Numerical Investigation of Richtmyer-Meshkov Instability Driven by Cylindrical Shocks

In this paper, a numerical method with high order accuracy and high resolution was developed to simulate the Richtmyer-Meshkov(RM) instability driven by cylindrical shock waves. Compressible Euler equations in cylindrical coordinate were adopted for the cylindrical geometry and a third order accurate group control scheme was adopted to discretize the equations. Moreover, an adaptive grid technique was developed to refine the grid near the moving interface to improve the resolution of numerical solutions. The results of simulation exhibited the evolution process of RM instability, and the effect of Atwood number was studied. The larger the absolute value of Atwood number, the larger the perturbation amplitude. The nonlinear effect manifests more evidently in cylindrical geometry. The shock reflected from the pole center accelerates the interface for the second time, considerably complicating the interface evolution process, and such phenomena of reshock and secondary shock were studied.

The cellular structure of a two-dimensional H2/O2/Ar detonation wave

In this paper, the cellular structure of a two-dimensional detonation wave in a low pressure H2/O2/Ar mixture calculated with a detailed chemical reaction model, high order scheme and high resolution grids is investigated. The regular cellular structure is produced about 1 ms after introducing perturbations in the reaction zone of a steady one-dimensional detonation wave. It is found from the present resolution study that the discrepancies concerning the structure type arising from the coarser grid employed can be resolved using a sufficiently fine grid size of 0.05 mm and below and shows a double-Mach-like strong-type configuration. During the structure evolution process, the structure configuration does not change much in the periods before and after the triple point collision. Through the triple point collision, three regular collision processes are observed and are followed by a quick change to the double-Mach-like configuration. The simulated structure tracks show that there are three different tracks associated with different triple points or the kink on the transverse wave. Comparisons with previous work and experiments indicate the presence of a strong structure for an ordinary detonation.

A lattice Boltzmann method for KDV equation

We prepose a 5-bit lattice Boltzmann model for KdV equation. Using Chapman-Enskog expansion and multiscale technique, we obtained high order moments of equilibrium distribution function, and the 3rd dispersion coefficient and 4th order viscosity. The parameters of this scheme can be determined by analysing the energy dissipation.

The Structure and Evolution of a Two-Dimensional H2/O2/Ar Cellular Detonation

This paper reports on two-dimensional numerical simulation of cellular detonation wave in a / / mixture with low initial pressure using a detailed chemical reaction model and high order WENO scheme. Before the final equilibrium structure is produced, a fairly regular but still non-equilibrium mode is observed during the early stage of structure formation process. The numerically tracked detonation cells show that the cell size always adapts to the channel height such that the cell ratio is fairly independent of the grid sizes and initial and boundary conditions. During the structural evolution in a detonation cell, even as the simulated detonation wave characteristics suggest the presence of an ordinary detonation, the evolving instantaneous detonation state indicates a mainly underdriven state. As a considerable region of the gas mixture in a cell is observed to be ignited by the incident wave and transverse wave, it is further suggested that these two said waves play an essential role in the detonation propagation.

Instability Analysis of Nonlinear Surface Waves in a Circular Cylindrical Container Subjected to a Vertical Excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier-Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated. (c) 2005 Elsevier SAS. All rights reserved.

Analysis of super compact finite difference method and application to simulation of vortex-shock interaction

Turbulence and aeroacoustic noise high-order accurate schemes are required, and preferred, for solving complex flow fields with multi-scale structures. In this paper a super compact finite difference method (SCFDM) is presented, the accuracy is analysed and the method is compared with a sixth-order traditional and compact finite difference approximation. The comparison shows that the sixth-order accurate super compact method has higher resolving efficiency. The sixth-order super compact method, with a three-stage Runge-Kutta method for approximation of the compressible Navier-Stokes equations, is used to solve the complex flow structures induced by vortex-shock interactions. The basic nature of the near-field sound generated by interaction is studied.

Scheme Construction with Numerical Flux Residual Correction (NFRC) and Group Velocity Control (GVC)

For simulating multi-scale complex flow fields like turbulent flows, the high order accurate schemes are preferred. In this paper, a scheme construction with numerical flux residual correction (NFRC) is presented. Any order accurate difference approximation can be obtained with the NFRC. To improve the resolution of the shock, the constructed schemes are modified with group velocity control (GVC) and weighted group velocity control (WGVC). The method of scheme construction is simple, and it is used to solve practical problems.

Evolution of three-dimensional coherent structures in compressible axisymmetric jet

A high order difference scheme is used to simulate the spatially developing compressible axisymmetric jet. The results show that the Kelvin-Helmholtz instability appears first when the jet loses its stability, and then with development of jet the increase in nonlinear effects leads to the secondary instability and the formation of the streamwise vortices. The evolution of the three-dimensional coherent structure is presented. The computed results verify that in axisymmetric jet the secondary instability and formation of the streamwise vortices are the important physical mechanism of enhancing the flow mixing and transition occurring.

