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Vorticity and Strain Dynamics for Vortex Ring Mixing Process

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Vorticity and Strain Dynamics for Vortex Ring Mixing Process 14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008 Paper No. 1271 VORTICITY AND STRAIN DYNAMICS FOR VORTEX RING MIXING PROCESS Yann Bouremel1, Michael Yianneskis2, Andrea Ducci3 Experimental and Computational Laboratory for the Analysis of Turbulence (ECLAT), Division of Engineering, King’s College London, Strand WC2R 2LS, UK; 1: [email protected] 2: [email protected] 3: [email protected] Abstract. Mixing is of paramount importance in the chemical, petrochemical and pharmaceu- tical industries, as almost half of their production lines involve mixing in stirred vessels, but further optimisation of related processes is hampered by incomplete understanding of the fluid mechanics of mixing. Consequently further investigations are necessary to obtain a better understanding of the dif- ferent mixing mechanisms taking place at different spatial length scales in order to optimise chemical processes and reactor design. Due to the complexity of mixing reactor design, a fundamental flow, such as that associated with a vortex ring, has been selected to develop a reliable procedure to gain some improved understanding of mixing mechanisms by assessing the competitive actions of the strain rate tensor, S, and vorticity vector, ω. 1 INTRODUCTION A vortex ring is a bounded region of vorticity whose streamlines delimit a torus ring. Vortex rings have generated considerable interest for a long time because of their intrinsic properties as well as their potential for technological applications. Vortex rings are very stable structures that are capable of propagating over long distances before being dissipated. Due to these characteristics, they can be used to deliver drugs. For example, vortex ring based devices are often preferred to nebulisers to inject medicated mists into the lungs. Other applications include non-lethal weapons and fire extinguishers to reach inaccessible spaces. Many examples of vortex rings can be found in nature: jellyfishes use vortex rings for propulsion (Shariff and Leonard, 1992), while dolphins blow vortex rings when socialising. Reviews of vortex ring studies can be found in Shariff and Leonard (1992) and Lim and Nickels (1995). There are several ways to produce a vortex ring, the most common one employs a piston pushing a column of fluid inside a tube which opens into an expansion chamber where the vortex ring is formed. As the piston pushes the column of fluid through the pipe, the tube boundary layers separate at the edge of the orifice and roll into a spiral generating the vortex ring. When the amount of energy absorbed by the vortex ring reaches a maximum, the vortex pinches-off and travels along the tank under its self-induced velocity. During this stage, the vortex ring goes through a laminar, then a transitional and finally a turbulent phase. A trailing jet at the 1 rear of the vortex ring is present only when the piston keeps moving after the vortex ring has pinched-off. The extra energy introduced into the flow is stored in a trailing jet similar to the Kelvin-Helmhotz instability. Vortex rings exhibit similarities with the trailing vortex structures emanating from the impellers of stirred vessels. As impeller-generated flows are rather complex, the vortex ring was selected as a simple fundamental flow in order to develop a reliable methodology to gain improved understanding of the mixing mechanisms taking place in vortical structures, by assessing the competitive actions of the strain rate tensor, S, and vorticity vector w. Gharib et al. (1998) studied the effects of the non-dimensional stroke L/D (where L is the piston stroke and D the internal pipe exit diameter) on the formation process and determined the range of stroke ratios (also called the formation number, (L/D)p−o) associated to the pinch-off stage. L can also be written as L = UpTe, with Up being the average piston velocity and Te the ejection time. The analysis of Gharib et al. (1998) shows that the the maximum amount of energy that can be absorbed by the vortex is obtained in a relatively narrow range of (L/D)p−o = (3.6−4.5). Within this range of L/D, the Reynolds number is defined according to equation (1), which is based on the slug-flow model (Shariff and Leonard, 1992): Γs−f Re = ν 1 (1) Γs−f = 2 LUp with Γs−f being the initial vortex ring circulation given by the slug-flow model. The current paper aims at studying the vorticity and strain dynamics of a vortex ring. From this prospective, a comparison between the enstrophy and scalar strain dynamics of a vortex ring flow has been made by Kenneth et al. (1991), who experimentally estimated the different terms of the instantaneous advection-diffusion equations of the scalar dissipation rate (∇ξ ·∇ξ). In particular they pointed out that the straining components affecting the dynamics of the enstrophy and of the scalar dissipation rate are always perpendicular to each other. 