Cavitation Instabilities

Transcription

1 Cavitation Instabilities Yoshinobu Tsujimoto Engineering Science, Osaka University, 1-3 Toyonaka, Machikaneyama, Osaka, , Japan Introduction It is well known that cavitating flows are generally unsteady mostly with periodic or non-periodic cavity shedding from the trailing edge of the cavities. The amplitude of the fluctuation may become significantly large under certain conditions with a definite frequency component. Here we treat such cases as Cavitation Instabilities. The cause and the mechanisms of the cavitation differ from case to case. The present manuscript is intended to review the cavitation instabilities focusing on their mechanisms. There are two types of cavitation instabilities. One is the system instability and the other is the intrinsic local flow instability. Since the volume of the cavities generally changes under cavitation instabilities, they are basically dependent on the flow system tested. However, there are certain instances where the total cavity volume is kept nearly constant and for such cases the instabilities do not depend on the system. Theoretical analyses started from the system instabilities associated with the POGO instability (Sack and Nottage, 1965) that occurs in the feed line of rocket engines. In the analysis, the effect of cavity is represented by the mass flow gain factor M defined as the decrease of cavity volume due to increase of inlet flow rate, and the cavitation compliance K defined as the decrease of the cavity volume due to the increase of the inlet pressure. This method succeeded to explain cavitation surge and rotationg cavitation. However, the method cannot be used to explain intrinsic cavitation instabilities for which the local flow effects around cavities is important. For the analysis of the local flow around cavities, a linear stability analysis of two-dimensional cavitating flow is proposed based on a closed cavity model with variable cavity length. This analysis clearly shows that the stability depends on the steady cavity length irrespective of the cavity thickness. The characteristic length is the chord length for isolated hydrofoils, and is the blade spacing for the case of cascades. Most of the cavitation instabilities in turbomachines occur in the range where the pressure 1

2 performance is not significantly affected by cavitation and even at design point. This makes the problem more serious leading to fatigue failure and cavitation damage of elements. Under this circumstance, the present text treats the problem as follows. 1. Fundamental treatmant of cavitating flow 1.1 Steady cavitation 1.2 Stability analysis of cavitating flow 1.3 Examples of the correlation of σ /2α with instabilities 2. Cavitating flow instabilities of isolated hydrofoils 2.1 Partial cavity oscillation 2.2 Transitional cavity oscillation 2.3 Discussions based on closed linear cavity model 2.4 Interaction of instabilities with forced oscillations 2.5 First observations of transitional and partial cavity oscillations 3. Cavitation instabilities in inducers 3.1 Experimental Apparatus 3.2 Cell number identification 3.3 Map of oscillating cavitations 3.4 Physical interpretation of each component 4. Cavitation instabilities of hydraulic systems including turbomachinery 4.1 Characteristics of components 4.2 One-dimensional instabilities (Surge and cavitation surge) 4.3 Two-dimensional instabilities (Rotating stall and rotating cavitation) 4.4 Mutual relation of flow instabilities 4.5 Examples of rotating stall and rotating cavitation 5. Non-linear, three dimensional, and compressibility effects on rotating cavitation 5.1 Non-linear effects 5.2 Three-dimensional effects 5.3 Flow compressibility effects 6. Cavitation characteristics-mass flow gain factor and cavitation compliance 6.1 Quasisteady evaluation of mass flow gain factor and cavitation compliance 6.2 Frequency dependence of mass flow gain factor and cavitation compliance 2

3 7. Cavitating flow instabilities of cascades 7.1 Alternate blade cavitation and stability analysis 7.2 Various modes of cavitation instabilities 7.3 Comparison with experiments 7.4 Cavitation instabilities associated with performance degradation-rotating choke 8. Suppression of cavitation instabilities 8.1 Leading edge sweep 8.2 Casing enlargement at the inlet 8.3 Alternate leading edge cutback 9. Tip leakage and inlet backflow vortex cavitation 9.1 Tip leakage cavitation 9.2 Inlet backflow cavitation 10. Further research needs 3

4 1. Fundamental treatment of cavitating flow 1.1 Steady cavitation Cavitation starts to occur when the minimum pressure in the flow reaches the vapor pressure p v at the operating temperature. The potential for cavitation is represented by the cavitation number σ = (p 1 p v )/(ρu 2 /2) where p 1 is the inlet pressure, ρ the density of the liquid, and U the free stream velocity. For turbomachines, the blade tip velocity U T is often used in place ofu as the representative velocity. If we decrease the cavitation number the cavitation starts to appear at a certain cavitation number σ i, which is called inception cavitation number. The inception cavitation number depends largely on the population of cavitation nuclei from which the cavitation starts. If we further decrease the cavitation number, the lift on the hydrofoil first increases for many cases and then starts to decrease. For turbomachines, the developed head starts to decrease. Those cavitation numberes are herein called critical cavitation number and represented by σ c. Further decrease of cavitation number results in significant decrease of lift or head and the cavitation number for this condition is called breakdown cavitation number and represented by σ b. Cavitation instabilities generally occur for σ i <σ < σ c. Cavitation itself can be a source of cavitation nuclei and the onset cavitation number of cavitation instabilities does not depend largely on the original population of cavitation nucleai. For many cases cavitation first appears in the tip leakage vortex or backflow vortices (Fig.1.1) for turbomachines or on the suction surface of the hydrofoils, both in the form of bubble cavitation. With further decrease of cavitation number, the population of the cavitation bubbles increases and eventually results in continuous cavitation. Continuous cavitation on the blade surface is called sheet cavitation. Even with continuous cavitation, cloud of bubble cavitations are shed from the trailing edge of continuous cavitations. So the cavitating flow is always unsteady but the averaged flow is considered to be steady unless the cavity shedding is more or less periodic. In the theoretical treatment in this manuscript only continuous cavitation is considered since most of the cavitation instabilities occur under continuous cavitation and only such treatment is possible at this moment. For continuous cavitation, the cavity surface is treated as a surface of constant pressure. 4

