Non-staggered Grid Research Articles

Smoothing analysis of two-color distributive relaxation for solving 2D Stokes flow by multigrid method

Smoothing properties of two-color distributive relaxation for solving a two-dimensional (2D) Stokes flow by multigrid method are theoretically investigated by using the local Fourier analysis (LFA) method. The governing equation of the 2D Stokes flow in consideration is discretized with the non-staggered grid and an added pressure stabilization term with stabilized parameters to be determined is introduced into the discretization system in order to enhance the smoothing effectiveness in the analysis. So, an important problem caused by the added pressure stabilization term is how to determine a suitable zone of parameters in the added term. To that end, theoretically, a two-color distributive relaxation, developed on the two-color Jacobi point relaxation, is established for the 2D Stokes flow. Firstly, a mathematical constitution based on the Fourier modes with various frequency components is constructed as a base of the two-color smoothing analysis, in which the related Fourier representation is presented by the form of two-color Jacobi point relaxation. Then, an optimal one-stage relaxation parameter and related smoothing factor for the two-color distributive relaxation are applied to the discretization system, and an analytical expression of the parameter zone on the added pressure stabilization term is established by LFA. The obtained analytical results show that numerical schemes for solving 2D Stokes flow by multigrid method on the two-color distributive relaxation have a specific convergence zone on the parameters of the added pressure stabilization term, and the property of convergence is independent of mesh size, but depends on the parameters of the pressure stabilization term.

An alternative full multigrid SIMPLEC approach for the incompressible Navier–Stokes equations

An alternative approach to solve the steady-state incompressible Navier–Stokes equations using the multigrid (MG) method is presented. The mathematical model is discretized using the finite volume method with second-order approximation schemes in a uniform collocated (nonstaggered) grid. MG is employed through a full approximation scheme-full MG algorithm based on V-cycles. Pressure-velocity coupling is ensured by means of a developed modified SIMPLEC algorithm which uses independent V-cycles for relaxing the pressure-correction and momentum equations. The coarser grids are used only internally in these cycles. All other original SIMPLEC steps can be performed only on the finest grid of the current full MG level. The model problem of the lid-driven flow in the unitary square cavity is used for the tests of the numerical model. Computational performance is measured through error and residual decays and execution times. Good performances were obtained for a wide range of Reynolds numbers, with speedups of orders as high as Linear relationships between execution times and grid sizes were observed for low and high Re values ( and 7,500). For intermediate Re values ( and 1,000), the linear trend was observed from more refined grids (512 onwards).

Summation-by-parts finite-difference shallow water model on the cubed-sphere grid. Part I: Non-staggered grid

Summation-by-Parts Finite Differences (SBP-FD) is an approach for approximation of differential operators satisfying a discrete analogue of integration by parts analytic property. SBP-FD allows one to build provably stable high-order spatial approximations. SBP-FD methods are widely used for approximation of partial differential equations in multiblock domains with logically-rectangular curvilinear mesh inside. The gnomonic cubed-sphere grid is an example of such a domain. However, applications of the SBP-FD approximation for the cubed-sphere grid in meteorological context are not a widespread research area.We present a SBP-FD based shallow water model using a non-staggered grid. The model is total-energy conserving, mass-conservative and has discrete analogues of other mimetic properties such as curl-free gradient property. The shallow water model is tested with the commonly used Williamson test suite supplemented with the Galewsky barotropic instability case. High-order convergence is shown for tests with analytic solutions. The SBP-FD shallow water model is more accurate than low-order mimetic finite-element counterparts, but slightly less accurate than high-order finite-volume, spectral-elements and discontinuous Galerkin schemes.

