Gerry E. Schneider

A MODIFIED STRONGLY IMPLICIT PROCEDURE FOR THE NUMERICAL SOLUTION OF FIELD PROBLEMS

A modified strongly implicit procedure for solving the system of algebraic equations that arise in the finite-difference or finite-analytic description of field problems is presented. The method is derived for a nine-point difference scheme and can readily be applied to the more conventional five-point scheme simply through the use of the five-point scheme coefficients. The method is demonstrated by application to several examples and a comparison is made between the performance of the modified procedure and that of the strongly implicit procedure, the alternating direction implicit method, and successive over-relaxation. In all cases examined the modified strongly implicit procedure offers superior results when the number of iterations required for convergence or the computational cost required for convergence is used as the measure of performance. The method is also less sensitive to control volume aspect ratio, relaxation parameters, and mesh subdivision than other available procedures. Savings in comp...

NUMERICAL SOLUTION OF PROBLEMS IN INCOMPRESSIBLE FLUID FLOW: TREATMENT OF THE VELOCITY-PRESSURE COUPLING

Coupling between the momentum and mass conservation equations for “ incompressible” flows is often the major cause of the slow convergence of iterative solution techniques. Several methods of handling this coupling, some of which are novel, are examined, and results of their application to a test problem are compared. The application is made in a manner that completely isolates the effect of the coupling and leads to a clearer understanding of how the methods perform. Several recommendations are made for potential users.

A SKEWED, POSITIVE INFLUENCE COEFFICIENT UPWINDING PROCEDURE FOR CONTROL-VOLUME-BASED FINITE-ELEMENT CONVECTION-DIFFUSION COMPUTATION

A skewed upwinding procedure is presented for application to the control-volume-based finite-element computation of convective-diffusive transport problems. The method is based on the application of sound physical arguments and further introduces a novel procedure for consideration of convecting flows that vary strongly in both magnitude and direction. Through its basis of development, the procedure inherently precludes the possibility of developing nonphysical spatial oscillations within the solution domain. The procedure is demonstrated by application to two test problems for which its performance has proven to be excellent. The method possesses relatively low levels of false diffusion, is relatively insensitive to grid orientation, demonstrates symmetric characteristics about the centerline of a step-change convective transport, and produces solutions completely free from undesirable spatial oscillations. This latter attribute, in conjunction with its very modest false diffusion levels, renders the pro...

FINITE-ELEMENT SOLUTION PROCEDURES FOR SOLVING THE INCOMPRESSIBLE, NAVIER-STOKES EQUATIONS USING EQUAL ORDER VARIABLE INTERPOLATION

The conventional finite-element formulation of the equations of motion (written in pressure-velocity variables) requires that the order of interpolation for pressure be one less than that used for the velocity components. This constraint is inconvenient and can be argued to be physically inconsistent when inertial effects are dominant. The origins of the constraint are discussed and three new finite-element formulations are advanced that permit equal order representation of pressure and velocity. Of these, the velocity correction scheme, similar to that commonly used in finite-difference procedures, offers superior performance for the examples examined in this paper.

Momentum Variable Procedure for Solving Compressible and Incompressible Flows

Navier-Stokes equations are solved for both compressible and incompressible flows using momentum component variables instead of the usual velocity variables as the dependent variables. The numerical procedure is developed in a control-volume-based, finite element context. The procedure is determined in a pressure-based algorithm rather than the density-based algorithms, which compressible methods normally use. The proper selection of the connections between the variables on control volume surfaces and the main nodal values allow the use of a collocated grid arrangement. The compressible and incompressible results of this algorithm are investigated by testing a number of test cases including the driven cavity, an entrance region flow, and a converging-diverging nozzle flow. The results indicate that the momentum component procedure is quite successful for solving compressible and incompressible flows within a single algorithm

Analogy-Based Method for Solving Compressible and Incompressible Flows

The different natures of compressible and incompressible governing equations of e uid e ow generally classify the solution methods into two main categories of compressible and incompressible methods. The main purpose of this paper is to introduce an analogy that extends incompressible methods to solve compressible e ows using the analogy of e ow equations. In this analogy, the selected momentum component variables play a signie cant role in transferring the individual characteristics of the two formulations to a common basis. To develop the methodology, a control-volume-based e nite element approach is used to solve the governing differential equations for a collocated grid distribution. The dee nition of two types of mass e ux components at the surfaces of the control volume removes the possibility of velocity ‐ pressure decoupling in the Euler limit. The analogy-based method is demonstrated by testing a number of selected cases. A highly recirculating problem, entrance e ow, and the nozzle e ow are among those for which results are presented here. These results demonstrate excellent performance of the analogy and the resulting methodology.

A COUPLED STRONGLY IMPLICIT PROCEDURE FOR VELOCITY AND PRESSURE COMPUTATION IN FLUID FLOW PROBLEMS

A coupled strongly implicit procedure (CSIP) is presented for application in solving the algebraic equation system that results from discrete modeling of incompressible fluid flow problems. The proposed procedure uses a derived pressure equation, representing conservation of mass, that is obtained through the substitution of discrete momentum equations into the discrete mass conservation equation. If a conservative formulation is used, the conservation property is not altered by the use of this pressure equation. The characteristics of the new procedure are examined through application to three test problems having significant inertial influences, in addition to the diffusion of momentum. It is observed that the proposed procedure is robust and stable over a wide range of its parameters. Cost comparisons indicate that while not optimized from a coding viewpoint, the proposed procedure is marginally more expensive than a fully optimized SIMPLE procedure. However, the significant benefits available with the...

APPLICATION OF AN ALL-SPEED FLOW ALGORITHM TO HEAT TRANSFER PROBLEMS

A fully implicit control-volume-based finite element method that solves flow at all speeds is extended and applied to fluid flow problems involving heat transfer. The method is verified by testing on both free and forced convection problems. Two heat transfer benchmark problems are chosen and solved for different Rayleigh or Reynolds numbers and include mesh refinement studies. The formulation permits solving the problems as either compressible or incompressible flows. The results of the two formulations are compared and discussed with respect to each other and those of the benchmark solutions.

COMPARISON OF PRESSURE-BASED VELOCITY AND MOMENTUM PROCEDURES FOR SHOCK TUBE PROBLEM

In this work, a one-dimensional investigation is performed to compare the performance of velocity-based and momentum-based procedures. Both procedures are formulated based on a control-volume approach with pressure as a dependent variable. The related integration point operators are derived based on incorporating the correct physical influence of the flow and other relevant couplings. The formulations are free from employing any explicit artificial viscosity and damping mechanism. They are applied to highly compressible flow by testing the shock tube problem in order to investigate two aspects of linearization and constant mass flux advantage within the developed procedures. The results show that the proposed momentum-based procedure (MBP) is more stable and accurate than the velocity-based procedure (VBP) without damping.

Numerical study of the flow behavior in the uniform velocity entry flow problem

A numerical investigation is presented to study the behavior of laminar flow in the developing zone of a channel from very low to high Reynolds numbers. In this regard, the full Navier-Stokes equations are solved using a control-volume-based finite element method with dual definitions for the momentum components at integration points in order to suppress the decoupling problem. The investigation shows that there is a peculiar overshoot behavior in the region whose limiting solution requires substantial mesh refinement. This behavior begins in the vicinity of the wall that affects wall shear stress calculations and, consequently, pressure drop computations. The study shows that the domain solution approaches limiting values with appropriate mesh refinement. The solutions for low-Reynolds-number flows also approach limiting solutions as the Reynolds number is decreased.