Shapes of steady slender axisymmetric ventilated cavities in ponderable liquid
The shapes of the slender steady axisymmetric ventilated cavities are calculated for up- and downward directions of water flow at different values of the Froude number and radii of the cylindrical hulls located inside the cavity. The ventilation is shown to increase the dimensions of the cavities pr...
Gespeichert in:
| Veröffentlicht in: | Гідродинаміка і акустика |
|---|---|
| Datum: | 2018 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут гідромеханіки НАН України
2018
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/174290 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Shapes of steady slender axisymmetric ventilated cavities in ponderable liquid / I.G. Nesteruk, B.D. Shepetyuk // Гідродинаміка і акустика. — 2018. — Т. 1, № 2. — С. 233-244. — Бібліогр.: 11 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-174290 |
|---|---|
| record_format |
dspace |
| spelling |
Nesteruk, I.G. Shepetyuk, B.D. 2021-01-11T16:18:26Z 2021-01-11T16:18:26Z 2018 Shapes of steady slender axisymmetric ventilated cavities in ponderable liquid / I.G. Nesteruk, B.D. Shepetyuk // Гідродинаміка і акустика. — 2018. — Т. 1, № 2. — С. 233-244. — Бібліогр.: 11 назв. — англ. 2616-6135 DOI: doi.org/10.15407/jha2018.02.233 https://nasplib.isofts.kiev.ua/handle/123456789/174290 532.528 The shapes of the slender steady axisymmetric ventilated cavities are calculated for up- and downward directions of water flow at different values of the Froude number and radii of the cylindrical hulls located inside the cavity. The ventilation is shown to increase the dimensions of the cavities produced after the conical cavitator and to decrease the length of base cavities. If the direction of water flow at infinity is opposite to that of the gravity, the injection rate cannot exceed some critical value for conical cavitators and cannot be lower than some critical value for base cavities. Рассчитаны формы тонких стационарных осесимметричных вентилируемых каверн для восходящего и нисходящего потоков воды при различных значениях исла Фруда и радиусов расположенных в каверне цилиндрических корпусов. Показано, что вентиляция увеличивает размеры каверн за коническим кавитатором и уменьшает длину донных каверн. Если направление потока воды на бесконечности противоположно силе тяжести, то интенсивность поддува не может превышать некоторое критическое значение для конических кавитаторов, а также не может быть меньше некоторого критического значения для донных каверн. Розрахованi форми тонких усталених осесиметричних вентильованих каверн для висхiдного та низхiдного потокiв води при рiзних значеннях числа Фруда i радiусiв розташованих у кавернi цилiндричних корпусiв. Показано, що вентиляцiя збiльшує розмiри каверн за конiчним кавiтатором i зменшує довжину донних каверн. Якщо напрямок потоку води на нескiнченностi протилежний до сили тяжiння, то iнтенсивнiсть пiддуву не може перевищувати деяке критичне значення для конiчних кавiтаторiв, а також не може бути меншою за деяке значення для донних каверн. en Інститут гідромеханіки НАН України Гідродинаміка і акустика Shapes of steady slender axisymmetric ventilated cavities in ponderable liquid Формы устойчивых тонких осесимметричных вентилируемых каверн в весомой жидкости Форми стійких тонких осесиметричних вентильованих каверн у важкій рідині Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Shapes of steady slender axisymmetric ventilated cavities in ponderable liquid |
| spellingShingle |
Shapes of steady slender axisymmetric ventilated cavities in ponderable liquid Nesteruk, I.G. Shepetyuk, B.D. |
| title_short |
Shapes of steady slender axisymmetric ventilated cavities in ponderable liquid |
| title_full |
Shapes of steady slender axisymmetric ventilated cavities in ponderable liquid |
| title_fullStr |
Shapes of steady slender axisymmetric ventilated cavities in ponderable liquid |
| title_full_unstemmed |
Shapes of steady slender axisymmetric ventilated cavities in ponderable liquid |
| title_sort |
shapes of steady slender axisymmetric ventilated cavities in ponderable liquid |
| author |
Nesteruk, I.G. Shepetyuk, B.D. |
| author_facet |
Nesteruk, I.G. Shepetyuk, B.D. |
| publishDate |
2018 |
| language |
English |
| container_title |
Гідродинаміка і акустика |
| publisher |
Інститут гідромеханіки НАН України |
| format |
Article |
| title_alt |
Формы устойчивых тонких осесимметричных вентилируемых каверн в весомой жидкости Форми стійких тонких осесиметричних вентильованих каверн у важкій рідині |
| description |
The shapes of the slender steady axisymmetric ventilated cavities are calculated for up- and downward directions of water flow at different values of the Froude number and radii of the cylindrical hulls located inside the cavity. The ventilation is shown to increase the dimensions of the cavities produced after the conical cavitator and to decrease the length of base cavities. If the direction of water flow at infinity is opposite to that of the gravity, the injection rate cannot exceed some critical value for conical cavitators and cannot be lower than some critical value for base cavities.
