We propose and analyse a novel, fully discrete numerical algorithm for the approximation of the generalised Stokes system forced by transport noise -- a prototype model for non-Newtonian fluids including turbulence. Utilising the Gradient Discretisation Method, we show that the algorithm is long-term stable for a broad class of particular Gradient Discretisations. Building on the long-term stability and the derived continuity of the algorithm's solution operator, we construct two sequences of approximate invariant measures. At the moment, each sequence lacks one important feature: either the existence of a limit measure, or the invariance with respect to the discrete semigroup. We derive an abstract condition that merges both properties, recovering the existence of an invariant measure. We provide an example for which invariance and existence hold simultaneously, and characterise the invariant measure completely. We close the article by conducting two numerical experiments that show the influence of transport noise on the dynamics of power-law fluids; in particular, we find that transport noise enhances the dissipation of kinetic energy, the mixing of particles, as well as the size of vortices.

翻译：本文提出并分析了一种新颖的全离散数值算法，用于逼近由输运噪声驱动的广义斯托克斯系统——这是包含湍流在内的非牛顿流体原型模型。利用梯度离散化方法，我们证明了该算法对于一大类特定梯度离散化方案具有长期稳定性。基于长期稳定性及所推导的算法解算子的连续性，我们构造了两组近似不变测度序列。目前，每个序列都缺少一个重要特征：要么是极限测度的存在性，要么是关于离散半群的不变性。我们推导了一个抽象条件，融合了这两个性质，从而恢复了不变测度的存在性。我们提供了一个不变性与存在性同时成立的示例，并完整刻画了该不变测度。文章最后通过两个数值实验展示了输运噪声对幂律流体动力学的影响；特别地，我们发现输运噪声增强了动能的耗散、粒子的混合以及涡旋的尺度。