Introduction to finite volume methods in computational fluid dynamics

The basic laws of fluid dynamics are conservation laws. They are statements that express the conservation of mass, momentum and energy in a volume closed by a surface. Only with the supplementary requirement of sufficient regularity of the solution can these laws be converted into partial differential equations. Sufficient regularity cannot always be guaranteed. Shocks form the most typical example of a discontinuous flow field. In case discontinuities occur, the solution of the partial differential equations is to be interpreted in a weak form, i.e. as a solution of the integral form of the equations. For example, the laws governing the flow through a shock, i.e. the Hugoniot-Rankine laws, are combinations of the conservation laws in integral form. For a correct representation of shocks, also in a numerical method, these laws have to be respected. There are additional situations where an accurate representation of the conservation laws is important in a numerical method. A second example is the slip line which occurs behind an airfoil or a blade if the entropy production is different on streamlines on both sides of the profile. In this case, a tangential discontinuity occurs. Another example is incompressible flow where the imposition of incompressibility, as a conservation law for mass, determines the pressure field. In the cases cited above, it is important that the conservation laws in their integral form are represented accurately. The most natural method to accomplish this is to discretize the integral form of the equations and not the differential form. This is the basis of a finite volume method. Further, in cases where strong conservation in integral form is not absolutely necessary, it is still physically appealing to use the basic laws in their most primitive form. The flow field or domain is subdivided, as in the finite element method, into a set of non-overlapping cells that cover the whole domain. In the finite volume method (FVM) the term cell is used instead of the term element used in the finite element method (FEM). The conservation laws are applied to determine the flow

[1]  David C. Slack,et al.  Time integration algorithms for the two-dimensional Euler equations on unstructured meshes , 1994 .

[2]  E. Süli,et al.  Numerical Solution of Ordinary Differential Equations , 2021, Foundations of Space Dynamics.

[3]  R. C. Swanson,et al.  On Central-Difference and Upwind Schemes , 1992 .

[4]  Philip L. Roe,et al.  Accelerated convergence of Jameson's finite-volume Euler scheme using Van der Houwen integrators , 1985 .

[5]  A. Jameson.  ANALYSIS AND DESIGN OF NUMERICAL SCHEMES FOR GAS DYNAMICS, 1: ARTIFICIAL DIFFUSION, UPWIND BIASING, LIMITERS AND THEIR EFFECT ON ACCURACY AND MULTIGRID CONVERGENCE , 1995 .

[6]  Arthur Veldman,et al.  Direct Numerical Simulation of Turbulence at Lower Costs , 1997 .

[7]  Zhi J. Wang,et al.  Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids. Basic Formulation , 2002 .

[8]  N. Ron-Ho,et al.  A Multiple-Grid Scheme for Solving the Euler Equations , 1982 .

[9]  Z. Wang,et al.  Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids , 2002 .

[10]  A. Jameson.  ANALYSIS AND DESIGN OF NUMERICAL SCHEMES FOR GAS DYNAMICS, 2: ARTIFICIAL DIFFUSION AND DISCRETE SHOCK STRUCTURE , 1994 .

[11]  James J. McGuirk,et al.  Finite Volume Discretization Aspects for Viscous Flows on Mixed Unstructured Grids , 1999 .

[12]  Leon Lapidus,et al.  Numerical Solution of Ordinary Differential Equations , 1972 .

[13]  D. Birchall,et al.  Computational Fluid Dynamics , 2020, Radial Flow Turbocompressors.

[14]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[15]  Chaowei Hu,et al.  No . 98-32 Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes , 1998 .

[16]  Arthur Rizzi,et al.  Numerical methods for the computation of inviscid transonic flkows with shock waves , 1981 .

[17]  K. W. Morton,et al.  Cell vertex methods for inviscid and viscous flows , 1993 .

[18]  J. Peraire,et al.  AN UPWIND UNSTRUCTURED GRID SOLUTION ALGORITHM FOR COMPRESSIBLE FLOW , 1993 .

[19]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[20]  R. Walters,et al.  Solution algorithms for the two-dimensional Euler equations on unstructured meshes , 1990 .

[21]  A. Jameson,et al.  Finite volume solution of the two-dimensional Euler equations on a regular triangular mesh , 1985 .

[22]  Arun S. Mujumdar,et al.  NUMERICAL HEAT TRANSFER: T.M. Shih Hemisphere. New York (1984) XVII+563 pp. , 1985 .

[23]  D. J. Mavriplis,et al.  Unstructured mesh algorithms for aerodynamic calculations , 1993 .

[24]  Dimitri J. Mavriplis,et al.  t ICASE Report No . 91-11 co ~ ICASE MULTIGRID SOLUTION OF COMPRESSIBLE TURBULENT FLOW ON UNSTRUCTURED MESHES USING A TWO-EQUATION MODEL , 2022 .

[25]  Philip L. Roe,et al.  A multidimensional generalization of Roe's flux difference splitter for the euler equations , 1993 .

[26]  Z. Lilek,et al.  A fourth-order finite volume method with colocated variable arrangement , 1995 .

[27]  Erik Dick,et al.  A flux-difference splitting method for steady Euler equations , 1988 .

[28]  K. Riemslagh,et al.  MULTI-STAGE JACOBI RELAXATION IN MULTIGRID METHODS FOR THE STEADY EULER EQUATIONS , 1995 .