Vol. 43 • No. 1 February 1996
ÜCHirFSliHIIK
M o t i o n S i m u l a t i o n o f a C y l i n d e r at t h e S u r f a c eo f a V i s c o u s F l u i d
b y L i o n e l Gentaz, B e r t r a n d Alessandrini and G e r a r d D e l h o m m e a n
F o u r i e r R e p r e s e n t a t i o n of N e a r - F i e l d F r e e - S u r f a c e F l o w s by Francis Noblesse and C h i Yang
A H i g h O r d e r P a n e l M e t h o d B a s e d o n S o u r c e D i s t r i b u t i o n a n d G a u s s i a n Q u a d r a t u r e
by Jen-Shiang K o u h and Clmn-Hsine Ho
T E C H N I S C H E I J M V E R S I T E r r Sctieepshydromecliaiiica A r c h i e f M e k e l w e g 2 , 2 6 2 8 C D D e l f t T e l : 0 1 5 - 2 7 8 6 8 7 3 / F a x : 2 7 8 1 8 3 6
Published by
Schiffahrts-Verlag „HANSA", Hamburg
D
a
s
F
a
c
h
b
u
c
h
f ü r S c h i f f b a u ,
S c h i f f s m a s c h i n e n b a u
u n d S c h i f f s t e c h n i l c
Herausgeber Prof. Dr.-Ing. H. Keil
338 Seiten, Format 14,5 x 21,5 cm,
zahireiche Skizzen und Tabellen,
Efalin, DM 98,40 zzgl.
Versandkosten, inkl. MwSt.
ISBN 3-87700-089-4
T E I L I
SCHIFFBAU - SCHIFFSMASCHINENBAU Betriebsfestigkeit schiffbaulicher Konstruktionen Prof. Dr.-Ing. H. Petershagen, Dr.-Ing. W. Fricke und Dr.-Ing. H. Paetzold
Fatigue Strength of Ship Structures
Prof. Dr.-Ing. H. Petershagen, Dr.-Ing. W. Fricke und Dr.-Ing. H. Paetzold
IVIoderne Schiffsiinien Dr.-Ing. G. Jensen l\1odern Ship Lines Dr.-Ing. G. Jensen
Wasserstrahiantriebe fiir Hochgeschwindigkeits-fahrzeuge
Prof. Dr.-Ing. C. F. L. Kruppa Waterjets for High Speed Propulsion
Prof. Dr.-Ing. C. F. L. Kruppa Schiffsgetriebe und -kuppiungen Dr.-Ing. W. Pinnekamp
Marine Gears, Couplings, and Clutches
Dr.-Ing. W. Pinnekamp
Abwarmenutzung auf Seeschiffen mit Dieselmotorenanlagen Dr.-Ing. K. Abel-Günther Waste Heat Recovery on Board Ships Dr.-Ing. K. Abel-Günther
Angewandte Schiffsakustik
Teil 1: Einführung in die Akustik, Schallabstrahlung ins Wasser, Zielpegel, ZieimaB
Dipl.-lng. K. Albrecht Applied Ship Acoustics Part I:
Dipl.-lng. K. Albrecht
T E I L I I
Verzeichnis der Schiffswerften der Bundesrepublik Deutschland
Technische Ausstattung und Baukapazitat Ausgeführte Neubauten in den letzten Jahren
T E I L I I I
Organisationen und Institutionen für den Schiffbau
B E S T E L L - C O U P O N
Ich / Wir bestelle(n)
Exemplare
Schiffahrts-Verlag
„Hansa"
C. Schroedter & Co.
P o s t f a c i i 9 2 0 6 5 5 D - 2 1 1 3 6 H a m b u r g Tel. (040) 7 9 7 1 3 - 0 2
U S t . - l d . - N r . D E 1 1 8 1 3 8 9 0 3
Handbuchi der Werften, Band XXII, DM 98,40 zuzüglich Versandkosten, inkl. Mehrwertsteuer
N a m e / F i r m a HanMnidi C)Ci' W c i f t c n Dar.ti X X I I Ihre USt.-ld.-Nr. Anschrift Datum/Unterschrift-X
ilIMIIIlMiiAlH
Journal for Research in Shipbuilding and Related Subjects
S H I P T E C H N O L O G Y R E S E A R C H / S C H I F F S T E C H N I K was f o u n d e d by K . W e n d e l i n 1952. I t is edited by H . S ö d i n g and V . B e r t r a m i n collaboration w i t h experts f r o m universities and m o d e l basins i n B e r l i n , D u i s b u r g and H a m b u r g , f r o m Germanischer L l o y d and other research organizations i n Germany.
