Level set方法在自适应Cartesian网格上的应用

采用基于自适应Cartesian网格的level set方法对多介质流动问题进行数值模拟。采用基于四叉树的方法来生成自适应Cartesian网格。采用有限体积法求解Euler方程,控制面通量的计算采用HLLC（Hartern, Lax, van Leer, Contact）近似黎曼解方法。level set方程也采用有限体积法求解,采用Lax-Friedchs方法计算通量,通过窄带方法来减少计算量,界面的处理采用ghost fluid方法。Runge-Kutta显式时间推进,时间、空间都是二阶精度。对两种不同比热比介质激波管问题进行数值模拟,其结果和精确解吻合;对空气/氦气泡相互作用等问题进行模拟,取得令人满意的结果。     

Numerical simulation of steady-state and transient natural convection in an isothermal open cubic cavity

In this work the numeric results, of the steady-state and transient heat transfer by natural convection in a horizontal isothermal open cubic cavity are presented. The most important assumptions in the mathematical formulation are two, the flow is laminar and the Boussinesq approximation is valid. The conservation equations in primitive variables are solved using the finite volume method and the SIMPLEC algorithm. The advective terms are approximated by the SMART scheme and the diffusive terms are approximated using the central differencing scheme. The results are obtained for a Rayleigh number range from 10⁴ to 10⁷.The numerical model predicted flow instabilities and Nusselt number oscillations for high Rayleigh numbers.     

A comparative study of high‐order variable‐property segregated algorithms for unsteady low Mach number flows

In this paper, five different algorithms are presented for the simulation of low Mach flows with large temperature variations, based on second‐order central‐difference or fourth‐order compact spatial discretization and a pressure projection‐type method. A semi‐implicit three‐step Runge–Kutta/Crank–Nicolson or second‐order iterative scheme is used for time integration. The different algorithms solve the coupled set of governing scalar equations in a decoupled segregate manner. In the first algorithm, a temperature equation is solved and density is calculated from the equation of state, while the second algorithm advances the density using the differential form of the equation of state. The third algorithm solves the continuity equation and the fourth algorithm solves both the continuity and enthalpy equation in conservative form. An iterative decoupled algorithm is also proposed, which allows the computation of the fully coupled solution. All five algorithms solve the momentum equation in conservative form and use a constant‐ or variable‐coefficient Poisson equation for the pressure. The efficiency of the fourth‐order compact scheme and the performances of the decoupling algorithms are demonstrated in three flow problems with large temperature variations: non‐Boussinesq natural convection, channel flow instability, flame–vortex interaction. Copyright © 2010 John Wiley & Sons, Ltd.     

基于蒙特卡罗随机有限元方法的随机多孔介质内流体自然对流不确定性研究

为分析孔隙率不确定性对多孔介质方腔内自然对流换热的影响,发展了一种基于KL（Karhunen-Loeve展开）-蒙特卡罗随机有限元算法的随机多孔介质内自然对流不确定性分析数理模型及有限元数值模拟程序框架。通过K-L展开及基于拉丁抽样法生成多孔介质孔隙率随机实现,并耦合多孔介质自然对流有限元程序,进行随机多孔介质内自然对流传热数值模拟,得出了多孔介质内流场与温度场平均值与标准偏差,并分析了孔隙率不确定性条件下Da数对Nu数的影响。结果表明,孔隙率不确定性对多孔介质方腔内自然对流有重要影响。随机多孔介质内流场及温度场与确定性条件下的流场及温度场存在一定偏差,Nu数标准偏差随着Da的增大先增大后减小。     

A higher resolution edge‐based finite volume method for the simulation of the oil–water displacement in heterogeneous and anisotropic porous media using a modified IMPES method

In this article, we present a higher‐order finite volume method with a ‘Modified Implicit Pressure Explicit Saturation’ (MIMPES) formulation to model the 2D incompressible and immiscible two‐phase flow of oil and water in heterogeneous and anisotropic porous media. We used a median‐dual vertex‐centered finite volume method with an edge‐based data structure to discretize both, the elliptic pressure and the hyperbolic saturation equations. In the classical IMPES approach, first, the pressure equation is solved implicitly from an initial saturation distribution; then, the velocity field is computed explicitly from the pressure field, and finally, the saturation equation is solved explicitly. This saturation field is then used to re‐compute the pressure field, and the process follows until the end of the simulation is reached. Because of the explicit solution of the saturation equation, severe time restrictions are imposed on the simulation. In order to circumvent this problem, an edge‐based implementation of the MIMPES method of Hurtado and co‐workers was developed. In the MIMPES approach, the pressure equation is solved, and the velocity field is computed less frequently than the saturation field, using the fact that, usually, the velocity field varies slowly throughout the simulation. The solution of the pressure equation is performed using a modification of Crumpton's two‐step approach, which was designed to handle material discontinuity properly. The saturation equation is solved explicitly using an edge‐based implementation of a modified second‐order monotonic upstream scheme for conservation laws type method. Some examples are presented in order to validate the proposed formulation. Our results match quite well with others found in literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Investigation of the accuracy of different methods of interferogram analysis for calculation of local free convection heat transfer coefficient on axisymmetric objects