5 From Bernoulli s equation, this condition suggests that the tangential velocity should be constant on the cavity surface. This condition would be violated at the rear stagnation point of the cavity. To overcome this difficulty and to obtain reasonable drag, various cavity closure models have been proposed as shown in Fig.1.2 (Brennen, 1995, Fig.8.2). Among them, we should mention about the reentrant jet model. In this model, a jet flows into the cavity from the cavity closure region and the rear stagnation point is shifted off the free surfaces into the body of the fluid. Re-entrant jets are often observed in real cavity flows, as discussed in Section 2.1. In the mathematical model the reentrant jet disappears into a second Riemann sheet. This represents a deficiency in the model since it implies an unrealistic removal of fluid from the flow. In reality the jet impacts on one of the cavity surfaces and is reentrained in the flow in an unsteady fashion, which often causes periodic cavity shedding. This will be discussed in Section Stability analysis of cavitationg flow We consider a cascade of flat plate blades as shown in Fig.1.3. Details of the analysis are discussed in Section 7. For simplicity, we assume that downstream conduit length is infinite and no velocity fluctuation occurs there. The upstream conduit length is assumed to be finite, L, in x -direction and the conduit is connected to a space with constant (static = total) pressure at the inlet AB. This is intended to determine if the predicted instability is system dependent or not. Since we consider a rotor, we assume that the velocity fluctuation at the inlet AB is normal to the cascade axis. For a stator it is only necessary to assume that the velocity fluctuation is in the direction of the free stream. We assume that all the velocity disturbances are small as compared with the uniform inlet velocity U at upstream infinity, based on the assumption of small incidence angle α and small cavity thickness. We also assume that the boundary condition on the cavity surface can be approximately applied on the blade surface. Under these approximations, the pressure boundary condition on the cavity surface can be represented by the velocity components parallel to the blades, without the normal components. This avoids the problem of cavity closure mentioned in Section 1.1 and a simple closed cavity model is possible. On the other hand, the flow near the cavity closure is only approximate and the detailed flow structure near the cavity closure such as reentrant jet and cavity shedding cannot be treated by the model. However, we adopt the linear closed cavity model to obtain the fundamental characteristics of the cavitation instabilities. 5

6 For the unsteady components, we assume disturbances with the time dependence of e jωt where ω = ω R + jω I is the complex frequency with ω R the frequency and ω I the damping rate, to be determined from the analysis. The velocity disturbance is represented by a source distribution q(s 1 ) on the cavity region, vortex distributions γ 1 (s 1 ) and γ 2 (s 2 ) on the blades, and the free vortex distribution γ t (ξ) downstream of blades, shed from the blades associated with the blade circulation fluctuation. We define the strength of these singularities using a coordinate fixed to the cavity to take account of cavity length fluctuation. If we divide the strength of those singularities and the cavity length into steady and unsteady components, we can represent the velocity with steady uniform velocity (U,Uα ), the steady disturbance ( u s, vs), and the unsteady disturbance ( u, v ): u = U + u s + u e jωt v = Uα + v s + v e jωt (1.1) We assume that α <<1, The boundary conditions are u, v << u s, v s << U and neglect higher order small terms. 1. The pressure on the cavity should equal vapor pressure. 2. The normal velocity on the wetted blade surface should vanish. 3. The cavity should close at (moving) cavity trailing edge. 4. The pressure difference across the blades should vanish at the blade trailing edge (Unsteady Kutta s condition). 5. Upstream and downstream conditions: Since the downstream flow rate fluctuation is suppressed owing to the infinite conduit length, the cavity volume fluctuation is related to the upstream fluctuation. The direction of the velocity fluctuation at the inlet AB is assumed as mentioned before. By specifying the strength of the singularity distributions at discrete points ( ) on the coordinates fixed to the fluctuating cavity as unknowns, we can represent the boundary conditions as follows. S ij For the steady component 6

7 A s (l s ) q s (S 11 )/Uα : γ 1 (S 11 )/Uα : = B s γ 2 (S 11 )/Uα : σ /2α (1.2) and for the unsteady component where A u (l s,ω) q (S 11 ) : γ 1 (S 11 ) : γ 2 (S 11 ) = 0 : u c α l N (1.3) A s (l s ) and A u (l s,ω) are coefficient matrices, is a constant vector. The steady flow can be determined from Eq. (1.2), which shows that the steady cavity length B s l s is a function of σ / 2α. Equation (1.3) is a set of linear homogeneous equations. For the cases with externally forced disturbances such as inlet pressure or flow rate fluctuations, we would have a non-zero vector on the right hand side representing the forced disturbances. For the present cases without any external disturbances, the determinant of the coefficient matrix [ A u (l s,ω)] should equal zero A u (l s,ω) = 0 (1.4) so that we have non-trivial solutions. The complex frequency ω = ω R + jω I is determined from this relation. This equation shows that the frequency ω R and the damping rate ω I as well as possible mode of instability depend only on the steady cavity length given. l s, or equivalently on σ / 2α, once the geometry and other flow conditions are 1.3. Examples of the correlation of σ /2α with instabilities Before going into a detailed examination of the stability analysis, several examples of the correlation are presented. Figure 1.4 shows a map of various oscillating cavitation types observed in a three bladed inducer (Tsujimoto et al., 1997) represented on a suction performance plot. The lines with constant σ / 2α are drawn in the figure. It can be 7

