Christian Pellone

Jean-Pierre Franc

Mickaël Perrin

Comptes Rendus Mécanique (CRM)

ple.on of the–Plessetn infinityicable toce dedemparable àestlution deC. R. Mecanique 332 (2004) 827–833Modelling of unsteady 2D cavity flows using the Logvinovichindependence principleChristian Pellone, Jean-Pierre Franc∗, Mickaël PerrinLaboratoire des écoulements géophysiques et industriels (INPG–CNRS, UMR 5519–UJF), BP 53, 38041 Grenoble cedex 9, FranceReceived 11 May 2004; accepted 11 June 2004Available online 18 August 2004Presented by René MoreauAbstractA simple model for two-dimensional cavity flows is presented. It is based upon the Logvinovich independence princiEach section of the cavity is assumed to behave independently of the neighbouring ones. The equation of evoluticavity interface is derived. It mainly takes into account an added mass effect and is similar to the well-known Rayleighequation relative to spherical bubbles. The dynamics of the 2D cavity is controlled by the pressure difference betweeand the cavity. The model proves to be in good agreement with Tulin’s solution for a steady cavity flow and easily applunsteady cavity flows.To cite this article: C. Pellone et al., C. R. Mecanique 332 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméModélisation des écoulements cavitants bidimensionnels basée sur le principe d’indépendance de Logvinovich. Unemodélisation simple des écoulements cavitants bidimensionnelsest proposée. Elle est basée sur le principe d’indépendanLogvinovich qui suppose que chaque section de cavité se comporte indépendamment des voisines. L’équation d’évolutionl’interface est présentée dans cette Note. Elle prend essentiellement en compte un effet de masse ajoutée et est col’équation de Rayleigh–Plesset qui régit l’évolution d’une bulle sphérique. La dynamique d’une cavité bidimensionnellecontrôlée par la différence entre la pression de cavité et la pression à l’infini. Le modèle est en bon accord avec la soTulin pour un écoulement supercavitant stationnaire et est facilement applicable à une configuration instationnaire.Pour cit rcet article : C. Pellone et al., C. R. Mecanique 332 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.Keywords: Computational fluid mechanics; Two-phase flows; CavitationMots-clés : Mécanique des fluides numérique ; Écoulements diphasiques ; Cavitation* Corresponding author.E-mail address: Jean-Pierre.Franc@hmg.inpg.fr (J.-P. Franc).1631-0721/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crme.2004.06.005828 C. Pellone et al. / C. R. Mecanique 332 (2004) 827–833cterizedction ofo intimateThisrge sizef cavitynaturalh-speedlyticalt methodre alsoipleIninfinity.al case,eloped tos inde-etweenantagesiscousinertia.ruember ispperity:1. IntroductionThis paper addresses the modelling of two-dimensional unsteady cavity flows. Cavity flows are charaby the coexistence of two phases, liquid and vapour. The latter can be due to cavitation or to the injenon-condensable gas into the liquid flow, as it is he case for ventilated supercavitating flows.The two phases are supposed to be separated by a well-defined interface. It is assumed that there is nmixing between them in the form of small bubbles, for instance, or at least that such a zone is limited in space.is especially the case for supercavitation, which is actually characterized by a vapour cavity of usually lawith respect to the cavitator. The interface of such a supercavity is clearly defined except in the vicinity oclosure where the cavity breaks up into smaller scale vapour structures. Fig. 1 shows a typical view of asupercavity generated by a two-dimensional circular cylinder. Such cavity flows can be found around higsupercavitating torpedoes as an example.Several techniques are available for the modelling of cavity flows. Apart from analytical or quasi-anamethods applicable to simple and often linearized configurations [1], we can mention the boundary elemenwhich is particularly powerful [2]. Other methods based on the resolution of Navier–Stokes equations aavailable with various types of cavitation models (see e.g. [3–5]).In this paper, a different model based on the Logvinovich independence principle is proposed. This princhas been widely used in Russia and Ukraine for the modelling of axisymmetric cavity flows ([6–9], see also [1]).axisymmetric configurations, a logarithmic singularity appears as the radial distance from the axis tends toIt is then necessary to limit the computational domain to an artificial