Under consideration for publication in J. Fluid Mech. 1 Velocity fluctuations resulting from an immersed sphere-wall collision By A. R U I Z - A N G U L O A N D R. Z E N I T Instituto de Investigaciones en Materiales Universidad Nacional Autónoma de México México D.F. 04510, México (Received ) The fluid motion around a sphere colliding with a wall immersed in a viscous fluid was studied experimentally. Using a PIV technique, the velocity field around the sphere was obtained for different times before and after the collision. For a given experiment the impact velocity of the sphere was kept constant using a motor control. Spheres of different diameters and densities were used in water, as well with water-glycerin mixtures. The range of the flow Reynolds number was from 85 to 400. It was found that, as a result of the collision, the viscous wake that was behind the sphere before contact detached and continued moving forward and interacted with the wall. Measures of the fluid agitation and the total vorticity caused by the collision were obtained from the velocity and vorticity fields. Both scale with particle size and collision velocity squared. The motion of the vortex ring that detaches from the sphere as a result of the collision is in agreement with experimental and numerical results from other authors. Currently at: Mechanical Engineering Department, California Institute of Technology, Pasadena CA 91125, U.S.A.
2 A. Ruiz-Angulo and R. Zenit 1. Introduction Although particulate two phase flows are prominent in many industrial applications and natural phenomena, their understanding is still far from being complete. The fluctuating nature of these flows is one of the important aspects that still lacks basic comprehension. It is well known that particulate two phase flows have turbulent-like behavior at Reynolds numbers which are lower than those observed in single phase turbulent flows. This characteristic is what makes these type of flows an attractive option for industrial applications (Nagata, 1975). Since the intersticial fluid must move around the inclusions that form the particulate phase, a velocity disturbance naturally arises in the continuos phase. This agitation has been discussed by many authors. Closure relations that can be used to predict fluctuations, from first principles without questionable assumptions, are scarce. One notable exception is the case of low Reynolds number suspensions. For such flows, models have been proposed to predict the hydrodynamic fluctuations both for the case of sedimenting particles (Brenner, 2000) and simple shear flows (Drazer et al. 2004). Another exception is the case of bubbly liquids for which the Reynolds number is large and the Weber number is small (Kang et al., 1997; Spelt and Sangani, 1998). Advances in the understanding of fluctuations have mostly been possible from experimental measurements. Cartellier and Riviere (2001) thoroughly discuss the induced agitation produced by bubbles in uniform bubbly liquids for small to moderate Reynolds numbers. They propose scaling laws for the fluctuations and also make comparisons with other particulate systems, like viscous sedimenting suspensions. There are many studies that address the creation of fluid fluctuations resulting from the motion of particles is solid-liquid flows. Two important experimental investigations are those by Parthasarathy and Faeth (1990) and Kenning (1996). In both investigations, in which dilute concentrations were studied, particles were released in a nearly stagnant fluid such that the
Velocity fluctuations resulting from a particle collision 3 velocity fluctuations were produced solely due to the motion of the particles. Both studies found that the liquid velocity fluctuations increased with particle loading and particle Reynolds number. Clearly, in such dilute flows, the particle collisions are infrequent; hence, collisions do not contribute significantly to the generation of velocity fluctuations. The objective of the present investigation is to provide a quantitative measure of the amount of agitation created in the suspending phase as a result of a collision of a single particle with a wall. This effect will become important for flows with non-dilute particle loading. It has been noted that when a particle moves at a sufficiently large Reynolds number towards a wall and collides against it, the wake originally in the rear of the particle detaches, continues to move forward