A fully implicit Navier-Stokes algorithm using an unstructured grid and flux difference splitting (original) (raw)
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34th Aerospace Sciences Meeting and Exhibit, 1996
An implicit algorithm is developed for the 2D compressible Favre-averaged Navier-Stokes equations. It incorporates the standard k-epsilon turbulence model of Launder and Spalding (1974) and the low-Reynolds-number correction of Chien (1982). The equations are solved using an unstructured grid of triangles with the flow variables stored at the centroids of the cells. The inviscid fluxes are obtained from Roe's flux difference split method. Linear reconstruction of the flow variables to the cell faces provides second-order spatial accuracy. Turbulent and viscous stresses as well as heat transfer are obtained from a discrete representation of Gauss's theorem. Interpolation of the flow variables to the nodes is achieved using a second-order-accurate method. Temporal discretization employs Euler, trapezoidal, or three-point backward differencing. An incomplete LU factorization of the Jacobian matrix is implemented as a preconditioning method. Results are presented for a supersonic turbulent mixing layer, a supersonic laminar compression corner, and a supersonic turbulent compression corner. (Author)
Study of Conservation on Implicit Techniques for Unstructured Finite Volume Navier-Stokes Solvers
2012
The work is an study of conservation on linearization techniques of time-marching schemes for unstructured finite volume Reynolds-averaged Navier-Stokes formulation. The solver used in this work calculates the numerical flux applying an upwind discretization based on the flux vector splitting scheme. This numerical treatment results in a very large sparse linear system. The direct solution of this full implicit linear system is very expensive and, in most cases, impractical. There are several numerical approaches which are commonly used by the scientific community to treat sparse linear systems, and the point-implicit integration was selected in the present case. However, numerical approaches to solve implicit linear systems can be non-conservative in time, even for formulations which are conservative by construction, as the finite volume techniques. Moreover, there are physical problems which strongly demand conservative schemes in order to achieve the correct numerical solution. The work presents results of numerical simulations to evaluate the conservation of implicit and explicit time-marching methods and discusses numerical requirements that can help avoiding such non-conservation issues.
An implicit finite-element method for high-speed flows
International Journal for Numerical Methods in Engineering, 1991
A fast algorithm is presented for constructing continuous lines, made up of element sides, which pass once through each node of a general unstructured triangular mesh and which are generally aligned in prescribed directions. The lines are used as the basis of an adaptive fully implicit unstructured grid procedure for the solution of two-dimensional problems of steady compressible inviscid and laminar viscous high-speed flows, where the equation system is solved by line relaxation using a block tridiagonal equation solver. For three-dimensional laminar viscous simulations it is proposed to utilize an implicitlexplicit finite-element formulation. In the vicinity of solid walls a grid exhibiting structure in the normal direction is employed while, away from this region, the grid will be totally unstructured. In the structured region, lines in the normal direction to the wall are readily identified, while lines in the surfaces parallel to the solid wall are constructed using the proposed two-dimensional procedure. The implicit algorithm is then used in the structured region and the equation solution is achieved via line relaxation. An explicit form of the solution algorithm is used elsewhere. To illustrate the performance of the proposed method, solutions are obtained for both transonic inviscid and transonic and hypersonic laminar viscous problems in two dimensions. The application of the proposed procedure to the solution of three-dimensional hypersonic laminar viscous flow over a double ellipsoid configuration is also described.
A Fast, Matrix-free Implicit Method for Compressible Flows on Unstructured Grids
Journal of Computational Physics, 1998
A fast, matrix-free implicit method has been developed to solve the three-dimensional compressible Euler and Navier-Stokes equations on unstructured meshes. An approximate system of linear equations arising from the Newton linearization is solved by the GMRES (generalized minimum residual) algorithm with a LU-SGS (lower-upper symmetric Gauss-Seidel) preconditioner. A remarkable feature of the present GMRES+LU-SGS method is that the storage of the Jacobian matrix can be completely eliminated by approximating the Jacobian with numerical fluxes, resulting in a matrix-free implicit method. The method developed has been used to compute the compressible flows around 3D complex aerodynamic configurations for a wide range of flow conditions, from subsonic to supersonic. The numerical results obtained indicate that the use of the GMRES+LU-SGS method leads to a significant increase in performance over the best current implicit methods, GMRES+ILU and LU-SGS, while maintaining memory requirements similar to its explicit counterpart. An overall speedup factor from eight to more than one order of magnitude for all test cases in comparison with the explicit method is demonstrated.
2018
This article deals with implementation of a high-order finite difference scheme for numerical solution of the incompressible NavierStokes equations on curvilinear grids. The numerical scheme is based on pseudo-compressibility approach. A fifth-order upwind compact scheme is used to approximate the inviscid fluxes while the discretization of metric and viscous terms is accomplished using sixth-order central compact scheme. An implicit Euler method is used for discretization of the pseudotime derivative to obtain the steady-state solution. The resulting block tridiagonal matrix system is solved by approximate factorization based alternating direction implicit scheme (AF-ADI) which consists of an alternate sweep in each direction for every pseudo-time step. The convergence and efficiency of the method are evaluated by solving some 2D benchmark problems. Finally, computed results are compared with numerical results in the literature and a good agreement is observed.
