UNCERTAINTY QUANTIFICATION FOR THERMOMAGNETIC CONVECTION OF PARAMAGNETIC FLUID IN RANDOM POROUS MEDIA BASED ON INTRUSIVE POLYNOMIAL CHAOS METHOD
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摘要: 目前流体流动与传热问题的研究大都基于确定性工况条件, 而现实流体流动与传热问题中存在着大量不确定性因素, 计算流体力学的不确定性量化提供了一种理解流体物性、边界条件与初始条件等不确定性因素对模拟结果影响的能力. 为揭示随机多孔介质内顺磁性流体热磁对流的传播规律与演化特征, 本文发展了一种基于侵入式多项式混沌展开法的热磁对流不确定性量化数理模型与算法程序. 该方法分别利用Karhunen-Loeve展开与多项式混沌展开表达输入随机参数与输出响应量, 同时利用伽辽金投影方法将随机热磁对流控制方程解耦为一组可以应用有限元修正方法求解的确定性控制方程, 并对输出响应量多项式混沌进行求解, 最后采用随机投影法求解相应的确定性控制方程中的混沌系数, 得到输出响应量的统计特征与混沌效应. 热磁对流不确定性量化表明多孔介质孔隙率不确定性通过控制方程演化, 进而影响着多孔介质方腔内顺磁性流体热磁对流, 顺磁性流体热磁对流呈现出显著的混沌效应. 与蒙特卡罗法预测结果相比, 两者计算结果吻合良好, 但侵入式混沌多项式展开法计算量显著减少.Abstract: At present, the research of fluid flow and heat transfer problems is mostly based on deterministic working conditions, but there are a large number of uncertain factors in real fluid flow and heat transfer problems. The uncertainty quantification of computational fluid dynamics provides an ability to understand the influence of uncertain factors such as fluid physical properties, boundary conditions and initial conditions on simulations results. In order to reveal the propagation law and evolution characteristics of thermomagnetic convection of paramagnetic fluid in random porous media, a mathematical model and algorithm program of uncertainty quantification for thermomagnetic convection were developed based on intrusive polynomial chaos expansion method. In this method, the input random parameters and output response were expressed by Karhunen-Loeve expansion and polynomial chaos expansion respectively. At the same time, the Galerkin projection method was adopted to decouple the stochastic control equations into a set of deterministic control equations which can be solved by finite element correction method, and each polynomial chaos of output response was solved. Finally, the stochastic projection method was used to solve the chaos coefficients in the corresponding deterministic control equations, and the statistical characteristics and chaos effect of the output response are obtained. The uncertainty quantification of thermomagnetic convection shows that the porosity uncertainty of porous media affects the thermomagnetic convection of paramagnetic fluid in a square cavity through the evolution of governing equations, and the thermomagnetic convection of paramagnetic fluid presents a significant chaos effect. The output response shows the characteristics of rapid convergence. The output response values in the first-order mode are at least one order of magnitude lower than the corresponding average values, while the output response values in the second-order mode are much smaller than those in the first-order mode. Compared with the Monte Carlo method, the two results agree well, but the computational cost of the intrusive polynomial chaos expansion method is significantly reduced.
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引 言
重力与磁场力相互作用引起的热磁对流现象广泛应用于提高晶体生产质量、驱动无机械元件及研究氧浓度传感器等[1-2]. 数值计算是研究热磁对流驱动作用机理的重要手段之一, 研究人员已开展了许多磁致顺磁性流体热磁对流方面的数值计算研究[3]. He等[4]提出了一种精确的分离谱元方法系统研究了不同磁感应强度与加热部件位置下矩形方腔内空气热磁对流的变化规律. Kaneda等[5]采用数值研究方法揭示了磁场力影响方腔内顺磁性流体热磁对流的作用机理. Zeng等[6]和姜昌伟等[7]数值研究了梯度磁场作用下多孔介质方腔内顺磁性或逆磁性流体的传热特性. Zhang等[8]应用Lattice-Boltzmann方法研究了四极磁场作用下多孔介质方腔内顺磁性流体的热磁对流与熵产.
目前, 梯度磁场下多孔介质内流体热磁对流的数值计算普通采用确定性数学与几何结构、边界条件与物性参数等. 然而现实中的热磁对流问题中数理模型、物性参数与边界条件等存在着大量的不确定性, 如果忽略这些不确定性因素将不能准确反映真实的热磁对流过程, 因此理论上讲必须考虑这些不确定性因素对热磁对流控制方程及其数值模拟结果的影响. 梯度磁场下多孔介质内流体热磁对流过程中, 由于多孔介质具有空间变异性, 造成多孔介质孔隙空间分布呈现出一定的随机性, 确定性热磁对流控制方程难以描述多孔介质孔隙率不确定性对热磁对流的影响, 因此需要发展出一种把孔隙率作为随机输入的热磁对流不确定性量化分析方法.
不确定性量化研究已成为近年来国际上热门的课题[9]. 传统的不确定性量化方法有蒙特卡罗(Monte-Carlo)法和摄动(Perturbation)法等. 叶坤等[10]基于不确定性与全局灵敏度分析方法分析了高超声速舵面气动热的不确定度. Wang和Qiu[11]基于摄动有限元法, 提出了模糊摄动有限元法, 用以预测材料性质、外部载荷与边界条件等具有模糊参数的不确定性温度场. 虽然蒙特卡罗法需要消耗大量的计算资源, 但由于该方法可以应用现有的求解器进行求解, 因此其广泛应用于计算流体力学不确定度量化, 同时其他计算流体力学不确定性分析方法均采用该方法作为基准方法; 而摄动法一般应用于随机参数变异系数比较小的场合.
近年来, 为解决传统不确定性分析方法高昂的计算成本, 研究者先后提出随机配点法、随机有限元法、谱方法及多项式混沌展开法等多种数值计算方法, 其中多项式混沌展开法应用最为广泛. 多项式混沌展开法最早由Wiener[12]提出, 早期在解决湍流问题中得到了较多应用. 根据与求解器的耦合方式不同, 多项式混沌展开法分为非侵入式(non-intrusive)不确定性量化方法和侵入式(intrusive)不确定性量化方法等.
