Nonequilibrium Gas and Generalized Billiards

Generalized billiards describe nonequilibrium gas, consisting of finitely many particles, that move in a container, whose walls heat up or cool down. Generalized billiards can be considered both in the framework of the Newtonian mechanics and of the relativity theory. In the Newtonian case, a generalized billiard may possess an invariant measure; the Gibbs entropy with respect to this measure is constant. On the contrary, generalized relativistic billiards are always dissipative,and the Gibbs entropy with respect to the same measure grows under some natural conditions. In this article, we find the necessary and sufficient conditions for a generalized Newtonian billiard to possess a smooth invariant measure, which is independent of the boundary action: the corresponding classical billiard should have an additional first integral of special type. In particular,the generalized Sinai billiards do not possess a smooth invariant measure. We then consider generalized billiards inside a ball, which is one of the main examples of the Newtonian generalized billiards which does have an invariant measure. We construct explicitly the invariant measure, and find the conditions for the Gibbs entropy growth for the corresponding relativistic billiard both formonotone and periodic action of the boundary.     

Multiple temperature model for the information preservation method and its application to nonequilibrium gas flows

The information preservation (IP) method has been successfully applied to various nonequilibrium gas flows. Comparing with the direct simulation Monte Carlo (DSMC) method, the IP method dramatically reduces the statistical scatter by preserving collective information of simulation molecules. In this paper, a multiple temperature model is proposed to extend the IP method to strongly translational nonequilibrium gas flows. The governing equations for the IP quantities have been derived from the Boltzmann equation based on an assumption that each simulation molecule represents a Gaussian distribution function with a second-order temperature tensor. According to the governing equations, the implementation of IP method is divided into three steps: molecular movement, molecular collision, and update step. With a reasonable multiple temperature collision model and the flux splitting method in the update step, the transport of IP quantities can be accurately modeled. We apply the IP method with the multiple temperature model to shear-driven Couette flow, external force-driven Poiseuille flow and thermal creep flow, respectively. In the former two cases, the separation of different temperature components is clearly observed in the transition regime, and the velocity, temperature and pressure distributions are also well captured. The thermal creep flow, resulting from the presence of temperature gradients along boundary walls, is properly simulated. All of the IP results compare well with the corresponding DSMC results, whereas the IP method uses much smaller sampling sizes than the DSMC method. This paper shows that the IP method with the multiple temperature model is an accurate and efficient tool to simulate strongly translational nonequilibrium gas flows.     

Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows

With the advent of state-of-the-art computers and their rapid availability, the time is ripe for the development of efficient uncertainty quantification (UQ) methods to reduce the complexity of numerical models used to simulate complicated systems with incomplete knowledge and data. The spectral stochastic finite element method (SSFEM) which is one of the widely used UQ methods, regards uncertainty as generating a new dimension and the solution as dependent on this dimension. A convergent expansion along the new dimension is then sought in terms of the polynomial chaos system, and the coefficients in this representation are determined through a Galerkin approach. This approach provides an accurate representation even when only a small number of terms are used in the spectral expansion; consequently, saving in computational resource can be realized compared to the Monte Carlo (MC) scheme. Recent development of a finite difference lattice Boltzmann method (FDLBM) that provides a convenient algorithm for setting the boundary condition allows the flow of Newtonian and non-Newtonian fluids, with and without external body forces to be simulated with ease. Also, the inherent compressibility effect in the conventional lattice Boltzmann method, which might produce significant errors in some incompressible flow simulations, is eliminated. As such, the FDLBM together with an efficient UQ method can be used to treat incompressible flows with built in uncertainty, such as blood flow in stenosed arteries. The objective of this paper is to develop a stochastic numerical solver for steady incompressible viscous flows by combining the FDLBM with a SSFEM. Validation against MC solutions of channel/Couette, driven cavity, and sudden expansion flows are carried out.     

