Cav03-GS-9-001 Fifth International Symposium on Cavitation (Cav2003) Osaka, Japan, November 1-4, 2003
A Numerical Study of an Unsteady Turbulent Wake Behind a Cavitating Hydrofoil Qiao Qin St. Anthony Falls Laboratory, University of Minnesota
[email protected]
Charles C.S. Song St. Anthony Falls Laboratory, University of Minnesota
[email protected]
Abstract A detailed numerical study of an unsteady turbulent wake behind a cavitating hydrofoil is carried out. A virtual singlephase cavitation model with barotropic flow assumption is used to analyze the unsteady structures in the far wake behind a cavitating hydrofoil. 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. Five types of unsteady vortex shedding mechanisms are identified in the wake of a cavitating hydrofoil. Introduction The unsteady turbulent wake behind bluff bodies such as cylinders and flat plates has been extensively investigated numerically as well as experimentally (Berger, 1972; Okajima, 1982; Huerre et al, 1990; Jackson, 1987; Kahawita and Wang, 2002). The turbulent wake generated by streamlined bodies such as hydrofoils has been found to be more interactive with the boundary layers formed on the surfaces of hydrofoil (Kornilov, 2002). It is very unsteady if boundary layer separation occurs on the suction surface of a hydrofoil. But the details of flow near the trailing edge are still poorly understood (McCroskey, 1982). The unsteady turbulent wake behind a cavitating hydrofoil is even more complex. This paper primarily focuses on the numerical investigation of the unsteady structures in the wake of a cavitating hydrofoil, especially various types of vortex shedding mechanism. Time-averaged results will also be presented and compared to those from experiments. Cavitation can be defined as the formation of the vapor phase in a liquid. The formation of individual bubbles and subsequent development of attached cavities, bubble clouds, etc., makes the unsteady structures in the wake of a cavitating hydrofoil fundamentally different from those in the wake of a non-cavitating hydrofoil. A detailed description and analysis of cavitation phenomena with different cavitation number and angle of attack can be found in Arndt et al (2000) and Song and Qin (2001). It is well known that cavitation inception occurs at composite parameter (σ/2α) of about 8.5, where σ is the cavitation number and α stands for angle of attack. When σ/2α is slightly lower than 8.5 (between 6 and 8), a bubble cavity will first appear near the leading edge of the suction side where there is a minimum pressure. This bubble slides down
Roger E.A. Arndt St. Anthony Falls Laboratory, University of Minnesota
[email protected]
along the suction surface and collapses near the trailing edge and the collapsed cavity in the form of a large eddy is transported into the wake. Further reducing σ/2α (between 4 and 6) results in the generation of sheet cavitation near the leading edge of the suction side. This cavity grows and breaks off to form a cloud cavity due to the reentrant jet. When σ/2α falls below 4, the flow pattern is changed significantly. The maximum cavity length exceeds the chord length and the cavity oscillates between partial cavity and super cavity. In this case the sheet cavity again starts from the leading edge but grows to the full chord length and eventually sheds to the wake. A selected case of a 2D NACA 0015 hydrofoil at 8 degree angle of attack, cavitation number 1.0 (σ/2α=3.6), Reynolds Number 106 will be used to investigate the details of unsteady structures in the wake. In the case of sheet/cloud cavitation, at least five types of discrete vortex systems are identified in this research. First a large negative vortex structure containing numerous bubbles forms, develops, breaks down and eventually sheds into the wake from the suction side. This is the primary vortex directly related to the cloud cavity, which is found to be rather periodic causing a sharp peak in frequency spectrum. The second vortex system is a positive vortex that is drawn from the pressure side of the foil by the primary negative vortex as it passes the trailing edge. These two vortices form a vortex pair in the wake and moves mainly in the downstream direction but with a slight upward velocity component. A third vortex system is a secondary negative vortex sheds from the suction side, which appears to be weaker in strength and less regular than the primary negative vortex. Its mechanism is not clearly understood. Whenever the secondary negative vortex passes by the trailing edge it also attracts a positive vortex from the pressure side to form a secondary vortex pair, which also moves downstream like the primary vortex pair. The fifth and last vortex system is due to the instability of the positive vortex sheet originating in the pressure side of the hydrofoil. This last vortex system consists only of positive vortices occurring between the vortex pairs, appearing like half the Karman vortex street. The mathematical model for non-cavitating and cavitating flows and their relationship, Large Eddy Simulation (LES) and Sub-Grid Scale (SGS) modeling, and the numerical method will first be summarized. The numerical results contain two parts: time-averaged quantities, which are mainly used for validation purpose and the unsteady characteristics, which are the main objective of this work. It is found that the lift coefficients collapse quite nicely if