非侵入式方法基于概率方法, 通过在随机空间内采样获得若干配置点, 然后在各配置点上调用现有的求解器运行求解, 并基于统计方法获得相关参数的统计特征. 非侵入式方法包括概率配点法、随机配置法、稀疏网格配点法等, 该方法有利于采用经过完好确认的CFD软件, 避免了修改控制方程给工业应用引入新误差的风险. 目前, 非侵入式方法广泛应用于流体流动与传热、燃烧与化学反应等领域的不确定性量化[13-17]. Narayan和Zhou[18]与Guo等[19]提出了非结构化网格随机配置与最小二乘法多项式近似两种不确定性量化方法, 并从理论上给出了这两种方法的稳定性与精度的结果, 显示出其在不确定性量化中的巨大潜力. 基于非侵入式方法, 黄明等开展了涡轮动叶凹槽状叶顶气热性能、高超声速吸气推进飞行器进气道性能、各向异性陶瓷基复合材料涡轮叶片热响应等的不确定性量化分析, 数值模拟与实验数据吻合一致验证了非侵入式方法的有效性, 为工程中的高维不确定性量化问题提供了很好的解决思路[20-24]. Rajabi等[25]对多孔介质方腔内双扩散自然对流基准问题进行了详细的不确定性传播分析及全局灵敏度分析以确定模型参数不确定性对腔体内流动、传热与传质过程的影响.
侵入式方法在随机维数较低的情况下具有一定的精度优势. 但该方法需要求解一个复杂的控制方程组, 同时需要重新开发新的求解器, 这增加了其在计算流体力学应用中的难度, 并且随机维数一旦增加, 这种优势立即消除. Xiu和Karniadakis[26]提出了广义多项式混沌展开法, 并给出了每个不确定性参数分布最佳正交基函数的证明. 王晓东和康顺[27]等将谱随机不确定性法与有限差分法相结合, 得到不确定性场控制方程的随机解耦方程组, 发展出用来随机自然对流模拟的不确定性量化方法. Li等[28]基于广义多项式混沌展开和Galerkin投影将随机对流扩散方程转化为一组耦合的确定性方程组, 研究了两类随机输入分布数值格式对不确定性量化结果的影响. Em-Alrani等[29]和Ma和Zabaras[30]提出了一种稳定化随机有限元方法来求解随机不可压缩Navier-Stokes方程与随机多孔介质内流体自然对流. Chakraborty和Chowdhury[31]提出了一种称为基于广义方差分析(ANOVA)的Galerkin投影新算法, 并成功应用于求解随机Navier-Stokes方程与随机稳态扩散问题. 由于侵入式方法需要修改控制方程及重新编写程序代码, 这给侵入式方法的工程应用带来了困难, 因此近年来侵入式方法的相关文献很少.
本文提出了一种基于侵入式多项式混沌展开法的随机多孔介质内顺磁性流体热磁对流不确定性量化方法, 构建出热磁对流不确定性量化的随机数理模型与有限元程序框架, 对随机多孔介质内顺磁性流体热磁对流进行不确定性量化.
1. 物理模型与控制方程
1.1 物理模型
图1给出热磁对流物理模型和坐标系统, 4块磁体置于充满顺磁性流体的多孔介质方腔四周. 多孔介质方腔左右侧壁面分别等温加热与冷却, 其他壁面绝热. 方腔无量纲尺寸、磁体无量纲尺寸、方腔与磁体无量纲距离分别为1, 5/6, 1/8.
1.2 控制方程
顺磁性流体热磁对流控制方程包括质量守恒方程、多孔介质动量守恒方程和能量守恒方程, 其无量纲控制方程如下.
质量守恒方程
$$ \nabla \cdot {\boldsymbol{U}} = 0 $$ (1) 动量守恒方程
$$ \begin{split} &\frac{{\partial {\boldsymbol{U}}}}{{\partial \tau }} + \frac{{\boldsymbol{U}}}{\varepsilon } \cdot \nabla {\boldsymbol{U}} = - \varepsilon \nabla P + Pr{\nabla ^2}{\boldsymbol{U}} -\hfill \\ & \quad \frac{{Pr}}{{Da}}\frac{{{{\left( {1 - \varepsilon } \right)}^2}}}{{{\varepsilon ^2}}}{\boldsymbol{U}} - \frac{{1.75\left\| {\boldsymbol{U}} \right\|}}{{\sqrt {150Da} }}\frac{{\left( {1 - \varepsilon } \right)}}{{{\varepsilon ^2}}}{\boldsymbol{U}} -\hfill \\ & \quad \varepsilon RaPr\theta {{\boldsymbol{e}}_{{g}}} - \varepsilon \gamma RaPr\nabla {{\boldsymbol{B}}^2}\theta \hfill \end{split} $$ (2) 能量守恒方程
$$ \frac{{\partial \theta }}{{\partial \tau }} + {\boldsymbol{U}} \cdot \nabla \theta = {\nabla ^2}\theta $$ (3) 热磁对流无量纲控制方程中磁场分布采用标量磁位法计算, 通过求解Maxwell方程组获得磁感应强度分布B
$$ \nabla \cdot {\boldsymbol{B}} = 0 $$ (4) $$ \nabla \times {\boldsymbol{H}} = {\bf{0}} $$ (5) 其中, B为无量纲磁感应强度, (Bx, By); H为无量纲磁场强度, (Hx, Hy); U为无量纲速度, (U, V); P为无量纲压力; θ为无量纲温度; τ为无量纲时间; ε为孔隙率; eg为重力方程的单位矢量; Pr为普朗特数; Ra为瑞利数; Da为达西数; γ为磁场力数.
