Beyond the Point Particle: LES-Style Filtering of Finite-Sized Particles

Transcription

1 ILASS Americas th Annual Conference on Liquid Atomization and Spray Systems, Chicago, IL, May 7 Beyond the Point Particle: LES-Style Filtering of Finite-Sized Particles Brooks Moses and Chris Edwards Department of Mechanical Engineering Stanford University, Stanford, CA 935 Abstract Multiphase LES-style spatial filtering provides a rigorous means of modeling flow over computationally-unresolved particles and droplets, without recourse to the point-particle limit. As such, it can be used to investigate the validity of point-particle models for particles of finite sizes, and to provide refinements that extend the range of validity of the models. We present results for the specific case of solid spherical particles, illustrating that a single-point-force model derived from the point-particle assumption produces reasonably accurate results at a particle Reynolds number of 1. for particles with diameters as large as 1/ of the filter radius, but causes significant errors in the filtered centerline velocity and resolved-scale viscous dissipation at Reynolds numbers of 1 and 1 for particles with diameters as small as 1/16th of the filter radius. Further, we show that the addition of an axial dipole component to the standard single-point force provides a substantially improved model. Corresponding Author

2 Introduction Traditionally, spray models use a point-particle assumption for calculating the drag force. Although the drag force on a particle or small droplet may be calculated with relatively sophisticated drag models, this force is then applied to the flowfield as a single-point force, with the assumption that the particle is sufficiently small that this is a valid approximation. However, there are few studies regarding when this approximation is valid, and in many models of the atomization region, the approximation is applied in cases where it would not reasonably be expected to be accurate. Alternative approaches, such as distributed- Lagrange-multiplier methods for solid particles [6], are typically based on assumptions that the particle is sufficiently large as to be well-resolved on the computational grid, or at least large enough to contain several grid cells. These, likewise, are inapplicable for particles or droplets that are within a size range between resolved and point-particle. An additional complication in many simulations of spray atomization is the inclusion of Large Eddy Simulation (LES) models of turbulence. Using an LES model implies that the simulated flowfield is a spatially-filtered version of the physical flowfield, and this should be taken into account in applying the particle force; even in the small-particle limit, a single-point force is no longer appropriate. In previous papers [3] [], we have proposed a framework based on spatially filtering the complete multiphase flow equations, which can represent particles and droplets over the entire range from resolved to unresolved in a manner which is consistent with an LES approach for the flow turbulence. The present paper demonstrates how this framework can be used to determine the range of particle sizes for which the pointparticle model is appropriate, and to extend the model to cover larger particles. Derivation of the Multiphase Equations The derivation of the spatially-filtered multiphase flow equations is covered in detail in a previous paper [3]; a brief review of the derivation will be given here, for the simplified case where there is only one fluid phase. For this work, we have also included a fictitious fluid within the particles. The fundamental idea of LES is to convolve the flow variables with a low-pass spatial filter, thus removing all of the features smaller than a given size (the filter size ). The evolution equations for these filtered variables are then obtained by performing a similar filtering on the original flow equations. The result is a set of equations and variables that can be completely resolved on a relatively coarse computational grid; the effects of the filtered-out features show up as subfilter-scale terms that then must be provided by closure models. We denote the spatial filtering operation by an overbar; for instance, if z is a variable (or term) in the original flow equations, z is its filtered equivalent. In the multiphase flow case, unlike single-phase LES, the filter must be applied to regions which contain portions of both the fluid and solid domains. Thus, in order to derive filtered equations which apply in these regions, we first need to define a single set of unfiltered equations which applies in both domains. We begin by defining a phase function f for the fluid; f is defined as 1 in the interior of the fluid, and elsewhere (including on the boundary). Thus, if some equation R = is true within the fluid, the equation fr = is true within the entire spatial domain. We also define fluid velocity and pressure fields, such that the velocity u f and the pressure p f are equal to the physical velocity and pressure at points within the interior of the fluid, and are continuous (including first derivatives) across all boundaries. Thus, if there is (for example) a slip at a fluid boundary, we can speak of the velocity for each fluid at that boundary, and define the slip as the difference between the two velocities on the boundary, rather than using limits from either side. In order to preserve the conservation properties of the equations, the filter needs to be applied to conserved quantities. The extended density and momentum fields, r and ru f, are not conserved because the extended portions are arbitrary. Instead, the appropriate primary variables are the phase-limited density and fluid momentum, fr and fru f, which contain only the physical density and momentum. Similarly, the momentum and continuity equations need to be derived in a form where each term represents a conserved quantity. We start by taking the momentum equation within a Newtonian fluid without assumptions of constant density and viscosity, and multiplying through by f to obtain a full-domain form: f (ru f) + f (ru f u f ) = f p f + f (l u) (1) + f µ u f +( u f ) T, where r, µ, and l are the fluid properties, which are either constant or fields defined within the fluid in a manner similar to u f and p f. In order to produce an equation that is appropriate for filtering, this needs to be rearranged into a conservation form, including the f factors. This can be done by including the surface stress condition, in which the fluid stress is given by t ˆn s = t surface ˆn s, () t = ˆd( p f + l u f )+µ u f +( u f ) T, (3)

