Parallel-multigrid computation of unsteady incompressible viscous flows using a matrix-free implicit method and high-resolution characteristics-based scheme

A three-dimensional parallel unstructured non-nested multigrid solver for solutions of unsteady incompressible viscous flow is developed and validated. The finite-volume Navier–Stokes solver is based on the artificial compressibility approach with a high-resolution method of characteristics-based scheme for handling convection terms. The unsteady flow is calculated with a matrix-free implicit dual time stepping scheme. The parallelization of the multigrid solver is achieved by multigrid domain decomposition approach (MG-DD), using single program multiple data (SPMD) and multiple instruction multiple data (MIMD) programming paradigm. There are two parallelization strategies proposed in this work, first strategy is a one-level parallelization strategy using geometric domain decomposition technique alone, second strategy is a two-level parallelization strategy that consists of a hybrid of both geometric domain decomposition and data decomposition techniques. Message-passing interface (MPI) and OpenMP standard are used to communicate data between processors and decompose loop iterations arrays, respectively. The parallel-multigrid code is used to simulate both steady and unsteady incompressible viscous flows over a circular cylinder and a lid-driven cavity flow. A maximum speedup of 22.5 could be achieved on 32 processors, for instance, the lid-driven cavity flow of Re = 1000. The results obtained agree well with numerical solutions obtained by other researchers as well as experimental measurements. A detailed study of the time step size and number of pseudo-sub-iterations per time step required for simulating unsteady flow are presented in this paper.     

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.     

Aspects of unsteady incompressible flow simulations

Since unsteady incompressible flow simulations involving complex geometry require long computing times, it is crucial to select computationally efficient solvers. For this purpose, two numerical procedures, one based on artificial compressibility and the other pressure projection method, are investigated for obtaining time-accurate solutions of the incompressible Navier–Stokes equations. The performance of the two methods is compared by obtaining unsteady solutions for the evolution of twin vortices behind a flat plate. Calculated results are compared with experimental and other numerical results. For an unsteady flow, which requires small physical time step, the pressure projection method was found to be computationally efficient since it does not require a subiteration procedure. It was also observed that the artificial compressibility method requires a fast convergence scheme at each physical time step in order to satisfy the incompressibility condition. This was obtained by employing a GMRES-ILU(0) solver in the artificial compressibility solver.     

Implementation of density-based solver for all speeds in the framework of OpenFOAM

In the framework of open source CFD code OpenFOAM, a density-based solver for all speeds flow field is developed. In this solver the preconditioned all speeds AUSM+(P) scheme is adopted and the dual time scheme is implemented to complete the unsteady process. Parallel computation could be implemented to accelerate the solving process. Different interface reconstruction algorithms are implemented, and their accuracy with respect to convection is compared. Three benchmark tests of lid-driven cavity flow, flow crossing over a bump, and flow over a forward-facing step are presented to show the accuracy of the AUSM+(P) solver for low-speed incompressible flow, transonic flow, and supersonic/hypersonic flow. Firstly, for the lid driven cavity flow, the computational results obtained by different interface reconstruction algorithms are compared. It is indicated that the one dimensional reconstruction scheme adopted in this solver possesses high accuracy and the solver developed in this paper can effectively catch the features of low incompressible flow. Then via the test cases regarding the flow crossing over bump and over forward step, the ability to capture characteristics of the transonic and supersonic/hypersonic flows are confirmed. The forward-facing step proves to be the most challenging for the preconditioned solvers with and without the dual time scheme. Nonetheless, the solvers described in this paper reproduce the main features of this flow, including the evolution of the initial transient.     

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.     

