Investigation of Cavitation Bubble Cloud with discrete Lagrangian Tracking

Transcription

1 Investigation of Cavitation Bubble Cloud with discrete Lagrangian Tracking Xiuxiu Lyu, Xiangyu Hu, Nikolaus A. Adams Chair of Aerodynamics and Fluid Mechanics, Department of Mechanical Engineering, Technical University of Munich 8578 Garching, Germany Abstract Here, to study more about the mechanism of the cavitating bubble cloud, we simulate numerically microbubble cloud inside bulk fluid with Lagrangian tracking. The bubble-flow mixture is modelled in a two-way coupled Eulerian- Lagrangian scheme, continuum description of the bulk fluid with discrete Lagrangian tracking of the gas bubbles. The Euler equations are solved based on the multi-resolution conservative compressible fluid method. Dynamics of the dispersed cavitating gas bubbles is modelled in a Lagrangian framework, describing the bubble location and bubble size variation. The sub-grid interface is allowed to diffuse on the computational cells around it within the kernel width. To close the system, the equations of state (EOS) of liquid-gas mixture are given in a stiff gas equation of state, along with one isobaric closure of the pressure. Stability and accuracy of the interface are studied by some classical cases. perturbation induced by the bubble cloud inside flow is simulated based on two-way coupling framework. Keywords: Bubble cloud, Euler-Lagrange method, Bubble dynamic, Cavitation. Introduction Bubbly flow has been investigated as applications in efficient diagnostic ultrasound treatment and chemistry process, where the dynamics of bubble cloud is of fundamental importance [,, ]. Our motivation comes from simulating numerically the bubbly cloud and at the same time tracking accurately the under-resolved or subgrid interface between micro-bubbles and bulk fluid. In literature, dynamics of interface flows has been normally cataloged into three major models: continuum model, level set method, and Lagrangian dispersed model. Continuum model solves the Euler equation coupled with transport equations [, 5, 6]. The effect of bubble cloud on the computational cells is represented by volume fraction. The actual position of the interface is recovered at the end of each computational step by an interface reconstruction process. In level set method introduced by Osher and Sethian [7], the interface is described as signed distance function, and it is popularly applied in computational sharp interface multi-phase fluid [8, 9]. This method is mostly applied for the numerical simulation of single bubble and suffers from drawback when the complexity increases to D. In the Lagrangian dispersed model [, ], the interface is discetized into particles which move dynamically at each computational time step. It is generally accurate in predicting the bubble dynamics and transportation when all bubbles are tracked with a much higher spatial resolution. Limitation of Lagrangian dispersed method is that equation of motion for each bubble has to be solved and larger number of bubbles can lead to higher computational cost especially when bubble-bubble interaction is taken into account. However, due to the simplicity in operation, low computational cost, Lagrangian dispersed model is quite suitable for the micro-bubble cloud simulation and meanwhile can be easily developed into D. Assuming that in each computational cell for the bubble-fluid mixture there s only one single velocity and pressure, the complete system for Eulerian-Lagrangian model includes the basic conservation law of mass, momentum and energy, one transport equation of the void fraction (bubble dynamics equation of each particle) and one effective Eulerian-Lagrangian coupling model. addresses: xiuxiu.lyu@tum.de (Xiuxiu Lyu), xiangyu.hu@tum.de (Xiangyu Hu), nikolaus.adams@tum.de (Nikolaus A. Adams)

2 In [], Darmana et al studied a new paralleled algorithm in application of to Euler-Lagrange in specific for dispersed gas-liquid flow. In Ref. [], Fuster and Colonius proposed a new model for bubbly cavitating flow, in which they solve the volume-averaged Eulerian equations for the bulk fluid coupled with Lagrangian equations for each gas bubble. Their results show that their method can capture the pressure disturbance induced in the liquid. However, the closure law in those models is quite easy, which is only one expression of the liquid pressure, not accurately the pressure of the mixture. In [], the mixture density is not an explicit function of the pressure of EOS. Here, we do the numerical simulation to study the dynamics of cloud of microbubbles to the pressure wave, in order to understand the detail mechanism of the bubble cloud under pressure wave and also the resulting stable bubble distributions afterwards. The background cavitating flow is treated as Eulerian flow with initially dispersed cavitating gas bubbles with radius of several micrometers.. Physical Models and numerical methods.. Single bubble dynamics The dynamics of the dispersed phase is treated in a Lagrangian framework, describing the volume and the motion of the gas bubbles. The bubbles are assumed to be spherical, as they are extremely small in the scale of nanometres or micrometers that the surface tension is large enough to maintain the spherical shape. Bubble-bubble interaction, bubble coalescence and collision are not considered. The dynamics and motions of bubbles is controlled by the bulk flow. The Rayleigh-Plesset equation proposed by Rayleigh [] and Plesset [5] has been widely used for the bubble dynamics modelling in hydrodynamic cavitations, multi-phase bubbly flows and underwater explosion bubbles. Based on Rayleigh-Plesset equation, dynamics of a spherical bubble surrounded by a weekly compressible liquid in infinite can be expressed as: ρ m Rd R dt + ( ) dr dt = p B p S R µ R dr dt, () n where ρ is the density of the liquid-gas mixture, R is the bubble radius, t is the time, S is surface tension, µ is the viscosity, p B is the pressure inside the bubble, p is the far-field pressure of the surrounding field. In general, the bubble inside the water contains vapour whose pressure is labelled as p v and some quantity of contaminant gas pressure p g at a reference bubble size R. And p v, is considered constant when the temperature T b is constant. We assume the mass of the gas m g in the bubble and its temperature T b remain the same, the pressure inside a bubble has the expression as flows: ( R ) γg, p B = p v + p g () R with γ g =.... Liquid equations We assume the bulk flow represents the liquid-gas mixture, which means the velocity of the mixture u equals to the velocity of the fluid u l. So that each computational cell is homogeneous and has only one velocity vector u and one pressure P. Also we assume the bubble-fluid mixture is inviscid and compressible. The conservation law of the mixture can be written as U + F (U) =, () t where U = (ρ m, ρ m u, E) T is the average local mixture density of the mass, momentum and energy, here E = ρ m e m + ρ mu, F represents the corresponding flux functions. To express the amount of the gas per unit volume, we define the void fraction of gas α g and the void fraction of the liquid α l, = α l + α g. () Also, according this definition, the averaged density and inner energy of the mixture can be written as ρ m = α l ρ l + α g ρ g, (5)

