a two-grid fictitious domain method for direct simulation of flows involving non-interacting...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2010; 63:1241–1255Published online 3 August 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.2119

A two-grid fictitious domain method for direct simulation of flowsinvolving non-interacting particles of a very small size

A. Dechaume1,2, W. H. Finlay2 and P. D. Minev1,∗,†

1Department of Mathematical and Statistical Sciences, University of Alberta Edmonton,Alberta, Canada T6G 2G1

2Department of Mechanical Engineering, University of Alberta Edmonton, Alberta, Canada T6G 2G8

SUMMARY

The full resolution of flows involving particles whose scale is hundreds or thousands of times smallerthan the size of the flow domain is a challenging problem. A naive approach would require a tremendousnumber of degrees of freedom in order to bridge the gap between the two spatial scales involved. Theapproach used in the present study employs two grids whose grid size fits the two different scalesinvolved, one of them (the micro-scale grid) being embedded into the other (the macro-scale grid). Thenresolving first the larger scale on the macro-scale grid, we transfer the so obtained data to the boundaryof the micro-scale grid and solve the smaller size problem. Since the particle is moving throughout themacro-scale domain, the micro-scale grid is fixed at the centroid of the moving particle and thereforemoves with it. In this study we combine such an approach with a fictitious domain formulation of theproblem resulting in a very efficient algorithm that is also easy to implement in an existing CFD code.We validate the method against existing experimental data for a sedimenting sphere, as well as analyticalresults for motion of an inertia-less ellipsoid in a shear flow. Finally, we apply the method to the flowof a high aspect ratio ellipsoid in a model of a human lung airway bifurcation. Copyright q 2009 JohnWiley & Sons, Ltd.

Received 8 January 2009; Revised 21 May 2009; Accepted 22 May 2009

KEY WORDS: fictitious domain; particles; direct simulation; respiratory; lung

1. INTRODUCTION

The algorithm described in this paper has been developed in order to study the motion of non-spherical rigid particles of a very small size, of order of several micrometers, in human lungairways whose characteristic size is of order of millimeters. Interest in this problem arises when

∗Correspondence to: P. D. Minev, Department of Mathematical and Statistical Sciences, University of AlbertaEdmonton, Alberta, Canada T6G 2G1.

†E-mail: [email protected]

Contract/grant sponsor: NSERC Special Research Opportunities Grant

Copyright q 2009 John Wiley & Sons, Ltd.

1242 A. DECHAUME, W. H. FINLAY AND P. D. MINEV

considering optimal delivery of aerosolized drugs to the human lungs in the form of high aspectratio particles for targeted treatment of certain lung diseases (see [1]), where the use of fibre-likeparticles to achieve local targeting via particle alignment has been proposed. Additional interestin such particles occurs in the occupational exposure assessment of asbestos fibres that may beunintentionally inhaled, and for which localized micro-dosimetry estimates would be useful. Theparticles in such aerosols have a fibre-like shape, which can be approximated by a long ellipsoid andthe distances between the particles are quite large in comparison to the particle length. Therefore,it is reasonable to study the deposition rate of such aerosols in the human lungs by studying thedynamics of a single rigid ellipsoid that is micrometers in length in bifurcating pipes that aremillimeters in diameter with a Reynolds number of the flow of the order of 1 (see [1]). One-waycoupled models for this problem, where the fluid is unaffected by the fibre, have been presentedby using analytic expressions for the fluid torque on the fibre (see [2]). However, we are interestedin the two-way coupled problem where the local fluid motion in the neighborhood of the fibre isinfluenced by the fibre motion and vice versa. Since the ratio between the two characteristic lengthscales involved in such problems is typically of the order of several hundred, it is not feasible toresolve the global flow on the same grid. The most natural approach is to use two different grids,each one fitting one of the length scales involved, and then solve two separate problems on eachof these grids by appropriately exchanging data between them. This approach is quite similar tothe so-called Chimera-grid method introduced by [3] (see also [4, 5] for more recent applications),which is analyzed within the framework of (fully overlapping) domain decomposition methodsin [6]. The fat boundary method of Maury [7] exploits the same idea, however, in the case of aPoisson equation in a domain with holes. In the present paper we apply the two-grid approach incombination with a fictitious domain method (FDM). This allows for the use of grids that do notchange in time, while the grid linked to the particle follows its trajectory. In addition, the Navier–Stokes equations are solved also in the domain occupied by the particle, hence the micro-scalegrid can be a simple grid in a cube, which allows for a simple computation on it, without theneed to update the system matrices each time the particle position changes. The particle motionis imposed using the FDM described in [8], which is derived from the method of Glowinski andco-workers (see [9] for example). The flow is first resolved on the macro-grid whose size is suchthat it cannot ‘feel’ the presence of the particle, then the velocity is interpolated on the boundaryof the micro-grid and the problem involving the particle is solved using the FDM. If the size ofthe particle or Reynolds number are such that it can influence the flow on the macro-grid, the twoproblems can be coupled iteratively in an implicit manner as discussed by [6] for example. Thefictitious domain approach is much more accurate than the so-called point-particle methods (seefor example [10, 11]) because it resolves directly all the interaction forces between the particle andthe fluid. In contrast, the point-particle methods assume rather than resolve the interaction forcesbecause the particle then is presumed to be concentrated at a point and therefore the interactionforces cannot be resolved by solving the equations that exactly represent the particle dynamicsand the fluid flow (see below). Typically these assumptions are that the drag force is either linearwith the difference of the fluid and particle velocity (Stokes drag) or some power of this differenceand these assumptions are valid only in flows of a certain type (usually the particle is assumed tobe fixed and the flow is uniform far away from it). In addition, they may or may not involve theadded-mass force and the Basset force but again, these forces are assumed to have a certain form.The rotational motion of the particle is completely ignored because the concept of angular velocitydoes not make sense if the particle is concentrated at one point. It is well known that the rotation ofthe particle significantly influences its dynamics and this is even more important if the shape of the

