Spectral analysis of energy transfer in turbulent flows laden with heated particles

In this study we consider particle-laden turbulent flows with significant heat transfer between the two phases due to sustained heating of the particle phase. The sustained heat source can be due to particle heating via an external radiation source as in the particle-based solar receivers or an exothermic reaction in the particles. Our objective is to investigate the effects of fluid heating by a dispersed phase on the turbulence evolution. An important feature in such settings is the preferential clustering phenomenon which is responsible for non-uniform distribution of particles in the fluid medium. Particularly, when the ratio of particle inertial relaxation time to the turbulence time scale, namely the Stokes number, is of order unity, particle clustering is maximized, leading to thin regions of heat source similar to the flames in turbulent combustion. However, unlike turbulent combustion, a particle-laden system involves a wide range of clustering scales that is mainly controlled by particle Stokes number. To study these effects, we considered a decaying homogeneous isotropic turbulence laden with heated particles over a wide range of Stokes numbers. Using a low-Mach-number formulation for the fluid energy equation and a Lagrangian framework for particle tracking, we performed numerical simulations of this coupled system. We then applied a high-fidelity framework to perform spectral analysis of kinetic energy in a variable-density fluid. Our results indicate that particle heating can considerably influence the turbulence cascade. We show that the pressure-dilatation term introduces turbulent kinetic energy at a range of scales consistent with the scales observed in particle clusters. This energy is then transferred to high wavenumbers via the energy transfer term. For low and moderate levels of particle heating intensity, quantified by a parameter $\unicode[STIX]{x1D6FC}$ defined as the ratio of eddy time to mean temperature increase time, turbulence modification occurs primarily in the dilatational modes of the velocity field. However, as the heating intensity is increased, the energy transfer term converts energy from dilatational modes to divergence-free modes. Interestingly, as the heating intensity is increased, the net modification of turbulence by heating is observed dominantly in divergence-free modes; the portion of turbulence modification in dilatational modes diminishes with higher heating. Moreover, we show that modification of divergence-free modes is more pronounced at intermediate Stokes numbers corresponding to the maximum particle clustering. We also present the influence of heating intensity on the energy transfer term itself. This term crosses over from negative to positive values beyond a threshold wavenumber, showing the cascade of energy from large scales to small scales. The threshold is shown to shift to higher wavenumbers with increasing heating, indicating a growth in the total energy transfer from large scales to small scales. The fundamental energy transfer analysis presented in this paper provides insightful guidelines for subgrid-scale modelling and large-eddy simulation of heated particle-laden turbulence.