Chebyshev finite spectral method with extended moving grids

A Chebyshev finite spectral method on non-uniform meshes is proposed. An equidistribution scheme for two types of extended moving grids is used to generate grids. One type is designed to provide better resolution for the wave surface, and the other type is for highly variable gradients. The method has high-order accuracy because of the use of the Chebyshev polynomial as the basis function. The polynomial is used to interpolate the values between the two non-uniform meshes from a previous time step to the current time step. To attain high accuracy in the time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme is used. To avoid numerical oscillations caused by the dispersion term in the Korteweg-de Vries (KdV) equation, a numerical technique on non-uniform meshes is introduced. The proposed numerical scheme is validated by the applications to the Burgers equation (nonlinear convectiondiffusion problems) and the KdV equation (single solitary and 2-solitary wave problems), where analytical solutions are available for comparisons. Numerical results agree very well with the corresponding analytical solutions in all cases.     

Treatment of numerical diffusion in strong convective flows

A three-dimensional extension of the QUICK scheme adapted for the finite volume method and non-uniform grids is presented to handle convection-diffusion problems for high Peclet numbers and steep gradients. The algorithm is based on three-dimensional quadratic interpolation functions in which the transverse curvature terms are maintained and the diagonal dominance of the coefficient matrix is preserved. All formulae are explicitly given in an appendix. Results obtained with the classical upwind (UDS), the simplified QUICK (transverse terms neglected) and the present full QUICK schemes are given for two benchmark problems, one two-dimensional, steady state and the other three-dimensional, unsteady state. Both QUICK schemes are shown to give superior solutions compared with the UDS in terms of accuracy and efficiency. The full QUICK scheme performs better than the simplified QUICK, giving even for coarse grids acceptable results closer to the analytical solutions, while the computational time is not affected much.     

CAD-consistent adaptive refinement using a NURBS-based discontinuous Galerkin method

This study concerns the development of a new method combining high-order computer-aided design (CAD)-consistent grids and adaptive refinement/coarsening strategies for efficient analysis of compressible flows. The proposed approach allows to use geometrical data from CAD without any approximation. Thus, the simulations are based on the exact geometry, even for the coarsest discretizations. Combining this property with a local refinement method allows to start computations using very coarse grids and then relies on dynamic adaption to construct suitable computational domains. The resulting approach facilitates interactions between CAD and computational fluid dynamics solvers and focuses the computational effort on the capture of physical phenomena, since geometry is exactly taken into account. The proposed methodology is based on a discontinuous Galerkin method for compressible Navier-Stokes equations, modified to use nonuniform rational B-Spline representations. Local refinement and coarsening are introduced using intrinsic properties of nonuniform rational B-Spline associated with a local error indicator. A verification of the accuracy of the method is achieved and a set of applications are presented, ranging from viscous subsonic to inviscid trans- and supersonic flow problems.     

A Comparison Study Between an Adaptive Quadtree Grid and Uniform Grid Upscaling for Reservoir Simulation

An adaptive quadtree grid generation algorithm is developed and applied for tracer and multiphase flow in channelized heterogeneous porous media. Adaptivity was guided using two different approaches. In the first approach, wavelet transformation was used to generate a refinement field based on permeability variations. The second approach uses flow information based on the solution of an initial-time fine-scale problem. The resulting grids were compared with uniform grid upscaling. For uniform upscaling, two commonly applied methods were used: renormalization upscaling and local-global upscaling. The velocities obtained by adaptive grid and uniformly upscaled grids, were downscaled. This procedure allows us to separate the upscaling errors, on adaptive and uniform grids, from the numerical dispersion errors resulting from solving the saturation equation on a coarse grid. The simulation results obtained by solving on flow-based adaptive quadtree grids for the case of a single phase flow show reasonable agreement with more computationally demanding fine-scale models and local-global upscaled models. For the multiphase case, the agreement is less evident, especially in piston-like displacement cases with sharp frontal movement. In such cases a non-iterative transmissibility upscaling procedure for adaptive grid is shown to significantly reduce the errors and make the adaptive grid comparable to iterative local-global upscaling. Furthermore, existence of barriers in a porous medium complicates both upscaling and grid adaptivity. This issue is addressed by adapting the grid using a combination of flow information and a permeability based heuristic criterion.     

Recent Development on the Conservation Property of Chimera

An estimate on the conservation error due to the non-conservative data interpolation scheme for overset grids is given in this paper. It is shown that the conservation error is a first-order term if second-order conservative schemes are employed for the Chimera grids and if discontinuities are located away from overlapped grid interfaces. Therefore in the limit of global grid refinement, valid numerical solutions should be obtained with a data interpolation scheme. In one demonstration case the conservation error in the original Chimera scheme was shown to affect flow even without discontinuities on coarse to medium grids. The conservative Chimera scheme was shown to give significantly better solutions than the original Chimera scheme on these grids with other factors being the same.     

