Staggered Mesh Scheme Research Articles

Transient sub-channel code development for lead-cooled fast reactor using the second-order upwind scheme

For its unique safety and economic advantages, the lead cooled fast reactor has become one of the most interesting candidate reactors for the Generation IV nuclear system. To study the thermal-hydraulics of the core under transient conditions, KMC-SUBtra--a sub-channel code for transient thermal-hydraulic analysis of lead cooled fast reactor has been developed. The code uses a modified pressure gradient method to solve the simultaneous equations of the fluid mass, momentum and energy containing cross flow and turbulent mixing, in which the axial pressure gradients are solved as pending variables of the simultaneous equations. A staggered mesh scheme is used for scalar and vector quantities and the second-order upwind scheme is adopted in the discretization of the convection term. The code was validated and verified by the experimental data and CFD simulation results on the steady and transient conditions so its capability for lead cooled reactors was confirmed. The transient flow and heat transfer in the fuel assembly with time-varying boundary conditions were studied, which revealed different transient thermal characteristics of the coolant and fuel rods. In addition, the results calculated using different order upwind schemes were compared and showed the second-order derivative of the axial flow is small.

Sub-grid properties and artificial viscous stresses in staggered-mesh schemes

We analyze the properties of a number of forms of artificial viscosity, when applied to the Lagrangian phase of a staggered grid Lagrange–remap hydrodynamics code, using a full von Neumann stability analysis. This allows us to derive the numerical dispersion relation for a single step of the scheme.We study the development of shocks within a weakly nonlinear acoustic wave in detail. The second-order von Neumann–Richtmyer artificial viscosity leads to significant post-shock ringing, while adding linear artificial viscosity reduces the accuracy of the scheme to first order and introduces a dependence on flow Mach number. The form of the numerical dispersion relation suggests that to control the post-shock ringing without reducing the accuracy of the solution, a higher-order artificial viscosity is required which will preferentially damp modes close to the mesh scale which are subject to significant phase error.The application of ideas based on the Large Eddy Simulation technique for the modelling of turbulent flows leads to a form of artificial viscous stress similar to that described by Schulz (1964). This artificial stress has several features which are beneficial for accurate modelling of compressible turbulent mixing processes, as demonstrated by a range of other test problems.We show that, in the limit of smooth flows, Christensen's approach of applying limiters to the components of a low-order artificial viscosity is analytically equivalent to a high-order scheme. However, the directionally-split implementation suggested in the original publication leads to damping of low Mach number Kelvin–Helmholtz roll up problems similar to that previously observed for finite-volume Godunov schemes.

Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations

The paper presents two contributions in the context of the numerical simulation of magnetized fluid dynamics. First, we show how to extend the ideal magnetohydrodynamics (MHD) equations with an inbuilt magnetic field divergence cleaning mechanism in such a way that the resulting model is consistent with the second law of thermodynamics. As a byproduct of these derivations, we show that not all of the commonly used divergence cleaning extensions of the ideal MHD equations are thermodynamically consistent. Secondly, we present a numerical scheme obtained by constructing a specific finite volume discretization that is consistent with the discrete thermodynamic entropy. It includes a mechanism to control the discrete divergence error of the magnetic field by construction and is Galilean invariant. We implement the new high-order MHD solver in the adaptive mesh refinement code FLASH where we compare the divergence cleaning efficiency to the constrained transport solver available in FLASH (unsplit staggered mesh scheme).

Semi-implicit staggered mesh scheme for the multi-layer shallow water system

We present a semi-implicit scheme for a two-dimensional multilayer shallow water system with density stratification, formulated on general staggered meshes. The main result of the present note concerns the control of the mechanical energy at the discrete level, principally based on advective fluxes implying a diffusion term expressed in terms of the gradient pressure. The scheme is also designed to capture the dynamics of low-Froude-number regimes and offers interesting positivity and well-balancing results. A numerical test is proposed to highlight the scheme's efficiency in the one-layer case.

Numerical Investigation of Flow in an Asymmetric Plane Diffuser Using a Multi-Scale Turbulence Model

The flow in an asymmetric plane diffuser was simulated using both the multiscale turbulence model based on the variable interval time average method and the standardk-εmodel based on the Reynolds average method. The numerical method used in this simulation is an unstructured staggered mesh scheme. The capability of the multi-scale model to simulate flow in an asymmetric plane diffuser has been validated. Pressure coefficient, frictional resistance coefficient and mean velocity profiles downstream are in agreement with experiments. Moreover, the results predicted by the multi-scale model are better than that predicted by the standardk-εmodel. The computational results show that the multiscale turbulence model can successfully simulate this type of separated flow.

