Computational Mechanics 32 (2003) 281–290 Ó Springer-Verlag 2003 DOI 10.1007/s00466-003-0485-5
Non-spherical bubble behavior in vortex flow fields J.-K. Choi, G. L. Chahine
281 Abstract The boundary element method (BEM) is applied to solve the unsteady behavior of a bubble placed in a vortex flow field. The steady vortex field is given in terms of the viscous core radius and the circulation, both of which may vary along the vortex axis. For this study, 2DYNAFSÓ, an axisymmetric potential flow code which has been verified successfully for diverse type of fluid dynamic problems, is extended. The modifications to accommodate the ambient vortex flow field and to model the extreme deformations of the bubble are presented. Through the numerical simulations, the time history of the bubble geometry and the corresponding pressure signal at a fixed field point are obtained. A special effort is made to continue the numerical simulation after the bubble splits into two or more sub-bubbles. Indeed, it is found that an elongated bubble sometimes splits into smaller bubbles, which then collapse with the emission of strong pressure signals. The behavior of the axial jets after the split is also studied in more detail. Keywords Bubble dynamics, Cavitation inception, Tip vortex cavitation, Cavitation noise
1 Introduction Many people have been interested in the bubble behavior in a vortex field because the earlier type of cavitation inception on hydrofoils and propeller blades takes place in the tip vortex. Despite several significant contributions to the study of bubble capture in a vortex, to our knowledge, no complete approach has yet been undertaken. The complexity of the problem has led the various contributors to neglect one or several of the factors in play, and therefore to only investigate the influence of a limited set of parameters. The first approaches to the problem were attempted independently at about the same time by Bovis [1] and Latorre [14]. Bovis considered the case where the flow velocities in the vortex flow are large enough to justify
J.-K. Choi (&), G. L. Chahine Dynaflow, Inc., 10621-J Iron Bridge Road, Jessup, MD 20794, USA e-mail:
[email protected] This work was conducted at DYNAFLOW, INC. (www.dynaflowinc.com). The work has been supported by the Office of Naval Research under the contract No. N0014–99-C-0369 monitored by Dr. Ki-Han Kim. This support is greatly appreciated.
the assumptions of inviscid potential flow. This simplification, valid for instance in tip vortex cavitation where very large tangential velocities come into play, and when the bubble is not too close to the vortex axis, allows one to consider other important effects. For instance, one can then consider in a consistent fashion important phenomena such as the modification of the vortex flow by the presence of the bubble and the volume change and shape deformation of the bubble [6]. On the other hand, Latorre [14] and the following studies [15], in a more pragmatic approach, considered real fluid effects to determine the bubble motion equation, neglecting bubble shape deformation and modification of the flow by the bubble behavior. They coupled these equations with a spherical bubble dynamics model to deduce noise emission in tip vortex cavitation. Later, Chahine [2] considered a broader approach where bubble deformation and motion were coupled while neglecting flow field modification by the bubble presence. His study showed that the pressure gradient across the bubble could lead to significant departure from a spherical bubble. This fact leads to the suggestion that the deformation and later splitting of the bubble during its motion towards the vortex center is, in addition to its volume change, a main source of noise in vortex cavitation. This appears to explain the reason for the location of tip vortex noise at cavitation inception very close to the blade [9], and is in agreement with later observations by Maines and Arndt [16] about bubble capture in tip vortex cavitation. A comprehensive review on the subject can be found in Chahine [3]. Recently, Hsiao et al. [10] successfully applied the dynamics of spherical bubbles, based on a modified Rayleigh–Plesset equation, to the prediction and the scaling of the tip vortex cavitation inception. Since the nucleus captured in the tip vortex flow field is very small in inception studies, the assumption of spherical bubble is justified as long as the bubble size during the dynamics remains very small compared to the viscous core size of the vortex field. However, for extended numerical simulations slightly beyond the inception, one has to consider the bubbles that deform into elongated shapes due to the pressure field induced by the vortex. In this study, the dynamics of the non-spherical bubbles are modeled by a more general method based on the potential flow theory and Green’s identity. The boundary element method (BEM) is applied to solve the unsteady behavior of a bubble placed in the vortex flow field. The steady vortex field is given in terms of the viscous core radius and the circulation, both of which may vary along
282
Fig. 1. The vortex field used in the current study (r ¼ 4:807); a variation of the core size and the circulation along the axis, b the tangential velocity (top) and the pressure (bottom) in the vortex field
the vortex axis. One can distinguish three phases in the interactive dynamics of bubbles and vortices; (a) bubble capture by the vortex, (b) interaction between the vortex and an initially quasi-spherical bubble on its axis, and (c) dynamics of elongated bubbles on the vortex axis. In this study, the last two phases of the dynamics are of interest and an axisymmetric code is utilized. The axisymmetric code, 2DYNAFSÓ, has been verified successfully for diverse type of fluid dynamic problems in the past [5]. For this study, the code has been extended to accommodate the ambient vortex flow field and to model the extreme deformation of the bubble. A special effort is made to continue the numerical simulation after the bubble splits into two or more sub-bubbles. The prime objective of this work is to identify the region where the spherical model is appropriate and to investigate the source of noise in tip vortex cavitation inception.