8 found that the boundaries between the cavitation types are nearly parallel to the constant σ / 2α curves. The next example is a surge instability of a marine propeller tested in a cavitation tunnel (Duttweiler and Brennen, 2000). Figure 1.5 shows the occurrence of the instability in a cavitation number and advance ratio map. Since the angle of attack in the vicinity of a propeller blade tip is approximately proportional to the difference, design advance ratio J 0 J 0 J, between the and the operating advance ratio J, a particular value of the parameter ξ = (J 0 J)/σ corresponds to a particular value of σ / 2α. Several lines of constant ξ are plotted in Fig.1.5 where it is clear that the transition between stable and unstable behavior corresponds quite closely to the particular value of ξ = 2.0. Thus the instability boundary corresponds to a particular configuration of steady cavity length on the propeller blade. The third example is the rotating cavitation in a centrifugal impeller of low specific speed (Friedriches and Kosyna, 2001). Figure 1.6 shows the suction performance curve in which the occurrence of rotating cavitation is also shown. The onset conditions of the rotating cavitation at various flow rates are correlated with σ / 2α in Table 1.1. This clearly shows that the rotating cavitation starts to occur atσ / 2α The above examples show that σ / 2α, or the steady cavity length, controls the onset of cavitation instabilities in various types of machines. References in Section 1 Sack, L.E., and Nottage, H.B., (1965), System Oscillations Associated with Cavitating Inducers, ASME Journal of Basic Engineering, December, pp Brennen, C.E., (1995), Cavitation and Bubble Dynamics, Oxford University Press Y.Tsujimoto, Y.Yoshida, Y.Maekawa, S.Watanabe and T.Hashimoto, Observations of Oscillating Cavitation of an Inducer, ASME Journal of Fluids Engineering, Vol.119, No.4, December (1997), pp Duttweiler, M.E., and Brennen, C.,E., (2002), Surge Instability on a Cavitating Propeller, J. Fluid. Mexh, Vol.458, pp Friedrichs, J., and Kosyna, G., (2002), Rotating Cavitation in a Centrigugal Pump Impeller of Low Specific Speed, ASME Journal of Fluids Engineering, Vol.124, No.2, pp

9 2. Cavitating flow instabilities of isolated hydrofoils 2.1 Partial cavity oscillation The cavitation instabilities treated in this section is often called reentrant jet instability or cloud cavitation since they are often accompanied with the reentrant jet and cloud caviy shedding, as shown below. However, it is more convenient here to call them partial cavity oscillation since we are classifying the instabilities based on the cavity length as suggested by the result of the linear stability analysis described in the preceding section. Figure 2.1 shows the cavity break-off cycle observed by De Lange and De Brun (1998) on NACA hydrofoil and the process can be explained as follows, corresponding to the frame number in the figure. 1. Following a break-off, a sheet cavity starts to grow. 2. At a certain length the the growth will cease and a strong reentrant jet is formed at the closure of the cavity. 3. This jet is directed towards the leading edge. The velocity of the jet front is of the same order of magnitude as the free stream velocity. 4. The reentrant jet reaches the cavity near the leading edge and impinges on the cavity interface. 5. The rear part of the sheet cavitaty breaks off and transforms into a bubble cloud which is more or less cylindrical in spanwise direction. Around the cloud there is a large amount of clockwise circulation, originating from the momentum of the jet, being opposite direction to the main flow. 6. The front part of the original cavity is reduced to a tiny sheet cavity, which will grow again and the cycle starts all over. The cloud will be transported in downstream direction. Figure 2.2 shows the map of various cavitation pattern for a plano-convex hydrofoil, in the cavitation number σ v vs incidence α plane Le, Franc and Michel, The cavity length l /c (c is the chord length) and the maximum cavity thickness e /c are also plotted in the figure. The zone of periodic cavity shedding is shown by the shading. This zone locates around the line of l /c = 0.5 and within the range of 2.5% < e /c < 7.5%. This result shows that the steady cavity length is the most important factor for the cavitation instability. The reentrant jet was observed also for this case and the jet traveling time was about 1/3 of the total period of oscillation. This 9