Numerical Investigation of Flow Past a Backward-Facing Step Using a Multi-Scale Turbulence Model

The flow past a backward-facing step was simulated using both the multiscale turbulence model based on the variable interval time average method and the standard k-ε model based on the Reynolds average method. The numerical method used in this simulation is an unstructured staggered mesh scheme. The capability of the multi-scale model to simulate flow past a backward-facing step has been validated. Both pressure coefficient and mean velocity profiles downstream are in agreement with experiments. Moreover, the results predicted by the multi-scale model are better than that predicted by the standard k-ε model. The computational results show that the multiscale turbulence model can successfully simulate this type of separated flow.

Numerical Investigation of Flow around a Triangle Cylinder Using a Multiscale Turbulence Model

The unsteady flow around a triangle cylinder was simulated using the multiscale turbulence model based on the variable interval time average method. The numerical method used in this simulation is an unstructured staggered mesh scheme. The computational results show that the multiscale turbulence model can successfully simulate vortex shedding characteristics. The Strouhal number and time-averaged velocity profiles also agree better with experiments than that of the standard k-ε model.

Numerical Investigation of Flow around a Triangle Cylinder Using a Multiscale Turbulence Model

The unsteady flow around a triangle cylinder was simulated using the multiscale turbulence model based on the variable interval time average method. The numerical method used in this simulation is an unstructured staggered mesh scheme. The computational results show that the multiscale turbulence model can successfully simulate vortex shedding characteristics. The Strouhal number and time-averaged velocity profiles also agree better with experiments than that of the standard k-ε model.

Theoretical study on the flow instability of supercritical water in the parallel channels

Super Critical Water (SCW) will experience considerable changes on the thermal and transport properties such as density, enthalpy, specific heat and thermal conductivity at normal operating condition. The flow instability in the parallel channels with SCW was studied in this paper. Mathematical and physical models were established to simulate the flow and the heat transfer characteristics of supercritical water with semi-implicit scheme and staggered mesh scheme. The flow instability of SCW was analyzed using the tiny perturbation method. Pseudo-subcooling number (NSPC) and pseudo-phase change number (NTPC) which can be used to distinguish the system instability were derived based on the property of SCW. The marginal stability boundary (MSB) was then obtained by using the NSPC and NTPC. The effects of different parameters, such as mass flow rate, heat flux, inlet temperature and system pressure, on the flow instability boundary were also investigated. When increasing the mass flow rate and the system pressure, decreasing the heat flux, the flow stability in the parallel channels increases. The effect of inlet temperature in the low pseudo-subcooling number region is different from that in high pseudo-subcooling number region.

Hybrid numerical methods to solve shallow water equations for hurricane induced storm surge modeling

In this paper an efficient numerical method based on hybrid finite element and finite volume techniques to solve hurricane induced storm surge flow problem is presented. A segregated implicit projection method is used to solve the 2D shallow water equations on staggered unstructured meshes. The governing equations are written in non-conservation form. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centered finite volume method. The nonlinear wave equation is solved by the node-based Galerkin finite element method. This staggered-mesh scheme is distinct from other conventional approaches in that the velocity components and auxiliary variables are stored at cell centers and vertices, respectively. The present model uses an implicit method, which is very efficient and can use a large time step without losing accuracy and stability.The hurricane induced wind stress and pressure, bottom friction, Coriolis effect, and tidal forcing conditions are implemented in this model. The levee overtopping option is implemented in the model as well. Hurricane Katrina (2005) storm surge has been simulated to demonstrate the robustness and applicability of the model.

A solution accurate, efficient and stable unsplit staggered mesh scheme for three dimensional magnetohydrodynamics

In this paper, we extend the unsplit staggered mesh scheme (USM) for 2D magnetohydrodynamics (MHD) [D. Lee, A.E. Deane, An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics, J. Comput. Phys. 228 (2009) 952–975] to a full 3D MHD scheme. The scheme is a finite-volume Godunov method consisting of a constrained transport (CT) method and an efficient and accurate single-step, directionally unsplit multidimensional data reconstruction-evolution algorithm, which extends Colella’s original 2D corner transport upwind (CTU) method [P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys. 87 (1990) 446–466]. We present two types of data reconstruction-evolution algorithms for 3D: (1) a reduced CTU scheme and (2) a full CTU scheme. The reduced 3D CTU scheme is a variant of a simple 3D extension of Collela’s 2D CTU method and is considered as a direct extension from the 2D USM scheme. The full 3D CTU scheme is our primary 3D solver which includes all multidimensional cross-derivative terms for stability. The latter method is logically analogous to the 3D unsplit CTU method by Saltzman [J. Saltzman, An unsplit 3D upwind method for hyperbolic conservation laws, J. Comput. Phys. 115 (1994) 153–168]. The major novelties in our algorithms are twofold. First, we extend the reduced CTU scheme to the full CTU scheme which is able to run with CFL numbers close to unity. Both methods utilize the transverse update technique developed in the 2D USM algorithm to account for transverse fluxes without solving intermediate Riemann problems, which in turn gives cost-effective 3D methods by reducing the total number of Riemann solves. The proposed algorithms are simple and efficient especially when including multidimensional MHD terms that maintain in-plane magnetic field dynamics. Second, we introduce a new CT scheme that makes use of proper upwind information in taking averages of electric fields. Our 3D USM schemes can be easily combined with various reconstruction methods (e.g., first-order Godunov, second-order MUSCL-Hancock, third-order PPM and fifth-order WENO), and a wide choice of 1D based Riemann solvers (e.g., local Lax–Friedrichs, HLLE, HLLC, HLLD, and Roe). The 3D USM-MHD solver is available in the University of Chicago Flash Center’s official FLASH release.

