Numerical Simulation for thermohaline multiple equilibrant system in non—rectangular domains

In this paper, incompressible, double-diffusive convection is simulated using finite-difference schemes. The Navier--Stokes equations are expressed in terms of stream function and vorticity. Because of the existence of large velocity, temperature and salinity gradients in boundary layers, a boundary-fitted coordinate system is used to concentrate the grid points near the wall and fit complex boundaries. The finite-difference methods used include the high-order accurate upwind difference scheme. It is shown that the scheme is a good candidate for direct simulations of double-diffusive convection flows. The proposed method is first applied to symmetry breaking and overturning states in thermohaline-driven flows in trapezoid basins. The basic phenomena agree well with those by Dijkstra and Molemaker (1997 {\em J. Fluid Mech.} {\bf 331} 169) and Quon and Ghil (1992 {\em J. Fluid Mech.} {\bf 245} 449), but symmetry breaking and overturning states can occur in an asymmetric geometrical region without perturbations. Then the method is applied to double-diffusive convections in a cavity with opposing horizontal temperature and concentration gradients at large thermal ($Rt$), solutal ($Rs$) Rayleigh numbers and Lewis number. There are three straight sides and a sine curve side in the cavity. Basically, numerical results are in agreement with those of Lee and Hyun (1990 {\em Int. J. Heat Mass Transfer} {\bf 33} 1619) qualitatively, but eddies mixing in the top left-hand corner near the curved wall affects the layered structure.     

Simple and efficient relaxation methods for interfaces separating compressible fluids,cavitating flows and shocks in multiphase mixtures

Numerical approximation of the five-equation two-phase flow of Kapila et al. [A.K. Kapila, R. Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Physics of Fluids 13(10) (2001) 3002–3024] is examined. This model has shown excellent capabilities for the numerical resolution of interfaces separating compressible fluids as well as wave propagation in compressible mixtures [A. Murrone, H. Guillard, A five equation reduced model for compressible two phase flow problems, Journal of Computational Physics 202(2) (2005) 664–698; R. Abgrall, V. Perrier, Asymptotic expansion of a multiscale numerical scheme for compressible multiphase flows, SIAM Journal of Multiscale and Modeling and Simulation (5) (2006) 84–115; F. Petitpas, E. Franquet, R. Saurel, O. Le Metayer, A relaxation-projection method for compressible flows. Part II. The artificial heat exchange for multiphase shocks, Journal of Computational Physics 225(2) (2007) 2214–2248]. However, its numerical approximation poses some serious difficulties. Among them, the non-monotonic behavior of the sound speed causes inaccuracies in wave’s transmission across interfaces. Moreover, volume fraction variation across acoustic waves results in difficulties for the Riemann problem resolution, and in particular for the derivation of approximate solvers. Volume fraction positivity in the presence of shocks or strong expansion waves is another issue resulting in lack of robustness. To circumvent these difficulties, the pressure equilibrium assumption is relaxed and a pressure non-equilibrium model is developed. It results in a single velocity, non-conservative hyperbolic model with two energy equations involving relaxation terms. It fulfills the equation of state and energy conservation on both sides of interfaces and guarantees correct transmission of shocks across them. This formulation considerably simplifies numerical resolution. Following a strategy developed previously for another flow model [R. Saurel, R. Abgrall, A multiphase Godunov method for multifluid and multiphase flows, Journal of Computational Physics 150 (1999) 425–467], the hyperbolic part is first solved without relaxation terms with a simple, fast and robust algorithm, valid for unstructured meshes. Second, stiff relaxation terms are solved with a Newton method that also guarantees positivity and robustness. The algorithm and model are compared to exact solutions of the Euler equations as well as solutions of the five-equation model under extreme flow conditions, for interface computation and cavitating flows involving dynamics appearance of interfaces. In order to deal with correct dynamic of shock waves propagating through multiphase mixtures, the artificial heat exchange method of Petitpas et al. [F. Petitpas, E. Franquet, R. Saurel, O. Le Metayer, A relaxation-projection method for compressible flows. Part II. The artificial heat exchange for multiphase shocks, Journal of Computational Physics 225(2) (2007) 2214–2248] is adapted to the present formulation.     

