Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier-Stokes equations is presented. The scheme is constructed on a staggered-mesh grid system. The convection terms are discretized with a fifth-order-accurate upwind compact difference approximation, the viscous terms are discretized with a sixth-order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth-order difference approximation on a cell-centered mesh. Time advancement uses a three-stage Runge-Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented.

A high order accurate difference scheme for complex flow fields

A high order accurate finite difference method for direct numerical simulation of coherent structure in the mixing layers is presented. The reason for oscillation production in numerical solutions is analyzed, It is caused by a nonuniform group velocity of wavepackets. A method of group velocity control for the improvement of the shock resolution is presented. In numerical simulation the fifth-order accurate upwind compact difference relation is used to approximate the derivatives in the convection terms of the compressible N-S equations, a sixth-order accurate symmetric compact difference relation is used to approximate the viscous terms, and a three-stage R-K method is used to advance in time. In order to improve the shock resolution the scheme is reconstructed with the method of diffusion analogy which is used to control the group velocity of wavepackets. (C) 1997 Academic Press.

Triple-region structure for turbulent-flow in a square duct - a finite-element approach

The Reynolds-averaged Navier-Stokes equations for describing the turbulent flow in a straight square duct are formulated with two different turbulence models. The governing equations are then expanded as a multi-deck structure in a plane perpendicular to the streamwise direction, with each deck characterized by its dominant physical forces as commonly carried out in analytical work using triple-deck expansion. The resulting equations are numerically integrated using higher polynomial (H-P) finite element technique for each cross-sectional plane to be followed by finite difference representation in the streamwise direction until a fully developed state is reached. The computed results using the two different turbulence models show fair agreement with each other, and concur with the vast body of available experimental data. There is also general agreement between our results and the recent numerical works anisotropic k-epsilon turbulence model.

Impact of difference accuracy on computational properties of vertical grids for a nonhydrostatic model

Starting from nonhydrostatic Boussinesq approximation equations, a general method is introduced to deduce the dispersion relationships. A comparative investigation is performed on inertia-gravity wave with horizontal lengths of 100, 10 and 1 km. These are examined using the second-order central difference scheme and the fourth-order compact difference scheme on vertical grids that are currently available from the perspectives of frequency, horizontal and vertical component of group velocity. These findings are compared to analytical solutions. The obtained results suggest that whether for the second-order central difference scheme or for the fourth-order compact difference scheme, Charny-Phillips and Lorenz ( L) grids are suitable for studying waves at the above-mentioned horizontal scales; the Lorenz time-staggered and Charny-Phillips time staggered (CPTS) grids are applicable only to the horizontal scales of less than 10 km, and N grid ( unstaggered grid) is unsuitable for simulating waves at any horizontal scale. Furthermore, by using fourth-order compact difference scheme with higher difference precision, the errors of frequency and group velocity in horizontal and vertical directions produced on all vertical grids in describing the waves with horizontal lengths of 1, 10 and 100 km cannot inevitably be decreased. So in developing a numerical model, the higher-order finite difference scheme, like fourth-order compact difference scheme, should be avoided as much as possible, typically on L and CPTS grids, since it will not only take many efforts to design program but also make the calculated group velocity in horizontal and vertical directions even worse in accuracy.

Impact of spatial resolution and spatial difference accuracy on the performance of Arakawa A-D grids

This paper alms at illustrating the impact of spatial difference scheme and spatial resolution on the performance of Arakawa A-D grids in physical space. Linear shallow water equations are discretized and forecasted on Arakawa A-D grids for 120-minute using the ordinary second-order (M and fourth-order (C4) finite difference schemes with the grid spacing being 100 km, 10 km and I km, respectively. Then the forecasted results are compared with the exact solution, the result indicates that when the grid spacing is I kin, the inertial gravity wave can be simulated on any grid with the same results from C2 scheme or C4 scheme, namely the impact of variable configuration is neglectable; while the inertial gravity wave is simulated with lengthened grid spacing, the effects of different variable configurations are different. However, whether for C2 scheme or for C4 scheme, the RMS is minimal (maximal) on C (D) grid. At the same time it is also shown that when the difference accuracy increases from C2 scheme to C4 scheme, the resulted forecasts do not uniformly decrease, which is validated by the change of the group A velocity relative error from C2 scheme to C4 scheme. Therefore, the impact of the grid spacing is more important than that of the difference accuracy on the performance of Arakawa A-D grid.

