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Pressure evolution equation for the particulate phase in inhomogeneous compressible disperse multiphase flows

Subramanian Annamalai*, S. Balachandar, P. Sridharan, and T. L. Jackson

  • Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, Florida 32611, USA

  • *subbu.ase@ufl.edu
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Phys. Rev. Fluids 2, 024301 – Published 3 February, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.024301

Abstract

An analytical expression describing the unsteady pressure evolution of the dispersed phase driven by variations in the carrier phase is presented. In this article, the term “dispersed phase” represents rigid particles, droplets, or bubbles. Letting both the dispersed and continuous phases be inhomogeneous, unsteady, and compressible, the developed pressure equation describes the particle response and its eventual equilibration with that of the carrier fluid. The study involves impingement of a plane traveling wave of a given frequency and subsequent volume-averaged particle pressure calculation due to a single wave. The ambient or continuous fluid's pressure and density-weighted normal velocity are identified as the source terms governing the particle pressure. Analogous to the generalized Faxén theorem, which is applicable to the particle equation of motion, the pressure expression is also written in terms of the surface average of time-varying incoming flow properties. The surface average allows the current formulation to be generalized for any complex incident flow, including situations where the particle size is comparable to that of the incoming flow. Further, the particle pressure is also found to depend on the dispersed-to-continuous fluid density ratio and speed of sound ratio in addition to dynamic viscosities of both fluids. The model is applied to predict the unsteady pressure variation inside an aluminum particle subjected to normal shock waves. The results are compared against numerical simulations and found to be in good agreement. Furthermore, it is shown that, although the analysis is conducted in the limit of negligible flow Reynolds and Mach numbers, it can be used to compute the density and volume of the dispersed phase to reasonable accuracy. Finally, analogous to the pressure evolution expression, an equation describing the time-dependent particle radius is deduced and is shown to reduce to the Rayleigh-Plesset equation in the linear limit.

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Phys. Rev. Fluids 2, 024301– Published 3 February, 2017
  • Received 17 June 2016
Article part of CHORUS

©2017 American Physical Society

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