Series solution of hydromagnetic flow and heat transfer with Hall effect in a second grade fluid over a stretching sheet

We examine the problem of flow and heat transfer in a second grade fluid over a stretching sheet [K. Vajravelu, T. Roper, Int. J. Nonlinear Mech. 34, 1031 (1999)]. The equations considered by Vajravelu and Roper [K. Vajravelu, T. Roper, Int. J. Nonlinear Mech. 34, 1031 (1999)], are found to be incorrect in the literature. In this paper, we not only corrected the equation but found a useful analytic solution to this important problem. We also extended the problem for hydromagnetic flow and heat transfer with Hall effect. The explicit analytic homotopy solution for the velocity field and heat transfer are presented. Graphs for the velocity field, skin friction coefficient, and rate of heat transfer are presented. Tables for the skin friction coefficient and rate of heat transfer are also presented. The convergence of the solution is also properly checked and discussed.     

Effects of mass transfer on MHD second grade fluid towards stretching cylinder: A novel perspective of Cattaneo–Christov heat flux model

The aim of this research is to analyze the effects of mass transfer on second grade fluid flow subjected to the heat transfer incorporated with the relaxation time to reach the state of equilibrium on or after the state of upheaval. A new heat model namely Cattaneo–Christov heat flux comprising the relaxation time is employed instead of very commonly used mundane model based on classical theory of heat flux. Flow is considered towards stretching cylinder in the existence of external magnetic field. Suitable transformations are first used to deduce the momentum, heat and concentration equations and are then solved analytically. The effects of physical quantities such as fluid parameter, magnetic field, Schmidt number, relaxation time, curvature parameter, Prandtl number and chemical reaction on momentum, temperature and concentration profile are examined graphically whereas for validation of results convergence analysis along with residual error are obtained numerically. A comparison of obtained results is also given with the existing literature as a limiting case of reported problem and are found an excellent agreement. The temperature profile indicates thinning effect for higher values of Prandtl number and relaxation time. It is also noted that the velocity increases with increasing values of fluid parameter whereas it declines for the case of magnetic field. This study can be used an application of central heating system and to measure the fast chemical reactions rates.     

超临界R134a水平圆管内冷却换热模拟研究

采用增强壁面函数的标准k-ε模型对超临界R134a水平圆管内冷却换热进行了模拟研究.分析了管内不同截面上流体温度、速度和湍动能的分布情况及对应关系。研究了质量流量和浮升力对换热系数的影响。结果表明,流体速度随着温度的降低而减小,并且最大速度处对应着最高温度和最小湍动能.换热系数随着质量流量的增加而增大,其峰值出现在准临界温度附近。浮升力在似液体区的影响较大,对流体换热起到增强的效果。     

太阳辐照对静止水面稳态蒸发的影响

针对低风速下静止水面的蒸发过程,考虑水面与气流的对流传热传质、水体对太阳能辐射的容积内光谱吸收及内部传热,建立分析太阳辐照作用的水面稳态蒸发模型.采用控制容积法结合蒙特卡洛法和谱带模型数值求解水体内部、表面、气流之间的能量传递与质交换.分析空气湿度、温度、流速及正逆温差下,太阳辐照强度对于水面蒸发的影响.结果表明,太阳辐照强度对低风速下静止水面蒸发的影响很大.     

Phase transitions of the energy-relative enstrophy theory for a coupled barotropic fluid–rotating sphere system

The statistical equilibrium of a coupled barotropic fluid–rotating solid sphere system is simulated using a energy-relative enstrophy spherical model in a wide range of parameter space by Monte Carlo (MC) methods [J.M. Hammersley, D.C. Handscomb, Monte Carlo Methods, Methuen & Co, London, Wiley, New York City, 1964; C.C. Lim, J. Nebus, Vorticity, Statistical Mechanics and Simulations, Springer, Berlin, 2006]. The energy-relative enstrophy model does not have the low temperature defect of the classical energy–enstrophy theory [R.H. Kraichnan, Statistical dynamics of two-dimensional flows, J. Fluid Mech. 67 (1975) 155–175] because of its microcanonical constraint on relative enstrophy. This model also differs from previous work in not fixing the angular momentum. A family of spin–lattice models are derived as convergent finite dimensional approximations to the total kinetic energy. MC simulations are used to calculate the mean nearest neighbor parity as order parameter or indicator of phase transitions in the system.     

方腔驱动流的数值模拟研究

利用CFD软件对方腔的温度场和速度场进行了数值模拟计算。采用有限元法对模型的连续性方程、动量方程和能量方程进行离散。对方腔内的流体受迫对流换热进行数值模拟,主要分析了方腔顶部流体在不同速度的情况下,对方腔内部流体的温度场和速度场以及对流换热量的影响,结果表明随着方腔顶部流速的增大方腔底部的温度场和换热量也增大。     

Thermodynamic States of the Mixed Spin 1/2 and Spin 1 Hexagonal Nanowire System Obtained from a Seven-Site Cluster Within an Improved Mean Field Approximation

The mean field approximation results in the mixedspin 1/2 Ising model and spin 1 Blume-Capel model, in the hexagonal nanowire system, are obtained from the Bogoliubov inequality. The Gibbs free energy, magnetization, and critical frontiers are obtained. Besides the stable branches of the order parameters, we obtain the metastable and unstable parts of these curves and also find phase transitions of the metastable branches of the order parameters. The classification of the stable, metastable, and unstable states is made by comparing the free energy values of these states.     

