Dynamics of a partially cavitating underwater vehicle
The purpose of the paper is to analyze the peculiarities of the partially cavitating vehicle dynamics. The research method is a computer simulation using the approximation model of the unsteady supercavity basing on G. V. Logvinovich's principle of independence of supercavity's sections ex...
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irk-123456789-1410652018-07-22T01:23:28Z Dynamics of a partially cavitating underwater vehicle Semenenko, V.N. Naumova, O.I. The purpose of the paper is to analyze the peculiarities of the partially cavitating vehicle dynamics. The research method is a computer simulation using the approximation model of the unsteady supercavity basing on G. V. Logvinovich's principle of independence of supercavity's sections expansion. A method of determination of the balanced motion parameters for the partially cavitating vehicle is elaborated. The examples of computer simulation of the partially cavitating vehicle motion are given, incluing the vehicle speeding-up and deceleration. The longitudinal motion of the balanced partially cavitating vehicle is is shown to be statically unstable on depth. In this case, the motion stability loss occurs in "hard" non-oscillatory manner, unlike the supercavitation case. The methods of stabilizing the partially cavitating vehicle motion are proposed. The effect of cavity ventilation on the dynamics of the cavitating vehicle is studied. Целью работы является анализ особенностей динамики частично кавитирующих аппаратов. Метод исследования - компьютерное моделирование с использованием аппроксимационной модели нестационарной суперкаверны, основанной на принципе независимости расширения сечений каверны Г. В. Логвиновича. Разработан метод нахождения равновесных значений параметров движения частично кавитирующего аппарата. Приведены примеры компьютерного моделирования движения частично кавитирущего аппарата, в частности при его разгоне и торможении. Показано, что продольное движение сбалансированного частично кавитирующего аппарата статически неустойчиво по глубине. При этом потеря устойчивости движения, в отличие от случая суперкавитации, происходит "жестким" неколебательным образом. Предложены способы стабилизации движения частично кавитирующего аппарата. Исследовано влияние вентиляции каверны на динамику кавитирующего аппарата. Метою роботи є аналіз особливостей динаміки частково кавітуючих апаратів. Метод дослідження- комп'ютерне моделювання з використанням апроксимаційної моделі нестаціонарної суперкаверни, яка базується на принципі незалежності розширення перерізів каверни Г. В. Логвиновича. Розроблено метод знаходження рівноважних значень параметрів руху частково кавітуючого апарата. Наведено приклади комп'ютерного моделювання руху частково кавітуючого апарата, зокрема при його розгоні і гальмуванні. Показано, що поздовжній рух збалансованого частково кавітуючого апарата статично нестійкий по глибині. При цьому втрата стійкості руху, на відміну від випадку суперкавітації, відбувається "жорстким" неколивальним чином. Запропоновано способи стабілізації руху частково кавітуючого апарата. Досліджено вплив вентиляції каверни на динаміку кавітуючого апарата. 2018 Article Dynamics of a partially cavitating underwater vehicle / V.N. Semenenko, O.I. Naumova // Гідродинаміка і акустика. — 2018. — Т. 1, № 1. — С. 70-84. — Бібліогр.: 20 назв. — англ. 2616-6135 http://dspace.nbuv.gov.ua/handle/123456789/141065 532.528 en Гідродинаміка і акустика Інститут гідромеханіки НАН України |
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The purpose of the paper is to analyze the peculiarities of the partially cavitating vehicle dynamics. The research method is a computer simulation using the approximation model of the unsteady supercavity basing on G. V. Logvinovich's principle of independence of supercavity's sections expansion. A method of determination of the balanced motion parameters for the partially cavitating vehicle is elaborated. The examples of computer simulation of the partially cavitating vehicle motion are given, incluing the vehicle speeding-up and deceleration. The longitudinal motion of the balanced partially cavitating vehicle is is shown to be statically unstable on depth. In this case, the motion stability loss occurs in "hard" non-oscillatory manner, unlike the supercavitation case. The methods of stabilizing the partially cavitating vehicle motion are proposed. The effect of cavity ventilation on the dynamics of the cavitating vehicle is studied. |
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Article |
| author |
Semenenko, V.N. Naumova, O.I. |
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Semenenko, V.N. Naumova, O.I. Dynamics of a partially cavitating underwater vehicle Гідродинаміка і акустика |
| author_facet |
Semenenko, V.N. Naumova, O.I. |
| author_sort |
Semenenko, V.N. |
| title |
Dynamics of a partially cavitating underwater vehicle |
| title_short |
Dynamics of a partially cavitating underwater vehicle |
| title_full |
Dynamics of a partially cavitating underwater vehicle |
| title_fullStr |
Dynamics of a partially cavitating underwater vehicle |
| title_full_unstemmed |
Dynamics of a partially cavitating underwater vehicle |
| title_sort |
dynamics of a partially cavitating underwater vehicle |
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Інститут гідромеханіки НАН України |
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2018 |
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http://dspace.nbuv.gov.ua/handle/123456789/141065 |
| citation_txt |
Dynamics of a partially cavitating underwater vehicle / V.N. Semenenko, O.I. Naumova // Гідродинаміка і акустика. — 2018. — Т. 1, № 1. — С. 70-84. — Бібліогр.: 20 назв. — англ. |
| series |
Гідродинаміка і акустика |
| work_keys_str_mv |
AT semenenkovn dynamicsofapartiallycavitatingunderwatervehicle AT naumovaoi dynamicsofapartiallycavitatingunderwatervehicle |
| first_indexed |
2025-07-10T11:52:37Z |
| last_indexed |
2025-07-10T11:52:37Z |
| _version_ |
1837260718477410304 |
| fulltext |
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 1. С. 7070–8484.