Papers and discussions proposed for publication should be sent to Prof. H. Söding, Institut f i i r Schiff-bau, Lammersieth 90, 22305 Hamburg, Germany; Fax +49 40 2984 3199; e-mail soeding@schiffbau. uni-hamburg.d400.de. Rules for authors, newest abstracts, keyword index and editors' software see under http://www.schiffbau.uni-hamburg.de
V o l . 43 • N o . 1 • F e b r u a r y 1996
L i o n e l Gentaz, B e r t r a n d Alessandrini and Gerard D e l h o m m e a u M o t i o n S i m u l a t i o n o f a C y l i n d e r a t t h e S u r f a c e o f a V i s c o u s F l u i d
Ship Technology Research 43 (1996), 3-18
A computer code t o determine the 2D viscous, l a m i n a r , incompressible flow a r o u n d a surface-piercing cylinder i n f o r c e d heave is described. T h e Navier-Stokes equations are discretised b y a finite difference m e t h o d . T h e free surface elevation is c o m p u t e d f r o m the k i n e m a t i c e q u a t i o n independently of t h e s ol ut ion of the t w o Unear systems i n velocity and pressure. T h e o r i g i n a l aspects of this code are b o t h consideration of the free surface b o u n d a r y conditions and re-g r i d d i n re-g of the m o v i n re-g physical d o m a i n at each i t e r a t i o n . H y d r o d y n a m i c forces and coefficients are calculated and compared w i t h experimental results and other numerical calculations i n perfect and viscous fluid. T h e interest of this t h e o r y fies i n the numerical c a l c u l a t i o n of viscous effects, w h i c h , at this t i m e , can only be obtained by experiments or e m p i r i c a l f o r m u l a s .
K e y w o r d s : C F D , added mass, d a m p i n g , Navier-Stokes, free surface, heave
Francis Noblesse and C h i Yang
F o u r i e r R e p r e s e n t a t i o n o f N e a r - F i e l d F r e e - S u r f a c e F l o w s Ship Technology Research 43 (1996), 19-37
A Fourier representation of free-surface effects, based on a decomposition i n t o wave a n d near-fleld components, w e l l suited f o r numerical evaluation i n the near field and the f a r field is given. T h i s Fourier representation of free-surface effects is valid f o r an a r b i t r a r y d i s t r i b u t i o n of sources a n d / o r dipoles and f o r a wide class of water waves i n c l u d i n g t i m e - h a r m o n i c and steady flows, w i t h or w i t h o u t f o r w a r d speed, i n deep water or i n u n i f o r m flnite water d e p t h . H l u s t r a t i v e applications t o wave d i f f r a c t i o n - r a d i a t i o n b y an offshore structure i n deep water and t o steady ship waves y i e l d m a t h e m a t i c a l representations useful f o r numerical and a n a l y t i c a l purposes.
K e y w o r d s : freesurface effect, Fourier analysis, f a r f i e l d wave, nearfield disturbance, d i f f r a c -t i o n , r a d i a -t i o n , source. Green f u n c -t i o n , p o -t e n -t i a l flow
T E C H N S S C H E U N I V E R S I T E I T Laboratorium voor S c h e s p s h y d r o m e c h a n i e a ^>rchief M e k e l w e g 2 , 2 6 2 3 C D D f el.: 015 786373 Fax: 015 T' , 3
Jen-Shiang K o u h and Chun-Hsine Ho
A H i g h O r d e r P a n e l M e t h o d B a s e d o n S o u r c e D i s t r i b u t i o n a n d G a u s s i a n Q u a d r a t u r e
Ship Technology Research 43 (1996), 38-47
I n panel m e t h o d s used t o compute p o t e n t i a l flows, the b o d y surface is t r a d i t i o n a l l y approx-i m a t e d by a number of panels descrapprox-ibed by planes or quadratapprox-ically curved surfaces, and the source d i s t r i b u t i o n on each panel is assumed as constant, linear or quadratic f u n c t i o n . I n the present m e t h o d , however, the surface integrals are de-singularized by a n a l y t i c a l m a n i p u l a t i o n t o allow a n u m e r i c a l q u a d r a t u r e by Gauss' m e t h o d . T h i s allows t o use the m a t h e m a t i c a l sur-face d e f l n i t i o n d i r e c t l y f o r the flow c o m p u t a t i o n , g i v i n g source s t r e n g t h , velocity and pressure at an a r b i t r a r y n u m b e r of Gauss points on each panel. T h e order of a p p r o x i m a t i o n can be made very h i g h , and i t can be adapted t o the c o m p l e x i t y of the panel. N u m e r i c a l results f o r a sphere, ellipsoids, and a W i g l e y h u l l show t h a t great accuracy is o b t a i n e d w i t h r e l a t i v e l y f e w c o m p u t i n g points.