2 Formulation 2.1 Tip vortex field The tip vortex field used in this study is based on the measured tip vortex field provided by Shen1 . From the measured vortex core radius and circulation distribution along the length of the tip vortex (Figure 1a), the so called Lamb vortex [13] (also known as Oseen vortex) is considered to construct the vortex flow field. The assumption behind applying the Lamb vortex field is that the momentum in the radial direction predominates those in the other directions. This assumption is valid along the length of the vortex except very near the tip where the vortex initiates with strong axial gradient in the circulation. If a steady state vortex field, that has the maximum tangential velocity at the local core radius ac , is considered, the tangential velocity vh around the vortex axis and 1
Shen Y (2000) Private communication with Naval Surface Warfare Center – Carderock Division.
the corresponding pressure field are given at the radial distance r by
C 1 en ; ð1Þ 2pr o 2 1:256qC2 n 1 n 1 en þ 2ðEi ðnÞEi ð2nÞÞ ; p ¼ p1 2 2 8p ac
vh ¼
ð2Þ where C is the local circulation around the vortex, q is the 2 liquid density, n ¼ 1:256ðr=a c Þ , and Ei is the exponential R1 integral defined as Ei ðnÞ ¼ n ðes =sÞ ds. Note that the ambient pressure p1 is related to the cavitation number r 2 by its definition, r ðp1 pv Þ= 12 qV1 , with the vapor pressure pv and the far field velocity V1 . The vortex fields used in this study are shown in Fig. 1. The ambient pressure p1 corresponds to the cavitation number r ¼ 4:807, which is the inception cavitation number predicted by the spherical model described later. Notice that the minimum pressure occurs near z ¼ 0:035 m on the vortex axis.
2.2 Spherical model (SAP) A spherical bubble model is valid as long as the pressure difference across the bubble due to the pressure gradient of the fluid around the bubble is negligible compared to the surface tension of the bubble. This is true if the bubble size is very small relative to the size of the vortex core [3]. The spherical model used in this study is based on Rayleigh–Plesset equation [17], but with some modifications as described in Hsiao et al. [10, 11]. For the completeness of this paper, the spherical model is very briefly explained below. In order to account for the pressure due to the difference between the liquid velocity u and the bubble velocity 2 up , an extra term of slip velocity, u up =4, is added to the classical Rayleigh–Plesset equation. We also adapt Gilmore’s [7] modification to account for the liquid compressibility which becomes important when bubble
surface velocity becomes comparable with the speed of sound in the liquid c.
3 R_ R_ _ 1 R_ R d 1 1þ þ RR€ þ R¼ 1 2 q c 3c c c dt " # 3k R0 2c 4l _ pv þ pg0 pðtÞ R R R R 2 1 þ u up ; ð3Þ 4 where R is the bubble radius, R0 is the initial bubble radius, l is the liquid viscosity, c is the surface tension parameter, pg0 is the initial gas pressure inside the bubble when R ¼ R0 with the polytropic gas constant k, and pðtÞ is the time varying pressure that the bubble encounters. The dots over R represent the time derivatives. In the surface averaged pressure (SAP) spherical model, pðtÞ is taken as the area weighted average of the outside liquid pressure over the bubble surface. This allows for a much more realistic description of the bubble behavior as it is captured by the vortex. That is, once the bubble reaches the vortex axis, the encounter pressure will increase as the bubble grows enabling a more realistic balancing of the bubble dynamics. The bubble trajectory during capture is predicted by an equation of motion given by Johnson and Hsieh [12]. 3 du 3 3 ¼ rp þ CD u up u up þ u up R_ ; dt q 4 R ð4Þ where the drag coefficient CD is determined from the empirical formula by Haberman and Morton [8]. With a prescribed pressure and velocity field, a Runge–Kutta fourth-order scheme is applied to integrate Eqs. (3) and (4) through time to provide the bubble trajectory and its volume variation during bubble capture by the tip vortex. Under the assumption of small bubble, the so called oneway coupling, which considers only the effect of the flow field on the bubble behavior, is applied.