UDC 532.528
DYNAMICS OF A PARTIALLY CAVITATING
UNDERWATER VEHICLE
V. N. Semenenko†, O. I. Naumova
Institute of Hydromechanics of NAS of Ukraine
Zhelyabov Str., 8/4, 03057, Kyiv, Ukraine
†E-mail: vnsvns60@gmail.com
Received 25.09.2017
The purpose of the paper is to analyze the peculiarities of the partially cavitating vehi-
cle dynamics. The research method is a computer simulation using the approximation
model of the unsteady supercavity basing on G. V. Logvinovich’s principle of indepen-
dence of supercavity’s sections expansion. A method of determination of the balanced
motion parameters for the partially cavitating vehicle is elaborated. The examples of
computer simulation of the partially cavitating vehicle motion are given, incluing the
vehicle speeding-up and deceleration. The longitudinal motion of the balanced par-
tially cavitating vehicle is is shown to be statically unstable on depth. In this case,
the motion stability loss occurs in “hard” non-oscillatory manner, unlike the super-
cavitation case. The methods of stabilizing the partially cavitating vehicle motion are
proposed. The effect of cavity ventilation on the dynamics of the cavitating vehicle is
studied.
KEY WORDS: partially cavitating vehicle, ventilated cavity, computer simulation
1. INTRODUCTION
The flow regime of planing within a supercavity, when a vehicle is fully located inside
the cavity, is the optimal regime of the cruise motion of high-speed supercavitating vehi-
cles (HSSV). The supercavitating vehicle dynamics has been studied for different control
strategies in our previous works [11–55].
The vehicle motion in the partial cavitation regime, when a cavity covers only the part
of the vehicle body, is possible when the vehicle speeding-up or deceleration, and also when
maneuvering and when arising the unsteady cavity oscillation (see Fig. 1Fig. 1). Non-symmetry
of the cavity shape and the line of the cavity closure on the body is caused by the inclination
of both the cavitator plane and the body axis relatively to mainstream, and by the gravity
action as well.
Many papers are devoted to studying the partial cavitation flow over hydrofoils and
3D bodies. In this case both the experimental and the numerical methods were applied.
Papers [66–99] are devoted to investigation of the partial cavitation flows over 3D bodies.
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To attain the required small value of the cavitation number 𝜎 for the specified motion
depth and velocity 𝐻, 𝑉 , the cavity pressure 𝑝𝑐 > 𝑝𝑣 must be usually ensured (where
𝑝𝑣 = 2.350 kPa is the saturated vapor pressure), i.e., the cavity must be ventilated.
As opposed to the fully developed
Fig. 1. Scheme of a partially cavitating vehicle
supercavities, the partial cavities al-
ways are essentially unsteady. There
are two different mechanisms of the
partial cavity instability. The first one
is caused with periodic formation and
destruction of a reentrant jet and oc-
curs for both the vapour and the venti-
lated cavities. It was experimentally established that blowing the gas into the cavity weakens
this instability type [99].
In practical calculations (in the mathematical model described below as well) this type of
“natural” non-stationarity is usually neglected. In this case the cavity shape and the rate of
gas loss from the cavity �̇�out are averaged by time, and the cavities are considered as steady
ones.
The second cause of unsteady behaviour of the partial cavities is connected with the sta-
bility loss and arising the ventilated cavity pulsation when exceeding some critical magnitude
of the similarity parameter 𝛽 = 𝜎𝑣/𝜎 (where 𝜎𝑣 is the vapour cavitation number) [1010, 1111].
The mathematical model described below takes this non-stationarity type into consideration.
Purpose of this article is to analyze peculiarities of dynamics of the high-speed underwater
vehicles moving in the partial cavitation regime. The research method is the computer
simulation of a cavitating body dynamics with using the approximation model of an unsteady
supercavity basing on the G. V. Logvinovich principle of independence of the supercavity
section expansion [1212]. Practically important cases of longitudinal motion of the partially
cavitating vehicle, its speeding-up and deceleration are considered.
2. MATHEMATICAL MODEL OF 3D MOTION OF A PARTIALLY CAVI-
TATING VEHICLE
We accept the following assumptions when formulating the problem of calculation of the
self-propelled cavitating vehicle dynamics:
∙ the vehicle body has a shape of an elongated body of revolution;
∙ the cavitator is a disk with diameter 𝐷𝑛;
∙ the vehicle mass 𝑚 and the engine thrust 𝐹pr are fixed;
∙ both the angle of attach 𝛼 and the sliding angle 𝛽 are always sufficiently small;
∙ the time dependence of all the forces acting is quasistationary.
The following forces and their moments relatively to the vehicle mass center must be
taken into consideration in the partial cavitation case:
∙ the hydrodynamic force acting onto the cavitator and its moment 𝐹𝑛, �⃗�𝑛;
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ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 1. С. 7070–8484.
∙ the gravity force of the vehicle 𝐹𝑔 = −𝑚𝑔 (it does not create moment);
∙ the hydrostatic buoyancy (Archimedean) force and its moment acting onto the wetted
part of the vehicle 𝐹𝑎, �⃗�𝑎;
∙ the skin friction drag and the transversal hydrodynamic force and its moment arising
in case of flowing the vehicle with angles of attack and/or sliding 𝐹𝑏, �⃗�𝑏;
∙ the inertial force and moment acting on the wetted part of the vehicle 𝐹𝑖, �⃗�𝑖;
∙ the thrust of the engine and its moment 𝐹cr, �⃗�cr.
