This paper studies the three-dimensional motion modeling for a super cavitating projectile. 6-degrees of freedom equation of the motion was constructed by defining the forces and moments acting on the supercavitating body while moving underwater.
Trang 1STUDY ON THE SPATIAL MOTION MODEL
OF UNDERWATER PROJECTILE
Nguyen Huu Thang*, Nguyen Hai Minh, Dao Van Doan
Abstract: This paper studies the three-dimensional motion modeling for a super
cavitating projectile 6-degrees of freedom equation of the motion was constructed
by defining the forces and moments acting on the supercavitating body while moving
underwater The impact force of the projectile tail with the cavity wall is determined by
the super-cavity size calculations when considering the effect of the reduction of the
super-cavity size The calculation results obtained the integrated motion of the super
cavitating projectile in the water environment, which was the basis for the study of
scattering for the underwater projectile
Keywords: Underwater projectile; Super-cavity; Supercavitating projectile; Planing force; Cavity model
1 INTRODUCTION
In literature, the dynamics of a supercavitating body is more complex than that
of moving body in a flow regime that does not separate in the air or in water The complexity causes are the cavity instability surrounding the body and the discrete impact between the body tail and the cavity wall At each moment, the force is determined from the relative position and relative motion of the body compared with the cavity and the cavity shape, formed in the first time Experimental studies
in [6] have shown that the stability of motion of supercavitating body can be achieved with all velocities ranging from 50-1450 (m/s) The analysis showed that the four different mechanisms of motion stabilization sequentially act at motion velocity increase (Fig.1)
Figure 1 Schemes of motion of supercavitating models
Two - cavity flow scheme (Fig.1a), V 50 (m/s): In this case the hydrodynamic drag center is placed behind the mass center, and stabilizing moment of the force Y2 acts to the model It means that the classic condition of stability is fulfilled
Sliding along the internal surface of the cavity (Fig 1b), V 70-200 (m/s): To compensate the buoyancy losses the body's tail part is planing along the lower internal cavity surface In this case, the motion may be stable as a whole
Impact interaction with cavity boundaries (Fig 1c), V 300-900 (m/s): Presence
of initial perturbations of the model attack angle and the angular velocity causes to the impact of the model tail part against the internal boundary of the cavity The mathematical modeling showed that after this impact the model can perform steady
Trang 2or damped oscillations accompanied by periodic impacts of the tail part alternately
against the upper and lower cavity walls Then the motion can remain stable as a whole
Aerodynamic interaction with vapor-splash in the cavity (Fig 1d), V 1000-1450
(m/s): At very high velocities the aerodynamic and splash forces of interaction with
vapor filling the cavity and splashes near the cavity boundaries have a considerable
effect on the body motion Since the clearance between the body surface and the
cavity boundary usually is small compared to the cavity radius, they apply the
known methods of near-wall aerodynamics to estimate the arising forces
The result of longitudinal inertial motion model (2D) was obtained on the basis
of the assumption the cavity has an axial symmetrical ellipse [1, 2, 3] Therefore,
this model does not mention the horizontal plane motion caused by the original
disturbance factors
In this paper, we establish the spatial motion model (3D) of an inertial motion
projectile considering the gravity effect, the reduction of the super-cavity size, the
impact force and the friction between the tail and the cavity wall, the effect of
movement around the projectile center to the motion trajectory of the center
2 MODELING OF SUPERCAVITATING PROJECTILE
2.1 The equations of 3D motion of supercavitating projectile
According to the general order, the mathematical model describes the motion
of the supercavitatingprojectile (SC-projectile), including the equations of the
solids of six degrees of freedom, equations for calculating the shape and size of
the cavity, the equations determine the relationship to calculate the interaction
forces between them
Figure 2 shows a scheme of the 6-DOF SC-projectile motion The body
coordinate system O1X1Y1Z1 and the flow coordinate system O1XVYVZV are
shown (Fig 2a), An origin of both the coordinate systems is placed at the projectile
mass center O1 O1XV of the flow coordinate system is directed along the velocity
vector V of the projectile mass center The axis O1X1 of the body coordinate
system is directed along the longitudinal body axis The axes O1Y1, O1Z1 together
with O1X1 axis form the positive triangle Also, we will use the fixed coordinate
system OXYZ and the semi-body coordinate system O1XgYgZg (Fig 2b) The
direction of the semi-body coordinate system axes coincides with the direction of
axes of the fixed coordinate system Oxyz at each time instant
a) b)
Figure 2 Scheme of forces acting onto SC- projectile and the coordinate systems
