Particle spin in a turbulent shear flow
P. H. Mortensen and H. I. Andersson
Department of Energy and Process Engineering, Norwegian University of Science and Technology, Trondheim 7491, Norway
J. J. J. Gillissen and B. J. Boersma
Laboratory for Aero and Hydrodynamics, TU-Delft, 2628 CA Delft, The Netherlands 共Received 19 January 2007; accepted 15 May 2007; published online 25 July 2007兲
The translational and rotational motions of small spherical particles dilutely suspended in a turbulent channel flow have been investigated. Three different particle classes were studied in an Eulerian-Lagrangian framework to examine the effect of the response times on the particle statistics. The results for the fluctuating particle velocities were consistent with earlier observations. The mean particle spin exceeded the mean fluid angular velocity in a region close to the channel walls. The fluctuating components of the particle spin were consistently lower than the corresponding fluid angular velocities for the heaviest particles. © 2007 American Institute of Physics.
关DOI:10.1063/1.2750677兴
The translational motion of small spherical particles in wall-bounded turbulent shear flows has received consider-able attention due to its widespread occurrence in nature and industry. The particle behavior in a turbulent channel flow and a turbulent boundary layer was investigated experimen-tally by flow visualization techniques and laser Doppler an-emometry by Kulick et al.1and Kaftori et al.,2,3respectively. During the past decade, further insights in the particle dy-namics in a turbulent background flow have been deduced from a number of computational studies. By means of large-eddy and direct numerical simulations, the physics of dilute suspensions of spherical particles has been further eluci-dated; see, e.g., Wang et al.,4 Kuerten,5 Vance et al.,6 and Marchioli et al.7
In spite of the considerable attention devoted to spherical particles suspended in a turbulent solvent, the focus has so far been on various aspects of the translational motion of the particles and on the accompanying turbulence modifications. The associated rotational motion or spin of the particles has not been addressed, except in the experimental study by Ye and Roco.8 They measured the rotational velocity of neu-trally buoyant polystrene spheres in a turbulent plane Cou-ette flow and observed that the particles rotated faster than the mean strain rate of the flow in the core region of the Couette flow, and that the particle spin exhibited a power-law decay with wall distance in the near-wall regions. They con-jectured that the fast particle rotation was a result of particle migration in the wall-normal direction.
The purpose of the present Communication is to report results for particle rotation in a turbulent shear flow. Data not only for the mean spin but also for the fluctuating spin com-ponents will be presented and compared with the corre-sponding fluid angular velocity. The study of spin of spheri-cal particles represents a first step towards a complete dynamical description of the motion of nonspherical particles in a turbulent flow field. In the latter case the particle orien-tations, and therefore also the particle spin vector, are essen-tial ingredients in a Lagrangian particle representation; see,
e.g., Goldstein.9A practical use of rotation of spheres is the vorticity optical probe invented by Frish and Webb,10 and further developed by Frish and Ferguson,11 as a means for direct measurements of fluid vorticity. This nonintrusive measurement technique relies on the hypothesis that the fluid viscosity makes small spherical particles suspended in the flow rotate with angular velocity accurately equal to half of the local vorticity. The results of the present work will show that this is not always the case.
In the present study, both translational and rotational mo-tions of small spheres in a particle-laden channel flow are investigated. The solvent into which the particles are injected is an incompressible, isothermal, and Newtonian fluid. Its motion is governed by the Navier-Stokes and continuity equations u t +共u · ⵜ兲u = − ⵜp + 1 Re* ⵜ2u, ⵜ · u = 0. 共1兲
In the above equations, u is the fluid velocity vector and p is the pressure. All variables are nondimensionalized with channel height h and friction velocity u*, and Re*denotes the
frictional Reynolds number defined as Re*= u*h /.
To describe the motion of the particles, a Lagrangian method is used to track the particle paths. It is assumed that the particles only feel the effect of the Stokes drag and the hydrodynamic torque. For a spherical particle, which is ho-mogeneous and possesses three mutually perpendicular sym-metry planes, according to Brenner12 the translational and rotational motion is given by the linear and angular momen-tum relations as dv dt = 1 t 共u − v兲, d dt = 1 r 共⍀ −兲, 共2兲
where v is the particle translational velocity vector, ⍀ = 1 / 2ⵜ ⫻u is the angular velocity vector of the fluid, and is the particle spin vector. The translational and rotational response times are defined as
PHYSICS OF FLUIDS 19, 078109共2007兲
1070-6631/2007/19共7兲/078109/4/$23.00 19, 078109-1 © 2007 American Institute of Physics
t= 2Da2
9 , r= Da2
15, 共3兲
where a is the particle radius and D is the density ratio be-tween the particle and the fluid. The response times provide a measure of the time the particles need to respond to changes in the flow field. After solving Eq.共2兲 for the new particle velocity, the particle position vector x is updated according to x =兰vdt.