Glassy phases in a layered ising model with random interlayer exchange

Thermodynamics of a layered Ising model with infinite-range ferromagnetic intralayer interaction and random nearest-neighbor interlayer coupling is considered. A detailed analysis of the model with vanishing average interlayer coupling is presented. The Gibbs free energy is found in the critical region, and the existence of many metastable states is demonstrated. Thermodynamic parameters of the system are found for periodic states. As the mean square interlayer coupling increases, the equilibrium state of the system undergoes an infinite sequence of first-order phase transitions, the number of magnetic planes and the distance between them change discontinuously, and so do both bulk magnetization and magnetic susceptibility.     

Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states

A simplified thermodynamic approach of the incompressible 2D Euler equation is considered based on the conservation of energy, circulation and microscopic enstrophy. Statistical equilibrium states are obtained by maximizing the Miller-Robert-Sommeria (MRS) entropy under these sole constraints. We assume that these constraints are selected by properties of forcing and dissipation. We find that the vorticity fluctuations are Gaussian while the mean flow is characterized by a linear [`(w)]-y\overline{\omega}-\psi relationship. Furthermore, we prove that the maximization of entropy at fixed energy, circulation and microscopic enstrophy is equivalent to the minimization of macroscopic enstrophy at fixed energy and circulation. This provides a justification of the minimum enstrophy principle from statistical mechanics when only the microscopic enstrophy is conserved among the infinite class of Casimir constraints. Relaxation equations towards the statistical equilibrium state are derived. These equations can serve as numerical algorithms to determine maximum entropy or minimum enstrophy states. We use these relaxation equations to study geometry induced phase transitions in rectangular domains. In particular, we illustrate with the relaxation equations the transition between monopoles and dipoles predicted by Chavanis and Sommeria [J. Fluid Mech. 314, 267 (1996)]. We take into account stable as well as metastable states and show that metastable states are robust and have negative specific heats. This is the first evidence of negative specific heats in that context. We also argue that saddle points of entropy can be long-lived and play a role in the dynamics because the system may not spontaneously generate the perturbations that destabilize them.     

非晶材料玻璃转变过程中记忆效应的热力学

低温下处于非平衡态的非晶材料升温到玻璃转变以上,要先后发生弛豫和回复最终达到平衡过冷液态,其中弛豫过程中释放的能量在回复过程中以等量的方式获取,表现出明显记忆行为.本文基于氧化物、金属与小分子等多种非晶形成体系,全面探讨了在围绕玻璃转变的一个冷却加热循环过程中的焓弛豫特征,建立了弛豫谱,发现弛豫焓在数值上与熔化焓密切相关.基于弛豫焓与非晶材料动力学Fragility之间的关联,展示了非晶体系在动力学极限(m=175)条件下的玻璃转变热力学基本特征,与热力学二级相变进行了对比.研究深化了对非晶弛豫与玻璃转变热力学的理解.     

Regularization of the Burnett equations for rapid granular flows via relaxation

In [J. Fluid Mech. 361 (1998) 41] Sela and Goldhirsch have used the Chapman–Enskog expansion to derive constitutive relations for the pressure deviator P, heat flux q, and rate of energy loss Γ for rapid flows of smooth inelastic spheres. Unfortunately as in the classical Chapman–Enskog expansion for elastic spheres any truncation of the expansion beyond Navier–Stokes order (n=1) will possess unphysical instabilities. In this paper we propose a visco-elastic relaxation approximation that eliminates the instability paradox for all wave numbers, and provide a system of local equations allowing robust numerical approximations of gas dynamics valid to the Burnett order. This system is weakly parabolic, has a linearly hyperbolic convection part, and is endowed with a generalized entropy inequality in the case of purely elastic collisions, thus it is linearly stable for all wave numbers. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman–Enskog expansion.     

