Tao Xing1 Zhenyin Li2 Steven H. Frankel3 School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907
1
Numerical Simulation of Vortex Cavitation in a ThreeDimensional Submerged Transitional Jet Vortex cavitation in a submerged transitional jet is studied with unsteady threedimensional direct numerical simulations. A locally homogeneous cavitation model that accounts for non-linear bubble dynamics and bubble/bubble interactions within spherical bubble clusters is employed. The velocity, vorticity, and pressure fields are compared for both cavitating and noncavitating jets. It is found that cavitation occurs in the cores of the primary vortical structures, distorting and breaking up the vortex ring into several sections. The velocity and transverse vorticity in the cavitating regions are intensified due to vapor formation, while the streamwise vorticity is weakened. An analysis of the vorticity transport equation reveals the influence of cavitation on the relative importance of the vortex stretching, baroclinic torque, and dilatation terms. Statistical analysis shows that cavitation suppresses jet growth and decreases velocity fluctuations within the vaporous regions of the jet. 关DOI: 10.1115/1.1976742兴
Introduction
Cavitation is a dynamic process involving the growth and collapse of gas cavities in liquids. The bubbles usually form and grow in regions of low pressure and then subsequently collapse when they are convected to regions of high pressure. If the bubble collapse occurs near a solid wall, asymmetries can result in the generation of extremely high pressure jets, which can cause considerable material damage or erosion. This problem is particularly important in hydraulic applications involving fluid machinery. Additionally, the alternating growth and collapse of bubbles associated with cavitation can result in high-frequency pressure fluctuations, the generation of excessive noise, and vibration. The ability to understand and predict cavitation flow physics is important in fluid engineering. It is well known that there is a strong link between cavitation inception and turbulent flow structures 关1兴. Evidence for this link exists in both wall-bounded flows, as well as free-shear flows 关2兴. Billard et al. 关3兴 presented further experimental evidence for this effect where they used a vortex 共turbulence兲 generator upstream of a venturi to produce preturbulence and observed a delay in cavitation inception and a reduction in noise. This behavior was attributed to the generation of small-amplitude, high-frequency pressure fluctuations affecting bubble behavior, in particular the morphology of the cavities. In another study, Baur and Ngeter 关4兴 studied the three-dimensional features of cavitation structures in a turbulent shear layer. They considered a channel flow with a rectangular sill mounted upstream on the bottom wall of the channel. This produced separated flow and a turbulent shear layer downstream in a similar manner to the turbulence generator used by Billard et al. 关3兴. The shear layer produced a region of high turbulence and horseshoe-shaped cavitation tubes that they associated with the bursting process in wall-bounded channel flows. 1
Ph.D., currently Postdoctoral Researcher at The University of Iowa Doctoral student Professor, Member ASME, corresponding author, E-mail:
[email protected] Contributed by the Fluids Engineering Division for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received by the Fluids Engineering Division May 2, 2003; final manuscript received April 7, 2005. Associate Editor: Steven Ceccio. 2 3
714 / Vol. 127, JULY 2005
For a given flow, the location of cavitation inception can be specified in different ways. In wall-bounded flows, such as Billard’s studies on venturi, minimum pressure occurs at or close to a solid surface, according to the maximum-modulus theorem. However, the most common exception to this rule is in vortex cavitation, where the unsteady effects and/or viscous effects associated with vortex shedding or turbulence cause deviation from the maximum-modulus theorem 关5兴. In vortex cavitation, often associated with separated flows, cavitation occurs in the cores of eddies formed in the shear layer emanating from the separation point 关2兴. This is due to these being regions of minimum pressure. Hence, flow separation and transition to turbulence can have a considerable effect on cavitation. Even without flow separation, intense pressure fluctuations due to turbulence can influence the position of cavitation. In both cases, the temporal pressure fluctuations dictate the cavitation process. Gopalan et al. 