Analytical investigation of Bénard-Marangoni convection heat transfer in a shallow cavity filled with two immiscible fluids

The mechanism of natural and Marangoni convection in a system with two stratified fluid layers without mass transfer at the interface is investigated. The basis of the analytical solution is an assumption of parallel flow over a large portion of the system. The two cases of heat fluxes through horizontal or vertical opposite walls are considered. It is demonstrated that four different patterns of convection can be observed in the present system. The zone of occurrence of these flow patterns are specified in terms of non-dimensional parameters. Velocity and temperature distributions, stream function and Nusselt number are presented over a wide range of the governing parameters. The results obtained are explained in terms of the basic physical mechanisms that govern these flows showing many interesting aspects of the complex interaction between the buoyant and surface tension mechanisms.     

Natural Convection Induced by a Centrifugal Force Field in a Horizontal Annular Porous Layer Saturated with a Binary Fluid

The Darcy Model with the Boussinesq approximation is used to study natural convection in a horizontal annular porous layer filled with a binary fluid, under the influence of a centrifugal force field. Neumann boundary conditions for temperature and concentration are applied on the inner and outer boundary of the enclosure. The governing parameters for the problem are the Rayleigh number, Ra, the Lewis number, Le, the buoyancy ratio, j{\varphi } , the radius ratio of the cavity, R, the normalized porosity, e{\varepsilon } , and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in a thin annular layer (R → 1), analytical solutions for the stream function, temperature and concentration fields are obtained using a concentric flow approximation and an integral form of the energy equation. The critical Rayleigh number for the onset of supercritical convection is predicted explicitly by the present model. Also, results are obtained from the analytical model for finite amplitude convection for which the flow and heat and mass transfer are presented in terms of the governing parameters of the problem. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical model and the numerical simulations.     

Numerical convergence studies of the mixed finite element method for natural convection flow in a fluid-saturated porous medium

This paper describes a numerical approximation scheme for the natural convection (NC) flow in a fluid-saturated porous medium. Our formulation of the problem is based on the mixed finite element method (FEM). Using the so-called consistent splitting scheme, a second-order accuracy in time and in space is verified by the numerical calculation. The resulting flow patterns and heat transfer for different Rayleigh numbers, Darcy numbers and porosities are numerically studied by the proposed scheme.     

Numerical simulation of laminar natural convection in a laterally heated vertical cylindrical enclosure: application to crystal growth

Laminar natural convection has been studied in a laterally heated vertical cylindrical enclosure with a free insulated surface and a centrally located constant temperature wall at the top. These conditions are a simplification of the conditions existing in a Czochralski crystal pulling system. The laminar, axisymmetric flow of a Newtonian, constant physical properties fluid under Boussinesq’s approximation has been considered. Governing equations in primitive variable form are solved numerically by control volume method. SIMPLE algorithm due to Patankar has been used for the numerical simulation. The effects of the constant wall heat flux boundary condition at the side wall have been investigated whereas the bottom wall is considered to be insulated. Streamlines and isotherms are presented for various Rayleigh numbers and Prandtl numbers. Heat flux vectors through the melt are plotted for selected cases. The axial velocity and temperature variations at different horizontal sections of the crucible have been presented graphically to explain the transport processes inside the crucible. It has been observed that in case of low Pr and high Ra, flow separation occurs at the vertical wall of the crucible which leads to an oscillatory flow as Ra increases. The investigation has been extended to the oscillatory regime of flow in the zone of supercritical Rayleigh numbers and some unsteady results are also presented. Finally a heat transfer correlation has been developed for steady-state case.     

Forced convection heat transfer in doubly connected ducts

Forced convection heat transfer in doubly connected ducts bounded externaly by a circle and internally by a regular polygon of various shapes is analysed using a finite element method. Hydrodynamically and thermally developed, steady, laminar flow of a constant property fluid is investigated. An insulated outer tube and constant heat flux at the inner core are considered. Temperature profiles as well as Nusselt numbers are presented. Salient characteristics of the temperature field in such passages are identified. Correlations for the Nusself number with aspect ratio are suggested.     

Laminar free convection heat transfer from isothermal convex bodies of arbitrary shape: a new dynamic model

Calculation of free convection from bodies of arbitrary shape has been investigated previously. The Body Gravity Function (BGF) which accounts for the geometry of each body shape was considered to be a constant value. In the present study, it is shown that BGF is not a constant value in a wide range of Rayleigh number. Instead, its value changes as Rayleigh number increases. Therefore, by analytical modeling of Dynamic BGF and derivation of a new parameter called Body Fluid Function, a novel method is proposed to calculate laminar free convection heat transfer from isothermal convex bodies of arbitrary shape. Results for 24 different body shapes are compared with the available experimental and numerical data. Excellent agreement shows that the present simple method accurately predicts laminar free convection heat transfer from isothermal convex bodies of arbitrary shape in the whole range of laminar flow and for fluids of any Prandtl number.     

