Multiblock Compressible Navier-Stokes Flow Solver Applied to Complex Launch Vehicles

The Monte Carlo method was used in modelling steady state heat conduction problems. The method uses the fixed and the floating random walks to determine temperature in the domain of the definition of the heat conduction equation, at a single point directly. A heat conduction problem with an irregular shaped geometry and prescribed boundary temperatures was analyzed. Computer programs for the analysis were coded using VAX FORTRAN on a VAX 4.4 mainframe. The fixed random walk and the floating random walk methods gave comparable results of 507.260K and 506.980K respectively for temperature at a point. The Monte Carlo method also gave a result of 50.580C , similar to that of 50.000C for a particular node, with the Finite Difference method (using Gauss- Seidel iteration) for a 2-dimensional regular shaped heat conduction problem. Keywords: Monte Carlo Method, Fixed Random Walk, Floating Random Walk, Modelling, Steady-State Heat Conduction African Research Review Vol. 2 (4) 2008: pp. 31-53

The article represents a new approximate analytical solution of the initial boundary value problem of one-dimensional heat conduction with boundary conditions of the third kind. It describes the temperature field of a homogeneous thin plate under thermal shock more accurately than previous solutions. The obtained solution was determined by the ANSYS mathematical package. The results of the analysis can be used to construct control laws for a small spacecraft taking into account the thermal shock of its solar panels.

An analytical solution to the problem of time-fractional heat conduction in a sphere consisting of an inner solid sphere and concentric spherical layers is presented. In the heat conduction equation, the Caputo time-derivative of fractional order and the Robin boundary condition at the outer surface of the sphere are assumed. The spherical layers are characterized by different material properties and perfect thermal contact is assumed between the layers. The analytical solution to the problem of heat conduction in the sphere for time-dependent surrounding temperature and for time-space-dependent volumetric heat source is derived. Numerical examples are presented to show the effect of the harmonically varying intensity of the heat source and the harmonically varying surrounding temperature on the temperature in the sphere for different orders of the Caputo time-derivative.

A new coarse-mesh computational method for the numerical solution of heat conduction and fluid flow problems is formally developed and applied to sample problems. The method is based upon formal use of Green's functions, which are defined locally over subdomains of the original system under consideration. The formal development of the local Green's function method for the solution of heat conduction problems is presented and discussed. Numerical solutions of sample problems for one-dimensional heat conduction with constant thermal conductivity, one-dimensional heat conduction with temperature-dependent thermal conductivity, and two-dimensional heat conduction with constant thermal conductivity are given, and these results are compared with results obtained using the finite difference and finite element methods. The formal development of the local Green's function method for the solution of fluid flow problems is then also presented and discussed; the numerical solution of a sample problem for simple one-dimensional incompressible fluid flow with viscous heating is also given.

The thermal conductivity models of Fourier, Cattaneo-Vernotte, Maxwell-Cattaneo-Luikov and models with fractional time derivatives of temperature and heat flux are considered. An algorithm for the numerical solution of the generalized problem of one-dimensional heat conduction is constructed. The algorithm comprises all the models listed above applying boundary conditions of the third kind and difference analogs of differential operators of the second order of accuracy in coordinate and time. The parameters (Biot number, time of thermal relaxation and thermal damping, indicators of fractional derivatives) that describe experimental transient processes are determined using the listed models applied for both in the center and on the surface of a low heat-conducting body.

ABSTRACTAn inverse analysis is used to simultaneously estimate the thermal conductivity and the boundary shape in steady-state heat conduction problems. The numerical scheme consists of a body-fitted grid generation technique to mesh the heat conducting body and solve the heat conduction equation – a novel, efficient, and easy to implement sensitivity analysis scheme to compute the sensitivity coefficients, and the conjugate gradient method as an optimization method to minimize the mismatch between the computed temperature distribution on some part of the body boundary and the measured temperatures. Using the proposed scheme, all sensitivity coefficients can be obtained in one solution of the direct heat conduction problem, irrespective of the large number of unknown parameters for the boundary shape. The obtained results reveal the accuracy, efficiency, and robustness of the proposed algorithm.

