Calibration relations for nonpolynomial splines

We construct wavelet decompositions and the corresponding decomposition – reconstruction algorithms in the case of an infinite flow (a grid on an open interval) and a finite flow (a grid on a segment) for a space of Lagrange type splines (in general, not polynomial). Bibliography: 11 titles.     

Algorithms of wavelet compression of linear spline spaces

Splines and wavelets have been finding increasing use in the theory of information. Wavelet decompositions are used in designing efficient algorithms for processing (compression) of large information flows. If one succeeds in establishing the embeddability of spaces of splines on a sequence of sparsing/refining grids, in representing the chain of embedded spaces as a direct sum of wavelet spaces, and in realizing the base functions with the minimum length of their support, then this suggests a wavelet decomposition of the information flow, leading, in turn, to substantial savings in the computational cost. This being so, it proves possible to resolve the initial information flow into components to single out the principal and refining information flows, depending on the needs. For uniform grids on the real line, wavelet decompositions are well known. In this case, there applies the powerful technique of harmonic analysis, as well as the lifting scheme or the wavelet scheme. However, many applications require considering bounded intervals and nonuniform grids. For example, for efficient compression of nonuniform flows of information (featuring singularities or rapidly fluctuating characteristics), it is expedient to employ an adaptive nonuniform grid, which takes account of the singularities of the flow being processed. This renders possible to improve approximation of functions without complicating the computations. The previously obtained results pertained to splines on infinite grids. Making both the grid and the corresponding numerical flow infinite renders theoretical studies simpler; however, in practice, one has to deal with finite flows. This paper continues the studies initiated for finite-dimensional spaces. The purpose of this work is to built a wavelet decomposition (compression) on a nonuniform grid and develop the corresponding decomposition and reconstruction algorithms for infinite flows (with a grid on an open interval) and finite flows (with a grid on a segment) for linear spaces of splines of Lagrange type.     

Performance comparison of the NWF and DC methods for implementing High-Resolution schemes in a fully coupled incompressible flow solver

This paper reports on the use of the Normalized Weighting Factor (NWF) method and the Deferred Correction (DC) approach for the implementation of High Resolution (HR) convective schemes in an implicit, fully coupled, pressure-based flow solver. Four HR schemes are realized within the framework of the NWF and DC methods and employed to solve the following three laminar flow problems: (i) lid-driven flow in a square cavity, (ii) sudden expansion in a square cavity, and (iii) flow in a planar T-junction, over three grid systems with sizes of 10⁴, 5 × 10⁴, and 3 × 10⁵ control volumes. The merit of both approaches is demonstrated by comparing the computational costs required to solve these problems using the various HR schemes on the different grid systems. Whereas previous attempts to use the NWF method in a segregated flow solver failed to produce converged solutions, current results clearly demonstrate that both methods are suitable for utilization in a coupled flow solver. In terms of CPU efficiency, there is no global and consistent superiority of any method over another even though the DC method outperformed the NWF method in two of the three test problems solved.     

Simulation of flows with strong shocks with an adaptive conservative scheme

In the present work, numerical simulations of unsteady flows with moving shocks are presented. An unsteady mesh adaptation method, based on error equidistribution criteria, is adopted to capture the most important flow features. The modifications to the topology of the grid are locally interpreted in terms of continuous deformation of the finite volumes built around the nodes. The arbitrary Lagrangian–Eulerian formulation of the Euler equations is then applied to compute the flow variable over the new grid without resorting to any explicit interpolation step. The numerical results show an increase in the accuracy of the solution, together with a strong reduction of the computational costs, with respect to computations with a uniform grid using a larger number of nodes.     