关6兴 found that the location and degree of cavitation was dependent on the nature of the vortical structures produced in a submerged water jet. If the jet was deliberately tripped, cavitation occurred in the cores of the vortex rings which formed downstream 共x / D = 2兲 of the nozzle. If the jet was not tripped, cavitation occurred in the cores of comparatively strong streamwise vortex tubes just downstream 共x / D = 0.55兲 of the nozzle. As observed by Arndt 关7兴, tripping the boundary layer of the jet apparently suppresses the secondary vortex cavitation. This mechanism of cavitation control is very similar to the one studied by Billard et al. 关3兴. Sridhar and Katz et al. 关8兴 have also shown that the presence of a few microscopic bubbles at very low void fraction can significantly affect the vortex dynamics within a water jet. Recent PIV 共particle imaging velocimetry兲 measurements in a cavitating turbulent shear layer reported by Iyer and Ceccio 关9兴 did not show a significant effect of cavitation on the vortical structure of the jet but did show an increase in streamwise velocity fluctuations and a decrease in the maximum cross-stream fluctuations and Reynolds stresses due to cavitation. They speculated that the presence of cavitation in the cores of the streamwise vortices decreased the coupling between the streamwise and cross-stream velocity fluctuations. The effects of cavitation on vortex dynamics in a twodimensional submerged planar laminar forced jet were studied
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numerically by Xing and Frankel 关10兴. A locally homogeneous cavitation model that accounts for nonlinear bubble dynamics and bubble/bubble interactions within spherical bubble clusters was employed 关11兴. The effects of varying key flow and cavitation model parameters on flow-cavitation interactions were investigated. The parameters varied included the cavitation number 共effectively the vapor pressure兲, the bubble number density, the bubble-cluster radius, and the Reynolds number 共limited by the two-dimensional assumption兲. The results showed cavitation occurring in the cores of primary vortical structures when the local pressure fell below the vapor pressure. Low levels of void fraction caused significant vortex distortion, with the details depending upon the model parameters. For higher Reynolds numbers and small values of bubble cluster radius, cavitation inhibited vortex pairing and resulted in vortex splitting and intensification. All of these observations were in good qualitative agreement with previous experimental and numerical studies. The vorticity transport equation was used to examine the mechanisms behind the effects of cavitation on vortex structures and it was found that both cavitation-induced dilatation and baroclinic torque terms played a role in vorticity generation. More details can be found in the recent paper by Xing and Frankel 关10兴. Three-dimensional simulations are required to simulate more complex cavitating flows. In the recent review article by Arndt 关7兴, in reference to several of the above studies, as well as others, it is stated that, “There is also mounting evidence that vortex cavitation is a dominant factor in the inception process in a broad range of turbulent flows.” He also states that, “while a vortex model for cavitation in jets does not exist, the mechanism of inception appears to be related to the process of vortex pairing.” Finally, he states that, “a new and important issue is that cavitation is not only induced in vortical structures but is also a mechanism for vorticity 共turbulence兲 generation.” Our previous two-dimensional results 关10兴 seem to bear this out but more work is needed before the ramifications of this become apparent. The objective of the present study is to conduct unsteady threedimensional direct numerical simulations of a round transitional jet and examine the effect of vapor formation due to cavitation on the vortical structure of the jet. Our simulations predict cavitation occurring in the core of the primary azimuthal vortical structures. The presence of low levels of vapor suppressed jet growth and decreased velocity fluctuations in the vicinity of and downstream of the cavitation regions of the jet. The rest of the paper presents the mathematical model, some details of the numerical methods, a problem description, results, and finally some conclusions.
