Non-staggered Grid Research Articles

Extension of Preissmann Scheme to Two-Dimensional Flows

A numerical solution to the finite difference of two-dimensional (2D) depth-averaged equations on nonstaggered grid points is proposed in this technical note. Following a locally one-dimensional procedure, the basic equations are split into a pair of one-dimensional equations. Therefore, the solution of a 2D problem is reduced to the solution of a sequence of two one-dimensional problems. The discretization of the split one-dimensional equations is obtained with the use of the original Preissmann operator. Using Fourier’s classic linear analysis, stability, dissipation and dispersion with frictional resistance are investigated for the variations of the Courant number and weighting time factor.

A Compact Momentum Interpolation Procedure for Unsteady Flows and Relaxation

An improved version of the momentum interpolation approach for calculating velocities at the cell faces in nonstaggered grids is proposed. The procedure is developed for unsteady flows and takes into account the inclusion of relaxation. Regarding the integration in time, the first-order Euler and the second-order Euler and Adams-Moulton schemes are analyzed. The proposed procedure results in a compact, easy-to-implement expression which is shown to provide the desirable performance: Spurious oscillations of pressure are avoided; converged, steady solutions are independent from relaxation coefficient and time-step size; and the accuracy of the discretization is not negatively affected.

A quasi-implicit time advancing scheme for unsteady incompressible flow. Part I: Validation

A quasi-implicit fractional-step method is presented for computing unsteady incompressible flow on unstructured grids. A non-staggered grid system is employed rather than a staggered grid system because of the simplicity and ease of extension to three-dimensional analysis. In this study, the momentum interpolation method, developed by Rhie and Chow [C.M. Rhie, W.L. Chow, Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA J. 21 (1983) 1525–1532] and further extended by Zang et al. [Y. Zang, R.L. Street, J.R. Koseff, A non-staggered grid, fractional-step method for time-dependent incompressible Navier–Stokes equations in curvilinear coordinates, J. Comput. Phys. 114 (1994) 18–33], is applied to problems on unstructured grids to resolve the pressure oscillation problem occurring in a non-staggered grid system. An implicit time advancing scheme is used in order to remove the time step restriction and to reduce the required CPU time for problems with complex geometries. The nonlinear equations resulting from this fully implicit scheme are linearized without deteriorating the overall time accuracy. The system matrices are solved using the CG family method, known with P-BiCGSTAB, for momentum equation and P-CG for pressure Poisson equation. The present numerical method is applied to solve four benchmark problems and the results show good agreement with previous experimental and numerical results.

Heat transfer due to double laminar slot jets impingement onto an isothermal wall within one side closed long duct

In this study, heat transfer due to double impinging vertical slot jets onto an isothermal wall was investigated numerically for laminar flow regime. Navier–Stokes and energy equations were discretized with a finite volume procedure on a non-staggered grid arrangement using SIMPLEM (SIMPLE-Modified) algorithm. The effect of the jet Reynolds number, the jet-isothermal bottom wall spacing, and the distance between two jets on heat transfer and flow field was examined. Air was chosen as the working fluid (Pr=0.71). It is found that multi-cellular flow is formed in the impingement region due to interaction between two jets and entrainment effects in the duct. The mean Nusselt number increases almost linearly with increasing of Reynolds number at isothermal surface. When Reynolds number of the first jet is higher than second one the heat transfer is enhanced significantly.

IMPROVEMENT OF SIMPLER ALGORITHM FOR INCOMPRESSIBLE FLOW ON STAGGERED GRID SYSTEM

In this paper, the updating of the intermediate velocity in the correction stage in SIMPLE-like algorithms on staggered grid is discussed in detail, based on the discussion CLEARER algorithm that is extended from the non-staggered grid to the staggered grid. The performance of SIMPLER and CLEARER algorithms are compared using two numerical examples with reliable solutions. The results show that CLEARER can predict the numerical results as accurately as SIMPLER, and it can also enhance the convergence rate greatly under low under-relaxation factors. In some cases CLEARER algorithm can only need 27% of iteration number required by SIMPLER to reach the same convergence criterion.