maximum radius. In the two-dimensionthis singularity is expected to be stronger and a special procedure, presented in this paper, has to be devovercome the difficulty.The Logvinovich independence principle consists in assuming that each cross-section of the cavity evolvependently of the neighbouring ones and that its evolution is mainly controlled by the pressure difference bthe cavity and the pressure at infinity or in other words by the cavitation numberσ (cf. Eq. (6) for a definitionof σ ). In comparison with Navier–Stokes techniques that are very time-consuming, one of the main advof this technique is its simplicity, in particular for unsteady cavity flows. It does not take into account veffects, which is generally not a serious limitation for supercavity flows since they are mainly governed byThe model presented below also requires that the slenderness of the cavity is small enough. This is generally tfor supercavitation since the cavity length increases more rapidly than its thickness as the cavitation nudecreased (cf. Eq. (10)).Fig. 1. Supercavity flow around a two-dimensional circular cylinder in the LEGI hydrodynamic tunnel. The dissymmetry between the uand lower part of the cavity is due to the free surface. (Cylinder diameter: 5 cm, flow height in undisturbed conditions: 40 cm, flow veloc12 m/s, cavitation number: 0.05.)C. Pellone et al. / C. R. Mecanique 332 (2004) 827–833 829aticallyatbece.quentreaseicteby:2. Equation of evolution of the cavity interfaceConsider the flow around a two-dimensional cavitator and its cavity in an infinite liquid medium as schemshown on Fig. 2. For simplicity, we consider a symmetric configuration with respect to the planey = 0. Theslender body approximation consists in assuming that the velocity does not differ significantly from the velocityinfinity U∞. In other words, the two componentsu andv of the flow velocity shown in Fig. 2 are supposed toeverywhere small with respect toU∞.Thev-component on the cavity interface (denotedvc) is deduced from the kinematic condition on the interfaThis condition infers that, if a fluid particle is on the interface at a given time, it will remain on it at any subsetime until it reaches the closure region where it will separate from the cavity. Hence,vc is given by:vc = ∂yc∂t+ U∞ ∂yc∂x(1)wherey = yc(x, t) is the equation of the cavity interface at timet .Furthermore, thev-component vanishes at infinity where the flow is uniform. Then, it is expected to decwhen moving away from the cavity. For a spherical bubble, the radial velocity behaves like 1/r2 1. In the axisym-metric case, the continuity equation shows that it decreases as 1/r. By integration, this leads to the logarithmsingularity for the pressure already mentioned. For the two-dimensional case, we assume that thev-componentbehaves like a given power 1/yn of the distancey to the plane of symmetry. It will be shown later that exponenndepends upon the cavitation numberσ and approaches 1 whenσ approaches 0. Nevertheless,n is supposed to balways greater than 1 so that no singular behaviour is expected. Thev-component is then assumed to be givenv(x, y, t) = vc[ycy]n=[∂yc∂t+ U∞ ∂yc∂x][ycy]n(2)In the slender body approximation and for an inviscid fluid, the momentum balance in they-dir ction writes:dvdt= − 1ρ∂p∂y(3)wherep is the pressure,ρ the liquid density and d/dt the transport derivative given by:ddt= ∂∂t+ U∞ ∂∂x(4)Fig. 2. Schematic view of a cavityflow and general notations.1 r is the distance to the bubble centre in the sphericalcase or to the cavity axis in the axisymmetric case.830 C. Pellone et al. / C. R. Mecanique 332 (2004) 827–833uref anbee cross-e wall atnane ofis iscavityn, theThe equation of evolution of the cavity interface is obtained by introducing expression (2) for thev-componentin the momentum balance (3) and integrating from the cavity interfaceyc where the pressure is the cavity presspc to infinity wherep is denotedp∞. We finally obtain the following equation:ycd2ycdt2+n(dycdt)2= −(n−1)U2∞σ2(5)In this equation d/dt is still the transport derivative (4) andσ is the usual cavitation number defined by:σ = p∞ − pc(1/2)ρU2∞(6)Eq. (5) is similar to the Rayleigh–Plesset equation2 which allows the computation of the time evolution ospherical bubble when submitted to a given pressure differencep∞ − pc. It suggests to follow any cross sectioof the 2D cavity in a Lagrangian way, as it is advected downstream at flow velocityU∞. During its downstreammovement, the temporal