Hydrodynamic analysis and modelling of ships Wave loading Harry B. Bingham Section for Coastal, Maritime & Structural Eng. Department of Mechanical Engineering Technical University of Denmark DANSIS møde om Maritim Aero- og Hydrodynamik May 13, 29
Wave loads on ships via potential flow theory Linear theory Wave resistance Motion response Structural loading Second-order effects Springing and Whipping Added resistance, mean drift forces Higher-order effects Parametric roll Moored ship resonance response
A potential flow approximation (no viscosity/vorticity) z U φtt + gφz = + N.L.T. on Sf η(x, y, t) n φ = on Sb x V = [u, v, w] = φ(x, t) 2 φ(x, t) = n φ = on Sh
The first-order solution Linearize the flow around a steady solution Φ (Neumann-Kelvin: Φ = Ux) Φ = Φ (x) + ǫφ 1 (x, t) +..., ǫ H L 1 (wave steepness) Newton s law - the equations of motion for the ship 6 k=1 t (M jk + a jk )ẍ k (t) + K jk (t τ)ẋ k (τ)dτ + C jk x k (t) = F jd (t) j = 1, 2,...,6 M jk, C jk : K jk, a jk : F jd : Body s inertia and hydrostatics Force due to radiation of waves by the body motions Force due to diffraction of the incident waves
Time-harmonic (frequency-domain) solution Let η(t) = R{A e iωt }, then x k (t) = R{ x k (ω)e iωt } and 6 { ω 2 } [M jk + A jk (ω)] + i ωb jk (ω) + C jk xk (ω) = F jd (ω) k=1 j = 1, 2,...,6 where the IRF s and the FRF s are related by Fourier transform: A jk (ω) = a jk 1 ω B jk (ω) = F jd (ω) = dt K jk (t)sinωt; dt K jk (t)cos ωt. dt F jd (t)e iωt
Numerical solution - 3D panel methods Green s theorem (φ n n φ): = ( 2 φg φ 2 G)dV = (φ n G φg n )ds V Where φ and G are two solutions to the Laplace equation. PDE in V integral equation over the boundary S φg n ds = φ n G ds An integral equation for φ on S. S S S
Motion response in waves (Free-Surface Green function) 1 1.5 1. Fn=. Fn=.1 Fn=.15 Fn=.2 exp. Fn=.2.5...5 1. 1.5 2. (L/λ) 1/2 7.5 5. Fn=. Fn=.1 Fn=.15 Fn=.2 exp. Fn=.2 2.5...5 1. 1.5 2. (L/λ) 1/2 1 Bingham et al, (1994). 2th Symp. Naval Hydrodynamics
Rankine-type panel methods, high-order panels Free-surface and bottom boundaries are gridded more flexibility Z X Y
Pressure distribution on the hull Z X Y
Added resistance in waves (2nd-order mean forces) 2 Neumann-Kelvin linearization 3 25 Experiment 1; ζ =.11m Experiment 2; ζ =.19m AEGIR; Momentum conservation AEGIR: Pressure integration 2 15 σ AW 1 5 5.5 1 1.5 2 λ/l 2 Joncquez et al, (28). 27th Symp. Naval Hydrodynamics
Added resistance in waves Double-body linearization 35 3 25 2 σ AW 15 1 5 5.2.4.6.8 1 1.2 1.4 1.6 1.8 2 λ/l
Other solution methods - Strip theory Two basic assumptions: 1. The body is thin: beam and draft << length. ds dξ dl ds dl dξ S b L C x 3-D forces as integrals (sums) of sectional force coefficients. 2. Encounter frequency is high: ω >> U( / x), zero speed free-surface condition is assumed valid. Much faster than 3D methods. Good performance for linear motions but not for added-resistance. 2nd-order strip theory has been successful for springing. 3 3 Vidic-Perunovic, J, and Jensen, JJ, (25). Marine Structures.
Parametric Roll Simple model with linear surge and heave motions and dynamic calculation of the roll GM, can predict the occurrence well. 4 4 Vidic-Perunovic, and Jensen (29). Submitted to Ocean Engineering.
A hybrid model for nonlinear moored ship motions 5 Assumptions Small wave steepness at the ship (linear hydrodynamics) Mooring system adds: Nonlinearity, and resonant horizontal modes Body motions are still small Harbor and ship problems are de-coupled 1. Wave forcing: Boussinesq-type model (MIKE 21BW) 2. Wave-body interaction: Panel method 3. Ship response: Time-domain equations of motion with nonlinear external forcing 5 Bingham (2). Coastal Engineering
Pictorial representation of the situation
Add a nonlinear forcing term to the equations of motion 6 k=1 t (M jk + a jk )ẍ k (t) + K jk (t τ)ẋ k (τ)dτ + C jk x k (t) = F jd (t) + F jnl (t); j = 1, 2,...,6 F jnl : Nonlinear forcing. Mooring lines, fender friction, viscous damping,... Wave forcing F jd (t) via linearization and interpolation from the Boussinesq grid to the panel method grid
Bi-linear interpolation and linearization.
Linearization of the Boussinesq solution Boussinesq solution η, [u, v ] = 1/(h + η) η h [u, v]dz p (x, t) = p(x,, t) = ρgη(x, t) w (x, t) = w(x,, t) = η(x, t) linear FSBC Fourier transform to get p (x, ω), ũ (x, ω), ṽ (x, ω), w (x, ω). { } cosh[k(z + h)] c(x, z, ω) = c (x, ω), c = u, v, p; cosh(kh) { } sinh[k(z + h)] w(x, z, ω) = w (x, ω) sinh(kh) ω 2 = gk tanh(kh) (ũ, ṽ ) = (ũ, ṽ ) kh tanh(kh)
A nonlinear test case: A ship moored in an L-shaped harbor 2 15 Wavemaker 1 #1 5 Ship -5-1 -15 Absorbing beach -2-2 -15-1 -5 5 1 15 2 Plan view of the experimental setup, lengths are in meters. The model- to full-scale factor is 8, and the water depth is.3 meters.
4 2 Quay Fenders Incident wave -2-2 2 4 The mooring arrangement with respect to the ship s waterline. The ship is an LPG tanker C b =.816. Lines and fenders are linear springs. (Shown at full scale.)
7 Measurements Calculations 1.4 Measurements Calculations 6 1.2 5 1 4.8 3.6 2.4 1.2 x 1 /A.5.1.15.2.25 4 3 2 η I at w.g. #1 Measurements Calculations with mooring Calculations without mooring x 2 /A.5.1.15.2.25 8 6 4 η I at the ship Measurements Calculations with mooring Calculations without mooring 1.5.1.15 f (s -1 ) Surge 2.5.1.15 f (s -1 ) Sway
.9 Measurements Calculations.5 Measurements Calculations.8.45.7.4.6.35.3.5.25.4.2.3.15.2.1.1.5.5.1.15.2.25 8 6 Heave Measurements Calculations with mooring Calculations without mooring.5.1.15.2.25 6 45 Pitch Measurements Calculations with mooring Calculations without mooring x 4 L/A 4 x 6 L/A 3 2 15.5.1.15 f (s -1 ).5.1.15 f (s -1 ) Roll Yaw
Conclusions Potential flow solutions to first- and second-order provide a good overall analysis of the most important wave induced loading factors for ships Hybrid models can efficiently include more physics in a mostly potential flow solution (mooring systems, empirical viscous damping forces, etc.)