Рассчитаны формы тонких стационарных осесимметричных вентилируемых каверн для восходящего и нисходящего потоков воды при различных значениях исла Фруда и радиусов расположенных в каверне цилиндрических корпусов. Показано, что вентиляция увеличивает размеры каверн за коническим кавитатором и уменьшает длину донных каверн. Если направление потока воды на бесконечности противоположно силе тяжести, то интенсивность поддува не может превышать некоторое критическое значение для конических кавитаторов, а также не может быть меньше некоторого критического значения для донных каверн.
Розрахованi форми тонких усталених осесиметричних вентильованих каверн для висхiдного та низхiдного потокiв води при рiзних значеннях числа Фруда i радiусiв розташованих у кавернi цилiндричних корпусiв. Показано, що вентиляцiя збiльшує розмiри каверн за конiчним кавiтатором i зменшує довжину донних каверн. Якщо напрямок потоку води на нескiнченностi протилежний до сили тяжiння, то iнтенсивнiсть пiддуву не може перевищувати деяке критичне значення для конiчних кавiтаторiв, а також не може бути меншою за деяке значення для донних каверн.
|
| issn |
2616-6135 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/174290 |
| citation_txt |
Shapes of steady slender axisymmetric ventilated cavities in ponderable liquid / I.G. Nesteruk, B.D. Shepetyuk // Гідродинаміка і акустика. — 2018. — Т. 1, № 2. — С. 233-244. — Бібліогр.: 11 назв. — англ. |
| work_keys_str_mv |
AT nesterukig shapesofsteadyslenderaxisymmetricventilatedcavitiesinponderableliquid AT shepetyukbd shapesofsteadyslenderaxisymmetricventilatedcavitiesinponderableliquid AT nesterukig formyustoičivyhtonkihosesimmetričnyhventiliruemyhkavernvvesomoižidkosti AT shepetyukbd formyustoičivyhtonkihosesimmetričnyhventiliruemyhkavernvvesomoižidkosti AT nesterukig formistíikihtonkihosesimetričnihventilʹovanihkavernuvažkíirídiní AT shepetyukbd formistíikihtonkihosesimetričnihventilʹovanihkavernuvažkíirídiní |
| first_indexed |
2025-11-27T00:48:35Z |
| last_indexed |
2025-11-27T00:48:35Z |
| _version_ |
1850789402968588288 |
| fulltext |
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 233233–244244.
UDC 532.528
SHAPES OF STEADY SLENDER AXISYMMETRIC
VENTILATED CAVITIES IN PONDERABLE LIQUID
I. G. Nesteruk1†, B. D. Shepetyuk2‡
1Institute of Hydromechanics of NAS of Ukraine
Zhelyabov Str., 8/4, 03057, Kyiv, Ukraine
†E-mail: inesteruk@yahoo.com
2Yuriy Fedkovych Chernivtsi National Univesity, Chernivtsi
vul. Kotsyubynskyi Str., 2, 58012, Chernivtsi, Ukraine
‡E-mail: shepetyukb@gmail.com
Received 17.12.2016
The shapes of the slender steady axisymmetric ventilated cavities are calculated for
up- and downward directions of water flow at different values of the Froude number
and radii of the cylindrical hulls located inside the cavity. The ventilation is shown
to increase the dimensions of the cavities produced after the conical cavitator and to
decrease the length of base cavities. If the direction of water flow at infinity is opposite
to that of the gravity, the injection rate cannot exceed some critical value for conical
cavitators and cannot be lower than some critical value for base cavities.
