全谱方法求解平行平板间非线性各向异性散射介质内辐射换热

Solution of thermal radiation heat transfer in a plane-parallel,nonlinear anisotropic scattering medium by full spectral method

提出采用全谱方法求解平行平板内,充满非线性各向异性散射参与性介质的辐射换热问题。在求解过程中,辐射传递方程中的非线性各向异性散射相函数采用Legendre多项式展开,空间微分项和散射积分项均采用Chebyshev多项式展开,在空间和角度上的辐射强度均采用谱方法进行离散,将原本复杂的微积分形式的辐射传递方程转化为矩阵形式的代数方程。选取边界为半透明边界,散射函数为空间函数的经典算例,检验全谱方法求解非线性各向异性散射介质内辐射换热的性能,并将其计算结果与解析解、最小二乘有限元和基于离散坐标的配置点谱方法的计算结果进行比较,表明全谱方法的计算结果与它们吻合得很好,且具有很高的计算精度。同时,全谱方法的计算误差随着节点数的增加呈指数降低;在较少计算节点的条件下,仍然可以获得较为满意的计算结果。

The full spectral method （FSM） is proposed to address the problem of radiation heat transfer in parallel planes with a nonlinear anisotropic scattering medium. In the solving process, the nonlinear anisotropic scattering phase function of radiation transfer equation is expressed by finite series of Legendre polynomials; the spatial differential term and the scattering integral term are expressed by finite series of Chebyshev polynomials; the spatial and angular distributions of radiation intensity are dis- cretized by the spectral method. Then, the integral-differential form of radiation transfer equation can be converted into matrix form of algebraic equation. Taking radiation heat transfer in space-dependent scattering medium with semitransparent boundary as an example, we test the performance of FSM for solving radiation heat transfer in nonlinear anisotropic scattering medium. Com- parisons of FSM results, analytic solutions, least square finite element solutions and spectral collocation method are performed based on discrete ordinates method （SCM-DOM） results. These comparisons indicate that FSM results are in good agreement with those results, and FSM has a good accuracy. Meanwhile, with the increase of collocation points, the computational error of FSM decreases with an exponential trend. FSM can also obtain satisfactory results even using a small number of nodes.