An explicit Chebyshev pseudospectral multigrid method for incompressible Navier-Stokes equations

The two-dimensional steady incompressible Navier-Stokes equations in the form of primitive variables have been solved by Chebyshev pseudospectral method. The pressure and velocities are coupled by artificial compressibility method and the NS equations are solved by pseudotime method with an explicit four-step Runge-Kutta integrator. In order to reduce the computational time cost, we propose the spectral multigrid algorithm in full approximation storage (FAS) scheme and implement it through V-cycle multigrid and full multigrid (FMG) strategies. Four iterative methods are designed including the single grid method; the full single grid method; the V-cycle multigrid method and the FMG method. The accuracy and efficiency of the numerical methods are validated by three test problems: the modified one-dimensional Burgers equation; the Taylor vortices and the two-dimensional lid driven cavity flow. The computational results fit well with the exact or benchmark solutions. The spectral accuracy can be maintained by the single grid method as well as the multigrid ones, while the time cost is greatly reduced by the latter. For the lid driven cavity flow problem, the FMG is proved to be the most efficient one among the four iterative methods. A speedup of nearly two orders of magnitude can be achieved by the three-level multigrid method and at least one order of magnitude by the two-level multigrid method.     

Numerical simulation of the regularized driven cavity flows at high Reynolds numbers

We present a numerical simulation of the unsteady incompressible flows in the unit cavity as well as in the rectangular cavity of aspect ratio 2 by using a second order projection scheme in time and a Chebyshev-Tau approximation for the space variables. The time-accurate discretization and the high accuracy of the spectral methods enable us to observe dynamical features not apparent in previous studies.     

基于预处理法非定常低速流动的数值模拟

本文基于预处理法,结合双时间步法,建立了应用高效隐式时间步进LU-SGS算法求解非定常低速流动问题的数值模拟方法．对典型的方腔顶盖瞬时启动驱动、周期振荡顶盖驱动等非定常低速流动问题进行了数值计算．结果表明,所建立的数值方法对非定常低速流动问题有较高的计算效率,并能有效的克服低速流动问题的系统刚性问题．     

Fractional step artificial compressibility schemes for the unsteady incompressible Navier-Stokes equations

We propose a novel, time-accurate approach for solving the unsteady, three-dimensional, incompressible Navier-Stokes equations on non-staggered grids. The approach modifies the standard, dual-time stepping artificial-compressibility (AC) iteration scheme by incorporating ideas from pressure-based, fractional-step (FS) formulations. The resulting hybrid fractional-step/artificial-compressibility (FSAC) method is second-order accurate and advances the Navier-Stokes equations in time via a two-step procedure. In the first step, which is identical to the convection-diffusion step in pressure-based FS methods, a preliminary velocity field is calculated, which is not divergence-free. In the second step, however, instead of deriving a pressure-Poisson equation as in FS methods, the projection of the velocity field into the solenoidal vector space is implemented using a dual-time stepping AC formulation. Unlike the standard dual-time stepping AC formulations, where the dual-time iterations are carried out with the entire non-linear system, in the FSAC scheme the convective and viscous terms are computed only once or twice per physical time step. Numerical experiments show that the proposed method provides second-order accurate solutions and requires considerably less CPU time than the widely used standard AC formulation. To demonstrate its ability to compute complicate problems, the method is also applied to a flow past cylinder with endplates.     

Numerical solution of incompressible Navier–Stokes equations using a fractional-step approach

A fractional step method for the solution of steady and unsteady incompressible Navier–Stokes equations is outlined. The method is based on a finite-volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (third and fifth) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when fifth-order upwind differencing and a modified production term in the Baldwin–Barth one-equation turbulence model are used with adequate grid resolution.     

三维非定常不可压涡量—速度Navier—Stokes方程组的有限差分法

提出了一种数值求解三维非定常涡量—速度形式的不可压Navier-Stokes方程组的有限差分方法,该方法在空间方向上具有二阶精度,并且系数矩阵具有对角占优性,因此适合高雷诺数问题的数值求解．同时,给出了适合的二阶涡量边界条件．通过对有精确解的狄利克雷边值问题和典型的驱动方腔流问题的数值实验,验证了本文格式的精确性、稳定性和有效性．     

Implicit multigrid computation of unsteady flows past cylinders of square cross-section

An implicit, multigrid scheme has been extended to treat unsteady flows using the concept of temporal subiteration (or dual-time stepping, as it sometimes is called). The efficiency and accuracy of the subiterated, multigrid approach has been discussed in a previous paper. Here, the scheme is applied to compute the unsteady flow past fixed cylinders of square cross-section at moderate Reynolds numbers. The observed pattern of periodic vortex shedding is computed and the dimensionless frequency of this phenomenon (the Strouhal number) is compared with experimentally determined values. Results of coupled aeroelastic computations also are presented that illustrate a hysteresis phenomenon as the structural frequency is varied.     

