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2012,24(6):951-958 DOI: 10.1016/S1001-6058(11)60323-5
NUMERICAL STUDY OF THE PITCHING MOTIONS OF SUPERCAVITA- TING VEHICLES*
YU Kai-ping, ZHANG Guang , ZHOU Jing-jun, ZOU Wang, LI Zhen-wang School of Aeronautics, Harbin Institute of Technology, Harbin 150001, China, E-mail: [email protected]
(Received March 14, 2012, Revised September 7, 2012)
Abstract: The pitching motions of supercavitating vehicles could not be avoided due to the lost water buoyancy. In order to have some insight for the design of the supercavitating vehicles, the fixed frequency and free pitching motions are investigated. A numerical predicting method based on the relative motion principle and the non-inertia coordinate system is proposed to simulate the free pitching motions of supercavitating vehicles in the longitudinal plane. Homogeneous and two fluid multiphase models are used to predict the natural and the ventilated supercavitating flows. In the fixed frequency pitching motions, a variety of working conditions are considered, including the pitching angular velocities and the supercavity scales and the results are found to beconsistent with the available experimental results in literature. The mesh deformation technology controlled by the moment of momentum equation is adopted to study the free pitching motions and finally to obtain the planing states proposed by Savchenko.The numerical method is validated for predicting the pitching motions of supercavitating vehicles and is found to enjoy better calculation efficiency as comparing with the mesh regeneration technology.
Key words: supercavitating flow, two fluid multiphase flow model, homogeneous model, pitching motion, mesh deformation
Introduction�
The natural supercavity is achieved when an underwater vehicle travels at a sufficient high speed, so that the fluid pressure drops to the saturated vapor pressure. When the vehicle is enveloped in the cavity, the viscous drag drops drastically thus the vehicle might move as in the air. Even for vehicles designed to travel at the natural supercavitating velocity, the drag must be firstly reduced by the ventilated super- cavitating to enable the vehicles to accelerate to the conditions at which the natural supercavity can be sus- tained. It is well proven that the ventilated cavitation can achieve a significant drag reduction and, therefore it has attracted a growing research attentions, especia- lly, by means of the CFD.
While the ventilated supercavity can dramatically reduce the drag of the vehicle, the lost water buoyancy due to the cavity enclosure makes the hydrodynamics quite different from the whole wet state. Besides the
* Project support by the Major National Natural Science Founation of China (Grant No. 10832007). Biography: YU Kai-ping (1968-), Male, Ph. D., Professor
lift on the cavitator and the fins, another lift from the interaction between the vehicle and the cavity boun- dary can not generally be avoided[1-3] due to the mo- ment balance, and the wet area is the main source of the stable restoring force[4,5]. Thus, a study of the res- toring force of the aft body and the further pitching motion of the vehicle is of great significance to the de- sign of supercavitating vehicles.
In the published literature, only a few papers con- cern with the force due to the interaction between the aft section of the vehicle and the cavity boundary and most papers deal with the trajectory control based on the Hassan’s theory that describes the forces and the moments experienced by a cylindrical body steadily planing on flat and cylindrical free surfaces[6]. No papers are found to study the pitching motions with CFD predictions. In papers[7,8], the interaction of the aft body of the vehicle with the cavity wall pitches in the supercavity was discussed based on experiments in water tunnel. In papers[9,10], a series of experiments and numerical simulations were conducted to investi- gate the hydrodynamic drag, lift and moment chara- cteristics at the aft section of underwater supercavita- ting bodies for different supercavity shapes. In papers[11,12] Kulkarn and Kirschner et al. studied the
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impact dynamics of a supercavitating projectile with its nose being prescribed to move in a straight line. Their models predicted that the motion of the proje- ctile consists of a series of impacts with the frequency increasing as the projectile slows down.
In this paper, a numerical method based on the relative motion principle and the non-inertia coordi- nate system is proposed to study the fixed frequency and free pitching motions of supercavitating vehicles. The method is validated by comparing the calculated results with those in papers[7,8]. At the same time, the method is used to successfully predict the planing state proposed by Savchenko. Therefore, the method can be used in the design of supercavitating vehicles.
