Kamiński Marcin, Ossowski Rafał Leszek
Faculty of Civil Engineering, Architecture and Environmental Engineering, Lodz University of Technology, 90-924 Łódź, Poland.
Entropy (Basel). 2025 Jan 14;27(1):67. doi: 10.3390/e27010067.
The main aim of this study is to achieve the numerical solution for the Navier-Stokes equations for incompressible, non-turbulent, and subsonic fluid flows with some Gaussian physical uncertainties. The higher-order stochastic finite volume method (SFVM), implemented according to the iterative generalized stochastic perturbation technique and the Monte Carlo scheme, are engaged for this purpose. It is implemented with the aid of the polynomial bases for the pressure-velocity-temperature (PVT) solutions, for which the weighted least squares method (WLSM) algorithm is applicable. The deterministic problem is solved using the freeware OpenFVM, the computer algebra software MAPLE 2019 is employed for the LSM local fittings, and the resulting probabilistic quantities are computed. The first two probabilistic moments, as well as the Shannon entropy spatial distributions, are determined with this apparatus and visualized in the FEPlot software. This approach is validated using the 2D heat conduction benchmark test and then applied for the probabilistic version of the 3D coupled lid-driven cavity flow analysis. Such an implementation of the SFVM is applied to model the 2D lid-driven cavity flow problem for statistically homogeneous fluid with limited uncertainty in its viscosity and heat conductivity. Further numerical extension of this technique is seen in an application of the artificial neural networks, where polynomial approximation may be replaced automatically by some optimal, and not necessarily polynomial, bases.
本研究的主要目的是求解具有高斯物理不确定性的不可压缩、非湍流和亚音速流体流动的纳维-斯托克斯方程的数值解。为此采用了基于迭代广义随机扰动技术和蒙特卡罗方法实现的高阶随机有限体积法（SFVM）。它借助于压力-速度-温度（PVT）解的多项式基来实现，加权最小二乘法（WLSM）算法适用于该多项式基。确定性问题使用免费软件OpenFVM求解，利用计算机代数软件MAPLE 2019进行最小二乘法局部拟合，并计算得到的概率量。利用该装置确定前两个概率矩以及香农熵空间分布，并在FEPlot软件中进行可视化。该方法通过二维热传导基准测试进行验证，然后应用于三维耦合驱动腔流动分析的概率版本。这种SFVM的实现方式被用于模拟具有有限粘度和热导率不确定性的统计均匀流体的二维驱动腔流动问题。该技术在人工神经网络的应用中得到了进一步的数值扩展，其中多项式逼近可能会被一些最优的（不一定是多项式的）基自动取代。