2
Mathematical Model
Cavitating flows feature both incompressible and compressible flow behaviors due to the presence of pure liquid in noncavitating region and mixed liquid and vapor in the cavitating regions. This makes the modeling and numerical methods challenging, especially when a pure compressible formulation is applied. Kubota 关11兴 avoids this problem by using the homogeneous incompressible formulation with variable density. This model was employed in this study for its significant improvement in the following three ways: nonlinear interaction between viscous flow and cavitation bubbles, consideration of the effects of bubble nuclei on cavitation inception and development, and ability to express unsteady characteristics of vortex cavitation. However, it should also be noted that surface tension, thermal, and viscous effects have been neglected in the model and the constant bubble number density per unit volume has restricted this model to be only valid for low void fraction flows. The form of the Navier-Stokes equations, the local homogeneous model 共LHM兲 equation, and the definition of the fluid properties, such as mixture density and viscosity, are identical to those applied in Kubota’s study 关11兴 and were presented more recently in Ref. 关10兴. The key equations are repeated here for completeness. Journal of Fluids Engineering
The following form of the Navier-Stokes equations was considered in this study:
+ ⵜ · 共V兲 = 0, t
共1兲
2 共V兲 + ⵜ · 共VV兲 = − ⵜ p + 2 ⵜ · 共S兲 − ⵜ 共 ⵜ · V兲, 3 t 共2兲 where S is the strain-rate tensor. The density of the liquid-vapor bubble mixture is defined as
= 共1 − f 兲ᐉ ,
共3兲
where ᐉ is the liquid density 共the vapor density is assumed to be negligible compared to the liquid density兲 and f , the local void fraction, is defined as 4 f = n R3 3
共4兲
with 0 ⬍ f ⬍ 1, n is the bubble number density, and R is the bubble radius. The mixture viscosity is evaluated using
= 共1 − f 兲ᐉ + f ,
共5兲
where ᐉ is the liquid viscosity and is the vapor viscosity, and both are assumed constant. The bubble number density is assumed constant in both space and time, which limits the accuracy of the model for large void fractions by neglecting bubble coalescence and splitting. The Rayleigh equation governs the dynamic behavior of a single bubble in a quiescent medium 关5兴. Because grid resolution in most numerical simulations is insufficient to resolve individual bubbles, Kubota et al. modified the Rayleigh equation to account for interactions between bubbles 共of the same radius, R兲, which may occur at scales below the grid scale. The final equation, referred to as the local homogeneous model 共LHM兲 equation is given as: 共1 + 2共⌬r兲2nR兲R + 2共⌬r兲2
冉
3 D 2R + 4共⌬r兲2nR + 2 Dt2
Dn 2 DR p − p , R = Dt Dt ᐉ
冊冉 冊 DR Dt
2
共6兲
where D / Dt is the material derivative, ⌬r is the bubble cluster radius 共distance over which bubbles may interact with each other in a given cluster兲, and p is the vapor pressure of the liquid for a given temperature. Notice as the bubble cluster radius goes to zero, the Rayleigh equation for a single bubble is recovered and bubble-bubble interactions are no longer included in the equation. Recently, Delale et al. 关12兴 have revised Kubota’s model and have addressed two important effects related to bubble/bubble interactions and viscous damping. In Kubota’s original model, the bubble cluster radius is chosen to be the grid size. In Delale et al., they related the bubble cluster radius to the radius of the bubbles within the cluster itself 共which is assumed the same for all bubbles within the cluster, but may grow or decay depending upon the local pressure兲 as follows: ⌬r = ⌳R,
共7兲
where ⌳ = constantⰇ 1 共if ⌳ = 1 then the classical Rayleigh equation is recovered兲. This model assumes that local number of bubbles within a cluster is proportional to the local volume of a bubble, and hence the local void fraction. To couple the LHM equation for the bubble radius to the Navier-Stokes equations, Kubota et al. derived a quasi-Poisson equation for the pressure which is given below 共for more details see Ref. 关11兴兲 JULY 2005, Vol. 127 / 715
Table 1 Description of the flow and cavitation parameters for the simulation cases
Fig. 1 Computational domain showing boundary surfaces and inflow plane
冉
ⵜ2 p + ⌰共p兲 = ⌽ V,V,
冊
R ,R + ⍀共V,V兲 t
共8兲
with p − p ⌰共p兲 = L4nR2 共1 + 2共⌬r兲2nR兲RL and
冉
⌽ V,V,
with
再 冉
冊
冊 冉 冊冎
R R R ,R = − L4nR R⌿ V,V, ,R + 2 t t t
冉
⌿ V,V,
冊
R 2R ,R = 2 − ⌸共p兲 t t
共9兲
Case
p
⌬r
n
Re
R0
1 2 3
1.80 0.60 0.50
0.10 0.70 0.75
1.0 1.0 1.0
106 106 106
1500 1500 1500
1.33⫻ 10−3 1.33⫻ 10−3 1.33⫻ 10−3
