A Simplified Two-Fluid Model Based on Equilibrium Closure for a Dilute Dispersion of Small Particles

Abstract

Two-fluid formalisms that fully account for all complex inter-phase interactions have been developed based on a rigorous ensemble-averaging procedure. Here, we apply equilibrium approximation to particle velocity to simplify two-phase flow equations for the case of a dilute dispersion of particles much smaller than the flow scales. First, we extend an earlier approach to consider the rotational motion of the particles and seek an equilibrium approximation for the angular velocity of the particulate phase. The resulting explicit knowledge of the particulate phase translational and rotational velocities in terms of fluid velocity eliminates the need to consider the momentum equations for the particulate phase. The equilibrium approximations also provide precise scaling for various terms in the governing equations of the two-fluid model, based on which a simplified set of equations is obtained here. Three different regimes based on the relative strength of gravitational settling are identified, and the actual form of the simplified two-phase flow equations depends on the regime. We present two simple examples illustrating the use of the simplified two-fluid formalism.

1. Introduction

Here, we consider a fully coupled two-fluid formulation suitable for the description of laminar, transitional, and turbulent flows laden with a dilute dispersion of small spherical particles. For simplicity, here, attention will be restricted to a monodisperse suspension of particles of uniform size, but the formalism can be extended to the polydisperse case. Also, we will restrict formalism to the case of a dilute suspension, where close encounters between particles, in terms of short-range hydrodynamic interaction and other related collisional mechanisms, play a secondary role and therefore can be ignored for the most part. Although we will consistently use the term “particles”, the results to be presented are equally valid for bubbles and drops, provided they can be considered rigid and spherical. Otherwise, allowances need to be made to account for internal motion and departure from spherical shape.

For dispersions of sufficiently small particles, an Eulerian approach has often been employed, where the particulate phase is considered as a second fluid with an Eulerian field representation for its properties. Reference [1] has established a rigorous definition of the particle Stokes number ($St$) as the ratio of the time scale of the particle response to the fluid time scale measured as the inverse of the maximal compressional strain rate. For $St<1$, an Eulerian particle velocity can be defined as the equilibrium particle velocity field to which the Lagrangian particle velocities will entrain exponentially rapidly. The restriction that $St\ll 1$ is satisfied by sufficiently small-sized particles and henceforth, such particles will be referred to as “small particles”. For such small particles, it has been shown that the particle velocity field can be expressed in terms of the surrounding fluid velocity field as an expansion, with the non-dimensional particle time scale as the small parameter [1,2,3,4]. The relative velocity between the particle and the surrounding fluid to the leading order is given by the temporal and spatial gradients of the macroscale fluid velocity. Higher-order approximations of the relative velocity can similarly be obtained from expansion in a systematic manner. In [1] it was shown that the leading order equilibrium approximation captures important physics associated with particle distribution, such as preferential accumulation in regions of high strain or vorticity.

In the Eulerian approach for the particulate phase, the equations for the conservation of mass, momentum, energy, etc., are obtained through an appropriate averaging process. A variety of averaging techniques—time, space, and ensemble averages—have been employed in the past (c.f. [5,6,7,8,9,10,11,12,13]). Here, we seek appropriate governing equations that are suitable for direct and large-eddy simulations of the two-phase flow at the macro scale. This requires that the averaging process leaves the range of scales associated with the unaffected macroscale dynamics and only averages out the randomness associated with the random microscale distribution of particles. Such selective averaging can be achieved when the flow scales are well separated from those associated with the particles. In this limit, we carefully interpret the averaging process (such as that employed by [9,12,13]) to occur over an ensemble of small-scale particle arrangements. The averaging process thus results in a set of partial differential equations for the macroscale dynamics of both the dispersed and the continuous phases. These equations are in general quite complex even in the limit of dilute dispersions. Furthermore, they involve higher-order statistical terms, such as the kinematic Reynolds stress arising from small-scale fluctuation, that are not closed (c.f. [12,13]). Closure assumptions are thus required to obtain a self-consistent set of equations for macroscale dynamics.

Here, we apply the equilibrium approximation to particle velocity in two primary ways to simplify the above-mentioned rigorous set of two-phase flow equations. Reference [1] discussed the equilibrium approximation for the translational velocity of particles. Here, we extend this approach to consider the rotational motion of the particles and seek an equilibrium approximation for the angular velocity of the particulate phase. Thus, firstly, the explicit knowledge of the particle’s translational and rotational velocities in terms of the fluid velocity eliminates the momentum equations for the particulate phase from consideration. Secondly, the equilibrium approximations provide precise scaling for the various terms in the governing equations of the two-fluid model. Based on the scaling analysis, a simpler set of equations appropriate for the case of a dilute dispersion of small spherical particles is obtained here. In particular, the simplified equations avoid the closure difficulty as the kinematic Reynolds stress term is shown to be of smaller order by the scaling analysis.

The simplified equations obtained here provide an accurate description of macroscale two-phase flow in the limit of dilute distribution of small particles and fully account for the back effect of particles on the flow. At sufficiently small Reynolds numbers for the macroscale flow, one can solve this set of macroscale equations accurately using direct numerical simulation (DNS). At higher Reynolds numbers, the macroscale flow can be strongly turbulent, in which case the macroscale equations can either be averaged over an ensemble of realizations of the macroscale turbulence to obtain the Reynolds-averaged formulation of the macroscale two-phase flow, or a spatial filter can be applied to obtain a large-eddy formulation of the macroscale two-phase flow.

The theoretical framework and the ensemble-averaging procedure are described in Section 2. The equilibrium formulation of the translational and rotational velocities of the particle is developed in Section 3 and is then applied to the ensemble-averaged mass and momentum equations of the fluid and particulate phases to obtain the reduced set of two-phase flow equations. A discussion then follows in Section 4, where the importance of inter-particle interaction is assessed. Concluding remarks are presented in Section 5. The Appendix A presents two simple examples illustrating the use of the simplified two-fluid formalism: particles settling in a shear flow and particles centrifuging in a circular Couette flow.

2. Theoretical Framework and Ensemble Average

The particle diameter, d, and the mean spacing between particles, l, are the two most important length scales that define the particulate phase. In the case of dilute dispersions, the volume fraction of particles, ${\phi }_p\propto (d/l{)}^3$, is small and as a result, $l>>d$. The particles, when they are not neutrally buoyant, tend to preferentially accumulate in regions of high strain or high vorticity (depending on the particle-to-fluid density ratio) and thus result in a nonuniform distribution [14]. The mean spacing between particles and the volume fraction must therefore, in general, be considered as local quantities, dependent on both time and space. Here, we consider the particle volume fraction, averaged over the entire domain, to be sufficiently small that even in regions where particles preferentially accumulate, the close-range interaction between particles is weak. If important, particle–particle interaction needs to be accounted for in terms of the screening effect of the neighboring particles [15,16] and interaction-induced migration of particles down a concentration gradient [17,18]. These issues pertaining to inter-particle interaction are addressed in more detail in Section Inter-Particle Interaction.

A wide range of length and time scales can characterize the flow. The presence of particles introduces perturbation to the otherwise undisturbed background flow with additional length scales of the order of a particle diameter. It is thus appropriate to address the flow scales in the absence of particles. Let η be the smallest relevant length scale of the undisturbed single-phase flow, which would exist in the absence of particles for the same external driving conditions as for the two-phase flow. By definition, all other flow scales are larger than η.

Here, we focus on the limit where all the relevant length scales of the undisturbed flow are much larger than the particle diameter, d (i.e., $\eta >>d$). This allows one to separate the macrodynamics that occur on the scale of the undisturbed background flow from the microdynamics that occur on the particle scale [16,19]. The dynamics of the flow at these two levels are interconnected. The flow at the macro scale, by advecting the particles, dictates their local concentration and also provides the large-scale (or far-field) background environment for the microscale flow around individual particles. The presence of particles and the associated perturbation flow at the micro scale (or small scale) contribute to added dissipation at the macro scale and also influence the effective transport properties at the macro scale. Here, we consider the regime where this back effect of particles on the surrounding flow is important and therefore, the macroscale two-phase flow with the particles significantly differs from the corresponding undisturbed single-phase macroscale flow. In this case, a fully two-way coupled formulation for the mixture that accounts for both the influence of the macroscale flow on the particles and the back effect of the particles on the macroscale flow is required.

The stochastic nature of the two-phase flow may now be precisely defined. The microscale flow around each particle can be chaotic and turbulent at sufficiently high particle Reynolds numbers, based on particle diameter and relative velocity. As will be shown later in Section 3.3, for the sufficiently small particles that are under consideration here, their Reynolds number is quite small, and the flow around each particle remains deterministic, given by the Stokes flow [20,21]. The overall flow at the small scale is, however, stochastic owing to the random location and velocity of the particles, which we collectively refer to as the small-scale particle arrangement. As will be shown below, the velocity of sufficiently small particles nearly follows the macroscale flow velocity. In this case, the small-scale randomness is primarily from the random particle position. Nevertheless, an average of all possible small-scale arrangements of the particles smears out the small-scale flow details, and one obtains the flow at the macro scale.

We wish to consider a general situation in which the macroscale flow is turbulent even in the absence of particles and therefore, η represents the Kolmogorov length scale of the macroscale turbulence, undisturbed by the presence of particles. In a turbulent two-phase flow, thus, the stochastic nature of the flow is partly due to macroscale turbulence and partly due to small-scale randomness. Note that in the present context, the term “turbulent two-phase flow” implies the presence of randomness at the macro scale. Provided that the macro and small scales are well separated (i.e., for $\eta >>d$), it will prove useful to consider the two stochastic sources separately in the construction of an ensemble of all possible states. An ensemble of macroscale turbulence realizations and an ensemble of small-scale particle arrangements for each realization of the macroscale turbulence can be considered to form the overall (or composite) ensemble of all possible states. To obtain appropriate governing equations for the description of the macroscale turbulence, only an average of an ensemble of all possible small-scale particle arrangements is warranted. An average of the overall (or composite) ensemble of all possible states is not required, as it will erase the details of the macroscale turbulence and thus will correspond to a Reynolds-averaged formulation for the two-phase flow turbulence.