Also, the control force on the fins and its moment 𝐹𝑓 , 𝑣𝑒𝑐𝑀𝑓 has to be added in problems
on vehicle maneuvering.
In this case a set of vector equations of the vehicle motion has the form [1313]:
𝑚(
𝑑�⃗�
𝑑𝑡
+ �⃗� × �⃗� ) = 𝐹𝑛 + 𝐹𝑏 + 𝐹𝑖 + 𝐹𝑎 + 𝐹𝑔 + 𝐹pr, (1)
𝑑�⃗�
𝑑𝑡
+ �⃗� × �⃗� = �⃗�𝑛 + �⃗�𝑏 + �⃗�𝑖 + �⃗�𝑎 + �⃗�pr, (2)
where �⃗� = {𝑉𝑥1, 𝑉𝑦1, 𝑉𝑧1} is the velocity vector of the body mass center and its components
in the body coordinates; �⃗� = {𝜔𝑥1, 𝜔𝑦1, 𝜔𝑧1} is the angular velocity vector relatively to the
body mass center; �⃗� is the body kinetic moment.
The model position relatively to the
Fig. 2. Scheme of forces acting onto a partially
cavitating vehicle in steady-state motion
Earth-fixed coordinates is defined by
the pitch angle 𝜓, the yaw angle 𝜙, and
the roll angle 𝜃. The model position
relatively to the flow coordinates is de-
fined by the angle of attack 𝛼 and the
sliding angle 𝛽. Three kinematic Eu-
ler’s relations [1313] and three differentia-
tion equations of motion of the vehicle
mass center relatively to the fixed co-
ordinates [44] have to be added to equa-
tions (1)(1), (2)(2).
The sum of the gravity and the hydrostatic force projections onto the axes of the body
coordinates is as follows:
𝐹𝑔𝑥 + 𝐹𝑎𝑥 = (𝐹𝑔 + 𝐹𝑎) sin𝜓,
𝐹𝑔𝑦 + 𝐹𝑎𝑦 = (𝐹𝑔 + 𝐹𝑎) cos𝜓 cos 𝜃,
𝐹𝑔𝑧 + 𝐹𝑎𝑧 = (𝐹𝑔 + 𝐹𝑎) cos𝜓 sin 𝜃,
(3)
where 𝐹𝑔 = −𝑚𝑔; 𝐹𝑎 = 𝜌𝑔𝑄𝑤; 𝑄𝑤 is the volume of the wetted vehicle part. The Archimedean
buoyancy force 𝐹𝑎 is applied to the center of the volume 𝑄𝑤 located on the distance 𝑥𝑎 from
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the cavitator. Then projections of the moment of the force 𝐹𝑎 relatively to the mass center
are equal to:
𝑀𝑎𝑦 = 𝐹𝑎𝑧𝑙𝑎, 𝑀𝑎𝑧 = 𝐹𝑎𝑦𝑙𝑎, 𝑙𝑎 = 𝑥𝑐 − 𝑥𝑎. (4)
The skin friction drag of the wetted vehicle part is calculated by formulae:
𝐹𝑏𝑥 = −𝜌𝑉
2
2
𝑆𝑤𝑐𝑓 , 𝑐𝑓 =
0.075
(lg Re𝑤 − 2)2
, Re𝑤 =
𝑙𝑤𝑉
𝜈
, (5)
where 𝑆𝑤 is the area of the wetted vehicle part surface; 𝑐𝑓 is the skin friction coefficient in
the turbulent boundary layer [1414]; Re𝑤 is the Reynolds number; 𝐿𝑐 is the average cavity
length; 𝑙𝑤 = 𝐿− 𝐿𝑐 is the average length of the wetted model part (see Fig. 2Fig. 2).
Components of the transversal hydrodynamic forces on the body 𝐹𝑏𝑦, 𝐹𝑏𝑧 for slender
partially cavitating body will be the same as in the non-cavitating flow [1212]:
𝐹𝑏𝑦 = 2𝛼
𝜌𝑉 2
2
𝑆𝑏, 𝐹𝑏𝑧 = −2𝛽
𝜌𝑉 2
2
𝑆𝑏, (6)
where 𝑆𝑏 is the model transom area. We have 𝛼 = 𝜓, 𝛽 = 𝜙 in the steady-state horizontal
motion. One considers that the force 𝐹𝑏 application point is located at the model tran-
som [1212]. Then projections of the moment of this force relatively to the model mass center
are equal to:
𝑀𝑏𝑧 = 𝐹𝑏𝑦𝑙𝑏, 𝑀𝑏𝑦 = −𝐹𝑏𝑧𝑙𝑏, 𝑙𝑏 = 𝑥𝑐 − 𝐿. (7)
Note that using the average length of the partial cavity in calculation is reasonable for
sufficiently small angles 𝛼, 𝛽 and not too small magnitudes of the Froude number. These
conditions are usually fulfilled in practice.
The more precise method of the force determination when the non-symmetric cavity
closes on the body basing on the method of plane section was proposed in monograph [1515].
In this case the authors stated that the partially cavitating body motion is stable if the
cavity length exceeds a half of the body length.
