Shape of Slender Axisymmetric Ventilated Supercavities
The integral-differential equation was obtained to simulate unsteady evolutions of the slender axisymmetric ventilated supercavity with the use of one-dimensional inviscid flow of the incompressible gas in the channel between the cavity surface and the body of revolution. For small ventilation rates, the solution of this equation was expressed as asymptotic series. In the steady case the nonlinear differential equation and its solutions were obtained. It was shown that the ventilation can increase and diminish the supercavity dimensions. Examples of calculations for different hull shapes are presented. At some critical values of the gas injection rate the cavity dimensions can become unbounded. Stability of steady and pulsating gas cavities was investigated in the case of the low gas injection rate.
The drag of high speed underwater vehicles can be reduced with the use of supercavitation. This idea was developed in many theoretical, numerical, and experimental investigations in a lot of countries. In 2013 we celebrated the 100 anniversary of the famous Ukrainian academician G. V. Logvinovich who sufficiently contributed to both the theoretical and experimental research of supercavitation and practical applications of this phenomenon. His principle of independence (Logvinovich ) is still the very efficient tool for calculating the shape of long 3D cavities. In this paper we will discuss its accuracy and areas of application.
To obtain small cavitation numbers at small vehicle velocities or at large movement depths, a gas ventilation inside the cavity is used (see, e.g., Logvinovich , Epshtein [2–4], Knapp et al. , Yegorov et al. , Levkovsky , Kuklinski et al. , Spurk , Wosnik et al. , Zhuravlev and Varyukhin , Matveev et al. , and Vlasenko and Savchenko ). In most cases the gas and vapor densities inside the cavity are much smaller than the density of water (approximately 800 times). This fact allows neglecting the influence of the gas flow inside the cavity and assuming the constant pressure on its surface. As a result, in the case of small gravity effects, the cavitation number ( is the water density, is the velocity of movement, is the water vapor pressure at ambient temperature, and and are pressures measured in the cross section of the cavity origin far away in the water flow and in the injected gas, resp.) is supposed to be constant over all of the cavity surface and the ventilated cavity shape is assumed to be coinciding with the vapor one. This fact is usually used both in theories of the ventilated cavities (e.g., Logvinovich , Epshtein [2–4], and Spurk ) and in experiments. For example, in tests by Wosnik et al.  and Vlasenko and Savchenko  the pressure inside the ventilated cavity was not measured, but the shape of the corresponding vapor cavity was used to estimate the cavitation number. Even in this case, when the effects of the gas flow inside the cavity and gravity are negligible, there is no complete theory for the cavity shape as a function of the gas supply rate, cavitation number, and the shape of the body located inside the cavity.
To improve the efficiency of the supercavitation in comparison with the attached flow pattern, the cavitation number has to be reduced and the cavity volume has to be used completely to locate the vehicle hull (see, e.g., Nesteruk ). It means that an injected gas must flow in a narrow channel between the cavity surface and the vehicle hull; see Figure 1. If the ventilation rate is large enough, the pressure on the cavity surface is no more constant and changes the cavity shape in comparison with the case of vapor cavitation. This complicated phenomenon could be investigated numerically with the use of viscous fluid equations (e.g., Zhuravlev and Varyukhin ). The ideal fluid approach and the slender body theory allow one to obtain simple equations for the shape of axisymmetric ventilated supercavity provided that the gas flow between the cavity surface and the body of revolution is one-dimensional inviscid and incompressible. Some interesting results were obtained in Manova et al.  and Nesteruk and Shepetyuk [16, 17] for steady flow of liquid without gravity effects.
Here the results of these papers are generalized for the unsteady vertical flows in the gravity field. The integral-differential equation is obtained to simulate unsteady evolutions of the slender axisymmetric ventilated supercavity. For small ventilation rates, a solution is expressed as asymptotic series. Examples of the steady cavity shape calculations for different hulls are presented. Peculiarities of the cavity closing near the hull contour discontinuities were investigated.
Stability and pulsations of ventilated cavities were investigated experimentally in Silberman and Song , Song , Michel , and Zou et al.  and by many other authors. The theories of such empty cavities were proposed by Woods , Paryshev [23, 24], Nesteruk , and Semenenko . Here we will try to construct simple stability theories for the steady and pulsating cavities with the use of the main term of the developed asymptotic solution. In comparison with the previous investigations we will take into account cases of the hull, located inside the cavity, different cavitator shapes, and the influence of the gravity.