Calculation of viscous and inviscid gas flows on unstructured grids using the AUSM scheme

An approach to the solution of the two-dimensional Navier-Stokes equations on triangular unstructured grids is considered. The method is based on the key idea of the Godunov scheme, namely, the advisability of solving the Riemann problem of arbitrary discontinuity breakdown. In the calculations the derivatives with respect to space are approximated with both the first and the second order. However, as distinct from the conventional Godunov method, in calculating the fluxes across the cell boundaries the Riemann problem is solved using the Advection Upstream Splitting Method (AUSM). The concepts involved in the AUSM scheme are discussed. The solution of the discontinuity breakdown problem obtained within the framework of this approach is compared with the results obtained using the Godunov method. Numerical solutions of some problems of viscous and inviscid perfect-gas flows obtained on unstructured grids of different fineness and those obtained on structured grids are also compared. The effect of the spatial approximation order on the accuracy of numerical solutions is studied.     

A fast nested multi‐grid viscous flow solver for adaptive Cartesian/Quad grids

A nested multi‐grid solution algorithm has been developed for an adaptive Cartesian/Quad grid viscous flow solver. Body‐fitted adaptive Quad (quadrilateral) grids are generated around solid bodies through ‘surface extrusion’. The Quad grids are then overlapped with an adaptive Cartesian grid. Quadtree data structures are employed to record both the Quad and Cartesian grids. The Cartesian grid is generated through recursive sub‐division of a single root, whereas the Quad grids start from multiple roots—a forest of Quadtrees, representing the coarsest possible Quad grids. Cell‐cutting is performed at the Cartesian/Quad grid interface to merge the Cartesian and Quad grids into a single unstructured grid with arbitrary cell topologies (i.e., arbitrary polygons). Because of the hierarchical nature of the data structure, many levels of coarse grids have already been built in. The coarsening of the unstructured grid is based on the Quadtree data structure through reverse tree traversal. Issues arising from grid coarsening are discussed and solutions are developed. The flow solver is based on a cell‐centered finite volume discretization, Roe's flux splitting, a least‐squares linear reconstruction, and a differentiable limiter developed by Venkatakrishnan in a modified form. A local time stepping scheme is used to handle very small cut cells produced in cell‐cutting. Several cycling strategies, such as the saw‐tooth, W‐ and V‐cycles, have been studies. The V‐cycle has been found to be the most efficient. In general, the multi‐grid solution algorithm has been shown to greatly speed up convergence to steady state—by one to two orders. Copyright © 2000 John Wiley & Sons, Ltd.     

Implementation of some higher-order convection schemes on non-uniform grids

A generalized formulation is applied to implement the quadratic upstream interpolation (QUICK) scheme, the second-order upwind (SOU) scheme and the second-order hybrid scheme (SHYBRID) on non-uniform grids. The implementation method is simple. The accuracy and efficiency of these higher-order schemes on non-uniform grids are assessed. Three well-known bench mark convection-diffusion problems and a fluid flow problem are revisited using non-uniform grids. These are: (1) transport of a scalar tracer by a uniform velocity field; (2) heat transport in a recirculating flow; (3) two-dimensional non-linear Burgers equations; and (4) a two-dimensional incompressible Navier-Stokes flow which is similar to the classical lid-driven cavity flow. The known exact solutions of the last three problems make it possible to thoroughly evaluate accuracies of various uniform and non-uniform grids. Higher accuracy is obtained for fewer grid points on non-uniform grids. The order of accuracy of the examined schemes is maintained for some tested problems if the distribution of non-uniform grid points is properly chosen.     

含泥质夹层油藏蒸汽注采过程的自适应网格算法研究

根据泥质夹层的低渗特性及空间分布，本文提出了一种含泥质夹层油藏网格渗透率的粗化计算方法，并在此基础上，将自适应网格算法应用于含泥质夹层油藏的数值模拟，提升其计算效率．在计算过程中，网格的动态划分仅依据流体物理量的变化，泥质夹层区域不全部采用细网格，仅针对流动锋面处的泥质夹层采用细网格，其余泥质夹层处采用不同程度的粗网格．相较于传统算法，网格数大幅下降．数值算例表明，自适应网格算法的计算结果精度与全精细网格一致，能够准确模拟出泥质夹层对于流体的阻碍作用，同时计算效率得到大幅提升，约为全精细网格算法的3~7 倍．     

NEAR-WAKE FLOW CALCULATIONS BY AN IMPLICIT 2D NAVIER-STOKES ROE RIEMANN SOLVER

Abstract

A flux formulation using a projected 2D Roe Riemann solver on unstructured grids (R2D Solver) is introduced for solving the Navier-Stokes equations and is applied to calculations of axisymmetric laminar near-wake flows behind a spherical-conical body. The numerical framework was first developed by P. L. Roe et al, in the late eighties. They looked for unsteady solutions to Euler's equations using a rather simple but exact three state linearization on triangular grids and decomposing the solution using some effective wave models. Our approach differs from their techniques by constructing a second order accurate and conservative flux functions under the well-known classical finite volume formulation. However, our Riemann Solver is obtained by a suitable linearization procedure upon all three prescribed nodal values given on each triangle. Our numerical method is applied to a Mach 4.3 flow problem for refined unstructured triangular grid behind the body. Numerical results indicate that our technique is stable, accurate and converges successfully to a stationary solution as the cell size is reduced from the coarse lo the finest grid.     

