SBIR program (Mechmath, LLC). The Minnesota. Supercomputing Institute (MSI), University of. Minnesota, generously provides the computational resources.
CONTROL OF CAVITATING FLOWS: A PERSPECTIVE Roger E. A. ARNDT1, Gary J. BALAS1 and Martin WOSNIK1 This is a summary of our research on unsteady partial cavitation and ventilated partially and supercavitating flows with special emphasis on control strategies. We draw upon our experience with a variety of projects ranging from the experimental and numerical description of sheet/cloud cavitation to the control of high-speed supercavitating bodies. In addition, our numerical simulations of partially cavitating hydrofoils indicate that there is a regime where the potential exists for feedback control of gust-induced oscillations. These efforts are in the incubation stage. The purpose of this presentation is to provide our perspective on the problems and opportunities for control of cavitation and to encourage a healthy interchange of ideas at this workshop. Key Words: cavitation, controls, feedback control, partial cavitation, supercavitation, ventilation 1. Introduction Cavitating flows have been studied extensively for more than 100 years. Most of the research effort has been directed toward the avoidance of cavitation, i.e. the physics of cavitation inception. As speeds of marine vehicles increased in the 1950s it became clear that cavitation would be unavoidable under certain circumstances. This has lead to a significant effort to investigate supercavitating flows. These researches have resulted in the development of supercavitating hydrofoils and propellers as well as the development of supercavitating inducers for high performance pumps. Since that time, supercavitating flow technology has languished in the US until recently, when there has been a renewed interest in high-speed underwater bodies (Ashley 2001). At the same time operation in a regime of partial cavitation has been considered for hydrofoils and is becoming common in a wide range of turbomachinery applications. This opens up a wide range of new considerations including the possibility of active control of these flows to eliminate or minimize unwanted vibration and unstable operation. 2. Attached Cavitation on Hydrofoils Various types of cavitation on surfaces can be found in practice, including bubble cavitation, sheet cavitation, cloud cavitation, and various forms of vortex cavitation. In spite of considerable research, there are still many features of the problem that have not been properly explored. Once cavitation occurs, a given flow field is significantly modified because the lowest pressure in the flow is typically limited to vapor pressure. Thus C pm = - σ
1
University of Minnesota
σ ≤σ i
(1)
Figure 1. Variation of lift, noise and spectral content of lift oscillations on a NACA 0015 hydrofoil (Kjeldsen et al. 2000). Since the lift coefficient of a hydrofoil scales with -Cpm, this parameter will decrease with decreasing σ as shown in Figure 1. The effect of cavitation on lift is directly related to the observed degradation of performance of turbomachinery due to cavitation. Cavitation is also a non-steady phenomenon. For example, steady flow over a hydrofoil at a value of σ below σi results in a highly dynamic form of sheet cavitation. Figure 1 also illustrates how the dynamics of lift oscillations can change dramatically with σ. A typical view of sheet/cloud cavitation on a NACA 0015 hydrofoil is shown in Figure 2a. Figure 2b contains a pictorial display of the various types of cavitating flow that were observed on this hydrofoil at various combinations of angle of attack, α, and cavitation number, σ. Several different cavitating
Figure 2a. Sheet/cloud cavitation on a NACA 0015 hydrofoil. Note the lateral extent of the wake.
regimes occur depending on the combination of σ and α. The demarcation between “inception” and Cpm (computed) varies such that σi is always less than -Cpm,, as expected. It was noted that sheet cavitation could be sub-divided into two regions such that at higher angles of attack (α ≥ 5°) the flow had a wider dynamic range. At low angle of attack, say less than 4°, only bubble cavitation occurred. At intermediate angles of attack and relatively high values of σ, cavitation inception is in the form of patchy cavitation. 3. Lift Oscillations Excellent agreement has been obtained between data acquired at SAFL and Obernach, Germany for a NACA 0015 hydrofoil. Cavitation induced lift oscillations were measured that have spectral characteristics that vary considerably over a range of 1.0 ≤ σ/2α ≤ 8.5, where σ is the cavitation number and α is the angle of attack, as shown in Figure 3. The fluctuations are associated with the periodic shedding of vortical clouds of bubbles into the flow, and their amplitude can exceed 100% of the steady state lift. Three types of oscillatory behavior are noted: I. 1.0 ≤ σ/2α ≤ 4: A strong spectral peak exists at a Strouhal number, fc/U, of about 0.25 that is independent of cavitation number. II. 4 ≤ σ/2α ≤ 6: A higher frequency, albeit weaker spectral peak dominates. The frequency of this peak is almost a linear function of cavitation number and corresponds to a constant Strouhal number, based on cavity length, of about 0.3. III. 6 ≤ σ/2α ≤ 8.5: Bubble/patch cavitation can occur. This induces a distinct, very low frequency, spectral peak.