after the collision and interacts with the wall. Zenit and Hunt (1999) and Zenit, Joseph and Hunt (1999) showed a series of preliminary visualizations of the flow involved around particles colliding against a wall. They discussed briefly the implications of the interaction of these flow structures on the coefficient of restitution. Eames and Dalziel (1999) presented a series of detailed visualizations of a very similar flow, and subsequently Eames and Dalziel (2000) presented a more complete analysis of the flow and its implication to dust resuspension. More recently, Leweke et al. (2004a) performed a more detailed visualization, again of the same flow, and Leweke et al. (2004b) complemented the study by performing direct numerical simulations. They analyzed the interaction of the detached vortex ring with the sphere and the wall and performed a stability analysis to predict the point at which the wake became unstable and lost its axisymmetric character. In this paper, the flow field generated by the collision of a sphere with a wall is analyzed quantitatively. The problem combines many subjects of interest in fluid mechanics: the detachment of a wake due to the unsteady motion of an object (Rockwell 1998), the interaction a vortex ring with a sphere and a wall (Walker et al. 1987) and the rebound
4 A. Ruiz-Angulo and R. Zenit of a particle colliding immersed in a liquid (Joseph et al. 2001; Gondret et al. 2002). By analyzing the fluid motion around colliding spheres, a quantitative measure of the collision-generated disturbance is measured in a direct manner. The aim of this study is to provide a basis to the understanding of generation of velocity fluctuations resulting from collisions in solid-liquid flows. To our knowledge, these type of basic quantitative estimations are lacking in the literature of two phase flows. 2. Experimental Setup The velocity field around a sphere during the collision process is studied. Figure 1 shows the experimental setup used in this investigation. It consisted of a rectangular glass container of 30 30 50 cm 3. At the bottom of the container, a thick glass plate was placed. The particle collisions occurred over this plate. On the lid of the container, the particle release mechanism was setup. Two fine Nylon threads were glued to the particle poles, as shown in the figure. By using two threads, the particle rotation was inhibited. The threads were slowly unwound from the shaft of a computer controlled DC motor. By controlling the rotational speed of the shaft, the linear speed of the particle could be varied. Different sizes of spheres, of different densities, were used. Most experiments were performed in water with spheres with a diameter of 2.54 cm; to corroborate the correct scaling of the results some experiments were performed with smaller spheres and/or with a water/glycerin mixture. The corresponding range of particle Reynolds number is shown on Table 1. The Reynolds number was calculated as Re = U p Dρ f /µ, where U p is the nominal particle velocity (prior to the contact with the wall), D is the particle diameter and µ and ρ f are the dynamic viscosity and density of the fluid, respectively. Values of 1.1 10 3 Pa s and 2.4 10 3 Pa s were assumed for the viscosity of water and the
Velocity fluctuations resulting from a particle collision 5 DC DC motor motor LASER LASER Viscous Viscous fluid fluid + + tracers tracers Sphere Sphere Fine Fine threads threads Glass Glass plate plate Controler Controler CCD CCD Camera Camera Processing Processing Storage Storage (PC) (PC) Figure 1. Experimental apparatus used to generate controlled collisions of a sphere with a wall. water-glycerin mixture (70-30 wt.), respectively. For this investigation, the results do not depend on the value of the particle Stokes number (St = Re/9ρ p /ρ f ). The motion of the particle is not controlled by its inertia as the velocity is controlled by slowly unwinding a thread; hence, the inertia of the particle does not affect the motion of the sphere and the fluid motion. This is not the case for particles settling freely (Joseph et al. 2000; Gondret et al. 2002). For a given experiment, the particle was placed originally at distance of approximately eight particle diameters from the bottom plate. The voltage applied to the motor was set to a constant value before energizing it. After a very small transient, the sphere began to descend at a constant velocity. For most cases studied here (with the exception of the Delrin particles), the particle Reynolds number was large such that no particle
