An explicit finite volume scheme on staggered grids for the Euler equations: Unstructured meshes,stability analysis,and energy conservation

We set up a numerical strategy for the simulation of the Euler equations, in the framework of finite volume staggered discretizations where numerical densities, energies, and velocities are stored on different locations. The main difficulty relies on the treatment of the total energy, which mixes quantities stored on different grids. The proposed method is strongly inspired, on the one hand, from the kinetic framework for the definition of the numerical fluxes, and, on the other hand, from the discrete duality finite volume (DDFV) framework, which has been designed for the simulation of elliptic equations on complex meshes. The time discretization is explicit and we exhibit stability conditions that guaranty the positivity of the discrete densities and internal energies. Moreover, while the scheme works on the internal energy equation, we can define a discrete total energy which satisfies a local conservation equation. We provide a set of numerical simulations to illustrate the behavior of the scheme.     

Finite volume scheme for the solution of fluid flow problems on unstructured non‐staggered grids

A recently developed non‐staggered methodology which uses the principle of applying fourth‐order dissipation to the governing pressure‐correction equation is developed so it can be applied to unstructured grids. A finite volume methodology is used for discretization. The fourth‐order dissipation term is found using second‐order gradient operators. This makes it straightforward to incorporate the dissipation term on unstructured grids. The new methodology is compared with solutions from a standard finite volume second‐order flow solver and is also tested for a standard laminar driven‐lid flow problem with grids systems that do not have a uniform structure. Finally, we demonstrate how the new methodology can be used to predict flow over a wavy boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

On the development of a triple‐preserving Maxwell's equations solver in non‐staggered grids

We present in this paper a finite difference solver for Maxwell's equations in non‐staggered grids. The scheme formulated in time domain theoretically preserves the properties of zero‐divergence, symplecticity, and dispersion relation. The mathematically inherent Hamiltonian can be also retained all the time. Moreover, both spatial and temporal terms are approximated to yield the equal fourth‐order spatial and temporal accuracies. Through the computational exercises, modified equation analysis and Fourier analysis, it can be clearly demonstrated that the proposed triple‐preserving solver is computationally accurate and efficient for use to predict the Maxwell's solutions. Copyright © 2009 John Wiley & Sons, Ltd.     

An overlapping control volume method for the Navier–Stokes equations on non‐staggered grids

An algorithm, based on the overlapping control volume (OCV) method, for the solution of the steady and unsteady two‐dimensional incompressible Navier–Stokes equations in complex geometry is presented. The primitive variable formulation is solved on a non‐staggered grid arrangement. The problem of pressure–velocity decoupling is circumvented by using momentum interpolation. The accuracy and effectiveness of the method is established by solving five steady state and one unsteady test problems. The numerical solutions obtained using the technique are in good agreement with the analytical and benchmark solutions available in the literature. On uniform grids, the method gives second‐order accuracy for both diffusion‐ and convection‐dominated flows. There is little loss of accuracy on grids that are moderately non‐orthogonal. Copyright © 1999 John Wiley & Sons, Ltd.     

Impact of the fourth‐order compact difference scheme on computational properties of 3D grids in a nonhydrostatic model

To investigate the potential of the fourth‐order compact difference scheme within the specific context of numerical atmospheric models, a linear baroclinic adjustment system is discretized, using a variety of candidates for practically meaningful staggered 3D grids. A unified method is introduced to derive the dispersion relationship of the baroclinic geostrophic adjustment process. Eight popular 3D grids are obtained by combining contemporary horizontal staggered grids, such as the Arakawa C and Eliassen grids, with optimal vertical grids, such as the Lorenz and Charney‐Phillip (CP) grids, and their time‐staggered versions. The errors produced on the 3D grids in describing the baroclinic geostrophic adjustment process relative to the differential case are compared in terms of frequency and group velocity components with the elimination of implementation error. The results show that by utilizing the fourth‐order compact difference scheme with high precision, instead of the conventional second‐order centered difference scheme, the errors in describing baroclinic geostrophic adjustment process decrease but only when using the combinations of the horizontally staggered Arakawa C grid and the vertically staggered CP or the C/CP grid, the time‐horizontally staggered Eliassen (EL) grid and vertically staggered CP or the EL/CP grid, the C grid and the vertically time‐staggered versions of Lorenz (LTS) grid or C/LTS grid, EL grid and LTS grid or EL/LTS grid. The errors were found to increase for specific waves on the rest grids. It can be concluded that errors produced on the chosen 3D grids do not universally decrease when using the fourth‐order compact difference scheme; hence, care should be taken when implementing the fourth‐order compact difference scheme, otherwise, the expected benefits may be offset by increased errors. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical simulation of the vertical structure of discontinuous flows

