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This paper pioneers the direct numerical simulation (DNS) and physical analysis in supersonic three-temperature carbon dioxide (CO2) turbulent channel flow. CO2 is a linear and symmetric triatomic molecular, with the thermal non-equilibrium three-temperature effects arising from the interactions among translational, rotational and vibrational modes under room temperature. Thus, the rotational and vibrational modes of CO2 are addressed. Thermal non-equilibrium effect of CO2 has been modeled in an extended three-temperature BGK-type model, with the calibrated translational, rotational and vibrational relaxation time. To solve the extended BGK-type equation accurately and robustly, non-equilibrium high-accuracy gas-kinetic scheme is proposed within the well-established two-stage fourth-order framework. Compared with the one-temperature supersonic turbulent channel flow, supersonic three-temperature CO2 turbulence enlarges the ensemble heat transfer of the wall by approximate 20%, and slightly decreases the ensemble frictional force. The ensemble density and temperature fields are greatly affected, and there is little change in Van Driest transformation of streamwise velocity. The thermal non-equilibrium three-temperature effects of CO2 also suppress the peak of normalized root-mean-square of density and temperature, normalized turbulent intensities and Reynolds stress. The vibrational modes of CO2 behave quite differently with rotational and translational modes. Compared with the vibrational temperature fields, the rotational temperature fields have the higher similarity with translational temperature fields, especially in temperature amplitude. Current thermal non-equilibrium models, high-accuracy DNS and physical analysis in supersonic CO2 turbulent flow can act as the benchmark for the long-term applicability of compressible CO2 turbulence., Comment: Carbon dioxide flow, Vibrational modes, Three-temperature effects, Supersonic turbulent channel flows

High-order gas-kinetic scheme (HGKS) with 5th-order non-compact reconstruction has been well implemented for implicit large eddy simulation (ILES) in nearly incompressible turbulent channel flows. In this study, the HGKS with higher-order non-compact reconstruction and compact reconstruction will be validated in turbulence simulation. For higher-order non-compact reconstruction, 7th-order normal reconstruction and tangential reconstruction are implemented. In terms of compact reconstruction, 5th-order normal reconstruction is adopted. Current work aims to show the benefits of high-order non-compact reconstruction and compact reconstruction for ILES. The accuracy of HGKS is verified by numerical simulation of three-dimensional advection of density perturbation. For the non-compact 7th-order scheme, 16 Gaussian points are required on the cell interface to preserve the order of accuracy. Then, HGKS with non-compact and compact reconstruction is used in the three-dimensional Taylor-Green vortex (TGV) problem and turbulent channel flows. Accurate ILES solutions have been obtained from HGKS. In terms of the physical modeling underlying the numerical algorithms, the compact reconstruction has the consistent physical and numerical domains of dependence without employing additional information from cells which have no any direct physical connection with the targeted cell. The compact GKS shows a favorable performance for turbulence simulation in resolving the multi-scale structures.

High-order gas-kinetic scheme (HGKS) has become a workable tool for the direct numerical simulation (DNS) of turbulence. In this paper, to accelerate the computation, HGKS is implemented with the graphical processing unit (GPU) using the compute unified device architecture (CUDA). To conduct the much large-scale DNS of turbulence, HGKS also be further upgraded with multiple GPUs using message passing interface (MPI) and CUDA architecture. The benchmark cases for compressible turbulence, including Taylor-Green vortex and turbulent channel flows, are presented to assess the numerical performance of HGKS with Nvidia TITAN RTX and Tesla V100 GPUs. For single-GPU computation, compared with the parallel central processing unit (CPU) code running on the Intel Core i7-9700 with open multi-processing (OpenMP) directives, 7x speedup is achieved by TITAN RTX and 16x speedup is achieved by Tesla V100. For multiple-GPU computation, the computational time of parallel CPU code running on 1024 Intel Xeon E5-2692 cores with MPI is approximately 3 times longer than that of GPU code using 8 Tesla V100 GPUs with MPI and CUDA. Numerical results confirm the excellent performance of multiple-GPU accelerated HGKS for large-scale DNS of turbulence. HGKS in GPU is also compiled with FP32 precision to evaluate the effect of number formats precision. Reasonably, compared to the computation with FP64 precision, the efficiency is improved and the memory cost is reduced with FP32 precision. For turbulent channel flows, difference in long-time statistical turbulent quantities is acceptable between FP32 and FP64 precision solutions. While the obvious discrepancy in instantaneous turbulent quantities can be observed, which shows that FP32 precision is not safe for DNS in compressible turbulence. The choice of precision should depended on the requirement of accuracy and the available computational resources.