Lax-Wendroff method for incompressible flow

The Lax-Wendroff scheme is extended to incompressible fluid flow problems within the framework of artificial compressibility method (ACM) utilizing a “cell-centered finite-volume” Δ-approximation on a non-orthogonal non-staggered grid. An ACM induces a transformation between conservative and primitive variables; the physical relevance of ACM is to bring about “matrix preconditionings”, provoking density perturbations. The coupled algorithm is pressure-based; it benefits from enhanced accuracy and greater flexibility using the “monotone upstream-centered schemes for conservation laws (MUSCL)” approach. Numerical experiments in reference to “buoyancy-driven” flows with strong sources illustrate that the entire strategy augments overall damping capability and robustness adhering to the factored pseudo-time integration method. Conclusively, the associated limiter function has little influence on the high resolution for selected test cases.

Three-Dimensional Hydrostatic Curved Channel Flow Simulations Using Non-Staggered Triangular Grids

Non-staggered triangular grids have many advantages in performing river or ocean modeling with the finite-volume method. However, horizontal divergence errors may occur, especially in large-scale hydrostatic calculations with centrifugal acceleration. This paper proposes an unstructured finite-volume method with a filtered scheme to mitigate the divergence noise and avoid further influencing the velocities and water elevation. In hydrostatic pressure calculations, we apply the proposed method to three-dimensional curved channel flows. Approximations reduce the numerical errors after filtering the horizontal divergence operator, and the approximation is second-order accurate. Numerical results for the channel flow accurately calculate the velocity profile and surface elevation at different Froude numbers. Moreover, secondary flow features such as the vortex pattern and its movement along the channel sections are also well captured.

Summation-by-Parts Finite-Difference Shallow Water Model on the Cubed-Sphere Grid. Part I: Non-Staggered Grid

Summation-by-Parts Finite-Difference Shallow Water Model on the Cubed-Sphere Grid. Part I: Non-Staggered Grid

Development of artificial compressibility method with elliptic relaxation approach

An artificial compressibility (AC) method resorting to the pressure-based algorithm is developed on a non-orthogonal non-staggered grid using a cell-centered finite-volume Δ-approximation for incompressible fluid flow problems. A pressure time-derivative is used to perturb the equation of continuity. This artifact provokes a density preconditioning which transforms conservative variables to primitive ones; the physical relevance of density preconditioning signifies to convoke compatible linearizations of residuals in collaboration with an AC parameter. The avoidance of pressure-velocity decoupling is enhanced by a consistently formulated cell-face dissipation approach, preserving an increased accuracy with a greater flexibility analogous to the MUSCL (monotone upstream-centered schemes for conservation laws) approach. An elliptic-relaxation scheme having accountability to promoting anisotropic diffusion coefficients is applied to smooth out the pressure residual. Numerical experiments dictate that pertaining to the pseudo-time integration method, the overall contrivance is benefited with an enhanced robustness having sensible oscillation damping characteristics.

Comparison of collocated methods of MCIM and PWIM for different solution algorithms in CVFEM

The pressure-velocity coupling method of MCIM (Mass Corrected Interpolation Method) and different solution algorithms of this method were previously developed and tested for steady and unsteady incompressible flows. But the effect of under-relaxation factors in cell-face velocities and the main equations never have been investigated. On the other hand, this method and its different algorithms never have been compared with similar and more common methods in the formulation of primitive variables and non-staggered grid arrangements. In this paper, the accuracy, performance, and computational efficiency of the different solution algorithms of MCIM are compared with the well-known PWIM (Pressure Weighted Interpolation Method) of Rhie & Chow. In this study, a fully coupled, segregated approach without any pressure correction and SIMPLE-like segregated algorithms are proposed for two methods of MCIM and PWIM on the control volume-based finite element (CVFEM) framework. Also, for each method, the proper under-relaxation factors for the main equations and cell-face velocities are introduced. The accuracy and performance of the proposed algorithms are demonstrated by solving different steady and unsteady benchmark problems, and the results of both methods for different algorithms are compared. The results indicate that the both methods have spatial 2nd-order accuracy approximately and are not very different from each other. The results also show that the numerical dissipation of the MCIM is much less than PWIM. Comparing the computational cost of the different algorithms of both methods shows that the coupled algorithm of the MCIM for steady and unsteady problems with different time steps and various grid numbers is at least 10% better than other algorithms.

