# Shape of Slender Axisymmetric Ventilated Supercavities

Authors: Igor Nesteruk
Journal: Journal of Computational Engineering
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1]) 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 [1], Epshtein [2–4], Knapp et al. [5], Yegorov et al. [6], Levkovsky [7], Kuklinski et al. [8], Spurk [9], Wosnik et al. [10], Zhuravlev and Varyukhin [11], Matveev et al. [12], and Vlasenko and Savchenko [13]). 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
(1)σ0=2(p∞-p0-pv)ρU∞2
(ρ is the water density, U∞ is the velocity of movement, pv is the water vapor pressure at ambient temperature, and p∞ and p0 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 [1], Epshtein [2–4], and Spurk [9]) and in experiments. For example, in tests by Wosnik et al. [10] and Vlasenko and Savchenko [13] 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 [14]). 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 [11]). 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. [15] and Nesteruk and Shepetyuk [16, 17] for steady flow of liquid without gravity effects.Figure 1
Axisymmetric supercavitation flow pattern with the gas ventilation in a circular channel between the vehicle hull and the cavity at the initial moment of timet=0.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 [18], Song [19], Michel [20], and Zou et al. [21] and by many other authors. The theories of such empty cavities were proposed by Woods [22], Paryshev [23, 24], Nesteruk [25], and Semenenko [26]. 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 [27]) the potential of the steady flow of ideal incompressible liquid was obtained by the method of matched asymptotic expansions. In Nesteruk [28], this slender body theory was generalized for the unsteady case. The potential of the external water flow can be written as follows:
(2)Φ(x,r,t,ε)=x+ε2ln⁡εA(x,t)+ε2{A(x,t)ln⁡r*+B(x,t)}+O(ε4ln⁡ε);(3)ε=Rmax⁡′L′;r*=rε;A(x,t)=F(x,t)(∂F∂x+τ∂F∂t);F(x,t)=R(x,t)ε;(4)Φ(x,r,t)=Φ′(x,r,t)L′U∞′(t);τ(t)=L′tx′U∞′(t);hhhhhhhhhhhhhhhhhhhhhhhhhhht=t′tx′;(5)B(x,t)=-A(x,t)ln⁡2-12∫∂A(ξ,t)∂ξsign⁡(x-ξ)ln⁡|x-ξ|dξ.
We use the symbols with “prime” for physical (dimensional) parameters; tx′ is the typical time of unsteadiness, for example, the pulsation period; R(x,t)=R′(x,t)/L′ is the body or the cavity radius; ε is a small parameter, a ratio of the maximum radius of the body Rmax⁡′ (or a cavitator-cavity system) to its length L′ at a fixed moment of time t; x,r 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 R(x,t) and the function U∞′(t) 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 radiusR(x,t) may be calculated with the use of pressure distribution p′(x,t) over the surface. Putting expression (2) into the Cauchy-Lagrange integral yields (Nesteruk [28, 29])
(6)ε2ln⁡εΖ(A)+ε2[Ζ(A)ln⁡F+Z(B)+0.5A2F-2]+O(ε4ln2ε)=-0.5Cp(x,t)±xFr-2(t),(7)Ζ(U)≡τ∂U∂t+∂U∂x+SU;S(t)=L′(U∞′)2dU∞′dt′;Fr(t)=U∞′(t)gL′,(8)Cp(x,t)≡2[p′(x,t)-p∞′(0,t)]ρU∞′2(t).
Here Fr(t) is the Froude number, g is the gravity acceleration, and p∞′(0,t) is pressure in liquid far from the body at the level x=0. The sign “−” in the last term of (6) corresponds to the gravitational acceleration vector directed along the x 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 U∞′(t).Nonlinear integral-differential equation (6) relates the slender body (or the cavity) radius and the pressure distribution in liquid over its surface p′(x,t). If the functions p′(x,t), p∞′(0,t), and U∞′(t) and the cavitator shape are given, the set of (3)–(8) allows one to calculate the unknown cavity radius R(x,t). The asymptotic solution of the nonlinear integral-differential equation (6) was obtained in Nesteruk [30].


## 3. Equation of the First Approximation and Logvinovich’s Principle of Independence

The leading term of (6) (of order ε2ln⁡ε) provides the equation of the first approximation (Nesteruk [28]):
(9)ε2ln⁡ε[τ∂A∂t+∂A∂x+SA]=±xFr-2(t)-0.5Cp(x,t).
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 [28].Equation (9) can be rewritten in the absolute coordinates xa, ra (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 xa and the cavitator movement are opposite and the pressure pc′(t) on the cavity surface depends on the time only. Then (9) attains the following form (Nesteruk [30]):
(10)ln⁡ε∂2R2∂t2=σat(t)±2xaFr02,(11)σat(t)=2[pat′-pc′(t)]ρU∞′2(t0),Fr0=U∞′(t0)gL′,tx′=L′U∞′(t0),
where t0 is some fixed moment of time and the atmospheric pressure pat′ on the undisturbed free water surface is assumed to be constant. The thickness parameter ε can be assumed to be coinciding with the ratio of the cavitator maximum radius to its length, for example, for slender conical cavitator ε=β=tg(θ) (2θ is the cone angle).According to (10), the evolution of every cross section of the cavity (at fixed value of xa) 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 ln⁡ε is used in the computer codes developed in the Institute of Hydromechanics NASU for the calculations of unsteady supercavities, for example, Semenenko and Naumova [31].It must be noted that the accuracy of the first approximation (9) or (10) is limited by value |ln⁡ε|-1, which yields only 44% for ε=0.1 and 22% for ε=0.01. 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 gaspv+pg(x,t) must be equal to the pressure in water on the cavity surface pc(x,t). 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
(12)Q(t)=Q′(t)(L′)2U∞′(t)=πV(x,t)W(x,t);W(x,t)=R2(x,t)-Rb2(x,t)
inside the cavity is only time dependent. Here the dimensionless gas velocity V(x,t) is based on U∞′(t); the dimensionless cavity and body radii (R(x,t) and Rb(x,t)) are based on L′. 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
(13)V(x,t)=∂Φg(x,t)∂x,Φg(x,r,t)=Φg′(x,r,t)L′U∞′(t)
and (12) as follows:
(14)Φg(x,t)=Q(t)π∫0xdxW(x,t)+ϕ1(t),
where ϕ1(t) is an arbitrary time dependent function.The Cauchy-Lagrange integral for the gas flow has the following form:(15)∂Φg′∂t′+(V′)22+pg′ρg′=[∂Φg′∂t′+(V′)22+pg′ρg′]x=0,
where ρg′ is the gas density. Gravity and inertial forces are neglected in (15), since ρ′/ρg′≈800 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
(16)Cp(x,t)=2[pc′(x,t)-p∞′(0,t)]ρ′U∞′2(t)=2[pg′(x,t)+pv-p∞′(0,t)]ρ′U∞′2(t)=-2ρg′τ(t)πρ′∂∂t[Q(t)∫0xdxW(x,t)]-σ0(t)+ρg′Q2(t)ρ′π2[W-2(0,t)-W-2(x,t)]-2ρg′Q(t)S(t)πρ′∫0xdxW(x,t).
Here R0(t)=R(0,t) and Rb0(t)=Rb(0,t) are the radii of the cavity and the body at the cross section of cavity origin x=0 (see Figure 1); the cavitator and body shapes can be changeable; σ0(t) is given by (1). Substitution of (16) into (6) yields the following nonlinear integral-differential equation for the ventilated cavity shape:
(17)ε2ln⁡εΖ(A)+ε2[Ζ(A)ln⁡F+Z(B)+0.5A2F-2]+O(ε4ln2ε)=0.5σ0(t)±xFr-2(t)+ρgτ(t)πρ∂∂t[Q(t)∫0xdxW(x,t)]+ρgQ(t)S(t)πρ∫0xdxW(x,t)-ρgQ2(t)2ρπ2[W-2(0,t)-W-2(x,t)].Even limited by the leading term of orderε2ln⁡ε, (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 (V=V′/U∞′≫1) the last term in (17) is proportional to V2 and is much greater than two previous terms proportional to V (see (12)). Thus, the shape of such slender intensively ventilated cavities can be estimated with the use of the following equation:
(18)ε2ln⁡εΖ(A)+O(ε2)=0.5σ0(t)±xFr-2(t)-ρgQ2(t)2ρπ2[W-2(0,t)-W-2(x,t)]
which does not contain any integral terms and corresponds to the Logvinovich independence principle.To solve both (17) and (18), the standard initial and boundary conditions in two domains can be used (Nesteruk [28]):
(19)x≥∫0tdtτ:R(x,0)=R1(x),∂R(x,0)∂t=R2(x);x≤∫0tdtτ:R(0,t)=R3(t),∂R(0,t)∂x=R4(t),
where R1, R2, R3, and R4 are given functions.


