ILASS-Americas 23rd Annual Conference on Liquid Atomization and Spray Systems, Ventura, CA, May 2011

Cavitation as Rapid Flash Boiling

Bradley Shields, Kshitij Neroorkar, and David P. Schmidt ∗

Department of Mechanical and Industrial Engineering

University of Massachusetts Amherst, Amherst, MA, 01003

Abstract

Diesel injector nozzles often experience cavitation due to regions of extremely low pressure. There arecomputational models that deal only with high temperature flash-boiling flow [1, 2, 3], as well as those thatfocus on the lower-temperature process of cavitation[4, 5]. The ideal code would have the ability to representboth high-temperature flash-boiling flows and lower temperature cavitating flows. The current work uses thehypothesis that cavitation can be modeled as flash-boiling with rapid heat transfer between the liquid andvapor phases. The following paper examines a multi-dimensional computational fluid dynamics approachbased on using an established flash-boiling model [6] to simulate cavitation in a fluid near room temperature.Coefficient of discharge is plotted against cavitation number, and the results are compared to the resultsof published cavitation code as well as experimental data. The flash-boiling model shows good agreementwith accepted values for discharge coefficient. The flash-boiling model is proposed as a tool to simulate bothflash-boiling and cavitation, and its accuracy is examined in non-cavitating cases.

NomenclatureA2 Nozzle outlet areaCc Contraction coefficientCd Coefficient of dischargeCp Specific heathfg Latent heat of vaporizationJa Jakob numberK Non-dimensional pressure ratio and cavitation parameterL/D Length/diameter ratiom Mass flow ratePc critical pressurePv Vapor pressureP1 Upstream pressureP2 Downstream pressurePsat Saturation pressureRe Reynolds numberri/D Inlet rounding∆T SuperheatU Flow velocityx Instantaneous qualityx Equilibrium qualityα Void fraction of vaporθ Quality relaxation time scaleρ Densityρ1 Inlet Densityρl Liquid densityρv Vapor densityτ Shear stress tensorφ Mass fluxψ Non-dimensional pressure ratio

∗Corresponding Author: schmidt@ecs.umass.edu

Introduction

The fuel injector is an extremely important com-ponent of the diesel engine, making the study of in-ternal flow effects essential to lowering emissions andto improving the understanding of the atomizationthat occurs with a given injector. Most importantly,the spray community needs to know how the injectornozzle impacts the downstream spray. The injectorcharacteristics represent the most significant param-eters for adjusting spray behavior.

Cavitation is the process by which liquid is con-verted to vapor by the low pressures within the noz-zle. With a sharp-edged orifice, as fluid rushes intothe nozzle, the flow often separates and contractswithin an annulus of vapor, known as the vena con-tracta [7]. By conservation of momentum, as theflow contracts and speeds up to enter the nozzle,pressure falls. Should this pressure drop below thevapor pressure of the fluid, liquid will convert to va-por in the flow. This vapor can be manifested asindividual bubbles in the flow or a foamy mixture ofgas and liquid.

Early modeling included the zero-dimensionalmodel developed by Nurick [8] in 1976. Nurick devel-oped a relation for a nozzle’s coefficient of discharge,

CD = CC

(

P1 − PV

P1 − P2

)1/2

(1)

where P1 is the upstream pressure, P2 is thedownstream or back pressure, and Pv is the liquid’svapor pressure. This expression relates the pressureratio and CC to the nozzle output. The variable CC

represents the fraction of the nozzle cross-sectionalarea that liquid passes through, the fraction of thenozzle that is not taken up by vapor.

The current study addresses slightly roundednozzles, with the assumption that the area occupiedby the vena contracta is constant in a given nozzlegeometry. By varying the pressure ratio and nozzlegeometry, Nurick observed hydraulic flip and cavi-tation. Nurick’s results showed that by varying thepressure ratio, thereby including or excluding cavi-tation, the discharge was directly affected. Schmidtand Corradini [7] compiled the work of several ex-perimentalists, as shown in Figure 1.