avoids the grid-scale 共odd-even兲 oscillations in the pressure field associated with pressure-velocity decoupling often encountered when using non-staggered grids. This is a popular method, especially when combined with the SIMPLE algorithm, because of its strongly elliptic nature which does not allow energy accumulation at the grid-scale wave number preventing grid-scale pressure oscillations. However, there are difficulties in discretizing non-linear terms on non-uniform staggered grids and the amount of programming and computational cost are more than with a non-staggered grid. The approach taken here employs a non-staggered grid and solves the pressure Poisson equation using a compact fourth-order accurate scheme which tends to suppress the spatial odd-even decoupling of the pressure-velocity fields 关13兴. 3.1 Fourth-Order Pressure-Correction Approach. Through the use of the fractional-step method formalism 共first-order accurate in time 关14兴兲, Dormy 关13兴 developed a new pressure correction approach. Dormy derived this new scheme for totally incom-
2
共10兲
共11兲
and ⌸共p兲 =
p − p 共1 + 2共⌬r兲2nR兲RL
共12兲
Maximum and minimum void fractions were specified to avoid pure liquid and vapor states with ⌰ = ⌽ = 0 and the bubble radius fixed. In the present implementation, a hyperbolic tangent function was used to ensure a smooth variation between these two extreme states. All quantities and equations were nondimensionalized by the liquid density and viscosity, the jet nozzle diameter, D, jet inlet velocity, U0, and dynamic pressure, ᐉU20. In the results, all non-dimensional quantities will be indicated by an asterisk, e.g., t* , x* , u*, etc.. The numerical methods employed in this study were modified from our previous work 关10兴 with the main difference here being related to how the quasi-Poisson equation for the pressure is solved in three dimensions. The details are described in the next section.
3
Numerical Methods
A well known difficulty in obtaining time-accurate solutions for an incompressible flow is the lack of a time derivative term in the continuity equation. Therefore, satisfying mass conservation is a key issue in solving the incompressible Navier-Stokes equations and a variety of different approaches have been pursued to address this issue. These approaches include the pressure-correction technique based on the SIMPLE algorithm, as well as approaches which directly solve a Poisson equation for the pressure via iterative methods. Many previous studies have employed a staggered grid where velocity and pressure nodes are not collocated. This 716 / Vol. 127, JULY 2005
Fig. 2 Instantaneous isosurface of Q-criterion magnitude at t* = 22. The isolevel shown is 3.2. „a… = 1.8, „b… = 0.6, „c… = 0.5.
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Fig. 3 Instantaneous isosurface of Q-criterion magnitude at t* = 26. The isolevel shown is 3.2. „a… = 1.8, „b… = 0.6, „c… = 0.5.
pressible flows. Here we attempt to apply Dormy’s methodology for the quasi-Poisson equation featured in the cavitation model of Kubota et al. 关11兴. Additional terms including the density and its time derivative are associated with cavitation effects, such as bubble growth and collapse, on the pressure field. By dropping these terms, the governing equations revert back to the form for a totally incompressible flow. Consider a velocity field 共V兲* which does not satisfy the continuity equation 共for instance, obtained by time advancing the Navier-Stokes equations without including the pressure gradient兲. The objective is to project this onto a divergence-free field by subtracting the gradient of a pressure-like variable such that
V = 共V兲* − ⵜh ,
共13兲
where V is the velocity field which satisfies the continuity equation. Taking the divergence of this equation yields ⌬2h = ⵜh · 共V兲* ,
共14兲
where ⌬2h is a second-order centered approximation of the Laplacian skipping the neighboring points. The sparse nature of this operator leads to pressure oscillations. The remedy proposed by Dormy is to introduce a fourth-order “compact equivalent” to the conservative discrete pressure equation. The original non-compact formulation of the three-dimensional Poisson equation can be written as Journal of Fluids Engineering
Fig. 4 Void fraction and vorticity plots at t* = 22. The isolevel shown is 3.2. „a… Contour plot showing void fraction „flood… and out-of-plane vorticity „ = 0.6…. „b… Q-criterion isosurface colored by the contour of void fraction „ = 0.6…. „c… Contour plot showing void fraction „flood… and out-of-plane vorticity „ = 0.5…. „d… Q-criterion isosurface colored by the contour of void fraction „ = 0.5….
