The transport of particles as solids, droplets or bubbles by a turbulent flow is a common enough feature in many natural and industrial processes; the mixing and combustion of pulverized coal in coal fired stations, and the dispersal of pollutants in the atmosphere and in rivers and estuaries are two of copious examples. In these sorts of flows the particles, while adverted by the mean flow, are dispersed by the random motion of the Turbulence. In many cases of interest, the particle size and density difference (inertia) are sufficiently large that the particles do not follow either the variations in mean carrier flow or the turbulence, so unlike the transport of a passive contaminant, particle transport does not generally obey the heat mass transfer analogy; this is especially so in a turbulent boundary layer, for example. Particle size and density can influence the transport in other ways. For instance, in many cases the mass loading of the dispersed phase is sufficiently high that it influences the motion of the carrier phase (two-way momentum coupling). To solve for the transport of an individual phase, it is necessary to solve for the transport of both phases coupled together. In the case of solid particles in a gas, this will occur when the volume fraction ≥ 10–5 [Elghobashi (1994)]. At higher volume fractions ≥ 10–3 the motion of a particle will be influenced by the motion of its neighbors either directly through inter-particle collisions or indirectly through their influence on the particle drag law, i.e., four way coupling between phases may occur. Even if such coupling does not occur, at sufficiently high inertia particle-wall interactions (which determine whether a particle will bounce or adhere or resuspend) may dominate the transport.
While particle transport refers in general to the way particle-turbulence interaction influences all transport properties in either phase, the concern here is with the influence of the turbulence on the spreading and mixing of particles and ultimately on their deposition on containment walls.
There are two methods, which are currently used to model particle transport. The first is the so-called "trajectory" approach where by solving their individual equations of motion, particles are tracked through a random flow field representative of the turbulence. [See, for example, Berlemont et. al. (1990)]. The principal advantages of this method are that it is very easy to implement and relatively easy to make changes in the physics, e.g., in the particle equation of motion and in the nature of the particle wall interactions, Sommerfeld (1990). The disadvantage is that even in very dilute flows it requires many realizations of the flow field to obtain an adequate picture of the transport. In general, the method is restricted to fairly simple types of random flight models and the influence of large scales structure in complex flows is generally not accounted for. In addition, some approximation has to be made for the Lagrangian timescale of the fluid seen by the particles.
The second approach known as the two-fluid approach treats both carrier (continuous) and dispersed phases as two fluids coexisting and interacting with one another, modeling each fluid in terms of a set of "continuum" equations which represent the net conservation of mass, momentum and energy of either species within some elemental volume of the mixture. [See Elghobashi (1994)]. The principal advantage of this approach over the trajectory approach is that it is computationally more efficient since it deals directly with the net transport property of interest. However, much controversy has surrounded the forms of these equations, not least of which concerns the type of averaging upon which they are based and the form of the constitutive relations that are necessary for their solution.
With the advent of supercomputers numerical simulation of particles in random flow fields typical of real turbulence has been crucial in furthering our understanding and modeling of particle transport. The application of direct numerical simulation (DNS) tracks particles through a flow field, each realization of which is a solution of the incompressible Navier Stokes equation. [See, for example, Squires and Eaton (1991a,b), Elghobashi and Truesdell (1992), and Mclaughlin (1989)]. The method is therefore relatively model free but limited to very low Reynolds number flow ~150. Despite these limitations, there is strong evidence that many of the main characteristics of real flows are being correctly predicted by DNS at least for moderate Reynolds numbers, especially in isotropic turbulence and turbulent boundary layers. The Reynolds number limitation has been overcome by Large Eddy Simulation (LES) resolving the 3-dimensional time dependent details of the largest scales of motion while using simple closure models for the smaller "sub-grid" scales. However, the method is only reliable with one way coupling, since in two way coupling it is the interaction between the sub-grid scales and the particles that dominates the coupling. Mention should also be made of simulations using so called kinematic turbulence. These are of two types: the first based on the Kraichnan's method of random Fourier modes [Fung et al. (1992)]; the second based on discrete vortex methods [Crowe et al. (1993)] used to examine the dispersion by large-scale organized structures in wakes and shear layers. Unlike DNS each realization of the flow field is not a solution of the Navier-Stokes equations so it is suitable only for studying dispersion. Nevertheless, useful information that does not depend upon the detailed dynamics can be obtained.