Numerical study of the unsteady flow of non-Newtonian fluid through differently shaped arterial stenoses

The problem of non-Newtonian and nonlinear pulsatile flow through an irregularly stenosed arterial segment is solved numerically where the non-Newtonian rheology of the flowing blood is characterized by the generalized Power-law model where both the shear-thinning and shear-thickening models of the streaming blood are taken into account. The combined influence of an asymmetric shape and surface irregularities (roughness) of the constriction has been explored in a study of blood flow with 48% areal occlusion. The vascular wall deformability is taken to be anisotropic, linear, viscoelastic, incompressible circular cylindrical membrane shell. The effect of the surrounding connective tissues on the motion of the arterial wall is also paid due attention. Results are obtained for a smooth stenosis model and also for a stenosis model represented by the cosine curve. The present analytical treatment has the potential to calculate the rate of flow, the resistive impedance and the wall shear stress without excessive computational complexity by exploiting the appropriate physiologically realistic prescribed conditions in nonuniform nonstaggered grids, and to estimate the effects of surface roughness as well as asymmetry of stenosis shape for both shear-thinning and shear-thickening models of Power-law fluid, representing the streaming blood through graphical representations in order to validate the applicability of the present improved mathematical model.

A divergence-free-condition compensated method for incompressible Navier–Stokes equations

The present study aims to develop a new formulation to effectively calculate the incompressible Navier–Stokes solutions in non-staggered grids. The distinguished feature of the proposed method, which avoids directly solving the divergence-free equation, is to add a rigorously derived source term to the momentum equation to ensure satisfaction of the fluid incompressibility. For the sake of numerical accuracy, dispersion-relation-preserving upwind scheme developed within the two-dimensional context was employed to approximate the convection terms. The validity of the proposed mass-preserving Navier–Stokes method is justified by solving two benchmark problems at high Reynolds and Rayleigh numbers. Based on the simulated Navier–Stokes solutions, the proposed formulation is shown to outperform the conventional segregated method in terms of the reduction of CPU time.

Interpolating Wavelets and Adaptive Finite Difference Schemes for Solving Maxwell's Equations: The Effects of Gridding

In this paper, we discuss the use of the sparse point representation (SPR) methodology for adaptive finite-difference simulations in computational electromagnetics. The principle of the SPR method is to represent the solution only through those point values indicated by the significant wavelet coefficients, which are used as local regularity indicators. Recently, two kinds of SPR schemes have been considered for solving Maxwell's equations: 1) staggered grids in the time-space domain are used for the discretization of the magnetic and electrical fields, as in the finite-differences time-domain (FDTD) scheme and 2) nonstaggered grids are used in combination with Runge-Kutta ODE solvers. In both cases, 1-D simulations of the SPR method leads to sparse grids that adapt in space to the local smoothness of the fields, and, at the same time, track the evolution of the fields over time with substantial gain in memory and computational speed. However, in the latter case, we found spurious oscillations in the simulations. Therefore, before extending the implementation of the SPR method to higher dimensions, we wanted to evaluate which of these two SPR strategies is more convenient. After a careful theoretical analysis of stability and numerical dispersion comparing the schemes in staggered and nonstaggered grids, we conclude that schemes for staggered grids seem to be preferable from the dispersion viewpoint, especially for low-order schemes and coarse grids. However, by adapting the grid density and increasing the order, SPR schemes for nonstaggered grids also show good performance. In our experiments, no spurious oscillations were detected. We observed that, for a given accuracy, the adaptive scheme on a nonstaggered grid requires less computational effort. Since the use of nonstaggered grids increases the stability range and facilitates the implementation of adaptive strategies, we believe that the SPR method in nonstaggered grids has a very good potential for computational electromagnetics

Numerical investigation of heat transfer and fluid flow characteristics inside a wavy channel

