An unstructured finite volume technique for the 3D Poisson equation on arbitrary geometry using a σ -coordinate system (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.
This paper presents a finite volume solver for the computation of three-dimensional viscous flows. A cell-centered approach is used and a quadratic reconstruction of the unknowns is performed to compute the advective fluxes on the cell faces. The gradients of the variables, necessary for the viscous fluxes, are constructed using Coirier's diamond path. A extended version of this method is proposed in this paper to ensure the consistency of the method whatever the distortion of the grid. A fully implicit time integration procedure is employed with preconditioned matrix-free GMRES solver.
SIAM Journal on Scientific Computing, 1998
In this paper, a fast direct solver for the Poisson equation on the half-staggered grid is presented. The Poisson equation results from the projection method of the finite difference solution of the incompressible Navier-Stokes equations. To achieve our goal, new algorithms for diagonalizing a semidefinite pair are developed. The fast solver can also be extended to the three-dimensional case. The motivation and related issues in using this half-staggered grid are also discussed.
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.
Computer Algebra in Scientific Computing, 2019
To increase the accuracy of computations by the method of collocations and least squares (CLS) a generalization of this method is proposed for the case of a non-uniform logically rectangular grid. The main work formulas of the CLS method on non-uniform grid, including the formulas implementing the prolongation operator on a non-uniform grid at the use of a multigrid complex are obtained with the aid of the computer algebra system (CAS) Mathematica. The proposed method has been applied for the numerical solution of two-dimensional stationary Navier-Stokes equations governing the laminar flows of viscous incompressible fluids. On a smooth test solution, the application of a nonuniform grid has enabled a 47-fold reduction of the solution error in comparison with the uniform grid case. At the solution of the problem involving singularities-the lid-driven cavity flow-the error of the solution obtained by the CLS method was reduced by the factors from 2.65 to 3.05 depending on the Reynolds number value.
Computers & Fluids, 2019
In this paper, we introduce a second-order time-and space-accurate technique, developed to solve in parallel free-surface flows in arbitrary three-dimensional geometries. The discretization is based on a second-order finite-volume technique on prisms elements, consisting of triangular grids on the horizontal and bounded by a free surface and an irregular bottom on the vertical. The equations are transformed vertically to the σ-coordinate system in order to obtain an accurate representation of top and bottom topography. The reconstruction of three presure/velocity decoupling methods using a Crank-Nicolson scheme formulation is proposed. The Momentum Interpolation Method (MIM) is combined with Local Extremum Diminishing (LED) second-order upstream scheme for convective terms is developed. The parallelization is designed by a block domain decomposition technique. The discretization results in nonsymmetric variable-coefficient linear systems which are solved using a parallel multi-color Successive Over-Relaxation algorithm. Several test cases of surface wave motion are used to demonstrate the capabilities, numerical stability and performance of the model.
A finite volume model on unstructured meshes and application in computing two dimensional flows
Journal of Computer Science and Cybernetics, 2012
Some methods in approximation of fluxes between adjacent cells have been proposed in context of the finite volume technique on unstructured meshes. A shallow water model has been developed for testing the proposed methods. The third order Adams-Bashforth scheme is used in integrating the governing equations. A filter is designed to remove spurious waves. The model is tested on unstructured triangular meshes with some examples in literature. Tóm tȃt. Bài báo trình bày mô. t số kỹ thuâ. t xấp xı ' thông lu. o. . ng giũ. a các nốt trên lu .ó. i phi cấu trúc trong kỹ thuâ. t thê ' tích hũ. u ha. n. Các phu. o. ng pháp này du. o. . c thu. ' nghiê. m vào mô. t mô hình nu .ó. c nông hai chiề u trên lu .ó. i tam giác phi cấu trúc. Mô hình này su. ' du. ng so. dồ tích phân thò. i gian Adams-Bashforth bâ. c ba. Mô. t bô. lo. c cũng du. o. . c du. a vào mô hình nhȃm loa. i bo ' các sóng nhiễu sinh ra trong quá trình tích phân. Các bài toán mẫu sẽ du. o. . c su. ' du. ng nhȃm kiê ' m tra kỹ nȃng mô pho 'ng cu 'a mô hình.
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.
A coupled finite volume solver for the solution of incompressible flows on unstructured grids
Journal of Computational Physics, 2009
This paper reports on a newly developed fully coupled pressure-based algorithm for the solution of laminar incompressible flow problems on collocated unstructured grids. The implicit pressure-velocity coupling is accomplished by deriving a pressure equation in a procedure similar to a segregated SIMPLE algorithm using the Rhie-Chow interpolation technique and assembling the coefficients of the momentum and continuity equations into one diagonally dominant matrix. The extended systems of continuity and momentum equations are solved simultaneously and their convergence is accelerated by using an algebraic multigrid solver. The performance of the coupled approach as compared to the segregated approach, exemplified by SIMPLE, is tested by solving five laminar flow problems using both methodologies and comparing their computational costs. Results indicate that the number of iterations needed by the coupled solver for the solution to converge to a desired level on both structured and unstructured meshes is grid independent. For relatively coarse meshes, the CPU time required by the coupled solver on structured grid is lower than the CPU time required on unstructured grid. On dense meshes however, this is no longer true. For low and moderate values of the grid aspect ratio, the number of iterations required by the coupled solver remains unchanged, while the computational cost slightly increases. For structured and unstructured grid systems, the required number of iterations is almost independent of the grid size at any value of the grid expansion ratio. Recorded CPU time values show that the coupled approach substantially reduces the computational cost as compared to the segregated approach with the reduction rate increasing as the grid size increases.
A multi-grid finite-volume method for free-surface flows
AD Publication, 2018
Abstract—A depth-averaged subcritical and/or supercritical, steady, free-surface flow numerical model is developed to calculate physical hydraulic flow parameters in open channels. The vertically averaged free-surface flow equations are numerically solved using an explicit finite-volume numerical scheme in integral form. The grid used may be irregular and conforms to the physical boundaries of any problem. A multi-grid algorithm has been developed and has subsequently been applied to accelerate the convergence solution. A grid clustering technique is also applied. The numerical approach is straight forward and the flow boundary conditions are easy enforced. The capabilities of the proposed method are demonstrated by analyzing subcritical flow in an abrupt converging-diverging open channel flume as well calculating supercritical flows in an expansion channel. The computed results are satisfactorily compared with available measurements as well as with other numerical technique results. Very coarse grid gives satisfactory comparison results. The explicit numerical code can be utilized, within the assumptions made about the nature of the flow, for various vertically averaged free-surface flow calculations. Scope is to simulate free-surface flows of practical interest in a straight forward way. It can be extended to channel designs.