The inertial force and its moment expressed in terms of the added masses of the transver-
sal flow 𝜆22, 𝜆33, etc. are essential in the unsteady flow over the vehicle in the partial
cavitation regime. For vehicles having a shape of an elongated body of revolution the six
coefficients of the generalized added masses are nonzero, and only three of them are inde-
pendent: 𝜆22 = 𝜆33, 𝜆26 = −𝜆35, 𝜆66 = 𝜆55 [1616]. The method of plane sections is used to
calculate the added masses:
𝜆22 = 𝜌𝜋
∫︁ 𝐿
𝐿𝑐
𝑅2(𝑠)𝑑𝑠, 𝜆26 = 𝜌𝜋
∫︁ 𝐿
𝐿𝑐
𝑅2(𝑠)(𝑠− 𝑥𝑐)𝑑𝑠, 𝜆66 = 𝜌𝜋
∫︁ 𝐿
𝐿𝑐
𝑅2(𝑠)(𝑠− 𝑥𝑐)
2𝑑𝑠, (8)
where 𝑅(𝑥) is the current radius of the vehicle wetted part. Projections of the inertial force
and moment onto the axes of the body coordinates are as follows [1616]:
𝐹𝑖𝑥 = −𝜆22(𝜔𝑦𝑉𝑧1 − 𝜔𝑧𝑉𝑦1) + 𝜆26(𝜔
2
𝑦 + 𝜔2
𝑧),
𝐹𝑖𝑦 = −𝜆22
𝑑𝑉𝑦1
𝑑𝑡
− 𝜆26
𝑑𝜔𝑧
𝑑𝑡
,
𝐹𝑖𝑧 = −𝜆22
𝑑𝑉𝑧1
𝑑𝑡
+ 𝜆26
𝑑𝜔𝑦
𝑑𝑡
,
(9)
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ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 1. С. 7070–8484.
𝑀𝑖𝑦 = −𝜆55
𝑑𝜔𝑦
𝑑𝑡
+ 𝜆26
𝑑𝑉𝑧1
𝑑𝑡
− 𝜆26𝑉𝑥1𝜔𝑦 + 𝜆22𝑉𝑥1𝑉𝑧1,
𝑀𝑖𝑧 = −𝜆55
𝑑𝜔𝑧
𝑑𝑡
− 𝜆26
𝑑𝑉𝑦1
𝑑𝑡
− 𝜆26𝑉𝑥1𝜔𝑧 − 𝜆22𝑉𝑥1𝑉𝑦1.
(10)
The added mass of the longitudinal flow of the cavitating vehicle 𝜆11 is small in comparison
with the vehicle mass 𝑚 and therefore it is not taken into consideration.
3. COMPUTER SIMULATION OF THE CAVITATING VEHICLE DYNAM-
ICS
We accept the following assumptions when formulating the problem of calculation of
unsteady behavior of the ventilated cavity:
∙ the first cavity instability type is not considered (see above);
∙ the cavity sections remain circular;
∙ the cavity gas expands isothermally;
∙ the cavity pressure 𝑝𝑐 is synchronously varied along the cavity.
A set of twelve equations of the solid body dynamics in projections onto the axes of
the body coordinates is integrated numerically together with two equations describing the
unsteady ventilated cavity evolution [1717,1818]:
𝜕2𝑆𝑐(𝜏, 𝑡)
𝜕𝑡2
= −𝑘1
2
𝑉 2(𝜏)𝜎(𝜏, 𝑡), (11)
𝑑
𝑑𝑡
[𝑝𝑐(𝑡)𝑄𝑔(𝑡)] = 𝑝∞
[︁
�̇�in − �̇�out(𝑡)
]︁
, (12)
where 𝑆𝑐 is the cavity section area; 𝜏 ≤ 𝑡 is the instant of the section formation; 𝑘1 is the
constant; 𝑄𝑔 = 𝑄𝑐 − 𝑄𝑏 is the cavity gas volume; 𝑄𝑐(𝑡) is the cavity volume; 𝑄𝑏(𝑡) is the
volume of the body within the cavity; �̇�in and �̇�out are volumetric rates of the gas supply
in the cavity and the gas loss from the cavity related to the water pressure 𝑝∞. The semi-
empirical formula is used to calculate the volumetric gas loss rate when the cavity closes on
a body:
�̇�out = 𝐶 ′
𝑒𝑉 𝜋𝐷
2
𝑏
𝐿𝑐
𝐷𝑛
tg
𝛼𝑏
2
, (13)
where 𝛼𝑏 is the average angle between the body contour and the cavity contour at the closure
point; 𝐶 ′
𝑒 = 0.28 ·10−2 is the empirical coefficient that weakly depends on both the cavitation
number and the body shape.
Also when calculation, the cavity axis curvature caused by the cavitator center displace-
ment, the cavitator inclination, and the gravity acting [22] is taken into consideration.
The model having a shape described in paper [22] was used as the calculation model of
the partially cavitating vehicle. Its main parameters are: the length 𝐿 = 5 m; the cylindrical
part diameter 𝐷𝑏 = 340 mm; the cavitator diameter 𝐷𝑛 = 70 mm; the mass 𝑚 = 600 kg;
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ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 1. С. 7070–8484.
the moments of inertia relatively to the vehicle axes 𝐼𝑥 = 8 kg·m2, 𝐼𝑦 = 𝐼𝑧 = 900 kg·m2.
The cruise motion parameters are: the velocity 𝑉cr = 120.0 m/s; the depth 𝐻 = 5.0 m;
the cavitation number 𝜎 = 0.02; the gas rate coefficient �̇�in = 0.0013; the cavity length
𝐿cr = 6.555 m; the engine thrust 𝐹cr = 23.213 kN.
