Semi-staggered Grid Research Articles

A simple and efficient finite difference scheme to the Cahn–Hilliard–Navier–Stokes system equations

Central difference schemes are usually not suited to convective terms in a transport equation due to its oscillation nature when convection effect is pronounced. However, this work found that applying the standard central difference scheme to the convective term along with another central difference scheme to the fourth order diffusive term in the Cahn–Hilliard equation can realize nearly an order of magnitude speed rise, in the framework of a fully explicit finite difference scheme. The discretization was done on a semi-staggered grid where pressure was stored at the cell center and other variables were stored at the cell corners. To accelerate computation, a simple parallelism based on OpenMP was used. The scheme was tested in a number of cases and was compared with both analytical and experimental outcomes. The results showed that the scheme is efficient compared with a previous fully explicit finite difference scheme for the Cahn–Hilliard equation, and that a time step more than five times larger can be employed.

An open-source, adaptive solver for particle-resolved simulations with both subcycling and non-subcycling methods

We present the IAMReX (incompressible flow with adaptive mesh refinement for the eXascale), an adaptive and parallel solver for particle-resolved simulations on the multi-level grid. The fluid equations are solved using a finite-volume scheme on the block-structured semi-staggered grids with both subcycling and non-subcycling methods. The particle-fluid interaction is resolved using the multidirect forcing immersed boundary method. The associated Lagrangian markers used to resolve fluid-particle interface only exist on the finest-level grid, which greatly reduces memory usage. The volume integrals are numerically calculated to capture the free motion of particles accurately, and the repulsive potential model is also included to account for the particle–particle collision. We demonstrate the versatility, accuracy, and efficiency of the present multi-level framework by simulating fluid-particle interaction problems with various types of kinematic constraints. The cluster of monodisperse particles case is presented at the end to show the capability of the current solver in handling multiple particles. It is demonstrated that the three-level AMR (Adaptive Mesh Refinement) simulation leads to a 72.46% grid reduction compared with the single-level simulation. The source code and testing cases used in this work can be accessed at https://github.com/ruohai0925/IAMR/tree/development. Input scripts and raw postprocessing data are also available for reproducing all results.

A Staggered Grid Method for Solving Incompressible Flow on Unstructured Meshes

A finite volume method based unstructured grid is presented to solve the two dimensional viscous and incompressible flow. The method is based on the pressure-correction concept and solved by using a semi-staggered grid technique. The computational procedure can handle cells of arbitrary shapes, although solutions presented in this paper were only involved with triangular and quadrilateral cells. The pressure or pressure-correction value was stored on the vertex of cells. The mass conservation equation was discretized on the dual cells surrounding the vertex of primary cells, while the velocity components and other scale variables were saved on the central of primary cells. Since the semi-staggered arrangement can’t guarantee a strong coupling relationship between pressure and velocity, thus a weak coupling relationship leads to the oscillations for pressure and velocity. In order to eliminate such an oscillation, a special interpolation scheme was used to construct the pressure-correction equation. Computational results of several viscous flow problems show good agreement with the analytical or numerical results in previous literature. This semi-staggered grid method can be applied to arbitrary shape elements, while it has the most efficiency for triangular cells.

Application of a new single staggered grid method to the heat transfer problems

In this paper, a new single staggered grid method is proposed to solve the fluid dynamic problems numerically. The advantages of the new grid method are analyzed in comparison with the classical grid algorithms such as the staggered grids, collocated grids and semi-staggered grids. The discretization of the basic equations for the fluid dynamics on the new single staggered grids is derived and the corresponding SIMPLE algorithm is introduced. As an example, the heat transfer problem of fluid-flow at a right angle is solved to prove the validity of the new single staggered grid method.

A phase-resolved, depth-averaged non-hydrostatic numerical model for cross-shore wave propagation

In this study, a phase-resolved and depth-averaged non-hydrostatic numerical model (SNH model) is developed. The non-incremental pressure-correction method is employed to solve the equation system in two successive steps. Firstly, an approximate Riemann solver in the framework of finite volume methods is employed to solve the hydrostatic shallow-water equations (SWE) on a collocated grid to obtain provisional solutions. Then, the intermediate solutions is updated by considering the non-hydrostatic pressure effect; a semi-staggered grid is used in this step to avoid predicting checkboard pressure field. A series of benchmark tests are used to validate the numerical model, showing that the developed model is well-balanced and describes the wetting and drying processes accurately. By employing a shock-capturing numerical scheme, the wave-breaking phenomenon is reasonably simulated without using any ad-hoc techniques. Compared with the SWE model, the wave shape can be well-preserved and the numerical predictions are much improved by using the SNH model.