2. Slender Body of Revolution with a Prescribed Pressure Distribution
The theory of supercavitation is very complicated due to the absence of exact solutions for 3D unsteady flows even in ideal fluid. But for long cavities similar to the slender bodies of revolution, the corresponding linear theories can be used. In monograph (Cole ) the potential of the steady flow of ideal incompressible liquid was obtained by the method of matched asymptotic expansions. In Nesteruk , this slender body theory was generalized for the unsteady case. The potential of the external water flow can be written as follows: We use the symbols with “prime” for physical (dimensional) parameters; is the typical time of unsteadiness, for example, the pulsation period; is the body or the cavity radius; is a small parameter, a ratio of the maximum radius of the body (or a cavitator-cavity system) to its length at a fixed moment of time ; are cylindrical coordinates shown in Figure 1; the integration in (5) must be carried out along the axis of symmetry from the nose till the tail of the body. If the body shape and the function are known, (2)–(5) allow one to calculate the flow field and the pressure distribution.
In supercavitating flows (or when the body shape is unknown), the radius may be calculated with the use of pressure distribution over the surface. Putting expression (2) into the Cauchy-Lagrange integral yields (Nesteruk [28, 29]) Here is the Froude number, is the gravity acceleration, and is pressure in liquid far from the body at the level . The sign “−” in the last term of (6) corresponds to the gravitational acceleration vector directed along the axis and the sign “+” corresponds to the case, when the directions of the body movement and gravity coincide (in order to have an axisymmetric cavity the direction of the gravity force is limited by these two cases). The dimensionless speeds are based on the current flow velocity .
Nonlinear integral-differential equation (6) relates the slender body (or the cavity) radius and the pressure distribution in liquid over its surface . If the functions , , and and the cavitator shape are given, the set of (3)–(8) allows one to calculate the unknown cavity radius . The asymptotic solution of the nonlinear integral-differential equation (6) was obtained in Nesteruk .
3. Equation of the First Approximation and Logvinovich’s Principle of Independence
The leading term of (6) (of order ) provides the equation of the first approximation (Nesteruk ): Thus the squared body (or cavity) radius can be found by successive integration of two linear differential equations (9) and (3) with partial derivatives of the first order. The corresponding general solution was obtained in Nesteruk .
Equation (9) can be rewritten in the absolute coordinates , (in which the liquid at infinity does not move and the body moves vertically). Let us suppose that the origin of this coordinate system is located on the undisturbed free water surface; the directions of the axis of symmetry and the cavitator movement are opposite and the pressure on the cavity surface depends on the time only. Then (9) attains the following form (Nesteruk ): where is some fixed moment of time and the atmospheric pressure on the undisturbed free water surface is assumed to be constant. The thickness parameter can be assumed to be coinciding with the ratio of the cavitator maximum radius to its length, for example, for slender conical cavitator ( is the cone angle).
According to (10), the evolution of every cross section of the cavity (at fixed value of ) is independent of the cavity behavior at other cross sections. This fact can be considered as one more theoretical support of Logvinovich’s principle of independence. An equation similar to (10) with an empirical constant instead of is used in the computer codes developed in the Institute of Hydromechanics NASU for the calculations of unsteady supercavities, for example, Semenenko and Naumova .
It must be noted that the accuracy of the first approximation (9) or (10) is limited by value , which yields only 44% for and 22% for . However these equations provide a nice qualitative description and were used to estimate the influence of many physical parameters such as gravity, capillarity, liquid compressibility, the cavitator shape, motion unsteadiness, and pressure gradients over the body surface; see Nesteruk [28–30, 32–34]. To improve the accuracy, the integral-differential equation (6) must be used, in which the independence of the cavity cross section is no more valid due to the integral term (5).
4. Shape of the Ventilated Axisymmetric Slender Cavity
In the case of ventilated cavity the total pressure of the water vapor and the gas must be equal to the pressure in water on the cavity surface . Here we neglect the capillarity forces and use the approach of one-dimensional flow of an ideal incompressible gas in the circular channel between the cavity surface and the hull located inside the cavity (see Figure 1). Then the continuity equation yields that the gas flux inside the cavity is only time dependent. Here the dimensionless gas velocity is based on ; the dimensionless cavity and body radii ( and ) are based on . By neglecting the longitudinal curvature of the circular channel (the body inside the cavity is assumed to be slender too), the potential of the one-dimensional gas flow can be determined from the relationship and (12) as follows: where is an arbitrary time dependent function.