The ensemble averages employed by [7,9,12,13] can be interpreted to be from all possible small-scale arrangements of the particles for a given macroscale flow. Thus, in the limit $\eta >>d$, these ensemble-averaged governing equations can be considered appropriate for the description of the macroscale two-phase flow, independent of the laminar or turbulent nature of the macroscale flow. The precise interpretation of the ensemble-averaged governing equations will however depend on the relative magnitude of the length scales, $\eta$ and $l$, or on the number of particles that constitute a small-scale arrangement.

Provided $\eta >>l$, a suitable local volume, V, large enough to contain many particles for a meaningful volume average, but small enough to ignore macroscale variation within it, can be defined. Then, a volume average of V can be used to approximate the ensemble average. The advantage then is that the net effect of the random small-scale particle arrangement on the macroscale motion becomes deterministic, in much the same way that the net effect of the molecular fluctuations is deterministic and accounted for by the viscous term in the Navier–Stokes equation. Thus, in the limit $\eta >>l>>d$, the ensemble-averaged equations can be considered to govern the deterministic evolution of the individual realizations of the macroscale two-phase flow.

As pointed out by Batchelor [15], the ensemble average remains applicable even if an intermediate averaging volume, V, cannot be found. In this limit ($\eta ~l$ and $\eta >>d$), the net effect of the small-scale particle arrangement in any individual realization is not deterministic and as a result, the evolution of the macroscale two-phase flow is fundamentally stochastic in nature. This situation is much like in the single-phase large-eddy equations, where the net subgrid-scale effect is not deterministic and therefore, the evolution of the resolved-scale flow is fundamentally stochastic [22,23]. The ensemble-averaged governing equations thus now represent the macroscale flow only in the mean sense (with the mean taken from all possible small-scale particle arrangements). Thus, the ensemble-averaged equations such as those given by [7,9,12,13] can be given somewhat different interpretations depending on the limits, whether $\eta >>l>>d$ or $\eta ~l>>d$.

3. Mathematical Formulation

Consider a local region of the particulate mixture centered at location ${\mathit{x}}_0$ containing rigid spherical particles of diameter d within it. The ensemble-averaging technique for multi-phase flow has been well established [9,24,25]. Define an indicator function

$$H(\mathit{x},t)=\left\{\begin{array}{c}0\\ 1\end{array}\hspace{1em}\begin{array}{c}\mathrm{if}\mathit{x}\mathrm{is}\mathrm{in}\mathrm{the}\mathrm{fluid}\\ \mathrm{if}\mathit{x}\mathrm{is}\mathrm{in}\mathrm{the}\mathrm{particle}.\end{array}\right.$$

(1)

Let $<.{>}_{\mathrm{ss}}$ denote an ensemble average of all possible small-scale particle arrangements for a given state of the macroscale flow and let $<.{>}_{\mathrm{ms}}$ denote an ensemble average of all states of the macroscale flow. Let $<.{>}_{\mathrm{ms}\text{-}\mathrm{ss}}$ be an average of all possible realizations constructed from infinite identical experimental trials. In the limit of scale separation between the macro scale and particle size (i.e., $\eta >>d$), $<.{>}_{\mathrm{ms}\text{-}\mathrm{ss}}$ can be conveniently separated into $<.{>}_{\mathrm{ss}}$ and $<.{>}_{\mathrm{ms}}$. The average of all possible small-scale particle arrangements, $<.{>}_{\mathrm{ss}}$, is then a conditional ensemble average for a fixed state of the macroscale flow. Here, we limit the averaging process to only cover all possible small-scale arrangements of the particles and thus preserve the details of the macroscale flow.

The particle volume fraction is given by

$${\phi }_p(\mathit{x},t)=\hspace{0.33em}<H(\mathit{x},t){>}_{\mathrm{ss}},$$

(2a)

and the corresponding fluid volume fraction is given by

$${\phi }_f(\mathit{x},t)=\hspace{0.33em}<1-H(\mathit{x},t){>}_{\mathrm{ss}}.$$

(2b)

These and other quantities to be discussed below are to be interpreted for an individual realization of the macroscale flow. If ${\mathit{U}}_c$ is the velocity of a flowing composite of fluid plus the particles (taking appropriate values in each phase), the macroscale fluid velocity can be expressed as

$${\phi }_f\mathit{u}(\mathit{x},t)=\hspace{0.33em}<(1-H){\mathit{U}}_c{>}_{\mathrm{ss}},$$

(3a)

and the corresponding particle velocity field can be expressed as

$${\phi }_p\mathit{v}(\mathit{x},t)=\hspace{0.33em}<H{\mathit{U}}_c{>}_{\mathrm{ss}}.$$

(3b)

The volume-averaged and mass-averaged composite velocities of the mixture can then be obtained as

$${\mathit{u}}_v(\mathit{x},t)={\phi }_p\mathit{v}+{\phi }_f\mathit{u}\mathrm{and}{\rho }_m{\mathit{u}}_m(\mathit{x},t)={\rho }_p{\phi }_p\mathit{v}+{\rho }_f{\phi }_f\mathit{u},$$

(4)

where ${\rho }_p$ and ${\rho }_f$ are the density of the particle and the fluid and

$${\rho }_m={\rho }_p{\phi }_p+{\rho }_f{\phi }_f$$

(5)

is the effective density of the mixture.

3.1. Particle Velocity Field

Consider a sufficiently small volume, of the order of ${\eta }^3$, of fluid centered around x₀, as shown schematically in Figure 1, where a good approximation of the macroscale fluid velocity is described by $\mathit{u}\left({\mathit{x}}_0\right)$ and its local temporal and spatial derivatives. The number and position of particles within this volume are clearly random variables. Based on the analysis of Ferry and Balachandar [1], it can be anticipated that particles smaller than the Kolmogorov scale follow the surrounding macroscale flow. In other words, there exists a unique equilibrium particle velocity that is entirely determined by the surrounding macroscale fluid velocity field and its local temporal and spatial derivatives. Certainly, in the limit of infinitesimally small tracer particles, they move perfectly with the carrier fluid and their velocity is the same as the local fluid velocity. For finite-sized particles, the equilibrium particle velocity is in general different from the local fluid velocity, and this difference arises from mechanisms such as differential gravitational settling and inertial response of particles to local temporal and convective acceleration of the fluid. However, the equilibrium particle velocity can be expressed explicitly in terms of the fluid velocity field.

Deviation in the particle velocity away from equilibrium, along with the random positioning of particles, will contribute to the stochastic nature of the problem at the micro scale. The initial velocity of the particle, at the time of injection or nucleation, can contribute to departure from the equilibrium velocity. Processes such as collision and close interaction between particles will also contribute to deviation in particle velocity from the equilibrium. For the equilibrium particle velocity field to be meaningful, perturbations away from it, arising from random initial velocity and occasional close interactions, must rapidly be forgotten. For large particles, such departures from equilibrium will persist for a very long time and therefore, an equilibrium description is meaningless. However, for sufficiently small particles, it can be anticipated that the dependence on initial conditions and other perturbations away from equilibrium will die off fast. Beyond this transient period, the particles can be considered to be in equilibrium with the macroscale fluid flow.

Ferry and Balachandar [1] established that the appropriate inverse time scale for the flow is given by $\left|\sigma \right|$, where $\sigma$ is the maximal compressional strain rate. Here, the maximum is taken from the entire volume of flow and time. Provided the particle time scale, defined as

$$\tau ={\displaystyle \frac{d^2(2\rho +1)}{36\nu }},$$

(6)

is smaller than the fluid time scale (i.e., provided $\left|\sigma \right|\tau <1$), it was shown that the assumption of an equilibrium particle velocity field is appropriate. In other words, given a macroscale flow field, $\mathit{u}(\mathit{x},t)$, there exists a unique Eulerian particle velocity field, $\mathit{v}(\mathit{x},t)$, to which the velocity of all particles will entrain exponentially fast. In the above equation, $\rho ={\rho }_p/{\rho }_f$ is the density ratio of the particle to the surrounding fluid and $\nu$ is the kinematic viscosity of the fluid. If we take ${\left|\sigma \right|}^{-1}$ to be approximately given by the Kolmogorov time scale ($t_k$), the condition for unique equilibrium particle velocity field can be restated as

$${\tau }_{+}={\displaystyle \frac{\tau }{t_k}}={\left({\displaystyle \frac{d}{\eta }}\right)}^2{\displaystyle \frac{(2\rho +1)}{36}}<1.$$

(7)

Here, ${\tau }_{+}$ can be interpreted as the particle Stokes number ($St$), introduced in the Introduction. As can be seen from the above equation, the requirement ${\tau }_{+}<1$ is consistent with the assumption of particles being sufficiently smaller than the Kolmogorov length scale. For such small-sized particles, the equilibrium velocity field is not only appropriate in the ensemble-averaged sense as defined in Equation (3b) but also describes the particle velocity in any realization. In each realization, the particle velocity is uniquely given in terms of the fluid velocity and therefore, the only source of small-scale randomness is the particle position. Note that as defined in (3b), an ensemble-averaged particle velocity field is admissible even for larger particles (see [9]); however, it will not be uniquely determined in terms of the fluid velocity field.

Particle Velocity Field at Equilibrium

The starting point for obtaining the equilibrium particle velocity field is the Lagrangian equation of motion. For particles of sizes smaller than the relevant length scales of the flow, the perturbation flow at the small scale, due to the relative motion of the particle with respect to the surrounding flow, approaches the Stokes flow limit. This will be the case irrespective of the nature of macroscale flow. The Reynolds number of the small-scale flow will later be verified through scaling analysis. The Lagrangian equation of motion for a rigid particle in the limit of small Re can be written as [26,27]

$$\begin{array}{c}{m}_p{\displaystyle \frac{d\mathit{v}}{dt}}=3\pi \mu d({\mathit{u}}^s-\mathit{v})+(m_p-m_f)\mathit{g}+m_f{\displaystyle \frac{D{\mathit{u}}^v}{Dt}}+{\displaystyle \frac{m_f}{2}}\left({\displaystyle \frac{D{\mathit{u}}^s}{Dt}}-{\displaystyle \frac{d\mathit{v}}{dt}}\right)\\ +{\displaystyle \frac{3\pi d^2\mu }{2\sqrt{\nu }}}{\displaystyle \frac{d^{1/2}({\mathit{u}}^s-\mathit{v})}{d{t}^{1/2}}}\end{array}.$$

(8)

As argued in the above section, for small particles satisfying (7), the particle velocity $\mathit{v}$ can be taken to be the equilibrium velocity and is the same as the ensemble-averaged particle velocity defined in Equation (3b). The first term on the right represents viscous drag on the particle and the second accounts for the gravitational force. Acceleration due to gravity is denoted by the vector $\mathit{g}$. The dynamic viscosity of the surrounding fluid is μ and the mass of the particle and that of the fluid of equal volume are represented by $m_p$ and $m_f$. The third and fourth terms on the right arise from the pressure gradient and added-mass effects.

