statistical properties of wave kinematics and related forces

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Statistical Properties of Wave Kinematics and Related Forces

Carl Trygve StansbergMARINTEK/CeSOS , Trondheim, Norway

CeSOS Workshop on Research Challenges in Probabilistic Load and Response Modelling,Trondheim, Norway, 19 December 2005

Test 1826 - Irr W - 721s k0A0=0.395

-0,5

0

0,5

0 0,5 1 1,5

Ux/Uref0

k0z

LinearSecond-orderGrue's methodWheeler (from linear)Wheeler (from measur)LDV experiment

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Contents

- Linear and second-order random wave kinematics modelling

- Case study, storm wave

- Simplified distribution model for free-surface velocity peaks

- Effects on forces


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Linear and second-order random wave kinematics modelling:

- Here: Focus on horizontal particle velocities u(t,z) at a given vertical level z, for long-crested irregular waves on deep water(for simplicity)

- Linear: u0(t,z) = i A(it) exp(jit) exp(kiz) z ≤ 0 i

- Second-Order:

utot(t,z) = u0(t,z) + u(2 sum)(t,z) + u(2 diff)(t,z) z ≤ 0

utot(t,z) = u0(t,z) +(∂u0/∂z│z=0)∙z + u(2 sum)(t,0) + u(2 diff)(t,0) z > 0

(light blue: in finite water only)

Here, u(2 sum) and u(2 diff) are given by their quadratic transfer functions (QTFs).

We shall in particular consider the free-surface velocity u(z=(t)) at the crest peaks Ac.


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Examples from numerical simulation, extreme events in random records:from Stansberg et al. (OMAE 2006)


Test 100 - Irr W - event #2 k0A0=0.378

-0,5

0

0,5

0 0,5 1 1,5

Ux/Uref0

k0z

Linear (up to z=0)Second-orderGrue's methodWheeler (from linear)Wheeler (from 2nd ord)

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Present case study

- Hs=16m, Tp=14s, Gamma=2.5 (steep 100-year storm sea state)

- Full-duration 3-hours storm record (1 realisation only, with 1000 wave cycles)

- Simulate linear and second-order

- Study probability distribution of crest heights Ac & velocity peaks Uc


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Sample time series, linear and second-order elevation


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Sample time series, linear and second-order free-surface velocity


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Sample time series, linear and second-order velocityat z=0


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Probability distributions from simulations, linear and second-order crest heights


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Probability distributions from simulations, linear and second-order velocity peaks


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Simplified second-order formula for maximum crest height:

E[Amax] = AR (1 + ½ kp AR )

(from Stansberg, (1998), based on Kriebel & Dawson (1993), Tayfun (1980))

whereAR [ (2 ln (M)) + 0.577/ (2 ln (M))]

kp = wave number = (2fp)2/g

This follows from modified Rayleigh distribution model for short-time statistics of nonlinear peaks a’:

P[A < a’] = 1 -exp [-a2/22]

where a = a’(1- ½ kpa) are the linear crests

This formulation is based on second-order regular wave theory.


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Simplified second-order formula for maximum crest height (cont.):

The simple formula compares surprisingly well with full second-order simulations, see e.g. Stansberg (1998):

although the negative difference-frequency effects are neglected.

Likely reason: Use of the spectral peak frequency fp probably leads to too long wave periods for the highest crests:

Forristall (2000) suggests fAmax = 1.05fp for the highest crests.

We think it should be even shorter, because in a random simulation it is locally shorter at a high peak than over the whole cyclus (found from Hilbert transform analysis of linear records). Thus we have found fAmax 1.15fp


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Similarly, we use the same approach in suggesting a simplified second-order formula for high velocity peaks Uc at the free surface

Assume regular waves:

utot(t,z) = u0(t,z) +(∂u0/∂z│z=0)∙z

which can be written, under the crest z=Ac :

uC = u0(1 + kp Ac )

and the peak value distribution function becomes:

P[u < uC ] = 1 - exp [- u02/2u

2] = 1 - exp [- uc2(1- kp Ac )2/2u

2]

(As for the crest heights before, we choose to use kp here, to compensate for the neglecting of difference-frequency terms).

Notice: Nonlinear term is twice as important as for crests


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Check of simplified crest height distribution(thin line), vs. simulations

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Check of simplified velocity peak distribution(thin line), vs. simulations

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Effect on velocity-determined wave forces(introductory study)

Type 1: Wave slamming

F ½ Cs A Uc2

(slamming coefficient Cs 3 – 6, depending on relative angle)

Type 2: Drag forces

F = ½ D CD u |u| (and then integrated up to the free surface)

Here we limit our study to look at properties of the peaks of free-surface velocity squared – indicates the statistical properties of local forces


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Sample record of u2, linear and second-order kinematics


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Total 3-hour record of u2, with linear and second-order kinematics (u > 6.5m/s)


Number of “events” increased by several 100%!

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Probability distributions of u2, linear and second-order kinematics, including comparison to exponential model


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Probability distribution of u2, with second-order kinematics,

compared to simplified distribution based on:

P[Y < y’] = 1 - exp[-y2/y2]

where

y = u02 (from linear velocity)

y’ = y(1 + kpAc)2 (from nonlinear velocity)

(simplified distribution shown with dashed line)


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Summary with conclusions

- Statistical properties of particle velocity peaks in steep random waves have been investigated.

- Second-order contributions lead to 30% increased maximum free-surface velocities (while crest heights are increased 15%)

- A simple distribution model, similar to Kriebel & Dawson’s for crests, compares well with the simulated results

- Preliminary studies of effects on resulting wave forces have been made by considering velocity squared.

- The results show considerable contributions from second-order kinematics on forces – almost 100% increase. A simplified, modified exponential distribution model compares well with simulations

- Further work recommended on statistical properties of integrated drag forces, and on related moments around z=0


statistical properties of wave kinematics and related forces

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