Modeling the Interaction between a Thermal Flow and a Liquid: Review and Future Eulerian-Lagrangian Approaches

Suspension Plasma Spraying is a complex process in which several physical mechanisms play a part. So the modeling and understanding of the interaction between a high-velocity and thermal flow and a liquid precursor phase is of major importance concerning the control and characterization of the process. The liquid droplet size distribution has a high influence on the kinetic properties of the as-sprayed nanometer particles before impacting on a target substrate. An overview of existing models is provided dealing with the penetration of the liquid phase into the thermal flame and the resulting fragmentation and vaporization of this phase before impact. The physical characteristics of the flow as well as existing Lagrangian and Eulerian modeling strategies are briefly discussed while paying attention to the physical parameters characterized and measured by numerical simulation. The potential of the various models and also their limits are intended to be highlighted. Future coupled Eulerian-Lagrangian modeling strategies are also proposed for a global and more exhaustive representation of the injection, fragmentation and dispersion part of the two-phase gas-liquid flow before particle impact on the substrate.


Introduction
Increasing higher efficiency rates or lifetimes of functional industrial parts re-How to cite this paper: Vincent, S., Meillot, E., Caruyer, C. and Caltagirone, J.-P. (2018) Modeling the Interaction between a Thermal Flow and a Liquid: Review and Fu-esses are concerned in order to heat and accelerate nanoparticle of ceramic materials [6]. Heat sources such as these techniques are required due to the high heat level to be transferred to the ceramic material. For example, Cold spray process is not adapted due to its concept based on only acceleration by high gas flow rate. Now, thermal spraying is up-to-date to build nanometre-scale structure. Nevertheless, modelling and simulation of the interaction between a thermal jet and a liquid precursor phase continue to be of major importance concerning the control and characterization in terms of size of − Carrier liquid droplet (size, distribution, position and velocity), − Distribution of nanometer particles before impact on a target substrate, − And thermokinetic properties of these solid particles carried by the liquid phase. The typical problem of interest is illustrated in Figure 1 whatever process is: the main difference between the two processes being the plume generator and the direct injection of the liquid inside it. Each technique can produce coatings to provide protection against high temperatures, corrosion, erosion or wear on materials [7]. For plasma spraying gun, around a cylindrical cathode, the operating gas (either pure, e.g., argon, or, more commonly, a mixture, e.g., argon/hydrogen, argon/hydrogen/helium) enters the torch and is heated by the electrical arc generated between a nozzle-shaped anode (of oxygen-free copper) and the cathode. This results in the transformation of the gas into plasma, which exits the torch as a free jet. HVOF is an internal combustion system that generates a supersonic flame jet.
Decreasing the deposit width (<100 µm) with nano-structured coatings leads to improve as-manufactured properties, but due to their low size and low inertia, nanometer particles are injected in the jet with a liquid precursor [8]. From a modelling point of view, several major informations can be extracted from this figure: − the heat and mass transfers are multi-scale in time and space, − the multi-physic characters of the flow as chemical, thermal, turbulent, multi-phase and electromagnetic features, − The coupling between fluid and solid mechanics.
These mechanisms have to be modelled depending on the considered zone of the problem (from hot flame inlet to impact of particles). General reviews have    [26] concerning the evolution of the liquid jet break-up in terms of the resulting droplet diameter (TAB or ETAB models) or temperature inside the thermal flow, based on measurements and theoretical developments. These global models have been used to predict the trajectories of droplets inside the flame [27]. In this work, the mean experimental velocity and temperature were utilized as inputs to the particle diameter and temperature models.   [33]. The energy conservation accounts for radiative effects through a local volume source term whereas the transport of species coming from the flame inlet are modelled by the resolution of all compounds [32] or by considering the ionized gas as a specific fluid [34]. With specific plasma conditions, leading to the restrike mode for example [35], the unsteadiness of the flow cannot be reproduced correctly by RANS approach [16]  The unsteady character of the thermal flow has been nicely recovered using LES turbulent modelling when compared to experiments [16] [36]. − The plume generator models: specific characters of this part of the suspension spraying process require the use of dedicated models for accounting of electromagnetic effects in a plasma torch [24] or of combustion effects in a HVOF gun [5] [10]. For the HVOF combustion, equations are in accordance with the combustive/fuel chemical reactions. For the plasma, several models have been proposed to simulate it, and their complexities evolve with increasing computational resources. The first simulations were limited to steady plasma flows and utilized temperature and velocity profiles as input data [12] [32] [37] [38]. The average result fields of the jet flowing though the ambient atmosphere do not take into account the transient plasma flow behavior which is becoming increasingly important for suspension spraying due to high frequency of the mechanisms appearing. Two routes can be employed for a non-stationary flow outside the torch:  A global approach, integrating the physical phenomena through the distinct equations of electromagnetic and fluid mechanics. These studies employed the same set of equations, based on the mass, momentum and energy conservation. This was coupled with the Maxwell equations for electromagnetism effects based on the local thermodynamic equilibrium (LTE) assumption for the gases, which supposes that all the species are at the same temperature. Last recent works are from [39]- [44]. These approaches lead to a better understanding of the jet generation inside the torch.  A simple approach based on a Joule effect in the arc column including correlations with experimental time-dependent voltage measurements [24] [45]. This approach leads to only investigate the unsteady flow outside the torch. The turbulence models and energy conservation models are the same as those described previously.
The article provides an overview of existing models for dealing with the penetration and the transport of the liquid phase within a thermal flow and the resulting fragmentation and vaporization of this phase before impact on the target substrate. In the two first sections, existing Lagrangian and Eulerian modelling strategies are briefly discussed while paying attention to the physical characteristics obtained by numerical simulation. Comparisons with existing experiments are provided in order to highlight the potential of the various models and also their limits. The final part is devoted to future coupled Eulerian-Lagrangian modelling strategies for a global and more exhaustive representation of the injection, fragmentation and dispersion of the two phase flame-liquid flow before impact on the substrate. Conclusions are finally drawn.

