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A simple algebraic multigrid (AMG) solver for linear equations is presented, and its performance compared with a conjugate gradient scheme. This multigrid method is extended to solve the discrete Navier—Stokes equations, obtained by applying a finite volume approach to three‐dimensional incompressible flow on a finite element mesh. The resulting multigrid solver is incorporated into a general purpose flow code (ASTEC), where it proves faster than the original solution algorithm, based upon SIMPLE. The linear AMG solver is both efficient and robust, but the extension to include coupling in the Navier—Stokes equations does not converge on all problems.

We study the sparse grid combination technique as an efficient method for the solution of fluid dynamics problems. The combination technique needs only O(h^–1_n(log(h^–1_n))^d–1) grid points for d‐dimensional problems, instead of O(h^–d_n) grid points used by the full grid method. Here, h_n = 2^–n denotes the mesh width of the grids. Furthermore, provided that the solution is sufficiently smooth, the accuracy (with respect to the L₂‐ and the L_∞‐norm) of the sparse grid combination solution is O(h^α_n(log(h^–1_n))^d–1), which is only slightly worse than O(h^α_n) obtained by the full grid solution. Here, α includes the order of the underlying discretization scheme, as well as the influence of singularities. Thus, the combination technique is very economic on both storage requirements and computing time, but achieves almost the same accuracy as the usual full grid solution. Another advantage of the combination technique is that only simple data structures are necessary. Where other sparse grid methods need hierarchical data structures and thus specially designed solvers, the combination method handles merely d‐dimensional arrays. Thus, the implementation of the combination technique can be based on any “black box solver”. However, for reasons of efficiency, an appropriate multigrid solver should be used. Often, fluid dynamics problems have to be solved on rather complex domains. A common approach is to divide the domain into blocks, in order to facilitate the handling of the problem. We show that the combination technique works on such blockstructured grids as well. When dealing with complicated domains, it is often desirable to grade a grid around a singularity. Graded grids are also supported by the combination technique. Finally, we present the first results of numerical experiments for the application of the combination method to CFD problems. There, we consider two‐dimensional laminar flow problems with moderate Reynolds numbers, and discuss the advantages of the combination method.

The Lattice Boltzmann (LB) method offers an alternative to conventional computational fluid dynamics (CFD) methods. However, its practical use for complex turbulent flows of engineering interest is still at an early stage. This paper aims to outline an LB wall-modeled large-eddy simulation (WMLES) solver.

The solver is dedicated to complex high-Reynolds flows in the context of WMLES. It relies on an improved LB scheme and can handle complex geometries on multi-resolution block structured grids.

Dynamic and acoustic characteristics of a turbulent airflow past a rod-airfoil tandem are examined to test the capabilities of this solver. Detailed direct comparisons are made with both experimental and numerical reference data.

This study allows assessing the potential of an LB approach for industrial CFD applications.

This paper describes the numerical solutions of type‐IV shock‐on‐shock interactions in hypersonic thermochemical nonequilibrium air flows around blunt bodies. The Navier‐Stokes equation solver for a chemically reacting and vibrationally relaxing gas mixture was applied to the present problem, where the concepts of the Advection Upstream Splitting Method (AUSM) and the Lower‐Upper Symmetric Gauss‐Seidel (LU‐SGS) method were basically employed along with the two‐temperature thermochemical model of Park. The aerodynamic heating with or without the shock‐on‐shock interaction to a sphere and circular cylinders are simulated for a hypersonic nonequilibrium flow. The numerical results show that typical type‐IV shock‐on‐shock interaction pattern with a supersonic jet structure is also formed in a high‐enthalpy thermochemical nonequilibrium flow, and that the contribution of convective heat flux in the translational/rotational mode to the total heat flux is dominant. Furthermore, the inherent unsteadiness of nonequilibrium type‐IV shock‐on‐shock interaction flowfield is discussed briefly.

The purpose of this paper is to develop Triple Finite Volume Method (tFVM), the author discretizes incompressible Navier-Stokes equation by tFVM, which leads to a special linear system of saddle point problem, and most computational efforts for solving the linear system are invested on the linear solver GMRES.

In this paper, by recently developed preconditioner Hermitian/Skew-Hermitian Separation (HSS) and the parallel implementation of GMRES, the author develops a quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation.

Computational results show that, the quick solver HSS-pGMRES-tFVM significantly increases the solution speed for saddle point problem from incompressible Navier-Stokes equation than the conventional solvers.

Altogether, the contribution of this paper is that the author developed the quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation.

The purpose of this paper is to propose an accurate and efficient technique for computing flow sensitivities by finite differences of perturbed flow fields. It relies on computing the perturbed flows on coarser grid levels only: to achieve the same fine-grid accuracy, the approximate value of the relative local truncation error between coarser and finest grids unperturbed flow fields, provided by a standard multigrid method, is added to the coarse grid equations. The gradient computation is introduced in a hybrid genetic algorithm (HGA) that takes advantage of the presented method to accelerate the gradient-based search. An application to a classical transonic airfoil design is reported.