High-order shock-fitting methods for direct numerical simulation of hypersonic flow with chemical and thermal nonequilibrium

In recent years, much progress has been made in the direct numerical simulation of laminar-turbulent transition of hypersonic boundary layer flow. However, most of the efforts at the direct numerical simulation of transition previously have been focused on the idealized perfect gas flow or “cold” hypersonic flows. For practical problems in hypersonic flows, high-temperature effects of thermal and chemical nonequilibrium are important and cannot be modeled by a perfect gas model. Therefore, it is necessary to include the real gas models in the numerical simulation of hypersonic boundary layer transition in order to accurately predict flow field parameters. Currently most numerical methods for hypersonic flow with thermo-chemical nonequilibrium are based on shock-capturing approach at relatively low order of accuracy. Shock capturing schemes reduce to first-order accuracy near the shock and have been shown to produce spurious oscillations behind curved strong shocks. There is a need to develop new methods capable of simulating nonequilibrium hypersonic flow fields with uniformly high-order accuracy and avoid spurious oscillations near the shock. This paper presents a fifth-order shock-fitting method for numerical simulation of thermal and chemical nonequilibrium in hypersonic flows. The method is developed based on the state-of-the-art real gas models for thermo-chemical nonequilibrium and transport phenomena. Shock-fitting approach is used because it has the advantage of capturing the entire flow field with high-order accuracy and without any oscillations near the shock. The new method has been tested and validated for a number of test cases over a wide span of free stream conditions. The developed method is applied for the study of receptivity of free stream acoustic waves over a blunt cone for hypervelocity flow. Some preliminary results of the computations of the high order shock fitting method for the above mentioned study have also been presented.     

Dynamic interaction of plasma flow with the hot boundary layer of a geomagnetic trap

The study of the interaction between collisionless plasma flow and stagnant plasma revealed the presence of an outer boundary layer at the border of a geomagnetic trap, where the super-Alfvén subsonic laminar flow changes over to the dynamic regime characterized by the formation of accelerated magnetosonic jets and decelerated Alfvén flows with characteristic relaxation times of 10–20 min. The nonlinear interaction of fluctuations in the initial flow with the waves reflected from an obstacle explains the observed flow chaotization. The Cherenkov resonance of the magnetosonic jet with the fluctuation beats between the boundary layer and the incoming flow is the possible mechanism of its formation. In the flow reference system, the incoming particles are accelerated by the electric fields at the border of boundary layer that arise self-consistently as a result of the preceding wave-particle interactions; the inertial drift of the incoming ions in a transverse electric field increasing toward the border explains quantitatively the observed ion acceleration. The magnetosonic jets may carry away downstream up to a half of the unperturbed flow momentum, and their dynamic pressure is an order of magnitude higher than the magnetic pressure at the obstacle border. The appearance of nonequilibrium jets and the boundary-layer fluctuations are synchronized by the magnetosonic oscillations of the incoming flow at frequencies of 1–2 mHz.     

尖化前缘的稀薄气体化学非平衡流动和气动加热相似律研究

以近空间尖前缘高超声速巡航飞行器的研制为背景, 作者在前一阶段采用模型理论分析方法, 陆续研究了沿微钝前缘驻点线的化学非平衡流动和气动加热相似律, 文章是上述研究的综合回顾和深化讨论.稀薄条件下, 驻点附近流动和传热出现一系列与连续流动模型不同的新特征, 超出了经典气动热预测理论的适用范围.作者建立了一个沿驻点线能量传递和转化的广义模型, 并分别推导了具有实际物理意义的边界层外离解非平衡流动判据和边界层内复合非平衡流动判据.基于这些判据构建了预测非平衡流动驻点气动加热的桥函数, 并讨论了稀薄非平衡真实气体流动和气动加热的相似律, 发现新型近空间尖前缘飞行器遭遇的气动热环境不同于传统大钝头航天器再入问题, 传统的天地换算相似准则将会失效.这些理论分析结果可为稀薄非平衡化学反应流及气动加热的实验和计算提供一个标模检验的手段.      