The last term accounts for the Basset history effect. The fractional derivative corresponds to the history integral with the standard one over square root decay for the kernel [28], which is appropriate for small values of the particle Reynolds number. Even in this limit, it is now well established that the decay of the history kernel is faster than one over square root over long times [29,30,31]. The fractional derivative is therefore only approximate, but will be sufficient for the present discussion. As we will see below, the history effect influences the equilibrium particle velocity field only at higher orders and therefore can be ignored. Here, $D/Dt$ and $d/dt$ represent total derivatives following the fluid and the particle, respectively.

In Equation (8), to the leading order, ${\mathit{u}}^s$ and ${\mathit{u}}^v$ are the same as the undisturbed fluid velocity at the particle location, and the higher-order terms include the Faxen correction as given below [26]:

$${\mathit{u}}^s=\mathit{u}+{\displaystyle \frac{d^2}{24}}{\nabla }^2\mathit{u}+\cdots \mathrm{and}{\mathit{u}}^v=\mathit{u}+{\displaystyle \frac{d^2}{60}}{\nabla }^2\mathit{u}+\cdots .$$

(9)

Here and in (8), all the fluid velocity and its spatial and temporal derivatives are evaluated at the particle location. The equation of particle motion can be non-dimensionalized to obtain

$$\begin{array}{c}{\tau }_{+}\left(1-{\displaystyle \frac{\beta }{3}}\right){\displaystyle \frac{d{\mathit{v}}_{+}}{d{t}_{+}}}=({\mathit{u}}_{+}^s-{\mathit{v}}_{+})+{\mathit{V}}_{s+}+{\displaystyle \frac{2}{3}}{\tau }_{+}\beta {\displaystyle \frac{D{\mathit{u}}_{+}^v}{D{t}_{+}}}+{\displaystyle \frac{{\tau }_{+}\beta }{3}}\left({\displaystyle \frac{D{\mathit{u}}_{+}^s}{D{t}_{+}}}-{\displaystyle \frac{d{\mathit{v}}_{+}}{d{t}_{+}}}\right)\\ +\sqrt{3{\tau }_{+}\beta }{\displaystyle \frac{d^{1/2}({\mathit{u}}_{+}^s-{\mathit{v}}_{+})}{d{t}_{+}^{1/2}}}\end{array},$$

(10)

where the Kolmogorov length, velocity, and time scales ($\eta$, $u_k$, $t_k=\eta /u_k$) have been used for non-dimensionalization and the non-dimensional quantities are denoted by the subscript ‘+’. Here, the non-dimensional settling velocity of the particle is given by ${\mathit{V}}_{s+}={\tau }_{+}(1-\beta ){\mathit{g}}_{+}$, where ${\mathit{g}}_{+}=\mathit{g}{t}_k^2/\eta$ measures the relative strength of acceleration due to gravity to the fluid acceleration at the Kolmogorov scale. The constant $\beta =3/(2\rho +1)$ with β = 0, 1, and 3, respectively, represents heavy particles, neutrally buoyant particles, and bubbles.

Equation (10) is clearly in the Lagrangian setting. For the present case of small particles, when an Eulerian representation for particle velocity is admissible, we follow [1] and first rewrite the equation of motion in the following form:

$$\left(1+\sqrt{3\beta {\tau }_{+}}{\displaystyle \frac{d^{1/2}}{d{t}_{+}^{1/2}}}\right)({\mathit{u}}_{+}^s-{\mathit{v}}_{+})=\left({\displaystyle \frac{d{\mathit{v}}_{+}}{d{t}_{+}}}-\beta {\displaystyle \frac{D{\mathit{u}}_{+}}{D{t}_{+}}}\right){\tau }_{+}-{\mathit{V}}_{s+}+O({\tau }_{+}^2).$$

(11)

Under equilibrium, a particle will fail to follow the local fluid due to two primary mechanisms. First, if the fluid locally undergoes temporal or convective acceleration, a particle of mass different from that of the displaced fluid will not respond the same way as the fluid. Second, the gravitational force will introduce the relative settling (or rise) of particles heavier (or lighter) than fluid. These two mechanisms are represented on the right-hand side of the above equation as the source of translational slip velocity between the particle and the ambient fluid. In the absence of these two mechanisms, the equilibrium particle velocity will be equal to the local fluid velocity.

For particles smaller than the Kolmogorov scale, the largest contribution to slip velocity arising from local fluid acceleration is primarily due to that of the smallest eddies. This justifies the choice of Kolmogorov scales for the non-dimensionalization of slip velocity and the different accelerations in Equation (10). The fluid and particle velocities by themselves will not scale with the Kolmogorov velocity; it is only the slip velocity that scales with $u_k$.

By inverting the operator on the left and rearranging terms (see [1,6] for details), we obtain the following expansion for the Eulerian particle velocity field:

$${\mathit{v}}_{+}\approx {\mathit{u}}_{+}^s+\left\{\begin{array}{ccc}-{\tau }_{+}(1-\beta ){\displaystyle \frac{D{\mathit{u}}_{+}}{D{t}_{+}}}& \mathrm{for}& {\mathit{V}}_{s+}\ll O\left({\tau }_{+}\right)\\ {\mathit{V}}_{s+}-{\tau }_{+}(1-\beta ){\displaystyle \frac{D{\mathit{u}}_{+}}{D{t}_{+}}}& \mathrm{for}& {\mathit{V}}_{s+}~O\left({\tau }_{+}\right)\\ {\mathit{V}}_{s+}-{\tau }_{+}\left\{\left(1-\beta \right){\displaystyle \frac{D{\mathit{u}}_{+}}{D{t}_{+}}}+{\mathit{V}}_{s+}·\nabla {\mathit{u}}_{+}\right\}& \mathrm{for}& {\mathit{V}}_{s+}~O\left(1\right)\end{array}\right.$$

(12)

The exact expression for $O\left({\tau }_{+}\right)$ particle velocity depends on the scaling of the settling velocity, and in the above equation, three different variants are shown for non-dimensional settling velocity ranging from much smaller than $O\left({\tau }_{+}\right)$ to O(1) quantities. Also note that ${\mathit{u}}_{+}^s={\mathit{u}}_{+}+{\displaystyle \frac{{\tau }_{+}\beta }{2}}{\nabla }_{+}^2{\mathit{u}}_{+}+O\left({\tau }_{+}^2\right)$ and therefore, the Faxen corrections need to be included in the estimation of the undisturbed fluid velocity seen by the finite-sized particle. References [32,33] have considered an expansion for particle velocity in the absence of added mass, Faxen, and history forces.

The leading effect of the pressure gradient force appears in the O(${\tau }_{+}$) correction term, while the Basset history and added-mass effects (the last two terms on the right-hand side of Equation (10)) appear at O(${\tau }_{+}^{3/2}$) and O(${\tau }_{+}^2$). Note, however, that the added-mass effect subtly appears in (12) through the definition of ${\tau }_{+}$. This at first can be surprising, since at small values of the particle Reynolds number, the Basset history force is expected to be stronger than the pressure gradient force. As indicated in Equation (10), the pressure gradient and history forces scale as ($\beta {\tau }_{+}$) and ($\sqrt{\beta {\tau }_{+}}$) in relation to the viscous drag force. Thus, for small ${\tau }_{+}$, the history force is expected to be more important than the pressure gradient force. In the case of heavy particles, since $\beta \to 0$, the relative strength of the history force further increases. The history force appearing only at O(${\tau }_{+}^{3/2}$) in Equation (12) is not at odds with this scaling argument. Equation (12) should correctly be interpreted as a statement on the equilibrium particle velocity. The source terms on the right-hand side of Equation (11) dictate the departure of the equilibrium particle velocity from the local fluid velocity. Thus, in the leading order, the difference between the fluid velocity and the particle velocity at equilibrium is given by this forcing, and the effects of history force and added mass have an influence only at a higher order. However, if particles are disturbed away from equilibrium, the standard scaling applies, and the contribution to the restoring force from the history effect will be more important than that from the pressure gradient force.

For neutrally buoyant particles ($\beta =1$), Equation (12) predicts ${\mathit{v}}_{+}\approx {\mathit{u}}_{+}+({\tau }_{+}/2){\nabla }_{+}^2{\mathit{u}}_{+}$. However, in Equation (10), only the gravitational force vanishes, and the effects of added mass and Basset history terms are clearly present. Again, the expansion in Equation (12) is a statement on the equilibrium particle velocity, and the velocity of neutrally buoyant particles, under equilibrium, is the same as the local fluid velocity experienced by the finite-sized particles (which are included in the Faxen correction). When disturbed from equilibrium, however, the added mass and history forces act along with the viscous drag to restore equilibrium. Provided the particles are sufficiently small, the equilibrium velocity is approached exponentially rapidly.

Higher-order extensions to Equation (12) can be obtained in a straightforward manner, but will not be considered here. The accuracy and convergence of the above expansion have been addressed in the context of particle motion in a turbulent channel flow [1,2]. The exact particle velocity as computed from the Lagrangian equation of particle motion (Equation (10)) was compared with the equilibrium particle velocity at O(1), O(${\tau }_{+}$), and O(${\tau }_{+}^2$) truncations of the expansion. It was shown that for small particles of ${\tau }_{+}$ < 1, the O(${\tau }_{+}$) correction given on the right-hand side of Equation (12) brings about significant improvement in the prediction of both one-time and long-time particle statistics.

In particular, the first-order correction can be shown to incorporate many important physics associated with inertial particle motion in turbulent flows [1]. In the limit of O(1) approximation, ${\mathit{v}}_{+}\approx {\mathit{u}}_{+}+{\mathit{V}}_{\mathrm{s}+}$, and the divergence of the particle velocity field is given by $\nabla ·{\mathit{v}}_{+}\approx \nabla ·{\mathit{u}}_{+}$. In this limit, an initially statistically well-mixed particle dispersion will remain so for all later times in an incompressible flow. With the O(${\tau }_{+}$) correction to the particle velocity, $\nabla ·{\mathit{v}}_{+}\ne 0$ and the preferential accumulation of heavier-than-fluid particles in regions of high strain and lighter-than-fluid bubbles in regions of high vorticity is accounted for [33]. Furthermore, the first-order correction can be shown to capture the turbophoretic migration of heavier-than-fluid particles towards the walls in the near-wall region and vice versa for bubbles [1].

