Experimental and Numerical Investigations of Cavitating Hydrofoils

22nd IAHR Symposium on Hydraulic Machinery and Systems June 29 – July 2, 2004 Stockholm – Sweden

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Experimental and Numerical Investigations of Cavitating Hydrofoils Author

Firm / Institution

City, Country

Lecturer (x)

Roger E.A. Arndt, Charles C.S. Song, Qiao Qin

Saint Anthony Falls Laboratory, University of Minnesota

Minneapolis, Minnesota, USA

x

Abstract This paper reviews our recent research aimed at gaining a better understanding of the problem of unsteady cavitating flows on lifting surfaces. We have explored this issue with extensive numerical and experimental investigations. This includes the development of new numerical models of cavitating flows including the effects of dissolved incondensable gas and ventilation. Experimental data are drawn from experiments carried out at two different scales in two different water tunnels; one in the US at the Saint Anthony Falls Laboratory (SAFL) and the other at the Versuchsanstalt für Wasserbau (VFW) in Obernach, Germany. Although several foil shapes have been examined, examples are drawn from our NACA 0015 hydrofoil experiments for the sake of brevity.

Résumé Cet article donne une vue d’ensemble des recherches récentes réalisées afin d’acquérir une meilleure compréhension de la problématique des écoulements cavitants instationnaires sur des hydrofoils. Cette thématique a été explorée de manière extensive en réalisant à la fois des expériences et des simulations numériques. Ces études incluent le développement de nouveaux modèles numériques des écoulements cavitants tenant compte des effets de la ventilation et des gaz dissous incondensables. Des données ont été extraites d’expériences réalisées à deux échelles différentes dans deux différents tunnels hydrodynamiques; celui du laboratoire de Saint Anthony Falls (SAFL) aux Etats-Unis d'Amérique et celui du Versuchsanstalt fur Wasserbau (VFW) à Obernai en Allemagne. Bien que de nombreuses formes de foils aient été examinées, seuls des exemples provenant de notre expérience hydrofoil NACA 0015 sont présentés par soucis de concision.

Numerical Simulations Cavitating flows are typically turbulent, highly dynamic and highly unstable two-phase (gas/liquid) flows. These features make numerical modeling a unique challenge. With the progressive understanding of the cavitation phenomenon, several modeling strategies have been developed. In earlier work, the main idea was to neglect the density of any vapor formed in the process and treat cavitating flows as free-surface potential flows of the liquid phase. A very significant contribution was made by Tulin (1958, 1964) by developing a smallperturbation (linearized) theory. With the advancement of computational methods, nonlinear analysis of cavitating flows emerged. Brennen (1969) employed a finite difference method to study axisymmetric flows. Recently, several potential-based or velocity-based boundary element methods had been developed by many investigators such as Kinnas & Fine (1993). In all of these studies, panels are placed on the cavity surface, which is determined iteratively by 1