基于Picard迭代的PN×PN-2谱元法求解定常不可压缩Navier-Stokes方程

A <em>P_N</em>×<em>P</em>_<em>N</em>-2 spectral element method based on the Picard linearized iteration was presented for the solution of 2D steady incompressible Navier-Stokes equations. Through the Picard iteration, the Navier-Stokes equations were converted to a series of Stokes-type equations to be solved with the <em>P_N</em>×<em>P</em>_<em>N</em>-2 spectral element method on the non-staggered grid in each iteration step. In order to eliminate the pseudo pressure mode, the pressure discretization is 2 orders lower than the velocity discretization, and the application of non-staggered grids makes the discretization of the equation convenient and avoids the interpolation error. The Stokes flow, the Kovasznay flow and the lid-driven cavity flow were simulated with the present method. The numerical results show that, the error converges with the spectral accuracy. In addition, avoidance of the pressure oscillation phenomenon indicates the accuracy and reliability of the present method.

Development of a Parallel 3D Navier–Stokes Solver for Sediment Transport Calculations in Channels

We propose a method to parallelize a 3D incompressible Navier–Stokes solver that uses a fully implicit fractional-step method to simulate sediment transport in prismatic channels. The governing equations are transformed into generalized curvilinear coordinates on a non-staggered grid. To develop a parallel version of the code that can run on various platforms, in particular on PC clusters, it was decided to parallelize the code using Message Passing Interface (MPI) which is one of the most flexible parallel programming libraries. Code parallelization is accomplished by “message passing” whereby the computer explicitly uses library calls to accomplish communication between the individual processors of the machine (e.g., PC cluster). As a part of the parallelization effort, besides the Navier–Stokes solver, the deformable bed module used in simulations with loose beds are also parallelized. The flow, sediment transport, and bathymetry at equilibrium conditions were computed with the parallel and serial versions of the code for the case of a 140-degree curved channel bend of rectangular section. The parallel simulation conducted on eight processors gives exactly the same results as the serial solver. The parallel version of the solver showed good scalability.

An unstructured finite volume method based on the projection method combined momentum interpolation with a central scheme for three-dimensional nonhydrostatic turbulent flows

This paper presents a three-dimensional nonhydrostatic model to solve the Navier–Stokes equations using an unstructured finite volume method. The physical domain could be geometrically arbitrary. To avoid the checkerboard problem caused by non-staggered grids, a momentum interpolation method is used by introducing face-normal velocities at the mid-points of the cell faces. As the Large Eddy Simulation (LES) requires at least second-order accuracy in time and in space for all the terms, a central scheme combined with an explicit Adams–Bashforth scheme is proposed in this model. The projection method is applied to decouple the velocity field and pressure. Several benchmark test cases are used to validate the second-order accuracy, the numerical stability and the performance of the model. Analysis on divergence noise using an unstructured collocated triangular grid, as well as on the ratio between vertical and horizontal spacing steps have been done to show the reliability of the model. The proposed model has been used to simulate backward-facing step flows, lid-cavity flows, turbulent open channel flows and the turbulent flows around a vertical cylinder. The convergence of the linear solver is analyzed in terms of the iterations and CPU time. The results are fairly in agreement with the references in the literature. The proposed model is able to correctly reproduce the characteristic flow features in all the test cases.

Mixing process in opposing flow junction with different angles and junction radii