K e y w o r d s : H i g h order, panel m e t h o d , p o t e n t i a l flow, Gaussian quadrature, C F D
From the editors' software collection: Discrete Fourier t r a n s f o r m a t i o n (revised)
Verlag;
Schiffahrts-Verlag „Hansa" C. Schroedter & Co. (GmbH & Co KG) Striepenweg 31,21147 Hamburg, Postfach 92 06 55, 21136 Hamburg Tel. (040) 7 97 13 - 02, Fax (040) 7 97 13 - 208,
Telegr.-Adr.: Hansapress Schriffleitung:
Prof. Dr.-Ing. H. Söding, Dr.-Ing. V. Bertram
Institut fflr Schiffbau, Lammersieth 90,22305 Hamburg Abonnementsveiwaltung:
Ingeborg Köhn
Tel. (040) 7 97 13 - 323, Fax (040) 7 97 13 - 324
Nachdruck, auch auszugsweise, nur mit Genehmigung des Verlages. Die SCHIITSrECHNIK erscheint viermaljahrUch. Abonnementspreise: Inland: jahdich D M 200,00 inkl. Mehnvertsteuer und Versandkosten; Ausland: jahrlich D M 200,00 zuzUghch Versandkosten. Der Abonnemenlspreis ist im voraus fallig. Zahlbar innerhalb 30 Tagen nach Rechnungseingang. Einzelpreis: D M 51,00 inkl. Mehnvertsteuer, inkl. Versandkosten. Abonnementskündi-gungen sind nur schriftlich mit einer Frist von 6 Wochen zum Ende eines Kalenderjahres beim Verlag möglich.
Höhere Gewalt entbindet den Verlag von jeder Lieferungsverpflichtung. -Erftillungsort und Gerichtsstand Hamburg.
Gesamtherstellung: Hans Kock, Buch- und Offsetdruck GmbH, Bielefeld
Publishen
Schiffahrts-Veriag "Hansa" C. Schroedter & Co. (GmbH & Co K G ) Striepenweg 31, 21147 Hamburg, Postfach 92 06 55,21136 Hamburg Tel. + 49 40 / 7 97 13 - 02, Fax + 49 40 / 7 97 13 - 208,
Telegraphic address: Hansapress Editor
Prof Dr.-Ing. H. Söding, Dr.-Ing. V. Bertram
Institut für Schiflbau, Lammersieth 90, D-22305 Hamburg Subscriptiondepartment:
Ingeborg Köhn
Tel. + 49 4 0 / 7 97 13 - 323, Fax + 49 4 0 / 7 97 13-324
All rights reserved. Reprmts prohibited without permission of the publisher. SHIP T E C H N O L O G Y R E S E A R C H is issued quarterly. Subscripüon price: D M 200,00 per year + mailing cost, to be payed in advance 30 days after receipt of mvoice. Cancellation of subscriptions at the end of a year only by written notice to the publisher 6 weeks in advance.
In case of force majeure the publisher is freed from delivery. Court competency and place of contract: Hamburg.
Production: Hans Kock, Buch- und Offsetdruck GmbH, Bielefeld
ISSN 0937-7255 ISSN 0937-7255
Motion Simulation of a Cylinder
at the Surface of a Viscous Fluid
L i o n e l G e n t a z , B e r t r a n d A l e s s a n d r i n i a n d G é r a r d D e l h o m m e a u , Ecole Centrale de Nantes^
1 I n t r o d u c t i o n
Nnmerous n u m e r i c a l studies have been carried out f o r the calculation of forces on a surface-piercing cylinder i n forced m o t i o n . Solutions i n perfect f l u i d using development of free surface conditions at different orders were developed first. Ursell (1949) gave a first-order s o l u t i o n f o r the circular cylinder i n heave. Lee (1968) and Parissis (1966) o b t a i n e d solutions at second order f o r cylinders i n heave w i t h a circular or Ushaped section. T h e y showed t h a t h y d r o d y -n a m i c seco-nd-order forces i-ncrease w i t h the freque-ncy of forced m o t i o -n a-nd ca-n be sig-nifica-nt compared t o first-order forces. Potash (1971) generaUsed this second-order s o l u t i o n t o coupled sway and r o l l m o t i o n s . Papanikolaou and Nowacki (1984) b u i l t a complete second-order the-o r y f the-o r sway, heave and rthe-oU mthe-otithe-ons f the-o r a r b i t r a r y sectithe-ons. A l l these meththe-ods use b the-o u n d a r y elements w h i c h are determined so t h a t the n o r m a l velocity on the b o d y is equal t o zero.
O n the other h a n d , Faltinsen (1977), besides others, used a direct n u m e r i c a l s i m u l a t i o n t o solve the f u l l y non-hnear p r o b l e m i n perfect fiuid. T h e b o d y and the free surface contours were meshed. T h e source i n t e n s i t y i n each b o u n d a r y element is the s o l u t i o n of an i n t e g r a l e q u a t i o n , and the geometry of t h e d o m a i n is u p d a t e d at each t i m e step. T h i s m e t h o d allows t o t r e a t various problems hke numerical t o w i n g t a n k , solitary wave and seakeeping problems.
