Hydrodynamics of a particle impact on a wall

The problem of a particle impacting on a wall, a common phenomenon in particle-laden flows in the minerals and process industries, is investigated computationally using a spectral-element method with the grid adjusting to the movement of the particle towards the wall. Remeshing is required at regular intervals to avoid problems associated with mesh distortion and the constantly reducing maximum time-step associated with integration of the non-linear convective terms of the Navier–Stokes equations. Accurate interpolation between meshes is achieved using the same high-order interpolation employed by the spectral-element flow solver. This approach allows the full flow evolution to be followed from the initial approach, through impact and afterwards as the flow relaxes. The method is applied to the generic two-dimensional and three-dimensional bluff body geometries, the circular cylinder and the sphere. The principal case reported here is that of a particle colliding normally with a wall and sticking. For the circular cylinder, non-normal collisions are also considered. The impacts are studied for moderate Reynolds numbers up to approximately 1200. A cylindrical body impacting on a wall produces two vortices from its wake that convect away from the cylinder along the wall before stalling while lifting induced wall vorticity into the main flow. The situation for a sphere impact is similar, except in this case a vortex ring is formed from the wake vorticity. Again, secondary vorticity from the wall and particle plays a role. At higher Reynolds number, the secondary vorticity tends to form a semi-annular structure encircling the primary vortex core. At even higher Reynolds numbers, the secondary annular structure fragments into semi-discrete structures, which again encircle and orbit the primary core. Vorticity fields and passive tracer particles are used to characterize the interaction of the vortical structures. The evolution of the pressure and viscous drag coefficients during a collision are provided for a typical sphere impact. For a Reynolds number greater than approximately 1000 for a sphere and 400 for a cylinder, the primary vortex core produced by the impacting body undergoes a short-wavelength instability in the azimuthal/spanwise direction. Experimental visualisation using dye carried out in water is presented to validate the predictions.     

Periodic interpolation and wavelets on sparse grids

Nested spaces of multivariate periodic functions forming a non-stationary multiresolution analysis are investigated. The scaling functions of these spaces are fundamental polynomials of Lagrange interpolation on a sparse grid. The approach based on Boolean sums leads to sample and wavelet spaces of significantly lower dimension and good approximation order. The algorithms for complete decomposition and reconstruction are of simple structure and low complexity. This revised version was published online in June 2006 with corrections to the Cover Date.     

Turbulent flow computations on 3D unstructured grids

An edge-based finite element method is presented for the simulation of compressible turbulent flows on unstructured tetrahedral grids. A two equation k–ω turbulence model is employed and the standard Galerkin approach is used for spatial discretisation. Stabilisation of the resulting procedure is achieved by the addition of an appropriate diffusion. An explicit multistage time-stepping scheme is used to advance the solution in time to steady state. The performance of the algorithm is demonstrated for the simulation of a high Reynolds number transonic separated flow over a wing.     

ANALYSIS OF MOVING MESH METHODS BASED ON GEOMETRICAL VARIABLES

1. IntroductionMany methods have been proposed for adapting the mesh to aChieve spatial resolution in thesolution of partial dthereatial equations. In addition to the capability of concedrating sufficientpoints about regions of rapid variation of the solution, a satisfactory mesh equatioll should besample, easy to program, and reasonably insensitive to the choice of its adjustable parameters.The earliest work on adaptive tecboques, based on moving trite element method (MFEM) wdone by Miller …     

On the convergence of curvilinear search algorithms in unconstrained optimization

The purpose of this paper is to unify the conditions under which curvilinear algorithms for unconstrained optimization converge. Particularly, two gradient path approximation algorithms and a trust region curvilinear algorithm are examined in this context.     

General method for producing a boundary-fitted orthogonal curvilinear mesh

A general method is described for computing an orthogonal mesh fitted to a two-dimensional physical domain with arbitrary closed boundary. The method allows optimum control of mesh spacing through the introduction of arbitrary (with weak constraints) ‘packing’ functions into the elliptic governing equations. Two particular aspects are addressed: first, the condition on a scaling factor which normalizes the mesh aspect ratio; second, the condition for avoiding run-out of the mesh beyond the boundaries of the physical domain.Conversion of the equations to finite difference form and appropriate iterative techniques are discussed. Finally applications of the method in the context of flow across a bundle of rods are presented.     

激光烧蚀瑞利－泰勒不稳定性模拟

 给出了激光烧蚀流体不稳定性计算程序EUL2D的物理方程,介绍了计算中使用的活动网格和一些技术问题处理。EUL2D程序的计算结果与Takabe公式、FAST2D程序和LASNEX程序的结果较好符合。数值计算日本大阪大学激光烧蚀瑞利－泰勒不稳定性实验,再现了实验结果。发现了横向电子热传导烧蚀在长波长扰动的非线性瑞利－泰勒不稳定性演变中起重要作用。     

A deterministic scheme for Smoluchowski's coagulation equation based on binary grid refinement

We present a deterministic scheme for the discrete Smoluchowski's coagulation equation based on a binary grid refinement. Starting from the binary grid Ω₀={1,2,4,8,16,. . .}, we first introduce an appropriate grid refinement by adding at each level 2^l grid points in every binary subsection of the grid Ω_l. In a next step we derive an approximate equation for the dynamic behavior on each level Ω_l based on a piecewise constant approximation of the right hand side of Smoluchowski's equation. Numerical results show that the computational effort can be drastically decreased compared to the corresponding complete integer grid. When considering unbounded kernels in Smoluchowski's equation we use an adaptive time step method to overcome numerical instabilities which may occur at the tails of the density function.     

Fast evaluation of trigonometric polynomials from hyperbolic crosses

The discrete Fourier transform in d dimensions with equispaced knots in space and frequency domain can be computed by the fast Fourier transform (FFT) in arithmetic operations. In order to circumvent the ‘curse of dimensionality’ in multivariate approximation, interpolations on sparse grids were introduced. In particular, for frequencies chosen from an hyperbolic cross and spatial knots on a sparse grid fast Fourier transforms that need only arithmetic operations were developed. Recently, the FFT was generalised to nonequispaced spatial knots by the so-called NFFT. In this paper, we propose an algorithm for the fast Fourier transform on hyperbolic cross points for nonequispaced spatial knots in two and three dimensions. We call this algorithm sparse NFFT (SNFFT). Our new algorithm is based on the NFFT and an appropriate partitioning of the hyperbolic cross. Numerical examples confirm our theoretical results.