Compressible effects modelling in turbulent cavitating flows Jean Decaix, Eric Goncalvès da Silva To cite this version: Jean Decaix, Eric Goncalvès da Silva. Compressible effects modelling in turbulent cav- itating flows. European Journal of Mechanics - B/Fluids, Elsevier, 2013, 39, pp.11-31. 10.1016/j.euromechflu.2012.12.001. hal-00768757 HAL Id: hal-00768757 https://hal.archives-ouvertes.fr/hal-00768757 Submitted on 12 Feb 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Compressible Effects Modelling in Turbulent Cavitating Flows Jean Decaix∗, Eric Goncalv`es∗ LEGI-Grenoble INP, 1025 rue de la Piscine, 38400 St Martin d’Heres, France Abstract A compressible, multiphase, one-fluid RANS solver has been developed to study turbulent cavitating flows. The interplay between turbulence and cav- itation regarding the unsteadiness and structure of the flow is complex and not well understood. This constitutes a determinant point to accurately sim- ulate the dynamic behaviour of sheet cavities. In the present study, different formulations including compressibility effects on turbulence are investigated. Numerical results are given for two partial cavities on Venturi geometries and comparisons are made with experimental data. Keywords: Cavitation, Homogeneous model, Compressibility effects and Turbulence model Nomenclature c speed of sound Cp,Cv thermal capacities E total energy ∗Corresponding author. Email address: [email protected] (Jean Decaix ) Preprint submitted to Elsevier September 10, 2012 e internal energy k turbulent kinetic energy Mt turbulent Mach number P static pressure Pk turbulent production Pvap vapour pressure Pr,Prt molecular and turbulent Prandtl numbers Q total heat flux ReL Reynolds number based on the length L T temperature u, v velocity components w conservative variables α void ratio γ ratio of thermal capacities ε dissipation rate λ, λt molecular and turbulent thermal conductivity , µt molecular and eddy viscosity ρ density σ cavitation number τ total stress tensor ω specific dissipation ()L liquid value ()V vapour value ()v viscous ()t turbulent 2 () Reynolds average for single-phase flow () phase average for two-phase flow ()˜ mass weighted average (Favre average) ()′ fluctuations with respect to the Reynolds and phase averages ()′′ fluctuations with respect to the mass weighted average 1. Introduction Cavitation is a significant engineering phenomenon that occurs in fluid machinery, fuel injectors, marine propellers, nozzles, underwater bodies, etc. In most cases, cavitation is an undesirable phenomenon, significantly degrad- ing performance, resulting in reduced flow rates, lower pressure increases in pumps, load asymmetry, vibrations, noise and erosion. Such flows are characterized by important variations of the local Mach number, compressibility effects on turbulence, and thermodynamic phase transition. For the simulation of these flows, the numerical method must accurately handle any Mach number. Moreover, the modelling of turbu- lence plays a major role in the correct simulation of unsteady behaviours. Sheet cavitation that appear on solid bodies are characterized by a clo- sure region which always fluctuates, with the presence of a re-entrant jet. This jet is mainly composed of liquid which flows upstream along the solid surface. Reynolds-Averaged Navier-Stokes (RANS) models are frequently used to simulate such unsteady cavitating flows. One fundamental problem with this approach is that turbulence models are tuned by non-cavitating 3 mean flow data. Moreover, the standard eddy-viscosity models based on the Boussinesq relation are known to over-produce eddy-viscosity, which reduces the development of the re-entrant jet and two-phase structure shedding [1]. The link to compressibility effects on turbulence is not clear. DNS of the supersonic boundary layer demonstrated a reduction in k production as a consequence of compressibility [2, 3, 4]. In cavitating flows, the supersonic regime is reached in the mixture area because of the drastic diminution of the speed of sound. The detailed mechanisms of the interaction between turbulent flows and cavitation have not yet been clearly revealed, especially for phenomena occurring at small scales. Different strategies have been investigated to limit or to correct standard turbulence models. An arbitrary modification was proposed by Reboud to reduce the turbulent viscosity [1], and has been used successfully by differ- ent authors [5, 6, 7, 8, 9, 10]. The Shear Stress Tensor (SST) correction proposed by Menter [11, 12] to reduce the eddy-viscosity in case of positive pressure gradient and a variant based on realizability constraints [13] were tested for unsteady cavitating flows [14]. Other corrections are based on the modelling of compressibility effects of the vapour/liquid mixture in the turbulence model. Correction terms