Of crucial importance to any model of particle transport is the form used for the particle equation of motion. Unfortunately no equation of motion exists appropriate for all conditions. Analytic forms exist for only a restricted range of flow conditions. Nevertheless, these have proven useful in establishing empirical models that extend the range of applicability. As a useful starting point the general aerodynamic force acting on a particle moving in an unsteady flow field can be seen as the sum of several forces, i.e.,
where is the inertial inviscid force that would act in the absence of viscocity, is the viscous force in the absence of inertial forces, and is the force due to the interaction of viscous and inertial effects. The inviscid force is composed of a force dependent upon the pressure gradient across the particle (and hence proportional to the local fluid acceleration) and an Added Mass term that accounts for the fact that a particle accelerating relative to the fluid is transferring momentum at a certain rate to the carrier flow: the increase in the apparent mass of the particle compared to the mass of the displaced fluid is commonly referred to as the added mass coefficient CM. (For a sphere CM = 1/2.) In addition, a particle will experience a lift force normal to the direction of its relative velocity and the local vorticity of the flow. [See Auton et al. (1988) for the first correct derivation of ]. Recent numerical simulations [Rivero et al. (1991)] for small to moderate Reynolds numbers in laminar flow indicate that it is probably correct under a wide range of conditions.
The viscous force can be represented as the sum of two components:
where is the viscous force as if the flow were steady and an extra force derived from the diffusion of vorticity generated at the surface of the particle at a rate proportional to the particle's relative acceleration. Since the diffusion rate is finite, this means that the force is dependent on the history of the particle motion.
In one particular case of unsteady Stokes flow, the equation of motion of a sphere has been derived exactly by Maxey and Riley (1983). For the low particle Reynolds number approximation is the same as that derived in Auton et al. (1988). The term corresponding to is due to Stokes steady drag while the remaining term is the history term commonly referred to as the Basset history term. Both terms differ from that in uniform steady flow in that they contain the influence of curvature in the ambient flow (evaluated at the center of the particle); these "extra forces" are known as Faxen forces. The form originally derived by Maxey and Riley implicitly assumes that the particle is introduced into the flow with the same velocity as the carrier flow. In general, this cannot be true so an extra term must be included to account for shear stresses induced by the formation of an intense thin vortex layer arising from the mismatch of initial and flow conditions [Maxey (1993)].
The extent of viscous-inertial interaction measured by had for some time been limited to the well-known corrections due to Oseen (1927) and Proudman and Pearson (1957) for moderate particle Reynolds numbers, with higher Reynolds number forms based on measurement. All forms are generally expressed in terms of a Drag Coefficient, CD , versus Reynolds Number, Rep. In recent times significant advances have been made in our understanding of the influence of inertial forces in unsteady uniform flows and in steady uniform shear flows. Mei et al. (1991b) and Lovelanti and Brady (1993) have shown that at long times when inertial effects become important the kernel of (associated with the diffusion of locally created vorticity) decays faster than t–1/2 based on simple diffusion theory. In a steady uniform shear a sphere, for instance, will experience a lift force [Saffman (1965, 1968)] as well as a rotating sphere in a uniform flow [Rubinow and Keller (1961)]. The latter is important near a wall where depending on the conditions of impact a rebounding particle will have some induced spin. In an unbounded flow with kinematic viscosity ν and shear gradient G the frequently used form of a lift derived by Saffman is limited to conditions of low particle Reynolds number in which the Saffman length (ν/G)1/2 << the Stokes length (ν/|vr|), where vr is the velocity of the particle relative to the local carrier flow velocity. In many flows of practical interest, especially in a turbulent boundary layer, these conditions are rarely satisfied. Very recently McLaughlin (1993) removed the restriction on Saffman length and extended the analytic forms to situations where the particle is near a wall. These lift predictions were consistent with those obtained numerically by Dandy and Dwyer (1990) who extended the range of lift for a stationary sphere to Reynolds numbers of about 100.
We examine transport of particles with low Reynolds number (Stokes flow) in homogeneous stationary turbulence in which the mean flow is either uniform or a simple shear. These are somewhat idealized flows but help us to define some useful parameters and highlight some simple effects that influence transport in real flows. A useful measure of the influence of particle size and density (particle inertia) is the particle relaxation or response time τp defined for a sphere of radius a and density ρp as:
So if G is the typical gradient of the mean flow and the integral timescale of the turbulence measured along a particle trajectory, particles will follow the mean flow if G τp << 1 and the turbulent fluctuations if . However, in the case of the turbulence the value is not prescribed in any way, depending upon the value τp itself and the Eulerian integral length scale, Le, and time scale, Tef, of the carrier flow turbulence. For instance, in the absence of drift as the ratio τp/Tef varies between 0 and ∞ the value of varies between (the fluid Lagrangian integral timescale) and Tef (the Eulerian integral timescale or eddy decay time).