A numerical method to solve the Reynolds‐averaged Navier–Stokes equations with the presence of discontinuities is outlined and discussed. The pressure is decomposed into the sum of a hydrostatic component and a hydrodynamic component. The numerical technique is based upon the classical staggered grids and semi‐implicit finite difference methods applied for quasi‐ and non‐hydrostatic flows. The advection terms in the momentum equations are approximated in order to conserve mass and momentum following the principles recently developed for the numerical simulation of shallow water flows with large gradients. Conservation of these properties is the most important aspect to represent near local discontinuities in the solution, following from sharp bottom gradients or hydraulic jumps. The model is applied to reproduce the flow over a step where a hydraulic jump forms downstream. The hydrostatic pressure assumption fails to represent this type of flow mainly because of the pressure deviation from the hydrostatic values downstream the step. Fairly accurate results are obtained from the numerical model compared with experimental data. Deviation from the data is found to be inherent to the standard k–ε model implemented. Copyright © 2001 John Wiley & Sons, Ltd.     

A comparison of spatial grids for numerical modelling of flows in near-coastal seas

Three-dimensional algorithms for the numerical computation of flows caused by tides or meteorological forcing are developed for four of Arakawa's spatial grid types using a spectral method in the vertical dimension. Three of the grids, in which the velocity components are computed at the same grid points, offer potential advantages over the commonly used C-grid. The computed results from the four grids are compared for three test problems based on the linearized hydrodynamical equations. It is concluded that the B-grid provides a viable alternative to the C-grid, with significant advantages when a spectral method is used.     

A study of properties of adaptive quadratic grids for three-dimensional near-surface field computations

Curved geometries and the corresponding near-surface fields typically require a large number of linear computational elements. High-order numerical solvers have been primarily used with low-order meshes. There is a need for curved, high-order computational elements. Typical near-surface meshes consist of hexahedral and/or prismatic elements. The present work studies the employment of quadratic meshes that are relatively coarse for field simulations. Directionally quadratic high-order elements are proposed for the near-surface field regions. The quadratic meshes are compared with the conventional low-order ones in terms of accuracy and efficiency. The cases considered include closed surface volume calculations, as well as computation of gradients of several analytic fields. A special method of adaptive local quadratic meshes is proposed and evaluated. Truncation error analysis for quadratic grids yields comparison with the conventional linear hexahedral/prismatic meshes, which are subject to typical distortions such as stretching, skewness, and torsion.     

用混合网格求解三维可压雷诺平均Navier—Stokes方程

用混合网格求解了三维紊流 N-S方程。在物面附近采用三棱柱网格 ,其它区域则采用四面体网格。方程的求解采用 Jamson的有限体积法 ,紊流模型采用两层 Baldwin-Lomax代数紊流模型。数值算例表明 ,用混合网格求解三维紊流 Navier-Stokes是非常有效的。     

A scheme for automatic generation of boundary-fitted depth- and depth-gradient- dependent grids in arbitrary two-dimensional regions

In this paper we present a scheme for the numerical generation of boundary-fitted grids that adapt to both water depth and depth gradient. The scheme can be used in arbitrary two-dimensional regions and is based on the application of the well-known control function approach to generate adaptive grids. The method includes the evaluation of water depths at the grid points from a known distribution of depth points and their associated depths plus a procedure for the numerical evaluation of depth gradients. It is demonstrated that the smoothness of the grid can be enhanced by introducing a suitable filtering technique.     

基于重叠网格与结构网格的圆柱绕流数值模拟

为研究重叠网格与结构网格在圆柱绕流数值模拟中的区别,以二维圆柱为例,利用有限元分析软件ANSYS 19.2中的DM与Mesh建立模型并划分重叠网格,利用ANSYS 19.2中的ICEM建立模型并划分结构网格。采用FLUENT 19.2中laminar模型模拟分析系统中的平均升力系数、平均阻力系数、斯特劳哈尔数St等流体动力特性。通过改变流体流速得到两种不同网格下各6组雷诺数Re,这6组雷诺数在60~160之间。结果表明:结构网格与重叠网格的St都随着Re的增加而增加,但相同雷诺数下重叠网格对应的St数值更大,St的增长速度更快;重叠网格与结构网格的平均升力系数与平均阻力系数随着Re的增加趋于稳定的速度都加快,但结构网格的平均升力系数与平均阻力系数趋于稳定的速度更快,且两种网格的平均升力系数与平均阻力系数趋于稳定速度的差距逐渐缩小,当Re=160时,两种网格的平均升力系数与平均阻力系数趋于稳定的速度几乎相同;当雷诺数在60~160之间时,采用重叠网格计算出来的斯特劳哈尔数比结构网格更加接近理论值;从升力功率谱密度分布曲线中可以看出,随着雷诺数的增加,两种网格下的频率逐渐变大,并且相同雷诺数下重叠网格的频率比结构网格大。     

The effect of using non-orthogonal boundary-fitted grids for solving the shallow water equations

The purpose of this paper is to examine the effects of using non-orthogonal boundary-fitted grids for the numerical solution of the shallow water equations. Two geometries with well known analytical solutions are introduced in order to investigate the accuracy of the numerical solutions. The results verify that a reasonable departure from orthogonality can be allowed when the rate of change of cell areas is kept sufficiently small (i.e. it is not necessary to create a strictly orthogonal grid when the grid is sufficiently smooth).     