The Cauchy-Lagrange integral for the gas flow has the following form: where is the gas density. Gravity and inertial forces are neglected in (15), since and they are very much smaller in comparison with the same forces in water flow, taken into account in (6). Putting expressions (12) and (14) into (15) yields Here and are the radii of the cavity and the body at the cross section of cavity origin (see Figure 1); the cavitator and body shapes can be changeable; is given by (1). Substitution of (16) into (6) yields the following nonlinear integral-differential equation for the ventilated cavity shape:
Even limited by the leading term of order , (17) remains integral-differential and the principle of independence is no more valid for unsteady ventilated cavities. In the case of a large velocity of the ventilated gas () the last term in (17) is proportional to and is much greater than two previous terms proportional to (see (12)). Thus, the shape of such slender intensively ventilated cavities can be estimated with the use of the following equation: which does not contain any integral terms and corresponds to the Logvinovich independence principle.
5. Steady Ventilated Supercavities
In steady case (17) limited by the leading term of order attains the following form: where all lengths are based on the cavity radius at its origin (see Figure 1). Relation (20) follows also from (18), but in the steady case (20) is valid at arbitrary values of ventilation rate. To characterize the influence of gas injection, other nondimensional parameters can be used instead of ; for example, where is the velocity of gas at . The dimensionless ventilation rate is the ratio of the pressure heads in the gas and water flows at . If the gravity forces can be neglected also in liquid flow (), nonlinear differential equation (20) coincides with the one obtained in Manova et al. .
With the use of parameter (20) can be rewritten as follows: The standard initial conditions at can be used to integrate (20) or (24), where is the derivative of the body radius at the cavity origin. It can be seen from (20) and (22) that the ventilation can strongly change the shape of the supercavity only at small values of or at the large ratio , since is very small.
Without ventilation () equation (20) or (23) coincides with the one proposed in Nesteruk . Its solution in this case can be easily obtained with the use of initial conditions (26): For ventilated supercavities the nonlinear differential equation (20) or (23) must be solved. It can be seen that gas injection strongly changes the cavity shape not only at great values of the parameter but also at very small values of the circular channel width .
The term in brackets in (20) equals zero at and can be both positive and negative at different cavitator and hull shapes. It means that the corresponding ventilated cavity can, respectively, be both larger and smaller than the vapor one (27). Examples of calculations can be found in Manova et al.  and Nesteruk and Shepetyuk  for the first case and in Nesteruk and Shepetyuk  for the second one.
When the gravity forces can be neglected, (20) or (23) demonstrates that two independent parameters which are the pressure inside the cavity () and the gas injection rate ( or or ) influence the cavity shape. The gas leakage from the stable artificial cavity is equal to the gas injection rate and depends on the cavity dimensions, cavity closing behaviour, and so forth. Therefore, the gas leakage can also be different at the fixed cavitation number (as the equal value of the gas injection rate). In particular, it is written in Logvinovich  that “it is impossible to suggest a universal method for the theoretical determination of the gas loss.” This conclusion was supported by experiments (see, e.g., Wosnik et al.  and Vlasenko and Savchenko ), where different gas leakage/injection rates were obtained at the fixed cavitation number. Nevertheless, there are some attempts to present the gas leakage from the empty cavity () at high Froude numbers as a function of cavitation number (see, e.g., Logvinovich  and Spurk ). At small Froude numbers in horizontal flow (the cavity has no more the axis of symmetry) the formulas representing the gas leakage rate as function of the cavitation and Froude numbers were proposed in Epshtein [2–4].
6. Different Cavitator Shapes and Restrictions of the Flow Parameters
The developed theory and equations can be applied for a different slender cavitator—a part of a hull wetted by water (). For example, the slope of the cavitator radius at the cavity origin section can be both positive (; see Figure 1, a slender conical cavitator) and nonpositive (; see Figure 2, a case of base cavities). In many applications the disc or nonslender conical cavitators are used to create a long supercavity. Equations (17), (20), and (23) can be used in the case of the nonslender cavitator too, provided that the initial part of the long cavity is known. In particular, in the steady case the Logvinovich empirical formula (Logvinovich ) is valid. In (29) all lengths are based on the radius of the disc cavitator. In particular, from (29) it follows that at the corresponding cavity radius and its slope are 3 and , respectively. If at the cavity is slender enough, we can use (20) with all lengths based on . In the initial condition (26) must be replaced by . It is difficult to expect a good accuracy from such calculations, but an estimation of the supercavity shape is possible. In particular, at , the vapor cavity maximum radius can be determined from formula (27) as follows: Calculations of the maximum radius of the supercavity created by the disc cavitator with the use of the Logvinovich empirical formula (Logvinovich ) differ from (30) less than 10% at .