10 can be explained if we assume that the traveling velocity of the jet is about 90% of the free stream velocity. Figure 2.3 compares the Strouhal number St = f L cav /U based on the maximum cavitaty length L cav (Kawanami, Kubota and Yamaguchi, 1998). Irrespective of the difference of the foil shape and the Reynolds number, as well as the facility tested, the Stroulal number falls within the range between 0.25 and In some cases the Strouhal number increases with the cavitation number as shown in the figure but in other cases the Strouhal number is kept nearly constant. The increase of Strouhal number with the cavitation number can be explained by the increase of reentrant jet velocity which can be approximated by U 1+σ. The contribution of the reentrant jet is clearly shown by an experiment by Kawanami et al They placed an obstacle of 2mm square cross section at 37% chord position from the leading edge, on the suction surface of a hydrofoil with the chord length c =150mm, to prevent the reentrant jet to proceed to the leading edge. Figure 2.4 shows the location of the obstacle, the picture of the cavity and the pressure fluctuation at 10% chord position. It is clearly shown that the large scale cavity shedding is prevented although there still exists a small scale cavity shedding. In order that the reentrant jet can reach the leading edge and cause large scale cavity shedding, the reentrant jet should have sufficient velocity at the cavity trailing edge to overcome the frictional force exerted by the blade surface. Although the jet velocity primarily depends on the free stream velocity and the cavitation number, it will be accelerated if we have adverse pressure gradient near the cavity trailing edge. In order to examine the effect of the pressure gradient, a series of experiments were carried out by Callenaere et al.,(2001) for the case of the cavity behind a diverging step as shown in Fig.2.5. The magnitude of the velocity fluctuation and the mean adverse pressure gradient is shown in Fig.2.6 in a plane of cavitation number and the step height. They correlate fairly well. In order to confirm the correlation, tests with decreased confinement and also with decreased divergence angle were carried out. For both cases the adverse pressure gradient is decreased and the region of periodical cavity shedding was significantly decreased, as expected. It is also necessary that the cavity is thick enough to prevent the reentrant jet to interfere with the cavity surface before the jet reaches the leading edge. The minimum cavity thickness e /c = 2.5% in Fig. 2.2 may suggest this restriction. The picture shown in Fig. 10

11 2.4 for the case with the obstacle shows a small scale cavity shedding caused by the interaction of the reentrant jet at the location of the obstacle. In summary, the following two condition will be required for the large scale cavity shedding (Franc, 2001) 1. The cavity must close in a region of large enough adverse pressure gradient favorable to the development of the reentrant jet. 2. The cavity must be thick enough to limit the interaction between the reentrant jet and the cavity interfere before the jet reaches the leading edge. The unsteady reentrant jet formation can be simulated numerically by using a boundary integral method (De Lange and De Bruin, 1998), as shown in Fig.2.7. The computation was aborted when the reentrant jet touched the cavity surface and the entire process of cavity shedding could not be simulated. With smaller angle of attack the reentrant jet touched the cavity surface at a location closer to the cavity trailing edge. The total time for the cavity and reentrantjet growth was about 1/3 of the period of cavity shedding estimated from the Strouhal number of Kubota, Kato and Yamaguchi (1992) succeeded in simulating the cavity break-off process by using a method called bubble two-phase method. The nuclei density is assumed to be constant and the bubble growth is computed from a modified Reyleigh-Plesset equation under the pressure history experienced by the bubble while it travels on the flow. This cavitation model provides the information on the void fraction to a Navier-Stokes solver which determines the velocity and pressure field. This model can simulate not only bubble cloud cavitation but also a sheet cavitation as a region of higher void fraction. Figure 2.8 shows the void fraction contour. A cloud of cavitation is shed at T=5.9. In this result a new separation cavity appears near the leading edge which cause a jet flow towards the blade surface between the old and new clouds. This jet flow blows off the old clouds. Unfortunately no discussions are made about the cavity shedding frequency. Song and He (1998) succeeded to simulate the whole cycle of the shedding process as shown in Fig.2.9. A barotropic model is employed in which a homogeneous liquid-vapor mixture is considered for which a state law of barotropic type is assumed. The foil shape is the same as for Fig.2.4 and a reentrant should be appearing in the experiment. However, no clear indication of the reentrant jet can be seen in the numerical results, perhaps caused by the limited resolution of the calculation. This may suggest that the flow arrangement which might cause reentrant jet is more important than 11

12 the reentrant jet itself. In all of the examples introduced here, the dependence of the instability on the hydraulic system was not studied systematically. However, the Strouhal numbers from different facilities falls within a small region as shown in Fig.2.3. This may suggest that the total volume of the cavities composed of sheet cavity on the hydrofoil and the shed cloud cavity is kept nearly constant. 2.2 Transitional cavity oscillation Linearized closed cavity models provide the relations σ 2 l /c l /c = 2α l /c(1 l /c) for partial cavity with l /c <1 (Acosta, 1955) and σ 2α = 1 l /c 1 (2.2) (2.1) for super cavity with l /c >1(Geurst, 1959) where l is the length of cavity and is the chord length of a flat plate hydrofoil. These results are plotted in Fig It is seen that σ /(2α) has a minimum value when the cavity-chord ratio l /c is about We also find that the value of σ /(2α) tends to infinity as we approach l /c =1 from both partial and super cavity sides. Under this situation, it is generally believed that the linear solutions of Eq.(2.1) and (2.2) are not applicable for 0.75 < l /c < some value l arger than 1. On the other hand, it is empirically known that the cavitation becomes unstable if the value of σ /(2α) is decreased below a value corresponding to the minimum of the curve of Eq.(2.1). In a series of tests on cascades with the solidity 1.25 and 0.625, Wade and Acosta (1967) reports that they observed unsteady cavitation in a region with 0.8 < l /c <1.2, irrespective of the value of the incidence angle 2deg < α < 4 deg. The Strouhal number based on the chord length was This value is significantly smaller than the value for the partial cavity instability treated in the last section. So, it would be appropriate to distinguish the transitional cavity oscillation from the partial cavity oscillation. The first example showing the difference is provided by Kjeldsen, Arndt and Efferts (2000) for NACA-0015 hydrofoil. Figure 2.11 shows the Strouhal number of the oscillation for the angle of attack α = 7deg. against the cavitation numberσ and the normalized cavity length l /c. Two types of flow unsteadiness are observed. One is for c 12

