Implementation of the IDEAL Algorithm on Nonorthogonal Curvilinear Coordinates for the Solution of 3-D Incompressible Fluid Flow and Heat Transfer Problems

Recently an efficient segregated algorithm for incompressible fluid flow and heat transfer problems, called IDEAL (Inner Doubly Iterative Efficient Algorithm for Linked Equations), has been proposed by the present authors. In the algorithm there exist inner doubly iterative processes for the pressure equation at each iteration level, which almost completely overcome the two approximations in the SIMPLE algorithm. Thus the coupling between velocity and pressure is fully guaranteed, greatly enhancing the convergence rate and stability of the solution process. In this article, the IDEAL algorithm is extended to the body-fitted collocated grid systems in 3-D nonorthogonal curvilinear coordinates. The extended IDEAL algorithm adopts two successful methods. One is that the interfacial contravariant velocity is calculated by the modified momentum interpolation method (MMIM); the other is that the interfacial contravariant velocity is improved by solving the pressure equation directly. Finally, three 3-D incompressible fluid flow and heat transfer problems are provided to compare the convergence rate and robustness between IDEAL and three other algorithms (SIMPLEM, SIMPLERM, and SIMPLECM). From the comparison it can be concluded that the IDEAL algorithm is more robust and efficient than the three other algorithms.     

NUMERICAL METHOD FOR THE PREDICTION OF INCOMPRESSIBLE FLOW AND HEAT TRANSFER IN DOMAINS WITH SPECIFIED PRESSURE BOUNDARY CONDITIONS

A variety of engineering applications involve incompressible flows in devices for which boundary pressures are known. The purpose of this article is to present a mathematical formulation and a computational method for the prediction of incompressible flow in domains with specified pressure boundaries. The computational treatment of specified pressure boundaries in complex geometries is presented within the framework of a nonstaggered technique based on curvilinear boundary-fitted grids. The construction of the discretization equations for unknown velocities on specified pressure boundaries and the solution of the discretization equations using the SIMPLE algorithm are discussed. The proposed method is applied for predicting incompressible forced flows in branched ducts and in buoyancy-driven flows. These examples illustrate the utility of the proposed method in predicting incompressible flows with specified boundary pressures encountered in practical applications.     

COMPLETE PRESSURE CORRECTION ALGORITHM FOR SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON A NONSTAGGERED GRID

Abstract

When Navier-Stokes equations for incompressible flow are solved on a nonstaggered grid, the problem of checkerboard prediction of pressure is encountered. So far, this problem has been cured either by evaluating the cell face velocities by the momentum interpolated principle III or by evaluating an effective pressure gradient in the nodal momentum equations [2] In this article it is shown that not only are these practices unnecessary, they can lead to spurious results when the true pressure variation departs considerably from linearity. What is required instead is afresh derivation of the pressure correction equation appropriate for a nonstaggered grid. The pressure correction determined from this equation comprises two components: a mass-conserving component and a smoothing component. The former corresponds to the pressure correction predicted by a staggered grid procedure, whereas the latter simply accounts for the difference between the point value of the pressure and the cell-averaged value of the pressure. The new pressure correction equation facilitates ( in a significant way) computer coding of programs written for three-dimensional geometries employing body-fitted curvilinear coordinate grids.     

A time marching strategy for solving parabolic and elliptic equations with Neumann boundary conditions

Abstract

In the community of computational fluid dynamics, pressure Poisson equation with Neumann boundary condition is usually encountered when solving the incompressible Navier–Stokes equations in a segregated approach such as SIMPLE, PISO, and projection methods. To deal with Neumann boundary conditions more naturally and to retain high order spatial accuracy as well, a sixth-order accurate combined compact difference scheme developed on staggered grids (NSCCD6) is adopted to solve the parabolic and elliptic equations subject to Neumann boundary conditions. The staggered grid system is usually used when solving the incompressible Navier–Stokes equations. By adopting the combined compact difference concept, there is no need to discretize Neumann boundary conditions with one-sided discretization scheme which is of lower accuracy order. The conventional Crank–Nicolson scheme is applied in this study for temporal discretization. For two-dimensional cases, D’yakonov alternating direction implicit scheme is adopted. A newly proposed time step changing strategy is adopted to improve convergence rate when solving the steady state solutions of the parabolic equation. High accuracy order of the currently proposed NSCCD6 scheme for one- and two-dimensional cases are shown in this article.     

