Rayleigh-Benard系统不稳定性的DSMC方法研究

本文应用DSMC方法初步研究了稀薄气体状态下Rayleigh-Benard系统的不稳定性特征,着重考察了气体密度梯度对Rayleihg-Benard系统不稳定性特征的影响。研究表明DSMC方法能够成功地模拟Rayleigh-Benard系统内涡的运动,得到了与外力方向相反的密度梯度会增加Rayleigh-Benard系统流动不稳定性的结果。     

On the validity of the Boltzmann–BGK model through relaxation evaluation

The Boltzmann-Bhatnagar-Gross-Krook（BGK）model is investigated for its validity regarding the collision term approximation through relaxation evaluation. The evaluation is based on theoretical analysis and numerical comparison between the BGK and direct simulation Monte Carlo（DSMC） results for three specifically designed relaxation problems. In these problems, one or half component of the velocity distribution is characterized by another Maxwellian distribution with a different temperature. It is analyzed that the relaxation time in the BGK model is unequal to the molecular mean collision time. Relaxation of component distribution fails to involve enough contribution from other component distributions, which makes the BGK model unable to capture details of velocity distribution, especially when discontinuity exists in distribution. The BGK model,however, predicts satisfactory results including fluxes during relaxation when the temperature difference is small. Particularly, the model-induced error in the BGK model increases with the temperature difference, thus the model is more reliable for low-speed rarefied flows than for hypersonic flows.     

Extension of the DSMC method to high pressure flows

A collision-limiter method, designated as equilibrium direct simulation Monte Carlo (eDSMC), is proposed to extend the DSMC technique to high pressure flows. The method is similar to collision-limiter schemes considered in the past with the important distinction that for inviscid flows, equilibrium is enforced in the entire flow by providing a sufficient number of collisions, based on pre-simulation testing. To test the method with standard DSMC and Navier–Stokes (NS) methods, axi-symmetric nozzle and embedded-channel flows are simulated and compared with experimental temperature data and pre-existing calculations, respectively. The method is shown to agree with third-order Eulerian nozzle flows and first-order channel flows. Chapman–Enskog theory is utilized to predict the range of initial conditions where eDSMC is potentially useful for modeling flows that contain viscous boundary layer regions. Comparison with supersonic nozzle data suggests that the eDSMC method is not adequate for capturing the large variation in flow length scales occurring in supersonic expansions into a vacuum. However, when eDSMC is used in combination with the baseline-DSMC method a near-exact solution is obtained with a considerable computational savings compared to the exact DSMC solution. Viscous flow channel calculations are found to agree well with an exact Navier–Stokes (NS) calculation for a small Knudsen number case as predicted by Chapman–Enskog theory.     

认识稀薄气体动力学

以通俗易懂的方式介绍了空气动力学当气体间断分子效应显著时发展起来的特殊分文——稀薄气体动力学、讨论了非平衡现象与稀薄气体动力学的关系．通过与8速度气体模型的间断Boltzmann方程的对比,解释了Boltzmann方程碰撞项的物理意义和数学困难,简要综述了其一般解法、讨论了分子在物体表面的反射和问题的边界条件,着重介绍了直接模拟Monte Carlo(DSMC)方法和为克服低速稀薄流动(如MEMS中流动)中模拟困难的信息保存(IP)方法。     

Gas-kinetic numerical method for solving mesoscopic velocity distribution function equation

A gas-kinetic numerical method for directly solving the mesoscopic velocity distribution function equation is presented and applied to the study of three-dimensional complex flows and micro-channel flows covering various flow regimes. The unified velocity distribution function equation describing gas transport phenomena from rarefied transition to continuum flow regimes can be presented on the basis of the kinetic Boltzmann–Shakhov model equation. The gas-kinetic finite-difference schemes for the velocity distribution function are constructed by developing a discrete velocity ordinate method of gas kinetic theory and an unsteady time-splitting technique from computational fluid dynamics. Gas-kinetic boundary conditions and numerical modeling can be established by directly manipulating on the mesoscopic velocity distribution function. A new Gauss-type discrete velocity numerical integration method can be developed and adopted to attack complex flows with different Mach numbers. HPF parallel strategy suitable for the gas-kinetic numerical method is investigated and adopted to solve three-dimensional complex problems. High Mach number flows around three-dimensional bodies are computed preliminarily with massive scale parallel. It is noteworthy and of practical importance that the HPF parallel algorithm for solving three-dimensional complex problems can be effectively developed to cover various flow regimes. On the other hand, the gas-kinetic numerical method is extended and used to study micro-channel gas flows including the classical Couette flow, the Poiseuille- channel flow and pressure-driven gas flows in two-dimensional short micro-channels. The numerical experience shows that the gas-kinetic algorithm may be a powerful tool in the numerical simulation of micro-scale gas flows occuring in the Micro-Electro-Mechanical System (MEMS). The project supported by the National Natural Science Foundation of China (90205009 and 10321002), and the National Parallel Computing Center in Beijing. The English text was polished by Yunming Chen.     