evolution of the cross-sectional area is given by Eq. (5). Initial conditions have tospecified at the instant of shedding of the cross section by the cavitator. They concern the initial size of thsection and its derivative which are connected respectively to the size of the cavitator and the slope of thcavity detachment.Eq. (5) expresses the Logvinovich princ ple according to which the temporalevolution of a given cross sectioof the cavity is independent of the neighbouring ones. The main effect that is taken into account in Eq. (5) isadded mass effect connected to the inertia of the surrounding liquid.3. DiscussionAt this step, the exponentis still unknown. In order to det rmine it, the steady case is examined. The shapthe cavity is given by the steady version of Eq. (5):d2(y2c)dx2= −(n − 1)σ (7)To get Eq. (7), we assumedn close to unity in the left-hand side of Eq. (5) only. It is checked below that tha good approximation. This equation has the followingelliptic solution for the shape of the cavity interface:y2c(n − 1)(σ/2)(2/4) +(x − /2)22/4= 1 (8)where is the cavity length. This solution is such that the cavity thickness is zero at both endsx = 0 andx = .The cavity slenderness is:δ =√(n − 1)σ2(9)In order to determinen, we compare the present solution to the classical one obtained by Tulin [10] for theflow about a symmetrical body in an infinite flow field (see also [1]). According to Tulin’s linearized solutiocavity is also elliptic and its slenderness is:δ = σ/21+ σ/2 (10)2 The left-hand side of the classical Rayleigh–Plesset equation would correspond ton = 3/2.C. Pellone et al. / C. R. Mecanique 332 (2004) 827–833 831of aby thered toTulin’swness andideulationchchsteps.in Eq. (5).ion of itsity orig-nessities. Theults.Fig. 3. Comparison of the present model with Tulin’s solution for the evolution of the cavity length with the cavitation number. Casesymmetrical wedge of chord lengthc and half vertex angle 8 degrees in an infinite flow field. The cavity length is non-dimensionedwedge chord length.The identification of Eqs. (9) and (10) allows the determination of exponentn:n = 1+ σ/2(1+ σ/2)2 (11)Exponentn is then always greater than unity and approaches unity whenσ tends to zero.Finally, the equation of evolution of the steady cavity is:d2(y2c)dx2= −2[σ/21+ σ/2]2(12)The solution of Eq. (12) in terms of the evolution of the cavity length with the cavitation number is compaTulin’s solution in Fig. 3 in the particular case of a wedge. The present solution is in good agreement withoriginal one. The systematic difference which is observed between the two is not so critical since it is well knothat, from an experimental viewpoint, cavity closure is affected by large fluctuations due to cavity unsteadinthat the cavity length is then not defined very accurately in practice.It has been checked that, if the exact value ofn given by Eq. (11) is taken instead of unity in the left-hand sof Eq. (12), this makes no significant difference in the range of cavitation numbers considered in Fig. 3.In the unsteady case, the temporal evolution of the cavity is governed by Eq. (5). The principle of the simof an unsteady cavity is then the following. At each time step, a new ‘slice’ of cavity is shed by the cavitator. Easlice is advected downstream independently and the evolution of its cross section is governed by Eq. (5). At eatime step, the whole cavity is reconstructed by piling up the different slices of cavity shed at previous timeAt time steps 1,2, . . . , k, the cavity is then made of 1,2, . . . , k slices respectively.Unsteadiness can have several origins. It can be caused by variations of pressure or cavitation numberAnother source of unsteadiness may be due to the movement of the cavitator or to the possible deformatwalls at constant cavitation number. If so, from a mathematical viewpoint, the unsteady behaviour of the cavinates in the initial conditions, at the instant of shedding ofeach cross section, and not in Eq. (5) itself. Unsteadican also be due to pressure oscillations inside the cavity, as observed for ventilated pulsating supercavpresent model is able to account for these various sources of unsteadiness.A difficulty arises as for the choice of exponentn in the unsteady case. It has been shown thatn is a function ofthe cavitation number for the steady case. It seems then reasonable to use the same determination ofn (Eq. (11)) inthe unsteady case. However, this is not fully justified and needs to be further validated against experimental res832 C. Pellone et al. / C. R. Mecanique 332 (2004) 827–833:ityetricaln of thehe linearf the veryencey lengthtimatedade upch leadscapablellapsess of two-yleigh–eented andFig. 