Proceedings of the ASME 2016 International Mechanical Engineering Congress & Exposition IMECE2016 November 13-19, 2016, Phoenix, Arizona, USA

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1 Proceedings of the ASME 2016 International Mechanical Engineering Congress & Exposition IMECE2016 November 13-19, 2016, Phoenix, Arizona, USA IMECE An Eulerian-Lagrangian Study of Cloud Dynamics near a Rigid Wall Jingsen Ma DYNAFLOW, INC J Iron Bridge Road, Jessup, MD, USA jingsen@dynaflow-inc.com Georges L. Chahine DYNAFLOW, INC J Iron Bridge Road, Jessup, MD, USA glchahine@dynaflow-inc.com Chao-Tsung Hsiao DYNAFLOW, INC J Iron Bridge Road, Jessup, MD, USA ctsung@dynaflow-inc.com ABSTRACT Bubble cloud collapse is one of the most destructive forms of cavitation. However, numerical modeling of this phenomenon is challenging and involves strong bubble dynamics, bubble-bubble, and bubble-wall interactions. Here we consider an initially spherical cloud near a rigid wall and treat the cloud bubbly mixture as a continuum and the bubbles in the cloud as discrete singularities. These two components are coupled through the local and instantaneous void fractions associated with the bubble volumes and locations. Simulations with sinusoidal pressure excitation of various frequencies and amplitudes for the same initial bubble cloud enable determination of the cloud resonance frequency. This condition results in wall pressures orders of magnitudes higher than the excitation pressure. As the amplitude of the driving pressure increases, the resonance frequency deviates significantly from the classical value obtained from linearized theory [1]. It becomes gradually smaller as the excitation amplitude increases then reaches a limit value. For high amplitude excitations, the peak wall pressure generated at the wall reaches a maximum when the ratio of the initial bubble radius to the maximum bubble radius is minimum. Too weak or too strong bubble interactions in the cloud inhibit strong collective effects. The effects of other parameters such as the initial distance between the cloud center and the wall are also discussed. INTRODUCTION The collapse of a cloud of bubbles near a rigid boundary is known as one of the most destructive forms of cavitation. It occurs in practical applications such as in cavitation on propellers [2 4], cavitating jets for cutting, cleaning, oxidation, disinfection [5,6], Shock Wave Lithotripsy (SWL) for kidney stone fragmentation [7 9], cavitation erosion testing [10] etc. Numerical modeling of such a problem involves multi-scale physics ranging from the micro scale of the individual bubbles to the overall scale of the cloud, and involves bubble-bubble and bubble-wall interactions [10 17]. Two-phase bubbly flows are usually modeled using approaches such as equivalent homogeneous continuum models, Eulerian two-fluid models, or Eulerian-Lagrangian approaches. Homogeneous models are useful for low void fractions, whereas Eulerian-Lagrangian approaches are more appropriate for higher void fractions [12 18]. In recent studies a continuum homogeneous model, an Eulerian multicomponent model, and an Eulerian-Lagrangian model were compared for the simulation of a bubble cloud interacting with a large bubble [29-32]. It was found that although all the models capture the average low-frequency dynamics, modeling the discrete bubbles is essential to a good understanding of the physics with a reasonable computational cost. Using a 3D two-way coupled Eulerian-Lagrangian model [22 24], we study here the dynamics near a rigid wall of a bubble cloud excited by a sinusoidal pressure. To do so, we treat the two-phase medium as a continuum and solve the corresponding Navier-Stokes equations with time and space varying density using a fixed grid. The bubbles in the cloud are modeled as singularities. The source terms (bubble volume variations) are obtained from modified Rayleigh-Plesset-Keller- Herring equations, while the dipole terms are obtained from bubble equations of motion. The two-way coupling between the Euler and Lagrange components is realized through the mixture 1 Copyright 2016 by ASME

