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N/DYNAMIC OF HIGH SPEED UNDERWATER PROJECTILE DONG Lire HOC CUA DAU DAN BAN DLTCJl Nl/dC
Dao Van Doan, Nguyen Van Hung Military Technical Academy
ABSTRACT
In this paper a simple model for dynamics of high-speed underwater projectile has been presented. In praticular we are interested in the nature and frequency of the impacts which occur as the projectile tail touches the supercavity wall. The results of the paper can be applied to study motion of high speed underwater projectiles.
Keywords: Dynamic; Underwater projectile; Supercavity.
TOM TAT
Bdi bdo da trinh bdy mpt md hinh dan gidn tinh todn dpng luc hoc ddu dgn bdn dudi nudc tdc dp cao. Ddc biet bdi bdo tap trung nghien cim bdn chdt vd tdn sd cua ddu dan khi dudi dgn va chgm vdo thdnh sieu khoang. Kit qud cua bdi bdo cd thi img dung nghien cuu chuyin dpng cua cdc thiit bi chuyen dpng dudi nudc tdc dp cao.
Tir khda: Dong luc hpc; Ddu dgn bdn duai nudc; Sieu khoang.
1. INTRODUCTION
When an underwater projectile moves through water at sufficient speed, the fluid pressure may drop locally below a level that sustains the liquid phase, and a low-density gaseous cavity forms. Flows exhibiting cavities enveloping a moving body entrirely are called "supercavitating"
(fig.l). In supercavitating flows, the liquid phase does not contact the moving body over most of its length, thus making skin drag almost negligible [1-5].
Figure I. Schematic diagram of a supercavitating projectile
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The presence of the cavity changes the nature of motion of the projectile. The projecrile while moving in the forward direction also starts rotating about its tip in the vertical plane. This rotational motion is imparted to the projectile due to disturbances that occur during firing. Because of this rotation, the tail of the projectile impacts on the cavity wall. It then bounces back and impacts on the opposite side of the cavity and this type of oscillatory motion continues till the diameter of the cavity becomes sufficiently small (fig. 2). Tail in position A moves to B where it impacts against the cavity wall. After impact, tail moves from B to C where it again impacts against the cavity wall, the diameter of which has become smaller. Subsequently, the projectile moves with the cavity boimdary in contact with the side surface and untimately the cavity disappears.In this paper, a simple model for calculating dynamic of high speed underwater projectile has been studied.
Figure 2. View of the underwater projectile from behind 2. MATHEMATICAL MODEL FOR THE UNDERWATER PROJECTILE MOTION
Our model is base on the following assumptions:
1. The effect of gravity on the dynamics of the projectile is negligible.
2. The projectile is assumed to rotate about the nose tip.
3. The cavitay and motion of projectile is confined to a plane.
4. In order to simplify the model, we assume that the projectile is not spinning about its longitudinal symmetry axis of the projectile.
2.1. Forces
The projectile is acted upon by a system of forces and moments corresponding to the interaction of the projectile with the cavity boundaries.
The resultant F of the forces acting on the underwater projectile can be written as:
F = F„+F,
Where F^ is the hydrodynamic force at the projectile nose generated by the interaction of projectile nose with the water, F, are the contact forces due to the interaction of the projectile tail
with the cavity. '^
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Similarly, the moments M can be written as: M = /^'^WG '^^I-^IG'^^I
Where d^^^, indicates a distance vector fix)m point N to pomt G, dIG indicates a distance vector from point I to point G, N is the nose location (Fig.4), I is the tail-cavity contact point, G is the center of mass, MI is the additional moment due to interaction of the projectile with the cavity.
2.1.1. The nose force
In the absence of impact, the only force on the projectile is due to the fluid force at the nose FN [^' ^3- ^N '^ resultant of lift force F^ and drag force F^. Lift and drag forces which are calculated by formula (1); i i
Fy=-pAv'C,iF,=-pAv'C, (1) Where: C^ is lift force coefficient. Cy =kcos^ /3
C^ is drag force coefficient. C^ = A: sin ^ cos fi
Where k is a nondimensional constant, p is angle between the projectile axis and the cavity axis, p is density of water, v is forward velocity, A is cross sectional area of the nose.