2.3 Shape factor The shape factor is introduced as an index that tells how well a spherical model can describe the bubble behavior. Suppose a bubble slightly deformed from its original spherical shape due to the pressure field of the vortex. If the deformed bubble is in static equilibrium with the surrounding pressure field, the local pressure balance can be written as pv þ pg ¼ plocal þ 2c=Rlocal , where plocal is the local liquid pressure and Rlocal is the local radius curvature. That is, the local radius can be expressed as Rlocal
2c ¼ : pv þ pg plocal
ð5Þ
283
Fig. 2. The bubble radius and the local pressures defined at three points on the bubble surface. The local pressures are different due to the pressure field of the vortex around its axis
liquid pressures, and the bubble will deform toward a static equilibrium in such a way that the local radii of curvature satisfy Eq. (5). The shape factor is defined as the ratio of this local equilibrium radius to the spherical bubble radius Rbub .
Rlocal : ð6Þ Rbub The shape factor can be calculated easily while the bubble is traced by the SAP spherical model. SF
2.4 Axisymmetric formulation For a more general description of the bubble deformation in a vortex field, non-spherical bubble geometry should be modeled. In the present study, an axisymmetric formulation is used to take advantage of the axisymmetric vortex field. Under the assumption of axisymmetry, only the bubble behavior after the bubble has reached the vortex axis is possible. However, this is not a strong restriction in practice because the spherical model can be used during the capture of the bubble where the bubble remains practically spherical until it reaches the vortex axis. An alternative approach is a fully three-dimensional method that is used in other studies [11]. Any velocity field utotal can be expressed via the Helmholtz decomposition as the sum of the gradient of a scalar potential / and the curl of a vector potential A, with r2 A ¼ x. utotal ¼ u þ v ¼ r/ þ r A
ð7Þ
The flow due to the bubble presence is assumed to be expressed by u ¼ r/ and the rotational flow field v of the tip vortex is assumed to remain not affected by the The pressure field near the vortex axis monotonically bubble presence and dynamics [4]. Since the potential decreases toward the center as depicted in Fig. 2. A nearly spherical bubble shown in the figure will experi- flow field due to the bubble presence satisfies the Laplace ence liquid pressures that change from point to point on equation r2 / ¼ 0, Green’s identity can be applied to construct the integral equation for the potential / and the bubble surface. For example, the three points 1, 2, the normal derivative of the potential o/=on. and 3 on the bubble surface will experience different
The boundary conditions on the bubble surface are (a) the continuity of the normal stresses (the dynamic condition) and (b) the condition that the fluid normal velocities should be equal to the interface normal velocities (the kinematic condition). On the bubble surface, o/=on is obtained as the solution of the integral equation, while / is given from the dynamic boundary condition. The potential / at time t þ dt is found by a simple Euler time stepping scheme.
284
o/ ðt Þ þ ðu þ vÞ r/ðtÞ dt : /ðt þ dt Þ ¼ /ðtÞ þ ot
ð8Þ Note that the terms in the braces include the Lagrangian change of the potential following a node on the bubble. To find an expression for o/=ot on the bubble surface, utotal of the Helmholtz decomposition (7) is substituted into the Navier–Stokes equation.
o 1 ðu þ vÞ ðu þ vÞ fr ðu þ vÞg þ rju þ vj2 ot 2 rp 2 þ mr ðu þ vÞ ; ¼ ð9Þ q where the pressure field p is the sum of the pressure px due to the vortex field and the pressure due to the bubble. Under the assumption that the predetermined vortex field satisfies the Navier–Stokes equation, Eq. (9) simplifies to the modified Bernoulli’s equation.
p px o/ 1 2 r þ þ jr/j þ v r/ q ot 2 : ¼ r/ ðr vÞ
ð10Þ
Here, Vo is the initial volume of the bubble and C is the local surface curvature given by C ¼ r n. Eqs. (12) and (13) are combined to obtain an expression for o/=ot, which then is substituted into Eq. (8).
