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.     

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.     

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.     

A p-multigrid spectral difference method for two-dimensional unsteady incompressible Navier-Stokes equations

This paper presents the development of a 2D high-order solver with spectral difference method for unsteady incompressible Navier-Stokes equations accelerated by a p-multigrid method. This solver is designed for unstructured quadrilateral elements. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. An artificial compressibility method (ACM) is employed in order to treat the inviscid fluxes using the traditional characteristics-based schemes. The viscous fluxes are computed using the averaging approach (Sun et al., 2007; Kopriva, 1998) [29] and [12]. A dual time stepping scheme is implemented to deal with physical time marching. A p-multigrid method is implemented (Liang et al., 2009) [16] in conjunction with the dual time stepping method for convergence acceleration. The incompressible SD (ISD) method added with the ACM (SD-ACM) is able to accurately simulate 2D steady and unsteady viscous flows.     

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.     

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.     

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.     

Parallel preconditioned WENO scheme for three-dimensional flow simulation of NREL Phase VI Rotor

The preconditioned weighted essentially non-oscillatory (P-WENO) solver for viscous flows (Huang et al. (2009) [9]) is extended to non-inertial reference frames. In the present scheme, patched multi-block grid system is employed and parallel computing is adopted as well. With the present parallel P-WENO solver, three-dimensional flows of the Phase VI Rotor from National Renewable Energy Laboratory (NREL) can be simulated and analyzed for different wind speeds. Our simulation results show good agreement with the numerical predictions based on incompressible Navier–Stokes (N–S) equations as well as the available wind tunnel data from NREL. The flow phenomena, including separation and attachment line, can be captured by the present scheme. The parallel strategy adopted is a block-domain decomposition method for the patched multi-block grid system. To balance the load among different computing nodes, a Tabu search algorithm is adopted for the parallelization. The parallel efficiency of the parallel P-WENO scheme is examined for node numbers ranging from 1 to 64. It is found that the parallel efficiency is monotonically decreased as the node number adopted is increased; the parallel efficiency is retained over 90% for all cases of different node numbers. Due to the high parallel efficiency, our parallel P-WENO solver is validated for applying to practical fluid problems from compressible to incompressible limits.     

CFD-based analysis and two-level aerodynamic optimization on graphics processing units

This paper presents the porting of 2D and 3D Navier–Stokes equations solvers for unstructured grids, from the CPU to the graphics processing unit (GPU; NVIDIA’s Ge-Force GTX 280 and 285), using the CUDA language. The performance of the GPU implementations, with single, double or mixed precision arithmetic operations, is compared to that of the CPU code.Issues regarding the optimal handling of the unstructured grid topology on the GPU, particularly for vertex-centered CFD algorithms, are discussed. Restructuring the existing codes was necessary in order to maximize the parallel efficiency of the GPU implementations. The mixed precision implementation, in which the left-hand-side operators are computed with single precision, was shown to bridge the gap between the single and double precision speed-ups. Based on the different speed-ups and prediction accuracy of the aforementioned GPU implementations of the Navier–Stokes equations solver, a hierarchical optimization method which is suitable for GPUs is proposed and demonstrated in inviscid and turbulent 2D flow problems. The search for the optimal solution(s) splits into two levels, both relying upon evolutionary algorithms (EAs) though with different evaluation tools each. The low level EA uses the very fast single precision GPU implementation with relaxed convergence criteria for the inexpensive evaluation of candidate solutions. Promising solutions are regularly broadcast to the high level EA which uses the mixed precision GPU implementation of the same flow solver. Single- and two-objective aerodynamic shape optimization problems are solved using the developed software.     

A block LU-SGS implicit unsteady incompressible flow solver on hybrid dynamic grids for 2D external bio-fluid simulations