6 A. Ruiz-Angulo and R. Zenit sphere material / fluid particle diameter, D, cm density ratio, ρ p/ρ f Re steel / water 2.54 7.80 100-400 steel / water 1.27 7.80 100-400 steel /water-glycerin (70-30 wt.) 2.54 7.80 85-340 aluminum / water 2.54 2.70 100-400 bronze /water 2.54 8.86 100-400 Delrin / water 2.54 1.44 100-400 Table 1. Fluid and particle properties. deceleration was observed before making contact with the wall (Joseph et al., 2001). For the range of Reynolds numbers studied here, the wake behind the sphere is not expected to become unsteady and detach before the collision occurs (Clift, Grace and Weber, 1978). 2.1. Particle image velocimetry To visualize and quantify the velocity field around the sphere during the collision process, an ordinary 2-D particle image velocimetry system was used. A Dantec Flowmap 1500 model system was used. The flow was illuminated using a pulsed laser sheet of approximately 0.5 mm of thickness. The laser sheet was perpendicular to the plane formed by the two strings that controlled the motion of the particle. Images of the laser illuminated plane were obtained with a 1000 1000 pixel digital camera. The laser and the camera were synchronized by a control unit, that allowed the adjustment of the time between frames, as well as the time between pairs of frames. The typical time between frames, used to calculate the velocities, was in the order of 10 ms. The time between pairs of photographs was the highest allowed by the system, 160 ms. The field of view of the
Velocity fluctuations resulting from a particle collision 7 Figure 2. Typical PIV image and velocity field. Re=300; d p=25.4 mm. The field of view is approximately 75x85 mm 2. camera was approximately 75 85 mm 2. The pixel resolution was 0.08 mm/pixel; hence, the uncertainty in the velocity measurement is of the order of 8 mm/s. A typical PIV image is shown in Figure 2. For all cases, the area of the particle was masked. Also, since the sphere blocked the laser light, only half of the flow field is shown and processed. For the conditions studied here, the flow remains axisymmetrical (Walker et al., 1995); hence, it is valid to assume that the flow field is the same at all azimuthal angles. In the figure, the velocity field obtained is superimposed on the typical PIV image. 3. Results Experimental measurements were performed for a range of particle sizes, fluid viscosities and impact velocities. Both the velocity and vorticity fields were calculated.
8 A. Ruiz-Angulo and R. Zenit 3.1. Velocity fields Figures 3 and 4 show the typical evolution of the velocity field around a sphere colliding with a wall. Each image shows a vector map which represents the direction and magnitude of the fluid velocity at each point. The times shown are presented in dimensionless terms according to t = tu p /D. Negative times correspond to instants before the particle has made contact with the wall. To ensure that the flow around the sphere remained steady and axisymmetric, all experiments were performed for Re<500. For the case shown, the Reynolds number is approximately 400; hence, a large steady wake is observed in the back side of the sphere. The time t = 0 approximately corresponds to the instant at which the sphere makes contact with the wall. It can be observed that after the collision the wake, originally in the back of the sphere, continues to move forward around the surface of the sphere (0 < t < 1.4). At t 2.0, the vortex reaches the wall and begins to spread radially. For t 4.0, the center of the ring does not continue to spread and remains fixed in space. For later times, the fluid motion decreases gradually, dissipated by the action of viscosity. The fluid motion around the sphere is qualitatively the same for all Reynolds numbers tested. It is important to note that experiments were not performed for Re>500. The velocity measurements obtained with PIV system used in the study are only 2D. Any deviations from an axisymmetric flow could not be captured with the present experimental arrangement. Clearly, for this Reynolds number range, there is a significant amount of fluid motion, or agitation, caused by the collision of the sphere with the wall. The dynamics of the fluid structures are rich and complex. This flow has been previously described qualitatively by Eames and Dalziel (2000) and more recently by Leweke et al. (2004b).
