Fluid velocity fluctuations in a collision of a sphere with a wall

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1 Fluid velocity fluctuations in a collision of a sphere with a wall Angel Ruiz-Angulo a Roberto Zenit a J. Rafael Pacheco b,c,, Roberto Verzicco d, a Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, México D.F , México b School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA c Environmental Fluid Dynamics Laboratories, Department of Civil Engineering and Geological Sciences, University of Notre Dame, Notre Dame, Indiana 46556, USA d Dipartimento di Ingegneria Meccanica, Universita di Roma Tor Vergata, Via del Politecnico 1, 033, Roma, Italy Abstract The fluid motion around a sphere colliding with a wall immersed in a viscous fluid is studied both experimentally and numerically. Using a particle image velocimetry (PIV) technique and a Navier-Stokes solver, the velocity field around the sphere was obtained for different times before and after the collision. Spheres of different diameters were displaced at constant speeds to achieve a range of Reynolds number from 50 to 400. In accordance with previous studies, we observed that the viscous wake (originally placed behind the sphere) detached and continued moving forward as a result of the collision. The motion of this vortical structure induces fluid velocity fluctuations in a region close to the contact point. We quantified the fluid motion induced by the collision considering a bulk measurement of the fluid disturbances. This fluid agitation is a measure of the kinetic energy and, therefore, it is readily associated with the added mass of the sphere. The measured agitation was found to increase weakly with Reynolds number and to approach the value inferred from potential flow for large Re. Furthermore, the temporal evolution of the total vorticity of the flow induced by the collision was studied. We found that, at first, the total vorticity increases rapidly resulting from the stretching of the detached wake as it overcomes the surface of the sphere. When the wake ceases to stretch radially, then the total vorticity begins to decay at an exponential rate as a result of viscous dissipation. The different components of the dissipation rate are analyzed to evaluate their relative importance during the dissipation process. In general, we found an excellent agreement between our experimental and numerical results. Preprint submitted to Elsevier Science 12 March 20

2 Key words: Collision, sphere, low Reynolds number, interaction PACS: , E-, Cb 1 Introduction Particulate two-phase flows are prominent in many industrial applications and natural phenomena. Despite their importance, a thorough understanding of the flow behavior is still lacking. The fluctuating nature of these flows is one of the important aspects that still lacks basic comprehension. Particulate twophase flows have turbulent-like behavior at lower Reynolds numbers than those observed in single-phase turbulent flows; this characteristic makes two-phase flows very attractive in industrial applications (Nagata, 1975). Since the interstitial fluid must move around the inclusions that form the particulate phase a velocity disturbance (agitation) naturally arises in the continuous 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. There are notable exceptions, e.g., the case of bubbly liquids at high Reynolds and at low Weber numbers (Kang et al., 1997; Spelt and Sangani, 1998) and the case of low Reynolds number suspensions, for which models have been proposed to predict the hydrodynamic fluctuations for both, sedimenting particles (Brenner, 20) and simple shear flows (Drazer et al., 2004). Advances in the understanding of fluctuations have been possible to a large degree from experimental measurements. Cartellier and Riviere (20) discussed the induced agitation produced by bubbles in uniform bubbly liquids for small to moderate Reynolds numbers. In their study, scaling laws for the fluctuations were proposed and comparisons were made with other particulate systems, i.e., viscous sedimenting suspensions. Martínez-Mercado et al. (2007) measured the velocity fluctuation in bubbly flow for a wide range of Re. They found that the scaling progressively changed for flows dominated by either viscous or inertial effects. There are many studies that addressed the creation of fluid fluctuations resulting from the motion of particles in solid-liquid flows. Two important experimental investigations were conducted by Parthasarathy and Faeth (1990) and Kenning (1996), in which dilute concentrations were investigated. The particles were released in a nearly stagnant fluid and therefore the velocity-fluctuations were the direct product of particle-motion. In both studies it was found that the velocity-fluctuations of the liquid increased Corresponding author. Address: ASU P.O. Box Tel: address: rpacheco@asu.edu (J. Rafael Pacheco). URL: rpacheco (J. Rafael Pacheco). 2

