We address the numerical challenge of solving the Hookean-type time-fractional Navier--Stokes--Fokker--Planck equation, a history-dependent system of PDEs defined on the Cartesian product of two $d$-dimensional spaces in the turbulent regime. Due to its high dimensionality, the non-locality with respect to time, and the resolution required to resolve turbulent flow, this problem is highly demanding. To overcome these challenges, we employ the Hermite spectral method for the configuration space of the Fokker--Planck equation, reducing the problem to a purely macroscopic model. Considering scenarios for available analytical solutions, we prove the existence of an optimal choice of the Hermite scaling parameter. With this choice, the macroscopic system is equivalent to solving the coupled micro-macro system. We apply second-order time integration and extrapolation of the coupling terms, achieving, for the first time, convergence rates for the fully coupled time-fractional system independent of the order of the time-fractional derivative. Our efficient implementation of the numerical scheme allows turbulent simulations of dilute polymeric fluids with memory effects in two and three dimensions. Numerical simulations show that memory effects weaken the drag-reducing effect of added polymer molecules in the turbulent flow regime.

翻译：我们致力于解决Hookean型时间分数阶Navier-Stokes-Fokker-Planck方程的数值求解挑战，这是一个定义在两个$d$维空间笛卡尔积上的历史依赖偏微分方程组，适用于湍流流动体系。由于其高维度、时间非局部性以及解析湍流所需的高分辨率，该问题具有极高的计算复杂度。为克服这些挑战，我们对Fokker-Planck方程的构型空间采用Hermite谱方法，将问题简化为纯宏观模型。通过分析可获得解析解的场景，我们证明了Hermite尺度参数最优选择的存在性。采用该参数时，宏观系统等价于求解耦合的微观-宏观系统。我们应用二阶时间积分及耦合项外推技术，首次实现了完全耦合时间分数阶系统的收敛速率与时间分数阶导数阶数无关的突破。该数值方案的高效实现使我们能够在二维和三维空间中对具有记忆效应的稀薄聚合物流体进行湍流模拟。数值模拟结果表明，在湍流流动体系中，记忆效应会削弱添加聚合物分子所产生的减阻效果。