2 density determined by the bubble distribution. This method has been successfully applied for two-phase bubble dynamics [16,17,33,34]. Using the same model and setup as in [24,25], the study is extended here to bubble clouds excited over a wide range of pressure excitation amplitudes and frequencies. EULER-LAGRANGE TWO-PHASE MODEL The 3D Eulerian-Lagrangian model employed in this study has been used for several applications described in detail in our previous studies. Applications included the investigation of the effects of a propeller flow on bubble size distribution in water [26,27], the modeling of propeller tip vortex cavitation inception [27,28], simulation of the bubbly flow in a bubble augmented propulsor [29], bubble entrainment in a plunging jet [30] and wave propagation in bubbly media [22 24,31], etc. The present paper presents a parametric study of bubble cloud dynamics using this model without introducing additional model development. We provide a brief summary of the key features of the model. Readers interested in more details on the model may refer to the references provided above. Continuum Model for Mixture Flow The two-phase bubbly mixture is treated as a continuum. [40,-43] with the following unsteady continuity and momentum conservation equations: m mu 0, (1) t Du 2 m p m u mg. (2) Dt u is the mixture velocity, m is the mixture density. and p is the mixture pressure. m is the mixture kinematic viscosity and g is the acceleration of gravity. The mixture density and viscosity are defined as, 1, 1, (3) m l g m l g where the subscripts l and g refer to liquid and gas properties respectively, while is the gas volume fraction. Lagrangian Modeling of Dispersed Bubbles All bubbles in the cloud are treated as point sources representing volume change and dipoles representing spheres translation. The volume variations are obtained using spherical bubble dynamics equations with the equivalent spherical bubble radius, R(t), obtained using a modified Rayleigh- Plesset-Keller-Herring equation, accounting for the nonuniform pressure field [11,27,35 38] u u 2 b R 3 R enc 1 RR (1 ) R cm 2 3cm 4 (4) 1 R R d 2 R 1 pv pg p 4. enc m m cm cm dt R R Here c m is the local sound speed in the mixture [16], p v is the liquid vapor pressure, pg0 is the initial gas pressure in the bubble, k is the gas polytropic constant, and is the surface tension. p enc and u enc are the mixture pressures and velocities averaged over the bubble surface and u b is the bubble center velocity. The added term (u enc - u b ) 2 /4 accounts for the slip between the bubble and the host medium. The bubble equation of motion used is: dub 3 3C p 2 g D u u + enc b u u enc b dt l 4 R 3 CL l uenc ub ω 3 uenc ub 2 R l ω R where C L is the lift coefficient [39], ω is the local vorticity, and C D is the drag coefficient [40]. The last term represents the coupling between volume oscillations and bubble motion [41]. Eulerian-Lagrangian Coupling Coupling between the continuum and the discrete bubbles is achieved in both ways. The dynamics of each bubble is in response to the local mixture flow field properties, while these properties depend on the overall bubble size and space distribution. A Gaussian distribution scheme is used to smoothly spread each bubble void effect over its neighboring fluid cells. This was found to significantly increase numerical stability and enable handling of violent cavitating flow simulations. Further details can be found in our previous publications such as [17] PROBLEM SETUP As illustrated in Fig. 1, we consider a bubble cloud with an overall initial radius, A 0, driven by a pressure function P(t): P t = P 0 +P amp sin (2 f t + ), (6) Here P 0 is the reference pressure at t = 0, P amp its amplitude, and f its frequency. The cloud center is initially at a distance X0 from the wall and is composed of bubbles with the same initial radius, R 0, and are randomly distributed in the sphere of radius A 0. This results in a quasi-uniform initial void fraction distribution α 0. At t = 0 all the bubbles are in equilibrium with P 0. In this paper we consider a cloud of bubbles under conditions and parametric ranges summarized in Table 1. P 0 (atm) P amp (atm) R 0 m) A 0 mm) α 0 (%) f khz) R (5) X 0 / A ~25 2~ ~25 0~1.0 Table 1: Key parameters of the simulation cases addressed below. 2 Copyright 2016 by ASME

