Analyzing and Modeling Turbulence Using Wavelet

Turbulence has traditionally been analyzed through a Fourier-scale framework. However, the finite spatial correlation of turbulent motions calls for a fundamentally different perspective—one in which scale and spatial interactions must be described simultaneously. This talk demonstrates how wavelet analysis can transform the traditional paradigm of turbulence analysis and modeling.

In turbulent thermal convection, a balance between inertial and thermal energy transfer leads to the Bolgiano–Obukhov scaling (BO59). However, most prior studies have reported Kolmogorov scaling (K41) except in near-wall regions. Wavelet multiresolution analysis (WMRA) is employed to seek direct evidence of BO59. While the linear correlation between cross-scale energy flux and thermal energy flux is weak, supporting K41, their mutual information is significant. This nonlinear statistical dependence suggests the potential existence of BO59 toward the center of the convection cell.

The good spectral and spatial resolution of wavelet is leveraged to discover subgrid-scale (SGS) models. In large-eddy simulation (LES), SGS models are typically prescribed a priori based on assumptions (e.g., eddy-viscosity hypothesis). Yet, turbulence generation, interscale transfer, and SGS dissipation vary substantially across flows, often leading to O(1) modeling errors. This talk presents a procedure for discovering the functional form of SGS models using high-fidelity direct numerical simulation (DNS) data. An optimization problem is formulated to weakly enforce a spectral optimality condition for the inertial energy transfer across the LES grid cutoff scale and the modeled SGS work. Results for homogeneous isotropic turbulence (HIT) and Burgers turbulence are presented, including a posteriori LES validation.