2 h2x 4 2 h2y 4 2 hz2 4 4 4 + 2 + 4 + O共hx 兲 + 2 + 4 + O共h y 兲 + 3 x 3 y x y z2 3 dz4 + O共hz4兲 = ⵜh · 共V兲* +
. t
共15兲
To solve this equation with fourth-order accuracy, Dormy proposed two-step approach that allows fourth-order accuracy at twice the computational cost of the second-order interpolated JULY 2005, Vol. 127 / 717
Fig. 5 Instantaneous minimum pressure within domain versus time. „a… = 1.8, p = 0.10, „b… = 0.6, p = 0.70, „c… = 0.5, p = 0.75
scheme. The above equation can be approximated with four-order accuracy using a two-step method. The first step is a second-order approximation to the second-order truncation terms written as:
冉
⌬h = h2x
4 4 4 2 2 4 + hy 4 + hz x y z4
冊冋
ⵜh · 共V兲* +
册
. t
共16兲
The fourth-order central differences scheme was applied to calculate the right hand side of the above equation as shown here for some function of f f xxxxi =
f i−2 − 4f i−1 + 6f i − 4f i+1 + f i+2 . ⌬x2
共17兲
At the boundaries, a third-order extrapolation was used f xxxxi =
6f i+1 + f i−1 − 4f i+2 + f i+3 − 4f i , ⌬x2
6f i−1 + f i+1 − 4f i−2 + f i−3 − 4f i . f xxxxi = ⌬x2 The second step uses as a correction term 718 / Vol. 127, JULY 2005
共18兲
共19兲
Fig. 6 Axial slices showing instantaneous contour plot of pressure at x* = 2.0 and t* = 22. „a… = 1.8, „b… = 0.6, „c… = 0.5.
冋
⌬h = ⵜh · 共V兲* +
册
1 − . 4 t
共20兲
Hence, the pressure was solved to fourth-order accuracy and the computational cost is exactly twice the cost of the second-order scheme as each of the steps requires the resolution of a sevenpoint compact Laplace operator 关13兴. Point successive overrelaxation was applied together with Dormy’s method to solve the Transactions of the ASME
Fig. 8 Instantaneous velocity vector at stream cross-section x* = 2.0 at t* = 22. The vector was subtracted by the vector of = 1.8. „a… = 0.6, „b… = 0.5.
Fig. 7 Axial slices showing instantaneous contour plot of pressure at x* = 4.0 and t* = 22. „a… = 1.8, „b… = 0.6, „c… = 0.5.
three-dimensional Poisson equation. The convergence criterion selected was max共兩⌬P兩兲 ⬍ 1 . D − 5,
共21兲
where ⌬P is the residual of pressure in the iterations. A stricter criterion was tried and no significant differences between the results were observed. It was also found that if the time step was sufficiently small, the Poisson equation converged quickly at each time step. Journal of Fluids Engineering
3.2 Discretization Schemes. Spatial derivatives appearing on the right hand side of the quasi-Poisson equation were discretized with a fourth-order central difference scheme, whereas all other spatial derivatives were discretized using a second-order central difference scheme. Time discretization was based on a simple Euler explicit forward difference with a conservative CFL number of 0.08 enforced to adequately resolve short time scales associated with small-scale vortical structures and bubble growth and collapse. A second-order accurate scheme based on Heun’s method was also considered to allow a larger time step. A comparison between the Euler and Heun methods showed similar results at the same time. For simplicity the Euler time-stepping method was used for the simulations in this study.