Figure 2b. Illustration of the various types of dynamics that can occur over a range of σ and α. The transition that occurs at σ/2α = 4 corresponds to a relative cavity length, l/c, of about 0.75. This behavior is predicted by Watanabe et al. (1998). They refer to type II modes as l/c < 0.75 and type I modes for l/c > 0.75. Two competing mechanisms are found for the induced shedding of cloud cavitation. At high values of σ/2α, reentrant jet physics dominate, with sheet cavity oscillations at a frequency, based on cavity length2, of fl/U ≈ 0.3. At low values of σ/2α, bubbly flow shock wave phenomena dominate with a constant Strouhal number based on chord length of fc/U ≈ 0.2. A significant effect on the wake structure is also noted. Frequency data collected from high-speed video of this flow are shown in Figure 4. These data agree very well with the lift data shown in Figure 3. Note that at approximately σ/2α ≈ 4, there is a sharp transition from one type of frequency trend to the other. In fact, Joint Frequency Time Analysis (JFTA) clearly indicates that there are two different mechanisms with different frequency characteristics in this range of σ/2α.
2
Using a simple reentrant jet model, Arndt et al (1995) were able to show that for cavity lengths, l, less than about 75 % of chord length, an estimate of frequency of fl 1 oscillation is given by = 1+σ U 4
Type I Oscillations
Type II Oscillations
Type III Oscillations
Figure 3. Measured Lift Spectra for a cavitating NACA 0015 hydrofoil. 4. Numerical Modeling of Unsteady Flow In order to investigate control strategies in detail, a virtual single-phase cavitation model utilizing a barotropic flow assumption is used to numerically simulate unsteady vortex shedding phenomena from a cavitating hydrofoil (Song and He, 1998, Song and Qin, 2001, Arndt et al, 2004). The model is capable of capturing the dynamic features outlined above. In terms of control, the periodic structure of the induced flow is of interest. Five types of discrete vortex shedding mechanisms are identified in this study. In particular, cloud cavitation (having a clockwise vortical structure when flow is from right to left) is shed from the suction side of the hydrofoil. This induces a counterclockwise vortex from the pressure side, forming a vortex pair that dominates the wake structure. The process is found to be substantially periodic, giving rise to periodic lift oscillations, in agreement with experimental results. The instantaneous velocity field is used to investigate the unsteady vortical structures shed from a cavitating hydrofoil and their evolution in the wake. To facilitate better analysis, the upstream-specified streamwise velocity was subtracted from the instantaneous velocity field of the whole computational domain. Figure 5 illustrates an instance of a large negative (clockwise rotation when flow is from right to left) 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 (counterclockwise rotation) vortex from the pressure side (type B vortex) that is being induced by the type A vortex. Within a short period of time, a vortex pair containing type A and type B vortices is well formed in the wake immediately downstream of the trailing edge. Since a
Figure 4. Composite plot of peak frequency, including data from SAFL, Obernach, and numerical computations. type A vortex is typically located downstream of a type B vortex, the induced velocities tend to move the vortex pair slightly upward in the cross streamwise direction. As a result, the vortex pair moves diagonally at a speed roughly equal to the free-stream speed, which results in a slightly slower convection speed of this vortex pair in the streamwise direction. 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 downstream, which is consistent with the wider spread of a cavitating wake than a non-cavitating wake (Arndt et al, 2000). There is also a secondary negative vortex (type C vortex) formed above the suction side near the trailing edge as is also shown in Fig. 5. This type C vortex is much weaker in strength than the type A vortex, but it also induces a corresponding secondary positive vortex (type D vortex) from the pressure side and eventually forms a secondary and weaker vortex pair in the wake (Figure. 