Computation of complex turbulent flow using matrix-free implicit dual time-stepping scheme and LRN turbulence model on unstructured grids

In this study, a matrix-free implicit dual time-stepping method has been developed. It is implemented, together with a low-Reynolds-number q-ω turbulence model, in a high-order upwind finite-volume solver on unstructured grids. Semi-implicit treatment of the source terms of the q and ω equations is also introduced to further stabilize the numerical solution. It has been found that these techniques provide strong stabilization in the computation of a supersonic flow with complex shock-boundary-layer interactions in a channel with a backward-facing step. The proposed method has a low-memory overhead, similar to an explicit scheme, while it shows good stability and computational efficiency as an implicit scheme. The method developed has been validated by comparing the computed results with the corresponding experimental measurements and other calculated results, which shows good agreement. Research is being done to extend the method to calculate unsteady turbulent flows.     

Computational technologies for simulation of complex near-wall turbulent flows using unstructured meshes

The work is devoted to the peculiarities of the implementation of hybrid Reynolds’ Averaged Navier-Stokes equations-Large Eddy Simulation (RANS-LES) approaches of the Detached Eddy Simulation (DES) family for simulation of complex near-wall turbulent flows using unstructured meshes. The problems of determining required geometric characteristics in the mesh nodes and adaptation of hybrid approaches to the used accurate numerical approximation scheme in space are considered. The classic benchmark problem of the decay of homogeneous isotropic turbulence and the results of the computation of a complex turbulent flow near the wall with the presence of flow separation and reattachment are considered to verify the implemented technique and to demonstrate its efficiency.     

Time-accurate Navier-Stokes simulation of vortex convection using an unstructured dynamic mesh procedure

A two-dimensional Navier-Stokes flow solver is developed for the simulation of unsteady flows on unstructured adaptive meshes. The solver is based on a second-order accurate implicit time integration using a point Gauss-Seidel relaxation scheme and a dual time-step subiteration. A vertex-centered, finite-volume discretization is used in conjunction with Roe’s flux-difference splitting. The Spalart-Allmaras one equation model is employed for the simulation of turbulence. An unsteady solution-adaptive dynamic mesh scheme is used by adding and deleting mesh points to take account of spatial and temporal variations of the flowfield. Unsteady viscous flow for a traveling vortex in a free stream is simulated to validate the accuracy of the dynamic mesh adaptation procedure. Flow around a circular cylinder and two blade-vortex interaction problems are investigated for demonstration of the present method. Computed results show good agreement with existing experimental and computational results. It was found that unsteady time-accurate viscous flows can be accurately simulated using the present unstructured dynamic mesh adaptation procedure.     

Gas-kinetic numerical study of complex flow problems covering various flow regimes

The Boltzmann simplified velocity distribution function equation, as adapted to various flow regimes, is described on the basis of the Boltzmann–Shakhov model from the kinetic theory of gases in this study. The discrete velocity ordinate method of gas-kinetic theory is studied and applied to simulate complex multi-scale flows. On the basis of using the uncoupling technique on molecular movements and collisions in the DSMC method, the gas-kinetic finite difference scheme is constructed by extending and applying the unsteady time-splitting method from computational fluid dynamics, which directly solves the discrete velocity distribution functions. The Gauss-type discrete velocity numerical quadrature technique for flows with different Mach numbers is developed to evaluate the macroscopic flow parameters in the physical space. As a result, the gas-kinetic numerical algorithm is established for studying the three-dimensional complex flows with high Mach numbers from rarefied transition to continuum regimes. On the basis of the parallel characteristics of the respective independent discrete velocity points in the discretized velocity space, a parallel strategy suitable for the gas-kinetic numerical method is investigated and, then, the HPF (High Performance Fortran) parallel programming software is developed for simulating gas dynamical problems covering the full spectrum of flow regimes. To illustrate the feasibility of the present gas-kinetic numerical method and simulate gas transport phenomena covering various flow regimes, the gas flows around three-dimensional spheres and spacecraft-like shapes with different Knudsen numbers and Mach numbers are investigated to validate the accuracy of the numerical methods through HPF parallel computing. The computational results determine the flow fields in high resolution and agree well with the theoretical and experimental data. This computing, in practice, has confirmed that the present gas-kinetic algorithm probably provides a promising approach for resolving hypersonic aerothermodynamic problems with the complete spectrum of flow regimes from the gas-kinetic point of view for solving the mesoscopic Boltzmann model equation.     