## 5. Steady Ventilated Supercavities

In steady case (17) limited by the leading term of order ε2ln⁡ε attains the following form:
(20)d2R2dx2=σ0ln⁡ε±2xFr2ln⁡ε+Δ[a-1(R2-Rb2)2],(21)Δ=-ρgQ2π2R04ρU∞2ln⁡ε,a=[1-Rb02R02]-2,
where all lengths are based on the cavity radius R0 at its origin x=0 (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 Δ; for example,
(22)Ve=-aΔln⁡ε=(ρgV02)(ρU∞2)orCQ=Q(U∞πR02),
where V0 is the velocity of gas at x=0. The dimensionless ventilation rate Ve is the ratio of the pressure heads in the gas and water flows at x=0. If the gravity forces can be neglected also in liquid flow (Fr→∞), nonlinear differential equation (20) coincides with the one obtained in Manova et al. [15].With the use of parameterVe (20) can be rewritten as follows:
(23)d2R2dx2=σeffln⁡ε±2xFr2ln⁡ε-ΔW2,(24)σeff=σ0-Ve=σ0-(ρgV02)(ρU∞2),(25)W(x)=R2(x)-Rb2(x).
The standard initial conditions at x=0(26)R=1,dRdx=β
can be used to integrate (20) or (24), where β 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 σ0±2x/Fr2 or at the large ratio V0/U∞, since ρg/ρ is very small.Without ventilation (Δ=Ve=CQ=0) equation (20) or (23) coincides with the one proposed in Nesteruk [32]. Its solution in this case can be easily obtained with the use of initial conditions (26):
(27)R2(x)=σ0x22ln⁡ε±x33Fr2ln⁡ε+2βx+1.
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 Δ but also at very small values of the circular channel width R-Rb.The term in brackets in (20)
(28)s2(x)=W-2(0)-W-2(x)
equals zero at x=0 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. [15] and Nesteruk and Shepetyuk [16] for the first case and in Nesteruk and Shepetyuk [17] 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 (σ0) and the gas injection rate (Δ or Ve or CQ) 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 [1] 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. [10] and Vlasenko and Savchenko [13]), 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 (Rb(x)≡0) at high Froude numbers as a function of cavitation number (see, e.g., Logvinovich [1] and Spurk [9]). 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 (x<0). For example, the slope of the cavitator radius at the cavity origin section x=0 can be both positive (β>0; see Figure 1, a slender conical cavitator) and nonpositive (β≤0; 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 [1])
(29)R(x)=(1+3x)1/3,x<3
is valid. In (29) all lengths are based on the radius of the disc cavitator. In particular, from (29) it follows that at x=xs=2 the corresponding cavity radius and its slope are Rs≈1.913 and βs≈0.273, respectively. If at x>xs the cavity is slender enough, we can use (20) with all lengths based on Rs. In the initial condition (26) β must be replaced by βs. It is difficult to expect a good accuracy from such calculations, but an estimation of the supercavity shape is possible. In particular, at β=ε, Fr→∞ the vapor cavity maximum radius can be determined from formula (27) as follows:
(30)Rm=Rs1-2βs2ln⁡βsσ0≈1.91σ0+0.193σ0.
Calculations of the maximum radius of the supercavity created by the disc cavitator with the use of the Logvinovich empirical formula (Logvinovich [1])
(31)Rm≈0.84σ0+1σ0
differ from (30) less than 10% at 0.001<σ0<0.1.Figure 2
Base steady ventilated cavity (β<0) on a cylindrical hull Rb(x)=Rb0=const.It is important to know which values of parametersσ0,Fr, and Ve 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 β>0 are presented in Figure 3 and attached to the case β<0 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.Figure 3
Dependences (27) for different values of the cavitation number at β>0, Fr→∞.Figure 4
Dependences (27) for different values of the cavitation number at β<0, Fr→∞.At first let us consider the caseβ>0. Since the parameter α=0.5σ0/ln⁡ε is negative at the positive cavitation numbers, then the branches of parabola (27) are directed to the bottom; accordingly at σ0<0 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 σ0<0. But already at zero cavitation number the stability disappears, because at σ0=0 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 β=0. So, according to the stability principle only the flows with the positive cavitation numbers can be realized for cavitators with β≥0. The solution at σ0=0 for such cavitators is unstable and has the theoretical meaning only.Forβ<0 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 β, 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 σ0=σmin⁡<0 corresponds to the case when the parabola (27) touches the axis x. At cavitation numbers, lesser than σmin⁡, cavity is unbounded. Application of the stability principle yields that at β<0 only flows with σ0<σmin⁡<0 can be realized. At β<0, Fr→∞ the minimum cavitation number may be defined from (27) as follows: σmin⁡=2β2ln⁡ε (Nesteruk [32]).In the case of liquid with gravity the limitations of values of parametersσ0,Fr follow from (27). Specifically, for the steady flow directed to the bottom, the coefficient 1/(3Fr2ln⁡ε) of the polynomial (27) is negative; therefore a cavity always has the bounded dimensions, and steadiness principle is valid for arbitrary values of σ0,Fr. 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 [32].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 lawσ0(t)=σ0(0)-a0t, is presented in Nesteruk [35]. In this flow the pressure inside the cavity is only time dependent and we can use (6) with Cp(x,t)=-σ0(t)=-σ0(0)+a0t or (17) with Q(t)=0. The solution of the equation of the first approximation (9) in this case can be written as follows (see Nesteruk [35]):
(32)R2(x,t)=σ(t)x22ln⁡ε+x3(a0-Fr-2)3ln⁡ε+2R0βx+R02.If the parametera0 satisfies the condition 0<a0<Fr-2, then a moment of time tc comes, when the cavity becomes unbounded for t>tc. The critical cavitation number σc=σ0(tc) can be also calculated. For example, for β=0 the formula σc=[-6(Fr-2-a0)2ln⁡ε]1/3 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 σ=σc.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 (a0>0), the cavity dimensions are finite both at the negative and at positive current values of the cavitation number σ0(t). 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 σ0<0.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 numberσ0). 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 (W(x,t)≪1) 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:(33)Rb(x)=αi(x-xi)2+2βi(x-xi)+Rbi2,xi≤x≤xi+1;i=0,1,2,…,k-1;x0=0,(34)Rb(x)=Rbk;x>xk.
Constants Rbi coincide with the values of the hull radius at =xii=0,1,2,…,k; that is, R(xi)=Rbi. Constants βi and αi can be chosen to ensure the continuity of the hull radius and its slope at x=xi; i=1,2,…,k-1. The hull shapes can be finite (Rbk=0) and infinite (Rbk>0). Examples of such shapes are presented by the following formulas and shown in Figures 5 and 6 by solid lines:
(35)Rb(x)={Rb0+x(Rb1-Rb0)x1,0≤x≤x1Rb1,x1≤x≤x2Rb12-(x-x2)2Rb12(x3-x2)2,x2≤x≤x30,x≥x3,(36)Rb(x)={Rb0-xRb0x1,0≤x≤x10,x≥x1.Figure 5
Hull shape (35) (Rb0=0.6; Rb1=1.5; x1=5; x2=7; x3=10; solid line) and critical infinite cavity shapes at zero cavitation number and Fr→∞ for two different cavitators (dashed line 1 corresponds to (37) at β=0.1; dashed line 2 corresponds to the disc cavitator (38)).Figure 6
Hull shape (36) (Rb0=0.6; x1=11; solid line) and critical base cavity shapes at minimal cavitation numbers and Fr→∞ for two different cavitators (dashed line 1 corresponds to σmin⁡=2β2ln⁡ε (39) at β=-0.1; dashed line 2 corresponds to (37) at β=0).By increasing the number ofk, 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 x=0?Let us consider the critical cavity shapes corresponding to the minimal possible values of the cavitation numberσ0=σmin⁡(Fr). In particular, in the flow without gravity (Fr→∞) the minimal possible cavitation number is zero at β≥0 and the corresponding critical cavity shape can be determined from (27) as follows:
(37)R(x)=2βx+1.
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 xs=2:
(38)R(x)≈{(1+3x)1/3,0≤x≤20.546(x-2)+1.9132,x≥2.
Examples of the critical cavity shape (37) are presented in Figure 5 by dashed line 1 (β=0.1) and in Figure 6 by dashed line 2 (β=0). 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 atβ<0, Fr→∞ the minimal possible cavitation number is σmin⁡=2β2ln⁡ε and the corresponding conical critical cavity shape can be determined from (27) as follows:
(39)R(x)=βx+1.
An example of the critical cavity shape (39) is presented in Figure 6 by dashed line 1 (β=-0.1).If the hull pierces the corresponding critical cavity shape (e.g., dashed lines 1 in Figures5 and 6), then it is impossible to cover such hull with one cavity starting at the cross section x=0. Neither the cavity pressure increase (diminishing the cavitation number σ0) 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 x≤x1 for case β=0.1 shown in Figure 5) or to use other cavities, for example, the ones starting at the cross sections x=x1 and x=x2 on the hull shown in Figure 5.For the vapor cavities on conical-cylindrical hulls (x2→∞ in (30)), a similar limitation was revealed in Nesteruk [36]. It was shown that a cavity created by a slender cone can close on the cylindrical part only if R01<2βx1+1 (in accordance with (37)). If the radius of the cylindrical part is greater than value 2βx1+1, 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 [36]).If the hull (35) touches the critical cavity shape only in one cross section x=x1, then the maximum length of the cavity closing on the conical part (or the minimal length of the cavity Lmin⁡ closing on the cylindrical part) is equal to x1 (the cavity length is calculated along the axis of symmetry). Then the relationship between x1=Lmin⁡ and the radius of the conical part Rb1 coincides with (37) or (38) after replacing x1 instead of x and Rb1 instead of R(x). In Varghese et al. [37] the cavities created by the disk cavitator were calculated with the use of nonlinear approach and the dependence of the minimum cavity length Lmin⁡ versus the cylindrical part radius Rb1 was revealed (no convergence was achieved at the values of cavity length smaller than Lmin⁡). The numerical results from the paper (Varghese et al. [37]) are shown in Figure 5 by markers. The rather good agreement between the data of Varghese et al. [37] 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 Figures5 and 6) can be entirely covered by a supercavity at zero gas injection rate Ve by increasing the pressure inside the cavity or by reducing the cavitation number σ0. If the cavitation number σ0 is not small enough, the corresponding supercavity closes on the hull.In many cases the ventilation increases the supercavity dimensions at fixedσ0 (see Section 5, Manova et al. [15], and Nesteruk and Shepetyuk [16]). If the increase of the pressure at x=0 (or diminishing σ0) 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 [16] show that it is possible only for small enough values of σ0. The ventilation decreases the effective cavity number (see (24)) and corresponding cavity must be larger (see (27) at σ0=σeff). On the other hand, according to the last term in (23) the ventilated cavity is smaller than the vapor one at σ0=σeff. Thus, covering the hull by the cavity at small values of the gas pressure depends on the values of the ventilation rate and σ0.Let us illustrate this conclusion by an example of the numerical solution of (23) obtained in Nesteruk and Shepetyuk [16] for conical-cylindrical hulls (x2→∞ in (30), β=ε=0.1, Rb0=0.8, and Fr→∞; see Figure 7). The values of x1=17 and Rb1=2 correspond to the hull located under the corresponding critical cavity shape R01<2βx1+1 (in accordance with (37)). At the small fixed cavitation number σ0=0.01 it is possible to have a cavity which closes on the cylindrical part. In this case the cavity length Lc can be rather good estimated with the use of linear theory—(27) at σ0=σeff. In particular, the discontinuity of Lc takes place (similar to the one revealed in Varghese et al. [37] and Nesteruk [36] for vapor cavities). If σeff→0 (or Ve→σ0) the cavity becomes unbounded. It means that ventilation rate is limited by the value
(40)Δcr=-σ0aln⁡εorVecr=σ0.Figure 7
The length of ventilated cavities on conic-cylindrical bodies atx1=17, Rb1=2, Rb0=0.8, β=ε=0.1, and Fr→∞ (Nesteruk and Shepetyuk [16]).Atσ0=0.1 the cavity closes only on conical part of the hull (Lc≤x1) 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 (Δcr≈0.00593 or Vecr≈0.105). Formally, a solution exists at greater values of ventilation, but the cavity becomes infinite at Δ≈0.00593 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 Fr→∞ 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 Fr→∞ and the hulls which can be presented or interpolated as (33). Substituting R2(x)=W(x)+Rb2(x) into (23) and using (33) yield
(41)d2Wdx2=σeffln⁡ε-2αi-ΔW2,xi<x<xi+1;hhihhhi=0,1,2,…,k-1;αk=0,x>xk.
The initial conditions (26) at x=0 can be rewritten as follows:
(42)W=1-Rb02,dWdx=2β-2β0.
Let us assume the function W(x) to be continuous at points x=xi; i=1,2,…,k. Its derivative dW/dx can be discontinuous at these points if the slope of the hull contour is discontinuous (e.g., at x=x1 for the hull shape shown in Figure 5). It means that, for every segment xi<x<xi+1, i=0,1,2,…,k-1, and the segment x>xk, the specific initial conditions should be used at the beginning of each segment x=xi, i=0,1,2,…,k:
(43)W=R2(xi)-Rbi2=Wi,dWdx=dR2dx|x=xi-2βi=ui.The order of (41) can be reduced by substitutions dW/dx=u and d2W/dx2=du/dx=udu/dW. With the use of initial conditions (43) the following first-order equation can be obtained:
(44)u2=2(σeffln⁡ε-2αi)(W-Wi)+2Δ[1W-1Wi]+ui2,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhxi<x<xi+1.Equation (44) can be integrated with the use of initial conditions (43):
(45)x-xi=±∫WiWξdξaiξ2+biξ+2Δ,hhhhhhhhhhhhhhhxi≤x≤xi+1,(46)ai=2(σeffln⁡ε-2αi);bi=ui2-2Wi(σeffln⁡ε-2αi)-2ΔWi.
Sign “+” or “−” must be used in (45) according to the value of u=dW/dx.The cavity closing corresponds to the zero value ofW(x), 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 W(x), 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 W (e.g., W=0.0001). 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 functionx=x(W) may be many-valued (see curves 4 and 5 in Figure 8). In this case it is necessary to integrate in (45) from Wi to Wex 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 Wex to W with the opposite sign in (45). The values of Wex can be easily calculated from the quadratic equation following from (44) at u=0:
(47)aiWex2+biWex+2Δ=0.Figure 8
The cavity shapes (27) at zero ventilation rate on a conic-cylindrical hull (x1=5, x2=8, Rb0=Rb1=0.5, Rb2=2.5, ε=0.1, and Fr→∞) for β=0.1 (curves 1–4, the corresponding values of σ0 are −0.2626, −0.046, 0, and 0.1, resp.), for β=0, σ0=0 (curve 5), and for β=-0.1 (curves 6–9, the corresponding values of σ0 are −0.2, −0.046, 0, and 0.2, resp.). The cavity shapes corresponding to the zero cavitation number (curves 3, 5, and 8) are shown by dashed lines.Ifai≠0, the maximum value of W can be calculated from (47) as follows:
(48)Wmax⁡=-bi-bi2-8Δai2ai.
An example of the cavity with the maximum value of W (for ai<0, bi>0) is presented in Figure 8 (curve 4). At ai≥0 the maximum of W exists only for bi<0. In particular, at ai=0 it follows from (22), (24), (46), and (47) that
(49)Wmax⁡=-2Δbi,σ0>ln⁡ε(2αi-ui22Wi).The minimum ofW exists only for ai>0 and bi<0 and can be calculated from (47) as follows:
(50)Wmin⁡=-bi+bi2-8Δai2ai.
An example of the cavity with the minimum value of W is presented in Figure 8 (curve 6).It must be noted that among the cavities presented in Figure8 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 x1→∞, it is possible to realize only cavities 4, 7, 8, and 9. Other cavities cannot exist in reality without conical part of the hull (x1≤x≤x2) 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 σ0 and at any ventilation rate all 3 cavitators (with β=0.1, 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)):
(51)d2Wdx2=σiln⁡ε-2αi+Δ(1Wi2-1W2),σi=σ0+Δln⁡ε(1W02-1Wi2);xi<x<xi+1;i=0,1,2,…,k-1.
If
(52)σiln⁡ε-2αi=0;ui=0,
then (51) has the obvious solution W(x)≡Wi=const at any value of the ventilation rate. The inverse function x=x(W) 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
(53)Rb(x)=0.5σix2ln⁡ε+2βix+Rbi2,xi<x<xi+1;i=0,1,2,…,k-1.
For example, for the part of the hull, corresponding to 0≤x≤x1 and shown in Figure 8, Rb(x)≡Rb0. Therefore (53) yields β=0 and σ0=0. The degenerate solution R(x)≡1 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 σ0 and β and can be refereed as neutral.Equation (20) demonstrates that for W(x)>W0 the ventilation increases the cavity dimensions. For example, at 0≤x≤x1 the dimensions of cavities 1–4 shown in Figure 8 for Δ=0 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., x1→∞). Formulas (46) and (48) allow calculating the critical ventilation rate corresponding to zero value of ai(54)Ve1(cr)=σ0-2αiln⁡εorΔ1(cr)=(2αiln⁡ε-σ0)(aln⁡ε).
If inequality (49) is valid, the cavity is bounded at the ventilation rates (54) but for ai>0 the value of bi2-8Δai in (48) can be zero or negative. The equation bi2-8Δ2(cr)ai=0 yields the second critical value, which with the use of (46) can be represented as follows:
(55)Δ2(cr)=Wi(ui2-2Wi(σ0/ln⁡ε))28ui2.
If Δ≤Δ2(cr), the corresponding cavity is bounded, but unbounded at Δ>Δ2(cr). It means that the ventilation rate is limited by the value Δ2(cr). For unbounded cylindrical hulls the critical ventilation rates were calculated in Manova et al. [15] and coincide with (54) and (55).On the other hand, forW(x)<W0 the ventilation decreases the cavity dimensions. For example, the dimensions of cavities 6–9 shown in Figure 8 for Δ=0 will decrease with increasing the ventilation rate at 0≤x≤x1. In particular, according to the formula (50) the Wmin⁡ decreases from value -bi/ai at Δ=0 to -0.5bi/ai at Δ=bi2/(8ai). The equation bi2-8Δ2(cr)ai=0 yields the critical value, presented by (55). If Δ≤Δ2(cr), 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 numberσ0 is fixed) and ui<0 (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 u0=2β-0=-0.2<0 and the increasing of the gas injection rate decreases the cavity dimensions at 0≤x≤x1. To calculate the cavities at x>x1 the values W1 and u1 were used according to formulas (43). The values u1 are also negative for all cavities shown in Figure 10.Figure 9
The cavity shapes on a conic-cylindrical hull (β=-0.1, x1=5, x2=8, Rb0=Rb1=0.5, Rb2=2.5, ε=0.1, and Fr→∞) for different values of the cavitation number (σ0 = −0.046, 0, and 0.2) and different gas injection rates (dashed lines correspond to Δ=0, solid lines to Δ=0.0005, and solid bold lines to Δ=0.002).Figure 10
The cavity shapes on a conic-cylindrical hull (β=-0.1, x1=5, x2=8, Rb0=Rb1=0.5, Rb2=2.5, ε=0.1, and Fr→∞) for the cavitation number σ0 = −0.2 and different gas injection rates (Δ=0, 0.005, 0.015, 0.0191, 0.025, 0.03, and 0.05 for curves 1–7, resp.). Curve 4 corresponds to the critical value of gas injection rate Δ=Δ2(cr) (see (55)).Curve 4 shown in Figure10 corresponds to the critical value of gas injection rate Δ=Δ2(cr)≈0.0191 calculated from (55). At greater values of Δ (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 β=0 and σ0=0 the hull presented in Figure 8 corresponds to the neutral shape (53) at 0≤x≤x1. It means that at 0≤x≤x1 the corresponding cylindrical cavity shape (shown in Figure 8 by curve 5) is the same at any values of ventilation. Since the value u1 is negative, at x>x1, at this segment the cavity radius diminishes as the gas injection rate increases. The examples of calculations are shown in Figure 11.Figure 11
The cavity shapes on a conic-cylindrical hull (β=0, x1=5, x2=8, Rb0=Rb1=0.5, Rb2=2.5, ε=0.1, and Fr→∞) for the cavitation number σ0 = 0 and different gas injection rates (Δ=0, 0.1, 0.5, and 1.0 for curves 1–4, resp.).Absolutely different behavior occurs in the caseui>0, 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 u0=2β-0=0.2>0. To calculate the cavities at x>x1 the values W1 and u1 were used according to formulas (43). At great enough values of the cavitation number σ0, 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 Δ. At σ0≥-0.046 the cavity shapes are concave (see, e.g., Figure 13). All the cavities cannot be extended for x>x2=8 (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 Δ(cr)≈0.148. At smaller values of σ0 the critical values of the gas injection rate diminish (e.g., at σ0=-0.046  Δ(cr)≈0.087 (see Figure 13) and at σ0=-0.2626  Δ(cr)≈0.00025). 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 x1 is restricted). For example, the cavity shape corresponding to the critical value of the ventilation Δ2(cr)≈0.02591 is shown in Figure 12 by curve 2.Figure 12
The cavity shapes on a conic-cylindrical hull (β=0.1,  x1=5,   x2=8,  Rb0=Rb1=0.5,  Rb2=2.5,   ε=0.1, and  Fr→∞) for the cavitation number σ0 = 0.1 and different gas injection rates Δ=0, 0.02591, 0.1, 0.126, and 0.148 (curves 1–5, resp.).Figure 13
The cavity shapes on a conic-cylindrical hull (β=0.1,   x1=5,  x2=8,  Rb0=Rb1=0.5,  Rb2=2.5,   ε=0.1, and Fr→∞) for the cavitation number σ0=-0.046 and different gas injection rates Δ=0, 0.01, 0.05, 0.064, and 0.087 (curves 1–5, resp.).