2. 侵入式不确定性量化方法
2.1 输出随机参数Karhunen-Loeve (KL)展开
考虑多孔介质孔隙率ε具有不确定性, 以
$\varepsilon \left( {{\boldsymbol{X}},\omega } \right)$ 代表输入随机参数孔隙率, 其中${\boldsymbol{X}} \in D$ (参数空间),$\omega \in \varOmega $ (概率空间). 输入随机参数可以分解为$\varepsilon \left( {{\boldsymbol{X}},\omega } \right) = $ $ \bar \varepsilon \left( {\boldsymbol{X}} \right) + \varepsilon '\left( {{\boldsymbol{X}},\omega } \right)$ , 其中$\bar \varepsilon \left( {\boldsymbol{X}} \right)$ 表示孔隙率均值,$\varepsilon '\left( {{\boldsymbol{X}},\omega } \right)$ 为孔隙率扰动. 通常采用协方差函数${R_{hh}}\left( {{{\boldsymbol{X}}_1},{{\boldsymbol{X}}_2}} \right) = $ $ \left\langle {\varepsilon '\left( {{{\boldsymbol{X}}_1},\omega } \right),\varepsilon '\left( {{{\boldsymbol{X}}_2},\omega } \right)} \right\rangle $ 来描述输入随机参数随机场的空间性质, 协方差函数可分解为[32]$$ {R_{hh}}\left( {{{\boldsymbol{X}}_1},{{\boldsymbol{X}}_2}} \right) = \sum\limits_{i = 1}^\infty {{\lambda _i}{\phi _i}\left( {{{\boldsymbol{X}}_1}} \right){\phi _i}\left( {{{\boldsymbol{X}}_2}} \right)} $$ (6) 式中
${\lambda _i}$ 和${\phi _i}$ 分别为特征值和特征函数. 特征值和特征函数可通过求解Fredholm方程的分析解得到$$ \int_D {{R_{hh}}\left( {{{\boldsymbol{X}}_1},{{\boldsymbol{X}}_2}} \right)} {\phi _i}\left( {{{\boldsymbol{X}}_2}} \right){\rm{d}}{{\boldsymbol{X}}_2} = {\lambda _i}{\phi _i}\left( {{{\boldsymbol{X}}_1}} \right) $$ (7) 由此,
$\varepsilon \left( {{\boldsymbol{X}},\omega } \right)$ 可表示为$$ \varepsilon \left( {{\boldsymbol{X}},\omega } \right) = \bar \varepsilon \left( {\boldsymbol{X}} \right) + \sum\limits_{i = 1}^\infty {\sqrt {{\lambda _i}} {\phi _i}\left( {\boldsymbol{X}} \right){{{\xi}} _i}\left( \omega \right)} $$ (8) 式中
${\xi _i}\left( \omega \right)$ 为0均值单位方差的正交高斯随机变量.将KL展开截断成有限项, 截取前
$m$ 项为KL展开最终表述形式为$$ \varepsilon \left( {{\boldsymbol{X}},\omega } \right) = \bar \varepsilon \left( {\boldsymbol{X}} \right) + \sum\limits_{i = 1}^m {\sqrt {{\lambda _i}} {\phi _i}\left( {\boldsymbol{X}} \right){{{\xi}} _i}\left( \omega \right)} $$ (9) 对于二维随机场, 协方差函数可写为
${R_{hh}}\left( {{{\boldsymbol{X}}_1},{{\boldsymbol{X}}_2}} \right) = $ $ {\sigma ^2}\exp \left( { - \left| {{X_1} - {X_2}} \right|/\eta - \left| {{Y_1} - {Y_2}} \right|/\eta } \right)$ , 其中${\sigma ^2}$ 与$\eta $ 分别为方差及相关长度.2.2 输出响应量多项式混沌展开
输出响应量分布特征未知, 可以采用多项式混沌展开描述输出响应量不确定性, 其紧凑格式可表示为
$$ F\left( \omega \right) = \sum\limits_{i = 0}^\infty {{F_i}{\varPsi _i}\left[ {{\boldsymbol{\xi}} \left( \omega \right)} \right]} $$ (10) 式中, F为随机输出场; ξ为矢量, 表示为
$[ {{{\xi}} _{{i_1}}}( \omega ), $ $ {{{\xi}} _{{i_2}}}( \omega ), \cdots , {{{\xi}} _{{i_n}}}( \omega ) ]^{\rm{T}}$ ;${\varPsi _i}\left[ {{{\xi}} \left( \omega \right)} \right]$ 为Hermite混沌基.实际应用中多项式混沌扩展通常截断成有限项. 因此, 方程(10)可简化为
$$ F\left( \omega \right) = \sum\limits_{i = 0}^P {{F_i}{\varPsi _i}\left[ {{\boldsymbol{\xi}} \left( \omega \right)} \right]} $$ (11) 扩展项总数(P + 1)由随机维数m与多项式展开最高阶数n决定
$$ P + 1 = \frac{{\left( {m + n} \right)!}}{{m!n!}} $$ (12) 因此控制方程中的速度、温度及压力通过多项式混沌展开可表示为
$$ \left.\begin{split} & {\boldsymbol{U}}\left( {{\boldsymbol{X}},\omega } \right) = \sum\limits_{i = 0}^P {{{\boldsymbol{U}}_i}\left( {\boldsymbol{X}} \right){{{\varPsi}} _i}\left( \omega \right)} \hfill \\ & P\left( {{\boldsymbol{X}},\omega } \right) = \sum\limits_{i = 0}^P {{P_i}\left( {\boldsymbol{X}} \right){{{\varPsi}} _i}\left( \omega \right)} \hfill \\ &\theta \left( {{\boldsymbol{X}},\omega } \right) = \sum\limits_{i = 0}^P {{\theta _i}\left( {\boldsymbol{X}} \right){{{\varPsi}} _i}\left( \omega \right)} \hfill \end{split} \right\}$$ (13) 2.3 输入随机参数非线性函数随机谱展开
孔隙率
$\varepsilon \left( {{\boldsymbol{X}},\omega } \right)$ 基于KL展开写成$$ \begin{split} \varepsilon \left( {{\boldsymbol{X}},\omega } \right) =& \bar \varepsilon \left( {\boldsymbol{X}} \right) + \sum\limits_{i = 1}^m {\sqrt {{\lambda _i}} {f_i}\left( {\boldsymbol{X}} \right){\xi _i}\left( \omega \right)} = \hfill \\ &\sum\limits_{i = 0}^P {{\varepsilon _i}\left( {\boldsymbol{X}} \right){\varPsi _i}\left( \omega \right)} \hfill \end{split} $$ (14) 此处,
$i = 0$ ,${\varepsilon _i}\left( {\boldsymbol{X}} \right) = \bar \varepsilon \left( {\boldsymbol{X}} \right)$ ;$i = 1, 2, \cdots ,m$ ,${\varepsilon _i}\left( {\boldsymbol{X}} \right) = $ $ \sqrt {{\lambda _i}} {f_i}\left( {\boldsymbol{X}} \right)$ ;$i > m$ ,${\varepsilon _i}\left( {\boldsymbol{X}} \right) = 0$ .控制方程(2)包含
$\varepsilon \left( {{\boldsymbol{X}},\omega } \right)$ 的非线性函数, 因此首先需要应用拉丁超立方抽样蒙特卡罗法将非线性函数表示为多项式基$$\left. \begin{split} & \frac{1}{{\varepsilon \left( {{\boldsymbol{X}},\omega } \right)}} = \sum\limits_{i = 0}^P {{{\bar \varepsilon }_i}\left( {\boldsymbol{X}} \right){{{\varPsi}} _i}\left( \omega \right)} \hfill \\ & \frac{{{{\left[ {1 - \varepsilon \left( {{\boldsymbol{X}},\omega } \right)} \right]}^2}}}{{\varepsilon {{\left( {{\boldsymbol{X}},\omega } \right)}^2}}} = \sum\limits_{i = 0}^P {{{\hat \varepsilon }_i}\left( {\boldsymbol{X}} \right){{{\varPsi}} _i}\left( \omega \right)} \hfill \\ & \frac{{1 - \varepsilon \left( {{\boldsymbol{X}},\omega } \right)}}{{\varepsilon {{\left( {{\boldsymbol{X}},\omega } \right)}^2}}} = \sum\limits_{i = 0}^P {{{\tilde \varepsilon }_i}\left( {\boldsymbol{X}} \right){{{\varPsi}} _i}\left( \omega \right)} \hfill \end{split} \right\}$$ (15) 2.4 基本谱分解技术解耦无量纲控制方程