Numerical simulation of 2D square driven cavity using fourth-order compact finite difference schemes

Fourth-order compact finite difference schemes are employed with multigrid techniques to simulate the two-dimensional square driven cavity flow with small to large Reynolds numbers. The governing Navier-Stokes equation is linearized in streamfunction and vorticity formulation. The fourth-order compact approximation schemes are coupled with fourth-order approximations for velocities and vorticity boundaries. Numerical solutions are obtained for square driven cavity flow at high Reynolds numbers and are compared with solutions obtained by other researchers using other approximation methods.     

Implementation of a transient adaptive sub-cell module for the parallel-DSMC code using unstructured grids

A method of transient adaptive sub-cells (TAS) suitable for unstructured grids that is modified from the existing one for the structured grids of DSMC is introduced. The TAS algorithm is implemented within the framework of a parallelized DSMC code (PDSC). Benchmarking tests are conducted for steady driven cavity flow, steady hypersonic flow over a two-dimensional cylinder, steady hypersonic flow over a cylinder/flare and the unsteady vortex shedding behind a two-dimensional cylinder. The use of TAS enables a reduction in the computational expense of the simulation since larger sampling cells and less simulation particles can be employed. Furthermore, the collision quality of the simulation is maintained or improved and the preservation of property gradients and vorticity at the scale of the sub-cells enables correct unsteady vortex shedding frequencies to be predicted. The use of TAS in a parallel-DSMC code allows simulations of unsteady processes at a level to be carried out efficiently, accurately and with acceptable computational time.     

Textbook multigrid efficiency for the incompressible Navier–Stokes equations: high Reynolds number wakes and boundary layers

Textbook multigrid efficiencies for high Reynolds number simulations based on the incompressible Navier–Stokes equations are attained for a model problem of flow past a finite flat plate. Elements of the full approximation scheme multigrid algorithm, including distributed relaxation, defect correction, and boundary treatment, are presented for the three main physical aspects encountered: entering flow, wake flow, and boundary layer flow. Textbook efficiencies, i.e., reduction of algebraic errors below discretization errors in one full multigrid cycle, are attained for second order accurate simulations at a laminar Reynolds number of 10,000.     

Comparison of finite-volume numerical methods with staggered and colocated grids

The paper presents a detailed comparison of two finite-volume solution methods for two-dimensional incompressible fluid flows, one with staggered and the other with colocated numerical grids. The staggered method is well-known and well-established, and it is used here as a standard against which the relatively new colocated approach is compared. Three test cases were considered, employing orthogonal rectilinear grids: lid driven cavity flow, backward facing step flow and flow through a pipe with sudden contraction. The results of the computations demonstrate that the convergence rate, dependency on under-relaxation parameters, computational effort and accuracy are almost identical for both solution methods. The colocated method converges faster in some cases, and has advantages when extensions such as multigrid techniques and non-orthogonal grids are considered.     

Numerical simulation of 3D fluid–structure interaction flow using an immersed object method with overlapping grids

The newly developed immersed object method (IOM) [Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady incompressible viscous flows around moving rigid bodies using an immersed object method with overlapping grids. J Comput Phys 2005; 207(1): 151–72] is extended for 3D unsteady flow simulation with fluid–structure interaction (FSI), which is made possible by combining it with a parallel unstructured multigrid Navier–Stokes solver using a matrix-free implicit dual time stepping and finite volume method [Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method. In: The second M.I.T. conference on computational fluid and solid mechanics, June 17–20, MIT, Cambridge, MA 02139, USA, 2003; Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method, Special issue on “Preconditioning methods: algorithms, applications and software environments. Comput Struct 2004; 82(28): 2425–36]. This uniquely combined method is then employed to perform detailed study of 3D unsteady flows with complex FSI. In the IOM, a body force term F is introduced into the momentum equations during the artificial compressibility (AC) sub-iterations so that a desired velocity distribution V₀ can be obtained on and within the object boundary, which needs not coincide with the grid, by adopting the direct forcing method. An object mesh is immersed into the flow domain to define the boundary of the object. The advantage of this is that bodies of almost arbitrary shapes can be added without grid restructuring, a procedure which is often time-consuming and computationally expensive. It has enabled us to perform complex and detailed 3D unsteady blood flow and blood–leaflets interaction in a mechanical heart valve (MHV) under physiological conditions.     