### 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 discontinuityLc≈xi as shown in Figures 14 and 15. If dR2/dx|x=xi<2βi or ui<0 (see (43)), the cavity length varies continuously. This case is shown in Figure 14. Absolutely different is the case ui>0 shown in Figure 15. At some critical values of parameters Δ and σ0 the cavity contour touches the hull at the point x=xi, and a “jump” in the cavity length is possible (see Figure 15).Figure 14
The cavities atΔ=0, Fr→∞, and β=ε=0.1 on the conic-conic hull (x1=10, Rb0=0.5, Rb1=1.3, and β1=0.052) for cavitation numbers σ0 = 0.05, 0.0603, and 0.07 (curves 1–3, resp.).Figure 15
The cavities atΔ=0, Fr→∞, and β=ε=0.1 on the conic-conic hull (x1=5, Rb0=0.5, Rb1=1.3, and β1=0.0065) for cavitation numbers σ0 = 0.05, 0.057, and 0.07 (curves 1–3, resp.).These two cases of cavity behaviour are separated by the conditionui=0 at Lc=xi or
(56)dR2dx|x=xi=2βi,(57)R2|x=xi=Rbi2.
At Δ=0 formulas (56) and (57) allow obtaining the relationships between the critical values of parameters xi, βi, and Rbi. With the use of (27), equation (56) can be written as follows:
(58)σcxiln⁡ε+2β=2βi,
where the critical value of the cavitation number σc may be determined from (57) as follows:
(59)σcxi22ln⁡ε+2βxi+1=Rbi2.
Finally, from (58) and (59) the following simple relationship can be obtained:
(60)Rbi2=(β+βi)xi+1.In the particular case of a conic-cylindrical hull (βi=0), (60) coincides with the relationship proposed in Nesteruk [36]. When the value of Rbi is smaller than the critical one (60) and Lc≈xi, 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 xi, βi, and Rbi depend on Δ and should be calculated with the use of (45). For the conic-cylindrical hull some examples can be found in Nesteruk and Shepetyuk [16] and are shown in Figure 7.In the caseui<0 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 Δ at x<xi, the negative value of ui leads to a decrease of the cavity length at x>xi. This fact is illustrated in Figure 16 by the cavities on the conic-cylindrical hull (x1=15, Rb0=1.15, and Rb1=2.2) created by a disc cavitator. The same cavitator and hull were tested in Vlasenko and Savchenko [13].Figure 16
The cavities atFr→∞ created by a disc cavitator located at x=-2 for the conic-cylindrical hull (x1=15, Rb0=1.15, and Rb1=2.2) used in Vlasenko and Savchenko [13]. At x<0 the cavity shape is represented by formula (29); at x>0 the values β=ε=βs≈0.273 are used in (45) for two cavitation numbers σ0 = 0.167 and σ0 = 0.06 and three values of gas injection rates Δ=0 (dashed lines), Δ=0.005 (solid lines), and Δ=0.01 (bold lines).The valueRb1=2.2 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 σ0=0.167 the corresponding values of u1 are negative and ventilation does not increase the cavity length significantly (see Figure 16). On the contrary, at the smaller cavitation number σ0 = 0.06 the values of u1 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 [13] the values of Δ 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 σ0) are the main reason of changes in the cavity length in the experiments (Vlasenko and Savchenko [13]). Unfortunately, the pressure inside the cavity was not measured in Vlasenko and Savchenko [13].If at a zero or a small value ofΔ the cavity covers a hull, its length approaches the value xk and Rb(x)=Rbk at x>xk (in particular, for a bounded hull Rbk=0), then the value uk 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. [10] and Vlasenko and Savchenko [13]).


