A New Co-Located Pressure-Based Discretization Method for the Numerical Solution of Incompressible Navier-Stokes Equations

There are two commonly used numerical strategies for the solution of incompressible Navier-Stokes equations in the context of the finite-volume method. These equations are either discretized using geometric interpolations and solved on staggered grids, or solved on co-located grids using geometric and momentum-based interpolations for convected and convecting velocities, respectively. This article presents an alternative finite-volume method for the solution of incompressible Navier-Stokes equations on co-located grids without resorting to two different interpolation formulas for convected and convecting velocities. The key idea for achieving physical coupling between pressure and velocity fields in the numerical model is the employment of proper closure equations during the discretization. These closure equations are used to convert the finite-volume balance equations to proper computational molecules at nodal points. A number of steady two-dimensional test problems are solved which show the applicability and excellent performance of the proposed method. The method does not have any inherent limitation and is extendable to three-dimensional flows as well.     

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.     

Development of a Compact and Accurate Discretization for Incompressible Navier-Stokes Equations Based on an Equation-Solving Solution Gradient,Part III: Fluid Flow Simulations on Staggered Polygonal Grids

A compact and accuracy discretization of incompressible Navier-Stokes equations on staggered polygonal grids is presented in this article. It is a sequel to our efforts in developing a feasible solution procedure to simulate incompressible flow problems in complicated domains. By taking advantage of the discretization procedure for the convection-diffusion equation described in our previous work, difference counterparts of the Navier-Stokes equations can be obtained on staggered polygonal grids. Additional ingredients of pressure–velocity coupling and boundary conditions for velocity gradients in the solution procedure are also described. Several test problems are solved to illustrate the feasibility of the present formulation. From the numerical results obtained, it is evident that the proposed scheme is a useful tools to simulate incompressible flow field in arbitrary domains.     

THE k-ε MODEL AND COMPUTED SPREADING RATES IN ROUND AND PLANE JETS

Several prior computational works that employ the steady, incompressible boundary layer (BL) equation have reported their inadequacy in predicting measured values of spreading rates in jets. This work employs a model that solves the complete set of transient conservation equations for multidimensional reacting flows and the k- k model without making the assumptions that lead to the steady, incompressible BL form of the equations. This work discusses the dependence of the computed spreading rates on the numerical resolution and on the additional terms that appear in the multidimensional equations, including the k- k model, relative to the BL equations.     

THREE-DIM ENSIONAL DIAGONAL CARTESIAN METHOD FOR INCOMPRESSIBLE FLOWS INVOLVING COMPLEX BOUNDARIES

A diagonal Cartesian method for the three-dimensional simulation of incompressible fluid flows over complex boundaries is presented in this article. The method is derived utilizing the superposition of the finite analytic solutions of a linearized two-dimensional convection-diffusion equation in Cartesian coordinates. The complex boundary is approximated with a structured grid in a series of calculation planes which are perpendicular to the x, y, and z coordinate axis. In the calculation plane, both Cartesian grid lines and diagonal line segments are used. It is observed that this geometry approximation is more accurate than the traditional sawtooth method. Mass conservation on complex boundaries is enforced with an appropriate pressure boundary condition. The method, which utilizes cell-vertex nodes on a staggered grid, uses boundary velocity information to avoid the specification of pressure values on boundaries. An enlarged control-volume method is introduced for the mass conservation and the pressure boundary condition on complex boundaries. The conservation of momentum on complex boundaries is enforced through the use of three-dimensional 19, 15, 11, or 7-point finite analytic elements. The proposed diagonal Cartesian method is verified by the solution of a rotated lid-driven cavity flow. It is shown that this diagonal Cartesian method predicts the fluid flow very well.     

A multidomain multigrid pseudospectral method for incompressible flows

Pseudospectral method has the merit of high accuracy and the defects of simple geometry suitability and low computational efficiency. To remedy the two defects, a multidomain multigrid Chebyshev pseudospectral method is proposed and validated through the numerical solution of two-dimensional incompressible Navier-Stokes equations in the primitive variable formulation. To facilitate the implementation of the multidomain multigrid method, the IP_N-IP_N method is utilized to approximate the velocity and pressure with the same degree of Chebyshev polynomials within each subdomain, and an interface/boundary condensation method is developed to implement the pseudospectral operators of multigrid at the interface/boundary of subdomains. The accuracy and efficiency of the proposed method are first validated by numerical solutions of the lid-driven cavity problem. The numerical results are in good agreement with the benchmark solutions, and the speeding up of multigrid is 4–9 compared against the single grid. Then the capability of the proposed method for even more complex geometries with a close/open boundary is demonstrated by numerical solutions of several typical problems. The proposed method is quite generic and can be extended to the high accuracy and efficiency solution of three-dimensional incompressible/compressible, unsteady/steady fluid flows and heat transfer problems.     

