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Evolution of Raindrop Spectra. Part I: Solution to the Stochastic Collection/Breakup Equation Using the Method of Moments

Graham FeingoldDepartment of Geophysics and Planetary Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Israel

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Shalvn Tzivion (Tzitzvashvili)Department of Geophysics and Planetary Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Israel

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Zev LevivDepartment of Geophysics and Planetary Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Israel

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Abstract

We present a solution to the stochastic collection/breakup equation (SCE/SBE) using our recently developed method of moments and the Low and List fragment distribution function. We prove that the collisional breakup equation conserves overall mass irrespective of the degree to which the fragment distribution function conserves mass for individual collisions. The method is compared with analytical solutions to the steady state collection/breakup equation, as well as with the breakup equation, for simple kernels. The proposed method produces better approximations to the analytical solutions for simple kernels, than those obtained using a single moment method. In writing the breakup terms, an approximation to the breakup kernel and fragment distribution function in discrete categories is used. This approach allows one to write the algorithm without prescribing the shape of the distribution function within the category. The approximation to the fragment distribution function in a discrete category is validated using existing single moment methods. In cases of breakup dominated evolution, one and two-moment solutions to the collecion/breakup equation are shown to differ little, though the solutions to collection dominated evolution are expected to differ appreciably. The method of moments thus represents a more universal, accurate approach to solving mass transfer equations.

Abstract

We present a solution to the stochastic collection/breakup equation (SCE/SBE) using our recently developed method of moments and the Low and List fragment distribution function. We prove that the collisional breakup equation conserves overall mass irrespective of the degree to which the fragment distribution function conserves mass for individual collisions. The method is compared with analytical solutions to the steady state collection/breakup equation, as well as with the breakup equation, for simple kernels. The proposed method produces better approximations to the analytical solutions for simple kernels, than those obtained using a single moment method. In writing the breakup terms, an approximation to the breakup kernel and fragment distribution function in discrete categories is used. This approach allows one to write the algorithm without prescribing the shape of the distribution function within the category. The approximation to the fragment distribution function in a discrete category is validated using existing single moment methods. In cases of breakup dominated evolution, one and two-moment solutions to the collecion/breakup equation are shown to differ little, though the solutions to collection dominated evolution are expected to differ appreciably. The method of moments thus represents a more universal, accurate approach to solving mass transfer equations.

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