Velocity fluctuations resulting from a particle collision 9 Figure 3. Velocity field around a sphere, during the collision with a wall. Each image corresponds to a different time instant. t = tu p/d = 0, approximately corresponds to the time at which the particle makes contact with the wall. The horizontal and vertical dimensions are scaled with D, the magnitude of the velocities is scaled with U p. For this case, Re=400. 3.2. Fluid agitation It is our interest to quantify the fluid disturbance caused by the collision of a single particle. From the velocity fields a measure of the fluid disturbance can be obtained,
10 A. Ruiz-Angulo and R. Zenit Figure 4. Velocity field around a sphere, during the collision with a wall (continuation of Fig. 3). defining an agitation quantity as: A(t) = (u 2 + v 2 )dv, (3.1) V where V is a control volume over which the agitation is measured, u and v are the fluid velocities in the horizontal and vertical directions respectively, in the measurement
Velocity fluctuations resulting from a particle collision 11 plane. This quantity can be calculated for each image pair, such that the progress of the agitation can be measured. Since we are assuming that the flow remains axisymmetric, the equation above can be simplified to : A(t) = πr (u 2 + v 2 )ds, (3.2) S where S is the surface area of the measurement plane, depicted by a dashed line box in Figure 2. Ideally, the area of the control volume should be chosen such that the measurement does not depend strongly on the choice of the area. As explained below, there are some experimental limitations on the size of the measuring are. For this case, the choice of the measurement control volume is arbitrary, as the magnitude of the agitation changes with the size of the control volume. Furthermore, a measure of the total agitation produced by a single collision event can be defined as A = 1 T T 0 A(t)dt (3.3) where T is the time over which the agitation is calculated, from before the particle has reached the wall until the fluid motion has nearly ceased, damped by the viscous effects. T is measured considering an agitation threshold of 1% of the maximum value produced during a given experiment. Figure 5 shows the dependance of the measured total agitation A on the size of the surface S, which was chosen to only account for the fluid disturbances resulting from the particle collision, excluding the fluid motion in the wake of the particle before the collision. The height of the surface was chosen to be of size D. As the width of the surface increases, the measurement of A changes. For large values of the width the measurement According to the second theorem of Pappus (Kern and Bland; 1948), V fdv = πr S fds, if f is axisymmetric. The volume of revolution V results from the generatrix area S; R is the distance from the origin to the centroid of S.
12 A. Ruiz-Angulo and R. Zenit 0.16 0.14 0.12 <A>/(U p 2 D 3 ) 0.1 0.08 0.06 0.04 0.02 0 10 20 30 40 50 60 70 V / (π D 3 / 6) Figure 5. Total fluid agitation as a function of the measuring volume. The agitation is made dimensionless diving by (U 2 p D 3 ); the volume is made dimensionless diving by the particle volume, πd 3 /6. The case shown is a typical experiment for which Re = 400. should not change significantly, because far from the center of the sphere the velocity of the fluid is negligibly small. However, as shown in the figure, for the maximum size of the measuring area A continues to increase. One set of experiments was performed with a smaller particle; therefore the accesible measuring area is larger (compared with the particle size). For that case, the total agitation does seem to reach an asymptotic value for large S. Thus, for the measurements presented here, an arbitrary value of the surface area was chosen. The size of S is 3D by D in the horizontal and vertical lengths, respectively. This choice corresponds to a value of the normalized measuring volume, V/(πD 3 /6), of 54.
0.8 Velocity fluctuations resulting from a particle collision 13 0.7 0.6 A* = A / (Up 2 D 3 ) 0.5 0.4 0.3 0.2 0.1 0 0.1 5 0 5 10 15 20 t * = t Up / D Figure 6. Fluid agitation as a function of time. Both the fluid agitation and the time are shown in dimensionless terms, A = A/Up 2 D 3 and t = tu p/d. Two experiments are shown, both performed at Re=400. 3.3. Evolution of fluid agitation Figure 6 shows a typical plot of the time evolution of the fluid agitation, as calculated using equation (3.1). The time origin has been displaced such that t = 0 corresponds to the time when the sphere makes contact with the solid surface. The figure shows two different experiments performed at the same nominal Reynolds number to demonstrate the repeatability of the results. Before the contact occurs the agitation increases rapidly as the fluid is pushed out from the gap between the particle surface and the wall. At contact, the fluid agitation increases to reach a maximum value. Immediately after the contact, the measure of A decreases, as the fluid in the gap between the particle and the wall has been squeezed into the
