Wavelet multiresolution analysis is extended to describe interphase, cross-scale interactions involving turbulence kinetic energy (TKE) of particle-laden turbulence. Homogeneous isotropic turbulence (HIT) suspended with inertial particles at the Stokes number of unity (critical particles) is analyzed. Direct numerical simulation is performed for decaying HIT coupled via the Stokes drag law in two ways with the dispersed phase. The effects of two-way coupling on spectral TKE transfer are examined. Clustering of the critical particles in thin, filamentlike regions is observed, and mean wavelet statistics are similar to those of the Fourier analysis. However, spatially local wavelet analysis demonstrates the complexities of the preferential concentration in interphase, interscale energy transfer and associated challenges in subgrid-scale (SGS) modeling of particle-laden turbulence. Two-way coupling enhances correlations between local particle concentration and local interphase TKE transfer. However, particle concentration alone does not indicate a definite direction of interphase energy transfer. Rather, particle clusters behave as an energy source or sink with similar probabilities. A similar argument is made for correlations between particle concentration and cross-scale energy transfer. Wavelet statistics conditioned on coarse-grained number density support the same conclusions. In addition, the joint statistics show the qualitative consistency of the SGS Stokes number in describing the two-way interactions, which should be considered in the SGS modeling of two-way coupled particle-laden turbulence.
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