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satisfying both the kinematic and dynamic boundary conditions. However, the difficulty of this approach is how to model the trailing edge where the cavity collapses. This is, of course, beyond the capability of potential flow. A cavity closure condition must be explicitly specified in developing this potential theory. Models dealing with this condition in the literature include the image model, the re-entrant jet model, the transition model of Wu, the singularity model of Tulin, and the open cavity model of Song etc. It should be pointed out that, from the physical point of view, there exists no closure model that satisfactorily describes the physical phenomena associated with cavitation. This is an important issue since the dynamics of oscillating partial cavitation on hydrofoils and blade sections are intimately coupled to the closure region. With further advancement of digital techniques, cavitating flows can now be numerically modeled through direct computation of the single-phase Navier-Stokes equations. A possible simplification of this type of complex flow is to assume the gas-liquid flow as a virtual single phase (Kubota et al 1992; Song et al 1997), with a sharp density change as soon as the pressure drops below some critical pressure. Some promising results have been obtained through this model (Song et al, 1997; Song et al 1998, Arndt et al 2000; Qin et al 2003a), which indicates that this model can capture the main physics of complex cavitating flows. Cavitation is also known to produce air bubbles due to incondensable gas coming out of solution in low pressure (supersaturated) regions of the flow (Arndt et al 2002). However, in all these models, the effect of incondensable gas on cavitation is neglected because a general quantitative correlation between cavitation and incondensable gas is still beyond our grasp. As has been observed in a previous investigation (Qin et al 2003a), the mean velocity distribution in the wake behind a cavitating hydrofoil fits well close to the trailing edge, but deviates systematically from the measurements further downstream (Arndt et al 2002). There is evidence that incondensable gas persists in the gaseous state after it comes out of solution. Hence some insidious effects on the distribution of velocity, pressure and density exist due to the production of bubbly flows. It is conjectured that the deviation of mean velocity distribution in the wake is one of the effects of dissolved gas that has come out of solution. To take the effect of incondensable gas into account, a two-phase cavitation model (revised virtual single-phase) model is now proposed. The calculated results show a significant improvement compared to those from the model that neglects the incondensable gas effect and are in good agreement with those from experiments. One feature of our current work is a detailed numerical study of an unsteady turbulent wake behind a cavitating hydrofoil. The complex wake features are illustrated in Figure 1. A virtual single-phase cavitation model with a barotropic flow assumption is used to analyze the unsteady Figure 1. View of cavitating hydrofoil structures in the far wake behind a cavitating

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hydrofoil. The numerical methodology utilizes Large Eddy Simulation (LES). This allows us to capture the complex dynamics of the flow with great fidelity. The model automatically becomes weakly compressible in the region where there is no cavitation. Time-averaged quantities are in good agreement with those from experiments, which mean that the model can capture the main dynamics of complex cavitating flows. The cavity flow simulated is highly unsteady, which strongly affects the wake flow. Original Single Phase Cavitating Model The general equation of state is ρ = ρ ( P, T ) , (1) where ρ is the density, P is the pressure and T is the temperature. When the thermal process is specified, such as isothermal or adiabatic process, the explicit dependence of ρ on T disappears from equation (1) and the resulting flow is called barotropic. The thermodynamic definition of sound speed a may be written as ∂p = a2 . ∂ρ

(2)

Substitution of equation (2) into the equation of continuity yields r ∂p r + V • ∇p + ρa 2 ∇ • V = 0 . ∂t

(3)

The above equation is a pressure transport equation containing a convection term and a production term. By combining with an equation of motion, the third term can also be shown to represent radiation. Dimensional analysis shows that the first term is in the order of M 2 S t while the second term is in the order of M2, where M is the Mach number and S t is the Strouhal number. The second term is negligible when M is small but the first term may not be negligible if S t is large. Song (1996) has shown that the fist term in equation (3) dominates the motion of highly unsteady flow even when Mach number is small. In this case, it can be shown (Yuan 1988) that the change in density and speed of sound is negligible as well. Song and Yuan referred to flows that satisfy the above condition as “weakly compressible”. Since most cavitating flows are highly unsteady, even water is assumed to be weakly compressible. Once the water vaporizes, the Mach number may no longer be small and the flow is treated as fully compressible. To simulate the vaporization phenomenon in cavitating flows, Song et al (1997) formulated a so-called virtual single-phase flow model 5

ρ = ∑ Ai p i i =0

for

pε < p < p c .