## 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 Fr→∞ and the hulls which can be presented or interpolated as (33). Substituting R2(x)=W(x)+Rb2(x) into (23) and using (33) yield
(41)d2Wdx2=σeffln⁡ε-2αi-ΔW2,xi<x<xi+1;hhihhhi=0,1,2,…,k-1;αk=0,x>xk.
The initial conditions (26) at x=0 can be rewritten as follows:
(42)W=1-Rb02,dWdx=2β-2β0.
Let us assume the function W(x) to be continuous at points x=xi; i=1,2,…,k. Its derivative dW/dx can be discontinuous at these points if the slope of the hull contour is discontinuous (e.g., at x=x1 for the hull shape shown in Figure 5). It means that, for every segment xi<x<xi+1, i=0,1,2,…,k-1, and the segment x>xk, the specific initial conditions should be used at the beginning of each segment x=xi, i=0,1,2,…,k:
(43)W=R2(xi)-Rbi2=Wi,dWdx=dR2dx|x=xi-2βi=ui.The order of (41) can be reduced by substitutions dW/dx=u and d2W/dx2=du/dx=udu/dW. With the use of initial conditions (43) the following first-order equation can be obtained:
(44)u2=2(σeffln⁡ε-2αi)(W-Wi)+2Δ[1W-1Wi]+ui2,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhxi<x<xi+1.Equation (44) can be integrated with the use of initial conditions (43):
(45)x-xi=±∫WiWξdξaiξ2+biξ+2Δ,hhhhhhhhhhhhhhhxi≤x≤xi+1,(46)ai=2(σeffln⁡ε-2αi);bi=ui2-2Wi(σeffln⁡ε-2αi)-2ΔWi.
Sign “+” or “−” must be used in (45) according to the value of u=dW/dx.The cavity closing corresponds to the zero value ofW(x), 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 W(x), 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 W (e.g., W=0.0001). 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 functionx=x(W) may be many-valued (see curves 4 and 5 in Figure 8). In this case it is necessary to integrate in (45) from Wi to Wex 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 Wex to W with the opposite sign in (45). The values of Wex can be easily calculated from the quadratic equation following from (44) at u=0:
(47)aiWex2+biWex+2Δ=0.Figure 8
The cavity shapes (27) at zero ventilation rate on a conic-cylindrical hull (x1=5, x2=8, Rb0=Rb1=0.5, Rb2=2.5, ε=0.1, and Fr→∞) for β=0.1 (curves 1–4, the corresponding values of σ0 are −0.2626, −0.046, 0, and 0.1, resp.), for β=0, σ0=0 (curve 5), and for β=-0.1 (curves 6–9, the corresponding values of σ0 are −0.2, −0.046, 0, and 0.2, resp.). The cavity shapes corresponding to the zero cavitation number (curves 3, 5, and 8) are shown by dashed lines.Ifai≠0, the maximum value of W can be calculated from (47) as follows:
(48)Wmax⁡=-bi-bi2-8Δai2ai.
An example of the cavity with the maximum value of W (for ai<0, bi>0) is presented in Figure 8 (curve 4). At ai≥0 the maximum of W exists only for bi<0. In particular, at ai=0 it follows from (22), (24), (46), and (47) that
(49)Wmax⁡=-2Δbi,σ0>ln⁡ε(2αi-ui22Wi).The minimum ofW exists only for ai>0 and bi<0 and can be calculated from (47) as follows:
(50)Wmin⁡=-bi+bi2-8Δai2ai.
An example of the cavity with the minimum value of W is presented in Figure 8 (curve 6).It must be noted that among the cavities presented in Figure8 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 x1→∞, it is possible to realize only cavities 4, 7, 8, and 9. Other cavities cannot exist in reality without conical part of the hull (x1≤x≤x2) 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 σ0 and at any ventilation rate all 3 cavitators (with β=0.1, 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)):
(51)d2Wdx2=σiln⁡ε-2αi+Δ(1Wi2-1W2),σi=σ0+Δln⁡ε(1W02-1Wi2);xi<x<xi+1;i=0,1,2,…,k-1.
If
(52)σiln⁡ε-2αi=0;ui=0,
then (51) has the obvious solution W(x)≡Wi=const at any value of the ventilation rate. The inverse function x=x(W) 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
(53)Rb(x)=0.5σix2ln⁡ε+2βix+Rbi2,xi<x<xi+1;i=0,1,2,…,k-1.
For example, for the part of the hull, corresponding to 0≤x≤x1 and shown in Figure 8, Rb(x)≡Rb0. Therefore (53) yields β=0 and σ0=0. The degenerate solution R(x)≡1 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 σ0 and β and can be refereed as neutral.Equation (20) demonstrates that for W(x)>W0 the ventilation increases the cavity dimensions. For example, at 0≤x≤x1 the dimensions of cavities 1–4 shown in Figure 8 for Δ=0 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., x1→∞). Formulas (46) and (48) allow calculating the critical ventilation rate corresponding to zero value of ai(54)Ve1(cr)=σ0-2αiln⁡εorΔ1(cr)=(2αiln⁡ε-σ0)(aln⁡ε).
If inequality (49) is valid, the cavity is bounded at the ventilation rates (54) but for ai>0 the value of bi2-8Δai in (48) can be zero or negative. The equation bi2-8Δ2(cr)ai=0 yields the second critical value, which with the use of (46) can be represented as follows:
(55)Δ2(cr)=Wi(ui2-2Wi(σ0/ln⁡ε))28ui2.
If Δ≤Δ2(cr), the corresponding cavity is bounded, but unbounded at Δ>Δ2(cr). It means that the ventilation rate is limited by the value Δ2(cr). For unbounded cylindrical hulls the critical ventilation rates were calculated in Manova et al. [15] and coincide with (54) and (55).On the other hand, forW(x)<W0 the ventilation decreases the cavity dimensions. For example, the dimensions of cavities 6–9 shown in Figure 8 for Δ=0 will decrease with increasing the ventilation rate at 0≤x≤x1. In particular, according to the formula (50) the Wmin⁡ decreases from value -bi/ai at Δ=0 to -0.5bi/ai at Δ=bi2/(8ai). The equation bi2-8Δ2(cr)ai=0 yields the critical value, presented by (55). If Δ≤Δ2(cr), 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 numberσ0 is fixed) and ui<0 (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 u0=2β-0=-0.2<0 and the increasing of the gas injection rate decreases the cavity dimensions at 0≤x≤x1. To calculate the cavities at x>x1 the values W1 and u1 were used according to formulas (43). The values u1 are also negative for all cavities shown in Figure 10.Figure 9
The cavity shapes on a conic-cylindrical hull (β=-0.1, x1=5, x2=8, Rb0=Rb1=0.5, Rb2=2.5, ε=0.1, and Fr→∞) for different values of the cavitation number (σ0 = −0.046, 0, and 0.2) and different gas injection rates (dashed lines correspond to Δ=0, solid lines to Δ=0.0005, and solid bold lines to Δ=0.002).Figure 10
The cavity shapes on a conic-cylindrical hull (β=-0.1, x1=5, x2=8, Rb0=Rb1=0.5, Rb2=2.5, ε=0.1, and Fr→∞) for the cavitation number σ0 = −0.2 and different gas injection rates (Δ=0, 0.005, 0.015, 0.0191, 0.025, 0.03, and 0.05 for curves 1–7, resp.). Curve 4 corresponds to the critical value of gas injection rate Δ=Δ2(cr) (see (55)).Curve 4 shown in Figure10 corresponds to the critical value of gas injection rate Δ=Δ2(cr)≈0.0191 calculated from (55). At greater values of Δ (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 β=0 and σ0=0 the hull presented in Figure 8 corresponds to the neutral shape (53) at 0≤x≤x1. It means that at 0≤x≤x1 the corresponding cylindrical cavity shape (shown in Figure 8 by curve 5) is the same at any values of ventilation. Since the value u1 is negative, at x>x1, at this segment the cavity radius diminishes as the gas injection rate increases. The examples of calculations are shown in Figure 11.Figure 11
The cavity shapes on a conic-cylindrical hull (β=0, x1=5, x2=8, Rb0=Rb1=0.5, Rb2=2.5, ε=0.1, and Fr→∞) for the cavitation number σ0 = 0 and different gas injection rates (Δ=0, 0.1, 0.5, and 1.0 for curves 1–4, resp.).Absolutely different behavior occurs in the caseui>0, 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 u0=2β-0=0.2>0. To calculate the cavities at x>x1 the values W1 and u1 were used according to formulas (43). At great enough values of the cavitation number σ0, 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 Δ. At σ0≥-0.046 the cavity shapes are concave (see, e.g., Figure 13). All the cavities cannot be extended for x>x2=8 (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 Δ(cr)≈0.148. At smaller values of σ0 the critical values of the gas injection rate diminish (e.g., at σ0=-0.046  Δ(cr)≈0.087 (see Figure 13) and at σ0=-0.2626  Δ(cr)≈0.00025). 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 x1 is restricted). For example, the cavity shape corresponding to the critical value of the ventilation Δ2(cr)≈0.02591 is shown in Figure 12 by curve 2.Figure 12
The cavity shapes on a conic-cylindrical hull (β=0.1,  x1=5,   x2=8,  Rb0=Rb1=0.5,  Rb2=2.5,   ε=0.1, and  Fr→∞) for the cavitation number σ0 = 0.1 and different gas injection rates Δ=0, 0.02591, 0.1, 0.126, and 0.148 (curves 1–5, resp.).Figure 13
The cavity shapes on a conic-cylindrical hull (β=0.1,   x1=5,  x2=8,  Rb0=Rb1=0.5,  Rb2=2.5,   ε=0.1, and Fr→∞) for the cavitation number σ0=-0.046 and different gas injection rates Δ=0, 0.01, 0.05, 0.064, and 0.087 (curves 1–5, resp.).


