R. H. Rangel

Numerical simulation of substrate impact and freezing of droplets in plasma spray processes

In the present study, deformation, interaction and freezing during substrate impact of molten droplets in plasma spray processes are numerically investigated. The numerical simulation is conducted on the basis of the full Navier-Stokes equations and the volume of fluid (VOF) function by using a two-domain method for the thermal field and freezing problem and a two-phase flow continuum model for the flow problem with a growing solid layer. Important processing parameters, such as droplet temperature, substrate temperature and droplet velocity are considered and their effects on flattening and freezing are discussed. The numerical results reveal that a droplet spreads and solidifies into a splat of diameter 1.6 to 3.8 times the initial droplet diameter within a time of 0.12 to 0.44 mu s. The droplet liquid separates from the solid/liquid interface when freezing occurs. Increasing initial droplet velocity and enhancing substrate temperature can significantly enhance the flattening extent of droplet. A fully liquid droplet impinging on a solid substrate may generate good contact between the splat and the substrate; a fully liquid droplet striking on another flattening splat produces ejection of the liquid; a fully liquid ring colliding with a flattening splat causes bounce of the liquid and formation of voids. A combination of a liquid droplet condition at a high initial velocity with a semi-solid or solid surface condition may produce good adhesion in sprayed deposits or coatings.

Nonlinear growth of Kelvin–Helmholtz instability: Effect of surface tension and density ratio

The nonlinear evolution of initially small disturbances at an interface separating two fluids of different density and velocity, including surface tension effects, is investigated with the use of the vortex‐sheet discretization approach. The location of the interface is tracked in time by following the motion of each vortex under the combined influence of all other vortices. The influence of surface tension and density discontinuity is incorporated in an equation governing the evolution of the circulation of each vortex. Increasing the surface tension or the density ratio is shown to reduce the growth of the disturbance. For density ratios larger than 0.2 a critical wavenumber exists that divides the unstable part of the spectrum into a region where a vorticity singularity can develop (with interface rollup) and a region where two finite vortical centers are formed (with partial or no rollup). For lower density ratios this bifurcation phenomenon is not observed.

The linear and nonlinear shear instability of a fluid sheet

A theoretical and computational investigation of the inviscid Kelvin–Helmholtz instability of a two‐dimensional fluid sheet is presented. Both linear and nonlinear analyses are performed. The study considers the temporal dilational (symmetric) and sinuous (antisymmetric) instability of a sheet of finite thickness, including the effect of surface tension and the density difference between the fluid in the sheet and the surrounding fluid. Previous linear‐theory results are extended to include the complete range of density ratios and thickness‐to‐wavelength ratios. It is shown that all sinuous waves are stable when the dimensionless sheet thickness is less than a critical value that depends on the density ratio. At low density ratios, the growth rate of the sinuous waves is larger than that of the dilational waves, in agreement with previous results. At higher density ratios, it is shown that the dilational waves have a higher growth rate. The nonlinear calculations indicate the existence of sinuous oscillat...

General solution of the particle momentum equation in unsteady Stokes flows

The general solution of the particle momentum equation for unsteady Stokes flows is obtained analytically. The method used to obtain the solution consists of applying a fractional-differential operator to the first-order integro-differential equation of motion in order to transform the original equation into a second-order non-homogeneous equation, and then solving this last equation by the method of variation of parameters. The fractional differential operator consists of a three-time-scale linear operator that stretches the order of the Riemann–Liouville fractional derivative associated with the history term in the equation of motion. In order to illustrate the application of the general solution to particular background flow fields, the particle velocity is calculated for three specific flow configurations. These flow configurations correspond to the gravitationally induced motion of a particle through an otherwise quiescent fluid, the motion of a particle caused by a background velocity field that accelerates linearly in time, and the motion of a particle in a fluid that undergoes an impulsive acceleration. The analytical solutions for these three specific cases are analysed and compared to other solutions found in the literature.

Modeling of molten droplet impingement on a non-flat surface

Numerical modeling of deformation and interaction of molten tungsten, nickel and titanium droplets impinging on a non-flat surface during thermal spraying has been performed in the present study. This research represents a significant advance relative to earlier published theoretical and numerical studies which were limited to a flat surface. Under the practical conditions in thermal spray processes, the complex physical phenomena of droplet spreading and consolidation take place at a microscopically rough and non-flat surface on a target substrate. The present study, therefore, provides a novel approach to the practical impingement behavior of molten droplets. On the basis of the present study, some fundamental mechanisms governing pore formation in spray processed materials may be established and the concomitant effects of important processing parameters may be determined.