The underwater vehicle motion is called as balanced, if sum of all the forces acting and
sum of all the moments are equal to zero. A method of numerical determination of balanced
values of the cavitator inclination angles 𝛿𝑧, the thrust deflection angle 𝜂𝑧, and the pitch angle
𝜓 was proposed in work [11]. Tab. 1Tab. 1 gives the balanced values of the angles 𝛿𝑧 and 𝜓 when
𝜂𝑧 = 0 for a number of positions of the vehicle mass center �̄�𝑐 = 𝑥𝑐/𝐿 for the supercavitating
vehicle motion in the cruise regime.
Table 1. Balanced angles 𝛿𝑧 and 𝜓 in the cruise regime (degrees)
�̄�𝑐 0.4 0.5 0.6
𝛿*𝑧 −8.760 −7.260 −5.779
𝜓* 0.3902 0.3718 0.3538
As is know, the longitudinal motion of the supercavitating vehicle in the regime of planing
within a cavity is unstable on depth [11]. In this case a scenario of a “soft” stability loss is
realized, when oscillation of the pitch angle 𝜓 and deflection of the model mass center from
the straight path 𝑦 arise and then increase. Then on the cruise part the supercavitating
vehicles need the motion depth-stabilization with an automatic error-closing control [11]:
𝛿𝑧(𝑡) = 𝛿*𝑧 + 𝑘1𝑦(𝑡− 𝑡1) + 𝑘2[𝜓(𝑡− 𝑡1) − 𝜓*] + 𝑘3�̄�(𝑡− 𝑡1). (14)
where 𝛿*𝑧 , 𝜓* are the regulated values of the parameters; 𝑘1, 𝑘2, 𝑘3 are the non-negative
coefficients of feedback (transfer ratios of a regulator); 𝑡1 is time of delay of the actuator
reaction; 𝑦 = 𝑦/𝐿; �̄� = �̇�𝐿/𝑉0; the angle are measured in radians. For this model in the
cruise regime, the following values of transfer ratios are used: 𝑘1 = 2.0, 𝑘2 = 5.0, 𝑘3 = 0.
4. BALANCING A PARTIALLY CAVITATING VEHICLE
A problem of determininig the balanced values of the angles 𝛿𝑧, 𝜂𝑧, and 𝜓 in the steady-
state motion of the vehicle in the partial cavitation regime is solved numerically. In this case
a set of equations for the model balance has the form:
𝐹𝑛𝑥 + 𝐹𝑏𝑥 + 𝐹𝑎𝑥 + 𝐹𝑔𝑥 + 𝐹pr cos 𝜂𝑧 = 0,
𝐹𝑛𝑦 + 𝐹𝑏𝑦 + 𝐹𝑎𝑦 + 𝐹𝑔𝑦 + 𝐹pr sin 𝜂𝑧 = 0,
𝑀𝑛𝑧 +𝑀𝑏𝑧 +𝑀𝑎𝑧 +𝑀pr𝑧 +𝑀𝑖𝑧 = 0,
(15)
where 𝑀𝑛𝑧 = 𝐹𝑛𝑦𝑥𝑐; 𝑀pr𝑧 = 𝐹pr sin 𝜂𝑧(𝐿 − 𝑥𝑐); 𝑀𝑖𝑧 = −𝜆22𝑉𝑥1𝑉𝑦1. Eliminating 𝐹pr from
equations (15)(15), we obtain a set of two functional equations with three unknowns:
𝐹1(𝛿𝑧, 𝜂𝑧, 𝜓) = 0, 𝐹2(𝛿𝑧, 𝜂𝑧, 𝜓) = 0. (16)
From these equations, fixing any of three angles 𝛿𝑧, 𝜂𝑧, 𝜓, one can determine the balanced
values of two other angles. Let, for example, we want to determine the balanced values of
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angles 𝛿*𝑧 , 𝜓* for fixed 𝜂𝑧. For numerical solving a set of equations (16)(16) we use the Newton’s
iteration algorithm: 𝛿
(𝑖+1)
𝑧 = 𝛿
(𝑖)
𝑧 + ∆𝛿𝑧, 𝜓
(𝑖+1) = 𝜓(𝑖) + ∆𝜓, where additions ∆𝛿𝑧, ∆𝜓 for
each iteration 𝑖 = 1, 2, . . . are solution of a set of two linear equations:
Fig. 3. Balanced values of angles 𝛿𝑧 and 𝜓
in the partial cavitation regime:
1 — �̄�𝑐 = 0.4, 2 — �̄�𝑐 = 0.5, 3 — �̄�𝑐 = 0.6
Fig. 4. Loss of the cavitating vehicle
motion stability:
1 — the planing within a cavity regime,
2 — the partial caitation regime
𝜕𝐹
(𝑖)
1
𝜕𝛿𝑧
∆𝛿𝑧 +
𝜕𝐹
(𝑖)
1
𝜕𝜓
∆𝜓 = −𝐹1
(︀
𝛿(𝑖)𝑧 , 𝜓
(𝑖)
)︀
,
𝜕𝐹
(𝑖)
2
𝜕𝛿𝑧
∆𝛿𝑧 +
𝜕𝐹
(𝑖)
2
𝜕𝜓
∆𝜓 = −𝐹2
(︀
𝛿(𝑖)𝑧 , 𝜓
(𝑖)
)︀
.
(17)
Derivatives in the left parts of equa-
tions (17)(17) are calculated numerically by the
finite differences.
Fig. 3Fig. 3 gives the calculated dependencies
𝛿*(�̄�𝑐) and 𝜓*(�̄�𝑐) for the fixed 𝜂𝑧 = 0 for
three position of the mass center �̄�𝑐 (the
angles are in degrees). Magnitudes �̄�𝑐 =
0.5 and �̄�𝑐 = 0.9 corresponds to the vehi-
cle velocity magnitudes 𝑉0 = 90 m/s and
𝑉0 = 110 m/s, respectively, when the mo-
tion depth is 𝐻 = 5 m.