将方程式(13) ~ 式(15)代入控制方程式(1) ~ 式(3), 可得方程式(16) ~ 式(18)
$$ \sum\limits_{i = 0}^P {\nabla \cdot {{\boldsymbol{U}}_i}{{\varPsi} _i} = 0} $$ (16) $$ \begin{split} & \sum\limits_{i = 0}^P {\frac{{\partial {{\boldsymbol{U}}_i}}}{{\partial \tau }}{{\varPsi} _i}} + \sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {\sum\limits_{l = 0}^P {{{\bar \varepsilon }_i}{{\boldsymbol{U}}_j} \cdot \nabla {{\boldsymbol{U}}_l}{{\varPsi} _i}{{\varPsi} _j}{{\varPsi} _l}} } } = \hfill \\ & \quad - \sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {{\varepsilon _i}} \nabla {P_j}{{\varPsi} _i}{{\varPsi} _j}} + Pr\sum\limits_{i = 0}^P {{\nabla ^2}{{\boldsymbol{U}}_i}{{\varPsi} _i}} - \hfill \\ & \quad \frac{{Pr}}{{Da}}\sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {{{\hat \varepsilon }_i}} {{\boldsymbol{U}}_j}{{\varPsi} _i}{{\varPsi} _j}} - \frac{{1.75\left\| {\boldsymbol{U}} \right\|}}{{\sqrt {150Da} }}\sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {{{\tilde \varepsilon }_i}} {{\boldsymbol{U}}_j}{{\varPsi} _i}{{\varPsi} _j}} - \hfill \\ & \quad RaPr\sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {{\varepsilon _i}} {\theta _j}{{\varPsi} _i}{{\varPsi} _j}{{\boldsymbol{e}}_{{g}}}} -\hfill \\ & \quad \gamma RaPr\nabla {{\boldsymbol{B}}^2}\sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {{\varepsilon _i}} {\theta _j}{{\varPsi} _i}{{\varPsi} _j}} \hfill \end{split} $$ (17) $$ \sum\limits_{i = 0}^P {\frac{{\partial {\theta _i}}}{{\partial \tau }}{{\varPsi} _i}} + \sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {{{\boldsymbol{U}}_i} \cdot \nabla {\theta _j}{{\varPsi} _i}{{\varPsi} _j}} } = \sum\limits_{i = 0}^P {{\nabla ^2}{\theta _i}{{\varPsi} _i}} $$ (18) 然后对方程式(16) ~ 式(18)进行Galerkin映射, 并基于多项式的正交性可得方程
$$ \nabla \cdot {{\boldsymbol{U}}_k} = 0 $$ (19) $$ \begin{split} & \frac{{\partial {{\boldsymbol{U}}_k}}}{{\partial \tau }} + \sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {\sum\limits_{l = 0}^P {{{\bar \varepsilon }_i}{{\boldsymbol{U}}_j} \cdot \nabla {{\boldsymbol{U}}_l}\frac{{\left\langle {{{\varPsi} _i}{{\varPsi} _j}{{\varPsi} _l}{{\varPsi} _k}} \right\rangle }}{{\left\langle {{\varPsi} _k^2} \right\rangle }}} } } = \hfill \\ & \quad - \sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {{\varepsilon _i}} \nabla {P_j}\frac{{\left\langle {{{\varPsi} _i}{{\varPsi} _j}{{\varPsi} _k}} \right\rangle }}{{\left\langle {{\varPsi} _k^2} \right\rangle }}} + Pr{\nabla ^2}{{\boldsymbol{U}}_k}- \hfill \\ & \quad \frac{{Pr}}{{Da}}\sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {{{\hat \varepsilon }_i}} {{\boldsymbol{U}}_j}\frac{{\left\langle {{{\varPsi} _i}{{\varPsi} _j}{{\varPsi} _k}} \right\rangle }}{{\left\langle {{\varPsi} _k^2} \right\rangle }}}- \hfill \\ & \quad \frac{{1.75\left\| {\boldsymbol{U}} \right\|}}{{\sqrt {150Da} }}\sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {{{\tilde \varepsilon }_i}} {{\boldsymbol{U}}_j}\frac{{\left\langle {{{\varPsi} _i}{{\varPsi} _j}{{\varPsi} _k}} \right\rangle }}{{\left\langle {{\varPsi} _k^2} \right\rangle }}} - \end{split}$$ $$ \begin{split} & \quad RaPr\sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {{\varepsilon _i}} {\theta _j}\frac{{\left\langle {{{\varPsi} _i}{{\varPsi} _j}{{\varPsi} _k}} \right\rangle }}{{\left\langle {{\varPsi} _k^2} \right\rangle }}{{\boldsymbol{e}}_{{g}}}} - \hfill \\ & \quad \gamma RaPr\nabla {{\boldsymbol{B}}^2}\sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {{\varepsilon _i}} {\theta _j}\frac{{\left\langle {{{\varPsi} _i}{{\varPsi} _j}{{\varPsi} _k}} \right\rangle }}{{\left\langle {{\varPsi} _k^2} \right\rangle }}} \end{split}$$ (20) $$ \frac{{\partial {\theta _k}}}{{\partial \tau }} + \sum\limits_{i = 0}^P {\sum\limits_{j = 0}^P {{{\boldsymbol{U}}_i} \cdot \nabla {\theta _j}\frac{{\left\langle {{{\varPsi} _i}{{\varPsi} _j}{{\varPsi} _k}} \right\rangle }}{{\left\langle {{\varPsi} _k^2} \right\rangle }}} } = {\nabla ^2}{\theta _k} $$ (21) 采用同样的方法可以得到速度与温度边界条件:
(1)速度边界
$$ {U_k} = {V_k} = 0 $$ (22) (2)温度边界
$$ \begin{split} &{\left. {\frac{{\partial {\theta _k}}}{{\partial Y}}} \right|_{Y = 0}} = {\left. {\frac{{\partial {\theta _k}}}{{\partial Y}}} \right|_{Y = 1}} = 0 \hfill \\ &{\left. {{\theta _0}} \right|_{X = 0}} = 1,{\text{ }}{\left. {{\theta _k}} \right|_{X = 0}} = 1,{\text{ }}k = 1,2, \cdots ,P \hfill \\ & {\left. {{\theta _0}} \right|_{X = 1}} = 0,{\text{ }}k = 0,1, \cdots ,P \hfill \end{split} $$ (23) 附录给出了m = 1, n = 2时解耦控制方程式(19) ~ 式(21)的多项式混沌展开式.