6). A detailed analysis based upon an instantaneous vorticity animation shows that the reentrant jet plays a key role in the formation of primary and secondary negative vortices. When the reentrant jet along the suction side of the hydrofoil starts to impinge on the developed sheet cavity, it usually cannot shear off the whole sheet cavity; instead the sheet cavity is broken by the reentrant jet somewhere close to the leading edge. The broken sheet cavity forms the primary negative vortex. The reentrant jet keeps developing in the upstream direction and eventually shears off the rest of the sheet cavity, which is the source of a secondary negative vortex. The primary vortex pair, type A and type B, are periodically shed into the wake while the formation of the secondary vortex pair, type C and type D, are less regular. There is typically one secondary vortex pair between two primary vortex pairs, but occasionally two secondary vortex pairs are observed. Between the primary and secondary shedding vortices, there is a certain time period in
Figure 5. A well formed primary vortex pair (type A and type B) in the wake immediately downstream of the trailing edge. Cavitation number 1.0 at 8o angle of attack, σ/2α = 3.58 which only positive vortices (named as type E vortices) are shed from the pressure side of the hydrofoil, shown in Figure 6. The lift of a cavitating foil oscillates in a manner that is highly correlated with the five types of vortices shed into the wake. The primary mode of lift oscillation correlates with the primary pair of vortex shedding. Its Strouhal number based on the chord length is approximately equal to 0.2 when σ/2α ≤ 4. The lift coefficient also oscillates periodically with Strouhal number approximately equal to 0.2, if the projected hydrofoil width is used as the reference length. This oscillation correlates perfectly with the positive vortex strings of the semi-Karman vortex street (Type E vortices). Lift oscillation due to the secondary pair of vortices is less regular but quite significant. For values of σ/2α > 4, the lift oscillation is dominated by reentrant jet physics and the oscillation Strouhal is about 0.3 based on cavity length. This is illustrated with a simulation corresponding to Mode I oscillations at σ/2α = 1.79. A portion of the time series of the lift coefficient at 8o angle of attack with cavitation number 0.5 is presented in Figure 7. A Fourier analysis of these data indicates that the Strouhal number St is 0.2, when the frequency is normalized by the chord length c and the upstream velocity U0. In comparing the time series data and the simulations it is found that instant of maximum lift (i) coincides with the time of maximum sheet cavity length. A short moment later, a reentrant jet will impinge on the cavity surface near the detachment point and shear off the cavity. This large piece of detached sheet cavity evolves into the primary negative vortex known to be a cloud cavity. The lift gradually decreases as the cloud cavity moves downstream along the hydrofoil surface, leaving a greater and greater area of the surface noncavitating. The pressure on the increasing noncavitating surface is higher than the critical pressure (vapor pressure) causing the lift to decrease at this
Figure 6. A well formed secondary vortex pair (type C and type D) in the wake. This event is followed by a relatively quiescent period (Type E vortex street). Cavitation number 1.0 at 8o angle of attack, σ/2α = 3.58 time. This primary negative vortex induces a large positive vortex when it arrives near the trailing edge. Point ii in Figure 7 corresponds to the instant in time when a type A vortex is just about to be shed from the trailing edge. A number of positive vortices behind the hydrofoil are the type E vortices that are equivalent to a semi-Karman vortex street. Its frequency is several times that of the cloud cavity causing secondary peaks in the lift curve indicated as point iii in Figure. 7.