A coupled element-based finite-volume method for the solution of incompressible Navier-Stokes equations

ABSTRACT

This article presents a new element-based finite-volume discretization approach for the solution of incompressible flow problems on co-located grids. The proposed method, called the method of proper closure equations (MPCE), employs a proper set of physically relevant equations to constrain the velocity and pressure at integration points. These equations provide a proper coupling between the nodal values of pressure and velocity. The final algebraic equations are not segregated in this study and are solved in a fully coupled manner. To show the applicability and performance of the method, it is tested on several steady two-dimensional laminar-flow benchmark cases. The results indicate that the method simulates the fluid flow in complex geometries and on nonorthogonal computational grids accurately. Also, it is shown that the method is robust in the sense that it does not require severe underrelaxation even at relatively high Reynolds numbers. In each test case, the required underrelaxation parameter, the number of iterations, and the corresponding CPU time are reported.     

Immersed boundary computations of shear- and buoyancy-driven flows in complex enclosures

Cartesian grids used with the immersed boundary method (IBM) offer an attractive alternative for simulating fluid flows in complex geometries. We present a ghost fluid method for incompressible flows solved with staggered grids. The primary feature is the satisfaction of local mass continuity for ghost pressure cells, rather than extrapolating the pressures from within the flow domain. The method preserves local continuity in each cell and also global continuity. As a result, no explicit mass sources or sinks are needed. We have applied the method to study shear- and buoyancy-driven flows in a number of complex cavities.     

Implementation of boundary conditions in the finite-volume pressure-based method—Part I: Segregated solvers

ABSTRACT

The paper deals with the formulation of a variety of boundary conditions for incompressible and compressible flows in the context of the segregated pressure-based unstructured finite volume method. The focus is on the derivation and the implementation of these boundary conditions and their relation to the various physical boundaries and geometric constraints. While a variety of boundary conditions apply at any of the physical boundaries (inlets, outlets, and walls), geometric constraints define the type of boundary condition to be used. The emphasis is on relating the mathematical derivation of the boundary conditions to the algebraic equations defined at each centroid of the boundary elements and their coefficients. All derived boundary conditions are validated through a set of test cases with comparison of computed results to available numerical and/or experimental data.     

Reevaluation of the Vertex Collocated Mesh for Primitive Variable Computation of Incompressible Fluid Flows by Enforcing Continuity-Based Boundary Conditions

Complementing a previous comparative study of the accuracy of the fundamental mesh structures for primitive variable computations of incompressible fluid flows, this article considers some alternative approaches to the closure of the pressure equations in the boundary nodes of the vertex collocated mesh. In the previous study, these boundary pressures were determined directly by a discretized Poisson equation; in the present article the pressure equations at these nodes are derived from specific continuity equations, obtained by mass balance on the half-cells and by unilateral parabolic approximation of the velocity component normal to the wall. The first approach reduces the accuracy of the vertex collocated mesh to first-order, while the second approach remains second-order-accurate, except in the traditional cavity problem, and provides better results than the Poisson equation approach for rough and moderate refinements. However, in comparison with the other types of mesh, the vertex collocated mesh remains the least accurate for refined meshes.     

An Equation-Solving Solution Gradient Strategy for High-Resolution and Compact Discretization of Hyperbolic Conservation Laws

The present study extends our previous formulation in simulating steady incompressible Navier-Stokes equations to solving unsteady compressible flows. It is designed by taking the solution gradient as an additional computational variable to build a high-resolution and compact discretization. Essential ingredients concerning the derivation procedure are detailed in the frame of general hyperbolic conservation laws. Numerical analyses on modeled problems are performed to reveal its stability criterion and formal accuracy. Several one- and two-dimensional problems are solved numerically, and the computational results are compared with those acquired by existing schemes to demonstrate their relative efficiency. It is found that the present formulation will be a useful tool to simulate general hyperbolic conservation laws.     

A novel immersed boundary/Fourier pseudospectral method for flows with thermal effects

ABSTRACT

A novel immersed boundary method (IBM) for flows with thermal effects is proposed, combining high accuracy and low computational cost, provided by the Fourier pseudospectral method (FPSM), for the possibility of handling complex and nonperiodical geometries using the IBM. With focus on incompressible flow problems modeled by Navier-Stokes, mass, and energy equations, the method of manufactured solutions is used for the numerical verification of Dirichlet boundary conditions imposed via the IBM. Then, the proposed method is applied on two different 2-D cases: (1) energy transfer due to natural convection in a square cavity, and (2) an annulus between horizontal concentric cylinders nonuniformly heated. Good agreement with available data in the literature has been achieved.     