Lagrangian Models for Liquid Precursor Interaction with a Thermal Flow
where F D , F b and F a are respectively drag, buoyancy and additional mass forces whereas Q r is the radiative flux to which are subjected the liquid precursor particles. The model (1) - (7) has been used by many authors to characterize the behavior of various precursor droplets in HVOF and plasma suspension processes [  Concerning the operating conditions before the impact zone forming the final coating, important information such as the particle temperature or velocity can be estimated by means of the Lagrangian modeling. It can be remarked that 10% differences are observed compared to experiments. They are in particular due to the assumption of the modeling itself which only considers an isolated droplet in the simulation, which is not the case in real processes. To finish with, it has to be noticed that the model (5) - (7) has been used without a RANS representation of the mass, momentum, energy and mass fraction transfer  However, they still rely on a scale separation between the size of the particles and the turbulent scales and they also need to define a priori particle-flow [26] and particle-particle interaction models which are difficult or even impossible to characterize experimentally. These limitations will be partly tackled with Eulerian approaches presented in the following section which are designed to solve a majority of time and space scales of the interfacial flow. The interest of Eulerian small scale modeling is to provide information for defining two-and four-way Lagrangian models in dense situations.

Modeling
The building of an Eulerian modeling for the interaction between a liquid pre-  (12) where τ is a characteristic time of the problem chosen equal to the numerical time step Δt used to discretize the time derivatives. The liquid volume fraction C is representative of the volume of liquid precursor in each elementary volume or grid cell. By definition, C = 1 in the liquid and C = 0 elsewhere. This approach is termed Volume of Fluid (VOF) method in the literature [54]. As explained in [34], the local characteristics ρ, μ, λ and C p of the fluids (air, plasma gas and liquid) are built according to the pressure, temperature, plasma gas concentration χ and liquid volume fraction C. The surface tension forces are determined according to C and integrated as a local source term in the cells cut by the interface through a volume force S ST [34]. The vaporization of the liquid is not accounted for as the Eulerian model is adapted to modeling the first instant interaction between the liquid jet and the plasma. The main difference between the RANS model (1) -(4) and the deterministic model (8) -(12) is the modeling of turbulence. In the latter, a LES approach is used to build the turbulent viscosity μ t [53] by assuming that the larger scales of the multi-phase flow are resolved while the smallest one are modeled by means of a diffusive behavior. Once μ t is known, the turbulent conductivity λ t and plasma gas diffusion coefficient D t are obtained by means of a turbulent Prandtl and Lewis analogy. With model (8) - (12), it is assumed that all the time and space scales of the liquid precursor interface are solved, meaning that the discretization grid has to be refined enough to capture all the interfacial structures generated by the interaction of the liquid jet with the thermal flow. Due to heat exchange between water and plasma, the phase change cannot be neglected for long time simulations. A phase change model is integrated; it is an adaptation of a Lagrangian model of [55]. However, the reduction of liquid volume is considered while the motion of water vapor created during the evaporation is not taken into account in the simulations. Each cell cut by the interface is considered as an equivalent sphere whose tradius is calculated as the local curvature radius C C ∇ ∇ ⋅ ∇ . The Equation (1) is applied on these equivalent spheres. The evaporated water volume is calculated and removed in the concerned cells and finally the phase function C is updated.
All variables are local and known in each cell of the mesh.