In uniform flows we consider motion in a frame of reference moving with the mean flow. Following Taylor's 1921 theory of diffusion by continuous movements, it is reasonable to assume that in an unbounded flow in the limit of diffusion times >> both and τp the net spatial number density of particles decays in time according to Fick's Law of Gradient Diffusion with long time diffusion coefficients Dij(∞) which denote the limiting values of Dij(t) where:
where vi(t) and yj(t) are the particle velocity and position at time t in the i and j directions and < .. > is a global ensemble average. In the case of a quasi-steady Stokes drag law one can show from the equation of motion that for long times the ratio of the particle to carrier flow (fluid element) long time diffusion coefficients is the ratio , where is the carrier flow Lagrangian timescale. In the absence of drift, approximate calculations [Reeks (1977)] of for an isotropic random stationary Gaussian velocity field show that it is > 1 with a maximum value for large particle that increases as the ratio of a structure parameter (the parameter indicates the importance of persistent or coherent structures on the dispersion). That is, the long time particle diffusion is actually greater than that for an equivalent fluid element, a result also true of DNS isotropic decaying [Elghobashi and Truesdell (1992)] and forced turbulence [Squires and Eaton (1991a)] and confirmed by the experiments of Wells and Stock (1983). However, in the DNS study of decaying isotropic turbulence, the maximum value of this ratio occurred at an intermediate value of the ratio of τp/Tef. This is further linked to the influence of structure on transport and to the trapping of particles within vortices: increasing particle inertia was found to produce a preferential bias in the particle trajectory towards regions of low vorticity and high strain rate [Squires and Eaton (1991b)]. Similar biasing of the particle concentration towards regions of high velocity for particles settling under gravity produces a higher settling velocity than that in the absence of gravity [Maxey (1987)].
However a far more significant effect of gravitational settling is its influence on particle dispersion. If particles are settling out under gravity with some drift velocity vg they see a shorter fluid timescale than those with zero drift because, of the shorter time they spend in an eddy. So is smaller than its value for zero drift. If the transit time across any eddy due to the particles relative mean motion >> the eddy decay time (i.e., vg >> the carrier flow rms velocity then this form for Dij can be approximated by:
where Leij is the Eulerian fluid length scale appropriate for the i-j directions, one of these directions being in the direction of vg. The influence of the mean motion on the diffusion coefficient is known as the crossing trajectory effect [Yudine (1959)]; the implication is that diffusion is reduced normal and parallel to the direction of gravity. In the case of isotropic turbulence the ratio of these length scales is a 1/2 implying a similar ratio of the corresponding particle Diffusion Coefficients, an effect also confirmed by the experiments of Ferguson and Stock (1994) and by DNS [Squires and Eaton (1991a) and Elghobashi and Truesdell (1992)].
Finally, particle transport in homogeneous turbulence can be used to introduce the concept of particle pressure and to evaluate the so-called equation of state. Pressure is used here in a general sense to denote the surface forces on an elemental volume due to the net momentum transferred across those surfaces from the turbulent motion of individual particles. So at equilibrium there will be a balance between the gradient of these surface forces and the net weight of the particles due to gravity. Using an approach similar to that used in classical thermodynamics to evaluate the virial of system of interacting molecules one can show [Reeks (1991)] that:
where pij are the components of the particle stress tensor, the net mass density of the particles and u(p)(t) the earner flow velocity along a particle trajectory. The first term on the RHS is the contribution from the particle Reynolds stresses and the second term from the interfacial momentum transfer due to the turbulence, which is the gradient of a surface force at position in the flow so that:
A similar result is given by Simonin et al. (1993). The equation of state also implies a fundamental relationship between the quantities on the right hand side of Equation (6) and the particle diffusion coefficient Dij(∞). Thus:
This relationship, valid in all turbulent flows, expresses a fundamental equivalence at equilibrium between the gradients of surface forces and diffusion fluxes.
Analysis based on Stokes drag of the long term dispersion of particles in a simple shear [Reeks (1993)] highlights certain deficiencies in the traditional two fluid model assumption that the dispersed phase behaves as a Newtonian Fluid; i.e., at equilibrium in a bounded simple shear flow the shear stresses, for example, are directly proportional to the symmetric mean rate of shearing of the dispersed phase (the so-called Boussinesq assumption). In reality, each component of the particle stress is the sum of two components: a homogeneous component having a form identical to that in homogeneous turbulence, i.e., explicitly independent of the shear; and a deviatoric component which depends upon the value of the shearing of both the carrier and dispersed phases. The deviatoric components are zero for fluid point motion but dominate over the homogeneous component as Gτp >> 1. Furthermore, unlike the homogeneous stresses, the deviatoric stresses are always asymmetric (arising from the inherent asymmetry in the interfacial momentum transfer surface forcer). There are no violations of Cauchy's second law since the net surface couple generated on an elemental volume is precisely compensated by an equal and opposite body couple. However this asymmetry does influence the dispersion: with the streamwise diffusion coefficients increasing faster than the cross-streamwise components an elemental volume of particles is stretched into a long narrow ellipsoid that rotates until its major axis is along the streamwise direction [Hyland, McKee, and Reeks (1993)].