Hybrid grid generation for two-dimensional high-Reynolds flows

An hybrid mesh generation algorithm for two-dimensional viscous flows at high-Reynolds number is presented. An advancing-front method is used close to solid surfaces and in the wake region(s). The boundary layer and wake grids possibly contain both highly stretched quadrilateral and triangular elements. The latter are inserted locally to improve the quality of the grid, thus circumventing some drawbacks of standard structured grid advancing-front methods. An advancing-front/Delaunay algorithm triangulates the remaining portion of the computational domain. An overview of both algorithms is given and results for viscous laminar and turbulent compressible flows around single and multi-element airfoils are shown to support the present approach.     

Large stencil viscous flux linearization for the simulation of 3D compressible turbulent flows with backward-Euler schemes

The purpose of this article is to study different approximate linearizations of the RANS equations viscous fluxes, for numerical simulations of compressible turbulent flows with backward-Euler schemes. The explicit convective flux under consideration is centred with artificial dissipation. The discrete viscous flux, calculated from cell-centred evaluation of the gradients, needs less computations and memory storage than other discretizations. But, in other respects, the balance of this numerical flux has a large stencil, which is not coherent with the 3-point per mesh direction stencil of classical implicit stages. Therefore 3-point and 5-point per mesh direction approximate linearizations are built from the thin layer flux formula. The stability condition of the corresponding backward-Euler schemes is given for a scalar linear equation (for the basic non-factored version of scheme and with LU-relaxation). Multigrid and monogrid computations of turbulent flow around two external configurations are performed with Wilcox’s k-ω turbulence model. The 5-point per mesh direction linearizations, coherent with the differential of the fluxes balance of thin layer approximation of explicit viscous fluxes, leads to the most efficient implicit stages.     

A unifying grid approach for solving potential flows applicable to structured and unstructured grid configurations

In this study, an efficient numerical method is proposed for unifying the structured and unstructured grid approaches for solving the potential flows. The new method, named as the “alternating cell directions implicit - ACDI”, solves for the structured and unstructured grid configurations equally well. The new method in effect applies a line implicit method similar to the Line Gauss Seidel scheme for complex unstructured grids including mixed type quadrilateral and triangle cells. To this end, designated alternating directions are taken along chains of contiguous cells, i.e. ‘cell directions’, and an ADI-like sweeping is made to update these cells using a Line Gauss Seidel like scheme. The algorithm makes sure that the entire flow field is updated by traversing each cell twice at each time step for unstructured quadrilateral grids that may contain triangular cells. In this study, a cell-centered finite volume formulation of the ACDI method is demonstrated. The solutions are obtained for incompressible potential flows around a circular cylinder and a forward step. The results are compared with the analytical solutions and numerical solutions using the implicit ADI and the explicit Runge-Kutta methods on single-and multi-block structured and unstructured grids. The results demonstrate that the present ACDI method is unconditionally stable, easy to use and has the same computational performance in terms of convergence, accuracy and run times for both the structured and unstructured grids.     

Numerical simulation of incompressible flows using adaptive unstructured meshes and the pseudo-compressibility hypothesis

The numerical simulation of incompressible viscous flows, using finite elements with automatic adaptive unstructured meshes and the pseudo-compressibility hypothesis, is presented in this work. Special emphasis is given to the automatic adaptive process of unstructured meshes with linear tetrahedral elements in order to get more accurate solutions at relatively low computational costs. The behaviour of the numerical solution is analyzed using error indicators to detect regions where some important physical phenomena occur. An adaptive scheme, consisting in a mesh refinement process followed by a nodal re-allocation technique, is applied to the regions in order to improve the quality of the numerical solution. The error indicators, the refinement and nodal re-allocation processes as well as the corresponding data structure (to manage the connectivity among the different entities of a mesh, such as elements, faces, edges and nodes) are described. Then, the formulation and application of a mesh adaptation strategy, which includes a refinement scheme, a mesh smoothing technique, very simple error indicators and an adaptation criterion based in statistical theory, integrated with an algorithm to simulate complex two and three dimensional incompressible viscous flows, are the main contributions of this work. Two numerical examples are presented and their results are compared with those obtained by other authors.     

An implicit low-diffusive HLL scheme with complete time linearization: Application to cavitating barotropic flows

