- Email: [email protected]

Technical Note

On the limiting parameters of artificial cavitation K.I. Matveev ∗ California Institute of Technology, Mail Code 301-46, Pasadena, CA 91125, USA Received 12 April 2002; received in revised form 13 August 2002; accepted 1 September 2002

Abstract Artificial cavitation, or ventilation, is produced by releasing gas into the liquid flow. One of the objectives of creating this multiphase flow is to reduce frictional and sometimes wave resistance of a marine vehicle completely or partially immersed in the water. Flows around surface ships moving along the water–air boundary are considered in this paper. It is favorable to achieve a negative cavitation number in the developed cavitating flow under the vessel’s bottom in order to generate additional lift. Cavities, formed in the flow, have limiting parameters that are affected by propulsion and lift-enhancing devices. Methods for calculating these influences and the results of a parametric study are reported. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Cavitation; Ventilation; Drag reduction

1. Introduction One of the classes of boundary layer control is the creation of a multiphase flow by injecting air into the liquid flow. It is known that artificially developed cavitating flows effectively reduce drag on surface ships (Matveev, 1999) and underwater rockets (Ashley, 2001). In the case of a surface vessel, called an Air Cavity Ship (ACS), air is supplied under the specially profiled bottom, so that a steady thin air cavity is generated. This cavity separates part of the hull surface from contact with the

∗

Fax: +1-626-395-8469. E-mail address: [email protected] (K.I. Matveev).

0029-8018/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0029-8018(02)00103-8

1180

K.I. Matveev / Ocean Engineering 30 (2003) 1179–1190

water, reducing hydrodynamic resistance. The conceptual scheme is given in Fig. 1, where the important parameters of the longitudinal hull geometry are designated. Pressure inside the cavity, filled with air, is higher than atmospheric pressure. Different from hovercraft and Surface Effect Ships, an ACS does not have flexible skirts, and its rigid bottom sections determine the shape of an air cavity. Consumption of air required to support the cavity on a properly designed ACS is approximately 2% of the power used for propulsion; and achieved drag reduction is 15–30%. Taking into account other positive features, such as low cost, small draft, and easy maintenance, it can be concluded that an ACS is a very promising concept in the family of high-performance marine vehicles. The idea of drag reduction by supplying air to wetted hull surfaces was proposed in the 19th century. However, most attempts to implement it have failed. The concept looks deceptively simple, but, without a deep physical understanding of the phenomenon, it is not possible to create a stable cavity with a low air injection rate. Butuzov (1967) pioneered and developed the idea that the ventilated flow under a ship’s bottom can be described using cavitation theory. A lot of research has been done in the fields of supercavitating and ventilated (aerated) flows; surveys can be found in Knapp et al. (1970); Brennen (1995); Savchenko (2001). However, most experiments and theories were devoted to flows with positive cavitation numbers and without upper solid boundary, contacting with the cavity. In the case of air lubrication under the ship’s bottom, it is desirable to have a flow with a negative cavitation number in order to support the vessel. Systematic research of such flows started in Russia in the 1960s and resulted in the creation of full-scale experimental ACSs (Butuzov et al., 1990). Since the 1990s, commercial and military ACSs with displacements up to 100 ton, such as shown in Fig. 2, have been produced in series. A review of several air lubrication developments and dimensional analysis of a planing ACS are given by Latorre (1997). The nature of a cavitating flow aimed at reducing drag can be understood from the example of ventilation behind a wedge attached to a horizontal wall in the presence of gravity (Fig. 3). The characteristic feature of cavity 1 is the formation of a pulsating re-entrant jet in the tail part of the cavity, while the cavity boundary close to the wedge remains stable. This flow is similar to usual supercavitation and ventilation with a positive cavitation number and without a wall (Knapp et al., 1970). Shape 2 is associated with a flow mode when no re-entrant jet is present, and the tail of the cavity attaches smoothly to the plate. In this case, the cavity-maintaining gas flow, as well as the cavitation drag, is theoretically equal to zero; pressure inside the cavity

Fig. 1.

Air cavity formed under the bottom of a fast ACS.

K.I. Matveev / Ocean Engineering 30 (2003) 1179–1190

Fig. 2.

1181

Landing boat Serna using air lubrication.