2.5 不确定性量化程序流程
基于侵入式多项式混沌展开法的热磁对流不确定性量化程序流程如图2所示, 该算法的计算流程如下:
(1)应用KL展开对已知协方差函数的输入随机参数进行展开, 应用拉丁超立方抽样蒙特卡罗法对多孔介质孔隙率非线性项进行多项式混沌展开;
(2)采用多项式混沌对输出响应量进行展开, 基于谱分解技术将热磁对流控制方程解耦成一组确定性控制方程;
(3)应用磁场控制方程求解热磁对流控制方程中的磁场力源项
$\nabla {{\boldsymbol{B}}^2}$ ;(4)通过插值方法对输入随机参数特征值与特征函数、磁场分布进行插值, 应用有限元求解解耦热磁对流控制方程获得输出响应量多项式混沌展开系数;
(5)利用所得多项式混沌展开系数, 评估输出响应量的统计特性、输出响应量的混沌效应和输出响应量的概率密度分布函数(PDF)与累积分布函数(CDF).
3. 结果分析与讨论
随机多孔介质内顺磁性流体热磁对流不确定性量化研究中固定Pr = 1, Ra = 1 × 105, Da = 1 × 10−3, γ = 25,
$\bar \varepsilon \left( {\boldsymbol{X}} \right) = 0.6$ , η = 1.0, σ = 0.1. 图3与图4分别给出了输入随机参数KL展开的特征值与特征函数. 由图3可知, 特征值随着随机维数增加而快速衰减, 取前6项特征值和特征函数即满足要求, 因此本研究中取m = 6. 表1给出了不同多项式混沌展开阶数下热壁面平均Nusselt数的平均值和标准偏差, 当n = 2时可以保证不确定性量化精度. 因此,$P + 1 = $ $ \left( {m + n} \right)!/\left( {m!n!} \right) = \left( {6 + 2} \right)!/(6!2!) = 28$ , 共需耦合求解28组控制方程, 即112个控制方程. 解耦热磁对流控制方程采用有限元求解, 网格划分数目为60 × 60.![]()
图 3 孔隙率KL展开中协方差核特征值Figure 3. Eigenvalues of covariance kernel used in the KL expansion of porosity![]()
图 4 孔隙率KL展开中协方差核特征函数Figure 4. Eigenfunction of covariance kennel used in the KL expansion of porosity表 1 多项式混沌展开阶数对平均Nusselt数均值与标准偏差的影响Table 1. Influence of order of polynomial chaos expansion on mean value and standard deviation of average Nusselt numberOrder of polynomial chaos expansion Mean value of average
Nusselt numberStandard deviation of average Nusselt number PCE MC relative error/% PCE MC relative error/% n = 1 3.3074 3.2161 2.84 0.4286 0.4187 2.36 n = 2 3.2029 3.2161 −0.41 0.4136 0.4187 −1.23 PCE: polynomial chaos expansion; MC: Monte Carlo ![]()
图 4 孔隙率KL展开中协方差核特征函数(续)Figure 4. Eigenfunction of covariance kennel used in the KL expansion of porosity (continued)3.1 输出响应量均值与标准偏差
图5与图6分别给出了基于侵入式多项式混沌展开法与蒙特卡罗法的输出响应量均值与标准偏差, 两种方法下的输出响应量均值与标准偏差非常吻合, 表明侵入式多项式混沌展开法能够有效地模拟多孔介质孔隙率不确定性在顺磁性流体热磁对流的演化与传播. 从图5可看出, 顺磁性流体受到磁浮升力与重力浮升力共同作用形成相互对持的局面, 方腔内顺磁性流体流动呈现分层流动现象, 方腔上部形成一个逆时针方向流动的漩涡而下部形成一个顺时针方向流动的漩涡. 温度场分布规律表明, 热壁面等温线稠密区出现上下部而冷壁面等温线稠密区出现在中部. 从图6可以看出, 由于孔隙率不确定性通过动量方程演化进而影响能量方程, 因此其对温度场的影响明显小于对速度场的影响. 由于腔体壁面的黏滞效应, 流函数最大标准偏差区域出现在方腔左侧中部, 而方腔壁面附近流函数标准偏差较小, 温度场最大标准偏差区域出现方腔的左右侧中部.
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图 5 不同不确定性量化方法下流函数与温度场均值Figure 5. Mean value of stream function and temperature field with different uncertainty quantification methods![]()
图 6 不同不确定性量化方法下流函数与温度场标准偏差Figure 6. Standard deviation of stream function and temperature field with different uncertainty quantification methods图7是基于侵入式多项式混沌展开法与蒙特卡罗法的热壁面局部Nusselt数分布均值与标准偏差比较, 可以看出两者吻合良好. 从热壁面局部Nusselt数均值变化规律来看, 顺磁性流体在重力与磁场力的相互作用下, 由于方腔左侧壁面下部区域磁场力与重力相互协同, 因此该区域传热能力增强; 而上部区域, 磁场力与重力相互对抗, 传热能力减弱. 热壁面局部Nusselt数标准偏差与均值具有类似的变化规律. 图8给出了两种方法下热壁面平均Nusselt数概率密度分布函数与累积分布函数, 两者吻合良好.