3 with particle loading and particle Reynolds number. Clearly, in such dilute flows, the particle collisions are infrequent and collisions do not contribute significantly to the generation of velocity fluctuations. More recently, Aguilar- Corona (2008) reported a full set of experimental measurements in dense liquid fluidized beds, using index matching techniques. The agitation is important in situations of 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 (1998) and Zenit et al. (1999) showed a series of visualizations of the flow around particles colliding against a wall. They briefly discussed 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 followed by a study in which dust resuspension was discussed (Eames and Dalziel, 2000). More recently, Leweke et al. (2004a) performed a detailed visualization of the same flow, and Leweke et al. (2004b) complemented the study by performing direct numerical simulations. Thompson et al. (2007) analyzed the interaction of the detached vortex ring with the sphere and the wall and performed a stability analysis to predict the threshold 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 of a particle colliding immersed in a liquid (Joseph et al., 20; Gondret et al., 2002). By analyzing the fluid motion around colliding spheres, a quantitative measure of the collision-generated disturbance was measured in a direct manner. In this investigation, we propose a norm for the amount of agitation created in the suspending phase as a result of a collision of a single particle with a wall. The aim of this combined experimental and numerical study is to provide a basis to the understanding of generation of velocity fluctuations resulting from collisions in solid-liquid flows. To the best of our knowledge, this basic quantitative estimation is lacking in the two-phase flows literature. The experimental techniques and numerical methods used in this paper are summarized in section 2, all of which have already been used in the analysis of different but related problems (Zenit et al., 1999; Joseph et al., 20; Orlandi and Verzicco, 1993), and only the salient aspects are presented. The results from numerical simulations and experiments for Re [50, 400] are analysed in section 3 which include a definition of the fluid agitation. Summary and conclusions are presented in section 4. 3

4 DC DC motor motor Fine Fine threads threads LASER LASER Viscous Viscous fluid fluid + + tracers tracers Sphere Sphere Glass Glass plate plate Controler Controler CCD CCD Camera Camera Processing Processing Storage Storage (PC) (PC) Fig. 1. Experimental apparatus used to generate controlled collisions of a sphere with a wall. 2 Experimental setup and numerical method Measurements of the velocity field around a sphere during the collision process were performed in a rectangular glass container of cm 3, as shown in figure 1. A thick glass plate was placed at the bottom of the container where the particle collisions occurred. The particle release mechanism was placed at the top, on the lid of the container. Two fine Nylon threads were glued to the particle poles to inhibit rotation, as shown in figure 1. The motion of the particle was controlled by slowly unwinding the threads from the shaft of a computer-controlled DC motor. Therefore, the inertia of the particle does not affect the motion of the sphere or the fluid motion, as opposed to the flows where particles are settling freely (Joseph et al., 20; Gondret et al., 2002). For most of the experiments spheres with diameters of 25.4mm were used in water at laboratory conditions (µ = 1.1mPa s). To corroborate the correctness of the results, some experiments were performed with smaller spheres (D = 12.7mm) in a water/glycerin mixture (µ = 2.4mPa s). The corresponding range of particle Reynolds number ranged between 50 and 400. Reynolds 4

5 number was defined 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, µ is the the dynamic viscosity of the fluid and ρ f is density. For a given experiment, the particle was placed originally at a distance of eight particle diameters from the bottom plate, and the voltage was set to a constant value before energizing the motor. After a very small transient, the sphere began to descend at constant velocity. Since the particle Reynolds number was large, a deceleration of the particle was not observed before it collided with the wall (Joseph et al., 20). 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 (Dantec Flowmap 1500 model) was used. The flow was illuminated with 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 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 160 ms, the highest allowed by the system. The field of view of the camera was approximately mm 2. And adaptive cross-correlation technique was used, with a final interrogation area of pixels, and an overlap of 50% in both directions. Subsequently, a peak validation, moving average and spacial filter routines were applied. The resolution of the optical array was 12.5 pixel/mm. Considering the uncertainty protocol of Lourenco and Krothapalli (1995), the velocity uncertainty was calculated to be about 5.3%. 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 was shown and processed. In the figure, the velocity field obtained was superimposed on the typical PIV image. The major drawback of the existing laboratory measurement techniques is that they cannot provide detailed information of all the fluid dynamics (velocity, pressure, and density) in the entire domain at the same time. Even with the existing scanning planar PIV systems (Boyer et al., 2006; Romani et al., 2007), it is extremely difficult to reconstruct a fully three-dimensional picture of the flow field that is crucial for deeper understanding of the particle collision. An- 5