3 where ˆd is the Kronecker delta tensor, and ˆn s is the inward normal vector on the fluid surface. Substituting this into the stress condition, and multiplying by f, gives p f f +(l u f ) f + µ u f +( u f ) T f = t surface f. () Adding this to the momentum equation allows us to combine terms such as f p f + p f f = (f p f ), converting the right-hand side to f (ru f) + f (ru f u f ) = (f p f )+ (fl u f ) + f µ u f +( u f ) T t surface f. (5) This can then be put into a conservation form by considering the phase evolution equation. At the surface, we have the boundary condition that u f ˆn s = u surface ˆn s u inflow, (6) where u surface is the surface velocity, u inflow is the rate of inflow (from the solid into the fluid) across the surface by vaporization or the equivalent. Thus, the phase evolution equation is f +(u f + u inflow ˆn s ) f =. (7) Multiplying this by ru f and using the identity ˆn s f = f gives ru f f + ru fu f f + ru f u inflow f =. (8) Combining this with the previous form of the momentum equation (5) gives the final conservation form of the equation, (fru f ) + (fru f u f ) = (f p f )+ (fl u f ) + f µ u f +( u f ) T t surface f ru f u inflow f. (9) A conservation form of the fluid continuity equation can be derived by a similar process, producing (fr) + (fru f )= ru inflow f. (1) This can then be spatially filtered. With a uniform linear filter, the filter commutes with derivatives, and thus the filtered forms are fru f + (fru f u f ) = f p f + (fl u f ) + [f µ ( u f +( u f ) T )] t surface f ru f u inflow f (11) and fr + (fru f )= ru inflow f. (1) As with the single-phase LES equations, the lefthand sides of these equations consist of time derivatives of the primary filtered variables fr and fru f, along with the usual nonlinear convection term. On the right-hand side, the pressure term (here a function of f p f rather than simply p f ) and viscous term are present in forms very similar to the single-phase version of the equations, and there are additional source terms due to the surface. In order to simplify the mathematics, at this point we will assume that the flow is incompressible and has constant density and viscosity, and that there is no mass flux across the surface. With these assumptions, the fluid properties can be taken outside the filters. There is one complication with this that is not present in the singlephase case, because the viscous term depends on f u f rather than (fu f ). In the incompressible constantproperty case, this can be expanded as µ [f ( u f +( u f ) T )] = µ fu f +( fu f ) T µ(u f f) µ(u f f), (13) which amounts to the usual viscous term in the bulk phase, plus an additional effect to cancel out the effects from the jump in fu f at the fluid surface. With a standard LES decomposition into terms calculated from the filtered fields and subfilter-scale model terms, the filtered momentum equation reduces to r fu f + r fu f fu f = f p f + µ fu f + µ ( fu f ) T µ(u f f) t surface f T sfs, µ(u f f) (1) where T sfs is the subfilter-scale convection model term, defined as T sfs = r (fu f u f ) r fu f fu f. (15) The t surface term and the surface-based viscous terms µ(u f f) and µ(u f f) combine to form a surface