O t h e r researchers use curvihnear meshes fitted t o the physical boundaries of the fluid t o solve conservation equations f o r the flow. T h e b o u n d a r y conditions are expressed i n the c u r v i h n e a r space, discretised b y finite differences. T h e r e s u l t i n g equation system is solved by an i t e r a t i v e procedure. W i t h this m e t h o d Telste (1985) treated a surface-piercing cyhnder i n perfect fluid. Shanks and Thompson (1977) solved the Navier-Stokes equations f o r l a m i n a r flow. Yeung and Ananthakrishnan (1991) developed a s o l u t i o n i n viscous fluid. T h e mesh is generated b y a p p l y i n g a v a r i a t i o n a l p r i n c i p l e , and the Navier-Stokes equations are solved b y a f o r m u l a t i o n using an a u x i h a r y velocity fleld. T h e y c o m p u t e d also viscous effects f o r a rectangular cyhnder i n heave (Yeung and Ananthakrishnan 1992). Nichols and Hirt (1977) presented results f o r two- and three-dimensional viscous flow and studied three-dimensional effects f o r a heaving b o d y , using an extension of the M a r k e r - a n d - C e l l ( M A C ) m e t h o d .
I n our m e t h o d the mesh is o b t a i n e d by a t r a n s f i n i t e i n t e r p o l a t i o n (Eriksson 1982). A p a r t i a l t r a n s f o r m a t i o n is used f o r t h e l a m i n a r Navier-Stokes and the c o n t i n u i t y e q u a t i o n . I n the c o m p u t a t i o n a l space, a l l the boundaries are on curvihnear co-ordinate hnes. T w o hnear systems s t e m m i n g f r o m the discretisation of the t r a n s p o r t equations and t h e mass conservation are solved b y the S I M P L E R a l g o r i t h m (Piquet and Visonneau 1991). T h e new free surface is o b t a i n e d f r o m the k i n e m a t i c free surface c o n d i t i o n , and the fiuid d o m a i n is r e g r i d d e d f o r the next i t e r a t i o n (Alessandrini and Delhommeau 1994).
2 E q u a t i o n s
T h e s u m m a t i o n convention over equally-named indices is used, w i t h i,j,k,l being 1 or 2.
^Division Hydrodynamique Navale, Laboratoire de Mécanique des Fluides, URA 1217 du CNRS, 1 Rue de la Noë, 44072 Nantes Cedex 03
f
2.1 P r i m i t i v e f o r m o f e q u a t i o n s
Navier-Stokes equations f o r a l a m i n a r flow are w r i t t e n i n the Cartesian Gahlean system ( O , x^,x'^). O is on the free surface at rest at the i n i t i a l p o s i t i o n of the centre of the cylinder.
is h o r i z o n t a l and x'^ is u p w a r d oriented. Unknowns are the Cartesian velocity components u^,u'^ and the t o t a l pressure P. T h e r e l a t i o n p — P + pgx'^, where p is t h e d y n a m i c pressure, allows g r a v i t y forces ( p o i n t i n g t o —x'^) t o be taken i n t o account, u is the k i n e m a t i c viscosity and p the fluid density.
Navier-Stokes and c o n t i n u i t y equations are
du" -du" 1 dp Ö^m" . , ^ ^
= 0. (2) dxi
2.2 T r a n s f o r m a t i o n i n t o c o m p u t a t i o n a l d o m a i n
A discretisation of the physical d o m a i n is made using curvihnear co-ordinates e^, e^. A l l the boundaries and mesh lines are located on = constant ( i = 1 or 2 ) . These co-ordinates allow a rectangular c o m p u t a t i o n a l d o m a i n t o remain fixed when the physical d o m a i n changes.
domain
F i g . 1: Physical and c o m p u t a t i o n a l d o m a i n
A t each time-step the m e t r i c is c o m p u t e d at each node. We have t o compute the components ttij of t h e covariant basis (whose vectors ai are tangent t o coordinates lines i n e'), t h e Jacobian J , t h e components bj of the contravariant basis (the vectors of w h i c h 6' are o r t h o g o n a l t o the co-ordinates lines i n e'), the contravariant m e t r i c tensor g^^, the g r i d c o n t r o l f u n c t i o n f'' and the displacement velocities of the mesh :
«^. = ^ , / = d e t ( a , , ) , bi = J{a,,)-\ g''= j-,b% f ' - ~ i { J 9 % vl = ^ . (3)
T h e p a r t i a l derivatives i n (1,2) are expressed i n the basis (e^,e^). A p a r t i a l t r a n s f o r m a t i o n is made so t h a t the vectors r e m a i n expressed i n the Cartesian basis. T h e n the convective f o r m of equations gives
+
.dl J ^ de>' ' J ' dek ,kl d^u" de^de^ dt de^ l^d_u^ dé 0. (4) (5)2.3 F r e e s u r f a c e b o u n d a r y c o n d i t i o n s
T h e free surface conditions comprise one Icinematic and t w o d y n a m i c conditions. T h e Icine-m a t i c c o n d i t i o n assures t h a t f l u i d particles on free surface stay o n this surface. T h i s c o n d i t i o n gives f o r the free surface elevation h ( w i t h the free surface at rest t a k e n as o r i g i n ) :
Dh o dh , dh 1 ^ " 2 fc\
+ v}— = u\ ( 6 )
T h e d y n a m i c conditions express the c o n t i n u i t y of stresses at the free surface. T h e stress tensor i n the fluid is w r i t t e n :
where 7 is the surface tension coefficient and r the m a i n curvature radius of free surface. U n i t tangent and n o r m a l vectors t o the free surface, t = {t^,t'^) and n - ( n ^ , n ^ ) , can be calculated w i t h the covariant and contravariant vectors ai = (011,012) and 6^ = {bl,bl):
ti = o i j / l a i l and rii = b'^/\b'^\. (8) T h e component i of the stress tensor T is given by T,- = (Tijiij, and the t w o d y n a m i c conditions
are o b t a i n e d by the p r o j e c t i o n of T on n and t. I f the pressure is supposed t o be zero above the free surface:
7 fdui du^\ b}b]
T . n = 0 < ^ , = , , . ^ + ^ + p . ( ^ ^ +
^ j 1^.
