Growth and Collapse of Cavitation Bubbles in Viscous Orifice Flow S

ILASS Americas 21th Annual Conference on Liquid Atomization and Spray Systems, Orlando, Florida, May 18-21 2008

Growth and Collapse of Cavitation Bubbles in Viscous Oriﬁce Flow

S. Dabiri1,∗ W. A. Sirignano1, D. D. Joseph1,2 1Department of Mechanical and Aerospace Engineering University of California, Irvine, CA 92697, USA 2Department of Aerospace Engineering and Mechanics University of Minnesota, Minneapolis, MN 55455, USA

Abstract Despite several studies on cavitation bubbles in inviscid flows, the effects of shear stress on inception of cavitation is still unknown. In order to study these effects, one might consider growth and collapse of a bubble in a shear flow. In this paper a numerical method based on level-set formulation has been developed to track a compressible bubble interface in a viscous flow. The full Navier-Stokes equations for multiphase flow are solved. The bubble center moves with the average velocity of the displaced fluid. The pressure and strain rate tensor are applied as time varying boundary conditions. The time dependency of these parameters comes from the calculation of these parameters following an element of liquid mass traveling in an injector. The results will be helpful in understanding the effects of cavitation on the atomization of jets.

∗Corresponding Author Introduction an injector. The results will be helpful in understanding Experiments have shown that occurrence of cavi- the effects of cavitation on the atomization of jets. tation in high-pressure liquid injectors leads to a bet- Governing Equations and Numerical Method ter atomization of the emerged jet (e.g., [1]). Cavita- In this study, we consider a cavitation bubble in a tion inception in the liquid injectors occurs behind the Couette flow that grows and collapses due to pressure sharp corners. At these points, both pressure drop and change in the flow. The liquid phase is considered to be high shear stress due to the viscous boundary layer are incompressible. On the other hand, the bubble is com- present. Depending on the cavitation number, the flow pressible due to change in the pressure at low Mach num- could show no cavitation, cavitation with traveling bub- ber. The bubble is assumed to consist of a noncondens- bles, cavitation with a fixed vapor bubble behind the cor- able gas with a uniform temperature in space and time. ner, or super-cavitation. Attempts to predict susceptibil- However, the formulation can be easily extended to the ity to cavitation in high-pressure orifice flows have been case with a mixture of liquid vapor and noncondensable advanced by Dabiri et al. [2],[3]. The analysis indi- gas in the bubble undergoing isentropic or polytropic ex- cates that cavitation is most likely is regions of high shear pansion and/or collapse. Governing equations for the stress. flow are the Navier-Stokes equations: It has been proposed that the collapse of traveling cavitation bubbles increases the disturbances inside the liquid flow. These disturbances later trigger the insta- ∂ρu + ∇ (ρuu) = −∇P + ∇ · T + σκδ(d)n (1) bilities in the emerged jet, and cause a shorter breakup ∂t distance. In this study, we have developed a numeri- 1 £ ¤ cal method to simulate growth and collapse of a com- T = µ ∇u + (∇u)T + λ(∇ · u)I (2) 2 pressible bubble in a viscous incompressible fluid. Full where u is the velocity, ρ and µ are the fluid density Navier-Stokes equations are solved for both liquid and and viscosity, respectively, which could be properties of gas phases. The deformation of bubbles, from a spher- either liquid or gas phase. T is the stress tensor and I ical shape, during growth and collapse in the flow de- is the unity tensor. λ is the bulk viscosity of the bubble. pends on the viscous stress, pressure, and the proxim- The last term in the momentum equation represents the ity to solid walls. The pressure inside the bubble is in- surface tension as a force concentrated on the interface. fluenced by both the pressure and the viscous stress in Here, σ is the surface tension coefficient, κ is the curva- the surrounding liquid. This will lead to either a faster ture of interface, δ is the Dirac delta function. d repre- growth in some cases or to a faster collapse, depend- sents the distance from the interface and n corresponds ing on the capillary number of the cavitation bubbles. to the unit normal vector at the interface. Richardson [4] did an analytical study of 2-D bubbles Continuity equation for the liquid phase is: in Stokes flow and showed the expansion of bubbles due to viscous shear stress only. Therefore, the presence of a ∇ · u = 0 (3) viscous shear stress could change the pressure threshold at which the flow will cavitate. Yu et al. [5] studied the and for the gas phase the volume could change due to the