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 different 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 Schäfer (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).

8 24

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.

4 ∗ ∗ ∗ The contour plots of the principal components of the strain rate tensor S11, S22 and S33 are shown in figures 4, 5 and 6, respectively. It should be noted that the translational velocity (Utr) of the vortex ring was subtracted from the velocity fields before estimating the strain ∗ rate tensor, and that the component S33 associated to the fluid strain rate perpendicular to the plane of measurement was obtained from the continuity equation of an incompressible fluid (i.e. ∂uz/∂z = −∂ux/∂x − ∂uy/∂y). A reference contour line delimiting the border of the vortex core is also shown in white in figures 4, 5 and 6. A comparison between the contour ∗ ∗ plots of figures 4 and 5 shows that the eigenvalues S11 and S22 assume solely positive and negative values, respectively, in the entire field of view, with the region in front of the vortex ∗ L being mainly subject to compression, -8

8 7.8 7.6 7.4

[-] 7.2

x/D 7 6.8 6.6 6.4 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 y/D [-]

0 1 2 3 4 5 6 ∗ L S11 [-] Up

∗ Figure 4: Plots of the contour of the eigenvalue S11 and of selected reference contour lines locally tangent to the corresponding eigenvector.

5 8 7.8 7.6 7.4

[-] 7.2

x/D 7 6.8 6.6 6.4 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 y/D [-]

-8 -7 -6 -5 -4 -3 -2 -1 0 ∗ L S22 [-] Up

∗ Figure 5: Plots of the contour of the eigenvalue S22 and of selected reference contour lines locally tangent to the corresponding eigenvector.

7.8

7.6

7.4 [-] 7.2

x/D 7

6.8

6.6

6.4 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 y/D [-]

-5 -4 -3 -2 -1 0 1 2 3 ∗ L S33 [-] Up ∗ Figure 6: Contour plot of the strain rate S33 directed orthogonally to the measurement plane.

∗ The contours of S33 (see figure 6) shows opposite behaviour at the front and rear sides of the vortex with fluid particles approached by the vortex being stretched in the direction orthogonal ∗ L to the measurement plane, 1.5 < S33 < 3.5, and fluid particles left behind the vortex being Up ∗ L compressed with −5 < S33 U < −2. Considering the axis-symmetry in the x direction of the p ∗ flow field investigated, this behaviour is expected, and the third eigenvalue S33 directed as z must display a similar trend to those eigenvalues that are directed as y at the front and rear part of the vortex. In other words the front (rear) side of the vortex is subject to bi-axial stretching (bi-axial compression) along the two directions perpendicular to the vortex axis associated to ∗ ∗ ∗ the eigenvalues S11 (S22) and S33 with local stretching rates (compression rates) of comparable

6 magnitude.

Utr

compression direction stretching direction vortex core

Figure 7: 2D sketch of the compression and stretching directions at the front and rear of a vortex ring.

The contour plot of the local energy dissipation rate obtained from equation (4) is shown ∗ ∗ ∗ in ﬁgure 8, while in ﬁgure 9 is represented the parameter S11 × S22 × S33 which is the third invariant of the strain rate tensor and is indicated by Davidson (2004) as a reference parameter to identify locally a region of higher dissipation.

8 7.8 7.6 7.4 [-] 7.2

x/D 7 6.8 6.6 6.4 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 y/D [-]

0.5 1 1.5 2 2.5 L −6 ǫ( 3 ) [-] ×10 Up

Figure 8: Contour plot of the local energy dissipation rate ǫ.

7 8 7.8 7.6 7.4 [-] 7.2

x/D 7 6.8 6.6 6.4 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 y/D [-]

-150 -100 -50 0 50 ∗ ∗ ∗ L 3 S11×S22×S33 ( ) [-] Up

∗ ∗ ∗ Figure 9: Contour plot of the the invariant of the strain rate tensor, S11 × S22 × S33.

As expected the non-dimensional dissipation rate assumes peak values of 2 − 2.5 × 10−6 at the border of the vortex core where the local strain and compression rates reach their absolute maxima. It is worth to mention that a similar trend is also displayed by the parameter ∗ ∗ ∗ S11 ×S22 ×S33 that, irrespective of the sign, shows peak variations, indicative of more signiﬁcant three-dimensional strain rates, in regions of higher dissipation.

