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Steady-state cavitation modeling in an open sourceframework: Theory and applied casesLucian Hanimann1*, Luca Mangani1, Ernesto Casartelli1, Matthias Widmer1
ON ROTATING MACH
TransportPhenomena andDynamics of
April 10-15, 2016
AbstractThis paper deals with the steady state cavitation modeling and associated numerical challenges. Althoughcavitation is most often an unsteady phenomenon [1, 2] it is necessary to have a robust and reliablesoftware able to represent the steady state (time averaged) flow patterns. Especially in the early designstate it is mandatory to be able to predict the occurrence of cavitation without time consuming transientcomputations.The authors will therefore give a short summary of the theory behind different cavitation models andtheir implementation into an in-house modified version of OpenFOAM R.The main focus of the paper will then point out the integration of mixture type cavitation modelsinto a pressure correction based steady-state solver. Different strategies have been tested and a stableformulation was found.
KeywordsCavitation Steady-State CFD
1Lucerne University of Applied Science and Arts, Lucerne, Switzerland*Corresponding author: firstname.lastname@example.org
Ap Matrix of Momentum CoefficientsCC Commercial Coded Characteristic Length [m]f Mass Fractionk Turbulent Kinetic Energy [m2s2]kl , kp , kv Model Coefficients for Merkle Modeln Vapor bubble density [m3]p Pressure [Pa]p Free Stream Pressure [Pa]R or S Mass Exchange Rate [kgm3s1]R Bubble Growth Radius [ms1]Rb Bubble Radius [m]re Equivalent Radius [m]t Time [s]t Free Stream Time Scale [s]u Velocity [ms1]U Free Stream Velocity [ms1]Vcell Cell Volume [m3] Gradient Operator Divergence Operator
c Condensatione Evaporation
i Phase Index (l or v)l Liquidsat Saturationt Turbulentv Vapor
Correction Previous Solution
Volume Fraction Partial Derivative Turbulent Dissipation [m2s3] or D Effective Exchange Rate [kgm1s1] Dynamic Viscosity [Pas1] Density [kgm3] Cavitation Number
Steady-state cavitation modeling in an open source framework: Theory and applied cases 2/10
In regions of strong acceleration the pressure has to drop toaccount for the increase in kinetic energy. If this pressure dropundergoes the saturation pressure, the phase changes. Suchliquid-to-vapor phase-changes are usually related to temper-ature changes and therefore associated with evaporation. Inthis special case it is based on a pressure change and calledcavitation.The existence of cavitation is mostly undesired since it leadsto vibrational fatigue, induced noise and increased erosion ofthe considered body [3, 4]. On the other hand it also rises aninteresting field for improvements since it can drastically re-duce the drag of a body [5, 6]. Several numerical studies havebeen conducted through the past years but still, the predictionof cavitation is a challenging task. Most often the inherentunsteady nature of the phenomena is tackled by time consum-ing transient simulation, although it may have a representativesteady-state solution.For industrial applications it is therefore necessary to enablesteady-state prediction of the cavitation phenomena. Thisrises several challenges to the numerical procedure. Cavi-tation models are empirical or semi-analytical correlations,often based on the Rayleigh-Plesset model for bubble growth. All these models have their constraints on the predictionof cavitation and investigation is needed on the global perfor-mance of these models.Most of the available publications focus on the physics of thecavitation solver but less on the numerical challenges. Fortransient simulations, which is the general case for cavitationprediction, this may not be of great importance, since with suf-ficiently small time step, the numerical solution may convergeanyway. However, this will lead to high computational costswhich makes the procedure useless for everyday engineeringapplications. This could be circumvented by using steadystate computations. To make this step, a deeper understandingand investigation of the numerics is needed.Literature survey has shown that, although often used, littleis described on the implementation of the published models.This is especially the case when the described model has to beported from the theoretical test cases onto real turbomachineryapplications. A stable, second order accurate implementationfor 3D, viscous flows has to be found, applicable to systemswith multiple frames of reference.The present paper will therefore first recapitulate the verybasic equations based on a consistent nomenclature and dis-cuss some of the most often used models and their variations.Based on the experience gained during the literature surveyand the validation, a toolbox will be presented. It should pro-vide the reader with an overview about different models andmodifications as well as enable to implement a reliable androbust numerical cavitation solver.First the very basic model, the homogeneous equilibriummodel, is introduced. In a next step the mass exchange be-tween the phases has to be addressed. Here, the phase fractionequation is introduced and the difficulties encountered duringliterature review commented and clarified.