A Parallel Multigrid Finite-Volume Solver on a Collocated Grid for Incompressible Navier-Stokes Equations

Multigrid techniques are widely used to accelerate the convergence of iterative solvers. Serial multigrid solvers have been efficiently applied to a broad class of problems, including fluid flows governed by incompressible Navier-Stokes equations. With the recent advances in high-performance computing (HPC), there is an ever-increasing need for using multiple processors to solve computationally demanding problems. Thus, it is imperative that new algorithms be developed to run the multigrid solvers on parallel machines. In this work, we have developed a parallel finite-volume multigrid solver to simulate incompressible viscous flows in a collocated grid. The coarse-grid equations are derived from a pressure-based algorithm (SIMPLE). A domain decomposition technique is applied to parallelize the solver using a Message Passing Interface (MPI) library. The multigrid performance of the parallel solver has been tested on a lid-driven cavity flow. The scalability of the parallel code on both single- and multigrid solvers was tested and the characteristics were analyzed. A high-fidelity benchmark solution for lid-driven cavity flow problem in a 1,024 × 1,024 grid is presented for a range of Reynolds numbers. Parallel multigrid speedup as high as three orders of magnitude is achieved for low-Reynolds-number flows. The optimal multigrid efficiency is validated, i.e., the computational cost is shown to increase proportionally with the problem size.     

AN IMPROVED SIMPLE METHOD ON A COLLOCATED GRID

A pressure-based method characterized by the SIMPLE algorithm is developed on a nonorthogonal collocated grid for solving two-dimensional incompressible fluid flow problems, using a cell-centered finite-volume approximation. The concept of artificial density is combined with the pressure Poisson equation that provokes density perturbations, assisting the transformation between primitive and conservative variables. A nonlinear explicit flux correction is utilized at the cell face in discretizing the continuity equation, which functions effectively in suppressing pressure oscillations. The pressure-correction equation principally consolidates a triplicate-time approach when the Courant number CFL > 1. A rotational matrix, accounting for the flow directionality in the upwinding, is introduced to evaluate the convective flux. The numerical experiments in reference to a few familiar laminar flows demonstrate that the entire contrivance executes a residual smoothing enhancement, facilitating an avoidance of the pressure underrelaxation. Consequently, included benefits are the use of larger Courant numbers, enhanced robustness, and improved overall damping properties of the unfactored pseudo-time integration procedure.     

A CALCULATION PROCEDURE FOR SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON CURVILINEAR NON STAGGERED GRIDS

A recently proposed pressure-correction algorithm for solution of incompressible Navier-Stokes equations on nonstaggered grids introduced the notion of smoothing pressure correction to overcome the problem of checkerboard prediction of pressure (9). The algorithm was derived for equations in Cartesian coordinates. In this article, the algorithm is extended to solution of Navier-Stokes equations in general curvilinear coordinates. By way of application, two cavity flow problems and two internal flow problems are solved. Comparisons with benchmark solutions or experimental data and (or) previous solutions employing staggered grids are made to validate the calculation procedure.     

Improved Formulations of the Discretized Pressure Equation and Boundary Treatments in Co-Located Equal-Order Control-Volume Finite-Element Methods for Incompressible Fluid Flow

Improved formulations of the discretized pressure equation and boundary treatments in co-located, equal-order, control-volume finite-element methods for the prediction of incompressible fluid flow are presented in the context of steady, planar two-dimensional problems. Three-node triangles and polygonal-cross-section control volumes, created by joining element centroids to midpoints of the sides, are used. The proposed improvements maintain the strength of coefficients in the discretized pressure equations, and help in keeping them diagonally dominant. Thus they facilitate convergence of iterative solution methods and the results do not display physically untenable features. Solutions of two test problems are presented and discussed.     

HIGHER-ORDER COMPACT SCHEMES FOR NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS,PART II: APPLICATIONS

Within the framework of the SIMPLE algorithm, a dual-dissipation scheme is proposed on nonorthogonal collocated grids for incompressible fluid flow problems, using a cell-centered finite-volume approximation. The dissipative mechanism employs dynamic limiters to control the amount of dissipation, preserving expediences of greater flexibility and increased accuracy in a way similar to the MUSCL approach. The artificial density concept is combined with the pressure Poisson equation, facilitating an avoidance of pressure underrelaxation. To account for the flow directionality in the upwinding, a rotational matrix is evoked to evaluate the convective flux.     

