An implicit gas-kinetic scheme for turbulent flow on unstructured hybrid mesh

In this study, an implicit scheme for the gas-kinetic scheme (GKS) on the unstructured hybrid mesh is proposed. The Spalart–Allmaras (SA) one equation turbulence model is incorporated into the implicit gas-kinetic scheme (IGKS) to predict the effects of turbulence. The implicit macroscopic governing equations are constructed and solved by the matrix-free lower-upper symmetric-Gauss–Seidel (LU-SGS) method. To reduce the number of cells and computational cost, the hybrid mesh is applied. A modified non-manifold hybrid mesh data(NHMD) is used for both unstructured hybrid mesh and uniform grid. Numerical investigations are performed on different 2D laminar and turbulent flows. The convergence property and the computational efficiency of the present IGKS method are investigated. Much better performance is obtained compared with the standard explicit gas-kinetic scheme. Also, our numerical results are found to be in good agreement with experiment data and other numerical solutions, demonstrating the good applicability and high efficiency of the present IGKS for the simulations of laminar and turbulent flows.     

A numerical method for a kinetic equation and its application to propagating shock waves

An efficient numerical method for kinetic equations and its application to analyses of moving shock wave problems are presented. The present study aims to give an efficient scheme for two-dimensional unsteady gas flows. An explicit MacCormack difference method is applied to solve a BGK-model equation. The efficiency and accuracy of the scheme are examined in an application to one-dimensional shock structure problems. Furthermore, the scheme is applied to a two-dimensional flow problem: nonstationary reflection of a shock wave at a wedge. The present scheme is found to be useful and efficient for the analyses of two-dimensional unsteady rarefied gas flows.     

Long time stability and convergence rate of MacCormack rapid solver method for nonstationary Stokes–Darcy problem

We propose and study a combination of two second-order implicit–explicit (IMEX) methods for the coupled Stokes–Darcy system that governs flows in karst aquifers. The first is a second-order explicit two-step MacCormack scheme and the second is a second-order implicit Crank–Nicolson method. Both algorithms only require the solution of two decoupled problems at each time step, one Stokes and the other Darcy. This combination so called the MacCormack rapid solver method is very efficient (faster, at least of second order accuracy in time and space) and can be easily implemented using legacy codes. Under time step limitation of the form $\Delta t\le Ch$ (where $h,$$\Delta t$ are mesh size and time step, respectively, and $C$ is a physical parameter) we prove both long time stability and the rate of convergence of the method. Some numerical experiments are presented and discussed.     

Numerical algorithms in computational fluid dynamics on vector computers

Modern vector computers tend to favour certain classes of algorithms (e.g. explicit, Jacobi-type) while other important algorithms, such as implicit ones or Monte Carlo, in their serial versions are not very suitable for these machines. However, restructuring of serial algorithms often enables the user to exploit fully the potential of vector machines, which will often result in remarkable performance improvements. In the following contribution, the vectorization of five well-known algorithms, the explicit-implicit MacCormack scheme, the implicit scheme of Beam and Warming, a boundary-layer algorithm, a Galerkin procedure and a Monte Carlo simulation, for the solution of problems in computational fluid dynamics is discussed and computation times are given.     

Simulation of unsteady compressible flow in a channel with vibrating walls - Influence of the frequency

This study deals with the numerical solution of a 2D unsteady flow of a compressible viscous fluid in a channel for low inlet airflow velocity. The unsteadiness of the flow is caused by a prescribed periodic motion of a part of the channel wall with large amplitudes, nearly closing the channel during oscillations. The channel is a simplified model of the glottal space in the human vocal tract and the flow can represent a model of airflow coming from the trachea, through the glottal region with periodically vibrating vocal folds to the human vocal tract.The flow is described by the system of Navier–Stokes equations for laminar flows. The numerical solution is implemented using the finite volume method (FVM) and the predictor–corrector MacCormack scheme with Jameson artificial viscosity using a grid of quadrilateral cells. Due to the motion of the grid, the basic system of conservation laws is considered in the Arbitrary Lagrangian–Eulerian (ALE) form.The authors present the numerical simulations of flow fields in the channel, acquired from a program developed exclusively for this purpose. The numerical results for unsteady flows in the channel are presented for inlet Mach number M_∞ = 0.012, Reynolds number Re_∞ = 4.5 × 10³ and the wall motion frequency 20 and 100 Hz.     