DSMC approach to nonstationary Mach reflection of strong incoming shock waves using a smoothing technique

The direct simulation Monte Carlo (DSMC) method is applied to simulation of nonstationary Mach reflection of strong shock waves. Normally the DSMC method is very time consuming in solving unsteady flow field problems especially for high Mach numbers, because of the necessity of iterative calculations to eliminate the inherent statistical fluctuation caused by a finite sample size. A central weighted smoothing technique is introduced to process the DSMC results, so that the iteration time can be significantly reduced. In spite of some relaxations of the shock wave structure, the smoothing technique is verified to be useful to estima te the flow fields qualitatively and even quantitatively by using a relatively small sample size. The comparison between the present approach and the kineticmodel approach (Xu et al. 1991a, 1991b) on the application to unsteady rarefied flow fields was also carried out.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Hypersonic rarefied flows DSMC analysis by a simplified chemical model

A simplified model for the analysis of molecular and reaction dynamics in hypersonic flows of rarefied gases is used to investigate the influence of oxygen dissociation in the prediction of aerothermal coefficients. The model is a direct simulation Monte Carlo method derived from the Larsen-Borgnakke method for the analysis of inelastic collisions. Tests performed in conditions near to re-entry conditions at an altitude of 115 km show significant effects of chemical reactions on the evaluation of the heat-transfer coefficient, while no real influence is found on lift and drag coefficients. Good agreement is found with results achieved using more sophisticated chemical models, as a 4 % maximum difference arises in the determination of surface quantities in re-entry conditions at 160 km.

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Sommario Si studiano gli effetti delle reazioni chimiche sui coefficienti aerodinamici di un' ala piana priva di spessore in regime di moto ipersonico in un gas rarefatto. A tal fine, si sono ottenute delle soluzioni numeriche della equazione di Boltzmann con reazioni chimiche mediante il cosiddetto metodo di Simulazione Diretta. Le collisioni reattive sono state trattate con un modello fortemente semplificato che però fornisce risultati molto vicini a quelli ottenuti da modelli più sofisticati.

     

Intrinsic instability of the lattice BGK model

Based on the stability analysis with no linearization and expansion, it is argued that instability in the lattice BGK model is originated from the linear relaxation hypothesis of collision in the model. The hypothesis stands up only when the deviation from the local equilibrium is weak. In this case the computation is absolutely stable for real fluids. But for flows of high Reynolds number, this hypothesis is violated and then instability takes place physically. By performing a transformation a quantified stability criteria is put forward without those approximation. From the criteria a sufficient condition for stability can be obtained and serve as an estimation of the limited Reynolds number as high as possible.     

A numerical method for the evaluation of hydrodynamic forces of translating bodies under a free surface

This paper presents a numerical method to evaluate the hydrodynamic forces of translating bodies under a free surface. Both steady and unsteady problems are considered. Analytical and numerical studies are carried out based on the Havelock wave‐source function and the integral equation method. Two main problems arising inherently in the proposed solution method are overcome in order to facilitate the numerical implementation. The first lies in evaluating the Havelock function, which involves integrals with highly oscillatory kernels. Particular integration contours leading to non‐oscillatory integrands are derived a priori so that the integrals can be evaluated efficiently. The second problem lies in evaluating singular kernels in the boundary integral equation. The corresponding non‐singular formulation is derived using some theorems of potential theory, including the Gauss flux theorem and the property related to the equipotential body. The subsequent formulation is amenable to the solution by directly using the standard quadrature formulas without taking another special treatment. This paper also attempts to enhance the computational efficiency by presenting an interpolation method used to evaluate matrix elements, which are ascribed to a discretization procedure. In addition to the steady case, numerical examples consist of cases involving a submerged prolate spheroid, which is originally idle and then suddenly moves with a constant speed and a constant acceleration. Also systematically studied is the variation of hydrodynamic forces acting on the spheroid for various Froude numbers and submergence depths. Copyright © 2000 John Wiley & Sons, Ltd.     

Investigation of slow monatomic gas flows past a circular cylinder

Slow low-Knudsen-number monatomic-gas flow past a circular cylinder is numerically investigated on the basis of a model kinetic equation. The gas flow is described by a new kinetic equation, from which the continuum equations for slow nonisothermal gas flows containing temperature stresses follow rigorously. It is shown that a closed convective-flow region arises near a nonuniformly heated cylinder in a slow gas flow if the flow impinges on the hot side of its surface. Using a new model of the Boltzmann equation makes it possible to study gas flows both in continuum and rarefied flow regimes.     