One should note that the range of val-
ues �̄�𝑐, for which the cavitating vehicle may
be balanced for the specified values of the
parameters 𝑚 and 𝜂𝑧, is decreased with de-
creasing the cavity length.
Also, the described algorithm is appli-
cable for determining the balanced parame-
ters for the case of the supercavitating vehi-
cle in the regime of planing within a cavity,
and it is even more effective than the earlier
method [11].
5. LONGITUDINAL MOTION OF A PARTIALLY CAVITATING VEHICLE
We consider first a case of the longitudinal motion of the balanced partially cavitating
vehicle when the engine thrust is equal to the total hydrodynamic drag: 𝐹pr = −𝐹𝑥.
Fig. 4Fig. 4 gives graphs of dependence of the pitch angle 𝜓 (in degrees) and the mass center
deflection 𝑦 = 𝑦/𝐿 from the flight path 𝑠 = 𝑠/𝐿 in the longitudinal motion of both the super-
cavitating and the partially cavitating models. In both cases the calculation was performed
for the initial balanced magnitudes of the parameters when �̄�𝑐 = 0.6.
As is seen, the longitudinal motion of the balanced partial cavitating model is stati-
cally unstable on depth. In this case the stability loss of the vehicle motion occurs in the
“hard” non-oscillatory manner, and the more is the average cavity length 𝐿𝑐, the more is
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the model “stability margin”. For the unbalanced vehicle, variations of the parameters 𝜓, 𝑦
are proportional to the starting deviations of parameters from their balanced values.
One can consider that when 𝐿𝑐 ≪ 𝐿, the partially cavitating vehicle dynamics is similar
to dynamics of a non-cavitating elongated body of revolution. As is known, it is statically
unstable owing to action of the overturning moment 𝑀𝑖𝑧. In practice, passive stabilization
of the elongated body motion in non-cavitating flow is realized with the tail fins. It seems,
the tail fins may be applied for passive stabilization of motion of the partially cavitating
vehicles as well.
The computer simulation has shown that one can successfully apply the automatic error-
closing 𝛿-stabilization on depth similar to one described in our paper [11] to active stabilize
the longitudinal motion of the partially cavitating vehicle. In this case the law of automatic
regulation of the cavitator inclination angle has the form:
𝛿𝑧(𝑡) = 𝛿*𝑧 + 𝑘2[𝜓(𝑡) − 𝜓*] + 𝑘3�̄�(𝑡), (18)
where 𝑘2 > 0, 𝑘3 > 0 are the regulator transfer ratios. It was established that, in difference
from the supercavitating vehicle case, one does need directly regulate deflection of the vehicle
mass center on depth 𝑦 in the case of partially cavitating vehicle (i.e., 𝑘1 ≡ 0 in (14)(14).
It should be noted that the optimal values of the transfer ratios 𝑘2, 𝑘3 may be varied in
dependence on a number of the parameters, especially 𝑥𝑐 and 𝐿𝑐. In paricular, the values
𝑘2 = 5.0, 𝑘3 = 20.0 ensure stable motion for a wide range of values 𝑥𝑐, 𝐿𝑐 for this calculated
model.
Also, computer simulation has shown that longitudinal motion of the partially cavitating
vehicles (i.e., when 𝛿𝑦 = 0 and 𝜙 = 0) is stable on course. The same is true for supercav-
itating vehicles. If the starting values 𝛿𝑦(0) ̸= 0 and/or 𝜙(0) ̸= 0, then variation of the
parameters 𝜙(𝑡), 𝑧(𝑡) in the vehicle lateral motion is proportional to the starting deviation
of the parameters 𝛿𝑦, 𝜙 from zero.
6. CAVITATING VEHICLE DYNAMICS DURING SPEEDING-UP
The cavity development up to its length 𝐿𝑐 corresponding to the cruise regime with the
velocity 𝑉cr must be ensured during speeding-up the cavitating vehicle from the low starting
velocity. In our paper [1919], the speeding-up process has been studied with the assumption
that the model path is straight-line (as when testing the models on the cable launching
tank [2020]) und the cavity is axisymmetric. Consider now the speeding-up process with
taking into account the dynamics of the partially cavitating vehicle by equations (1)(1), (2)(2). In
this case calculation of possible deviations of the parameters 𝜓, 𝜙, 𝑦, 𝑧 on the speeding-up
part and elaboration of methods of their stabilization are of the practical interest.
Paper [1919] shows that a starting engine with the thrust 𝐹𝑠𝑡 exceeding in about five times
the cruise engine thrust 𝐹cr is required to accelerate this cavitating vehicle model with the
admissible maximal overload �̄� = 𝑎/𝑔 ≈ 20. The following starting parameters were accepted
in calculation: 𝑉 (0) = 10 m/s; 𝑝𝑐(0) = 2.350 kPa, in this case we have 𝜎(0) = 2.894;
𝐿𝑐 = 0.130 m.
Computer simulation has shown that when speeding-up the balanced model its “stability
margin” usually is sufficient up to the acceleration part end, and in this case deviations of the
parameters 𝜓, 𝑦 are very small. After finishing the speeding-up, the system of stabilization
of the balanced cruise parameters (14)(14) must be turned on.
