A Numerical Method for Detailed Simulations of Atomization in Non-Isothermal Environments

The atomization process of turbulent liquid jets is not well understood. Detailed numerical simulations can help study the fundamental mechanisms in regions where experimental access and analysis is virtually impossible. However, simulating atomization accurately is a huge numerical challenge since time and length scales vary over several orders of magnitude, the phase interface is a material discontinuity, and surface tension forces are singular. The Refined Level Set Grid (RLSG) method presented here is one numerical approach to simulate the primary breakup process of liquid jets and sheets in detail. With it, the liquid/gas phase interface is tracked by a level set method using an auxiliary high resolution equidistant Cartesian grid. This not only allows for application of higher-order WENO schemes retaining their full order of accuracy both for advecting and reinitializing the level set scalar, but it also provides the necessary high resolution of the phase interface geometry during topology change events in an efficient manner. When atomization occurs in non-isothermal environments, such as in combustion devices, thermal fluctuations can be significant on length scales associated with the liquid atomization process. Since the surface tension force is a function of local temperature, these thermal fluctuations may result in large local variations of the surface tension force, thereby potentially impacting the atomization process. Here, we present a numerical technique to incorporate these thermal Marangoni forces into the balanced force algorithm, together with verification and validation test cases geared towards testing the applicability of the proposed methods to the case of atomization. Introduction Thermal fluctuations can have a significant impact on the dynamics of liquid/gas interfaces because, for most gas/liquid combinations, both surface tension and phase transition depend strongly on the local temperature. An important technical application where both surface tension forces and phase transition are dominant is the atomization of liquid fuels for combustion processes. For combustion to occur, the liquid fuel needs to be atomized, evaporated, and mixed with an oxidizer. The ensuing chemical reactions can generate temperature variations on the order of 10K over small length scales of the order of 10−3m. Due to the strong dependence of surface tension on temperature, the temperature fluctuations result in Marangoni forces that impact the flowfield at the gas/liquid interface, which in turn alter the interfacial temperature distribution via the induced interfacial flow [1]. Although the ratio of global inertial to surface tension forces is typically large in these applications, atomization, which is the first in the sequence of processes leading to combustion, always occurs on small scales (involving droplets which are many orders of magnitude smaller than the diameter of the initial liquid jet). At these scales, surface tension forces are dominant. Variations in temperature, resulting in variations in surface tension forces, can thus influence atomization significantly. However, to the knowledge of the authors, there exists no detailed experimental data set analyzing the phase interface dynamics during atomization in this situation. Numerical simulations, on the other hand, can study the impact of thermally induced variations in surface tension forces and can thus help determine whether they play a role in the atomization outcome. To this end, this paper presents a numerical method for thermal Marangoni forces based on the Refined Level Set Grid (RLSG) method [2]. However, due to the lack of validation data in actual atomization situations, we study in this paper another example where the temperature-dependent surface tension influences the dynamics of phase interfaces: the thermocapillary ∗Corresponding Author: [email protected] ICLASS 2009 A Numerical Method for Detailed Simulations of Atomization in Non-Isothermal Environments motion of drops and bubbles. The study of the thermocapillary motion was first reported by [3], who determined the terminal velocity of a spherical drop in the creeping flow limit. A number of experiments have been conducted in drop towers, sounding rockets and aboard space shuttles; see the extensive review of [4]. Some of the more recent experiments have noted complicated transients and time-dependent behavior due to the effects of finite viscosity and thermal diffusivity [5–7]. In the microgravity experiments the viscous and thermal timescales of the two fluids differ by up to two orders of magnitude within a single experimental run, and so complicated temporal behavior is not surprising. All of the presently available theoretical or numerical results assume constant fluid properties, except for the surface tension which is assumed to vary linearly with temperature. The review article [4] concluded that the most important theoretical problem that needs to be addressed for an isolated drop is the consideration of the fully transient problem incorporating the dependence of physical properties on temperature. In this paper, a numerical method for finite-volume flow solvers to consistently incorporate the Marangoni forces is verified and validated using both theoretical and experimental data. Governing equations and numerical technique Consider a spherical drop of one fluid with radius r0 placed in an initially quiescent bulk fluid with an imposed (typically positive) linear temperature gradient GT in the vertical direction. The two fluids are immiscible with, in general, different densities, viscosities and thermal properties. We shall use the initial radius of the drop, r0, as the length scale and GT r0 as the temperature scale. For the thermocapillary motion of drops and bubbles, it is customary to use U = σTGT r0/μb as the velocity scale, where μb is the dynamic viscosity of the bulk phase and σT is the surface tension derivative with respect to temperature. This then gives μb/σTGT as the time scale. The surface tension is scaled by σ0, which for the problems considered here is the surface tension at the initial temperature at the center of the drop. Throughout, subscript d refers to properties of the drop phase and subscript b to those of the bulk phase. We shall assume that the surface tension σ between the two fluids varies linearly with slope σT (which for most fluids of interest is negative). With these scalings, the non-dimensional linear equation of state becomes σ(T ) = 1 + Ca(T − T0) , (1) where T0 is the non-dimensional initial temperature at the center of the drop, and the capillary number, which gives the relative importance of the tangential to normal stresses at the drop interface, is Ca = μbU σ0 = σTGT r0 σ0 . (2) The non-dimensional Navier–Stokes equations governing the motion of an unsteady, incompressible, immiscible, two-fluid system, are