Influence of Local Magnetic Fields on P-Doped Si Floating Zone Melting Crystal Growth in Microgravity

A two-dimensional axisymmetric numerical model is presented to study the influence of local magnetic fields on P-doped Si floating zone melting crystal growth in microgravity. The model is developed based on the finite difference method in a boundary-fitted curvilinear coordinate system. Extensive numerical simulations are carried out, and parameters studied include the curved growth interface shape and the magnetic field configurations. Computed results show that the local magnetic field is more effective in reducing the impurity concentration nonuniformity at the growth interface in comparison with the longitudinal magnetic field. Moreover, the curved growth interface causes more serious impurity concentration nonuniformity at the growth interface than the case with a planar growth interface.

对流扩散方程的变步长摄动有限差分格式

摄动有限差分(PFD)方法是构造高精度差分格式的一种新方法。变步长摄动有限差分方法是等步长摄动有限差分方法的发展和推广。对需要局部加密网格的计算问题，变步长PFD格式不需要对自变量进行数学变换，且和等步长PFD格式一样，具有如下的共同特点：从变步长一阶迎风格式出发，通过把非微商项(对流系数和源项)作变步长摄动展开，展开幂级数系数通过消去摄动格式修正微分方程的截断误差项求出，由此获得高精度变步长PFD格式。该格式在一、二和三维情况下分别仅使用三、五和七个基点，且具有迎风性。文中利用变步长PFD格式对对流扩散反应模型方程，变系数方程及Burgers方程等进行了数值模拟，并与一阶迎风和二阶中心格式及其问题的精确解作了比较。数值试验表明，与一阶迎风和二阶中心格式相比，变步长PFD格式具有精度高，稳定性与收敛性好的特点。变步长PFD格式与等步长PFD格式相比，变步长PFD解在薄边界层型区域的分辨率得到了明显的提高。

Numerical simulation of rock failure and earthquake process on mesoscopic scale

On the basis of the lattice model of MORA and PLACE, Discrete Element Method, and Molecular Dynamics approach, another kind of numerical model is developed. The model consists of a 2-D set of particles linked by three kinds of interactions and arranged into triangular lattice. After the fracture criterion and rules of changes between linking states are given, the particle positions, velocities and accelerations at every time step are calculated using a finite-difference scheme, and the configuration of particles can be gained step by step. Using this model, realistic fracture simulations of brittle solid (especially under pressure) and simulation of earthquake dynamics are made.

双曲守恒型方程的二阶摄动有限差分格式

对双曲守恒型方程，将其一阶迎风格式空间差商的常系数摄动展开为时间步长和空间步长的幂级数，通过确定幂级数系数而获得二阶精度的摄动有限差分（PFD）格式。进而从双曲守恒型方程的通量分裂型一阶迎风格式出发，通过娄似的摄动展开方法，获得空间精度为二阶的通量分裂形式的摄动有限差分（FPFD）格式。这两类格式保留了一阶守恒迎风格式的简洁结构形式，使用三节点即可达到二阶精度，又避免了三点二阶格式的非物理数值振荡。并将这两类格式推广应用到双曲守恒型方程组，最后通过模型方程和一维激波管流动的数值算例验证了格式的高精度、高分辨率性质。

NS方程激波计算的摄动有限差分方法

摄动有限差分（PFD）方法从一阶迎风差分格式出发，将差分系数展开为网格步长的幂级数，通过提高修正微分方程的逼近精度来获得更高精度的差分格式。由于格式基于一阶迎风格式，因此具有迎风效应、网格节点少等特点。本文首先通过对Burgers方程的摄动差分格式的推导，将摄动有限差分格式引入时间相关法的计算，并构造了守恒形式的摄动有限差分格式，然后推广到一维Navier-Stokes方程组的计算。数值比较研究表明：本文构造的NS方程摄动有限差分格式具有比一阶迎风较高的精度和分辨率，而且保持了一阶迎风格式的无振荡性质。

Direct Numerical Simulation Of Hypersonic Boundary-Layer Transition Over A Blunt Cone