On some approaches to solve CFD problems

This paper presents two efficient methods for spatial flows calculations. In order to simulate of incompressible viscous flows, a second-order accurate scheme with an incomplete LU decomposed implicit operator is developed. The scheme is based on the method of artificial compressibility and Roe flux-difference splitting technique for the convective terms. The numerical algorithm can be used to compute both steady-state and time-dependent flow problems. The second method is developed for modeling of stationary compressible inviscid flows. This numerical algorithm is based on a simple flux-difference splitting into physical processes method and combines a multi level grid technology with a convergence acceleration procedure for internal iterations. The capabilities of the methods are illustrated by computations of steady-state flow in a rotary pump, unsteady flow over a circular cylinder and stationary subsonic flow over an ellipsoid.     

Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method

The development and validation of a parallel unstructured tetrahedral non-nested multigrid (MG) method for simulation of unsteady 3D incompressible viscous flow is presented. The Navier-Stokes solver is based on the artificial compressibility method (ACM) and a higher-order characteristics-based finite-volume scheme on unstructured MG. Unsteady flow is calculated with an implicit dual time stepping scheme. The parallelization of the solver is achieved by a MG domain decomposition approach (MG-DD), using the Single Program Multiple Data (SPMD) programming paradigm. The Message-Passing Interface (MPI) Library is used for communication of data and loop arrays are decomposed using the OpenMP standard. The parallel codes using single grid and MG are used to simulate steady and unsteady incompressible viscous flows for a 3D lid-driven cavity flow for validation and performance evaluation purposes. The speedups and efficiencies obtained by both the parallel single grid and MG solvers are reasonably good for all test cases, using up to 32 processors on the SGI Origin 3400. The parallel results obtained agree well with those of serial solvers and with numerical solutions obtained by other researchers, as well as experimental measurements.     

An efficient solver for the RANS equations and a one-equation turbulence model

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. Using the algebraic turbulence model of Baldwin and Lomax, this scheme has been used to solve the compressible Reynolds-averaged Navier–Stokes (RANS) equations for transonic and low-speed flows. In this paper we focus on the convergence of the RK/Implicit scheme when the effects of turbulence are represented by the one-equation model of Spalart and Allmaras. With the present scheme the RANS equations and the partial differential equation of the turbulence model are solved in a loosely coupled manner. This approach allows the convergence behavior of each system to be examined. Point symmetric Gauss-Seidel supplemented with local line relaxation is used to approximate the inverse of the implicit operator of the RANS solver. To solve the turbulence equation we consider three alternative methods: diagonally dominant alternating direction implicit (DDADI), symmetric line Gauss-Seidel (SLGS), and a two-stage RK scheme with implicit preconditioning. Computational results are presented for airfoil flows, and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and a transport-type equation for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses RK/Implicit schemes.     

An implicit compact scheme solver for two-dimensional multicomponent flows

A 2D implicit compact scheme solver has been implemented for the vorticity-velocity formulation in the case of nonreacting, multicomponent, axisymmetric, low Mach number flows. To stabilize the discrete boundary value problem, two sets of boundary closures are introduced to couple the velocity and vorticity fields. A Newton solver is used for solving steady-state and time-dependent equations. In this solver, the Jacobian matrix is formulated and stored in component form. To solve the system of linearized equations within each iteration of Newton’s method, preconditioned Bi-CGSTAB is used in combination with a matrix-vector product computed in component form. The almost dense Jacobian matrix is approximated by a partial Jacobian. For the preconditioner equation, the partial Jacobian is approximately factored using several methods. In a detailed study of several preconditioning techniques, a promising method based on ILUT preconditioning in combination with reordering and double scaling using the MC64 algorithm by Duff and Koster is selected. To validate the implicit compact scheme solver, several nonreacting model problems have been considered. At least third order accuracy in space is recovered on nonuniform grids. A comparison of the results of the implicit compact scheme solver with the results of a traditional implicit low order solver shows an order of magnitude reduction of computer memory and time using the compact scheme solver in the case of time-dependent mixing problems.     

Application of the Boltzmann kinetic equation to the eddy problems

The main purpose of this work is to investigate the feasibility of applying a kinetic approach to the problem of modeling turbulent and unstable flows. First, initial value problems with the Taylor–Green (TG) type and isotropic velocity conditions for compressible flow in two-dimensional (2D) and three-dimensional (3D) periodic domains are considered. Further, 3D direct numerical simulation of decaying isotropic turbulence is performed. Macroscopic flow quantities of interest are examined. The simulation is based on the direct numerical solution of the Boltzmann kinetic equation using an explicit–implicit scheme for the relaxation stage. Comparison with the solution of the Bhatnagar–Gross–Krook (BGK) model equation obtained by using an implicit scheme is carried out for the decaying isotropic turbulence problem and demonstrates a small difference. For the TG initial condition results show a fragmentation of the large initial eddies and subsequently the full damping of the system. Numerical data are close to the analytic solution of TG problem. A dependence of the kinetic energy on the wave number is obtained by means of the Fourier expansion of velocity components. A power-law exponent for the kinetic energy spectrum tends to the theoretical value “−3” for 2D turbulence in 2D case and to the famous Kolmogorov value “−5/3” in 3D case.     