(4)

The coefficients Ai are so chosen that the resulting pressure-density curve has a desirable shape. Integration of equation (2) yields p − p0 = a02 ( ρ − ρ 0 ) , (5) Where the subscript “0” represents a reference quantity. This pressure-density relationship is used wherever flows are weakly compressible, that is, wherever the pressure is greater than the critical pressure pc. Equations (4) and (5) join at pc to cover the entire liquid-water vapor phase of the flows. This model has proved to be very efficient compared to models that devote to tracing rapidly moving and deforming free surfaces. Modified Virtual Single-Phase Cavitation Model 3


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One of the deficiencies in the above model is that any cavitating flow of interest follows a perfectly reversible process between liquid phase and vapor (gas) phase. This is against the experience from experiments. In engineering applications, the operating liquid contains certain amount of incondensable gas in dissolved state. This portion of incondensable gas will remain in gaseous state for a considerably longer period after it comes out of solution. It expends the flow and leads to the existence of local gas volume fraction, which will, in turn, impact on local density, velocity and pressure distributions. To take into account this effect, a revision to the original virtual single-phase model is necessary. It is assumed that the liquid contains certain amount of incondensable gas in dissolved state with mass fraction f, and its corresponding volume fraction α 0 in gaseous state can be deduced from f as ρ α0 = f , (6) ρg Where ρg is the density of incondensable gas and ρ is the composite density of the mixture and is defined as follows (Singhal et al, 2002) 1

ρ

=

f

ρg

+

1− f

ρw

,

(7)

Where ρ w is the density of water. Once the pressure in the flow falls bellow the critical pressure pc, the liquid vaporizes. It is assumed that the fraction of liquid that vaporizes also release its portion of the incondensable gas, that is, α 0 . As the volume fraction of water vapor at that time is much greater than that of the incondensable gas, the volume fraction of incondensable gas can, therefore, be neglected. When the pressure rises above the critical pressure pc, the portion of vaporized water is assumed again to follow the reversible process and condenses to water. However, the incondensable gas will remain gaseous and follow an isothermal process P

ρg

= const .

(8)

The equation of state has the similar form as (5) except that the composite density instead of water density is used here ρ = (1 − α ) ρ w + αρ g , (9) where α is the volume fraction of the incondensable gas after it comes out of solution. It is initialized to α 0 and its subsequent evolution will follow its own equation of continuity r ∂αρ g + ∇ • (αρ gV ) = 0 . (10) ∂t

The general conservation form of equations of continuity and motion is as follows Where,

r ∂U +∇•Q = S , ∂t

(11)

     ρa 2 v ρa 2 w  ρa 2u  p     2     ρu   ρvu − τ yx   ρu + p − τ xx   ρwu − τ zx    U =  ρv  , Q1 =  ρuv − τ xy  , Q2 =  ρv 2 + p − τ yy  , Q3 =  ρwv − τ zy  ,      2     ρvw − τ yz   ρuw − τ xz   ρw + p − τ zz   ρw       αρ g w  αρ g u αρ g  αρ g v  

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 ∂a 2 ∂a 2 ∂a 2  + ρv + ρw  ρu  ∂x ∂y ∂z   0   S= . 0     0   0

(12)

An extra equation is added to compute the incondensable gas volume fraction, α, at the end of each time step. Computational Methodology A Large Eddy Simulation (LES) is used. The technique is described in several of our papers e.g. Qin et al, (2003a, b). The finite volume method with a 2nd order accurate MacCormack’s predictor-corrector numerical method is used. A three dimensional LES code is used to simulate a two dimensional NACA 0015 hydrofoil, at 6 and 8 degrees angle of attack, respectively. Five grids (including two phantom grids) are used in the span wise direction. Since the flow close to the hydrofoil surface and the wake behind the foil are the main interest in the current study, the mesh structure of the computational domain deliberately reflects this concern by heavily clustering the mesh close to the solid surface of the hydrofoil so that the fine mesh encloses the foil and covers the core of the wake behind the foil. The commonly used far-field boundary conditions are implemented at upstream and downstream boundaries. In addition, the inflow stream wise velocity is assigned and the downstream cross-sectional-averaged pressure is controlled as the reference pressure. These conditions are summarized as follows Upstream ∂u i

Downstream

∂x

=

∂P = 0 (i=1,2,3), ∂x

and

u1=U0,

∂ui ∂P ∂C g = 0 (i=1,2,3) and P = Preference . = = ∂x ∂x ∂x

(13)