Direct Numerical Simulation of Supersonic Turbulent Boundary Layer Flow

Direct numerical simulations of a spatially evolving supersonic flat-plate turbulent boundary layer flow with free Mach number M = 2.25 and Reynolds number Re = 365000/in are performed. The transition process from laminar to turbulent flow is obtained by solving the three-dimensional compressible Navier-Stokes, equations, using high-order accurate difference schemes. The obtained statistical results agree well with the experimental and theoretical data. From the numerical results it can be seen that the transition process under the considered conditions is the process which skips the Tolimien-Schlichting instability and the second instability through the instability of high gradient shear layer and becomes of laminar flow breakdown. This means that the transition process is a bypass-type transition process. The spanwise asymmetry of the disturbance locally upstream imposed is important to induce the bypass-type transition. Furthermore, with increasing the time disturbance frequency the transition will delay. When the time disturbance frequency is large enough, the transition will disappear.

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.

A Multimoment Finite-Volume Shallow-Water Model On The Yin-Yang Overset Spherical Grid

A numerical model for shallow-water equations has been built and tested on the Yin-Yang overset spherical grid. A high-order multimoment finite-volume method is used for the spatial discretization in which two kinds of so-called moments of the physical field [i.e., the volume integrated average ( VIA) and the point value (PV)] are treated as the model variables and updated separately in time. In the present model, the PV is computed by the semi-implicit semi-Lagrangian formulation, whereas the VIA is predicted in time via a flux-based finite-volume method and is numerically conserved on each component grid. The concept of including an extra moment (i.e., the volume-integrated value) to enforce the numerical conservativeness provides a general methodology and applies to the existing semi-implicit semi-Lagrangian formulations. Based on both VIA and PV, the high-order interpolation reconstruction can only be done over a single grid cell, which then minimizes the overlapping zone between the Yin and Yang components and effectively reduces the numerical errors introduced in the interpolation required to communicate the data between the two components. The present model completely gets around the singularity and grid convergence in the polar regions of the conventional longitude-latitude grid. Being an issue demanding further investigation, the high-order interpolation across the overlapping region of the Yin-Yang grid in the current model does not rigorously guarantee the numerical conservativeness. Nevertheless, these numerical tests show that the global conservation error in the present model is negligibly small. The model has competitive accuracy and efficiency.

Shallow Water Model On Cubed-Sphere By Multi-Moment Finite Volume Method

A global numerical model for shallow water flows on the cubed-sphere grid is proposed in this paper. The model is constructed by using the constrained interpolation profile/multi-moment finite volume method (CIP/MM FVM). Two kinds of moments, i.e. the point value (PV) and the volume-integrated average (VIA) are defined and independently updated in the present model by different numerical formulations. The Lax-Friedrichs upwind splitting is used to update the PV moment in terms of a derivative Riemann problem, and a finite volume formulation derived by integrating the governing equations over each mesh element is used to predict the VIA moment. The cubed-sphere grid is applied to get around the polar singularity and to obtain uniform grid spacing for a spherical geometry. Highly localized reconstruction in CIP/MM FVM is well suited for the cubed-sphere grid, especially in dealing with the discontinuity in the coordinates between different patches. The mass conservation is completely achieved over the whole globe. The numerical model has been verified by Williamson's standard test set for shallow water equation model on sphere. The results reveal that the present model is competitive to most existing ones. (C) 2008 Elsevier Inc. All rights reserved.

A Cip/Multi-Moment Finite Volume Method For Shallow Water Equations With Source Terms

A novel finite volume method has been presented to solve the shallow water equations. In addition to the volume-integrated average (VIA) for each mesh cell, the surface-integrated average (SIA) is also treated as the model variable and is independently predicted. The numerical reconstruction is conducted based on both the VIA and the SIA. Different approaches are used to update VIA and SIA separately. The SIA is updated by a semi-Lagrangian scheme in terms of the Riemann invariants of the shallow water equations, while the VIA is computed by a flux-based finite volume formulation and is thus exactly conserved. Numerical oscillation can be effectively avoided through the use of a non-oscillatory interpolation function. The numerical formulations for both SIA and VIA moments maintain exactly the balance between the fluxes and the source terms. 1D and 2D numerical formulations are validated with numerical experiments. Copyright (c) 2007 John Wiley & Sons, Ltd.

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.

Features of the supercritical transition in the wake of a circular cylinder

The features of the wake behind a uniform circular cylinder at Re = 200, which is just beyond the critical Reynolds number of 3-D transition, are investigated in detail by direct numerical simulations by solving 3-D incompressible Navier-Stokes equations using mixed spectral-spectral-element method. The high-order splitting algorithm based on the mixed stiffly stable scheme is employed in the time discretization. Due to the nonlinear evolution of the secondary instability of the wake, the spanwise modes with different wavelengths emerge. The spanwise characteristic length determines the transition features and global properties of the wake. The existence of the spanwise phase difference of the primary vortices shedding is confirmed by Fourier analysis of the time series of the spanwise vorticity and attributed. to the dominant spanwise mode. The spatial energy distributions of various modes and the velocity profiles in the near wake are obtained. The numerical results indicate that the near wake is in 3-D quasi-periodic laminar state with transitional behaviors at this supercritical Reynolds number.