For the first time, the non-equilibrium time-relaxation kinetic model (NTRKM) is proposed for compressible turbulence modeling on unresolved grids. Within the non-equilibrium time-relaxation framework, NTRKM is extended in the form of modified Bhatnagar-Gross-Krook model. Based on the first-order Chapman-Enskog expansion, NTRKM connects with the six-variable macroscopic governing equations. The first five governing equations correspond to the conservative laws in mass, momentum and total energy, while the sixth equation governs the evolution of unresolved turbulence kinetic energy Kutke. The unknowns in NTRKM, including turbulent relaxation time and source term, are determined by essential gradient-type assumption and standard dynamic modeling approach. Current generalized kinetic model on unresolved grids consequently offers a profound mesoscopic understanding for one-equation subgrid-scale turbulence kinetic energy Ksgs model in compressible large eddy simulation. To solve NTRKM accurately and robustly, a non-equilibrium gas-kinetic scheme is developed, which succeeds the well-established gas-kinetic scheme for simulating Navier-Stokes equations. Three-dimensional decaying compressible isotropic turbulence and temporal compressible plane mixing layer on unresolved grids are simulated to evaluate the generalized kinetic model and non-equilibrium gas-kinetic scheme. The performance of key turbulent quantities up to second-order statistics confirms that NTRKM is comparable with the widely-used eddy-viscosity Smagorinsky model (SM) and dynamic Smagorinsky model (DSM). Specifically, compared with the DNS solution in temporal compressible plane mixing layer, the performance of NTRKM is much closer with DSM and better than SM. This study provides a workable approach for compressible turbulence modeling on unresolved grids., Comment: 35 pages

In this paper, a high-order gas-kinetic scheme in general curvilinear coordinate (HGKS-cur) is developed for the numerical simulation of compressible turbulence. Based on the coordinate transformation, the Bhatnagar-Gross-Krook (BGK) equation is transformed from physical space to computational space. To deal with the general mesh given by discretized points, the geometrical metrics need to be constructed by the dimension-by-dimension Lagrangian interpolation. The multidimensional weighted essentially non-oscillatory (WENO) reconstruction is adopted in the computational domain for spatial accuracy, where the reconstructed variables are the cell averaged Jacobian and the Jacobian-weighted conservative variables. The two-stage fourth-order method, which was developed for spatial-temporal coupled flow solvers, is used for temporal discretization. The numerical examples for inviscid and laminar flows validate the accuracy and geometrical conservation law of HGKS-cur. As a direct application, HGKS-cur is implemented for the implicit large eddy simulation (iLES) in compressible wall-bounded turbulent flows, including the compressible turbulent channel flow and compressible turbulent flow over periodic hills. The iLES results with HGKS-cur are in good agreement with the refereed spectral methods and high-order finite volume methods. The performance of HGKS-cur demonstrates its capability as a powerful tool for the numerical simulation of compressible wall-bounded turbulent flows and massively separated flows., Comment: high-order gas-kinetic scheme, general curvilinear coordinate, implicit large eddy simulation, wall-bounded turbulent flows, compressible turbulence

The direct numerical simulation (DNS) of compressible isotropic turbulence up to the supersonic regime $Ma_{t} = 1.2$ has been investigated by high-order gas-kinetic scheme (HGKS) [{\it{Computers}} \& {\it{Fluids, 192, 2019}}]. In this study, the coarse-grained analysis of subgrid-scale (SGS) turbulent kinetic energy $K_{sgs}$ budget is fully analyzed for constructing one-equation SGS model in the compressible large eddy simulation (LES). The DNS on a much higher turbulent Mach number up to $Ma_{t} = 2.0$ has been obtained by HGKS, which confirms the super robustness of HGKS. Then, the exact compressible SGS turbulent kinetic energy $K_{sgs}$ transport equation is derived with density weighted filtering process. Based on the compressible $K_{sgs}$ transport equation, the coarse-grained processes are implemented on three sets of unresolved grids with the Box filter. The coarse-grained analysis of compressible $K_{sgs}$ budgets shows that all unresolved source terms are dominant terms in current system. Especially, the magnitude of SGS pressure-dilation term is in the order of SGS solenoidal dissipation term within the initial acoustic time scale. Therefore, it can be concluded that the SGS pressure-dilation term cannot be neglected as the previous work. The delicate coarse-grained analysis of SGS diffusion terms in compressible $K_{sgs}$ equation confirms that both the fluctuation velocity triple correlation term and the pressure-velocity correlation term are dominant terms. Current coarse-grained analysis gives an indication of the order of magnitude of all SGS terms in compressible $K_{sgs}$ budget, which provides a solid basis for compressible LES modeling in high Mach number turbulent flow., Comment: High-order gas-kinetic scheme, supersonic isotropic turbulence, compressible $K_{sgs}$ transport equation, coarse-grained budget analysis. arXiv admin note: text overlap with arXiv:1911.10966 by other authors

