A Numerical Investigation over the Cavitating Flow Regime of a 2D-Hydrofoil

A Numerical Investigation over the Cavitating Flow Regime of a 2D-Hydrofoil Luca Bonfiglio 1, Stefano Brizzolara 1 1. Massachusetts Institute of Technology, Mechanical Engineering Department, Sea Grant
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A Numerical Investigation over the Cavitating Flow Regime of a 2D-Hydrofoil Luca Bonfiglio 1, Stefano Brizzolara 1 1. Massachusetts Institute of Technology, Mechanical Engineering Department, Sea Grant College Program, Cambridge, Massachussets, USA KEY WORDS Super-Cavitating; Hydrofoil; RANSE; OpenFOAM; Parkin INTRODUCTION The advances in sailing boat races have been greatly proven in the recent America s Cup competition. Sailing boats have reached speeds above 40 knots with a simple concept: the wetted surface of the hull is minimized and the required displacement is obtained through a lifting force produced by submerged hydrofoils working at very high speeds. This is a well-known concept in naval architecture that has been exploited since the beginning of the 20th Century. Hydrofoils used in sailing boat races are yet not designed for cavitating flow, but major changes in the design will be needed in case speed increases above 50 knots. When high speed crafts (including fast sailing boats) operate significantly above the planing threshold speed, the convenience of completely or partially supporting their weight by lifting hydrofoils is evident (Du Cane (1964)). A very low pressure field induced by high inflow speed triggers water vaporization at ambient temperature: cavitation cannot be avoided and foil shape has to be designed with the goal of maintaining a stable flow regime eventually compromising the lift. When craft speed arise above 50 knots, the design philosophy for the basic section of the lifting hydrofoil has to radically change and turn to super-cavitating hydrofoils (Auslaender (1962)) being the final goal addressed towards the delay and stabilization of the cavity shape over the hydrofoil surface. In super-cavitating regimes the suction surface of the hydrofoil is fully enveloped in the cavity which (typically) detaches at the leading edge of the foil and closes in the wake well aft the trailing edge. The pressure side of the hydrofoil is the only responsible for lift generation thus the main design target is represented by the shape of the foil surface. Several simplified theories assuming steady state potential flow (mentioned later in this introduction) were developed in the past to deal with this essential design problem. The design of super-cavitating hydrofoils, though, must consider also off-design, certainly even more rigorously than the case of partial or sub-cavitating hydrofoils. The shape (length and thickness primarily) of the super-cavity is in fact highly sensitive to a variation of speed and angle of attack that can occur in different operating conditions, with the serious hazard to lose the stable super-cavitating regime abruptly falling into different type of cavitation (e.g. partial or base vented) which can be highly unstable and can cause oscillations and hysteresis effects on the generated lift force over time. It is clear that a computational tool able to predict the hydrodynamic characteristics of cavitating foil is essential in the design process of such type of hydrofoils. From a fluid dynamic point of view, the flow around supercavitating hydrofoil is a complex phenomenon which involves many different challenges. The early prediction methods were formulated under linear theories (Tulin (1953), Acosta (1955)) but major weaknesses were experienced in the cavity shape prediction especially for thick profiles. Uhlman (1987) used a nonlinear surface vorticity method to prove that the cavity size decreases with the increase of profile thickness, in contrast with linear theories findings. Since then many non-linear potential flow based methods were formulated in the context of a boundary element approach both in 2D (Kinnas and Fine (1991)) and 3D (Fine and Kinnas (1993), Kinnas and Fine (1992) and Kim and Lee (1996)). Among these studies Young and Kinnas (2001) developed a surface panel method to predict cavitating flows in unsteady conditions. The main problem related to cavitating flows is that the cavity shape can be found only through the solution of the flow around the profile which depend on the shape of the cavity itself. Panel methods treat the cavity as a domain boundary where specific dynamic and kinematic boundary conditions are imposed. They are based on potential flow formulation and they need to be formulated in order to iteratively solve for the cavity shape. Kinnas et al. (1994) developed a non-linear BEM for partially and super-cavitating hydrofoils including a viscous model based on the boundary layer-theory. The validity of this method has been proven for steady-state flows, but it cannot be applied for unsteady cavitation. Recently a potential flow based method has been proposed by Celik et al. (2014) who