Stochastic Backscatter Model for Unstructured Solvers

TL;DR

The paper tackles extending stochastic backscatter to unstructured CFD solvers by analytically deriving the spatial correlation induced by Laplacian smoothing. It develops the 3D Green's function for $(1+b^2\nabla^2)^{-1}$ via Fourier methods and expresses the smoothed field as a convolution with this Green's function, yielding a closed-form, exponentially decaying spatial correlation kernel with decay length $b$. The key result is $\langle dW_i(\mathbf x,t)dW_j(\mathbf y,s)\rangle = \frac{8}{\pi}\delta_{ij}\delta(t-s) e^{-|\mathbf x-\mathbf y|/b}$, establishing analytically tractable spatial statistics for unstructured grids. This work enables applying SBS in unstructured solvers and provides a principled link between the LES filter width and the stochastic forcing's spatial correlation.

Abstract

The Stochastic Backscatter Model involves the generation of a set of random variables characterised by prescribed correlations in space and time. These variables are obtained by smoothing an initially uncorrelated random field, which produces an exponentially decaying spatial correlation. The smoothing is applied implicitly and sequentially along the three coordinate directions of the computational domain, making the approach suitable only for structured CFD solvers. To extend the method to unstructured solvers, implicit Laplace smoothing can be employed instead. However, the spatial correlation resulting from this alternative approach must be derived analytically. This constitutes the main objective of the present dissertation.