Analytical solution of an Eulerian dilute particle-laden flow problem

Abstract

Dilute particle-laden flows are encountered in various natural processes and manmade applications. To reduce the computational resources used to simulate such flows, the particle phase could be formulated using the Eulerian approach, resulting in a continuum hydrodynamic model. The aim of this paper is to present the analytical solution of a steady two-dimensional flow problem using that model. The dispersed solid particles, considered in this problem, are immersed in a uniform fluid flow field. The solid phase is coupled to the fluid using a linear Stokes-drag, which is valid for low slip velocities. The general solution of the solid phase velocity field is obtained by solving its quasi-linear hyperbolic momentum equations using the method of characteristics. For the case of a uniform inlet particle velocity, the solid phase velocity field is obtained in terms of Lambert W function. Subsequently, this velocity field is substituted in the solid phase continuity equation; transforming it to a semi-linear hyperbolic partial differential equation, which is solved to obtain the solid phase volume fraction field.

Effects of interphase coupling in one-dimensional particle-laden flow

“…The one-way coupling (1WC) model is used in the literature when the particle concentration (used interchangeably in this paper with "particle phase volume fraction") is too low to affect the gas phase [6,9]. Thus, the governing equations of the fluid phase are decoupled from that of the particle phase, which means they can be solved independently to obtain the fluid phase flow field.…”

“…Thus, the governing equations of the fluid phase are decoupled from that of the particle phase, which means they can be solved independently to obtain the fluid phase flow field. The particle phase, on the other hand, is coupled to the fluid phase so that its governing equations are [6] where is the particle phase volume fraction, p is the particle material density, u pi is the particle phase velocity, u fi is the fluid phase velocity, and is the reciprocal of the particle relaxation time which is constant for low particle Reynolds number and given by [6,9] (1)…”

“…This approximation allows the fluid phase to be described as a continuous media in the Eulerian framework. The dispersed particle phase, on the other hand, could be modelled using either the discrete Lagrangian approach [3][4][5] or the continuum Eulerian approach [6][7][8][9]. In the former, the equations of motion for each particle are solved in order to obtain its positions and velocities throughout its time of motion whereas in the latter, the particle phase is assumed to behave as a pseudo-fluid so that its variables are analogous to the hydrodynamic ones .…”

“…The Lagrangian approach requires much more computational resources to calculate the trajectories and velocities of the massive number of particles encountered in realistic flow situations [10]. This led to extensive research effort in the development of Eulerian models [7,[11][12][13][14][15][16] and their application to different flow problems [7,9,[17][18][19][20][21][22].…”

“…Since the summation of the volume fractions of the two phases ought to be unity, the value of the volume fraction of either phase is used to formulate the governing equations because it determines how the two phases are coupled to each other. The oneway coupling model is often employed when the particle volume fraction (concentration) is very low compared to that of the fluid phase [6,8,9]. This implies that the volume fraction of the fluid phase remains constant at unity.…”