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individual lift coefficient is normalized with its own corresponding non-cavitating lift coefficient and is plotted versus the composite parameter σ 2α , regardless of cavitation number and angle of attack. Finally conclusions are drawn based on these results. Governing Equations Song (1996) showed that the compressibility effect may dominate the dynamics of highly unsteady flows even when Mach number is small. Using the following equation for sound speed a, ∂p = a2 , (1) ∂ρ the equation of continuity for compressible fluid may be written as r ∂p r + V • ∇p + ρa 2∇ • V = 0 . (2) ∂t By dimensional analysis, it can be shown that the first term is proportional to M 2 St while the second term is
proportional to M 2 , where M is the Mach number and St is the Strouhal number. The second term is negligible if Mach number is small, but the first term may not be negligible if the Strouhal number is large. When Mach number is small, both density and sound speed are nearly constant so that the equation of continuity may be written as r ∂p + ∇ • ρ 0 a02V = 0 , (3) ∂t where the subscript “0” represents a reference quantity. Equation (3) is called the equation of continuity for weakly compressible flow, which is valid for steady as well as highly unsteady flows as long as the Mach number is small. Cavitation is usually highly unsteady and noisy; the liquid flow should be regarded as weakly compressible rather than incompressible. When cavitation occurs, the density drops sharply as soon as the pressure is below the critical pressure. To simulate this vaporization phenomenon, Song and He (1998) formulated a so-called virtual single-phase flow model as follows 5
ρ = ∑ Ai p i i =0
for
pε < p < pc .
(4)
The coefficients Ai are so chosen that the resulting pressuredensity curve has a desirable shape. Equation (4) smoothly joins with the weakly compressible flow pressure-density relationship at the critical pressure pc p − p0 = a0 (ρ − ρ 0 ) for pc ≤ p . (5) Since the Mach number of the flow inside a cavity may not be small, the fully compressible flow model with barotropic assumption is used. Evaporation requires large amount of latent heat so that the energy equation may be needed if the thermodynamic aspect is of interest. However, as we focus on the cavitation characteristics in the macro sense at this time, thermodynamic effects remain excluded. The general conservation form of equations of continuity and motion can therefore be summarized as follows r ∂U +∇•Q= S , (6) ∂t where,
p ρa 2 u ρu 2 ρu + p − τ xx , Q1 = , U= ρuv − τ ρv xy ρuw − τ xz ρw ρa 2 v ρa 2 w ρvu − τ yx ρwu − τ zx , Q2 = 2 Q = , 3 ρwv − τ zy ρv + p − τ yy 2 ρvw − τ yz ρ w + p − τ zz ∂a 2 ∂a 2 ∂a 2 + ρv + ρw ρu ∂x ∂y ∂z 0 . S= (7) 0 0
Large-Eddy Simulation Approach Turbulent flows are composed of eddies. The scales of eddies ranges from those comparable to the domain of interest down to the Kolmogorov dissipation scale. To resolve all the scales in the flow of engineering interest remains impractical with current digital resources, and likely to be for the next decade. There is evidence that large scales of motion have vigorous interaction with the mean flow and they strongly depend on the flow geometry and the nature of the flow, while the small scales (or Sub-grid Scale, SGS) of the motion as the production of a nonlinear interaction of large eddies tend to be more universal and homogeneous and have less influence on mean flows. The essence of Large-Eddy Simulation (LES) is to explicitly resolve the large scales of the motion and, in the meanwhile, model the small scales (SGS) of the motion in the flow. The LES approach is becoming increasingly popular in the community of CFD because of this nature. After applying a Favre-filtering operation to equation (6), and neglecting the non-linearity of the viscous stresses (Piomelli, 1999), an extra non-linear term occurs, τ Sij = ρ (ui u j − ui u j ) , (8) which are called Sub-grid Scale (SGS) stresses and must be modeled. Here quantities with overbars stand for filtered quantities, or large scale quantities. The overbars will be dropped hereafter for convenience. A popularly used SGS model is the eddy-viscosity model of the form (9) τ Sij = −2 ρυ t S ij , that relates the SGS stresses τ Sij to the Favre-filtered strainrate tensor Sij. In most cases the equilibrium assumption is made to further simplify the problem and an algebraic model for the eddy viscosity is obtained υ t = (C s ∆ ) 2 | S | S ij , (10) where ∆ is the grid size and Cs is the Smagorinsky constant and takes values between 0.1 and 0.23, depending on the flow. This model is also known as the Smagorinsky model. However, the Smagorinsky constant Cs must be decreased in the presence of shear, near solid boundaries or in transitional flows. This can be done by incorporating another damping
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where Cr is the Courant stability factor, a numerical value that is between zero and one.
