Implementation of an explicit density-based solver for

9th OpenFOAM Workshop 23-26 June 2014 in Zagreb, Croatia

Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM Christian Deimel∗1 and Romuald Skoda1 1

Ruhr-Universit¨ at Bochum, Chair of Hydraulic Fluid Machinery, Universit¨ atsstr. 150, 44801 Bochum, Germany, [email protected]

Key words: Cavitation, density-based method, Godunov scheme, barotropic model

Abstract For the simulation of cavitating flows, methods in commercially available CFD codes are commonly based on a simplified formulation of the Rayleigh-Plesset equation. In practical application, these models need to be tuned specifically for each problem, which makes their use difficult for predicting cavitation behaviour. As an alternative, approaches with a barotropic thermodynamical model can be used which do not need any empirical parameters to describe cavitation. In the work presented, we will show the implementation of an explicit, density-based approach with such a model currently only available in research CFD codes by using the open source framework OpenFOAMr . Validation cases are presented as well, showing that both implementations lead to similar results in terms of time evolution of the vapour fraction as well as shedding frequency.

1

Introduction

For the simulation of cavitating flows, methods in commercially available CFD codes are commonly based on a simplified formulation of the Rayleigh-Plesset equation which is derived from the dynamics of a single vapour bubble [3]. Based on this equation, models for the mass transfer between the two distinct phases have been developed [12, 13, 20]. To incorporate these mass transfer models, an additional transport equation for the vapour phase is needed. Commonly, this approach is referred to as volume of fluid-method (VOF) with RayleighPlesset model. 1

Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM

Although these models are widely used, several problems exist. First of all, they are usually implemented in combination with pressure-based codes which are incompressible in the sense of wave dynamics. Because of this, these models are not suitable for capturing the driving mechanisms behind cavitation erosion, like shock wave formation and high pressure peaks occuring on nano-second timescales, however these phenomena are of particular interest for prediction of the lifetime of machine parts. Second, these models rely on several tuning parameters. For correct simulation, often a time consuming process is needed to calibrate these parameters against measured data. Furthermore, the prediction of the transient behaviour of cavitation shedding is highly dependent on the turbulence model used. As turbulent kinetic energy is usually overpredicted in the two-phase region, an artificial reduction of the turbulent viscosity is needed, basically disabling the turbulence model in this region [15]. An alternative to the VOF approach in combination with a Rayleigh-Plesset model is to capture the compressible wave dynamics of cavitating flows by using a density-based code. By assuming a homogenous mixture of both phases (this also applies to the VOF approach), the two-phase flow can be modeled by an equation of state for the mixture density. This approach is implemented in our in-house code hydRUB, which has been successfully applied to several problems (e.g. [23, 24]). As hydRUB is an academical CFD code which was never intended to be released to the public, the downside is the lack of interoperability with common software tools as well as a low level of user-friendliness. Therefore, the obvious choice was to reimplement the method in a way which makes it easier accessible for every day CFD users. Thus, by implementing our method in OpenFOAMr , we strive to make it available to a broader user base. We call this implemenentation hydRUBFoam. In this paper, a short decription of the underlying numerical method will be given.

2

Physical model

Since the driving effects behind cavitation shedding are known to be inertia driven [5], it is justified to neglect viscous effects and consider the compressible Euler equations for mass and momentum. As another simplification, both phases are assumed to be in thermodynamic equilibrium, which means that inter-phase heat and mass transfer occur instantaneously. Applying these assumptions, the phase change follows an isentropic path, hence the energy equation can be neglected as well. As a result, the density of the fluid is only dependant on the pressure (and vice versa). This is commonly called a barotropic model (fig. 1). As equation of state, it is either possible to use analytical relations like the Tait equation for the liquid state, perfect gas equation for the state of pure vapour and a relation for the two-phase region in between, or use lookup tables based on measurements or reference equations like the so called Oldenbourg polynominals [17] or the industrial standard formulation IAPWS-IF97 [26] for water. In our simulations, we employ the latter approach, obtaining the needed thermodynamic quantities from the IAPWS equations. However, the speed of sound in the two-phase region cannot be determined this way. Thus, we use the so called Wallis formula [27] 2

Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM

T

liquid

barotropic EOS T0 Tsat(s0)

p= sat co ura ns tion t. line

p0

ρliq(s0)

Tc, pc

vapour

2-phase region

ρ(p,s0)

s

Figure 1: Barotropic EOS in the T,s-diagram

1 = ρmix · c2

γ 1−γ + 2 ρvap · cvap ρliq · c2liq

! (1)

with cliq obtained from the reference equation and cvap via the relation for an ideal gas c2vap = κ · Rvap · T.