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图 7 不同不确定性量化方法下热壁面局部Nusselt数分布Figure 7. Distribution of local Nusselt number at the hot wall with different uncertainty quantification methods![]()
图 8 不同不确定性量化方法下热壁面平均Nusselt数概率密度函数和累积分布函数Figure 8. Probability density function and cumulative distribution function of average Nusselt number at hot wall with different uncertainty quantification methods3.2 输出响应量混沌效应
图9给出了一阶模式下的流函数, 第1模式流函数与流函数平均值具有相似的轮廓, 而流函数第2 ~ 6个模式呈现出混沌效应. 图10给出了一阶模式下的温度场, 一阶模式下的温度场在温度场不确定性中占主导地位, 第1模式下的温度场与温度场标准偏差相类似, 并且呈现出与温度场平均值明显的差异, 而温度场第2 ~ 6个模式具有一定的混沌效应. 从控制方程可知, 多孔介质孔隙率不确定性主要通过动量方程影响着方腔内顺磁性流体流动, 而顺磁性流体流动通过能量方程演化导致温度场波动, 即温度场不确定性主要由速度场控制, 这就造成孔隙率不确定性对温度场的影响小于对速度场的影响. 最后, 应该注意的是流函数与温度场的平均值至少比一阶模式下的流函数与温度场高一个数量级. 图11给出了二阶模式下的部分流函数与温度场. 二阶模式下的流函数与温度场呈现出显著的混沌效应, 一阶模式下的流函数与温度场数值远远高于二阶模式下的流函数与温度场数值, 反映出随机输出响应量多项式混沌展开具有快速收敛的特性.
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图 11 流函数与温度场二阶模式Figure 11. Second-order modes of stream function and temperature field4. 结论
提出了一种基于侵入式多项式混沌展开的随机多孔介质内顺磁性流体热磁对流不确定性量化方法, 研究了孔隙率不确定性在热磁对流控制方程中的传播规律与演化特征, 主要结论如下.
(1) 基于侵入式多项式混沌展开实现了随机多孔介质内顺磁流性流体热磁对流的不确定性量化. 通过谱分解将热磁对流随机控制方程转化为一组解耦确定性控制方程并编制了相应的计算程序. 该方法为计算流体力学不确定性量化问题的求解提供了一条有效的途径.
(2) 量化了随机多孔介质内顺磁性流体热磁对流的传播规律与统计特征. 输入随机参数通过随机热磁对流控制方程传播与演化, 进而影响着顺磁性流体热磁对流, 顺磁性流体热磁对流呈现出显著的混沌效应. 输出响应量呈现出快速收敛的特性, 一阶模式下流函数与温度场数值至少比相应的平均值低一个数量级, 而二阶模式下流函数与温度场数值远小于一阶模式的数值.
(3) 比较了基于侵入式多项式混沌展开法与蒙特卡罗法两种方法下随机多孔介质内顺磁性流体热磁对流问题的统计特征, 统计特征非常吻合. 计算效率方面, 与蒙特卡罗法相比, 侵入式多项式混沌展开法在相同的精度下具有较小的计算量.
附录
$ \;$ 随机维数m = 1, 多项式混沌展开最高阶数n = 2时控制方程式(19) ~ 式(21)的多项式混沌展开式如下
模式0 (方程式(A1) ~ 式(A3))
$$ \nabla \cdot {{\boldsymbol{U}}_0} = 0 \tag{A1}$$ $$ \begin{split} & \frac{{\partial {{\boldsymbol{U}}_0}}}{{\partial \tau }} + \left( {{{\bar \varepsilon }_0}{{\boldsymbol{U}}_0} \cdot \nabla {{\boldsymbol{U}}_1} + {{\bar \varepsilon }_0}{{\boldsymbol{U}}_1} \cdot \nabla {{\boldsymbol{U}}_1} + 2{{\bar \varepsilon }_0}{{\boldsymbol{U}}_2} \cdot \nabla {{\boldsymbol{U}}_2}} \right.+ \hfill \\ & \quad {{\bar \varepsilon }_1}{{\boldsymbol{U}}_0} \cdot \nabla {{\boldsymbol{U}}_1} + {{\bar \varepsilon }_1}{{\boldsymbol{U}}_1} \cdot \nabla {{\boldsymbol{U}}_0} + 2{{\bar \varepsilon }_1}{{\boldsymbol{U}}_1} \cdot \nabla {{\boldsymbol{U}}_2}+ \hfill \\ & \quad 2{{\bar \varepsilon }_1}{{\boldsymbol{U}}_2} \cdot \nabla {{\boldsymbol{U}}_1} + 2{{\bar \varepsilon }_2}{{\boldsymbol{U}}_0} \cdot \nabla {{\boldsymbol{U}}_2} + 2{{\bar \varepsilon }_2}{{\boldsymbol{U}}_1} \cdot \nabla {{\boldsymbol{U}}_1}+ \hfill \\ & \quad \left. { 2{{\bar \varepsilon }_2}{{\boldsymbol{U}}_2} \cdot \nabla {{\boldsymbol{U}}_0} + 8{{\bar \varepsilon }_2}{{\boldsymbol{U}}_2} \cdot \nabla {{\boldsymbol{U}}_2}} \right) = - \left( {{\varepsilon _0}\nabla {P_0} + {\varepsilon _1}\nabla {P_1}} \right. +\hfill \\ & \quad \left. { 2{\varepsilon _2}\nabla {P_2}} \right) + Pr{\nabla ^2}{{\boldsymbol{U}}_0} - \frac{{Pr}}{{Da}}\left( {{{\hat \varepsilon }_0}{{\boldsymbol{U}}_0} + {{\hat \varepsilon }_1}{{\boldsymbol{U}}_1} + 2{{\hat \varepsilon }_2}{{\boldsymbol{U}}_2}} \right)- \hfill \\ & \quad \frac{{1.75\left\| {{{\boldsymbol{U}}_0}} \right\|}}{{\sqrt {150Da} }}\left( {{{\tilde \varepsilon }_0}{{\boldsymbol{U}}_0} + {{\tilde \varepsilon }_1}{{\boldsymbol{U}}_1} + 2{{\tilde \varepsilon }_2}{{\boldsymbol{U}}_2}} \right) -\hfill \\ & \quad RaPr\left( {{\varepsilon _0}{\theta _0} + {\varepsilon _1}{\theta _1} + 2{\varepsilon _2}{\theta _2}} \right){{\boldsymbol{e}}_{{g}}} -\hfill \\ & \quad \gamma RaPr\nabla {{\boldsymbol{B}}^2}\left( {{\varepsilon _0}{\theta _0} + {\varepsilon _1}{\theta _1} + 2{\varepsilon _2}{\theta _2}} \right) \hfill \end{split}\tag{A2} $$ $$ \frac{{\partial {\theta _0}}}{{\partial \tau }} + \left( {{{\boldsymbol{U}}_0} \cdot \nabla {\theta _0} + {{\boldsymbol{U}}_1} \cdot \nabla {\theta _1} + 2{{\boldsymbol{U}}_2} \cdot \nabla {\theta _2}} \right) \hfill = {\nabla ^2}{\theta _0} \hfill \tag{A3} $$ 模式1 (方程式(A4) ~ 式(A6))