The numerical solution of the unsteady natural convection flow in a square cavity at high Rayleigh number using SADI method

The unsteady natural convection flow in a square cavity at high Rayleigh number Ra=10 ⁷ and 2×10 ⁷ has been computed using cubic spline integration. The required solutions to the two dimensional Navier-Stokes and energy equations have been obtained using two alternate numerical formulations on non-uniform grids. The main features of the transient flow have been briefly discussed. The results obtained by using the present method are in good agreement with the theoretical predictions ^[1,2].The steady state results have been compared with accurate solutions presented recently for Ra=10 ⁷.     

A bounded convection scheme for the overlapping control volume approach

This paper introduces a flux-limited scheme FLOCV for the overlapping control volume (OCV) approach to 2D steady and unsteady convection–diffusion problems on structured non-orthogonal grids. FLOCV switches from second- to first-order interpolation in the presence of extrema. Smooth switching between the two is ensured by weighted average second- and first-order upwind differencing, with the weights being dynamically determined. Five convective test problems are solved using this scheme and results are compared with known analytical solutions. It is found that FLOCV approximately retains second-order accuracy of the base discretization scheme on uniform grids and smooth non-uniform orthogonal grids. It is also found effective in removing oscillations for problems with discontinuities on both orthogonal and non-orthogonal grids, with little degradation of accuracy. © 1997 John Wiley & Sons, Ltd.     

Stream function–potential function coordinates for aeroacoustics and unsteady aerodynamics

‘Stream function as a coordinate approach’ (SFC) combined with compact high-order finite difference schemes has been developed and applied to aeroacoustics and unsteady aerodynamics problems. Straightforward implementation of SFC creates coarse grids at the vicinity of stagnation points that smears high-order numerical computations. Grid clustering is employed to resolve coarse grid near stagnations points. The agreement between numerical results and particle image velocimetry (PIV) measurements for flapping airfoil shows the robustness of the current approach for performing high-order computations.     

Assessment of ghost-cell based cut-cell method for large-eddy simulations of compressible flows at high Reynolds number

Large-eddy simulations (LES) of high Reynolds number flows are performed using a non-body conformal method in conjunction with a wall model. We use a simple wall function to model the wall-shear stress and the truncation error of the numerical discretization to model the sub-grid scale turbulence (implicit LES), although these can be easily replaced if necessary. The validation cases are: turbulent flow through an inclined channel, turbulent flow over a wavy surface, and supersonic flow over a circular cylinder. Since the near-wall grids are naturally coarse, the key is to use a method that is capable of capturing the flow dynamics accurately in the vicinity of the interface. Towards the purpose, we develop a Cartesian cut-cell method, referred to as the ghost-cell based cut-cell method (GC-CCM), in the context of fully compressible solutions of Navier–Stokes equations. This method employs ghost-cells inside the solid interface such that the local spatial reconstruction remains consistent everywhere including in the vicinity of the boundary. In order to capture the near-wall flow behavior more accurately with coarse grids, this method decomposes cell faces of merged cells and computes fluxes through each decomposed segment separately. The objective of this work is to qualify whether the proposed method can accurately represent the high Reynolds number flows in the vicinity of immersed interfaces. To analyze the performance of the proposed method, we compare the results to the corresponding numerical results from the two other non-body conformal methods, namely the ghost-cell based immersed boundary method (GCIBM) and standard cut-cell method (S-CCM), that are implemented in the same numerical solver. The comparison demonstrates that the proposed method is capable of capturing near-wall flows relatively accurately with coarse grids.     

Simulation seismic wave propagation in topographic structures using asymmetric staggered grids

A new 3 D finite- difference ( FD ) method of spatially asymmetric staggered grids was presented to simulate elastic wave propagation in topographic structures. The method approximated the first-order elastic wave equations by irregular grids finite difference operator with second-order time precise and fourth-order spatial precise. Additional introduced finite difference formula solved the asymmetric problem arisen in non-uniform staggered grid scheme, The method had no interpolation between the fine and coarse grids. All grids were computed at the same spatial iteration. Complicated geometrical structures like rough submarine interface, fault and nonplanar interfaces were treated with fine irregular grids. Theoretical analysis and numerical simulations show that this method saves considerable memory and computing time, at the same time, has satisfactory stability and accuracy.     