A hybrid dynamic grid generation technique for two-dimensional (2D) morphing bodies and a block lower-upper symmetric Gauss-Seidel (BLU-SGS) implicit dual-time-stepping method for unsteady incompressible flows are presented for external bio-fluid simulations. To discretize the complicated computational domain around 2D morphing configurations such as fishes and insect/bird wings, the initial grids are generated by a hybrid grid strategy firstly. Body-fitted quadrilateral (quad) grids are generated first near solid bodies. An adaptive Cartesian mesh is then generated to cover the entire computational domain. Cartesian cells which overlap the quad grids are removed from the computational domain, and a gap is produced between the quad grids and the adaptive Cartesian grid. Finally triangular grids are used to fill this gap. During the unsteady movement of morphing bodies, the dynamic grids are generated by a coupling strategy of the interpolation method based on ‘Delaunay graph’ and local remeshing technique. With the motion of moving/morphing bodies, the grids are deformed according to the motion of morphing body boundaries firstly with the interpolation strategy based on ‘Delaunay graph’ proposed by Liu and Qin. Then the quality of deformed grids is checked. If the grids become too skewed, or even intersect each other, the grids are regenerated locally. After the local remeshing, the flow solution is interpolated from the old to the new grid. Based on the hybrid dynamic grid technique, an efficient implicit finite volume solver is set up also to solve the unsteady incompressible flows for external bio-fluid dynamics. The fully implicit equation is solved using a dual-time-stepping approach, coupling with the artificial compressibility method (ACM) for incompressible flows. In order to accelerate the convergence history in each sub-iteration, a block lower-upper symmetric Gauss-Seidel implicit method is introduced also into the solver. The hybrid dynamic grid generator is tested by a group of cases of morphing bodies, while the implicit unsteady solver is validated by typical unsteady incompressible flow case, and the results demonstrate the accuracy and efficiency of present solver. Finally, some applications for fish swimming and insect wing flapping are carried out to demonstrate the ability for 2D external bio-fluid simulations.     

Applications of stencil-adaptive finite difference method to incompressible viscous flows with curved boundary

This paper investigates the applicability of the stencil-adaptive finite difference method for the simulation of two-dimensional unsteady incompressible viscous flows with curved boundary. The adaptive stencil refinement algorithm has been proven to be able to continuously adapt the stencil resolution according to the gradient of flow parameter of interest [Ding H, Shu C. A stencil adaptive algorithm for finite difference solution of incompressible viscous flows. J Comput Phys 2006;214:397-420], which facilitates the saving of the computational efforts. On the other hand, the capability of the domain-free discretization technique in dealing with the curved boundary provides a great flexibility for the finite difference scheme on the Cartesian grid. Here, we show that their combination makes it possible to simulate the unsteady incompressible flow with curved boundary on a dynamically changed grid. The methods are validated by simulating steady and unsteady incompressible viscous flows over a stationary circular cylinder.     

Transonic drag prediction on a DLR-F6 transport configuration using unstructured grid solvers

A second international AIAA Drag Prediction Workshop (DPW-II) was organized and held in Orlando Florida on June 21-22, 2003. The primary purpose was to investigate the code-to-code uncertainty, address the sensitivity of the drag prediction to grid size and quantify the uncertainty in predicting nacelle/pylon drag increments at a transonic cruise condition. This paper presents an in-depth analysis of the DPW-II computational results from three state-of-the-art unstructured grid Navier-Stokes flow solvers exercised on similar families of tetrahedral grids. The flow solvers are USM3D - a tetrahedral cell-centered upwind solver, FUN3D - a tetrahedral node-centered upwind solver, and NSU3D - a general element node-centered central-differenced solver. Overall, grid refinement did not consistently improve the correlation with experimental data for either the wing/body or the wing/body/nacelle pylon configuration. Although, the range in total drag for the wing/body fine grids was only 5 counts, a code-to-code comparison of surface pressures and surface restricted streamlines indicated that the three solvers were not all converging to the same flow solutions- different shock locations and separation patterns were evident. Similarly, the wing/body/nacelle/pylon solutions did not appear to be converging to the same flow solutions. Although the absolute values of total drag predicted by two of the solvers for the medium and fine grids did not compare well with the experiment, the incremental drag predictions were within ±3 counts of the experimental data. Although, the sources of code-to-code variation in force and moment predictions for the three unstructured grid codes have not yet been identified, the current study reinforces the necessity of applying multiple codes to the same application to assess uncertainty.     

Comparison of different solution algorithms for collocated method of MCIM to calculate steady and unsteady incompressible flows on unstructured grids

In this paper, the collocated method of MCIM (Mass Corrected Interpolation Method) is employed for solving two-dimensional incompressible flows on unstructured grid systems. A control-volume-based finite element (CVFEM) approach has been taken in which conservation equations are formed for each control volume throughout the solution domain. In this study, three different algorithms are proposed and investigated. In the first algorithm a fully coupled formulation is used, however in the second algorithm a segregated approach without pressure correction is employed. In the third solution algorithm, the scheme of MCIM is implemented to SIMPLE-like segregated algorithm. The linear system of equations obtained in these algorithms is solved using a direct sparse solver. The accuracy and performance of the proposed algorithms are demonstrated by solving different steady and unsteady benchmark problems.     