3 A T (t) x O P(t) A(t) R (t) r the effects of the driving frequency. The other parameters of the problem are kept the same P 0 =10 atm, P amp =9 atm, R 0 = 50 μm, α 0 = 5%, A 0 = 1.5 mm and X 0 = A 0. Fig. 2 and Fig. 3 compare, for different driving frequencies, the pressure versus time monitored at the wall center. The figures show that the cloud behavior is very sensitive to the driving frequency and that for this case 8 khz generates the highest pressure peak of almost 300 atm, thirty times higher than the excitation pressure. This indicates that controlling the driving frequency is critical in practical use of cloud cavitation, e.g., in SWL [7 9]. A B (t) X(t) Fig. 1: Schematic of the problem of an initially spherical bubble cloud near a rigid wall. SIMULATION RESULTS & DISCUSSION Effects of Driving Pressure Frequency Fig. 3: Variation of the maximum pressure at the wall center with the driving frequency, f. P 0 =10 atm, P amp =9 atm, R 0 =50 μm, α 0 =5% and X 0 =1.5 mm. Fig. 2: Pressure at the wall center created by a cloud of radius A 0 =1.5mm excited by pressure of different frequencies P 0 =10atm, P amp =9atm, R 0 =50μm, α 0 =5% and X 0 =1.5mm. As illustrated in our recent work [4], the tuning between the cloud characteristic frequency and the pressure field frequency is essential to generating very high cavitation impulsive loads. When the frequency of the driving pressure wave matches the characteristic frequency of a bubble cloud, strong collective behaviour occurs and creates extremely high pressures on the wall. This, however, is not seen when these two frequencies are not tuned well. Motivated by this we consider here, for the same amplitude of the imposed pressure oscillations (9 atm), Resonance Frequency of Bubble Cloud It is important to note that the resonance frequency of 8 khz observed in Fig. 3 is much smaller than the reported natural frequency of a cloud, f cloud, derived for small amplitude oscillations and linear bubble dynamics theory [16]. The expression for the natural frequency of the cloud away from boundaries is given by: A R0 10 fcloud f 1, (7) 1 P g 0 2 with f0 3, 2 2 R 0 Pg 0R 0 which gives a value of f cloud = 23 khz. Two factors may cause this discrepancy: one is the presence of the rigid wall and the other is the fact that the excitation amplitude is high resulting in strong non-linear effects not accounted for in (7). To investigate which of these two factors contributes the most, two sets of simulations are shown below. The first set of simulations considers the effect of the driving frequency when the driving pressure amplitude and all the other parameters are kept the same but the nearby wall is 3 Copyright 2016 by ASME

4 removed. The second set considers the effect of the driving frequency also in absence of a nearby wall but when the pressure amplitude is much smaller, P amp = 0.1 atm. The resonance frequency for these two sets of simulations can be deduced from Fig. 4 which shows the peak pressures computed at the wall center as a function of the driving frequency. The presence of the wall is seen to move the resonance frequency from 10 khz to 8 khz. This effect is much smaller than the effect of the driving amplitude. At the low amplitude excitation, 0.1 atm, the numerically computed resonance frequency matches very well the linear theory predictions, f cloud ~ 23 khz. The shift from 23kHz to 8 khz is therefore mostly due to nonlinear effects. more violently, as evidenced by the changes of the bubble sizes visualized in the figure by the color or each bubble. 