2 FLOW CONFIGURATION AND EXPERIMENTAL APPARATUS PIV measurements were obtained to study the vorticity and strain dynamics of the vortex ring; for the trailing vortex results, the LDA velocity measurements of Sch¨afer (2001) were employed to calculate the vortex dynamics of interest for the present work. The vortex ring flow was produced by a piston pushing a column of fluid inside a tube of diameter D =28.6 mm which opened into an expansion chamber where the vortex ring was formed. A sketch of the experimental set-up used to generate the vortex ring is shown in figure 1. 2 Figure 1: Sketch of the experimental rig employed to generate vortex ring. The experiments for the vortex strain dynamics presented in this paper were carried for −1 piston stroke and velocity of L = 4D and Up=7.5 cm s , respectively. To prevent the formation of a trailing jet and piston vortices behind the vortex ring the piston stroke was terminated inside the pipe at a distance of S/D = 1 from the tube exit. The flow field produced by the vortex ring was investigated at an axial distance centered around x = 7D from the pipe exit when the vortex is propagated forward by its self induced velocity (Shariff and Leonard, 1992; Rosenfeld et al., 1998). An estimation of the vortex velocity of translation can be obtained from figure 2, where the normalised axial coordinate of the vortex core, xvc/L, is plotted against the normalised piston displacement, Upt/D. These data refer to experiments that were initially carried to characterise the flow during the vortex formation stage and the early phase of self- −1 propagation suddenly after pinch off (L/D=6 and Up=7.5 cm s ). In figure 2 pinch off takes place for a piston displacement Upt/D=4, when the slope of the curve becomes linear and the vortex propagates with a constant velocity (2.7 cm s−1 for the experimental data shown in figure 2). 0.36 Vortex center Xcoordinates pinched-off vortex trajectory fit line 0.30 0.24 [-] 0.18 /L vc x 0.12 0.06 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Upt/D [-] Figure 2: Variation of the vortex axial coordinate, xcv/L, with piston displacement Upt/D, −1 (L/D=6, Up=7.5 cm s ). 3 3 DEFORMATION RATE TENSOR The stresses cause deformations of the fluid elements that are determined by the spatial varia- ∂ui ∂ui tions of the velocities ui: called the rate-of-deformation tensor. The fluid deformation ∂xj ∂xj is decomposed into a pure rate-of-deformation tensor called the rate-of-strain tensor and a pure rate-of-rotation tensor. ∂ui 1 ∂ui ∂uj 1 ∂ui ∂uj = + + − = Sij + Ωij (2) ∂xj 2 ∂xj ∂xi 2 ∂xj ∂xi with Sij being the rate-of-strain tensor and Ωij being the rate-of-rotation tensor. The rate-of- deformation tensor ∂ui can be decomposed along the principal axes of the S tensor to visualise ∂xj and quantify the main direction of stretching and compression (Davidson, 2004). ∗ ∗ ∗ S11 0 0 0 −ω3 ω2 ∗ ∗ ∗ ∇u = 0 S2 0 + ω3 0 −ω1 (3) ∗ ∗ ∗ 0 0 S33 −ω2 ω1 0 ∗ ∗ ∗ ∗ ∗ ∗ with S11, S22, S33 being the eigenvalues of S and ω1, ω2, ω3 being the components of the vorticity ∗ along the principal axes of S for each point. Each Sii and the corresponding eigenvector define ∗ the local intensity and direction of stretching or compression; if Sii > 0, the fluid element ∗ considered is stretched, whereas if Sii < 0, the fluid is compressed. Furthermore, according to this coordinate system, the energy dissipation ǫ can be expressed in ∗ terms of the stretching and compression eigenvalues (Sii) as shown in equation 4 (Davidson, 2004): ∗2 ∗2 ∗2 ǫ = 2νSijSij = 2ν(S11 + S22 + S33 ) (4) 4 VORTEX RING STRAIN DYNAMICS The strain dynamics controlling the vortex ring during the propagation stage are discussed in this section. The vorticity contour and velocity field shown in figure 3 were obtained by averaging the instantaneous flow fields of an interrogation area of size 2.5D × 2.46D moving with the vortex for 60 consecutive frames taken with a frequency of 235 Hz. The symmetry of the vortex ring is evident from the cross section shown in figure 3 with two counter-rotating vortex cores with opposite vorticity of equal magnitude (ωmax = kωmink = 32Up/L). 32 8 24 16 7.5 8 L [-] 0 ω [-] 3Up x D 7 -8 -16 6.5 -24 6 -32 -1 -0.5 0 0.5 1 y D [-] Figure 3: Vorticity contour and velocity vector plots of the vortex ring flow at x/D = 7.
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