of the VOMM model is comprehensively examined in LES of three turbulent flows, including the forced homogeneous isotropic turbulence, decaying homogenous isotropic turbulence, and temporally evolving turbulent mixing layer. The VOMM model outperforms the dynamic Smagorinsky model, dynamic mixed model (DMM), and approximate deconvolution model in predictions of various turbulence statistics, including the velocity spectrum, structure functions, statistics of velocity increments and vorticity, temporal evolutions of the turbulent kinetic energy, dissipation rate, momentum thickness and Reynolds stress, as well as the instantaneous vortex structures at different grid resolutions and times. In addition, the VOMM model only takes up 30% time of the DMM model for all flow scenarios. These results demonstrate that the proposed VOMM model improves the numerical stability of LES and has high a posteriori accuracy and computational efficiency by incorporating the a priori information of turbulence statistics, highlighting that the VOMM model has a great potential to develop advanced SGS models in the LES of turbulence.

Hydrodynamic modeling of coaxial confined particle-laden turbulent flow

A new particle subgrid scale model is proposed to consider the effect of gas flow on particle motions. Multiphase gas-particle turbulent flow is modeled by a second-order moment two-phase turbulence model involving a four-way coupling strategy to describe the interactions among gas-particle, particle-gas and particle-particle collisions. A large eddy simulation algorithm is developed to solve the hydrodynamic parameters of confined swirling and non-swirling particle-laden flow in coaxial chamber. Results show that predictions are well agreed with experimental data. Vortex structures and vortices distributions of gas and particle flow are quietly different. Coherent structures of swirling particle flows are not observed, and length of recirculation region is almost one quarter of non-swirling flow. Compared to developed flow region of non-swirling flow, standard deviation values of granular temperature at near entrance decreased by 5.5 times. Dominant frequencies of particle number density of non-swirling and swirling flows are 18 Hz and 10 Hz. Particle dispersions exhibit anisotropic characteristics, and their distributions are not in accordance with the normal distribution. The 2D turbulence model needs to be further improved due to the failure of describing vortex stretch in this job.

Four-way coupled modelling of swirling particle-laden flow in Methane-central coaxial jets

A novelty particle subgrid scale model is established to model the effect of gas flow on particle motion behaviors and a four-way coupling method to describe the particle-particle collisions by granular temperature equation is also developed. Menthane-central jets of particle-laden swirling flow in coaxial chamber are modeled by a multiphase second-order moment model and numerically predicted by the large-eddy simulation. Simulation results of two-phase hydrodynamics are in good agreement with experimental measurement. Evolution of coherent structure, recirculation region, periodic vortices formation, breakup, coalescence and pair, stronger mixing are captured and analyzed successfully. Compared to gas-annular flow, it is benefit for the stabilization of flame due to the strongly radial spread of particle dispersions and mixing momentum exchanges. The core-annular flow structures of particle concentration are observed, and instantaneous particle velocity are complied with the normal distribution. At shear layers of core-jetting regions of X/R=0.56, correlations of axial-tangential gas and particle fluctuation velocities are approximately 3.0 and 6.0 times larger than those of far downstream flows.

Assessment of subgrid-scale models in wall-modeled large-eddy simulations of turbulent channel flows

Assessment of subgrid-scale models in wall-modeled large-eddy simulations of turbulent channel flows