Fig. 3.

Ventilated flow behind a wedge.

exceeds that in the undisturbed flow, i.e. the cavitation number is negative (Butuzov, 1967). The peculiarity of shape 3 is that in theory the cavity pierces the plate at its aft end (as shown by the dashed line). During tests, strong pulsations are observed all over the cavity in this case, as in experiments on over-ventilated flows with positive cavitation number (Silberman and Song, 1961). This regime is realized at high gas consumption. The formation of an unclosed cavity 4 is also possible at certain conditions (Butuzov, 1967); however, the power needed for air injection is too high to make this regime attractive for practical drag reduction. Thus, the flow mode that produces cavity 2 is the most promising. As shown by calculations and verified in experiments (Butuzov, 1967), the cavity length is proportional to the square of the flow velocity. Cavity geometrical characteristics and the cavitation number, corresponding to this most favorable situation, are called the limiting parameters; successful ACSs are designed to operate in this regime. If the ship length is large and the speed is not sufficiently high, the entire bottom of the vessel cannot be covered by a single cavity. This explains unsuccessful attempts to reduce drag by supplying gas through only a single nozzle in low speed regimes. Several air cavities must be created on a slow ACS. One of the difficulties in ACS design is related to the dependence of the cavity limiting parameters on the influences from prolusion systems and lift-enhancing devices, which are difficult to take into account accurately in model testing. In this paper, this influence is studied numerically for the simplified configuration of a rear-

1182

K.I. Matveev / Ocean Engineering 30 (2003) 1179–1190

ward-facing step on the lower surface of a horizontal wall. Choosing properly the locations and characteristics of the propulsive and lifting devices when designing an ACS, it is possible to achieve high performance on a full-scale vessel.

2. Mathematical formulation A two-dimensional cavitating flow around a wedge on the lower side of the horizontal wall (Fig. 4) is considered following Butuzov (1967). A generalized Riabouchinsky scheme is applied, using a fictitious contour introduced at the cavity tail. The liquid is assumed to be ideal and incompressible; the flow is potential and steady. Pressure in the cavity, including its boundary, is considered uniform and equal to pc. From Bernoulli’s theorem it follows that at the cavity boundary streamline pc ⫽

rU2 ru2c ⫹ p0⫺ ⫹ rgyc, 2 2

(1)

where p0 is the pressure in the undisturbed flow at y ⫽ 0. Eq. (1) can be re-written in non-dimensional form as s⫽⫺

冉冊

uc 2 2yc ⫺1, ⫹ 2 Fr b U

(2)

where s is the cavitation number and Fr is the Froude number, defined by expressions s⫽

p0⫺pc rU2 / 2

Fr ⫽ U / 冑gb

, .

(3) (4)

Cavities applied for drag reduction are usually thin, so Eq. (2) can be linearized as follows yc u1 s ⫽⫺ 2 ⫹ 2 Fr b U

,

(5)

where u1 is the velocity perturbation. In linear theory, a wedge and a cavity can be

Fig. 4.

Scheme of the cavitating flow.

K.I. Matveev / Ocean Engineering 30 (2003) 1179–1190

1183

simulated by a distribution of sources with intensities proportional to the derivative of the obstacle ordinate q(x) ⫽ 2Uy⬘.

(6)

Velocity perturbations induced by the sources are computed by L ⫹ b1

1 u1(x) ⫽ ⫺ 2π

冕

q(x)dx . x⫺x

(7)

⫺b

Combining Eqs. (5)–(7), a final equation for the cavity shape yc is

冕 L

s yc(x) 1 y⬘c(x)dx ⫹ ⫽ ⫺ ⫹ Fcont(x) ⫹ Fdist(x), Fr2b π x⫺x 2

(8)

0

where

冕 0

L ⫹ b1

1 h’0(x)dx 1 ⫺ Fcont(x) ⫽ ⫺ π x⫺x π ⫺b

冕

h⬘1(x)dx ; x⫺x

(9)