3

(2)

Numerical method and implementation

The numerical method implemented in our codes is a Low-Mach number consistent Godunov type flux formulation which has been developed especially for cavitating flows [18]. By defining the state at the cell faces using the following equations   1 pL − pR · ρL · uL + ρR · uR + (3) uface = ρL + ρR cface 1 · (pL + pR ) 2 = max(cL ,cR ),

pface =

(4)

cface

(5)

one can define the Euler flux as 

1





0



 uL/R  pface     F~ = ρL/R · uface ·   vL/R  +  0  . wL/R 0 3

(6)

Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM

Note that for the sake of simplicity, the flux formulation is shown for a face with its normal vector parallel to the first component of the velocity vector. Listing 1: Code snippet from hydRUBFoam, showing the relevant part of the implementation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

// Reconstruct left and right state from primitive variables su rf ac eS ca la rF ie ld rho_R ( fvc :: interpolate ( rho , right , " reconstruct ( rho ) " ) ); su rf ac eS ca la rF ie ld rho_L ( fvc :: interpolate ( rho , left , " reconstruct ( rho ) " ) ); su rf ac eV ec to rF ie ld U_R ( fvc :: interpolate (U , right , " reconstruct ( U ) " ) ); su rf ac eV ec to rF ie ld U_L ( fvc :: interpolate (U , left , " reconstruct ( U ) " ) ); // left and right state for pressure and speed of sound are determined using // an upwind scheme su rf ac eS ca la rF ie ld cSf ( max ( cSf_R , cSf_L ) ); su rf ac eS ca la rF ie ld pSf ( 0.5*( p_L + p_R ) ); // determine transport velocity at cell faces su rf ac eS ca la rF ie ld phiv_R ( U_R & mesh . Sf () ) ; su rf ac eS ca la rF ie ld phiv_L ( U_L & mesh . Sf () ) ; su rf ac eS ca la rF ie ld phivSf ( " phivSf " , ( rho_L * phiv_L + rho_R * phiv_R - mesh . magSf () *( p_R - p_L ) / cSf ) /( rho_L + rho_R ) ); // determine fluxes at cell faces phi = max ( phivSf , v_zero ) * rho_L + min ( phivSf , v_zero ) * rho_R ; su rf ac eV ec to rF ie ld phiUp ( ( max ( phivSf , v_zero ) * rho_L * U_L + min ( phivSf , v_zero ) * rho_R * U_R ) + pSf * mesh . Sf () ); solve ( fvm :: ddt ( rho ) + fvc :: div ( phi ) ) ; solve ( fvm :: ddt ( rhoU ) + fvc :: div ( phiUp ) ) ;

For high resolution reconstruction of density and velocity at the cell faces, TVD schemes are 4

Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM

Listing 2: Excerpt from fvSchemes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

timeStepping ddtSchemes { default }

RK4 ; // RK4 / Euler

multistageEuler dtCoeff ;

interpolationSchemes { default reconstruct ( rho ) reconstruct ( U ) ... } gradSchemes { default grad ( U ) grad ( rho ) }

none ; Minmod ; // upwind MinmodV ; // upwind

none ; cellMDLimited Gauss linear 1; cellLimited Gauss linear 1;

employed. In OpenFOAMr , this is done by using the second order interpolation schemes included in the standard distribution. Additional schemes which are not available in the standard distribution, e.g. the SMART scheme [7], can be implemented easily by writing them in the form introduced by Sweby [25]. Constructing higher order schemes like WENO, which use more than only the values in the direct neighbour cells of a given face, is much more difficult for unstructured, three dimensional meshes with arbitrary elements. To implement these schemes, non-trivial changes to the internal code structure of OpenFOAMr are required when parallel computation is involved (thus, for all real-world problems, as they are usually too large to be solved on a single processor core in reasonable time). So far, this has only been done as part of research projects, but is not commonly available [14]. To specify the schemes to be used in hydRUBFoam, entries in the fvSchemes dictionary have to be made accordingly. To reconstruct density and velocities using the MINMOD scheme, for example, one would specify settings as shown in listing 2. Note that for the velocity, the version of the scheme taking into account the direction of the field needs to be used. Furthermore, gradient schemes need to be specified, as the gradient of the values between two neighbouring cells determines if and how much limitation is applied. The use of the cellLimited/cellMDlimited gradient schemes is advised as they are least diffusive. In hydRUB, our common choice is MINMOD for the reconstruction of the density and SMART for the reconstruction of the velocity vector. As the SMART scheme is not available in OpenFOAMr , we used MINMOD for the velocity as well in both codes for the following simulations. Pressure and speed of sound on the cell faces are not reconstructed but obtained from the values in the adjacent cell centres. 5

Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM

Time integration is performed using a low storage four stage Runge-Kutta scheme (RK4) with the coefficients β1 = 0.11, β2 = 0.2766, β3 = 0.5, β4 = 1. Using this scheme, it is possible to resolve even the smallest time scales occuring in cavitation flows, which are in the range of nanoseconds. This leads to accurate reproduction of the compressible wave dynamics involved. As the implemented scheme is an explicit scheme, the size of the time step ∆t is restricted by the CFL (Courant-Fredrichs-Lewy) condition, here only shown for the x-direction [22]:

CFLx =

∆t λx ∆x

with λx = |u| + c

(7)

For first order time integration, the CFL number is required to be less than 1, while the RK4 scheme remains stable for a CFL number up to about 2. It is important to note that in hydRUB, the CFL number is calculated at the cell centres, while in hydRUBFoam the calculation is performed at the cell faces using interpolated values for the velocity and speed of sound. This is due to the fact that hydRUB is a structured code, making the calculation a rather trivial task, as the cell size in every spatial direction can be calculated easily. On the other hand, hydRUBFoam uses an unstructured approach, which supports a wide range of cell shapes, thus making the calculation at the cell faces the straight-forward choice. Trying out different ideas for the calculation of the CFL number is a possible subject for further research, however this is of minor importance. With the current implementation, small differences in the timestep size occur between hydRUB and hydRUBFoam, which can be compensated by slight adjustment to the target CFL number. For the equation of state, a new thermodynamic model derived from the basicThermo class has been implemented. This thermodynamic model, called tabularBaroThermo, relies on the use of lookup tables to determine all relevant thermodynamic quantities (pressure, speed of sound, in future also viscosity) from the density in a given cell by linear interpolation.

4

Validation cases

To validate the correct implementation, various comparisons of calculation results have been conducted. First of all, not included in this study, a comparison was done for three different Riemann problems: Liquid water non-cavitating, air modeled as ideal gas and liquid water cavitating (see also [23]). For further validation on more complex flow problems, the flow around two different hydrofoils was examined.

6

Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM

(a) Overview of computational domain

25

50

pressure tappings (d=0.7 mm) p1 p2 p3 p4 14.6 26.6

axis of inclination

35.1 107.9

(b) Close-up view of O-Grid

(c) Top view of hydrofoil CLE, position of pressure tappings used for dynamic wall pressure distribution measurements (after [16])

Figure 2: Validation test case CLE profile 4.1

CLE hydrofoil

The CLE1 -foil is a 2D hydrofoil which has been subject of various experimental PhD theses at TU Darmstadt [2, 9]. Furthermore, simulation of the cavitating flow around this hydrofoil, using the Sauer-Schnerr mass transfer model, has been done by Frobenius [6]. However, experimental as well as numerical investigations were conducted for cavitation numbers higher than in this study. Hence, no comparison to these results will be made. A cavitation number2 of σ = 1.0 was chosen to compare both codes against each other as lower cavitation numbers lead to a larger cavity, which helps in evaluating the results. The computational domain can be seen in fig. 2. The angle of attack in this case is 5◦ . Although cavitation is a three-dimensional phenomenon, it can be shown that a 2D simulation is sufficient to capture the main features of the flow. The O-Grid of the mesh used for this study consists of 197 × 36 nodes, which is a rather coarse mesh. The far field velocity, imposed on the inlet, is u∞ = 13 m s−1 (leading to a Reynolds number of Re = 1.3 × 106 based on the chord length). The lookup table used for the calculations is water at a temperature of T = 293 K, which leads to a downstream pressure of p∞ = 86 818 Pa. As we are neglecting viscous effects, top and bottom walls of the computational 1 2

circular leading edge p∞ −pvap σ = 0.5ρu with pvap = vapour pressure, u∞ , p∞ = freestream conditions 2 ∞