$$ \nabla \cdot {{\boldsymbol{U}}_1} = 0 \tag{A4}$$ $$ \begin{aligned} & \frac{{\partial {{\boldsymbol{U}}_1}}}{{\partial \tau }} + \left( {{{\bar \varepsilon }_0}{{\boldsymbol{U}}_0} \cdot \nabla {{\boldsymbol{U}}_1} + {{\bar \varepsilon }_0}{{\boldsymbol{U}}_1} \cdot \nabla {{\boldsymbol{U}}_0}} \right. + 2{{\bar \varepsilon }_0}{{\boldsymbol{U}}_1} \cdot \nabla {{\boldsymbol{U}}_2}+ \hfill \\ & \quad 2{{\bar \varepsilon }_0}{{\boldsymbol{U}}_2} \cdot \nabla {{\boldsymbol{U}}_1} + {{\bar \varepsilon }_1}{{\boldsymbol{U}}_0} \cdot \nabla {{\boldsymbol{U}}_0} + 2{{\bar \varepsilon }_1}{{\boldsymbol{U}}_0} \cdot \nabla {{\boldsymbol{U}}_2} +\hfill \\ & \quad 3{{\bar \varepsilon }_1}{{\boldsymbol{U}}_1} \cdot \nabla {{\boldsymbol{U}}_1} + 2{{\bar \varepsilon }_1}{{\boldsymbol{U}}_2} \cdot \nabla {{\boldsymbol{U}}_0} + 10{{\bar \varepsilon }_1}{{\boldsymbol{U}}_2} \cdot \nabla {{\boldsymbol{U}}_2} +\hfill \\ & \quad 2{{\bar \varepsilon }_2}{{\boldsymbol{U}}_0} \cdot \nabla {{\boldsymbol{U}}_1} + 2{{\bar \varepsilon }_2}{{\boldsymbol{U}}_1} \cdot \nabla {{\boldsymbol{U}}_0} + 10{{\bar \varepsilon }_2}{{\boldsymbol{U}}_1} \cdot \nabla {{\boldsymbol{U}}_2} +\hfill \\ & \quad \left. { 10{{\bar \varepsilon }_2}{{\boldsymbol{U}}_2} \cdot \nabla {{\boldsymbol{U}}_1}} \right) = - \left( {{\varepsilon _0}\nabla {P_1} + {\varepsilon _1}\nabla {P_0} + 2{\varepsilon _1}\nabla {P_2}} \right.+ \end{aligned}$$ $$ \begin{split} & \quad \left. { 2{\varepsilon _2}\nabla {P_1}} \right) + Pr{\nabla ^2}{{\boldsymbol{U}}_1} - \frac{{Pr}}{{Da}}\left( {{{\hat \varepsilon }_0}{{\boldsymbol{U}}_1} + {{\hat \varepsilon }_1}{{\boldsymbol{U}}_0}} \right.+ \hfill \\ & \quad \left. { 2{{\hat \varepsilon }_1}{{\boldsymbol{U}}_2} + 2{{\hat \varepsilon }_2}{{\boldsymbol{U}}_1}} \right) - \frac{{1.75\left\| {{{\boldsymbol{U}}_1}} \right\|}}{{\sqrt {150Da} }}\left( {{{\tilde \varepsilon }_0}{{\boldsymbol{U}}_1} + {{\tilde \varepsilon }_1}{{\boldsymbol{U}}_0} + 2{{\tilde \varepsilon }_1}{{\boldsymbol{U}}_2}} \right.+ \hfill \\ & \quad \left. { 2{{\tilde \varepsilon }_2}{{\boldsymbol{U}}_1}} \right) - RaPr\left( {{\varepsilon _0}{\theta _1} + {\varepsilon _1}{\theta _0} + 2{\varepsilon _1}{\theta _2} + 2{\varepsilon _2}{\theta _1}} \right){{\boldsymbol{e}}_{{g}}}- \hfill \\ & \quad \gamma RaPr\nabla {{\boldsymbol{B}}^2}\left( {{\varepsilon _0}{\theta _1} + {\varepsilon _1}{\theta _0} + 2{\varepsilon _1}{\theta _2} + 2{\varepsilon _2}{\theta _1}} \right) \hfill \end{split} \tag{A5}$$ $$ \begin{split} &\frac{{\partial {\theta _1}}}{{\partial \tau }} + \left( {{{\boldsymbol{U}}_0} \cdot \nabla {\theta _1} + {{\boldsymbol{U}}_1} \cdot \nabla {\theta _0} + 2{{\boldsymbol{U}}_1} \cdot \nabla {\theta _2}} \right.+ \hfill \\ & \quad \left. { 2{{\boldsymbol{U}}_2} \cdot \nabla {\theta _1}} \right) = {\nabla ^2}{\theta _1} \hfill \end{split} \tag{A6}$$ 模式2 (方程式(A7)~式(A10))
$$ \nabla \cdot {{\boldsymbol{U}}_2} = 0 \tag{A7}$$ $$ \begin{split} & \frac{{\partial {{\boldsymbol{U}}_2}}}{{\partial \tau }} + \left( {{{\bar \varepsilon }_0}{{\boldsymbol{U}}_0} \cdot \nabla {{\boldsymbol{U}}_2} + {{\bar \varepsilon }_0}{{\boldsymbol{U}}_1} \cdot \nabla {{\boldsymbol{U}}_1} + {{\bar \varepsilon }_0}{{\boldsymbol{U}}_2} \cdot \nabla {{\boldsymbol{U}}_0}} \right.