Figure 7. Time series of lift coefficient at 8o angle of attack with cavitation number 0.5 (σ/2α = 1.79). 5. Minimizing Gust Response: A Control Possibility The physics summarized above point to several possibilities for control of lift oscillations. An excellent database is available since excellent agreement has been found between numerical simulations and experimental data over the entire range of σ/2α. The focus of the following discussion is on bubble/patch cavitation (Type III oscillations). When the angle of attack is small, the boundary layer in the non-cavitating flow is thin and attached to the foil everywhere except at the trailing edge. If the
pressure region. The increase in pressure eliminates the cavitation, lift increases, the pressure decreases again cavitation is induced, starting a new cycle. This natural cycle can be applied to the concept of selective ventilation near the suction peak to minimize the gust response of a hydrofoil. The ventilation (i.e. artificial cavitation) would produce the same effect. When a hydrofoil encounters a gust, lift increases. This response could be dampened through ventilation. A feedback control system based on this concept could be designed that would smooth the passage of a hydrofoil through an unsteady flow field. One example would be the passage below a wave train where the orbital motion induced by the surface waves serves to induce lift oscillations. 6. Ventilated Hydrofoils
Figure 8. Numerical simulations of Type III oscillations due to bubble/patch cavitation. The lower curve displays lift fluctuations. The upper three curves are pressure at various values of x/c. cavitation number is gradually decreased, then a bubble cavity will appear first near the nose where the pressure is minimum. The expansion due to a bubble cavity induces boundary layer separation as indicated by the instantaneous vorticity field. Observation of the corresponding pressure field indicates that the bubble is located at the middle of a large separation eddy. The computed pressure at three points on foil surface and lift coefficient as functions of time are shown in Figure. 8. The lift coefficient curve exhibits a sawtooth form indicating that the time of bubble cavity formation coincides with the time of maximum lift. The bubble slides along the foil and collapses near the trailing edge as the lift decreases to the minimum. As the collapsed cavity in the form of a large eddy is being transported in the wake, the boundary layer gradually recovers its original form and the lift increases to a maximum. A new bubble cavity is formed when the lift is a maximum and the pressure is minimum again. The computed Strouhal number for this case is, S = fc/U = 0.1 which corresponds to Type III oscillations. Bubble/patch cavitation is shown to have a natural oscillation cycle that consists of a pressure minimum during non-cavitating lift that induces cavitation followed by a reduction in lift due to cavitation which in turn increases the pressure in the minimum
In another application, thin flat plate theory for partially cavitating and supercavitating flows indicates that there is a regime of flow in the range of 3 ≤ σ/2α ≤ 5 in which the possibility of high lift to drag ratios exist, cf. Figure 9. This potential can be realized with finite thickness hydrofoils with a special thickness distribution. Unfortunately the peak lift to drag ratio is very close to the point where substantial flow induced lift oscillations are possible. The objective here is to eliminate or minimize the lift oscillations with ventilation while retaining the favorable performance characteristics. Some early data of a novel hydrofoil design are illustrated in Figure 10, showing promising lift to drag ratios for 4.2 ≤ σ/2α ≤ 6.2 (0.95 > σ > 1.3). There are several technical issues that need to be addressed before this concept can reach full potential.
Figure 9. Comparison between the lift coefficient of a flat plate, a NACA 0015 hydrofoil and a specially designed hydrofoil.
region of focus
Figure 10. Comparison of lift to drag ratios of a conventional hydrofoil (NACA 0015) and a special ventilated hydrofoil design. 7. Control of Supercavitating and Ventilated, High-Speed Underwater Bodies As speeds of marine vehicles increased in the 1950s it became clear that cavitation would be unavoidable under certain circumstances. This lead to a significant effort to investigate supercavitating flows. These researches lead to the development of supercavitating hydrofoils and propellers as well as to the development of supercavitating inducers for high performance pumps. Recently there has been interest in this technology for use in high-speed underwater bodies. When a vapor or gas filled cavity exceeds body dimensions, it is classified as a supercavity. Generally speaking, the shape and dimensions of vapor filled and ventilated cavities (sustained by air injection) are the same when correlated with the cavitation number based on cavity pressure. Ventilated cavities require a certain quantity of ventilation gas in order to be maintained. The issues involved are complex. This has been discussed recently by Schauer (2003) and Wosnik et al (2004). Figure 11 illustrates the relationship between air demand, cavitation number and cavity dimensions. The control of supercavitating bodies is a technical challenge. For example a fully cavitating underwater vehicle has only small regions at the nose (cavitator) and on the afterbody in contact with water. Therefore it experiences significantly reduced drag compared to a fully wetted vehicle and can reach very high speeds. Such a vehicle can be controlled with deflection of front (cavitator) or aft control surfaces (fins), thrust vectoring, or a combination thereof. Due to greatly reduced buoyancy, the cavitator and control surfaces near the rear must also provide enough lift to support the vehicle’s weight under cruise conditions.