The experimental data validated the accuracy ofthe model. However, some of the experimental datashow a coefficient of discharge that increases linearlyfrom K=1 to K=2 but then falls as K increases. Thepressure ratio K is a form of cavitation parameterwhich appears in Eqn. 1, and is defined as:

K =

(

P1 − PV

P1 − P2

)

(2)

Figure 1. Nurick Theory vs. Experiments. Dataare plotted on log-log axes from [8, 9, 10, 11, 12, 13,14, 15]

As the value of K exceeds a threshold of about1.7 to 1.9, the cavitating flow transitions to non-cavitating flow, and Schmidt notes that the variabil-ity of the data could indicate other effects, such asReynolds number dependency. These other effectsmay include the scale of Hiroyasu’s [10] nozzles (in-dicating scale-dependant factors) or manufacturingimperfections.

Som [16] breaks modern cavitation modelinginto two major groups: single fluid/continuum mod-els, and two fluid models. Single fluid/continuummodels utilize the vapor volume fraction to aver-age vapor and liquid phase properties of a fluid.Schmidt et al [17] is an example of the pseudo-fluidapproach, assuming liquid and vapor to be in a ther-mal equilibrium, evenly mixed in each cell, and withno-slip conditions between the phases. The sepa-rate phases and mixture were treated as compress-ible. The model’s two-phase sound speed was ap-proximated using Wallis’ HEM closure [18].

Two-phase approaches are those that handleeach phase with its own set of conservation equa-tions. Som [16] breaks these down further intotwo categories, Eulerian-Eulerian approaches andEulerian-Lagrangian approaches. Eulerian-Eulerianmodels, such as that proposed by Singhal et al [4],are similar to single fluid models in that they havefluid density as a function of vapor mass fraction.The vapor mass fraction is found through a trans-port equation that includes mass and momentumconservation equations. Source terms define va-por generation/condensation rates, and stem fromflow parameters and fluid properties. A generalizedRayleigh-Plesset equation is used to derive the bub-

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ble dynamics equation.Eulerian-Lagrangian models instead view the

liquid flow in a Eulerian sense, but follow the bub-bles of the vapor phase as Lagrangian particles, asin Gavaises and Arcoumanis’ [19]. The bubbles arehandled with the nonlinear Rayleigh-Plesset equa-tion and offer predictions of bubble dynamics. Usu-ally assumptions or empirical estimates of bubblenumber density are required.

The vaporization process that occurs throughcavitation is very similar to that of flash-boiling,with a few important caveats. Where cavitation rep-resents the vapor formed through a constant tem-perature system experiencing a drop in pressure,flash boiling represents the same system with a lowerpressure drop but elevated temperatures. In flashboiling, the availability of energy required for phasechange is limited by the speed of interphase heattransfer while cavitation is often inertially domi-nated [17, 20]. This can be shown by the Jakobnumber,

Ja =ρlCp∆T

ρvhfg(3)

where ρ represents density, and Cp representsspecific heat at constant pressure, ∆T is the amountof superheat, and hfg is the latent heat of vaporiza-tion. As Ja 6 1, more energy per unit volume is re-quired for vaporization than is available in the formof sensible heat. As temperature increases, so doesthe ρvhfg term. Therefore, at elevated temperaturesthe energy required for vaporization increases, in-creasing the time required for heat transfer betweenphases, even approaching flow transit time throughthe nozzle. In contrast, in cavitating flows, the timerequired for vaporization is very small, ensuring thatvaporization is essentially instantaneous.

Another important consideration is that phasechange is a continuous process in flash-boiling noz-zles. Every fluid molecule will experience a localpressure that is less than the vapor pressure priorto exiting the nozzle. In cavitating flow, it is possi-ble for an annular vapor region to form, after whichliquid need not further change phase. Consequentlyneglecting the temporal nature of the heat transferprocess is erroneous when Pv > P2, and requiresdeliberate consideration. Schmidt [21] investigatedthe accuracy of using a cavitation model to simu-late Reitz’s [22] flash-boiling experiment using twosets of assumptions, thermal equilibrium and ther-mal non-equilibrium. As temperature increased, theequilibrium model’s results became erroneous, whilethe non-equilibrium model remained accurate.