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2010; 63:1241–1255DOI: 10.1002/fld

A TWO-GRID FDM FOR DIRECT SIMULATION OF FLOWS 1243

particle is non-spherical. In addition, the point-particle methods can only represent the influenceof the particle on the fluid locally, at one point, and such effects as particle wakes and vorticitygeneration due to particle rotation are ignored. Thus, the point-particle methods are closer to theaveraged models of multiphase flows while the FDM approximates the solution of the system ofequations that is known to exactly describe the fluid-particle dynamics at the level of continuummechanics. In the present algorithm, we assume that the flow outside the micro-grid is unaffectedby the particle motion (and typically, the micro-grids that we used are significantly larger than theparticle size); however, the flow inside the micro-grid is resolved directly including the full particledynamics and the only influence of the macro-grid solution is through the boundary conditions onthe micro-grid domain. It is not difficult to have a two-way coupling between the flows on the twogrids and we demonstrate in an example below what the effect of this two-way coupling is. Sincethe particle concentration in the lungs is very low, it is very reasonable to assume that trackingone particle (with a given initial position) is enough to represent its dynamics since the probabilityfor interaction with other particles is very low. The algorithm can be extended to include multipleparticles, however, the computational price would be high and will certainly require the use of aparallel cluster. A single-grid version of the multi-particle algorithm is presented in [8].

The algorithm is tested on a standard sedimentation problem for which reliable experimentaldata are available, and on the so-called Jeffery’s solution for an inertia-less ellipsoid in a linearshear flow. In addition, numerical results for the flow of a small ellipsoidal particle in a lung airwaybifurcation are presented.

The remainder of the paper is organized as follows. In the next section we present the numericalalgorithm including a brief description of the FDM. In Section 3 we present the validation results,and we finish with brief conclusions.

2. FORMULATION AND DISCRETIZATION

The fluid is assumed to be incompressible and Newtonian, with density � f and kinematic viscosity�, and the particle is perfectly rigid, with a relative density �r =�p/� f (�p is its density). Theproblem has been non-dimensionalized with respect to the characteristic length L and velocity U.Therefore, the equations of motion of the fluid are

DuDt

=−∇ p+ 1

Re∇2u and ∇·u=0 in � f (1)

with D/Dt being the full derivative in time (including the advection terms), � f denoting the domainoccupied by the fluid and Reynolds number is defined by Re=UL/�. The equations governing themotion of the particle are given by

dUdt

= �r −1

�r

1

Freg+ 1

�r

1

VF (2)

d(Ix)

dt=T (3)

dXdt

=U,dh

dt=x (4)



where U, x, X, h and I are, respectively, the velocity, angular velocity, position of the center ofmass, angular position and inertia tensor of the rigid particle. The unit vector in the direction ofgravity is eg and Fr =U2/gL is the Froude number with g being the gravity acceleration, and Vis the volume of the particle. The total hydrodynamic force and torque acting on the particle areF=∫��p

r·nds and T=∫��p(x−X)×(r·n)ds, respectively, where r=−pd+(∇u+(∇u)T)/Re

is the stress tensor of the fluid, d is the Kronecker tensor and n is the unit outward normal withrespect to the particle.