An adaptive multilevel multigrid formulation for Cartesian hierarchical grid methods

A Cartesian grid method with adaptive mesh refinement and multigrid acceleration is presented for the compressible Navier-Stokes equations. Cut cells are used to represent boundaries on the Cartesian grid, while ghost cells are introduced to facilitate the implementation of boundary conditions. A cell-tree data structure is used to organize the grid cells in a hierarchical manner. Cells of all refinement levels are present in this data structure such that grid level changes as they are required in a multigrid context do not have to be carried out explicitly. Adaptive mesh refinement is introduced using phenomenon-based sensors. The application of the multilevel method in conjunction with the Cartesian cut-cell method to problems with curved boundaries is described in detail. A 5-step Runge-Kutta multigrid scheme with local time stepping is used for steady problems and also for the inner integration within a dual time-stepping method for unsteady problems. The inefficiency of customary multigrid methods on Cartesian grids with embedded boundaries requires a new multilevel concept for this application, which is introduced in this paper. This new concept is based on the following novelties: a formulation of a multigrid method for Cartesian hierarchical grid methods, the concept of averaged control volumes, and a mesh adaptation strategy allowing to directly control the number of refined and coarsened cells.     

Application of single-level,pointwise algebraic,and smoothed aggregation multigrid methods to direct numerical simulations of incompressible turbulent flows

Single- and multi-level iterative methods for sparse linear systems are applied to unsteady flow simulations via implementation into a direct numerical simulation solver for incompressible turbulent flows on unstructured meshes. The performance of these solution methods, implemented in the well-established SAMG and ML packages, are quantified in terms of computational speed and memory consumption, with a direct sparse LU solver (SuperLU) used as a reference. The classical test case of unsteady flow over a circular cylinder at low Reynolds numbers is considered, employing a series of increasingly fine anisotropic meshes. As expected, the memory consumption increases dramatically with the considered problem size for the direct solver. Surprisingly, however, the computation times remain reasonable. The speed and memory usage of pointwise algebraic and smoothed aggregation multigrid solvers are found to exhibit near-linear scaling. As an alternative to multi-level solvers, a single-level ILUT-preconditioned GMRES solver with low drop tolerance is also considered. This solver is found to perform sufficiently well only on small meshes. Even then, it is outperformed by pointwise algebraic multigrid on all counts. Finally, the effectiveness of pointwise algebraic multigrid is illustrated by considering a large three-dimensional direct numerical simulation case using a novel parallelization approach on a large distributed memory computing cluster.     

Incompressible flow computations over moving boundary using a novel upwind method

In this paper a novel method for simulating unsteady incompressible viscous flow over a moving boundary is described. The numerical model is based on a 2D Navier–Stokes incompressible flow in artificial compressibility formulation with Arbitrary Lagrangian Eulerian approach for moving grid and dual time stepping approach for time accurate discretization. A higher order unstructured finite volume scheme, based on a Harten Lax and van Leer with Contact (HLLC) type Riemann solver for convective fluxes, developed for steady incompressible flow in artificial compressibility formulation by Mandal and Iyer (AIAA paper 2009-3541), is extended to solve unsteady flows over moving boundary. Viscous fluxes are discretized in a central differencing manner based on Coirier’s diamond path. An algorithm based on interpolation with radial basis functions is used for grid movements. The present numerical scheme is validated for an unsteady channel flow with a moving indentation. The present numerical results are found to agree well with experimental results reported in literature.