The present work attempts to confirm the ac-

curacy of the flash boiling code’s ability to quan-tify cavitating flows. The flash boiling CFD codedescribed by Schmidt et al. [6] is employed undercavitating conditions and evaluated for its accuracyin predicting coefficient of discharge. This cavitat-ing regime is outside of the applicability for whichthe model was designed. To test the accuracy ofthe non-equilibrium model, simulation results werecompared to accepted cavitating and non-cavitatingcoefficient of discharge data.

Methodology

A parametric study was conducted, using a 2Daxisymmetric nozzle 3 mm in diameter with an L/Dratio of 4. The inlet rounding of the nozzle, ri/D,was 1/40. The nozzle was examined at upstreampressures ranging from 6 to 200 MPa, with a con-stant backpressure of 5 MPa. The working fluid waswater at an average temperature of 18°C, to ensurea low enough temperature for instantaneous heattransfer between the fluid’s liquid and gas phases.The properties of the working fluid were providedby the REFPROP database and code library. REF-PROP uses the Wagner and Pruss equation of statefor water. The HRMFoam model was used to pre-dict mass flow rate, and thus discharge coefficient,for the axisymmetric flow.

Derived from Bernoulli’s Equation evaluated atthe nozzle inlet and outlet, the ideal nozzle massflowrate is given by

mideal = A2

√

2ρ1(P1 − P2) (4)

The symbol A2 is the nozzle outlet area, ρ1 isthe inlet density, and P1 and P2 are upstream anddownstream pressure, respectively. The ratio of thenumerically computed flowrate to ideal output flowis the coefficient of discharge.

The CFD code, HRMFoam, was created using apseudo-fluid paradigm. The full description is givenin Schmidt et al. [6] and is only summarized here.The governing equations considered in this case in-clude the continuity and momentum equations de-noted by Eqns. 5 and 6 respectively.

∂ρ

∂t+ ▽·φ = 0 (5)

∂ρU

∂t+ ▽· (φU) = −∇p+ ∇· τ (6)

The term φ represents the mass flux and is given as

φ = ρU (7)

The variable τ represents the shear stress tensor.At present, no turbulence model has been incorpo-rated and laminar flow is assumed. The reason for

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this assumption is that our main focus here is un-derstanding the effect of flash boiling/cavitation onthe flow. The study of those phenomena coupledwith turbulence is left for future work. The pressureequation is

ρ∇ · (H(U)

ap)−ρ∇ ·

1

ap∇p+

∂ρ

∂x|p,h

Dx

Dt= 0 (8)

where the subscript p represents the computationalcell under consideration. The variable a representsthe contributions from the specific cells. The opera-tor H is defined as

H(U) = r −∑

N

aNUN (9)

where r is the contribution from the source termsto the linear system matrix, and N represents theneighboring cells. H is a convenient replacement forthe off-diagonal and source term contributions. Theflash boiling model is used to calculate the last termof Eqn. 8.

The homogeneous relaxation model was used toprovide closure to the above mentioned system ofequations. This model describes the rate at whichthe instantaneous quality, the mass fraction of vaporin a two-phase mixture, will tend towards its equi-librium value. The simple linearized form proposedby Bilicki and Kestin [23] for this rate equation isshown in Eqn 10

Dx

Dt=x− x

θ(10)

In the above equation, x represents the instanta-neous quality, x represents the equilibrium qualityand θ represents the time scale over which x relaxesto x. Eqn. 10 is an approximation to the extremelycomplicated processes that are associated with theflash boiling process. It can be noted that the HRMequation is inserted into the last term of the pres-sure equation formulation, Eqn. 8. The value ofx is obtained from a look-up table as a function ofpressure and enthalpy. The instantaneous quality iscalculated from the void fraction as shown below

x =αρv

ρ(11)

where α is the void fraction of vapor, and ρv rep-resents saturated vapor density. The void fraction iscalculated as follows

α =ρl − ρ

ρl − ρv(12)