4
Problem Description
Three-dimensional direct numerical simulations of a submerged, round water jet exhausting into a water-filled chamber were conducted. A rectangular box computational domain was chosen to represent the chamber. Top hat velocity profiles were specified at the inlet with nondimensional momentum thickness of 0.3. A small amplitude sinusoidal disturbance was added to this base profile with the Strouhal number corresponding to the primary frequency chosen as 0.223 and the secondary frequency chosen as half this value. The jet Reynolds number was based on the nozzle diameter of the jet. The cavitation number was defined here JULY 2005, Vol. 127 / 719
Fig. 9 Instantaneous velocity vector at stream cross-section x* = 4.0 at t* = 22. The vector was subtracted by the vector of = 1.8. „a… = 0.6, „b… = 0.5.
in terms of dimensional quantities as = 共p⬁ − p兲 / 0.5ᐉU20, where p⬁ is the chamber pressure. The cavitation number was varied by changing the vapor pressure. The Reynolds number was defined as ᐉhU0 / ᐉ and serves to determine the liquid viscosity. The ratio of the vapor to liquid viscosity was 0.00912 following Kubota et al. A sketch of the computational domain is given in Fig. 1. The computational domain extended from −2 艋 y * 艋 2 in the spanwise direction, −2 艋 z* 艋 2 in the transverse direction, and 0 艋 x* 艋 8.9 in the streamwise direction. The domain was discretized using a uniform Cartesian mesh with 120⫻ 100⫻ 100 points. The use of a uniform mesh avoids problems related to transforming the quasiPoisson equation, involving computer memory and numerical accuracy, when using a non-uniform mesh. No-slip velocity boundary conditions were enforced on the sides of the box, and the inflow plane except for the nozzle opening. Fourth-order extrapolation was used for the pressure on the walls and outflow boundary pimax,j = 共104pimax−1,j − 114pimax−2,j + 56pimax−3,j − 11pimax−4,j兲/35. 共22兲 The convective boundary condition was used for velocity at the outflow boundary 720 / Vol. 127, JULY 2005
Fig. 10 Axial slices showing instantaneous contour plot of streamwise vorticity at x* = 2.0 and t* = 22. „a… = 1.8, „b… = 0.6, „c… = 0.5.
V V + u0 = 0. t x
共23兲
For all cases the Reynolds number based on the jet diameter, inlet velocity, and liquid viscosity was fixed at 1500. The effects of the bubble number density 共fixed at 106兲 and bubble-cluster radius 共fixed at 1.0兲 have already been studied in the twodimensional submerged laminar jet simulations 关10兴 and are not studied further here. The only parameter related to the cavitation Transactions of the ASME
Fig. 12 Isosurface showing magnitude of vortex stretching term at t* = 22. The isolevel shown is 3.7. „a… = 1.8, „b… = 0.6, „c… = 0.5.
grid independence through comparisons of flow statistics obtained on both the current mesh and with approximately twice the number of grid points and differences of less than 2% were observed. Hence, the simulations presented next can be considered grid independent.
5
Results and Discussion
The effects of cavitation on the instantaneous jet vortical structure can be observed in Figs. 2 and 3, where pairs of isosurfaces of the magnitude of Q, defined as 关15兴 1 Q = 共⍀ij⍀ij − SijSij兲, 2 Fig. 11 Axial slices showing instantaneous contour plot of streamwise vorticity at x* = 4.0 and t* = 22. „a… = 1.8, „b… = 0.6, „c… = 0.5.
model varied here is the cavitation number 共or equivalently the vapor pressure, p兲. Three values of the cavitation number will be used to consider jets under non-cavitating 共 = 1.8, p = 0.1兲 and cavitating conditions 共 = 0.6, p = 0.7 and = 0.5, p = 0.75兲. All cases are summarized in Table 1 and were tested on IBM RISC/ 6000 SP POWER3+ thin nodes with 375 MHz CPU and 4 GB memory. The results to be presented in the next section were checked for Journal of Fluids Engineering
共24兲
where ⍀ij = 共ui,j − u j,i兲 / 2 and Sij = 共ui,j + u j,i兲 / 2 are the antisymmetric and symmetric components of ⵜu, are plotted for all three cases. General features of the round jet transition process, from nominally laminar conditions near the nozzle exit, through the vortex roll-up and pairing process, formation of streamwise vortex tubes, and eventual breakdown into small-scale structures, are shared by all cases. Careful examination of the plots reveals distortion of the primary vortex ring structures located at approximately x* = 2 and x* = 4, with a weakening of the streamwise vortex tube structures downstream of these locations for the = 0.6 and = 0.5 cases as compared to the = 1.8 case. These modifications to the jet vortex structure can be attributed to vapor formation in these regions due JULY 2005, Vol. 127 / 721
Fig. 14 Momentum thickness versus axial distance
Fig. 13 Isosurface showing magnitude of the dilatation term in „a… and „b… and the baroclinic torque term in „c… and „d… at t* = 22. The isolevel shown is 1.5 for the dilatation term and 0.005 for the baroclinic torque term. „a… = 0.6, „b… = 0.5, „c… = 0.6, „d… = 0.5.
