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On Numerical Simulation of Cavitating Flows Under Thermal Regime International Journal of Heat and Mass Transfer 105 (2017) 411–428 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt On numerical simulation of cavitating flows under thermal regime ⇑ D. Colombet a,b,c, , E. Goncalvès Da Silva b,c,1, R. Fortes-Patella b,c a CNES, DLA, 52 rue Jacques Hillairet, 75612 Paris Cedex, France b Univ. Grenoble Alpes, LEGI, F-38000 Grenoble, France c CNRS, LEGI, F-38000 Grenoble, France article info abstract Article history: In this work, we investigate closure laws for the description of interfacial mass transfer in cavitating flows Received 3 April 2016 under thermal regime. In a first part, we show that, if bubble resident time in the low pressure area of the Received in revised form 15 September flow is larger than the inertial/thermal regime transition time, bubble expansion are no longer monitored 2016 by Rayleigh equation, but by heat transfer in the liquid phase at bubbles surfaces. The modelling of inter- Accepted 23 September 2016 facial heat transfer depends thus on a Nusselt number that is a function of the Jakob number and of the bubble thermal Péclet number. This original approach has the advantage to include the kinetic of phase change in the description of cavitating flow and thus to link interfacial heat flux to interfacial mass flux Keywords: during vapour production. The behaviour of such a model is evaluated for the case of inviscid cavitating Bubble Mixture model flow in expansion tubes for water and refrigerant R114 using a four equations mixture model. Compared Cavitation with inertial regime (Rayleigh equation), results obtained considering thermal regime seem to predict Heat/mass transfer lower local gas volume fraction maxima as well as lower gradients of velocity and gas volume fraction. Numerical simulation It is observed that global vapour production is closely monitored by volumetric interfacial area (bubble size and gas volume fraction) and mainly by the Jakob number variations. It is found that, in contrast with phase change occurring in common boiling flow, Jakob number variation is influenced by phasic temper- /ðq =q ÞD ature difference but also by density ratio variation with pressure and temperature (Ja L G T). Ó 2016 Elsevier Ltd. All rights reserved. 1. Introduction and model based on inertial [4] or thermal bubble growth [5].To estimate locally vapour volume fraction, one first approach is to Cavitation is involved in various flow applications such as assimilate the gas–liquid mixture to a barotropic fluid. In other hydraulic turbines, pumps, rocket turbopump inducers, fuel injec- words, the density of the gas–liquid mixture is considered to be tors, marine propellers, underwater bodies, etc. In most of cases, a function of the local static pressure in the flow. For the simulation cavitation is an undesirable phenomenon, significantly degrading of a cavitating flow through a venturi, [1] proposed a sinus barotro- performance, resulting in lower pressure head of pumps, asymmet- pic law considering a direct link between the gas volume fraction, ric load on turbomachinery blades, vibrations, noise and erosion. In phasic densities, local pressure and vapour saturation pressure. For industrial applications, cavitating flows usually take form as a tur- the simulation of cavitating flows in tubopump inducers of spatial bulent vapour polydispersed bubbly flow with phase change, bub- rockets, [6] proposed a sinus barotropic law with a vapour satura- ble break-up and coalescence. In the literature, various gas–liquid tion pressure calculated from local temperature in the flow. Those mixture or two-fluid models have been developed to investigate robust approaches have provided interesting results for the simu- isothermal and non-isothermal cavitating flows. According to the lation of hydrofoils [7], venturies [8], turbopump inducers [9–11], assumptions made, those models differ on two main points: equa- pump-turbines [12] or fuel injectors [13]. Although the simplicity tions solved and description of phase change. of this modelling approach, this model enable to study complex Among cavitation models, different approaches can be found to industrial applications. However, the adaptability of such model describe phase change due to cavitation: barotropic model [1], for thermosensitive liquids, where temperature gradients are sig- short relaxation time model [2], velocity divergence model [3] nificant, seem to suffer from a lake of physical descriptions of mass and heat transfers induced by phase change at bubbles surfaces. ⇑ Corresponding author at: Univ. Grenoble Alpes, LEGI, F-38000 Grenoble, France. One second approach is to express explicitly mass and heat E-mail addresses: [email protected] (D. Colombet), eric. transfer terms and to consider that interfacial transfers are [email protected] (E. Goncalvès Da Silva), [email protected] instantaneous. In that case, as proposed by [2] or [14], by introduc- (R. Fortes-Patella). ing infinite relaxation parameters (or infinite global transfer 1 Present address: ENSMA, Poitiers, France. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.09.070 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved. 412 D. Colombet et al. / International Journal of Heat and Mass Transfer 105 (2017) 411–428 Nomenclature Tk tempreature of phase k,K À1 List of symbols u stretching velocity, m s À1 3 aI volumetric interfacial area, m V b bubble volume, m v ; À1 B B factor k velocity of phase k ms v gas liquid mixture velocity, m sÀ1 Cpk specific heat capacity of phase k at constant pressure, m À1 À1 v Jkg K r mean relative velocity between phases, v ¼ v À v ; msÀ1 Cvk specific heat capacity of phase k at constant volume, r G L À À Jkg 1 K 1 À1 cw Wood’s speed of sound, m s Greek symbols À1 cEOS speed of sound based on sinus equation of state, m s ak volume fraction of phase k ; À1 c ; c ¼ = ck speed of sound of phase k ms k heat capacity ratio for phase k k Cpk Cvk À3 À1 db mean Sauter diameter of the bubble size distribution, m Ck mass transfer term of phase k; kg m s th ; th ¼ k =ðq Þ; 2 À1 k ; À1 À1 Dk thermal diffusivity of phase k Dk k kCpk m s k conductivity of phase k Wm K À1 l ek specific internal energie of phase k; Jkg k dynamic viscosity of phase k,Pas À À1 2 1 mk kinematic viscosity of phase k; m s Ek total energy of phase k; Jkg À1 pk internal energy reference in stiffened gas equation of Em gas liquid mixture total energy, J kg À1 ; À2 À1 state, J kg HL heat transfer coefficient in phase k Wm K À À q ; 3 1 k density of phase k kg m hk specific enthalpie of phase k; Jkg À À s time for inception of thermal regime, s J local mass flux, kg m 2 s 1 loc s0 time for inception of thermal regime with effect of the Ja Jakob number, Ja ¼ q C DT=ðq LÞ L pL G relative velocity, s l mean length of bubble path in the low pressure area s resident time of bubbles in the low pressure area, s where p < psat; m res À1 L latent heat of vaporisation L ¼ hG À hL; Jkg Supercripts NuL Nusselt number I at the bubble surface NuL0 Nusselt number wihout wlip p pressure, Pa sat at saturation 1 T at triple point pk pressure reference in stiffened gas equation of state, Pa ¼ = th Pe thermal Péclet number, Pe Udb DL Pec critical thermal Péclet number, Subscripts ¼ m = G gas phase Pr Prandtl number, Pr L DLth 00 ; À3 À1 k phase k qk interfacial heat flux from phase k Jm s R bubble radius, m L liquid phase ¼ q v =l m gas liquid mixture Re bubble Reynolds number, Re L rdb L À s surface tension of the fluid, Nm 1 0 initial value 2 Sb bubble surface, m t time, s coefficients), momentum, mass and heat transfers can be evaluated injectors [31]. For the simulation of cavitating flow in cryogenic considering very short equilibrium relaxation times between pha- fluids, [32,33] use the formulation proposed by [21]. sic pressures, velocities, temperatures, Gibbs free energy. This Various authors have attempted to take into account the effect approach, initially devoted for the simulation of diphasic detona- of liquid phase thermal gradients in the flow on cavitation. For bar- tion waves [15], is considered to be valid for cavitating flow at very otropic approach [34,6] as well as for inertial controlled growth high velocity. As shown recently by [3], very similar results can be model [33,35,36], it has been done meanly by calculating the satu- achieved considering simply that the mass transfer term is propor- rated vapour pressure as a function of the local temperature tional to the mixture velocity divergence. This approach has been (psatðTÞ) and by estimating the bubble temperature variation using recently used for non isothermal cavitation by [16] for the 2D sim- energy balance at the bubble scale. In the same time, a few numer- ulations of cavitating flow through a venturi. ical works evoked that vapour production can be driven by thermal To describe cavitation, one last approach is to consider finite controlled growth of vapour bubbles [37–39,5].In[37], authors rate mass transfer and to express explicitly mass transfer exchange recall that bubble growth follow two steps. The increase of bubble term due to phase change. In the literature, a large number of such volume is initially controlled by the liquid inertial (inertial growth) cavitation models consider that vapour production in a cavitating and then is controlled by heat transfer at bubble surface (thermal flow is only driven by inertial controlled growth of vapour bubbles.
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