Hybrid finite element/volume method for 3D incompressible flows with heat transfer

An implicit hybrid finite element (FE)/volume solver has been extended to incompressible flows coupled with the energy equation. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centred finite volume (FV) method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centres and the auxiliary variable at vertices. The Generalized Minimal Residual (GMRES) matrix-free strategy is adapted to solve the governing equations in both FE and FV methods. The presented 2D and 3D numerical examples show the robustness and accuracy of the numerical method.

An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics

We introduce an unsplit staggered mesh scheme (USM) for multidimensional magnetohydrodynamics (MHD) that uses a constrained transport (CT) method with high-order Godunov fluxes and incorporates a new data reconstruction–evolution algorithm for second-order MHD interface states. In this new algorithm, the USM scheme includes so-called “multidimensional MHD terms”, proportional to ∇ · B , in a dimensionally-unsplit way in a single update. This data reconstruction–evolution step, extended from the corner transport upwind (CTU) approach of Colella, maintains in-plane dynamics very well, as shown by the advection of a very weak magnetic field loop in 2D. This data reconstruction–evolution algorithm is also of advantage in its consistency and simplicity when extended to 3D. The scheme maintains the ∇ · B = 0 constraint by solving a set of discrete induction equations using the standard CT approach, where the accuracy of the computed electric field directly influences the quality of the magnetic field solution. We address the lack of proper dissipative behavior in the simple electric field averaging scheme and present a new modified electric field construction (MEC) that includes multidimensional derivative information and enhances solution accuracy. A series of comparison studies demonstrates the excellent performance of the full USM–MEC scheme for many stringent multidimensional MHD test problems chosen from the literature. The scheme is implemented and currently freely available in the University of Chicago ASC FLASH Center’s FLASH3 release.

Pressure-Based Finite-Volume Methods in Computational Fluid Dynamics

Pressure-based finite-volume techniques have emerged as the methods of choice for a wide variety of industrial applications involving incompressible fluid flow. In this paper, we trace the evolution of this class of solution techniques. We review the basics of the finite-volume method, and trace its extension to unstructured meshes through the use of cell-based and control-volume finite-element schemes. A critical component of the solution of incompressible flows is the issue of pressure-velocity storage and coupling. The development of staggered-mesh schemes and segregated solution techniques such as the SIMPLE algorithm are reviewed. Co-located storage schemes, which seek to replace staggered-mesh approaches, are presented. Coupled multigrid schemes, which promise to replace segregated-solution approaches, are discussed. Extensions of pressure-based techniques to compressible flows are presented. Finally, the shortcomings of existing techniques and directions for future research are discussed.

Coriolis weighting on unstructured staggered grids

There is an increasing interest to move ocean codes from classical Cartesian staggered mesh schemes to unstructured staggered grids. By using unstructured grid models one may construct meshes that follow the coastlines more accurately, and it is easy to apply a finer resolution in areas of special interest. In this paper we focus on how to approximate the Coriolis terms in such unstructured staggered grid models using equivalents of the Arakawa C-grid for the linear equations governing the propagation of the inertia-gravity waves. We base the analysis on a Delaunay triangulation of the region in question and use the Voronoi points and the midpoints on the triangle edges to define a staggered grid for the sea elevation and the velocity orthogonal to the edges of the triangles. It is shown that a standard method for the Coriolis weighting may create unphysical growth of the numerical solutions. A modified Coriolis weighting that conserves the total energy is suggested. In real applications diffusion is often introduced both for physical reasons, but often also in order to stabilise the numerical experiments. The growing modes associated with the unstructured staggered grids and equal weighting may force us to enhance the diffusion more than we would like from physical considerations. The modified weighting offers a simple solution to this problem.