4 source term, which should be provided by the particle model. The continuity equation likewise simplifies to f + (fu f)=. (16) The first term is a surface source term corresponding to the volume displacement of the particle, which again is provided by the particle model. Fictitious Fluid in the Particle Domain A number of the particle model terms in (1) and (16) can be treated conveniently by including a fictitious fluid within the particle domain. This fictitious fluid for a particle i is taken to have a velocity u i equal to the particle velocity at every point within the particle, and a density and viscosity equal to that of the surrounding fluid, and is subject to a body force F fict. that maintains this velocity constraint. The pressure of the fictitious fluid can be chosen arbitrarily; for rigid particles a value of p i = p,i 1/rw i r i (where w i is the particle s rotational velocity and r i the distance from the center of rotation) is convenient, as then F fict. only accounts for acceleration of the particle. In this formulation, the primary solution variables are the combined fluid and solid velocity field ũ, and the combined fluid and solid pressure field p, defined as ũ = (fu f )+Âf i u i (17) i and p = (f p f )+Âf i p i. (18) i where f i is the phase function for particle i, defined in a manner analogous to the fluid phase function f. The same derivation of the momentum and continuity equations can be applied to the fictitious fluid, substituting u i, p i, and f i for u f, p f, and f. When the resulting fictitious-fluid equations are added to the real-fluid equations to obtain equations for the combined fields, many of the surface terms cancel because u f = u i and f = f i at the surface of particle i. The final form of the equations is thus ũ = (19) and r ũ + r (ũũ) = p + µ ũ t surface f T sfs + F fict., () where the subfilter-scale convection term is now h i T sfs = r f i u i u i r (ũũ) (1) fu f u f + i and the fictitious-fluid force term is F fict. = r  i apple a i + r i dw i dt. () Particle Models For a particle in steady state, there are thus two remaining model terms to be determined to close the equations: the net surface stress term, t surface f, and the subfilter-scale convection term, T sfs. Physically, the surface stress term represents the net force from the particle on the fluid, and thus will integrate to the net drag force. The subfilter-scale convection term represents the transfer of momentum between the resolved scales and unresolved scales within the fluid, and thus over a large domain where the flow is completely resolved at the boundaries, momentum conservation requires that it integrate to zero. An exact form of these model terms can be calculated from a detailed flowfield simulation by numerically filtering the exact velocity field and surface stresses, and these exact model terms can then be used as a basis for simpler approximations. For the set of results presented here, we started with a set of axisymmetric calculations of flow around a sphere provided by Brian Helenbrook [], which were computed using the spectralelement method described in [1]. The filtering procedure is nominally a three-dimensional integral for each point in the filtered space, but for axisymmetric fields and sufficiently simple filters this can be reduced to a two-dimensional quadrature with the integral in the third dimension performed analytically. A representative example of the surface-stress and subfilter-scale convection fields for a relatively large unresolved particle is shown in Figure 1. In the small-particle limit, the surface terms are reduced to a single location in physical space; thus, in the limit they form a filtered point force in the computational space, with a strength equal to the net particle drag force. Furthermore, in the non-limit case, the extent of the force term is limited by the sum of the particle radius and filter radius, and thus a filtered point force remains a good approximation for particle sizes up to a significant fraction of the filter size. When the computational grid spacing is the same as the filter size, this is approximately equivalent to the usual point-particle representation as a singlepoint force distributed to the neighboring grid points, but it differs in cases where the grid is more refined relative to the filter. The subfilter-scale convection term has a net integral of zero; because momentum is conserved, all of the momentum that is transferred to the unresolved scales (in particular, the disturbed flow around the particle) upstream of the particle must be transferred back to the resolved scales downstream as the wake diffuses back into uniform flow. Thus, this term cannot be modeled by a point force. Also, because it is a result of effects within the bulk flow rather than on the particle surface, it is not necessarily confined to a region within a filter-