The performance of high-order gas-kinetic scheme (HGKS) has been investigated for the direct numerical simulation (DNS) of isotropic compressible turbulence up to the supersonic regime. Due to the multi-scale nature and coupled temporal-spatial evolution process, HGKS provides a valid tool for the numerical simulation of compressible turbulent flow. Based on the domain decomposition and message passing interface (MPI), a parallel HGKS code is developed for large-scale computation in this paper. The standard tests from the nearly incompressible flow to the supersonic one, including Taylor-Green vortex problem, turbulent channel flow and isotropic compressible turbulence, are presented to validate the parallel scalability, efficiency, accuracy and robustness of parallel implementation. The performance of HGKS for the nearly incompressible turbulence is comparable with the high-order finite difference scheme, including the resolution of flow structure and efficiency of computation. Based on the accuracy of the numerical solution, the numerical dissipation of the scheme in the turbulence simulation is quantitatively evaluated. As a mesoscopic method, HGKS performs better than both lattice Boltzmann method (LBM) and discrete unified gas-kinetic scheme (DUGKS), due to its high-order accuracy. Meanwhile, based on the kinetic formulation HGKS shows advantage for supersonic turbulent flow simulation with its accuracy and robustness. The current work demonstrates the capability of HGKS as a powerful DNS tool from the low speed to supersonic turbulence study, which is less reported under the framework of finite volume scheme., Comment: High-order gas-kinetic scheme, direct numerical simulation of turbulence, parallel computation. arXiv admin note: text overlap with arXiv:1905.03007; text overlap with arXiv:1911.10966 by other authors

This paper aims at the simulation of multiple scale physics in the system of radiation hydrodynamics. The system couples the fluid dynamic evolution equations with the radiative heat transfer. The coupled system is solved by the gas-kinetic scheme (GKS) for the compressible viscous and heat conducting flow and the unified gas-kinetic scheme (UGKS) for the non-equilibrium radiative transfer, together with the momentum and energy exchange between these two phases. For the radiative transfer, due to the possible large variation of fluid opacity in different regions, the transport of photons in the flow system is simulated by the multiscale UGKS, which is capable of naturally capturing the transport process from the free streaming to the diffusive propagation.Since both GKS and UGKS are finite volume methods,all unknowns are defined inside each control volume and are discretized consistently for the hydrodynamic and radiative variables. For the coupled system, the scheme has the asymptotical preserving (AP) property, such as recovering the equilibrium diffusion limit for the radiation hydrodynamic equations in the optically thick region, where the cell size is not limited by photon's mean free path. A few test cases, such as radiative shock wave problems, are used to validate the current approach., Comment: Radiation hydrodynamics, asymptotic preserving, gas kinetic scheme, unified gas kinetic scheme, radiative shock wave