6 Fig. 2. Typical PIV image and velocity field at Re=300; D=2.54 mm. The field of view is approximately 75x85 mm 2. other challenge presented by this unsteady flow is the accurate quantification of the vorticity deduced from PIV measurements. The values of vorticity may have large uncertainties depending on the method of calculation, noise, and spatial resolution. Therefore, one is obliged to rely on numerical simulations to gather additional information of the three-dimensional structure of the flow. Furthermore, direct comparison between experiments and numerical results serve as validation procedures for both techniques. 2.2 Navier-Stokes equations and the numerical scheme Consider the flow in a completely filled cylinder of fluid with kinematic viscosity ν(= µ/ρ f ) of radius R and height H. The walls are stationary and the flow is driven by the motion of the sphere which is impulsively started from rest at constant speed normal to the bottom wall, and stops after traveling a distance h. To non-dimensionalize the system, the same quantities as in the experiment have been adopted, i.e., the diameter of the sphere D is used as the length scale, the velocity scale is the constant velocity of the sphere before impact U p and the time scale is the inertial D/U p. Figure 3 shows a schematic 6

7 R D U p 00 h H Fig. 3. Schematic of the flow apparatus. The inset shows the initial position of the sphere and the vorticity contour of azimuthal vorticity ω θ for t > 0 after the sphere has touched the wall. of the flow. In cylindrical coordinates, (r, θ, z), the non-dimensional velocity vector, pressure and time are denoted by u = (u, v, w), p and t, respectively. The governing equations are the (non-dimensional) incompressible Navier-Stokes equations u/ t + (u )u = p + Re 1 u, u = 0. (1) The boundary conditions for the velocity field are stress-free at the top and no-slip on the side/bottom walls. On the sphere, u and v are zero for all times and w = 1 for t < 0 (prior to impact) and w = 0 for t 0 (perfect inelastic collision). Although the computational domain is a cylinder while the tank of the laboratory is a parallelepiped the diameter and height of the cylindrical matched the width and height of the experimental apparatus and the boundaries are anyway far enough not to interfere with the flow dynamics. The system of equations (1) are written in a cylindrical coordinate frame and discretized on a staggered grid with the velocities at the faces and all the scalars in the center of the computational cell; the resulting system of equations is solved by a fractional-step method. The discretization of both, 7

8 viscous and advective terms, is performed by central second-order accurate finite-difference approximations. The discretization of both viscous and advective terms is performed by second-order-accurate central finite-difference approximations. The elliptic equation, necessary to enforce incompressibility, is solved directly using trigonometric expansions in the azimuthal direction and the tensor-product method (Lynch, Rice, and Thomas, 1964) for the other two directions. Temporal evolution is via a third-order Runge Kutta scheme which calculates the nonlinear terms explicitly and the viscous terms implicitly. The stability limit due to the explicit treatment of the convective terms is CFL < 3, where CFL is the Courant, Friedrich and Levy number. A useful feature of this scheme is the possibility to advance in time by a variable time step, without reducing the accuracy or introducing interpolations. We have varied δt in all the simulations in this paper such that the local CFL 1.5, where CFL = ( u /δr + v /(rδθ) + w /δz) δt, with the velocity components averaged at the center of each computational cell, then the smallest such determined local δt is used for time advancement. The numerical method and validation tests are described in (Verzicco and Orlandi, 1996; Verzicco and Camussi, 1997, 1999, 2003; Smirnov et al., 2009; Pacheco et al., 2009) where more in-depth analysis can be found. The immersed boundary method (IB) is used of this study to simulate the sphere. The main advantage of using the IB consists in solving flows bounded by arbitrarily complex geometries without resorting to body-conformal grids for which the motion is prescribed, and therefore the solution technique essentially has the same ease of use and efficiency as that of simple geometries. The method is second-order in space and this technique has already been implemented in many different scenarios and grid layouts, e.g. laminar and turbulent convection (Pacheco et al., 2005; Pacheco-Vega et al., 2007; Stringano et al., 2006), turbulent flows and particle collision (Kang et al., 2009; Kim et al., 20; Cristallo and Verzicco, 2006; Uhlmann, 2005), biological devices (de Tullio et al., 2009). The three-dimensional simulations were conducted using using the immersed boundary method of Fadlun et al. (2000) with uniform grids in the azimuthal direction and non-uniform grids in the radial direction and axial direction. At least 400 grid points were placed inside the sphere with a clustering of grids at the z-axis and at the bottom wall of the cylinder. Numerical simulations were repeated with different grid sizes to verify the grid-independence results and to test the adequacy of a coarser grid in resolving all the relevant flow scales. For most of the runs, grid points have been used in the azimuthal radial and axial directions respectively. For the simulations requiring the highest resolution, several runs have been made with resulting in a maximum difference in velocities of less than 1%. The flow evolution was followed for 60 time units. We have also performed similar checks as in Verzicco et al. (1997) and Smirnov et al. (2009) who studied different but related problems of baroclinic instabilities in the presence of rotation and stratification. 8