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A full-slip boundary condition is used at both the upper and lower walls as well as at the artificial boundaries at the ends of the span to avoid the occurrence of viscous boundary layers. Although the mesh is heavily clustered near the solid surface, it is still not fine enough to resolve the thin boundary layer in the non-cavitating scenario. A partial-slip boundary condition (Song 1999) is therefore applied. When cavitation occurs, however, the boundary layer becomes thicker and hence the partial-slip automatically becomes no-slip. In either case, the pressure condition on foil solid wall is specified in the following way ∂P = 0. ∂n

(15)

Experimental Methods The experiments were carried out at two different scales in two different water tunnels. Tests at the Saint Anthony Falls Laboratory (SAFL) were made in a 19 cm square high-speed water tunnel. A complementary set of tests using a geometrically similar configuration was carried

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out in the 30 cm square water tunnel at the Versuchsanstalt für Wasserbau (VAO) in Obernach, Germany. The use of two different facilities permits an investigation of size effects and a determination of whether any of the data are influenced by the dynamic response of the test facility (Svingen et al, 2002). The two facilities also have very different water quality characteristics. Because of its special design, the water quality in the SAFL tunnel is more stable over time during cavitation testing. The water quality in the Obernach facility responds more quickly to changes that are induced by cavitation experimentation. This proved to be an advantage in obtaining new information on the effects of water quality on fully cavitating flows (Arndt and Keller, 2003). The SAFL test setup is shown in Figure 2. In addition to a lift balance, instrumentation included an array of static pressure ports and an array of piezoelectric film transducers, quartz crystal transducers, and miniature accelerometers. The pressure transducers were manufactured from circular sections of polyvinylidine fluoride film.

Figure 2. SAFL Test Setup

Figure 3. Hydrofoil mounted on force balances in the Obernach tunnel

The SAFL facility was also equipped for acoustic studies, Laser Doppler Anemometry (LDA) And Particle Imaging Velocimetry (PIV) for the study of wake characteristics and Phase Doppler Anemometry (PDA) for measurement of bubble size distribution in the wake. Both LDA and PDA capabilities are incorporated into a single system manufactured by TSI Inc. Acoustic measurements were made with the aid of Bruel and Kjaer Type 8103 miniature hydrophones mounted in the dead water space above the test section. The Obernach Cavitation Tunnel has a test section with a cross section of 30 cm x 30 cm and is 2 m long. A specially designed piezoelectric force balance was used to measure unsteady lift and drag. A hydrofoil of 30 cm span and 12.8 cm chord length is mounted on circular plugs in the sidewalls of the test section, allowing for a setting at any desired angle of attack. The plugs are instrumented with piezoelectric force sensors (Kistler) for lift and drag measurements (Figure 3). The force balance has very high frequency response (natural frequency when mounted on flanges f0(x, y) ≈ 3 kHz). The force balances and recording systems were calibrated as a unit using weights. Repeatability was observed to be within 1 percent A Bruel&Kjaer Type 8100 hydrophone is used to measure the noise signal from cavitating test bodies.

Simulation Results

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Mean Wake Data Mean velocity profiles of the wake at various angles of attack at different downstream locations under both non-cavitating and cavitating conditions are presented in Figure 4. Classical theory suggest a similarity profile of the form (U ref − u ( y )) U ref

 y  x = f  c  xc 

(16)

Where x is the distance from the trailing edge, y is measured normal to the flow direction and Uref is the reference velocity. Using this formulation, Figure 4a represents a typical viscous wake behind a non-cavitating hydrofoil. Mean velocity distributions at different downstream locations indeed collapse quite nicely. The spread of the wake due to viscous effects is quite narrow at 8-degrees angle of

b) With cavitation

a) Without cavitation Figure 4. Calculated mean velocity profile at 8° angle of attack with σ = 0.5

b) With a cavitation number of 1.2

a) Without cavitation Figure 5. Measured mean velocity profile at 7-degree angle of attack

attack. When the hydrofoil cavitates, large vortical structures containing numerous bubbles of different sizes are shed into the wake as shown in Figure 1. The spread of the wake becomes considerably wider than those from non-cavitating wakes (Figure 6b). This is because these clouds of bubbles extend much further in the cross-stream direction than the viscous wake associated with non-cavitating flow (Kjeldsen et al, 2000). The experimental data at a 7degree angle of attack are also included in Figure 5 for validation. Note here that the definition of positive y in Figure 7 is the negative direction computationally. It is easy to 7