It is important to know which values of parameters , and are possible in real flows. To demonstrate the influence of the cavitator shape let us consider a vapor cavity in a liquid without gravity. The dependences (27) corresponding to are presented in Figure 3 and attached to the case in Figure 4. According to the stability principle, a small change in the parameters determining a solution ought to give rise to small changes in this solution.
At first let us consider the case . Since the parameter is negative at the positive cavitation numbers, then the branches of parabola (27) are directed to the bottom; accordingly at they are directed upward. For zero cavity number the parabola (27) degenerates into linear dependence (see Figure 3). Such character of solution testifies about its steadiness at . But already at zero cavitation number the stability disappears, because at formula (27) gives an infinite cavity, and at any positive cavitation number (which can be very near to zero) the cavity dimensions are bounded. The same character of instability takes place for . So, according to the stability principle only the flows with the positive cavitation numbers can be realized for cavitators with . The solution at for such cavitators is unstable and has the theoretical meaning only.
For the direction of parabola (27) branches does not change and it similarly degenerates into the linear dependence at zero cavitation number (see Figure 4). But due to the negative value of parameter , the zero cavitation number corresponds to a bounded cavity. The bounded cavities exist also for some negative cavitation numbers (small enough for module (see Figure 4)). Critical value of corresponds to the case when the parabola (27) touches the axis . At cavitation numbers, lesser than , cavity is unbounded. Application of the stability principle yields that at only flows with can be realized. At , the minimum cavitation number may be defined from (27) as follows: (Nesteruk ).
In the case of liquid with gravity the limitations of values of parameters follow from (27). Specifically, for the steady flow directed to the bottom, the coefficient of the polynomial (27) is negative; therefore a cavity always has the bounded dimensions, and steadiness principle is valid for arbitrary values of . For the opposite direction of the flow, this coefficient changes its sign, and the situation is similar to the case shown in Figure 4. The dependences of the minimum cavitation number versus the Froude number are calculated in Nesteruk .
The stability principle can be used to analyze the parameter restrictions for unsteady supercavity flows too. An example of a slender axisymmetric cavitator located in the steady flow of liquid with gravity directed upwards with the cavitation number, which varies according to the linear law , is presented in Nesteruk . In this flow the pressure inside the cavity is only time dependent and we can use (6) with or (17) with . The solution of the equation of the first approximation (9) in this case can be written as follows (see Nesteruk ):
If the parameter satisfies the condition , then a moment of time comes, when the cavity becomes unbounded for . The critical cavitation number can be also calculated. For example, for the formula can be obtained from (32). Thus, the diminishing of the cavitation number to the value lesser than critical is impossible. Probably, the supercavity flow pattern is lost in the moment when .
It is interesting to emphasize that such flow of a liquid without gravity is always stable. Really, it follows from (32) that, when the cavitation number diminishes (), the cavity dimensions are finite both at the negative and at positive current values of the cavitation number . According to (32) the cavity dimensions diminish continuously when the cavitation number increases. In particular, there are possible unsteady supercavity flows with a conical cavitator at negative cavity numbers, while steady flows are impossible at .
In this example we considered the evolutions of the cavity shape forced by the changes of the pressure inside the cavity only (or by the variations of the cavitation number ). Solution (32) is independent of the shape of the hull located inside the cavity. At small values of the area of the circular channel between the cavity surface and the hull () equation (17) must be used. In particular, we show in the next sections that for the flows with ventilated supercavities their parameters restrictions depend on the pressure, ventilation rate, and the shape of the hull located inside the cavity.
7. Cavities on Different Hulls
Let us assume the shape of the hull located in the cavity to be a combination of functions as follows: Constants coincide with the values of the hull radius at ; that is, . Constants and can be chosen to ensure the continuity of the hull radius and its slope at ; . The hull shapes can be finite () and infinite (). Examples of such shapes are presented by the following formulas and shown in Figures 5 and 6 by solid lines:
By increasing the number of , arbitrary time independent hull shape can be interpolated by formulas (33) with a rather good accuracy. We will use these formulas in the next section to obtain a steady problem solution. Here we will try to answer the very important question: is it possible to cover any given hull by one steady cavity with the origin at the cross section ?