Equation (12) thus provides a good approximation to the velocity of particles of sufficiently small Stokes numbers. The advantage of the explicit representation of the particle velocity field in terms of the fluid velocity is clear. Complete knowledge of the macroscale fluid velocity is sufficient to extract the associated particle velocity field. Unlike in the classical two-fluid models, additional momentum equations for the particle velocity field need not be solved. In the classical two-fluid approach, the solution of such particulate phase’s momentum equation is particularly cumbersome for very small particles as the equations become increasingly stiff with decreasing particle ${\tau }_{+}$. In fact, it is precisely in this limit of small ${\tau }_{+}$ that the expansion (Equation (12)) is expected to work the best. The accuracy of the equilibrium approximation has been sufficiently tested in various turbulent flows [34,35,36,37,38].

3.2. Particle Angular Velocity Field

The above line of reasoning can be followed for the Eulerian field representation of the angular velocity of the particles. The equation for the angular motion of a particle in non-dimensional form can be written as [26,39]

$${\tau }_{\mathsf{\Omega }+}{\displaystyle \frac{d{\mathsf{\Omega }}_{+}}{d{t}_{+}}}=({\displaystyle \frac{1}{2}}{\mathit{\omega }}_{+}^s-{\mathsf{\Omega }}_{+})+{\displaystyle \frac{{\tau }_{\mathsf{\Omega }+}}{2\rho }}{\displaystyle \frac{D{\mathit{\omega }}_{+}^v}{D{t}_{+}}}+\sqrt{{\displaystyle \frac{{\tau }_{\mathsf{\Omega }+}}{\rho }}}{L}_{+}({\displaystyle \frac{1}{2}}{\mathit{\omega }}_{+}^s-{\mathsf{\Omega }}_{+}),$$

(13)

where ${\mathsf{\Omega }}_{+}$ is the non-dimensional angular velocity of the particle. Here, ${\mathit{\omega }}_{+}^s$ and ${\mathit{\omega }}_{+}^v$ are the local macroscale vorticity of the surrounding fluid as seen by the particle and they include the Faxen correction as given below:

$$\begin{array}{c}{\mathit{\omega }}_{+}^s={\mathit{\omega }}_{+}+{\displaystyle \frac{3}{10}}{\tau }_{+}\beta \left({\nabla }_{+}^2{\mathit{\omega }}_{+}\right)+O\left({\tau }_{+}^2\right)\\ {\mathit{\omega }}_{+}^v={\mathit{\omega }}_{+}+{\displaystyle \frac{3}{14}}{\tau }_{+}\beta \left({\nabla }_{+}^2{\mathit{\omega }}_{+}\right)+O\left({\tau }_{+}^2\right)\end{array}$$

(14)

where ${\mathit{\omega }}_{+}={\nabla }_{+}\times {\mathit{u}}_{+}$ is the vorticity of the undisturbed ambient flow evaluated at the center of the particle. The above equation is similar in form to Equation (10). The non-dimensional rotational time scale of the particle is given by

$${\tau }_{\mathsf{\Omega }+}={\displaystyle \frac{{\tau }_{\mathsf{\Omega }}}{t_k}}={\displaystyle \frac{\rho }{60}}{\left({\displaystyle \frac{d}{\eta }}\right)}^2={\displaystyle \frac{3-\beta }{10}}{\tau }_{+}.$$

(15)

The first term on the right of (13) is the viscous torque due to the relative rotation of the particle with respect to the background fluid, and it is analogous to the Stokes drag term. The second term on the right will be present even in the absence of the particle and it is analogous to the pressure gradient term in the equation of translational motion. The last term is the Basset-like viscous history effect on the particle rotation. Here, $L_{+}$ is a linear integrodifferential operator, and its form for the case of particle rotation in a rotational ambient fluid can be found in Gatignol [26]. There is no term analogous to the added-mass effect, i.e., no added moment of inertia [19]. Furthermore, it is assumed that there is no externally imposed couple on the particles analogous to the gravitational force in the equation of translational motion. The above equation is valid in the limit of very small rotational Reynolds numbers, defined as ${\mathit{Re}}_{\omega }=d^2\left|\mathit{\omega }\right|/\nu$. In this limit, to the leading order, the angular velocity of the particle is the same as that of the surrounding fluid (i.e., $\mathsf{\Omega }\approx \mathit{\omega }/2$) and therefore, the Reynolds number based on particle rotation, ${\mathit{Re}}_{\Omega }=d^2\left|\mathsf{\Omega }\right|/\nu$, is also small.

As with the translational motion, the existence of a unique “equilibrium” angular velocity field for the particles requires that all transients, arising from initial conditions and from other close-range inter-particle and particle–wall interactions, must decay fast. If the rapid decay of the transients can be guaranteed, a unique “equilibrium” angular velocity field for the particle is meaningful. Such rapid decay of transients and approach to an equilibrium angular velocity field can be expected for sufficiently small particles. How small the particles need to be depends on the state of the macroscale flow. For particles whose rotational time scale is smaller than some characteristic time scale of the fluid, one can expect the transients to decay fast and the particle angular velocity to entrain to the equilibrium angular velocity. From Equation (15), it is clear that the time scales of translational and rotational motion are of the same order and therefore, it can be expected that the condition in (7) is appropriate even for the existence of an equilibrium angular velocity field for the particles.

Particle Angular Velocity Field at Equilibrium

Here, we try to express the angular velocity of the particle as an expansion in terms of the vorticity field of the surrounding flow. We expect the angular velocity of small particles to be almost the same as the local angular velocity of the surrounding fluid and any difference to be $O\left({\tau }_{\mathsf{\Omega }+}\right)$ or smaller. First, we rewrite Equation (13) in the following form:

$$\left(1+\sqrt{{\displaystyle \frac{{\tau }_{\mathsf{\Omega }+}}{\rho }}}{L}_{+}\right)\left({\displaystyle \frac{1}{2}}{\mathit{\omega }}_{+}^s-{\mathsf{\Omega }}_{+}\right)={\tau }_{\mathsf{\Omega }+}\left({\displaystyle \frac{d{\mathsf{\Omega }}_{+}}{d{t}_{+}}}-{\displaystyle \frac{1}{2\rho }}{\displaystyle \frac{D{\mathit{\omega }}_{+}^v}{D{t}_{+}}}\right).$$

(16)

The implication of the above equation is that the deviation of equilibrium particle angular velocity away from the local macroscale angular velocity of the fluid is driven by the inability of the particle to respond to the local angular acceleration of the surrounding fluid. In the absence of this driving mechanism, the equilibrium value of particle angular velocity has no reason to be different from that of the macroscale flow as seen by the particle.

For small ${\tau }_{\mathsf{\Omega }+}$, the operator on the left can be inverted, and we finally obtain the following expansion for particle angular velocity:

$${\mathsf{\Omega }}_{+}\approx \left\{\begin{array}{ccc}{\displaystyle \frac{1}{2}}{\mathit{\omega }}_{+}^s-{\displaystyle \frac{{\tau }_{\mathsf{\Omega }+}}{2}}\left({\displaystyle \frac{\rho -1}{\rho }}{\displaystyle \frac{D{\mathit{\omega }}_{+}}{D{t}_{+}}}\right)& \mathrm{for}& {\mathit{V}}_{\mathrm{s}+}\ll O\left(1\right)\\ {\displaystyle \frac{1}{2}}{\mathit{\omega }}_{+}^s-{\displaystyle \frac{{\tau }_{\mathsf{\Omega }+}}{2}}\left({\displaystyle \frac{\rho -1}{\rho }}{\displaystyle \frac{D{\mathit{\omega }}_{+}}{D{t}_{+}}}+{\mathit{V}}_{\mathrm{s}+}·\nabla {\mathit{\omega }}_{+}\right)& \mathrm{for}& {\mathit{V}}_{\mathrm{s}+}=O\left(1\right)\end{array}\right..$$

(17)

The history effect influences the equilibrium angular velocity of the particle only at the next level and therefore does not appear explicitly in the above equation. When disturbed away from this equilibrium, however, the history effect along with the viscous torque plays an important role in restoring the equilibrium.

By taking the curl of the particle velocity field (Equation (12)) and combining it with the above equation, it can be seen that

$$2{\mathsf{\Omega }}_{+}\approx \nabla \times {\mathit{v}}_{+}.$$

(18)

Thus, under the equilibrium condition, the dispersed particulate matter in the leading order does indeed behave like a (second) fluid. With the inclusion of the higher-order terms, departure from the true fluid-like behavior becomes apparent. For a true fluid, the translational and rotational motion are interdependent. As can be seen from the above equation, for the particulate phase, the translational and rotational velocities remain independent degrees of freedom, which is consistent with the Lagrangian rigid body behavior of particles. The departure from a true fluid-like behavior is partly due to the fact that the translational and rotational time scales of the particle, although of the same order, are not identical. Furthermore, the added-mass effect has no counterpart in the rotational motion of the particle. The vortex stretching mechanism, which plays an important role in the dynamics of fluid, is clearly absent in the case of rigid particles. These differences begin to manifest in the higher-order terms.

As with translational velocity, the advantage of the above explicit representation of a particle’s angular velocity is quite clear. Additional differential equations for the angular velocity of the particulate phase need not be solved. Computation of the fluid velocity field, ${\mathit{u}}_{+}(\mathit{x},t)$, is sufficient, from which both the fluid vorticity field ${\mathit{\omega }}_{+}(\mathit{x},t)$ and subsequently, from Equation (17), the particle angular velocity field, ${\mathsf{\Omega }}_{+}(\mathit{x},t)$, can be obtained without much effort.

