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