Direct numerical simulation of transition How over a blunt cone with a freestream Mach number of 6, Reynolds number of 10,000 based on the nose radius, and a 1-deg angle of attack is performed by using a seventh-order weighted essentially nonoscillatory scheme for the convection terms of the Navier-Stokes equations, together with an eighth-order central finite difference scheme for the viscous terms. The wall blow-and-suction perturbations, including random perturbation and multifrequency perturbation, are used to trigger the transition. The maximum amplitude of the wall-normal velocity disturbance is set to 1% of the freestream velocity. The obtained transition locations on the cone surface agree well with each other far both cases. Transition onset is located at about 500 times the nose radius in the leeward section and 750 times the nose radius in the windward section. The frequency spectrum of velocity and pressure fluctuations at different streamwise locations are analyzed and compared with the linear stability theory. The second-mode disturbance wave is deemed to be the dominating disturbance because the growth rate of the second mode is much higher than the first mode. The reason why transition in the leeward section occurs earlier than that in the windward section is analyzed. It is not because of higher local growth rate of disturbance waves in the leeward section, but because the growth start location of the dominating second-mode wave in the leeward section is much earlier than that in the windward section.

Receptivity To Free-Stream Disturbance Waves For Hypersonic Flow Over A Blunt Cone

A high-order shock-fitting finite difference scheme is studied and used to do direction numerical simulation (DNS) of hypersonic unsteady flow over a blunt cone with fast acoustic waves in the free stream, and the receptivity problem in the blunt cone hypersonic boundary layers is studied. The results show that the acoustic waves are the strongest disturbance in the blunt cone hypersonic boundary layers. The wave modes of disturbance in the blunt cone boundary layers are first, second, and third modes which are generated and propagated downstream along the wall. The results also show that as the frequency decreases, the amplitudes of wave modes of disturbance increase, but there is a critical value. When frequency is over the critial value, the amplitudes decrease. Because of the discontinuity of curvature along the blunt cone body, the maximum amplitudes as a function of frequencies are not monotone.

Thermocapillary Migration Of Nondeformable Drops

In this paper, we present a numerical study on the thermocapillary migration of drops. The Navier-Stokes equations coupled with the energy conservation equation are solved by the finite-difference front-tracking scheme. The axisymmetric model is adopted in Our simulations, and the drops are assumed to be perfectly spherical and nondeformable. The benchmark simulation starts from the classical initial condition with a uniform temperature gradient. The detailed discussions and physical explanations of migration phenomena are presented for the different values of (1) the Marangoni numbers and Reynolds numbers of continuous phases and drops and (2) the ratios of drop densities and specific heats to those of continuous phases. It is found that fairly large Marangoni numbers may lead to fluctuations in drop velocities at the beginning part of simulations. Finally, we also discuss the influence of initial conditions on the thermocapillary migrations. (C) 2008 American Institute of Physics.

Dns Of Compressible Turbulent Boundary Layer Around A Sharp Cone

Direct numerical simulation of the turbulent boundary layer over a sharp cone with 20 degrees cone angle (or 10 degrees half-cone angle) is performed by using the mixed seventh-order up-wind biased finite difference scheme and sixth-order central difference scheme. The free stream Mach number is 0.7 and free stream unit Reynolds number is 250000/inch. The characteristics of transition and turbulence of the sharp cone boundary layer are compared with those of the flat plate boundary layer. Statistics of fully developed turbulent flow agree well with the experimental and theoretical data for the turbulent flat-plate boundary layer flow. The near wall streak-like structure is shown and the average space between streaks (normalized by the local wall unit) keeps approximately invariable at different streamwise locations. The turbulent energy equation in the cylindrical coordinate is given and turbulent energy budget is studied. The computed results show that the effect of circumferential curvature on turbulence characteristics is not obvious.

MILU-CG method and the numerical study on the flow around a rotating circular cylinder

A hybrid finite difference method and vortex method (HDV), which is based on domain decomposition and proposed by the authors (1992), is improved by using a modified incomplete LU decomposition conjugate gradient method (MILU-CG), and a high order implicit difference algorithm. The flow around a rotating circular cylinder at Reynolds number R-e = 1000, 200 and the angular to rectilinear speed ratio alpha is an element of (0.5, 3.25) is studied numerically. The long-time full developed features about the variations of the vortex patterns in the wake, and drag, lift forces on the cylinder are given. The calculated streamline contours agreed well with the experimental visualized flow pictures. The existence of critical states and the vortex patterns at the states are given for the first time. The maximum lift to drag force ratio can be obtained nearby the critical states.