A Parallel Overset Grid High-Order Flow Solver for Large Eddy Simulation

This work describes the development and validation of a parallel high-order compact finite difference Navier–Stokes solver for application to large-eddy simulation (LES) and direct numerical simulation. The implicit solver can employ up to sixth-order spatial formulations and tenth-order filtering. The parallelization of the solver is founded on the overset grid technique. LES were then performed for turbulent channel flow with Reynolds numbers ranging from Re _τ=180 to 590, and flow past a circular cylinder with a transitional wake at Re _D =3900. The channel flow solutions were obtained using both an implicit LES (ILES) approach and a dynamic sub-grid scale model. The ILES method obtained virtually identical solutions at half the computational cost. The original vector and new parallel solvers produce indistinguishable mean flow solutions for the circular cylinder. Repeating the cylinder simulation on a much finer mesh resulted in significantly better agreement with experimental data in the near wake than the coarse grid solution and other previous numerical studies.     

A high order Discontinuous Galerkin Finite Element solver for the incompressible Navier–Stokes equations

The paper presents an unsteady high order Discontinuous Galerkin (DG) solver that has been developed, verified and validated for the solution of the two-dimensional incompressible Navier–Stokes equations. A second order stiffly stable method is used to discretise the equations in time. Spatial discretisation is accomplished using a modal DG approach, in which the inter-element fluxes are approximated using the Symmetric Interior Penalty Galerkin formulation. The non-linear terms in the Navier–Stokes equations are expressed in the convective form and approximated through the Lesaint–Raviart fluxes modified for DG methods.Verification of the solver is performed for a series of test problems; purely elliptic, unsteady Stokes and full Navier–Stokes. The resulting method leads to a stable scheme for the unsteady Stokes and Navier–Stokes equations when equal order approximation is used for velocity and pressure. For the validation of the full Navier–Stokes solver, we consider unsteady laminar flow past a square cylinder at a Reynolds number of 100 (unsteady wake). The DG solver shows favourably comparisons to experimental data and a continuous Spectral code.     

Incompressible flow calculations with a consistent physical interpolation finite volume approach

The computation of incompressible three-dimensional viscous flow is discussed. A new physically consistent method is presented for the reconstruction for velocity fluxes which arise from the mass and momentum balance discrete equations. This closure method for fluxes allows the use of a cell-centered grid in which velocity and pressure unknowns share the same location, while circumventing the occurrence of spurious pressure modes. The method is validated on several benchmark problems which include steady laminar flow predictions on a two-dimensional cartesian (lid driven 2D cavity) or curvilinear grid (circular cylinder problem at Re = 40), unsteady three-dimensional laminar flow predictions on a cartesian grid (parallelopipedic lid driven cavity) and unsteady two-dimensional turbulent flow predictions on a curvilinear grid (vortex shedding past a square cylinder at Re = 22,000).     

非结构粘性直角网格并行算法建立与应用

开展了基于粘性直角非结构网格的并行CFD解算软件的开发研究,工作分两部分：网格分区实现和解算器并行实施,文章介绍了关于CFD解算器并行的实施情况,给出了并行过程中的操作流程,并对一些关键问题进行了讨论。并行计算结果表明项目所采用的并行途径和方法有效,计算结果可靠。     

Compact third-order multidimensional upwind discretization for steady and unsteady flow simulations

We propose a new third-order multidimensional upwind algorithm for the solution of the flow equations on tetrahedral cells unstructured grids. This algorithm has a compact stencil (cell-based computations) and uses a finite element idea when computing the residual over the cell to achieve its third-order (spatial) accuracy. The construction of the new scheme is presented. The asymptotic accuracy for steady or unsteady, inviscid or viscous flow situations is proved using numerical experiments. The new high-order discretization proves to have excellent parallel scalability. Our studies show the advantages of the new compact third-order scheme when compared with the classical second-order multidimensional upwind schemes.     

A Parallel Domain Decomposition Method for 3D Unsteady Incompressible Flows at High Reynolds Number

Numerical simulation of three-dimensional incompressible flows at high Reynolds number using the unsteady Navier–Stokes equations is challenging. In order to obtain accurate simulations, very fine meshes are necessary, and such simulations are increasingly important for modern engineering practices, such as understanding the flow behavior around high speed trains, which is the target application of this research. To avoid the time step size constraint imposed by the CFL number and the fine spacial mesh size, we investigate some fully implicit methods, and focus on how to solve the large nonlinear system of equations at each time step on large scale parallel computers. In most of the existing implicit Navier–Stokes solvers, segregated velocity and pressure treatment is employed. In this paper, we focus on the Newton–Krylov–Schwarz method for solving the monolithic nonlinear system arising from the fully coupled finite element discretization of the Navier–Stokes equations on unstructured meshes. In the subdomain, LU or point-block ILU is used as the local solver. We test the algorithm for some three-dimensional complex unsteady flows, including flows passing a high speed train, on a supercomputer with thousands of processors. Numerical experiments show that the algorithm has superlinear scalability with over three thousand processors for problems with tens of millions of unknowns.