The particle parameters used in the simulation are shown in TableI. In all particle cases共A, B, and C兲, the particle size a+ and the number of particles, Np, are kept constant. In order to study the effect of different particle inertia 共i.e., response time兲, the only varying parameter is the density ratio D. Further, the coupling between the particles and the fluid is one-way. The translational response times used
herein compare closely with those used by others, e.g., Mar-chioli et al.7 In a typical laboratory experiment in a h = 10 cm wide channel at Re*= 360, these particles have a
radius a = 100m.
A direct numerical simulation is used to compute a tur-bulent Poiseuille flow at a Reynolds number Re*= 360. The
size of the computational domain is 6h in the streamwise direction, 3h in the spanwise direction, and h in the wall-normal direction. Periodic boundary conditions are applied in the homogeneous streamwise共x兲 and spanwise 共y兲 direc-tions, respectively. In the wall-normal direction 共z兲, no-slip conditions are enforced at both walls 共z=0 and z=h兲. The computation is carried out with 192⫻192⫻192 grid points in x, y, and z directions. This corresponds to a resolution with ⌬x+⬇11.3 and ⌬y+⬇5.6 in the homogeneous
direc-tions. In the wall-normal direction the grid is slightly refined towards the wall such that⌬z+varies between approximately
0.9 and 2.86 walls units. The same numerical algorithm as that used by Gillissen et al.13 is employed for solving the Navier-Stokes equations关Eq. 共1兲兴.
The particles are released randomly in a fully developed turbulent flow field. The initial translational and angular par-ticle velocities equal the corresponding fluid velocities at the position of the particles. The particle equations of motion 关Eq. 共2兲兴 are integrated in time via a second-order Adams-Bashforth scheme. First, however, the fluid velocity u and the fluid vorticityⵜ⫻u were interpolated from the staggered grid onto the particle positions. The interpolation scheme is of second-order accuracy. The time step is the same as that used for the Navier-Stokes equations, namely,⌬t+= 0.036 in wall units, which is substantially smaller than the particle
FIG. 1.共Color online兲 Rms values of fluctuating fluid and particle veloci-ties. Fluid共open circle兲; case A 共dashed line兲; case B 共solid line兲; case C 共dash-dot line兲. 共a兲 Streamwise, 共b兲 spanwise, and 共c兲 wall-normal intensities.
FIG. 2.共Color online兲 Mean spanwise fluid and particle spin. Fluid 共open circle兲; case A 共dashed line兲; case B 共solid line兲; case C 共dash-dot line兲. 共a兲 Fluid and particles, and共b兲 fluid and conditionally averaged fluid at particle positions.
TABLE I. Particle properties for the study of the effect oft+andr+. The
superscript + indicates that the quantity is scaled by wall variables based on u*and.
Case t+ r+ a+ D Np
A 1 0.3 0.36 35 1.0⫻106
B 5 1.5 0.36 174 1.0⫻106
C 30 9 0.36 1041 1.0⫻106
078109-2 Mortensen et al. Phys. Fluids 19, 078109共2007兲
response times. The particle boundary conditions are peri-odic in the two homogeneous directions. If a particle hits the wall, it elastically bounces back while carrying its previous spin and velocity; besides that, the wall-normal velocity component switches sign. This is indeed a crude wall model, but the number of wall collisions in each particle case is very low and it is expected that the overall particle statistics gath-ered over 80 000 time steps are practically unaffected.
Figure 1 shows the rms values of the fluctuating fluid and particle velocities. In Fig.1共a兲it is seen that, close to the wall, the streamwise intensities increase with increasing re-sponse timet+. In all three cases the particle intensities ex-ceed the corresponding fluid intensity. The spanwise and wall-normal intensities are shown in Figs.1共b兲 and1共c兲, re-spectively. It is seen that the intensities of the particles de-crease monotonically with increasing response time in these directions. Both the increase in streamwise intensity and de-crease in spanwise and wall-normal intensity with increasing response time are in agreement with the recent results of Marchioli et al.7It is well known that there is a net transfer of particles towards the wall regions and that particles tend to concentrate in preferential regions of low streamwise fluid velocity; see, e.g., Refs.14and15. The enhanced streamwise particle intensity is due to this preferential concentration of particles. Also, inertia effects become more dominant with increasing particle response times. Thus, the particles have longer memory, and while there is a net particle transport towards the walls, the mean fluid velocity gradient gives rise to increased streamwise particle intensity in the vicinity of the wall. The reduction of particle intensities in the spanwise
and wall-normal directions is also a combined result of pre-ferred particle concentration and inertia.