Crossing trajectories will also influence the particle dispersion: so as in homogeneous flow, it is significantly reduced when the particles are settling under gravity. They can also play a part when at any point in shear the mean particle velocity is different from that of the carrier phase [Huang and Stock (1994)].
A thorough investigation of the particle transport using an equation of motion which includes added mass and history forces (see the Maxey and Riley equation) is currently restricted to kinematic simulation carried in a stationary random isotropic Gaussian homogeneous velocity fields, see Mei et al. (1991). While particle RMS velocities are reduced, the particle dispersion coefficients in the long term are surprisingly little different from their values based on Stokes drag. This has some connection with the result that the particle stop distance is the same as its Stokes drag value. Wang and Stock (1988) and Mei et al. (1991) have also evaluated the influence of non-Stokes drag on particle dispersion in the same flow field: they found that the long term diffusion coefficient for large particles in the absence of gravity was slightly greater than the equivalent value based on Stokes drag. As to the dispersion in a simple shear only the influence of lift forces has so far been investigated. Ounis & Ahamdi (1991) have shown that the principle influence of lift is to increase the long term diffusion coefficient in the cross-streamwise direction compared to its Stokes drag value which is independent of the local shear. Hyland (1994) has confirmed this result and shown that the diffusion coefficient in the other directions are enhanced by the lift with similar trends in behaviour with increasing particle inertia. He has also considered the evolution of a point source of particles released at the center of the shear, observing a similar stretching and rotation of an evolving ellipsoid but at a greater rate to that for Stokes drag. The dispersion is in general described by the telegraph equation, which approaches the normal diffusion equation as t → ∞.
The simple analytic theories based on simple diffusion that we have used to quantify and analyze transport in the simple flows are no longer adequate in non uniform flows. However they form a basis for developing more general approaches. There are two approaches worthy mentioning, namely, the approaches by Reeks (1992) and Simonin et al. (1993) which in the end yield very similar results and generate the correct behavior in the simple flows discussed above. Both use the two-fluid model approach. Thus in the case of a simple Stokes drag the net momentum equation, for example, would be found by averaging the instantaneous equations for the momentum based on the equation of motion. Using density weighted (Favre) averages you obtain:
where is the density weighted mean particle velocity and the instantaneous carrier flow velocity is expressed as the sum of a mean and fluctuating component . The essential problem is then finding closed equations for the Reynolds stresses and the turbulent interfacial momentum transfer term . Reeks uses a pdf approach similar to that used in kinetic theory whereby the constitutive relations for the dispersed phase are derived from approximate solutions of a pdf equation representing the probability density that a particle has a certain velocity and position at time t (the analogue of the Boltzmann equation) with a term representing the net force experienced by a particle due to its interaction with random turbulent eddies. For a particle with velocity and position the component of this net force per unit volume is given by:
where is the instantaneous phase space density and the dispersion coefficients μij and λij and γi are dispersion coefficients dependent upon displacements in the velocity and position of a particle starting out at time zero and arriving at at time t. The net interfacial momentum term is then simply the value of integrated over all particle velocities, namely
A similar result has been derived by Simonin et al. (1993) by recognizing that the fluctuating interfacial term is responsible for an extra fluid-particle drift over and above that due to the mean relative velocity . It is the presence of this drift velocity in particle settling under gravity in homogeneous turbulence that causes the globally averaged settling velocity to be higher than in it is in still fluid [Maxey (1987)]. The form derived for this drift term gives an expression which assuming an exponential decaying autocorrelation for the fluid is identical to the value derived by Reeks. The form is an obvious generalization of the results derived for simple flows with suitable averages defined about a specific point in space, rather than a global average associated with the whole ensemble. We note that in addition there is an extra term, which is zero in homogeneous flows equivalent to a body force, which will tend to move particles from high to low regions of turbulence. In weakly inhomogeneous flows in the long time limit when the inertial acceleration term can be ignored the momentum equation reduces to a simple diffusion equation with value for Dij etc. given by the fundamental Equation (8) and a drift velocity (derived from the body force and the gradient of the particle co-variance). The latter is referred to as the turbophoretic velocity [Reeks (1992)] indicating that particles will drift from high to low regions of turbulence. This behavior has been observed in experiment [Young and Hanratty (1991)]. This has important implications for the behavior of particles in turbulent boundary layers [Reeks et al. (1993)].
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