KEY WORDS: supercavitation, ventilated cavities, slender body theory
1. INTRODUCTION
The drag of the high-speed underwater vehicles can be reduced by decreasing the area
wetted with water, i.e., by the use of supercavitation [11–33]. To obtain small cavitation
numbers at small velocities or at large depths, a gas ventilation inside the cavity is used
(see, e.g., [44]). The ventilation is also very important in experiments, since the velocities in
water tunnels are usually much smaller than in the case of real vehicles. Limited velocities
of water tunnels increase the influence of gravity on the cavity shape and dimensions.
Theoretical and numerical investigations of ventilated cavities are very limited. Even in
the 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. If an injected gas flows in a
narrow channel between the cavity surface and the vehicle hull, the pressure on the cavity
surface is no longer constant and changes the cavity shape in comparison with the case of
vapor cavitation. This complicated phenomenon was investigated numerically with the use
of viscous fluid equations [55]. The ideal fluid approach and the slender body theory allow one
to obtain simple equations for the shape of axisymmetric ventilated supercavity provided the
233
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 233233–244244.
gas flow between the cavity surface and the body of revolution is one-dimensional inviscid
and incompressible. Some interesting results were obtained in [66–88] for a steady flow of liquid
without gravity effects.
In [99] the results of these papers are generalized for unsteady vertical flows in the gravity
field. In particular, the first approximation equation for the radius 𝑅(𝑥) of steady axisym-
metric ventilated cavity was proposed
𝑑2𝑅2
𝑑𝑥2
=
𝜎0
ln 𝜖
+
2𝑘𝑥
Fr2 ln 𝜖
+ ∆
[︂
𝑎− 1
(𝑅2 −𝑅2
𝑏)
2
]︂
, (1)
where all lengths are dimensionless (refered to the cavity radius at its origin 𝑅0), 𝑘 = 1
corresponds to the case, when the directions of the water flow at infinity and the gravitational
acceleration coincide; 𝑘 = −1 corresponds to the case, when the directions of these vectors
are opposite (in order to have an axisymmetric cavity, the direction of the gravity force is
limited by these two cases).
The constant parameters 𝜎0, Fr, ∆ and 𝑎 are given by formulas:
𝜎0 =
2(𝑝∞ − 𝑝𝑣 − 𝑝0)
𝜌𝑈2
, Fr =
𝑈√
𝑔𝑅0
,
∆ = − 𝜌𝑔𝑄
2
𝜋2𝑅4
0𝜌𝑈
2 ln 𝜖
, 𝑎 =
[︂
1 − 𝑅2
𝑏0
𝑅2
0
]︂−2
,
where 𝜌 is the water density; 𝑈 is the constant velocity of the water flow at infinity; 𝑝𝑣 is the
water vapor pressure at ambient temperature; 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 respectively;
𝜌𝑔 is the constant density of injected gas; 𝑄 is the volumetric gas flow rate; 𝑅𝑏, 𝑅𝑏0 are the
radii of the hull at points 𝑥 and 𝑥 = 0; 𝜖 is a small parameter, the ratio of the maximum
diameter of the system cavity-cavitator to its length.
In this papper we shall concentrate on the numerical solutions of equation (1)(1) at different
values of the Froude number Fr and parameter 𝑘. We shall calculate shapes and dimensions
of ventilated cavities and analyse critical values of the injection rate.
2. NUMERICAL PROCEDURE AND EXAMPLES OF THE VENTILATED
CAVITY SHAPE CALCULATIONS
To integrate the differential equation (1)(1), the standard initial conditions at the cavity
origine 𝑥 = 0
𝑅 = 1,
𝑑𝑅
𝑑𝑥
= 𝛽
and the 4-th order Runge—Kutta method were used.
The calculations showed that for coinciding directions of the gravity and the flow at
infinity (𝑘 = 1), the cavity dimensions are limited at any gas injection rate. The situation is
similar to the case of natural cavitation, when the solution of equation (1)(1) has the following
form (see [1010]):
𝑅2(𝑥) =
𝜎0𝑥
2
2 ln 𝜖
+
𝑘𝑥3
3Fr2 ln 𝜖
+ 2𝛽𝑥 + 1. (2)
234
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 233233–244244.
