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

Abstract

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.

Constant-energetics physical-space forcing methods for improved convergence to homogeneous-isotropic turbulence with application to particle-laden flows

“…Similar conclusions are obtained by using Eq. (10), in that the constant dissipation is established rapidly in Fig. 1(b) leading to oscillations in the turbulent kinetic energy which are nonetheless comparable to those obtained using (6) and much less accentuated than those arising from (5).…”

“…The instantaneous kinetic-energy spectra obtained at short times using the proposed coefficients (7) and (10), evaluated at t = 5.7t ℓ,∞ and t = 7.1t ℓ,∞ , respectively, lie within the envelope represented by the maximum and minimum values resulting from employing (5) during long periods of time t = 43t ℓ,∞ − 72t ℓ,∞ , as shown in Fig. 1(c).…”

“…The Navier-Stokes equations (1) and (2) are solved with a finite-difference energy-conserving discretization of second order central in space and fourth-order Runge-Kutta in time. 10,11 A. Convergence in physical and spectral spaces Comparisons are made in Fig. 1 between DNS performed using the forcing coefficients outlined above, showing that the proposed coefficients (7) and (10) generally lead to steadier statistics and faster convergence in comparison to the methods (5) and (6) from Refs.…”

“…In the simulations, Eqs. (1) and (2), along with the Lagrangian particle equations (18) and (19), are integrated subjected to an initial velocity field of homogeneous-isotropic turbulence extracted at t = 43t ℓ,∞ from the DNS performed in Section III for each one of the four forcing approaches (5), (6), 7, and (10). A mid-plane cross section of the simulations is presented in Fig.…”

Spatially localized multi-scale energy transfer in turbulent premixed combustion

“…Elaborate spectral descriptions of compressible turbulent flows in Fourier space exist under the framework of Craya-Herring, Helmholtz or Moyal's decompositions of the velocity field into solenoidal and irrotational components by convenient projections of the velocity field onto parallel and perpendicular directions to the wavenumber vector (Moyal 1952;Craya 1958;Sagaut & Cambon 2008;Pouransari et al 2017). The pursuit of these classic alternatives in spectral space, however, offers no advantage with the types of wavelet bases regularly employed in turbulent flow analyses, in that the solenoidal or irrotational conditions do not translate into any useful relationship for the wavelet-transformed velocity field.…”