In a next step, the SIMPLE-based pressure correction equa-tion in its non-zero velocity divergence form is introduced.Having now the complete set of Navier-Stokes equations andthe additional fraction equation, the models for the transferrate are introduced.We will then continue with some of the best known modelsand discuss some of the modifications, suggested by otherauthors during the years.The validation will be carried out by two well known testcases. For the analysis of stability and performance a firstsimple test case was chosen, the blunt-body test case of Rouseand McKnown. To prove the applicability and stability of theimplemented procedure a marine propeller, distributed at theSecond International Symposium on Marine Propulsors 2011(smp11) , was chosen.
1. METHODSIn this section we will introduce step-by-step the final cavita-tion solver, starting with the homogeneous equilibrium model,introducing the phase fraction and modified Navier-Stokesequations and finally discussing the phase exchange modelsand their modifications.
1.1 Homogeneous equilibrium modelThe homogeneous equilibrium model states that the velocity,temperature and pressure between the two phases are equal.This assumption is based on the belief that momentum, energyand mass transfer are fast enough to reach equilibrium.Mixture density (Eq. 1) and viscosities (Eq. 2) are thereforeintroduced in the Navier-Stokes equations (Momentum andContinuity).
= v v + l l (1) = v v + l l (2)
Where v and l are the vapor and liquid volume fractionrespectively.
1.2 Phase Fraction EquationSince only one set of NS-equations is solved, the contribu-tion of the phase change is accounted by empirical or semi-analytical equations and an additional transport equation forthe phase fraction. The general transport equation can eitherbe stated in form of mass (Eq. 3) or volume fraction (Eq. 5).
+ ( u f i ) = ( f i ) + Ri (3)
With the definition of the mass fraction as in Eq. 4, thetransport equation can be rewritten in terms of volume fraction(Eq. 5).
f i =ii (4)
+ (iui ) = (ii ) + Ri (5)
Steady-state cavitation modeling in an open source framework: Theory and applied cases 3/10
Where Ri is the mass exchange rate and generally ex-pressed as Rv = Re Rc if stated for the vapor fraction.Therefore Rv = Rl . is usually known as the effective exchange rate coefficient.Although often presented in literature [9, 10, 11] it was dif-ficult to find a description of this contribution and was onlyfound in one single paper so far . According to these au-thors it is the diffusive mass flux of vapor penetrating acrossthe cluster of an equivalent radius re into the surroundingfluid. This is a well known approach to increase the spread-ing rate of a transported quantity.The equation in  was given as shown in Eq. 6.
+ (vuv ) D = S (6)
With the effective exchange coefficient:
D = Rre (7)
The equivalent radius is defined as:
re describes the radius inside which also surrounding cellsshould be influenced by the phase change mass exchange.R is the bubble growth radius and can be derived from theRayleigh-Plesset equation for bubble dynamics. The resultingequation, Eq. 9, is a first order approximation after drop-ping the non-linear acceleration term, the surface tension andviscous contribution from the original formulation.
R = sign(psat p)
Compared to the original formulation stated by Singhal et al. in (Eq. 3) this would imply that Eq. 10 holds.
Dv = v
Dv = vv
Dv = v(v 1 + 1v
It is clear that there must be a difference between the imple-mentations described in literature and it is not clear whichimplementation is correct or represents the reality in a betterway. Since most authors did not account for this effective massexchange coefficient it was decided to neglect this diffusion-like contribution in the implemented model.The final form of the phase fraction equation for steady statecomputations was implemented for the liquid phase given inEq. 12:
ull =Rc Rel
With representing the conserved volume flux through thecell faces including Rhie-Chow correction.It is important to