Motivated by the modeling of the spatial structure of the velocity field of three-dimensional turbulent flows, and the phenomenology of cascade phenomena, a linear dynamics has been recently proposed able to generate high velocity gradients from a smooth-in-space forcing term. It is based on a linear Partial Differential Equation (PDE) stirred by an additive random forcing term which is delta-correlated in time. The underlying proposed deterministic mechanism corresponds to a transport in Fourier space which aims at transferring energy injected at large scales towards small scales. The key role of the random forcing is to realize these transfers in a statistically homogeneous way. Whereas at finite times and positive viscosity the solutions are smooth, a loss of regularity is observed for the statistically stationary state in the inviscid limit. We here present novel simulations, based on finite volume methods in the Fourier domain and a splitting method in time, which are more accurate than the pseudo-spectral simulations. We show that the novel algorithm is able to reproduce accurately the expected local and statistical structure of the predicted solutions. We conduct numerical simulations in one, two and three spatial dimensions, and we display the solutions both in physical and Fourier spaces. We additionally display key statistical quantities such as second-order structure functions and power spectral densities at various viscosities.

翻译：受三维湍流速度场空间结构建模及级联现象现象学的启发，近期提出了一种能从空间光滑强迫项产生高速度梯度的线性动力学机制。该机制基于一个由时间δ相关加性随机强迫项驱动的线性偏微分方程。其所提出的确定性机制对应于傅里叶空间中的输运过程，旨在将大尺度注入的能量向小尺度传递。随机强迫项的关键作用在于以统计均匀的方式实现这些能量传递。尽管在有限时间及正粘度条件下解是光滑的，但在无粘极限下统计稳态会观测到正则性丧失。本文提出基于傅里叶域有限体积方法及时间分裂方法的新型数值模拟，其精度优于伪谱模拟。我们证明该新算法能准确再现预测解的局部结构与统计特性。我们在1维、2维及3维空间中进行数值模拟，并展示物理空间与傅里叶空间中的解。此外，我们展示了关键统计量，如不同粘度下的二阶结构函数与功率谱密度。