超临界压力CO2在水平圆管内流动传热数值分析

采用SST k-w低雷诺数湍流模型对加热条件下超临界压力CO2在内径di=22.14 mm,加热长度Lh=2440 mm水平圆管内三维稳态流动与传热特性进行了数值计算.通过超临界CO2在水平圆管内的流动传热实验数据验证了数值模型的可靠性和准确性.首先,研究了超临界压力CO2在水平圆管内的流动传热特点,基于超临界CO2在类临界温度Tpc处发生类液-类气“相变”的假设,揭示了水平圆管顶母线和底母线区域不同的流动传热行为.然后,分析了热流密度qw和质量流速G对水平圆管内超临界压力CO2流动换热的影响,通过获取流体域内的物性分布、速度分布和湍流分布等详细信息,重点解释了不同热流密度qw和质量流速G下顶母线内壁温度Tw,i分布产生差异的传热机理,分析结果确定了类气膜厚度d、类气膜性质、轴向速度u和湍动能k是影响顶母线壁温分布差异的主要因素.研究结果可以为超临界压力CO2换热装置的优化设计和安全运行提供理论指导.     

Advanced Study of Unsteady Heat and Chemical Reaction with Ramped Wall and Slip Effect on a Viscous Fluid

This paper investigate the effect of slip boundary condition, thermal radiation, heat source, Dufour number,chemical reaction and viscous dissipation on heat and mass transfer of unsteady free convective MHD flow of a viscous fluid past through a vertical plate embedded in a porous media. Numerical results are obtained for solving the nonlinear governing momentum, energy and concentration equations with slip boundary condition, ramped wall temperature and ramped wall concentration on the surface of the vertical plate. The influence of emerging parameters on velocity,temperature and concentration fields are shown graphically.     

Statistical mechanics of a nonequilibrium steady-state classical particle system driven by a constant external force

A classical particle system coupled with a thermostat driven by an external constant force reaches its steady state when the ensemble-averaged drift velocity does not vary with time. In this work, the statistical mechanics of such a system is derived solely based on the equiprobability and ergodicity principles, free from any conclusions drawn on equilibrium statistical mechanics or local equilibrium hypothesis. The momentum space distribution is determined by a random walk argument, and the position space distribution is determined by employing the equiprobability and ergodicity principles. The expressions for energy, entropy, free energy, and pressures are then deduced, and the relation among external force, drift velocity, and temperature is also established. Moreover, the relaxation towards its equilibrium is found to be an exponentially decaying process obeying the minimum entropy production theorem.     

振荡流共轭换热格子-Boltzmann模拟

振荡流共轭换热现象广泛存在于热声热机等工程应用中.基于双分布格子-Boltzmann模型,对平行平板间振荡流共轭换热进行了数值模拟.通过假定共轭界面处流体和固体的未知内能分布函数均为对应的平衡态滑移修正格式,提出了一种处理共轭换热边界的新方法.模拟结果表明,该方法可以保证共轭界面上温度连续和热流连续.分析了不同流体与固体导热系数比情况下振荡流共轭换热的速度场、温度场以及热流分布的特点.     

非共沸工质R134a /R245fa的管内冷凝换热特性研究

为研究非共沸工质的冷凝换热特性,本文基于Nusselt理论,建立了竖直圆管内非共沸混合蒸汽的冷凝模型,研究了不同质量比例的R134a/R245fa在不同条件下的冷凝换热特性,结果表明:混合蒸汽质量比例不同,两种组分的露点温度不同,混合物的冷凝特性不同,低沸点组分的气-液相份额差是表征传质阻力的关键因素;混合蒸汽质量比例、质量流速、壁面温度、压力是影响非共沸混合工质冷凝换热的重要因素。     

An immersed boundary-thermal lattice Boltzmann method using an equilibrium internal energy density approach for the simulation of flows with heat transfer

In present paper, a novel immersed boundary-thermal lattice Boltzmann method by the name of “an equilibrium internal energy density approach” is proposed to simulate the flows around bluff bodies with the heat transfer. The main idea is to combine the immersed boundary method (IBM) with the thermal lattice Boltzmann method (TLBM) based on the double population approach. The equilibrium internal energy density approach based on the equilibrium velocity approach [X. Shan, H. Chen, Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E 47 (1993) 1815] is used to combine IBM with TLBM. The idea of the equilibrium internal energy density approach is that the satisfaction of the energy balance between heat source on the immersed boundary point and the amount of change of the internal energy density according to time ensures the temperature boundary condition on the immersed boundary. The advantages of this approach are the simple concept, easy implementation and the utilization of original governing equation without modification. The simulation of natural convection in a square cavity with various body shapes for different Rayleigh numbers has been conducted to validate the capability and the accuracy of present method on solving heat transfer problems. Consequently, the present results are found to be in good agreement with those of previous studies.