9 3 Results Experimental measurements and numerical simulations were performed for a range of particle Reynolds numbers to determine the velocity and vorticity fields. Since the velocity measurements obtained with the PIV were twodimensional, any deviations from the axisymmetric state without swirl could not be determined with the present experimental setup. Therefore, all the laboratory experiments were conducted for Re 400 and h = 8 in order to minimize the possibility of a three-dimensional flow around the sphere (Walker et al., 1987). Numerically, the magnitude of the azimuthal Fourier modes of the velocity field were monitored in time. In all the cases considered in this study, the azimuthal perturbations remained close to zero (to machine accuracy). The excellent agreement between the numerical results and the experimental measurements (to be shown in the following sections) suggests that considering an axisymmetric flow in the experimental setup (in the neighborhood of the sphere and without swirl) is a reasonable assumption. Figure 4 shows a typical evolution of the velocity field around a sphere colliding with a wall. Each image shows a contour representing the azimuthal vorticity ω θ (= u/ z w/ r) superimposed with a vector map representing the direction and magnitude of the fluid velocity at each point. The numerical results are shown on the plane θ = 0 and the results from the laboratory experiments on the plane θ = π. From figure 4, it can be observed that after the collision, the wake (originally in the back of the sphere) moves forward around the surface of the sphere (0 < t < 1). At t 2 the vortex hits the wall and begins to spread radially. For t 4, the center of the ring remains fixed in space and the fluid motion decreases gradually due to the viscous dissipation. The nature of the motion of the wake described above is qualitatively similar for the range of Re used in this investigation. These observations are in agreement with other studies (Eames and Dalziel, 2000; Leweke et al., 2004a,b; Thompson et al., 2007). 3.1 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 by defining an agitation quantity A(t) = (u 2 + v 2 + w 2 ) dv, (2) V where V is a control volume over which the agitation is measured, (u, v, w) are the vector components of fluid velocity in the radial, azimuthal and vertical 9

10 (a) t = (b) t = (c) t = (d) t = (e) t = 2.96 (f) t = 4.13 Fig. 4. Velocity and vorticity fields around a sphere during the collision with the bottom wall for Re = 400. Results from experiments and numerical simulations are shown on the meridional plane separated by and angle π. The contours represent the vorticity ω θ. Each image corresponds to a different time instant. At inertial time t = 0 the particle makes contact with the wall. directions respectively. This definition of fluid agitation is related to the kinetic energy imposed on the fluid as a result of the collision, T c = 1/2ρ f  (the fluid agitation is written in dimensional form). Furthermore, we can infer an effective virtual mass m considering 1 2 m U 2 p = 1 2 ρ fâu 2 p D 3. (3) Hence, the collision virtual mass coefficient K c would be: K c = m π = 6 A. (4) 6 D3 ρ f π 10