3.3. Simple Estimates

For the present case of a dilute dispersion of rigid spheres of diameter much smaller than the relevant length scales of the surrounding macroscale flow, one can obtain the appropriate measures of particle time scale and Reynolds numbers. We recall the following standard scaling for the turbulent small-scale (Kolmogorov scale) dissipative eddies:

$$\mathrm{Length}\mathrm{scale}:\eta ={\displaystyle \frac{{\nu }^{3/4}}{{\epsilon }^{1/4}}};\mathrm{Velocity}\mathrm{scale}{v}_k=(\nu \epsilon {)}^{1/4};\mathrm{Time}\mathrm{scale}{t}_k=(\nu /\epsilon {)}^{1/2}.$$

(19)

If the flow on the macro scale is inhomogeneous, then the above estimates must be regarded as local, and the local small-scale dissipation, $\epsilon \propto <{\mathit{u}}^{\prime }·{\mathit{u}}^{\prime }{{>}_{\mathrm{ms}}}^{3/2}/L$, is dictated by the energy cascade from the energy-containing eddies [40]. Here, the angle brackets represent an ensemble average of all the macroscale turbulent states and ${\mathit{u}}^{\prime }=\mathit{u}-<\mathit{u}{>}_{\mathrm{ms}}$ denotes macroscale fluid velocity perturbation away from its ensemble average. L is the appropriate length scale for the energy-containing eddies. The turbulence Reynolds number defined in terms of the energy-containing eddies can then be expressed in terms of the length scale ratio as

$${\mathit{Re}}_L={\displaystyle \frac{<{\mathit{u}}^{\prime }·{\mathit{u}}^{\prime } {{>}_{\mathrm{ms}}}^{1/2}L}{\nu }}\propto {\left({\displaystyle \frac{L}{\eta }}\right)}^{4/3}.$$

(20)

The non-dimensional translational and rotational time scales of the particle have already been established in Equations (7) and (15) with the Kolmogorov time scale as the representative fluid scale. Instead of the Kolmogorov scale, if any other macroscale fluid time scale were to be used, the corresponding non-dimensional translational and rotational time scales of the particle will further decrease. Therefore, for sufficiently small particles, ${\tau }_{+}$ and ${\tau }_{\mathsf{\Omega }+}$ will be less than unity and rapid convergence of the equilibrium expansions for translational and rotational particle velocity (Equations (12) and (17)) can be expected. In the case of heavier-than-fluid particles ($\rho >>1$), the $O\left({\tau }_{+}\right)$ correction can play a significant role in determining the relative motion between the fluid and the particles and in dictating the preferential accumulation of particles. In the case of near-neutrally buoyant and lighter-than-fluid particles and bubbles of small size, the corresponding ${\tau }_{+}$ and ${\tau }_{\mathsf{\Omega }+}$ will be much smaller than unity and therefore, the first-order corrections are of lesser importance.

Estimates of particle Reynolds numbers will be obtained next, for which first we need to establish the appropriate scaling for relative velocity. In the limit of a strong external gravitational force on the particle, its terminal velocity, ${\mathit{V}}_{s+}$, in the leading order determines the translational slip between the particle and the surrounding fluid, from which the estimate for the translational particle Reynolds numbers can be obtained as

$${\mathit{Re}}_v={\displaystyle \frac{d\left|\mathit{u}-\mathit{v}\right|}{\nu }}\propto {\displaystyle \frac{{\mathit{V}}_s}{<{\mathit{u}}^{\prime }·{\mathit{u}}^{\prime }{{>}_{\mathrm{ms}}}^{1/2}}}\left({\displaystyle \frac{d}{\eta }}\right){Re}_L^{1/4}.$$

(21)

Thus, provided the terminal velocity is of the order of macroscale turbulent velocity fluctuation or less, the assumption of small particle Reynolds numbers is well justified for small particles. In the limit when gravitational settling is not strong, the translational slip is $O\left({\tau }_{+}\right)$. In this limit, the corresponding estimate of the particle Reynolds number can be expected to be lower as well. If we assume $D\mathit{u}/Dt$ to scale as $\eta /t_k^2$, the estimate for translational slip velocity can then be obtained from Equation (12) as $\tau \eta /t_k^2$. From this, the estimate of the particle Reynolds number can be obtained as

$${\mathit{Re}}_v\propto {\displaystyle \frac{(2\rho +1)}{36}}{\left({\displaystyle \frac{d}{\eta }}\right)}^3.$$

(22)

Similarly, if we take $\mathit{\omega }$ to scale as $1/t_k$ the estimate of the rotational Reynolds number can be obtained as

$${\mathit{Re}}_{\omega }\propto {\left({\displaystyle \frac{d}{\eta }}\right)}^2.$$

(23)

The Reynolds number based on rotational slip between the particle and the surrounding flow will be even smaller. Thus, for small particles, both ${Re}_v$ and ${Re}_{\omega }$ are guaranteed to be much smaller than 1. In fact, the translational Reynolds number is smaller than the non-dimensional time scales by a factor of $d/\eta$. In the above equations, the estimates of relative translational velocity and macroscale vorticity are taken to be those based on the Kolmogorov scale. It can be easily verified that if, instead, estimates were to be made based on larger scales of motion, the corresponding Reynolds numbers would decrease further. Thus, the estimates in Equations (22) and (23) are the upper bound.

The validity of the various assumptions made in Section 3.1 and Section 3.2 can now be verified. The use of Equations (8) and (13) for the translational and rotational motion of the particle is appropriate. In them, the low-Reynolds-number form of the history kernel is similarly justified. In summary, the use of expansions (12) and (17) for the translational and rotational motion of the particle is appropriate for the present case.

3.4. Conservation Equations

3.4.1. Mass Conservation

Here, for simplicity, we will assume that there is no mass transfer between the continuous and the discrete phases and also assume the particles to be of fixed size and shape. Any volume change in response to local pressure change, which might occur in the case of bubbles, is ignored. Then, the phase indicator function is a material variable and is always one following the particle and zero following the fluid (see Equation (1)). In other words,

$${\displaystyle \frac{\partial (1-H)}{\partial t}}+{\mathit{U}}_{\mathit{c}}·\nabla (1-H)=0\mathrm{and}{\displaystyle \frac{\partial H}{\partial t}}+{\mathit{U}}_{\mathit{c}}·\nabla H=0.$$

(24)

We also note that $\nabla ·{\mathit{U}}_{\mathit{c}}=0$. By taking an ensemble average of the first result of Equation (24), we obtain the mass balance for the fluid phase, which can be expressed as

$$\nabla ·\mathit{u}={\displaystyle \frac{\partial {\phi }_p}{\partial t}}+\nabla ·\left({\phi }_p\mathit{u}\right)$$

(25)

Understandably, even in the case of an incompressible fluid, the divergence of macroscale flow depends on the local change in the particle volume fraction. In the limit of a small volume fraction, ${\phi }_p\to 0$, one might ignore this effect and simplify the continuity equation to the standard incompressibility condition: $\nabla ·\mathit{u}=0$.

For the particulate phase, by taking the ensemble average of the second of Equation (24), we obtain

$${\displaystyle \frac{\partial {\phi }_p}{\partial t}}+\nabla ·\left(\mathit{v}{\phi }_p\right)=0.$$

(26)

It is the advection of particles following the particle velocity field, $\mathit{v}(\mathit{x},t)$, that results in the preferential accumulation of particles in regions of high strain rate or high rotational rate, depending on whether they are heavier than or lighter than the surrounding fluid. The mass balance in terms of volume-averaged and mass-averaged composite velocities of the mixture is

$$\nabla ·{\mathit{u}}_v=0\mathrm{and}{\displaystyle \frac{\partial {\rho }_m}{\partial t}}+\nabla ·({\rho }_m{\mathit{u}}_m)=0.$$

(27)

Note that the volume-averaged mixture velocity remains divergence-free.

3.4.2. Momentum Equation

The appropriate momentum equation for the macroscale fluid motion can be obtained by starting with the Navier–Stokes equation for the region outside the particles, with appropriate boundary conditions on all the particles and at the boundaries of the large-scale domain of interest, supplemented with the equation of translational and rotational motion for each of the particles. This mathematical formulation of the problem is fundamental in that it describes the evolution of the flow both at the macro scale and at the small scale faithfully in each realization. For the present case of a dilute distribution of small particles, the clear separation of particle scale from the undisturbed flow scale allows for an average description of the macroscale motion. Substantial simplification arises by taking an ensemble average over all possible probabilistic distributions of particles at the small scale to obtain the net effect of the microscale detail on the macroscale motion. The process of obtaining the ensemble-averaged momentum equations [9] and their closure has been presented in great detail by [13,41] (see also Prosperetti [42]). The resulting macroscale momentum equation for the flow can be expressed as

$$\begin{array}{c}{\displaystyle \frac{\partial {\phi }_f{\rho }_f\mathit{u}}{\partial t}}+\nabla ·({\phi }_f{\rho }_f\mathit{u}\mathit{u})-{\phi }_f{\rho }_f\mathit{g}+\nabla p_m=\nabla ·\left(2{\mu }_{eff}{\mathit{E}}_v\right)\\ -n{\mathbf{F}}_{\mathrm{hyd}}+\nabla ·\left({\phi }_f{\mathit{R}}_f\right).\end{array}$$

(28)

In the above equation, $p_m$ is the mixture pressure, which is taken to be that of the fluid, since particles rapidly equilibrate to this pressure [43]. In the above equation, n is the number density of the particles, defined as

$$n={\displaystyle \frac{6{\phi }_p}{\pi d^3}}.$$

(29)

Note that in the macroscale momentum equation, instead of the term $\nabla p_m$, the pressure effect is often represented in terms of the continuous-phase pressure as ${\phi }_f\nabla p_f$. The relation between the two forms has been explored in detail by Marchioro et al. [41]. The difference between the two is asymptotically small for the present topic of the dilute dispersion of small particles. The assumption of pressure equilibrium must be carefully considered in inviscid and compressible multi-phase flows. The hyperbolicity of the multi-phase flow equations is often enforced with the introduction of an interfacial pressure that differs from the local fluid pressure [44]. Recent research [45] has however shown that by introducing particle–fluid–particle stress [46,47] in the governing equations, well-posedness can be recovered.