Heat conduction [1] is usually modeled as a diffusion process embodied in heat conduction equation. The traditional numerical methods [2, 3] for heat conduction problems such as the finite difference or finite element are well developed. However, these methods are based on discretized mesh systems, thus they are inherently limited in the geometry treatment. This chapter describes the Monte Carlo method that is based on particle transport simulation to solve heat conduction problems. The Monte Carlo method is “meshless” and thus can treat problems with very complicated geometries. The method is applied to a pebble fuel to be used in very high temperature gas-cooled reactors (VHTGRs) [4], which is a next-generation nuclear reactor being developed. Typically, a single pebble contains ~10,000 particle fuels randomly dispersed in graphite– matrix. Each particle fuel is in turn comprised of a fuel kernel and four layers of coatings. Furthermore, a typical reactor would house several tens of thousands of pebbles in the core depending on the power rating of the reactor. See Fig. 1. Such a level of geometric complexity and material heterogeneity defies the conventional mesh–based computational methods for heat conduction analysis. Among transport methods, the Monte Carlo method, that is based on stochastic particle simulation, is widely used in neutron and radiation particle transport problems such as nuclear reactor design. The Monte Carlo method described in this chapter is based on the observation that heat conduction is a diffusion process whose governing equation is analogous to the neutron diffusion equation [5] under no absorption, no fission and one speed condition, which is a special form of the particle transport equation. While neutron diffusion approximates the neutron transport phenomena, conversely it is applicable to solve diffusion problems by transport methods under certain conditions. Based on this idea, a new Monte Carlo method has been recently developed [6-8] to solve heat conduction problems. The method employs the MCNP code [9] as a major computational engine. MCNP is a widely used Monte Carlo particle transport code with versatile geometrical capabilities. Monte Carlo techniques for heat conduction have been reported [10-13] in the past. But most of the earlier Monte Carlo methods for heat conduction are based on discretized mesh systems, thus they are limited in the capabilities of geometry treatment. Fraley et al[13] uses a “meshless” system like the method in this chapter but does not give proper treatment to the boundary conditions, nor considers the “diffusion-transport theory correspondence” to be described in Section 2.2 in this chapter. Thus, the method in this chapter is a transport theory treatment of the heat conduction equation with a methodical boundary correction. The

We propose an efficient boundary collocation method for the solution of certain two- and three-dimensional problems of steady-state heat conduction in isotropic bimaterials. In particular, in two dimensions we consider the case where a circular region composed of one material is coated with an annular region of another material. In three dimensions, we examine the corresponding case for axisymmetric domains. The proposed method involves the use of a domain decomposition technique in conjunction with a matrix decomposition algorithm. The circulant structure of the matrices appearing in this method is exploited by using fast Fourier transforms. The method is tested numerically on several problems.

The problem of determination of the stationary temperature field in a body with plane sheet of heat sources is reduced to the solution of an integral equation of the first kind. A method for the determination of the set of solutions to this equation is proposed. The components of the vector of elastic displacements and the components of the tensor of temperature stresses are found according to the known temperature field by using the equations of thermoelasticity. We study the distributions of thermal displacements and stresses for any axially symmetric distribution of temperature in a circular domain. Numerous elements of contemporary engineering constructions operate under conditions of nonuniform heating that promote the formation of temperature gradients and thermal stresses. For the comprehensive analysis of the strength of structures, it is necessary to know the levels of thermal stresses and the character of their action. The investigation of the thermoelastic state of heat-generating elements is of high importance for practical purposes [9]. The presence of thin heat-generating inclusions in a body leads to the formation of tensile or compressive stresses. In the first case, the action of these stresses, together with the action of the other factors, may lead to crack initiation and propagation. In the second case, the action of stresses may decrease the intensity of stresses in the vicinity of the existing crack under force loading and promote the formation of conditions inhibiting the process of crack growth [10]. In the investigation of the stressed state of a body with thermally active cracks (with temperature or heat fluxes given on these cracks), the determination of stresses acting at the sites of the cracks is an intermediate stage and then the stresses acting upon the crack surfaces are removed. The major part of investigations of the stress-strain state of bodies with circular thermally active or heatinsulated cracks are performed for axially symmetric problems by the method of integral Hankel transformation and dual integral equations [11–16]. In [3], a method of two-dimensional boundary integral equations based on the theory of the Newton potential is developed for the solution of three-dimensional problems of stationary heat conduction and thermoelasticity for bodies with cracks. According to this method, the densities of potentials in the problem of heat conduction are the intensities of heat sources at the sites of the cracks determined, for given temperatures on the cracks, from the integral equations with polar kernels. The temperature field and stressed state of a body with given temperature or heat flow in a circular domain described by polynomials of the third degree are investigated in [4]. In the axially symmetric case, for polynomials of any degree, this problem is studied in [5]. According to the classic statement of the problems of stationary heat conduction with thermally active inclusions [4, 5], the presence of heat sources is postulated solely in the domain of the inclusion, which leads to singular distributions of the heat flows on its edges.

