WP1 Hydrodynamics - Overview - Aalborg Universitet · WP1 Hydrodynamics - Overview Harry B. Bingham...
Transcript of WP1 Hydrodynamics - Overview - Aalborg Universitet · WP1 Hydrodynamics - Overview Harry B. Bingham...
WP1 Hydrodynamics - Overview
Harry B. BinghamDTU Mechanical Engineering
April 26, 2012
WP1 Hydrodynamics - Overview
Schematic of the hydrodynamic problem
f
S
S
f
Deep water waves
Wave forcing at the device
Refraction from the bottom
Diffraction from coastal boundaries
Wave farm
Triad interaction
WP1 Hydrodynamics - Overview
S
f
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f
S
> 100m
10 − 50m
WP1 Hydrodynamics - Overview
The Standard Wave-To-Wire-ModelNewton’s 2nd law
Mjk xk (t) = f HDj + f ML
j + f PTOj , j = 1, 2, . . . , 6.
WP4 WP1 WP2 WP3
WP1 Hydrodynamics - Overview
WP-1 The Hydrodynamic Forces
1 The Incident Wave climate at the device location (10 - 50m waterdepth)
Spectral description of long-term wave conditions (DHI)Deterministic modelling of extreme conditions (DTU)
2 Mildly-nonlinear wave-structure interaction - Optimization andlong-term loading
Linear hydrodynamic coefficientsNonlinear hydrostatics, mooring system, and PTO forces includedin the equations of motion (DHI/DTU)Model under development to compute linear hydrodynamics on avariable depth, possibly semi-enclosed domain (DTU)
3 Strongly nonlinear wave-structure interaction - Final designanalysis
Fully nonlinear inviscid flow solver (PhD - DTU)Viscous flow solver (DHI)
WP1 Hydrodynamics - Overview
WP-1 The Hydrodynamic Forces
1 The Incident Wave climate at the device location (10 - 50m waterdepth)
Spectral description of long-term wave conditions (DHI)Deterministic modelling of extreme conditions (DTU)
2 Mildly-nonlinear wave-structure interaction - Optimization andlong-term loading
Linear hydrodynamic coefficientsNonlinear hydrostatics, mooring system, and PTO forces includedin the equations of motion (DHI/DTU)Model under development to compute linear hydrodynamics on avariable depth, possibly semi-enclosed domain (DTU)
3 Strongly nonlinear wave-structure interaction - Final designanalysis
Fully nonlinear inviscid flow solver (PhD - DTU)Viscous flow solver (DHI)
WP1 Hydrodynamics - Overview
The Exact Equations of Motion for a Floating Structure
Consider two coordinate systems:
x = [x , y , z] Fixed in space
x = [x , y , z] Fixed to the structure
The frames are related by:
xt = [x1, x2, x3] A position vector, and
xr = [x4, x5, x6] A rotation vector
WP1 Hydrodynamics - Overview
A general vector in the two frames is related by
x = L(x − xt ) x = xt + LT x
where
L =
c6c5 s6c5 −s5
−s6c4 + c6s4s5 c6c4 + s4s5s6 s4c5
s6s4 + c6c4s5 −c6s4 + s6s5c4 c5c4
and the sequence of rotations is: yaw (x6), pitch (x5), roll (x4); withci = cos (xi) and si = sin (xi).The angular velocity of the WEC in the body-fixed frame is:
ω = [−x6s5 + x4, x6c4c5 + x5c4, x6c4c5 + x5s4]
which is related to the time rate of change of the Euler angles by
xr = Wω =
1 s4s5/c5 c4s5/c5
0 c4 −s4
0 s4/c5 c4/c5
ω.
WP1 Hydrodynamics - Overview
Newton’s law at the body Center of Gravity (CG)
E[
xt˙ω
]
= q
where
E =
[
m −[rg × m] LT
(rg × m)LT Ig [rg × (rg × m)]LT
]
and
q =
[
F − m[ω × (ω × rg)]
M − m{rg × [ω × (ω × rg)]} − LT (ω × Igω)
]
with F and M the total applied external force and moment(hydrodynamic, hydrostatic, mooring system, PTO...).
WP1 Hydrodynamics - Overview
The Exact Hydrostatic Forcing
Fs + w = ρg∫ ∫
Sb(t)z n dS − mgk
Ms = (rg × w) + (rb × Fs)
with n the normal vector to the body surface and k the unit vector inthe vertical.
WP1 Hydrodynamics - Overview
Test Case: A Heaving Sphere
The grids for the static (left) and dynamic (right) calculations.
−5
0
5 −5
0
5−5
−4
−3
−2
−1
0
1
2
3
4
5
yx
z
−5
0
5 −5
0
5−5
−4
−3
−2
−1
0
yx
z
WP1 Hydrodynamics - Overview
Preliminary Results
Heave response to increasingly larger waves:
0 1 2 3 4 5 6 7 8 9 10−2
−1
0
1
2
3
4
t/T
x 3/A
LinearH/L=0.020475H/L=0.051187H/L=0.07678
WP1 Hydrodynamics - Overview
The Next Steps
In Progress:
Validate the hydrostatic forces using a forced motion
“Publish” to WP4
Compare with experimental measurements for the WaveStarFloater
Longer Term Developments:
Use the waterline defined by the incident wave rather than thestill water plane
Split the diffraction forcing into incident (Froude-Krilov) andscattered components and compute exact Froude-Krilov forces.
WP1 Hydrodynamics - Overview