A technique of flux reconstruction at the interfaces of nonconforming grids for aeroacoustic simulations

A flux reconstruction technique is presented to perform aeroacoustic computations using implicit high-order spatial schemes on multiblock structured grids with nonconforming interfaces. The use of such grids, with mesh spacing discontinuities across the block interfaces, eases local mesh refinements, simplifies the mesh generation process, and thus facilitates the computation of turbulent flows. In this work, the spatial discretization consists of sixth-order finite-volume implicit schemes with low-dispersion and low-dissipation properties. The flux reconstruction is based on the combination of noncentered schemes with local interpolations to define ghost cells and compute flux values at the grid interfaces. The flow variables in the ghost cells are calculated from the flow field in the grid cells using a meshless interpolation with radial basis functions. In this study, the flux reconstruction is applied to both plane and curved nonconforming interfaces. The performance of the method is first evaluated by performing two-dimensional simulations of the propagation of an acoustic pulse and of the convection of a vortex on Cartesian and wavy grids. No significant spurious noise is produced at the grid interfaces. The applicability of the flux reconstruction to a three-dimensional computation is then demonstrated by simulating a jet at a Mach number of 0.9 and a diameter-based Reynolds number of 4×10⁵ on a Cartesian grid. The nonconforming grid interface located downstream of the jet potential core does not appreciably affect the flow development and the jet sound field, while reducing the number of mesh points by a factor of approximately two.     

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

It 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.     

Algebraic turbulence models for the computation of two-dimensional high-speed flows using unstructured grids

The incorporation of algebraic turbulence models in a solver for the 2D compressible Navier–Stokes equations using triangular grids is described. A practical way to use the Cebeci–Smith model and to modify it in separated regions is proposed. The ability of the model to predict high-speed perfect-gas boundary layers is investigated from a numerical point of view.     

Octree partitioning of hybrid grids for parallel adaptive viscous flow simulations

A parallel finite volume method for the Navier–Stokes equations with adaptive hybrid prismatic/tetrahedral grids is presented and evaluated in terms of parallel performance. A new method of domain partitioning for complex 3D hybrid meshes is also presented. It is based on orthogonal bisection of a special octree corresponding to the hybrid mesh. The octree is generated automatically and can handle any type of 3D geometry and domain connectivity. One important property of the octree-based partitioning that is exploited is the octree's ability to yield load-balanced partitions that follow the shape of the geometry. This biasing of the octree results in a reduced number of grid elements on the interpartition boundaries and thus fewer data to communicate among processors. Furthermore, the octree-based partitioning gives similar quality of partitions for very different geometries, while requiring minimal user interaction and little computational time. The partitioning method is evaluated in terms of quality of the subdomains as well as execution time. Viscous flow simulations for different geometries are employed to examine the effectiveness of the octree-based partitioning and to test the scalability of parallel execution of the Navier–Stokes solver and hybrid grid adapter on two different parallel systems, the Intel Paragon and the IBM SP2. © 1998 John Wiley & Sons, Ltd.     

Computational aspects of a pseudo-elastic constitutive model for muscle properties in a soft-bodied arthropod

In this paper we discuss the computational implementation of a new constitutive model that describes the muscle properties in a soft-bodied arthropod. Qualitatively, the muscle tissues behave similar to particle-reinforced rubber and are capable of large non-linear elastic deformations, show a hysteretic behavior, and display stress softening during the first few cycles of repeated loading. Such behavior can be described by the framework of pseudo-elastic transversely isotropic hyperelasticity. The computational model assumes compressible overall response, and is based upon a multiplicative split of the deformation gradient tensor into volumetric and isochoric parts. Details regarding the implementation of the computational model in the context of an implicit finite element solution procedure are presented. In particular, an explicit expression is provided for the material tangent stiffness tensor. Results obtained utilizing the new implementation are also presented.     

Stability of theoretical model for catastrophic weather prediction

Stability related to theoretical model for catastrophic weather prediction, which includes non-hydrostatic perfect elastic model and anelastic model, is discussed and analyzed in detail. It is proved that non-hydrostatic perfect elastic equations set is stable in the class of infinitely differentiable function. However, for the anelastic equations set, its continuity equation is changed in form because of the particular hypothesis for fluid, so "the matching consisting of both viscosity coefficient and incompressible assumption" appears, thereby the most important equations set of this class in practical prediction shows the same instability in topological property as Navier-Stokes equation, which should be avoided first in practical numerical prediction. In light of this, the referenced suggestions to amend the applied model are finally presented.