An improved model for droplet solidification on a flat surface

An existing model of the deformation and solidification of a single droplet impinging on a cold surface has been revised and improved. The original model is based on a two-dimensional axisymmetric flow approximation of the velocity field, the Neumann solution to the one-dimensional Stefan solidification problem, and an integral mechanical energy balance. The improved model features a more appropriate velocity field which satisfies the no-shear boundary condition at the free surface, and an accurate derivation of the dissipation term from the mechanical energy equation. This equation has been solved numerically. Comparisons of the original and the improved models have been performed. Results show that the original model over-estimates the final splat size by about 10%. The discrepancy is more pronounced at larger Weber numbers, where viscous effects dominate. The effects of the Weber number, We, the Reynolds numbers, Re, and the solidification parameter have been investigated through detailed numerical calculations. Two regimes of spreading/solidification have been identified. If Re/We is small, the process is one of dissipation of the incident droplet kinetic energy; whereas for large values of Re/We the process can rather be characterized as a transfer between kinetic and potential energy. In the latter case, the variations of the final splat size versus the solidification constant exhibit a non-monotonic behaviour. This indicates that, for a given material, the deposition process can be optimized. Correlations relating the final splat size to the process parameters are given.

Numerical investigation of particle–particle and particle–wall collisions in a viscous fluid

The dynamics of particle-particle collisions and the bouncing motion of a particle colliding with a wall in a viscous fluid is numerically investigated. The dependence of the effective coefficient of restitution on the Stokes number and surface roughness is analysed. A distributed Lagrange multiplier-based computational method in a solid-fluid system is developed and an efficient method for predicting the collision between particles is presented. A comparison between this method and previous collision strategies shows that the present approach has some significant advantages over them. Comparison of the present methodology with experimental studies for the bouncing motion of a spherical particle onto a wall shows very good agreement and validates the collision model. Finally, the effect of the coefficient of restitution for a dry collision on the vortex dynamics associated with this problem is discussed.

Numerical simulation of impingement of molten Ti, Ni, and W droplets on a flat substrate

Thermal spraying has been widely applied to process thin protective coatings on preshaped parts and to manufacture metallic preforms of a variety of geometries. The quality of materials produced by thermal spraying depends critically on the impact conditions of the droplets. In the present study, the deformation behavior and interaction of molten droplets impinging onto a flat substrate during thermal spraying have been numerically simulated. The calculated results reveal that a droplet spreads uniformly in the radial direction during impingement and eventually forms a thin splat with final diameter and thickness up to 11.3 times and down to 0.02 times the impact diameter, respectively. The final splat diameter increases rapidly with increasing impact velocity and melt density or decreasing melt viscosity. For the processing conditions of interest, the final splat diameter and the spreading time may be approximated by correlations: ds/do = l.04Re0.2 and ts/(do/uo) = 0.62Re0.2, where ds/do is the dimensionless splat diameter; ts/(do/uo) is the dimensionless spreading time; and Re is the Reynolds number. A fully liquid droplet impinging onto a flat solid substrate leads to good contact between the splat and the substrate. Multiple fully liquid droplets striking simultaneously onto other flattening, fully liquid splats cause ejection and rebounding of the liquid, as well as formation of voids within the liquid.

Single droplet vaporization including thermal radiation absorption

Total hemispherical absorption distributions for n-decane and water droplets irradiated by a blackbody under spherically symmetric conditions are used in the calculation of transient droplet heating and vaporization. The gas model used is based on the extended film theory. Three liquid-phase models have been extended to include thermal radiation absorption. The results show that radiation absorption can be as important as, or more important than, the choice of liquid-phase model. Based on the effective-conductivity model, two dimensionless parameters expressing the ratios of radiation absorption and gas-liquid heat transfer to liquid conductive and convective heat transfer are defined. It is shown that the first parameter determines the droplet heating regime. Over 200 numerical calculations using the effective-conductivity model and various initial and ambient conditions have been performed for water and w-decane droplets. It is shown that the ratio of the two dimensionless parameters evaluated at the initial time can be used to correlate and predict the radiation absorption influence on the droplet lifetime. All cases investigated correspond to the slow heating regime. Under the conditions analyzed, the nonuniformity of the radiation absorption has little effect on the overall droplet heating and vaporization process. CD = cp = F =

Numerical analysis of the deformation and solidification of a single droplet impinging onto a flat substrate

An existing model has been modified to explore the deformation and solidification of a single droplet impinging on a substrate. The modification accounts for possible solid fraction of material at impact. Numerical results predict that the kinetic energy dominates the process at impinging velocities greater than about 100 m s−1. In addition, the thermal diffusivity of the solidifying material controls the process, but the temperature of the substrate relative to the melting temperature of the material must be considered when comparing materials. It is believed that droplets solidifying into thinner, wider discs would reduce porosity; therefore, dense materials accelerated to high speed would solidify into masses with the highest bulk density.