1. Numerical methods
1.1 Basic governing equations of two-fluid model In the multiphase system considered in this paper,
the fluid is assumed to be isothermal and the densities of the fluid are functions of the pressure but not the temperature. Under this assumption, only the conti- nuity and the momentum equations are used, but not the energy equations.
To simulate the ventilated cavitation flows is to solve the standard 3-D Navier Stokes equations, the turbulence equations, the gas state equation and the volume fraction equation.
The continuity equation for the single phase is
( ) + ( ) = 0Vt
� �� � �
� � � ���
�, = 1,2� (1)
The momentum equation for the single phase is
( )+ [ ( )] =
V V Vt
� � �� � � �
� �� �
�� �
�
T+ ( ) +V V p� � � � � �� � �� � � � � �� �
+ , = 1M g� � � ,2� � � (2)
For a ventilation cavitating flow, � represents the gas phase or the liquid phase, for a n ural cavita- ting flow
at� represents the vapor phase or the liquid
phase. The volume fraction equation is
(3)
The gas state equation is
=1= 1
N
��
��
( , )P T� �� (4)
where � is the density of the air, P is the local pressur T is the temperature which keeps constant in the sim lations. Formulas (1)-(4) are the whole governing equations.
e,u
.2 Cavitation model Zwart-Gerber-Belamri model is
1In this paper, the
used to predict the natural supercavitation.
3 2=3
vv vc
B l
p pm F
R� �
�� �
� ,
+ 3 (1 ) 2=3
vnuc v ve
B
p prm FR
� ����
�l
(5)
here subscripts , respectively represent the w va
ls,vapor and liquid ph se nucr is the volume fraction
at the nucleation sites, BR represents the radius of the nucleation sites, whil F is an empirical factor which may differ for conde ation and vaporization, p is the pressure in the liquid surrounding the bubble,
d vp is the saturated vapor pressure with the value of 3 170 Pa in this paper.
e ns
an
1.3 Turbulence model due consideration of the com-
vantage of the standard
In this paper, withputational expense, a steady computation with the Shear Stress Transport (SST) turbulence model is carried out to give the initial data for the unsteady cal- culations in which the Detached Eddy Simulation (DES) turbulence model is utilized. A simple introdu- ction of the two turbulence models is given below.
.3.1 SST turbulence model 1The predominant disad
k �� turbulence model is that the model is designed gh Reynolds number flows thus it performs
poorly for low Reynolds number cases and in the near wall regions. The Wilcox k
for hi
�� turbulence model was firstly introduced by Kolm rov, some improve- ments were made with the most popular suggestions made by Wilcox, who formulated a low Reynolds number alternative to the standard k
ogo
�� turbulence model.
The kdeficiency of the �� turbulence model stems from the sensitivity of model to the � . One possible solution is to use a combination of the k ��model equations implemented near wall region the k
s and�� model to be employed in the bulk flow re-
gion, w ch leads Menter to formulate the SST turbu- lence model. The details can be found in the paper[13].
.3.2 DES turbulence model
hi
In order to improve the predictive capabilities of 1
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turbulence models in highly separated regions, Spalar prop
rical resolutions The simulations are based on 3-D calculations
zation of these equations is used
osed a hybrid approach, which combines features of classical RANS formulations with elements of Large Eddy Simulations (LES) methods. The concept is termed the DES and it is based on the idea of cove- ring the boundary layer by a RANS model and swi- tching the model to a LES mode in detached regions. Ideally, the DES can predict the separation line from the underlying RANS model, but which means to cap- ture the unsteady dynamics of the separated shear layer with the resolution of the developing turbulent structures. Compared to the classical LES methods, the DES saves the computing resources by orders of magnitude for high Reynolds number flows. Though this is due to the moderate costs of the RANS model in the boundary layer region, the DES still offers some of the advantages of an LES method in separated re- gions.
1.4 Nume
and a finite volume discreti. A solver of the coupled conservation equations
of mass and momentum is adopted, with an implicit time scheme and the multigrid technology.
The transient term is treated with a second-order implicit scheme.