11 Fig. 5. Total fluid agitation as a function of the measuring volume. The case shown is a numerical run at Re = 400. In the experiments, A(t) can be calculated for each image pair such that the progress of the agitation can be measured. For an axisymmetric flow without swirl, the equation above can be simplified to A(t) = 2π (u 2 + w 2 )rdrdz, (5) S where S is the surface of the measurement plane. The area of the surface is related to the volume by the expression V = 2π rs, where r is the distance from the centroid of the surface S to the origin (see the dashed line box in figure 2). A measure of the total agitation produced by a single collision event can be defined as T A = 1 A(t)dt (6) T 0 where T is the time over which the agitation is calculated, i.e., from the time prior to the particle touching the wall to the time where the fluid motion has nearly ceased. The quantity A const. as T. Thence the time interval T is selected so that the integral in (6) reaches 99% the value it would have at T during a given experiment, which is similar to the definition of the boundary layer thickness on a flat plate. The height of the surface (particle diameter) was chosen to account for the fluid disturbances resulting from the particle collision alone, thence excluding the fluid motion in the wake of the particle before the collision. As the the 11

12 size of the surface increases (keeping the height constant), the experimental measurement of A should reach a constant value, because far from the center of the sphere, the velocity of the fluid is negligible. Ideally the measurement of the agitation should be independent of the magnitude of the surface S. Due to experimental limitations regarding the size of the measuring area (the field of view of camera was approximately 3 3 dimensionless units) we relied on the numerical verification for the proper dimensions of the control volume (see figure 2). Figure 5 shows the dependence of the total agitation A on the size of the volume V obtained numerically. As shown in the figure, A attains a constant value for V > 10. The size of the surface S used in the measurements was 3 1 which corresponds to a value of the measuring volume V = 54. Therefore, this size of V was used to estimate the agitation for both experiments and numerical simulations. 3.2 Evolution of fluid agitation The flow evolution of the agitation from (2) is shown in figure 6. The figure shows two different experiments performed at Re = 400 to demonstrate the repeatability of the experiments along with the numerical results under the same conditions. The time origin has been displaced such that t = 0 corresponds to the time when the sphere makes contact with the solid surface. 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 followed by a sharp decay, as the fluid in the gap between the particle and the wall is being squeezed into the 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, as the viscous dissipation begins to damp out the fluid disturbance generated by the collision. Figure 7 shows the evolution of the agitation obtained from the numerical simulation for Re [50, 600]. Notice the strength of the viscous effects (Re < 200) on the inertia of the detached wake. The trend shown in the figure suggests that the magnitude of the agitation increases and the rate of decay of the fluid disturbance decreases slightly as the Re increases. The experiments show a similar trend, within experimental uncertainty. 12

13 Fig. 6. Fluid agitation as a function of time for Re = 400 for two different experiments and the numerical solution. The two results from experiments are indicated with symbols and, and the numerical solution with solid line. Fig. 7. Fluid agitation as function of time for several Reynolds numbers., Re = 600;, Re = 400;, Re = 300;, Re = 200;, Re = 100;, Re = 50 13

14 Fig. 8. Total fluid agitation as a function of Re. 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 dashed line shows the curve-fit of the average value of the agitation from the experiments for Re 200 and the solid line indicates the numerical results. The dashed dotted line at the right shows the value expected for potential flow from equation (4). 3.3 Total agitation The total amount of energy transferred to the fluid during a single collision can be presented in terms of the total agitation A defined by equation (6). Figure 8 shows the total agitation measured for all the experiments performed in this investigation together with the numerical results. The variability is particularly large at low Reynolds numbers (Re 200) because the summation of many velocity vectors over a large area amplifies the uncertainty. The experimental data for Re 200 can be fitted to a straight-line < A >= mre+b where m = The solid line in the figure represents the numerical results of total agitation for Re [50, 600]; the slope is approximately m = The trend suggests that the total agitation scales linearly (albeit weakly) with Re. We note in passing that for this particular experimental setup the total agitation does not depend on the Stokes number because the particles are not moving freely. Considering equation (3) we can expect the value of A to reach the limit for very large Re (potential flow limit) which corresponds to K c = 1/2. Our measurements are in reasonable agreement with this upper bound value. 14