In this paper, we intend to address the high-order gas-kinetic scheme (HGKS) in the direct numerical simulation (DNS) of compressible isotropic turbulence up to the supersonic regime. With the consideration of robustness and accuracy, the WENO-Z scheme is adopted for spatial reconstruction in the current higher-order scheme. Statistical quantities are compared with the high-order compact finite difference scheme to determine the spatial and temporal criterion for DNS. According to the grid and time convergence study, it can be concluded that the minimum spatial resolution parameter $\kappa_{max} \eta_0 \ge 2.71$ and the maximum temporal resolution parameter $\Delta t_{ini}/\tau_{t_0} \leq 5.58/1000$ are adequate for HGKS to resolve the compressible isotropic turbulence, where $\kappa_{max}$ is the maximum resolved wave number, $\Delta t_{ini}$ is the initial time step, $\eta_0$ and $\tau_{t_0}$ are the initial Kolmogorov length scale and the large-eddy-turnover time. Guided by such criterion, the compressible isotropic turbulence from subsonic regime $Ma_t=0.8$ to supersonic one $Ma_t=1.2$, and the Taylor microscale Reynolds number $Re_{\lambda}$ ranging from $10$ to $72$ are simulated. With the high initial turbulent Mach number, the strong random shocklets and high expansion regions are identified, as well as the wide range of probability density function over local turbulence Mach number. All those impose great challenge for high-order schemes. In order to construct compressible large eddy simulation models at high turbulent Mach number, the ensemble budget of turbulent kinetic energy is fully analyzed. The solenoidal dissipation rate decreases with the increasing of $Ma_t$ and $Re_{\lambda}$. Meanwhile, the dilational dissipation rate increases with the increasing of $Ma_t$, which cannot be neglected for constructing supersonic turbulence model., Comment: High-order gas-kinetic scheme, direct numerical simulation, compressible isotropic turbulence, supersonic regime

In recent years, coupled with traditional turbulence models, the second-order gas-kinetic scheme (GKS) has been used in the turbulent flow simulations. At the same time, high-order GKS has been developed, such as the two-stage fourth-order scheme (S2O4) GKS, and used for laminar flow calculations. In this paper, targeting on the high-Reynolds number engineering turbulent flows, an implicit high-order GKS with Lower-Upper Symmetric Gauss-Seidel (LU-SGS) technique is developed under the S2O4 framework. Based on Vreman-type LES model and $k - \omega$ SST model, a turbulent relaxation time is obtained and used for an enlarged particle collision time in the implicit high-order GKS for the high-Reynolds number turbulent flows. Numerical experiments include incompressible decaying homogeneous isotropic turbulence, incompressible high-Reynolds number flat plate turbulent flow, incompressible turbulence around NACA0012 airfoil, transonic turbulence around RAE2822 airfoil, and transonic high-Reynolds number ARA M100 wing-body turbulence. Comparisons among the numerical solutions from current implicit high-order GKS, the explicit high-order GKS, the implicit second-order GKS, and experimental measurements have been conducted. Through these examples, it is concluded that the high-order GKS has high accuracy in space and time, especially for smooth flows, obtaining more accurate turbulent flow fields on coarse grids compared with second-order GKS. In addition, significant acceleration on computational efficiency, as well as super robustness in simulating complex flows are confirmed for current implicit high-order GKS. This study also indicates that turbulence modeling plays a dominant role in the capturing physical solution, such as in the transonic three-dimensional complex RANS simulation, in comparison with numerical discretization errors., Comment: implict high-order GKS, LU-SGS, two-stage fourth-order scheme, computational efficiency, time-relaxation turbulence simulation. arXiv admin note: text overlap with arXiv:1810.04486

For increasingly rarefied flowfields, the Navier-Stokes (NS) equations lose accuracy partially due to the single temperature approximation. To overcome this barrier, a continuum multi-temperature model based on the Bhatnagar-Gross-Krook (BGK) equation coupled with the Landau-Teller-Jeans relaxation model has been proposed for two-dimensional hypersonic non-equilibrium multi-temperature flow computation. In recent study, a two-stage fourth-order gas-kinetic scheme (GKS) has been developed for equilibrium flows, which achieves a fourth-order accuracy in space and time as well as high efficiency and robustness. In this paper, targeting for accurate and efficient simulation of multi-temperature non-equilibrium flows, a high-order three-dimensional multi-temperature GKS is implemented under the two-stage fourth-order framework, with the fourth-order Simpson interpolation rule for the newly emerged source term. Simulations on decaying homogeneous isotropic turbulence, low-density nozzle flow, rarefied hypersonic flow over a flat plate, and type IV shock-shock interaction are used to validate the multi-temperature model through the comparison with experimental measurements. The unified gas kinetic scheme (UGKS) results, and the Direct simulation Monte Carlo (DSMC) solutions will be used as well in some cases for validation. Computational results not only confirm the high-order accuracy and quite robustness of this scheme, but also show the significant improvement on computational efficiency compared with UGKS and DSMC, especially in the near continuum flow regime., Comment: Multi-temperature kinetic model; Gas-kinetic method; High-order tem- poral discretization; Non-equilibrium flow computation