Heat transfer in drop-laden turbulence

Abstract

Heat transfer by large deformable drops in a turbulent flow is a complex and rich-in-physics system, in which drop deformation, breakage and coalescence influence the transport of heat. We study this problem by coupling direct numerical simulation (DNS) of turbulence with a phase-field method for the interface description. Simulations are run at fixed-shear Reynolds and Weber numbers. To evaluate the influence of microscopic flow properties, like momentum/thermal diffusivity, on macroscopic flow properties, like mean temperature or heat transfer rates, we consider four different values of the Prandtl number, which is the momentum to thermal diffusivity ratio: $Pr=1$ , $Pr=2$ , $Pr=4$ and $Pr=8$ . The drop volume fraction is $\varPhi \simeq 5.4\,\%$ for all cases. Drops are initially warmer than the turbulent carrier fluid and release heat at different rates depending on the value of $Pr$ , but also on their size and on their own dynamics (topology, breakage, drop–drop interaction). Computing the time behaviour of the drops and carrier fluid average temperatures, we clearly show that an increase of $Pr$ slows down the heat transfer process. We explain our results by a simplified phenomenological model: we show that the time behaviour of the drop average temperature is self-similar, and a universal behaviour can be found upon rescaling by $t/Pr^{2/3}$ . Accordingly, the heat transfer coefficient $\mathcal {H}$ (respectively its dimensionless counterpart, the Nusselt number $Nu$ ) scales as $\mathcal {H}\sim Pr^{-2/3}$ (respectively $Nu\sim Pr^{1/3}$ ) at the beginning of the simulation, and tends to $\mathcal {H}\sim Pr^{-1/2}$ (respectively $Nu\sim Pr^{1/2}$ ) at later times. These different scalings can be explained via the boundary layer theory and are consistent with previous theoretical/numerical predictions.