(9)T h e free surface b o u n d a r y conditions (11,12,13) are now expressed i n the c o m p u t a t i o n a l space:
f + = w i t h A' = ^j{u'-vl). (11)
, = , , . ^ + ^ + 2 , . | ^ - ^ ^ (12)
{b)b]au^b]b]a,,f^,=^. (13)
3 D i s c r e t i s a t i o n a n d n u m e r i c a l s o l u t i o n 3.1 M e s h i n g o f t h e p h y s i c a l d o m a i n
T h e flow a r o u n d the heaving cylinder is supposed s y m m e t r i c t o the x'^ axis; thus t h e physical d o m a i n is bounded by one h a l f of the cylinder, the free surface, the axis of s y m m e t r y and the outer boundary. T h e size of the d o m a i n is sufficient t o ensure the i n f i n i t e d e p t h hypothesis ( i n linear t h e o r y ) and no reflection at the e x t e r n a l border. A n algebraic t r a n s f i n i t e i n t e r p o l a t i o n m e t h o d (Eriksson 1982) is used t o o b t a i n an i n i t i a l s t r u c t u r e d monoblock mesh. T h e m e t h o d consists of an i n t e r p o l a t i o n allowing t o define the mesh i n the inner d o m a i n i f t h e d i s t r i b u t i o n of nodes on the boundaries is k n o w n . T h e mesh is refined near the b o d y and the free surface t o take b o u n d a r y layer effects i n t o account. A t every t i m e step a new mesh is generated b y c o m p u t i n g the intersection of the free surface w i t h every hne = constant of t h e i n i t i a l mesh v e r t i c a l l y t r a n s l a t e d . O n each of these lines, the nodes are spread f o l l o w i n g t h e i r i n i t i a l spacing. T h e velocities of the mesh take i n t o account displacements of the nodes (see 2.2). F i g . 2 shows
an example of the mesh d e f o r m a t i o n . T h i s k i n d of mesh is used i n chapter 4.2. f o r c o m p u t i n g h y d r o d y n a m i c coefficients.
F i g . 2. E v o l u t i o n of the physical d o m a i n at the intersection of the cyhnder and the free surface
3.2 D i s c r e t i s a t i o n o f e q u a t i o n s
Discrete unknowns (velocities, pressure and free surface elevation) are located on the nodes of the g r i d .
T h e t r a n s p o r t equations (4) are r e w r i t t e n according t o the first and second derivatives except the cross derivatives:
9
A f t e r d i v i s i o n by e*' = 6 * / eqs. (14) w i t h ( f ) = become
^ £ . 1 ^ . 1 + ^ e * 2 £ » 2 = 2A\(t)^*i + 2A14>^*2 + -<j)i + S,f,.
du" 1 du" 1 _^jk dp ^ki d^u"
(14)
(15)
Eqs. (15) are hnearised i n velocities by c o m p u t i n g the convection terms A * , A 2 and t h e source terms 5",^ (except the pressure g r a d i e n t ) at the previous t i m e step.
F i g . 3. M u l t i - e x p o n e n t i a l scheme
2-1+
A discretisation of the linearised t r a n s p o r t equation at node P ( F i g . 3) is o b t a i n e d w i t h a m u l t i - e x p o n e n t i a l scheme using the s y m m e t r i c decomposition of eq. (15) f o l l o w i n g each o f the c u r v i l i n e a r co-ordinates:
= 2Al(l},,i + G l , ( ^ , , 2 , . 2 = 2A*2^,.2 + G2 w i t h Gi + G2 = ^4>t + S^. (16) Terms A* and Gi are supposed t o be constant on the five points g r i d , so the previous system is equivalent t o t w o first-order d i f f e r e n t i a l equations w i t h constant coefficients
<^,..,.. = 2 A * + Gi =^ - ^ ( < ^ p - C,-4>i- - Ci+<f>i+) = -Gi. (17)
I f we use a first-order u p w i n d finite difference scheme f o r the unsteady t e r m , t h e discrete t r a n s p o r t equation i n ( f ) at node P becomes
( c ^ + ^ + ^ ) ^S-"' - t ( C . - * . - + C . . * . . ) - + = 0. ( 1 8 )
A discretisation of t l i e c o n t i n u i t y equation (5) a n d terms of the pressure gradient c o m i n g f r o m source terms (see eqs. 14,15) by centred second-order schemes gives checkerboard oscillations (Visonneau 1993). Non-centred f i r s t order schemes ( w i t h 3 nodes) are therefore used. T h e c o n t i n u i t y e q u a t i o n is decentred u p w i n d and the pressure gradient d o w n w i n d (Alessandrini and Delhommeau 1994).
