Prediction of radiative heat transfer in 3D complex geometries using the unstructured control volume finite element method

In this paper, a 3D algorithm for the treatment of radiative heat transfer in emitting, absorbing, and scattering media is developed. The numerical approach is based on the utilization of the unstructured control volume finite element method (CVFEM) which, to the knowledge of the authors, is applied for the first time to simulate radiative heat transfer in participated media confined in 3D complex geometries. This simulation makes simultaneously the use of the merits of both the finite element method and the control volume method. Unstructured 3D triangular element grids are employed in the spatial discretization and azimuthal discretization strategy is employed in the angular discretization. The general discretization equation is presented and solved by the conditioned conjugate gradient squared method (CCGS). In order to test the efficiency of the developed method, several 3D complex geometries including a hexahedral enclosure, a 3D equilateral triangular enclosure, a 3D L-shaped enclosure and 3D elliptical enclosure are examined. The results are compared with the exact solutions or published references and the accuracy obtained in each case is shown to be highly satisfactory. Moreover, this approach required a less CPU time and iterations compared with those of even parity formulation of the discrete ordinates method.     

Finite element method for radiation heat transfer in multi-dimensional graded index medium

In graded index medium, ray goes along a curved path determined by Fermat principle, and curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectories, a finite element method based on discrete ordinate equation is developed to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Two particular test problems of radiative transfer are taken as examples to verify this finite element method. The predicted dimensionless net radiative heat fluxes are determined by the proposed method and compared with the results obtained by finite volume method. The results show that the finite element method presented in this paper has a good accuracy in solving the multi-dimensional radiative transfer problem in semitransparent graded index medium.     

A new differencing scheme for the discrete-ordinates method in complex geometries

This paper is an assessment of a new discrete-ordinates algorithm recently developed by the authors for the numerical treatment of radiative participating media in both two- and three-dimensional enclosures. The algorithm is based on the utilization of general characteristic relations instead of the traditional differencing schemes for the spatial marching procedure. It is ideally suited for the treatment of complex geometries, the grid being formed from triangles (2D) or tetrahedra (3D). The method is exempt of any numerical oscillation and may be readily interfaced with the finite-element method for the solution of problems involving other modes of heat transfer. The mathematical derivation is detailed in the text and several examples are given for complex enclosures. The method proves to be very accurate and of good flexibility.     

Control volume finite element method for radiation

In this paper a new methodology is presented by the authors for the numerical treatment of radiative heat transfer in emitting, absorbing and scattering media. This methodology is based on the utilisation of Control Volume Finite Element Method (CVFEM) and the use, for the first time, of matrix formulation of the discretized Radiative Transfer Equation (RTE). The advantages of the proposed methodology is to avoid problems that confronted when previous techniques are used to predict radiative heat transfer, essentially, in complex geometries and when there is scattering and/or non-black boundaries surfaces. Besides, the new formulation of the discretized RTE presented in this paper makes it possible to solve the algebraic system by direct or iterative numerical methods. The theoretical background of CVFEM and matrix formulation is presented in the text. The proposed technique is applied to different test problems, and the results compared favourably against other published works. Moreover this paper discusses in detail the effects of some radiative parameters, such as optical thickness and walls emissivities on the spatial evolution of the radiant heat flux. The numerical simulation of radiative heat transfer for different cases using the algorithm proposed in this work has shown that the developed computer procedure needs an accurate CPU time and is exempt of any numerical oscillations.     

Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems

In graded index medium, the ray goes along a curved path determined by Fermat principle, and the curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectory, the methods not based on ray-tracing technique need to be developed for the solution of radiative transfer in graded index medium. For this purpose, in this paper the streaming operator along a curved ray trajectory in original radiative transfer equation for graded index medium is transformed and expressed in spatial and angular ordinates and the radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems are derived. The conservative and the non-conservative forms of radiative transfer equation for three-dimensional graded index medium are given, which can be used as base equations to develop the numerical simulation methods, such as finite volume method, discrete ordinates method, and finite element method, for radiative transfer in graded index medium in cylindrical and spherical coordinate systems.     

Benchmark numerical solutions for radiative heat transfer in two-dimensional medium with graded index distribution

In graded index media, the ray goes along a curved path determined by Fermat principle. Generally, the curved ray trajectory in graded index media is a complex implicit function, and the curved ray tracing is very difficult and complex. Only for some special refractive index distributions, the curved ray trajectory can be expressed as a simple explicit function. Two important examples are the layered and the radial graded index distributions. In this paper, the radiative heat transfer problems in two-dimensional square semitransparent with layered and radial graded index distributions are analyzed. After deduction of the ray trajectory, the radiative heat transfer problems are solved by using the Monte Carlo curved ray-tracing method. Some numerical solutions of dimensionless net radiative heat flux and medium temperature are tabulated as the benchmark solutions for the future development of approximation techniques for multi-dimensional radiative heat transfer in graded index media.     