+1( ) 1.5 n�� �� +1 1 12 + 0.5=n n n n n
t t� � � � �� ��
� �(6)
where stands for , or , the three omponents.
e s re uler. The advection schemes can be cast
in th
(7)
where is the v
� u , v w velocity c
Th diffusive term a calc ated in a central di- fference mann
e form
= +ip up� � � �� �r
up�
chem
alue at the upwind node, r is the vector from the upwind node to ip and a hi reso- lution s e is adopted which uses the a spe al non- linear recipe
ghci
� at each node, be g computed to be as close to 1 as possible without introducing new ex- trema. The recipe
in
� is based on the boundedness principles.
2. Computational models and grids The velocity components, the volume fractions,
scale are speci- fied
ig.
ig.2 Two d
ig.3 T
out fins are applied in the imulations as shown in Fig.2. The computational
grids
ers
)Axial�radial�
circumferential
the turbulence intensity and the lengthat the velocity inlet boundary and extrapolated to
the pressure outlet or the opening boundaries. The mass inlet boundary is defined at the blowhole. The pressure distribution is specified at the pressure outlet
boundary and extrapolated to the inlet boundaries. At the external boundary, the pressure and the volume fractions are extrapolated and the opening boundary condition is specified. The details are shown in Fig.1.
F 1 The boundary conditions
F ifferent models in simulations
F he computational grids
Models with and withs
used in this paper are shown in Fig.3. Before carrying out the formal computations, different meshes (shown in Table 1) were used to predict the pressure coefficient on the cone and the cylinder sec- tions of the body to test the influence of the grids on the computational results.
Table 1 Different grid paramet
Gridnumber
Grid size of the first layer on the model (m
Grid-1 0.00025 210�48�68
Grid-2 0.00025 225�58�78
Grid-3 0.00025 245�70�88
e 4 show mputationa the ressure coefficient on the cone and the cylinder sec-
tions
sultsThe following calculations mainly include two
Figur s the co l results ofp
for model 1 with three kinds of grids. It can be seen that the differences between grid-2 and grid-3 are small, and with due consideration of the calculation accuracy and expense, the grid-2 is finally used for all subsequent calculations.
. The computational re3
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parts: the fixed frequency pitching motions and the free
i t dis- tributions on the model 1 with different grid
ig.5 The sketch map of measurement point
ig
ionsThe computational conditions in paper are as
the
pitching motions that are controlled by the mo- ment of momentum equation in the longitudinal plane. The fixed frequency pitching motions are calculated preliminarily to determine the influencing factors on the cavity scale and the lift characteristics of the body and the computational results can be used to compare with the experimental results in paper[8] to validate the numerical method.
F g.4 Comparison of predicted surface pressure coefficien
F
F .6 The change law of pressure at measuring point
3.1 The results of fixed frequency pitching mot[8]
follows: The upstream velocity V� is 10.25 m/s, upstream pressure P� is 33 800 Pa, the total ventila- tion rate 2= /QC Q V D is 0.144, and the ventilated cavitation number is 0.0864. The pitching angle follo- ws the la max sin ( )t
�
=w: � � � , where max = ( /� �180) rad is the maximal pitching angle, =10� � The time step in the tation is 0.0
te of all simulations in this pap teady calculated result with the pitching angle = 0 rad
un dystea compu 05 s. The is a sinitial sta er
� andthe vehicle is fully enveloped in the supercavity.
Figure 6 shows the pressure evolutio ea- suring point as shown in Fig.5 during the fixedquen
n at the m-fre-
motions. r velo- city
cy pitching motion. The simulation results show that the pressure on the point varies significantly, which is consistent with the experimental results in paper[8] and shows that the numerical method is valid. In Fig.6 = /t t T� is a dimensionless variable.
Now we consider the influencing factors in the pitching The influences of the angula
on the cavity shape and the hydrodynamics are as follows. The pitching angle law: = [1/�30sin ( )] radt� , = 4 , 6 , 8� � � � , the free stream velocity is increased to 25 m/s, and the a mbientpressure turns to 2.5 atm.
ig.7 of
hen
F The change of the supercavity shape in the processvibration
Figure 7 shows the change of the cavity shape = 4w � � , and the mass flow rate is
0.01the fro
ig.8 angular velocity
3456 kg/s. It can be seen that the supercavity is stable in nt part of the supercavitating vehicle, and large amounts of gas escape from its tail because of the periodical tail-clap of its vehicle.