3.3 F i n a l p r e s s u r e - v e l o c i t y l i n e a r s y s t e m s
A t each node P , eq. (18) gives a discrete r e l a t i o n between velocity and pressure u n k n o w n s on P and adjacent nodes. T h e c o n t i n u i t y equation (5) gives another r e l a t i o n between u n k n o w n velocities on P and adjacent nodes. Such relations must also be defined on t h e boundaries of t h e d o m a i n . For a l l boundaries except the free surface, D i r i c h l e t or N e u m a n n conditions o n velocities or pressure are used. O n the free surface, t h e n o r m a l d y n a m i c c o n d i t i o n (12) gives a r e l a t i o n f o r t h e pressure, the RHS being calculated at t h e previous step. T h e surface tension t e r m is neglected. T h e t a n g e n t i a l dynamic c o n d i t i o n (13) gives a r e l a t i o n f o r h o r i z o n t a l velocity
A n o t h e r e q u a t i o n m u s t be f o u n d f o r the v e r t i c a l velocity u'. A r e l a t i o n c o m i n g f r o m t h e c o n t i n u i t y e q u a t i o n (5) at t h e free surface w i h be p r e f e r r e d , i f possible, t o a D i r i c h l e t or a N e u m a n n c o n d i t i o n . T h e choice between D i r i c h l e t or N e u m a n n c o n d i t i o n is made t o a v o i d non-zero values outside t h e diagonal of the matrices.
T h e discretisation of the t r a n s p o r t and c o n t i n u i t y equation and the b o u n d a r y conditions finally give t h e t w o linear systems
{E-A)JJ + GF = f and DV = g. (20) Vector U contains t h e velocities M ^ U ^ at each node, vector P the pressure p . £ is a diagonal
m a t r i x , A a m a t r i x w i t h zero on t h e diagonal. D and G are the matrices s t e m m i n g f r o m t h e discretisation of t h e divergence of the velocity and of t h e pressure gradient. Source terms a n d cross terms of second order are i n the RHS ƒ .
3.4 S o l u t i o n o f t h e l i n e a r s y s t e m s
T h e pressure equation D(E- A ) " ^ ( ƒ - GP) = g o b t a i n e d b y c o m b i n i n g eqs. (20) cannot be solved d i r e c t l y because of t h e p r a c t i c a l i m p o s s i b i h t y t o i n v e r t t h e f u l l m a t r i x D{E - A ) ~ ^ G . Instead, t h e c o m b i n e d e q u a t i o n is solved by the i t e r a t i v e S I M P L E R a l g o r i t h m (Piquet and
Visonneau 1991). I t uses as a p p r o x i m a t e inverse of E - A t o o b t a i n the pressure e q u a t i o n {DE-^G)P = DE-\AU + f ) - 9 .
K n o w i n g t h e velocity field C/C^-i) at the previous time-step, S I M P L E R determines t h e velocity U^''^ a n d pressure P^'') field t o be f o u n d b y
C/ = P - 1 ( A [ / M + / ) ( e l ) ; Pi^^) = iDE-^G)-\DÜ - g) (e2);
U* = {E - A ) - \ f - GP^^^) (e3); P'= {DE-'G)-\DU* - g) (e4); (21) C/W = U*- E-^GP' (e5).
I n ( e l ) we c o m p u t e an advective velocity field. T h e divergence of t h e velocity field o b t a i n e d by (e3) is 7^ 0, so a pressure correction P ' is c o m p u t e d i n (e4) and used i n (e5). P ' does n o t change t h e new pressure field computed i n (e2). One i t e r a t i o n is sufficient t o c o m p u t e velocity a n d pressure fields w i t h a quite good accuracy. T h e matrices E - A a n d DE'^G have t o be i n v e r t e d . T h e p o s i t i o n of non-zero coefficients i n these matrices is precisely k n o w n i n a s t r u c t u r e d mesh. T h e m a t r i x E-A is well c o n d i t i o n e d a n d has a d o m i n a n t diagonal, c o n t r a r y t o
DE~^G. T h e C G S T A B a l g o r i t h m (Van der Vorst 1992)a.iid an incomplete L U p r e c o n d i t i o n i n g is t h e n used. T h i s a l g o r i t h m is a variant o f t h e bi-conjuguate gradient a l g o r i t h m w i t h a greater robustness.