5 6 - Flow Direction Flow Direction (a) Surface force term, t surface, i f i (streamwise component). (b) Subfilter-scale term, T sfs (streamwise component). Figure 1: Terms to be modeled, as computed by numerically filtering the results of a fully-resolved flow simulation for a spherical particle with Re = 1 and r filter =.D particle. The scales are nondimensionalized; a value of 1 corresponds to a force density of ru. The solid line indicates the true surface location, and the dashed lines represent the outermost extents of the filtered surface location. width of the particle; however, for unresolved spherical particles, it appears that most of the effect is within the near-particle region. In the very smallest particles, the assumption of the point-particle model is that the surfaceforce term dominates the subfilter-scale term, and thus that the latter can be neglected entirely. For the development of simple models for small particles, it can be useful to ignore the distinction between the surface-force term and the subfilter-scale convection term, and instead produce a model for the net combined force. The point-particle model can be thought of as the beginning of a series of models matching successively higher moments of this net force, as it matches the zeroth moment. The next models in this series would add a filtered point dipole contribution to match the first moment in either the axial or radial direction; further refinements in this direction would add dipoles in both directions, and then higher-order terms. Here, we will only consider a model which matches the first axial moment. For comparison, we can also consider the best possible model: an exact full-field representation of the surface and subfilter-scale terms. This is useful as a baseline to compare against; any practical model should produce a simulation with an accuracy that is somewhere between the accuracy of the point-particle model and that of the exact model. Numerics for the Filtered-Flow Calculations LES calculations are usually performed on a computational grid which has a spacing comparable to the filter size. Although this is a reasonable resolution for practical simulations, where efficiency dictates using the coarsest possible grid, it is inappropriate for model development because the numerical error can be a significant fraction of the modeling error, and is strongly dependent on the characteristics of the numerical method. Thus, for these calculations, we have used a computational grid with a resolution much finer than the filter size. This decouples the grid from the filter, and we can speak of grid-independent solutions for a finite filter size, which are independent of the underlying numerics. For these calculations, the flow simulations were done with a incompressible steady-state SIMPLE-based flow solver written using the OpenFOAM CFD libraries [5], on a uniform three-dimensional cartesian grid. Due to the use of a fictitious fluid within the particle region, the flow equations are equivalent to those for a singlephase fluid except for the addition of a body force term comprising the particle model, and thus the implementation is relatively unremarkable. Results: Qualitative Comparisons The results printed here cover three possible particle models: a point-force model as would be obtained from the point-particle assumption, a model consisting of a point force plus a point dipole, and the exact model. These have been considered for particle Reynolds numbers of 1, 1, and 1, over a range of filter sizes. The calculations were performed on a grid with symmetry plane sidewalls intersecting at the particle center, and the particle center located 3 grid cells downstream of the inlet. Details of the filter sizes, grid resolution, and domain size are given in Table 1. Figures, 3, and illustrate the flowfields that result

6 Filter radius Grid Spacing Domain width Table 1: Filter sizes and computational grids used in the simulations, in units of the particle diameter. Because the domain is symmetric about the particle axis, the effective channel width is equal to twice the domain width. from the single-point-force (i.e., point-particle) model, the combined point-force and streamwise-point-dipole model, and the exact model, as compared to the flowfield obtained by filtering a detailed calculation. In all three of these cases, the exact model almost perfectly reproduces the filtered flowfield from the detailed calculation. This is as we would expect; the discrepancies are solely due to numerical error (which should be very small) and effects from the presence of the domain boundaries. The closeness of the results indicates that these errors have a relatively small effect on the other computed flowfields, and thus that most of the variation is indeed a result of using approximate particle models. The high-reynolds-number, large-particle case shown in Figure is the more challenging case to model, since the unfiltered flow around the particle has a significant amount of small-scale structure with boundary layers and a recirculation zone, and the wake extends for a large distance downstream of the particle location. The point-force model dramatically underpredicts the velocity defect at the particle location, and entirely misses the localized velocity minimum immediately downstream of the particle. The dipole model corrects for these flaws, but perhaps too much; the structure of the filtered flow around the particle is concentrated into the region within the extent of the filtered dipole, making the velocity defect slightly too pronounced. The case shown in Figure 3, with the same Reynolds number but a particle of one-fourth the diameter, shows that the situation is only marginally improved with a smaller particle. The quantitative errors appear to be reduced, but there is still near-particle structure that is not present in the results using the point-force model. On the other hand, the case shown in Figure, which returns to the particle size in the first case but has a much lower Reynolds number, indicates that at low Reynolds numbers the details of the model have a much smaller effect. Here, the unfiltered flow around the particle is much simpler, without thin boundary layers or a long wake, and the convective terms in general have only a small effect on the flow. One measure of the significance of the velocity inaccuracies is their effect on the resolved-scale viscous energy dissipation in the flow. The rate of viscous energy dissipation, D, at a point is given by D = t ij u i x j. (3) This is calculated (with the OpenFOAM libraries) using the same numerical derivatives that are used to calculate the flow equations. Results: Quantitative Comparisons Figures 5, 6, and 7 provide a more quantitative picture of the comparison among the various models. Here, we have plotted the filtered velocity along the particle s axis and the crosswise integral of the viscous dissipation at each streamwise location, for various filter sizes and a range of Reynolds numbers. In the case with a Reynolds number of 1 and a filter size of.d particle, shown in Figure 5a and 5b, the comparison between the exact model and the detailed calculations again shows that the numerical errors are insignificant, although the velocity in the wake is slightly increased due to the presence of the domain boundaries. The velocity phenomena from the previous plots can be seen here more clearly; the largest velocity deficit at the particle location is signficantly underpredicted by the point-force model, and it appears that the point-force model only produces the wake effects, but not the near-particle effects. However, the velocity deficit in the wake is also underpredicted somewhat by the point-force model. This can also be seen in the dissipation plots; there is a large two-humped peak in the dissipation near the particle, which is almost completely missed by the point-force results, and the point-force results also somewhat underpredict the dissipation in the wake. The addition of the axial point dipole improves the near-particle situation considerably (although, again, we see that the velocity minimum is exaggerated), but the results in the wake region are essentially unaffected. This is unsurprising, as the model only affects the nearparticle region, and the integrated force along streamlines will be essentially zero outside that region. The cases with smaller particles in the remainder of Figure 5 show similar effects, but the deviations between the models are smaller at the smaller particle sizes, as the near-particle effects become less and less significant compared to the wake. (Also, in the case with the smallest particle, the errors in the wake velocity are somewhat larger due to increased effects from the domain boundaries.) However, even with the smallest particles, there is an error of about % in the point-force dissipation results that is significantly improved in the near-particle