## 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 discontinuityLc≈xi as shown in Figures 14 and 15. If dR2/dx|x=xi<2βi or ui<0 (see (43)), the cavity length varies continuously. This case is shown in Figure 14. Absolutely different is the case ui>0 shown in Figure 15. At some critical values of parameters Δ and σ0 the cavity contour touches the hull at the point x=xi, and a “jump” in the cavity length is possible (see Figure 15).Figure 14
The cavities atΔ=0, Fr→∞, and β=ε=0.1 on the conic-conic hull (x1=10, Rb0=0.5, Rb1=1.3, and β1=0.052) for cavitation numbers σ0 = 0.05, 0.0603, and 0.07 (curves 1–3, resp.).Figure 15
The cavities atΔ=0, Fr→∞, and β=ε=0.1 on the conic-conic hull (x1=5, Rb0=0.5, Rb1=1.3, and β1=0.0065) for cavitation numbers σ0 = 0.05, 0.057, and 0.07 (curves 1–3, resp.).These two cases of cavity behaviour are separated by the conditionui=0 at Lc=xi or
(56)dR2dx|x=xi=2βi,(57)R2|x=xi=Rbi2.
At Δ=0 formulas (56) and (57) allow obtaining the relationships between the critical values of parameters xi, βi, and Rbi. With the use of (27), equation (56) can be written as follows:
(58)σcxiln⁡ε+2β=2βi,
where the critical value of the cavitation number σc may be determined from (57) as follows:
(59)σcxi22ln⁡ε+2βxi+1=Rbi2.
Finally, from (58) and (59) the following simple relationship can be obtained:
(60)Rbi2=(β+βi)xi+1.In the particular case of a conic-cylindrical hull (βi=0), (60) coincides with the relationship proposed in Nesteruk [36]. When the value of Rbi is smaller than the critical one (60) and Lc≈xi, 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 xi, βi, and Rbi depend on Δ and should be calculated with the use of (45). For the conic-cylindrical hull some examples can be found in Nesteruk and Shepetyuk [16] and are shown in Figure 7.In the caseui<0 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 Δ at x<xi, the negative value of ui leads to a decrease of the cavity length at x>xi. This fact is illustrated in Figure 16 by the cavities on the conic-cylindrical hull (x1=15, Rb0=1.15, and Rb1=2.2) created by a disc cavitator. The same cavitator and hull were tested in Vlasenko and Savchenko [13].Figure 16
The cavities atFr→∞ created by a disc cavitator located at x=-2 for the conic-cylindrical hull (x1=15, Rb0=1.15, and Rb1=2.2) used in Vlasenko and Savchenko [13]. At x<0 the cavity shape is represented by formula (29); at x>0 the values β=ε=βs≈0.273 are used in (45) for two cavitation numbers σ0 = 0.167 and σ0 = 0.06 and three values of gas injection rates Δ=0 (dashed lines), Δ=0.005 (solid lines), and Δ=0.01 (bold lines).The valueRb1=2.2 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 σ0=0.167 the corresponding values of u1 are negative and ventilation does not increase the cavity length significantly (see Figure 16). On the contrary, at the smaller cavitation number σ0 = 0.06 the values of u1 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 [13] the values of Δ 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 σ0) are the main reason of changes in the cavity length in the experiments (Vlasenko and Savchenko [13]). Unfortunately, the pressure inside the cavity was not measured in Vlasenko and Savchenko [13].If at a zero or a small value ofΔ the cavity covers a hull, its length approaches the value xk and Rb(x)=Rbk at x>xk (in particular, for a bounded hull Rbk=0), then the value uk 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. [10] and Vlasenko and Savchenko [13]).