Finally, a no-slip boundary condition is imposed on the boundary ��p of the particle

u=U+x×(x−X) on ��p

The boundary conditions on the outer boundary of � f depend on the particular flow of interest.There are two main approaches to the discretization of PDEs in domains whose shape changes in

time. The first one includes the Largangian and the arbitrary Lagrangian–Eulerian discretizations,which are closely related. The main feature of these discretizations is that the grid is fittedto the boundary of the moving domain and therefore can optimally approximate the solution.Unfortunately, the price to be paid in this procedure is quite high since it requires frequent re-meshing and updates of the matrices of the discrete problem (see [12, 13] for the case of flowswith rigid particles and [14] for the case of fluid particles). The other possibility, the Eulerianapproach, is by far the most popular because it utilizes a fixed grid and tracks in various waysthe moving boundaries within this grid. Then, in order to account for the change of propertiesacross the moving boundaries, various methods either modify accordingly the discrete operators(see [15] for the immersed interface method, [16, 17] for the level set methods, [18] for a localbasis enrichment and many other techniques, which we do not summarize here), or impose thechange of properties (rheological properties, density, etc.) as a side constraint either using Lagrangemultipliers (see [9] for the FDM) or directly modifying the right-hand side of the discrete system(see [19] for the immersed boundary method, [8, 20] for a non-iterative FDM formulation). Becauseof its computational efficiency, the FDM was selected for the resolution of the flow around theparticle in the present study and the particular version used in it is briefly summarized below.

2.1. Fictitious domain formulation

As shown in [8, 21], the FDM reformulates the system (1–4) by extending the Navier–Stokesequations to the entire domain �=� f ∪�p. The resulting system reads

DuDt

=−∇ p+ 1

Re∇2u+(�r −1)(g−f) in � (5)

∇·u=0 in � (6)

u=U+x×(x−X) in �p (7)

dUdt

= 1

V

∫�p

fd�, x= 1

2V

∫�p

∇×ud� (8)

dXdt

=U,dh

dt=x (9)



where

g=⎧⎨⎩

1

Freg in �p

0 in � f

(10)

represents the gravity acting on the particle and

f=

⎧⎪⎨⎪⎩

1

Freg+ 1

�p−1

(−DuDt

+ 1

Re∇2u−∇ p

)in �p

0 in � f

(11)

represents the interaction force between the two phases. Since the fluid has been extended inside�p, the conditions on the boundary of the particle are no longer necessary and we impose only theboundary conditions on the boundary of �. The equation for the angular velocity in (9) followsfrom the restriction of (7) to the boundary of �p taking into account the Stokes theorem. It workswell in the case of spherical particles, but our numerical results for ellipsoidal particles demonstratedthat the results produced by means of this equation differ from the available analytical/experimentaldata. Much better results are produced using the time derivative of the expression proposed in [22](their Equation (24b))

d(Ix)

dt= 1

�r

∫�p

(x−X)×DuDt

d� (12)

which also follows from (7), however, using it over the entire �p (not just its restriction on theboundary of �p). While the reason for this discrepancy is unclear, it seems to be due to the factthat the rigid body constraint (7) is not very strictly imposed at a discrete level. Conceivably, thetwo results should agree better if the rigid body constraint is more strictly imposed on the particleboundary as proposed by [23] for example. Alternatively, the angular velocity can be computedfrom an equation that stems from the discretization of the angular momentum equation of theparticle (see for example Equation (7.34) in [24]). The results presented in the following sectionare computed using (12).

2.2. Discretization

The set of coupled equation is discretized in time using a second-order pressure-correction operatorsplitting procedure:

Advection–diffusion sub-step: The position and orientation of the particle are predictedexplicitly by

Xp =Xn−1+2�tUn, hp =hn−1+2�txn

where �t is the time step. Then we solve for u∗ from

�0u∗− 1

Re∇2u∗ =−�1un−�2un−1−∇ pn+(�r −1)g in �

u∗ =0 on ��



where �0=3/(2�t), �1=−2/�t , �2=1/(2�t) and un , un−1 are the velocities from time levels nand n−1, which have been advected alongside an approximation of the characteristics (see [21]for details).