Kinetic energy conservation issues associated with the collocated mesh scheme for incompressible flow

The collocated-mesh scheme is often favored over the staggered-mesh scheme for turbulence simulation in complex geometries due to its simpler form in curvilinear coordinates. The collocated mesh scheme does not conserve kinetic energy however, and few careful checks of the impact of these errors have been made. In this work, analysis is used to identify two sources of kinetic energy conservation error in the collocated-mesh scheme: (1) errors arising from the interpolations used to estimate the velocity on the cell faces, and (2) errors associated with the slightly inconsistent pressure field used to ensure mass conservation for the cell face volume fluxes. It is shown that the interpolation error can be eliminated through the use of first-order accurate centered interpolation operators with mesh-independent weights. The pressure error appears to be intrinsic to the scheme and it is shown to scale as O(Δ t 2Δ x 2). The effects of the conservation errors are investigated numerically through simulations of inviscid flow over an airfoil and in large eddy simulations of turbulent channel flow. Neither the interpolation error nor the pressure error appear to lead to significant problems in the channel flow simulations where viscous dissipation is present and where the Cartesian mesh is stretched in only one direction. The inviscid airfoil simulations performed on a curvilinear mesh show a much greater sensitivity to the interpolation error. The standard second-order centered interpolation is shown to lead to severe numerical oscillations, while the kinetic energy-conserving first-order centered interpolation produces solutions that are almost as smooth as those obtained with a third-order upwind interpolation. When compared in channel flow simulations, however, the first-order centered interpolation is shown to be far superior to the third-order upwind interpolation, the latter being adversely affected by numerical dissipation. Only slight differences were noted when comparing channel flow simulations using first-order and second-order centered interpolations. These results suggest that numerical oscillations can be controlled in curvilinear coordinates through the use of properly-constructed non-dissipative centered interpolations.

Analysis of an Exact Fractional Step Method

An exact fractional step or projection method for solving the incompressible Navier–Stokes equations is analyzed. The method is applied to both structured and unstructured staggered mesh schemes. There are no splitting errors associated with the method; it satisfies the incompressibility condition to machine precision and reduces the number of unknowns. The exact projection technique is demonstrated on a two-dimensional cavity flow and a multiply connected moving domain with a free surface. Its performance is compared directly to classic fractional step methods and shown to be roughly twice as efficient. Boundary conditions and the relationship of the method to streamfunction-vorticity methods are discussed.

Accuracy and Conservation Properties of a Three-Dimensional Unstructured Staggered Mesh Scheme for Fluid Dynamics

A three-dimensional unstructured mesh discretization of the rotational from of the incompressible Navier–Stokes is presented. The method uses novel and highly efficient algorithms for interpolating the velocity vector and constructing the convention term. The resulting discretization is shown to conserve mass, kinetic energy, and vorticity to machine precision both locally and globally. The spatial accuracy of the method is analyzed and found to be second order on regular meshes and first order on irregular meshes. The numerical efficiency, accuracy, and conservation properties of the method are tested on three-dimensional meshes and found to be in agreement with theory.

Conservation Properties of Unstructured Staggered Mesh Schemes

Classic Cartesian staggered mesh schemes have a number of attractive properties. They do not display spurious pressure modes and they have been shown to locally conserve, mass, momentum, kinetic energy, and circulation to machine precision. Recently, a number of generalizations of the staggered mesh approach have been proposed for unstructured (triangular or tetrahedral) meshes. These unstructured staggered mesh methods have been created to retain the attractive pressure aspects and mass conservation properties of the classic Cartesian mesh method. This work addresses the momentum, kinetic energy, and circulation conservation properties of unstructured staggered mesh methods. It is shown that with certain choices of the velocity interpolation, unstructured staggered mesh discretizations of the divergence form of the Navier–Stokes equations can conserve kinetic energy and momentum both locally and globally. In addition, it is shown that unstructured staggered mesh discretizations of the rotational form of the Navier–Stokes equations can conserve kinetic energy and circulation both locally and globally. The analysis includes viscous terms and a generalization of the concept of conservation in the presence of viscosity to include a negative definite dissipation term in the kinetic energy equation. These novel conserving unstructured staggered mesh schemes have not been previously analyzed. It is shown that they are first-order accurate on nonuniform two-dimensional unstructured meshes and second-order accurate on uniform unstructured meshes. Numerical confirmation of the conservation properties and the order of accuracy of these unstructured staggered mesh methods is presented.

Inf-sup conditions for finite-difference approximations of the stokes equations

AbstractInf-sup conditions are proven for three finite-difference approximations of the Stokes equa-tions. The finite-difference approximations use a staggered-mesh scheme and the schemes resulting from the backward and the forward differencings.