Effect of temperature gradient within a solid particle on the rotation and oscillation modes in solid-dispersed two-phase flows

Liquid–solid two-phase flow with heat transfer is simulated, and the effect of temperature gradient within a solid particle on the particle behaviour and heat transfer is studied. The interaction between fluid and particles is considered with our original immersed solid approach on a rectangular grid system. The local heat flux at the fluid–solid interface is described with an anisotropic heat conductivity matrix, and the governing equation of temperature is time-updated with an implicit treatment for the diffusion term. The method is applied to a 2-D natural convection flow of a relatively low Rayleigh number including multiple particles. Heat transfer and particle behaviours are studied for different solid heat conductivities (ratio to the fluid conductivity ranging between 10⁻³ and 10³) and solid volume fractions. Under a condition of relatively low heat conductivity ratio, the particles show a simple circulating flow. By increasing the heat conductivity ratio, a transition of the particulate flow is observed to oscillation mode around the domain centre due to the buoyancy force as a restitution force. The oscillation period is found to vary with the heat conductivity ratio, and it is related to the time scales for the heat transfer via fluid and solid.     

Streamline upwind finite element method for conjugate heat transfer problems

This paper presents a combined finite element method for solving conjugate heat transfer problems where heat conduction in a solid is coupled with heat convection in viscous fluid flow. The streamline upwind finite element method is used for the analysis of thermal viscous flow in the fluid region, whereas the analysis of heat conduction in solid region is performed by the Galerkin method. The method uses the three-node triangular element with equal-order interpolation functions for all the variables of the velocity components, the pressure and the temperature. The main advantage of the proposed method is to consistently couple heat transfer along the fluid-solid interface. Three test cases, i.e. conjugate Couette flow problem in parallel plate channel, counter-flow in heat exchanger, and conjugate natural convection in a square cavity with a conducting wall, are selected to evaluate the efficiency of the present method. The English text was polished byYunming Chen.     

Lattice Boltzmann investigation for enhancing the thermal conductivity of ice using Al2O3 porous matrix

An enthalpy-based Lattice Boltzmann method (LBM) with double-distribution function (DDF) model is used to investigate numerically the effects of inserting a porous matrix on the heat transfer performance of the phase change material (PCM). Simulations are carried out for melting of ice in saturated Al₂O₃ porous matrix encapsulated in a concentric annulus. The process is considered as a conduction/convection controlled phase change problem at a representative elementary volume (REV) scale. The present results are validated by previous published numerical simulations of melting with and without porous media. In this research paper, the effects of decreasing the porosity on the temperature contours, flow patterns within the melt zone, complete melting time of the PCM and average Nusselt number are discussed qualitatively and quantitatively.     

Numerical simulation of solidification processes in enclosures

The present work deals with the development and application of numerical models for the simulation of solidification problems liquid/solid taking diffusion and convection into account. For the calculation of the thermal coupled flow process the finite element method is applied. In order to improve the numerical stability of the free convection problems, the streamline-upwind/Petrov–Galerkin method is used. Solidification processes are moving boundary problems. Three different models are set up which consider latent heat at the solidification front respectively in the mixed zone during the phase transition. Moreover, numerical methods are investigated in order to describe the behaviour of the flow at the boundary of the moving phase. Three examples serve illustrations; the technical example – casting of a transport and storage container – was provided by the company Siempelkamp Gießerei GmbH.     

Stability of equilibrium of a double-layer system with a deformable interface and a prescribed heat flux on the external boundaries

The stability of mechanical equilibrium of a system of two horizontal immiscible-liquid layers with similar densities is studied. The problem is solved for a prescribed heat flux on the external boundaries. Within the framework of a generalized Boussinesq approximation, which takes the interface deformation correctly into account, the onset of convection caused by heating the system from above or below is considered. Two long-wave instability modes attributable to the presence of the deformable interface and the given heat flux on the external boundaries are detected. The system response to monotonic and oscillatory disturbances with finite wavelengths is investigated. A complete stability map is constructed.     

Throughflow and g-jitter effects on binary fluid saturated porous medium

A non-autonomous complex Ginzburg-Landau equation (CGLE) for the finite amplitude of convection is derived, and a method is presented here to determine the amplitude of this convection with a weakly nonlinear thermal instability for an oscillatory mode under throughflow and gravity modulation. Only infinitesimal disturbances are considered. The disturbances in velocity, temperature, and solutal fields are treated by a perturbation expansion in powers of the amplitude of the applied gravity field. Throughflow can stabilize or destabilize the system for stress free and isothermal boundary conditions. The Nusselt and Sherwood numbers are obtained numerically to present the results of heat and mass transfer. It is found that throughflow and gravity modulation can be used alternately to heat and mass transfer. Further, oscillatory flow, rather than stationary flow, enhances heat and mass transfer.     