Genetic optimization algorithm hybridized with classical gradient-based search techniques; usage of fast and accurate gradient computation technique.

The new variant of the prolongation operator with weighting terms based on the volume of grid cells improves the accuracy of the MAFD method for turbulent viscous flows. The hybrid GA is capable to efficiently handle and compensate for the error that, although very limited, is present in the multigrid-aided finite-difference (MAFD) gradient evaluation method.

The proposed new variants of HGA, while outperforming the simple genetic algorithm, still require tuning and validation to further improve performance.

Significant speedup of CFD-based optimization loops.

Introduction of new multigrid prolongation operator that improves the accuracy of MAFD method for turbulent viscous flows. First application of MAFD evaluation of flow sensitivities within a hybrid optimization framework.

The suitability of a coupled scheme based on parabolic/elliptic Navier‐Stokes equations for calculating film cooling flows and heat transfer downstream of flush, angled injection slots is explored. The coupled algorithm that combined the coarse mesh ‘outer’ Navier‐Stokes and fine grid ‘inner’ parabolic Navier‐Stokes codes makes retention of the current high resolution model desirable because an acceptable accuracy and economy of computation time are attainable using only mini‐computer resources. The ‘inner‐code’ includes the FLARE approximation to permit small reverse flow. The inner and outer codes are coupled by adopting an approach analogous to classical multigrid methods. It is found that for high blowing mass flow rate of 1.0 with the case of greater than 40° injection angle, the fully parabolic procedure is unable to cope with an extensive separation region immediately downstream of the slot; the present coupling methodology is crucial. The study involves the calculation of heat transfer rates on the surface downstream of the angled slot. Predicted film cooling effectiveness distribution together with the effects of governing parameters are described and show close agreement with the experimental data.

An optimized multigrid method (NSFLEX‐MG) for the NSFLEX‐code (Navier‐Stokes solver using characteristic flux extrapolation) of MBB (Messerschmitt Bolkow Blohm GmbH) is described. The method is based on a correction scheme and implicit relaxation procedures and is applied to two‐dimensional test cases. The principal feature of the flow solver is a Godunov‐type averaging procedure based on the eigenvalues analysis of the Euler equations by means of which the inviscid fluxes are evaluated at the finite volume faces. Viscous fluxes are centrally differenced at each cell face. The performance of NSFLEX‐MG is demonstrated for a large range of Mach numbers for compressible inviscid and viscous (laminar and turbulent) flows over a RAE‐2822 airfoil and over a NACA‐0012 airfoil.

This study aims to feature the application of the artificial compressibility method (ACM) for the numerical prediction of two-dimensional (2D) axisymmetric swirling flows.

The respective academic numerical solver, named IGal2D, is based on the axisymmetric Reynolds-averaged Navier–Stokes (RANS) equations, arranged in a pseudo-Cartesian form, enhanced by the addition of the circumferential momentum equation. Discretization of spatial derivative terms within the governing equations is performed via unstructured 2D grid layouts, with a node-centered finite-volume scheme. For the evaluation of inviscid fluxes, the upwind Roe’s approximate Riemann solver is applied, coupled with a higher-order accurate spatial reconstruction, whereas an element-based approach is used for the calculation of gradients required for the viscous ones. Time integration is succeeded through a second-order accurate four-stage Runge-Kutta method, adopting additionally a local time-stepping technique. Further acceleration, in terms of computational time, is achieved by using an agglomeration multigrid scheme, incorporating the full approximation scheme in a V-cycle process, within an efficient edge-based data structure.

A detailed validation of the proposed numerical methodology is performed by encountering both inviscid and viscous (laminar and turbulent) swirling flows with axial symmetry. IGal2D is compared against the commercial software ANSYS fluent – by using appropriate metrics and characteristic flow quantities – but also against experimental measurements, confirming the proposed methodology’s potential to predict such flows in terms of accuracy.

This study provides a robust methodology for the accurate prediction of swirling flows by combining the axisymmetric RANS equations with ACM. In addition, a detailed description of the convective flux Jacobian is provided, filling a respective gap in research literature.

The purpose of this paper is to describe a pragmatic parallelization of a publicly available serial code aimed for direct numerical simulations of turbulent flow fields. The code solves the full Navier‐Stokes equations in a cylindrical coordinate system.

The parallelization is performed by a single program multiple data approach using the Message‐Passing Interface (MPI) Library for processor communication.

In order to maintain the original coding of the subroutines, two obstacles had to be overcome. First, special attention had to be given to the inversion of the sparse matrixes from the linear terms in the Navier‐Stokes equations solved by an implicit scheme. Second, the serial FFT‐routines, needed for the direct Poisson‐solver, had to be replaced by parallel versions. Two directions of parallelization were tested. Parallelization in the axial direction turned out to be more efficient than parallelization in the circumferential direction.

This paper presents a pragmatic parallelization of an open source finite difference code and should be useful to researchers in the field of numerical methods for fluid flow who need to parallelize a numerical code.