Particle-laden flow

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). In such flows, the mass and volume fractions of the dispersed phase are typically very low, and the dynamics are governed primarily by the continuous 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 dynamics of particle-laden flows are complex, and a prudent choice of governing equations depends largely on the extent to which the dynamics are affected by the presence of the dispersed phase. Typically, the governing momentum equation of the carrier phase is a 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 dispersed 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.

${\frac {dv_{i}}{dt}}={\frac {1}{\tau _{p}}}(u_{i}-v_{i})+g_{i}$

where $u_{i}$ represents the carrier phase velocity, $v_{i}$ represents the particle velocity and $g_{i}$ represents a body force (typically gravity). $\tau _{p}$ 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. 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:

$\tau _{p}={\frac {\rho _{p}d_{p}^{2}}{18\mu }}(1+0.15Re_{p}^{0.687})^{-1}$

where $Re_{p}$ is the particle Reynolds number.

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 carrier phase are assumed to be unaffected by the presence of the particles. How 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 mass fraction 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 is sensible.

Modeling

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 Direct numerical simluations (DNS) indicate that an important phenomenon to model is preferential accumulation. Particles are known to accumulate in regions of high shear and low vorticity (such as turbulent boundary layers) and also are known to migrate down turbulence intensity gradients (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 much higher).

Further reading

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