14 A. Ruiz-Angulo and R. Zenit measuring area. Shortly after, the agitation is observed to increase again as the detached wake, originally in the rear of the sphere, enters the measuring surface. After reaching a local maximum, the agitation is observed to decrease monotonically. Clearly, the viscous dissipation begins to damp out the fluid disturbance generated by the collision. A similar evolution is observed for all the Reynolds numbers tested in this investigation. Figure 7(a) shows the evolution of the agitation for experiments for several Reynolds numbers. The same characteristics can be observed in all the experiments for the range studied here, with the exception of the case of the smallest Re shown (Re=100), in which the viscous dissipation effect may presumably be more important. Clearly, the magnitude of the agitation trace increases with Reynolds number. Also, the rate of decay of the agitation appears to be approximately the same for the cases shown in the figure. Figure 7(b) shows the same data as in Fig. 7(a) but in dimensionless terms. The dimensionless agitation is calculated as the ratio A/(Up 2 D 3 ). The time is made dimensionless considering t = tu p /D. From this figure, it can be said that the measurements scale well with this choice of dimensionless variables. It can noted that the rate of decay of the fluid disturbance decreases slightly as the Re increases. 3.4. Total agitation To quantify the total agitation that occurs during a single collision, the results can be presented in terms of the total agitation, A, defined above in equation (3.3). Figure 8 shows the total dimensionless agitation measured for all the experiments performed in this investigation. Although some variability is evident, probably resulting from experimental uncertainty, the trend is clear. The dimensionless agitation does scale well with Reynolds number. The total dimensionless agitation is nearly constant for values of the Reynolds number larger than 150; for smaller values of Re, the agitation is smaller. It must be noted that for this particular experimental arrange the total agitation does not depend
Velocity fluctuations resulting from a particle collision 15 A(t) = (π R ) S (u 2 + v 2 ) ds 10 8 10 9 10 10 10 11 10 12 Re=100 Re=150 Re=200 Re=250 Re=300 Re=350 Re=400 10 13 10 0 10 20 30 40 50 60 time [s] A* = A / (Up 2 D 3 ) 10 0 10 1 10 2 (a) dimensional quantities Re=100 Re=150 Re=200 Re=250 Re=300 Re=350 Re=400 10 3 10 4 5 0 5 10 15 20 t * = t Up / D (b) dimensionless quantities Figure 7. Fluid agitation as a function of time for several Re. on the Stokes number; since the particles are not moving freely, their inertia does not come into play.
16 A. Ruiz-Angulo and R. Zenit Dimensionless total agitation, <A>/(U p 2 D 3 ) 0.25 0.2 0.15 0.1 0.05 0 0 100 200 300 400 500 Re=U p D ρ f / µ f Figure 8. Total dimensionless fluid agitation as a function of Reynolds number. The symbols show the average values obtained from different particle-fluid combinations. The error bars depict the variability of the measurements obtained from the same nominal parameters. The horizontal dashed line shows the average value of the agitation for Re> 150. 3.5. Vorticity fields Form the velocity measurements presented above it is straight-forward to calculate the vorticity of the flow. For this case, since the flow is axisymmetric, the only non-zero vorticity component is the azimuthal one, ω ψ. From the velocity components in the plane of measurement, u and v, the vorticity is calculated directly by ω ψ = u y v x (3.4) Figures 9 and 10 show the vorticity field for different times for a typical experiment. These vorticity plots correspond to the velocity fields presented in Figs. 3 and 4. The structure of the vorticity field before the particle has made contact with the wall is in accordance to what is expected for the flow around a sphere at such Reynolds number (Clift, Grace and Weber, 1978).
Velocity fluctuations resulting from a particle collision 17 Figure 9. Vorticity field around a sphere, during the collision with a wall. These vorticity fields were obtained from the velocity fields shown in Figs. 3 and 4. The horizontal and vertical dimensions are scaled with D, the magnitude of the vorticity is scaled with U p/d. For this case, Re=400. It can be observed that the vorticity that was confined in the wake is shed forward when the sphere makes contact with the wall. The vortical structures move around the
18 A. Ruiz-Angulo and R. Zenit Figure 10. Vorticity field around a sphere, during the collision with a wall (continuation of Fig. 9). surface of the sphere and interact with the horizontal wall. Some additional vorticity is produced as the fluid moves around the sphere and over the wall. The main vortex reaches the wall after certain time and it appears remain fixed in a stationary position and slowly dissipates.