L

h0 and h1 are the ordinates of the main (wedge) and closing contours, respectively. Fdist(x) is the disturbance induced by other objects in the flow, such as propulsive and lift-enhancing devices. When their sizes are small in comparison with other characteristic lengths in the problem, these devices can be approximately modeled by hydrodynamic singularities; a sink for a water-jet inlet, a dipole for a propeller, a source for a trim tab or an interceptor, and a vortex for a hydrofoil. It is assumed that a propeller, a water-jet inlet, and a trim tab are positioned on the wall line; and a hydrofoil is located below the air cavity. Components of Fdist(x) are expressed as follows: Fsource(x) ⫽

1 q1 ; 2πU x1⫺x

(10)

Fdipole(x) ⫽

1 d ; 2πU (x1⫺x)2

(11)

Fvortex(x) ⫽

g y1 ; πU (x⫺x1)2 ⫹ y21

(12)

where q1, d and g are the strengths of the singularities, and x1 and y1 are their coordinates. In the vortex case, the mirror effect is taken into account. Eq. (8) contains an unknown parameter L; in addition, the inclination of a fictitious contour to the horizontal axis b is not known in advance. In order to complete the formulation, two additional equations are needed. One is the condition of the closed contour comprising a cavity and a wedge:

1184

K.I. Matveev / Ocean Engineering 30 (2003) 1179–1190 L ⫹ b1

冕

y⬘(x)dx ⫽ 0.

(13)

⫺b

Another equation is the equality of the inclinations of the cavity and a fictitious wedge at their contact point: y⬘c(L) ⫽ ⫺b.

(14)

Thus, the cavitation problem is reduced to an integral–differential equation (8) together with constraints (13) and (14). A numerical procedure for solving a similar problem is discussed by Butuzov (1967). Cavity length L is considered as a known parameter; cavitation number s and angle b are unknown variables. The integration distance is divided in small intervals, where the shape of the cavity yc(x) is approximated by second-order polynoms. The obtained system of linear algebraic equations is solved numerically.

3. Results and discussion On actual ACSs, the wedge generating an air cavity is often a part of the hull with an inclination angle near zero, resembling a rearward-facing step (Fig. 5). In order to obtain numerical data for such a configuration, calculations are carried out in the limit as b→⬁, while the step height is kept constant. The computed dependencies for reduced closure angle bL/h and reduced cavitation number sL/h versus reduced cavity length f ⫽ gL / U2 are shown in Fig. 6. Two values of the fictitious wedge length b1/L are used: 0.01 and 0.1. These two closing contours produce significantly different closing angles in the regimes distant from the limiting case, which corresponds to b ⫽ 0. This discrepancy at low f can be attributed to the presence of the re-entrant jet (Fig. 7). The smaller the fictitious contour, the higher the value of the closing angle. In reality, the flow at the cavity closure is unsteady; a pulsating re-entrant jet carries the pockets of gas away from the cavity in the form of bubbles and foam. The physics of a re-entrant jet is discussed by Knapp et al. (1970); a recent experimental study, investigating in detail the properties

Fig. 5.

Limiting ventilated flow behind a step.

K.I. Matveev / Ocean Engineering 30 (2003) 1179–1190

1185

Fig. 6. Dependence of the closure angle (a) and the cavitation number (b) on the reduced cavity length: b1 / L ⫽ 0.1 (dashed line); b1 / L ⫽ 0.01 (dash-dotted line).

Fig. 7.

Flow with a re-entrant jet in the vicinity of the cavity tail.

of the formation and dynamics of a re-entrant jet, is reported by Callenaere et al. (2001). When the limiting regime is achieved, the length of the cavity does not depend on the size of the fictitious contour, because the closing angle is zero in both cases. Therefore, the results of modeling aimed at finding the limiting parameters are not sensitive to the choice of the closing wedge. The cavitation number, corresponding to the limiting regime, is negative. From Eq. (3) it follows that the pressure inside the cavity exceeds the pressure in the undisturbed flow, hence, a lift force is generated, favorable for surface vessels. The smaller the closing angle b (at fixed b1), the