Finite element approach for radiative transfer in multi-layer graded index cylindrical medium with Fresnel surfaces

Because the optical plane defined by the incidence and reflection direction at a cylindrical surface has a complicated relation with the local azimuthal angle and zenith angle in the traditional cylindrical coordinate system, it is difficult to deal with the specular reflective boundary condition in the solution of the traditional radiative transfer equation for cylindrical system. In this paper, a new radiative transfer equation for graded index medium in cylindrical system (RTEGCN) is derived based on a newly defined cylindrical coordinate system. In this new cylindrical coordinate system, the optical plane defined by the incidence and reflection direction is just the isometric plane of the local azimuthal angle, which facilitates the RTEGCN in dealing with cylindrical specular reflective boundaries. A least squares finite element method (LSFEM) is developed for solving radiative transfer in single and multi-layer cylindrical medium based on the discrete ordinates form of the RTEGCN. For multi-layer cylindrical medium, a radial basis function interpolation method is proposed to couple the radiative intensity at the interface between two adjacent layers. Various radiative transfer problems in both single and multi-layer cylindrical medium are tested. The results show that the present finite element approach has good accuracy to predict the radiative heat transfer in multi-layer cylindrical medium with Fresnel surfaces.     

周期性渐扩-渐缩通道层流流动与换热特性研究

以渐扩-渐缩通道内周期性充分发展的层流流动与换热为研究对象,采用SIMPLE算法,适体坐标网格及Amano周期性边界条件的实施方案对之进行数值模拟,计算了在层流范围内不同Re数下的流动与换热规律.结果表明,在Re=100~1000范围内,与平行平板通道相比,阻力增强了(10~200)%,换热增强了(40~320)%.     

Three-dimensional radiative transfer modeling using the control volume finite element method

In the present study, a three-dimensional algorithm for the treatment of radiative heat transfer in emitting, absorbing and scattering media is developed. The approach is based on the utilization of control volume finite element method (CVFEM) which, to the knowledge of the authors, is applied at the first time to 3D radiative heat transfer in participating media. The accuracy of the present algorithm is tested by comparing its predictions to other published works. Comparisons show that CVFEM produces good results. Moreover, this approach permits compatibility with other numerical methods used for computational fluids mechanics problems.     

Finite element method for modeling radiative transfer in semitransparent graded index cylindrical medium

Both Galerkin finite element method (GFEM) and least squares finite element method (LSFEM) are developed and their performances are compared for solving the radiative transfer equation of graded index medium in cylindrical coordinate system (RTEGC). The angular redistribution term of the RTEGC is discretized by finite difference approach and after angular discretization the RTEGC is formulated into a discrete-ordinates form, which is then discretized based on Galerkin or least squares finite element approach. To overcome the RTEGC-led numerical singularity at the origin of cylindrical coordinate system, a pole condition is proposed as a special mathematical boundary condition. Compared with the GFEM, the LSFEM has very good numerical properties and can effectively mitigate the nonphysical oscillation appeared in the GFEM solutions. Various problems of both axisymmetry and nonaxisymmetry, and with medium of uniform refractive index distribution or graded refractive index distribution are tested. The results show that both the finite element approaches have good accuracy to predict the radiative heat transfer in semitransparent graded index cylindrical medium, while the LSFEM has better numerical stability.     

非规则形状介质内辐射-导热耦合传热的间断有限元求解

采用间断有限元法(discontinuous finite element method,DFEM)求解非规则形状介质内的辐射导热耦合传热问题,得到了典型非规则形状介质内辐射导热耦合传热问题的高精度数值结果.和传统连续型有限元方法不同,DFEM将计算区域划分成相互独立的离散单元,形函数的构造、未知量的加权近似以及控制方程的求解均在每一个离散单元上进行.通过在单元之间施加迎风格式的数值通量,DFEM保证了整个计算区域的连续性,因此这种方法兼具良好的几何灵活性和局部守恒性.推导了辐射传输方程和能量扩散方程的射导热耦合传热问题,得到了典型非规则形状介质内辐射导热耦合传热的高精度数值结果.     

Hybrid finite volume/ finite element method for radiative heat transfer in graded index media

The rays propagate along curved path determined by the Fermat principle in the graded index medium. The radiative transfer equation in graded index medium (GRTE) contains two specific redistribution terms (with partial derivatives to the angular coordinates) accounting for the effect of the curved ray path. In this paper, the hybrid finite volume with finite element method (hybrid FVM/FEM) (P.J. Coelho, J. Quant. Spectrosc. Radiat. Transf., vol. 93, pp. 89–101, 2005) is extended to solve the radiative heat transfer in two-dimensional absorbing-emitting-scattering graded index media, in which the spatial discretization is carried out using a FVM, while the angular discretization is by a FEM. The FEM angular discretization is demonstrated to be preferable in dealing with the redistribution terms in the GRTE. Two stiff matrix assembly schemes of the angular FEM discretization, namely, the traditional assembly approach and a new spherical assembly approach (assembly on the unit sphere of the solid angular space), are discussed. The spherical assembly scheme is demonstrated to give better results than the traditional assembly approach. The predicted heat flux distributions and temperature distributions in radiative equilibrium are determined by the proposed method and compared with the results available in other references. The proposed hybrid FVM/FEM method can predict the radiative heat transfer in absorbing-emitting-scattering graded index medium with good accuracy.     