Figure 2. Computational Grid

The most important consideration while usingthe HRM model is the formulation of the time scale.Downar-Zapolski [1] used the pressure profile andmass flux from the Moby Dick experiments of Re-ocreux [24] and combined their governing equationsto derive an equation for the time scale. They foundthat in all cases, the time scale was a monotonicallydecreasing function of the void fraction and a nondimensional pressure. Based on their observations,they proposed the following equation for the timescale θ

θ = θ0α−0.54ψ−1.76 (13)

The value of the coefficient is θ0 = 3.84 ·10−7 [s]and

ψ =Psat − P

Pc − Psat(14)

where Pc is the critical pressure. This empiricalequation (Eqn. 13) is being used beyond the rangeand fluids for which it was formulated. Though thereis no guarantee that the model will be accurate un-der such conditions, previous studies have producedvery encouraging results [17, 25, 26, 27, 6]. Addi-tionally, the same time scale correlation was usedfor all the fluids considered.

The flow was represented by a coarse mesh of12000 cells, with greater cell density near the inletcorner and along the wall above the inlet (Figure2). The mesh is made up of mainly quadrilateralprisms. The cases were re-run with a 500% finermesh as well, yielding a mean change of 4.51% incoefficient of discharge. The coarse mesh requiresfurther refinement to be verified as convergent.

Results

The findings of this parametric study were com-pared to the results of several experimentalists aswell as Singhal’s Full Cavitation Model (Figure 3).

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Figure 3. HRMFoam vs. experimental and computational results. Data plotted on log-log axes from[14, 8, 15, 4]

The accepted onset of cavitation is K = 1.7 [7];larger values of K indicate non-cavitating flow, whilesmaller values indicate cavitating flow. HRMFoammodels the flow accurately, with coefficients of dis-charge differing from the Nurick trend by an averageof 0.022, and good agreement with the Full Cavi-tation model. Part of this difference from Nurick’stheory is the challenge of discretizing the region nearthe sharp inlet corner. Whereas a more roundedcorner allows for velocity and pressure to transitionsmoothly (if rapidly) to their downstream values, asharp inlet does not. It represents a discontinuity

where velocity changes instantaneously.In non-cavitating cases, the flow clearly exhibits

non-trivial transient vapor formation due to periodicvortex shedding (Figure 4). As this region repre-sents essentially single phase flow, HRMFoam shouldnot indicate significant generation of vapor. As timegoes on, vortices form just inside the nozzle inlet(Figure 4, inset), growing in number as time con-tinues. Periodically, one of these vortices is pinchedoff by the flow and separates, moving down the noz-zle. The shedded vortex is carried out through thenozzle exit. The velocity changes the vortices intro-

Figure 4. Vortex streamlines highlighted against nozzle flow [m/s] Inset: vena contracta close-up, imme-diately inside nozzle inlet

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Figure 5. Nozzle side view of nozzle throat, center-line to outer radius. Top: Void fraction Bottom:

Pressure [Pa]

duce to the flow and the vapor they generate arepossible sources of error. This phenomenon has alsobeen reported by Canino and Heister [28]. Caninoand Heister noted a reduction in CD as the radius ofa nozzle’s inlet corner approached perfect sharpness.

Figures 4, 5, and 6 show the (non-cavitating) K= 2.4 case. Figure 5 is an image of the nozzle mir-rored over the horizontal axis, showing alpha andpressure on the top and bottom, respectively. Thelow pressure regions are vortex centers, which dropwell below the vapor pressure of the working fluid.The regions of low pressure correspond to the re-gions of vapor formation, at the “eye” of each vor-tex. Figure 6 shows the high velocity gradient andmagnitude at the entrance to the nozzle. The redregion at the inlet corner is a product of the largegradient there.

Figure 6. Flow velocity at inlet [m/s]

Conclusion

A parametric study was conducted, testing theability of the Homogenous Relaxation Model to ac-curately depict cavitating flow conditions. It re-mains to be seen if vortex shedding and the errorit causes is a physical reality or error in numericalapproximation. However, HRMFoam shows reason-able accuracy when dealing with either cavitatingor flash boiling flows. The results presented here areslightly mesh dependent, and further mesh conver-gence study is left to future work.

Acknowledgments

We thank General Motors Research Center forsupporting this research.

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