International Journal for Numerical Methods in Fluids

WU, JIANKANG

doi: 10.1002/(SICI)1097-0363(19970315)24:5<423::AID-FLD503>3.0.CO;2-#pmid: N/A

The wave equation model, originally developed to solve the advection–diffusion equation, is extended to the multidimensional transport equation in which the advection velocities vary in space and time. The size of the advection term with respect to the diffusion term is arbitrary. An operator‐splitting method is adopted to solve the transport equation. The advection and diffusion equations are solved separate ly at each time step. During the advection phase the advection equation is solved using the wave equation model. Consistency of the first‐order advection equation and the second‐order wave equation is established. A finite element method with mass lumping is employed to calculate the three‐dimensional advection of both a Gaussian cylinder and sphere in both translational and rotational flow fields. The numerical solutions are accurate in comparison with the exact solutions. The numerical results indicate that (i) the wave equation model introduces minimal numerical oscillation, (ii) mass lumping reduces the computational costs and does not significantly degrade the numerical solutions and (iii) the solution accuracy is relatively independent of the Courant number provided that a stability constraint is satisfied. © 1997 by John Wiley & Sons, Ltd.

BACA, R. G.; CHUNG, J. N.; MULLA, D. J.

doi: 10.1002/(SICI)1097-0363(19970315)24:5<441::AID-FLD501>3.0.CO;2-9pmid: N/A

A new computational method is developed for numerical solution of the Richards equation for flow in variably saturated porous media. The new method, referred to as the mixed transform finite element method, employs the mixed formulation of the Richards equation but expressed in terms of a partitioned transform. An iterative finite element algorithm is derived using a Newton–Galerkin weak statement. Specific advantages of the new method are demonstrated with applications to a set of one— dimensional test problems. Comparisons with the modified Picard method show that the new method produces more robust solutions for a broad range of soil– moisture regimes, including flow in desiccated soils, in heterogeneous media and in layered soils with formation of perched water zones. In addition, the mixed transform finite element method is shown to converge faster than the modified Picard method in a number of cases and to accurately represent pressure head and moisture content profiles with very steep fronts. © 1997 by John Wiley & Sons, Ltd.

JIANG, HONG; FORSYTH, PETER A.

doi: 10.1002/(SICI)1097-0363(19970315)24:5<457::AID-FLD504>3.0.CO;2-Ipmid: N/A

In this paper, numerical methods for solving the transonic full potential equation are developed. The governing equation is discretized by a flux‐biasing finite volume method. The resulting non‐linear algebraic system is solved by using a continuation method with full Newton iteration. The continuation method is based on solving a highly ‘upstream‐weighted’ discretization and then gradually reducing the upstream weighting. A general PCG‐like sparse matrix iterative solver is used to solve the Jacobians at each non‐linear step. Various types of incomplete LU (ILU) preconditioners and ordering techniques are compared. Numerical results are presented to demonstrate that these methods are efficient and robust for solving the transonic potential equation in the workstation computing environment. © 1997 by John Wiley & Sons, Ltd.

GERVAIS, J. J.; LEMELIN, D.; PIERRE, R.

doi: 10.1002/(SICI)1097-0363(19970315)24:5<477::AID-FLD500>3.0.CO;2-Spmid: N/A

We present results of a stability analysis of the lid‐driven cavity flow based on classical C0 finite element discretizations of the Navier–Stokes system. Using arc length continuation and subspace iteration to compute the eigenvalues of the tangent operator, we study the dependence of the bifurcation diagram and of the spectrum on the chosen discretization. © 1997 by John Wiley & Sons, Ltd.

GUPTA, MAHESH; HIEBER, C. A.; WANG, K. K.

doi: 10.1002/(SICI)1097-0363(19970315)24:5<493::AID-FLD502>3.0.CO;2-Wpmid: N/A

A simulation of planar 2D flow of a viscoelastic fluid employing the Leonov constitutive equation has been presented. Triangular finite elements with lower‐order interpolations have been employed for velocity and pressure as well as the extra stress tensor arising from the constitutive equation. A generalized Lesaint–Raviart method has been used for an upwind discretization of the material derivative of the extra stress tensor in the constitutive equation. The upwind scheme has been further strengthened in our code by also introducing a non‐consistent streamline upwind Petrov–Galerkin method to modify the weighting function of the material derivative term in the variational form of the constitutive equation. A variational equation for configurational incompressibility of the Leonov model has also been satisfied explicitly. The corresponding software has been used to simulate planar 2D entrance flow for a 4:1 abrupt contraction up to a Deborah number of 670 (Weissenberg number of 6·71) for a rubber compound using a three‐mode Leonov model. The predicted entrance loss is found to be in good agreement with experimental results from the literature. Corresponding comparisons for a commercial‐grade polystyrene, however, indicate that the predicted entrance loss is low by a factor of about four, indicating a need for further investigation. © 1997 by John Wiley & Sons, Ltd.

EMVIN, PETER; DAVIDSON, LARS

doi: 10.1002/(SICI)1097-0363(19970315)24:5<519::AID-FLD530>3.0.CO;2-Mpmid: N/A

This paper presents a local mesh refinement procedure based on a discretization over internal interfaces where the averaging is performed on the coarse side. It is implemented in a multigrid environment but can optionally be used without it. The discretization for the convective terms in the velocity and the temperature equation is the QUICK scheme, while the HYBRID‐UPWIND scheme is used in the turbulence equations. The turbulence model used is a two‐layer k–ϵ model. We have applied this formulation on a backward‐facing step at Re=800 and on a three‐dimensional turbulent ventilated enclosure, where we have resolved a geometrically complex inlet consisting of 84 nozzles. In both cases the concept of local mesh refinements was found to be an efficient and accurate solution strategy. © 1997 by John Wiley & Sons, Ltd.