Discrete Continuity Equation Research Articles

A Cell-Integrated Semi-Lagrangian Semi-Implicit Shallow-Water Model (CSLAM-SW) with Conservative and Consistent Transport

Abstract A Cartesian semi-implicit solver using the Conservative Semi-Lagrangian Multitracer (CSLAM) transport scheme is constructed and tested for shallow-water (SW) flows. The SW equations solver (CSLAM-SW) uses a discrete semi-implicit continuity equation specifically designed to ensure a conservative and consistent transport of constituents by avoiding the use of a constant mean reference state. The algorithm is constructed to be similar to typical conservative semi-Lagrangian semi-implicit schemes, requiring at each time step a single linear Helmholtz equation solution and a single application of CSLAM. The accuracy and stability of the solver is tested using four test cases for a radially propagating gravity wave and two barotropically unstable jets. In a consistency test using the new solver, the specific concentration constancy is preserved up to machine roundoff, whereas a typical formulation can have errors many orders of magnitude larger. In addition to mass conservation and consistency, CSLAM-SW also ensures shape preservation by combining the new scheme with existing shape-preserving filters. With promising SW test results, CSLAM-SW shows potential for extension to a nonhydrostatic, fully compressible system solver for numerical weather prediction and climate models.

Consistency problem with tracer advection in the Atmospheric Model GAMIL

The radon transport test, which is a widely used test case for atmospheric transport models, is carried out to evaluate the tracer advection schemes in the Grid-Point Atmospheric Model of IAP-LASG (GAMIL). Two of the three available schemes in the model are found to be associated with significant biases in the polar regions and in the upper part of the atmosphere, which implies potentially large errors in the simulation of ozone-like tracers. Theoretical analyses show that inconsistency exists between the advection schemes and the discrete continuity equation in the dynamical core of GAMIL and consequently leads to spurious sources and sinks in the tracer transport equation. The impact of this type of inconsistency is demonstrated by idealized tests and identified as the cause of the aforementioned biases. Other potential effects of this inconsistency are also discussed. Results of this study provide some hints for choosing suitable advection schemes in the GAMIL model. At least for the polar-region-concentrated atmospheric components and the closely correlated chemical species, the Flux-Form Semi-Lagrangian advection scheme produces more reasonable simulations of the large-scale transport processes without significantly increasing the computational expense.

A non‐diffusive, divergence‐free, finite volume‐based double projection method on non‐staggered grids

AbstractSecond‐order accurate projection methods for simulating time‐dependent incompressible flows on cell‐centred grids substantially belong to the class either of exact or approximate projections. In the exact method, the continuity constraint can be satisfied to machine‐accuracy but the divergence and Laplacian operators show a four‐dimension nullspace therefore spurious oscillating solutions can be introduced. In the approximate method, the continuity constraint is relaxed, the continuity equation being satisfied up to the magnitude of the local truncation error, but the compact Laplacian operator has only the constant mode. An original formulation for allowing the discrete continuity equation to be satisfied to machine‐accuracy, while using a finite volume based projection method, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by solving a second elliptic field for a scalar field obtained by prescribing that its additional discrete gradients ensure discrete continuity based on the previously adopted linear interpolation of the velocity. The resulting numerical scheme is applied to several flow problems and is proved to be accurate, stable and efficient.This paper has to be considered as the companion of: 'F. M. Denaro, A 3D second‐order accurate projection‐based finite volume code on non‐staggered, non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows. IJNMF 2006; 52(4):393–432. Now, we illustrate the details and the rigorous theoretical framework. Copyright © 2006 John Wiley & Sons, Ltd.

Positive-Definite and Monotonic Limiters for Unrestricted-Time-Step Transport Schemes

Abstract General positive-definite and monotonic limiters are described for use with unrestricted-Courant-number flux-form transport schemes. These limiters are tested using a time-split multidimensional transport scheme. The importance of minimizing the splitting errors associated with the time-split operator and of the consistency between the transport scheme and the discrete continuity equation is demonstrated.