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observe that the mean velocity distribution becomes systematically narrower as the flow develops further downstream in a cavitating wake; in the meanwhile, the magnitudes of the velocity deficit are greater than that of experimental data. This is believed to be the effect of dissolved gas. A new model that is able to account for the dissolved gas effect has been developed and is discussed below (Qin et al, 2003b). Unsteady Vortical Structures in the Wake and their Evolution Time mean velocity data obscure the complex physics of the wake flow. A series of instantaneous numerical snapshots are shown below to illustrate this point. Five types of unsteady vortex shedding mechanisms are identified in the wake of a cavitating hydrofoil. This is illustrated in Figure 6. To facilitate better analysis, the upstream-specified stream-wise velocity has been removed from the whole velocity field. The top figure illustrates the instance of a large negative (counter clockwise rotation) vortex structure containing a large amount of bubbles being shed into the wake near the trailing edge of the suction side (type A vortex) and a positive (clockwise) vortex from the pressure side (type B vortex) that is being induced by the type A vortex. The vortex pair containing type A and type B vortices moves downstream at a rate that is slower than the free stream velocity. This velocity difference depends on a number of factors, such as cavitation number, angle of attack, etc. Since a type A vortex is located downstream of a type B vortex, the induced velocities tend to move the vortex pair upward in the cross-stream wise direction. As a result, the vortex pair moves diagonally at a speed roughly equal to the free-stream speed. Another observation is that the type A vortex is stronger than the type B vortex so that the relative position of the pair tends to become more parallel to the free stream as they move (middle figure). The above observation is consistent with the wider spread of cavitating wake. There is also a secondary negative vortex (type C) formed above the suction side near the trailing edge. This type C vortex is weaker in strength than the type A vortex, but it also induces a corresponding secondary positive vortex from the pressure side and eventually forms a secondary and weaker vortex pair in the wake (lower figure). Another observation is that the strength of the vortices becomes weaker as they move further downstream. As will be analyzed later, the primary shedding vortex pair of type A and type B is rather periodic while the secondary vortex pair of type C and type D is less regular. A complete understanding of the underlying physics is not at hand. Between two periods of primary vortex shedding, there is certain time period in which only positive vortices (type E) shed from pressure side (also shown in Figure 8). This is somewhat similar to a semi Karman´ vortex street. The type E vortices are weaker in strength than the primary vortices, but occur at higher frequency. Effect of Dissolved Gas Mean velocity profiles in the wake were calculated at two different angles of attack at the same downstream locations as Qin et al (2003a) under the same cavitating conditions. The incondensable gas level in ambient flow is set to 15 ppm. The mean velocity profiles in the wake without inclusion of the effect of non-condensable gas were shown in Figure 4. It is clearly shown in Figure 7 that two improvements are made with the consideration of the effect of incondensable gas. First, the deviation of the mean velocity profile has been greatly alleviated and the mean velocity profile starts to collapse quite nicely for different locations. Secondly the peak of mean velocity defect is getting smaller compared to those data without the consideration of the effect of incondensable gas. This is more realistic compared to the measurements. 8


Type B

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Type A

Figure 7. Calculated mean velocity profile at 8degree angle of attack, cavitation number 0.5 with the effect of incondensable gas (15 ppm)

Type A Type C

Type B

Type C Type D

Type E

Figure 6. Details close to trailing edge illustrating the generation of large negative vortex structure on the suction side and its induced positive vortex from the pressure side. Three instants in time are shown.

