A mathematical model constructed within a theoretical framework applicable to direct numerical simulation (DNS) predicts the behavior of evaporating liquid drops entrained in a turbulent shear layer.

In the model, liquid drops are assumed to be dispersed at low volume fraction (though not necessarily low mass fraction) in a carrier gas. All chemical species are assumed to be calorically perfect. Gravitation is neglected. It is assumed that values of the viscosity, thermal conductivity, and species diffusivity of the gas phase can be prescribed, independently of the local mixture fraction.

The compressible conservation equations for mass, momentum, and energy for the gas phase are formulated in an Eulerian reference frame and include terms to account for exchanges of mass, momentum, and energy with the drops. The drops are assumed spherical and their internal temperature is assumed uniform.

Each drop is tracked in a time-accurate manner in a Lagrangian reference frame; these equations include terms for the drag exerted on each drop by the surrounding flowing gas. Each drop is assumed to exchange heat with the gas phase through convection and conduction only, since this study is performed at low temperature. Evaporation is represented by the nonequilibrium Langmuir-Knudsen law. The model accounts for complete two-way phase coupling (both gas-to-liquid and liquid-to-gas) of mass, momentum, and energy based on a thermodynamically self-consistent specification of vapor enthalpy, internal energy, and latent heat of vaporization.

The Growth of the Mixing Layer was found to be increasingly attenuated with increasing ML in the range 0 ≤ML ≤ 0.35, but not appreciably affected by changes in St0. On these plots, U0denotes the magnitude of each of the opposing free-stream velocities, tdenotes time, δw,0represents a specified initial value of vorticity thickness, and δw represents the vorticity thickness at any given time.

The model has been used to simulate the behavior of a three-dimensional, temporally developing, initially isothermal gas mixing layer formed by the merging of an airstream with a gas stream laden with hydrocarbon drops. Effects of the initial liquid-mass-loading ratio (ML), the initial Stokes number (St0), the initial temperature of the drops, and the three-dimensionality of the flow on the evolution of the mixing layer were examined. The dominant parameter affecting the flow was found to be ML (for example, see figure). The laden stream was found to become saturated before evaporation was complete, at all but the smallest values of ML. Drops in the mixing layer were observed to be centrifuged out of regions of high vorticity and to migrate toward regions of high strain in the flow, with resultant formation of concentration streaks in spanwise braid regions wrapped around peripheries of secondary streamwise vortices. Persistent regions of positive and negative slip velocity and slip temperature were identified. Other characteristics examined included variances of liquid- and gas-phase velocities and relationships among gas-velocity, drop-number-density, and thermodynamic profiles. From considerations of first and second order statistics, a comprehensive picture of the mixing layer is described.

This work was done by Josette Bellan and Richard S. Miller of Caltech for NASA's Jet Propulsion Laboratory. NPO-20434

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Mathematical modeling of two-phase flow with evaporation

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This article first appeared in the October, 1999 issue of NASA Tech Briefs Magazine.

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