I t e r a t i o n s o f the S I M P L E R a l g o r i t h m can only i m p r o v e the velocity-pressure c o u p l i n g . T o handle n on lin e a r ities , supplementary iterations (called nonlinear i t e r a t i o n s ) can be made i n each time-step. T h u s a f t e r c o m p u t i n g new velocity and pressure fields, the m e t r i c coefficients (eq. 3 ) , coefficients Al,A2, S4, of ( 1 5 ) , discretised terms of the c o n t i n u i t y equation or pressure gradient (i.e. a l l e x p h c i t terms of discretized equations) can be c o m p u t e d again w i t h the n e w velocities. T h e n new linear systems (20) are solved t o o b t a i n another velocity a n d pressure field. So the influence of iterations on nonlinearities wiU be evaluated i n each c o m p u t a t i o n case.
3.5 U p d a t e o f t h e f r e e s u r f a c e e l e v a t i o n
Once the new velocity and pressure fields are k n o w n , the free surface elevation h is c o m p u t e d f r o m k i n e m a t i c c o n d i t i o n (11). T h e t i m e derivative of h is discretised b y a non-centred scheme of first order, and the spatial derivative by a Dawson 4-point u p s t r e a m scheme w h i c h reduces the phase s h i f t of first or second-order finite difference schemes. T h e sign of the coefficient A^ gives the d i r e c t i o n of the stencil t o ensure s t a b i l i t y :
^ j = -Cohi - Cihi+i - C2hi+2 - Cg/ii+s. (22)
Coefficients d are: Co = 5 / 3 , C i = - 5 / 2 , C2 = 1, Cg = - 1 / 6 .
Near t o the boundaries where Dawson's scheme cannot be used, u p s t r e a m 2-point schemes are used. A n e x p l i c i t i t e r a t i v e solution w i t h a f r a c t i o n a l t i m e step is preferred t o an i m p l i c i t s o l u t i o n . T h e free surface elevation on the cylinder (hi), w h i c h cannot be c o m p u t e d f r o m t h e k i n e m a t i c c o n d i t i o n , is l i n e a r l y e x t r a p o l a t e d .
T h u s , at each t i m e step ( k ) we have t o r e - g r i d the fiuid d o m a i n w i t h the l o c a t i o n of t h e cyhnder at step ( k ) and the free surface elevation at step ( k - 1 ) ; t o calculate the m e t r i c f o r t h e new mesh; t o discretise the convectiondiffusion terms, source t e r m s , coefficients of the m u l t i -e x p o n -e n t i a l sch-em-e and discr-etis-ed v-elocity div-erg-enc-e and pr-essur-e gradi-ent; t o c o n s t i t u t -e a n d solve the pressure-velocity hnear systems; and finally t o compute the new free surface elevation.
4 N u m e r i c a l r e s u l t s
4.1 P r i n c i p l e o f d e t e r m i n i n g f o r c e a n d h y d r o d y n a m i c c o e f f i c i e n t s o f a c y l i n d e r Forced heaving m o t i o n s of a surface-piercing cyhnder are simulated i n viscous fiuid. T h e b o d y is never e n t i r e l y immersed or emersed. T h e heaving m o t i o n is given b y ydt) = A s i n w i . A h c o m p u t a t i o n s (except the first case i n 4.5) are done w i t h a k i n e m a t i c viscosity v = l O - ^ m V s .
H y d r o d y n a m i c forces per u n i t l e n g t h are o b t a i n e d by a d d i n g pressure force Rpy{t) a n d f r i c t i o n force R f y ( t ) . Rpy{t) includes buoyancy. B o t h forces are i n t e g r a t e d at each t i m e step along the a c t u a l l y i m m e r s e d cylinder contour:
Rpyit) = - [ Pnydl = - [ \ p - pgx')bl de'; (23) Jcyl Jo
Uy is t l i e v e r t i c a l component of the e x t e r n a l u n i t n o r m a l (defined b y the mesh hne = 0 ) . For c a l c u l a t i o n of h y d r o d y n a m i c coefficients, we have t o take i n t o account the mean and the time-variable h y d r o s t a t i c forces Fb and Rhy{t), respectively For a circular cyhnder we have
Fb = pgA = pg7rr'/2, (25) Ri^y(t) = -pg{6r' + ycit)rcos6) w i t h ê = &Tcsm{ycit)/r). (26)
where A is the w e t t e d surface of the cyhnder at rest. , free surface at rest
F i g . 4. Parameters used i n c o m p u t i n g h y d r o d y n a m i c a ! forces
A d d e d mass coefficients CM22 are i n phase w i t h the acceleration, d a m p i n g coefficients CA22 w i t h the velocity. For heaving they are f o u n d by a Fourier analysis:
+00
Rpy{t) + Rfy{t) -Fb- Rhy{t) = E( a „ cos nut + ö„ sin nut). (27) n=0
We have
CM22 = -Au^O.bpnr^ Au^O.bpirr'^ T ^ ^ 2 CA22 = - ^ ^ ^ 2 • (28)
F o l l o w i n g Tasai and Koterayama (1976) and Lee (1968), the forces P i " ^ and phases Sn at different orders n are defined by
-1-00
Rpyit) + Rfvit) -Kb = Y ^ K COS nut + b'^ sin nut)
= 2pgr' ' + £ 2pgr' " i ^ i " ) sin(na;i + 5 „ ) . (29)
T h e difference between Fa{l) and Si on the one h a n d , and CM22 and CA22 on the other h a n d , is due t o Rhyit) w h i c h is not ~ s i n w i . I n practice, terms of higher order of Rhy{t) are neghgible. T h u s , c o m p u t i n g F^°\Fa{n) and S^ f o r n > 2 w i t h a „ and 6„ or w i t h a'^ and b'^ gives nearly t h e same results; the only difference greater t h a n 5% is i n the t h i r d - o r d e r force. M o r e o v e r , i n perfect fluid the v e r t i c a l d r i f t force Fa{0) c o m p u t e d f r o m the Fourier coefficient at n = 0 is t h e constant p a r t of the second order t e r m F^"^^.