used a BEM based on source and doublet distribution on the foil and the cavity with Dirichlet boundary conditions. They verified their method with numerical results obtained with a Reynolds Averaged Navier-Stokes code and they validated it on a series of NACA 16 hydrofoils in steady sheet cavitation. Their method is limited to steady cavitaton occurring on the back of 2D profiles and its application to NACA 16 hydrofoils of different thickness, showed some weaknesses when used for thin profiles at different angles of attack. Their work confirms that in case of complex turbulent flows where hydrofoils are operated at high angles of attack a more realistic modeling of the physics need to be involved. The solution of the Navier-Stokes equations coupled with a suitable cavitation model recently brought extraordinary improvements in cavitating flow predictions. Recovering viscous effects and vortex shedding due to separation in the context of a nonlinear method to capture the cavity boundary has considerably improved the analysis of the unsteady characteristics of cavitation like the re-entrant jet prediction. However, one of the main challenges to face when solving Navier-Stokes equations in very high speed flows is the modeling of turbulent effects, responsible for the random fluctuations of the main flow variables. In fact the boundary layer flow interacts with the cavity interface, contributing to its development (Ji et al. (2015)). Different methodologies of increasing accuracy could be applied for the solution of turbulent cavitating flows. RANSE technique is based on the solution of the mean flow characteristics after the averaging of Navier-Stokes equations; several successful applications were presented by Oprea and Bulten (2009), LI DQ and Lindell (2009) and Hoekstra and Vaz (2009). A more accurate turbulence modeling was presented by Bensow and Bark (2010) who analyzed the cavitating flow around a NACA-0015 foil using a Large Eddies Simulation technique (LES) which computes the large scale anisotropic vortical structures modeling the smaller subgrid scales. Many of these applications have been proposed for partially cavitating hydrofoils, but only few of them consider super-cavitating conditions. In the present paper we present a numerical study on simple super-cavitating foil geometries, with the goal to verify and validate a computational method based on Unsteady Reynolds Averaged Navier-Stokes Equations (URANS) which at the moment represents the best compromise between accuracy and computational effort. The open source libraries of OpenFOAM represent the core of a comprehensive fluid dynamic analysis tool developed by Bonfiglio and Brizzolara (2015). The CFD suite includes a preprocessing package for mesh generation and case set-up as well as a post-processing tool for results analysis. The present study represents a first step for the validation and verification of a cavitating solver meant to be included in the more general CFD analysis tool for hydrofoils. The peculiarity of cavitating hydrofoils is the multiphase nature of the flow, which in this study is solved using a volume of fluid approach without involving any boundary condition on the cavity surface. The vaporization and condensation of the liquid is numerically solved using the Schnerr and Sauer cavitation model, fully resolving the vapor flow allowing for a better solution of the pressure recovery at the cavity closure. NUMERICAL SET UP Modeling a cavitating flow implies the addition of a set of equations describing the thermophysical behavior of the fluid. Cavitation is a well-known phenomena in which very high flow velocities cause low pressure fields responsible for the phase-change from liquid to vapor. Lord Rayleigh (1917) is considered the pioneer of the physical description of cavitation. He solved the problem of the collapse of a cavity in an incompressible, inviscid fluid and, with the goal to investigate over cavitation damage behind screw propellers, he developed the first model for growth and collapse of bubble. Rayleigh derived the motion equation of a spherical bubble with radius R(t) in any point of a liquid r giving an external pressure P(t) and an internal pressure p(r). Following a potential flow approach, thus neglecting viscosity and surface tension effects, the velocity potential is: φ=rṙ r From Bernoulli equation, it can be concluded that: Ṙ= 3 p(r) P(t) 2 ρ Rayleight equations were generalized later on by Plesset (1949) to include viscosity and surface tension effects. However the cavitation model used in the present study is the one proposed by Sauer and Schnerr (2001) which uses equation (2) to model the the bubble growth process. In this study, the numerical solution of the flow governing equations is obtained using a finite volume approach. The unsteady cavitating viscous flow simulations are performed using interphasechangefoam: a multiphase flow solver designed for two incompressible isothermal immiscible fluids with phase-change. This solver uses a volume of fluid phase-fraction