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a
b
Fig. 5. Variation of motion parameters during speeding-up:
a — without stabilization, b — with 𝛿-stabilization on depth
On the contrary, at presence of starting imbalance of the model motion the deviations
on the acceleration part may become inadmissibly large. Fig. 5Fig. 5a gives graphs of varying the
parameters 𝑉 = 𝑉 (𝑡)/𝑉cr, �̄�𝑐 = 𝐿𝑐(𝑡)/𝐿cr, 𝜓, 𝑦 on the interval 0 < 𝑡 < 1 s. The following
initial values of the parameters were accepted in calculation: 𝛿𝑧 = 0, 𝜓 = 0. The starting
engine with thrust 𝐹𝑠𝑡 = 150 kN is turned off at instant 𝑡 = 0.5 s, then the cruise engine is
turned on.
As is seen, in this case the cruise values 𝐿cr, 𝑉cr are not attained, what is explained by
increasing the motion depth 𝐻 = 𝐻0 − 𝑦. The graphs of variation of the same parameters
for the same conditions but when operating 𝛿-stabilization on depth (14)(14) is given in Fig. 5Fig. 5b.
In this case the complete speeding-up the cavitating vehicle occurs on the same interval 0.5
s with the small deviations of the parameters 𝜓, 𝑦.
The cavity development during speeding-up of the cavitating vehicle may be accelerated
by increasing the rate of gas-supply into the cavity �̇�in. However, in this case the cavity
evolution will be increasingly influenced by dynamic properties of gas-filled cavities. As
was shown in [1010, 1111], when exceeding the some critical value of the similarity parameter
𝛽 = 𝜎𝑣/𝜎 (where 𝜎𝑣 is the vapour cavitation number) the cavity loses stability and begins
to pulsate intensively. In this case the rate of gas-loss from the cavity �̇�out increases, and
that prevents next increasing the average cavity length. It was concluded in work [1919] that
the effectiveness of the increased gas-supply during speeding-up the supercavitating vehicle
is weak, and it decreases when the starting engine thrust increases.
7. CAVITATING VEHICLE DYNAMICS DURING DECELERATION
The cavity evolution law and law of the vehicle velocity reduction after turning off the
cruise engine is of practical interest. At transition from the supercavitating regime to the
partial cavitation regime the vehicle becomes imbalanced in the general case. In this case,
the non-stationary transversal forces begin to act on the vehicle, so that the deviations of
the motion parameters from specified ones may rapidly exceed the admissible values.
Fig. 6Fig. 6 gives graphs of varying the model velocity 𝑉 , the cavity length 𝐿𝑐, and the angle
of attack 𝛼 (in radians) in the model motion on inertia after the cruise engine turning off.
It is assumed in this case that the gas supply into the cavity is not changed, and the depth
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stabilization system continues to operate.
In this calculation, the balanced param-
Fig. 6. Variation of motion parameters
after the cruise engine turning-off
eter values for the cruise regime are used as
the initial data (see Tab. 1Tab. 1, �̄�𝑐 = 0.6). On
the time interval 0 < 𝑡 < 2.0 s the model en-
ters on the regime of steady oscillation of the
pitch angle with the frequency 6.562 Hz. At
instant 𝑡 = 2.0 s the cruise engine is turned
off, after this the model moves on inertia.
In this case both the velocity and the cavity
length decrease. The calculation was per-
ormed to instant 𝑡 = 2.71 s, when the model
angle of attack exceeds the admissible value
𝛼 = 7∘.
As is seen, the motion on inertia of the
partially cavitating vehicle without fins rapidly loses the static stability. Computer simula-
tion has shown that in this case the motion can not be stabilized by means of the automatic
𝛿-control (18)(18). It seems, in practice the motion may be stabilized by opening or extending
the tail fins from the vehicle body.
8. EFFECT OF CAVITY VENTILATION ON THE CAVITATING VEHICLE
DYNAMICS
As it was said above, in practice one has to blow gas into a cavity (i.e., to ventilate a
cavity) for ensuring required cavity dimensions for specified magnitudes of the velocity and
depth 𝑉 , 𝐻 of the supercavitating vehicle motion. During speeding-up the vehicle, the cavity
development may be accelerated by increasing the gas-supply rate �̇�in.
Obvious effect of the gas-supply rate onto the cavitating vehicle dynamics is explained by
the cavity dimensions and shape variation at the cavity ventilation. In this case the forces
of interaction of the vehicle with water are changed for both the partially cavitating vehicles
and the supercavitating vehicles moving in the planing within a cavity regime.
Consider the cavitating vehicle motion with the same depth 𝐻 and the same cavitation
number 𝜎, but with different velocities 𝑉 . In these cases, the dynamical similarity of flow will
be disturbed because of the cavity shape distortion under gravity action owing to changing
the Froude number Fr, and also owing to possible unsteady behavior of the ventilated cavity.
In this case the balanced values of the angles 𝛿𝑧, 𝜓, 𝜂𝑧 and other parameters will be changed,
Table 2. Parameters of the partially cavitating vehicle steady-state motion:
𝐻 = 5.0 m; 𝜂𝑧 = 0; 𝜎 = 0.03
𝑉, m/s 𝛿*𝑧 𝜓* Fr 𝛽 �̄�in
50.0 −10.905 0.8835 60.35 3.880 0.2000
70.0 −6.262 0.4463 84.49 1.966 0.2181
90.0 −3.770 0.2711 108.6 1.189 0.2255
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and this leads to additional cavity shape distortion (see Tab. 2Tab. 2).
The table has nomenclature: Fr = 𝑉/
√
𝑔𝐷𝑛 is the Froude number; �̄�in = �̇�in/𝑉 𝐷
2
𝑛 is
the coefficient of the gas-supply rate.