7 region by the addition of the axial point dipole. With a Reynolds number of 1, in Figure 6, the results are similar but less pronounced. Here, the pointforce model does predict most of the velocity deficit at the particle location, and some of the associated peak in the dissipation. Adding the axial dipole force improves the results in the near-particle region, but leaves the wake region unaffected; in fact, the dissipation in the wake is slightly worse with the dipole force added, despite the overall improvement. The errors are again reduced as the particle size decreases, but remain even for the smaller particles. Finally, in the case with a Reynolds number of 1, the differences in the velocity fields between the various models have become largely insignificant. The error between the exact model and the filtered detailed results has grown somewhat, however, due to the increased effect of the domain boundaries caused by the increased viscosity. Some underprediction of the dissipation by the point-force model remains, though it is of similar order to the domain-boundary effects. Here, this error is only partly removed by the addition of the axial point-dipole term. Conclusions The multiphase flow equations can be spatially filtered across domains that contain embedded rigid particles, by extending the single-phase LES approach. This extension leads to a rigorous formalism for including immersed solid boundaries within spatially-filtered calculations, regardless of whether those boundaries are resolved or unresolved. In particular, this derivation does not depend on assuming that the small-particle limit holds, and thus can be used to investigate the accuracy of this assumption. This formalism shows that a model for a partlyresolved or unresolved particle should consist of two components a term containing the forces from the surface of the particle onto the fluid, and a term containing the subfilter-scale convection effects from the small-scale velocity perturbations around the particle and provides a method for calculating these components from a resolved calculation of the same flowfield. Pointparticle models, by contrast, contain only the surfaceforce term and neglect the subfilter-scale convection term. We have shown that, for spherical particles in steady flow, the accuracy of the point-particle assumption is strongly dependent on the Reynolds number. At a particle Reynolds number of 1, the point-particle model produces relatively accurate results, even for particles with diameters as large as 1/ the filter radius. However, at a Reynolds number of 1 or 1, the point-particle model produces significant errors in the filtered centerline velocity and in the viscous dissipation caused by the particle. These errors are reduced when the particle size is decreased, but persist to particles with diameters 1/16th of the filter radius and smaller. The point-particle model can be extended by adding components, such as a filtered point dipole, to match higher-order moments of the exact surface-force and subfilter-scale convection terms. The addition of just one of these, a filtered axial dipole component, provides a significant reduction in the centerline-velocity and dissipation errors in the near-particle region at higher Reynolds numbers. Nomenclature f Phase function r Density µ Viscosity u Velocity p Pressure ˆn Surface normal vector t Stress tensor T Subfilter-scale model term r Radius D Diameter An overbar, as in f, denotes a spatial filtering. References [1] B. T. Helenbrook, A Two-Fluid Spectral-Element Method. Comput. Methods Appl. Mech. Engrg., 191:73 9, 1. [] B. T. Helenbrook, simulations of steady flow around spheres, private communication, 6. [3] B. Moses and C. F. Edwards. Development of a Fluid Interaction Model for Partly-Resolved Particles. 16th Annual Conference on Liquid Atomization and Spray Systems, Monterey, California, May 3. html [March 19, 7]. [] B. Moses and C. F. Edwards. LES-Style Filtering and Partly-Resolved Particles. 18th Annual Conference on Liquid Atomization and Spray Systems, Irvine, California, May 5. research/ilass5.html [March 19, 7]. [5] OpenFOAM CFD Toolbox, version 1., 5. http: //prdownloads.sourceforge.net/foam [March 19, 7]. [6] N. A. Patankar. A Formulation for Fast Computations of Rigid Particulate Flows. Stanford Center for Turbulence Research Annual Research Briefs, 1. pdf [March 19, 7].