## 9. Asymptotic Solution at Small Values of the Ventilation Rate

### 9.1. Basic Equations

If the gas injection rate is small enoughQ(t)≪1, the solution of the nonlinear equation (17) can be expressed as follows:
(61)R2(x,t)≈R(0)2(x,t)+R(1)2(x,t),R(0)2(x,t)≫R(1)2(x,t),
where R(0)(x,t) is the solution of (17) at zero ventilation rate Q(t)=0,
(62)ε2ln⁡εΖ(A0)+ε2[Ζ(A0)ln⁡F0+Z(B0)+0.5A02F0-2]=0.5σ0(t)±xFr-2(t),(63)A0(x,t)=F0(x,t)(∂F0∂x+τ∂F0∂t);F02(x,t)=R(0)2(x,t)ε2;B0(x,t)=-A0(x,t)ln⁡2hhhhhhhh-12∫∂A0(ξ,t)∂ξsign⁡(x-ξ)ln⁡|x-ξ|dξ.
Function R(1)(x,t) is a solution of the following equations:
(64)ε2ln⁡εΖ(A1)=ρgτ(t)πρ∂∂t[Q(t)∫0xdxW0(x,t)]+ρgQ(t)S(t)πρ∫0xdxW0(x,t)-ρgQ2(t)2ρπ2[W0-2(0,t)-W0-2(x,t)],(65)W0(x,t)=R(0)2(x,t)-Rb2(x,t),(66)A1(x,t)=F1(x,t)(∂F1∂x+τ∂F1∂t);F12(x,t)=R(1)2(x,t)ε2.
To solve the nonlinear integral-differential equation (62) the asymptotic series obtained in Nesteruk [30] and the standard initial and boundary conditions (19) can be used for the function R(0)(x,t). If the accuracy restrictions are not very severe, the first approximation equation (similar to (9) or (10) with main terms of order ε2ln⁡ε only) can be easily solved. Knowing the function R(0)(x,t) the right hand part of (64) and (65) can be calculated and the function R(1)(x,t) 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 termR(0)(x,t) 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 p0. According to the first approximation of (62) the cavity radius R0(x)≈R(0)(x) is given by formula (27). Let us consider small changes in gas pressure Δp=p-p0, where p is a new constant gas pressure inside the cavity. According to (27) and (1) the new cavity radius can be written as follows:
(67)R2(x)=σ0x22ln⁡ε-Δpx2ρU∞2ln⁡ε±x33Fr2ln⁡ε+2βx+1.
Here we neglect the changes in the vapor pressure.Integration of formulas (27) and (67) allows calculating the changes in the cavity volume ΔG1:
(68)ΔG1≈-ΔpπL033ρU∞2ln⁡ε,
where L0 is the initial cavity length, which is measured along the axis x and can be found from (27) as follows:
(69)Rb2(L0)=σ0L022ln⁡ε±L033Fr2ln⁡ε+2βL0+1.
Here we suppose the body radius Rb(x) to be time independent; therefore ΔG1 yields also the changes in the volume of gas.On the other hand for a fixed mass of gas its pressure and volumeG are related according to the polytropic relationship:
(70)pGγ=p0G0γ,
where γ=const (in particular, for isothermal process γ=1); the initial volume of gas can be calculated with the use of (27) as follows:
(71)G0=π[σ0L036ln⁡ε±L0412Fr2ln⁡ε+βL02+L0-∫0L0Rb2(x)dx].
Putting in (70) p=p0+Δp and G=G0+ΔG2 and linearization yield
(72)ΔG2≈-ΔpG0γp0.
Values ΔG1 and ΔG2 have opposite signs (see (68) and (72)) and can compensate each other, if |ΔG2|>|ΔG1|. Otherwise the cavity becomes unstable. The critical value of the gas pressure can be obtained from |ΔG2|=|ΔG1|, (68), and (72):
(73)p0(cr)=-3G0ρU∞2ln⁡εγπL03.
At smaller values of the gas pressure the cavity is stable. If p0>p0(cr), the cavity becomes unstable. The existence of the critical gas pressure value was revealed in experiments (e.g., Michele [20]) 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):
(74)σv=σ0(cr)-6G0ln⁡εγπL03,
where the vapor cavitation number
(75)σv=2(p∞-pv)ρU∞2
and the critical value of the cavitation number can be obtained from (69) as follows:
(76)σ0(cr)=2ln⁡εL02[Rb2(L0)∓L033Fr2ln⁡ε-2βL0-1].
It follows from (71), (73), and (76) that
(77)σv=2(γ-1)ln⁡εγL02[Rb2(L0)∓L033Fr2ln⁡ε-2βL0-1]-6ln⁡εγL03[βL02+L0±L0412Fr2ln⁡ε-∫0L0Rb2(x)dx].
Formula (77) allows calculating the critical value of σv for every value of the cavity length. The corresponding critical cavitation number can be calculated with the use of (76). Thus the dependence σ0(cr)(σv) can be obtained. In particular, for an empty isothermal cavity at large Froude numbers (Rb≡0, γ=1, and Fr→∞) it follows from (77) and (76) that
(78)σv=-6(βL0+1)ln⁡εL02,σ0(cr)=-2(2βL0+1)ln⁡εL02.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 σ0(cr) (σv) depends on β in comparison with the relationship
(79)σ0(cr)σv≈0.37
obtained by Paryshev [23] for the isothermal cavities created by nonslender cavitators with the use of semiempirical formulas for the cavity shape. In the cases of long cavities (L0→∞) and for the base cavities with β=0 the relationships (78) yield the formulas
(80)σ0(cr)σv≈23;σ0(cr)σv≈13,
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 σ0(cr)/σv≈0.2 was measured by Michel [20]. It can be seen from Figure 17 that the largest stability regions correspond to the base cavities (β≤0). In particular, stable cavities with negative cavitation number are possible at β<0. It follows from (78) that the cavitation number is limited by value 2β2ln⁡ε, obtained in Nesteruk [32].Figure 17
The dependencesσ0(cr) (σv) for the empty isothermal cavity (Rb≡0 and γ=1) at Fr→∞. Curves 1–3 correspond to the cavities, created by conical cavitators with β=0.05, 0.1, and 0.273, respectively; ε=β. Paryshev’s theoretical linear dependence (79) is represented by dashed line 4. Base cavities for β=0, −0.1 are represented by curves 5 and 6, respectively (ε=0.1). The restriction σ0(cr)=σv is shown by the bold line 7.The influence of the gravity forces is shown in Figure18. For the conical cavitators the presence of gravity diminishes the stability regions, if its direction coincides with the direction of the ambient flow (see curves 1 and 2). In particular, there are no stable cavities at some small values of σv. Opposite behaviour is typical for the base cavities (see curves 9 and 10). If the directions of the ambient flow and the gravity are opposite the minimal cavitation number exists (this fact was already discussed in Section 6). The critical cavity shapes are similar to the solid line shown in Figure 4. For such lines dL/dσ0→∞; therefore the minimum values of the cavitation number can be calculated from (76) with the use of the condition dσ0/dL0=0. These minimum possible values of the cavitation number correspond to the ends of the curves 4–7 (see Figure 18) and yield additional stability restrictions.Figure 18
The dependencesσ0(cr) (σv) for the empty isothermal cavity (Rb≡0, γ=1, and ε=0.1) at different values of the Froude number. Curves 1–5 correspond to the cavities, created by the conical cavitator (β=0.1), and curves 6–10 correspond to base cavities (β=-0.1). The direction of the flow coincides with the direction of the gravity for the curves 1, 2, 9, and 10 and is opposite for the curves 4–7. The cases Fr→∞ are shown by lines 3 and 8; Fr=5 corresponds to lines 1, 5, 6, and 10; Fr=10 corresponds to lines 2, 4, 7, and 9. The restriction σ0(cr)=σv is shown by the bold line 11.The cavities with the hulls located inside have smaller stability regions in comparison with the empty cavities (due to the smaller value ofG0 (see (73)). The same conclusion has been drawn out by Paryshev [23]. To illustrate this fact, let us take the neutral hull shape:
(81)Rb2(x)=σ0x22ln⁡ε±x33Fr2ln⁡ε+2βx+Rb02.
For such hull formula (27) is a solution of (20) at any value of the gas injection rate; that is, (27) is not only the approximation of the steady cavity shape at a small ventilation but also the exact solution at any gas injection rate. If Fr→∞, (81) coincides with the relationship (53). The length of the neutral hull Lb can be obtained from (81) as a solution of Rb2(Lb)=0. Let us assume that Rb2≡0 at x>Lb. Then Rb2(L0)=0 and for the isothermal gas processes inside the cavity, (77) can be rewritten as follows:
(82)σv=-6ln⁡εL03[β(L02-13Lb2)+L0-23LbRb02±L04+Lb4/312Fr2ln⁡ε].Let us consider at first the case of very small area of the circular channel between the cavity and the hull surfaceRb0→1. Then Lb→L0 and formulas (82) and (76) yield
(83)σv≈-2ln⁡εL02[2βL0+1±L033Fr2ln⁡ε]=σ0(cr).
Therefore, cavities with the very small area of the circular channel between the cavity and the hull can be stable at the cavitation numbers which are very close to maximum possible value, σv. Thus such cavities are mostly unstable.Let us consider the case of the maximal area of the circular channel, that is,Rb0→0. For the base cavities (β≤0 and Rb0→0) the hull volume tends to zero. The stability of the empty base cavities has been already investigated before and illustrated in Figures 17 and 18. It can be seen that the stability regions are rather large. In the case β>0 and Rb0→0, the corresponding cavities are not empty. Let us investigate their stability at Fr→∞. It follows from (76) and (81) that
(84)Lb=-4βln⁡εσ0=2βL022βL0+1.
Substituting (84) into (82) yields that at fixed value of σ0(cr) the corresponding value of σv is smaller by the value Δσv=8β2L0ln⁡ε/(2βL0+1)2; therefore the corresponding stability regions are smaller in comparison with the case of empty cavities (shown in Figure 17 by curves 1–3). The examples of calculations based on (78), (81), and (82) are shown in Figure 19. It can be seen that grater values of the volume of the hull part located inside the cavity (larger values of Lb/L0) decrease the stability region. The most stable are the empty cavities presented in Figure 19 by dashed lines 7 and 8.Figure 19
The dependencesσ0(cr) (σv) for the isothermal cavity (γ=1) on the neutral shaped hull (81) at Fr→∞ and ε=0.1. Curves 1–4 correspond to β=0.1 and curves 5–8 correspond to β=-0.1. Solid lines 1–3 and 5–7 correspond to the values Lb/L0=0.5, 0.7, and 0.9, respectively. For comparison the cases of the empty cavity are shown by dashed lines 4 and 8. The restriction σ0(cr)=σv is shown by the bold line 9.It must be noted that the experiments at standard room temperature and at values of the pressure close to the atmospheric one yield rather grate values ofσv (e.g., σv>2 at U∞<10 m/s). At these values of σv the corresponding critical cavitation number is rather large and cavity length is rather small. Thus, it is difficult to obtain long stable empty cavities in the low speed experiments. Vlasenko and Savchenko [13] revealed stable cavities closing on the conical part of the hulls. Their length is restricted by the value L0≈9, obtained at some gas injection rate for the hull shown in Figure 16. Further increasing of the ventilation leads to the drastic increase of the cavity length. Since the values of gas injection rate are too small to change the cavity shape significantly, such behaviour can be treated as a loose of stability.Let us try to estimateσv corresponding to the experimental critical value L0=9 with the use of (77) and compare with the experimental value σv≈2.52 (Vlasenko and Savchenko [13]). Since the experiments were performed for the disc cavitator, we need to use formula (29) for the initial part of the cavity at x<2 and take into account the corresponding volume of the gas ΔG0=π[0.2(75/3-1)-2]. We assume that Rb=1 at x<2, ε=β=0.273, γ=1, and Fr→∞. The calculations based on the modified formula (77) yield σv≈2.54 which is very close to the experimental value.The obtained results can also be applied for the separated bubbles. In incompressible liquidγ→∞ and according to (72) such bubble is always unstable. In the case of a gas, the same gas is present in the external flow and inside the separated zone (there is no more a huge difference in densities as for a gas cavity located in the water flow). Nevertheless, the small velocities inside the separated zone allow supposing the pressure to be constant and using the developed theory. In this case the vapor cavitation number coincides with the Euler number as follows:
(85)σv=Eu=2p∞ρgU∞2.
According to the presented analysis, is difficult to obtain long enough stable separated bubbles for Eu>2, but at smaller values of the Euler number the separated zones can be stable. Therefore, the Euler number must be taken into account in simulations of the gas flow separation (as mentioned in Nesteruk [25]).


### 9.3. Stability of the Pulsating Ventilated Cavities

In the previous section it was shown that steady stable cavities at low speeds (U∞<10 m/s) have to be very short. In order to cover long hulls it is possible to use pulsating cavities which were investigated experimentally in Silberman and Song [18], Song [19], Michel [20], and Zou et al. [21] and by many other authors. The theories of such empty cavities were proposed by Woods [22], Paryshev [23, 24], Nesteruk [25], and Semenenko [26]. Here we consider the small oscillations of the pressure inside the cavity which covers the hull.Let us suppose that the gas injection rates are small enough to use only the main termR(0)(x,t) in (61) and to consider the pressure inside the cavity to be only time dependent. The corresponding changes of the cavitation number can be represented as follows:
(86)σ0(t)=σ-0+b1sin(2πt),
where σ-0 is the average value of the cavitation number and the dimensionless time is based on the period of oscillations T.Let us suppose the velocity of the ambient flow and the hull shape to be time independent and use the radiusR0 of the cavity and the body at the cross section of cavity origin x=0 (see Figure 1) for dimensionless lengths and to calculate the parameters (4) and (7). Then
(87)τ=R0TU∞;S(t)=0;Fr=U∞gR0.
Let us take only the main term of order ε2ln⁡ε in (62) and use the analytical solution obtained in Nesteruk [28]. In particular, for the harmonic low (86) the following formula was obtained (Nesteruk [33]):
(88)R2(x,t)=σ-0x22ln⁡ε±x33Fr2ln⁡ε+2βx+1+b2F2(x,t),(89)F2(x,t)=2πxτcos⁡[2π(t-xτ)]+sin[2π(t-xτ)]-sin⁡(2πt);b2=b14π2τ2ln⁡ε.
Formula (88) shows that for the pulsating cavity R2(x,t) differs from R2(x) corresponding to the steady cavity at the mean value of the cavitation number σ-0 by function b2F2(x,t). Formula (89) corresponds to a propagating wave with the amplitude increasing downstream. The length of the wave—the minimal distance between maxima or minima of the function (89)—is equal to 1/τ. The same behaviour was revealed in experiments (Song [19]).Integration of (88) allows calculating the cavity volume:
(90)G(t)=π∫oL(t)R2(x,t)dx=πL[σ-0L26ln⁡ε±L312Fr2ln⁡ε+βL+1]+b3[F3(L,t)-F3(0,t)],(91)F3(s,t)=cos⁡[2π(t-sτ)](πτ)-s{sin[2π(t-sτ)]+sin⁡(2πt)};b3=b14πτ2ln⁡ε,
where the cavity length L(t) may be obtained from
(92)Rb2(L)=σ-0L22ln⁡ε±L33Fr2ln⁡ε+2βL+1+b2F2(L,t).
At small values of b1 (or b2) the difference between L(t) and the length L- of the steady cavity at the cavitation number σ-0 must be small, since L- is a solution of (92) at b1=b2=0. Here we do not take into account the cases of discontinuous changes in the cavity length discussed in Section 8.5 and shown in Figure 15.By puttingΔL=L-L- into (90) the pulsations of the cavity volume can be calculated:
(93)ΔG(t)=G(t)-G-≈b3[F3(L-,t)-F3(0,t)],
where G- is the average volume of the steady cavity at the cavitation number σ-0 and can be obtained from (90) at b1=b3=0. With the use of (93) the average value of the cavity volume pulsations can be calculated:
(94)ΔG-=∫o1ΔG(t)dt=b4F4(τL-),b4=b18π3τ3ln⁡ε,(95)F4(y)=sin[2π(1-y)]-sin⁡(2πy)+πy{cos⁡[2π(1-y)]-cos⁡⁡(2πy)}.
Equations (94) and (95) show that the average value of the cavity volume pulsations is proportional to the amplitude of the gas pressure pulsations b1 and can be diminished, when F4(y)≈0. It can be expected that the pulsating cavities with zero average value of the volume pulsations are stable; that is, they can exist during a long time. According to the formula (95) the zero values of F4 are possible at
(96)τL-=L-′T′U∞′=Sh=n;n=1,2,3,….
Equation (96) shows that stable oscillating cavities occur only at integer values of the Strouhal number Sh or at integer number of waves located on the average cavity length. This fact was revealed in experiments on empty 2D cavities (Song [19]) and supported by the theories developed by Woods [22] and Semenenko [26] for 2D cavities and by Paryshev [23, 24] and Nesteruk [25] for the axisymmetric empty cavities. Presented analysis illustrates that the same behaviour is typical also in the case of rigid hulls located inside the cavity. If the average cavity length is fixed, the corresponding frequencies of pulsations in the cavity and in its wake are limited by (96).Formula (96) can also be applied for the separation bubbles both in gases and in liquids. In particular, let us compare (96) with the experimental Strouhal numbers for Karman vortex street, measured for circular cylinders of diameter d′ at large Reynolds numbers (see, e.g., Theodore von Kármán [38]):
(97)ShK=d′T′U∞′≈0.2.
Since the cylinder diameter d′ is few times smaller than the average length of its separated zone, formulas (96) and (97) are in good agreement.In order to investigate the influence of the ventilation rate on the pulsating cavities, (64) must be solved. But some conclusions can be drawn out with the use of simple physical considerations. At the local minima of the circular channel area the pressure in gas flow and cavity radius decrease and in maxima the pressure in gas flow and cavity radius increase. It leads to the increasing of the pulsation amplitude with the increasing of the ventilation rate. This fact was revealed by experiments (see, e.g., Silberman and Song [18]).


