Investigation of numerical viscosities and dissipation rates of shock-capturing schemes for implicit large-eddy simulation
Date of Award
Doctor of Philosophy (PhD)
Mechanical and Aerospace Engineering
Turbulence--Mathematical models., Eddies--Mathematical models.
In recent times, the limitations of popular subgrid-scale models for Large-Eddy Simulation have led to the development of a number of sophisticated modeling approaches. An approach that utilizes the numerical viscosity arising from the non-linear discretization of convective fluxes to model the smaller more isotropic eddies has received considerable attention. This method is referred to as Implicit Large-Eddy Simulation (ILES). The goal of this dissertation is to advance current understanding of a class of ILES techniques based on shock-capturing schemes. This is achieved through intricate quantitative insights into the stability and diffusive characteristics of Roe-MUSCL-based shock-capturing scheme. The first step towards achieving the goal involved developing a formulation to quantify the numerical viscosity of the Roe-MUSCL scheme for the case of non-linear advection equation. The expression derived is a function of the flux limiter employed, distance between cell centers on either side of a face, face-normal velocity and a scaling factor. The significance of the scaling factor is revealed when the Roe-MUSCL scheme, originally developed for 1-D scenarios, is applied to 2-D scalar advection problems. It is seen that without the scaling factor, the MUSCL scheme may not necessarily be monotonic in multi-dimensional scenarios. In the next step, Roe's original shock-capturing scheme developed for 1-D Euler equations is extended to three dimensions in a manner that is consistent with the finite volume framework. After extending the scheme to second-order using van Leer's MUSCL extrapolation technique, an expression for numerical viscosity is derived following the procedure outlined for the advection equation case. To minimize the excessive dissipation of Roe-MUSCL for turbulent flows, the high numerical viscosity is mitigated using a z-factor that depends on the local Mach number. In order to demonstrate the performance of Roe-MUSCL in capturing the physics of turbulent flows, a detailed investigation is conducted for decaying homogeneous isotropic turbulence with varying degrees of compressibility. The spectral profiles of numerical viscosity and dissipation rate demonstrate the effectiveness of the z-factor both in narrowing the wavenumber range in which dissipation occurs, and in shifting the location of the dissipation peak closer to the cut-off wavenumber. A number of statistical parameters for mesh resolution greater than 32^3 showed good agreement with prior direct numerical simulation and experimental studies. In the final step, Roe-MUSCL together with three Runge-Kutta temporal schemes is investigated for the Taylor-Green vortex transitional flow problem. The spectral profiles of numerical dissipation rate show that the Runge-Kutta scheme of Shu-Osher is the most stable and together with Roe-MUSCL accurately captures the physics of vortex stretching and subsequent production of dissipative eddies.
Bidadi, Shreyas, "Investigation of numerical viscosities and dissipation rates of shock-capturing schemes for implicit large-eddy simulation" (2015). Dissertations. 80.