Particle-laden flows

Particle-laden flows refers to a class of two-phase fluid flow, in which one of the phases is continuously connected (referred to as the continuous or carrier phase) and the other phase is made up of small immiscible particles (referred to as the dispersed or particle phase). Fine aerosol particles in air is an example of a particle-laden flow; the aerosols are the dispersed phase, and the air is the carrier phase.

The modeling of two-phase flows has a tremendous variety of engineering and scientific applications: pollution dispersion in the atmosphere, fluidization in combustion processes, aerosol deposition in spray medication along with many others.

Governing equations

The starting point for a mathematical description of almost any type of fluid flow is the classical set of Navier–Stokes equations. To describe particle-laden flows, we must modify these equations to account for the effect of the particles on the carrier, or vice versa, or both - a suitable choice of such added complications depend on a variety of the parameters, for instance, how dense the particles are, how concentrated they are, or whether or not they are chemically reactive. In most real world cases, the particles are very small and occur in low concentrations, hence the dynamics are governed primarily by the continuous phase. A possible way to represent the dynamics of the carrier phase is by the following modified Navier-Stokes momentum equation:

\[ \frac{\partial \rho u_i}{dt} + \frac{\partial \rho u_i u_j}{\partial x_j} = - \frac{\partial P}{\partial x_i} + \frac{\partial \tau_{ij}}{\partial x_j} + S_i \]

where \(S_i\) is a momentum source or sink term, arising from the presence of the particle phase. The above equation is an Eulerian equation, that is, the dynamics are understood from the viewpoint of a fixed point in space. The dispersed phase is typically (though not always) treated in a Lagrangian framework, that is, the dynamics are understood from the viewpoint of fixed particles as they move through space. A usual choice of momentum equation for a particle is\[ \frac{d v_i}{dt} = \frac{1}{\tau_p} (u_i - v_i)\]

where \( u_i \) represents the carrier phase velocity and \( v_i \) represents the particle velocity. \( \tau_p \) is the particle relaxation time, and represents a typical timescale of the particle's reaction to changes in the carrier phase velocity - this value can be thought of as the particle's inertia with respect to the dispersed phase. The interpretation of the above equation is that particle motion is hindered by a drag force. In reality, there are a variety of other forces which act on the particle motion (such as gravity, Basset history and added mass). However, for many physical examples, in which the density of the particle far exceeds the density of the medium, the above equation is sufficient. A typical assumption is that the particles are spherical, in which case the drag is modeled using Stokes drag assumption\[ \tau_p = \frac{\rho_p d_p^2}{18 \mu} \]

\(d_p\) is the particle diameter, \(\rho_p\), the particle density and \(\mu\), the dynamic viscosity of the carrier phase. More sophisticated models contain the correction factor\[ \tau_p = \frac{\rho_p d_p^2}{18 \mu} (1 + 0.15 Re_p^{0.687})^{-1} \]

where \(Re_p\) is the particle Reynolds number, defined as\[ Re_p = \frac{| \vec{u} - \vec{v} | d_p}{\nu} \]

Coupling

If the mass fraction of the dispersed phase is small, then one-way coupling between the phases is a reasonable assumption; that is, the dynamics of the particle phase are affected by the carrier phase, but the reverse is not the case. However if the mass fraction of the dispersed phase is large, the interaction of the dynamics between the two phases must be considered - this is two-way coupling.

A problem with the Lagrangian treatment of the dispersed phase is that once the concentration of particles becomes large, it becomes computationally expensive to track the sufficiently large sample of particles required for statistical convergence. In addition, if the particles are sufficiently light, they behave essentially like a second fluid. In this case, an Eulerian treatment of the dispersed phase is sensible.

Modeling

Like all fluid dynamics-related disciplines, the modelling of particle-laden flows is an enormous challenge for researchers - this is because most flows of practical interest are turbulent.

Direct numerical simulations (DNS) for single-phase flow, let alone two-phase flow, are computationally very expensive; the computing power required for models of practical engineering interest are far out of reach. Since one is often interested in modeling only large scale qualitative behavior of the flow, a possible approach is to decompose the flow velocity into mean and fluctuating components, by the Reynolds-averaged Navier-Stokes (RANS) approach. A compromise between DNS and RANS is large eddy simulation (LES), in which the small scales of fluid motion are modeled and the larger, resolved scales are simulated directly.

Experimental observations, as well as DNS indicate that an important phenomenon to model is preferential accumulation. Particles (particularly those with Stokes number close to 1) are known to accumulate in regions of high shear and low vorticity (such as turbulent boundary layers), and the mechanisms behind this phenomenon are not well understood. Moreover, particles are known to migrate down turbulence intensity gradients (this process is known as turbophoresis). These features are particularly difficult to capture using RANS or LES-based models since too much local information is removed by the application of the time-averaging operator.

Due to these difficulties, existing turbulence models tend to be ad hoc, that is, the range of applicability of a given model is usually suited toward a highly specific set of parameters (such as geometry, dispersed phase mass loading and particle reaction time), and are also restricted to low Reynolds numbers (whereas the Reynolds number of flows of engineering interest tend to be very high).

Further reading

• Mashayek, F. and Pandya, R. V. R. (1921), Progress in Energy and Combustion Science 20, 196, 196–212.