Self-similar Heat Transfer in a Turbulent Particle-laden Free Flow

The inherently multiscale nature of fluid turbulence plays a major role in most natural and engineered systems, so that understanding and modeling turbulence at different scales is crucial for many applications, ranging from improving aerodynamic performance to optimizing industrial processes. When particles are suspended in a fluid, their interactions with the turbulent flow structures result in modified flow patterns, altered energy transfer mechanisms, and the emergence of new scales, making the understanding and modeling of particle-laden turbulent flows a challenging and active area of research. Present study focuses on analyzing the fluid-particle thermal interaction in a particle-laden turbulent flow, which recently has received attention [1, 2, 3], and for which experiments cannot provide a detailed picture. Specifically, we investigate a basic archetypal non-isothermal flow configuration, where an initial temperature step between two isothermal regions is advected by a turbulent, isotropic solenoidal velocity field. In this scenario, the flow is seeded with a set of identical inertial particles smaller than the Kolmogorov length scale. Utilizing the formalism introduced by Pousanrari and Mani [2] to represent particle dynamics, we demonstrate that the advection by the turbulent velocity field produces a self-similar evolution of both fluid and particle statistics for any particle inertia. Remarkably, when rescaled with a single length scale deduced from the mean temperature, all temperature and velocity-temperature moments collapse. This length scale shows a t^1/2 diffusive growth. The outcome of such analysis with a set of Direct Numerical Simulations (DNS) at moderate Reynolds number, carried out by using the Eulerian-Lagrangian point particle model, valid for small heavy sub-Kolmogorov particles, in both one-way coupling and two-way thermal coupling regimes. We considered a wide range of particle Stokes numbers, from 0.1 to 3, and a Taylor microscale Reynolds number up to 124. The thermal Stokes number to Stokes number ratio, which depends only on the ratio between the particle specific heat and fluid specific heat, is kept constant and equal to 4.43, which is representative of water droplets in air. The simulations allow to confirm self-similarity up to third order moments and the t^1/2 scaling, as in [4]. Moreover, the DNS data allow to complement the self-similar analysis by providing quantitative values for the phenomenological coefficients of the unclosed terms in Pouransari and Mani approach. The implications on turbulent modelling will be discussed. Overall, our analysis allows to quantify the role of particle inertia and thermal inertia on the ability of turbulence to transport heat, and to highlight the effect of the modulation of the carrier flow temperature fluctuations by particle thermal feedback.