ICONE19-43114 ON A GLOBAL MASS PRESERVING INCOMPRESSIBLE TWO-PHASE FLOW SOLVER

An interface capturing scheme, implemented within the framework of the conservative phase-field method, is applied to simulate the incompressible two-phase flows with large density ratio and surface tension. For accurately predicting the solution, the finite difference dispersionrelation-preserving advection schemes are utilized to accommodate the true dispersion relation. The solutions computed from the resulting dispersion-relation-preserving advection schemes can minimize the dispersion error. The surface tension force is calculated by the continuum surface tension force (CSF) formulation of Brackbill et al.. (Brackbill, J. U., et al.., 1992) For the sake of programming simplicity, the proposed incompressible two-phase flow solver will be discretized on the semi-staggered grids without incurring velocity-pressure checkerboard oscillations. To verify the proposed method, four benchmark problems, including Rayleigh-Taylor instability, bubble rising and coaxial/oblique bubble merging problems, are numerically investigated. All predicted results have been shown to compare fairly well with the experimental and/or other numerical results. It also shows that the satisfactions of mass conservations are guaranteed for the proposed two-phase flow solver. Finally, the two-phase flow solver is applied to numerically predict the Kevin-Helmholtz instability in a tilted horizontal channel. The predicted results show the applicability to the stratified two-phase flow analysis in the nuclear engineering field, like the slug flow, droplet impacting or counter-current flow problems

Method for efficient prevention of gravity wave decoupling on rectangular semi-staggered grids

Generation of short gravity wave noise often occurs on semi-staggered rectangular grids as a result of sub-grid decoupling when there is a strong forcing in the mass field. In this study a numerical scheme has been proposed to prevent the generation of the gravity wave decoupling. The proposed numerical method provides efficient communication between decoupled gravity wave finite-difference solutions by a simple averaging of a term in the height tendency in the continuity equation. The proposed method is tested using a shallow water sink model developed for the purpose of this study. It has been demonstrated that this method outperforms other existing approaches. The new scheme is time efficient, based on explicit time integration and can be easily implemented. The proposed method is applicable in hydrodynamic models specified on semi-staggered B or E grids.

Fast orthogonalization to the kernel of the discrete gradient operator with application to Stokes problem

We obtain a simple tensor representation of the kernel of the discrete d -dimensional gradient operator defined on tensor semi-staggered grids. We show that the dimension of the nullspace grows as O ( n d - 2 ) , where d is the dimension of the problem, and n is one-dimensional grid size. The tensor structure allows fast orthogonalization to the kernel. The usefulness of such procedure is demonstrated on three-dimensional Stokes problem, discretized by finite differences on semi-staggered grids, and it is shown by numerical experiments that the new method outperforms usually used stabilization approach.

Uzawa method on semi-staggered grids for unsteady Bingham media flows

A finite difference scheme on semi-staggered grids is used for numerical simulation of unsteady flows of an incompressible viscoplastic Bingham medium. The Duvaut–Lions variational inequality is considered as a mathematical model. The convergence of the Uzawa method is proved. The efficiency of the method is demonstrated on a model problem with a known exact solution (plane Poiseuille flow) and on the unsteady driven cavity problem.

An adaptive mesh redistribution method for the incompressible mixture flows using phase-field model

A phase field model which describes the motion of mixtures of two incompressible fluids is presented by Liu and Shen [C. Liu, J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D 179 (2003) 211–228]. The model is based on an energetic variational formulation. In this work, we develop an efficient adaptive mesh method for solving a phase field model for the mixture flow of two incompressible fluids. It is a coupled nonlinear system of Navier–Stokes equations and Allen–Cahn phase equation (phase-field equation) through an extra stress term and the transport term. The numerical strategy is based on the approach proposed by Li et al. [R. Li, T. Tang, P.-W. Zhang, Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys. 170 (2001) 562–588] to separate the mesh-moving and PDE evolution. In the PDE evolution part, the phase-field equation is numerically solved by a conservative scheme with a Lagrange multiplier, and the coupled incompressible Navier–Stokes equations are solved by the incremental pressure-correction projection scheme based on the semi-staggered grid method. In the mesh-moving part, the mesh points are iteratively redistributed by solving the Euler–Lagrange equations with a parameter-free monitor function. In each iteration, the pressure and the phase are updated on the resulting new grid by a conservative-interpolation formula, while the velocity is re-mapped in a non-conservative approach. A simple method for preserving divergence-free is obtained by projecting the velocity onto the divergence-free space after generating the new mesh at the last iterative step. Numerical experiments are presented to demonstrate the effectiveness of the proposed method for solving the incompressible mixture flows.