Let us consider the critical cavity shapes corresponding to the minimal possible values of the cavitation number . In particular, in the flow without gravity () the minimal possible cavitation number is zero at and the corresponding critical cavity shape can be determined from (27) as follows: In the case of the disc cavitator the critical cavity shape can be estimated with the use of (29) and (27) and continuity condition at : Examples of the critical cavity shape (37) are presented in Figure 5 by dashed line 1 () and in Figure 6 by dashed line 2 (). An example of the critical cavity shape (38) for the disc cavitator is presented in Figure 5 by dashed line 2.
For the base cavities at , the minimal possible cavitation number is and the corresponding conical critical cavity shape can be determined from (27) as follows: An example of the critical cavity shape (39) is presented in Figure 6 by dashed line 1 ().
If the hull pierces the corresponding critical cavity shape (e.g., dashed lines 1 in Figures 5 and 6), then it is impossible to cover such hull with one cavity starting at the cross section . Neither the cavity pressure increase (diminishing the cavitation number ) nor increasing the gas injection rate can create a cavity covering all the hull. It is possible only to cover the part of the hull located upstream to the piercing region (e.g., at for case shown in Figure 5) or to use other cavities, for example, the ones starting at the cross sections and on the hull shown in Figure 5.
For the vapor cavities on conical-cylindrical hulls ( in (30)), a similar limitation was revealed in Nesteruk . It was shown that a cavity created by a slender cone can close on the cylindrical part only if (in accordance with (37)). If the radius of the cylindrical part is greater than value , all the cavities close on the conical part. For such cavities both the positive and the negative cavitation numbers are possible (the stability occurs in all these cases) and their shapes can be both convex and concave (Nesteruk ).
If the hull (35) touches the critical cavity shape only in one cross section , then the maximum length of the cavity closing on the conical part (or the minimal length of the cavity closing on the cylindrical part) is equal to (the cavity length is calculated along the axis of symmetry). Then the relationship between and the radius of the conical part coincides with (37) or (38) after replacing instead of and instead of . In Varghese et al.  the cavities created by the disk cavitator were calculated with the use of nonlinear approach and the dependence of the minimum cavity length versus the cylindrical part radius was revealed (no convergence was achieved at the values of cavity length smaller than ). The numerical results from the paper (Varghese et al. ) are shown in Figure 5 by markers. The rather good agreement between the data of Varghese et al.  and dashed line 2 supports the conclusions of the presented analysis and illustrates the accuracy of the Logvinovich formula (29) and the first approximation (27).
Any hull of bounded dimensions located under the corresponding critical cavity shape (like dashed lines 2 in Figures 5 and 6) can be entirely covered by a supercavity at zero gas injection rate by increasing the pressure inside the cavity or by reducing the cavitation number . If the cavitation number is not small enough, the corresponding supercavity closes on the hull.
In many cases the ventilation increases the supercavity dimensions at fixed (see Section 5, Manova et al. , and Nesteruk and Shepetyuk ). If the increase of the pressure at (or diminishing ) is impossible we could try to cover the hull by the cavity entirely only by increasing the gas injection rate. Equations (23) and (24) and a numerical solution of (23) obtained in Nesteruk and Shepetyuk  show that it is possible only for small enough values of . The ventilation decreases the effective cavity number (see (24)) and corresponding cavity must be larger (see (27) at ). On the other hand, according to the last term in (23) the ventilated cavity is smaller than the vapor one at . Thus, covering the hull by the cavity at small values of the gas pressure depends on the values of the ventilation rate and .
Let us illustrate this conclusion by an example of the numerical solution of (23) obtained in Nesteruk and Shepetyuk  for conical-cylindrical hulls ( in (30), , .8, and ; see Figure 7). The values of and correspond to the hull located under the corresponding critical cavity shape (in accordance with (37)). At the small fixed cavitation number it is possible to have a cavity which closes on the cylindrical part. In this case the cavity length can be rather good estimated with the use of linear theory—(27) at . In particular, the discontinuity of takes place (similar to the one revealed in Varghese et al.  and Nesteruk  for vapor cavities). If (or ) the cavity becomes unbounded. It means that ventilation rate is limited by the value
At the cavity closes only on conical part of the hull () and there is a large difference between the linear theory and nonlinear calculations based on (23). The gas injection rate is limited by some other critical value ( or ). Formally, a solution exists at greater values of ventilation, but the cavity becomes infinite at and such flow cannot be real.