Projection sub-step:

�0(u∗∗−u∗)=−∇(pn+1− pn) in �

∇·u∗∗ =0 in �

u∗∗ ·n=0 on ��

n being the outward normal to ��.Rigid body constraint: The rigid body velocity is first predicted by

�0Up+�1Un+�2Un−1= 1

V

∫�p

p

(�0u∗∗+�1un+�2un−1)d�

Then the Lagrange multiplier f is eliminated from the set of equations, which yields the end-of-step centroidal velocity and angular velocity

Un+1= 1

V

1

�r

∫�p

p

u∗∗ d�+ �r −1

�rUp

xn+1= (Ip)−1

�0

(1

�r

∫�p

p

(x−X)×(DuDt

)n+1

d�−�1(Ix)n−�2(Ix)n−1

)

where Ip is the predicted inertia tensor and (Du/Dt)n+1=�0u∗∗+�1un+�2un−1 the fluid accel-eration. The equation satisfied by the end-of-step fluid velocity is

un+1={u∗∗ in � f

xn+1×(x−Xp)+Un+1 in �pp

which could be imposed point-wise or in the L2 sense. Our tests showed negligible differencesbetween those two ways of imposing the rigid body constraint, but as the imposition in the L2

norm requires inverting the velocity mass matrix, it is much slower than the alternative way. Forthis reason, all the results presented in this paper have been computed with imposing the rigidbody motion point-wise. Finally, the position and orientation of the particle are corrected using

Xn+1=Xn+ �t

2(Un+1+Un), hn+1=hn+ �t

2(xn+1+xn)

The spatial discretization employs P2−P1 tetrahedral finite elements and the resulting linearsystems are solved using a conjugate gradient solver.

2.3. Two-grid procedure

As already mentioned above, the two very different spatial scales involved in the problem underconsideration are resolved by means of an algorithm employing two embedded grids (see Figure 1).The overall algorithm proceeds as follows.



Figure 1. Representation of the two-grid setting, with the macro-grid GH embeddingthe micro-one Gh with boundary �P .

2.3.1. One-way coupling between the macro- and micro-grids.

1. Resolve the flow on the macro-grid GH having a grid size of order of H . Since H �h,h being the scale of the micro-grid (further denoted by Gh), the particle is not taken intoaccount in this step. The flow is resolved using the same procedure as in 2.2 skipping thelast step in which the rigid body constraint is to be imposed.

2. Interpolate the so-computed velocity at the boundary nodes of the grid Gh (the initial condi-tions are computed the same way but for the entire domain), we denote the boundary of theregion covered by Gh by �P . Typically this grid covers a parallelepiped P centered at thecentroid of the particle and advected together with it. The interpolation uses the standard P2approximation in each tetrahedron. We prescribe zero velocity at any of the boundary nodesof Gh that are outside the computational domain covered by GH .

3. Since in general the resulting velocity on the boundary, u�P violates the incompressibilitycondition (that the integral of its normal component over the boundary is zero), we apply acorrection by solving the following problem:

u�P ·n= u�P ·n+L on �P (13)∫�P

u�P ·ndS=0 (14)

As the second equation constitutes only one constraint, we need only one (constant) Lagrangemultiplier L in order to enforce it, and this is a simple computational problem. We expandu�P ·n over the restrictions of the P2 basis functions on the boundary element faces and solvethe Galerkin formulation of the problem. Since the resulting mass matrix is non-diagonalwe lump it by means of a quadrature based on the six nodes of the surface elements and apiecewise linear approximation.

4. Resolve the flow on the grid Gh with the boundary conditions for the velocity given byu�P . If any of the nodes of this grid is outside the region meshed by GH , zero velocity isprescribed at it. This is done under the assumption that the micro-grid can cross only no-slip



boundaries and can be easily adapted to cases when non-zero Dirichlet or Neumann boundaryconditions are to be prescribed on this boundary. The particle motion is accounted for bymeans of the discrete FDM procedure described in 2.2.

5. Advance in time and advect the grid Gh together with the centroid of the particle. The gridcan also be rotated with the angular velocity of the particle, which would be useful if theparticle aspect ratio is very large. The results presented in the following section are producedwithout such a rotation.