A PRESSURE-CORRECTION METHOD FOR THE SOLUTION OF INCOMPRESSIBLE VISCOUS FLOWS ON UNSTRUCTURED GRIDS

An unstructured grid, finite volume method is presented for the solution of two-dimensional viscous, incompressible flow. The method is based on the pressure-correction concept implemented on a semi-staggered grid. The computational procedure can handle cells of arbitrary shape, although solutions presented herein have been obtained only with meshes of triangular and quadrilateral cells. The discretization of the momentum equations is effected on dual cells surrounding the vertices of primary cells, while the pressure-correction equation applies to the primary-cell centroids and represents the conservation of mass across the primary cells. A special interpolation scheme s used to suppress pressure and velocity oscillations in cases where the semi-staggered arrangement does not ensure a sufficiently strong coupling between pressure and velocity to avoid such oscillations. Computational results presented for several viscous flows are shown to be in good agreement with analytical and experimental data reported in the open literature.

A PRESSURE‐CORRECTION METHOD FOR THE SOLUTION OF INCOMPRESSIBLE VISCOUS FLOWS ON UNSTRUCTURED GRIDS

An unstructured grid, finite volume method is presented for the solution of two-dimensional viscous, incompressible flow. The method is based on the pressure-correction concept implemented on a semi-staggered grid. The computational procedure can handle cells of arbitrary shape, although solutions presented herein have been obtained only with meshes of triangular and quadrilateral cells. The discretization of the momentum equations is effected on dual cells surrounding the vertices of primary cells, while the pressure-correction equation applies to the primary-cell centroids and represents the conservation of mass across the primary cells. A special interpolation scheme s used to suppress pressure and velocity oscillations in cases where the semi-staggered arrangement does not ensure a sufficiently strong coupling between pressure and velocity to avoid such oscillations. Computational results presented for several viscous flows are shown to be in good agreement with analytical and experimental data reported in the open literature.

Development of a Non-linear Levels Model and its Application to Bora-driven Circulation on the Adriatic Shelf

The present paper describes a non-linear levels numerical model. The model is written in the semi-staggered grid with the forward-backwards numerical scheme. It is applied to reproduce the bora-driven circulation on the Adriatic shelf in order to relate to previous numerical studies where processes of advection of momentum and local generation of turbulence were not considered. Starting from a hydrostatic state the wind-stress produces a marked seiche and inertial oscillation in the numerical solutions. These oscillations are discussed with regard to the theory of the Adriatic seiche.

Computed rotational energy spectra of two energy and enstrophy conserving schemes on semi-staggered grids

Rotational energy spectra are examined in experiments with two energy and enstrophy conserving schemes defined on the semi-staggered grid. One of the schemes conserved enstrophy as defined on an equivalent staggered grid, and the other conserved enstrophy as defined on the semi-staggered grid. As predicted by the theory (Janjic, 1984), the rotational energy corresponding to the scheme conserving enstrophy as defined on an equivalent staggered grid decreases faster with increasing intensity of the wave number vector than that of the scheme conserving enstrophy as defined on the semistaggered grid. Moreover, the spectrum obtained with the scheme conserving the equivalent staggered grid enstrophy resembles the observed atmospheric spectra in a large part of the admissible wave number range.

Fourth-Order Horizontal Advection Schemes on the Semi-staggered Grid

Abstract Horizontal advection schemes on the semi-staggered grid E are presented, which in their linearized versions have a fourth-order space accuracy. The first is a scheme for evaluation of the horizontal nonlinear terms in the momentum equation. It prevents false energy cascade in the manner of the Janjic scheme, i.e., by conserving rational C grid energy and enstrophy. However, it is derived by transformation of the fourth-order rather than of the second-order Arakawa Jacobian. In an extension of the case which includes the divergent part of the flow, care is taken to achieve the conservation of the physically important integral properties of the general flow. The other is a scheme for advection of a passive scalar variable. It is derived by a modification of the former scheme. The momentum equation scheme is tested in a long-term integration of the shallow water equations. As a consequence of the imposed conservation constraints, it is able to simulate general characteristics of the flow up to rough...

Nonlinear Advection Schemes and Energy Cascade on Semi-Staggered Grids

Abstract A common problem with nonlinear advection schemes is the false accumulation of energy at the smallest resolvable scales. To keep this process under control, following Arakawa (1966), a number of energy and enstrophy conserving schemes for staggered and semi-staggered grids have been designed. In this paper, it is demonstrated that, in contrast to the staggered grid, the conservation of energy and enstrophy on the semi-staggered gods does not guarantee that the erroneous transport of energy from large to small scales will be effectively restricted. Using a new approach to the application of the Arakawa Jacobian, a scheme for a semi-staggered grid which exactly reflects the Arakawa theory for nondivergent flow is obtained for the first time. This is achieved by conservation of energy and enstrophy as defined on the staggered grid. These two quantities are of higher accuracy and cannot be calculated directly from the dependent variables on the semi-staggered grid. It is further demonstrated that the...