function. Moin and Kim (1982) proposed the following modification υ t = (C s D f ∆ ) 2 | S | S ij , (11)
Computational Setup and Boundary Conditions A three dimensional LES code is used to simulate a two dimensional NACA 0015 hydrofoil, at 6 and 8 degrees angle of attack, respectively. Only three grids are used in the span wise direction. The flow configuration is shown in Fig. 1. 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 (Fig. 2). 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 ∂P = = 0 (i=1,2,3), and u1=U0, (14) ∂x ∂x Downstream ∂u i ∂P = = 0 (i=1,2,3) and P = Preference . (15) ∂x ∂x 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 noslip. In either case, the pressure condition on foil solid wall is specified in the following way ∂P =0. (16) ∂n
where Df is a nondimensional damping factor. This damping factor will only affect the turbulent viscosity near the solid wall. Despite the lack of universality of the Smagorinsky constant and some other defects, the eddy viscosity model is still a very useful model for LES. To dynamically obtain the Smagorinsky coefficient, Germano (1991) proposed a dynamic eddy viscosity model. In the dynamic model, the Smagorinsky coefficient is determined as the calculation progresses, based on the energy content of the smallest resolved scale. This is accomplished by introducing a secondary test filter whose width is larger than the grid filter width. This is accomplished by introducing a secondary test filter whose width is larger than the grid filter width. Two drawbacks, however, exist behind the dynamic model. One is that the filtering process of the second test filter is considerably more expensive. The other is that the dynamic model is proved not dissipative enough near the wall while Smagorinsky model is too dissipative. The dynamic model, therefore, goes to the other extreme, compared to the Smagorinsky model, in this sense. A more sophisticated SGS model needs to be investigated later on. Numerical Method The finite volume method with a 2nd order accurate MacCormack’s predictor-corrector numerical scheme is used. By integrating equation (6) over each finite volume, and using the divergence theorem, one can get r r ∂U 1 + n • QdA = S . (12) ∂t ∀ A Here U is volume averaged quantity placed at the center of the volume. To meet the numerical stability requirement, the time increment is limited by the following: ∀ ∆t ≤ C r r r , (13) | u • A | +a | A |
∫∫
y 5c c
1.5c
Downstream 0
4c
Upstream
x
Fig. 1 Computational flow configuration
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(a) The grid system enclosing the hydrofoil
(b)
(c)
Fig. 2 Grid system surrounding the NACA 0015 hydrofoil at 6 degree angle of attack (b) A close-up near the leading edge, (c) A close-up near the trailing edge Simulation Results Time-averaged computational results Lift Coefficient. The lift coefficient of a 2D hydrofoil is defined as L Cl = , (17) 1 2 ρU 0 c 2 where L equals the net lift force per unit span working on the hydrofoil, U0 is the free stream velocity and c is the hydrofoil chord length. Different representations of calculated time-averaged lift coefficients at 6 and 8 degree angles of attack, respectively, are presented in figures 3 and 4. Figure 3 simply shows lift coefficient as a function of cavitation number. Noncavitating lift coefficients are calculated quite accurately and found agree very well with experimental data indicating that the weakly compressible flow model works quite well for a true single phase flow, as long as the Mach number is small. Some deviations exist between calculated and measured lift coefficients under cavitating conditions, however. It is interesting and somewhat surprising to note that two lift coefficients collapse as long as cavitation occurs, or in another words, the lift coefficient is independent of the angle of attack after hydrofoil cavitates. The composite parameter ( σ 2α ) appears to be valid for hydrofoils of finite thickness although it is originally from cavitating flat plate theory. When the lift coefficient is normalized by its own non-cavitating lift coefficient and plotted as a function
of the composite parameter ( σ 2α ), it is found that the lift coefficients collapse markedly, regardless of the cavitation number and the angle of attack (fig. 4).