Physical/numerical duality of explicit/implicit subgrid-scale modelling

The development of implicit approaches has prompted debate on the actual usefulness of any explicit subgrid-scale modelling in large-eddy simulation. This question is addressed here by considering two generic turbulent flows: (i) the Taylor-Green vortex problem; (ii) the pipe flow. For both flow configurations, implicit modelling is found to overtake the very popular Smagorinsky model. To understand this robust observation, an analysis in the Fourier space is presented for the Taylor-Green vortex problem. The concept of spectral eddy viscosity, widely used in the pioneer work of Marcel Lesieur in two-point closure and subgrid-scale modelling, is revisited in a general framework based on explicit/implicit subgrid-scale modelling. In particular, the essentially anisotropic nature of implicit modelling is exhibited, as a favourable feature in terms of consistency with the computational mesh. Smagorinsky's model, considered as a generic explicit subgrid-scale model in the framework of Boussinesq's hypothesis, is found to be highly sensitive to numerical errors. Removing the latter is easy but makes computationally inefficient this type of explicit modelling. Comparisons between a priori and a posteriori spectral eddy viscosities show that neither Smagorinsky's model nor implicit modelling can mimic the expected spectral behaviour. Smagorinsky's model is observed to be weakly scale-selective with a poor ability to actually filter the solution. The feature of scale-selectivity is well replicated by implicit modelling which exhibits excellent capabilities for filtering. However, its lack of influence at the largest scales is against the expected behaviour for the spectral eddy viscosity at low wavenumber through the establishment of a non-zero plateau value. This lack of consistency of implicit LES could be overcome thanks to an extra explicit modelling but the attempt to mix Smagorinsky's model and implicit LES is not successful in this study. The potential of implicit large-eddy simulation is also exhibited for the accurate computation of near-wall turbulence inside a pipe flow despite the use of a regular Cartesian mesh with an immersed boundary method. Interestingly, the resulting coarse wall-normal resolution in the near-wall direction does not prevent the reliable prediction of statistical profiles up to the capture of subgrid-scale details. It is suggested that the regularisation associated with implicit modelling is a necessary condition to reach numerical accuracy. However, to faithfully represent the large-scale dynamics, present results confirm that non-local triad interactions must be taken into account as widely discussed in the inspiring textbook Lesieur [Turbulence in fluids. 4th ed. Springer; 2008] of Marcel Lesieur.

Machine learning building-block-flow wall model for large-eddy simulation

A wall model for large-eddy simulation (LES) is proposed by devising the flow as a combination of building blocks. The core assumption of the model is that a finite set of simple canonical flows contains the essential physics to predict the wall shear stress in more complex scenarios. The model is constructed to predict zero/favourable/adverse mean pressure gradient wall turbulence, separation, statistically unsteady turbulence with mean flow three-dimensionality, and laminar flow. The approach is implemented using two types of artificial neural networks: a classifier, which identifies the contribution of each building block in the flow, and a predictor, which estimates the wall shear stress via a combination of the building-block flows. The training data are obtained directly from wall-modelled LES (WMLES) optimised to reproduce the correct mean quantities. This approach guarantees the consistency of the training data with the numerical discretisation and the gridding strategy of the flow solver. The output of the model is accompanied by a confidence score in the prediction that aids the detection of regions where the model underperforms. The model is validated in canonical flows (e.g. laminar/turbulent boundary layers, turbulent channels, turbulent Poiseuille–Couette flow, turbulent pipe) and two realistic aircraft configurations: the NASA Common Research Model High-lift and NASA Juncture Flow experiment. It is shown that the building-block-flow wall model outperforms (or matches) the predictions by an equilibrium wall model. It is also concluded that further improvements in WMLES should incorporate advances in subgrid-scale modelling to minimise error propagation to the wall model.

Stretch Effects in Large Eddy Simulation of Turbulent Premixed Bunsen Flames

ABSTRACT The stretch effects in large eddy simulation (LES) of a turbulent lean propane-air premixed Bunsen flame in the wrinkled flamelet regime are investigated. A simple approach to model the subgrid-scale flame stretch is proposed. It is based on the analysis which shows that, when the surface-filtered strain rate in a subgrid-scale stretch model is approximated using the volume-filtered velocity field solved for in LES, heat release and associated gas expansion in the filtered flame brush tend to artificially alter the wrinkling of resolved flame fronts. To minimize such artifacts, it is suggested that the volume-filtered strain rate on the unburned side of the filtered flame brush be used to approximate the surface-filtered strain rate and projected through the filtered flame brush. The results show the importance of mitigating the artificial heat release effects when considering the strain effects in LES. The relative importance of the curvature stretch is also investigated in terms of the mean and local effects. For a Bunsen flame with a positive Markstein number, the mean flame curvature effect tends to increase the total burning rate by enhancing the laminar flame speed near the flame tip, while the local flame curvature effect tends to decrease the total burning rate by suppressing the wrinkling of the resolved flames. It is found that the competition between the two makes the overall curvature effects not influence the total burning rate much for the flame investigated here, as compared with the strain effects.