Fig. 7Fig. 7 gives graphs of varying the parameters 𝜓, 𝑦 for the range 0 < 𝑡 < 10 s. The
calculation was performed for starting magnitudes of the parameters 𝑉 , 𝐻, 𝛿𝑧, 𝜓 from
Tab. 2Tab. 2 with turned-on 𝛿-stabilization on depth (14)(14).
In the first case we have 𝛽 > 𝛽cr, therefore the ventilated cavity rapidly loses stability and
begins to pulsate (see [1010,1111]). As a result, the “hard” instability of the partially cavitating
vehicle motion is developed in spite of the 𝛿-stabilization. In the second case and in the third
case we have 𝛽 < 𝛽cr, and perturbations arising at the 𝛿-stabilization system operation are
damped with the lapse of time.
9. CONCLUSIONS
The mathematical model of 3D dynam-
Fig. 7. Variation of motion parameters
at different gas-supply rates:
1 — 𝑉 = 50.0 m/s, 2 — 𝑉 = 70.0 m/s,
3 — 𝑉 = 90.0 m/s
ics of the self-propelled partially cavitating
underwater vehicle has been developed.
The method of determination of the bal-
anced motion parameters of the partially
cavitating vehicle is elaborated. Examples
of computer simulation of the partially cav-
itating vehicle motion (especially for both
the vehicle speeding-up and deceleration)
are given.
The performed computer simulation has
shown that the longitudinal motion of the
balanced partially cavitating vehicle is stat-
ically unstable on depth. In this case the
motion stability loss occurs in “hard” non-
oscillatory manner unlike the supercavitation case.
The deviations of parameters 𝜓, 𝜙, 𝑦, 𝑧 during the speeding-up depend on degree of
the starting imbalance of the cavitating vehicle and can become unacceptably large. Thus,
the partially cavitating vehicle motion requires in additional depth stabilization like the
supercavitating vehicle motion. It has been shown that the same error-closing control system
as for the supercavitating vehicles may be successfully applied to stabilize the partially
cavitating vehicle motion for both the fixed velocity and the speeding-up.
It has been shown that the ventilated cavity shape distortion and their unsteady be-
haviour with the same cavitation number but with different gas-supply rates can influence
essentially onto the cavitating vehicle dynamics. This conclusion is valid for both the par-
tially cavitating vehicles and the supercavitating vehicles moving in the planing within a
cavity regime.
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В. М. Семененко, О. I. Наумова
Динамiка частково кавiтуючого пiдводного апарата
Метою роботи є аналiз особливостей динамiки частково кавiтуючих апаратiв. Ме-
тод дослiдження — комп’ютерне моделювання з використанням апроксимацiйної
моделi нестацiонарної суперкаверни, яка базується на принципi незалежностi роз-
ширення перерiзiв каверни Г. В. Логвиновича. Розроблено метод знаходження рiв-
новажних значень параметрiв руху частково кавiтуючого апарата. Наведено при-
клади комп’ютерного моделювання руху частково кавiтуючого апарата, зокрема
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ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 1. С. 7070–8484.
при його розгонi i гальмуваннi. Показано, що поздовжнiй рух збалансованого час-
тково кавiтуючого апарата статично нестiйкий по глибинi. При цьому втрата стiй-
костi руху, на вiдмiну вiд випадку суперкавiтацiї, вiдбувається “жорстким” неко-
ливальним чином. Запропоновано способи стабiлiзацiї руху частково кавiтуючого
апарата. Дослiджено вплив вентиляцiї каверни на динамiку кавiтуючого апарата.
КЛЮЧОВI СЛОВА: частково кавiтуючий апарат, вентильована каверна, стiй-
кiсть, комп’ютерне моделювання
В. Н. Семененко, О. И. Наумова
Динамика частично кавитирующего подводного аппарата
Целью работы является анализ особенностей динамики частично кавитирующих
аппаратов. Метод исследования — компьютерное моделирование с использованием
аппроксимационной модели нестационарной суперкаверны, основанной на прин-
ципе независимости расширения сечений каверны Г. В. Логвиновича. Разработан
метод нахождения равновесных значений параметров движения частично кавити-
рующего аппарата. Приведены примеры компьютерного моделирования движения
частично кавитирущего аппарата, в частности при его разгоне и торможении. По-
казано, что продольное движение сбалансированного частично кавитирующего ап-
парата статически неустойчиво по глубине. При этом потеря устойчивости движе-
ния, в отличие от случая суперкавитации, происходит “жестким” неколебательным
образом. Предложены способы стабилизации движения частично кавитирующего
аппарата. Исследовано влияние вентиляции каверны на динамику кавитирующего
аппарата.
КЛЮЧЕВЫЕ СЛОВА: частично кавитирующий аппарат, вентилируемая каверна,
устойчивость, компьютерное моделирование
84
INTRODUCTION
MATHEMATICAL MODEL OF 3D MOTION OF A PARTIALLY CAVITATING VEHICLE
COMPUTER SIMULATION OF THE CAVITATING VEHICLE DYNAMICS
BALANCING A PARTIALLY CAVITATING VEHICLE
LONGITUDINAL MOTION OF A PARTIALLY CAVITATING VEHICLE
CAVITATING VEHICLE DYNAMICS DURING SPEEDING-UP
CAVITATING VEHICLE DYNAMICS DURING DECELERATION
EFFECT OF CAVITY VENTILATION ON THE CAVITATING VEHICLE DYNAMICS
CONCLUSIONS
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