Opposing flow junctions are considered as important elements in thermal and hydraulic equipment. This study numerically investigates the effects of angles and junction radii on coherent flow structures at opposing flow junctions with subcritical conditions. Hence, the three-dimensional unsteady Reynolds-averaged Navier–Stokes equations are solved with the k–e turbulence model on a non-staggered grid using the indirect addressing treatment. After the verification of the numerical model, several numerical simulations are conducted for angles 35°, 40°, 45°, 50°, 60°, 70°, 80°, and 90° with different upstream Froude numbers and junction radii. The streamwise-oriented vortical cells only elongate into the branch channel of the opposing flow junction with angle 90°. By decreasing the angle between the main channel and confluent tributary, these cells decay in the main channel width, and one of the separation zones is gradually eliminated, as that does not form in the opposing flow junctions with angle 40°. The enhancement of junction radii decreases the dimensions of the separation and stagnation zones. Numerical simulation results of a curved-edge opposing flow junction with angle 80° indicate that any of the streamwise-oriented vortical cells in the main channel and separations zones in the branch channel are not formed. In such a condition, three vortical cells formed along the branch without decay, two cells are located near the side wall, and another cell is near the free surface.

A Dynamic Coarse Grain Discrete Element Method for Gas-Solid Fluidized Beds by Considering Particle-Group Crushing and Polymerization

The discrete element method (DEM) coupled with computational fluid dynamics (CFD) is used extensively for the numerical simulation of gas-solid fluidized beds. In order to improve the efficiency of this approach, a coarse grain model of the DEM was proposed in the literature. In this model, a group of original particles are treated as a large-sized particle based on the initial particle distribution, and during the whole simulation process the number and components of these particle-groups remain unchanged. However, collisions between particles can lead to frequent crushing and polymerization of particle-groups. This fact has typically been ignored, so the purpose of this paper is to rationalize the coarse grain DEM-CFD model by considering the dynamic particle-group crushing and polymerization. In particular, the effective size of each particle-group is measured by a quantity called equivalent particle-group diameter, whose definition references the equivalent cluster diameter used by the energy-minimization multi-scale (EMMS) model. Then a particle-group crushing criterion is presented based on the mismatch between the equivalent diameter and actual diameter of a particle-group. As to the polymerization of two colliding particle-groups, their velocity difference after collision is chosen as a criterion. Moreover, considering the flow heterogeneity induced by the particle cluster formation, the EMMS drag force model is adopted in this work. Simulations are carried out by using a finite volume method (FVM) with non-staggered grids. For decoupling the Navier-Stokes equations, the semi-implicit method for pressure linked equations revised (SIMPLER) algorithm is used. The simulation results show that the proposed dynamic coarse grain DEM-CFD method has better performance than the original one.

Computational Modeling of Turbulent Flows

The article presents an improved numerical method developed for calculating turbulent flows basing on the algorithm SIMPLE, Rhie–Chow interpolation on non-staggered grids, and monotonic second and third order accurate TDV schemes. The use of high-order difference schemes for calculation of viscous flows enables achievement of significantly better resolution of the algorithm on coarse grids, which decisively affects the efficiency of two-dimensional and especially three-dimensional calculations.

Numerical investigation of two-phase flow in a horizontal porous evaporator with localised heating using non-Darcian flow and two equations model

In this article, a new mathematical model, based on the modified enthalpy formulation of Two-Phase Mixture Model (TPMM), is developed by considering non-Darcy flow and Local Thermal Non-Equilibrium (LTNE) conditions to describe numerically the two-dimensional problems of the complete evaporation process inside an asymmetrically heated porous evaporator. The governing equations have been discretised using Finite Volume Method (FVM) on a fixed non-staggered grid layout. The in-house code has been validated against experimental data and the results show good agreement between them. Three different models have been employed for the partitioning of the wall heat flux. Only the effect of the heat flux has been analysed, while all other parameters and properties have been kept fixed during the present investigation. The numerical simulations indicated that the locally high heat flux significantly influences the working fluid flow performances, and lead to extend the superheated vapour region towards the exit of the evaporator. The results also show that the two-phase zone significantly expanded in the axial direction and does not occupy a large portion inside the channel due to the energy transport through the two-phase region is significantly higher in the flow direction as compared to the transverse direction, caused by the decrease in the effective diffusion coefficient in this region and higher axial velocity. Numerical analysis reveal that the different models for the partitioning of the wall heat flux have almost no influence on the overall predictions, except in the vicinity of the wall. The results clearly demonstrate that the numerical simulations with the modified h-formulation by considering non-Darcy law and LTNE model are indispensable for the problem of complete evaporation process, particularly under the high both heat flux and inlet velocity conditions.