Thus, covering the hulls located under the corresponding critical cavity shapes by ventilated cavities must be investigated with the use of solutions of nonlinear equation (20) or (23). In the next section it will be shown that in the case the resolving (20) can be reduced to the calculation of simple integrals.
8. Solution of the Steady Problem at High Froude Numbers
8.1. Integral Formula for the Solution
The nonlinear differential equation (20) or (23) can be solved numerically at any value of the Froude number and for any shape of the hull located inside the cavity. Let us consider the case and the hulls which can be presented or interpolated as (33). Substituting into (23) and using (33) yield The initial conditions (26) at can be rewritten as follows: Let us assume the function to be continuous at points ; . Its derivative can be discontinuous at these points if the slope of the hull contour is discontinuous (e.g., at for the hull shape shown in Figure 5). It means that, for every segment , , and the segment , the specific initial conditions should be used at the beginning of each segment , :
The cavity closing corresponds to the zero value of , but from the physical point of view it is impossible to have the zero cross section area of gas flow in the steady ventilated cavity and from the mathematical point of view it is impossible to achieve the zero value of , since the differential equation (41) has a discontinuity. To avoid these difficulties we will suppose that the cavity “closure” corresponds to some small positive value of (e.g., ). Then the cavity length (along the axis of symmetry) can be also calculated from integral relations (45).
8.2. Maximum and Minimum Values of the Circular Channel Area
The inverse function may be many-valued (see curves 4 and 5 in Figure 8). In this case it is necessary to integrate in (45) from to corresponding to the maximum or the minimum value of the gas flow cross section area in the circular channel between the cavity surface and the hull and then to integrate from to with the opposite sign in (45). The values of can be easily calculated from the quadratic equation following from (44) at :
If , the maximum value of can be calculated from (47) as follows: An example of the cavity with the maximum value of (for , ) is presented in Figure 8 (curve 4). At the maximum of exists only for . In particular, at it follows from (22), (24), (46), and (47) that
It must be noted that among the cavities presented in Figure 8 only curves 4 and 9 have classical elliptical shapes. Cavities 3 and 8 are parabolic, 2 and 7 are conical, 1 and 6 are hyperbolic, and 5 is cylindrical. If , it is possible to realize only cavities 4, 7, 8, and 9. Other cavities cannot exist in reality without conical part of the hull () according to the stability principle mentioned above. The cavity shapes corresponding to the zero cavitation number (dashed curves 3, 5, and 8) do not cover the hull shown in Figure 8. It means that at any value and at any ventilation rate all 3 cavitators (with , 0, and −0.1) cannot create a cavity large enough to cover the hull.
8.3. Degenerate Solutions, Neutral Hull Shapes, and Critical Values of the Ventilation Rate
Equation (41) can be rewritten as follows (with the use of (22), (24), and (43)): If then (51) has the obvious solution at any value of the ventilation rate. The inverse function is many-valued and such degenerate solution cannot be obtained with the use of (45). Relationship (33) in view of (43) and (52) allows calculating the corresponding body shape For example, for the part of the hull, corresponding to and shown in Figure 8, . Therefore (53) yields and . The degenerate solution is shown in Figure 8 by straight line 5. The hull (53) ensures the same cavity shape at any ventilation rate. This shape coincides with the shape of the vapor cavity at the same values of parameters and and can be refereed as neutral.
Equation (20) demonstrates that for the ventilation increases the cavity dimensions. For example, at the dimensions of cavities 1–4 shown in Figure 8 for will increase with increasing the ventilation rate. The maximum cavity radius can be also infinite, if the cavity covers all the hull or the hull is unbounded (e.g., ). Formulas (46) and (48) allow calculating the critical ventilation rate corresponding to zero value of If inequality (49) is valid, the cavity is bounded at the ventilation rates (54) but for the value of in (48) can be zero or negative. The equation yields the second critical value, which with the use of (46) can be represented as follows: If , the corresponding cavity is bounded, but unbounded at . It means that the ventilation rate is limited by the value . For unbounded cylindrical hulls the critical ventilation rates were calculated in Manova et al.  and coincide with (54) and (55).
