Numerical simulation of the linear and second-order surface flows around circular cylinders in random waves

Deift University of

Technology

Ship HydrornechafliCS laboratory

Library

Mekelweg 2

21282 CD DeIft

Phone: -i-31 (0)15 2786873 E-mail: p.w.déheer©tUdelft.flI

Numerical Simulation of the Linear and SecondaOrder Surface Flows

around Circular Cylinders in Pndom Waves

Yougbwan Kirn1, P.». Sclavoirnas2

1Aii

Burean of Shipping, 24g Institute of Tedinology

Introduction

Fhdd flow simulatiOn near ohore suctures iii waves has been studied many limes. Most waiks wate

r

monochromatic orbjthiuw8tiC W5V. In the pzesent study, muIti-freq1Cy wave is considered, aiming the prediction

cf the multiple linear and quadratic txans fimctions with a single run and finally the application to realistic ocom spectrum. The method of solution is a Rankine Panel Method, Which has been developed at MIT. Ifi order to eicfract

the components of arbitruiy frequencies, the Fourier ansm was carried out using not FT bat an integral màthod. The computatiOn was cancestrated on the diffiactlön prthlein and the linear and quadratic transfer fendions were

with other beathmark4est renda

BomwIayValue Problems

The linear and second-order diffiadion prcbl are quite well known. 'fie velocity potential ofrantkvnor

multi-component incidOnt wave can be written as

j=P.e{çSj(x,y,z)e'}

(1)

J

= Re(

E(4,(x,y,z)e'' + Ø(x,y,z)e'"'])

(2)

where the subseript 1,2 means the order of problem. In the second-order wave,there w-e sum- and dirence-frequoncy components, being applied to numerical computation as an analyticibm or being generated during the numerical

mnn1atiOn of the second-order wave flow. The free-surface boundary conditions can bewiiilcuas follows

=

onz=O

(3)

where 6, means the delta fimdion which becomm unit when the second-order prOblem is considered, i.e. k"2.

Therefore, at a certain time, the linear solution should be known in order to solve the secOnd-order problem. A zero-flux condition must be imposed on the body surface and other rigid boundary.Furthermore, we need the radiation ccm&iti for the irniqn of solution.

Numerical Method

The method of solution is the Rankine pane! method, which has been developed by Sclavonnos(1988) for steady forward-speed problem and Nakos(1993) fir the unsteady ship motion, and in particular by lCim(1997) for the Sond order diffi-áctkar problem. Panels are flat, but the representation of the.physical quantities is approximated using a

B-spline basiS foflctico, which takes them as below

B(Z; i1)E Bu=

b5"(j,)

X

b5()

(4)

where (p) and (a) are the orders of Bspline fonction en twocoordinates along panel surface. Time integration sthe is a modified Euler method, so that the free-surface boundary conditions are wciden as follows:

'?k

=

oF(r'r1)

+

=

Zk"*

(cz)r1, 1) (5) where F, and P,mean the fCing terras of equation (3). In additica, the velocity pth an solid boundary and

normal flux on free surface can be obtained frani Grom's identit. The radiation condition is imposed using the

artificial damping zone, in which the kinematic free-surfa boundary condition is modified.

lanue

The same idea with Nakos(1993) 1997) can be estanded to the stability anaIyss fur an ibt«ay order of

B-spline fimction. For a certain disturbance on the deserstied free surface With canstintspacing panels, .& and y,

the dispersion relation in the disorate &mitin am be

wrii

as thliow

W=e''-(2-gAt2S)e+l=O

(6)

where t indicates time segment and S(vv) is the function of the wave number u and w , which is related with the Fourier tansfonnatien of basis fimction and Rankme somce. For the arbitraiy orders of basis flmctio,i dethied in equa$on(4),

(uv)

becomes

i

(-1}'»

vs)"

(-I)

s(rth..r}"

S(u,v)

= _____

Ju2_+v +(v&y+2, 1.I(uAs)2 +(v&v+2,r)2 _(u

+2

(us+2x)2+(vi)2

+ -

(-l)'2&c(us)(v&r)"

+

(uM+

2x)(v& + 2,r)

I(utg+ 2e)2+(vAs + 2e)2 when s =.O(&x,4y) O, and it

approaches the inverse of continuous

wave number.. Thereibre, equalinn (6)

recovers the continuous dispsioei

relarionas ,j-iO.Figure3thowsthe

comparison of disorote and continuous

dispsian relationS fur different pand

q, Le. the order of basis fiincsion. Here

ft

means4Ax/g&. It is interesting

that the diserepancy for bi-quadratic and

bi-4e order fimctions is not significant. A condition fur totflpotaliy neutral stability

cazibederivadasasixnplefrrm,

()

Figure 2 shows the contour plots of Stx I4ß2 when the aspect

ratio of panel is unit and u=w. Computation will be stable when

I SI tx/4fi2 i. in this figure, the basis fiinc*icms are the

bi-quadratic and bi-3M order flmdions, showing almost identical

borders. Therefore, we may conclude that the order of basis

function higher than bi-quadratic doesn't provide significant

irnpravnent in the viewpoint of the consistency and stability of

rumierical schome.

