On the short term distribution of the individual maxima of non-linear hydrodynamic forces

On the Short-Term Distribution of the Individual Maxima

of Non-Linear Hydrodynamic Forces

by

Tor Vinje, Assistent Professor, Division of Shiphydrodynamics, University of Trondheim,

Norwegian Ins titu te of Technology, Trondheim

Abstract

The short-term distribution of maxima of hydrodynamic forces calculated according to Morison's formula: F =kjI+k2 UI UI, where U isa stationary,

Gaussian stochastic process with zero mean, is performed. This distribution is found to coincide with the Rayleigh distribution for OKFmax<Fo and Prob (Fm> f) is found to approach zero exponentially for Fmax> F0, where

2

F =-'

0

Var()

2k2 Var(U)

The exact solution is found to coincide with these functions for the entire domain of Fmax, Fmax>O, with an error of an order expected from the

numerical-integration formulae used.

1. Introduction

The hydrodynamic forces acting on bodies situated in moving fluid are often calculated according to the so-called Morison's formula:

F=k1

+k2UIUJ (1.1)

where U is the velocity of the fluid at one representative point (when the body is assumed not to be present). The

formula (1.1) is restricted to the case when the body itself is at rest.

For circular cylinders the coefficients k1 and k2 in Eq. (1.1) are given as:

irD2

k1 = Cp

(1.2)

k2 = Cp-

D (1.3)

where D is

the diameter of the cylinder, and the

dirnentionless coefficients Cm and Cd are of order 1 (Cm

= 2 and Cd 0,8 are representative values). For other types

of cylinders D is

taken as one characteristic dinìension of the cylinder, and Cm and Cd are given

according to this length. Some examples of values of Cm

and Cd for different types of cylinders are given in

Myers & al /6/. When the velocity, U(t), is due to ocear waves, it has been shown that U(t) becomes a stationary, Gaussian distributed stochastic process with zero mean. The calculations in the rest of this paper will be based on

this assumption. In addition the wave-spectrum will be

assumed to be "narrow", which is shown to be fairly well satisfied for most observed ocean wave spectra.

The calcualtion of the statistical distribution of the local maxima of the stationary stochastic process F(t)

will be done according to a method developed by

Huston & Skopinski /1/. This method is thoroughly

discussed by Lin /2!, to whom the reader is referred for more details about the method.

2. The formulation of the problem.

According to ¡1/ and /2/ the frequency of positive crossing through a level T? of F is given as:

00

N(T?)

= f

f(77, )td

(2.1)

o

where Pft is the joint probability density function of F and F. The dot stands for differentiation with respect to time.

In turn the probability that the maxima of F(t) will

exceed r is, for "narrow band" spectra, approximately

given as:

Of(77)

_{Prob(Fmax>ii) = N(i)IN(0)}

(2.2)

Hence: Whenever Pît is known, Of can be calculated

either by means of analytic or numerical methods. From Eq. (1.1) it is found that:

Deift Univer$ity of Technotoy

Sai? HVromccncs Laboratorj

Library

Mekelweg 2- 2ê28 CD OeIft

The Netherlands

Xl =

X2 = U/\/o

x3=

(U+U)

'vi4

where m4m0 - m22 ni0 m4 and 00 n mn

= f

w S(w)dw

o

where S(w) is the spectrum of the variable U(t). It is

easily stated that

_{(X1, X2, X3) are}

_{stochast.ically} independent and Gaussian distributed with zero mean and variance equal to unity. Introducing Eq. (2.4)

- (2.6) into Eq. (1.1) and (2.3) one can define the

following new, dimensionless, variables:

= k1/

= 1 +;iX2IX2J (2.9)

- XI

(2.4)

The coefficient e, defined according to Eq. (2.7), tells (2.5) whether the spectrum is "narrow" or not. Cartwright &

Longuet-Higns /3/ have shown that

for "narrow"

spectra e = O and following Z2 is rewritten in the much simpler way:

Z2 = X2+2pX1IX2Ì (2.12)

Now the problem is transferred from finding Pfj into (2.7) _{finding Pz1}

z2 from the relatively simple equations (2.9) and (2.12) 3. The calculation of_Pz1

_z,

(2.6) (2.8) and

Z2 = 'z

k1m2 m2 where k2mo

P r

k1m2

Fig. (3. 1) Sketch of Eq. (2.9) and Eq. (2. 12) for p = 0.25

=

2 XsX2+2PX1IX2I

m2

(2.10)

(2.11)

lt is clear from the equations (2.1) and (2.2) that only the case when (Z1 >0 & Z2 >0) is of interest, and this is, for simplicity, assumed in the rest of the paper.

To calculate Pz1z2 it is necessary to find the roots of Eq. (2.9) and (2.12) and the domain in the

(X1, X2)-plane where (Z1 <z1 & Z2 <z2)

On Fig.