Our numerical simulations also indicate that the foil surface upstream of the detached cloud cavity is fully wetted and non-cavitating for a fraction of the oscillation cycle. Another interesting phenomenon can be observed from the simulations as well. They show that a new cavity is induced by the primary cloud cavity at the pressure side near the trailing edge by the arriving primary cavity. Careful observation reveals that the pressure side cavity originates at the trailing edge and spreads upstream against the flow. This means only the radiating pressure wave can explain the phenomenon. In other words, the compressibility effect is shown to be quite important in modeling cavity flows. This phenomenon has also been confirmed experimentally. Finally, the solution for void fraction of incondensable gas can be superimposed on the solution for the wake structure as shown in Figure 8. These results agree qualitatively with experimental visualization. Lift Oscillations Negative vortex shedding due to the sheet/cloud cavitation on the suction side

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.

Figure 8 Evolution of the vortex pair in the wake. The void fraction of incondensable gas is shown in color.

Figure 9 Time series of lift coefficient at 8degree angle of attack, cavitation number 0.5

(type A) and its induced positive vortex shedding from the pressure side (type B) is highly periodic. This is clearly reflected in the time series of lift coefficient. As an example, the time series of lift coefficient at 8 degrees angle of attack is presented in Figure 9. Spectral analysis indicates a very strong tone at a Strouhal number of about 0.2, if the frequency is normalized by chord length and the upstream velocity, St = fc/U. This compares well with the lower branch of oscillation data obtained at Obernach data shown in Figure 10. However, the SAFL spectral data of Kjeldsen et al (2000) indicate a lower value. However, these data do indicate a weaker peak at about the right frequency. Whether or not this is a facility dependent anomaly is not clear (Svingen, et al, 2002).

Figure 10. Frequency of oscillation determined at Obernach with high-speed video (left) and with lift measurements (right). Adapted from Arndt et al (2000)

Summary and Conclusions The unsteady lift and vortex structure behind a cavitating hydrofoil was first investigated using a virtual single-phase cavitating model with a barotropic flow assumption. The wake structure is 10


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found to be quite complex. Five types of unsteady vortex shedding are identified. A pair of primary vortices consisting of a cloud cavity (type A negative vortex) and an induced positive vortex (type B) are shed periodically into the wake region. Between two pairs of primary vortices, there are secondary pairs of vortices shed in less periodical manner. There are also strings of positive vortices formed between primary and secondary pairs of vortices. This fifth type of vortices appears like a semi Karman´ vortex street. The lift coefficient of the cavitating foil oscillates in a manner highly correlated with the 5 types of vortices. The primary mode of lift oscillation correlates with the primary pair of vortex shedding. Its Strouhal number based on the averaged cavity length is approximately equal to 0.2. The lift coefficient also oscillates periodically with Strouhal number approximately equal to 0.2, if the projected foil width is used as the reference length. This oscillation correlates perfectly with the positive vortex string or the semi Karman´ vortex street. Lift oscillation due to the secondary pair of vortices is less regular but quite significant. However, the mean velocity profile in the wake of a cavitating hydrofoil is found to systematically deviate from the measurements further downstream using the previously developed virtual single-phase cavitation model. It is conjectured that this systematic deviation is due to the effect of dissolved gas that has come out of solution. A revised virtual single-phase cavitation model is therefore proposed and the calculated results show a significant improvement compared to those from the model that neglects the incondensable gas effect. Good agreement with experiments was obtained. The numerical model permits detailed observations with great fidelity. This allows the following additional conclusions to be made: a) The cavitation simulated is a highly unsteady phenomenon. The entire sheet cavity including the vapor and gas within it detaches and is carried away periodically to produce a cloud cavity. Therefore, there is no steady state cavity surface across which dissolved gas can be carried into the cavity by diffusion. b) Incondensable gas can serve as a good flow visualization medium in cavitating flow. The importance of compressibility even of water at small Mach number is demonstrated.

