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.     

A Hybrid Boundary/Finite Element Method for Simulating Viscous Flows and Shapes of Droplets in Electric Fields

A mixed boundary element and finite element numerical algorithm for the simultaneous prediction of the electric fields, viscous flow fields, thermal fields and surface deformation of electrically conducting droplets in an electrostatic field is described in this paper. The boundary element method is used for the computation of the electric potential distribution. This allows the boundary conditions at infinity to be directly incorporated into the boundary integral formulation, thereby obviating the need for discretization at infinity. The surface deformation is determined by solving the normal stress balance equation using the weighted residuals method. The fluid flow and thermal fields are calculated using the mixed finite element method. The computational algorithm for the simultaneous prediction of surface deformation and fluid flow involves two iterative loops, one for the electric field and surface deformation and the other for the surface tension driven viscous flows. The two loops are coupled through the droplet surface shapes for viscous fluid flow calculations and viscous stresses for updating the droplet shapes. Computing the surface deformation in a separate loop permits the freedom of applying different types of elements without complicating procedures for the internal flow and thermal calculations. Tests indicate that the quadratic, cubic spline and spectral boundary elements all give approximately the same accuracy for free surface calculations; however, the quadratic elements are preferred as they are easier to implement and also require less computing time. Linear elements, however, are less accurate. Numerical simulations are carried out for the simultaneous solution of free surface shapes and internal fluid flow and temperature distributions in droplets in electric fields under both microgravity and earthbound conditions. Results show that laser heating may induce a non-uniform temperature distribution in the droplets. This non-uniform thermal field results in a variation of surface tension along the surface of the droplet, which in turn produces a recirculating fluid flow in the droplet. The viscous stresses cause additional surface deformation by squeezing the surface areas above and below the equator plane.     

Coupled heat transfer and viscous flow,and magnetic effects in weld pool analysis

A finite element formulation and analysis is developed to study coupled heat transfer and viscous flow in a weld pool. The thermal effects generate not only buoyancy forces but also a variation in the surface tension which acts to drive the viscous flow in the molten weld pool. A moving phase boundary separates molten and solid material. Numerical experiments reveal the nature of the highly convective flow in the weld pool and the associated thermal profiles. The relative importance of buoyancy, surface tension, phase change, convection, etc. are examined. We also consider the sensitivity of the solution to the finite element mesh and related non-linear numerical instabilities. Of particular interest is the coupling of the thermal and viscous flow fields for the case when radial flow is inward or outward.     

Soret and Dufour effects on unsteady MHD flow past an infinite vertical porous plate with thermal radiation

The objective of the present study is to investigate the effect of flow parameters on the free convection and mass transfer of an unsteady magnetohydrodynamic flow of an electrically conducting, viscous, and incompressible fluid past an infinite vertical porous plate under oscillatory suction velocity and thermal radiation. The Dufour （diffusion thermo） and Soret （thermal diffusion） effects are taken into account. The problem is solved numerically using the finite element method for the velocity, the temperature, and the concentration field. The expression for the skin friction, the rate of heat and mass transfer is obtained. The results are presented numerically through graphs and tables for the externally cooled plate （Gr 〉 0） and the externally heated plate （Gr 〈 0） to observe the effects of various parameters encountered in the equations.     

Effect of viscous dissipation on the heat transfer in sinusoidal profile finned dissipators

In the present work, the efficiency of finned heat dissipators cooled by laminar flow is studied. The analysis is carried out by varying certain sizing parameters in correspondence with different viscous dissipation conditions. The velocity distribution in the fluid and the temperature distribution in the dissipator and in the fluid are determined by means of a finite element method. The model allows to study the variation of the heat transfer coefficient due to the fluid dynamic conditions imposed by the fin profile. Lastly, a comparison between the optimum fin shapes obtained under different viscous dissipation effects is carried out.     

A spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for conjugate heat transfer applications

We present a spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for fluid–solid conjugate heat transfer applications. Using the discrete Boltzmann equation, we propose a numerical scheme for conjugate heat transfer applications on unstructured, non‐uniform grids. We employ a double‐distribution thermal lattice Boltzmann model to resolve flows with variable Prandtl (Pr) number. Based upon its finite element heritage, the spectral‐element discontinuous Galerkin discretization provides an effective means to model and investigate thermal transport in applications with complex geometries. Our solutions are represented by the tensor product basis of the one‐dimensional Legendre–Lagrange interpolation polynomials. A high‐order discretization is employed on body‐conforming hexahedral elements with Gauss–Lobatto–Legendre quadrature nodes. Thermal and hydrodynamic bounce‐back boundary conditions are imposed via the numerical flux formulation that arises because of the discontinuous Galerkin approach. As a result, our scheme does not require tedious extrapolation at the boundaries, which may cause loss of mass conservation. We compare solutions of the proposed scheme with an analytical solution for a solid–solid conjugate heat transfer problem in a 2D annulus and illustrate the capture of temperature continuities across interfaces for conductivity ratio γ > 1. We also investigate the effect of Reynolds (Re) and Grashof (Gr) number on the conjugate heat transfer between a heat‐generating solid and a surrounding fluid. Steady‐state results are presented for Re = 5?40 and Gr = 10⁵?10⁶. In each case, we discuss the effect of Re and Gr on the heat flux (i.e. Nusselt number Nu) at the fluid–solid interface. Our results are validated against previous studies that employ finite‐difference and continuous spectral‐element methods to solve the Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

基于特征分裂有限元准隐格式的共轭传热整体耦合数值模拟方法

共轭传热现象在科学和工程领域中大量存在. 随着计算能力的发展, 对共轭传热现象进行准确有效的数值模拟, 成为科学研究和工程设计上的重要挑战.共轭传热数值模拟的方法可以分为两大类: 分区耦合和整体耦合.本文采用有限元法对共轭传热问题进行整体耦合模拟. 固体传热求解采用标准的伽辽金有限元方法.流动求解采用基于特征分裂的有限元方法. 该方法是一种重要的求解流动问题的有限元方法, 可以使用等阶有限元. 该方法的准隐格式与其他格式相比, 具有时间步长大的特点. 将稳定项中的时间步长与全局时间步长分开, 改进了准隐格式的稳定性. 基于改进的特征分裂有限元方法的准隐格式, 发展了一种层流共轭传热数值模拟的整体耦合方法. 采用这种方法可以将流体计算域和固体计算域作为一个整体划分有限元网格, 并且所有变量都可以采用相同的插值函数, 从而有利于程序的实现. 通过对典型问题的模拟, 验证了这种方法的准确性. 本工作还研究了固体区域时间步长对定常共轭传热问题数值模拟收敛性的影响.      

Conjugate conduction-natural convection in an enclosure with time-periodic sidewall temperature and inclination

The unsteady conjugate conduction-natural convection in enclosure is of great theoretical significance and is widely encountered in engineering applications in the areas of fluid dynamics and heat transfer. However, there are relatively few efforts to investigate the unsteady flow physics and heat transfer characteristics in the inclined enclosure of finite thickness walls. In the present work, this problem is numerically investigated by a high accuracy multidomain temporal-spatial pseudospectral method. The enclosure is filled with Boussinesq fluid and is bounded by four finite thickness and conductive walls; one of the vertical sidewall is exposed to time-periodic temperature environment while the opposite sidewall holds constant temperature; the top and bottom walls are assumed to be adiabatic. Particular efforts are focused on the effects of three types of influential factors: the wall thermophysical properties, the time-periodic temperature patterns and the inclination, and the time-periodic flow patterns and heat transfer characteristics are presented. Numerical results reveal that within the present parameter range, the heat transfer rate increases almost linearly with the thermal conductivity ratio and thermal diffusivity ratio but decreases with the inclination angle. Moreover, the heat transfer could be enhanced or weakened by selecting different temperature pulsating period in the case of finite thickness wall, while it is always enhanced if the walls are zero thickness. The back heat transfer and heat transfer resonance phenomena are observed, and their relationships with the time-periodic flow patterns and temperature distributions are analyzed. The findings are helpful to the understandings of the fluid flow and heat transfer mechanisms in the related enclosure configurations, and may be of engineering use in thermal design improvement.     

Conjugate natural convection of non-Newtonian hybrid nanofluid in wavy-shaped enclosure

The present study concerns the modelization and numerical simulation for the heat and flow exchange characteristics in a novel configuration saturated with a nonNewtonian Ag-MgO hybrid nanofluid. The wavy shaped enclosure is equipped with onequarter of a conducting solid cylinder. The system of equations resulting from the mathematical modeling of the physical problem in its dimensionless form is discretized via the higher-order Galerkin-based finite element method(GFEM). The dependency of vario...     

Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid Past a Vertical Plate Immersed in a Porous Medium

The effects of viscous dissipation on unsteady free convection from an isothermal vertical flat plate in a fluid saturated porous medium are examined numerically. The Darcy–Brinkman–Forchheimer model is employed to describe the flow field. A new model of viscous dissipation is used for the Darcy–Brinkman–Forchheimer model of porous media. The simultaneous development of the momentum and thermal boundary layers are obtained by using a finite difference method. Boundary layer and Boussinesq approximation have been incorporated. Numerical calculations are carried out for various parameters entering into the problem. Velocity and temperature profiles as well as local friction factor and local Nusselt number are shown graphically. It is found that as time approaches infinity, the values of friction factor and heat transfer coefficient approach steady state.     

Biothermomechanics of skin tissues

Biothermomechanics of skin is highly interdisciplinary involving bioheat transfer, burn damage, biomechanics and neurophysiology. During heating, thermally induced mechanical stress arises due to the thermal denaturation of collagen, resulting in macroscale shrinkage. Thus, the strain, stress, temperature and thermal pain/damage are highly correlated; in other words, the problem is fully coupled. The aim of this study is to develop a computational approach to examine the heat transfer process and the heat-induced mechanical response, so that the differences among the clinically applied heating modalities can be quantified. Exact solutions for temperature, thermal damage and thermal stress for a single-layer skin model were first derived for different boundary conditions. For multilayer models, numerical simulations using the finite difference method (FDM) and finite element method (FEM) were used to analyze the temperature, burn damage and thermal stress distributions in the skin tissue. The results showed that the thermomechanical behavior of skin tissue is very complex: blood perfusion has little effect on thermal damage but large influence on skin temperature distribution, which, in turn, influences significantly the resulting thermal stress field; the stratum corneum layer, although very thin, has a large effect on the thermomechanical behavior of skin, suggesting that it should be properly accounted for in the modeling of skin thermal stresses; the stress caused by non-uniform temperature distribution in the skin may also contribute to the thermal pain sensation.     

Effect of thermal dispersion on free convection in a fluid saturated porous medium

The present article considers a numerical study of thermal dispersion effect on the non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium. Forchheimer extension is considered in the flow equations. The coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. The non-dimensional governing equations are solved by the finite element method (FEM) with a Newton–Raphson solver. Numerical results for the details of the stream function, velocity and temperature contours and profiles as well as heat transfer rates in terms of Nusselt number are obtained. The study shows that the increase in thermal dispersion coefficient of the porous medium results in more heat energy to disperse away in the normal direction to the wall. This induces more fluid to flow along the wall, enhancing the heat transfer coefficient particularly near the wall.     

A fast and robust fictitious domain method for modelling viscous flows in complex mixers: The example of propellant make‐down

This work deals with the development of a fast three‐dimensional numerical strategy for the simulation of viscous fluid flow in complex mixing systems. The proposed method is based on a distributed Lagrange multiplier fictitious domain method and the use of the low‐cost MINI finite element. Contrary to the previous fictitious domain method developed by our group a few years ago, the underlying partial differential equations are solved here in a coupled manner using a consistent penalty technique. The method is discussed in detail and its precision is assessed by means of experimental data in the case of an agitated vessel. A comparison made with our existing fictitious domain method and its decoupled Uzawa‐based solver clearly shows the advantages of resorting to the MINI finite element and fully coupled solution strategy. The new technique is then applied to the simulation of the flow of a Newtonian viscous fluid in a three‐blade planetary mixer in the context of the production of solid propellants. Copyright © 2008 John Wiley & Sons, Ltd.     

Conjugate forced convection and heat conduction in circular microchannels

The effects of axial heat conduction in the solid walls of microchannels of circular cross-sections are analyzed here. A systematic approach is adopted, with the aim of pointing out the influence of geometrical parameters and of solid wall thermal conductivity on microchannel heat transfer. The reliability of a commonly adopted criterium, based on the so-called axial conduction number, to assess the relevance of axial heat conduction is also discussed. Numerical simulations concern the simultaneously developing laminar flow of a constant property fluid in microchannels of different length, wall thickness and wall material, heated with a uniform heat flux at the outer surface, for different values of the Reynolds number. Moreover, since often in experimental tests the two end sections of the microchannel wall are not perfectly insulated, the effects of heat losses through these sections are also considered. A hybrid finite element procedure, which implies the step-by-step solution of the parabolized momentum equations in the fluid domain, followed by the solution of the energy equation in the entire domain, corresponding to both the solid and the fluid parts, is used for the numerical simulations.     