8kHz P amp = 9 atm near wall P amp = 9 atm no wall P amp = 0.1 atm no wall 10kHz 23kHz Fig. 4: Comparison of the peak pressures in a range of frequencies obtained for both an isolated bubble cloud and for a bubble cloud near a cloud for two driving pressure amplitudes. Effects of Driving Pressure Amplitude In order to examine further the effects of the nonlinear bubble dynamics on the cloud resonance frequency, additional simulations addressing the influence of the driving pressure amplitude on the cloud behavior are shown below. Fig. 5 shows for increasing amplitudes: P amp = 4 atm, 9 atm and 16 atm, snapshots of the bubble cloud dynamics at different times during the first period of oscillation of the cloud. The different images from top to bottom in each column correspond to different stages during the first cycle of oscillation. It is seen that overall, in all cases, the bubbles in the cloud grow and collapse in response to the driving pressure. The collapse of the bubble cloud exhibits in all the cases a cascading sequence with the bubbles farthest from the wall collapsing first and those closest to the wall collapsing last. However, the details of the cloud behavior are different between cases and are strongly dependent on the amplitude of the driving pressure. When this amplitude is low, the bubbles respond weakly and oscillate with small amplitudes. As the amplitude increases the bubbles grow and collapse more and Fig. 5: Time sequences (from top to bottom) of bubble locations and pressures in a bubble cloud driven by pressure excitations with amplitude of 4 atm (Left), 9 atm (Middle) and 16 atm (Right). The bottom row is an angled view of the wall pressure distribution. The bubbles are represented by spheres. The color on the bubbles and wall represents pressure where blue denotes 0 atm. and red denotes 150 atm. The other parameters are: P 0 =10 atm, R 0 =50 μm, A 0 =1.5 mm, α 0 =5%, f= 23 khz and X 0 =1.5 mm. Accordingly, the resulting pressure at the wall due to the bubble cloud collapse increases significantly when the driving pressure amplitude increases as seen in the last row of Fig. 5. This is made further clear in Fig. 6 which shows the peak pressure as a function of driving frequency for different values of the relative amplitude: P amp. P0 (8) 4 Copyright 2016 by ASME

5 Each curve for a fixed ξ enables one to deduce the corresponding resonance frequency. The figure clearly shows that the value of the resonance frequency drops significantly as the relative driving pressure amplitude, ξ, increases until recovering the classical small perturbation frequency provided by Equation (7) as illustrated by the dashed line connecting the resonance frequency for each ξ. Fig. 8: Effects of ξ on the peak pressure at resonance of a bubble cloud. P 0 =10atm, R 0 =50μm, A 0 =1.5mm, α 0 =5% and X 0 =1.5mm. Fig. 6: Peak pressure monitored on the wall center under different frequencies, excited by different relative driving pressure amplitude ξ. P 0 =10atm, R 0 =50μm, A 0 =1.5mm, α 0 =5% and X 0 =1.5mm. Symbols of a different color correspond to a different ξ. This is further illustrated in Fig. 7 which plots the resonance frequency versus ξ, and Fig. 8, which shows the corresponding peak pressure versus ξ. this clearly indicates that for small ξ (i.e., corresponding to linear bubbly dynamics conditions) the peak frequency is close to 23 khz. This resonance frequency gradually reduces to about 5 khz when ξ is close to 1. On the other hand, the peak pressure corresponding to the resonance frequency increases from ~ 10 atm to ~1,000 atm as ξ increases from 0.01 to 1.5. Effects of Initial Bubble Size Fig. 7: Effects of ξ on the resonance frequency of a bubble cloud. P 0 =10atm, R 0 =50μm, A 0 =1.5mm, α 0 =5% and X 0 =1.5mm. Fig. 9: Peak pressure generated on the wall center as a function of different initial bubble sizes in the cloud. P 0 =10atm, P amp =15atm, A 0 =1.5mm, α 0 =5% and X 0 =1.5mm. 5 Copyright 2016 by ASME