1 1 /ðt þ dt Þ ¼ /ðtÞþ jr/ðtÞj2 þ fpx pv 2 q )# k Vo þ cC dt : pg0 V ðt Þ
ð14Þ
In axisymmetric problems, the flow variables are independent of the angular coordinate h in a cylindrical system (r; h; z), and thus the integration in h-direction can be carried out explicitly. The bubble surface is modeled by N straight-line segments on a meridian plane. The potential / is assumed to vary linearly over each segment, while o/=on is assumed constant over each segment. The integral equation is then collocated at the center of each segment, and the resulting matrix vector equation is solved by using a standard LU decomposition technique. To advance each node on the bubble surface, the velocity at each node is required. The normal velocity is known from the solution of the integral equation while the tangential velocity is obtained by numerical differentiation of / along the arc length on the meridian plane. The nodes are then advanced according to the sum of this local velocity and the velocity of the ambient vortex field, and the new bubble geometry is obtained. The time step size dt is controlled by an adaptive scheme that ensures that smaller time steps are chosen when the potential changes rapidly. The upper limit of the / change within a time step is prescribed by a user specified parameter d/max . Then the time step size is determined by
In the present study, the vorticity in the vortex field is d/max dt ¼ ; ð15Þ further assumed to be predominant only in the axial 2 1 þ 12 Vmax direction ez . With r v ¼ xz ez , this results in where Vmax is the normalized maximum nodal velocity at o p px o/ 1 2 the concerned time t. This prescription ensures that the ð11Þ þ þ jr/j þ v r/ ¼ 0 : q oz ot 2 maximum change in / is of the order of d/max within a time step. Because the disturbances due to the bubble, i.e. / and Once the solution is obtained at any time step, the p px , decay to zero at infinity, pressure signal at given field points can be calculated by o/ p px 1 using the Green identity and the unsteady Bernoulli ð12Þ equation. First the Green identity is used to calculate the jr/j2 v r/ : ¼ q ot 2 potential at a given field point, and then the velocity is The pressure inside each bubble is assumed homogeneous, obtained from numerical differentiation. Finally, the and the gas inside each bubble is assumed to be composed pressure can be calculated by the Bernoulli equation. of both vapor of the liquid and non-condensable gas. The A bubble placed on the vortex axis usually elongates pressure at any instant is given by the sum of the partial along this axis. Oftentimes the elongation is so extreme pressures of the liquid vapor and of the non-condensable that the bubble eventually splits into smaller sub-bubbles. gas. Vaporization of the liquid is assumed to occur at a fast When this happens, a special treatment is necessary to enough rate so that the vapor pressure inside the bubble continue the simulation. A schematic view of a bubble remains equal to the equilibrium liquid vapor pressure at about to split is shown in Fig. 3. When a node approaches the ambient temperature. The non-condensable gas is as- the axis within a specified small distance, three nodes sumed to satisfy a polytropic law with an exponent k, and including two neighboring nodes are tested to find the two thus pV k remains constant regardless of the bubble volume nodes that are closest to the axis. The segment connecting VðtÞ that varies with time. these two nodes is detected as the segment to split. This k detection procedure can be complex if multiple nodes Vo p ¼ pv þ pg0 cC : ð13Þ approach the axis simultaneously or the bubble splits into V ðt Þ more than three sub-bubbles at the same time step. Once a
285 Fig. 3. Schematic view of a bubble that splits into two sub-bubbles
segment to split is found, the segment is removed by placing the two end nodes of the segment exactly on the axis. Many variables including potential and velocities need to be extrapolated to the new position and the node and segment indices as well as the bubble index are updated. The gas pressure must remain constant through the split2 . The gas pressure before the split can be expressed as pg;B ¼ pg0 ðVo =VB Þk , where the subscript B represents the quantity just before the split. Then the gas pressure after the split can be calculated by
pg ðtÞ ¼ pg;B
k
Vi;A Vi ðtÞ
¼ pg0
Vo VB
k
Vi;A Vi ðtÞ
k ;
ð16Þ
where Vi;A is the volume of the ith sub-bubble just after the split and Vi ðtÞ is the volume of the ith sub-bubble at arbitrary later time t. The sum of Vi;A over all sub-bubbles should be equal to VB , unless the process involves gas loss which is not considered in the present study.