The periodically fully developed laminar heat transfer and fluid flow characteristics inside a two-dimensional wavy channel in a compact heat exchanger have been numerically investigated. Calculations were performed for Prandtl number 0.7, and Reynolds number ranging from 100 to 1,100 on non-orthogonal non-staggered grid systems, based on SIMPLER algorithm in the curvilinear body-fitted coordinates. Effects of wavy heights, lengths, wavy pitches and channel widths on fluid flow and heat transfer were studied. The results show that overall Nusselt numbers and friction factors increase with the increase of Reynolds numbers. According to the local Nusselt number distribution along channel wall, the heat transfer may be greatly enhanced due to the wavy characteristics. In the geometries parameters considered, friction factors and overall Nusselt number always increase with the increase of wavy heights or channel widths, and with the decrease of wavy lengths or wavy pitches. Especially the overall Nusselt number significantly increase with the increase of wavy heights or channel widths, where the flow may become into transition regime with a penalty of strongly increasing in pressure drop.

Fractional step artificial compressibility schemes for the unsteady incompressible Navier–Stokes equations

We propose a novel, time-accurate approach for solving the unsteady, three-dimensional, incompressible Navier–Stokes equations on non-staggered grids. The approach modifies the standard, dual-time stepping artificial-compressibility (AC) iteration scheme by incorporating ideas from pressure-based, fractional-step (FS) formulations. The resulting hybrid fractional-step/artificial-compressibility (FSAC) method is second-order accurate and advances the Navier–Stokes equations in time via a two-step procedure. In the first step, which is identical to the convection–diffusion step in pressure-based FS methods, a preliminary velocity field is calculated, which is not divergence-free. In the second step, however, instead of deriving a pressure-Poisson equation as in FS methods, the projection of the velocity field into the solenoidal vector space is implemented using a dual-time stepping AC formulation. Unlike the standard dual-time stepping AC formulations, where the dual-time iterations are carried out with the entire non-linear system, in the FSAC scheme the convective and viscous terms are computed only once or twice per physical time step. Numerical experiments show that the proposed method provides second-order accurate solutions and requires considerably less CPU time than the widely used standard AC formulation. To demonstrate its ability to compute complicate problems, the method is also applied to a flow past cylinder with endplates.

A time-splitting method on a nonstaggered grid in curvilinear coordinates for implicit simulation of non-hydrostatic free-surface flows

The numerical solution of flows with a freely moving boundary is of great importance in practical application such as ship hydrodynamics. Details are given of the development of a two-dimensional vertical numerical model for simulating unsteady and steady free-surface flows on a nonstaggered grid in curvilinear coordinates, using a non-hydrostatic pressure distribution. In this model, Reynolds equation and the kinematic free-surface boundary condition are solved simultaneously, so that the water-surface elevation can be integrated into the solution and solved for, together with the velocity and pressure field. In computational space, the Cartesian velocity components and the pressure are defined at the center of a control volume, while the volume fluxes are defined at the midpoint on their corresponding cell faces. Detailed numerical results are presented for the wave generation above an obstacle and resonant motion standing wave. The results show that the numerical algorithm described is able to produce accurate predictions and is also easy to apply.Key words: free-surface simulation, nonstaggered grid, time-splitting method, unsteady flow, turbulent flow.

An Improved Numerical Scheme for the SIMPLER Method on NonOrthogonal Curvilinear Coordinates: SIMPLERM

In this article, an improved numerical algorithm named SIMPLERM is proposed for incompressible fluid flow computations on the nonstaggered and nonorthogonal curvilinear grid system. In the proposed algorithm, the contravariant velocities are chosen as the cell face velocities and the Cartesian components as the primary variables. The velocity under-relaxation factor is incorporated into the momentum interpolation, and special treatment is adopted to avoid the underrelaxation factor dependence of the velocity solution. In addition, a 1 − δ pressure difference is introduced into the interfacial contravariant velocity determination. Compared with the existing implementation methods of the SIMPLE family on non-staggered and nonorthogonal grids, the SIMPLERM algorithm can guarantee the coupling between velocity and pressure, underrelaxation independence of the solution, and satisfaction of the conservation law, while still possessing sufficient robustness.