13 l /c > 0.7 and the Strouhal number St b = fc /U based on the chordlength is kept nearly constant, This value is very close to the value 0.14 observed by Wade and Acosta (1967) it is considered to be the transitional cavity oscillation. The other is for l /c < 0.7 and the Strouhal number based on the chordlength increases as the decrease of l /c. This is considered to be the partial cavity oscillation. The Strouhal number based on the cavity length St c = fl /U is kept nearly constant, 0.3. Figure 2.12 shows the Strouhal number for various angles of attack for the case of transitional cavity oscillation. It is kept nearly constant irrespective of the large difference in the angle of attack α. We should note that bubble cavitation was observed for α < 4 deg, while sheet cavitation was observed for α > 4 deg. The partial cavity oscillation was successfully simulated by the barotropic model by Song and He (1998). In order to examine the dependence of the partial and transitional cavity oscillations on the hydrauric system, a series of tests was conducted (Sato et al., 2002) using the test section as shown in Fig The cross section of the conduit is reduced with the area ratio 4.65 upstream of the test section and the location of the contraction has been changed. The distance between the leading edge of the test foil and the contraction is called the inlet conduit length and represented by L. A flat hydrofoil with a sharp leading edge as shown in Fig.2.14 has been used with the flat surface as the suction side and the angle of attack α is defined as the angle between the flow and the flat suction surface. Figure 2.15 shows the spectrum of inlet pressure fluctuation measured at 1.7c upstream for various normalized cavity length L c * = l /c for α =1.5deg and α = 5.0 deg. The Strouhal number St b = fc /U is based on the chord length. For both angle of attack we find a sharp peak for longer cavities with L c * > The Strouhal number S is kept nearly constant and is This value is quite close to the case of Wade and Acosta (1967) and Kjeldsen et al.(2000) and the instability corresponds to the transitional cavity oscillation. For shorter cavities with L c * < 0.75, we observe a broadband component with a peak frequency which increases as the cavity length is decreased. This is very similar to the partial cavity oscillation shown in Fig.2.3. The frequency is larger for the case with α =1.5deg. The Strouhal numbers St b = fc /U and St c = fl /U (average cavity length l is used here) are shown for various cases in Fig.2.16 and For α =1.5deg (Fig.2.16), we observe no significant dependence on the inlet conduit length, for the transitional and partial cavity oscillations. The value of St c, about 0.1, for partial d t b 13

14 cavity oscillation is significantly smaller than the normal value shown in Fig.2.3. For α = 5.0deg (Fig.2.17) the frequency of partial cavity oscillation is higher than the case with α =1.5deg and the value of St c varies from 0.1 to 0.5. The frequency of the partial cavity oscillation is higher for shorter inlet conduit and approximately scales with 1/ L d /c, which is obtained by assuming a constant cavitation compliance. The frequency of the transitional cavity oscillation does not depend on the inlet conduit length significantly. Figures 2.18 and 2.19 compare the inlet pressure fluctuation and the fluctuation of cavity length. For the transitional cavity oscillation shown in Fig.2.18 they correlate fairly well without any phase difference. For the partial cavity oscillation shown in Fig.2.19, we observe a small phase delay of cavity length fluctuation behind the pressure fluctuation and the poor periodicity as compared with the case of transitional cavity oscillation. Reentrant jets were observed both for partial and transitional cavity oscillations. In order to see the effects of the reentrant jets, a bar with 3mm*3mm square cross section (chord length: c = 70mm ) was placed at (1/ 3)c downstream of the leading edge. Figure 2.20 shows the location of the reentrant jet and the location of the end of the shed cavity, for the cases with and without the bar. We find that the cavity is shed when the reentrant jets reach the leading edge or the bar. The spectrum of the inlet pressure fluctuation is shown in Figs.2.21 and We observe that the frequency is the same for the cases with and without the bar and the magnitude is even larger for the case with the bar. This is quite different from the case of partial cavity oscillation shown in Fig.2.4, and shows that the reentrant jet does exist but does not play an important role for the case of transitional cavity oscillation. 2.3 Discussions based on closed linear cavity model Stability analysis To obtain the fundamental characteristics of cavitation instabilities with isolated hydrofoils, the linear stability analysis described in Section 1.2 has been carried out for a cascade with the solidity C / h = 0.1 and β = 74deg (Watanabe et al., 1998). Here, it is assumed that the disturbance has the same phase for all the blades. The relation between the parameter σ /(2α) and the steady cavity length l s is shown in Fig. 2.23(a). The minimum of σ /(2α) is found at l s / C = The Strouhal number St =ωl s /U of 14

15 amplifying modes are shown in Fig. 2.23(b). Mode I appears for l s /C > 0.78 and the frequency is exactly to zero. In order to understand this result, we consider a one dimensional flow in the inlet conduit with the length L, assuming the uniform velocity fluctuation δu 1 in the conduit and the pressure fluctuation δp 1 at the inlet of the cascade or the exit of the inlet conduit. If we assume no pressure fluctuation at the conduit inlet, the momentum equation can be represented as ρl dδu 1 dt = δp 1 If we assume that there is no velocity fluctuation in the downstream of the cascade, the continuity relation requires hcosβδu 1 = dv c dt where V c is the volume of the cavity per blade. The cavitation compliance K is defined as K = ρu 2 /2 h 2 V c p 1 By combining the above relations, we obtain L d 2 δu 1 dt U 2 cosβ δu 1 h K = 0 Here, it has been assumed that the cavity volume V c is a function of inlet pressure δp 1 since the angle of attack is assumed to be constant here. This equation shows that the fluctuation increases exponentially (the frequency is zero) if the cavitation compliance K is negative. Figure 2.23(c) shows the cavitation compliance. The value of the compliance becomes negative for l s /C > This result shows that Mode I shows the statical instability caused by the negative cavitation compliance, which is much the same as the instability of a mass-spring system with a negative spring constant. We also have Mode II and III whose Strouhal numbers based on the cavity length are constant and appearing for shorter partial cavities. However, the Strouhal numbers using the frequency in place of the angular frequencies are about 1.25 and 2.28 for mode II and III respectively. They are much larger than the value of for the partial cavitation instabilities. As a conclusion, the linear stability analysis based on a closed cavity model fails to predict both partial and transitional cavity oscillations. However, it clearly shows that 15