A 3D second‐order accurate projection‐based Finite Volume code on non‐staggered, non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows

AbstractIt is well known that exact projection methods (EPM) on non‐staggered grids suffer for the presence of non‐solenoidal spurious modes. Hence, a formulation for simulating time‐dependent incompressible flows while allowing the discrete continuity equation to be satisfied up to machine‐accuracy, by using a Finite Volume‐based second‐order accurate projection method on non‐staggered and non‐uniform 3D grids, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by first solving an elliptic equation on a compact stencil that is by performing a standard approximate projection method (APM). In such a way, three sets of divergence‐free normal‐to‐face velocities can be computed. Then, a second elliptic equation for a scalar field is derived by prescribing that its additional discrete gradient ensures the continuity constraint based on the adopted linear interpolation of the velocity. Characteristics of the double projection method (DPM) are illustrated in details and stability and accuracy of the method are addressed. The resulting numerical scheme is then applied to laminar buoyancy‐driven flows and is proved to be stable and efficient. Copyright © 2006 John Wiley & Sons, Ltd.

Flooding and Drying in Discontinuous Galerkin Finite-Element Discretizations of Shallow-Water Equations. Part 1: One Dimension

Free boundaries in shallow-water equations demarcate the time-dependent water line between ``flooded'' and ``dry'' regions. We present a novel numerical algorithm to treat flooding and drying in a formally second-order explicit space discontinuous Galerkin finite-element discretization of the one-dimensional or symmetric shallow-water equations. The algorithm uses fixed Eulerian flooded elements and a mixed Eulerian--Lagrangian element at each free boundary. When the time step is suitably restricted, we show that the mean water depth is positive. This time-step restriction is based on an analysis of the discretized continuity equation while using the HLLC flux. The algorithm and its implementation are tested in comparison with a large and relevant suite of known exact solutions. The essence of the flooding and drying algorithm pivots around the analysis of a continuity equation with a fluid velocity and a pseudodensity (in the shallow water case the depth). It therefore also applies, for example, to space discontinuous Galerkin finite-element discretizations of the compressible Euler equations in which vacuum regions emerge, in analogy of the above dry regions. We believe that the approach presented can be extended to finite-volume discretizations with similar mean level and slope reconstruction.

Fully conservative finite difference scheme in cylindrical coordinates for incompressible flow simulations

A new finite difference scheme on a non-uniform staggered grid in cylindrical coordinates is proposed for incompressible flow. The scheme conserves both momentum and kinetic energy for inviscid flow with the exception of the time marching error, provided that the discrete continuity equation is satisfied. A novel pole treatment is also introduced, where a discrete radial momentum equation with the fully conservative convection scheme is introduced at the pole. The pole singularity is removed properly using analytical and numerical techniques. The kinetic energy conservation property is tested for the inviscid concentric annular flow for the proposed and existing staggered finite difference schemes in cylindrical coordinates. The pole treatment is verified for inviscid pipe flow. Mixed second and high order finite difference scheme is also proposed and the effect of the order of accuracy is demonstrated for the large eddy simulation of turbulent pipe flow.

Numerical Analysis of Saturated Soil via Finite Strain EFGM Strategy

A formulation of the Element-Free Galerkin Method (EFGM) for saturated soil with pore water is presented under Cam-clay model to overcome the difficulties which appear in the FE analysis. A weak form of the set of equations for the equilibrium of total nominal stress rate and the continuity of pore water is derived and its discretization, based on the usual EFGM strategy, e.g., the moving least square method (MLSM), is fumished within the framework of finite strain. Numerical discretization of the weak form leads to an updated Lagrangean scheme. Such a numerical strategy, in which the nonlocal characteristics incorporated is into the discretized continuity equation, is expected to be advantageous in avoiding the dependency of mesh type and size on numerical solution of strain localization problem. The stiffness matrix is numerically evaluated by the fourth order of the Gauss integration, and boundary conditions are treated with the help of the penalty method. To trace the bifurcation phenomena in the relation of load and deflection, several patterns of initial imperfection are given along the surface of soil specimen. Shear banding within a soil specimen can be introduced while avoiding the numerical difficulties which depend on mesh size in the corresponding FE analyses.

