On scale-invariance based models of hydrodynamic turbulence

A. Scotti
PhD Thesis, The Johns Hopkins University
August 1997
Baltimore MD

ABSTRACT: This thesis explores some consequences of employing the concept of scale-invariance in turbulent flows to parametrize subgrid scales in Large Eddy Simulation (LES). The thesis is divided in two parts. In the first part, we study the problem of performing LES on complex, anisotropic grids in conjunction with the dynamic eddy viscosity model. The idea of scale--invariance is used in this context to relate the unknown coefficients at different scales, via the Germano identity. The effects of the computational grid's anisotropy are considered explicitly, and it is shown that the model coefficients acquire a dependence on the aspect ratios of the grid. The formulation is tested in LES of steady isotropic turbulence on highly anisotropic discretizations. Among others, the simulations show that partial two dimensionalization of the flow in the case of pencil-like anisotropic grids causes significant problems for the dynamic model, but that isotropic test-filtering can be used to remedy the problem. In the second part of the thesis, the scale-invariance hypothesis is extended much further, and a new class of LES closures is developed, based on the construction of synthetic, fractal subgrid- scale fields. The relevant mathematical tool is known as fractal interpolation, which interpolates the resolved velocity with fields that have fluctuations down to much smaller scales. The idea is first applied to a 1-D case, namely the forced Burgers equation, and is then extended to cover the 3-D case. First, the extension of fractal interpolation to the 3-D case is derived (in a way that allows a trivial extension to n dimensions), then it is used to formulate a subgrid closure. The model is applied to LES of both steady and freely decaying isotropic turbulence. We find that the assumption of fractality per se is not enough to yield physically meaningful results, and explore several variants of the model in which the rules to generate the synthetic fields explicitly incorporate the condition that energy dissipation take place. Good results are obtained only once the fractal dimension is allowed to vary in different eigendirections of the resolved strain-rate tensor so as to (nearly) maximize energy dissipation.