7

Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM

(a) Pressure signals and corresponding frequencies for hydRUB

(b) Pressure signals and corresponding frequencies for hydRUBFoam

Figure 3: Comparison of hydRUB and hydRUBFoam, σ = 1.0, 1st order in space, 2nd order in time domain as well as the hydrofoil surface are treated as slip walls. To ensure stability and accuracy of the solution, a CFL-numer of CF L = 0.95 is prescribed for both codes. To evaluate the frequency of the cavitation shedding, the method used in the experimental investigation is adapted for this study. Shown in fig. 2(c) are the positions of pressure tappings [16]. In this study, the pressure at four distinct points corresponding to these tappings, close to the hydrofoil surface is sampled over time. To determine the frequencies of the periodic shedding, a fast fourier transformation of these sampled values is done. Shown in figs. 3 and 4 are the results for first order and second order accuracy in space. The Runge-Kutta time integration scheme is used in all cases. Comparing the results, one can see that both codes lead to results in good agreement with each other. From the FFT results, a dominant frequency of fCLE ≈ 32 Hz can be determined, which leads to a Strouhal number Stc =

f ·c ≈ 0.26 u∞ 8

(8)

Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM

(a) Pressure signals and corresponding frequencies for hydRUB

(b) Pressure signals and corresponding frequencies for hydRUBFoam

Figure 4: Comparison of hydRUB and hydRUBFoam - CLE Profile, σ = 1.0, second order in space and time based on the chord length. As already mentioned, no experimental results exist for the simulated cavitation number. However, in a recent diploma thesis conducted at our chair a numerical result of fCLE ≈ 42 Hz was determined by the use of a VOF-method and the SauerSchnerr mass transfer model [4, 8] after careful parameter adjustment. Considering the fact that viscous terms, turbulence and the energy equation were neglected in our calculation, we conclude that our code performs quite well in predicting the dominant shedding frequency without the need for adjusting empirical parameters. 4.2

NACA0015

The NACA0015 hydrofoil can be considered a standard test case for cavitating flows, as it has been widely examined numerically as well as experimentally. Simulations using RayleighPlesset models were conducted for example by Kim [10], Kubota et al. [12], Schnerr and Sauer [20], using a barotropic model by Koop [11], Schnerr et al. [21] and experiments for example by Arndt et al. [1]. It was shown that three different main types of cavitation 9

Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM

(a) Overview of the mesh used for the simulation

(b) Closeup view of the mesh

Figure 5: Validation test case NACA15 occur, depending on the cavitation number and angle of attack. Thus, the flow around thin hydrofoils can be characterized by introducing σ/(2 · α) as a parameter. For small values of σ/(2 · α), a Strouhal number of Stc ≈ 0.2 based on the chord length is obtained (bubbly flow shock wave phenomena as dominating mechanism), whereas for higher values of σ/(2·α), the Strouhal number is Stl = f · lc /U ≈ 0.3 based on the cavity length (dominated by reentrant jet physics)[1]. In our simulations, the prescribed inlet velocity is u∞ = 12 m s−1 . For a chosen cavitation number of σ = 1, at the outflow a pressure of p∞ = 74 200 Pa is imposed. Analogous to the simulations of the CLE hydrofoil, slip wall boundary conditions are used for the hydrofoil surface and the top and bottom channel walls. The angle of attack of the flow is α = 6◦ , which leads to σ/(2·α) ≈ 4.775. The mesh used has a resolution of 181×31 nodes (see fig. 5), which is the same mesh resolution as used by Sezal [22]. To restrict the time step size, a CFL number of CF L = 0.95 is used for hydRUBFoam and CF L = 0.9 for hydRUB respectively. This choice leads to nearly identical step sizes in both simulations (see section 3) due to slight differences in the CFL number calculation. In fig. 6, the time evolution of the global vapour volume fraction as well as its FFT are shown for both codes. Again, comparable results are achieved. Examining the frequency plot, we can see a dominant frequency of fN ACA ≈ 9 Hz. This corresponds to the frequency found by Sezal [22], which was to be expected because of the similarities in the mesh as well as the numerical method. However, using a finer mesh, Koop [11] found a shedding frequency of f ≈ 24 Hz for the same flow conditions. This corresponds to the second peak which can be seen in the frequency plots. It is noteworthy that Schmidt et al. [19] found the occurence of low frequency components in the time evolution of the vapour volume fraction on coarse meshes, which vanish if a sufficiently fine mesh is used. Also worth trying is to evaluate the frequency not for the whole simulated time range (as shown in fig. 6), but only for a small fraction (i.e. few shedding cycles). Additionally, employing the same technique as in the evaluation of the CLE hydrofoil could lead to further clarification which of the main frequencies found from 10

Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM

(a) Time evolution of vapour volume fraction and corresponding frequencies for hydRUB

(b) Time evolution of vapour volume fraction and corresponding frequencies for hydRUBFoam

Figure 6: Comparison of hydRUB and hydRUBFoam - NACA0015 Profile, σ = 1.0, second order in space and time the evaluation of the vapour volume fraction is the actual shedding frequency.

5

Conclusion and outlook

The numerical method which so far was only available in academical CFD codes was successfully implemented in OpenFOAMr . By using two different hydrofoils as example, it has been shown that both codes lead to comparable results. Furthermore, although viscous effects and the energy equation were neglected, there is a good agreement to experimental results and numerical results from simulations employing a Rayleigh-Plesset model respectively. In the future, the extension of the code to solve the energy equation as well is planned. Furthermore, the viscous terms will also be included to incorporate the effects of friction. Another important addition, which is expected to be achievable rather easily because of 11

Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM

existing OpenFOAMr infrastructure, is the inclusion of mesh movement, as this is needed for many industrial applications. Finally, we can conclude that using open source software like OpenFOAMr as a basis for own development is a promising alternative to commercial CFD codes, where the option for extensions by the user are rather limited. Also, because the internal structure commercial codes is unavailable for obvious reasons, implementation of own features can be a rather cumbersome task if debugging is involved when results turn out to be not as expected.

References [1] Roger E.A. Arndt, C.C.S. Song, M. Kjeldsen, J. He, and A. Keller. Instability of Partial Cavitation: A Numerical/Experimental Approach. In TwentyThird Symposium on Naval Hydrodynamics. National Academies Press, 2000. URL http://purl.umn.edu/49781. [2] Ralph B¨ohm. Erfassung und hydrodynamische Beeinflussung fortgeschrittener Kavitationszust¨ande und ihrer erosiven Aggressivit¨at. PhD thesis, TU Darmstadt, Darmstadt, 1998. [3] Christopher E. Brennen. Fundamentals of multiphase flow. Cambridge University Press, Cambridge [England] and New York, 2005. ISBN 0521139988. [4] Christian Deimel, Markus G¨ unther, Romuald Skoda, Tom´aˇs V´ıt, Petra Danˇcov´a, and Petr Novotn´ y. Application of a pressure based CFD code with mass transfer model based on the Rayleigh equation for the numerical simulation of the cavitating flow around a hydrofoil with circular leading edge. EPJ Web of Conferences, 67:02018, 2014. [5] Jean-Pierre Franc and Jean-Marie Michel, editors. Fundamentals of cavitation, volume 76 of Fluid mechanics and its applications. Kluwer Academic Publishers, Dordrecht, 2004. ISBN 1402022336. [6] Moritz Frobenius. Numerische Simulation kavitierender Str¨omungen in hydraulischen Str¨omungsmaschinen. PhD thesis, TU M¨ unchen, M¨ unchen, 2004. [7] P. H. Gaskell and Lau, A. K. C. Curvature-compensated convective transport: SMART, A new boundedness- preserving transport algorithm. International Journal for Numerical Methods in Fluids, 8(6):617–641, 1988. [8] Markus G¨ unther. Numerische Simulation der Str¨omung um ein Tragfl¨ ugelprofil mit OpenFOAM. Diploma thesis, Ruhr-Universit¨at Bochum, Bochum, 2013. [9] Michael Hofmann. Ein Beitrag zur Verminderung des erosiven Potentials kavitierender Str¨omungen. PhD thesis, TU Darmstadt, Darmstadt, 2001.