σ Fig. 3 Lift Coefficient as a function of Cavitation Number Pressure Distribution. Pressure coefficient is defined as P − P0 , (18) Cp = 1 ρU 02 2 where P0 is the upstream reference pressure. And the cavitation number is simply defined as the negative of the pressure coefficient, e.g. σ = −C p .
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Wake Velocity Distribution. 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 Figures 7 through 11. White (1974) demonstrates that the growth of the width b of both laminar and turbulent wakes is proportional to the square root of the distance downstream of the trailing edge, while the decay of the minimum velocity in the wake is proportional to x-1/2. That is b( x ) ∝ x , U ref − u ( y ) ∝
σ/2α Fig. 4 Normalized lift coefficient as a function of σ 2α The time averaged pressure coefficient under non-cavitating condition is plotted in Figure 5, together with the theoretical data (Abbott et al, 1959). One point that needs to be stressed here is that the pressure coefficient is one at the trailing edge in Abbott, which turned out to be an error. Batcher (1967) proved that the pressure at the trailing edge is approximately zero instead of one. The calculated data agrees quite well with theoretical data. Figure 6 is the time averaged pressure coefficients under cavitating condition. Because of the nature of this cavitation model (a rather simplified densitypressure relationship instead of true equation of state), the time averaged quantities, such as lift and pressure coefficients, are only good in a qualitative sense.
Fig. 5 Time averaged pressure distribution on the suction side for non-cavitating flow
Fig. 6 Time averaged pressure distribution on both sides of the foil at 8 degree angle of attack, cavitation number 0.5
(19) 1
, (20) x where x is the distance from the trailing edge and Uref is the reference velocity. To follow the White’s idea, plots 7 through 10 are organized with the following definitions y Y'= , (21) xc (U ref − u ( y )) x U'= . (22) U ref c Figures 7 and 8 represent typical viscous wake behind a non-cavitating hydrofoil. The mean velocity distributions at different downstream locations indeed collapse quite nicely. The spread of the wake due primarily to viscosity is quite narrow at 6 and 8 degrees angle of attack. When the hydrofoil cavitates and large vortical structure containing numerous bubbles of different sizes is shed into the wake, the spread of the wake becomes much wider than those from non-cavitating wakes (figures 9 and 10). This is because these clouds of bubbles extend much further in the cross stream direction than the viscous wake associated with noncavitating flow (Kjeldsen et al, 2000). The experimental data at 7 degree angle of attack are also included in figures 11 and 12 for validation. Note that the definition of positive y experimentally is the negative direction computationally. It is easy to observe that the mean velocity distribution becomes systematically narrower as the flow develops further downstream in a cavitating wake, and also 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 the results are presented in Qin et al (2003b).
Fig. 7 Calculated mean velocity profile at 6 degree angle of attack without cavitation
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Fig. 8 Calculated mean velocity profile at 8 degree angle of attack without cavitation
Fig. 9 Calculated mean velocity profile at 6 degree angle of attack, cavitation number 0.5
Fig. 10 Calculated mean velocity profile at 8 degree angle of attack, cavitation number 0.5
(a) With cavitation number of 1.2
(b) without cavitation Fig. 11 Measured mean velocity profile at 7 degree angle of attack (Reprinted from Arndt et al, 2002) Unsteady Structures Unsteady Vortical Structures in the Wake and their Evolution. To facilitate better analysis, the upstreamspecified stream-wise velocity has been removed from the whole velocity field. Figure 14 illustrates the instance of a shed large negative (defined as the normal direction into the paper) vortex structure containing large amount of bubbles in the wake near the trailing edge of the suction side (named as type A vortex) and a positive (defined as the normal direction out of the paper) vortex from the pressure side (named as type B vortex) that is being induced by type A vortex. A few hundred time steps later, a vortex pair containing type A and type B vortices is formed in the wake right after the trailing edge (fig. 15). Further analysis shows that the moving velocity of this vortex pair in the stream wise direction 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 type A vortex is located downstream of 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 the speed roughly equal to the free-stream speed. But, since a finite width channel is considered, the vortex pair eventually will lose its upward speed as it approaches the upper wall. 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. The above observation is consistent with the wider spread of cavitating wake. Figure 16 shows this trend (exactly the same scale as in figure 15). There is also a secondary negative vortex (named type C) formed above the suction side near the trailing edge in figures 15 and 16. This type C vortex is weaker in strength than 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 (figure 17). Another observation is that the strength of vortices becomes weaker when it moves further downstream as shown by figure 18. 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. The mechanism behind it is not well understood. Between two periods of primary vortex shedding, there is certain time period in which only positive vortices (named type E) shed from pressure side (figure 19). This is somewhat similar to half Karman Vortex Street. Also
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type E vortex is weaker in strength than the primary vortices,
but occurs at higher frequency.