(3.1.) where Eq. (2.9) and (2.12) are

sketched, the three possible roots are indicated as P1, P2

and P3. It is easily shown that for z1

small enough

(z1 <

-) only one root, P1, will exist. For z1 > the condition that only P1 exists depends on z2. The

Ê = k1 + 2k2 ÚJUI (2.3)

and since U(t) is a stationary, Gaussian process, U is

stochastically independent of both U and U, but U and

U on the other hand, are coupled. Due to the

con-venience of using stochastic independently and

shaded area on Fig. (3.1) indicates where

(Z1 <z1 &Z2 <z2)

For z1 <Pz1z2 is easily found as:

a(x1,x2) PZIZ2

p'

i,z2) PX,X2

,,,)

abs _{a(z z}

)I)(3.l)

where x1 , and x2 2 have to be introduced.

(Eri,n) are the coordinates of the point P,.

When the roots P2 and P3 exist, Pz1z2 is found as:

Pz1z2 - Pi _az1az2a2 (3.2)

where lis the integral of Px,x2 over the hatched area on Fig. (3.1). 1f we formally write Eq. (2.12) as:

X2 =

(X1,Z1,Z2)

Pz1z2 can be found: aE2 òis

= p1(zl,z2)+p12(E2,2)

az1 aE3

ap

PX1x2(E3,3)

where the index P means that the parenthesis has to be calculated for (xi ,x2) = (En,ri)

Hence Pz,z2 is found as:

Pz,z2 =exp((E+))

1

(3.5)

1+2hz, +6J.L2

when only P1 exists, and as:

exp(} (E +))

2 _2ir i

- exp(-(+))

_2ir

_{1-2pz +6p2}

i

I (E +))

_+6p2

+ -

exp(-2ir

when all the three roots exist.

The roots, P, now can be found as follows: Eliminating X1 from Eq. (2.9) and (2.12) gives

X+2i2

1

- X2--

_p

P1 is found as the root of Eq. (3.7) which is real for any value of Z1 > O (and X2 <O). This is given, according to

any standard textbook on equations of the third degree, as:

For X2 > O P2 and P3 are found as the two roots which become non-real when Z1 becomes small enough. These two roots are given as:

The restrictions on Z1 and Z2, which ensure that P2 and P3 will exist, are

z2 <f(2iZ1l)312!12p

(3.19)

Z1>

(3.20)

which indicates when Eq. (3.6) has to be taken into account.

4. The calculation of û ()

The integral in Eq. (2.1) is easily found through Eq. (2.9) and (2.12) and Pfdfd = Pzjz2 dl1 dz2 as m21

N(i) =

Z,Z222

Tm'1

-where y - = w is

the specified frequency of the narrow Spectrum.

Before discussing the calculation of Eq. (4.2), the following two asymptotic expansions are examined:

N(17)/N(0) when

i-0

and

77-3.

It is self-evident that for ? - Oonly the linear term of Eq. (1.1) is significant and hence N(77) becomes:

N(i) =

- exp(r2/2)

(4.3) (3.12)

= 2\/

cos(2ir/3/3)

E2 = Z,P

and

2v'jcos(J3)

(3.13) (3.14)

= Z1 i

where

p = (l-2,uZ1)/6p2

and is given by:

(3.15) (3.16) cos

= q/,/

,

0<5<ir

(3.17) q = Z2J4p2 (3.18)

cl =

(3.8) E1 = Zi -'-pE2 (3.9)

where q and p are given as:

q = Z2 I4p2 (3.10) p = (1+2pZ1)/6p2 (3.11) (3.3) (3.4) (3.6) (3.7) (4A) (4.2)

And 0(n)=N(n)/N(0) is given according to the Rayleigh

distribution as:

0()-* exp(-z2 ¡2)

(4.4)

for 77 - O

where z

_{/k1 '/}

For n the non-linear term gets dominant and the solution of Eq. (2.9) and (2.13) has to be close to the

solution of:

where and are found approximately as:

and i i Pzjz2

exp(-(+j))

_1-2pz1+6p Z2+

₁₊

z2 2p 2p which gives: I i I Pz1 z2 _{4iz exp (-gt}

and can be recognized as a point near to P. Hence the asymtotic expansion b) gives:

(4.8)

z1 z2

-y)exP(-i,,r--) (4.11)

Integration then gives:

00

1 Z

N(ni) ¿13

f

_{Pziz2(ZIZ2)Z2dZ2}

= exp(---- --)

(4.12)

2n 8p 2p

where z= 77/k1 '/

It is clear that N(0) cannot be calculated by putting 770 in Eq. (4.12), but rather found from the asymptotic expansion a);

N(0)=

-.