3 Numerical results 3.1 Non-spherical bubble behavior in a constant vortex line In this section, a vortex line with a constant core radius, ac ¼ 5:08 mm and a constant circulation Co ¼ 0:439838 m2 /s is considered. The pressure at infinity is p1 ¼ 192; 281 Pa, and the axial velocity in the vortex field is uniformly Vz ¼ 13:78 m/s. This condition corresponds to a foil of chord 0.3 m at an incident angle of 2.3 degrees advancing at 13.78 m/s. At time t ¼ 0, a bubble of initial radius Ro ¼ 200 lm with gas pressure pg0 ¼ 189; 944 Pa (equilibrium pressure) is released at the origin (r, z)=(0, 0) on the vortex axis. The simulated bubble behavior is shown in Fig. 4a. The bubble grows more or less spherically up to the radius of about 1 mm, then it elongates up to 6 mm as it travels downstream along the axis. The mid part of the bubble becomes concave, and the bubble eventually splits into two sub-bubbles. Both of the subbubbles develop axial jets from the split, one propagates 2
Alternatively, one could prescribe a given pressure loss during the split.
Fig. 4. The bubble behavior in a vortex flow field. a a spherical bubble of 200 lm released at the origin, and b a bubble started 0.224 ms later with initial radius 947 lm and initial radial velocity 1.968 m/s
upstream and the other downstream. The jets later touch the other side of the sub-bubble on the axis. In order to validate the code for non-zero initial radial velocity R_ , the behavior of the same bubble under the same condition is simulated starting from the initial condition corresponding to t ¼ 0:224 ms of the previous simulation. The bubble at this time is still within the stage of the initial spherical growth. From the simulation by the spherical model, it is found that R ¼ 0:995 mm and R_ ¼ 1:968 m/s at t ¼ 0:224 ms. The bubble behavior starting from this initial condition is shown in Fig. 4b. The spherical bubble in the middle corresponds to the initial bubble of this case. For a better view of the bubble geometry, the plot is made as the observer is moving along the axis with the constant axial velocity of the ambient vortex field Vz . In general, the bubble behavior agrees with that of the simulation that starts from t ¼ 0 as further described below. In Figure 5, the bubble volumes are shown for two runs with different time step sizes, a run that started 0.224 ms later, two runs from the spherical models, and lastly one from a fully three-dimensional code. The volume curves for the two time step sizes used in these computations agree well with each other. The bubble splits into half near t =1.4 ms, after which halved volume continue to collapse independently. These volume curves agree well with the fully three-dimensional prediction by 3DYNAFSÓ, which simulates only until the split. The implementation of the initial condition also works properly because the volume curve of the bubble started in the middle agrees with the others. The bubble volume from the SAP spherical model and that from non-spherical calculation agree during the growth phase, but deviate from each other as the bubble grows larger. The maximum volume of the non-spherical computation is larger than the spherical one because of the non-spherical bubble elongation. Notice that the time
286
Fig. 5. Comparison of the bubble volumes predicted by the axisymmetric methods with two different time step sizes, by a mid-start at t = 0.224 ms, by two spherical models, and by a three-dimensional method
Fig. 6. Bubble behavior predicted by the SAP spherical model. (R: bubble radius, Penc: encounter pressure, Pfp: field point pressure at (z, r) = (0, 0.3) m, Pg: gas pressure inside the bubble)
between the beginning and the split for the non-spherical bubble is approximately equal to one period of the spherical bubble oscillation. Also, note the major disagreement between all the runs and the classical spherical model which significantly overpredicts the bubble size.