F The final cavity shape with different pitching = 4 , 6 , 8� � � �
Figure 8 how l cavity shape of model 1 s s the finafor different angular velocities when the pitching
955
angular = 0 rad� after the cavity reaches a relatively stable state, and the three pitching motions have the same initial state. It can be seen that the pitching angular velocity has a great influence on the gas lea- kage. The pitching motions with higher frequencies make the gas leak more quickly.
Fi angular
i itching
variati with different pitching angular eloci
g.9 The lift coefficient with different angular pitching velocity
F g.10 The drag coefficient with different angular pangular velocity
Figure 9 and Fig.10 demonstrate the lift and drag ons of model 1
v ties. It can be seen that the larger the pitching angular velocity, the larger the lift and the drag be- come, which shows that the impact velocity between the aft body and the cavity boundary would affect the lift and the drag. When the angular velocity = 4� � ,the lift and the drag vary regularly from beginning to end, because the aft body hardly affects the leakage of gas and on the other hand, when = 6� � and =�8� , the lift and the drag change significantly in the process of the pitching motion from the initial state, however, finally, the maximal lift and drag come to a close value, which demonstrates that the cavity shape or the closure position of the cavity has a greater in- fluence on the hydrodynamics of the body as compa- ring with that of the pitching angular velocity. On the other hand, the maximal and minimum lift coefficients show an asymmetric feature, that is why the asymme- tric cavity shape affects the closure angle of the cavity on the body. The lift coefficient and the drag coeffi-
cient are defined as
2=0.5
ll
FCv S�
, 2= dd
FC (8)0.5 v S�
3.2 The result f free pitchingIn paper ] suggested four possible modes of
ehicle motions according to the vehicle velocity and which two
mod
ig.
The body reference frame is built on the center of the longitudi-
al plane in the body coordinate are
y
s o motions[14
vthe supercavity development, among
es are demonstrated in what follows. The velocity of the first motion mode is about 70 m/s and that of the other is in the range of 50 m/s-200 m/s, as shown in Fig.11. In this paper, the velocity is different, although the motion modes are the same, as obtained by adjusting the ambient pressure. Two free velocities adopted in this paper are 100 m/s and 110 m/s when the vehicle is enveloped by a complete natural super- cavity. The other two velocities are 25 m/s and 38 m/s.
F 11 Two possible modes of vehicle motion
the cavitator, the dynamics equations inn
2( ) =g xm u rv r x F� �� (9)
m( + + ) =gv ru rx F� � (10)
+ + =z g g zI r mx v mx ru� � M (11)
where and are the ctions of
u v velocities along the dire- x and in thy e body reference frame, gx
is the c rdinate of the center of gravity, r is the pit- ching angular velocity of the body. In this paper, the motion of he cavitator center is limited to the di - ction of eX in the inertial coordinate j st like the axial restrained model experiments in paper[15] so that Eqs.(9) and (10) can not be solved due to the un- known for s
oo
t
ce
reu
xF and yF , only the moment of mo- mentum Eq.(11) is solved and if Eqs.(9) and (10) are also solved then we will have the uncontrolled traje- ctory.
The kinematics equations are
956
= cos sineX u v� ��� (12)
=eY sin + cosu v� � � (13)
here W � is the pitching angle. It is assumed that and = c teX� ons = 0 , so we have = coseu XeY� ��
and cosev rX= �� �� . Substitute u and v� into the final ation is
z zI r �� (14)
Eq.(11), and equ
here
= 0M
w 2= 1zI .34 kgm , = 8.8 kgm , = 0.3 mgx . zM by numerical simulations. The initial is
s calculated
f the vehicle is veloped b the supercavity ro pitching angle and the ste numerical pre-
ig. when
ig.
pitching motion the subsequent relatively stable state,
ith the vehicle being enveloped by the natural super- cavi
ich can be seen in Fig.15 as it
rad.
ig. when
ig.
osi
further affect the gesture f th
rate
ig.1
ly
tate o en ywith ze adydicting results are used. The time step of the unsteady computation is 0.001 s.