1186

K.I. Matveev / Ocean Engineering 30 (2003) 1179–1190

less air escapes from the cavity (Gurevich, 1979), and consequently, the power needed for air injection is lower. In the limiting case, this angle is zero, as is the theoretical air flow rate needed to maintain the air cavity. The theory applied in this study was verified by the experiments on a wedge of a finite angle a (Butuzov, 1967). Fig. 8 demonstrates the comparison between experimental data (averaged) and numerical results for the non-dimensional combinations of the cavitation number, reduced wedge length f1 ⫽ gb / U, Froude number based on cavity length Fr21 ⫽ U2 / gL, and the cavity thickness. The agreement is satisfactory, considering the strong assumptions imposed. Experiments do show that the air flow rate in the limiting regime is very small. Gas escapes from the cavity through two narrow strings at the lateral sides of the cavity; this effect is threedimensional and cannot be modeled with the two-dimensional theory considered here. The process of gas leakage in the limiting regime is similar to that observed in the case of the stable ventilated cavities in the absence of a wall (Epstein, 1970). In the limiting regime, the area covered by the cavity is weakly sensitive to the significant increases in the air supplied. In the regime distant from the limiting case, e.g. when no cavity is present and air is supplied to the rear side of the wedge (flow velocity fixed), a high air flow rate is required initially to achieve the limiting cavity, since a strong pulsating re-entrant jet is present when the cavity length is much smaller than a limiting one. It happened in the shipbuilding that after successful tests of a model with an air

Fig. 8. Comparison of experimental data (dashed line; Butuzov, 1967) and theoretical results (solid line) for non-dimensional combinations characterizing the cavitation number and the cavity length and thickness.

K.I. Matveev / Ocean Engineering 30 (2003) 1179–1190

1187

cavity (but without a propulsor and a control ride system) the full-scale vessel did not deliver the expected performance, namely the air cavity characteristics (bottom area covered by air, air flow rate, and cavitation number) were not as good as at the model scale; such a case is mentioned by Tudem (2002). The most probable reason is due to the negative influence of the propulsor on the cavitating flow. This effect can be estimated with the present model: the dipole resembles the propeller action, and the sink corresponds to the water-jet inlet. The dipole and sink disturbances are introduced via Eqs. (10) and (11). In this study they are located on the wall line behind the cavity, and their non-dimensional intensities, q1/Uh and d/ULh, are taken to be one. For practical purposes the limiting cavity is of the most importance. The dependencies of the reduced cavity length and cavitation number in this regime on the location of singularities are shown in Fig. 9. Positioning the propulsor close to the aft end of the cavity may significantly degrade air cavity performance, reducing the limiting length and the pressure inside the cavity. For considered intensities of singularities, their influence is small when the distance from the cavity end is equal to one cavity length. This is in accordance with observation, when an incorrectly positioned propulsor can lead to the significant reduction of the cavity size. Using mathematical modeling, this effect can be estimated at the design stage, and special measures can be taken to minimize negative interactions between the air cavity and the propulsor. The influence from a source, simulating trim tabs and interceptors located behind the cavity, is also shown in Fig. 9. The limiting cavity length is

Fig. 9. Dependence of the cavity length (a) and the cavitation number (b) on locations of a sink (solid line), a source (dashed line), and a dipole (dash-dotted line).

1188

K.I. Matveev / Ocean Engineering 30 (2003) 1179–1190

enhanced and the cavitation number is decreased, which means that the usage of these devices at the transom can actually improve the limiting cavity characteristics. At the present time, hydrofoils are a popular means of ride control and lift-enhancing systems on advanced marine vehicles. The proper application of both hydrofoils and artificial cavitation on one vessel can significantly improve its performance. The influence from a vortex modeling a hydrofoil on the air cavity characteristics is studied in this work. The vortex disturbance is expressed by Eq. (12); its vertical coordinate is chosen to be equal to y1 ⫽ 0.2L, and its reduced circulation is |g| / Uh ⫽ 0.5. The horizontal coordinate and the sign of circulation are the variable parameters. A positive circulation corresponds to the clock-wise case. The vortex is positioned in front of, along and behind the cavity. The results for the reduced cavity length and cavitation number are given in Fig. 10. The location of a positive vortex near the wedge is accompanied by a profound growth of the limiting cavity length and a small increment of the cavity pressure. When a positive vortex is positioned closer to the cavity aft end, the cavity length becomes much smaller than an undisturbed value, accompanied by a large drop in the cavity pressure. The effect from the vortex of negative circulation is the opposite. It seems that the best choice for the location of a hydrofoil with positive circulation is in the vicinity of the wedge, since in this case both the limiting cavity length and cavity surplus pressure increase; also, this hydrofoil produces an upward force. However, on a fast ACS the flow velocity is usually high enough to create a large single air cavity covering a significant portion of the hull, so an increase of the

Fig. 10. Dependence of the cavity length (a) and the cavitation number (b) on the vortex location of positive (䊊) and negative (왕) circulation.