collapse of cavitation bubble in shear flows. They ob- pressure: 1 ∂ρ served that for sufficiently large shear the collapse rate ∇ · u = − − u · ∇ρ (4) of the bubble will increase and the re-entrant jet will dis- ρ ∂t appear. Experiments also have shown that the high shear We assume that the density variation in space inside stress can cause cavitation in the liquid even at high pres- the bubble is negligible. The temperature is also assumed sures. For example, Kottke et al. [6] observed cavitation to be constant. Therefore the density will be proportional in creeping flow at pressures much higher than the vapor to the pressure for an ideal gas. Thus, the continuity pressure. equation reduces to: In order to study this phenomenon, the full Navier- Stokes equations for multiphase flow are solved. A level- 1 ∂P ∇ · u = − (5) set formulation is used to track the interface and model P ∂t the surface tension effects. The flow properties includ- P is the average pressure inside the bubble. The ing the bubble shape are resolved. The bubble center average pressure in the bubble is used instead of local moves with the average velocity of the displaced fluid. pressure in order to suppress the acoustic waves in the The pressure and strain rate tensor are applied as time bubble. varying boundary conditions. The time dependency of Later we will combine the two continuity equations these parameters comes from the calculation of these pa- for liquid and gas (eq. (3) and (5)) using the level set rameters following an element of liquid mass traveling in formulation. The numerical solution of the unsteady Navier- Now using the level-set definition, the fluid proper- Stokes equations is performed using the finite-volume ties can be defined as: method on a staggered grid. The convective term is discretized using the Quadratic Upwind Interpolation ρ = ρliq + (ρgas − ρliq)Hε(θ) (10) for Convective Kinematics (QUICK) (by Hayase [7]). µ = µliq + (µgas − µliq)Hε(θ) (11) The Semi-Implicit Method for Pressure-Linked Equation (SIMPLE), developed by Patankar [8], is used to solve where Hε is a modified Heaviside function that has a the pressure-velocity coupling. The time integration smooth jump: is accomplished using the second-order Crank-Nicolson  scheme.  0 θ < −ε H = (θ + ε)/(2ε) + sin(πθ/ε)/(2π) |θ| ≤ ε ε  Interface Tracking and Level-Set Formulation 1 θ > ε We are considering flow of two immiscible fluids. (12) The interface between these fluids moves with the local where ε represents the thickness of the interface and has velocity of flow field. To track the motion of the inter- the value of 1.5h where h is the cell size. This Heaviside face, the level-set method is used which has been devel- function corresponds to a delta function that can be used oped by Osher and coworkers (e.g., [9] and [10]). The to evaluate the force caused by surface tension: level-set function, denoted by θ, is defined as a signed ½ distance function. It has positive values on one side of [1 + cos(πθ/ε)]/(2ε) |θ| ≤ ε δ = (13) the interface (gas phase), and negative values on the other ε 0 otherwise side (liquid phase). The magnitude of the level-set at each point in the computational field is equal to the short- The last term in the momentum equation (1) in- est distance from that point to the interface. cludes the normal unity vector and the curvature of the The level-set function is being convected by the flow interface which can be calculated as follows: as a passive-scalar variable: ∇θ n = , κ = −∇ · n (14) ∂θ |∇θ| + u · ∇θ = 0 (6) ∂t The level set function can help us to combine two It is obvious that, if the initial distribution of the continuity equations (eq. (3) and (5)) for liquid and gas level-set is a signed distance function, after a finite time phase as follow: of being convected by a nonuniform velocity field, it will not remain a distance function. Therefore, we need to 1 ∂P ∇ · u = −Hε(θ) (15) re-initialize the level-set function so it will be a distance P ∂t function (with property of |∇θ| = 1) without changing Results and Discussion the zero level-set (value at the interface). In order to validate the accuracy of the numeri- Suppose θ (x) is the level-set distribution after 0 cal method, a comparison is performed between current some time step and is not exactly a distance function. method and solution of Rayleigh-Plesset equation for This can be reinitialized to a distance function by solv- Oscillation of a spherical bubble. Volume variation in ing the following partial differential equation [9]: time is shown in Figure 1. The bubble is initially at equi- ∂θ0 librium when the pressure drops to a smaller value for = sign(θ )(1 − |∇θ0|) (7) ∂τ 0 a short period of time. This causes the bubble to grow and then oscillate around the equilibrium as the pressure with initial conditions: recovers its initial value.