F 12 The initial and final state of the pitching motion= 100 m /seX�
F 13 The change process of pitching angle when 100 m /s
=eX�
Figure 12 shows the initial state of the free - and
wty. It can be seen that the vehicle moves with a
two-cavity state, and the initial state is obtained by ad- justing the ambient pressure. Figure 13 shows the variations of the pitching angle with time, which shows that after about 0.6 s, the variations of the pit- ching angle become stable, in the range between �0.004 rad and 0.004 rad.
However, with the initial state as shown in Fig.14 with the free velocity = 110 m /seX� , the vehicle fina-
lly keeps a planing state whis shown in Fig.11(b) and the final pitching angle
is smaller than �0.004
F 14 The initial and final state of the pitching motion= 110 m /seX�
F 15 The change process of pitching angle when 110 m /s
=eX�
The above analysis shows that the closure p - tion of the cavity would seriously affect the lift chara- cteristics of the vehicle thuso e vehicle and the similar conclusions are obtained in the fixed frequency pitching motions.
In order to understand the free pitching motions with a low free stream velocity, we let the free velo- city = 25 m /seX� and = 38 m /seX� , the mass flow
be 0.0134 kg/s, and consider the free pitching mo- tion with ventilated supercavity.
F 6 The change process of pitching angle when 25 m/s, 38 m/s
=eX�
Figures 16 and 17 show that the vehicle final enters the planing state, when the free velocity =eX�
957
38 m /n
velocity
ig.17
ig.18 Th
ig.19 Th
In or
and asethod, w
flow rate
s , the vehicle reaches the stable state sooner
tream
also maller scal
o verify the numerical elocity be 25 m/s, the mass
used to simulate the ven- lated supercavity and the natural supercavity, respe-
turbulence model is used to carry out the s
that the numerical method is valid
s leak more quickly and the final cavity scale
ates are successfully predi- cted,
method has an ob- viou
IRSCHNER I. N., KRING D. C. and STOKES A. W. et al. Control strategies for supercavitating vehicles[J].
Vibration and Control, 2002, 8(2): 219-
[4]
tha when = 25 m /seX� , and the final stable pitching angle is smaller, which shows that the free s
indeed affects the lift of the aft body.
F The final state of the .0134 kg/s
motion state wheis 0.0045 kg/s
evolution of pitchinmass flow rate is 0.0045
der to know wh
e let the free v
pitching motion when mass flowrate is 0
F e n , mass flow rate = 25 m /seX�
F e g angle when kg/s
ether the vehicle might
= 25 m /seX� ,
reach a stable state when the cavity has a s e a different closure way t
mbe 0.0045kg/s, the initial and calculated ter-
mination states of the body are shown in Fig.18. The vehicle finally does not reach a steady state in the pro- cess of the vehicle motion. From Fig.16 and Fig.19, it can be seen that although the free velocity is the same, the vehicle has a different gesture due to the different cavity shape and closure way of the cavity on the body thus it has a different lift.
Therefore, it can be seen that when the vehicle enters the planing state, the lift on the aft body is stable, so that the lift characteristics can be studied in water tunnel experiments.
4. ConclusionsIn this paper, a two fluid multiphase model and
the homogeneous model aretictively. The SST
teady calculation as the initial data of the subse- quent unsteady simulations in which the DES turbule- nce is assumed. On this basis, the fixed frequency and free pitching motions under different working condi- tions are investigated.
Firstly, the pressure measuring points on the vehicle are selected to obtain the variations of the pre- ssure, which are found to be consistent with the con- clusions in paper[8], so
ated. Secondly, the pitching angular velocity of the
vehicle is found to have a great influence on the gas leakage. The pitching motions with higher frequencies make the ga
becomes smaller. Finally, the free pitching motions are investigated
by restricting the motion using the equations of the moment of momentum and the mesh deformation technology. The planing st
including the states enclosed by the natural supe- rcavity and the ventilated supercavity. The cavity scale and the closure position of the cavity are found to seriously affect the lift on the aft body and maybe lead to the trajectory instability.
The numerical method in this paper can predict all kinds of gestures of the vehicle under different working conditions in the longitudinal plane, and can even predict the trajectory. The
s calculation advantage comparing with the mesh regeneration technology. The whole trajectory predi- ction in the longitudinal plane will be carried out in future.
References
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