proposed by Wilcox [15] in the case of compressible flows were tested for unsteady periodic cavitating flows [6]. A sensitivity analysis of constants Cε1 and Cε2, which directly influence the production and dissipation of turbulence kinetic energy, was conducted for a k − ε model and a cavitating hydrofoil case [16]. A k − ℓ model including a scale-adaptive term [17] was developped for cavitating flows [18]. This term allows the turbulence model to recognized the resolved scales in the flow and 4 to adjust the eddy-viscosity as a consequence. Finally, a filter-based method was investigated [19] by which the sub-filter stresses are constructed directly using the filter size and the k − ε turbulence closure. The present work is part of research aimed at developing a numerical tool devoted to cavitating flows. A particular emphasis is placed on the investi- gation of closures for additional terms appearing on the mixture turbulence transport equations. These terms are due to the fact that the divergence of the phase velocity fluctuations is not null in two-phase flow, even if pure phases are assumed to be incompressible. Such terms (pressure-dilatation, mass flux, compressible dissipation) are similar to those involved in the com- pressible formulation of single-phase turbulence models. For high-speed aero- dynamic flows, additional terms in the turbulent kinetic energy (TKE) equa- tion have been modelled by Zeman [20, 21] and Sarkar [22, 23, 24]. It was showed that these dilatational terms act as sinks and, for sufficiently large turbulent Mach number, could lead to reduction of the turbulence levels. The compressible dissipation have been modelled by Sarkar and Wilcox [25, 15]. A synthesis of these different closures in the case of aerodynamic flows are presented in [26, 27, 28]. An in-house finite-volume code solving the Reynolds-Averaged Navier- Stokes (RANS) compressible equations was developed with a homogeneous approach [29, 14, 30]. The cavitation phenomenon is modelled by a barotropic liquid-vapour mixture equation of state (EOS). The closures for additional terms proposed by Sarkar and Wilcox based on a turbulent Mach number 5 are studied and compared. The influence of the cut-off turbulent Mach num- ber is investigated for the Wilcox model. Various unsteady simulations are performed to compute two partial cavities for which the dynamic behaviour is different. Moreover, comparisons with results obtained with the Reboud compressibility correction, specially developped for cavitating flows are pro- posed. Our goals in this study are to investigate the effect of various com- pressibility corrections and turbulent closures for flows involving a cavitation sheet. This would help in gaining an insight into the range of predictions that we can expect with these models. The paper is organized as follows: the theoretical formulation is sum- marized, including physical models and elements of the numerical methods. This is followed by sets of results on Venturi geometries and discussions. 2. Governing equations and models The numerical simulations are carried out using an in-house code solv- ing the one-fluid compressible RANS system. The homogeneous mixture approach is used to model two-phase flows. The phases are assumed to be sufficiently well mixed and the disperse particle size are sufficiently small thereby eliminating any significant relative motion. The phases are strongly coupled and moving at the same velocity. In addition, the phases are as- sumed to be in thermal and mechanical equilibrium: they share the same temperature T and the same pressure P . The evolution of the two-phase flow can be described by the conservation laws that employ the representa- tive flow properties as unknowns just as in a single-phase problem. 6 To obtain the one-fluid equations, we use the phase average of a phase quantity φk defined by Ishii [31]: 1 φ = φ dτ (1) k T k k where Tk is the time period of phase k. The mass-weighted average or Favre average is commonly used for compressible flows. In the following, an over- tilde denotes a Favre average. Using this definition and starting from the instantaneous conservation equations, the equations for each phase can be expressed. The equation for mixture quantities are obtained by the summa- tion of the separate equations of the phase quantities. We introduce αk the void fraction or the averaged fraction of presence of phase k. The density ρ, the center of mass velocity u and the internal energy e for the mixture are defined by: ρ = αkρk (2) k ρui = αkρkuk,i (3) k ρe = αkρkek (4) k The mass gain per unit volume due to phase change (evaporation or con- densation) is noted Γk = ρk(uI − uk).nkδI where δI is a Dirac distribution having the different interfaces as a support, uI the interface velocity and nk the vector normal to the interface directed outward from phase k.