Helicity and Physical Fidelity in Turbulence Modeling

This thesis is a study of physical fidelity in turbulence modeling. We first consider conservation laws in several popular turbulence models and find that of the Leray, Leray-deconvolution, Bardina and Stolz-Adams approximate deconvolution model (ADM), all but the Bardina model conserve a model energy. Only the ADM conserves a model helicity. Since the ADM conserves a model energy and helicity, we then investigate a joint helicity-energy spectrum in the ADM. We find that up to a filter-dependent length scale, the ADM cascades energy and helicity jointly in the same manner as the Navier-Stokes equations.We also investigate helicity treatment in discretizations of turbulence models. For inviscid, periodic flow, we implement energy conserving discretizations of the ADM, Leray, and Leray-deconvolution models as well as the Navier-Stokes equations (NSE) and observe helicity treatments. We find that of none of the models conserve helicity (or model helicity) in the discretizations. Since the Leray-deconvolution model of turbulence is newly developed, our implementation is new and thus we analyze the trapezoidal Galerkin scheme that we implement and compare it to the usual Leray model.Lastly, we develop an energy and helicity conserving trapezoidal Galerkin scheme for the Navier-Stokes equations. We prove conservation properties for the scheme, stability, and show the scheme does not lose asymptotic accuracy compared to the usual trapezoidal Galerkin scheme. We also present numerical experiments that compare the energy and helicity conserving scheme to more typical schemes