A Fully Coupled Navier-Stokes Solver for Fluid Flow at All Speeds

This article deals with the formulation and testing of a newly developed, fully coupled, pressure-based algorithm for the solution of fluid flow at all speeds. The new algorithm is an extension into compressible flows of a fully coupled algorithm developed by the authors for laminar incompressible flows. The implicit velocity–pressure–density coupling is resolved by deriving a pressure equation following a procedure similar to a segregated SIMPLE algorithm using the Rhie-Chow interpolation technique. The coefficients of the momentum and continuity equations are assembled into one matrix and solved simultaneously, with their convergence accelerated via an algebraic multigrid method. The performance of the coupled solver is assessed by solving a number of two-dimensional problems in the subsonic, transsonic, supersonic, and hypersonic regimes over several grid systems of increasing sizes. For a desired level of convergence, results for each problem are presented in the form of convergence history plots, tabulated values of the maximum number of required iterations, the total CPU time, and the CPU time per control volume.     

Pressure Correction–Based Iterative Scheme for Navier-Stokes Equations using Nodal Integral Method

A novel numerical scheme is developed for steady-state, 2-D incompressible Navier-Stokes equations using the nodal integral method (NIM). The NIM-based schemes have been used to solve partial differential equations in different areas of physics and are known to have very high accuracy compared to conventional numerical schemes. The scheme is implemented using a SIMPLE (Semi-Implicit Method for Pressure-Linked Equations)-like algorithm for pressure and velocity correction instead of solving the exact pressure Poisson equation. To verify the results, two well-known test problems, the lid-driven cavity and natural convection of air in a square cavity, are chosen. For both cases, results are in good agreement with the benchmark solutions even for quite coarse grids.     

NUMERICAL SOLUTION OF NAVIER-STOKES EQUATIONS WITH NONSTAGGERED GRIDS USING FINITE ANALYTIC METHOD

A new method referred to here as the momentum weighted interpolation method (MWIM) is introduced to solve the incompressible Navier-Stokes equations with nonstaggered grids. This new method is compared to other methods that employ nonstaggered and staggered grids, in terms of computational effort, number of iterations, and accuracy. It is found that MWIM with nonstaggered grids is as accurate as the staggered grid methods, is easier to implement, and is computationally more efficient. The finite analytic method is used to discretize the governing equations of the fluid flow.     

A novel fixed-grid interface-tracking algorithm for rapid solidification of supercooled liquid metal

Abstract

In this study, we present a novel fixed-grid interface-tracking method using finite volume method to simulate multidimensional rapid solidification (RS) of under-cooled pure metal. The discretized advection equation for solid fraction function is solved using the THINC/WLIC method, which is a VOF method. The governing equations for fluid flow are solved numerically using pressure-velocity coupling SIMPLE algorithm in a 2-D model with incompressible Newtonian fluid. The energy equation is modeled using an enthalpy-based formulation. The nonequilibrium solidification kinetics, interface tracking, undercooling, nucleation, heat transfer, and movement of liquid are included in the presented RS model.     

A COMPARATIVE ASSESSMENT WITHIN A MULTIGRID ENVIRONMENT OF SEGREGATED PRESSURE-BASED ALGORITHMS FOR FLUID FLOW AT ALL SPEEDS

This article deals with the evaluation of six segregated high-resolution pressure-based algorithms, which extend the SIMPLE, SIMPLEC, PISO, SIMPLEX, SIMPLEST, and PRIME algorithms, originally developed for incompressible flow, to compressible flow simulations. The algorithms are implemented within a single grid, a prolongation grid, and a full multigrid method and their performance assessed by solving problems in the subsonic, transonic, supersonic, and hypersonic regimes. This study clearly demonstrates that all algorithms are capable of predicting fluid flow at all speeds and qualify as efficient smoothers in multigrid calculations. In terms of CPU efficiency, there is no global and consistent superiority of any algorithm over the others, even though PRIME and SIMPLEST are generally the most expensive for inviscid flow problems. Moreover, these two algorithms are found to be very unstable in most of the cases tested, requiring considerable upwind bleeding (up to 50%) of the high-resolution scheme to promote convergence. The most stable algorithms are SIMPLEC and SIMPLEX. Moreover, the reduction in computational effort associated with the prolongation grid method reveals the importance of initial guess in segregated solvers. The most efficient method is found to be the full multigrid method, which resulted in a convergence acceleration ratio, in comparison with the single grid method, as high as 18.4.     

SOLUTION OF NAVIER-STOKES EQUATIONS ON NONSTAGGERED GRID AT ALL SPEEDS

The present author recently devised a pressure correction algorithm for solution of incompressible Navier-Stokes equations on a nonstaggered grid [6]. This algorithm introduced the notion of smoothing pressure correction to overcome the problem of checkerboard prediction of pressure. In this article, the algorithm is extended to prediction of compressible flows with and without shocks. The predictions show that the algorithm yields results that compare extremely favorably with previous ones [6] obtained using a staggered grid. Accurate shock capturing on coarse grids, however, requires use of total variation diminishing ( TVD) discretization of the covective terms coupled with measures for stabilisation of the iteration process.     