Nonequilibrium Thermodynamics of Polymeric Liquids via Atomistic Simulation

The challenge of calculating nonequilibrium entropy in polymeric liquids undergoing flow was addressed from the perspective of extending equilibrium thermodynamics to include internal variables that quantify the internal microstructure of chain-like macromolecules and then applying these principles to nonequilibrium conditions under the presumption of an evolution of quasie equilibrium states in which the requisite internal variables relax on different time scales. The nonequilibrium entropy can be determined at various levels of coarse-graining of the polymer chains by statistical expressions involving nonequilibrium distribution functions that depend on the type of flow and the flow strength. Using nonequilibrium molecular dynamics simulations of a linear, monodisperse, entangled C₁₀₀₀H₂₀₀₂ polyethylene melt, nonequilibrium entropy was calculated directly from the nonequilibrium distribution functions, as well as from their second moments, and also using the radial distribution function at various levels of coarse-graining of the constituent macromolecular chains. Surprisingly, all these different methods of calculating the nonequilibrium entropy provide consistent values under both planar Couette and planar elongational flows. Combining the nonequilibrium entropy with the internal energy allows determination of the Helmholtz free energy, which is used as a generating function of flow dynamics in nonequilibrium thermodynamic theory.     

纳米尺度流动速度计算的新方法

采用流体力学中流量与流速的计算和分子动力学相结合的方法,模拟液态氩在纳米通道内的三维Poiseuille流动和驱动方腔流动,计算流体流速.结果表明:平板形纳米通道内,该方法求得的流速与传统分子动力学方法所求流速基本吻合,可以用该方法计算不同壁面情况下的流速;对于纳米方腔通道内的流体,在不同模型下两种方法计算出的流速分布大致相同,但是其边界速度差别很大,在边界的速度计算方面新方法的精确度更高,收敛速度比传统方法快.     

Time-dependent and outflow boundary conditions for Dissipative Particle Dynamics

We propose a simple method to impose both no-slip boundary conditions at fluid-wall interfaces and at outflow boundaries in fully developed regions for Dissipative Particle Dynamics (DPD) fluid systems. The procedure to enforce the no-slip condition is based on a velocity-dependent shear force, which is a generalized force to represent the presence of the solid-wall particles and to maintain locally thermodynamic consistency. We show that this method can be implemented in both steady and time-dependent fluid systems and compare the DPD results with the continuum limit (Navier-Stokes) results. We also develop a force-adaptive method to impose the outflow boundary conditions for fully developed flow with unspecified outflow velocity profile or pressure value. We study flows over the backward-facing step and in idealized arterial bifurcations using a combination of the two new boundary methods with different flow rates. Finally, we explore the applicability of the outflow method in time-dependent flow systems. The outflow boundary method works well for systems with Womersley number of O(1), i.e., when the pressure and flowrate at the outflow are approximately in-phase.     

Influence of wall motion on particle sedimentation using hybrid LB-IBM scheme

We integrate the lattice Boltzmann method (LBM) and immersed boundary method (IBM) to capture the coupling between a rigid boundary surface and the hydrodynamic response of an enclosed particle laden fluid. We focus on a rigid box filled with a Newtonian fluid where the drag force based on the slip velocity at the wall and settling particles induces the interaction. We impose an external harmonic oscillation on the system boundary and found interesting results in the sedimentation behavior. Our results reveal that the sedimentation and particle locations are sensitive to the boundary walls oscillation amplitude and the subsequent changes on the enclosed flow field. Two different particle distribution analyses were performed and showed the presence of an agglomerate structure of particles. Despite the increase in the amplitude of wall motion, the turbulence level of the flow field and distribution of particles are found to be less in quantity compared to the stationary walls. The integrated LBM-IBM methodology promised the prospect of an efficient and accurate dynamic coupling between a non-compliant bounding surface and flow field in a wide-range of systems. Understanding the dynamics of the fluid-filled box can be particularly important in a simulation of particle deposition within biological systems and other engineering applications.     