Controlling buffet through local mechanical action and heat addition near a lambda-shaped shock

Unsteady transonic flow past a two-dimensional airfoil with heat and momentum addition is numerically investigated. The flow analysis is based on the solution of the unsteady Reynolds equations closed by the k-ω SST turbulence model. The equations are integrated using the finite volume method. Several positions and shapes of the heat and momentum addition zones are considered for the purpose of determining an optimal means for controlling buffet. It is established that the most considerable variations in the buffet parameters are achieved when heat addition and mechanical action are realized on the upper wing surface. The thermal energy supply always increases the buffet frequency, while the mechanical action can both increase and reduce it.     

NUMERICAL INVESTIGATION ON FLOW STABILITY OF RAYLEIGH-B?NARD CONVECTION OF COLD WATER IN A RECTANGULAR CAVITY COOLED AND HEATED SYMMETRICALLY RELATIVE TO THE TEMPERATURE OF DENSITY MAXIMUM

为了解具有密度极值流体瑞利-贝纳德对流特有现象和规律,利用有限容积法对长方体腔内关于密度极值温度对称加热-冷却时冷水瑞利-贝纳德对流的分岔特性进行了三维数值模拟,得到了不同条件下的对流结构型态及其分岔序列,分析了密度极值特性、瑞利数、热边界条件以及宽深比对瑞利-贝纳德对流的影响. 结果表明:具有密度极值冷水瑞利-贝纳德对流系统较常规流体更加稳定,且流动型态及其分岔序列更加复杂;相同瑞利数下多种流型可以稳定共存,各流型在相互转变中存在滞后现象;随着宽深比的增加,流动更易失稳,对流传热能力增强;系统在导热侧壁时比绝热侧壁更加稳定,对流传热能力有所减弱;基于计算结果,采用线性回归方法,得到了热壁传热关联式.     

An Enthalpy Control Volume Method for Transient Mass and Heat Transport with Solidification

A single domain enthalpy control volume method is developed for solving the coupled fluid flow and heat transfer with solidification problem arising from the continuous casting process. The governing equations consist of the continuity equation, the Navier–Stokes equations and the convection–diffusion equation. The formulation of the method is cast into the framework of the Petrov–Galerkin finite element method with a step test function across the control volume and locally constant approximation to the fluxes of heat and fluid. The use of the step test function and the constant flux approximation leads to the derivation of the exponential interpolating functions for the velocity and temperature fields within each control volume. The exponential fitting makes it possible to capture the sharp boundary layers around the solidification front. The method is then applied to investigate the effect of various casting parameters on the solidification profile and flow pattern of fluids in the casting process.     

A stabilized Nitsche‐type extended embedding mesh approach for 3D low‐ and high‐Reynolds‐number flows

This paper presents a stabilized extended finite element method (XFEM) based fluid formulation to embed arbitrary fluid patches into a fixed background fluid mesh. The new approach is highly beneficial when it comes to computational grid generation for complex domains, as it allows locally increased resolutions independent from size and structure of the background mesh. Motivating applications for such a domain decomposition technique are complex fluid‐structure interaction problems, where an additional boundary layer mesh is used to accurately capture the flow around the structure. The objective of this work is to provide an accurate and robust XFEM‐based coupling for low‐ as well as high‐Reynolds‐number flows. Our formulation is built from the following essential ingredients: Coupling conditions on the embedded interface are imposed weakly using Nitsche's method supported by extra terms to guarantee mass conservation and to control the convective mass transport across the interface for transient viscous‐dominated and convection‐dominated flows. Residual‐based fluid stabilizations in the interior of the fluid subdomains and accompanying face‐oriented fluid and ghost‐penalty stabilizations in the interface zone stabilize the formulation in the entire fluid domain. A detailed numerical study of our stabilized embedded fluid formulation, including an investigation of variants of Nitsche's method for viscous flows, shows optimal error convergence for viscous‐dominated and convection‐dominated flow problems independent of the interface position. Challenging two‐dimensional and three‐dimensional numerical examples highlight the robustness of our approach in all flow regimes: benchmark computations for laminar flow around a cylinder, a turbulent driven cavity flow at Re = 10000 and the flow interacting with a three‐dimensional flexible wall. Copyright © 2016 John Wiley & Sons, Ltd.     

The influence of a magnetic field on free convection in a finite vertical channel

An approximate analytical solution is presented for developing free convection flows of electrically conducting fluids between finite vertical channels which are subjected to a uniformly applied transverse magnetic field. Specifically, the basic approximation lies in the linearization of the governing boundary layer type of equations. It is demonstrated that the application of a transverse magnetic field reduces the induced flow rate in the channel and the heat transfer to the fluid.     

Deformation and strain analysis for high extrusion ratios and elevated temperatures

A method is described for obtaining full-field distribution of strain during axisymmetric extrusion. Strains are determined from changes in the shapes of flowlines as observed through the deformation of an ink-stamped grid on an axial plane of the extruded billet. Numerical methods are described for approximation and differentiation of the materialflow functions. The method is applied to extrusion of aluminum at 430°C with an extrusion ratio of 12.4 to 1. Good agreement is found between calculated and measured flow patterns.