Results
Typical results obtained with the Eulerian LES model are presented in Figure 3.  The evolution of the jet injection into ArH 2 plasma can be seen in Figure 3. Two steps can be analyzed. The first one concerns the first instants (less than 50 µs) during which the liquid column penetrates the plasma flow. The destruction of the liquid starts just at the tip of the column with appearing of ligaments after a short run in the flow: only one mm of run is necessary to create them. Simultaneously, instabilities appear at the liquid-shape surface and grow up to the close limit of the plasma jet (Bottom in Figure 3). Then the filaments break into droplets in the continuity. As seen in the second picture (Figure 3 Bottom), the balance is achieved in less than 100 µs. Instabilities expand on the jet surface and stay active as soon as the jet reaches the plasma zone (around 0.003 mm). Now, a lot of ligaments are formed at the periphery of the jet and are stretched and broken up into large droplets (primary fragmentation). According to the low introduction pressure (0.25 MPa) the penetration length is about 1.3 mm above the torch axis.
An overpressure zone is well observed in the upstream direction to the jet (Figure 4 Bottom).
The plasma temperature decreases strongly with the liquid injection testifying from air engulfment inside the plasma (Figure 4 Upper): puffs of cold air can be seen around big droplets flowing downstream.
Two calculations types have been done with and without phase change. To differentiate the droplet behavior and the impact of the phase change, the droplet number has been calculated at the same time (100 µs) in the whole field. With and without the phase change model, the jet behavior does not seem different, but the droplet number decreases due to the water vaporization ( Figure 5). The distribution of the droplet number according to their diameters is represented in Figure 5.
If the biggest size droplets stay in the same number range (more than 38 µm) due to their thermal inertia, the lowest diameter droplets diminish in number because of the evaporation. The lost mass can be estimated to 10% that is not negligible in a so short time, less than 50 µs, required time to get the system in balance.
The domain is cut in several parts from the center line to the border of the plasma jet ( Figure 6 according to [57]) in order to determine the droplet distribution along the plasma radius.
It is observed that smaller droplets (10 μm) are especially present in the zones close to the torch axis. In this place, temperature and velocity are higher; thus droplets are more evaporated or broken-up. On the contrary, droplets with a diameter of the order of 30 μm are more numerous in the peripheral zone ( Figure 7).

Mixed Eulerian-Lagrangian Models for the Interaction between a Thermal Flow and a Liquid Precursor
As recommended by Cetegen and Basu [9], with the advances in parallel computational facilities and understanding of models, multi-scale models are     It is proposed to use a deterministic LES compressible model (8) -(12) everywhere on a computational grid whose finest grid cell is of the size of the smaller droplets in the secondary break-up zone (namely d d1 in Figure 8). As   (5) -(7), are applied. The correct numerical method for implementing such a multi-scale Eulerian-Lagrangian model is the VOF-Sub Mesh approach [54]. Indeed, this method can describe at the same time an Eulerian VOF function C and also Lagrangian markers (Lagrangian droplets in our multi-scale model) whose volume, velocity and temperature can be modeled differently to those applied to C. The first implementation of the VOF-SM multi-scale model, including Lagrangian modeling of the droplet vaporization (with diameters in the range d d2 ), has been first presented by Caruyer et al. [58]. With the VOF-PLIC method [59], the droplet size cannot be inferior to the cell size. It implies the artificial generation of large liquid precursor droplets to conserve the liquid volume over time and thus a truncation of the droplet distribution. In order to counter that, the new hybrid method VOF-SM [57], can track the sub-mesh interface by means of Lagrangian markers. As seen in Figure 9, with the VOF-PLIC method, larger droplets (>20 μm) are overestimated, as expected, to the detriment of the smallest.

Conclusions and Prospects
Thermal spray technologies, such as HVOF or plasma spraying, are able to pro- − High resolution of the turbulence with LES turbulence models and large interfacial scales, − High number of physical identities (solid particle, droplet) treated by Eulerian methods.
In fact, the interactions between the liquid and high velocity and high temperature gas flows, such as those generated by HVOF or DC plasma guns, lead to such a large range of droplet sizes, size which evolves with time during evaporation, that the taking-into-account of numerous and small physical entities can only be treated by mixed Eulerian-Lagrangian models. That is the main strategy proposed here as conclusion and advocated by the authors to analyze in depth the behavior of nanoparticles injected in thermal spray processes.