Figure 11. Air entrainment coefficient, Q/Ud2, versus cavitation number for a ventilated cavity (Schauer 2003). The discontinuities between the vehicle-cavity and vehicle-fluid interaction represent a challenge for control engineers. The interaction of the vehicle with the cavity boundary results in large impulse forces on the body due to the fluid. The problem is compounded by the fact that the dynamic model of the vehicle in contact with the fluid differs from its model in the cavity. Hence it is likely a high bandwidth, nonlinear control system will be needed to reject wall interaction disturbances and track desired trajectory commands. To date, we have investigated a supercavitating vehicle with cavity-piercing fins as aft control surfaces. A 12-degree of freedom nonlinear simulation model of a supercavitating vehicle has been developed to better understand the control design issues. The model has a disk cavitator at the front of the vehicle and 4 fin actuators that impinge into the water aft. The cavitator is actuated in the pitch and yaw directions. Two models of a supercavitating vehicle have been proposed in the literature (Dzeil et. al. 2003, Kirschner et. al. 2002). The model used is based on the supercavitating vehicle model developed by Kirschner. The model includes a nonlinear, time-dependent cavity dynamics model, cavitator and fin force and moment data that are functions of speed, angle-of-attack, sideslip and fin immersion ratio. The sensor suite is assumed to be able to measure pitch, roll and yaw acceleration, rate and angle. In the simulation model, the sensitivity of the vehicle dynamic model to variations in the fin immersion ratio, center of gravity location, speed, cavitator force and cavitation number were investigated. This study found that the vehicle dynamics were most sensitive to variations in the fin immersion ratio and speed and relatively insensitive to changes in cavitation number, cavitator and fin force variables. From a controls point of view, the speed of the vehicle can be easily
Figure 12. Supercavitation control experiment. sensed. Therefore speed can be used as a feedback variable in the control design. Fin immersion ratio strongly affects the system dynamics and is difficult (if not impossible) to sense. Therefore, the control design must be robust to variations in the fin immersion ratio to achieve desired objectives. Unfortunately the physical fin immersion model is crude. It is based on a steady-state equilibrium flow assumption and neglects the interaction with fluid dynamics and the cavity. Therefore a simple experiment has been developed to better understand the interaction of the fins with the cavity and fluid. Knowledge of how changes in the supercavity affect torque and forces on the control surface is needed to properly design a control law. This is being investigated using the nonlinear supercavitation vehicle simulation and an experimental setup with a single fin. An experiment with a semi-axisymmetric, ventilated cavity and a single fin was carried out on the floor of the high-speed water tunnel at St. Anthony Falls Laboratory (Figure 12). Initially, different cavitator shapes were tested (two cones, 30 and 45 deg. half-angle, a fore-shortened 45 deg. cone and a sharp-edged disk) and found to have significant effect on cavity size as well as cavity interface instabilities. With a wedge-shaped, swept, cavitypiercing fin the interaction of control surface and supercavity was studied. Hysteresis in cavity shape (cavitation number) versus ventilation flow rate was found for both the main, semi-axisymmetric supercavity and the supercavitating fin. For increasing fin angles of attack, ventilation air demand increases for constant cavitation number, or, conversely, re-entrant jets will penetrate further upstream if ventilation rate is kept constant. Currently fin forces for different ventilation rates (cavitation numbers) and angles of attack are being measured, and a simple open-loop control experiment with fin response to an upstream/cavity disturbance is being carried out. Experiments with this facility are planned to better understand the interaction between