A conservative model for unsteady flows in deformable closed pipes and its implicit second-order finite volume discretisation

We present a one-dimensional model for compressible flows in a deformable pipe which is an alternative to the Allievi equations and is intended to be coupled in a “natural way” with the shallow water equations to simulate mixed flows. The numerical simulation is performed using a second-order linearly implicit scheme adapted from the Roe scheme. The validation is performed in the case of water hammer in a rigid pipe: we compare the numerical results provided by an industrial code with those of our spatial second-order implicit scheme. It appears that the maximum value of the pressure within the pipe for large CFL numbers and a coarse discretisation is accurately computed.     

Control of numerical effects of dispersion and dissipation in numerical schemes for efficient shock-capturing through an optimal Courant number

Composite schemes consist of several steps of a dispersive scheme followed by one step of a dissipative scheme [Liska Richard, Wendroff Burton. Composite schemes for conservation laws. SIAM J Numer Anal 1998;35(6):2250-71]. The latter [Liska Richard, Wendroff Burton. 2D shallow water equations by composite schemes. Int J Numer Meth Fluids 1999;30:461-79] acts as a filter reducing oscillations in regions of discontinuity. Liska and Wendroff have derived the composite Lax-Wendroff/Lax-Friedrichs (LWLF) [Liska Richard, Wendroff Burton, 1998] scheme which blends the Lax-Wendroff (LW) scheme with the 2-step Lax-Friedrichs (LF) scheme. The formulation of the 2-step Lax-Friedrichs scheme [Liska Richard, Wendroff Burton, 1998] is different from that of the classic Lax-Friedrichs scheme and has been devised by Liska [Liska Richard, Wendroff Burton, 1998]. In this work, we propose to replace LW scheme by MacCormack (MC) scheme since the latter is less dispersive. We obtain a new composite scheme in 1-D and in 2-D by blending the MacCormack scheme with the 2-step Lax-Friedrichs scheme which we term as the composite MacCormack/Lax-Friedrichs (MCLF) scheme. This is followed by analytical work on the effective amplification factor (EAF) and the relative phase error (RPE) for both families of schemes in 1-D and 2-D: LWLFn and MCLFn, consisting of (n − 1) steps of the dispersive scheme (LW or MC) and 1 step of the dissipative LF scheme. We introduce a new concept, baptised as Curbing of Dispersion by Dissipation for Efficient Shock-capturing, CDDES in which a cfl number is computed whereby dissipation curbs dispersion. This cfl number is termed as optimal in this work. We conduct a comparative study based on numerical experiments in 2-D namely: contact-discontinuity problem [Ould Kaber SM. A legendre pseudospectral viscosity method. J Comput Phys 1996;128:165-80], rotating hill problem [Ould Kaber SM, 1996] and the deformative flow of Smolarkiewicz [Dabdub Donald, Seinfeld John H. Numerical advective schemes used in air quality models-sequential and parallel implementation. Atmos Environ 1994;28(20):3369-85, Ghods A, Sobouti F, Arkani-Hamed J. An improved second order method for solution of pure advection problems. Int J Numer Meth Fluids 2000;32:959-77, Nguyen K, Dabdub D. Two-level time-marching scheme using splines for solving the advection equation. Atmos Environ 2001;35:1627-37] to show that the MacCormack/Lax-Friedrichs (MCLF) scheme is more efficient than LWLF scheme to capture shocks in regions of discontinuity. We also show that better results are obtained at optimal cfl numbers for some variants of LWLFn and MCLFn schemes, with n = 2, 3, 4 and 5.     

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.     

On optimal node spacing for immersed boundary–lattice Boltzmann method in 2D and 3D

A computational study on optimal spacing of Lagrangian nodes discretizing a rigid and immobile immersed body boundary in 2D and 3D is presented in order to show how the density of the Lagrangian points affects the numerical results of the Immersed Boundary–Lattice Boltzmann Method (IB–LBM). The study is based on the implicit velocity correction-based IB–LBM proposed by Wu and Shu (2009, 2010) that allows computing the fluid–body interaction force. However, the (original) method fails for densely spaced Lagrangian points due to ill-conditioned or even singular linear systems that arise from the derivation of the method. We propose a modification that improves the solvability of the linear systems and compare the performance of both methods using several benchmark problems. The results show how the spacing of the Lagrangian points affects the numerical results, mainly the permeability of the discretized body boundary in applications to fluid flows over rigid obstacles and blood flows in arteries in 2D and 3D.     