Fig. 1. Shapes of the ventilated cavities created by slender conical cavitator (𝛽 = 0.1)
in the water flow directed upwards (𝑘 = −1, Fr = 20) for different values of the gas flow rate.
The hull is cylindrical, 𝑅𝑏0 = 0.9, 𝜎0 = 0.1
Since the value ln 𝜖 is negative, the polynomial (2)(2) can attain zero values at large enough
values of 𝑥 at any Froude number. Thus, the cavity length and diameter are always limited.
Another behavior occurs at opposite directions of the gravity and velocity of the ambient
flow (𝑘 = −1). At some critical value of the Froude number the graph of the polynomial (2)(2)
touches the 𝑥 axes. Increasing the Froude number yields unlimited cavities, which cannot
be realized according to the stability principle [1111]. Corresponding critical Froude numbers
were calculated in [1010] for natural cavities and 𝑅𝑏0 = 0. The results of the ventilated cavity
shapes calculations are presented in Figs. 1Figs. 1 to 33 for 𝑘 = −1, different cavitators (𝛽 = 0.1, 0,
−0.1) and injection rates.
For the slender conical cavitator 𝛽 > 0, the dimensions of ventilated cavities increase with
increasing the ventilation rate (see Fig. 1Fig. 1). At some critical value of ventilation parameter
(∆ ≈ 0.0003925) the cavity touches the cylindrical hull and becomes infinite. Since a small
change in the flow parameters must cause small changes in the flow (according to the stability
principle, see, e.g., [1111]), the flow rate cannot exceed this critical value. An unrealistic infinite
cavity is shown by the dashed line. Similar limited values of the ventilation rate occur on
cylindrical hulls in liquid without gravity [66].
The length of the base cavities (𝛽 ≤ 0) decreases with increasing the ventilation rate
(see Figs. 2Figs. 2 and 33). At some critical value of ventilation parameter the cavity touches the
cylindrical hull and becomes infinite. Since a small change in the flow parameters must cause
small changes in the flow (according to the stability principle, see, e.g., [1111]), the flow rate
cannot be smaller than this critical value. Unrealistic infinite cavities are shown in Figs. 2Figs. 2
and 33 by dashed lines. Similar limited values of the ventilation rate occur on cylindrical hulls
in liquid without gravity [77].
235
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 233233–244244.
Fig. 2. Shapes of the base ventilated cavities, 𝛽 = 0 in the water flow
directed upwards (𝑘 = −1, Fr = 8.6) for different values of the gas flow rate.
The hull is cylindrical, 𝑅𝑏0 = 0.5, 𝜎0 = 0.05
Fig. 3. Shapes of the base ventilated cavities, 𝛽 = −0.1 in the water flow
directed upwards (𝑘 = −1, Fr = 1.3) for different values of the gas flow rate.
The hull is cylindrical, 𝑅𝑏0 = 0.5, 𝜎0 = 0.1
236
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 233233–244244.
3. DIMENSIONS OF THE VENTILATED CAVITIES, CREATED BYA SLEN-
DER CONICAL CAVITATOR
In the case of slender conical cavitator with 𝛽 = 0.1, the maximal cavity radius and
the length of the ventilated cavities versus injection rate are presented in Figs. 4Figs. 4 to 77. The
dimensions of the ventilated cavity increase with the increasing of the ventilation rate at
fixed values of the cavitation and Froude numbers and the radius of the cylindrical hull.
Figs. 4Figs. 4 and 55 show that the cavity dimensions decrease with the increasing of the Froude
number in the water flow directed upwards (𝑘 = −1) at fixed values of the cavitation number,
flow rate and 𝑅𝑏0. This behavior is typical for zero gas injection rates too (see equation (2)(2)).
With increasing ∆ the influence of the Froude number increases. The critical values of the
injection rates and corresponding cavity dimensions (see the tops of the curves shown in
Figs. 4Figs. 4 and 55) also increase with the increasing Froude number. The increase of the radius
of the cylindrical hull (parameter 𝑅𝑏0) drastically diminishes the critical values of ∆.
