Parameterization of the Evolving Drop-Size Distribution Based on Analytic Solution of the Linearized Coalescence-Breakup Equation

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  • 1 Mathematics Department, Trinity College, Hartford, Connecticut
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Abstract

Analytic solution of the linearized coalescence-breakup equation is used as a basis for parameterizing the evolving drop-size distribution. The linearized coalescence-breakup equation is formulated using only a small number of drop-size bins for the sake of computational efficiency but at the sacrifice of considerable detail. The low-resolution analytic solutions are then enhanced with detail provided by a high-resolution representation of the equilibrium distribution. In this step, the assumption is made that the drop distribution approaches high-resolution equilibrium form in a manner consistent with the temporal behavior of the low-resolution analytic solution. The low-resolution analytic solution and the high-resolution enhancement comprise the parameterization. Two specific parameterizations, based on two-bin and four-bin model solutions, are presented. The two-bin parametric solution is easy to compute but approaches equilibrium monotonically and thereby fails to undergo the pronounced fluctuation that characterizes high-resolution model results. The four-bin parametric solution involves more computation but produces an evolving drop distribution that closely resembles the fluctuating distributions obtained by detailed numerical solution of the coalescence-breakup equation.

Abstract

Analytic solution of the linearized coalescence-breakup equation is used as a basis for parameterizing the evolving drop-size distribution. The linearized coalescence-breakup equation is formulated using only a small number of drop-size bins for the sake of computational efficiency but at the sacrifice of considerable detail. The low-resolution analytic solutions are then enhanced with detail provided by a high-resolution representation of the equilibrium distribution. In this step, the assumption is made that the drop distribution approaches high-resolution equilibrium form in a manner consistent with the temporal behavior of the low-resolution analytic solution. The low-resolution analytic solution and the high-resolution enhancement comprise the parameterization. Two specific parameterizations, based on two-bin and four-bin model solutions, are presented. The two-bin parametric solution is easy to compute but approaches equilibrium monotonically and thereby fails to undergo the pronounced fluctuation that characterizes high-resolution model results. The four-bin parametric solution involves more computation but produces an evolving drop distribution that closely resembles the fluctuating distributions obtained by detailed numerical solution of the coalescence-breakup equation.

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