16 the branch of steady cavitation longer than 78% of the chord is statically unstable caused by the negative cavitation compliance, K < 0, in that region Time marching calculations By considering the unsteadiness of the cavities, Tulin (1980) obtained the solutions for growing and collapsing cavities for various growth rates Ý V of cavity volume, as shown in Fig.2.24 (C L = 2πα for two dimensional flow). If we assume a disturbance for the case of V Ý = 0 and l /c > 0.75, the cavity will keep growing ( V Ý > 0) to super cavity or shrink ( V Ý < 0) to the other branch with l /c < On the other hand, the results suggest that the branch l /c < 0.75 with Ý V = 0 is stable. hysteresis schematically shown in Fig.2.25 is suggested. Based on this result, a To confirm the above mechanism for the transitional cavity oscillation, a time marching calculation was carried out (Watanabe et al., 2001). In this calculation, a linear closed cavity is assumed but the cavity length is allowed to oscillate with finite amplitude. Using mirror images, a hydrofoil in a channel with a constant height H and a finite upstream length L is considered (Fig.2.26). Velocity fluctuation is not allowed in the downstream assuming that the downstream channel length is infinite. The steady cavity length under this geometry is shown in Fig.2.27 by a solid line. Figure 2.28 shows the time histories of cavitation number at duct inlet σ L, cavity length l /C, cavity volume V c /C 2 and the pressure coefficient at the leading edge which is obtained by adjusting the pressure difference caused by the inertia effects of the fluid in the duct on σ L, for the case when the inlet cavitation number σ L is increased from to at t /(C /U) = 0. It is shown that the cavity length tends to a new equilibrium value of 0.197, after a few cycles of oscillation. This shows that the shorter branch of the steady solution is stable. On the other hand, Fig shows the results when the inlet cavitation number σ L is decreased from 0.35 to 0.2 within t = 0 30C /U. We have a large amplitude limit cycle oscillation. The direction of the trajectory agrees with that suggested by the result of Fig To consider the cause of the transitional cavity oscillation, the cavitation compliance is shown in Fig.2.30 also for super cavities for the present case. The cavitation compliance is negative also for short super cavities with l /c <1.15. For the explanation of transitional cavity oscillation, the static instability of short super cavitation caused by negative compliance would be more important than the instability of longer branch of the steady partial cavitation. That is, the cavitation compliance is negative C p 16

17 ( K < 0 ) for cavities with 0.75 < l /C <1.15 and there is no statically stable steady solution for the cavitation number between those for l /C = 0.75 and l /C =1.15. Figure 2.31 compares the frequencies obtained by the present simulation with that of experiments shown in Figs We have a reasonable agreement. For partial cavity oscillations, we obtained only damping oscillations in the simulation as shown in Fig In order to study about the damping oscillations, a stability analysis similar to Fig.2.23 was carried out including damping modes. Figure 2.32 shows the complex frequencies of the modes including damping ones, for various cavity lengths, after normalizing them using the mean cavity length l and the free-stream velocity U. Those modes obtained are named Mode 0- Mode VII, depending on the value of their frequency ω R. Mode 0, with frequency zero, becomes amplifying for l /C > 0.73, corresponding to the static instability of longer partial cavityes, or transitional cavity oscillation. Figure 2.33 shows the comparison of the frequency of the damping mode, Mode I with that of the damping oscillation shown in Fig They agree reasonably for the cases of various inlet duct length. This shows that the partial cavity oscillation observed in experiments may correspond to the damping mode. As shown in Fig.2.31, the frequencies of the partial cavity oscillations and the transitional cavity oscillation obtained by the time marching calculations depend on the inlet conduit length. However, the frequency observed in the experiment does not depend on the conduit length at least for transitional cavity oscillations. This is perhaps caused by the cavity shedding which was neglected in the calculations. In summary, partial and transitional cavity oscillations cavitations are explained as follows. Partial cavity oscillation: Partial cavitation is basically stable for disturbances but it does have a damping mode. One of the damping modes is destabilized by the reentrant jet or cavity shedding. However, the mechanism of the destabilization is not cleat at this moment. Also it is not explained why the amplifying modes predicted by the stability analysis have not been observed in experiments. Transitional cavity oscillation: Shorter super cavitation is statically unstable caused by the negative cavitation compliance, as well as the longer branch of partial cavitation. That is, there is no stable steady solution for the cavities 0.75 < l /C <1.15 and only unsteady cavitation is 17

