The ability to control the distribution of particles in a fluid is generally regarded as a factor of great importance in a variety of fields (manufacturing processes, biomedical applications, materials engineering and various particle separation processes, to cite a few). The present study considers the hitherto not yet addressed situation in which solid spherical particles are dispersed in a non-isothermal fluid undergoing turbulent vibrationally-induced convection (chaotic thermovibrational flow in a square cavity due to vibrations perpendicular to the imposed temperature difference). Although the possibility to use laminar thermovibrational flows (in microgravity) and turbulent flows of various types (in normal gravity conditions) to induce the accumulation of solid mass dispersed in a non-isodense fluid is already known, the interplay of finite-size finite-mass particles with chaotic flow in weightlessness conditions has never been considered. In the present study this subject is tackled through direct numerical solution of the fluid and particle tracking equations in the framework of a one-way coupling approach. Results are presented for relatively wide ranges of vibrational amplitude, particle size and density.

Many natural and industrial processes are known to depend on the delicate interplay of two or more phases. Such interactions, of a liquid-solid, gas-solid, gas-liquid and in some cases, gas-liquid-solid nature, typically result in multiphase flows [

The mixing (or segregation) of the involved phases can depend on the type of fluid motion associated with the considered problem. As many natural and industrial processes are intrinsically turbulent, a vast literature exists where such phenomena have been investigated in these circumstances. In particular, most existing studies have focused on the preferential clustering of either isodense or non-neutrally buoyant solid particles in the case of

More recently, other studies have been conducted considering turbulent natural (buoyancy) convection, which can no longer be considered isotropic. This type of flow can be further distinguished into two different variants according to the relative direction of gravity and the imposed temperature difference (a temperature gradient parallel [

The present study focuses on the so-called thermovibrational convection, that is, a variant of standard buoyancy convection where steady gravity is replaced by, or superimposed onto, a sinusoidal displacement in time (i.e., vibrations). As shown by recent numerical efforts on this subject [

Despite these efforts and a few other relevant investigations [

A turbulent thermovibrational flow can be achieved when the amplitude of vibrations (or the related vibrational Rayleigh number) exceeds a given value and the fluid ‘response’ (in terms of induced velocity) becomes non-synchronous with the forcing of the system [

In line with many works cited in the introduction and owing to its simplicity, this work focuses on a classical differentially heated two-dimensional (2D) square cavity with length

with

^{−1}, ^{2}/s) and ^{2}/s), _{T} ^{−1}) and Δ^{2}/α, α/L, ρα^{2}/L,

In addition to solving the fluid flow itself, additional mathematical modeling is obviously required for the dispersed phase, which is accounted for through the solution of the Maxey-Riley equation, i.e.,

This may be regarded as an equation accounting for all the forces acting on the generic solid particle (with the exception of the Basset force, which can be considered negligible for the conditions considered in the present work) and the related balance in a Lagrangian frame of reference.

The additional non-dimensional parameters appearing in this equation are: ξ, i.e., the density ratio between the carrier fluid and the dispersed phase (ξ_{particle}/ρ_{fluid}_{particle}

The one-way coupling approach adopted in this work signifies that the dispersed phase has no effect on the fluid flow, which implicitly indicates that the present simulations refer (or are valid for) a dilute system.

The present work has been carried out using the computational platform OpenFOAM on uniform grids. It is worth recalling that, for this type of grids and turbulent flows, according to well-known requirements based on arguments related to the so-called Kolmogorov length scale, the recommended number of divisions across the computational domain can be defined as:

Here, a maximum value of _{ω} ^{9} is considered (along with Pr = 15), which according to this criterion would return a value of _{div}

The related outcomes shown in

A backward differencing in time has been used together with upwind and central differencing schemes in space for the convective term and diffusion term appearing in

The results presented in this section can be used to identify the salient factors contributing to the behavior of particles under the effect of vibrations and related turbulent fluid flow. They have been obtained by changing the influential parameters in a segregated manner in order to reveal the influence of each of them (namely, St and ξ accounting for the inertial effects, and γ and _{ω}^{4}.

We wish to start from an important premise. The accumulation of particles in turbulent flows comes primarily as a result of the inertial nature of the particles coupled with presence of eddies in the fluid. The investigation of Maxey et al. [

Preferential accumulation is typically observed in regions of strong vorticity for bubbles and regions of strong strain-rate for particles, meaning that for cases where the dispersed phase is less dense than the continuous phase (ξ < 1) the particles will cluster in the center of the eddies/vortices and vice versa for the cases where the dispersed phase is denser than the continuous phase (ξ > 1).

The present results, however, reveal that, when subjected to vibrations, the displacement of the cavity gives rise to an external (‘additional’ with respect to the effects discussed before) force that influences the trajectory of the particles, whereby they are continuously pushed from one side of the cavity to another.

Notably, this causes the dispersed phase to accumulate periodically along the walls parallel to the y-axis (as the vibrations are imposed along the x-axis) and upon detachment from the wall, form filament type structures. As these filaments migrate periodically from wall to wall, they curve and bend around the eddies present in cavity (due to the chaotic nature of the flow). This delicate formation mechanism is illustrated in

The formation and survival of the small structures described in _{ω} _{ω}

This is also evident in _{ω,}

The inertial properties of the particles can also influence greatly the formation (if any) of the structures. This is illustrated in

In particular, the light particles (ξ = 0.3) are represented in purple and the heavy particles (ξ = 2), in orange. Remarkably, it can be seen that the spaces occupied by the clouds formed for ξ = 0.3 and ξ = 2, are for the most part complimentary to each other; moreover, the light particles are prone to form stronger filaments than the heavy particles.

Another important influential factor, already identified in the earlier study by Lappa [_{ω}

Towards the end of unifying the previously segregated fields represented by the study of particle behavior in terrestrial turbulent flows and that concerned with the high-regular aggregates formed by particles interacting with laminar vibrational flow in microgravity conditions, particle dynamics have been investigated in conjunction with chaotic (turbulent) thermovibrational flow.

It has been shown that circumstances still exist for which particles (initially uniformly distributed in the entire physical domain) de-mix from the fluid and form recognizable (well defined) structures. As opposed to the perfect morphology of clusters emerging in laminar flow, however, when thermovibrational flow is chaotic the topology of the structures is relatively irregular and time-dependent.

Nevertheless, precise trends and relationships can be established if specific problem ‘statistics’ are connected to the behavior of the temporally evolving structures. As an example, a lack of filament formation due to a decrease in

An exciting prospect for the future is to conduct an extensive analysis of these interdependences using the present relevant mathematical and numerical framework.

Vibration amplitude

Pressure

Prandtl number

Rayleigh number

Temperature

Time

Velocity

Horizontal coordinate

Vertical coordinate

Particle Stokes Number

Vibrational period

Thermal diffusivity

_{T}

Thermal expansion coefficient

Kinematic viscosity

Non-dimensional amplitude of vibrations

Density

Angular frequency

Non-dimensional angular frequency

Temperature difference

Fluid/particle density ratio

Non-dimensional Kolmogorov length scale

Characteristic time scale

Hot

Cold

Particle

Flow

Divisions