Gabor mode enrichment in large eddy simulation of turbulent flows
Keywords: Aeronautics and Astronautics.

Buoyed by the rise in computational capabilities, eddy resolving simulation methodologies such as Large Eddy Simulation (LES), first developed in 1970s, have now come to dominate academic research in numerical simulation of turbulent flows. Wall-modeled LES (WMLES) which alleviates the no-slip boundary condition at walls with a stress condition computed using either a physics based or a mathematically derived boundary condition based on the filtering kernel, has further enabled the use of LES in applications involving very high Reynolds number wall bounded flows. Motivated by this growing success and push for use of WMLES in both engineering and geophysics communities, we address a new emerging challenge in this field: Given a robust characterization of the large scales, can we come up with a reasonably consistent synthesis of the unresolved/subfilter scales at low computational cost? This is a particularly pertinent question in wall bounded flows since several applications that involve unsteady loads, surface vibrations, acoustic radiation, etc. require near wall representation of turbulent scales with a spatio-temporal bandwidth substantially larger than that implied by WMLES. This problem naturally arises in wind energy applications where the Planetary Boundary Layer (PBL) turbulence at extreme Reynolds numbers can only be resolved up to scales much larger than the chord lengths of wind turbine blades. Other multiphysics applications such as particle laden turbulence also require spectral resolution of the explicitly resolved fluid turbulence to be larger than what is typically available in LES. The method developed in this thesis is based on combining a scale-resolving turbulence simulation (such as WMLES) with limited bandwidth, with an efficient, physics-based scale-enrichment model. The enrichment scheme generates a physically consistent realization of the finer-scale eddies locally which dynamically respond to the resolved larger scales and enable statistical predictions of physical quantities with a substantially larger bandwidth. This physics based model is carefully constructed to obtain small scales with accurate second-order space-time correlations. By representing the small scales using \emph{Gabor modes/wavepackets}, there is a substantial compression in degrees of freedom. Furthermore, the use of WKB-variant of Rapid Distortion Theory (RDT) along with a spectral viscosity closure allows for temporal evolution of the Gabor modes via a set of ODEs described in an x-k frame. Use of spatially and spectrally localized Gabor modes ensures correct inter-scale energy transfer and allows for treatment of spatial inhomogeneities. The transform of physical field variables from their Gabor modes representation to the physical space is accomplished using modern non-uniform FFT algorithms thereby achieving an overall nlog(n) computational cost. Robustness of this new method is assessed on various wall-bounded flow configurations (including a problem with Coriolis acceleration and stratification) using a representation of large scales obtained using both ideal WMLES (filtered DNS) and realistic/true WMLES simulations at high Reynolds numbers. While the enrichment of large scales obtained via idealized LES is excellent, the results for realistic LES are also very promising. Finally, we discuss a highly idealized problem to study the interaction of free-stream (isotropic) turbulence with a turbulent wake generated by an actuator disk. Beyond characterizing the flow physics in detail, we present evidence suggesting that the recovery and entrainment in the wake is much more sensitive to the integral length scale of the ambient turbulence, than its intensity. Furthermore, the incident flow seen by a downstream object present in the wake, has no low-rank behavior in terms of space-time coherence. We subsequently demonstrate that a truncated reconstruction of the flow field using space-time POD modes can be enriched with fine scales using Gabor modes to recover the full bandwidth of the high resolution simulation.