Fully coupled flow-induced vibration of structures under small deformation with GMRES method

Lagrangian-Eulerian formulations based on a generalized variational principle of fluid-solid coupling dynamics are established to describe flow-induced vibration of a structure under small deformation in an incompressible viscous fluid flow. The spatial discretization of the formulations is based on the multi-linear interpolating functions by using the finite element method for both the fluid and solid structures. The generalized trapezoidal rule is used to obtain apparently non-symmetric linear equations in an incremental form for the variables of the flow and vibration. The nonlinear convective term and time factors are contained in the non-symmetric coefficient matrix of the equations. The generalized minimum residual （GMRES） method is used to solve the incremental equations. A new stable algorithm of GMRES-Hughes-Newmark is developed to deal with the flow-induced vibration with dynamical fluid-structure interaction in complex geometries. Good agreement between the simulations and laboratory measurements of the pressure and blade vibration accelerations in a hydro turbine passage was obtained, indicating that the GiViRES-Hughes-Newmark algorithm presented in this paper is suitable for dealing with the flow-induced vibration of structures under small deformation.     

A note on a derivative scheme for the finite volume method applied to incompressible viscous fluid

In the last two decades, there have been significant advances in application of the finite volume method to a wide spectrum of physical phenomena, ranging from heat transfer and compressible/incompressible fluid flow to solid mechanics. The finite volume method requires for different applications similar approximations of derivatives at the control surface. Within this framework, this study discusses a derivative scheme used in the finite volume method for incompressible viscous fluids. The numerical scheme is based on an implicit technique associated with the SIMPLE method to attain pressure–velocity coupling. The present work addresses simulation of two-dimensional flows in plane channels with and without contractions. The results show velocities and pressure fields with good agreement when compared to analytical results.     

Unsteady conjugate problem of a dissipative fluid in a horizontal channel with a periodic variation temperature

The unsteady two-dimensional transient heat transfer problem referring to a fully laminar flow developing in a parallel-plane channel exposed to a periodic variation surface temperature with distance is numerically studied. The effects of channel thickness, Péclet number, wall-to-fluid conductivity ratio, thermal diffusivity ratio, angular frequency and the viscous dissipation parameter are determined in the solutions. The non-linear equations are discretized by means an implicit finite difference scheme and the electric analogy to the resulting system is applied to convert these equations into a network-electrical model that was solved using a computer code (electric circuits simulator). In this scheme, only spatial discretization is necessary, while time remains as a real continuous variable, and its programming does not require manipulation of the sophisticated mathematical software that is inherent in other numerical methods. The network simulation method, which satisfies the conservation law for the heat flux variable and the uniqueness law for temperature, also permits the direct visualization of the local and/or integrated transport variables at any point or section of the medium.     

非稳态分数阶Oldroyd-B流体绕楔形体流动研究

研究了非稳态分数阶Oldroyd-B流体在多孔介质中通过楔形拉伸板的驻点流动问题。基于分数阶Oldroyd-B流体的本构模型建立了动量方程,并在其中引入了浮升力和驻点流动特征。此外,考虑了具有热松弛延迟时间的修正的分数阶Fourier定律,并将其应用于能量方程和对流换热边界条件。接着,采用与L1算法相结合的有限差分法求解控制偏微分方程。最后,分析了相关物理参数对流动的影响。结果表明,随着楔角参数的增加,流体受到的浮升力增大,导致速度加快;达西数越大,介质的孔隙度变大,流体的流动越快;此外,温度分布先略有上升后明显下降,这表明Oldroyd-B流体具有热延迟特性。     

Effect of Viscous Dissipation on the Boundary Layer Flow and Heat Transfer Past a Permeable Stretching Surface Embedded in a Porous Medium with a Second-Order Slip Using Chebyshev Finite Difference Method

This article investigates a theoretical and numerical study for the effect of viscous dissipation on the steady flow with heat transfer of Newtonian fluid toward a permeable stretching surface embedded in a porous medium with a second-order slip and thermal slip. The governing nonlinear partial differential equations are converted into nonlinear ordinary differential equations (ODEs) using similarity variables. The resulting ODEs are successfully solved numerically with the help of Chebyshev finite difference method. Graphically results are shown for non-dimensional velocities and temperature. The effects of the porous parameter, the suction (injection) parameter, Eckert number, first- and second-order velocity slip parameter, the thermal slip parameter and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and Nusselt numbers are presented. A comparison of numerical results is made with the earlier published results under limiting cases.