Numerical multi-block grids in coastal ocean circulation modeling

In coastal ocean modeling, one desires to capture the evolution and interaction of multi-scales of physical phenomena in a complicated physical domain. With limited computer resources, an appropriate choice of the numerical grid has a key role in determining the quality of the solution of a numerical coastal ocean model. Traditionally, single-block rectangular grids have been most commonly used in coastal ocean modeling for their simplicity. An effective coastal ocean model represents the dynamics of the coastal ocean flow on a numerical grid, including the effects of complicated features such as coastlines, bottom topography (submarine canyons, seamounts, narrow straits), and multi-scale physical phenomena. These problems require a model grid system more efficient than a traditional single-block rectangular grid. The model grids must give better resolution of coastlines and boundary conditions, multi-scale physical phenomenon, and save computer resources. These grids can also easily increase horizontal resolution in a subregion of the model domain without increasing computer expense with high resolution over the entire domain. The multi-block numerical generation grid technique is used in developing a coastal ocean system applied to the Mediterranean Sea (MED) with complicated coastlines, bottom topography and multi-scale physical features. The MED coastal ocean system consists of the MED model based on the Princeton Ocean Model, numerical grid generation routines, and a grid package which allows the model to be coupled with model grids. The traditional, nine-block orthogonal grid, and eight-block curvilinear nearly orthogonal coastline-following grid are used in the study. The numerical solutions with the three grids are compared in term of effectiveness. The numerical simulations show some MED basic physical features.     

A fast running numerical model based on the implementation of volume forces for prediction of pressure drop in a fin tube heat exchanger

Numerical based design of geometrical structures is common when studying systems involving heat exchangers, a central component in several fields, such as industrial, vehicle and household systems. The geometrical structure of heat exchangers is generally comprised by closely placed fins and tube bundles. The creation of a mesh grid for a geometrically compact heat exchanger will result in a dense structure, which is not feasible for personal computer usage. Hence, volume forces were created based on Direct Numerical Simulations (DNS) on a Flow Representative Volume (FRV) of a tube fin heat exchanger in an internal duct system of a heat pump tumble dryer. A relation of the volume averaged velocity and the volume averaged force was established in two different FRV models with a finite element simulation in COMSOL. This relation was subsequently used to create flow resistance coefficients based on volume averaged expressions of fluid velocity and volume forces. These flow resistance coefficients were implemented in two respective porous models, which represent the entire heat exchanger except the interior arrangements of fins and tube bundles. Hence, the computation time was reduced thanks to the absence of a dense mesh grid. Experimental results of the entire heat exchanger showed good agreement with the second porous model in terms of pressure drop and volume flow rate.     

A Multigrid Algorithm for Hybrid Joint PDF Simulations in Complex Geometries

In hybrid joint probability density function (joint PDF) algorithms for turbulent reactive flows the equations for the mean flow discretized with a classical grid based method (e.g. finite volume methods (FVM)) are solved together with a Monte Carlo (particle) method for the joint velocity composition PDF. When applied for complex geometries, the solution strategy for such methods which aims at obtaining a converged solution of the coupled problem on a sufficiently fine grid becomes very important. This paper describes one important aspect of this solution strategy, i.e. multigrid computing, which is well known to be very efficient for computing numerical solutions on fine grids. Two sets of grid based variables are involved: cell-centered variables from the FVM and node-centered variables, which denote the moments of the PDF extracted from the particle fields. Starting from a given multiblock grid environment first a new (refined or coarsened) grid is defined retaining the grid quality. The projection and prolongation operators are defined for the two sets of variables. In this new grid environment the particles are redistributed. The effectiveness of the multigrid algorithm is demonstrated. Compared to solely solving on the finest grid, convergence can be reached about one order of magnitude faster when using the multigrid algorithm in three stages. Computation time used for projection or prolongation is negligible. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

High performance computation for DNS/LES

The paper is focused on high-order compact schemes for direct numerical simulation (DNS) and large eddy simulation (LES) for flow separation, transition, tip vortex, and flow control. A discussion is given for several fundamental issues such as high quality grid generation, high-order schemes for curvilinear coordinates, the CFL condition for complex geometry, and high-order weighted compact schemes for shock capturing and shock–vortex interaction. The computation examples include DNS for K-type and H-type transition, DNS for flow separation and transition around an airfoil with attack angle, control of flow separation by using pulsed jets, and LES simulation for a tip vortex behind the juncture of a wing and flat plate. The computation also shows an almost linear growth in efficiency obtained by using multiple processors.     