Trang 3Writes a set of equations of the 6-DOF motion of a solid body in projections on the axes of the body coordinate system O1X1Y1Z1, which are the principal axis of inertia of the body:
1
1
1
1
1
1
(1)
1 ( cos
o
x
y
z
x
y
z
dV
dt dV
dt dV
dt d
dt d
dt d
dt d
dt
d
M
M
d
M
t
s
dx
V dt
dy
V d
d
dt
t dz
V dt
Where:V V x1, y1 ,V z1 - is the projection of the velocity vector of the body mass center in the body coordinate system; x, y, z
- is the angular velocity vector relative to the body mass center; m- is the body mass;I I x, y I z- are the moments of inertia relatively to the axes O1X1, O1Y1, O1Z1, respectively; F x1,F y1,F z1, M x1, M y1,M z1- are the projections of the composite force vector and the main moment on the same axis;
,, - are the yaw, pitch, and roll angle, respectively; , - are the angle of attack and sliding angle, respectively, they are present in the relationship between the velocity coordinate system O1XvYvZv and the body coordinate system O1X1Y1Z1 (figure 2a) In this case, the following relations are valid:
x
1 sin cos ;
y
V V V z1 Vsin
We accept the following assumptions for the formulation of the general problem of the SC-Projectile dynamics:
Trang 41 The SC- projectile is a slender body of revolution and symmetrical through
the x-axis, in this case, Iy = Iz;
2 The disk-shaped projectile nose with Dn diameter is the Cavitator shape;
3 The mass m, the mass center position xc, and the vehicle moments of inertia
Ix, Iz do not vary during motion
The Components of force acting on the projectile consist: the projections of the
gravity force mg
, the hydrodynamic force on the cavitator F n F nx1, F ny1 ,F nz1 and the force of interaction between the projectile tail and the cavity wall
F F F F
: F x1F nx1F px1mgsin ;
F F F mgcos c s
1 1 1 sin
F F F mgcos
2.2 Cavity model
The cavity is a major component of the SC-projectile system The behavior of the
cavity bubble around the projectile affects the body immersion The cavitator
continuously creates a cavity while the projectile is moving (Fig 3a) The cavity shape
model is taken from a solution presented in Lovinovich’s work (1972) The shape and
size of the cavity depending on the motion parameters of the projectile To determine
the shape and size of the cavity, an approximate method is given in the theory of G.V
Logvinovich [10]: Divide the entire cavity into finite sections; each section is
cylindrical, limited by the cross-section perpendicular to the cavity axis (Fig 3b) Each
part of the cavity is formed from the moment the water environment begins to separate
from the surface of the Cavitator center Due to the effect of inertial force, the cavity
expands in a direction perpendicular to the orbit of the Cavitator After the cavity
reaches its maximum size, it starts to contraction and disappears due to hydrostatic
pressure pressing on the outer surface of the cavity The important parameter
representing the characteristics of the cavity is the cavitation number:
2
2(p p c) / ( V )
(2)
where: p- is the ambient pressure the cavity;
c
p - is the pressure inside the cavity;
V - projectile velocity; - density of water
Let the projectile starts a motion with the arbitrary angle, hydrostatic pressure
varies with depth The hydrostatic pressure around of the cavity at each moment is
determined by the expression:
( ) atm ( ) sin ( ) cos (t)
p t p g H S t t (3) where:p atm- atmospheric pressure at the surface;
H0 - is initial depth;
Trang 5( )
S t - is the trajectory of the cavitator center formed at the time instant t;
sin ( ) t andcos (t) - is the deflection of the cavitator center trajectory line
at each time for the axis O1Xg
a)
b)
Figure 3 Scheme of symmetrical axis and sections of cavity
According to G.V Logvinovich, the cavity sections expand and shift independently of each other, radius of cavity section at the distance x from cavitator is determined [10]:
2
(4) and the cavity contraction rateR cis:
1
2
20 (1 ) 4, 5 (0,82 ) (1 )
1, 92 ( 3)
c
R
K
Here, K1, K2 are two functions of , that defined to represent the cavity model:
1 1
2 1, 92 ( ) 1
n
l K
d
;
0,5 40 17
4, 5
1 1
1
The projectile's nose (cavitator) is a fundamental part of the projectile; it creates
a cavity bubble around the body and generates force components on projectile’s nose The forces acting on a disk-type cavitator in steady flow are well understood
To a very high accuracy, it may be assumed that the force vector acts through the center of the disk and normal to its wetted face The drag, lift and side force are then given by [8]:
Trang 6
8
n
F (6)
8
n
F (7)
8
n
F (8) Where: n z x c/V x1- is the cavitator attack angle; n y x c/V x1- is
the cavitator sliding angle, dn- is the disk cavitator diameter; CD0=0.82- is the
drag coefficient of the cavitator at an attack angle of zero and a Cavitation
number of zero
2.3 Planing forces model
The interactive force(F plane)on the wet surface of the projectile tail affects the
projectile speed and the stability of motion So that, the stability of the SC-
projectile depends on the moments associated with the nose and the aft The
contact of the projectile tail with the cavity wall creates the planing force [8], that
interaction force is nonlinear The shape and size of the cavity and the relative
position between the bullet and the cavity will determine immersion The
interactive force included planing force(F p)and friction force(F f)acting at the