based interface capturing approach, thus solving momentum equation for a single fluid mixture whose density and viscosity depend on the local concentration of vapor and water defined through the indicator function γ for which a transport equation is solved together with Navier-Stokes equations (Open- FOAM Foundation (2014)). The unsteady prediction of the cavity shape relies on the accurate prediction of the volume of fluid variable. A compressive convection scheme increases the accuracy of the interface prediction using an additional velocity field U r to steepen the gradient of volume fraction function γ close to the cavity boundary. The conservation equation for γ is therefore written as: (1) (2) γ t + (γu)+ [γ(1 γ)u r]= ṁ ρ V (3) Where the right hand side term represents a source term that will be discussed later in this section. U r represents the relative velocity between vapor and water: large values correspond to sharper interface but they might lead to nu- Bonfiglio A Numerical Investigation over the Cavitating Flow Regime of a 2D-Hydrofoil 2 Figure 1: Cavity Shape. In blue calpha = 0, in green calpha = 0.5, in red calpha = 1 and in orange calpha = 1.5 Time = 0.1s, σ v = 0.141, α=0 deg, Re=4.52e5 merical problems, U r is defined as: [ ( )] ϕ U r = n f min C f S f,max ϕ S f being n f the face normal flux, S f the ( face ) area vector and ϕ the ϕ volume metric flux. The term max S f represents the maximum face velocity of the control volume. The additional term [γ(1 γ)u r ] is indeed called artificial compression term and it is only effective when 0 γ 1 (at the interface). (4) no significant impact on global performances (drag coefficient in figure 2). In the closure region, where the cavity thickness gradually reduces, the cavity becomes more unstable with calpha greater than 1, for this reason calpha=0 has been selected for all the following simulations in the present study. In this approach the cavity surface is modeled as a fictitious interface where the VoF fraction function γ assumes an intermediate value between 1 (liquid) and 0 (vapor): γ = 0.5. The solution of equation 3 leads to the determination of the physical characteristic of the fluid mixture: ρ=(1 γ)ρ L + γρ V (5) µ=(1 γ)µ L + γµ V (6) c D CPU Time [s] The source term in (3) represents the vapor production, determined on the basis of thermophysical consideration: ṁ + ρ = C V ρ L V ρ ṁ= γ(1 γ) R 3 2(pV p) 3ρ L if p p V ṁ ρ = C V ρ L C ρ γ(1 γ) (7) 3 2(p pv ) R 3ρ L if p p V CFD c D 500 CPU Time EFD c D calpha Figure 2: Drag Coefficient predictions at different interface compression values (calpha). Time = 0.1s, σ v = 0.141, α=0 deg, Re = 4.52e5, number of nucleii 8.5e The bubble radius is given in terms of n 0 which represents the nuclei concentration per unit of volume of pure liquid: ( γ R= 1 γ ) (8) 4π n 0 It is possible to control the relative velocity and thus the interface compression tuning C f (calpha): a sharp interface generally leads to results more consistent with the physics, but it might cause numerical instability, while when the artificial compression is not used (calpha= 0), the VOF transition from 1 to 0 is smeared over a larger number of cells. For this reason the type of numerical grid has been specially designed to study cavitating and super-cavitating hydrofoil, including a refinement region of unstructured cells downstream the trailing edge where the cavity is supposed to develop. Figure 1 shows the influence of the artificial compression term on cavity shapes: increasing the value of calpha leads to cavity instability generally requiring higher computational time with Figure 3: Schnerr Sauer Model: Velocity Contour at Time = 0.1s, σ v = 0.141, γ=0 deg, Re=4.52e5 Bonfiglio A Numerical Investigation over the Cavitating Flow Regime of a 2D-Hydrofoil 3 Figure 4: Schnerr Sauer model: σ k contour and velocity vector field in the cavity at Time = 0.1s, σ v = 0.141, γ = 0 deg, Re = 4.52e5 Limiter with Explicit Solution (MULES) algorithm to avoid unbounded solution for the variable γ: if a cell is completely filled with a certain phase, it cannot be further filled with any other phases (Kissling et al. (2010)). The flow simulated in the present study is characterized by a relatively low Re number, thus no turbulence modeling is required. The numerical set-up described above leads to the solution of the pressure and velocity fields as well as the VoF scalar function. Following Parkin (1956) approach for the analysis of the experimental results, different cavitating flow conditions can be distinguished through the definition of a nominal cavitation index σ V (or K V ) (based on the fluid saturation temperature) and an actual cavitation index based on the pressure value measured inside the cavity σ K (or K K ) CFD c D EFD c D σ V = p 0 p V 1 2 ρv σ 2 K = p 0 p K 1 2 ρv (9) 2 c D e+11 1e+12 1e+13 1e+14 1e+15 1e+16 1e+17 number of nucleii Figure 5: Schnerr Sauer model: Drag coefficient over nuclei concentration in 1 m 3 of water. Nuclei Diameter: d nuc = 2E 6. σ v = 0.141, γ=0 deg, Re=4.52e5 The