A numerical method for generic barotropic flows is presented, together with its application to the simulation of cavitating flows. A homogeneous-flow cavitation model is indeed considered, which leads to a barotropic state equation. The continuity and momentum equations for compressible flows are discretized through a mixed finite-element/finite-volume approach, applicable to unstructured grids. P1 finite elements are used for the viscous terms, while finite volumes for the convective ones. The numerical fluxes are computed by shock-capturing schemes and ad-hoc preconditioning is used to avoid accuracy problems in the low-Mach regime. A HLL flux function for barotropic flows is proposed, in which an anti-diffusive term is introduced to counteract accuracy problems for contact discontinuities and viscous flows typical of this class of schemes, while maintaining its simplicity. Second-order accuracy in space is obtained through MUSCL reconstruction. Time advancing is carried out by an implicit linearized scheme. For this HLL-like flux function two different time linearizations are considered; in the first one the upwind part of the flux function is frozen in time, while in the second one its time variation is taken into account. The proposed numerical ingredients are validated through the simulations of different flow configurations, viz. the Blasius boundary layer, a Riemann problem, the quasi-1D cavitating flow in a nozzle and the flow around a hydrofoil mounted in a tunnel, both in cavitating and non-cavitating conditions. The Roe flux function is also considered for comparison. It is shown that the anti-diffusive term introduced in the HLL scheme is actually effective to obtain good accuracy (similar to the one of the Roe scheme) for viscous flows and contact discontinuities. Moreover, the more complete time linearization is a key ingredient to largely improve numerical stability and efficiency in cavitating conditions.     

The unconditionally positive-convergent implicit time integration scheme for two-equation turbulence models: Revisited

The unconditionally positive-convergent implicit scheme for two-equation turbulence models, originally developed by Mor-Yossef and Levy, is revisited. A compact, simple, and uniform reformulation of the method for the use of both structured and unstructured grid based flow solvers is presented. An analytical proof of the scheme revision is given showing that positivity of the turbulence model solutions and convergence of the turbulence model equations are guaranteed for any time step. Numerical experiments are conducted, simulating two test cases of three-dimensional complex flow fields using structured and hybrid unstructured grids. To demonstrate the overall scheme’s robustness, it is applied to non-linear k-ω and non-linear k- turbulence models. Results from the numerical simulations show that the scheme exhibits very good convergence characteristics, is robust, and it always preserves the positivity of the turbulence model dependent variables, even for an infinite time step.     

Assessment of multiscale resolution for hybrid turbulence model in unsteady separated transonic flows

This paper presents results of a computational study conducted to assess the multi-scale resolution capabilities of a hybrid two-equation turbulence model in predicting unsteady separated high speed flows. Numerical solutions are obtained using a third order Roe scheme and the SST (shear-stress-transport) two-equation-based hybrid turbulence model for three-dimensional transonic flow over an open cavity. A detailed assessment of the effects of the computational grid and the hybrid turbulence model coefficient is presented for the unsteady flow field. Computed results are presented for both the resolved and the modeled turbulent kinetic energy (TKE) and for the predicted sound pressure level (SPL) spectra, which are compared to available experimental data and large Eddy simulation (LES) results. The comparison shows that the predicted SPL spectra agree well with the experimental results over a frequency range up to 2500 Hz, and that hybrid turbulence effectively models the shorter wavelengths. The results demonstrate improved agreement with experimental SPL spectra with increased grid resolution and a reduced hybrid turbulence model coefficient. In addition, they show that energy dissipation of the unresolved scales is over-predicted at low resolutions and that the hybrid coefficient influences the grid resolution requirements.     

Implicit linearity preservation type schemes for moving meshes

We are concerned with the accurate implicit approximation of compressible flows in a fixed and moving mesh context, such as piston engine flows. Geometries are commonly complex and flows compressible. Therefore, it is convenient to develop the numerical approach in the context of a space-time finite-volume formulation for unstructured meshes. The hyperbolic flux is obtained by a generalized Riemann solver taking into account the mesh motion. Using the linearity preservation property we propose a new class of stable implicit schemes developing low numerical viscosity. These schemes can be viewed as a correction of the usual MUSCL flux, induced by the time derivative and mesh motion. Accurate numerical results are obtained for transonic (shock tube) as well as low Mach number flows (diesel engine). It is numerically proved, that for large time steps, those approximations can be as accurate as some explicit schemes. The proposed schemes, due the compactness of the stencils, are well adapted for parallelization strategy.     

Higher-order time accurate numerical methods for singularly perturbed parabolic partial differential equations

This article presents two numerical methods for singularly perturbed time-dependent reaction-diffusion initial–boundary-value problems. The spatial derivative is replaced by a hybrid scheme, which is a combination of the cubic spline and the classical central difference scheme in both the methods. In the first method, the time derivative is replaced by the Crank–Nicolson scheme, whereas in the second method the time derivative is replaced by the extended-trapezoidal scheme. These schemes are applied on the layer resolving piecewise-uniform Shishkin mesh. Some numerical examples are carried out to show the accuracy and efficiency of these methods.     

High-resolution finite difference schemes using curvilinear coordinate grids for DNS of compressible turbulent flow over wavy walls

Direct numerical simulations of compressible turbulent flow over wavy wall geometries have been carried out by solving N–S equations on general curvilinear coordinates. A 6th order WENO scheme with minimized dispersion and controllable dissipation is employed to compute the inviscid fluxes, a 4th order central difference scheme is applied to compute the viscous fluxes, and a 6th order conservative compact scheme is used for computing the geometrical metrics. An implicit LU-SGS method is used for time integration to improve computational efficiency over the explicit schemes such as the Runge–Kutta approach. The validity and applicability of the present algorithm is confirmed by comparing our results with laboratory experimental measurements and DNS results in the literature.     