3 where the subscripts l and g denote respectively the liquid and the gas. ρ m e m = α l ρ l e l + α g ρ g e g, (6).. The isobaric closure law For the computation of gas-liquid mixture, we apply the expressions of pressure in multi-component five-equation model in consideration of void fraction. The stiffed-gas EOS for each fluid is written as P i = (γ i ) ρ i e i γ i B i (7) involving only constants γ and reference pressure B for each fluid. In our simulation, for water γ l = 7., B l = MPa, for gas γ g =., B g =.MPa. As mentioned before, the mixture pressure must apply the property: mixing of two fluids with the same pressure should result in a mixture equilibrium pressure with the same value. Failing to make this equilibrium will generate numerical vibration near the location of interface. The EOSs for liquid-gas mixture must follow Introducing ξ g = P l = P g = P m. (8) γ g, ξ l = γ l and ξ m = α l ξ l + α g ξ g, an implicit expression for P is given as ρ m e m = P m ξ m + α l γ l B l ξ l + α g γ g B g ξ g (9) which yields the expression P m = ρ me m ξ m ξ m ( αl γ l B l ξ l + α g γ g B g ξ g ). ().. Euler-Lagrange coupling The void fraction distribution is derived from the instantaneous bubble sizes and locations. The concept of kernel function has been applied widely in the smooth particle hydrodynamics (SPH) [6]. And to smooth the discontinuities and improve stability of the vapor void faction in the computational grids, we apply a Gaussian kernel volume distribution scheme. The Gaussian interpolation function is given by ( ) ( ) ζ σ x,xb = ( ) exp k= xk x b,k σ π σ () where σ is the kernel width, x k and x b,k are the grid cell center location and bubble location respectively. In order to enforce mass conservation, the kernel function is normalized over the volume of integration by V cv ζ σ ( xcv,x b ) dv =. () Using the above kernel function, volume fraction of the gas bubbles in every controlled volume can be calculated as: α g = N i= πr i ζ σ. () In this formulation, the volume occupied by the gas bubble in a fluid control volume is accounted by α g and fluid void fraction α l (α g + α l = ).. Results and Discussion.. Simulation set-up In our simulation, we solve the Euler equations on a Cartesian grid, with the details referred in Ref. [8, 7]. The equations are discretized with the fifth-order WENO method [8] and the second-order TVD Runge-Kutta scheme [9]. All the computations are carried out with the CFL number of..

4 Figure : Void fraction α l along the bubble radius under different resolutions. Figure : wave propagation induced from an oscillating gas bubble in the bulk fluid... One bubble oscillating With the variable time-step numerical algorithm in Ref. [], we can solve the bubble dynamics even when the bubble size has extreme size variation ratio at one time step. Here, we consider a single gas bubble placed inside water in a D domain. The bubble radius change dynamically in time as R = R ( εsinωt), where ε is the perturbation magnitude and ω is the frequency. We set R = µm, ε =. and ω = khz. The D dimension size is L, where L = R. The width of the support of the kernel δ =.R. Figure. indicates the void fraction of water along the radius away from the bubble center under different resolutions. Here, the bubble oscillate actively only in the first round, afterwards, the volume of the bubble goes back to the equilibrium state. The results show the convergence of our method. Figure. shows the pressure wave induced by the oscillating bubble at t = t f =.,.,. and 5.. The pressure wave is well resolved and propagates to left and right sides. It indicates that the local variables are well predicted based on our method for a single sub-grid bubble... Multi-bubbles under pressure pulse Here in this section, we simulate the dynamics of a D cylindrical bubble interacting with a sinusoidal pulse in the bulk flow. We investigate a D bubble cloud with bubbles randomly radius distribution between µm and 5µm, the radius of the cluster being R c = mm. The sinusoidal pressure wave moving from left to right is set to ( be p = p + psin π x ), whose wave length is λ = R c and wave amplitude is p =.p with pressure at infinity λ =.atm. The initial pressure wave center locates.cm right side of the bubble cloud center. The negative p pressure will reach the bubble cloud firstly, which makes the bubbles expand firstly and collapse secondly. The kernel width for each gas bubble is σ =.R i, R i is the radius of the bubble i. The fluid void fraction at initial ranges from.557 to., with a gray scale contour in Figure. a. The computational domain size is set to be (.cm,.cm) with the finest resolution (, 5). As shown in Figure., the v-velocity in the mixture is plotted at.µs,.µs,.µs, and.µs after the sinusoidal pulse reaches the cloud. Simultaneously, pressure field at.µs,.µs, and.µs are also shown respectively in Figure.. As shown in this figure, the pressure pulse inside the bulk flow induces the bubbles vibration, in reverse, the vibration of the gas bubbles induces the pressure noise in the bulk flow.