Figure2共a兲shows the mean spanwise angular velocity or spin of the fluid and the particles scaled by the inverse of the viscous time scale t*=/ u*
2
. The fluid spin, i.e., 0.5dU / dz, exceeds the particle spin in the immediate vicinity of the wall. Away from the wall, there exists a region 共10⬍z+
⬍80兲, where the opposite is true. In each particle case, the mean particle spin takes on higher values than the mean fluid spin, just as observed experimentally in a turbulent Couette flow by Ye and Roco.8It is noteworthy that the mean span-wise spin in case B is higher in this region, while there is hardly any differences between cases A and C.
The physical mechanism behind the increased particle spin can be explained from Fig.2共b兲. This figure shows the conditionally averaged fluid spin at the particle positions. First, it should be noted that particle inertia does not directly influence the particle mean spin for the present particle pa-rameters. However, the particle inertia in case C has a sig-nificant influence on the mean streamwise translational ve-locity. Particle inertia causes particles to concentrate in preferential regions 共regions of high rate of strain兲 in the turbulent flow field. When particles accumulate in these re-gions共not shown here兲, they adapt relatively fast to the local strain rate. By comparing the results in Figs.2共a兲and2共b兲, we observe no discernible differences between the particle mean spanwise spin and the corresponding conditionally av-eraged fluid spin. The interesting results in Fig. 2共a兲 are caused by preferential accumulation of particles and are thus only an indirect effect of inertia.
FIG. 3. 共Color online兲 Streamwise rms values of fluid and particle spin. Fluid共open circle兲; case A 共dashed line兲; case B 共solid line兲; case C 共dash-dot line兲. 共a兲 Fluid and particles, and 共b兲 fluid and conditionally averaged fluid at particle positions.
FIG. 4.共Color online兲 Spanwise rms values of fluid and particle spin. Fluid 共open circle兲; case A 共dashed line兲; case B 共solid line兲; case C 共dash-dot line兲. 共a兲 Fluid and particles, and 共b兲 fluid and conditionally averaged fluid at particle positions.
078109-3 Particle spin in a turbulent shear flow Phys. Fluids 19, 078109共2007兲
The fluid and particle spin intensities共rms values of the angular velocity fluctuations兲 in the streamwise, spanwise, and wall-normal directions are shown in Figs. 3–5, respec-tively. Particle inertia did not lead to any significant devia-tions between the conditionally averaged fluctuating fluid spin and the corresponding particle spin intensities in cases A and B. As far as translational motions were concerned, only case A particles adapted fully to the conditionally averaged fluid velocities. Since the rotational response timer
+
for case B particles is only modestly larger than the translational re-sponse timet
+
for case A particles, i.e., 1.5 vs 1.0, it is not surprising that case B particles adjust their angular velocity to that of the surrounding fluid. In the region 10⬍z+⬍80,
the particles seem to correlate with regions of relatively large spanwise vorticity fluctuations, and this tendency is most evident for case B particles. This also explains the remark-edly high spanwise mean spin in this case, as observed in Fig.2. On the contrary, all particles tend to avoid regions of strong streamwise vorticity and Fig.3shows that the condi-tionally averaged streamwise fluid intensities are substan-tially lower than the streamwise fluid intensities. Case C par-ticles, due to their larger translational inertia, more easily penetrate regions of relatively strong streamwise vorticity. This is consistent with the view advocated by Eaton and Fessler,14that the heaviest particles will not be preferentially concentrated and they cannot respond rapidly to vortex mo-tion. The observation that the streamwise particle spin in
case C lags that of cases A and B in the near-wall region 共z+ⱗ16兲 is a consequence of particle rotational inertia and
not of preferential concentration. The same arguments can be used also for the wall-normal particle spin.
The angular velocity 共or spin兲 of spherical particles in directly simulated channel-flow turbulence has been reported for the first time. In keeping with measurements by Ye and Roco,8 the mean spanwise spin exceeds the mean angular fluid velocity in the near-wall region. We ascribe this phe-nomenon to the preferential particle concentration, which is an indirect effect of particle inertia. Root-mean-square values of the particle spin fluctuations have also been reported. The lightest particles tend to passively follow the local flow con-ditions, whereas the heaviest particles are severely affected by inertia. The streamwise spin component is observed to be substantially smaller than the corresponding fluid vorticity for all particles considered. This is due to preferential con-centration of the lightest particles. The heaviest particles, on the other hand, pave their way into the coherent near-wall structures, but due to their long response time they are not spun up by the surrounding fluid.
This work has been supported by the Research Council of Norway through the PETROMAKS programme.
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Fluid共open circle兲; case A 共dashed line兲; case B 共solid line兲; case C 共dash-dot line兲. 共a兲 Fluid and particles, and 共b兲 fluid and conditionally averaged fluid at particle positions.
078109-4 Mortensen et al. Phys. Fluids 19, 078109共2007兲