Nonstaggered APPLE Algorithm for Incompressible Viscous Flow in Curvilinear Coordinates

The NAPPLE algorithm for incompressible viscous flow on Cartesian grid system is extended to nonorthogonal curvilinear grid system in this paper. A pressure-linked equation is obtained by substituting the discretized momentum equations into the discretized continuity equation. Instead of employing a velocity interpolation such as pressure-weighted interpolation method (PWIM), a particular approximation is adopted to circumvent the checkerboard error such that the solution does not depend on the under-relaxation factor. This is a distinctive feature of the present method. Furthermore, the pressure is directly solved from the pressure-linked equation without recourse to a pressure-correction equation. In the use of the NAPPLE algorithm, solving the pressure-linked equation is as simple as solving a heat conduction equation. Through two well-documented examples, performance of the NAPPLE algorithm is validated for both buoyancy-driven and pressure-driven flows.

Exact charge conservation scheme for Particle-in-Cell simulation with an arbitrary form-factor

The new method of local electric current density assignment in the Particle-in-Cell code in Cartesian geometry is presented. The method is valid for an arbitrary quasi-particle form-factor assuming that quasi-particle trajectory over time step is a straight line. The method allows one to implement the PIC code without solving Poisson equation. The presented formula for the current density associated with the motion of a single quasi-particle is the unique linear combination of form-factor differences in consistency with the discrete continuity equation. The computation scheme is demonstrated in 2D and 3D.

Dilation-Free Solutions for the Incompressible Flow Equations on Nonstaggered Grids

The present method employs a pressure Poisson formulation that has the unique capability of satisfying the discrete continuity equation exactly while producing smooth pressure on nonstaggered grids. A fourth-order accurate discretization is employed for the pressure gradient operator in both the momentum equation and the pressure Poisson equation. The use of higher-order accurate approximations for the pressure derivatives results in the enlargement of Laplace operator, which incorporates both odd and even grid points and thus spatially couples the pressure field. The numerical solutions are compared to solutions obtained from the classic pressure Poisson approach, which uses artificial dissipation to couple the pressure field, as well as benchmark results of a stream function-vorticity formulation (which automatically produces dilatation-free solutions).

Incompressible navier-stokes solutions with a new primitive variable solver

A new primitive variable RNS/NS formulation for incompressible viscous flow is presented. This formulation, which has been applied for both two- and three-dimensional flows, is a semi-implicit, time-marching method using standard primitive variables. For this scheme the discrete continuity equation is satisfied directly at each time step, without artificial compressibility or pressure Poisson concepts. All the linear terms (continuity, pressure gradients, diffusion) are treated implicitly and all the non-linear convection terms are treated explicitly. Since all acoustic behavior is implicit, a convection-only CFL stability condition results even for incompressible flow. This is true without a P t term added to the continuity equation. The formulation is robust and provides accurate solutions that compare well with experimental data.

Climate Simulations with a Semi-Lagrangian Version of the NCAR Community Climate Model

Abstract A semi-Lagrangian version of the National Center for Atmospheric Research Community Climate Model is developed. Special consideration is given to energy consistency aspects. In particular, approximations are developed in which the pressure gradient in the momentum equations is consistent with the energy conversion term in the thermodynamic equation. In addition, consistency between the discrete continuity equation and the vertical velocity ω in the energy conversion term of the thermodynamic equation is obtained. Simulated states from multiple-year simulations from the semi-Lagrangian and Eulerian versions are compared. The principal difference in the simulated climate appears in the zonal average temperature. The semi-Lagrangian simulation is colder than the Eulerian at and above the tropical tropopause. The terms producing the thermodynamic balance are examined. It is argued that the semi-Lagrangian scheme produces less computational smoothing of the temperature at the tropopause than the first...

Finite Element Analysis of LSI Packages with Three-Dimensional Steady Heat Transfer and Fluid Flow by the SIMPLE Algorithm Formulation.

A finite element procedure based on the SIMPLE algorithm is presented for the solution of steady-state Navier-Stokes equations for three-dimensional forced convection. The finite element formulation for pressure correction is derived from the discretized continuity equation. The computational results obtained by the procedure reveal reasonable accuracy with rather small computation time for a converged steady-state solution than by the unsteady method. By using this procedure the parametric study of cooling of a LSI module on a double-sided card is performed. The effects of thermal resistance of a multilayer circuit board on the temperature distribution of LSI module are intensively examined. It is found that the thermal leak through the circuit board cannot be ignored for memories on multilayer double-sided cards, on the other hand, it can be ignored for those on single-layer double-sided cards.