transom of the sliding section is shown in Fig.4
Figure 4 Scheme of the interaction force between the projectile tail with a cavity wall
Trang 7In planing force modeling methods, it is assumed that the planing force depends entirely on the vertical velocity Hassan has given the following set of equations for the planing force and moment for cylindrical body planing on a curved surface as [11]:
2
2
(9)
2
2( 2) cos2
2
b
h
(10)
where: R R - is the radius of the body and the cavity at the immersion b, c
position, respectively;
lp, l - are the wet length of the projectile tail and the total length of the
projectile, respectively;
p
- is the immersion angle;
h - is the immersion depth
Since the planing force is a non-linear function, an appropriate method is to calculate the planing force along the plane of sliding planing (not necessarily along the Y or Z directions) by determining the immersion angle along that direction This force can then be that separated into the Y and Z directions (see Fig 4)
Since the Projectile's tail is in contact with the liquid at high speed, so the friction force is generated [11]:
1
cos 2
F V S C (11)
4 1 arctan( ) ( ) arcsin 1
p
R
; 1
7
0.031
p
p
C ul k
; c
p
h
u
;
2
R
p
- is the distance between centers of cross-section of cavity and projectile at the transom
The planing force in the body frame are obtained through the projections of the plane of immersion into the X-Y and X-Z:
1 1
(12)
Because the shape of the cavity at the tail is closely related to the cavitator angle
of attacknand cavitator sliding anglen, in the steady-state regime, by assuming
Trang 8instantaneous cavity formation and cylindrical cavities, the total cavitator angle of
attack expressed in polar coordinate system as: t n2n2 , it determines the
planing direction arctan( n)
p
n
defines the angle between the plane of immersion and the vertical symmetry axis of the cavity cross section at immersion
location seen in Figure 4
From the size of the cavity and the relative position of the projectile to the
cavity (Figure 4), it is possible to determine when the bullet tail impacts the cavity
wall through the immersion depth The parameter h in equation (9) is expressed as:
h
(13)
and the immersion angle at body tail with considering the effect of contraction
rate of the cavity determined as:
p
(14)
3 SIMULATION RESULTS AND DISCUSSION
The differential equation system (1) combines the formula for calculating the
cavity size (4), (5), the impact condition (13) and the immersion angle (14), solved
by the numerical integration method, using the algorithm Runge Kuta math- 4
Applying calculations for underwater projectile parameters in table 1 We received
results in the variation of change of dynamics parameters and the motion trajectory
on figure 5-8
Table 1 Initial parameters of the simulation
Distance from tail to mass center of projectile (lp) 14,31.10-3 (m)
)
Initial center velocity in the X- direction (V0x) 350 (m/s)
Initial center velocity in the Y- direction (V0y) 5 (m/s)
Initial center velocity in the Z- direction (V0z) 5 (m/s)
Trang 9a) b)
Figure 5 Time evolution of angular velocity (a) and angular velocity y z (b)
Under the gravity effect, the lift force at the nose of the projectile and the condition of the loss of Archimedes force, the projectile moves around the center
in the cavity with rules angular velocity in figure 5 Due to the movement of projectile around the center, the oscillates angular velocity around the value “0” When the impact of the projectile tail with the cavity wall happens, the moment of the impact force serves as the stable moment of the projectile Thereby creating the projectile movement is stable through three phases: the movement does not involve the force with the cavity wall, the movement involves the impact with the cavity wall and the movement slides on the “wet road” After that, when the projectile totally soaks, a monotonous increase of the angle of attack leads to the projectile instability on the flight path
Figure 6 Time evolution of angle of attack (a) and sliding angle (b)
Due to movement around the center of mass, the angle of attack and sliding angle begins to oscillate around the value "0" (Figure 6) Under the influence of stable moment, the bullets move stably through three stages: motion does not affect
Trang 10on the wall of super-cavity, the movement has an effect on the wall of super-cavity
and sliding motion is on "wet road" When the bullets are completely wetted, angle
of attack and of sliding angle increase monotonically That leads to unstable
ammunition on the trajectory
The variation of the projectile center velocity in the X, Y and Z directions is
shown in Figure 7 and Figure 8a Accordingly, the value of the X- directional
speed decreases monotonically, the speed value in the Y and Z directions fluctuates
around the value "0" and increases monotonically at the stage of instability on the
flight path
Figure 7 Time evolution of velocity of mass center in the X- direction (a) and
Y- direction (b)
The trajectory of the projectile mass center in three-dimensional space is shown
in Figure 8b,including the value of the range (X), the height (Y) and the drift (Z)
Figure 8 The velocity in the Z- direction (a)
and the trajectory of the mass center in space (b)