18 possible. This is the main reason of the transitional cavity oscillation. Reentrant jet do exist for transitional cavity oscillations but it does not play an important role for this instability. 2.4 Interaction of instabilities with forced oscillations In order to study the interaction of the cavitation instabilities with the forced oscillation of a foil, pitching oscillation was given to the foil treated in Section 2.2, with the center of the pitching motion at midchord (Hashimoto, et al., 2001). The apparatus and the foil are shown in Figs and Figure 2.34 shows the Stroulal number St b = fc /U and the amplitude of the pressure fluctuation A p * = 2 p/ ρu 2 of cavitation instabilities measured at the inlet (see Fig.2.14) with the foil fixed at an angle of attack α =1.5 deg. The amplitude becomes maximum at the mean cavity length L cm l /C =1.0. Figure 2.35 shows the spectra of inlet pressure fluctuation with the foil fixed ( St f = f f C /U = 0) or pitched around the mean angle of attack α =1.5 deg, with the amplitude α =±1deg and with the non-dimensional frequency St f = f f C /U shown in the figure. When the forcing frequency St f is smaller ( St f = ) than the transitional cavity oscillation frequency St b = 0.13, the amplitude of the transitional cavity oscillation at St b = 0.13 is significantly decreased and the components with the forcing frequency St f and its hamonics appear. The magnitude of the harmonics becomes maximum near the frequency half of the transitional cavity oscillation frequency St b. When the forcing frequency St f is closer to the cavity oscillation frequency St b ( St f = 0.105, 0.14 ), the component caused by the transitional cavity oscillation totally disappears. It reappears when the forcing frequency ( St f = 0.42) than St b. St f is significantly larger Figure 2.36 shows the fluctuation of cavity length L c * = l(t)/c with various forcing frequency St f, with the mean cavity length L cm * = l m /C =1.0. The amplitude of the cavity length fluctuation first slightly increases ( St f =0.035 to 0.105) and then decreases, and the phase delays as we increase the forcing frequency. Figure 2.37 shows the amplitude of the cavity length fluctuation A Lc * = l c /C and the phase β α Lc of the cavity length oscillation. β α Lc < 0 means the phase delay of cavity length oscillation behind the pitching oscillation. Except for the case with L cm * =1.0, the amplitude is monotonically decreased with the increase of the forcing frequency St f. 18

19 For the case of L cm * =1.0, the amplitude becomes maximum at St f = 0.1. This is considered to be caused by the interaction with the transitional cavity oscillation but the amplitude is smaller at St f = 0.15 which is closer to the transitional cavity oscillation frequency of St b = The phase is generally delayed as we increase the forcing frequency. Figure 2.38 shows the amplitude and the phase of inlet pressure fluctuation. For partial cavitation ( L cm * = 0.3, 0.5, 0.7 ) the amplitude increases monotonically and becomes almost constant at St f = 0.3. This frequency St f = 0.3 corresponds to the partial cavity oscillation frequency shown in Fig With L cm * = 1.0, the amplitude becomes maximum at St f = 0.15, which is caused by the interaction with transitional cavity oscillation. The phase β α p delays monotonically from 180 deg. For super cavities ( L cm * =1.5, 2.0 ), both amplitude and the phase are nearly constant. Two types of calculations are performed. The first is a liniar calculation assuming the cavity length fluctuation to be small, similar to the stability analysis treated in Section 1.2. With the foil oscillation, we will have a non-zero term on the right hand side of Eq. (1.3) representing the effect of foil oscillation. The unsteady flow components are determined by solving Eq. (1.3) by providing the frequency of the forced frequency St f. This method can be applied only to partial cavitation with L cm * < 0.75 or super cavitation with L cm * > 1.0, since the mean flow within the range 0.75< L cm * < 1.0 is unrealistic in the linearized calculations. For the range with transitional cavity oscillation, a time marching calculations described in Section are carried out. For both cases, a foil in a semi-infinite duct as shown in Fig is considered by using mirror images of the singularities. Figure 2.39 compares the amplitude and phase of cavity length and inlet pressure fluctuation of partial cavitation obtained from the experiment and from the linear calculations. Althogh the amplitude of the cavity length fluctuation is overpredicted, the general tendency is reasonably simulated by the model. We should note that the saturation of the pressure amplitude around St f = 0.3, which might be associated with the partial cavity oscillation, is well simulated by the calculation. The peak frequency of the inlet pressure fluctuation by the calculation increases as we decrease the mean cavity length L cm *in the calculation. Figure 2.40 compares the results of experiments and linear calculations for super 19

20 cavities. Linear calculations show a singularity near St f = 0.1 but such behavior is not found in experiment. Except for the singular behavior, the linear calculations can simulate the general tendency observed in experiments. In order to examine the interaction between the transitional cavity oscillation and the forced pitching oscillation, time marching calculations similar to those in Section have been carried out. Figures 2.41 and 2.42 show the cavity length fluctuation for the case of L cm * =1.0, α 0 =1.5deg, α =1deg. The wave form is composed of the forced oscillation and the transitional cavity oscillation. The time marching calculations generally overpredict the transitional cavity oscillation. At the resonance condition St f = 0.14 the calculation diverged. Figures 2.43 and 2.44 show the results with smaller amplitude of foil oscillation, L cm * =1.0, α 0 =1.5deg, α = 0.25deg. In this case the calculation at St f = 0.14 was possible. Athough the magnitude of the cavity length oscillation is much larger than experiment, the frequency and the phase relation at St f = 0.14 agree with experiment. In the experiment, the amplitude of the transitional cavity oscillation is kept nearly contant. This might be caused by the cavity break-off which is neglected in the calculations. 2.5 First observations of transitional and partial cavity oscillations Prior to the work of Wade and Acosta (1967) on cascade, detailed observations of transitional cavity oscillation of a plano-convex hydrofoil has been made by the same authors (1966). This is the first detailed report on the transitional cavity oscillation and many important aspects of the oscillation, as well as the general characteristics of steady cavitation are reported. Cavity oscillations similar to the partial cavity oscillation have been observed by Knapp (1955) on bodies of revolution with a semicircular nose, a parallel midsection, and an ogive afterbody. Since these pioneering works provides most of the important characteristics of the oscillations, major results of the reports are reproduced in this section Transitional cavity oscillation (Wade and Acosta, 1966) Steady Cavitation Figure 2.45 shows the plano-convex foil shape and the definitions of the forces on the foil. The chord c and the span s of the foil was 2.77 in and 2.85 in, respectively. Fig shows the lift coefficient C = L/(csρV 2 L /2), drag coefficient C = D/(csρV 2 D /2), 20