Following: 2ir

0(77)-exp(- --)

(4.14)

for

The exact value of 0(77) has been calculated from Eq. (4.2) by quadrature, using a 16 points Gauss-quadrature formula for the contribution from P2 and P3 and a 15 points Laguerrequadrature formula for the contnbution

from P1. The formuI are taken from Abramowitz and

Stegun /4/. These integrals are compared to the

asvmp-totic expansions Eq. (4.3) and Eq. (4.14), as shown in

Fig. (4.1), and the correspondence is remarkably good: The relative error is found to be less than 3%, which is of )he same order as the expected error from the numerical integration. 0.2

exact solution

calculated

points. 1 2 3 4 5 6

Fig. (4.1) oasafunctionofzfor=a2

One surprising thing is that Eq. (4.4) and Eq.

(4.14) coincide for z = -and that the computed values of 0(77) coincide with Eq. (4.3) for z K -- and with Eq. (4.14) for

>

i.

In addition 0(77) given either by Eq. (4.3) or Eq. (4.14) has the same derivative (i.e. proba-bility density function) for z= -. _{lt looks as though the}

exact solution of the problem is found in closed form,

and the author is not able toprove thisrigorously, due

to the complicated mathematical formfor Eq. (3.5) and

(3.6)(when introducing Eq. (3.8)- (3.15)).

5. Discussion

Due to the fact that both Eq. (4.4) and Eq. (4.14) are of the exponential type (according to Gumbel /5/), the "most probable largest value" of F,P,can be calculated, in terms of z, as =

'2 in N when

<

_(5M and (4.5)

Z1 =pX

Z2 = 2pX1X2 which is written: X2 r=\/Zl/bL (4.6) (4.7) Xi = Z2/2pX2

=

2p1nN+1when

>

4ji 2ji

The solutions of

J2 in N =

-for N = iO3, i0 and iO are given below as:

N=

= 0.1386

N = io4 =0.1165 N = io5 = 0.1042

which are the,l'owest values of ji that permit calculation

ofaccording to Eq. (5.2).

Introducing Eq. (1.2) and (1.3) into Eq. (-.11) one gets:

2 _Cd m0

¡1=

_r

Cm Dm2

(5.5)

Bringing into mind that U(t) is given terms of a narrow wavespectrum, p is rewritten:

2 Cd U

P--:

_ii;

t5

where 02 is the variance of the wave.elevation of the

fluid particles at the point where F is calculated.

Assuming that F is calculated at the surface, a is given

by means of the

significant wave height, H113,

a = H113, and hence:

Cd H113

-c;:1

D Putting ji equal to , one gets:

For

D/ H1,3

>

has to be calculated according to Eq. (5.1) For

D/H113

<

has to be calculated according to Eq. (5.2)

These results are based on: Cm = 2, Cd = 0.8. and p 8 (According to Eq. (5.4)).

The transformation fromto is done according to Eq. (2.9):

=k/2z

2Hv3p7rD2Cm

.z

16 (5.8) (5.6) (5.7) 6. Concluding remarks.

It has

been shown that

in the case when the

wave-spectrum can be regarded as "narrow" (e = O) the distribution of local maxima can be calculated according

to the relatively simple formulae: Eq. (4.4) and Eq.

(4.14). If these formulae are going to be used for other

spectra, one has to show that the curve in Fig. (3.1)

forms an upper limit to the curves for e * O, parallel to what Cartwright & Longuet-Higgins /3/ showed for the Rayleigh distribution for linear variables. Though this is not proved, Cartwright & Longuet-Higgins' work indi-cates that it is expected to be so in this case, too.

The rigorous proof can be given according to a

method outlined by Lin/2/, where both F, F and F are taken into account, and N(ri), which in this case is the frequency of maxima above the level 17 is calculated as:

N(7?)

= f

pf(7?,0,f)jfldf

(6.1) and 0(77) as: dO(7?) d7? NO7)/NT where NT

7

N(7?)d7? (6.3)

It is easily seen this calculation is more complicated that the one performed in this paper. In addition more

parameters (than p) will be involved, which makes a systematic variation of parameters more time con-suming.

References.

/1/ Huston, W. B. and Skopinski, T. M.:

Probability and frequency characteristics of some flight buffer loads.

NACA, TN, 3733, August 1956. ¡2/ Lin, Y. K.:

Probabilistic theory of structural dynamics. Mc. Graw-Hill, N.Y.

/3/ Cartwright, D. E. and Longuet-Higgings, M.S.: The statistical distribution of maxima of a random

function.

Proc.Roy.Soc.vol. 237A (1956) /4/ Abramowitz, M. and Stegun, L.A.:

Handbook of mathematical functions. Dover, N.Y., 1970.

/5/ Gumbel, E.J.

Appi. Math, ser. U.S. Bur.stand. no 33 (1954) /6/ Myers, J. J. and Holm, C. M. and Mc.Allister, R. F.:

Handbook of ocean and underwater engineering.

Mc. Graw-l-lill, N.Y.

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Type %iarter: Ja-Du Offset-Service. Nesodden. Lay.ouf: Bryn/ui! Saastad. Nesodden, Printed by: Euro Trykk A /5, Oslo, Norway.

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