Adaptive ANOVA decomposition of stochastic incompressible and compressible flows

Realistic representation of stochastic inputs associated with various sources of uncertainty in the simulation of fluid flows leads to high dimensional representations that are computationally prohibitive. We investigate the use of adaptive ANOVA decomposition as an effective dimension–reduction technique in modeling steady incompressible and compressible flows with nominal dimension of random space up to 100. We present three different adaptivity criteria and compare the adaptive ANOVA method against sparse grid, Monte Carlo and quasi-Monte Carlo methods to evaluate its relative efficiency and accuracy. For the incompressible flow problem, the effect of random temperature boundary conditions (modeled as high-dimensional stochastic processes) on the Nusselt number is investigated for different values of correlation length. For the compressible flow, the effects of random geometric perturbations (simulating random roughness) on the scattering of a strong shock wave is investigated both analytically and numerically. A probabilistic collocation method is combined with adaptive ANOVA to obtain both incompressible and compressible flow solutions. We demonstrate that for both cases even draconian truncations of the ANOVA expansion lead to accurate solutions with a speed-up factor of three orders of magnitude compared to Monte Carlo and at least one order of magnitude compared to sparse grids for comparable accuracy.     

高超声速化学非平衡流动MHD效应的数值模拟

对偶极子磁场作用下的三维钝头体高超声速黏性绕流的化学非平衡流动进行了数值模拟.计算结果表明,在偶极子磁场强度为1.353 T时,与无磁场作用时相比,激波的脱体距离增加,增幅约为50%;壁面压力系数下降,局部下降最大达37%;壁面摩擦系数减小,局部最大可减小20%.同时,计算结果与磁场作用下的冻结流进行了比较,结果显示化学非平衡流中激波脱体距离比冻结流中小50%,滞止点温度低一半.     

圆截面直流道中微粒黏弹性聚焦机理研究

微粒黏弹性聚焦技术近年来受到了广泛的研究重视, 但影响粒子聚焦特性的关键参数调控机理仍不清楚. 基于此目的, 本文量化研究了圆截面直流道中非牛顿流体诱导微粒黏弹性聚焦的行为, 给出了流速和流道长度对粒子聚焦特性的调控机理. 具体而言: 首先, 对比分析不同黏度牛顿流体(水和22 wt%甘油水溶液)和非牛顿流体(8 wt%聚乙烯吡咯烷酮水溶液)中粒子横向迁移行为, 发现非牛顿流体中粒子将在弹性力主导下聚焦至流道中心区域, 而牛顿流体中粒子则在惯性升力主导下迁移形成Segré-Silberberg圆环. 其次, 量化分析粒子尺寸和驱动流速对黏弹性聚焦效果的影响, 发现随着流速的增加, 粒子聚焦效果逐渐变好并最终趋于稳定, 且大粒子较小粒子具有更好的聚焦效果. 最后, 研究粒子沿流道长度的动态聚焦过程, 推导并验证了粒子聚焦所需安全流道长度的数学模型, 发现大粒子聚焦所需安全流道长度显著短于小粒子. 上述研究结果对于提升粒子黏弹性聚焦机理和过程的理解, 实现微粒聚焦特性的灵活控制具有非常重要的意义.     

Discrete Boltzmann Method with Maxwell-Type Boundary Condition for Slip Flow

The rarefied effect of gas flow in microchannel is significant and cannot be well described by traditional hydrodynamic models. It has been known that discrete Boltzmann model(DBM) has the potential to investigate flows in a relatively wider range of Knudsen number because of its intrinsic kinetic nature inherited from Boltzmann equation.It is crucial to have a proper kinetic boundary condition for DBM to capture the velocity slip and the flow characteristics in the Knudsen layer. In this paper, we present a DBM combined with Maxwell-type boundary condition model for slip flow. The tangential momentum accommodation coefficient is introduced to implement a gas-surface interaction model.Both the velocity slip and the Knudsen layer under various Knudsen numbers and accommodation coefficients can be well described. Two kinds of slip flows, including Couette flow and Poiseuille flow, are simulated to verify the model.To dynamically compare results from different models, the relation between the definition of Knudsen number in hard sphere model and that in BGK model is clarified.     