Numerical analysis of the micro-Couette flow using a non-Newton–Fourier model with enhanced wall boundary conditions

Non-equilibrium effects exist extensively in microfluidic flows, and the accurate simulation of the Knudsen layer behind them is rather challenging for the linear Newton–Fourier model. In this paper, a high-order reduced model (nonlinear coupled constitutive relations) from Eu’s generalized hydrodynamic equations is applied for the investigation of the micro-Couette flows of diatomic nitrogen and monatomic argon as well as Maxwell and hard-sphere molecules using the MacCormack scheme. In order to simulate the confined flows accurately, a set of enhanced wall boundary conditions based on this model are derived with respect to the degree of non-equilibrium. Both the 1st-order Maxwell–Smoluchowski model and the Langmuir slip model are also investigated. For a large range of Knudsen numbers, the results show that the enhanced boundary conditions make a significant improvement in the prediction of flow profiles, especially the temperature profile. The reason behind that is analyzed in detail. The numerical predictions obtained from the high-order model in conjunction with the enhanced boundary conditions are also compared with DSMC, regularized 13 moment equations, Burnett-type equations as well as Navier–Stokes solutions, which highlight its excellent capability in describing the underlying mechanism of the Knudsen layer in the Couette flow.     

Application of Second Order TVD and WENO Schemes in Internal Aerodynamics

We deal with the comparison of several finite volume TVD schemes and finite difference ENO schemes and we describe a second order finite volume WENO scheme which was developed for the case of general unstructured meshes. The proposed second order WENO reconstruction is much simpler than the original ENO scheme introduced in [Harten and Chakravarthy 1991]. Moreover, the proposed WENO method is very easily extendible for unstructured meshes in 3D. All above mentioned schemes are applied for the solution of 2D and 3D transonic flows in the turbines and channels and the numerical solution is compared to experimental results or to the results obtained by other authors.     

三维激光烧蚀流体界面不稳定性程序的并行化

在共享存储并行机和MPP并行机上,基于MPI(MessagePassingInterface)并行编程环境,本文研究三维激光烧蚀界而不稳定性程序(Lared-S)的并行实现.三维激光烧蚀的数值模拟采用分裂方法,其90％以上的计算负载存在于流体方程和热传导方程的求解(流体方程的求解采用分裂显格式,热传导方程的求解采用分裂隐格式).本文给出基于三维分裂格式的交替平面数据通信模式.分裂隐格式的求解转化为三对角方程组的求解,其并行实现采用块流水线并行算法.数值实验结果表明交替平面数据通信策略和块流水线并行算法是有效且可扩展的.在共享存储并行机上,应用64台处理机获得93％以上的并行效率;在MPP并行机上,应用128台处理机获得90％以上的并行效率.     

A hybrid numerical method and its application to inviscid compressible flow problems

An improved version of the artificially upstream flux vector scheme, is developed to efficiently compute inviscid compressible flow problems. This numerical scheme, named AUFSR (Tchuen et al. 2011), is obtained by hybridizing the AUFS scheme with Roe’s solver. This approach handles difficulties encountered by the AUFS scheme, in the case where the flux vector does not check the homogeneous property. The present scheme for multi-dimensional flows introduces a certain amount of numerical dissipation to shear waves, as Roe’s splitting. The AUFSR scheme is not only robust for shock-capturing, but also accurate for resolving shear layers. Numerical results for 1D Riemann problems and several 2D problems are investigated to show the capability of the method to accurately compute inviscid compressible flow when compared to AUFS, and Roe solvers.     

A two-level finite element method for the Allen–Cahn equation

We consider the fully implicit treatment for the nonlinear term of the Allen–Cahn equation. To solve the nonlinear problem efficiently, the two-level scheme is employed. We obtain the discrete energy law of the fully implicit scheme and two-level scheme with finite element method. Also, the convergence of the two-level method is presented. Finally, some numerical experiments are provided to confirm the theoretical analysis.     

A discontinuous Galerkin algorithm for the two-dimensional shallow water equations

A new high-resolution finite element scheme is introduced for solving the two-dimensional (2D) depth-integrated shallow water equations (SWE) via local plane approximations to the unknowns. Bed topography data are locally approximated in the same way as the flow variables to render an instinctive well-balanced scheme. A finite volume (FV) wetting and drying technique that reconstructs the Riemann states by ensuring non-negative water depth and maintaining well-balanced solution is adjusted and implemented in the current finite element framework. Meanwhile, a local slope-limiting process is applied and those troubled-slope-components are restricted by the minmod FV slope limiter. The inter-cell fluxes are upwinded using the HLLC approximate Riemann solver. Friction forces are separately evaluated via stable implicit discretization to the finite element approximating coefficients. Boundary conditions are derived and reported in details. The present model is validated against several test cases including dam-break flows on regular and irregular domains with flooding and drying.