5 TRAILING VORTEX STRAIN DYNAMICS

The methodology described earlier is applied in this section to the flow field of the trailing vortices in the impeller stream region of a vessel stirred by a Rusthon turbine. The LDA velocity data of Schäfer (2001) were employed for this purpose. The current investigation of the strain dynamics occurring in the trailing vortex region has focused to a phase angle of φ = 15◦ behind the blade when the local energy dissipation rate is maximum (Ducci and Yianneskis, 2005) and therefore local strain rates should be more significant. In the current section a cylindrical coordinate system is considered with its axis aligned with the vessel one and its origin set at the bottom of the vessel. The plots of the vorticity contour and velocity field associated to the trailing vortices for φ = 15◦ are shown in figure 10. The two counter- rotating trailing vortices, positioned at r/D ≈ 0.5, are nearly symmetric with respect to the −1 radial direction with local absolute vorticity maxima of kω3(πN) k=0.3.

8 1.2

1.1 [-] 1 z/D 0.9

0.8 −1 ωθ(πN) 0.7 0 0.2 0.4 0.6 0.8 r/D [-]

-0.2 -0.1 0 0.1 0.2 0.3

Figure 10: Vorticity contour and velocity vector plots of the trailing vortices for φ = 15◦.

A direct comparison between the strain dynamics occurring in the vortex ring and trailing vortices can be obtained from the plots shown in figures 11 (a) and (b), where the coefficients Q and R are the second and third invariants, respectively, of the deformation rate tensor, ∇u. According to Davidson (2004) flow regions subject to vortex stretching, vortex compression, bi-axial strain and axial strain can be identified by points in the first, second, third and fourth quadrants, respectively, of the Q − R plane. In figure 11 (a) red and black diamonds denote selected points at the front and rear parts of the vortex ring cores, while blue and green crosses points ahead and behind, respectively. As expected the red and black diamonds are situated in ′ 2 1 R 3 the upper part of the Q−R plane, associated to highly rotational regions (Q > Q = −3( 4 ) ), and, in agreement with the discussion made for figure 6, the front part of the vortex core is subject to vortex stretching (R > 0) and the rear part to vortex compression (R < 0). The green and blue crosses corresponding to fluid outside of the vortex ring are mainly positioned around the Q′ lines, indicating that fluid rotation is negligible in these regions. In front of the vortex ring, the blue crosses associated to mainly negative R denote flow dynamics dominated by bi-axial strain. This is in agreement with the results presented in section 4 where it was shown that the fluid ahead of the vortex is stretched along two directions and compressed along the third one. On the contrary, the points behind the vortex ring (green crosses) are mainly subject to axial strain (R > 0), with the fluid being stretched along one direction and compressed along the other two (see section 4). A similar analysis was carried for the trailing vortices and their Q − R coefficients are shown in figure 11 (b). Points in the trailing vortex cores (red diamonds) are subject to vortex stretching with R> 0 and Q > Q′, while points in front of the trailing vortices (green crosses) are dominated by axial strain (R> 0).

9 200 0.03

150 0.02 100 [-] [-]

2 0.01 2 ) − p ) L 50 U ( Q πN

( 0 0 Q

-50 -0.01

-100 -600 -500 -400 -300 -200 -100 0 100 200 300 400 -0.02 L 3 -1 0 1 2 3 R ( U ) [-] − −3 p R(πN) 3 [-] ×10 (a) (b)

Figure 11: Plots of Q and R for the two ﬂows investigated: (a) vortex ring; (b) trailing vortices.

6 CONCLUSION

The strain dynamics of a vortex ring were investigated by determining the eigenvalues and eigenvectors of the strain rate tensor. This methodology allowed to obtain an improved and simpler visualisation of the local deformation rate which can be decomposed in terms of compression and stretching rates directed as a local reference system oriented along the eigenvector directions. It was found that compression and stretching rates oriented as the vortex ring axis are dominant in the front and rear parts of the vortex, respectively, as fluid approached by the vortex is squeezed in the axial direction while fluid left behind the vortex is subject to stretching. The maximum energy dissipation rate was located at the border of the vortex core where the strain rate assumed peak values. A comparison was also made between the strain dynamics affecting the vortex ring and those present in the trailing vortices of a stirred vessel.

ACKNOWLEDGEMENTS

Financial support for the work reported here was provided by the Engineering and Physical Sciences Research Council (EPSRC) of the UK, grant EP/D032539. The authors gratefully acknowledge provision of the stirred vessel LDA velocity data by Dr. Marcus Sch¨afer (Sch¨afer, 2001) acquired as part of the BRITE-EURAM project BRPR-CT96-0185.

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