Modelling free surfaces in oscillating pipe flows

An oscillating pipe flow with a free surface is investigated numerically and experimentally. The pipe diameter is 12mm. Due to this small diameter capillary forces play an important role. Therefore special attention has to be paid to the flow field near the free surface. The numerical model is based on the fundamental flow equations. The free surface is resolved according to the volume-of-fluid method. The model equations are solved on a moving grid. In the experiment, pictures of the flow field are taken near the free surface. The effects occuring near the interface will be presented here. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

First-order accurate finite difference schemes for boundary vorticity approximations in curvilinear geometries

In this work we develop first-order accurate, forward finite difference schemes for the first derivative on both a uniform and a non-uniform grid. The schemes are applied to the calculation of vorticity on a solid wall of a curvilinear, two-dimensional channel. The von Mises coordinates are used to transform the governing equations, and map the irregular domain onto a rectangular computational domain. Vorticity on the solid boundary is expressed in terms of the first partial derivative of the square of the speed of the flow in the computational domain, and the derived finite difference schemes are used to calculate the vorticity at the computational boundary grid points using combinations of up to five computational domain grid points. This work extends previous work (Awartani et al., 2005) [3] in which higher-order schemes were obtained for the first derivative using up to four computational domain grid points. The aim here is to shed further light onto the use of first-order accurate non-uniform finite difference schemes that are essential when the von Mises transformation is used. Results show that the best schemes are those that use a natural sequence of non-uniform grid points. It is further shown that for non-uniform grid with clustering near the boundary, solution deteriorates with increasing number of grid points used. By contrast, when a uniform grid is used, solution improves with increasing number of grid points used.     

Application of the multigrid approach for solving 3D Navier-Stokes equations on hexahedral grids using the discontinuous Galerkin method

The Galerkin method with discontinuous basis functions is adapted for solving the Euler and Navier-Stokes equations on unstructured hexahedral grids. A hybrid multigrid algorithm involving the finite element and grid stages is used as an iterative solution method. Numerical results of calculating the sphere inviscid flow, viscous flow in a bent pipe, and turbulent flow past a wing are presented. The numerical results and the computational cost are compared with those obtained using the finite volume method.     

An Experimental/Numerical Investigation on Drag Reduction by Dimples

The paper is concerned with experimental and numerical investigations of the turbulent flow over dimpled surfaces. Shallow dimples distributed regularly over the wall of a plane channel with large aspect ratio are used to study their effect on the skin-friction drag. The resulting pressure drop in the channel was measured for smooth and dimpled walls. In addition to these investigations on internal flows, an external flow study was performed and boundary-layer profiles were measured using a Pitot-tube rake. Complementary to the measurements, direct numerical simulations for the internal flow configuration with and without dimples were carried out for two different grid resolutions and analyzed in detail. The objective was to clarify whether or not dimples cause reduction of the skin-friction drag. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A multilevel finite volume method with multiscale-based grid adaptation for steady compressible flows

An implicit multilevel finite volume solver on adaptively refined quadtree meshes is presented for the solution of steady state flow problems. The nonlinear problem arising from the implicit time discretization is solved by an adaptive FAS multigrid method. Local grid adaptation is performed by means of a multiscale-based strategy. For this purpose data of the flow field are decomposed into coarse grid information and a sequence of detail coefficients that describe the difference between two refinement levels and reveal insight into the local regularity behavior of the solution. Here wavelet techniques are employed for the multiscale analysis. The key idea of the present work is to use the transfer operators of the multiscale analysis for the prolongation and restriction operator in the FAS cycle. The efficiency of the solver is investigated by means of an inviscid 2D flow over a bump.     