21 and the moment coefficient C M = D/(c 2 sρv 2 /2)under non-cavitating condition, plotted against the angle of attack α. It is seen that at about 1 deg angle of attack, there is a slight stalling effect in the lift curve with a corresponding increase in the drag. This effect is characteristics of certain sharp-nose aerofoils and is due to the type of boundary-layer separation occurring on the foil. This wave in the lift curve comes about because of the type of laminar separation of the boundary layer at the leading edge and its subsequent turbulent re-attachment. This hump in the lift curve can be removed by increasing the Reynolds number to approximately or by increasing the nose surface roughness. Figs.2.47 and 2.48 show, respectively, the variation of the lift-to-drag ratio and the center-of-pressure location with angle of attack, the kinks in these curves being due, once again, to the boundary-layer separation. Figure 2.49 shows the comparison of the measured cavitation number K = (p p k )/(ρv 2 /2) based on the cavity pressure K v = (p p v )/(ρv 2 /2) based on the vapor pressure the discrepancy is caused by the fact that the cavity pressure is always higher than the vapor pressure due to the contribution of non-condensable gas pressure. For cavitating flow, the values of the force coefficients as a function of the measured cavitation number are shown in Figs , each graph being for a different angle of attack. The subsequent photographs indicate the degree of cavitation occurring on the hydrofoil at a few different cavitation numbers which are marked on the graphs. We should note that the lift coefficient becomes maximum when the cavity length l becomes nearly equal to the chord length c. This is because the cavity provides additional camber to the foil, viewed from the external flow. Figs show graphs of the cavitation number divided by angle of attack as a function of cavity length. The solid points are those occurring in the unsteady flow regime. As can be seen, the unsteady region occurs over a region of approximately 0.6 l /c to 1.2 l /c, regardless of the angle of attack. This agrees with the results of the experiments by Kjeldsen et al., (2000) and Sato et al., (2002), described in Section 2.2, and shows that the cavity length, or the parameter K /( 2α) ( or σ /(2α) in the notation of the present manuscript) is the parameter controlling the cavitation instabilities. Theoretical curves obtained from linearized free-streamline theory in the regions of super-cavitation (Wu, 1956) and partial cavitation (Acosta, 1955) on a flat-plate hydrofoil are also shown in Figs We see that for fully cavitating flow the agreement is v p k, and the cavitation number p. They correlate fairly well and 21

22 better than for the partial cavitating case. This, however, is to be expected since, in the former case, the hydrofoil acts exactly like a flat plate whereas, in partial cavitation, camber and thickness effects play a role. Unsteady Cavitation The general behavior for all angles of attack equal to and greater than 4 deg is similar, and the general development of the nonsteady process is the same for all angles. From the fully wetted condition to a cavity length of 60 percent chord, the cavities are steady in the mean; the cavity is not glassy clear, however, but is filled with a frothy mixture of air and water, and has no definite structure, such as a reentrant jet, for example. Although audible noise is generated, the resulting force oscillations are small as measured by the embedded strain gauge. The cavity retains its frothy character until just before oscillation commences, at which point the portion of the cavity near the leading edge becomes clear and glassy. Shortly thereafter the cavity begins to oscillate. These initial oscillations are of small amplitude, both in extent and force, and are relatively high in frequency, ranging Hz. The frequency of this oscillation is more or less independent of the tunnel speed, which is different from the characteristics of partial cavity oscillation. This raises the possibility that the oscillation is related to the dynamics of the the force balance or of the tunnel. To investigate this point, the force balance with model attached was shock-excited but no evidence of the Hz was observed. This stage of oscillation is rather transitory; and with a slight decrease in tunnel pressure, the oscillation changes over into a more characteristic low-frequency, large amplitude disturbance. The oscillations then typically have a double amplitude of about one half chord. Typical frequencies in this stage were about 12 to 25 Hz depending on velocity and angle of attack. One cycle of the cavity oscillation is shown in Fig The flow is left to right with the leading edge being at the upper left. The sequence starts the upper left and and time increases downward. Starting at the minimum cavity length, the cavity grows smoothly and, as it approaches the end of the hydrofoil, a reentrant jet is seen to form and gradually fill the rearward portion of the cavity. On reaching the end of the foil, the cavity surface becomes uneven and irregular and small vortices may be shed from the end of the cavity, causing small fluctuations in the force on the hydrofoil. The flow within the cavity then appears to become unstable and a large volume of cavity is abruptly shed into the stream; the cycle is then repeated. This sequence of events is 22