Fig. 7. Shape factors calculated by the SAP spherical model. (RBUB : spherical bubble radius, SF1;2 : shape factors defined by Eq. (6) for points 1 and 2 shown in Fig. 2)
along the vortex axis, it experiences a sudden growth after t = 2 ms, followed by multiple collapses and rebounds with the emission of strong pressure pulses. In Fig. 7, the shape factors calculated from the SAP spherical model are shown. The shape factors remain within a couple of percent range from unity until the bubble starts to grow suddenly. The initial deviation before t =1 ms is due to the off center location of the bubble center during its capture. From the history of the spherical bubble behavior, four bubble conditions are selected as the initial conditions for the non-spherical simulations. That is, the radius R, the radial velocity of the bubble R_ , and the gas pressure inside the bubble pg are taken from the spherical bubble behavior when it passes at z = 0.0, 0.013, 0.030, and 0.035 m, which correspond to t = 0.0, 0.994, 2.282, and 2.661 ms, respectively. The results from these four simulations are shown in Fig. 8. In Fig. 8a, the bubble behavior and the pressure at a field point for the bubble released at z = 0.0 are shown. Even though the spherical model starts from an off center location, the non-spherical computation starts from the same initial bubble placed on the axis due to the limitation of the axisymmetric method. The bubble behavior shown on the left figure is viewed from an observer moving with the constant axial velocity of the vortex field, Vz ¼ 13 m/s. While the bubble oscillates, it translates downstream faster than the axial velocity of the vortex field due to the pressure gradient along the axis. The bubble oscillates for a while3 until the numerical instability finally turns into an axial jet that touches down on the other side of the bubble when the computation stops at t = 0.15 ms or z = 0.002 m. The pressure signal, shown on the right figure, also presents the oscillation of the bubble in the period of 0.92 ls, which agrees with the theoretical natural period of 0.94 ls under the given condition. The amplitude of the oscillation increases as the bubble moves downstream because the ambient pressure becomes lower at downstream locations. This 2DYNAFSÓ simulation shows that the code can continue
3.2 Bubble behavior along the tip vortex In this section, a spherical bubble of radius 5 lm released at z = 0 and r = 0.1645 mm off axis in the tip vortex field shown in Figure 1 is considered. Note that this vortex field corresponds to the cavitation number, r ¼ 4:807, which is predicted as the cavitation inception point by the SAP spherical model. The bubble is in static equilibrium at the release location with the initial gas pressure pg0 ¼ 434499 Pa. First, the SAP spherical model is applied to predict the bubble behavior as shown in Fig. 6. The bubble released at off axis location is captured by the vortex, arriving at the 3 The total simulation time, 0.15 ms, is long relative to the bubble axis at about 1 ms after the release. As the bubble moves oscillation period of 0.92 ls.
287
Fig. 8. The bubble behaviors (left) and the pressure signals (right) as predicted by 2DYNAFSÓ in the tip vortex, r = 4.807. a the initial bubble released on the vortex axis at z = 0, b the spherical bubble released at z = 0.013 m, c at z = 0.03 m, and
d at z = 0.035 m. For simulations in b, c, and d, the spherical bubbles as predicted by the SAP spherical model at corresponding locations are used as the initial conditions
the simulation for a long enough time before the numerical instability takes over the simulation. Within the duration of this simulation, the SAP spherical model may well be applied without any further assumption because the bubble essentially remains spherical. In Fig. 8b and c, the bubbles that started at downstream locations are shown. The bubble at z =0.013 m develops a weak jet along the axis, which touches down later after the bubble becomes flatter perpendicular to the axis. During this deformation, the oscillation frequency remains almost the same with greater pressure amplitude after the bubble becomes flatter. The bubble at z = 0.03 m fluctuates only two cycles while it deforms into a shape with a neck in the middle. During the second collapse, a jet is developed along the axis, which eventually touches down on the other end of the bubble. Knowing that the shape factors at z = 0.03 m are SF1 ¼ 1:01 and SF2 ¼ 0:98, one can conclude only an initial 1% deviation of the shape factor from unity can
cause the bubble to deform into a very non-spherical shape. Although the bubbles released at z = 0.013 and 0.03 m deform to non-spherical shapes, the far field pressure signal may well be predicted by the SAP spherical model because the major physics behind the phenomenon is still the oscillating volume source. However, the spherical model would miss the pressure signal produced when the jet finally touches down on the other side of the bubble. Note that the 2DYNAFSÓ simulation can be continued after the touch down, even though such continuation is not attempted in the present study. In Fig. 8d, the bubble behavior from z = 0.035 m is shown. The shape factors at this location are SF1 ¼ 1:13 and SF2 ¼ 0:81. It is interesting that the bubble elongates extremely without any cyclic fluctuations. At the maximum elongation, the length to radius ratio reaches more than 10. The maximum radial growth is 0.085 mm, which is more than twice the maximum radius, 0.039 mm, predicted by
288
Fig. 9. The effect of time step size (left) and the effect of the number of segments (right) on the jump in the pressure signal
the SAP spherical model. The bubble growth in the radial direction is still very small compared to the local core radius of the vortex, 0.56 mm. The elongated bubble forms two necks, of which the downstream one breaks resulting in two sub-bubbles. Later, the downstream sub-bubble collapses, when the simulation stops. The pressure signal shows a discontinuity at the bubble split and a large peak toward the collapse of the downstream sub-bubble. The intermediate bubble behaviors shown in Fig. 8 can also be interpreted as if the spherical bubble is directly captured by the vortex at corresponding axial locations, instead of the upstream capture and translation along the axis.