求解粘性流体和热迁移联立方程的迎风局部微分求积法

微分求积方法(DQM)已成功地应用于数值求解流体力学中的许多问题．但是已有的工作大多限于正规区域的流动问题,同时缺少用迎风机制来描述流体流动的对流特性．该文对一个不规则区域中的不可压缩层流和热迁移的耦合问题给出了一种具有迎风机制的局部微分求积方法,对通过边界和坐标不平行的收缩管道中的流体,只用少数网格点得到了比较好的数值解．和有限差分方法（FDM）相比较,这一方法具有计算工作量少、存储量小和收敛性好等优点．     

Pseudospectral method for transient viscoelastic flow in an axisymmetric channel

A new pseudospectral method for simulating transient viscoelastic flows is presented. The governing equations are a system of seven first-order equations of mixed type. The essential features of the method are (i) all seven independent flow variables are represented on a common Chebyshev-Gauss-Lobatto grid; (ii) the pressure is treated in such a way as to give a globally divergence-free velocity field, i.e., the divergence of the velocity field vanishes globally within the region, and (iii) different time scales pertaining within the hyperbolic constitutive equations are treated using the splitting technique of LeVeque and Yee originally proposed in a finite-difference context. The method is applied to transient axisymmetric flow of an Oldroyd B fluid in a channel formulated in two ways: (I) as an initial boundary-value problem, and (II) as a body-force problem. © 1993 John Wiley & Sons, Inc.     

Fourier analysis of a robust multigrid method for convection-diffusion equations

Summary. We consider a two-grid method for solving 2D convection-diffusion problems. The coarse grid correction is based on approximation of the Schur complement. As a preconditioner of the Schur complement we use the exact Schur complement of modified fine grid equations. We assume constant coefficients and periodic boundary conditions and apply Fourier analysis. We prove an upper bound for the spectral radius of the two-grid iteration matrix that is smaller than one and independent of the mesh size, the convection/diffusion ratio and the flow direction; i.e. we have a (strong) robustness result. Numerical results illustrating the robustness of the corresponding multigrid -cycle are given. Received October 14, 1994     

A numerical method to compute solidification and melting processes

This paper deals with thermodynamically consistent numerical predictions of solidification and melting processes of pure materials using moving grids. Till date, enthalpy-porosity-based formulations of numerical codes have been generally the popular choice, although because of an artificial numerical smearing of the interface, it is virtually impossible to reproduce a sharp melting/solidification front that is supposed to exist for phase changes of pure substances. Numerical techniques based on moving grid methods have been relatively less used as they rely on complex and time-consuming adaptive grid generations. Using the moving grid approach, the authors present a method to solve solidification and melting problems. A simple linear interpolation is used to slide grid nodes along the interface to handle the otherwise obtained grid skewness near the interface. The numerical approach employed is validated with standard test cases available in the literature. The capability of capturing very complex flow field structures and the superiority of the present approach over enthalpy-porosity-based formulations is discussed. The authors also demonstrate the ability of the set-up computer code to solve complex thermofluid processes such as occur during crystal growth in Czochralski reactors.     

Steady incompressible flow around objects in general coordinates with a multigrid solution method

Steady incompressible flow around objects in general coordinates is investigated. First, an overview of the popular approaches to discretize incompressible flow problems in general coordinates is given. It has been chosen to solve the equations on a staggered grid with contravariant flux unknowns and pressure as primitive variables. A solution method multigrid is used, with a line smoother able to deal with stretched cells. For flow problems around objects solved with a single block discretization periodic boundary, conditions are prescribed and adaptations for the discretization and the multigrid method are given. Steady flow around a circular cylinder and around an ellipse are presented. © 1994 John Wiley & Sons, Inc.     

An adaptive grid method for navier-stokes flow computation II: grid addition

A further investigation has been conducted for a previously proposed multiple one-dimensional adaptive grid method for Navier-Stokes flow calculations. The method is based on the equidistribution concept and designed by utilizing the different characteristics of the velocity and pressure fields. It is demonstrated that one major advantage of this method is its flexibility in adding the nodal points along any coordinates if desirable. As the adaptive readjustment of grid distribution proceeds from the initial grid system, it is shown that not only the overall error is reduced but the error distribution is more uniform. This finding is consistent with the one-dimensional analysis by Babuska and Rheinboldt. Furthermore, for the flow problem tested, the resulting adaptive grid system yielded by the present method is close to optimum after two to three stages of adaption.