Direct numerical simulation of the evolution of a two-dimensional vortex system in a rarefied gas

A two-dimensional problem with Taylor-Green initial conditions and periodic boundary conditions in a viscous compressible weakly-rarefied gas is solved within the framework of the kinetic approach. The aim of the study is to model the evolution of a given vortex system on the basis of the direct numerical solution of the Boltzmann equation. For this purpose, the discrete ordinates method is used with the collision integral calculated by the Cheremisin conservative projection method which conserves the density, momentum, and energy. The solution obtained makes it possible to trace the evolution of a vortex system given by the initial conditions and to determine the spectral properties of the flow. The flow parameter distributions are presented for successive moments of time.     

An asymptotic iteration method for the numerical analysis of near-critical free-surface flows

Satisfying the boundary conditions at the free surface may impose severe difficulties to the computation of turbulent open-channel flows with finite-volume or finite-element methods, in particular, when the flow conditions are nearly critical. It is proposed to apply an iteration procedure that is based on an asymptotic expansion for large Reynolds numbers and Froude numbers close to the critical value 1.The iteration procedure starts by prescribing a first approximation for the free surface as it is obtained from solving an ODE that has been derived previously by means of an asymptotic expansion (Grillhofer and Schneider, 2003). The numerical solution of the full equations of motion then gives a surface pressure distribution that differs from the constant value required by the dynamic boundary condition. To determine a correction to the elevation of the free surface we next solve an ODE that is obtained from the asymptotic analysis of the flow with a prescribed pressure disturbance at the free surface. The full equations of motion are then solved for the corrected surface, and the procedure is repeated until criteria of accuracy for surface elevation and surface pressure, respectively, are satisfied.The method is applied to an undular hydraulic jump as a test case.     

A Petrov-Galerkin method for the numerical solution of the Bradshaw-Ferriss-Atwell turbulence model

The Bradshaw-Ferriss-Atwell model for 2D constant property turbulent boundary layers is shown to be ill-posed with respect to numerical solution. It is shown that a simple modification to the model equations results in a well-posed system which is hyperbolic in nature. For this modified system a numerical algorithm is constructed by discretizing in space using the Petrov-Galerkin technique (of which the standard Galerkin method is a special case) and stepping in the timelike direction with the trapezoidal (Crank-Nicolson) rule. The algorithm is applied to a selection of test problems. It is found that the solutions produced by the standard Galerkin method exhibit oscillations. It is further shown that these oscillations may be eliminated by employing the Petrov-Galerkin method with the free parameters set to simple functions of the eigenvalues of the modified system.     

DSMC方法与稀薄气流计算的发展

直接模拟ＭｏｎｔｅＣａｒｌｏ（ＤＳＭＣ）方法是依赖物理的概率模拟方法．在求解过渡领域流动的众多解析、数值和模拟方法中,目前只有ＤＳＭＣ方法是可以模拟三维复杂真实气体流动的方法．从宏观参量到细观速度分布函数的水平上,该方法均能得到实验的支持．本文综述了ＤＳＭＣ方法的最近发展,包括我们在检验模型、处理内能松弛模拟和化学反应模拟以及求解三维绕流的通用方法等方面的工作。     

Shock wave structure in a diatomic gas based on a kinetic model

The structure of a normal (direct) shock in a gas for the parameters corresponding to nitrogen is investigated with allowance for the rotational degrees of freedom on the basis of a model kinetic equation. For various Mach numbers the structure is compared with both the known experimental results and the solutions of the Navier-Stokes approximation within the framework of two-temperature hydrodynamics. The possibility of assuming the constancy of the fraction of excited rotational degrees of freedom is studied.     

Direct Monte Carlo Simulation of Transverse Supersonic Rarefied Gas Flow around a Cylinder

Supersonic flow around a cylinder is investigated using the direct simulation Monte Carlo method over a wide rarefaction range: from the Knudsen number Kn = 0.1 to free-molecular flow. The effect of the cylinder temperature on the region of sharp nonequilibrium near the cylinder and the heat flux is studied.     

Kinetic analysis of longitudinal Couette flow between coaxial cylinders

On the basis of a model kinetic equation, the rarefied gas flow between coaxial circular cylinders, of which the outer one is at rest while the inner one travels along its symmetry axis at a constant velocity, is studied. The problem is solved numerically in both the linear and nonlinear formulations by an implicit conservative method of second-order accuracy. The effect of the rarefaction, the cylinder radius ratio, and the inner cylinder velocity on the flow parameters is investigated. The limits of applicability of the linearized kinetic equation are established.