3.3 Convergence study on the pressure jump The jump in the pressure signal at the splitting of the bubble is investigated in more detail. The case of the 5 lm bubble released at 0.035 m downstream in the tip vortex field is studied. The effect of the time step size is investigated by varying the factor d/max =am , which is proportional to the time step size, from 0.1 to 0.01, and 0.005 as shown on the left of Fig. 9. The spikes at the time of the split are believed to be only numerical behaviors because very high pressure values are recorded only for a couple of time steps. The artificial loss of one segment at the splitting moment would be responsible for these spikes. As the time step size becomes smaller, the pressure jump becomes larger. The effect of number of segments is shown on the right of Fig. 9, where the numbers of segments are varied from 32 to 64 and 128. As the number of segments increases, the jump becomes less. Since the convergence with respect to both the number of segments and time step size is not consistently established in the present study, it is unclear if the pressure jump is real or numerical. However, it is worth to notice that the pressures from the runs with 64 and 128 segments have strong signal just after the split, prior to the final spikes at the reentering jet touchdown of the downstream bubbles.
symbols are drawn at every 100 time steps, and the bubble shapes at five representative time steps are also shown in the figure. At the splitting moment, an artificial spike and a pressure jump across the split are observed. It is clear from the figure that the N-shaped pressure signal after the split is closely related to the formation of the two axial jets moving in opposite directions. Since the total volume of the bubbles remains more or less constant through this quick process, the acoustic source of the pressure signal is not the fluctuating volume but mainly the movement of the fluid mass due to the jets or the change in the volume of liquid ‘‘sucked’’ into the jets in the sub-bubbles. In Fig. 11, the bubble induced pressure fields at the 18900th and the 19300th time step are shown. High pressures of magnitude greater than 50 kPa are observed near the two jets.
4 Conclusions Bubble behaviors in tip vortex flows are studied with the use of an axisymmetric boundary element method. In the present work, DYNAFLOW’s code 2DYNAFSÓ was extended to handle extreme deformations of the bubble such as its splitting into sub-bubbles. Results from the extended code are compared with the predictions by the SAP spherical model. The bubble behavior in tip vortex flow is first obtained from the spherical model during the capture by the vortex. Then, the axisymmetric method is applied to obtain non-spherical behavior at four stages along the axis. In the early stages, the bubble is indeed spherical while it oscillates at its natural frequency. However, when the bubble reaches the axis near the beginning of sudden growth, it develops an axial jet on the upstream side of the bubble shooting downstream. This nonspherical behavior is seen even when the initial shape factor based on enforcing sphericity deviates less than 1% from unity. Even at this stage, the spherical model is expected to provide useful information because the bubble is more or less spherical except that there is a relatively small jet on the axis. The bubble behavior 3.4 becomes very non-spherical for the last stage, when it Jet behavior A magnified view of the pressure signal from the simulation elongates extremely to reach a length to radius ratio of more than 10. Then the bubble splits into two emitting a with 128 segments is shown in Figure 10. The circular
289
Fig. 10. Pressure signal at the field point (0.0 m, 0.3 m) and the bubble shapes at five moments from 2DYNAFSÓ simulation with 128 segments. Symbols are drawn at every 100 time steps
Fig. 11. Pressure fields (in Pa) just after the jet development in the sub-bubbles. Arrows represent the bubble induced velocity vectors
spike of pressure signal followed by another strong pressure signal when one of the sub-bubbles collapses. In the computation with 128 segments, two axial jets originating from the split and a strong pressure signal during the formation of the jets are observed. This behavior supports the hypothesis that the noise at the inception of the vortex cavitation may come from bubble splitting and/or the jets formed after the splitting. Further numerical and experimental studies are required to understand the phenomenon better.
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