Research on Effects of Obstacles on Heat Transfer and Fluid Flow in Backward Facing Step Flow

An extensive research has been done on heat transfer and pressure drop characteristics in micro-channel using liquid water. A baffle has been introduced downstream the sudden expansion zone to enhance the rate of heat transfer. The height and the location of the baffle were varied for the Reynolds number range 50≤ Re ≤ 200, which is a laminar flow. Twodimensional flow domain with non-staggered grid arrangement was taken and the two-dimensional mass, momentum and energy equation was solved using finite volume method in ANSYS 16.2. This study reports that the presence of baffle in the microchannel increased the rate of heat transfer. The skin friction coefficient has been calculated and the parameters influencing the heat transfer augmentation have been optimized

Evaluation of Non-Staggered Body-Fitted Grid Based Solution Method in Application to Supercritical Fluid Flows

Evaluation of Non-Staggered Body-Fitted Grid Based Solution Method in Application to Supercritical Fluid Flows

Back and forth error compensation and correction method for linear hyperbolic systems with application to the Maxwell's equations

We study the Back and Forth Error Compensation and Correction (BFECC) method for linear hyperbolic systems and in particular for the Maxwell's equations. BFECC has been applied to schemes for scalar advection equations to improve their stability and order of accuracy. Similar results have been established in this paper for linear hyperbolic systems with constant coefficients. We apply BFECC to the central difference scheme, Lax-Friedrichs scheme and a combination of them for the Maxwell's equations and obtain second order accurate schemes with large CFL numbers (greater than 1 in one or two dimensions). The method is further applied to schemes on non-orthogonal unstructured grids. The new BFECC schemes for the Maxwell's equations operate on a single non-staggered grid and are simple to implement on unstructured grids. Numerical examples are given to demonstrate the effectiveness of the new schemes.

Mixed convection in a square enclosure with a rotating flat plate

Mixed convection in a square enclosure arising from thermal buoyancy force and a rotating flat plate is investigated in the paper. The moving boundary problem is solved with the implicit virtual boundary method on a fixed non-staggered Cartesian grid system. Computations are performed for Prandtl number 0.71 at various Rayleigh numbers Ra and rotational Reynolds numbers Re. For a rotor of length d=0.6 rotating at Re=430, the numerical results show that thermal oscillation occurs when the Rayleigh number exceeds 0.55×106. There is a critical Rayleigh number (0.13×106). Below that the rotor enhances the heat transfer. By contrast, the rotor would suppress the heat transfer beyond the critical Rayleigh number. In the case of Ra=106, the Nusselt number oscillates 23 cycles when the rotor makes 29 revolutions. For a longer rotor d=0.9 rotating at Re=430, thermal oscillation exists for all Rayleigh numbers, and the critical Rayleigh number increases to 0.38×106. In the case of Ra=106, the Nusselt number oscillates twice when the rotor makes one revolution.

Effects of initial stage of dam-break flows on sediment transport

Experimental and numerical studies of dam-break flows over sediment bed under dry and wet downstream conditions are investigated and their effects on sediment transport and bed change on flow are illustrated. Dam-break waves are generated by suddenly lifting a gate inside the flume for three different upstream reservoir heads. The flow characteristics are detected by employing simple and economical measuring technique. The numerical model solves the two-dimensional Reynolds-Averaged Navier–Stokes (RANS) equations with k-ε turbulence closure using the explicit finite volume method on adaptive, non-staggered grid. The model is validated with laboratory data and is extended for simulating non-equilibrium sediment transport and bed evolution process. The volume of fluid technique is used to track the evolution of the free surface, satisfying the advection equation. The comparison study reveals that the current model is capable of defining the dam-break flow and improves the accuracy of determining morphological changes at the initial stages of the dam-break flow. A good agreement between the model solutions and the experimental data is observed.