Entropic lattice Boltzmann method for turbulent flow simulations: Boundary conditions

We introduce a new concept of boundary conditions for realization of the lattice Boltzmann simulations of turbulent flows. The key innovation is the use of a universal distribution function for particles, analogous to the Tamm–Mott-Smith solution for the shock wave in the classical Boltzmann kinetic equation. Turbulent channel flow simulations demonstrate that the new boundary enables accurate results even with severely under-resolved grids. Generalization to complex boundary is illustrated with an example of turbulent flow past a circular cylinder.     

A Higher-Order Boundary Treatment for Cartesian-Grid Methods

The Euler equations describe the flow phenomena of compressible inviscid gas dynamics. We simulate such flows using a higher-order Cartesian-grid method, together with a special treatment for the cells cut by the boundary of an object. A new method for the treatment of the boundary is described where these cut boundary cells are maintained as whole cells rather than as cut cells, thus avoiding stability problems. The method is second-order accurate in one dimension and higher-order accurate in two dimensions but not strictly conservative; however, we show that this error in the conservation does not lead to spurious phenomena on some representative test calculations. The advantages of the new boundary treatment are that it is higher-order accurate, that it is independent of the applied method, and that it is simple.     

基于流动显示的压缩拐角流动结构定量研究

在Ma = 3.0的超声速风洞中, 采用NPLS技术对上游边界层为层流的25° 压缩拐角进行了流动显示实验, 获得了压缩拐角的精细流动结构, 边界层、剪切层和激波等结构清晰可见. 基于流动显示数据, 采用间歇性、空间相关性和分形分析对流动结构进行了定量研究, 计算了边界层和分离区的间歇因子分布, 获取了边界层中拟序结构和结构角的大小, 给出了边界层分形维数的分布, 并与Ringuette和Bookey等的实验结果进行比较, 阐述了压缩拐角流动结构的定量特征.     

The sharkskin instability of polymer melt flows

Flows of polymeric liquids undergo instabilities whose origins are quite different from those of Newtonian flows, due to their elastic character and the complexity of the fluid/solid boundary condition. This article reviews recent studies of one such instability, the sharkskin phenomenon observed during extrusion of many linear polymers. Key experimental observations are summarized; one important fact that has become clear is the importance of the interaction between the molten polymer and the solid walls of the flow channel, especially near the contact line at the exit of the channel. Recent developments in understanding the relationship between wall slip and disentanglement of wall-adsorbed polymers from the bulk flow are briefly described, and putative heuristic mechanisms relating the instability to slip and contact line motion are presented. Finally, we review mathematical analyses of the stability of viscoelastic shear flows with slip boundary conditions. Some recent analyses yield instability predictions that are consistent with experiments, but further work is required to discriminate between the various mechanisms that have been proposed. (c) 1999 American Institute of Physics.     

Development of the diffusion-inertia model of particle deposition in turbulent flows

There is presented a modification of the diffusion-inertia model that describes the distribution and deposition of low-inertia particles in turbulent near-wall flows. For the transport equation of the dispersed phase concentration, there is proposed a new wall function that takes into account the nonequilibrium effects and nonlocality of the turbulent transport of the dispersed phase in the near-wall zone caused by the particles’ inertia. This allowed widening the applicability limits of the diffusion-inertia model even for particles with a relaxation time with a magnitude of several hundred. The calculation results for the rate of the particles’ deposition from the turbulent flow to the walls in a round pipe are in good accord with the literature experimental data and the data of direct numerical simulation.     

Automated Digital Image Based Measurement of Boundary Fractal Dimension for Complex Nanoparticles

There is a growing realization that complex nanoparticles produced by combustion reaction, precipitation, and spray technology using supercritical fluids, are fractally structured. The boundary fractal dimension is linked to the flow, packing and consolidation dynamics of nanopowders. It also contains information on the formation dynamics of the nanoparticles produced by various methods. Extraction of the fractal dimension information embodied in the nanoparticle's fractal structure is hampered by the lack of automated characterization algorithms for processing images of particles. This paper describes an efficient algorithm for analyzing digitized images of fractally structured nanoparticles and presents a computer program that automates the procedure using digital image processing techniques. The program functionality is demonstrated and discussed using digital images of typical pigment, ceramic and pharmaceutical powders.