4. Unsteady behaviour of the cavity past a symmetrical wedge in thecase of sinusoidal oscillations of the cavitation number (chord lengthc = 1, half vertex angle: 8 degrees, time step:t = 0.1, flow velocity: U∞ = 1, space step:x = U∞t = 0.1). Variation of the appliedcavitation numberσ with time t : σ = 0.2 − 0.15sint . Top view of the cavity at different time steps between 0 and 200, coloured by cavthickness (maximum thickness in white). The origin of space steps corresponds to the basis of the wedge from which the cavity detaches.An example of solution is given in Fig. 4 for an oscillatory cavitation number in the case of the symmwedge already considered in Fig. 3. Fig. 4(b) presents a collection of top views of the cavity as a functiodistance from the wedge basis at different times. The grey level is representative of the cavity thickness. Tpart at the bottom of diagram 4(b) corresponds to the starting stage and more precisely to the advection ofirst cross section of the cavity shed at the first time step.Here, the oscillation frequency has been chosen so that it is comparable to the advection time/U∞ based onthe maximum cavity length. Significant unsteady effects arethen expected. This can be seen by the phase differbetween the oscillations in pressure and cavity length. When the cavitation number is minimum, the cavitis not maximum and, moreover, the maximum cavity length is much smaller than that which would be esfrom Fig. 3 assuming a quasi-steady behaviour.It can be also observed on Fig. 4 that at some instants (typically around time step 125), the cavity is mof two separate parts. This is characteristic of the development of an undulation of the cavity interface whito a local pinching of the cavity and the subsequent separation of part of the cavity. The model is thenof predicting the break-off of the cavity and the production of a large scale vapour structure which codownstream.The present model appears to have the capabilities of predicting the steady and unsteady behaviourdimensional cavity flows. It is based upon an Eq. (5) which can be considered as a 2D version of the RaPlesset equation and which can easily be solved numerically. The model appears to be intermediate betwanalytical techniques which are limited to simple and usually steady configurations and more sophisticatime-consuming techniques as the resolution of Navier–Stokes equation together with a cavitation model.References[1] J.P. Franc, J.M. Michel, Fundamentals of Cavitation, Kluwer Academic, 2004.C. Pellone et al. / C. R. Mecanique 332 (2004) 827–833 8331,, C. R.n,ion,s,tute/RTO[2] S.A. Kinnas, The prediction of unsteady sheet cavitation, in: J.M. Michel, H. Kato (Eds.), Proc. 3rd Int. Symp. on Cavitation, vol.Grenoble, France, 1998, pp. 19–36.[3] L. Diéval, C. Pellone, J.P. Franc, M. Arnaud, Une technique de suivi d’interfaces pour la modélisation de la cavitation par pocheAcad. Sci. Paris, Sér. IIb 328 (2000) 809–812.[4] Workshop on physical models and CFD tools for computation of cavitating flows, in: Y. Tsujimoto (Ed.), 5th Int. Symp. on CavitatioOsaka, Japan, 2003.[5] A. Kubota, H. Kato, H. Yamaguchi, A new modelling of cavitating flows: a numerical study of unsteady cavitation on a hydrofoil sectJ. Fluid Mech. 240 (1992) 59–96.[6] G.V. Logvinovich, Hydrodynamics of Free SurfaceFlows, Nauvoka Dunka, Kiev, 1969 (in Russian).[7] G.V. Logvinovich, V.V. Serebryakov, The Methods for the Calculation of the Shape of Slend r Axisymmetric Cavities, HydromechanicNauvoka Dunka, Kiev, 1975 (in Russian).[8] V.N. Semenenko, Artificial Supercavitation. Physics and Calcultion, VKI/RTO Special Course on Supercavitation, Von Karman Instifor Fluid Dynamics, Brussels, 2001.[9] A. Vasin, The Principle of Independence of the Cavity Sections Expansion as the Basis for Investigation on Cavitation Flows, VKISpecial Course on Supercavitation, Von Karman Institute for Fluid Dynamics, Brussels, 2001.[10] M.P. Tulin, Steady two-dimensional cavity flows about slender bodies, DTMB Rept. 834, 1953.

Modelling of unsteady 2D cavity flows using the Logvinovich independence principle

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Modelling of unsteady 2D cavity flows using the Logvinovich independence principle

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