In the above momentum equation, ${\mathit{E}}_v$ is the strain rate of the mixture, defined based on the volume-averaged mixture velocity, ${\mathit{u}}_v$, as

$${\mathit{E}}_v={\displaystyle \frac{1}{2}}\left\{\nabla {\mathit{u}}_v+{\left(\nabla {\mathit{u}}_v\right)}^T\right\}.$$

(30)

In the derivations of Marchioro et al. [41], it was shown rigorously that the appropriate stress that should appear in the above form of the momentum equation is the mixture stress. In Equation (28), the first two terms on the right arise from the mixture stress. Even in the limit of a dilute dispersion of small particles considered here, the expression for mixture stress, as given in [41], includes additional terms. For clarity, such additional terms that arise from the rigorous small-scale ensemble-averaging procedure are ignored here, as they will be shown below to be of smaller order.

The velocity fluctuation at the micro scale, upon ensemble averaging of all microscale arrangements, gives rise to the kinematic Reynolds stress, defined as

$${\mathit{R}}_f=-{\rho }_f<\tilde{\mathit{u}}\tilde{\mathit{u}}{>}_{\mathrm{ss}},$$

(31)

where $\tilde{\mathit{u}}$ is the local small-scale perturbation velocity field away from the macroscale flow due to the presence of particles. (Note: $\tilde{\mathit{u}}$ is not to be confused with ${\mathit{u}}^{\prime }$. The latter is a perturbation flow on the macro scale with the smallest length scale given by η, while the former represents perturbation flow on the scale of the particle diameter arising purely due to the presence of particles.) For the present case of a dilute distribution of small particles, we expect this contribution to the stress field to be small (as will be confirmed in Section 3.4.4).

${\mathit{F}}_{\mathrm{hyd}}$ is the hydrodynamic force per particle, and it is taken to include all the forces that act on the particles, except for the external gravitational force acting on the particle. The viscous and inertial forces exerted by the surrounding fluid are taken to be included in ${\mathit{F}}_{\mathrm{hyd}}$. The equation of motion for the particulate phase can then be written as

$${\rho }_p{\phi }_p\left[{\displaystyle \frac{\partial \mathit{v}}{\partial t}}+\mathit{v}·\nabla \mathit{v}\right]={\rho }_p{\phi }_p\mathit{g}+n{\mathit{F}}_{\mathrm{hyd}}+{\rho }_p\nabla ·\left({\phi }_p{\mathit{R}}_p\right),$$

(32)

where ${\mathit{R}}_p$ is the analogous kinematic Reynolds stress arising from the actual particle velocities deviating from $\mathit{v}$. Within the framework of equilibrium particle velocity, the only source of randomness is in their position, since their velocity is explicitly known as a function of their macroscale location. Perturbation away from the equilibrium may arise due to inter-particle interaction, which can be considered to be rare in a dilute dispersion. Thus, all particles in some neighborhood, in the leading order, experience the same hydrodynamic force exerted on them by the surrounding macroscale flow. The particles collectively exert an opposite force on the surrounding fluid, and the back effect on the flow is represented by the last-but-one term on the right-hand side of Equation (28). At the order of accuracy represented by the equilibrium translational velocity given in Equation (12), the history and lift force terms can be neglected and the hydrodynamic component of the force can be written as

$${\mathit{F}}_{\mathrm{hyd}}=\left\{\begin{array}{ccc}3\pi \mu d(\mathit{u}-\mathit{v}+{\displaystyle \frac{d^2}{24}}{\nabla }^2\mathit{u})+m_f{\displaystyle \frac{D\mathit{u}}{Dt}}& \mathrm{for}& {\mathit{V}}_{\mathrm{s}+}\ll O\left({\tau }_{+}\right)\\ 3\pi \mu d(\mathit{u}-\mathit{v}+{\displaystyle \frac{d^2}{24}}{\nabla }^2\mathit{u})-m_f\mathit{g}+m_f{\displaystyle \frac{D\mathit{u}}{Dt}}& \mathrm{for}& {\mathit{V}}_{\mathrm{s}+}\ge O\left({\tau }_{+}\right)\end{array}\right.$$

(33)

The above hydrodynamic force, when substituted into the particulate phase momentum equation given in (32), is fully consistent with the equation of motion (8), except for the kinematic particulate Reynolds stress, which will be shown below to be negligible. Note that the hydrodynamic force term in (32) includes the pressure gradient force.

In a similar manner, ${\mathit{T}}_{\mathrm{hyd}}$ is the hydrodynamic torque acting on a particle from the surrounding fluid, and the collective action of the relative rotational motion of all the particles in a local volume is represented by the second term on the right-hand side of (28), where $\epsilon$ is the third-rank Levi–Civita tensor. Note that the differential rotation of the particles can be thought of as contributing to an antisymmetric component of the stress field. To the order of accuracy represented by the equilibrium rotational velocity given in Equation (17), the history term can be neglected, and the hydrodynamic component of the torque can be written as

$${\mathit{T}}_{\mathrm{hyd}}=\pi \mu d^3\left({\displaystyle \frac{\mathit{\omega }}{2}}-\mathsf{\Omega }\right)+{\displaystyle \frac{\pi d^5}{120}}{\rho }_f{\displaystyle \frac{D\mathit{\omega }}{Dt}}+{\displaystyle \frac{\pi d^5}{80}}\mu {\nabla }^2\mathit{\omega }.$$

(34)

3.4.3. Enhanced Dissipation

The form of the terms on the right-hand side of the fluid momentum in Equation (28) can be investigated with consideration of the associated dissipative effect on the macroscale motion. For the case of particles smaller in size than the flow scales considered here, the macroscale flow impressed upon the particles located in a small volume around ${\mathit{x}}_0$ (see Figure 1) can be represented as

$$\mathit{u}(\mathit{x},t)\approx \mathit{u}({\mathit{x}}_0,t)+(\mathit{x}-{\mathit{x}}_0)·{\left.\nabla \mathit{u}\right|}_{\mathit{x}={\mathit{x}}_0}+{\displaystyle \frac{1}{2}}(\mathit{x}-{\mathit{x}}_0)(\mathit{x}-{\mathit{x}}_0):{\left.\nabla \nabla \mathit{u}\right|}_{\mathit{x}={\mathit{x}}_0}.$$

(35)

The far-field velocity experienced by the particle, ignoring the effect of all other particles in its neighborhood, is thus approximated to be quadratically varying. To be asymptotically consistent, it is important to consider not only the local relative motion with respect to the surrounding fluid, ${\mathit{u}}_r({\mathit{x}}_0,t)=\mathit{u}({\mathit{x}}_0,t)-\mathit{v}({\mathit{x}}_0,t)$, but also the local strain rate, $\mathit{E}({\mathit{x}}_0,t)$, and rotation rate, $\mathit{W}({\mathit{x}}_0,t)$, of the macroscale flow ($\nabla \mathit{u}=\mathit{E}+\mathit{W}$) and the quadratic variation [13,48].

The motion of particles located at ${\mathit{x}}_0$ can be represented as

$$\mathit{v}({\mathit{x}}_0,t)+\mathsf{\Omega }({\mathit{x}}_0,t)\times (\mathit{x}-{\mathit{x}}_0),$$

(36)

and they influence the macroscale flow in three significant ways. These contributions arise (a) from the relative translational motion between the particle and the surrounding flow (i.e., $\mathit{u}({\mathit{x}}_0,t)-\mathit{v}({\mathit{x}}_0,t)$), which exerts a net force on the fluid; (b) from the relative rotational motion of the particle with respect to the surrounding flow (i.e., ${\displaystyle \frac{1}{2}}\mathit{\omega }({\mathit{x}}_0,t)-\mathsf{\Omega }({\mathit{x}}_0,t)$), which exerts a net torque on the fluid, here, the local vorticity of the macroscale flow is defined in terms of the rotation rate tensor as ${\mathit{W}}_{ij}=-{\displaystyle \frac{1}{2}}{\epsilon }_{ijk}{\mathit{\omega }}_k$; and (c) from the macroscale strain rate field ($\mathit{E}({\mathit{x}}_0,t)$), for which the corresponding stress field is altered due to the presence of the particles.

The effect of small-scale fluid motion, arising from the relative translational slip of the particles, on the macroscale motion contributes to added dissipation of energy. The increase in kinetic energy of the fluid at the macro scale due to the force exerted on it (third term on the right-hand side of Equation (28)) from the particles per unit volume of the mixture is $-n({\mathit{F}}_{\mathrm{vis}}·\mathit{u})$, while the increase in kinetic energy of the particles within a unit volume of the mixture is $n({\mathit{F}}_{\mathrm{vis}}·\mathit{v})$, where ${\mathit{F}}_{\mathrm{vis}}$ is the viscous component of the hydrodynamic force, ${\mathit{F}}_{\mathrm{hyd}}$. The difference between the two contributes to the kinetic energy of the small-scale fluid motion and to the leading order can be written as

$${\displaystyle \frac{{\phi }_p{\rho }_f(2\rho +1)}{2\tau }}{\left|\mathit{u}-\mathit{v}\right|}^2.$$

(37)

This non-negative flow of energy at the small scale is therefore dissipative as far as the macroscale motion is concerned. The contributions to dissipation from the Faxen term and the Basset history force appear only at a higher order. The inertial contributions to total hydrodynamic force (added mass and pressure gradient forces), ${\mathit{F}}_{\mathrm{ine}}$, also give rise to an energy exchange to the small-scale fluid motion, which can be written as

$$n{\mathit{F}}_{\mathrm{ine}}·\left(\mathit{u}-\mathit{v}\right).$$

(38)

Unlike (37), the above can take both positive and negative values, indicating the flow of energy both to and from the small-scale fluid motion, and therefore is not dissipative in nature.

The presence of particles in the macroscale strain field results in enhanced stress, even in the absence of any relative translational or rotational motion. In the case of the random distribution of spherical particles, without any preferred small-scale structure or arrangement of particles, the stress–strain relation remains isotropic in the limit $\mathit{Re}\to 0$ and ${\mathit{Re}}_{\omega }\to 0$. The net effect of particles is to increase the effective viscosity of the mixture above that of the pure fluid by Einstein’s correction factor: ${\mu }_{eff}=\mu (1+5{\phi }_p/2)$ [15]. Here, the argument is standard: the rate of work done by the fluid through the symmetric component of stress per unit volume of the mixture is $\nabla ·\left(2{\mu }_{eff}\mathit{u}·{\mathit{E}}_v\right)$, while the associated rate of increase in the kinetic energy of the fluid is $\mathit{u}·\nabla ·\left(2{\mu }_{eff}{\mathit{E}}_v\right)$ and the difference accounts for energy dissipation. In addition to the effective viscosity, there is additional contribution to the stress field arising from small-scale momentum flux. The velocity fluctuation at the micro scale gives rise to the kinematic Reynolds stress, defined in (31), which is also accounted for in the ensemble-averaged fluid momentum (Equation (28)).