Fourier Transform

FF is Widely tised in the Fourier transfurin of timesignal. However, it may be not suitable when the frequencies of intetest are

not equally spad. In the present study, a Firier-traosfixnis$on

81

0.1 3 3 IA 0.5

ux/2 uàz/2Z

Figure 1 Diserete and continuous dispersion relations

(l1'+U') A/2

Fig.2 Cuutourof

_ISI&t4

program based on an integral concept was developed in order to actthe_{arbiltaty frequoncy amipanisite, whith a}

user wants to sele. The idea is simple. For a eu1aw ûmatian, whi _{is 'ritton as the sum of a constant and the sones} of exponontial fimdion, the fillowing integral is satisfied.

W

f(t) = C, +

e'

_J

fit) e

dt=C, J e

_{+ LC* J}

e«a7.h)r (9)

Here theweight_{fonction is an exponential function with frequoncy, u_ The integrals in right side are ivial and we}

knowtheanalytic solution, 'while the left integral must be obtained using rannoncal menIion. When we apply N+l freqjiendes. whi& are equal to the basis frequencies of

f(t)

a matrix equation for ithknown c can be assembled. In tbe second-order prqblem, the sinn- and difference-frequmicias can be applied In equation (9). b the resi nwnmil

coinputalion, there are e unexpected components, for example saw-tooth wave or slowly-decaying transient mode. Therethre in order to minimiz0 the numerical error, seme diumny fr tiendes we recommended to be included. These dummy frequencies must cover a certain bandwidth, minhmizlug an aliasing error.

NumericalCoiupulxtlOn& Results

The compUtation was carried for single cdinders, being bOEnmonnted and bmcated. Polar grid systesi with

proper stretthmg near the body was apphed. In the simulation of multi-component wav the solution grid must be fine enough to resolve the.shcrtest

wave and the computational

domain should be large _f

enougbtontaiflthalongest

_{

sf1

wave. Usually, in the second.

''

J order problem, the former 4

comes from the sian-frequency components, while

the later frOm the

dIfl'e-frequency. Fig. 3 showe the time signnk of the linear and

second-order surgéfrce co a o bottcan-mconted circular IO

cIinder. Four components are mixed in the linear

cign so that the 16(4X4)

mm-frequency and 16(4X4) Figure 3 The linear and 2 order serge forces on a bottom-mounted c34ilzder

diatice-frequcticy compon- radius(aYdepth(d)4).4, kaI.O,1.2,l.4,L6 ente are in the second-order

O.eod Ordar

i

Figure 4 Grid denden on QTF matrices: the same case withFlg. 3

signal. Quadratic transfor functions (QT!s) can be extracted from the second-ordercIgni using the Fourier transform

82

CO t(gA'6

H I.

described above. Fig.4 shows QTF matrices obtained from the time signais with different grid numbers, and the grid

dependency in this case is not significant in this case. The same computational

method can be extended to

random ocean spectra. Fig. 5

shows the instantaneous linear

and second-order wave profiles

near a truncated circular

cylinder with 10m radius. In

this case, I1TC spectrum of

Sea State 5 was adopted. The significant wave height is 3.25 m and the modal wave period is 9.7 sec. In the second-order

profile, local waves are significant near the body. Fig.

6 shows the force signals in Figure 5 Instantaneous wave profile near a truncated cylinder

random wave, and Sea State 6 IUC spectrum, sea state 5, d/a=4, Elevations are magnified.

wasapplied.lnSeaState6, the

significant wave height is 5.Om and the modal wave period is 114m. The second-order surge force is not significant compared with the linear force, while the second-order heave force cannot be ignored. Microseim effect may be a major source of the second-order heave force. The analysis of the second-order force is not simple sm a lot of sum-and difference-frequency components are mixed. The Fourier transform described above is not enough in this case. Therefore, signal-procesaing techniques will be very useful to analyze the nonlinear statistical characteristics of the

second-order random signaL

Suxg. forc.

A A

A&h,1

AL

& A L

lineaTJ'!

2nd

io

io

i8o 1k 280 210 2%0 280 HeaVe force

ib

io

i8o iòo 260 210 2.0 280 280 360 310 s80 _t(g/a)1" Figure 6 The linear and2nd order throeson the cylinder in Figure 5, 1TTC spectrum, Sea State 6

[i] Sclavounos, P.D. and Nakos, D.E., Stability analysis of panel methods for free surfuce flows with forward speed, Froc. of1 ONR, bague, 1988

(2] Nakso, D.E., Stability of transient gravity waves on a discrete free surfuce MIT report, 1993

[3] Kim, Y. and et al., Linear and nonlinear interactions surfu waves bodies L a three-dimensional Rnnkme panel

methods, Applied Ocean Research, VoL ¡9,1997