Figs. 6Figs. 6 and 77 show that the cavity dimensions increase with the increasing of the Froude
number in the water flow directed downwards (𝑘 = 1) at fixed values of the cavitation
number, flow rate and 𝑅𝑏0. This behavior is typical for zero gas injection rates too (see
equation (2)(2)). The dimensions of the ventilated cavity increase with the increasing of the
ventilation rate but remain limited. The increase of the radius of the cylindrical hull (pa-
rameter 𝑅𝑏0) drastically increases the cavity dimensions.
4. LENGTH OF THE BASE VENTILATED CAVITIES
In the case of the base cavities (𝛽 ≤ 0) on the infinite cylindrical hull, the maximum
radius of the cavity coincide with the radius of the cavitator in the section of the cavity
origin 𝑥 = 0, i.e., 𝑅𝑚𝑎𝑥 = 1. The lengths of the ventilated cavities versus injection rate are
presented in Figs. 8Figs. 8 and 99 for 𝛽 = 0 and Fig. 10Fig. 10 and 1111 for 𝛽 = −0.1. It can be seen that the
length diminishes with the increase of the ventilation rate for both directions of the water
flow (𝑘 = 1 and 𝑘 = −1). The increase of the radius of the cylindrical hull (parameter 𝑅𝑏0)
drastically diminishes the length of the cavity.
Figs. 8Figs. 8 and 1010 show that the cavity dimensions decrease with the increasing of the Froude
number in the water flow directed upwards (𝑘 = −1) at fixed values of the cavitation number,
flow rate and 𝑅𝑏0. This behavior is typical for zero gas injection rates too (see equation (2)(2))
and for the case of conical cavitator (see Fig. 5Fig. 5). At zero values of the gas flow rate, the
cavities shown in Figs. 8Figs. 8 and 1010 are closed. It means that there are no limitations for ∆ for
the parameters used to calculate these plots.
The Froude numbers of 8.6 and 1.3 used in Figs. 2Figs. 2 and 33 respectively were too small
to have closed vapor cavities. In these cases only ventilated cavities can be realized at the
ventilation rates grater than critical ones.
Figs. 9Figs. 9 and 1111 show that the cavity dimensions increase with the increasing of the Froude
number in the water flow directed downwards (𝑘 = 1) at fixed values of the cavitation
number, flow rate and 𝑅𝑏0. This behavior is typical for zero gas injection rates too (see
equation (2)(2)) and for the case of conical cavitator (see Fig. 7Fig. 7). At zero values of the gas flow
rate the cavities are always closed in the water flow directed downwards (𝑘 = 1). It means
that there are no limitations for the parameter ∆.
237
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 233233–244244.
Fig. 4. Maximum radius of the ventilated cavity, created by slender conical cavitator
with 𝛽 = 0.1 in the water flow directed upwards (𝑘 = −1), versus injection rate
at different Froude numbers and 𝑅𝑏0 (𝜎0 = 0.1)
Fig. 5. Length of the ventilated cavity, created by slender conical cavitator
with 𝛽 = 0.1 in the water flow directed upwards (𝑘 = −1), versus injection rate
at different Froude numbers and 𝑅𝑏0 (𝜎0 = 0.1)
238
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 233233–244244.
Fig. 6. Maximum radius of the ventilated cavity, created by slender conical cavitator
with 𝛽 = 0.1 in the water flow directed downwards (𝑘 = 1), versus injection rate
at different Froude numbers and 𝑅𝑏0 (𝜎0 = 0.05)
Fig. 7. Length of the ventilated cavity, created by slender conical cavitator
with 𝛽 = 0.1 in the water flow directed downwards (𝑘 = 1), versus injection rate
at different Froude numbers and 𝑅𝑏0 (𝜎0 = 0.05)
239
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 233233–244244.
Fig. 8. Length of the base ventilated cavity for 𝛽 = 0
in the water flow directed upwards (𝑘 = −1), versus injection rate
at different Froude numbers and 𝑅𝑏0 (𝜎0 = 0.05)
Fig. 9. Length of the base ventilated cavity for 𝛽 = 0
in the water flow directed downwards (𝑘 = 1), versus injection rate
at different Froude numbers and 𝑅𝑏0 (𝜎0 = 0.05)
240
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 233233–244244.