6. Go to step 2.

Obviously this algorithm is not a complete domain decomposition procedure because we assumethat the particle scale is much smaller that the scale of the flow domain and therefore neglect itspresence in the first step and avoid the iteration between the solutions on the two grids. If thisassumption is not valid, the algorithm is modified as follows.

2.3.2. Two-way coupling between the macro- and micro-grids.

1∗. Resolve the flow on the macro-grid GH having a grid size of order of H . The advection–diffusion sub-step is modified so as if the foot of the characteristic corresponding to aparticular node of the macro-grid is inside the micro-grid (as positioned at the previoustime level) then the previous time level advected velocity un is produced interpolatingthe velocity un on the micro-grid (see [21] for details on the interpolation procedureand computation of the foot of the characteristic). The same is done for the advectedvelocity un−1. This provides the feedback for the solution on the macro-grid accountingfor the presence of the particle.

2∗–5∗. The same as steps 2–5 of the one-way coupling algorithm above.6∗. Go to step 1∗.

3. VALIDATION RESULTS

In order to validate this method, four test cases have been considered: the tumbling motion of anellipsoid in a linear shear flow, the sedimentation of a sphere in a tank with one-way and two-waycoupling and the migration of a small ellipsoid in a flow in a bifurcating pipe. These results arepresented and analyzed in the following sub-sections.

3.1. Sedimentation

3.1.1. One-way coupling. In [25], the sedimentation of a sphere in a tank has been studied exper-imentally for a range of Reynolds and Froude numbers. The particle diameter is the characteristiclength L, the sedimentation velocity of a sphere in an infinite medium is the characteristic velocityU and the relative density of the particle �r =1.16. The tank dimensions are 20

3 × 203 × 32

3 , thesphere is released from rest at a distance 8 from the bottom of the tank. For the simulations, wefirst consider the case of one-way coupling between the macro- and micro-grids. The macro-gridhas been set large enough to contain the micro-grid at any time, with a bottom wall at zero heightwhere the pressure and velocity have been set to zero. The micro-grid, centered on the particlecentroid, is a cube of dimensions 20

3 × 203 × 20

3 . The simulation has been run at Reynolds andFroude numbers Re=1.5 and Fr =0.0098, the micro-grid contains 185 000 elements with more



0 2 4 6 8 10

-1

-0.5

0

exp. set 1exp. set 2

simulation

Figure 2. Sedimentation velocity vz as a function of the time t , for a spherical particleof diameter 1 and relative density �r =1.16 in a tank of dimensions 20

3 × 203 × 32

3 .The signs + and × represent 2 sets of data from the experiments at Re=1.5 in

[25], with Fr =0.0098. The solid line is the simulation result for �t=0.001.

refinement where the particle is and the time step is �t=0.001. Extending the micro-grid in thesedimentation direction, increasing the mesh resolution or decreasing the time step were found tohave a negligible effect on the results shown (less than 1%).

In Figure 2, the time evolution of the sedimentation velocity vz is shown as a function ofthe time t . It can be seen that the results from the simulation are in good agreement with theexperimental data. The micro-grid crosses the bottom of the tank at the time t=4.82. At subsequenttimes, we impose zero velocity at the nodes of the micro-grid that are outside the tank.

3.1.2. Two-way coupling. We consider here the case of a two-way coupling between the macro-and micro-grids. With the same physical set-up, we used the same numerical parameters but afiner macro-grid mesh. For the one-way coupling simulations, the macro-grid mesh was just aparallelepiped with four tetrahedral elements, enough to represent a flow field at rest (zero velocityand pressure). Here, we used three different uniform meshes for the macro-grid with 10 000, 20 000and 47 000 elements. On Figure 3, the difference �vz between the sedimentation velocity of thetwo-way coupling for each of the macro-grids and the velocity of the one-way coupling simulationis shown. Far from the region where the particle reaches the bottom of the tank, for the coarsestmesh, the relative difference is less than 5%; for the other finer meshes it is less than 1%. As theparticle comes closer to the bottom of the tank, the difference increases to 15% for the coarsestgrid and less than 4% for the other ones. The complexity of the fluid flow between the particleand the bottom wall is probably the reason for this discrepancy. As compared with the mesh with20 000 elements, it can be seen that improving the mesh has virtually no impact on the quality ofthe solution. Simulations with a twice smaller micro-grid domain, not presented here, showed adifference of about 30%. Such a situation would require a much finer macro-grid mesh to simulate