Type A Type B
Fig. 14 Generation of large negative vortex structure on the suction side and its induced positive vortex from the pressure side Type A
Type B
Fig. 15 A formed primary vortex pair (type A and type B) in the wake
Type C
Fig. 16 The evolution of the vortex pair in the wake and a type C vortex
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Type C Type D
Fig. 17 A formed secondary vortex pair (type C and type D) in the wake
Type B
Type A Fig. 18 An instance of weakened primary vortex pair (type A and type B) further downstream in the wake
Type E
Fig. 19 Type E positive vortex chain formed in the wake
Type E Fig. 20 A close-up of type E vortex chain Spectral Characteristics. Negative vortex shedding due to its induced positive vortex shedding from the pressure side the sheet/cloud cavitation on the suction side (type A) and (type B) is highly periodic. This is clearly reflected in the
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time series of lift coefficient. The time series of lift coefficient at 6 and 8 degrees angle of attack are presented in figures 21 and 23. Their corresponding results from spectral analysis are presented in figures 22 and 24. It is easily seen that Strouhal number is around 0.2, if the frequency is normalized by the cavity length and the upstream velocity fl St = cav , (23) U where lcav is the cavity length. What is not so clear in Figures 22 and 24 but is evident in figures 21 and 23 is that there are secondary lift fluctuations superimposed in both lift coefficients. This is type E vortex shedding frequency due to the instability of positive vortex sheet. If this frequency is normalized by the projected chord length normal to the stream wise direction fc sin α St = , (24) U the Strouhal number is in the order of 0.2, which is very close to the flow separation phenomenon found in the wake of circular cylinder. Therefore the type E vortex can be regarded as a half Karman Vortex Street. The composite plot of peak frequency at various angles of attack with different cavitation numbers are presented in figure 25. These results are in a good agreement with those from experiments (Kjeldsen et al, 2000).
Fig. 23 Time series of lift coefficient at 8 degree angle of attack, cavitation number 0.5
Fig. 24 Frequency spectrum of the lift coefficient, 8 degree angle of attack, cavitation number 0.5
Fig. 21 Time series of lift coefficient at 6 degree angle of attack, cavitation number 0.5
Fig. 22 Frequency spectrum of lift coefficient, 6 degree angle of attack, cavitation number 0.5
Fig. 25 Composite plot of peak frequency at 2 different angles of attack The spectral characteristic of the shedding vortex pair can be illustrated in another way here. Each figure (26 and 27) contains instantaneous pair of pressure/vorticity fields. The dark color in the pressure plot stands for low pressure while the bright color for high pressure. For the vorticity plot dark color stands for the negative vorticity while the bright color for the positive vorticity. The color upstream of the leading edge pretty much stands for zero vorticity. See the legend on the right side of each figure for detailed reference. Figure 26 illustrates an instance where the newly generated large negative vortex structure from the suction side and its induced positive vortex from the pressure side are about to
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shed into the wake while the previous shed vortex pair is ready to leave the computational domain. Clearly positive vortex follows the negative vortex but both of them have negative pressure so that they are hardly distinguished in the pressure plot. An instance similar to figure 26 at 6 degree angle of attack is presented at figure 27. The distance between two cycles is roughly five times the chord length, making the vortex shedding due to sheet/cloud
cavitation about 0.2, which is in a very good agreement with those from the experiments (Kjeldsen et al, 2000). In addition, the vortex shedding due to the instability of the positive vortex sheet is also apparent between two major vortex pair cycles, which is mainly viscous effects and very similar to the wake behind the circular cylinder.
