Hydrodynamics and Torpedoes
Torpedoes are self-propelled underwater weapons designed to strike enemy targets, whereby understanding and manipulating key hydrodynamic phenomena is vital in optimising a torpedo's capability, speed, and efficiency.
The motion and hydrodynamic characteristics of torpedoes can be modelled by classical kinematics, often using spherical polar coordinates to describe the torpedo’s position in space. These coordinates describe the translational (x,y,z) and angular (φ,θ,γ) motion of the system under water: [1]
where Xg, Yg, Zg represent the projections of velocity components onto the fixed coordinate system.
With respect to this coordinate system, the translational velocity of the torpedo can be characterised via classical mechanics as follows.
These derivations allow the components of the velocity vector to be projected onto each respective axis of the system.
The same can be done for angular components in a coupled system, where:
Hydrodynamics and Laminar Flow:
These components of a modelled torpedo can be used to both model its motion and further understand the effects of hydrodynamic forces acting upon the system during its time-of-flight.
A torpedo in motion within a real fluid with quantifiable viscosity is subject to tangential stress caused by the flow of fluid particles relative to the surface. [2] This phenomenon can be characterised simply by Newton’s formula which quantifies the internal forces of friction by relating the relates the stress acting tangentially to the surface (t) to the velocity gradient of the system, (dV/dt):
where μ represents the coefficient of viscosity of the water.
The velocity gradient causes a hydrodynamic pressure along the hull of the torpedo subject to Bernoulli’s Law:
Computational Fluid Dynamics and Turbulence
Hydrodynamic analysis of torpedoes plays an important role in their functionality as destructive naval weapons with recent computational developments making it possible to optimise their characteristics and dimensions via the Myring Equations: [3]
The motion of fluid around the nose and forebody of the torpedo itself can be determined by the Navier-Stokes equations which arise from applying Newton’s Second Law to fluid motion: [4]
Incompressible Navier-Stokes equation
This particular form of the Navier-Stokes equation excludes external body forces and governs the momentum of the Newtonian Fluid by balancing forces acting (notably the pressure gradient (∇P), the gravitational force per unit volume (ρg) and the viscous forces related to diffusion (μ∇ V)) on a fluid element.
A recent computational fluid dynamics (CFD) study conducted by Sudheendra Prabhu et al. analysed the behaviour of fluid over conventional torpedoes as well as the pressure distribution on the surface (see Navier-Stokes equation). [5] The so-called Spallart-Allmaras (SA) turbulence model is a simplified Navier-Stokes model that solves a single translational motion equation for eddy viscosity, such as that acting on the surface of a torpedo. This permits insight into the torpedo’s drag reduction and wake structures, properties which affect the stealth and acoustic characteristics of the system. The SA model is particularly efficient in flow separation which is important for vehicles travelling at speed underwater.
ρ = density of water
v = molecular kinematic viscosity
v̂ = unsteady viscosity
μ = dynamic viscosity
Ŝ = mean flow shear
σ = diffusion coefficient
This specific differential form of the SA model relates the time dependence of the system's viscosity gradients to factors which affect the turbulence of the torpedo, including diffusive and convection contributions. All other variables are constant and remain specific to the system under investigation. Ultimately, CFD simulations can optimise the dimensions of the torpedo hull to improve drag coefficients and stability caused by turbulence around the system via the SA model.
2
Computational Fluid Dynamics study on the velocity field and pressure of a torpedo hull optimised by the Myring equations. Image taken from Sudheendra et al. [5]
Cavitation Phenomena
Cavitation is a physical phenomena which occurs at low pressures and high velocities, involving the formation of gaseous bubbles around an object submerged within a liquid medium. In the context of torpedoes, supercavitating systems utilise this to reduce skin friction drag, whereby the bubble formed can encompass the entirety of the torpedos nose and hull. [6] Given the significant reduction in resistance, supercavitating torpedoes can travel much faster at speeds beyond the limits of conventional systems.
The degree of cavitation is best described by the so-called (dimensionless) cavitation number (σ).Optimised torpedo systems aim to minimise this value: [7]
where P, ρ and V have their usual physical meanings.
Scheme of supercavitation phenomenon on torpedo's nose - Taken from https://de.wikipedia.org/wiki/Datei:Superkavitation_schema.png
[1] Redrawn from G. M. Podobriy et al. Theoretical Principles of Torpedo Weapons, U.S. Joint Publication Research Service (1976), p.6
[2] Principles of Naval Architecture, The Society of Naval Architects and Marine Engineers, 1988. https://navalifpe.wordpress.com/wp-content/uploads/2011/09/principles-of-naval-architecture-vol-1-sname.pdf (Accessed December 2024)
[3] Redrawn from D.-H. Ji, H.-S. Choi, J.-I. Kang, H.-J. Cho, M.-G. Joo and J.-H. Lee, Design and Control of Hybrid Underwater Glider, Advances in Mechanical Engineering (2019), 11(5), p.1-9
[4] M. T. da Gama and Rodrigo Coelho, The Navier Stokes Equation, Ciências U. Lisboa. https://fenix.ciencias.ulisboa.pt (Accessed January 2025)
[5] K. Sudheendra Prabhu and G. Srinivas, Hydrodynamic Performance Enhancement of Torpedo-Shaped Underwater Gliders Using Numerical Techniques, F100Research (2024), 13, p.1274
[6] E. Alyanak, R. Grandhi and R. Penmetsa, Optimum Design of a supercavitating torpedo considering overall size, shape, and structural configuration. International Journal of Solids and Structures (2006), 43, 642-657. DOI: 10.1016/j.ijsolstr.2005.05.040.
[7] A. May, Water Entry and the Cavity-Running Behaviour of Missiles (1975). Navsea Hydroballistics Advisory Committee
Video:
Watch a torpedo crack a boat in half. Available at: https://www.youtube.com/watch?v=O736ClrS-K8 (accessed 21. Feb . 2025)
Torpedo launch animation F21. Available at: https://www.youtube.com/watch?v=pPok1DWjCNM (accessed 21. Feb. 2025)