K.I. Matveev / Ocean Engineering 30 (2003) 1179–1190

1189

limiting length is not important. The positioning of a hydrofoil with negative circulation under the downstream half of the cavity significantly enhances the pressure in the cavity. Hence, a naval architect must optimize the hydrofoil–cavity system to deliver the best total performance. In the case of a relatively slow cargo ship, the cavity length is of great importance, since it defines the number of wedges employed and the overall hull area covered by air cavities. A hydrofoil with positive circulation located under the first-half of the cavity and one with negative circulation in the second-half can produce significant enlargement of the cavity. However, since several cavities are usually employed on a slow ship, the influence of a hydrofoil on the neighboring cavities is unfavorable. Taking in consideration that hydrofoils are not very effective at low speeds, their application on a slow ACS to improve air cavity characteristics is questionable. Notice that a hydrofoil is treated as a vortex in this paper. In reality, when its chord is comparable with the distance to the hull and the cavity boundary, this approximation may produce significant errors. More appropriate distribution of the sources and vortices simulating a hydrofoil are required for accurate estimations.

4. Concluding remarks The influence of singularities, simulating a propulsor and lifting surfaces on the limiting parameters of a cavity formed under the bottom of a surface vessel, are studied with a two-dimensional linearized model for an ideal liquid. It was found that the results for the limiting cavity are not sensitive to the size of a fictitious closing contour located at the aft end of the cavity. Propulsion units positioned in the vicinity of the cavity end negatively affect cavity characteristics, while the trim tabs and interceptors at the transom improve them. The influence of a hydrofoil, located under the cavity, depends on its horizontal location. The strategies for modification of the artificially cavitating flows by proper hydrofoil positioning are discussed for ACSs operating in fast and slow regimes.

References Ashley, S., 2001. Warp drive underwater. Scientific American 284, 70–79. Brennen, C.E., 1995. Cavitation and Bubble Dynamics. Oxford University Press, Oxford. Butuzov, A.A., 1967. Artificial cavitation flow behind a slender wedge on the lower surface of a horizontal wall. Fluid Dynamics 3, 56–58. Butuzov, A.A., Gorbachev, Y.N., Ivanov, A.N., Kalyuzhny, V.G., Pavlenko, A.N., 1990. Reduction of ship resistance using ventilated gas cavities. Shipbuilding 11, 3–6 (in Russian). Callenaere, M., Franc, J.-P., Michel, J.-M., Riondet, M., 2001. The cavitation instability induced by the development of a re-entrant jet. Journal of Fluid Mechanics 444, 223–256. Epstein, L.A., 1970. Methods of the Dimensional Analysis and Similarity Theory in Ship Hydromechanics. Sudostroenie, Leningrad (in Russian). Gurevich, M.I., 1979. Theory of the Jets in Ideal Fluids. Nauka, Moscow (in Russian). Knapp, R.T., Daily, J.W., Hammit, F.G., 1970. Cavitation. McGraw-Hill, New York. Latorre, R., 1997. Ship hull drag reduction using bottom air injection. Ocean Engineering 24, 161–175.

1190

K.I. Matveev / Ocean Engineering 30 (2003) 1179–1190

Matveev, K.I., 1999. Modeling of vertical plane motion of an air cavity ship. In: Proceeding of the Fifth International Conference on Fast Sea Transportation, SNAME, Seattle, WA., pp. 463–470. Savchenko, Y.N., 2001. Supercavitation—problems and perspectives. In: Proceedings of the Fourth International Symposium on Cavitation, California Institute of Technology, Pasadena, CA. Silberman, E., Song, S., 1961. Instability of ventilated cavities. Journal of Ship Research 5, 13–33. Tudem, U.S., 2002. The challenge of introducing innovative Air Lifted Vessels to the commercial market. In: Proceedings of the 18th Fast Ferry Conference, Nice, France.