Fig. 10. Length of the base ventilated cavity for 𝛽 = −0.1
in the water flow directed upwards (𝑘 = −1), versus injection rate
at different Froude numbers and 𝑅𝑏0 (𝜎0 = 0.05)
Fig. 11. Length of the base ventilated cavity for 𝛽 = −0.1
in the water flow directed downwards (𝑘 = 1), versus injection rate
at different Froude numbers and 𝑅𝑏0 (𝜎0 = 0.05)
241
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 233233–244244.
5. CONCLUSIONS
With the use of known differential equation of the first approximation, the shapes of
the slender steady axisymmetric ventilated cavities were calculated for up- and downwards
directions of the water flow for different values of the Froude number and the radius of
the cylindrical hulls located inside the cavity. It was shown that ventilation increases the
dimensions of the cavities created by conical cavitator and decreases the length of the base
cavities. When the direction of the water flow at infinity is opposite to the direction of the
gravity, the injection rate cannot exceed some critical value for conical cavitators and cannot
be lower than some critical value for the base cavities.
Since the very simple one-dimensional model was used for the gas flow in the circular
channel between the cavity surface and the hull, the results must be verified by experiments,
applying CFD methods and using the viscous models for gas and water flows.
REFERENCES
[1] Логвинович Г. В. Гидродинамика течений со свободными границами. — Киев : На-
укова думка, 1969. — С. 215.
[2] Savchenko Y. N. Perspectives of the supercavitation flow applications // International
Conference on Superfast Marine Vehicles Moving Above, Under and in Water Surface
(SuperFAST’2008). –– St. Petersburg, Russia, 2008.
[3] Nesteruk I. G. Drag drop on high-speed supercavitating vehicles and supersonic submari-
nes // Прикладна гiдромеханiка. — 2015. — Т. 17(89), № 4. — С. 52–57.
[4] Vlasenko Y. D., Savchenko G. Y. Study of the parameters of a ventilated supercavity
closed on a cylindrical body // Supercavitation / Ed. by I. Nesteruk. –– Springer, 2012. ––
P. 201–214.
[5] Zhuravlev Y. F., Varyukhin A. V. Numerical simulation of interaction gas jets flowing
into water cavity with its free surfaces simulation // International Conference on Super-
fast Marine Vehicles Moving Above, Under and in Water Surface (SuperFAST’2008). ––
St. Petersburg, Russia, 2008.
[6] Манова З. I., Нестерук I. Г., Шепетюк Б. Д. Оцiнки впливу iнтенсивної вентиляцiї
на форму тонких осесиметричних каверн // Прикладна гiдромеханiка. — 2011. — Т.
13(85), № 2. — С. 44–50.
[7] Нестерук I. Г., Шепетюк Б. Д. Особливостi форми донних штучних осесиметричних
каверн // Прикладна гiдромеханiка. — 2011. — Т. 13(85), № 3. — С. 69–75.
[8] Нестерук I. Г., Шепетюк Б. Д. Форма штучних осесиметричних каверн при до-
та надкритичних значеннях iнтенсивностi пiддуву // Прикладна гiдромеханiка. —
2012. — Т. 14(86), № 2. — С. 53–60.
[9] Nesteruk I. Shape of slender axisymmetric ventilated supercavities // Journal of Com-
putational Engineering. –– 2014. –– Vol. 2014. –– P. 501590(1–18).
242
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 233233–244244.
[10] Nesteruk I. On the shape of a slender axisymmetric cavity in a ponderable liquid //
Fluid Dynamics. –– 1979. –– Vol. 14, no. 6. –– P. 923–927.
[11] Нестерук И. Г. Об ограничениях на параметры кавитационных течений // При-
кладная математика и механика. — 1986. — Т. 50, № 4. — С. 584–588.
REFERENCES
[1] G. V. Logvinovich, Hydrodynamics of flows with free boundaries. Halsted Press, 1973.
[2] Y. N. Savchenko, “Perspectives of the supercavitation flow applications,” in Interna-
tional Conference on Superfast Marine Vehicles Moving Above, Under and in Water
Surface (SuperFAST’2008), (St. Petersburg, Russia), 2008.
[3] I. G. Nesteruk, “Drag drop on high-speed supercavitating vehicles and supersonic sub-
marines,” Applied Hydromechnics, vol. 17(89), no. 4, pp. 52–57, 2015.