Diagonal Cartesian Method for the Numerical Simulation of Flow and Suspended Sediment Transport over Complex Boundaries

There is increasing demand for simulation tools of flow and suspended sediment transport over complex boundaries in hydraulic engineering. The diagonal Cartesian method, which approximates complex boundaries using both Cartesian grid lines and diagonal lines segments, is presented in the paper to simulate the complex boundaries of two-dimensional shallow-water turbulence equations and nonequilibrium suspended sediment transport equation. The method, which utilizes cell-centered nodes on a nonstaggered grid, uses boundary velocity information at the wall boundary to avoid the specification of water level. An enlarged finite-difference method is introduced for momentum and suspended sediment equations on the complex boundary. This paper describes an application of the diagonal Cartesian method to calculate the tidal current and suspended sediment concentration of Quanzhou Bay in the Fujian province of China. The results show that the method predicts the flow and suspended sediment concentration well, and the calculations agree well with the measurement.

Development of a continuity-preserving segregated method for incompressible Navier–Stokes equations

The present study aims to develop a new method for obtaining the non-oscillatory incompressible Navier–Stokes solutions on the non-staggered grids. Within the segregated grid framework, the divergence-free equation is chosen to replace one of the momentum equations so as to preserve the fluid incompressibility. For the sake of numerical accuracy, the five-point stencil convection–diffusion–reaction scheme is developed to obtain the nodally exact solution for this chosen momentum equation. The validity of the proposed mass-preserving Navier–Stokes method is justified by solving the three problems which are amenable to analytical solutions. The simulated solution quality is shown to outperform that of the conventional segregated approach, besides gaining a very high spatial rate of convergence.

On the Numerical Solution of the Incompressible Navier-Stokes Equations in Primitive Variables Using Grid Generation Techniques

In this paper presents a new model procedure for the solution of the incompressible Navier-Stokes equations in primitive variables, using grid generation techniques. The time dependent momentum equations are solved explicitly for the velocity field using the explicit marching procedure, the continuity equation is implied at each grid point in the solution of pressure equation, while the SOR method is used for the Neumann problem for pressure. Results obtained for the model problem of driven flow in a square cavity demonstrate that the method yields accurate solutions. The results of the numerical computations in a driven cavity, which are presented for the history of the residues at several Reynolds numbers Re=100,1000,4000 and 5000 all the computed results are obtained without any artificial dissipation. This feature of the present procedure demonstrates its excellent convergence and stability characteristics. Numerically results obtained for the steady state static pressure in the driven cavity are presented for the first time at Re=4000 and 5000 using non-staggered grid.

CFL-Violating Numerical Schemes for a Two-Fluid Model

In this paper we propose a class of linearly implicit numerical schemes for a two-phase flow model, allowing for violation of the CFL-criterion for all waves. Based on the Weakly Implixit Mixture Flux (WIMF) approach [SIAM J. Sci. Comput., 26 (2005), pp. 1449---1484], we here develop an extension denoted as Strongly Implicit Mixture Flux (SIMF). Whereas the WIMF schemes are restricted by a weak CFL condition which relates time steps to the fluid velocity, the SIMF schemes are able to break the CFL conditions corresponding to both the sonic and advective velocities. The schemes possess some desirable features compared to current industrial pressure-based codes. They allow for sequential updating of the momentum and mass variables on a nonstaggered grid by solving two sparse linear systems. The schemes are conservative in all convective fluxes and consistency between the mass variables and pressure is formally maintained. Numerical experiments are presented to shed light on the inherent differences between the WIMF and SIMF families of schemes. In particular, we demonstrate that the WIMF scheme is able to give an exact resolution of a moving contact discontinuity. The SIMF schemes do not possess the "exact resolution" property of WIMF, however, the ability to take larger time steps can be exploited so that more efficient calculations can be made when accurate resolution of sharp fronts is not essential, e.g. to calculate steady state solutions.