On the other hand, for the ventilation decreases the cavity dimensions. For example, the dimensions of cavities 6–9 shown in Figure 8 for will decrease with increasing the ventilation rate at . In particular, according to the formula (50) the decreases from value at to at . The equation yields the critical value, presented by (55). If , the corresponding cavity has a minimum, but increasing the ventilation rate over the critical value diminishes the cavity dimensions and its shape has no minimum.
8.4. Examples of the Ventilated Cavity Shape Calculations
If the pressure in the cavity at some cross section is fixed (e.g., the cavitation number is fixed) and (the area of the circular gas channel diminishes), the increase of the gas injection rate yields the diminishing of the cavity dimensions (since local pressure in the cavity diminishes). This fact is illustrated by the calculation examples shown in Figures 9 and 10. In both cases and the increasing of the gas injection rate decreases the cavity dimensions at . To calculate the cavities at the values and were used according to formulas (43). The values are also negative for all cavities shown in Figure 10.
Curve 4 shown in Figure 10 corresponds to the critical value of gas injection rate calculated from (55). At greater values of (curves 5–7) cavity shape has no minimum. At smaller and zero values of the gas injection rate (curves 1–3) cavity shapes have a minimum point and some of them (curves 2 and 3) have also a maximum located near the closing on the conical part of the hull. For and the hull presented in Figure 8 corresponds to the neutral shape (53) at . It means that at the corresponding cylindrical cavity shape (shown in Figure 8 by curve 5) is the same at any values of ventilation. Since the value is negative, at , at this segment the cavity radius diminishes as the gas injection rate increases. The examples of calculations are shown in Figure 11.
Absolutely different behavior occurs in the case , when the area of the circular gas channel increases. The dimensions of the ventilated cavities increase as the gas injection rate increases. This fact is illustrated by the calculation examples shown in Figures 12 and 13. In both cases . To calculate the cavities at the values and were used according to formulas (43). At great enough values of the cavitation number , the cavity shapes transform from convex with the maximum thickness section at small values of the gas injection rates (see Figure 12, curves 1 and 2) to the concave at greater values of . At the cavity shapes are concave (see, e.g., Figure 13). All the cavities cannot be extended for (according to the stability principle); thus the values of gas injection rate are limited. For example, for the case shown in Figure 12 the critical value . At smaller values of the critical values of the gas injection rate diminish (e.g., at (see Figure 13) and at ). It must be noted that for the hull shape shown in Figures 8–13 the cavities corresponding to the critical values (54) and (55) are bounded (since is restricted). For example, the cavity shape corresponding to the critical value of the ventilation is shown in Figure 12 by curve 2.
8.5. Cavity Closing Near the Hull Contour Discontinuities
A special attention should be paid to the cases when the cavity end passes through the hull contour discontinuity as shown in Figures 14 and 15. If or (see (43)), the cavity length varies continuously. This case is shown in Figure 14. Absolutely different is the case shown in Figure 15. At some critical values of parameters and the cavity contour touches the hull at the point , and a “jump” in the cavity length is possible (see Figure 15).
These two cases of cavity behaviour are separated by the condition at or At formulas (56) and (57) allow obtaining the relationships between the critical values of parameters , , and . With the use of (27), equation (56) can be written as follows: where the critical value of the cavitation number may be determined from (57) as follows: Finally, from (58) and (59) the following simple relationship can be obtained:
In the particular case of a conic-cylindrical hull (), (60) coincides with the relationship proposed in Nesteruk . When the value of is smaller than the critical one (60) and , the cavity length varies continuously as shown in Figure 14. Otherwise the cavity length has a discontinuity as shown in Figure 15. For the ventilated cavities the critical values of parameters , , and depend on and should be calculated with the use of (45). For the conic-cylindrical hull some examples can be found in Nesteruk and Shepetyuk  and are shown in Figure 7.
In the case a rather weak dependence of the cavity length on the ventilation rate should be expected. Even in the case when cavity dimensions increase with increasing at , the negative value of leads to a decrease of the cavity length at . This fact is illustrated in Figure 16 by the cavities on the conic-cylindrical hull (, , and ) created by a disc cavitator. The same cavitator and hull were tested in Vlasenko and Savchenko .