0 As can be seen on Figure 1 there is a good agreement θ (x, 0) = θ0(x) (8) with analytical results even though a relatively coarse grid is used. where  Cavitation in liquid injectors could result in im-  -1 if θ < 0 proved spray quality [1]. However, it would be compu- sign(θ) = 0 if θ = 0 (9)  tationally expensive to simulate the orifice and resolve 1 if θ > 0 the cavitation bubbles at the same time. Therefore, the and τ is a pseudo time. The steady solution of equa- first approach that we take is to resolve the orifice flow tion (7) is the distance function with property |∇θ| = 1 without the presence of bubble. Then, the pressure and and since sign(0)=0, then θ0 has the same zero level-set shear in the flow is measured as a mass element travels as θ0. through the injector. Now, we assume if there is a bubble 1.8 1 Rayleigh-Plesset Current method 0.8 1.6 (a) 0.6 1.4

0.4 1.2 p C 0.2 (b) Volume 1 0

0.8 (c) -0.2

0.6 -0.4

0 0.5 1 1.5 2 0 2 4 6 8 10 12 14 time tU/D

Figure 1. Comparison between numerical results on a Figure 3. History of the pressure coefficient at a fluid P −Pd 64×64×64 grid with solution of Rayleigh-Plesset equa- element traveling through the orifice. Cp = 1 2 . (a), 2 ρU tions for oscillation of a spherical bubble. (b) and (c) correspond to instants at which the bubble shape is shown in Figure 4.

sure. As the element goes closer to the orifice entrance, flow accelerates and pressure drops. A minimum in the pressure occurs as the element passes the corner of the r x orifice. Then, the pressure increases and finally reaches Axis of symmetry the downstream pressure. In the next step, this time varying pressure is applied Figure 2. Geometry of the axisymmetric orifice and the as the boundary condition to a bubble in a Couette type boundary fitted grid. (flow from left to right) flow. The rate of strain of the background Couette flow is equal to the average strain rate of the traveling mass element in the orifice. The flow field is two-dimensional but the initial bubble shape is spherical resulting in a three- in the flow it will travel with local velocity of the flow dimensional flow field and bubble distortion. field. Then, the shear stress and pressure history of the Figure 4 shows the shape of the cavitation bubble mass element is applied to the bubble and the growth and and the velocity field at different times as the bubble ex- collapse of bubble under these conditions is studied. pands in the Couette flow. Elongation at an angle and Figure 2 shows the orifice geometry and the compu- and increase in bubble volume occurs. tational domain in which the flow is solved. More details can be found in Dabiri et al. [2]. Figure 3 shows the time Conclusion variation of the pressure coefficient, Cp for octane flow- ing through an orifice of 240µm diameter at Reynolds A numerical method based on the level-set formula- number of Re = 8000. tion is developed to consider the volume change of a cav- The time variation of the pressure coefficient has itation bubble due to pressure change in the surrounding been measured by following a fluid element in the in- liquid. The method is used to study the growth and col- jector. The pressure coefficient is defined as lapse of cavitation bubbles as they travel inside a fuel injector. The results could help in understanding the effects P − P of shear stress on cavitation and also the contribution of C = d (16) p 1 ρU 2 cavitation to disturbances in the orifice flow and atom- 2 ization of the emerged jet. The results presented here are where, Pd is the pressure at downstream of the orifice preliminary results and further investigation will be con- and U is the nominal velocity of the liquid in the orifice. ducted to determine how the cavitation will be affected Initially the pressure is close to upstream reservoir pres- by the shear in the flow and how it changes the flow field. Nomenclature D orifice diameter I the unity tensor n normal unit vector P pressure P average pressure in the bubble Re the Reynolds number t time u velocity vector U average jet velocity (a) κ curvature of the interface λ bulk viscosity µ viscosity ρ density σ surface tension θ level-set function

Subscripts gas gas phase liq liquid phase References (b) [1] H. Hiroyasu. Atomization and Sprays, 10(3- 5):511–527, 2000. [2] S. Dabiri, W. A. Sirignano, and D. D. Joseph. Physics of Fluids, 19(7):072112, 2007. [3] S. Dabiri, W. A. Sirignano, and D. D. Joseph. Jour- nal of Fluid Mechanics, 2008. In press. [4] S. Richardson. Journal of Fluid Mechanics, 33(3), 1968. (c) [5] P. W. Yu, L. Ceccio, and G. Tryggvason. Physics of Fluids, 7(11), 1995. Figure 4. Snapshots of bubble shape and velocity field [6] P. A. Kottke, S. S. Bair, and W. O. Winer. AICHE as the bubble grows in shear flow. Journal, 51(8):2150–2170, 2005. [7] T. Hayase, J. A. C. Humphrey, and R. Greif. J. Comput. Phys., 98:108–118, 1992. [8] S. V. Patankar. Numerical heat transfer and fluid flow. Hemisphere, Washington, DC/New York, 1980. [9] M. Sussman, E. Fatemi, P. Smereka, and S. Osher. Computers and Fluids, 27:663–680, 1998. [10] S. Osher and R. P. Fedkiw. J. Comput. Phys., Acknowledgment 169:436, 2001. This research has been supported by the US Army Research Office through grant No. W911NF-06-1-0225, with Dr. Kevin McNesby and Dr. Ralph Anthenien hav- ing served sequentially as program managers. D. D. Joseph was also supported by NSF grant No. CBET- 0302837.