An experimental study using a Mach–Zehnder interferometer has been carried out in order to investigate the accuracy of four different methods of interferogram analysis for obtaining free convection heat transfer coefficient on heated axisymmetric objects. Different methods of reference fringe interferogram analysis can be categorized as classical and transform methods. In transform methods, a mathematical transform method is applied to solve the governing integral equation while in the classical methods the integral equation is approximated by a finite summation. Classical methods are also divided into two groups according to the equations which are based upon. Experiments have been carried out on a vertical isothermal cylinder in air with three different surface temperatures. Four methods of interferogram analysis which are three classical and a transform method have been used to calculate the temperature distributions and the local free convection heat transfer coefficients at different cross sections. In order to investigate the accuracy of the methods, experimental values of the local heat transfer coefficient have been compared with the numerical solution and the mean relative error of each method has been obtained. Results show that the transform method is the most accurate one with the shortest solution time. It is also shown that assuming more complex functions for variation of index of refraction in the classical methods could lead to more accurate results.     

Conjugate mixed convection with surface radiation from a vertical electronic board with multiple discrete heat sources

The problem of combined conduction-mixed convection-surface radiation from a vertical electronic board provided with three identical flush-mounted discrete heat sources is solved numerically. The cooling medium is air that is considered to be radiatively transparent. The governing equations for fluid flow and heat transfer are converted from primitive variable form to stream function-vorticity formulation. The equations, thus obtained, are normalised and then are converted into algebraic form using a finite volume based finite difference method. The resulting algebraic equations are then solved using Gauss–Seidel iterative method. An optimum grid system comprising 151 grids along the board and 111 grids across the board is chosen. The effects of various parameters, such as modified Richardson number, surface emissivity and thermal conductivity on temperature distribution along the board, maximum board temperature and relative contributions of mixed convection and radiation to heat dissipation are studied in detail. Further, the contributions of free and forced convection components of mixed convection to board temperature distribution and peak board temperature are brought out. The exclusive roles played by surface radiation and buoyancy in the present problem are clearly elucidated.     

A depth‐integrated non‐hydrostatic finite element model for wave propagation

This paper presents a two‐dimensional finite element model for simulating dynamic propagation of weakly dispersive waves. Shallow water equations including extra non‐hydrostatic pressure terms and a depth‐integrated vertical momentum equation are solved with linear distributions assumed in the vertical direction for the non‐hydrostatic pressure and the vertical velocity. The model is developed based on the platform of a finite element model, CCHE2D. A physically bounded upwind scheme for the advection term discretization is developed, and the quasi second‐order differential operators of this scheme result in no oscillation and little numerical diffusion. The depth‐integrated non‐hydrostatic wave model is solved semi‐implicitly: the provisional flow velocity is first implicitly solved using the shallow water equations; the non‐hydrostatic pressure, which is implicitly obtained by ensuring a divergence‐free velocity field, is used to correct the provisional velocity, and finally the depth‐integrated continuity equation is explicitly solved to satisfy global mass conservation. The developed wave model is verified by an analytical solution and validated by laboratory experiments, and the computed results show that the wave model can properly handle linear and nonlinear dispersive waves, wave shoaling, diffraction, refraction and focusing. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Reliability of Numerical Solutions of Brine Transport in Groundwater: Analysis of Infiltration from a Salt Lake

The density dependent flow and transport problem in groundwater is solved numerically by means of a mixed finite element scheme for the flow equation and a mixed finite element-finite volume time-splitting based technique for the transport equation. The proposed approach, spatially second order accurate, is used to address the issue of grid convergence by solving on successively refined grids the salt lake problem, a physically unstable downward convection with formation of fingers. Numerical results indicate that achievement of grid convergence is problematic due to ill-conditioning arising from the strong nonlinearities of the mathematical model.     