to cavitation inception, as evidenced by superimposing instantaneous void fraction and vorticity as shown in Fig. 4. The formation of vapor in the cores of the primary vortex rings is clearly visible in Figs. 4共a兲 and 4共b兲. Evidence of vapor formation downstream of the primary vortex rings, in the vicinity of streamwise vortex tubes, can be seen in Fig. 4共c兲, but not in Fig. 4共d兲 at the simulation time shown. Further evidence of cavitation inception and the unsteady bubble dynamics that result can be seen in Fig. 5, which plots a time history of the minimum pressure within the computational domain. This figure clearly shows that the local fluid pressure drops below the specified vapor pressure for the two cavitating cases, p = 0.7 and 0.75, at several times during the time span shown, including the times corresponding to the instantaneous results shown previously. The high-frequency fluctuations for the minimum pressure are only observed in the two cavitating cases— 722 / Vol. 127, JULY 2005
not in the noncavitating case. This is caused by the alternative growth and decay of bubbles inside the cavitating regions within very short periods and is similar to observations in our previous two-dimensional cavitating jet study 关10兴. Also, the periodic shedding of vortex rings caused the jet to behave in an alternating cavitating and noncavitating mode. The increase of cavitation number delays the inception of cavitation and also reduces the duration time for the jet to be cavitating. In order to quantify the effect of cavitation on vortex dynamics, instantaneous contour plots of pressure at x* = 2 共Fig. 6兲 and x* = 4 共Fig. 7兲 are shown at t* = 22. For the noncavitating case, the pressure field is nearly uniform in the core of the vortex ring. In contrast, there are several regions where the pressure has dropped below the vapor pressure for the two cavitating cases. The formation of vapor tends to break the vortex ring into several sections in the azimuthal direction. Evidence for local flow acceleration in the cavitating cases can be seen by comparing instantaneous velocity vector plots at t* = 22 in Figs. 8 and 9. This local flow acceleration can be explained by the continuity equation 关Eq. 共1兲兴. When vapor forms within the center of a vortex, the local density decreases with respect to time. As a result, the spatial derivatives of velocity become positive based on conservation of mass law. For the same cavitation number, further downstream, greater acceleration can be observed. For the same axial location, the smaller the cavitation number the greater the flow acceleration. This is because a smaller cavitation number corresponds to greater vapor pressure, thus larger zones within the vortex will cavitate and the derivative of fluid density with respect to time inside those zones becomes larger. Local distortion and intensification of vortical structures due to vaporous cavitation can be seen in instantaneous contour plots of streamwise vorticity in Figs. 10 and 11. The vorticity of a fluid particle can be altered by vortex stretching, dilatation, baroclinic torque, and viscous diffusion, according to the vorticity transport equation in a three-dimensional, variable density flow, ⵜ ⫻ ⵜ p 1 2 D 共ⵜ 兲. 共25兲 + = 共 · ⵜ 兲V − 共ⵜ · V兲 + Re Dt 2 The magnitude of the vortex stretching term has increased by close to a factor of 4 due to cavitation as seen in Fig. 12. The dilatation and baroclinic torque terms are identically zero in the non-cavitating jet, but are nonzero in the vapor-containing regions of the cavitating jet as seen in Fig. 13. The baroclinic torque term is over two orders of magnitude smaller than the dilatation term, consistent with our previous two-dimensional cavitating jet simulations. However, the maximum values of dilatation and baroclinic torque terms in the current simulation are two orders of magnitude larger than in the two-dimensional cavitating jet simulations. The increase of baroclinic torque terms can be explained by considerTransactions of the ASME
Fig. 16 Profiles of Reynolds stress component / Uc2 at axial stations for all cases. Solid line is = 1.8, long dashed line is = 0.6 and short dashed line is = 0.5. „a… x* = 2.0, „b… x* = 4.0, „c… x* = 6.0.