Parallel simulation of three-dimensional complex flows: Application to two-phase compressible flows and turbulent wakes

In this paper, we present parallel simulations of three-dimensional complex flows obtained on an ORIGIN 3800 computer and on homogeneous and heterogeneous (processors of different speeds and RAM) computational grids. The solver under consideration, which is representative of modern numerics used in industrial computational fluid dynamics (CFD) software, is based on a mixed element-volume method on unstructured tedrahedrisations. The parallelisation strategy combines mesh partitioning techniques, a message-passing programming model and an additive Schwarz algorithm. The parallelisation performances are analysed on a two-phase compressible flow and a turbulent flow past a square cylinder.     

基于结构网格的高超声速流动大规模并行数值模拟研究

本文采用MPI消息传递模式自主开发出适用于高超声速流动数值模拟的并行计算软件,该软件以三维Navier-Stokes方程为基本控制方程来求解层流问题,应用基于结构网格的有限体积法对计算域进行离散,采用AUSMPW+格式求解对流通量,利用MUSCL插值方法获得高阶精度,时间格式上采用LU-SGS方法进行时间迭代以加快求解定常流动的收敛过程。在高性能计算机上针对不同高超声速流动进行大规模并行计算的结果表明,所开发的CFD并行计算软件具有较高的并行计算效率,为高超声速飞行器气动力/热的准确预测提供了高效工具。     

Second-Order Accurate Godunov Scheme for Multicomponent Flows on Moving Triangular Meshes

This paper presents a second-order accurate adaptive Godunov method for two-dimensional (2D) compressible multicomponent flows, which is an extension of the previous adaptive moving mesh method of Tang et al. (SIAM J. Numer. Anal. 41:487–515, 2003) to unstructured triangular meshes in place of the structured quadrangular meshes. The current algorithm solves the governing equations of 2D multicomponent flows and the finite-volume approximations of the mesh equations by a fully conservative, second-order accurate Godunov scheme and a relaxed Jacobi-type iteration, respectively. The geometry-based conservative interpolation is employed to remap the solutions from the old mesh to the newly resulting mesh, and a simple slope limiter and a new monitor function are chosen to obtain oscillation-free solutions, and track and resolve both small, local, and large solution gradients automatically. Several numerical experiments are conducted to demonstrate robustness and efficiency of the proposed method. They are a quasi-2D Riemann problem, the double-Mach reflection problem, the forward facing step problem, and two shock wave and bubble interaction problems.     

Implicit Large Eddy Simulation of non-wall-bounded turbulent flows based on the multiscale properties of a high-order finite volume method

In this paper it is presented the application of a higher-order finite volume method based on Moving Least Squares approximations (FV-MLS) to the resolution of non-wall-bounded compressible turbulent flows. Our approach is based on the monotonically implicit Large Eddy Simulation (MILES). The main idea of MILES methodology is the absence of any explicit subgrid scale (SGS) model in the numerical algorithm to solve turbulent flows. In the case of the FV-MLS method, we take advantage of the multiresolution properties of Moving Least Squares Approximations, and we show that they can be used as an implicit SGS model. The numerical results are encouraging. The third-order FV-MLS method is able to reproduce the inertial subrange, and it obtains better results than other usual numerical schemes in LES computations, such as the MUSCL scheme. We note that in the present state of this research, the numerical method is not yet suited for wall-bounded flows. This paper is the first step in the application of the FV-MLS method to general turbulent flows.     

The discontinuous Galerkin finite element method for the 2D shallow water equations

A high-order finite element method, total variational diminishing (TVD) Runge–Kutta discontinuous Galerkin method is investigated to solve free-surface problems in hydraulic dynamics. Some cases of circular dam and rapidly varying two-dimensional flows are presented to show the efficiency and stability of this method. The numerical simulations are given on structured rectangular mesh for regular domain and on unstructured triangular mesh for irregular domain, respectively.     

Thermal Investigation for Solid Rocket Motor Nozzle Inserts Material

A time-resolved numerical computational approach, involving the combustion of double-base propellant is performed on thermal protection material for SRM nozzle. An implicit Navier-Stokes Solver is selected to simulate two-dimensional axial-symmetric unsteady turbulent flow of compressible fluid. The governing equations are discredited by using the finite Volume method. S-A turbulence model is employed. CFD scheme is implemented to investigate the temperature distribution causes at nozzle throat inserts comp...