A primitive variable method for the solution of three-dimensional incompressible viscous flows

In this paper we present a new primitive variable method for the solution of the three-dimensional, incompressible, Reynolds averaged Navier-Stokes equations in generalized curvilinear coordinates. The governing equations are discretized on a non-staggered grid and the discrete continuity equation is replaced by a discrete pressure-Poisson equation. The discrete pressure equation is designed in such a way that: (i) the compatibility condition for the Poisson-Neumann problem is automatically satisfied, and (ii) the discrete incompresibility constraint is satisfied to, at least, truncation error accuracy while the computed pressure is smooth. The momentum equations are integrated in time using the four-stage Runge-Kutta algorithm while the pressure equation is solved using the point-successive relaxation technique. The method is applied to calculate the turbulent flow field over a ship model. The computed results are in very good agreement with the experimental data.

Iterative solution of Navier-Stokes dual variable difference equations

The dual transformation applied to implicit finite difference approximations of the Navier-Stokes equations reduces the number of unknowns by a factor of three, removes the pressures from the discrete equations and produces velocities which satisfy the discrete continuity equation exactly. New iterative methods for the solution of the unsymmetrical dual variable system are developed and are proven to converge for a large class of problems. These iterative methods involve a sequence of discrete Laplacian systems whose solutions converge to the solution of the dual variable system. They take advantage of the special structure of the dual variable coefficient matrix, are very fast compared to the direct methods currently used, are less memory intensive and can be more easily vectorized and parallelized.

The discrete continuity equation in primitive variable solutions of incompressible flow

The use of a non-staggered computational grid for the numerical solutions of the incompressible flow equations has many advantages over the use of a staggered grid. A penalty, however, is inherent in the finite-difference approximations of the governing equations on non-staggered grids. In the primitive-variable solutions, the penalty is that the discrete continuity equation does not converge to machine accuracy. Rather it converges to a source term which is proportional to the fourth-order derivative of the pressure, the time increment, and the square of the grid spacing. An approach which minimizes the error in the discrete continuity equation is developed. Numerical results obtained for the driven cavity problem confirm the analytical developments.

The discrete continuity equation in primitive variable solutions of incompressible flow

The discrete continuity equation in primitive variable solutions of incompressible flow

Investigation into the finite element analysis of enclosed turbulent diffusion flames

A complete finite element model of enclosed turbulent diffusion flames is presented. A velocity-pressure formulation is preferred in the analysis. A mixed interpolation is employed for the velocity and pressure. In the solution of the Navier-Stokes equations, a segregated formulation is adopted, where the pressure discretization equation is obtained directly from the finite element discretized continuity equation considering the relationships established between the velocity and pressure in the finite element discretized momentum equations. The state of turbulence is defined by a k-∈ model of turbulence. Near solid boundaries, a wall-functions approach is employed. The combustion rates are estimated by the eddy dissipation concept. The expensive direct treatment of the integrodifferential equations of radiation is avoided by employing the moment method, which allows the derivation of an approximate elliptic differential equation for the radiation intensity. The proposed finite element model is verified by investigating several examples. The results are compared with experiments and finite difference predictions.

Finite element solution of an enclosed turbulent diffusion flame

AbstractA finite element formulation of enclosed turbulent diffusion flames is presented. A primitive variables approach is preferred in the analysis. A mixed interpolation is employed for the velocity and pressure. In the solution of the Navier‐Stokes equations, a segregated formulation is adopted, where the pressure discretization equation is obtained directly from the discretized continuity equation, considering the velocity‐pressure relationships in the discretized momentum equations. The state of turbulence is defined by a κ–ϵ model. Near solid boundaries, a wall function approach is employed. The combustion rates are estimated using the eddy dissipation concept. The expensive direct treatment of the integrodifferential equations of radiation is avoided by employing the moment method, which allows the derivation of an approximate local field equation for the radiation intensity. The proposed finite element model is verified by investigating a technical turbulent diffusion flame of semi‐industrial size, and comparing the results with experiments and finite difference predictions.