0 2 4 6 8

-0.1

-0.05

0

0.05

macro-grid 10000macro-grid 20000macro-grid 47000

Figure 3. Difference �vz between the sedimentation velocity of the two-way coupling and the onefrom the one-way coupling as a function of the time t . The number of elements in each macro-gridsis, respectively, 10 000, 20 000 and 47 000. All other parameters are the same as the ones used for

the one-way coupling simulation.

adequately the flow in a region that is closer to the particle, and consequently would have a muchhigher computational cost.

3.2. Jeffery orbit

In [26], the motion of an inertia-less ellipsoidal particle in a shear flow has been studied analytically.Let us consider a flow in the direction x , sheared in the direction y, with a shear rate �. The lengthof the major axis of the ellipsoid is chosen to be the characteristic length L, and the characteristicvelocity is given by U=�L. For vanishing Re, it has been shown in [26] that the ellipsoid undergoesan orbiting motion, convected by the local undisturbed flow and satisfying the following equation:

�=−a2r sin2�+cos2�

a2r +1

where � is the angle between the direction x and the projection in the plane (x, y) of the unitvector along the ellipsoid major axis, � is its time derivative and ar the aspect ratio of the ellipsoid.The period of an orbit is T =2�(a2r +1)/ar .

For the numerical simulations, we considered the following parameters: Re=0.1, ar =2, �r =1.The macro-grid has been chosen to be large enough that it contains the micro-grid at any time, thepressure is set to 0 and the velocity corresponds to a linear shear flow with �=1. The micro-gridis set to a cube with a size of 4. The initial conditions for the particle are as follows: its positionis chosen so that the undisturbed fluid velocity is 1 at its centroid, which is the value of theinitial velocity, the orientation is aligned with the flow with an angular velocity set to that ofthe Jeffery solution, �0=−0.2. The initial position and velocity have been selected so that themicro-grid has a significant velocity with respect to the macro-grid. The time step is �t=0.01and the micro-grid contains 215 000 elements with more refinement near the particle. Extending



202-

-0.8

-0.6

-0.4

-0.2

Jeffery orbit

simulation

Figure 4. Angular velocity � as a function of the angle � for an ellipsoidal particle with an aspect ratioar =2: the continuous line represents the Jeffery analytic solution, and the + signs − the simulationwith Re=0.1, �t=0.01. In order to make the figure clearer, the simulation data displayed are an evenly

spaced 110 th sample of the actual full data set.

the micro-grid, increasing the mesh resolution or decreasing the time step were found to have anunnoticeable effect on the results shown.

The period, averaged over 6 revolutions, is 15.78 as compared with its analytic value of 15.71giving a relative error of 0.5%, while the particle centroid velocity in the direction of the flowstays within 0.1% of the undisturbed flow velocity. As can be seen in Figure 4, the results fromthe simulation are in very good agreement with Jeffery’s solution, the tumbling motion being wellcaptured by the numerical procedure.

3.3. Particle in a bifurcating pipe

As a final example, we consider the flow in a bifurcation as shown in Figure 5, with an inletradius L=250�m, outlet radii both equal to 0.75L and a bifurcation angle of 45 degrees.These dimensions are typical of the distal bronchial regions. The flow at the inlet is set to aPoiseuille profile with a maximum velocity U=0.064ms−1 and the fluid has a kinematic viscosity�=1.6×10−5m2 s−1. The dimensionless numbers associated with this flow are Re=1 andFr =1.64 with gravity pointing in the z direction. At the outlets, homogeneous Dirichlet conditionson the pressure and homogeneous Neumann conditions on the velocity are imposed. The lengthof the mother and daughters tubes has been chosen long enough not to influence the flow in thebifurcation, with a total of 340 000 elements in the macro-mesh.

The particle is a 0.02L long prolate ellipsoid with an aspect ratio of ar =10 and a relativedensity �r =1000. It is released from the dimensionless position (0.5,0,−3.18), where the originO is located at the carina of the bifurcation, with an initial velocity equal to the velocity ofthe macro-flow at the particle position, without any initial angular velocity and oriented in thez direction.



Figure 5. Geometry of the bifurcation, the origin O is located at the carina marked by a + sign.