Large-eddy simulation of airflow dynamics around a cluster of buildings

The wind flow around the buildings with different rooftops is studied numerically using large-eddy simulation (LES). The filtered Navier-Stokes equations in LES are used to compute the large eddies, whereas the dynamic Smagorinsky subgrid-scale (SGS) model calculates the small eddies. A turbulent spot method is applied for the synthesis of artificial turbulent fields at the inlet. In this separated and reattached flow, our aim is to analyze the wakes’ vortical structure along with the buildings of different heights and shapes. A large number of engineering applications involve precise predictions of the airflow around buildings to ensure performance and safety. To validate the numerical solver, a comparison is performed with the earlier reported numerical and experimental data. The turbulent flow characteristics are discussed in terms of instantaneous flow structure and time-averaged statistical flow quantities. All the cases are simulated for a Reynolds number [Formula: see text] to understand the turbulent airflow patterns. The flow shows a separable bubble at the leading edge of the rooftop of the building that leads to recirculation at the lee side of the first row of buildings. Due to the recirculation, the suction arises near the leading edge of the building to assure a continuous flow of air around the obstacles. Further, Reynold stresses show high momentum fluxes in the frontal region of the buildings.

An accurate and practical numerical solver for simulations of shock, vortices and turbulence interaction problems

Numerical simulations of high-speed compressible flows remain challenging in engineering as the appearance of shock waves poses difficulties for high-resolution schemes as well as explicit large eddy simulation methods. Therefore, in this work, we propose a practical numerical solver for simulations of shock-vortex and shock-turbulence problems with the implicit large eddy simulation approach and newly developed low-dissipative schemes. In order to handle the multi-scale raised in the shock-vortex and shock-turbulence problems, it is required that the flow solver can simultaneously solve the large-scale flow structures such as shock waves with numerical oscillation-free, the resolvable flow structures above the grid scale with low-dissipation, and the under-resolve isotropic sub-grid scale with numerical stability. To this end, a low-dissipative, structure-preserving scheme with rigorously adjusted numerical dissipation is employed in the proposed solver. The structure-preserving property of this scheme can ensure that the large-scale structure is solved without numerical oscillations and the low-dissipative property of this scheme can produce high-resolution results for the resolvable flow scales. Moreover, the sub-grid scale is solved and stabilized by the inherent numerical dissipation in the shock-capturing scheme. The proposed numerical solver is then applied to simulate a wide range of shock-vortex and shock-turbulence interaction problems including supersonic planar jets, transonic flows past a deep cavity and impingement of a supersonic jet on a cone mounted on a flat plate. A comparison is also made with the solver using conventional total variation diminishing schemes. The numerical results of the supersonic planar jet have demonstrated that the proposed solver can resolve the small-scale structure such as Kelvin–Helmholtz instability involved in shock and shear layer with high-resolution. Through the results of transonic flow past a deep cavity and comparisons with the experimental data, it is verified that the proposed solver can reproduce the turbulence statistical data. Furthermore, the proposed solver resolves the complex turbulent fluid mechanical phenomena in the impingement of a supersonic jet on a cone, which demonstrates the proposed solver can robustly handle irregular geometry. The proposed solver also enjoys simplicity without using the explicit sub-grid scale model and involving the complexity of very high-order schemes. Thus, this work provides an accurate and practical numerical solver for shock-vortex and shock-turbulence problems in high-speed flows.