## 9.1. Basic Equations

If the gas injection rate is small enoughQ(t)≪1, the solution of the nonlinear equation (17) can be expressed as follows:
(61)R2(x,t)≈R(0)2(x,t)+R(1)2(x,t),R(0)2(x,t)≫R(1)2(x,t),
where R(0)(x,t) is the solution of (17) at zero ventilation rate Q(t)=0,
(62)ε2ln⁡εΖ(A0)+ε2[Ζ(A0)ln⁡F0+Z(B0)+0.5A02F0-2]=0.5σ0(t)±xFr-2(t),(63)A0(x,t)=F0(x,t)(∂F0∂x+τ∂F0∂t);F02(x,t)=R(0)2(x,t)ε2;B0(x,t)=-A0(x,t)ln⁡2hhhhhhhh-12∫∂A0(ξ,t)∂ξsign⁡(x-ξ)ln⁡|x-ξ|dξ.
Function R(1)(x,t) is a solution of the following equations:
(64)ε2ln⁡εΖ(A1)=ρgτ(t)πρ∂∂t[Q(t)∫0xdxW0(x,t)]+ρgQ(t)S(t)πρ∫0xdxW0(x,t)-ρgQ2(t)2ρπ2[W0-2(0,t)-W0-2(x,t)],(65)W0(x,t)=R(0)2(x,t)-Rb2(x,t),(66)A1(x,t)=F1(x,t)(∂F1∂x+τ∂F1∂t);F12(x,t)=R(1)2(x,t)ε2.
To solve the nonlinear integral-differential equation (62) the asymptotic series obtained in Nesteruk [30] and the standard initial and boundary conditions (19) can be used for the function R(0)(x,t). If the accuracy restrictions are not very severe, the first approximation equation (similar to (9) or (10) with main terms of order ε2ln⁡ε only) can be easily solved. Knowing the function R(0)(x,t) the right hand part of (64) and (65) can be calculated and the function R(1)(x,t) 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 termR(0)(x,t) 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 p0. According to the first approximation of (62) the cavity radius R0(x)≈R(0)(x) is given by formula (27). Let us consider small changes in gas pressure Δp=p-p0, where p is a new constant gas pressure inside the cavity. According to (27) and (1) the new cavity radius can be written as follows:
(67)R2(x)=σ0x22ln⁡ε-Δpx2ρU∞2ln⁡ε±x33Fr2ln⁡ε+2βx+1.
Here we neglect the changes in the vapor pressure.Integration of formulas (27) and (67) allows calculating the changes in the cavity volume ΔG1:
(68)ΔG1≈-ΔpπL033ρU∞2ln⁡ε,
where L0 is the initial cavity length, which is measured along the axis x and can be found from (27) as follows:
(69)Rb2(L0)=σ0L022ln⁡ε±L033Fr2ln⁡ε+2βL0+1.
Here we suppose the body radius Rb(x) to be time independent; therefore ΔG1 yields also the changes in the volume of gas.On the other hand for a fixed mass of gas its pressure and volumeG are related according to the polytropic relationship:
(70)pGγ=p0G0γ,
where γ=const (in particular, for isothermal process γ=1); the initial volume of gas can be calculated with the use of (27) as follows:
(71)G0=π[σ0L036ln⁡ε±L0412Fr2ln⁡ε+βL02+L0-∫0L0Rb2(x)dx].
Putting in (70) p=p0+Δp and G=G0+ΔG2 and linearization yield
(72)ΔG2≈-ΔpG0γp0.
Values ΔG1 and ΔG2 have opposite signs (see (68) and (72)) and can compensate each other, if |ΔG2|>|ΔG1|. Otherwise the cavity becomes unstable. The critical value of the gas pressure can be obtained from |ΔG2|=|ΔG1|, (68), and (72):
(73)p0(cr)=-3G0ρU∞2ln⁡εγπL03.
At smaller values of the gas pressure the cavity is stable. If p0>p0(cr), the cavity becomes unstable. The existence of the critical gas pressure value was revealed in experiments (e.g., Michele [20]) 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):
(74)σv=σ0(cr)-6G0ln⁡εγπL03,
where the vapor cavitation number
(75)σv=2(p∞-pv)ρU∞2
and the critical value of the cavitation number can be obtained from (69) as follows:
(76)σ0(cr)=2ln⁡εL02[Rb2(L0)∓L033Fr2ln⁡ε-2βL0-1].
It follows from (71), (73), and (76) that
(77)σv=2(γ-1)ln⁡εγL02[Rb2(L0)∓L033Fr2ln⁡ε-2βL0-1]-6ln⁡εγL03[βL02+L0±L0412Fr2ln⁡ε-∫0L0Rb2(x)dx].
Formula (77) allows calculating the critical value of σv for every value of the cavity length. The corresponding critical cavitation number can be calculated with the use of (76). Thus the dependence σ0(cr)(σv) can be obtained. In particular, for an empty isothermal cavity at large Froude numbers (Rb≡0, γ=1, and Fr→∞) it follows from (77) and (76) that
(78)σv=-6(βL0+1)ln⁡εL02,σ0(cr)=-2(2βL0+1)ln⁡εL02.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 σ0(cr) (σv) depends on β in comparison with the relationship
(79)σ0(cr)σv≈0.37
obtained by Paryshev [23] for the isothermal cavities created by nonslender cavitators with the use of semiempirical formulas for the cavity shape. In the cases of long cavities (L0→∞) and for the base cavities with β=0 the relationships (78) yield the formulas
(80)σ0(cr)σv≈23;σ0(cr)σv≈13,
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 σ0(cr)/σv≈0.2 was measured by Michel [20]. It can be seen from Figure 17 that the largest stability regions correspond to the base cavities (β≤0). In particular, stable cavities with negative cavitation number are possible at β<0. It follows from (78) that the cavitation number is limited by value 2β2ln⁡ε, obtained in Nesteruk [32].Figure 17
The dependencesσ0(cr) (σv) for the empty isothermal cavity (Rb≡0 and γ=1) at Fr→∞. Curves 1–3 correspond to the cavities, created by conical cavitators with β=0.05, 0.1, and 0.273, respectively; ε=β. Paryshev’s theoretical linear dependence (79) is represented by dashed line 4. Base cavities for β=0, −0.1 are represented by curves 5 and 6, respectively (ε=0.1). The restriction σ0(cr)=σv is shown by the bold line 7.The influence of the gravity forces is shown in Figure18. For the conical cavitators the presence of gravity diminishes the stability regions, if its direction coincides with the direction of the ambient flow (see curves 1 and 2). In particular, there are no stable cavities at some small values of σv. Opposite behaviour is typical for the base cavities (see curves 9 and 10). If the directions of the ambient flow and the gravity are opposite the minimal cavitation number exists (this fact was already discussed in Section 6). The critical cavity shapes are similar to the solid line shown in Figure 4. For such lines dL/dσ0→∞; therefore the minimum values of the cavitation number can be calculated from (76) with the use of the condition dσ0/dL0=0. These minimum possible values of the cavitation number correspond to the ends of the curves 4–7 (see Figure 18) and yield additional stability restrictions.Figure 18
The dependencesσ0(cr) (σv) for the empty isothermal cavity (Rb≡0, γ=1, and ε=0.1) at different values of the Froude number. Curves 1–5 correspond to the cavities, created by the conical cavitator (β=0.1), and curves 6–10 correspond to base cavities (β=-0.1). The direction of the flow coincides with the direction of the gravity for the curves 1, 2, 9, and 10 and is opposite for the curves 4–7. The cases Fr→∞ are shown by lines 3 and 8; Fr=5 corresponds to lines 1, 5, 6, and 10; Fr=10 corresponds to lines 2, 4, 7, and 9. The restriction σ0(cr)=σv is shown by the bold line 11.The cavities with the hulls located inside have smaller stability regions in comparison with the empty cavities (due to the smaller value ofG0 (see (73)). The same conclusion has been drawn out by Paryshev [23]. To illustrate this fact, let us take the neutral hull shape:
(81)Rb2(x)=σ0x22ln⁡ε±x33Fr2ln⁡ε+2βx+Rb02.
For such hull formula (27) is a solution of (20) at any value of the gas injection rate; that is, (27) is not only the approximation of the steady cavity shape at a small ventilation but also the exact solution at any gas injection rate. If Fr→∞, (81) coincides with the relationship (53). The length of the neutral hull Lb can be obtained from (81) as a solution of Rb2(Lb)=0. Let us assume that Rb2≡0 at x>Lb. Then Rb2(L0)=0 and for the isothermal gas processes inside the cavity, (77) can be rewritten as follows:
(82)σv=-6ln⁡εL03[β(L02-13Lb2)+L0-23LbRb02±L04+Lb4/312Fr2ln⁡ε].Let us consider at first the case of very small area of the circular channel between the cavity and the hull surfaceRb0→1. Then Lb→L0 and formulas (82) and (76) yield
(83)σv≈-2ln⁡εL02[2βL0+1±L033Fr2ln⁡ε]=σ0(cr).
Therefore, cavities with the very small area of the circular channel between the cavity and the hull can be stable at the cavitation numbers which are very close to maximum possible value, σv. Thus such cavities are mostly unstable.Let us consider the case of the maximal area of the circular channel, that is,Rb0→0. For the base cavities (β≤0 and Rb0→0) the hull volume tends to zero. The stability of the empty base cavities has been already investigated before and illustrated in Figures 17 and 18. It can be seen that the stability regions are rather large. In the case β>0 and Rb0→0, the corresponding cavities are not empty. Let us investigate their stability at Fr→∞. It follows from (76) and (81) that
(84)Lb=-4βln⁡εσ0=2βL022βL0+1.
Substituting (84) into (82) yields that at fixed value of σ0(cr) the corresponding value of σv is smaller by the value Δσv=8β2L0ln⁡ε/(2βL0+1)2; therefore the corresponding stability regions are smaller in comparison with the case of empty cavities (shown in Figure 17 by curves 1–3). The examples of calculations based on (78), (81), and (82) are shown in Figure 19. It can be seen that grater values of the volume of the hull part located inside the cavity (larger values of Lb/L0) decrease the stability region. The most stable are the empty cavities presented in Figure 19 by dashed lines 7 and 8.Figure 19
The dependencesσ0(cr) (σv) for the isothermal cavity (γ=1) on the neutral shaped hull (81) at Fr→∞ and ε=0.1. Curves 1–4 correspond to β=0.1 and curves 5–8 correspond to β=-0.1. Solid lines 1–3 and 5–7 correspond to the values Lb/L0=0.5, 0.7, and 0.9, respectively. For comparison the cases of the empty cavity are shown by dashed lines 4 and 8. The restriction σ0(cr)=σv is shown by the bold line 9.It must be noted that the experiments at standard room temperature and at values of the pressure close to the atmospheric one yield rather grate values ofσv (e.g., σv>2 at U∞<10 m/s). At these values of σv the corresponding critical cavitation number is rather large and cavity length is rather small. Thus, it is difficult to obtain long stable empty cavities in the low speed experiments. Vlasenko and Savchenko [13] revealed stable cavities closing on the conical part of the hulls. Their length is restricted by the value L0≈9, obtained at some gas injection rate for the hull shown in Figure 16. Further increasing of the ventilation leads to the drastic increase of the cavity length. Since the values of gas injection rate are too small to change the cavity shape significantly, such behaviour can be treated as a loose of stability.Let us try to estimateσv corresponding to the experimental critical value L0=9 with the use of (77) and compare with the experimental value σv≈2.52 (Vlasenko and Savchenko [13]). Since the experiments were performed for the disc cavitator, we need to use formula (29) for the initial part of the cavity at x<2 and take into account the corresponding volume of the gas ΔG0=π[0.2(75/3-1)-2]. We assume that Rb=1 at x<2, ε=β=0.273, γ=1, and Fr→∞. The calculations based on the modified formula (77) yield σv≈2.54 which is very close to the experimental value.The obtained results can also be applied for the separated bubbles. In incompressible liquidγ→∞ and according to (72) such bubble is always unstable. In the case of a gas, the same gas is present in the external flow and inside the separated zone (there is no more a huge difference in densities as for a gas cavity located in the water flow). Nevertheless, the small velocities inside the separated zone allow supposing the pressure to be constant and using the developed theory. In this case the vapor cavitation number coincides with the Euler number as follows:
(85)σv=Eu=2p∞ρgU∞2.
According to the presented analysis, is difficult to obtain long enough stable separated bubbles for Eu>2, but at smaller values of the Euler number the separated zones can be stable. Therefore, the Euler number must be taken into account in simulations of the gas flow separation (as mentioned in Nesteruk [25]).


