Fluid dynamic turbulence is one of the most challenging computational physics problems because of the extremely wide range of time and space scales involved, the strong nonlinearity of the governing equations, and the many practical and important applications. While most linear fluid instabilities are well understood, the nonlinear interactions among them makes even the relatively simple limit of homogeneous isotropic turbulence difficult to treat physically, mathematically, and computationally. Turbulence is modeled computationally by a two-stage bootstrap process. The first stage, direct numerical simulation, attempts to resolve the relevant physical time and space scales but its application is limited to diffusive flows with a relatively small Reynolds number (Re). Using direct numerical simulation to provide a database, in turn, allows calibration of phenomenological turbulence models for engineering applications. Large eddy simulation incorporates a form of turbulence modeling applicable when the large-scale flows of interest are intrinsically time dependent, thus throwing common statistical models into question. A promising approach to large eddy simulation involves the use of high-resolution monotone computational fluid dynamics algorithms such as flux-corrected transport or the piecewise parabolic method which have intrinsic subgrid turbulence models coupled naturally to the resolved scales in the computed flow. The physical considerations underlying and evidence supporting this monotone integrated large eddy simulation approach are discussed.

New insights into large eddy simulation

Implicit High-Order Gas Kinetic Scheme for Turbulence Simulation

Three dimensional high-order gas-kinetic scheme for supersonic isotropic turbulence I: Criterion for direct numerical simulation

A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured mesh

In this paper, a new mesoscopic approach with both the adjustable Prandtl number and the ratio of bulk to shear viscosity has been developed to simulate three-dimensional compressible decaying homogeneous isotropic turbulence under the framework of discrete unified gas kinetic scheme (DUGKS). In the new approach, two reduced model Boltzmann equations with newly designed source terms are solved. In the continuum limit, the Navier–Stokes–Fourier system can be recovered by applying the Chapman–Enskog analysis. A three-dimensional DUGKS code has been developed, incorporating the fifth-order weighted essentially non-oscillatory scheme to better reconstruct the particle distribution functions at the cell interfaces. In addition, a new lattice velocity model with 77 discrete particle velocities is applied to ensure that the accuracy of the Gauss–Hermite quadrature is up to the ninth-order, and as such, the heat flux can be accurately evaluated. To validate our code, we simulate two cases with different initial turbulent Mach numbers and Taylor microscale Reynolds numbers. The simulation results converge with the increase in resolution and agree well with the results from the literature. As a direct application of our DUGKS, we briefly study the influence of bulk viscosity on turbulence statistics and flow structures. Our results show that the DUGKS is a reliable tool for simulating compressible decaying isotropic turbulence at low and moderate turbulent Mach numbers. More parametric studies are needed in the future to further explore the full capabilities of this specific mesoscopic method. © 2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0029424., s

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Simulation of three-dimensional compressible decaying isotropic turbulence using a redesigned discrete unified gas kinetic scheme

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.

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.

Guiyu Cao

Papers

Implicit High-Order Gas Kinetic Scheme for Turbulence Simulation

Three dimensional high-order gas-kinetic scheme for supersonic isotropic turbulence I: Criterion for direct numerical simulation

Implicit High-Order Gas Kinetic Scheme for Turbulence Simulation

Physical Modeling and Numerical Studies of Three-dimensional Non-equilibrium Multi-temperature Flows

Physical modeling and numerical studies of three-dimensional non-equilibrium multi-temperature flows