The formulation of a general numerical method for two-dimensional incompressible flow and heat transfer in irregular-shaped domains is presented. The calculation domain is first divided into six-node macroelements. Then each macroelement is divided into four three-node triangular subelements. Polygonal control volumes are associated with the nodes of these elements. All dependent variables other than pressure are stored at the nodes of the subelements, and they are interpolated by functions that respond appropriately to an element Peclet number and the direction of an element-averaged velocity vector. The pressure is stored only at the vertices of the macroelements and is interpolated linearly in these elements. The discretization equations are obtained by deriving algebraic approximations to integral conservation equations applied to the polygonal control volumes. An iterative procedure akin to SIMPLER is used to solve the discretization equations.

A control volume finite-element method for two-dimensional fluid flow and heat transfer

: The authors discuss the creation of a finite element program suitable for solving incompressible, viscous free surface problems in steady axisymmetric or plane flows. For convenience in extending program capability to non-Newtonian flow, non-zero Reynolds numbers, and transient flow, a Galerkin formulation of the governing equations is chosen, rather than an extremum principle. The resulting program is used to solve the Newtonian die-swell problem for creeping jets free of surface tension constraints. The authors conclude that a Newtonian jet expands about 13%, in substantial agreement with experiments made with both small finite Reynolds numbers and small ratios of surface tension to viscous forces. The solutions to the related stick-slip problem and the tube inlet problem, both of which also contain stress singularities, are also given. (Modified author abstract)

The solution of viscous incompressible jet and free-surface flows using finite-element methods

A novel computational procedure for the prediction of incompressible fluid flow using primitive variables is presented. The formulation retains the geometric flexibility of the finite-element method and derives the governing discrete algebraic equations by using a conservation balance applied to discrete control volumes distributed throughout the domain. A novel method of closure, to relate the control volume surface values to the nodal point values, is introduced whereby a local discrete analog to the governing differential equation is formed at the control volume Surfaces. From this discrete equation analog the control surface values are determined in terms of the nodal values that represent the discrete problem unknowns. The manner in which this discrete equation is formed, solved, and used permits resolution of two longstanding problems in computational fluid dynamics, namely accurate convection modeling and preclusion of pressure field decoupling. A new and general boundary condition specifi...

Control Volume Finite-Element Method for Heat Transfer and Fluid Flow Using Colocated Variables— 1. Computational Procedure

A control volume-based finite-element method for solving the Navier-Stokes equations using equal-order velocity-pressure interpolation is presented. Unlike the unequal-order-type methods that compute pressure at much fewer grid points than velocity, the proposed equal-order method calculates velocity and pressure at all the grid points in the domain. The validity of the method is established by applying it to solve some test problems

A control volume-based finite-element method for solving the navier-stokes equations using equal-order velocity-pressure interpolation

The numerical accuracy of the fluid-dynamics code FIDAP is investigated by means of test computations on two-dimensional flows. The theoretical basis of the algorithm is briefly reviewed, and results for laminar flow in a wall-driven cavity and for both laminar and turbulent flows over a backward-facing step are presented in extensive tables and graphs and characterized in detail. Good agreement with published experimental data and previous computations is obtained for laminar flows using a version of FIDAP without streamlined upwinding (STU). For turbulent flows, the addition of STU is found to be useful, although it causes separation-zone reattachment length to be underpredicted.

Evaluation of fidap on some classical laminar and turbulent benchmarks

General tinite-element models of compre ol>e local approximations of the 'elocit~· field, the density. and the tempemture for compressible fluids and the ,'e1ocity. temperature. and prl'S'i,re ror incompressihlc tluids. Thcuries or local solenoidal approximl.tions and mixed tiniteclement models ror coml,n'ssihle film lIn' lIerh·ed. A numhl'r or cumpulallonal s..ht'mes are lIe"elopt'll ror Ihe uumerical solulion or hoth lransient and stl':IlI~'nOllllniformtlow prohlems illml\'illg illcol1lpressihlelIuids. Numericlll resulls ohtllinro rrom seH~rallest problel1lsllre gh'el1. It is Shllllil Ihat Ihe finile elel1lenl ml,thod ha" grelll llolenlial for U'ie ill tlo" prohll'I1I".and represenls II p,,"erful new 1001 for Ihe llllalysis of I'iwous flows. ..

L. Carter Wellford

Papers

Analysis of Flow of Viscous Fluids by the Finite-Element Method