## 9.3. Stability of the Pulsating Ventilated Cavities

In the previous section it was shown that steady stable cavities at low speeds (U∞<10 m/s) have to be very short. In order to cover long hulls it is possible to use pulsating cavities which were investigated experimentally in Silberman and Song [18], Song [19], Michel [20], and Zou et al. [21] and by many other authors. The theories of such empty cavities were proposed by Woods [22], Paryshev [23, 24], Nesteruk [25], and Semenenko [26]. Here we consider the small oscillations of the pressure inside the cavity which covers the hull.Let us suppose that the gas injection rates are small enough to use only the main termR(0)(x,t) in (61) and to consider the pressure inside the cavity to be only time dependent. The corresponding changes of the cavitation number can be represented as follows:
(86)σ0(t)=σ-0+b1sin(2πt),
where σ-0 is the average value of the cavitation number and the dimensionless time is based on the period of oscillations T.Let us suppose the velocity of the ambient flow and the hull shape to be time independent and use the radiusR0 of the cavity and the body at the cross section of cavity origin x=0 (see Figure 1) for dimensionless lengths and to calculate the parameters (4) and (7). Then
(87)τ=R0TU∞;S(t)=0;Fr=U∞gR0.
Let us take only the main term of order ε2ln⁡ε in (62) and use the analytical solution obtained in Nesteruk [28]. In particular, for the harmonic low (86) the following formula was obtained (Nesteruk [33]):
(88)R2(x,t)=σ-0x22ln⁡ε±x33Fr2ln⁡ε+2βx+1+b2F2(x,t),(89)F2(x,t)=2πxτcos⁡[2π(t-xτ)]+sin[2π(t-xτ)]-sin⁡(2πt);b2=b14π2τ2ln⁡ε.
Formula (88) shows that for the pulsating cavity R2(x,t) differs from R2(x) corresponding to the steady cavity at the mean value of the cavitation number σ-0 by function b2F2(x,t). Formula (89) corresponds to a propagating wave with the amplitude increasing downstream. The length of the wave—the minimal distance between maxima or minima of the function (89)—is equal to 1/τ. The same behaviour was revealed in experiments (Song [19]).Integration of (88) allows calculating the cavity volume:
(90)G(t)=π∫oL(t)R2(x,t)dx=πL[σ-0L26ln⁡ε±L312Fr2ln⁡ε+βL+1]+b3[F3(L,t)-F3(0,t)],(91)F3(s,t)=cos⁡[2π(t-sτ)](πτ)-s{sin[2π(t-sτ)]+sin⁡(2πt)};b3=b14πτ2ln⁡ε,
where the cavity length L(t) may be obtained from
(92)Rb2(L)=σ-0L22ln⁡ε±L33Fr2ln⁡ε+2βL+1+b2F2(L,t).
At small values of b1 (or b2) the difference between L(t) and the length L- of the steady cavity at the cavitation number σ-0 must be small, since L- is a solution of (92) at b1=b2=0. Here we do not take into account the cases of discontinuous changes in the cavity length discussed in Section 8.5 and shown in Figure 15.By puttingΔL=L-L- into (90) the pulsations of the cavity volume can be calculated:
(93)ΔG(t)=G(t)-G-≈b3[F3(L-,t)-F3(0,t)],
where G- is the average volume of the steady cavity at the cavitation number σ-0 and can be obtained from (90) at b1=b3=0. With the use of (93) the average value of the cavity volume pulsations can be calculated:
(94)ΔG-=∫o1ΔG(t)dt=b4F4(τL-),b4=b18π3τ3ln⁡ε,(95)F4(y)=sin[2π(1-y)]-sin⁡(2πy)+πy{cos⁡[2π(1-y)]-cos⁡⁡(2πy)}.
Equations (94) and (95) show that the average value of the cavity volume pulsations is proportional to the amplitude of the gas pressure pulsations b1 and can be diminished, when F4(y)≈0. It can be expected that the pulsating cavities with zero average value of the volume pulsations are stable; that is, they can exist during a long time. According to the formula (95) the zero values of F4 are possible at
(96)τL-=L-′T′U∞′=Sh=n;n=1,2,3,….
Equation (96) shows that stable oscillating cavities occur only at integer values of the Strouhal number Sh or at integer number of waves located on the average cavity length. This fact was revealed in experiments on empty 2D cavities (Song [19]) and supported by the theories developed by Woods [22] and Semenenko [26] for 2D cavities and by Paryshev [23, 24] and Nesteruk [25] for the axisymmetric empty cavities. Presented analysis illustrates that the same behaviour is typical also in the case of rigid hulls located inside the cavity. If the average cavity length is fixed, the corresponding frequencies of pulsations in the cavity and in its wake are limited by (96).Formula (96) can also be applied for the separation bubbles both in gases and in liquids. In particular, let us compare (96) with the experimental Strouhal numbers for Karman vortex street, measured for circular cylinders of diameter d′ at large Reynolds numbers (see, e.g., Theodore von Kármán [38]):
(97)ShK=d′T′U∞′≈0.2.
Since the cylinder diameter d′ is few times smaller than the average length of its separated zone, formulas (96) and (97) are in good agreement.In order to investigate the influence of the ventilation rate on the pulsating cavities, (64) must be solved. But some conclusions can be drawn out with the use of simple physical considerations. At the local minima of the circular channel area the pressure in gas flow and cavity radius decrease and in maxima the pressure in gas flow and cavity radius increase. It leads to the increasing of the pulsation amplitude with the increasing of the ventilation rate. This fact was revealed by experiments (see, e.g., Silberman and Song [18]).


## 10. Conclusions

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, a solution was expressed as asymptotic series. In steady case the nonlinear differential equation and its solutions were obtained. It was shown that the cavity shape depends strongly on the values of the ventilation rate and the cavitation number.The ventilation can strongly increase and decrease the cavity dimensions. In particular, for the cavities created by the slender conical cavitator the increase of the ventilation rate increases the cavity dimensions. The cavity could become unbounded at some critical values of the ventilation rate. On the other hand, the ventilation can strongly decrease the base cavity length. Some specific neutral hulls ensure no changes in the cavity shape at any value of the ventilation.Examples of the steady cavity shape calculations for different hulls are presented. It was shown that the cavities can have classical elliptical shapes and be also conical, hyperbolic, and cylindrical. They may also have the minimum of the radius. Peculiarities of the cavity closing near the hull contour discontinuities were investigated.Simple stability theories for the steady and pulsating cavities at small ventilation rates were proposed. The results are in good agreement with the known theoretical and experimental findings of other authors and generalize them for the cases of the hull, located inside the cavity, and different cavitators and take into account the influence of the gravity. In particular, it was shown that the cavities with a very small area of the circular gas channel between the hull and the cavity surface are unstable and that gravity and the cavitator shape strongly influence the stability. The fact that stable pulsating cavities are possible, when an integer number of waves are located on the average cavity length, is also valid for nonempty cavities.More complicated models are necessary to simulate the gas flow inside the cavity. In particular, the viscosity of gas has to be taken into account.



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