Figure 12a. Detail of cavitation on the nose. the fin deflection, deflection rates and immersion ratio with the cavity and fluid dynamics. These experimental results will be used to refine the 12 degree-of-freedom nonlinear simulation model. A full envelope inner-loop controller and autopilot will be synthesized that is robust to modeling error and dynamics not treated in the model. Trajectories for the augmented vehicle will be designed to achieve desired mission and tracking requirements. Simulation results will likely drive new water tunnel experiments to help access the interaction between the robustness of the closed-loop system and achievable performance. 8.Conclusions Control of cavitation has, in the past, been limited to avoiding cavitation by designing for the lowest possible inception index. However, many applications involve unavoidable cavitation. Numerical and experimental research with partially cavitating hydrofoils has identified three regimes of periodic flow where active control could be used to modify or minimize flow induced lift and drag oscillations to eliminate or minimize unwanted vibration and unstable operation. We have also found that controlled ventilation offers a possible new method for controlling gust response. In addition we have developed a control model for high-speed ventilated supercavitating bodies. We have identified several issues with fin control that are being studied in a water tunnel. Hysteresis in cavity shape (cavitation number) versus ventilation flow rate was found for both the main, semi-axisymmetric supercavity and the supercavitating fin. For increasing fin angles of attack, ventilation air demand increases for constant cavitation number, or, conversely, re-entrant jets will penetrate further upstream if ventilation rate is kept constant. Currently fin forces for different ventilation rates
(cavitation numbers) and angles of attack are being measured, and a simple open-loop control experiment with fin response to an upstream/cavity disturbance is being carried out. 9. Acknowledgments
Dzielski, J, Kurdila, A (2003) A Benchmark control problem for supercavitating vehicles and an initial investigation of solutions. J. Vibration and Control, 9:791-804
This project is sponsored by the Office of Naval Research (Dr. Kam Ng), the National Science Foundation (Dr. Michael Plesniak) and the DARPA SBIR program (Mechmath, LLC). The Minnesota Supercomputing Institute (MSI), University of Minnesota, generously provides the computational resources. Dr. Qiao Qin provided the numerical simulations. Dr. Andreas Keller and Dr. Morten Kjeldsen assisted with the experiments at Obernach, Germany.
Kirschner, IN, Kring DC, Stokes, Fine, EN (2002) Control strategies for supercavitating vehicles. J. Vibration and Control, 8: 219-242 Kjeldsen M., Arndt, R.E.A., Effertz, M. (2000) Spectral Characteristics of Sheet/Cloud Cavitation. J. Fluids Engineering, 122, 481-487
10. References Arndt REA, Ellis C, Paul S (1995) Preliminary investigation of the use of air injection to mitigate cavitation erosion. J. Fluids Engineering. 117:498-592. Arndt REA, Song CCS, Qin Q (2004) Experimental and numerical investigations of cavitation. 22nd IAHR Symposium on Hydraulic Machinery and Systems, June 29 – July 2, Stockholm, Sweden Arndt REA, Song CCS, Kjeldsen M, He J, Keller A (2000) Instability of partial cavitation; A numerical/ experimental approach. Proceedings 23rd Symposium on Naval Hydrodynamics. Val de Reuil, France Ashley S (2001) “Warp Drive Underwater” Scientific American, May
Schauer TJ (2003) An experimental study of a ventilated supercavitating vehicle. MS Thesis, Department of Aerospace Engineering and Mechanics, University of Minnesota Song, C.C.S. and He J (1998) Numerical Simulation of Cavitating Flows by a Single-Phase Flow Approach. Third International Symposium on Cavitation, Grenoble, France Song, C.C.S. and Qin, Q (2001) Numerical Simulation of Unsteady Cavitating Flow. 4th International Symposium on Cavitation, Pasadena, CA Watanabe S, Tsujimoto Y, Franc J-P, Michel J-M (1998) Linear analyses of cavitating instabilities. Proc. of 3rd International Symposium on Cavitation, 1, Grenoble, 347-352 Wosnik M, Schauer T, Arndt REA (2004) Experimental investigation of the turbulent bubbly wake in a ventilated flow. In Advances in Turbulence X, Proceedings of the Tenth European Turbulence Conference, Andersson HI, Krogstad PA, eds, CIMNE, Barcelona, 657-660