15 3.4 Vorticity fields From the velocity measurements presented above it is straight-forward to calculate the vorticity of the flow. Plots of the vortex structure visualized with the Q-criterion (Hunt et al., 1988) and the vorticity field for different times for a typical experiment are depicted in figure 9. These plots correspond to the velocity/vorticity fields presented in figure 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 et al., 1978). It can be observed that the vortical structures move around the 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 remains in a stationary position slowly dissipating. Movies 1 and 2, available with the online version of the paper, are animations of the vortex structure and contours of azimuthal vorticity ω θ demonstrating the axisymmetric nature of the flow as well as the motion of the wake as a result of the collision. 3.5 Total vorticity A measure of the total amount of vorticity (enstrophy) produced as a result of the collision can be obtained from W (t) = 1 ω ωdv, (7) 2 V whereas the rate of change of the total amount of vorticity dw (t)dt can be shown to satisfy (Batchelor, 1967) { dw dt V = ω ω u + 1 } Re 2 ω dv, = ω (ω u) dv + 1 V 2 Re Γ( ω 2 ) dγ 1 ( ω : ω) dv, Re V 3 = H i. (8) i=1 The left-hand-side of (8) represents the total amount of vorticity change due to (H 1 ) vortex stretching; (H 2 ) viscous difussion through the boundary; and (H 3 ) viscous dissipation (it is always a sink). The vortex stretching term vanishes for all two-dimensional flows, since both the velocity and the velocity gradient vanish in the direction of the vorticity vector (i.e., normal to the plane of 15

16 (a) t = 0.52 (b) t = 0.64 (c) t = 1.80 (d) t = 2.96 (e) t = 4.13 (f) t = 5.92 (g) t = 6.45 (h) t = 7.61 Fig. 9. Vortex structure visualized with the Q-criterion and vorticity field around a sphere during the collision with the bottom wall for Re = 400. Results from the numerical simulations are shown on the meridional plane separated by and angle π. The contours represent the vorticity ω θ. Movie 1, available in the online version, shows an animation of the falling sphere at a rate of 6 frames per second. 16

17 Fig. 10. Time evolution of total vorticity W (t) as a function of the inertial time. The symbols are the experimental data for Re = 300 ( ) and Re = 400 ( ). The lines are the numerical data for Re = 300 ( ) and Re = 400 ( ). motion). For an axisymmetric flow without swirl (rv = 0), the vorticity and velocity vectors are everywhere orthogonal, but the vortex stretching term is not zero since the velocity gradient does not vanish in the direction of the vorticity vector. For an axisymmetric flow W (t) = 1/2 ω 2 θdv, therefore (8) reduces to dw dt V = u ωθ 2 dv + 1 ( ) (ω 2 θ ) r 2Re Γ r n r + (ω2 θ) z n z dγ 1 ( ) ( ) ωθ 2 2 ( ) 2 ωθ ωθ + + dv, (9) Re V r r z where n r and n z are the components of the unit-vector normal to the surface. The time-evolution of W (t) for Re = (300, 400) is depicted in figure 10. An influx of vorticity can be observed in figure 9 and movie 1 as the particle and its wake enter the measuring volume (t 0). Notice the vorticity contour at the leading edge of the sphere and the shape of the vortex structure around it, shown in figure 9(a) at t = At around t 0, the total vorticity (shown in figure 10) begins to increases due to the combined effect of diffusion and advection of vorticity through the boundary. At around t 0, there is a peak in the curve as the sphere is suddenly arrested. The amount of vorticity carried by the leading portion of the sphere is suddenly brought to zero causing a sharp decline in W in a very short period of time (0 t 1). The amount of total vorticity now begins to increase due to a combined effect flux of vorticity through the upper portion of the control volume caused by the shedding of 17