3.4.4. Scaling Analysis

The fluid momentum (Equation (28)) along with the expressions for the hydrodynamic force and torque (33) and (34) remains formidable for analysis and computations. Fortunately, in the limit of a dilute dispersion of small particles, all of the contributions are not of equal importance. The real advantage of the equilibrium expansion for the translational and rotational velocities is that precise scaling of the various terms of the momentum equation can now be established. Scaling arguments can be used to identify terms of lesser importance, which can then be ignored to simplify the momentum equation. In particular, we consider both the non-dimensional particle time scale, ${\tau }_{+}$, and the particle volume fraction, ${\phi }_p$, to be small quantities and therefore retain only terms up to $O({\tau }_{+},{\phi }_p)$ and systematically ignore all higher-order terms.

In Section 3.2, it has already been argued that, while the macroscale fluid and particle velocities are of the order of rms turbulent velocity fluctuation, the translation slip velocity scales differently. When the gravitational effect is not strong (i.e., ${\mathit{V}}_{\mathrm{s}+}\le O\left({\tau }_{+}\right)$), the dimensional translational slip velocity scales as $\tau \eta /t_k^2$. Under a stronger gravitational effect (i.e., ${\mathit{V}}_{\mathrm{s}+}\gg O\left({\tau }_{+}\right)$), the translational slip in the leading order is dictated by the settling velocity. However, it must be noted that this leading-order slip velocity, ${\mathit{v}}_{+}-{\mathit{u}}_{+}\approx {\mathit{V}}_{\mathrm{s}+}$, is a constant dependent only on the density ratio, particle size, etc., and therefore, its spatial and temporal derivatives are identically zero.

We will now examine the relative scaling of the various terms in the momentum equation with respect to the inertial terms. We will continue to use the Kolmogorov scaling. The gravitational body force in Equation (28) can be combined with the drag force on the fluid due to the gravitational settling of the particles and with the buoyancy force to obtain the following net direct influence of gravity:

$${\phi }_f{\rho }_f\mathit{g}+3\pi n\mu d{\mathit{V}}_s+n{m}_f\mathit{g}={\rho }_m\mathit{g}$$

(39)

where ${\rho }_m$ is the effective mixture density as defined in Equation (5). Only perturbation in the mixture density away from the mean, arising from fluctuations in the volume fraction of particles, is dynamically significant. The relative scaling of the gravitational force in the fluid momentum equation is then ${\phi }_p(\rho -1){\mathit{V}}_{\mathrm{s}+}{\tau }_{+}^{-1}$. The dimensional grouping ${\phi }_p(\rho -1)$ can be identified as the local non-dimensional mass loading of the particles, which for a dilute dispersion is guaranteed to be small for the case of bubbles and near-neutrally buoyant particles, and can be O(1) for heavier-than-fluid particles. In the limit of ${\mathit{V}}_{\mathrm{s}+}\ll O\left({\tau }_{+}\right)$, it is clear that the effect of gravity can be entirely neglected. In the limit of ${\mathit{V}}_{\mathrm{s}+}\sim O\left({\tau }_{+}\right)$, the relative influence of gravity is proportional to the mass loading. For the case of ${\mathit{V}}_{\mathrm{s}+}\gg O\left({\tau }_{+}\right)$, we will restrict the mass loading to be $M\le O({\tau }_{+}/{\mathit{V}}_{\mathrm{s}+})$, so that the effect of gravity is only as strong as necessary to modify the turbulence. Otherwise, the gravitational settling of the particles will be the dominant source of turbulence.

In the viscous term, the strain rate of the mixture greatly simplifies under most conditions. As long as ${\mathit{V}}_{\mathrm{s}+}\le O\left({\tau }_{+}\right)$, the strain rate of the mixture can be approximated as ${\mathit{E}}_v\approx \mathit{E}={\displaystyle \frac{1}{2}}(\nabla \mathit{u}+(\nabla \mathit{u}{)}^T)$ with the difference being $O\left({\phi }_p\sqrt{{\tau }_{+}}\right)$ or smaller. In the case of ${\mathit{V}}_{\mathrm{s}+}\sim O\left(1\right)$, the effect of the gravitational settling of particles cannot be ignored, and the difference between ${\mathit{E}}_v$ and $\mathit{E}$ is $O\left({\phi }_p\right)$ and therefore must be accounted for.

From the equilibrium expansion for the rotational velocity of the particles given in (17), it can be estimated that the appropriate scaling of the rotational slip between the particles and the surrounding fluid is ${\tau }_{\mathsf{\Omega }}/t_k^2$. Based on this, the relative contribution of the three hydrodynamic torque terms in (34) to the fluid momentum (Equation (28)) can be estimated to be $O\left({\phi }_p{\tau }_{\mathsf{\Omega }+}\right)$ or less and therefore can be ignored at the level of approximation considered here. The above estimate is independent of the scaling of the settling velocity of the particles.

Precise evaluation and closure of the kinematic Reynolds stress term are not possible, but an estimate of its scaling can be obtained. The magnitude of local small-scale perturbation velocity, $\tilde{\mathit{u}}$, can be taken to scale with the translational slip velocity, and the spatial extent of this perturbation can be assumed to be $O\left({\phi }_p\right)$. Thus, the appropriate scaling for the relative importance of the kinematic Reynolds stress goes as follows: $O\left({\tau }_{+}^2{\phi }_p^2\right)$ and $O\left({\phi }_p^2\right)$, respectively, for the cases of ${\mathit{V}}_{\mathrm{s}+}\le O\left({\tau }_{+}\right)$ and ${\mathit{V}}_{\mathrm{s}+}\sim O\left(1\right)$. Thus, under all conditions of present interest, the contribution to fluid momentum balance from the kinematic Reynolds stress term can be ignored.

All contributions other than those outlined above are of importance in the fluid momentum balance. Substituting (33) into (28) and after ignoring smaller terms and some arrangement, we obtain the following hierarchy of momentum equations:

$${\rho }_m{\displaystyle \frac{D\mathit{u}}{Dt}}=\left\{\begin{array}{ccc}-\nabla p_m+\nabla ·\left(2{\mu }_{\mathrm{eff}}\mathit{E}\right)& \mathrm{for}& {\mathit{V}}_{\mathrm{s}+}\ll O\left({\tau }_{+}\right)\\ {\rho }_m\mathit{g}-\nabla p_m+\nabla ·\left(2{\mu }_{\mathrm{eff}}\mathit{E}\right)& \mathrm{for}& {\mathit{V}}_{\mathrm{s}+}\sim O\left({\tau }_{+}\right)\\ {\rho }_m\mathit{g}-\nabla p_m+\nabla ·\left(2{\mu }_{\mathrm{eff}}{\mathit{E}}_v\right)-{\phi }_p({\rho }_p-{\rho }_f)\tau \mathit{g}·\nabla \mathit{u}& \mathrm{for}& {\mathit{V}}_{\mathrm{s}+}\sim O\left(1\right)\end{array}\right.,$$

(40)

where the definition of the mixture density, ${\rho }_m$, is given in Equation (5). Also, in the limit of ${\mathit{V}}_{\mathrm{s}+}\sim O\left(1\right)$, the strain rate of the mixture can be expressed to $O({\tau }_{+},{\phi }_p)$ accuracy as

$${\left[{\mathit{E}}_v\right]}_{ij}={\left[\mathit{E}\right]}_{ij}+{\displaystyle \frac{1}{2}}\left({\mathit{V}}_{si}{\displaystyle \frac{\partial {\phi }_p}{\partial x_j}}+{\mathit{V}}_{\mathrm{sj}}{\displaystyle \frac{\partial {\phi }_p}{\partial x_i}}\right)\mathrm{for}{\mathbf{V}}_{\mathrm{s}+}\sim O(1).$$

(41)

The continuity equation also simplifies greatly if we consider only terms of $O({\tau }_{+},{\phi }_p)$:

$$\nabla ·\mathit{u}=\left\{\begin{array}{ccc}0& \mathrm{for}& {\mathit{V}}_{\mathrm{s}+}\le O\left({\tau }_{+}\right)\\ -{\mathit{V}}_s·\nabla {\phi }_p& \mathrm{for}& {\mathit{V}}_{\mathrm{s}+}\sim O\left(1\right)\end{array}\right..$$

(42)

4. Discussion

The simplified two-way coupled two-fluid model is given by the continuity Equation (42), the fluid momentum Equation (40), and the equation of particle volume fraction (26). In this formalism, the momentum equation for the particulate phase is not needed; instead, the particle velocity is given by the equilibrium expansion (Equation (12)). The above set of equations is appropriate for the complete mathematical description of the macroscale motion of a dilute dispersion of small particles. It must be emphasized that this formalism is asymptotically accurate and provides a complete description of the multi-phase flow to only $O({\tau }_{+},{\phi }_p)$ accuracy. The advantage, however, is that the above system is considerably simpler than the general two-fluid model.

The general resemblance of the simplified continuity and momentum equations to the dusty gas formulation is clear [49,50]. The details of the simplified two-fluid equations, however, depend on the relative strength of gravitational settling. The limit ${\mathit{V}}_{\mathrm{s}+}\ll O\left({\tau }_{+}\right)$ corresponds to situations where acceleration due to gravity is much weaker than the acceleration of Kolmogorov eddies. In this limit, the gravitational influence can be entirely ignored in the dynamics, and the fluid can be considered to be incompressible to $O({\tau }_{+},{\phi }_p)$ accuracy. The influence of the particulate phase on the fluid momentum equation is simple: the effective density and viscosity are those of the mixture. Nevertheless, in a turbulent flow, the impact on the flow field and its statistics can be far from simple, mainly due to the preferential accumulation of particles and the resulting spatially complex distribution of density and viscosity.

With the increasing importance of gravitational settling, for ${\mathit{V}}_{\mathrm{s}+}\sim O\left({\tau }_{+}\right)$, the direct effect of gravity coupled with the spatially varying density field cannot be ignored. However, the fluid can still be considered incompressible to $O({\tau }_{+},{\phi }_p)$ accuracy. With a further increase in the importance of gravity, the limit ${\mathit{V}}_{\mathrm{s}+}\sim O\left(1\right)$ corresponds to situations where acceleration due to gravity is much stronger than the acceleration of Kolmogorov eddies. In this limit, the momentum equation includes two additional effects. First, the strain rate of the mixture in the viscous term can no longer be approximated as that of the pure fluid. The effect of the particulate phase must be included as defined in Equation (41). The second effect is captured by the last term on the right-hand side of (40c), whose role is clearly illustrated in the first example problem presented in the Appendix A. Furthermore, the effect of gravitational settling must also be accounted for in the continuity equation as particles settle in a region of a vertical concentration gradient. The application of the above-presented simplified two-fluid formalism to two simple example problems is illustrated in the Appendix A. In the present work, by restricting attention to incompressible flows in the constant property limit, we have avoided addressing thermal effects. A compressible or non-isothermal version of the model must be thermodynamically consistent and satisfy the second law of thermodynamics.

Inter-Particle Interaction

Owing to the dilute nature of the suspension under consideration, we have consistently avoided addressing the influence of interaction between particles. However, even in such dilute systems, the hydrodynamic interaction between the particles can influence the flow in a measurable way and the importance of various influences must be carefully evaluated. Foremost among these is the screening effect on the motion of a particle due to the presence of all other particles in its neighborhood. The standard Stokes drag for an isolated particle in an otherwise uniform flow has to be modified here with a volume fraction-dependent function ${\chi }^{-1}=(1-\phi )/(1-6.55\phi )$ in order to account for the screening effect of all other particles present in its neighborhood [51], and accordingly, the first term on the right-hand side of Equation (8) must be modified as $3\pi \mu d\left(\mathit{u}-\mathit{v}\right)/\chi$. The factor $(1-6.55\phi {)}^{-1}$ arises from the familiar problem of reduction in the terminal velocity of a dispersion of particles, and the additional factor $(1-\phi )$ accounts for the corresponding counter motion of the surrounding fluid, and thus, the correction is in terms of relative velocity. The screening effect thus influences the equilibrium translational slip velocity and the force between the fluid and particulate phases only at O(${\tau }_{+}{\phi }_p$) and therefore can be neglected within the present formalism.

A key assumption involved in the above form of the screening correction to Stokes drag is that the particles locally form a free random distribution within the fluid [52,53]. Owing to the preferential accumulation of particles, a non-random macro structure is expected, even in isotropic turbulence. However, for the standard screening correction to apply, it is required that there should be no preferred structure for particle arrangement at the small scale, i.e., within the averaging volume V. This assumption seems reasonable since the particles are after all driven by the turbulent flow, and there may be no preferred particle arrangement below the flow scales.

The rotational equation of motion (Equation (13)) is appropriate for an isolated small particle in an undisturbed ambient flow. The presence of other particles in the neighborhood can be expected to have an influence on the rotational equation of motion as well. However, the screening effect on rotary motion is likely to be far weaker for the primary reason that, while the effective Stokeslet arising from the relative translational motion between the particle and the surrounding flow decays as 1/r, the corresponding rotlet arising from the relative angular motion decays faster as 1/r³. In any case, we are not aware of any work on the screening effect on rotary motion (and corresponding modification to viscous torque), analogous to that for relative translational motion.

The effective viscosity of the mixture is also based on the stress field around a single particle in the absence of all others. The inter-particle interaction can be shown to modify effective viscosity only at $O\left({\phi }_p^2\right)$. Furthermore, the stresslet arising from the macroscale strain field decays as 1/r³, and therefore, the neglect of particle interaction in the effective viscosity of the mixture can be justified.

The advection of particles following the particle velocity field, $\mathit{v}(\mathit{x},t)$, results in the preferential accumulation of particles in regions of high strain rate or high rotational rate, depending on whether they are heavier than or lighter than the surrounding fluid. In principle, Equation (26), along with the equilibrium particle translational velocity field given in Equation (12), is sufficient to follow the time evolution of the particle volume fraction over time, from any given initial condition. In practice, however, this approach faces severe difficulty, especially in the case of turbulent flows [34]. The difficulty arises from the fact that in certain regions of the flow where particles begin to concentrate, the particle volume fraction continues to increase for long periods reaching very large values. Such large local values in volume fraction can lead to sharp gradients, which in computational settings can demand a very fine grid resolution. Furthermore, on physical grounds, the particle volume fraction cannot exceed that of a closely packed arrangement.

Even in the case of a dilute dispersion of particles, as the local volume fraction of particles begins to build up, the inter-particle interaction cannot be ignored any longer. This local close-range hydrodynamic interaction of particles will eventually limit the uncontrolled growth of ${\phi }_p$ that could otherwise be possible with Equation (26). Several approaches to modeling the effect of such small-scale interaction in macroscale particle concentrations can be found in the literature [17,54,55]. One of the simplest approaches is to model this effect as a diffusion process of particles migrating down a concentration gradient. This effect, when included, results in the following advection–diffusion equation for the volume fraction of particles:

$${\displaystyle \frac{\partial {\phi }_p}{\partial t}}+\nabla ·({\phi }_p\mathit{v})=\nabla ·(D\nabla {\phi }_p),$$

(43)

where $D$ is the effective diffusivity, which is generally tensorial in nature. The diffusivity can be expected to be dependent on the particle volume fraction. For the same value of concentration gradient, inter-particle interaction and hence diffusion may be weak in regions of low concentration, while in regions of higher concentration, inter-particle interaction can lead to a strong diffusion of particles down the concentration gradient. Thus, the overall diffusion process is essentially nonlinear in nature.

Shear-induced gradient diffusivity scales as $D\propto {\phi }_p^2$ [54]. It might therefore appear that this effect is weak and that the right-hand side can be ignored compared to the other terms in (42). For a dilute system, this is indeed valid over the bulk of the flow; however, localized regions of high concentration gradient may develop within the flow to necessitate the inclusion of the concentration gradient diffusion. In addition to diffusion down the concentration gradient, there can also be diffusion down the shear gradient [17,56,57,58].

The additional diffusion term in Equation (43) implies that instead of definition (3b) for the particle velocity, we now have

$${\displaystyle \frac{1}{{\phi }_p}}<H{\mathit{U}}_c{>}_{\mathrm{ss}}=\mathit{v}-D{\displaystyle \frac{1}{{\phi }_p}}\nabla {\phi }_p.$$

(44)

where the first term on the right corresponds to the equilibrium particle velocity. The second term is diffusion velocity arising from close interaction between particles, and therefore, it corresponds to the net effect of departure from equilibrium. The corresponding equation for the mass balance for the fluid phase now becomes

$$\nabla ·\mathit{u}={\displaystyle \frac{\partial {\phi }_p}{\partial t}}+\nabla ·({\phi }_p\mathit{u})-\nabla ·(D\nabla {\phi }_p).$$

(45)

Corresponding mass balances in terms of volume-averaged and mass-averaged composite velocities of the mixture are

$$\nabla ·{\mathit{u}}_v=0\mathrm{and}{\displaystyle \frac{\partial {\rho }_m}{\partial t}}+\nabla ·({\rho }_m{\mathit{u}}_m)=\nabla ·(D\nabla {\rho }_m).$$

(46)

Note that even with the inclusion of the diffusion term, the volume-averaged mixture velocity correctly remains divergence-free.

5. Conclusions

Eulerian–Eulerian approaches for two-phase flows, where the dispersed particulate phase is also treated as a continuum, have been in development for several decades. In these approaches, the equations for the conservation of mass, momentum, energy, etc., are obtained through an appropriate averaging process, and a variety of averaging techniques have been employed in the past. For example, through a rigorous ensemble-averaging process, Zhang and Prosperetti [12,13] have obtained a complete set of governing equations for the fluid and the particulate phases, fully accounting for the interaction between the phases. These equations are, however, quite complex, even in the limit of a dilute dispersion, and furthermore, they involve higher-order statistical terms that require closure assumptions.

Here, we apply the equilibrium approximation to particle velocity, developed by Ferry and Balachandar [1], to simplify the above-mentioned rigorous set of two-phase flow equations. First, we extend the approach presented by [1] to consider the rotational motion of the particles and seek an equilibrium approximation for the angular velocity of the particulate phase. The resulting explicit knowledge of the particulate phase’s translational and rotational velocities in terms of the fluid velocity eliminates the need to consider the momentum equations for the particulate. The equilibrium approximations also provide precise scaling for the various terms in the governing equations of the two-fluid model, based on which a simpler set of equations appropriate for the case of a dilute dispersion of small spherical particles are obtained here.

The simplified two-fluid formalism developed here is given by the continuity Equation (42), the fluid momentum Equation (40), the equation of particle volume fraction (26), and an explicit expression for the velocity of the particulate phase (Equation (12)), and together, they provide the complete mathematical description of the flow field at the macro scale for a dilute dispersion of small particles. The formalism is however only asymptotically accurate, and it provides a complete description of the two-phase flow to only $O({\tau }_{+},{\phi }_p)$ accuracy. The advantage, however, is that the above system is considerably simpler than the general two-fluid model. The actual form of these equations depends on the relative strength of the gravitational settling velocity of the particles. Three different regimes of ${\mathit{V}}_{\mathrm{s}+}\ll O\left({\tau }_{+}\right)$, ${\mathit{V}}_{\mathrm{s}+}\sim O\left({\tau }_{+}\right)$, and ${\mathit{V}}_{\mathrm{s}+}\sim O\left(1\right)$ are identified, and they correspond to situations where acceleration due to gravity in comparison to the acceleration of Kolmogorov eddies is much weaker, of the same order, and much larger, respectively. These simplified sets of equations also avoid the closure difficulty of the complete two-fluid equations.

We consider a regime where the flow scales (of the corresponding single phase undisturbed by the particles) are well separated from those associated with the particles. In this limit, we carefully interpret the averaging processes, such as those employed by Joseph et al. [9], and Zhang and Prosperetti [12,13], to apply to an ensemble of small-scale particle arrangements. Thus, the resulting two-fluid formalism is suitable for the detailed description of the flow at the macro scale, and we envision direct and large-eddy simulations of these equations for laminar, transitional, and turbulent two-phase flows. Finally, in the Appendix A, we present two simple examples illustrating the use of the simplified two-fluid formalism: particles settling in a shear flow and particles centrifuging in a circular Couette flow.