A staggered grid arrangement for solving incompressible flows with hybrid unstructured meshes

This work presents an alternative to the discretization of the Navier–Stokes equations using a finite volume method for hybrid unstructured grids with a staggered grid arrangement of variables. It has developed a numerical scheme, analogous to the element-based finite volume method, for the solution of 2-D incompressible fluid flow problems using several coupling strategies. All velocity components are stored at each face of the elements (pressure control volumes), following the usual procedure of staggering velocity and pressure. With this staggered arrangement, the balance of mass and momentum is satisfied, simultaneously, for the same set of variables, rendering numerical stability when compared to the nonstaggered arrangement.     

Simulation of coherent structures in turbulent boundary layer using Gao–Yong equations of turbulence

The equations of incompressible turbulent flow developed by the Gao–Yong turbulence model have two important features. First, they do not contain any empirical coefficients or wall functions. Second, the series representation of turbulence energy equation reflects multi‐scale structures of the nonlinearity of turbulence, and, therefore, is capable of describing both statistical mean flows and the coherent structures. This paper presents some simulation results of a two‐dimensional turbulent boundary layer with zero pressure gradient based on Gao–Yong equations of turbulence. With a staggered grid arrangement, an incompressible SIMPLE code was used in the simulations. The simulated coherent structures have verified the adaptability of the newly derived equations to real intermittent turbulent flows. The effect of the orders of the energy equation and computational grid scales on the detection of coherent structures is also investigated. © 2004 Wiley Periodicals, Inc. Heat Trans Asian Res, 33(5): 287–298, 2004; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20019     

Avoiding under-relaxations in SIMPLE algorithm

Abstract

The SIMPLE algorithm is devised by interpolating the mass continuity and non-advective momentum equations, provoking apparent simplicity and clarity in the formulation. The SIMPLE variant entitled as the SIMPLE-AC scheme convokes an artificial compressibility (AC) parameter to augment the diagonal dominance of discretized pressure-correction equation. Both methods are characteristically pressure-based, employing a cell-centered finite-volume Δ-formulation on a non-orthogonal collocated grid. A dual-dissipation scheme accompanied by limiting factors, repressing local extrema into the cell-face velocity and pressure, is used to unravel the issue of pressure-velocity decoupling; both SIMPLE and SIMPLE-AC schemes maintain an equivalent scaling (e.g., primary and auxiliary pseudo-time steps remain the same) with the cell-face dissipation and nodal influence coefficients. The phenomenal progress embedded in both contrivances facilitate an avoidance of the pervasive velocity/pressure under-relaxation. However, the SIMPLE-AC algorithm is benefited with using a higher CFL number, enhanced robustness, and convergence compared with the SIMPLE method.     

A Numerical Computation of Heat and Fluid Flow in L-Shaped Curved Channels

A numerical analysis of steady, hydrodynamically and thermally fully developed, incompressible laminar flow with constant physical properties is presented. Two different geometrical cases of L-shaped channels are considered: right-curved and left-curved. The solution of the discretized continuity, momentum, and energy equations was obtained by using an elliptic Fortran program based on the SIMPLE algorithm. Solutions are obtained for Dean number ranges from 4.1 to 210.6 with dimensionless radius of curvature of 100, and Prandtl number of 0.7. The secondary flow streamlines, the isotherms, average Nusselt numbers, and friction factors are presented depending on Dean number and L-shaped channel orientation. As a result, it is observed that the secondary flows resulting from centrifugal forces change the distribution of the velocity and the temperature fields. In addition, it is determined that channel orientation has a profound effect on the flow and temperature fields.     

STABILIZED FLUX-BASED FLOW / THERMAL FINITE ELEMENT REPRESENTATIONS WITH APPLICATIONS TO CONVECTIVELY COOLED STRUCTURES

Stabilized flux-based finite element representations for steady two-dimensional incompressible flow / thermal problems with emphasis on subsequently applying such techniques to convectively cooled structures are described in this article. First, the discretized equations are derived from a mixed formulation using both primary and flux variables in conjunction with the Streamline-Upwind-Petrov-Galerkin and Pressure-Stabilizing-Petrov-Galerkin features that are used to stabilize the solutions. The constitutive equations are then introduced into the discretized representations and the equations are finally solved for the primary variables. Equal-order linear quadrilateral interpolation functions are used for the velocities, pressure, and temperature. Numerical results are presented for a variety of situations, and finally emphasis is placed on applications to convectively cooled structures that are subjected to intense localized heating.