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High-speed underwater projectiles modeling: a new empirical approach

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Abstract

Supercavitating projectiles can achieve high speeds; however, this will pose technical challenges on their stability and flight performances. One of the most important issues which a high-speed underwater projectile (HSUP) deals with is the so-called planing force. In an ideal supercavitating flight scenario, the entire vehicle is considered to fly within the cavity walls. Nevertheless, in practice, disturbances can cause the vehicle to impact on the liquid–gas boundary. In such situations, the forces generated at the cavity boundary are referred to as the planning forces. The present paper discusses the in-flight dynamics of the HSUP. Equations of motion are developed for the projectile movements in the cavity while the tail impacts on the cavity wall. Dominant nonlinearities associated with planing forces are well thought-out in the modeling. Two available models and a new empirical model for prediction of the planning force are described. By using and combining these models, four methods are introduced and compared, through the simulation runs of supercavitated projectile flight, with two available experimental test cases.

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Abbreviations

α :

Immersion angle

α n :

Local angle of attack in the projectile nose

A n :

Cavitator area

\(C_{{A_{{0_{\text{c}} }} }}\) :

Axial force coefficient of the cavitator in zero angle of cavitator

\(C_{{A_{{\delta_{\text{c}} }} }}\) :

Axial force coefficient due to change in angle of cavitator

\(C_{{A_{{\delta_{\text{a}} }} }}\) :

Axial force coefficient due to change in aileron angle

C D :

Drag coefficient of the cavitator

C D0 :

Drag coefficient of the cavitator at zero incidence

\(C_{{L\delta_{\text{a}} }}\) :

Rolling moment coefficient due to change in aileron angle

\(C_{{M\delta_{\text{c}} }}\) :

Pitching moment coefficient due to change in angle of cavitator

\(C_{{z\delta_{\text{c}} }}\) :

Normal force coefficient due to change in angle of cavitator

D :

Reference length

δ a :

Aileron angle

δ c :

Cavitator angle

\(F_{A}^{B}\) :

Hydrodynamic force in body coordinate

\(F_{\text{g}}\) :

Gravitational force

F n :

Cavitator force

F p :

Planing force

g :

Gravitational acceleration

\(h^{\prime}\) :

Immersion depth

(Ixx, Iyy, Izz):

Moments of inertia about each axis

(Ixy, Iyz, Izx):

Products of inertia

(L, M, N):

Component of hydrodynamic moments acting on supercavitating vehicle along each axis

l :

Total length of projectile

l max :

Maximum length of the cavity

l ' :

Distance between the tail and the center of gravity of projectile

\(M_{A}^{B}\) :

Hydrodynamic moment in body coordinate

M p :

Planing moment

M s :

Mass

(p, q, r):

Angular rates

P c :

Cavity pressure

(φ, θ, ψ):

Euler angles

P ∞ :

Ambient pressure

q :

Dynamic pressure

R :

Projectile radius

R 1 :

Initial cavity radius

R c :

Cavity radius

\(\dot{R}_{c}\) :

Contraction rate of cavity radius

ρ :

Density

R max :

Maximum radius of the cavity

R n :

Cavitator radius

S :

Reference area

σ :

Cavitation number

t max :

Time for a cavity section to achieve R max

τ :

Delay time

(u, v, w):

Component of supercavitating vehicle velocity along each axis

V :

Velocity magnitude

(X, Y, Z):

Component of hydrodynamic forces acting on supercavitating vehicle along each axis

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Authors and Affiliations

  1. Department of Mechanical Engineering, Shiraz University, Mollasadra Square, Shiraz, Iran

    Mojtaba Mirzaei, Mohammad Mehdi Alishahi & Mohammad Eghtesad

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  1. Mojtaba Mirzaei
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  2. Mohammad Mehdi Alishahi
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  3. Mohammad Eghtesad
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Correspondence to Mojtaba Mirzaei.

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Technical Editor: Marcelo A. Savi.

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Mirzaei, M., Alishahi, M.M. & Eghtesad, M. High-speed underwater projectiles modeling: a new empirical approach. J Braz. Soc. Mech. Sci. Eng. 37, 613–626 (2015). https://doi.org/10.1007/s40430-014-0190-7

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