Experimental determination of hydrodynamic coefficients

Transcription

1 Experimental determination of hydrodynamic coefficients 1

2 Needs Analytical results for few geometries only Analytical damping: Linear (radiation only) Coefficients dependent of submergence Coefficients dependent of amplitude? Derivative of added mass needed in water entry equations

3 Methods Free oscillation (decay) tests. Forces oscillations. (Fixed structure in waves) 3

4 Free decay: Set-up for tests at moderate frequencies 4 Side view Front view

5 Free decay tests: Set-up for tests at low frequencies 5

6 Free decay curve 1 x n-1 x n+1 Linear damping: ( ζ ω ϕ) ζωt xt () Xe cos 1 t = + x/x.5 ω T n X(t) Relative damping: B ζ = = B cr B Mω -.5 x(t) x n ω *t Added mass estimate m t ( a) A ω = 1 + m ω s ω ω ( a) m s m t Natural frequency in air Natural frequency in water Structural mass (test object) Efficient mass of test rig 6

7 Damping estimate Amplitude ratio x x ln n n 1 ζωt e = = e π ζω t+ + ω x x n e = ζπ = πb Mω n+ 1 Quadratic damping: ζπ Linearization equal dissipated energy: T / T / n F ( xdx ) = Bxdx D n Linear + quadratic damping term: 16 x B = B + B e n 1 3 Tn e F ( D x ) = Bx 1 + Bx x 7

8 Plotting experimental data free decay p i Bi = M p p 1 8

9 Forced oscillation Methods: Sinusoidal motion Zero crossing Curve fitting Model equation must reflect physics Damping / drag forces only Inertia forces only x,f tot ωt 9

10 Testing in waves Measure forces directly Difficult to extract added mass and damping Relies on correct relative motion 1

11 Sensitivity to free surface / bottom proximity Circular, horizontal disk Added mass solid disk, R=1 m, H/R=5 T=4.19 sec T=3.5 sec A 33 /ρr D/R Free surface bottom

12 Sensitivity to interaction. Coupling Moonpool motion -structure.5 Non-dimensional added-mass A 99 for coupled system Surface-elevation at field-point #1, position (x,y)=(,). Waveheading: A 99 /ρl k (L=L pp / and k=3,4 or 5) draft:3.1m draft:4.69m draft:7.5m draft:1.5m draft:15m draft:1m draft:67.5m ζ/ζ a [-] 1 5 draft:3.1m draft:5.63m draft:1.5m draft:18m draft:67.5m Frequency [rad/s] Frequency [rad/s]

13 13 Perforated & ventilated structures, Oseberg sør template,.

14 14 Protection cover (approx 14 * 19m)

15 SIMO Modelling of subsea structures 15

16 16 Results from model tests: added mass vs motion amplitude

17 Outline of solution 3 domains z = d z = h 3.. z a R Disk 1. Domain 1: R a, z h. Domain : R a, z d. Domain 3: R a, d z h Boundary value problem considered for the velocity potential i t Φ = Re{ ϕe ω }: Laplace equation valid in fluid domain: ϕ =. ϕ The linearized free surface condition: g ωϕ= on z = h z No flow through bottom: ϕ = on z = z A radiation condition as R. On the disk two cases are considered: ϕ a) A solid disk in harmonic heave motion: = ω X A, z I.e. the standard no through-flow condition. b) A perforated disk in harmonic motion. The pressure drop over the disk is considered: 17 1 τ μτ Δ p( R) = ρ V ( R) V ( R) [1] r r

18 Pressure drop 1 τ Δ p R = ρ μτ ( ) V ( R) V ( R) r r τ μ : Perforation ratio : Discharge coefficient 18

19 Matching. Principles In each domain: Potential written as sum of eigenfunctions satisfying the boundary conditions at z=, z = h and R= infinity. Unknown coefficients found from matching potential and radial velocities at R=a. 19

20 Definition of hydrodynamic force harmonic oscillation F = A X + B X = A ω X cos( ωt) + B ωx sin( ωt) 33 A 33 A 33 A 33 A [ cos( ) sin( )] = ω X ρr C ωt + C ωt 3 A A B C C A B = = A 33 3 ρr B 33 ρωr 3 Zero perforation, infinite water: A33 8 CA = = 3 ρr 3 C B B = = ρωr 33 3

21 Keulegan Carpenter number Ordinary: Porous: KC = π X A R KC por = 1 τ μτ X R A 1

22 Added mass solid disk Added mass solid disk, R=1 m, H/R=5 T=4.19 sec T=3.5 sec A 33 /ρr D/R

23 Damping solid disk Damping solid disk, R=1 m, H/R=5 1.9 T=4.19 sec T=3.5 sec.8.7 B 33 /ρωr D/R 3

24 Added mass solid disk 1 8 Added mass solid disk, R=1 m, H/R=5 T=6.93 sec T=5. sec 6 A 33 /ρr D/R 4

25 Damping solid disk Damping solid disk, R=1 m, H/R=5 5 T=6.93 sec T=5. sec 4 B 33 /ρωr D/R

26 Added mass solid disk 1 9 Added mass solid disk, R=1 m, H/R=5 T= sec T=1.3 sec 8 7 A 33 /ρr D/R

27 Damping solid disk Damping solid disk, R=1 m, H/R= T= sec T=1.3 sec 1 B 33 /ρωr D/R

28 Added mass circular disk versus measured values for rectangular plate (fully submerged).5 Linearized A 33, R=.1913 m, H/R=6.137, D/R= T=1.4 sec T=1.6 sec T=1.9 sec A 33 /ρr KC por

29 Damping for circular disk versus measured values for rectangular plate (fully submerged) 3.5 Linearized B 33, R=.1913 m, H/R=6.137, D/R= T=1.4 sec T=1.6 sec T=1.9 sec B 33 /ρωr Reason For Deviation? KC por

30 Estimated and measured damping including estimate on the effect of the edge vortex Total Measured Measured Measured perforation Edge effect Hatch 18 C b μkc por

31 Estimated and measured damping including estimate on the effect of the edge vortex. 3.5 Total Measured Measured Measured Perforation Edge effect Hatch C b μkc por

32 Total force on hatch 18.9 Hatch F t /ω ρr μkc por

33 Total force on hatch..8 Hatch F t /ω ρr μkc por 33