0 0.5 1 1.5 2 2.5 3

-4

-3

-2

-1

0

1

2

micro mesh 5micro mesh 10micro mesh 20

Figure 6. The trajectory of the particle in the x, z plane in the case of a micro-mesh coveringa cube whose edge length is 5 times (solid line), 10 times (dotted line) and 20 times(dashed line) the length of the particle. The parameters of the simulation are �t=0.001,

Re=1, Fr =1.64, �r =1000, ar =10 with a particle length of 0.02.

It can be seen in Figure 6 that the effect of the size of the micro mesh on the trajectory of theparticle is small. The micro meshes for this set of simulations cover a cube centered at the particlecentroid, with sizes 5, 10 and 20 times greater than the length of the particle and contain about215 000 elements with more refinement near the particle; the time step is �t=0.001.

In order to study the effect of the time step on the accuracy of the numerical solution we useda micro-grid, which covers a cube 10 times larger than the particle length and varied the timestep between 0.001 and 0.0001. The time step change has a dramatic effect on the trajectory of



0 0.5 1 1.5 2 2.5 3

-4

-3

-2

-1

0

1

2

Figure 7. The trajectory of the particle at varying time steps. The micro-mesh covers a cubewhose edge length is 10 times the length of the particle. The parameters of the simulationare �t={0.005,0.002,0.001,0.0005,0.0002,0.0001}, Re=1, Fr =1.64, �r ={1,1000},

ar =10 with a particle length of 0.02.

the particle if the relative density of the particle is large, �r =1000 (see Figure 7). In such case,at �t=0.005, the particle even deposits on the wall (solid line), while at smaller time steps thetrajectory becomes closer and closer to the trajectory of a neutrally buoyant particle. On the otherhand, in the neutrally buoyant case, the trajectory is very well resolved at a much larger time step,�t=0.001. On Figure 8, the evolution of the rotation vector of the particle projected in the (x, z)plane, �, is plotted for the case �r =1000 and �t=0.0001. During its course, the particle rotatesin the (x, z) plane, the maximum absolute value of the other components of the rotation vectorbeing smaller than 0.04. It can be seen that the particle orientation is closely correlated with thedirection of the local flow at the particle position, and most of the rotational motion takes placeas the particle passes from the mother tube to the daughter one. The tumbling motion induced bythe shear flow is relatively weak for such a dense and small ellipsoid.

4. CONCLUSIONS

The two-grid FDM presented in this paper is an efficient technique for resolution of flows involvingtwo very different spatial scales and different physical models (a fluid and rigid body in the presentcase; other models can be treated in the same manner). It is also easy to implement in existingCFD codes and relatively simple for implementation on parallel computers. Two possible couplingsbetween the data on the two grids have been proposed. In the first case, the one-way couplingalgorithm, the solution on the macro-grid does not ‘feel’ the presence of the particle. In the second,the two-way coupling algorithm, the solution on the micro-grid is used to interpolate the velocityat the foots of the characteristics of the nodes of the macro-grid. Numerical experiments on a



0 0.2 0.4 0.6 0.8

-4

-3

-2

-1

0

1

2

Figure 8. Evolution of the angle � between the particle major axis and the direction z, for a micro-meshwhose edge length is 10 times the length of the particle, �t=0.0001, Re=1, Fr =1.64, �r =1000,

ar =10 with a particle length of 0.02.

sedimentation problem show that if the scale of the particle is much smaller than the scale ofthe whole computational domain, the two couplings yield very similar results. The applicationthat triggered the development of this procedure is the simulation of the particle depositionsin the middle portion of the human airways where the particle concentration is very small andtherefore the particle–particle interactions are negligible. In the case of denser suspensions, theparticle–particle interactions have to be modeled, as in [8]. Since in such case, each particle willintroduce one micro-grid, the overall computational expense would be much greater and thereforethe algorithm would benefit from parallelization. This is beyond the scope of the present paper.

The method has been validated on flows involving very small particles in channels of a muchlarger size. Our validation results confirm the accuracy and efficiency of the present technique.Application of the method to the flow and motion of a fibre-like particle in a human lung airwaydemonstrates the utility of the approach.

ACKNOWLEDGEMENTS

The research in this paper has been supported by an NSERC Special Research Opportunities Grant.Computing resources have been provided by the Western Canada Research Grid.

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