A Computational Study of Viscoelastic Droplet Collisions

A computational study of oblique binary viscoelastic droplet collisions is performed. The free surface is modeled by a Lagrangian moving-mesh interface tracking method which is validated against analytical solutions and experimental data for binary Newtonian droplet collisions. Interpolation error due to cell deformation is minimized with a local topological mesh adaptation algorithm. This method was then applied to the collision of viscoelastic droplets through a finite volume implementation of the linearized Phan-ThienTanner constitutive stress model. The extensional nature of an oblique droplet collision highlights the effects of the large Trouton ratios typically found in this class of fluids though increased stabilization of fluid ligament structures and expansion of the coalsence regime. By altering the spray kinematics in this manner, the spray characteristics as a whole are altered by the viscoelastic rheological effects. The fully three dimensional computational study investigated the effect of Weber number, and impact parameter on collision outcomes of viscoelastic droplets. Corresponding Author: [email protected] Introduction Atomizing sprays are encountered regularly in many industrial applications including inkjets, coatings, and combustion. The coalescence and separation modes of the atomized colliding droplets can greatly affect downstream spray kinematics due to changes in size and velocity distributions [1]. Difficulties arise when experimentally quantifying this phenomenon, even with dense spray diagnostics, due to the wide range of time and length scales inherent in an atomization process. Numerical simulations can shed light into the nature of binary droplet collisions as well as complement sub-grid spray models. Binary collisions of Newtonian droplets have been studied extensively in experimental settings. One of the earliest works on the topic by Ashgriz and Poo [2] precisely determined collision outcome regimes of water droplets using relevent dimentionless groups. Qian and Law [3] continued in a similar manner with a study of hydrocarbon droplet collisions in ambient nitrogen producing a series of high resolution time resolved photographs. These images have since become a common benchmarking tool for many CFD packages that involve free surface or interfacial physics simulation. Additionaly, viscosity has been shown by Willis [4] and Dai [5] to affect the maximum deformation of the binary system. Through experimentation, Motizigemba et al. [6] found that the time to reach maximum deformation of a shear-thinning droplet pair was independant of viscosity characteristics but the magnitude of that deformation was not. Additionally, they showed through simulation that elongational velocity gradients generally dominate shear gradients for χ = 0 collision systems. In theory this elongational flow dominance could manifest itself in fluids with strain dependant Troutan ratios. The work by G. S. de Paulo et al. [7] successfully coupled the marker and cell (MAC) interface tracking method to PTT viscoelastic rheology and were able to simulate both jet buckling and extrudate die swell. In the field of droplet collisions Yue [8] modeled coalescence and retraction of two dimensional viscoelastic droplets using the OldroydB stress model but the range of Weber and Reynolds numbers explored were lower than those typically encountered in an atomizing spray [9] which is part of the novelty of this study. Methodology This computational study solved the incompressible Cauchy equation of momentum conservation on a cell centered finite volume mesh. Conservation of momentum, Eqn.1 was solved on moving, deforming control volumes. ρ( ∂u ∂t +∇ · φu) = −∇p+∇ · σ (1) The variable φ is the volumetric flux, u is velocity, and p is the pressure, all defined at cell faces. ρ is the density and σ represents the deviatoric stress tensor before application of a constituative stress model. In this work there are two stress models employed to solve for a stress tensor τ : classical Newtonian and the Phan-Thien-Tanner viscoelastic model. For Newtonian cases τ is defined as τ = μ[∇u + (∇u)] (2) where μ is the dynamic Newtonian viscosity of the fluid. Among the models compiled in the viscoelastic CFD library by J. Fávero [10] (the library used in this study) is the Phan-Phien-Tanner consituative stress model by Tanner [11] which is of the form: f(tr(τ ))τ + λ ∇ τ = 2ηD (3) The symbol ∇ (.) represents the upper-convected derivative and λ is the stress relaxation time of the fluid. ∇ τ is then defined as ∇ τ = Dτ Dt − τ · L − L · τ (4) L represents the effective velocity gradient