A finite volume method to solve the Navier–Stokes equations for incompressible flows on unstructured meshes

A method to solve the Navier–Stokes equations for incompressible viscous flows and the convection and diffusion of a scalar is proposed in the present paper. This method is based upon a fractional time step scheme and the finite volume method on unstructured meshes. A recently proposed diffusion scheme with interesting theoretical and numerical properties is tested and integrated into the Navier–Stokes solver. Predictions of Poiseuille flows, backward-facing step flows and lid-driven cavity flows are then performed to validate the method. We finally demonstrate the versatility of the method by predicting buoyancy force driven flows of a Boussinesq fluid (natural convection of air in a square cavity with Rayleigh numbers of 10 ³ and 10 ⁶ ).     

Unsteady MHD free convection of a micropolar fluid between two parallel porous vertical walls with convection from the ambient

The present work is concerned with the unsteady free convection flow of an incompressible electrically conducting micropolar fluid, bounded by two parallel infinite porous vertical plates submitted to an external magnetic field and the thermal boundary condition of forced convection. The governing equations are solved using a numerical technique based on the electrical analogy, where only previous spatial discretization is necessary to obtain a stable and convergent solution with very low computational times. To solve the system of algebraic equations with time as continuous function, an electric circuit simulator is used. This method permits the direct visualization of the local and/or integrated transport variables (temperatures and velocities) at any point or section of the medium. Numerical results for temperature, velocity and microrotation are illustrated graphically.     

Variational multiscale methods for premixed combustion based on a progress-variable approach

In this study, novel computational techniques for the numerical simulation of premixed combustion based on a progress-variable formulation are proposed. Two new variational multiscale methods within a finite element framework are developed for the system of mass, momentum and progress-variable equations: a purely residual-based variational multiscale method and an algebraic variational multiscale-multigrid method. The proposed methods are tested for the numerical example case of a flame–vortex interaction using Arrhenius chemical kinetics. This actually laminar reactive flow problem may serve as a model problem for interactions of turbulent flows and (premixed) flames. The results obtained from this test case show that both methods are capable of accurately predicting the features expected during the progression of the flame–vortex interaction. The evolution of both a pocket of unburned gas and a secluded, drop-like structure, which detaches itself and moves upwards, are accurately predicted already for a relatively coarse discretization.     

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     

COMPUTATIONS OF MULTIPHASE FLOWS WITH SURFACE TENSION USING AN ADAPTIVE FINITE ELEMENT METHOD

An adaptive finite element method for solving incompressible steady-state axisymmetric free surface flow problems is presented. While the methodology is applicable to most types of multiphase flows, we are particularly interested in modeling laminar free surface flows such as free and impinging jets, which are usually modeled in a Lagrangian framework. An Eulerian strategy is used to capture the interface. Surface tension is included in the model in order to study its influence on the topology of the free surface. A stabilized finite element discretization is used to help solve the coupled system of partial differential equations. An adaptive methodology helps optimize the accuracy of the computed solution. The proposed adaptive free surface capturing methodology is competitive with interface tracking techniques, while being more flexible. The verification and validation of the methodology completes the article.     

Aerodynamic analysis of HAWTs operating in unsteady conditions

This article presents a numerical method for predicting unsteady aerodynamics of horizontal axis wind turbines (HAWTs). In this method the flow field is described by the unsteady incompressible Navier–Stokes equations. The rotor and tower are idealized respectively as actuator disc and flat plate permeable surfaces on which external normal surficial forces are balanced by fluid pressure discontinuities. The external forces exerted by the rotor and tower on the flow are prescribed according to blade element theory. Dynamic behaviour of the rotor aerodynamic characteristics is simulated using either the Gormont or the Beddoes–Leishman model. The resulting mathematical formulation is solved using a control volume finite element method. The fully implicit scheme is used for time discretization. In general, the proposed method has demonstrated its capability to adequately represent the field data. It has been demonstrated that the accuracy of the predicted results depends primarily on the dynamic stall model as well as on the turbulence model employed. Copyright © 2001 John Wiley & Sons, Ltd.     

A CONTROL-VOLUME FINITE-ELEMENT METHOD (CVFEM) FOR UNSTEADY,INCOMPRESSIBLE, VISCOUS FLUID FLOWS

A control-volume finite-element method (CVFEM), to simulate unsteady, incompressible, and viscous fluid flows, using nine-noded quadrilateral elements, has been developed. The Navier-Stokes equations in primitive variables u-v-p are the mathematical model of the flows. A technique of upwind, known as MAW, Mass-Weighted Interpolation, was extended for the finite element employed in this work. The set of nonlinear partial differential equations was integrated, and after using interpolation functions and time discretization, the algebraic system of equations was solved by using the frontal method of solution. Results obtained for some benchmark problems compared favorably with available results from the literature.     

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.