6 strongest when X 0 =0 since the bubble grows to a larger maximum size then collapses to a smaller minimum size. Fig. 10: Peak pressure generated on the wall center as a function of the ratio of maximum over initial bubble size. P 0 =10atm, P amp =15atm, A 0 =1.5mm, α 0 =5% and X 0 =1.5mm. In this section, the importance of the bubble cloud size, for the same void fraction, on the cloud collapse is investigated. Initial bubble sizes from 2 to 60 μm are considered for the following conditions: P 0 =10 atm, P amp =15atm, A 0 =1.5mm α 0 =5%, f=15khz and X 0 =1.5mm. The resulting wall peak pressure values are shown in Fig. 9. The peak pressure generated at the wall has a sharp maximum for a value of the initial bubble radius of about 7 m. This actually occurs when the cloud dynamics is such that the maximum bubble radius achieved provides the largest ratio of maximum over initial bubble size. Another important observation is that too weak or too strong bubble interactions in the cloud inhibit strong collective effects. This is clearly evidenced in Fig. 10 which shows that the generated pressure can reach several thousands of atmospheres when the ratio between maximum and initial bubble size is about 20, while it is negligible when this ratio is low. Effects of Initial Distance from Wall In this section we examine the effects of the initial distance between the wall and cloud center and vary this distance between one cloud radius and zero. All other parameters are kept the same with P 0 =10 atm, P amp =9 atm, R 0 =50 μm, A 0 =1.5 mm, α 0 =5% and f=12 khz. In order to do so, the bubbles in the portion of the spherical cloud that is located inside the wall are disabled. The pressure versus time created at the wall center for three standoff distances are compared in Fig. 11. For these conditions, the peak pressure is weaker for the smallest X 0. We can attribute this to the fact that the total number of bubbles is reduced as X 0 decreases and eventually the bubble number is halved when the cloud becomes a half cloud with X 0 =0. In this case the magnitude of the peak pressure is only reduced by about 20%. This could be explained by Fig. 12 where the temporal size variation of the central bubble is compared for different X 0. The figure shows that the dynamics of central bubble is the Fig. 11: Pressure at the wall center created by a bubble cloud starting at different initial distances to the wall. P 0 =10atm, P amp =9atm, R 0 =50μm, A 0 =1.5mm α 0 =5%, f=12khz. Fig. 12: Size Variations of the bubble initially at cloud center for the cloud with different initial standoff distance X 0. P 0 =10atm, P amp =9atm, R 0 =50μm, A 0 =1.5mm α 0 =5%, f=12khz. 6 Copyright 2016 by ASME

7 CONCLUSION The dynamics of a bubble cloud subjected to an imposed pressure time variation and the resulting pressures on a nearby rigid wall is studied utilizing a 3D two-way coupled Euler- Lagrange model. The two-phase medium is modeled using an Eulerian continuum model while the dynamics and motion of the bubbles are tracked in a Lagrangian fashion. Simulations involving an initially spherical cloud of bubbles were conducted under a range of conditions. We first assessed the effect of the frequency of the driving pressure on the bubble cloud behavior and highlighted strong nonlinear bubble dynamics effects for large amplitude excitation. The resonance frequency, which results in the highest collective bubble behavior, is identified and results in a pressure peak much higher than the excitation pressure. This optimum frequency, which results in the strongest cloud collapse, strongly depends of the amplitude of the driving pressure. At low amplitudes, the classical linear oscillation frequency of a bubble cloud is recovered. As the excitation amplitude increases the resonance frequency gradually drops and reaches a limit value. The peak pressure generated at the wall reaches a maximum for an optimum value of the initial bubble radius, which corresponds to when the ratio of the maximum bubble radius to the initial bubble radius is maximum. Too weak or too strong bubble interactions in the cloud inhibit strong collective effects. Further discussion presented also included the effects of other parameters such as the initial distance between the bubble cloud center and the wall, which require further detailed study. ACKNOWLEDGEMENT This work was supported by the Department of Energy under SBIR Phase II DE-SC The authors would also like to acknowledge support from the Office of Naval Research under contract N C-0382 monitored by Dr. Ki-Han Kim. 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