+ \hfill \\ & \quad 4{{\bar \varepsilon }_0}{{\boldsymbol{U}}_2} \cdot \nabla {{\boldsymbol{U}}_2} + {{\bar \varepsilon }_1}{{\boldsymbol{U}}_0} \cdot \nabla {{\boldsymbol{U}}_1} + {{\bar \varepsilon }_1}{{\boldsymbol{U}}_1} \cdot \nabla {{\boldsymbol{U}}_0} +\hfill \\ & \quad 5{{\bar \varepsilon }_1}{{\boldsymbol{U}}_1} \cdot \nabla {{\boldsymbol{U}}_2} + 5{{\bar \varepsilon }_1}{{\boldsymbol{U}}_2} \cdot \nabla {{\boldsymbol{U}}_1} + {{\bar \varepsilon }_2}{{\boldsymbol{U}}_0} \cdot \nabla {{\boldsymbol{U}}_0}+ \hfill \\ & \quad 4{{\bar \varepsilon }_2}{{\boldsymbol{U}}_0} \cdot \nabla {{\boldsymbol{U}}_2} + 5{{\bar \varepsilon }_2}{{\boldsymbol{U}}_1} \cdot \nabla {{\boldsymbol{U}}_1} + 4{{\bar \varepsilon }_2}{{\boldsymbol{U}}_2} \cdot \nabla {{\boldsymbol{U}}_0}+ \hfill \\ & \quad \left. { 30{{\bar \varepsilon }_2}{{\boldsymbol{U}}_2} \cdot \nabla {{\boldsymbol{U}}_2}} \right) = - \left( {{\varepsilon _0}\nabla {P_2} + {\varepsilon _1}\nabla {P_1} + {\varepsilon _2}\nabla {P_0}} \right. +\hfill \\ & \quad \left. { 4{\varepsilon _2}\nabla {P_2}} \right) + Pr{\nabla ^2}{{\boldsymbol{U}}_2} -\hfill\\ & \quad \frac{{Pr}}{{Da}}\left( {{{\hat \varepsilon }_0}{{\boldsymbol{U}}_2} + {{\hat \varepsilon }_1}{{\boldsymbol{U}}_1} + {{\hat \varepsilon }_2}{{\boldsymbol{U}}_0} + 4{{\hat \varepsilon }_2}{{\boldsymbol{U}}_2}} \right) -\hfill \\ & \quad \frac{{1.75\left\| {{{\boldsymbol{U}}_2}} \right\|}}{{\sqrt {150Da} }}\left( {{{\tilde \varepsilon }_0}{{\boldsymbol{U}}_2} + {{\tilde \varepsilon }_1}{{\boldsymbol{U}}_1} + {{\tilde \varepsilon }_2}{{\boldsymbol{U}}_0} + 4{{\tilde \varepsilon }_2}{{\boldsymbol{U}}_2}} \right)- \hfill \\ & \quad RaPr\left( {{\varepsilon _0}{\theta _2} + {\varepsilon _1}{\theta _1} + {\varepsilon _2}{\theta _0} + 4{\varepsilon _2}{\theta _2}} \right){{\boldsymbol{e}}_{{g}}} -\hfill \\ & \quad \gamma RaPr\nabla {{\boldsymbol{B}}^2}\left( {{\varepsilon _0}{\theta _2} + {\varepsilon _1}{\theta _1} + {\varepsilon _2}{\theta _0} + 4{\varepsilon _2}{\theta _2}} \right) \hfill \end{split} \tag{A8}$$ $$ \begin{split} & \frac{{\partial {\theta _2}}}{{\partial \tau }} + \left( {{{\boldsymbol{U}}_0} \cdot \nabla {\theta _2} + {{\boldsymbol{U}}_1} \cdot \nabla {\theta _1}} + {{\boldsymbol{U}}_2} \cdot \nabla {\theta _0} \right. +\hfill \\ & \quad \left. 4{{\boldsymbol{U}}_2} \cdot \nabla {\theta _2} \right) = {\nabla ^2}{\theta _2} \hfill \end{split} \tag{A9}$$ -
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图 3 孔隙率KL展开中协方差核特征值
Figure 3. Eigenvalues of covariance kernel used in the KL expansion of porosity
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图 4 孔隙率KL展开中协方差核特征函数
Figure 4. Eigenfunction of covariance kennel used in the KL expansion of porosity
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图 4 孔隙率KL展开中协方差核特征函数(续)
Figure 4. Eigenfunction of covariance kennel used in the KL expansion of porosity (continued)
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图 5 不同不确定性量化方法下流函数与温度场均值
Figure 5. Mean value of stream function and temperature field with different uncertainty quantification methods
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图 6 不同不确定性量化方法下流函数与温度场标准偏差
Figure 6. Standard deviation of stream function and temperature field with different uncertainty quantification methods
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图 7 不同不确定性量化方法下热壁面局部Nusselt数分布
Figure 7. Distribution of local Nusselt number at the hot wall with different uncertainty quantification methods
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图 8 不同不确定性量化方法下热壁面平均Nusselt数概率密度函数和累积分布函数
Figure 8. Probability density function and cumulative distribution function of average Nusselt number at hot wall with different uncertainty quantification methods
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图 11 流函数与温度场二阶模式
Figure 11. Second-order modes of stream function and temperature field
表 1 多项式混沌展开阶数对平均Nusselt数均值与标准偏差的影响
Table 1 Influence of order of polynomial chaos expansion on mean value and standard deviation of average Nusselt number
Order of polynomial chaos expansion Mean value of average
Nusselt numberStandard deviation of average Nusselt number PCE MC relative error/% PCE MC relative error/% n = 1 3.3074 3.2161 2.84 0.4286 0.4187 2.36 n = 2 3.2029 3.2161 −0.41 0.4136 0.4187 −1.23 PCE: polynomial chaos expansion; MC: Monte Carlo 
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