Unsteady natural convective flow past a moving vertical cylinder with heat and mass transfer

Transient natural convection boundary layer flow of an incompressible viscous fluid past an impulsively moving semi- infinite vertical cylinder is considered. The temperature and concentration of the cylinder surface are taken to be uniform. The unsteady, nonlinear and coupled governing equations of the flow are solved using an implicit finite difference scheme. The finite difference scheme is unconditionally stable and accurate. Numerical results are presented with various sets of parameters for both air and water. Transient effects of velocity, temperature and concentration profiles are analyzed. Local and average skin friction, rates of heat and mass transfer are shown graphically. Received on 1 November 1999     

Laminar natural convection of Cu-water nanofluid in concentric annuli with radial fins attached to the inner cylinder

Laminar natural convection of Cu-water nano-fluid between two horizontal concentric cylinders with radial fins attached to the inner cylinder is studied numerically. The inner and outer cylinders are maintained at constant temperature. The governing equations in the polar two-dimensional space with the respective boundary conditions are solved using the finite volume method. The hybrid-scheme is used to discretize the convection terms. In order to couple the velocity field and the pressure in the momentum equations, the well known semi-implicit method for pressure linked equation reformed algorithm is adopted. Using the developed code, a parametric study is undertaken, and the effects of the Rayleigh number, Number of fins, length of the fins and the volume fraction of nano-particles on the fluid flow and heat transfer inside the annuli are investigated. In this study, two cases with different number of fins are considered. It is observed from the results that the average Nusselt number increases with increasing both the Rayleigh number and the volume fraction of the nano-particles. Moreover, the average Nusselt number decreases by increasing the fins’ length and the number of fins. Heat transfer rate increases by increasing the fins’ length at all Rayleigh numbers, but it increases by increasing the number of fins at high Rayleigh numbers.     

A combined finite volumes - finite elements method for a low-Mach model

In this paper, we develop a finite volumes - finite elements method based on a time splitting to simulate some low-Mach flows. The mass conservation equation is solved by a vertex-based finite volume scheme using a τ-limiter. The momentum equation associated with the compressibility constraint is solved by a finite element projection scheme. The originality of the approach is twofold. First, the state equation linking the temperature, the density, and the thermodynamic pressure is imposed implicitly. Second, the proposed combined scheme preserves the constant states, in the same way as a similar one previously developed for the variable density Navier-Stokes system. Some numerical tests are performed to exhibit the efficiency of the scheme. On the one hand, academic tests illustrate the ability of the scheme in term of convergence rates in time and space. On the other hand, our results are compared to some of the literature by simulating a transient injection flow as well as a natural convection flow in a cavity.     

Natural convection flow from a continuously moving vertical surface immersed in a thermally stratified medium

 Natural convection boundary layer flow over a continuously moving isothermal vertical surface immersed in a thermally stratified medium has been investigated here. The non-linear coupled partial differential equations governing the non-similar flow have been solved numerically using an implicit finite difference scheme. For small values of the streamwise distance the partial differential equations are solved by using a perturbation expansion procedure and also using the Shanks transformation. The results indicate that the thermal stratification significantly affects both the surface shear stress and the surface heat transfer. The buoyancy parameter and the Prandtl number increase significantly, both the surface shear stress and heat transfer. Also the buoyancy force gives rise to an overshoot in the velocity profile. Received on 1 February 2000     

MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure

This study looks at MHD natural convection flow and heat transfer in a laterally heated enclosure with an off-centred partition. Governing equations in the form of vorticity–stream function formulation are solved using the polynomial differential quadrature (PDQ) method. Numerical results are obtained for various values of the partition location, Rayleigh, Prandtl and Hartmann numbers. The results indicate that magnetic field significantly suppresses flow, and thus heat transfer, especially for high Rayleigh number values. The results also show that the x-directional magnetic field is more effective in damping convection than the y-directional magnetic field, and the average heat transfer rate decreases with an increase in the distance of the partition from the hot wall. The average heat transfer rate decreases up to 80% if the partition is placed at the midpoint and an x-directional magnetic field is applied. The results also show that flow and heat transfer have little dependence on the Prandtl number.     

Buoyancy-driven convection of water near its density maximum with time periodic partially active vertical walls

The effect of density inversion on transient natural convection heat transfer of cold water in a square cavity with partially active vertical walls is studied numerically. The governing equations are solved by control volume method with power law scheme. In the hot location the temperature is varied sinusoidally and in the cold location uniform temperature is maintained. Nine different positions of the active zones are considered. Results are discussed for various values of the amplitude, period and different Grashof numbers and presented graphically in the form of isotherms, streamlines, mid-height velocity profile and average Nusselt number. It is found that density inversion of water affects natural convection flow and heat transfer. Heat transfer rate is enhanced upto 80% when the heating location is in the middle of the hot wall.     

