An implicit high order cell-centered finite volume scheme for the solution of three-dimensional Navier–Stokes equations on unstructured grids (original) (raw)
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European Conference on Computational Fluid …, 2006
In this work, a new method was described for spatial discretization of threedimensional Navier Stokes equations in their primitive form, on unstructured, staggered grids. Velocities were placed on the cell faces and pressure in cell centers and were linked with the projection method. Thanks to the variable arrangement, no stabilization procedure was needed to avoid spurious pressure/velocity elds. A way around the deferred correction was also described and used in this work. Several laminar cases were computed to show the validity of the method. Computation of velocities on the cell faces and the ability to integrate in time with projection method without any stabilization procedure make the proposed method a good candidate for large eddy simulation (LES) of turbulence in complex geometries.
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)
Journal of Applied Fluid Mechanics
In this paper, A Novel Alternating Cell Direction Implicit Method (ACDI) is researched which allows implementation of fast line implicit methods on quadrilateral unstructured meshes. In ACDI method, designated alternating cell directions are taken along a series of contiguous cells within the unstructured grid domain and used as implicit lines similar to Line Gauss Seidel Method (LGS). ACDI method applied earlier for the solution of potential flows is extended for the solution of the incompressible Navier-Stokes equations on unstructured grids. The system of equations is solved by using the Symmetric Line Gauss-Seidel (SGS) method along the alternating cell directions. Laminar flow fields over a single element NACA-0008 airfoil are computed by using structured and unstructured quadrilateral grids, and inviscid Euler flow solutions are given for the NACA-23012b multielement airfoil. The predictive capability of the method is validated against the data taken from the experimental or the other numerical studies and the efficiency of the ACDI method is compared with the implicit Point Gauss Seidel (PGS) method. In the selected validation cases, the results show that a reduction in total computation between 18% and 23% is achieved by the ACDI method over the PGS. In general, the results show that the ACDI method is a fast, efficient, robust and versatile method that can handle complicated unstructured grid cases with equal ease as with the structured grids.
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.
A fully implicit Navier-Stokes algorithm using an unstructured grid and flux difference splitting
Applied Numerical Mathematics, 1994
An implicit algorithm is developed for the two-dimensional, compressible, laminar Navier-Stokes equations using an unstructured grid of triangles. A cell-centered data structure is employed with the flow variables stored at the centroids of the triangles. The algorithm is based on Roe's flux difference split method for the inviscid fluxes, and a discrete representation of the viscous fluxes and heat transfer using Gauss' Theorem. Linear reconstruction of the flow variables to the cell faces, employed for the inviscid terms, provides second-order spatial accuracy. Interpolation of the flow variables to the nodes is achieved using a second-order accurate method. Temporal discretization employs Euler, trapezoidal or 3-point backward differencing. The complete, exact Jacobian of the inviscid and viscous terms is derived. The algorithm is applied to the Riemann Shock Tube problem, a supersonic laminar boundary layer on a flat plate, and subsonic viscous flow past an NACA0012 airfoil. Results are in excellent agreement with theory and previous computations.
2010
The fluid dynamic equations are discretized by a high-order spectral volume (SV) method on unstructured tetrahedral grids. We solve the steady state equations by advancing in time using a backward Euler (BE) scheme. To avoid the inversion of a large matrix we approximate BE by an implicit lower-upper symmetric Gauss-Seidel (LU-SGS) algorithm. The implicit method addresses the stiffness in the discrete Navier-Stokes equations associated with stretched meshes. The LU-SGS algorithm is then used as a smoother for a pmultigrid approach. A Von Neumann stability analysis is applied to the two-dimensional linear advection equation to determine its damping properties. The implicit LU-SGS scheme is used to solve the two-dimensional (2D) compressible laminar Navier-Stokes equations. We compute the solution of a laminar external flow over a cylinder and around an airfoil at low Mach number. We compare the convergence rates with explicit Runge-Kutta (E-RK) schemes employed as a smoother. The effects of the cell aspect ratio and the low Mach number on the convergence are investigated. With the p-multigrid method and the implicit smoother the computational time can be reduced by a factor of up to 5-10 compared with a well tuned E-RK scheme.
37th Aerospace Sciences Meeting and Exhibit, 1999
A multiblock unstructured grid approach is presented for solving three-dimensional incompressible inviscid and viscous turbulent flows about complete configurations. The artificial compressibility form of the governing equations is solved by a node-based, finite volume implicit scheme which uses a backward Euler time discretization. Point Gauss-Seidel relaxations are used to solve the linear system of equations at each time step. This work employs a multiblock strategy to the solution procedure, which greatly improves the efficiency of the algorithm by significantly reducing the memory requirements by a factor of 5 over the singlegrid algorithm while maintaining a similar convergence behavior. The numerical accuracy of solutions is assessed by comparing with the experimental data for a submarine with stern appendages and a high-lift configuration.
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.
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.
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.