The value is smaller than the critical one based on (60); therefore no “jumps” in cavity length (as shown in Figure 15) are expected. At the greater value of the cavitation number the corresponding values of are negative and ventilation does not increase the cavity length significantly (see Figure 16). On the contrary, at the smaller cavitation number = 0.06 the values of become positive and the same values of the ventilation rate drastically increase the cavity length (see Figure 16). In the experiments performed in Vlasenko and Savchenko  the values of were smaller than 0.0001. The calculations show that the cavity shapes are very close to the vapor ones (shown in Figure 16 by dashed lines) at these small gas injection rates. Thus, the changes in the pressure inside the cavity (or different values of ) are the main reason of changes in the cavity length in the experiments (Vlasenko and Savchenko ). Unfortunately, the pressure inside the cavity was not measured in Vlasenko and Savchenko .
If at a zero or a small value of the cavity covers a hull, its length approaches the value and at (in particular, for a bounded hull ), then the value is negative (according to formula (43)). In this case the cavity length depends on the ventilation rate very slightly. This fact was revealed also in experiments (see Wosnik et al.  and Vlasenko and Savchenko ).
9. Asymptotic Solution at Small Values of the Ventilation Rate
9.1. Basic Equations
If the gas injection rate is small enough , the solution of the nonlinear equation (17) can be expressed as follows: where is the solution of (17) at zero ventilation rate , Function is a solution of the following equations: To solve the nonlinear integral-differential equation (62) the asymptotic series obtained in Nesteruk  and the standard initial and boundary conditions (19) can be used for the function . If the accuracy restrictions are not very severe, the first approximation equation (similar to (9) or (10) with main terms of order only) can be easily solved. Knowing the function the right hand part of (64) and (65) can be calculated and the function can be determined as a solution of the set of linear differential equations with the partial derivatives (64) and (66) and zero initial and boundary conditions. Thus the presentation (61) allows avoiding the nonlinear integral-differential equations.
9.2. Stability of the Steady Ventilated Cavities
Let us use the main term in (61) and the first approximation in (62) to analyze the stability of ventilated cavities at small values of the gas injection rates. Let us consider the steady cavity with a constant gas pressure inside the cavity . According to the first approximation of (62) the cavity radius is given by formula (27). Let us consider small changes in gas pressure , where is a new constant gas pressure inside the cavity. According to (27) and (1) the new cavity radius can be written as follows: Here we neglect the changes in the vapor pressure.
Integration of formulas (27) and (67) allows calculating the changes in the cavity volume : where is the initial cavity length, which is measured along the axis and can be found from (27) as follows: Here we suppose the body radius to be time independent; therefore yields also the changes in the volume of gas.
On the other hand for a fixed mass of gas its pressure and volume are related according to the polytropic relationship: where (in particular, for isothermal process ); the initial volume of gas can be calculated with the use of (27) as follows: Putting in (70) and and linearization yield Values and have opposite signs (see (68) and (72)) and can compensate each other, if . Otherwise the cavity becomes unstable. The critical value of the gas pressure can be obtained from , (68), and (72): At smaller values of the gas pressure the cavity is stable. If , the cavity becomes unstable. The existence of the critical gas pressure value was revealed in experiments (e.g., Michele ) and supported by the theory developed by Paryshev [23, 24].
Equation (73) can be rewritten in the dimensionless form taking into account the definition of the cavitation number (1): where the vapor cavitation number and the critical value of the cavitation number can be obtained from (69) as follows: It follows from (71), (73), and (76) that Formula (77) allows calculating the critical value of for every value of the cavity length. The corresponding critical cavitation number can be calculated with the use of (76). Thus the dependence () can be obtained. In particular, for an empty isothermal cavity at large Froude numbers (, , and ) it follows from (77) and (76) that
Examples of calculations based on (78) are presented in Figure 17. Stable cavities correspond to regions located above the corresponding curves. In the case of the slender cavitators, the function () depends on in comparison with the relationship obtained by Paryshev  for the isothermal cavities created by nonslender cavitators with the use of semiempirical formulas for the cavity shape. In the cases of long cavities () and for the base cavities with the relationships (78) yield the formulas respectively. The difference between the first relationships (80) and (79) can be explained by the limited accuracy of both the first approximation equation (27) and the semiempirical formulas used by Paryshev. In the experiments with 2D cavities the value was measured by Michel . It can be seen from Figure 17 that the largest stability regions correspond to the base cavities (). In particular, stable cavities with negative cavitation number are possible at . It follows from (78) that the cavitation number is limited by value , obtained in Nesteruk .