Research On The Collocation Spectral Method For Radiative Heat Transfer In Cylindrical System

Radiative heat transfer is one of the three fundamental modes of heat transfer.It exists extensively in many fields such as industry,military and medicine.The behavior of radiation is governed by the integro-differential radiative transfer equation(RTE),Solving the RTE accurately is the premise to analyze the process of radiative heat transfer theoretically.For the investigation of radiative heat transfer in axisymmetric systems,one can adopt the cylindrical coordinates to simplify matters.However,present methods,including the traditional discrete ordinates method(DOM),for the RTE in cylindrical coordinates are expensive with low accuracy.Developing the efficient and accurate methods to solve the RTE in cylindrical coordinates is the pivotal challenge to be solved.The collocation spectral method(CSM)has the infinite order accuracy for the smooth function.It has been widely used in the analysis of various fluid flow and heat transfer problems.In recent years,the CSM for solution of radiative heat transfer in Cartesian coordinates has also gained increasingly interest.In this thesis,in order to reduce the difficulty in the application of CSM,the explicit expressions are firstly deduced,and the high efficient iterative matrix solver is developed based on the Schur decomposition.These works lay the theoretical foundation for the successfully developing collocation spectral solver for the RTE in cylindrical coordinates,and provide the technical support for the future works on the instability analysis of radiation hydrodynamics and radiation magnetohydrodynamics in cylindrical systems.Secondly,for the sake of constructing benchmark solutions in the one-dimensional cylinder,the CSM is adopted to solve the radiative integro-differential transfer equation,which is independent on the angular variable.The segments integration method combined with the interpolation is proposed to deal with the non-smooth of integrand.The results show that this method is much better than others,and it can produce benchmark solutions exceeding seven significant digits efficiently.Again,the effects of various factors on the numerical accuracy for the solution of RTE in cylindrical coordinates by the CSM and DOM are investigated based on the benchmarks.The results show that the accuracy of CSM greatly depends on the form of equation and the collocation type of radial grid.Using the non-conservative form can produce results much higher accurate than the conservative form.The radial computational domain should use the diameter rather than the radius.The accuracy of DOM depends on the pole condition and the chosing of angular ordinates.The axisymmetric pole condition suggested in the literature actually results in greater error than the the pole condition corresponding to specular reflection.Choosing the ordinates to pass through the centroid of the solid angles is more recommended than other schemes.Then,with the effects of various parameters considered,the performance of CSM is assessed by comparing to the DOM.The results show that the program of CSM is robust as that of DOM.The accuracy of two methods decreases with the decreasing of emissivity and the increasing of optical thickness.In the same grid number,the computational cost and accuracy of CSM are both higher than those of DOM.Whereas,to achieve the same accuracy,the computational cost and grid requirement of CSM are both lower than those of DOM.Finally,the changes on the performance of CSM and DOM from the one-dimensional case to the two-dimensional case are investigated.The results show that,in the one-dimensional case,the CSM and DOM have the fifth-and second-order convergence accuracy,respectively.In the two-dimensional case,both two methods may suffer from severe ray effect,and the ray effect can be eliminated by the analytical determination of wall related component of the radiative intensity.However,even in the case without ray effect,the accuracy of two methods decays severely,and they have the only second-and first-order convergence accuracy,respectively.In conclusion,the CSM can be an alternative for the solution of radiave heat transfer in cylindrical coordinates,and it outperforms the traditional DOM.However,it losts the property of high order convergence for the multi-dimensional problems,thus the performance should be improved further.