18 Fig. 11. Time evolution of total vorticity W (t) as a function of the viscous time t ν. The symbols are the experimental data for Re = 300 ( ) and Re = 400 ( ). The lines are the numerical data for Re = 300 ( ) and Re = 400 ( ). vorticity from the wake of the sphere (1 < t < 2, figures 9(b-c)-10) and fluid elements moving radially outward elongating in the direction of the vorticity (figures 9(c-h)-10, t 2). Similar evolutions of W (t) were observed for the range of Reynolds numbers considered in the experiments. We can estimate the rate of change of total vorticity using (9). From an orderof-magnitude analysis, we first neglect viscous effects to estimate the growth rate of W (t) by considering that the vortex stretching term (H 1 ) dominates for 0 < t < 2. If we assume that u is proportional to r in the the wake (for that time lapse) we can write V uω 2 θ r dv τ 1W (t), (10) where τ 1 is a proportionality constant. Therefore, (9) can be simplified to dw/dt = τ 1 W. Under these assumptions, we have W (t) exp(t/τ 1 ). (11) In figure 10, the exponential increase of W (t) is observed for both experimental and numerical results for t < 2. Now, when the vortex stretching mechanism (H 1 ) no longer dominates, that is when the vortex has reached the wall and has overcome the surface of the sphere (t > 4), the total vorticity begins to decrease. Neglecting both the vortex stretching (H 1 ) and the viscous diffusion through the surface Γ (H 2 ) in 18

19 (8), we can write dw dt 1 Re V ( ωθ r ) ( ) 2 2 ωθ + + r ( ) 2 ωθ dv. (12) z From figure 10, for times t > 4, an exponential decay of W can be observed. Such a dependence would be expected if the integrand of the right-hand side of (12) is proportional to ( ωθ r ) ( ) 2 2 ωθ + + r ( ) 2 ωθ ˆω2 θ z D, (13) 2 where ˆω θ denotes the dimensional azimuthal vorticity. In fact, from our numerical results we have verified that this is indeed the case. Furthermore, we found that ω θ r ω θ z ω θ r (14) within the measuring volume. Hence, (12) can be simplified to dw/dt = 1/ReW. Considering a diffusive time-scale, t ν = t/re, the rate of change of total vorticity is W (t) exp( t ν /τ 2 ), (15) where τ 2 is a proportionality constant. Figure 11 shows the behavior of W as a function of t ν. Clearly, the rate of decay is fitted closely by a decaying exponential. A certain dependence of the rate of decay is expected for different values of Re. Finally, for our numerical results, it is possible to evaluate the size of each of the components in the right-hand side of (8) Therefore, it is possible to evaluate the relative importance of each term during the evolution of W. Figure 12(a) shows the measurement of dw d/t and W from a typical simulation result. Figure 12(b) shows the value of each of the terms (H i ) in the right-hand side of (8). It can be observed that, prior to the contact with the wall, the rate of change of W is positive because the vortex stretching and diffusion dominate over dissipation. Shortly after the moment of contact, the diffusion and rate of stretching terms begin to decrease in value, while the viscous dissipation remains relatively constant. Hence, a decrease in dw/dt is observed. For longer times, the viscous dissipation term dominates the process, leading to the exponential decay of the total vorticity observed in figure

20 (a) Fig. 12. (a) Time evolution of: (a) the left-hand-side and right-hand-side of the rate of change of total vorticity dw/dt ( ), ΣH i -terms ( ) and total vorticity W ( ) for Re = 400; (b) H i -terms and the rate of change of total vorticity dw/dt. Vortex stretching H 1 ( ); viscous diffusion H 2 ( ); viscous dissipation H 3 ( ); rate of change of total vorticity dw/dt ( ) for Re = Summary and Conclusions (b) 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 the Reynolds number linearly, i.e., ( A Re) for 50 < Re < 400. Since the concept of fluid agitation is a measure of the kinetic energy of the fluid, its definition is readily associated with the well-known concept of added mass. Our measurements are in good agreement with the value of agitation expected from potential flow (infinite Re). Therefore, the fluid agitation resulting from a single particle collision can be approximated by the added mass coefficient. A description of the production and dissipation of the vorticity associated with the collision was also 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); Thompson et al. (2007) and our own numerical simulations for the same flow-conditions. 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. (20) and Gondret et al. (2002) obtained experimental 20

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