Co‐located equal‐order control‐volume finite element method for two‐dimensional axisymmetric incompressible fluid flow

The formulation of a control-volume-based finite element method (CVFEM) for axisymmetric, two-dimensional, incompressible fluid flow and heat transfer in irregular-shaped domains is presented. The calculation domain is discretized into torus-shaped elements and control volumes. In a longitudinal cross-sectional plane, these elements are three-node triangles, and the control volumes are polygons obtained by joining the centroids of the three-node triangles to the mid-points of the sides. Two different interpolation schemes are proposed for the scalar-dependent variables in the advection terms: a flow-oriented upwind function, and a mass-weighted upwind function that guarantees that the discretized advection terms contribute positively to the coefficients in the discretized equations. In the discretization of diffusion transport terms, the dependent variables are interpolated linearly. An iterative sequential variable adjustment algorithm is used to solve the discretized equations for the velocity components, pressure and other scalar-dependent variables of interest. The capabilities of the proposed CVFEM are demonstrated by its application to four different example problems. The numerical solutions are compared with the results of independent numerical and experimental investigations. These comparisons are quite encouraging.

[1]  Suhas V. Patankar,et al.  A CONTROL VOLUME-BASED FINITE-ELEMENT METHOD FOR SOLVING THE NAVIER-STOKES EQUATIONS USING EQUAL-ORDER VELOCITY-PRESSURE INTERPOLATION , 1984 .

[2]  B. Rabi Baliga,et al.  SOLUTION OF THREE-DIMENSIONAL CONVECTION-DIFFUSION PROBLEMS USING TETRAHEDRAL ELEMENTS AND FLOW-ORIENTED UPWIND INTERPOLATION FUNCTIONS , 1986 .

[3]  C. Rhie,et al.  Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation , 1983 .

[4]  B. R. Baliga,et al.  A CONTROL VOLUME FINITE-ELEMENT METHOD FOR TWO-DIMENSIONAL FLUID FLOW AND HEAT TRANSFER , 1983 .

[5]  D W Crawford,et al.  Measurement and prediction of flow through a replica segment of a mildly atherosclerotic coronary artery of man. , 1986, Journal of biomechanics.

[6]  Andreas Acrivos,et al.  Buoyancy-driven convection in cylindrical geometries , 1969, Journal of Fluid Mechanics.

[7]  Andreas Lahrmann,et al.  An element formulation for the classical finite difference and finite volume method applied to arbitrarily shaped domains , 1992 .

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

[9]  Vaughan R Voller,et al.  STREAMLINE UPWIND SCHEME FOR CONTROL-VOLUME FINITE ELEMENTS, PART I. FORMULATIONS , 1992 .

[10]  B. R. Baliga,et al.  A NEW FINITE-ELEMENT FORMULATION FOR CONVECTION-DIFFUSION PROBLEMS , 1980 .

[11]  D W Crawford,et al.  Effect of mild atherosclerosis on flow resistance in a coronary artery casting of man. , 1984, Journal of biomechanical engineering.

[12]  C. Prakash,et al.  Examination of the upwind (donor-cell) formulation in control volume finite-element methods for fluid flow and heat transfer , 1987 .

[13]  Gerry E. Schneider,et al.  A SKEWED, POSITIVE INFLUENCE COEFFICIENT UPWINDING PROCEDURE FOR CONTROL-VOLUME-BASED FINITE-ELEMENT CONVECTION-DIFFUSION COMPUTATION , 1986 .

[14]  R. A. Nicolaides,et al.  Discretization of incompressible vorticity–velocity equations on triangular meshes , 1990 .

[15]  J. Gillis,et al.  Laminar flow in a pipe at low and moderate reynolds numbers , 1968 .

[16]  G. D. Raithby,et al.  Skew upstream differencing schemes for problems involving fluid flow , 1976 .

[17]  B. R. Baliga,et al.  Evaluation and Enhancements of Some Control Volume Finite-Element Methods - Part 2. Incompressible Fluid Flow Problems , 1988 .

[18]  J. Z. Zhu,et al.  The finite element method , 1977 .

[19]  S. Patankar.  Numerical Heat Transfer and Fluid Flow , 2018, Lecture Notes in Mechanical Engineering.

[20]  C. Prakash,et al.  AN IMPROVED CONTROL VOLUME FINITE-ELEMENT METHOD IFOR HEAT AND MASS TRANSFER, AND FOR FLUID FLOW USING EQUAL-ORDER VELOCITY-PRESSURE INTERPOLATION , 1986 .

[21]  Gerry E. Schneider,et al.  Control Volume Finite-Element Method for Heat Transfer and Fluid Flow Using Colocated Variables— 1. Computational Procedure , 1987 .

[22]  Yassin A. Hassan,et al.  A Stable Mass-Flow-Weighted Two-Dimensional Skew Upwind Scheme , 1983 .

[23]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[24]  R. Camarero,et al.  Simulating two-dimensional turbulent flow by using the κ-ε model and the vorticity-streamfunction formulation , 1992 .

[25]  P. Hood,et al.  A numerical solution of the Navier-Stokes equations using the finite element technique , 1973 .

[26]  R. Kessler,et al.  Comparison of finite-volume numerical methods with staggered and colocated grids , 1988 .

[27]  F. Durst,et al.  Investigations of laminar flow in a pipe with sudden contraction of cross sectional area , 1985 .

[28]  R. J. Schnipke,et al.  An equal-order velocity-pressure formulation that does not exhibit spurious pressure modes , 1986 .

[29]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .