Correction scheme for point-particle models applied

Correction scheme for point-particle models applied to a nonlinear drag law in simulations of particle-fluid interaction J. A. K. Horwitz, A. Mani∗ Department of Mechanical Engineering Stanford University

Abstract Drag laws for particles in fluids are often expressed in terms of the undisturbed fluid velocity, defined as the fluid velocity a particle sees before its disturbance develops in the fluid. In two-way coupled point-particle simulations the information from the undisturbed state is not available and must be approximated using the disturbed velocity field. (Horwitz, J. A. K. & Mani, A. 2016 Accurate calculation of stokes drag for point-particle tracking in two-way coupled flows. Journal of Computational Physics 318, 85–109) recently developed a procedure to estimate the undisturbed velocity for particles moving at low Reynolds number and obeying the linear Stokes drag law. Using this correction, convergence of numerical simulations was demonstrated to match the expected physical behavior for a range of canonical settings. In this paper we examine this correction scheme for particles moving at finite Reynolds number, by considering the nonlinear Schiller-Naumann drag law. Our results indicate that a linear correction can significantly improve prediction of drag force up to particle Reynolds number of about 10. Additionally, we present an investigation of the impact of this correction when applied to simulations of forced homogeneous turbulent particle-laden flows to further demonstrate the importance of modelling the undisturbed fluid velocity. While the shapes of particle acceleration and Reynolds number pdfs are not sensitive to correcting for the undisturbed fluid velocity, we show that an uncorrected scheme can result in significant under-prediction of the mean particle Reynolds number and standard deviation of particle acceleration. Furthermore, examination of particle radial distribution function reveals modest enhancements in preferential concentration predicted by the correction scheme (for St > 1). Our investigations of turbulent flows indicate the undisturbed fluid velocity correction to be more important for particles with larger size, while the Stokes number dependence is more complicated. The correction procedure shows a greater difference in particle slip velocity at higher Stokes numbers but more enhancement in preferential concentration at lower Stokes numbers. Finally, we propose a regime diagram to guide scheme selection for point-particle modelling.

1. Introduction Many natural and industrial processes comprise the transport of small solid particles by a gas. Sand storms can pose health risks for Earth communities as well as challenges to sustainable human habitat on Mars (Kok et al., 2012). The ash ejected during volcanic eruptions (Lavallee et al., 2015) can be hazardous to air traffic. In addition, small particulate matter O(µm) created during the combustion of gasoline and coal has been linked to increases in daily mortality (Laden et al., 2000). Another application concerns solar receiver technologies that utilize absorbing particles; one such design is under investigation by the PSAAP2 program (Pouransari & Mani, 2017; Farbar et al., 2016). A channel flow of air is to be seeded with O(10µm) nickel particles and irradiated. The goal is to increase the outlet gas temperature by having particles act as intermediaries, absorbing the solar radiation and quickly convecting their internal energy volumetrically to the surrounding translucent air. ∗ Corresponding

Author, email: [email protected]

Preprint submitted to Elsevier

January 1, 2018

A commonly used tool to study particle-laden flows is via Euler-Lagrange numerical simulation. In this methodology, the carrier fluid phase is simulated in a static frame while each piece of dispersed phase material is tracked in a mesh-free approach. The fluid equations comprise the Navier-Stokes equations augmented by appropriate body forces (e.g. gravity) and particle forces. Particles obey Newton’s second law and experience a change in their velocity owing to the hydrodynamic interactions with the fluid. However, resolving the detailed interaction around each particle for most applications is prohibitively expensive so that the point-particle assumption is often adopted. Extensive discussion on point-particle models and their physical predictions can be found in recent review papers, (Eaton, 2009; Balachandar, 2009; Soldati & Marchioli, 2009; Kuerten, 2016). Under the point-particle assumption, instead of calculating the coupling force by integrating the fluid stress over the surface of a particle, a force model is specified by the user which is commonly based on an exact solution or curve-fit to measurements of a simplified model setting. When the mass loading of particles is small, it is common to approximate the system as one-way coupled. In this limit, while particles experience resistance owing to the fluid, the fluid does not feel the presence of particles. However, when the mass loading of particles becomes O(1), then the particles can play a substantial role in modifying fluid statistics (Ferrante & Elghobashi, 2003; Kulick et al., 1994; Nakhaei & Lessani, 2017), and reaction of the drag on the fluid phase must be explicitly accounted for. Recently, the effect of two-way coupling on particle dynamics has been the subject of several works (Horwitz & Mani, 2016; Subramaniam et al., 2014; Gualtieri et al., 2015; Ireland & Desjardins, 2017). These studies were explored to remedy the observation that two-way coupled point-particles create disturbance flows which contaminate the fluid velocity in the near-field of the particles (Boivin et al., 1998). This creates a numerical problem because calculation of the drag force often requires knowledge of the fluid velocity in the absence of the disturbance introduced by the presence of the particle. A similar issue arises in large eddy simulation of flows with particles because only a spatially filtered form of the fluid velocity field is available, while the particle equation of motion, also requires knowledge of the subgrid-scale velocity (Marchioli, 2017; Park et al., 2017). This idea is illustrated in Figure 1. Standard drag laws are a function of the difference between the particle’s velocity vp and the undisturbed fluid velocity at the particle location u ˜p . While the particle velocity is readily obtained from the Lagrangian particle equations, the undisturbed flow is not directly accessible. When particles are two-way coupled to the fluid, the drag force experienced by the particle is fed back to the fluid. This creates a disturbance flow in the fluid which means the fluid velocity interpolated to the particle location, up , is not equal to the undisturbed fluid velocity, u ˜p . As the disturbance flow develops in the fluid, the difference between disturbed, up , and undisturbed, u ˜p , fluid velocity grows meaning that numerical implementations relying on the disturbed fluid velocity will be in error. Horwitz & Mani (2016) showed this error was of the order of the particle size relative to the grid spacing, Λ = dp /dx. Under the assumption that particles obey Stokes drag, Fd = 3πµdp (˜ up − vp ), accurate predictions for particle settling velocity were found when the undisturbed fluid velocity was well-predicted. The methods developed in Gualtieri et al. (2015), and Ireland & Desjardins (2017) are based on analytical solutions to a regularized point-force while the scheme developed in Horwitz & Mani (2016) also includes specific effects of numerical discretization in their correction scheme. Having developed accurate schemes to estimate the undisturbed fluid velocity for two-way coupled Stokesian point-particles is an important step towards validation of point-particle simulations in more complicated regimes with higher fidelity approaches (fully resolved particles and experiments.) See Akiki et al. (2017) and Segura (2004) as examples. However, the assumption that Stokes drag is the coupling force confines the validity of these schemes to low particle Reynolds numbers, which is a limiting condition considering practical applications. To extend the utility of two-way coupled point-particle methods, it is necessary to verify such methods outside the Stokes limit. In this work, we apply the correction scheme of Horwitz & Mani (2016) developed for particles obeying the Stokes drag law to particles settling at higher Reynolds numbers in an otherwise motionless fluid. We assume the particles obey the Schiller-Naumann correlation, Fd = FStokes · (1 + 0.15Re0.687 ) (Clift et al., 1978). p We explore two primary questions: is the linear scheme developed in Horwitz & Mani (2016) to compute u ˜p appropriate when the drag law is not Stokesian, and if so, over what range of Reynolds numbers can that correction scheme be used? Though the correction scheme in Horwitz & Mani (2016) assumes certain Stokesian symmetries in the near-field of the particle, it seems the deviation from this ideal scenario changes 2

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Figure 1: Comparison of different velocities in Euler-Lagrange simulation of two-way coupled flow: vp represents the particle velocity, u is the fluid velocity field, and u ˜ is the undisturbed velocity field (color online). slowly with Reynolds number. This is explained qualitatively by the analytical solution to the NavierStokes equations subject to a point force (Batchelor, 1967). Secondly, we explore the consequences of this verifiable scheme on statistics of particle slip velocity and radial distribution function in forced particle-laden homogeneous turbulence. 2. Methods The fluid and particle solvers are based on the code developed by Pouransari et al. (2015). The NavierStokes equations augmented by a numerically regularized point-force are solved using 2nd order finite differences on a staggered grid. The resulting Poisson equation to enforce incompressiblity is solved directly with fast Fourier transforms. Particle positions and velocities are tracked in their respective Lagrangian frames. Both fluid and particle equations are updated with explicit 4th order Runge-Kutta time stepping. Particle momentum is coupled to fluid momentum via the drag force that occurs in their respective dynamic equations. Once the drag force model is chosen, the algorithm is closed by computing the undisturbed fluid velocity, u ˜p , and projecting the Lagrangian force to the fixed Eulerian grid. The undisturbed fluid velocity is estimated by the formula u ˜p = up + Cp dx2 ∇2 up . In the previous formula, each of the quantities, up , 2 Cp ,and ∇ up are interpolated to the particle location. The fluid velocity and its Laplacian are calculated at the control volume cell faces and then interpolated to the particle position. The correction coefficient C is based on the interpolation of an auxiliary field whose spacing is twice as fine as the fluid grid. Once the particle is located within the C-field grid, the correction coefficient Cp is computed by interpolating from the surrounding C-gridpoints to the particle location. The C-coefficients are independent of time and depend on the non-dimensional particle diameter to grid spacing, Λ = dp /dx. The C coefficients can be found in Horwitz & Mani (2016). Therefore, for a given fluid grid and particle size, the C-coefficients can be loaded into the simulation to serve as a lookup table. Once the drag force is calculated using the undisturbed 3

velocity, the force is projected to the fluid grid using reverse trilinear interpolation. For further details on the point-particle algorithm, the reader is referred to the extensive discussion found in Horwitz & Mani (2016). 2.1. Specification of coupling force In this study, we release a point-particle from rest in a viscous fluid and allow it to settle under gravity. We assume the particle obeys the Schiller-Naumann drag correlation. This drag law takes the form: Fd =

mp (˜ up − vp )(1 + 0.15Re0.687 ) p τp

(1)

Here, Rep = |(˜ up − vp )|dp /ν, where u ˜p , vp , dp , and ν are respectively the undisturbed fluid velocity evaluated at the particle location, the particle velocity, diameter, and fluid kinematic viscosity. In (1), mp and τp are respectively the particle mass and inertial relaxation time. Newton’s 2nd law gives the particle equation in the Lagrangian frame: u ˜ p − vp dvp = (1 + 0.15Re0.687 ) + g(1 − ρf /ρp ) p dt τp

(2)

In (2), g is gravity, and ρf and ρp are respectively the fluid and particle density. The Schiller-Naumann steady-state settling velocity, us , (with u ˜ = 0), satisfies:   u d 0.687  s p = gτp (1 − ρf /ρp ) = ustokes (3) us 1 + 0.15 ν Two methods of evaluating u ˜p are compared. The first method assumes that u ˜p is equal to the disturbed fluid velocity evaluated at the particle location. This velocity is calculated using trilinear interpolation, which was found to be the most accurate method among conventional interpolation schemes in the absence of accounting for the undisturbed velocity (Horwitz et al., 2016). The second method uses the procedure described in Horwitz & Mani (2016) to estimate the undisturbed fluid velocity (zero for this flow) from discretely available velocity points in the neighborhood of the particle. No change is made to the C coefficients reported in that work. Because the fluid-particle system is treated as two-way coupled, the negative of equation (1), normalized by the local fluid volume, is applied to the right hand side of the fluid momentum equation as a force density. The disturbance velocity field created by the force density is what gives rise to the difference between the computed (disturbed velocity field), and the undisturbed velocity field in the absence of the particle. We compare the Schiller-Naumann particle settling histories to the settling histories assuming the particles obey Stokes drag. All calculations are performed on a 1283 grid (unless otherwise noted) which was found to be sufficient for this type of verification problem (Horwitz & Mani, 2016). 2.2. Dimensionless Parameters We vary three parameters, the terminal particle Reynolds number Rep , the non-dimensional particle size Λ, and the particle Stokes number, St∆ = τp /τvisc . Because there is no imposed undisturbed flow in this problem, only the particle Reynolds number is a free physical parameter. However, two-way coupled pointparticle simulation necessarily introduces numerical parameters owing to the fact that a singular source term is being numerically regularized. In Horwitz & Mani (2016), it was demonstrated that prediction of particle settling velocity in the absence of estimating the undisturbed fluid velocity depends strongly on Λ. The second numerical parameter, (which can become a physical parameter in unsteady or turbulent flows), is the Stokes number, defined as the ratio of the particle relaxation time τp to the resolved viscous time of the grid, τvisc = dx2 /ν. In direct numerical simulation, the smallest fluid scales in turbulence are the same order as the grid spacing, so that this definition of the Stokes number carries the same amount of information as a Kolmogorov based Stokes number (Pope, 2000). The reason for exploring a Stokes number dependence in this study is that while the Stokes number in the physical problem only controls the non-dimensional time at which a particle assumes some fraction of its terminal velocity, this parameter also 4

represents numerically the transit time of the particle compared with the diffusion time of the disturbance field created by the particle. Only when St∆ is large can one assume the disturbance flow establishes quickly and is quasi-steady, allowing the time-static correction scheme of Horwitz & Mani (2016) to be justified. Conversely, when the Stokes number is order unity or less, the transit of the particle may couple to the disturbance flow in an unsteady way. In this study, we assume a steady drag correlation (which is a common assumption in the literature), and it is therefore important to test in what regimes the physical assumption of quasi-steadiness is consistent with the results of numerical calculations employing this assumption. 3. Results 3.1. Verification In this settling particle study, we examine four particle Reynolds numbers, Rep = [0.1, 1, 5, 10], three Stokes numbers, St∆ = [0.5, 5, 25], and three non-dimensional particle sizes Λ = [0.1, 0.5, 1]. The results of the Reynolds number study are shown in Figure 2. Shown are the numerical predictions for correction and trilinear schemes applied to a two-way coupled point-particle obeying the Schiller-Naumann correlation. Schiller-Naumann settling velocity histories based on numerical solutions of (2) (with u ˜p = 0) along with analytical Stokes settling solutions are shown for comparison. All results are normalized by the SchillerNaumann steady-state settling velocity. For all of these cases, the correction scheme is superior to trilinear interpolation. While the correction coefficients presented in Horwitz & Mani (2016) were developed for the zero Reynolds number limit, we observe that the same coefficients offer reasonable prediction of the undisturbed fluid velocity up to Rep of about ten. The maximum steady-state error for the correction scheme is about 9% for Rep = 5. We may ask why the correction scheme developed in Horwitz & Mani (2016) is able to reasonably predict particle settling outside the Stokesian regime. The point-particle equations are simply the Navier-Stokes equations augmented with a numerically regularized point force. As the point-particle moves, so too does the point force. A related problem is the steady jet flow which is a solution to the full steady and axisymmetric Navier-Stokes equations subject to a stationary point force (Batchelor, 1967). The solution is not unique and depends on a free non-dimensional parameter α which controls the strength of the jet, and can be defined in such a way as to predict the analytical force F felt by a stationary sphere of diameter dp held fixed in a uniform flow of velocity U in a fluid whose kinematic viscosity is ν. In the low Reynolds number limit F ∼ 16πµν/α, so that a consistent expression for α is α = (16/3)Re−1 p . In other words, this definition of α implies F = 3πµdp U as Rep → 0, where Rep = U dp /ν. The salient observations from this solution are firstly, in the high α (low Rep limit), the stream function at large distances is Stokesian, independent of the nature of how the force is applied, and dependent only on the force magnitude. In other words, the streamlines far away from a point force are identical to those which would be created by a uniform flow over a finite sized particle in the low Reynolds number limit. Secondly, the deviation from Stokesian flow can be qualitatively measured by the nature of the streamlines. An angle can be constructed between the point at which streamlines are a minimum distance from the axis of symmetry, as in Figure 3 (a). The formula is: cosθo = (1 + α)−1 . In the limit α  1, (Rep  1), the streamlines are symmetric, so that θo = π/2, (90o ). The normalized minimum stream function angle vs. Reynolds number is shown in Figure 3 (b). We can see that the stream function angle is a smoothly varying function of the Reynolds number. At unity Reynolds number, the variation in the stream function angle is about 9o with respect to the Stokes regime. Despite this deviation from Stokesian symmetry, the employed linear correction scheme evidently matches the analytical solution with low error, allowing satisfactory predictions of the settling velocity even at a relatively high Reynolds number, Rep = 10. One explanation is that the Laplacian operator in the correction scheme samples points in all directions from a particle’s surroundings and thus errors due to asymmetry alone are cancelled when extrapolating the undisturbed flow velocity. With regard to the applicability of two-way coupled point-particles in more complex applications, these observations suggest that the “contamination” effect, that is having numerically dependent features in the near field of point-particles, interacting with resolved features in the far field, may be of second order importance to accurate momentum/energy coupling. In Mehrabadi et al. (under review), the authors find 5

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Figure 4: Effect of non-dimensional particle size on settling velocity histories, Rep = 1, St∆ = 5 (color online). that point-particles obeying the Schiller-Naumann correlation calculated with the undisturbed fluid velocity reproduce well the TKE, dissipation, and particle kinetic energy, of that predicted in particle-resolved simulation of decaying particle-laden homogeneous isotropic turbulence. This suggests that two-way coupled point-particle simulations may have predictive power for integral statistics when the details of particle near fields are free from recirculation, below say, the formation of a wake region around Rep ≈ 24 (Taneda, 1956). One area that should be explored more is how the artificial structures created by point-particles affects more sensitive statistics like particle radial distribution function (we explore this in the next section) and Lagrangian structure functions. The main take away here is that the correction scheme compares reasonably well with the reference solution at moderate Reynolds numbers (especially compared with the trilinear prediction in Figure 2). Figure 4 demonstrates the robustness of the developed scheme in predicting the correct settling velocity over a range of dimensionless particle sizes Λ. It is clear that the proposed correction performs better than trilinear interpolation at both Λ = 0.5 and Λ = 0.1. The steady state errors for the correction scheme are respectively ≈ 3% and 0% while for the trilinear scheme, the respective errors are ≈ 26% and 2%. We next examine the effect of Stokes number on the settling velocity history. These results are shown in Figure 5. For both St∆ = 0.5 and St∆ = 25, simulations that utilize the correction scheme match the reference solution much better when compared with the trilinear scheme. While the agreement for the high Stokes case is excellent, we see some undershoot in the settling velocity history predicted by the correction scheme for the low Stokes number case (Figure 5 (a)). It should be noted that simulations at low Stokes number, St∆ , but finite Λ, imply moderate to low particle to fluid density ratio. Therefore, the neglect of history effects (Lovalenti & Brady, 1993), (Maxey & Riley, 1983) over steady drag alone is not strictly justified, and the analytical solutions here have neglected history effects. In fact, inclusion of history effects at low density ratio (at least in the low Rep limit) is known to increase the time it takes for the particle to reach terminal velocity (Coimbra & Rangel, 1998). While the undisturbed flow is identically zero (and therefore steady), the scheme used to estimate u ˜p is becoming more sensitive to numerical history effects owing to a particle seeing different parts of its own disturbance field within a grid cell, as the disturbance field is evolving in time. The same behavior was also observed in Horwitz & Mani (2016). We show in Figure 5 (c) for a smaller non-dimensional grid size of Λ = 0.5, and density ratio of 36, the oscillations and undershoot in the steady state settling velocity are dramatically reduced. It appears the correction scheme is well-suited for the gas-solid flow regime where the density ratio is much larger than unity. However, even in cases well outside the gas-solid regime, e.g. ρp /ρf = 36, the correction scheme performs satisfactorily. In the limit of small Stokes number and high density ratio, (St  1) and (ρp /ρf  1), the non-dimensional particle size will inevitably be small (Λ  1), so that the difference between the disturbed and undisturbed velocity will also be small. 7

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Figure 5: Effect of Stokes number (density ratio) on settling velocity histories, Rep = 1 (color online).

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Rep Figure 6: Settling velocity for different terminal Reynolds numbers, Λ = 1, blue symbols: Trilinear, green symbols: Corrected, squares: St∆ = 25, stars: St∆ = 5, triangles: St∆ = 0.5, dash line: Stokes, solid line: Schiller-Naumann, (color online). The verification results are summarized in Figure 6. For brevity, only data for Λ = 1.0 (the worst case) are shown. For all particle Reynolds and Stokes numbers considered, it is clear the proposed correction scheme is superior to trilinear interpolation for this verification problem. 3.2. Model Performance in Homogeneous Isotropic Turbulence Having verified the correction scheme’s ability to accurately predict the undisturbed fluid velocity for a settling particle over a wide range of particle parameters, we turn to a more complicated flow to understand the consequences of modelling the undisturbed fluid velocity. We examine forced homogeneous isotropic turbulence laden with small solid particles. The mass loading of the system, Φm = 0.025, is small so that turbulence modification is negligible. In addition, the volume fraction of the system, Φ ≤ O(10−5 ), so that collisions are ignored. Though the latter non-dimensional parameters characterize the system as one-way coupled, the goal of this section is to elucidate some consequences of accurately modelling the undisturbed fluid velocity in a mean stationary turbulent environment which will help inform modellers who study systems with stronger momentum coupling. For this problem, we examine statistics of slip velocity and radial distribution function of the particle phase comparing the predictions of the correction scheme to those using trilinear interpolation. The effect of the undisturbed velocity on turbulence modification has been examined in a recent work (Mehrabadi et al., under review). The present simulations are initialized with Pope’s model spectrum (Pope, 2000) using Rogallo’s procedure (Rogallo, 1981). The particles are seeded at the beginning of the simulation with the local fluid velocity. The system is then sustained through the linear forcing scheme of Rosales & Meneveau (2005) whereby a forcing term, Aρui , is added to the right-hand side of the fluid momentum equation. The forcing coefficient A is chosen to yield a turbulence intensity based on the Taylor Reynolds number of Reλ ≈ 33. We study two non-dimensional particle sizes, Λ = [0.25, 0.5], and three Stokes numbers: Stη ≈ [3.47, 10.4, 31.3]. The non-dimensional particle sizes correspond to the fluid domain being discretized with N 3 = 1283 and N 3 = 2563 gridpoints, respectively. For a fixed Λ, the Stokes number is varied by changing the particle to fluid density ratio. The corresponding density ratios are, ρp /ρf = [1000, 3000, 9000]. Once the system becomes statistically stationary, particle statistics are collected approximately every forcing time-scale (A−1 ) and averaged over at least 50 forcing time scales. 9

Before presenting the results, it is worth commenting on the applicability of using a correction developed for an isolated particle in an otherwise quiescent flow in a regime containing other particles and in a nonuniform flow. The diluteness of the suspension ensures that particle-particle screening (particles seeing the disturbance flow created by other particles) is a second order effect compared with the order unity effect of calculating the drag accurately using the undisturbed fluid velocity estimated as the flow velocity at the location of each particle in the absence of all other particles. However, when the inter-particle separation becomes comparable to the particle size, then the effect of the disturbances created by neighbors should be explicitly taken into account in the drag formulation (Batchelor, 1972; Akiki et al., 2017). In (Mehrabadi et al., under review), it is shown that point-particles obeying the Schiller-Naumann correlation using the undisturbed velocity presented here well-predict the particle acceleration pdfs and kinetic energy predicted in a non-dimensionally equivalent particle-resolved simulation, even though the screening was not explicitly modelled in the point-particle simulation. That work considered decaying homogeneous isotropic turbulence at Φ = 0.001. Though preferential concentration exists in these systems, its mean effect on screening is thought to be small in the dilute limit. The second comment is with regard to the non-uniformity of the flow (both owing to the presence of particle disturbances, and because the underlying flow is turbulent). When particles are sufficiently large compared to the fluid scales, they are able to see the local curvature in the velocity field. Then the steady drag formula requires a physical modification in the form of a Faxen correction. The Faxen correction is ∼ d2p ∇2 u = O(d2p u0 /λ2 ), where u0 is the rms fluid velocity and λ is the Taylor microscale. The ratio of the Faxen correction to uncorrected Stokes drag (for example) is then O(d2p /λ2 ) 1. 2.5

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Figure 9: Radial distribution function (RDF) for corrected (green) and uncorrected (blue) schemes, Λ = 0.5, plotted for different Stokes numbers. Insets show the enhancement in preferential concentration at small separations predicted by the correction scheme is retained at high Stokes numbers (color online).

4. Discussion The previous observations motivate a notional regime diagram for point-particle modelling (Figure 10). In Horwitz & Mani (2016), it was shown that the error in calculation of the undisturbed fluid velocity was O(Λu0 ). (Note the precise error will depend upon the projection stencil, the drag correlation, and the fluid numerics.) Using 10% as an arbitrary boundary (notwithstanding the observations of the previous section), the space of particle size and Stokes number can be divided into two regions: for Λ < 0.1, the undisturbed fluid velocity is comparable to the disturbed fluid velocity, so a correction to account for this difference may not be necessary. For Λ > 0.1, two-way coupling (TWC) effects are significant in the near-field of the particle, so it is important that a correction for the undisturbed velocity be used, consistent with the drag model and projection scheme that are used. The parameter space may also be divided into two physical regimes based on density ratio. For ρp /ρf ≥ 1000, the fluid-dispersed phase system is likely in the gas-solid 12

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10 1

dp =2, $

Figure 10: Regime diagram for modelling point-particles. regime. For smaller density ratios, the system is likely in another multiphase flow state such as gas-liquid, liquid-solid, liquid-liquid, or other more complicated mixture. In the gas-solid regime, steady drag (Stokes or Reynolds corrected) is likely to be dominant over other drag terms. For example, non-dimensionalization of the Maxey-Riley equation reveals the fluid acceleration, added mass, and history terms are small at high density ratio. Nevertheless, even at lower order density ratios, steady drag can still dominate history effects. A combined experimental and numerical investigation of nearly neutrally buoyant particles settling in cellular flows up to order unity Reynolds number found numerical simulations where particles experienced Schiller-Naumann drag well captured the experimental particle trajectories while augmentation of Stokes drag with the history term did not agree with experimental results at finite Reynolds number (Bergougnoux et al., 2014). It is important to note that there is conflicting evidence as to the importance of the history term at high density ratio. Coimbra & Rangel (1998) applied analytical Maxey-Riley solutions to particles moving in simple flows with and without the history term. At large density ratio, these two cases showed similar velocities, while Daitche (2015) explored particle-laden turbulent flow and showed, consistent with the analysis of Ling et al. (2013), that the critical parameter governing the importance of the history term was the particle size relative to the Kolmogorov scale. The respective results of Daitche (2015) regarding particle slip velocity and Olivieri et al. (2014) concerning the radial distribution function of particles in turbulence with and without history effects show modest variation at a density ratio of 1000, but significant variation at a density ratio of 10. In view of the aforementioned scaling analyses, these observations may be confounded by the fact that the maximum particle diameter for the high density ratio particles was about one-tenth the Kolmogorov scale. Based on these observations, the regime diagram in Figure 10 has a cut off at a non-dimensional length-scale of unity, suggesting that, even in the case of gas-solid flow, unsteady drag may become significant owing to the particle size being comparable to or larger than characteristic fluid scales. In contrast, in non-gas-solid flows, where the density of the dispersed phase is not much larger than that of the fluid phase, other drag terms are likely to be important. In these regimes, the particle equation of motion requires modelling of both the undisturbed fluid velocity and the undisturbed fluid acceleration. Proper implementation of this force requires a suitable correction scheme. As we have seen in the previous results, while the verification problem we have explored is the simple settling of a particle under gravity in an 13

otherwise quiescent flow, the lack of undisturbed fluid time scales does not preclude numerical unsteadiness entering the solution. For the low Stokes number (low density ratio) settling cases, we saw undershooting as well as oscillations in the particle settling velocity. What our correction assumes, consistent with a high Stokes number or density ratio particle, is that a particle’s disturbance flow has fully established, and then it begins to move. In other words, the disturbance field is always in equilibrium. This is what it means for a drag formula to be called “steady”. When particle Stokes number is small or density ratio is low, now there is a coupled interaction since the particle begins to move before its disturbance flow has fully developed. Though the undisturbed flow has no interesting physics for a particle settling in an otherwise stagnant fluid, the numerically strange behavior is an alert that the physical assumptions provided for using the steady drag law were not justified at early times. So too then must modellers be aware that simply using a drag formula whose motivation is based on the undisturbed flow physics may fail to coupled interactions explicitly captured in the time history of the fluid phase (the ∂u/∂t term is always solved for) and not explicitly captured in the particle model. 5. Conclusion In this work, we demonstrated the effectiveness of a method used to estimate the undisturbed fluid velocity created by Stokesian point-particles in a regime where the point-particles obey the Schiller-Naumann correlation. For different particle Reynolds numbers, non-dimensional sizes, and Stokes numbers, particles were released from rest in an otherwise stationary fluid and allowed to settle under gravity. When compared with trilinear interpolation to evaluate the fluid velocity found in the drag formula, the correction scheme offered significant improvement in predicting the particle settling velocity history for all of the parameters considered. The correction scheme was found to perform best in the high Stokes limit, but good agreement was also observed at low Stokes number with increasing particle to fluid density ratio. The correction scheme was found to predict the steady state settling velocity with reasonable accuracy (< 10%), for Reynolds numbers up to ten. While the correction scheme (Horwitz & Mani, 2016) was developed under the assumption of linear Stokesian flows, its ability to be used at finite particle Reynolds number in nonlinear correlations allows it to be used in a larger variety of applications. We then examined forced homogeneous isotropic turbulence laden with small particles to study the effect that modelling the undisturbed fluid velocity has on the predicted particle acceleration and Reynolds number. The discrepancy between correction and trilinear schemes is seen to increase both with increasing non-dimensional particle size and Stokes number. Though the trilinear scheme captures the qualitative shape of the particle acceleration and Reynolds number pdfs, we demonstrated that in the absence of the correction, the particle acceleration standard deviation and mean Reynolds numbers can be significantly under-predicted. In examining the particle rdf, the correction scheme also predicts a modest increase in preferential concentration at small separations. We also propose a regime diagram to aid point-particle modellers. Its aim is to be used in conjunction with physical regime maps like that proposed by Elghobashi (2006). Assuming that a modeller’s problem falls in the two-way or four-way coupling regime, we suggest for what regimes the undisturbed fluid quantities require modelling, and which terms in the particle equation of motion should be included. In the case of four-way coupling, additional modelling concerns include collisions, Faxen, and lubrication effects owing to high volume fraction. With regard to the latter point concerning effects of volume fraction, the point-particle equations adopted here are based on those originally formulated by Saffman (Saffman, 1973) where the finite size of particles is neglected in place of the dominant coupling owing to momentum transfer between the dispersed and continuous phase. Such equations are exact up to O(Φ), where the length scale associated with the volume fraction must be large enough for Φ to be considered a transportable quantity (Batchelor, 1967). So too should energy statistics associated with the fluid phase derived from the point-particle formulation be considered exact up to O(Φ) (Mehrabadi et al., under review). Explicit inclusion of volume fraction transport in the governing equations is possible by means of filtering (Anderson & Jackson, 1967; Capecelatro & Desjardins, 2013). The error introduced in defining a “local volume fraction” and accounting for an “undisturbed local volume fraction” as Λ approaches unity is no serious concern. In the continuous sense, the Landau-Squire 14

1.4

1.2

1.2

1 0.8 Schiller-Naumann Level 1 Level 2 Level 3

0.8 0.6

u=us

u=us

1

Schiller-Naumann Level 1 Level 2 Level 3

0.6 0.4

0.4

0.2

0.2 0

0 0

2

4

6

8

10

0

2

4

6

t==p

t==p

(a) Rep = 0.1

(b) Rep = 10

8

10

Figure 11: Effect of C-field refinement on settling velocity histories, Λ = 1, St∆ = 5 (color online). C-field levels refer to coefficients given in Appendix B of Horwitz & Mani (2016). solution (Batchelor, 1967) creates no source of mass, while in the discrete sense, errors associated with the undisturbed local volume fraction appear to be small in comparison to errors associated with the undisturbed fluid velocity (see Table 1 in (Ireland & Desjardins, 2017)). Other applications of this work including evaporating droplets, where a mass transfer term couples the rate of change of droplet volume to that of the fluid. When the droplet size is comparable to the grid, similar errors in mass transfer rate have been observed (Salman & Soteriou, 2004). The present work suggests a similar procedure could be constructed for mass transfer by holding a droplet fixed in an otherwise quiescent flow and measuring a disturbance mass transfer rate. By calibrating the measured disturbed mass transfer rate against the analytical rate, a procedure could be developed to more accurately model the undisturbed mass transfer rate which could then be incorporated into dynamic simulations. The type of correction scheme explored in this work also has relevance to multiphase problems in the presence of heat transfer. In these systems, the heat source would depend on the difference between the undisturbed fluid temperature and the particle’s temperature. In this scenario, a correction scheme would be required to remove the temperature disturbance as seen by the particle that created it. Such a procedure would find applications in particle-laden solar receivers (Pouransari & Mani, 2017) where the particle concentration field can couple to solar radiation heating. A correction for the undisturbed temperature field would also be useful in predicting particle settling velocity in heated systems (Frankel et al., 2016) where particle heating of the gas can couple to dilitation modes with scale comparable to the particle size (Pouransari et al., 2017). Acknowledgements This work was funded by the United States Department of Energy through the Predictive Science Academic Alliance Program 2 (PSAAP2) at Stanford University. Jeremy Horwitz has also been supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-114747. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Appendix We examine the effect of C-field refinement on the settling velocity histories. These results are shown in Figure 11. For Rep = 0.1, the settling velocity converges in steady-state to the reference settling velocity. This observation is consistent with the streamlines being only slightly perturbed from the Stokesian streamlines when the Reynolds number is small. Alternatively, this is why C-field coefficients which assumed Stokesian symmetries were able to provide good predictions of the particle settling velocity even though a nonlinear drag correlation was used. In contrast however, at Rep = 10, it is clear the correction developed for Stokes drag does not converge to the steady state settling velocity of the nonlinear drag correlation. This 15

is a consequence of C-fields based on Stokesian symmetries breaking down in the near-field of the particle. Since the C-field coefficients are tuned under the assumption of zero Reynolds number, convergence can only be expected in that limit. For precise prediction of the undisturbed fluid velocity outside the Stokes regime, it would be necessary to introduce Reynolds number dependent C-coefficients. Nevertheless, as we have shown, C-coefficients tuned in the zero Reynolds number limit yield reasonable results up to Rep = 10. Akiki, G., Jackson, T. L. & Balachandar, S. 2017 Pairwise interaction extended point-particle model for a random array of monodisperse spheres. Journal of Fluid Mechanics 813, 882–928. Anderson, T. B. & Jackson, R. 1967 A fluid mechanical description of fluidized beds. equations of motion. Industrial and Engineering Chemistry Fundamentals 6, 527–539. Balachandar, S. 2009 A scaling analysis for pointparticle approaches to turbulent multiphase flows. 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