8 (a) Exact velocity field, determined by numerically filtering the well-resolved simulation (b) Velocity field resulting from a filtered-point-force model (c) Velocity field resulting from a model combining a filtered point force and a filtered point dipole (d) Velocity field resulting from the exact surface-force and subfilter-scale convection models Figure : Axial velocity contours for a large-particle, high-re case with Re = 1 and r filter =.D particle, comparing various surface-force and subfilter-scale convection models to the exact filtered velocity field. The solid line indicates the true surface location, and the dashed lines represent the outermost extents of the filtered-point and filtered-dipole models.

9 (a) Exact velocity field, determined by numerically filtering the well-resolved simulation (b) Velocity field resulting from a filtered-point-force model (c) Velocity field resulting from a model combining a filtered point force and a filtered point dipole (d) Velocity field resulting from the exact surface-force and subfilter-scale convection models. Figure 3: Axial velocity contours for a small-particle, high-re case with Re = 1 and r filter = 16.D particle, comparing various surface-force and subfilter-scale convection models to the exact filtered velocity field. The solid line indicates the true surface location, and the dashed lines represent the outermost extents of the filtered-point and filtered-dipole models.

10 (a) Exact velocity field, determined by numerically filtering the well-resolved simulation (b) Velocity field resulting from a filtered-point-force model (c) Velocity field resulting from a model combining a filtered point force and a filtered point dipole (d) Velocity field resulting from the exact surface-force and subfilter-scale convection models Figure : Axial velocity contours for a large-particle, low-re case with Re = 1 and r filter =.D particle, comparing various surface-force and subfilter-scale convection models to the exact filtered velocity field. The solid line indicates the true surface location, and the dashed lines represent the outermost extents of the filtered-point and filtered-dipole models.

11 E-5 E-5 (a) Centerline velocity, r filter =.D particle. (b) Viscous dissipation, r filter =.D particle E-8 E-8 5 (c) Centerline velocity, r filter = 16.D particle. (d) Viscous dissipation, r filter = 16.D particle E-9 E-9 5 (e) Centerline velocity, r filter = 3.D particle. (f) Viscous dissipation, r filter = 3.D particle. Figure 5: Results for Re = 1 particle, comparing the centerline velocity and viscous dissipation predicted by the filtered-point-force model, the combined point-force and point-dipole model, and the exact model to the filtered velocity and dissipation from the detailed calculation.

12 (a) Centerline velocity, r filter =.D particle. (b) Viscous dissipation, r filter =.D particle E-5 (c) Centerline velocity, r filter = 8.D particle. (d) Viscous dissipation, r filter = 8.D particle E-6 E-6 5 (e) Centerline velocity, r filter = 16.D particle. (f) Viscous dissipation, r filter = 16.D particle. Figure 6: Results for Re = 1 particle, comparing the centerline velocity and viscous dissipation predicted by the filtered-point-force model, the combined point-force and point-dipole model, and the exact model to the filtered velocity and dissipation from the detailed calculation.

13 (a) Centerline velocity, r filter =.D particle. (b) Viscous dissipation, r filter =.D particle (c) Centerline velocity, r filter =.D particle. (d) Viscous dissipation, r filter =.D particle (e) Centerline velocity, r filter = 8.D particle. (f) Viscous dissipation, r filter = 8.D particle. Figure 7: Results for Re = 1 particle, comparing the centerline velocity and viscous dissipation predicted by the filtered-point-force model, the combined point-force and point-dipole model, and the exact model to the filtered velocity and dissipation from the detailed calculation.