Fig. 26 A new large vortex structure formed after previous shedding Upper half is the instantaneous pressure distribution and the lower part is the corresponding vorticity, 8 degree angle of attack, cavitation number 0.5
Fig. 27 A new large vortex structure formed after previous shedding Upper half is the instantaneous pressure distribution and the lower part is the corresponding vorticity, 6 degree angle of attack, cavitation number 0.5 Conclusions The unsteady lift and vortex structure behind a cavitating hydrofoil have been investigated using the virtual singlephase cavitating model with a barotropic flow assumption. The time averaged quantities are in good agreement with experimental data and theoretical result indicating that this model can capture the major dynamics of cavitating flows. A closer look at the mean velocity distributions in the cavitating wake reveals that, without consideration of the dissolved gas, the mean velocity distribution becomes systematically narrower than that indicated by experimental data. The time averaged lift coefficients for two different
angle of attack normalized with the value of non-cavitating condition nearly collapses on a straight line when plotted against σ / 2α . 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 half Karman vortex street.
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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 half Karman vortex street. Lift oscillation due to the secondary pair of vortices is less regular but quite significant. Acknowledgements The National Science Foundation (Dr. Michael Plesniak) and the Office of Naval Research (Dr. Kam Ng) sponsor this project. The Minnesota Supercomputing Institute (MSI), University of Minnesota, generously provides the computational resources. References [1] Abbott, I.H. and Doenhoff, A.E.V., 1959, Theory of Wing Sections, Dover Publications, Inc. New York [2] Arndt, R.E.A., 2002, Cavitation in Vortical Flows, Annual Review of Fluid Mechanics, 34, 143-175 [3] 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 [4] 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 [5] Batchelor, G.K., 1967, An Introduction to Fluid Dynamics, Cambridge University Press [6] Berger E., 1972, Periodic Flow Phenomena, Annual Review of Fluid Mech., 4, 313-340 [7] Bourgoyne D.A., Ceccio S.L., Dowling D.R., Park S.J.J., Brewer W., Pankajakshan R., 2000, Hydrofoil Turbulent Boundary Layer Separation at High Reynolds Numbers, Naval Hydrodynamic Symposium [8] Huerre, P. and Monkewitz P., 1990, Local and Global Instabilities in Spatially Developing Flows, Annual Review of Fluid Mech., 22, 473-537 [9] Jackson C.P., 1987, A Finite-element Study of the Onset of Vortex Shedding in Flow past Variously Shaped Bodies, J. Fluid Mech., 182, 23-45 [10] Kahawita, R. and Wang P., 2002, Numerical Simulation of the Wake Flow Behind Trapezoidal Bluff Bodies, Computers & Fluids, 31, 99-112 [11] Kjeldsen, M., Arndt, R.E.A., Effertz, M., 2000, Spectral Characteristics of Sheet/Cloud Cavitation, Journal of Fluids Engineering, 122, 481-487 [12] Kornilov, V.I., Pailhas, G. and Aupoix, B., 2002, Airfoil-Boundary Layer Subjected to a Two-Dimensional Asymmetrical Turbulent Wake, AIAA Journal, 40(8), 15491558 [13] Kubota, A., Kato H. and Yamaguch, H., 1992, A New Modeling of Cavitating Flows: A Numerical Study of Unsteady Cavitation on a Hydrofoil Section, J. Fluid Mech., 240, 59-96 [14] McCroskey, W.J., 1982, Unsteady Airfoils, Annual Review of Fluid Mechanics, 14, 285-311
[15] Piomelli, U., 1999, Large-Eddy Simulation: Achievements and Challenges, Progress in Aerospace Sciences 35(4), 335-362 [16] Okajima, A., 1982, Strouhal Numbers of Rectangular Cylinders, J. Fluid Mech., 123, 379-398 [17] 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. 92101. [18] Song, C.C.S., He J., Zhou F. and Wang G., 1997, Numerical simulation of cavitating and non-cavitating flows over a hydrofoil, SAFL project report no. 402, University of Minnesota [19] Song, C.C.S., Qin, Q., 2001, Numerical Simulation of Unsteady Cavitating Flows, 4th International Symposium on Cavitation, Pasadena, California [20] Song C.S.S. and Yuan, M., 1988, A weakly compressible flow model and rapid convergence methods, Journal of Fluids Engineering, Vol. 110 [21] White, F.M., 1974, Viscous Fluid Flow, McGraw-Hill, New York [22] Wu, T.Y., 1972, Cavity and Wake Flows, Annual Review of Fluid Mech., 4, 243-284 [23] 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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