The filtered advection-diffusion equation : Lagrangian methods and modeling

Citation

Moeleker, Piet (2000) The filtered advection-diffusion equation : Lagrangian methods and modeling. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/2q17-4g27. https://resolver.caltech.edu/CaltechTHESIS:10062010-114741410

Abstract

This research focuses on the incompressible scalar advection-diffusion equation. After applying a Gaussian filter, an infinite series expansion is found for the advection term to obtain a closed equation. Only the first two terms in this expansion are retained yielding the tensor-diffusivity subgrid model. This model can be interpreted as a tensor-diffusivity term which is proportional to the rate-of-strain tensor of the large-scale filtered velocity field. Due to the negative diffusion in the stretching directions, care needs to be taken in the choice of a numerical method. The scalar field is decomposed in a collection of anisotropic or axisymmetric Gaussian particles. Equations of motion for the location and the shape/size of the particles are derived using an expansion in Hermite polynomials. A novel, accurate remeshing scheme was found resulting in explicit expressions for the amplitudes of the new set of particles. A stagnation flow was used for illustrative purposes and validation. Using a 2D time-dependent velocity field yielding chaotic advection, both axisymmetric and anisotropic particles yield good agreement with filtered direct numerical simulations and compare favorably with the Smagorinsky subgrid model. Computational efficiency makes axisymmetric particles the preferred choice. A literature study using a 3D stationary one-parameter chaotic velocity field was used to validate model and particle-method in 3D. For highly chaotic fields good agreement was obtained with this study. Computations have been performed for 3D forced isotropic periodic turbulence to study scalar mixing. Comparisons with literature are made. It was shown that when the unfiltered velocity field is known, the most accurate results are obtained by moving particles using this field. It was concluded that a good subgrid model modifies the equation of motion to get a good approximation to the unfiltered velocity field.

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