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Research Papers: Multiphase Flows

Euler–Lagrange Simulations of Bubble Cloud Dynamics Near a Wall

[+] Author and Article Information
Jingsen Ma

Dynaflow, Inc.,
10621-J Iron Bridge Road,
Jessup, MD 20794
e-mail: jingsen@dynaflow-inc.com

Chao-Tsung Hsiao, Georges L. Chahine

Dynaflow, Inc.,
10621-J Iron Bridge Road,
Jessup, MD 20794

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 12, 2014; final manuscript received October 13, 2014; published online December 3, 2014. Assoc. Editor: Mark R. Duignan.

J. Fluids Eng 137(4), 041301 (Apr 01, 2015) (10 pages) Paper No: FE-14-1128; doi: 10.1115/1.4028853 History: Received March 12, 2014; Revised October 13, 2014; Online December 03, 2014

We present in this paper a two-way coupled Eulerian–Lagrangian model to study the dynamics of clouds of microbubbles subjected to pressure variations and the resulting pressures on a nearby rigid wall. The model simulates the two-phase medium as a continuum and solves the Navier–Stokes equations using Eulerian grids with a time and space varying density. The microbubbles are modeled as interacting singularities representing moving and oscillating spherical bubbles, following a modified Rayleigh–Plesset–Keller–Herring equation and are tracked in a Lagrangian fashion. A two-way coupling between the Euler and Lagrange components is realized through the local mixture density determined by the bubbles' volume change and motion. Using this numerical framework, simulations involving a large number of bubbles were conducted under driving pressures at different frequencies. The results show that the frequency of the driving pressure is critical in determining the overall dynamics: either a collective strongly coupled cluster behavior or nonsynchronized weaker multiple bubble oscillations. The former creates extremely high pressures with peak values orders of magnitudes higher than that of the excitation pressure. This occurs when the driving frequency matches the natural frequency of the bubble cloud. The initial distance between the bubble cloud and the wall also affects significantly the resulting pressures. A bubble cloud collapsing very close to the wall exhibits a cascading collapse, with the bubbles farthest from the wall collapsing first and the nearest ones collapsing last, thus the energy accumulates and this results in very high pressure peaks at the wall. At farther cloud distances from the wall, the bubble cloud collapses quasi-spherically with the cloud center collapsing last.

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References

Figures

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Fig. 1

Illustration of the void fraction computation [33]

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Fig. 2

Schematic of a bubble cloud with an equivalent radius A(t) and distributed bubbles of radius rb(t) near a rigid wall. Ox is the axis of symmetry and r is the radial distance from the axis of symmetry. The distance between the center of the cloud and the wall is X(t). AT(t) and AB(t) indicate the distances between the center of the cloud and the top and bottom of the cloud, respectively. The cloud is subjected to an imposed pressure function at its edge, P(t).

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Fig. 3

Time sequences of bubble locations and pressures in a bubble cloud excited by a pressure wave as listed in Table 1: (a) case 1, (b) case 2, and (c) case 3. The bubbles are represented by spheres colored by the gas pressure inside each of them. Lengths are scaled by bubble radii in the top row corresponding to 50 μm. Velocity vectors are represented by arrows. Only velocity vectors larger than 2 m/s extend beyond the bubble representation and can be seen in the snapshots. The bottom row shows only the velocity vectors on the bubbles in each case at t = T.

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Fig. 4

Variations of the total volume of all bubbles in the cloud for the three cases described in Table 1

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Fig. 5

Variations of the outer cloud size for the three cases in Table 1. AB and AT are the distances from the center of the cloud to the top and bottom of the cloud as defined in Fig. 2.

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Fig. 6

Void fraction distributions within the cloud at different times during the first cycle for cases 2 and 3, i.e., without and with a wall, α0 = 5%, ω = 10 kHz. (Note that due to the selected Gaussian spreading radius of A0 (see Eq. (9)) a nonzero value of the void fraction exists here, beyond the cloud radius, i.e., approximately between −2A0 and 2 A0.)

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Fig. 7

Isosurface of 5% void fraction at t = 0.9 T for case 3 (presence of a wall). The arrows denote the translational velocities of the bubbles located at different locations in the cloud.

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Fig. 8

Comparison of the size variation of bubbles at different cloud locations for the three different test cases described in Table 1 (a) ω = 100 kHz, no wall; (b) ω = 10 kHz, no wall; and (c) ω = 10 kHz, with wall

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Fig. 9

Comparison of the pressures at different cloud locations for the three test cases described in Table 1: (a) ω = 100 kHz, no wall; (b) ω = 10 kHz, no wall; and (c) ω = 10 kHz, with wall

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Fig. 10

Size variation of bubbles on the axis of symmetry at seven radial locations in the cloud for case 3 of α0 = 5%, ω = 10 kHz, with wall. The numbers 1–7 on each line indicate the order in which each bubble starts to collapse. Negative values of x correspond to bubble locations in the lower half of the cloud closer to the wall.

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Fig. 11

Variations of the axial (along x in Fig. 2) component of bubble translation velocity for bubble on the axis of symmetry at seven radial locations in the cloud for case 3 of α0 = 5%, ω = 10 kHz, with wall

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Fig. 12

Time evolution of the pressure at different radial locations on the wall for case 3 of α0 = 5%, ω = 10 kHz, with wall

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