This paper introduces a mathematical model of a thermal protection system incorporating the Riemann-Liouville fractional derivative. The system is considered as a three-layer structure, where the temperature distribution in the first two layers follows the classical heat conduction equation. In contrast, the third layer, characterized by its porous nature, is modeled using a fractional-order heat conduction equation. The thermal contact resistances between the layers are taken into account. The external surface is subjected to a boundary condition of the second kind, incorporating an aerothermal heat flux derived from NASA Langley Research Center data, while the internal surface is governed by a Dirichlet boundary condition. Additionally, the temperature-dependent material properties are considered. A significant contribution of this study is the development of a numerical scheme for a three-layer thermal protection system model, in which one layer is porous and described using the Riemann-Liouville fractional derivative. The proposed approach allows for accurate simulation of heat conduction in systems with complex material structures. The influence of the fractional derivative order β on the temperature profile was investigated, showing that variations in β significantly affect the thermal response of the system. Furthermore, a mesh refinement study was conducted to assess the impact of spatial discretization on the numerical results. These findings establish the model as a valuable tool for computer simulations and provide a basis for further development and optimization of mathematical and computational approaches in the analysis of thermal protection systems.

A linear inverse model for the temperature-dependent thermal conductivity determination in one-dimensional problems

Introduction.- Mathematical Preliminaries.- Regularization of moment problems by trancated expansion and by the Tikhonov method.- Backus-Gilbert regularization of a moment problem.- The Hausdorff moment problem: regularization and error estimates.- Analytic functions: reconstruction and Sinc approximations.- Regularization of some inverse problems in potential theory.- Regularization of some inverse problems in heat conduction.- Epilogue.- References.- Index.

This paper presents a multi-domain boundary face method (MD-BFM) for solving steady-state heat conduction problems on large-scale engineering structures. In the BFM, both boundary integration and variable approximation are performed in the 2-D parametric space of the boundary surfaces of the body in question. The surface parameters are obtained through the boundary representation data structure (B-rep) implanted in all CAD packages. The geometric quantities in the boundary integrals of the BIE are computed directly in the parametric spaces of the surfaces to avoid geometric errors. On the other hand, a MDBFM is necessary for structures of complex shapes or constituted of different kinds of materials. In this work, the multi-domain BFM is developed and applied for heat conduction analysis of a real gravity dam. Numerical results have demonstrated that our method can achieve comparable accuracy than other methods (e.g. the FEM), but at much lower costs in terms of both computer

The problem of heat conduction in a multiphase medium is a free boundary problem in which the free boundary is the phase transition line. The solution to this problem is highly irregular in that the temperature gradient is discontinuous at the free boundary. Two‐dimensional, discontinuous, space–time finite elements are introduced to obtain a finite element, free boundary formulation for this problem. The jump in heat flux and position of the phase transition line become dependent variables in the finite element model. The resulting models give accurate results while allowing the calculation of free boundary problems with a fixed finite element mesh.

The modern state of approximate integral methods used in applications, where the processes of heat conduction and heat and mass transfer are of first importance, is considered. Integral methods have found a wide utility in different fields of knowledge: problems of heat conduction with different heat-exchange conditions, simulation of thermal protection, Stefantype problems, microwave heating of a substance, problems on a boundary layer, simulation of a fluid flow in a channel, thermal explosion, laser and plasma treatment of materials, simulation of the formation and melting of ice, inverse heat problems, temperature and thermal definition of nanoparticles and nanoliquids, and others. Moreover, polynomial solutions are of interest because the determination of a temperature (concentration) field is an intermediate stage in the mathematical description of any other process. The following main methods were investigated on the basis of the error norms: the Tsoi and Postol’nik methods, the method of integral relations, the Gudman integral method of heat balance, the improved Volkov integral method, the matched integral method, the modified Hristov method, the Mayer integral method, the Kudinov method of additional boundary conditions, the Fedorov boundary method, the method of weighted temperature function, the integral method of boundary characteristics. It was established that the two last-mentioned methods are characterized by high convergence and frequently give solutions whose accuracy is not worse that the accuracy of numerical solutions.