Influence of heat transfer on the dynamic response of a spherical gas/vapour bubble

The standard approach to analyse the bubble motion is the well known Rayleigh–Plesset equation. When applying the toolbox of nonlinear dynamical systems to this problem several aspects of physical modelling are usually sacrificed. Particularly in vapour bubbles the heat transfer in the liquid domain has a significant effect on the bubble motion; therefore the nonlinear energy equation coupled with the Rayleigh–Plesset equation must be solved. The main aim of this paper is to find an efficient numerical method to transform the energy equation into an ODE system, which, after coupling with the Rayleigh–Plesset equation can be analysed with the help of bifurcation theory. Due to the strong nonlinearity and violent bubble motions the computational effort can be high, thus it is essential to reduce the size of the problem as much as possible. In the first part of the paper finite difference, Galerkin and spectral collocation methods are examined and compared in terms of efficiency. In the second part free and forced oscillations are analysed with an emphasis on the influence of heat transfer. In the case of forced oscillations the unstable branches of the amplification diagrams are also computed.     

MHD natural convection in porous media-filled enclosures

In this work, the magnetohydrodynamics （MHD） natural convection heat transfer problem inside a porous medium filled with inclined rectangular enclosures is investigated numerically. The boundary conditions selected on the enclosure are two adiabatic and two isothermal walls. The governing equations, continuity, and Forchheimer extension of the Darcy law and energy are transformed into dimensionless forms by using a set of suitable variables, and then solved by using a finite difference scheme. The governing parameters are the magnetic influence number, the Darcy Rayleigh number, the inclination angle, and the aspect ratio of the enclosure. It is found that the magnetic influence number and the inclination angle have pronounced effects on the fluid flow and heat transfer in porous media-filled enclosures.     

Non-Linear Two Dimensional Double Diffusive Convection in a Rotating Porous Layer Saturated by a Viscoelastic Fluid

Double diffusive convection in a rotating anisotropic porous layer, saturated by a viscoelastic fluid, heated from below and cooled from above has been studied making linear and non-linear stability analyses. The fluid and solid phases are considered to be in equilibrium. In momentum equation, we have employed the Darcy equation which includes both time derivative and Coriolis terms. The linear theory based on normal mode method is considered to find the criteria for the onset of stationary and oscillatory convection. A weak non-linear analysis based on minimal representation of truncated Fourier series analysis containing only two terms has been used to find the Nusselt number and Sherwood number as functions of time. We have solved the finite amplitude equations using a numerical scheme. The results obtained, during the above analyses, have been presented graphically and the effects of various parameters on heat and mass transfer have been discussed. Finally, we have drawn the steady and unsteady streamlines, isotherms, and isohalines for various parameters.     

A numerical study of heat island flows: Stationary solutions

We present two‐dimensional numerical simulations of a natural convection problem in an unbounded domain. The flow circulation is induced by a heat island located on the ground and thermal stratification is applied in the vertical direction. The main effect of this stably stratified environment is to induce the propagation of thermal perturbations in the horizontal direction far from the local thermal source. Numerical stationary solutions at Ra?10⁵ are computed in large elongated computational domains: convergence with respect to the domain sizes is investigated at different resolutions. On fine grids, with mesh size , a thermal sponge layer is added at the vertical boundaries: this local damping technique improves the convergence with respect to the domain length. Boussinesq equations are discretized with a second‐order finite volume scheme on a staggered grid combined with a second‐order projection method for the time integration. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical study of evaporation by mixed convection of a binary liquid film flowing down the wall of two coaxial cylinders

Evaporation by mixed convection of a binary liquid film flowing down the external wall of a vertical cylinder has been investigated numerically. Two cases were considered: one where the cylinder wall is soaked with a liquid, and another where a liquid film flows along this wall. Heat, mass and momentum transfer in the liquid film and the vapor phase are modelled by mixed convection equations. In order to locate the liquid–vapor interface, a suitable coordinate transformation is carried out with suitable variables. The discretization of the dimensionless equations by an implicit difference scheme leads to a system of algebraic equations, which are solved by using Gauss algorithm for the momentum conservation equations and Thomas algorithm for the energy and diffusion conservation equations. The film thickness is calculated by the Newtons method. Results show, in particular, that the film thickness cannot be neglected and that the latent heat transfers are increasingly significant as the liquid film components become more volatile.