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Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM

[10] Sung-Eun Kim. A Numerical Study of Unsteady Cavitation on A Hydrofoil. In Proceedings of the 7th International Symposium on Cavitation, 2009. [11] Arjen Hemmy Koop. Numerical Simulation of Unsteady Three-Dimensional Sheet Cavitation. PhD thesis, University of Twente, Enschede, 2008. [12] Akihiro Kubota, Hiroharu Kato, and Hajime Yamaguchi. A new modelling of cavitating flows: a numerical study of unsteady cavitation on a hydrofoil section. Journal of Fluid Mechanics, (240):59–96, 1992. [13] Charles L. Merkle, JinZhang Feng, and Phillip E.O. Buelow. Computational Modeling of the Dynamics of Sheet Cavitation. In Third International Symposium on Cavitation (CAV1998), 1998. [14] Pringuey. High Order Schemes on Three-Dimensional General Polyhedral Meshes Application to the Level Set Method. Communications in Computational Physics, 2012. [15] Jean-Luc Reboud, Benoit Stutz, and Olivier Coutier. Two-phase flow structure of cavitation: experiment and modelling of unsteady effects. In Third International Symposium on Cavitation (CAV1998), 1998. [16] Jean-Luc Reboud, Regiane Fortes-Patella, Michael Hofmann, Henrik Lohrberg, Gerhard Ludwig, and Bernd Stoffel. Numerical and experimental investigations on the selfoscillating behaviour of cloud cavitation: Part 2: Dynamic pressures. In Proceedings of the 3rd ASME/JSME Joint Fluids Engineering Conference, 1999. [17] Ernst Schmidt and Ulrich Grigull. Properties of water and steam in SI-units. Springer, Berlin and Heidelberg and New York, 4th enlarged printing edition, 1989. ISBN 3540096019. [18] Steffen J. Schmidt, Ismail H. Sezal, G¨ unter H. Schnerr, and Matthias Thalhammer. Riemann Techniques for the Simulation of Compressible Liquid Flows with Phase-transition at all Mach numbers - Shock and Wave Dynamics in Cavitating 3-D Micro and Macro Systems. In 46th AIAA Aerospace Sciences Meeting and Exhibit, 2008. [19] Steffen J. Schmidt, Matthias Thalhammer, and G¨ unter H. Schnerr. Inertia controlled instability and small scale structures of sheet and cloud cavitation. In Proceedings of the 7th International Symposium on Cavitation, 2009. [20] G¨ unter H. Schnerr and J¨ urgen Sauer. Physical and Numerical Modeling of Unsteady Cavitation Dynamics. In 4th International Conference on Multiphase Flow, 2001. [21] G¨ unter H. Schnerr, Steffen J. Schmidt, Ismail H. Sezal, and Matthias Thalhammer. Shock and Wave Dynamics of Compressible Liquid Flows with Special Emphasis on Unsteady Load on Hydrofoils and on Cavitation in Injection Nozzles: Invited Lecture. In Sixth International Symposium on Cavitation, 2006.

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Implementation of an explicit density-based solver for the simulation of cavitating flows in OpenFOAM

[22] Ismail H. Sezal. Compressible Dynamics of Cavitating 3-D Multi-Phase Flows. PhD thesis, TU M¨ unchen, M¨ unchen, 2009. [23] Romuald Skoda, Uwe Iben, Alexander Morozov, Michael Mihatsch, Steffen J. Schmidt, and Nikolaus A. Adams. Numerical simulation of collapse induced shock dynamics for the prediction of the geometry, pressure and temperature impact on the cavitation erosion in micro channels. In WIMRC 3rd International Cavitation Forum 2011, 2011. [24] Romuald Skoda, Uwe Iben, Martin G¨ untner, and Rudolf Schilling. Comparison of compressible explicit density-based and implicit pressure-based CFD methods for the simulation of cavitating flows. In Eigth International Symposium on Cavitation (CAV2012), 2012. [25] P. K. Sweby. High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws. SIAM Journal on Numerical Analysis, 21(5):995–1011, 1984. URL http://www.jstor.org/stable/2156939. [26] Wolfgang Wagner and Hans-Joachim Kretzschmar. International steam tables: properties of water and steam, based on the industrial formulation IAPWS-IF97. Springer, Berlin and Heidelberg, 2nd edition, 2008. ISBN 3540742344. [27] Graham B. Wallis. One-dimensional Two-phase Flow. McGraw-Hill, New York, 1969.

Nomenclature c f p t u,v,w x,y,z Rvap T α β γ κ ρ σ Re St

Speed of sound m s−1 Frequency Hz Pressure Pa Time s Velocity m s−1 Cartesian coordinates m −1 −1 Specific gas constant of water vapour (287 J kg K ) Temperature K Angle of attack Runge-Kutta coefficients Vapour volume fraction Heat capacity ratio Density kg m−3 Cavitation number Reynolds number Strouhal number

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