Acknowledgements This project is sponsored by the National Science Foundation (Dr. Michael Plesniak, Program Manager) and the Office of Naval Research (Dr. Kam Ng, Program Manager). The computational resources are generously provided by the Minnesota Supercomputing Institute (MSI), University of Minnesota. Dr. Andreas Keller and Dr. Morten Kjeldsen assisted with the experiments at Obernach.

References Arndt, R.E.A. and Keller, A., 2003, “A Case Study of International Cooperation: 30 Years of Collaboration in Cavitation Research” (with AP Keller) Keynote Paper, Proceedings of 4th ASME_JSME Joint Fluids Engineering Conference, Honolulu, Hawaii

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Arndt, R.E.A., Kjeldsen, M., Song, C.C.S., Keller, A., 2002, Analysis of Cavitation Wake Flows, Proceedings of the Hydraulic Machinery and the Systems 21st IAHR Symposium, September 9-12, Lausanne Arndt, R.E.A, Song, C.C.S., Kjeldsen, M., He, J., Keller, A., 2000, Instability of Partial Cavitation: a Numerical/Experimental Approach, 23rd Symposium on Naval Hydrodynamics, September, Rouen, France Brennen, C., 1969, A Numerical Solution of Axisymmetric cavity flows, Journal of Fluid Mechanics 37, 671-688 Kinnas, S.A., Fine N.E., 1993, A Numerical Nonlinear Analysis of the Flow around Two- and Three-dimensional Partially Cavitating Hydrofoils, Journal of Fluid Mechanics 254, 151-181 Kjeldsen, M., Arndt, R.E.A., Effertz, M., 2000, Spectral Characteristics of Sheet/Cloud Cavitation, Journal of Fluids Engineering, 122, 481-487 Kubota, A., Kato H. and Yamaguchi, H., 1992, A New Modeling of Cavitating Flows: A Numerical Study of Unsteady Cavitation on a Hydrofoil Section, J. Fluid Mech., 240, 59-96 Qin, Q., Song, C.S.S., Arndt, R.E.A. A, 2003a, Numerical Study of Unsteady Turbulent Wake behind a Cavitating Hydrofoil, 5th International Symposium on Cavitation, Osaka, Japan Qin, Q., Song, C.S.S., Arndt, R.E.A., 2003b, Incondensable Gas Effect on Unsteady Turbulent Wake behind a Cavitating Hydrofoil, 5th International Symposium on Cavitation, Osaka, Japan Singhal, A.K., Athavale, M.M., Li, H. and Jiang, Y., 2002, Mathematical Basis and Validation of the Full Cavitation Model, Journal of Fluids Engineering, 124, 617-623 Song, C.C.S., 1996, Compressibility Boundary Layer Theory and its Significance in Computational Hydrodynamics, Journal of Hydrodynamics, series B Vol. 8, No. 2, pp 92-101. Song, C.C.S., He J., Zhou F. and Wang G., 1997, Numerical simulation of cavitating and noncavitating flows over a hydrofoil, SAFL project report no. 402, University of Minnesota Song, C.C.S. and He, J., 1998, Numerical Simulation of Cavitating Flows with a Single-Phase Approach, Proceedings of 3rd International Symposium on Cavitation, Grenoble, France Svingen, B., Kjeldsen, M. And Arndt, R.E.A., 2002 “Dynamics of Closed Circuit Hydraulic Model Loops” Proceedings ASME Fluids Engineering Summer Meeting, Montreal, Canada Tulin, M.P., 1958, New Development in the Theory of Supercavitating Flows, Proc. Second Symp. On Naval Hydrodynamics, ONR/ACR-38, 235-260 Tulin, M.P., 1964, Supercavitating Flows – Small Perturbation Theory, Journal of Ship Research 7, 16-37 Yuan, M., 1988, Weakly Compressible Flow Model and Simulations of Vortex-shedding Flows about a Circular Cylinder, PhD thesis, University of Minnesota

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