Computation of turbulent flow past an array of cylinders using a spectral method with Brinkman penalization

A major difficulty in computing engineering flows at high Reynolds number is the need for non-uniform grids adapted to solid boundaries that may be moving or changing shape. These non-uniform grids are expensive to calculate and cannot be used with the most accurate or efficient numerical schemes. We present one solution to this problem: a Brinkman (volume) penalization of the obstacle which allows an efficient pseudo-spectral method to be used to solve the Navier–Stokes equations on a Cartesian grid. Although this is the most severe test of the penalization (due to the global support of the Fourier basis), it is shown that the method still yields reasonable results. We also present an analytical solution of Stokes flow calculated using the penalization which illustrates the error and continuity properties of the approach. Work is currently underway to implement the penalization approach in a wavelet basis.     

Approximate Wall Boundary Conditions in the Large-Eddy Simulation of High Reynolds Number Flow

The near-wall regions of high Reynolds numbers turbulent flows must be modelled to treat many practical engineering and aeronautical applications. In this review we examine results from simulations of both attached and separated flows on coarse grids in which the near-wall regions are not resolved and are instead represented by approximate wall boundary conditions. The simulations use the dynamic Smagorinsky subgrid-scale model and a second-order finite-difference method. Typical results are found to be mixed, with acceptable results found in many cases in the core of the flow far from the walls, provided there is adequate numerical resolution, but with poorer results generally found near the wall. Deficiencies in this approach are caused in part by both inaccuracies in subgrid-scale modelling and numerical errors in the low-order finite-difference method on coarse near-wall grids, which should be taken into account when constructing models and performing large-eddy simulation on coarse grids. A promising new method for developing wall models from optimal control theory is also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

A simple algorithm to improve the performance of the WENO scheme on non-uniform grids

This paper presents a simple approach for improving the performance of the weighted essentially nonoscillatory(WENO) finite volume scheme on non-uniform grids. This technique relies on the reformulation of the fifthorder WENO-JS(WENO scheme presented by Jiang and Shu in J. Comput. Phys. 126:202–228, 1995) scheme designed on uniform grids in terms of one cell-averaged value and its left and/or right interfacial values of the dependent variable.The effect of grid non-uniformity is taken into consideration by a proper interpolation of the interfacial values. On nonuniform grids, the proposed scheme is much more accurate than the original WENO-JS scheme, which was designed for uniform grids. When the grid is uniform, the resulting scheme reduces to the original WENO-JS scheme. In the meantime,the proposed scheme is computationally much more efficient than the fifth-order WENO scheme designed specifically for the non-uniform grids. A number of numerical test cases are simulated to verify the performance of the present scheme.     

Non-conservative stability of multi-step non-uniform columns

The non-conservative stability of non-uniform columns under the combined action of concentrated and variably distributed forces is solved analytically. Two types of follower force system are considered: (i) concentrated follower forces and variably distributed follower forces, (ii) concentrated follower forces and variably distributed conservative forces. The exact solutions for stability of four kinds of one-step non-uniform columns subjected to the two types of follower force system are derived for the first time. Then a new exact approach, which combines the exact solutions of one-step columns and the transfer matrix method, is presented for the non-conservative stability analysis of multi-step non-uniform columns. The advantage of the proposed method is that the resulting eigenvalue equation for a multi-step non-uniform column with any kinds of two end support configurations, an arbitrary number of spring supports and concentrated masses can be conveniently determined from a second order determinant. The decrease in the determinant order, as compared with previously developed procedures, leads to significant savings in the computational effort. A numerical example shows that the results obtained from the proposed method are in good agreement with those determined from the finite element method (FEM), but the proposed method takes less computational time than FEM.     

Using Vorticity as an Indicator for the Generation of Optimal Coarse Grid Distribution

An improved vorticity-based gridding technique is presented and applied to create optimal non-uniform Cartesian coarse grid for numerical simulation of two-phase flow. The optimal coarse grid distribution (OCGD) is obtained in a manner to capture variations in both permeability and fluid velocity of the fine grid using a single physical quantity called “vorticity”. Only single-phase flow simulation on the fine grid is required to extract the vorticity. Based on the fine-scale vorticity information, several coarse grid models are generated for a given fine grid model. Then the vorticity map preservation error is used to predict how well each coarse grid model reproduces the fine-scale simulation results. The coarse grid model which best preserves the fine-scale vorticity, i.e. has the minimum vorticity map preservation error is recognized as an OCGD. The performance of vorticity-based optimal coarse grid is evaluated for two highly heterogeneous 2D formations. It is also shown that two-phase flow parameters such as mobility ratio have only minor impact on the performance of the predicted OCGD.