Theoretical study of combined radiative conductive and convective heat transfer in semi-transparent porous medium in a spherical enclosure

A new method for the solution of the radiative transfer equation in spherical media based on a modified discrete ordinates method is extended to study radiative, conductive and convective heat transfer in a semi-transparent scattering porous medium. The set of differential equations is solved using the fourth-order Runge-Kutta method. Various results are obtained for the case of combined radiative and conductive heat transfer, as well as for the interaction of those modes with convection. The effects of some radiative properties of the medium on the heat transfer rate are examined.     

Modeling of radiative heat transfer in an axisymmetric cylindrical enclosure with participating medium

Radiative heat transfer in an axisymmetric enclosure with absorbing, emitting, and scattering medium is studied here by using the different methods such as MDOM, FVM, and MFVM with emphasis on the treatment of angular derivative term, which appears in curvilinear coordinates due to angular redistribution. After final discretization equation for MFVM is introduced by using the step scheme and directional weights, the present approach is validated by applying it to three different benchmarking problems with absorbing, emitting, and scattering medium. All of the results presented here support its accuracy as well as moderate efficiency. Finally, the present approaches are applied to a truncated cone-shaped enclosure as a body-fitted geometry case.     

Discontinuous finite element method for radiative heat transfer in semitransparent graded index medium

To avoid the complicated and time-consuming computation of curved ray trajectories, a discontinuous finite element method based on discrete ordinate equation is extended to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Two cases of radiative heat transfer in two-dimensional rectangular gray semitransparent graded index medium enclosed by opaque boundary are examined to verify this discontinuous finite element method. Special layered and radial graded index distributions are considered. The predicted dimensionless net radiative heat fluxes and dimensionless temperature distributions are determined by the discontinuous finite element method and compared with the results obtained by the curved Monte Carlo method in references. The results show that the discontinuous finite element method has a good accuracy in solving the multi-dimensional radiative transfer problem in a semitransparent graded index medium.     

Non-gray radiative transfer in plane-parallel media with reflecting boundaries

A generalized equation of radiative transfer in the two-group picket-fence model is analyzed for a plane parallel, emitting, absorbing and isotropically scattering medium containing uniform heat sources and having boundary surfaces which are diffuse emitters and diffuse reflectors and are maintained at uniform but arbitrary temperatures. The solution of the general problem is expressed by the superposition of simpler problems which are solved by the application of the normal-mode-expansion technique. Highly accurate numerical results are presented for the temperature distribution and the radiative heat flux in the medium.     

用射线踪迹法解黑表面矩形介质耦合换热

本文用射线踪迹-节点分析法研究了二维黑体表面矩形、各向同性散射半透明介质内辐射与导热瞬态耦合换热。采用全隐格式的有限差分法离散二维瞬态微分能量方程,用辐射传递系数来表示辐射源项,结合谱带模型并采用射线踪迹法求解辐射传递系数。采用Patankar线性化方法将辐射源项及不透明边界条件线性化,并采用附加源项法处理边界条件,运用ADI方法求解名以上的线性化方程组,从而解得二维矩形介质内的瞬态温度分布。     

Numerical visualization of free surface oscillation predicted with arbitrary Lagrangian-Eulerian method

A numerical prediction method has been proposed to predict non-linear free surface oscillation in a three-dimensional container. The fluid motions are numerically predicted with Navier-Stokes equations discretized in a Lagrangian scheme with sufficient numerical accuracy. The profile of a free surface is precisely represented with three-dimensional body-fitted coordinates (BFC), which are regenerated in each computational step on the basis of the arbitrary Lagrangian-Eulerian (ALE) formulation. The computational method was applied to non-linear sloshings and transitions from sloshing to swirling motions. The predicted free surface motions were visualized as sequential image files and animations to understand their dynamic futures     

Least-squares finite element method for radiation heat transfer in graded index medium

To avoid the complicated and time-consuming computation of curved ray trajectories, a least-squares finite element method based on discrete ordinate equation is extended to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Four cases of radiative heat transfer are examined to verify this least-squares finite element method. Linear and nonlinear graded index are considered. The predicted dimensionless net radiative heat fluxes are determined by the least-squares finite element method and compared with the results obtained by other methods. The results show that the least-squares finite element method is stable and has a good accuracy in solving the multi-dimensional radiative transfer problem in a semitransparent graded index medium, while the Galerkin finite element method sometimes suffers from nonphysical oscillations.     

圆筒形半透明介质内非稳态复合导热与辐射的研究

本文研究细长电加热体在圆筒形半透明介质中的温度响应，通过对控制非稳态导热与辐射复合传热过程的积分-微分方程直接进行数值求解，分析了热辐射对内部径向热流及温度变化的影响．模拟计算结果对热线法测量半透明介质导热系数的研究具有理论指导意义。