The magnitude nose force FN is calculated by formula (2):
-^-= -pAv^k cos B (2) F 1
cos;9 2 ^ ^
Fz
Projectile's
Figure 3. Hydrodynamic forces acting on the projectile nose
2.1.2. The tail force
The tail forces FI are generated by the interaction of projectile with the cavity walls. Components of the tail forces are tail lift force FIY and tail drag force FIZ which are calculated by formula (3) [3,4]:
\F^=pBv\\~cosfi)
\F^=PBV'sin fi
Figure 4. The tail forces
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Where B is cross sectional area of water layer:
B = 15D (4) D is diameter of projectile at tail, Ai? is thickness of water layer (fig. 5):
S = lsinfi + ^cosfi-h (5)
Where I is length of projectile, h = h(v) is distance from cavity axis to cavity wall at tail section.
Projectile axis I I^^^~~~~-~~,,,______^ Cavity boundary
N Figure 5. Geometry of impact
2.2. Equations of motion
To describe the motion of the projectile, a body fixed coordinate system as shown in figure 6 is chosen. The Z axis measure the horizontal distance of the center of mass from a fixed datum.
The p angle is angle between the projectile axis and the cavity axis.
Figure 6. Coordinate z am 2.2.1. Forward motion
During the priod of motion between successive impacts, the only force on the projectile is the nose force which is directed along the projectile axis (shown in figure 3).
The forward motion of the projectile is calculated by the following equation (6):
Y^F = mz = -F^= —pAz-k cos^ fi (6)Q
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For small value of P this becomes:Where x = ^— • Equation (7) has the solution; pAk 2m
(7)
z(r)=-ln(l + j V ) (8) Z
Where Z is the projectile's initial velocity. The velocity is given by:
z(t) = -r^^ (9)
2.2.2. Rotatory motion
The angle P changes when the projectile tail impacts on the cavity boundary and the rotatory motion of the projectile is determinedly by changing of angle p.
The rotatory motion of the projectile can be written as: ^M = lfi = -Fj^asin fi - F^a cos fi Where a is distance from the center of mass to point of application tail forces, I is transverse moment of inertia of the projectile about its center of mass.
Using equation (3), this becomes Ifi = ~pBz^asinfi
Substituting for B, using equation (5) and assuming small angles p, we obtain equation (10) which describe the rotatory motion of the projectile:
J3+P^(Lfi^^-hyfi = 0 (10) 3. NUMERICAL RESULTS
The following values have been used in the numerical simulation:
m = 0,026 kg, p - 1000 kg/m^ A - 7c(0.001)^ m^ D = 0,0057 m, L - 0,14 m,a = - = 0.07m, / = — - = 0.0000425 kgm^ muzzle velocity i^ = 245m /s\X= 0.5, k - 0.9.
The simple simulation is carried out for Is. The results are shown in fig. 7.
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fetocitv of Pmjeclile (m/s) Distana' of forward moUon (m)
1 1 ( s e c )
Figure 7. Velocity and distance of forward motion jrom a sample simulation
It can be seen from the fig.7 that the velocity decreases rapidly with time. This rapid decrease is due to the drag force acting at the nose of the projectile. The projectile, released wdth muzzle velocity 245 m/s, travels about 50 m in 1 sec, at point its velocity is about 40m/s. on the other hand, from calculated result, we obtained the first 9 successive inpacts which occurred in the first second of simulated flight. Each impact corresponds to a progressively smaller cavity size.
4. CONCLUSIONS
In this paper a simple model for dynamics of high-speed underwater projectile has been presented. Our model predicts that as the projectile races through the water, it rotates like a moment- free rigid body about the point where its nose touches the water. For ly/pical initial conditions, this leads to impacts of its tail with the cavity walls. These impacts become more frequent as the body slows down, since the diameter of the cavity decreases with a decrease in forward speed. The number of impacts achieved in a given time interval (e.g. in one second) depends on tbe initial conditions of the body. The larger the initial angular velocity, the greater the number of impacts. The dynamics are also found to depend on the initial deflection angle. •
References:
[1]. Dao Van Doan, NguySn Van Hung; Xdy dung mo hinh tinh todn thugt phong ngodi dgn bdn dudi nudc.
Tap chi Khoa hoc va Ky thuat. Hoc vien Ky thuat Quan sir, s6 thang 8-2014.
[2]. G.V. Lognovich, Hydrodynamics of Free Boundary Flows, IPST, Jerusalem, 1972, p.l04.
[3]. Savchenko Yu.N, Investigation of High-speed supercaviating underwater motion of bodies, AGARD report 827 "High speed body motion in water", 1998, Kiev.
[4]. Hassan, Analysis of hydrodynamic planing forces associated with cavity riding vehicles., private communication, 2004.
[5]. TKiceniuk, An experiemntal study of the hydrodynamical forces acting on a family of cavity producing conical bodies of revolution inclined to the flow, Cal.Inst.Tech, Hydrodynamics lab. Report No.E-I2.17, 1954.
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