Velocity fluctuations resulting from a particle collision 19 3.6. Motion of the wake as a result of the collision To quantify the extent of the disturbance produced by the collision, the motion of the detached vortex ring was tracked as a function of time. The highest vorticity point, which in this case is the center of the main vortex ring, can be located on the vorticity fields for each time. Figure 11 shows the evolution of the horizontal, r, and vertical, z, positions of the center of the main vortex ring as a function of time for a range of Reynolds numbers. As it was observed in Figs. 9 and 10, before the collision the primary ring has a fixed r position approximately equal to 0.6 r/d. At contact, at z/d 1, the ring begins to spread radially and continues to descend. At t 2 the ring reaches the wall. At later times, the center of the ring appears to bounce slightly for Re>300. This is in accordance to what has been observed when a vortex ring collides with a wall (Walker et al. 1987). The radial position of the vortex continues to increase rapidly up to t = 7.5. For greater times, both the z and r position of the vortex remains approximately fixed. It could be argued that the radial position of the ring continues to increase; however, for such long times the uncertainty of the measurements is large since the vorticity is diffused due to viscous effects. For comparison, the results obtained by Leweke et al. (2004b) for a collision at Re = 800 are shown along with the measurements. Clearly, the correlation between the present data set and Leweke s is very good. The two regimes described by Leweke et al. are clearly observed in the three experimental results shown in the figure. 3.7. Total vorticity Since for this flow the vorticity cannot increase as a result of the well-known vortex stretching mechanism, the total flow vorticity can change due to: convection, production due to flow-wall interaction and viscous dissipation. As it was observed above, the wake detaches from the rear side of the sphere and continues to move around it. As the ring
20 A. Ruiz-Angulo and R. Zenit 2 1.5 Re200 Re300 Re400 Leweke Re800 e z * = z / D 1 0.5 0 0.5 5 0 5 10 15 t* = t Up / D 1.6 (a) z-position 1.4 1.2 e r = r / D 1 0.8 0.6 0.4 5 0 5 10 15 t* = t Up / D (b) r-position Re200 Re300 Re400 Leweke Re800 Figure 11. Position of the center of the main vortex ring as a function of time. The positions are shown in dimensionless terms as a function of tu p/d. moves around the sphere, the vorticity of the flow should increase slightly, or at least balance the viscous dissipation. For later times, when the vortex ring ceases to spread radially, the vorticity should decay monotonically resulting from viscous dissipation. To quantify the total amount of vorticity produced as a result of the collision, a measure
Velocity fluctuations resulting from a particle collision 21 of total vorticity can be calculated W (t) = 1 ω ωdv = πr 2 2 V S ω 2 ψds (3.5) considering the control volume, V defined in the previous section. S is the measuring area from Fig. 2 and ω ψ is the vorticity in the azimuthal direction. Figure 12(a) shows the total vorticity considering the same control volume used before to calculate the fluid agitation, for several experiments at different Re. As in the case of the agitation, the total vorticity increases rapidly instants before the particle has reached the wall. At the time at which the contact occurs the total vorticity does not reach a maximum value, as seen for the case of the agitation; the measurement continues to increase for a brief period, presumably because a significant amount of vorticity is created as the fluid moves around the sphere and over the wall. Clearly, for this part of the flow, the vorticity generation and convection is larger than the viscous dissipation. It can be noted that the time period over which the total vorticity continues to increase corresponds to the period during which the detached vortex ring is moving around the sphere. When the vortex ring reaches the wall and ceases to spread radially (t 2, for Figs. 10 and 12), the total vorticity reaches a maximum and then begins to decrease monotonically. For the experiments shown in the figure, the evolution of the total vorticity is similar for all the cases shown. The maximum amount of total vorticity increases with Reynolds number. The rate of decay does vary slightly with Re, being smaller as Re increases. In Fig.12(b) the same data shown in (a) is presented but in dimensionless terms. Clearly, the choice of non-dimensionalization parameters scales the measurements well. The rate of change of the total amount of vorticity in a volume (Batchelor, 1967) can