[4] Y. D. Vlasenko and G. Y. Savchenko, “Study of the parameters of a ventilated super-
cavity closed on a cylindrical body,” in Supercavitation (I. Nesteruk, ed.), pp. 201–214,
Springer, 2012.
[5] Y. F. Zhuravlev and A. V. Varyukhin, “Numerical simulation of interaction gas jets
flowing into water cavity with its free surfaces simulation,” in International Confer-
ence on Superfast Marine Vehicles Moving Above, Under and in Water Surface (Super-
FAST’2008), (St. Petersburg, Russia), 2008.
[6] Z. I. Manova, I. G. Nesteruk, and B. D. Shepetyuk, “Estimations of intensive venti-
lation influence on the slender axisymmetric cavity shape,” Applied Hydromechnics,
vol. 13(85), no. 2, pp. 44–50, 2011.
[7] I. G. Nesteruk and B. D. Shepetyuk, “Shape peculiarities of the base artificial axisym-
metric cavities,” Applied Hydromechnics, vol. 13(85), no. 3, pp. 69–75, 2011.
[8] I. G. Nesteruk and B. D. Shepetyuk, “Shape of artificial axisymmetric cavities at sub-
and supercritical values of the ventilation rate,” Applied Hydromechnics, vol. 14(86),
no. 2, pp. 53–60, 2012.
[9] I. Nesteruk, “Shape of slender axisymmetric ventilated supercavities,” Journal of Com-
putational Engineering, vol. 2014, pp. 501590(1–18), 2014.
[10] I. Nesteruk, “On the shape of a slender axisymmetric cavity in a ponderable liquid,”
Fluid Dynamics, vol. 14, no. 6, pp. 923–927, 1979.
[11] I. Nesteruk, “The restrictions of the parameters of cavitational flow,” PMM Journal of
Applied Mathematics and Mechanics, vol. 50, no. 4, pp. 446–449, 1986.
243
ISSN 2616-6135. ГIДРОДИНАМIКА I АКУСТИКА. 2018. Том 1(91), № 2. С. 233233–244244.
I. Г. Нестерук, Б. Д. Шепетюк
Форми стiйких тонких осесиметричних вентильованих каверн у важкiй
рiдинi
Розрахованi форми тонких усталених осесиметричних вентильованих каверн для
висхiдного та низхiдного потокiв води при рiзних значеннях числа Фруда i радiусiв
розташованих у кавернi цилiндричних корпусiв. Показано, що вентиляцiя збiльшує
розмiри каверн за конiчним кавiтатором i зменшує довжину донних каверн. Якщо
напрямок потоку води на нескiнченностi протилежний до сили тяжiння, то iнтен-
сивнiсть пiддуву не може перевищувати деяке критичне значення для конiчних
кавiтаторiв, а також не може бути меншою за деяке значення для донних каверн.
КЛЮЧОВI СЛОВА: суперкавiтацiя, вентильованi каверни, теорiя тонкого тiла
И. Г. Нестерук, Б. Д. Шепетюк
Формы устойчивых тонких осесимметричных вентилируемых каверн в
весомой жидкости
Рассчитаны формы тонких стационарных осесимметричных вентилируемых ка-
верн для восходящего и нисходящего потоков воды при различных значениях чис-
ла Фруда и радиусов расположенных в каверне цилиндрических корпусов. Пока-
зано, что вентиляция увеличивает размеры каверн за коническим кавитатором и
уменьшает длину донных каверн. Если направление потока воды на бесконечно-
сти противоположно силе тяжести, то интенсивность поддува не может превышать
некоторое критическое значение для конических кавитаторов, а также не может
быть меньше некоторого критического значения для донных каверн.
КЛЮЧЕВЫЕ СЛОВА: суперкавитация, вентилируемые каверны, теория тонкого
тела
244
INTRODUCTION
NUMERICAL PROCEDURE AND EXAMPLES OF THE VENTILATED CAVITY SHAPE CALCULATIONS
DIMENSIONS OF THE VENTILATED CAVITIES, CREATED BY A SLENDER CONICAL CAVITATOR
LENGTH OF THE BASE VENTILATED CAVITIES
CONCLUSIONS
|