A mass-conserving vorticity–velocity formulation with application to nonreacting and reacting flows

In a commonly implemented version of the vorticity–velocity formulation, the governing equations for the fluid dynamics are expressed as two Poisson-like velocity equations together with the vorticity transport equation. However, for some flows with large vorticity gradients, spurious mass loss or gain can be observed. In order to conserve mass, a modification to the vorticity–velocity formulation is proposed, involving the substitution of the kinematic definition of vorticity in certain terms of the fluid-dynamic equations. This modified formulation results in a broader computational stencil when the equations are in a second-order-accurate discretized form, and a stronger coupling between the predicted vorticity and the curl of the predicted velocity field. The resulting system of elliptic equations – which includes the energy and species transport equations for the reacting flow case – is discretized with finite differences on a nonstaggered grid and is then solved using Newton’s method. Both the unmodified and modified vorticity–velocity formulations are applied to two problems with high vorticity gradients: (1) incompressible, axisymmetric fluid flow through a suddenly expanding pipe and (2) a confined, axisymmetric laminar flame with detailed chemistry and multicomponent transport, generated on a burner whose inner tube extends above the burner surface. The modified formulation effectively eliminates the spurious mass loss in the two test cases to within an acceptable tolerance. The two cases demonstrate the broader range of applicability of the modified formulation, as compared with the unmodified formulation.

Three-dimensional numerical simulation of viscoelastic flow through an abrupt contraction geometry

AbstractA numerical scheme for simulating three-dimensional viscoelastic flow was developed. The three-dimensional finite volume method (FVM) based on a non-staggered grid and the semi-implicit method for pressure-linked equations (SIMPLE) were adopted to solve the continuity and momentum equations. As we used a non-staggered grid, the momentum interpolation method (MIM) was employed to avoid checkerboard type pressure fields. Viscoelastic properties of the fluid were described by the Phan-Thien and Tanner (PTT) model, which was treated by the compressive interface capturing scheme for arbitrary meshes (CICSAM). The elastic viscous split stress scheme (EVSS) was used to decouple the velocity and the stress fields. Algebraic equations obtained by the above schemes were handled by an iterative solver and a multi-grid method was applied to accelerate convergence. In order to verify the scheme developed in this study, Newtonian flow in a rectangular duct was studied and the resulting velocity fields were compared with the analytical flow fields. Then viscoelastic flow in 4:1 contraction geometry, one of the most frequently used benchmarking problems, was predicted by applying the fully three-dimensional finite volume method.

Development of a Dispersion Relation-Preserving Upwinding Scheme for Incompressible Navier–Stokes Equations on NonStaggered Grids

ABSTRACT In this article a scheme which preserves the dispersion relation for convective terms is proposed for solving the two-dimensional incompressible Navier–Stokes equations on nonstaggered grids. For the sake of computational efficiency, the splitting methods of Adams-Bashforth and Adams-Moulton are employed in the predictor and corrector steps, respectively, to render second-order temporal accuracy. For the sake of convective stability and dispersive accuracy, the linearized convective terms present in the predictor and corrector steps at different time steps are approximated by a dispersion relation-preserving (DRP) scheme. The DRP upwinding scheme developed within the 13-point stencil framework is rigorously studied by virtue of dispersion and Fourier stability analyses. To validate the proposed method, we investigate several problems that are amenable to exact solutions. Results with good rates of convergence are obtained for both scalar and Navier–Stokes problems.

An improved method for calculating flow past flapping and hovering airfoils

A method is reported here for calculating unsteady aerodynamics of hovering and flapping airfoil for two-dimensional flow via the following improved methodologies: (a) a correct formulation of the problem using stream function (ψ) and vorticity (ω) as dependent variables; (b) calculating loads and moment by a new method to solve the governing pressure Poisson equation (PPE) in a truncated part of the computational domain on a nonstaggered grid; (c) accurate solution using high accuracy compact difference scheme for the vorticity transport equation (VTE) and (d) accelerating the computations by using a high-order filter after each time step of integration. These have been used to solve Navier–Stokes equation for flow past flapping and hovering NACA 0014 and 0015 airfoils at typical Reynolds numbers relevant to the study of unsteady aerodynamics of micro air vehicle (MAV) and insect/bird flight.