5 .5cm Figure : Bubble cluster interacting with a sinusoidal pulse, the v velocity field induced by the bubble cluster..5 cm Figure : The pressure field induced by the bubble cluster when bubble cluster interacting with a sinusoidal pulse. 5

6 . Conclusion In this paper, a Lagrangian-Eulerian coupling scheme is implemented based on one sharp interface multi-resolution conservative Euler solver for bubbly flow. The main conclusion of this paper can be drawn as follows: the current method convergences; the pressure pulse inside bulk flow induces the bubbles vibration, in reverse, the vibration of the gas bubbles induces the pressure noise in the bulk flow; the pressure wave induced by the oscillating bubble can be predicted by our method. 5. References [] Y. Matsumoto, J. S. Allen, S. Yoshizawa, T. Ikeda, Y. Kaneko, Medical ultrasound with microbubbles, Experimental thermal and fluid science 9 () (5) [] C. C. Coussios, R. A. Roy, Applications of acoustics and cavitation to noninvasive therapy and drug delivery, Annu. Rev. Fluid Mech. (8) 95. [] E. Vlaisavljevich, K. W. Lin, M. T. Warnez, R. Singh, L. Mancia, A. J. Putnam, E. Johnsen, C. Cain, Z. Xu, Effects of tissue stiffness, ultrasound frequency, and pressure on histotripsy-induced cavitation bubble behavior, Physics in medicine and biology 6 (6) (5) 7. [] C. W. Hirt, B. D. Nichols, Volume of fluid (vof) method for the dynamics of free boundaries, Journal of computational physics 9 () (98) 5. [5] G. Allaire, S. Clerc, S. Kokh, A five-equation model for the simulation of interfaces between compressible fluids, Journal of Computational Physics 8 () () [6] F. Örley, T. Trummler, S. Hickel, M. Mihatsch, S. Schmidt, N. Adams, Large-eddy simulation of cavitating nozzle flow and primary jet break-up, Physics of Fluids 7 (8) (5) 86. [7] S. Osher, J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on hamilton-jacobi formulations, Journal of computational physics 79 () (988) 9. [8] X. Y. Hu, B. C. Khoo, N. A. Adams, F. L. Huang, A conservative interface method for compressible flows, Journal of Computational Physics 9 () (6) [9] E. Lauer, X. Y. Hu, S. Hickel, N. A. Adams, Numerical modelling and investigation of symmetric and asymmetric cavitation bubble dynamics, Computers & Fluids 69 () 9. [] E. Delnoij, F. Lammers, J. Kuipers, W. Van Swaaij, Dynamic simulation of dispersed gas-liquid two-phase flow using a discrete bubble model, Chemical Engineering Science 5 (9) (997) [] D. Darmana, N. G. Deen, J. Kuipers, Parallelization of an euler lagrange model using mixed domain decomposition and a mirror domain technique: Application to dispersed gas liquid two-phase flow, Journal of Computational Physics () (6) 6 8. [] D. Fuster, T. Colonius, Modelling bubble clusters in compressible liquids, Journal of Fluid Mechanics 688 () [] J. Ma, C.-T. Hsiao, G. L. Chahine, Numerical study of acoustically driven bubble cloud dynamics near a rigid wall, Ultrasonics sonochemistry (8) [] L. Rayleigh, Viii. on the pressure developed in a liquid during the collapse of a spherical cavity, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science () (97) [5] M. S. Plesset, The dynamics of cavitation bubbles, Journal of applied mechanics 6 (99) [6] J. J. Monaghan, Why particle methods work, SIAM Journal on Scientific and Statistical Computing () (98). [7] L. H. Han, X. Y. Hu, N. A. Adams, Adaptive multi-resolution method for compressible multi-phase flows with sharp interface model and pyramid data structure, Journal of Computational Physics 6 () 5. [8] G. S. Jiang, C. W. Shu, Efficient implementation of weighted eno schemes., Tech. rep., DTIC Document (995). [9] C. W. Shu, S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics 77 () (988) 9 7. [] H. Alehossein, Z. Qin, Numerical analysis of rayleigh-plesset equation for cavitating water jets, International Journal for Numerical Methods in Engineering 7 (7) (7)