Application of macroscopic forcing method (MFM) for revealing turbulence closure model requirements
Keywords: Mechanical Engineering.

For many engineering applications, it is computationally efficient and sufficient to model the system based only on averaged quantities. The Reynolds-Averaged Navier-Stokes (RANS) equations govern the evolution of flow momentum fields in this averaged space. However, these equations are not fully determined due to the closure problem associated with the turbulent advective transport of momentum. To utilize RANS equations, one needs models that express the Reynolds stresses in terms of an explicit or implicit function of the mean velocity field. In this thesis, we utilize the Macroscopic Forcing Method (MFM) (Mani and Park 2021), a novel computational method for directly measuring the closure operator associated with transport of scalar and vector fields via flow advection. We choose Homogeneous Isotropic Turbulence (HIT) as the canonical setting to apply MFM to study the transport of both passive scalar and vector momentum fields. From the obtained closure operator for each problem, after some mathematical manipulation, we can determine the eddy diffusivity operator. This differential operator acts on the mean field gradients and outputs the unclosed fluxes associated with the advective transport of the fields of interest. Our results indicate that the eddy diffusivity operator in HIT is scale-dependent and non-local. In the limit of large scales, when the wavenumber associated with the mean field is much smaller than the large eddy wavenumber, the eddy diffusivity operator is constant and consistent with the Boussinesq approximation. However, in the limit of large wavenumbers, corresponding to scales smaller than the large eddy, the eddy diffusivity vanishes inversely proportional to the wavenumber. An extension of our analysis of the closure operator is also developed to determine temporal non-locality. The resulting operator is appropriate for ensemble-averaged transport processes involving time-varying transient effects. We consider the leading terms in a Kramers-Moyal expansion of the eddy diffusivity operator for passive scalars and quantitatively determine a relaxation time scale over which the mean turbulent flux of the scalar quantity relaxes to the quasi-steady values determined by the prior steady analysis. This time scale is wavenumber dependent and is found to be inversely proportional to the wavenumber in the limit of small scales. Further, we use MFM as a lens through which we can assess different large eddy simulation (LES) operators. We use MFM to project the LES and Direct Numerical Simulation (DNS) equations onto the averaged space and obtain the respective RANS closure operators of both systems. Contrasting the RANS closure operators with DNS and LES fields, we show that while the standard Smagorinsky model performs reasonably at low wavenumbers, it is overly dissipative in the high-wavenumber limit. Inspired by the aforementioned RANS results, we tested a modified LES closure operator with proper scale dependent form and demonstrated substantial improvements in capturing mean mixing at high wavenumbers. To build confidence in our MFM results, we conducted a series of simulations of HIT and channel flow to investigate the time integration requirements that allow reliable reporting of the statistical error associated with finite time simulations. We show that in the absence of these requirements, and under traditionally adopted temporal integration windows, the reported confidence interval can be mispredicted by about a factor of two. Enforcing the criteria obtained in our analysis eliminates this discrepancy and allows for accurate confidence intervals for simulations of turbulent flows