numerical prediction of cavitating flows is achieved through the solution of the continuity and the momentum equations expressed in terms of a velocity and a pressure equation: the former solved using a smoothsolver with a Gauss Seidel smoother suitable for symmetric algebraic system, while the latter using a multigrid method (GAMG - Geometric Algebraic Multi Grid) with a Diagonal Incomplete Cholesky based smoother. The outer iterations required for the solution of the non-linear partial differential system of equations rely on the PISO algorithm (Issa (1985)). Being the nature of cavitation essentially unsteady, it is particularly important to accurately solve the problem in time domain. For this reason an implicit Euler scheme time discretization has been selected with a variable time-step dynamically adapted in order to keep the maximum local Courant number under a certain threshold (Co 1). The solution of the volume of fluid fraction function is achieved using a smoothsolver with a Gauss Seidel smoother with a Multidimensional Universal Figure 3 shows the contour of the velocity field downstream the trailing edge of a base-cavitating wedge-hydrofoil at 0 deg of angle of attack: cavity detachment is experienced at the sharp edges of the hydrofoil blunt trailing edge. As expected the vapor wake flow is stagnant right after the trailing edge while the velocity increases getting closer to the cavity closure point. The velocity vector field of the vapor phase is plotted in figure 4 along with the contour of the cavitation number K K. As expected a flow recirculation is experienced close to the sharp edges. The pressure inside the cavity is equal to the nominal saturation pressure. The velocity vector field of the vapor phase predicted by the numerical method is plotted in figure 4 along with the contour of the cavitation number K K. As expected a flow re-circulation is experienced close to the sharp edges. The pressure inside the cavity is equal to the nominal saturation pressure. A sensitivity study aimed to understand the influence of the nuclei concentration in the drag coefficient prediction has been performed and the results are presented in figure 5. A proper tuning of the arbitrary physical parameters of the cavitation model may lead to a closer match between the numerical prediction and the experiments performed by Parkin (1956) as described in the following section. The increase of nuclei concentration in the fluid leads to a larger vapor concentration at the hydrofoil surface which results in smaller time-averaged drag values. On the other hand the larger number of cavitation nuclei increases the flow instability, leading to larger standard deviation values of the forces (indicated through the error bar in figure 5). Figure 5 shows results obtained with a particularly coarse grid of dimensionless wall distance y+ of about 3. A comprehensive grid independence study have been performed and results will be presented along with the description of the mesh topology, in the following section. A nuclei concentration of 8.5e13 has been selected for every simulations presented in the following sections of the present paper. Bonfiglio A Numerical Investigation over the Cavitating Flow Regime of a 2D-Hydrofoil 4 HYDROFOIL GEOMETRY AND MESH TOPOLOGY The geometry that has been selected for the verification and validation study presented in this paper is the one investigated by Parkin (1956). This shape is particularly interesting for the presence of sharp leading and trailing edges which promote cavitation inception at relatively low speeds. Sharp edges are often used in super-cavitating hydrofoil design when performances need to be ensured in a large range of operating conditions: since in a wide range of angle of attacks the cavity detaches from the edges, hydrofoil performances (lift and drag) do not depend on the translation of detachment points along the foil surface. The problem of cavity detachment and closure is one of the main challenges in cavitation prediction. In presence of sharp edges the detachment position is known a priori, but for hydrofoils not specifically designed for cavitating regimes numerical procedures need to be used. Armstrong (1953) and Wu (1967) developed a theory based on irrotational inviscid fluid hypothesis which has been used to demonstrated that the curvature of the streamlines become finite at separation points and the cavity detachment occurs when their curvature becomes equal to the curvature of the solid body. Brennen (1969) and Arakeri (1975) showed through an experimental campaign that the cavity detachment point depends on viscous effects, surface tension and surface roughness. In conventional super-cavitating hydrofoils detachment is typically experienced at the leading edge and at the two edges of the blunt trailing edge. The aim is to induce the detachment of the supercavity at the leading edge at low cavitation indexes and to induce separation and base cavitation at relatively high cavitation
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