Computer Methods in Applied Mechanics and Engineering, 2007
This paper introduces the use of Moving Least-Squares (MLS) approximations for the development of high order upwind schemes on unstructured grids, applied to the numerical solution of the compressible Navier-Stokes equations. This meshfree interpolation technique is designed to reproduce arbitrary functions and their succesive derivatives from scattered, pointwise data, which is precisely the case of unstructured-grid finite volume discretizations. The Navier-Stokes solver presented in this study follows the ideas of the generalized Godunov scheme, using Roe's approximate Riemann solver for the inviscid fluxes. Linear, quadratic and cubic polynomial reconstructions are developed using MLS to compute high order derivatives of the field variables. The diffusive fluxes are computed using MLS as a global reconstruction procedure. Various examples of inviscid and viscous flow are presented and discussed.
Engineering Applications of Computational Fluid Mechanics, 2012
An unsteady incompressible numerical method for the solution of Navier-Stokes equations is presented. The finite volume solver adopts the method of artificial compressibility, using an implicit dual time stepping scheme for time accuracy. The 2D solver operates on general hybrid meshes containing triangles and quadrilaterals, while the 3D solver operates on hybrid meshes containing tetrahedra, pyramids, prisms and hexahedra. The developed algorithms for spatial discretization and time integration are mesh transparent. An upwind spatial discretization scheme is used for the convective terms and a central scheme for the diffusive terms. Efficient calculation of flow fluxes is implemented in an edge-wise fashion. A new combined method for efficient and accurate evaluation of variable gradients is achieved by using an averaging technique and by avoiding multiple spatial integration of the same element of the mesh. The results obtained agree well with numerical solutions obtained by other researchers.
CFR: A Finite Volume Approach for Computing Incompressible Viscous Flow
Journal of Applied Fluid Mechanics
An incompressible unsteady viscous two-dimensional Navier-Stokes solver is developed by using "Consistent Flux Reconstruction" method. In this solver, the full Navier-Stokes equations have been solved numerically using a collocated finite volume scheme. In the present investigation, numerical simulations have been carried out for unconfined flow past a single circular cylinder with both structured and unstructured grids. In structured grid, quadrilateral cells are used whereas triangular elements are used in unstructured grid. Simulations are performed at Reynolds number (Re) = 100 and 200. Flow simulation over a NACA0002 airfoil at Re = 1000 using unstructured grid based solver is also reported. The vortex shedding phenomena is mainly investigated in the flow. It is observed that the nature of flow depends strongly on the value of the Reynolds number. The present results are found to be in satisfactory agreement with several numerical results and a few experimental results available from literature.
Comparison of several spatial discretizations for the Navier–Stokes equations
2000
Grid convergence studies for subsonic and transonic flows over airfoils are presented in order to compare the accuracy of several spatial discretizations for the compressible Navier-Stokes equations. The discretizations include the following schemes for the inviscid fluxes: (1) second-order-accurate centered differences with third-order matrix numerical dissipation, (2) the second-order convective upstream split pressure scheme (CUSP), (3) third-order upwind-biased differencing with Roe's flux-difference splitting, and (4) fourth-order centered differences with third-order matrix numerical dissipation. The first three are combined with second-order differencing for the grid metrics and viscous terms. The fourth discretization uses fourthorder differencing for the grid metrics and viscous terms, as well as higher-order approximations near boundaries and for the numerical integration used to calculate forces and moments. The results indicate that the discretization using higher-order approximations for all terms is substantially more accurate than the others, producing less than two percent numerical error in lift and drag components on grids with less than 13,000 nodes for subsonic cases and less than 18,000 nodes for transonic cases. Since the cost per grid node of all of the discretizations studied is comparable, the higher-order discretization produces solutions of a given accuracy much more efficiently than the others.
Proceedings of 10th World Congress on Computational Mechanics, 2014
The development of a CFD tool based on Discontinuous Galerkin discretization is reported. This tool solves the compressible, Reynolds-Averaged Navier-Stokes equations for three-dimensional, hybrid unstructured meshes. This tool is aimed at complex aerospace applications, thus requiring advanced turbulence models and an efficient numerical framework to enhance computational performance and numerical accuracy for high Reynolds number, high Mach number flows. Inviscid fluxes are computed by upwind Roe or HLLC schemes and viscous fluxes are computed using BR1 or BR2 formulations. A 2nd-order accurate, 5stage, explicit Runge-Kutta time-stepping scheme is used to march the equations in time. The solver computational efficiency, convergence, accuracy and parallel scalability are addressed through flow simulations over typical validation test cases. Convergence rates for increasing degrees of freedom are shown to be asymptotic, with numerical errors compatible to DG schemes. Parallelism is shown to be conformant with the expected scalability behaviour. For the aerospace applications considered in this paper, acceptable agreement with theoretical or experimental results is obtained at adequate computational costs.