Fig. 15 Profiles of Reynolds stress component / Uc2 at axial stations for all cases. Solid line is = 1.8, long dashed line is = 0.6 and short dashed line is = 0.5. „a… x* = 2.0, „b… x* = 4.0, „c… x* = 6.0.
ing the current void fraction is ⬇25 times larger than in the twodimensional simulations. This will increase the local pressure and density gradient. The increase of the dilatation term suggests local vorticity has been intensified by bubbles in the cavitating regions. Instantaneous data from eight flowthrough times 共defined as x* / 具U0典兲 were averaged to obtain mean and rms velocity and Reynolds stress profiles. The axial variation of the jet momentum thickness, shown in Fig. 14, indicates that cavitation suppressed Journal of Fluids Engineering
jet growth. Radial profiles of streamwise, cross-stream, and shear components of the Reynolds stress tensor at axial stations corresponding to the main cavitation regions and just downstream of these regions are shown in Figs. 15–17, respectively. These plots suggest that cavitation suppressed velocity fluctuations downstream of the cavitation regions, consistent with the suppressed jet growth and weaker streamwise vortices noted earlier. The suppression of cross-stream and shear Reynolds stress components due to cavitation is consistent with the recent PIV 共particle imaging velocimetry兲 measurements of Iyer and Ceccio 关9兴 for a cavitating shear layer and it is greatest for the case with the lowest cavitation number. They observed an increase in streamwise velocity fluctuations near the outer edge of the shear layer, and a decrease near the shear layer center. The location of maximum JULY 2005, Vol. 127 / 723
observed. Analysis of various terms in the vorticity transport equation reveals a decrease in the magnitude of the vortex stretching term and the presence of non-zero dilatation and baroclinic torque terms in the main cavitation regions. Statistical analysis of the flow field demonstrates that cavitation tends to suppress jet growth and reduce Reynolds stresses. These effects are in qualitative agreement with previous experimental studies of cavitating shear flows and bubble-flow interactions. Future efforts should focus on extending the cavitation model to apply to larger void fraction flows by relaxing the constant bubble number density assumption, accounting for bubble size distributions, and considering thermal nonequilibrium effects. Finally, inclusion of a subgridscale turbulence model to facilitate large eddy simulations of cavitating flows would allow higher Reynolds number and more complex geometries, and hence scaling effects, to be considered.
Acknowledgments The authors would like to gratefully acknowledge Caterpillar Inc., Peoria, IL, and the Indiana 21st Century Research and Technology Fund for support of this research.
Nomenclature D f n p p⬁ R Re S t U0 V Uc u0 (x, y, z) ⌬r ⵜ ⌳ Subscripts Fig. 17 Profiles of Reynolds stress component / Uc2 at axial stations for all cases. Solid line is = 1.8, long dashed line is = 0.6 and short dashed line is = 0.5. „a… x* = 2.0, „b… x* = 4.0, „c… x* = 6.0.
velocity fluctuations is not at the center of the shear layer, also consistent with Iyer and Ceccio 关9兴. Further downstream the influence of cavitation on the Reynolds stresses becomes more significant, especially for the case with the lowest cavitation number.
6
Conclusions and Future Work
Numerical simulations of low Reynolds number, transitional submerged round jets reveal that cavitation occurs within the cores of the primary vortex rings formed just downstream of the nozzle exit when the local fluid pressure drops below the vapor pressure. Cavitation tends to distort and breakup the vortex rings and also weakens the secondary streamwise vortex tubes. Evidence for local flow acceleration and intensification of spanwise vorticity at the expense of streamwise vorticity due to cavitation is 724 / Vol. 127, JULY 2005
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
nozzle diameter void fraction bubble number density pressure chamber pressure bubble radius Reynolds number strain rate tensor time maximum velocity at inlet velocity vector centerline velocity convective velocity at outlet Cartesian coordinate system molecular viscosity vorticity density bubble cluster radius cavitation number gradient operator constant
ᐉ ⫽ liquid ⫽ vapor
Superscripts *
⫽ on-dimensional quantity
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