22 A. Ruiz-Angulo and R. Zenit W(t) = (π R / 2) S ω ψ 2 ds 10 1 10 2 10 3 10 4 Re=200 Re=250 Re=300 Re=350 Re=400 10 5 5 0 5 10 15 20 25 30 35 40 time [s] W* = W / (D Up 2 ) 10 4 10 3 10 2 (a) dimensional total vorticity Re=200 Re=250 Re=300 Re=350 Re=400 10 1 be shown to satisfy 10 0 5 0 5 10 15 20 t * = t Up / D (b) dimensionless total vorticity Figure 12. Total vorticity as a function of time for a several Re d dt V 1 ω ωdv = 2 V Using the vorticity conservation equation we can write d dt V ω D ω dv (3.6) Dt 1 ω ωdv = ω {( ω) v + ν 2 ω}dv. (3.7) 2 V After a few non-trivial steps of vector algebra and using the divergence theorem we can
write, in index notation, d dt V Velocity fluctuations resulting from a particle collision 23 ( ) 1 2 ω u i ωi ω i iω i dv = ω i ω j dv ν dv + 12 V x j V x j x ν (ω i ω i ) n j dγ (3.8) j Γ x j where Γ is the material surface surrounding V. The second and third terms of Eqn. 3.8 represent the viscous dissipation and diffusion, respectively. The first term of the equation is different from zero in locations where the fluid rate of extension is in the same direction as the vorticity. In the absence of this vortex-stretching mechanism, the change of the total amount of vorticity is given by d dt W (t) = d dt V 1 2 ω iω i dv = 1 2 ν Γ which for an axisymmetric flow reduces to: ( ) (ω i ω i ) ωi ω i n j dγ ν dv, (3.9) x j V x j x j d dt W (t) = d πr ω dt 2 ψds 2 = 1 ( (ω 2 S 2 ν ψ ) n r + (ω2 ψ ) ) Γ r z n z dγ νπr S ( ) 2 ωψ + r ( ) 2 ωψ ds. (3.10) z The first term on the right hand side of the equation above represents the diffusion of vorticity through the surface Γ, while the second term represents the dissipation rate. Clearly, W (t) can increase only as a result of the first term, which accounts for the vorticity diffusion through the boundary of the measuring volume or for the creation of new vorticity on the surface of the sphere and the wall. On the other hand, the dissipation term can only lead to a reduction of the total vorticity. An influx of vorticity can be observed in Figures 9 and 10, as the particle and its wake enter the measuring volume (until t 2). For this time period the total vorticity, shown in Fig. 12, continues to increase and reaches a maximum value. Subsequently the total vorticity begins to decrease, which is an indication of the prevalence of the viscous dissipation. Therefore, it can be expected that the rate of change of the total vorticity will first increase as a result of diffusion influx through the boundaries of V (first term
24 A. Ruiz-Angulo and R. Zenit dominant) to then steadily decrease with time (second term dominant), as observed in Fig.12. 4. Summary and Conclusions In this investigation, a measurement of the fluid agitation induced by single collisions is reported. To our knowledge, such detailed measurements do not exist in the literature. Making use of a PIV system, the fluid motion around a sphere colliding with a wall at relatively large Reynolds number was observed. It was found that the collision induced agitation increases with the volume of the particle and the square if its velocity, for Re>150. Moreover, a description of the production and dissipation of the vorticity associated with the collision was presented. The motion of the vortex ring that detached from the sphere as a result of the collision was tracked for several Re. A good agreement was found with the experimental and numerical results of Leweke et al. (2004b). The determination of the collision induced agitation is of importance to various subjects of current interest. Although computers nowadays permit the performance of detailed simulations, in many cases parts of the flow are modelled rather than solved. For instance, instead of resolving the entire fluid motion around particles undergoing collisions, it is usual to model the collision process through a global lumped parameter, the coefficient of restitution, ɛ. Both Joseph et al. (2001) and Gondret, Lance and Petit (2002) obtained experimental measurements of the coefficient of restitution for the case of liquid immersed collisions. They found that the coefficient of restitution scales with the particle Stokes number. This relatively simple relation can be used to model the particle collision interaction in models or simulations of particulate two-phase flows (see for example Hadinoto and Curtis, 2004). In the same manner, the velocity fluctuations
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