4.2 H y d r o d y n a m i c c o e f f i c i e n t s o f t h e c i r c u l a r c y l i n d e r
F i r s t the case A/r = 0.2 ( r = l m ) and u = 3.16 r a d / s was simulated w i t h 2 meshes (coarse and fine) of 5000 nodes (100 on the free surface, 50 on the c y l i n d e r ) and one mesh of 7200 nodes (120-60). T h e coarse 5000-node mesh had a mesh size d/r = 0.03 of the first r o w near the cyhnder, the fine one had d/r = W''^, and the 7200-node mesh had d/r = IQ-'^ there.
O n l y the last t w o meshes resolve the b o u n d a r y layer a n d thus give correct viscous forces Rfy{t). T h e c o m p u t a t i o n s w i t h these t w o meshes showed some convergence problems (see F i g . 5, h t t l e v e r t i c a l segments e.g. at t = 2.1 and 4.1s). T h e C P U t i m e on a 40 M f l o p c o m p u t e r f o r 12s s i m u l a t i o n t i m e was 18h f o r the fine 5000-node mesh, 25h f o r the 7200-node mesh ( t i m e steps 2 • 10~^s), a n d 3.5h f o r the coarse mesh ( t i m e step 10~^s).
Differences i n Rpy{t) f o r the three meshes are not very significant. M a x i m u m and m i n i m u m values of h y d r o d y n a m i c coefficients were obtained w i t h the t w o 5000-node meshes (see t a b l e ) by e v a l u a t i n g the results between 6 and 12 s after the s t a r t of t h e s i m u l a t i o n .
- Rpy(t)-Fb : computation with a fine grid of 7200 nodes - Rpy(t)-Fb : computation with a fine grid of 5000 nodes - Flpy(t)-Fb : computation with a coarse grid
F i g . .5. Comijaxison of forces f o r tliree meslies
Heaving m o t i o n of a circular cylinder: ijc{t) = As'mojt, A/r = 0.2, OJ - 3.16 r a d / s .
_ Rpy(t)-Fj3 : usual computation with a coarse grid
Rpy(t)-Fb '• computation with a coarse grid and 1 iteration on non linearities Rpy(t)-Fb : computation with a coarse grid and 3 iterations on non linearities
t(s)
F i g . 6. Comparison of pressure force f o r different immbers of iterations on nonlinearities H e a v i n g m o t i o n of circular cylinder: yc{t) = A s i n w i , A/r - 0.2, u) = 3.16 r a d / s
CM O N a v i e r - S t o k e s calc. A / r = 0 . 2 N a v i e r - S t o k e s calc. A / r = 0 . 4 Y a m a s l i i t a e x p . A / r = 0 . 2 Y a m a s h i t a e x p . A/r-OA P a p a n i k o l a o u calc. A / r = 0 . 2 P a p a n i k o l a o u calc. A / r = 0 . 4 K r = ü ^ / g F i g . 8. A d d e d mass CM22 C\J C\i 1.0 h 0.5 0.0 V O 0,6 . N a v i e r - S t o k e s calc. A / r = 0 . 2 . N a v i e r - S t o k e s calc. A / r = 0 . 4 Y a m a s l i i t a e x p . A / r = 0 . 2 Y a m a s h i t a e x p . A/r-OA T a s a i & al. e x p . A / r = 0 . 2 T a s a i & a l . e x p . A / r = 0 . 4 P a p a n i k o l a o u calc. A / r = 0 . 2 - P a p a n i k o l a o u calc. A / r = 0 . 4 l . U 1 . 0 O K r = a ^ / g F i g . 9. D a m p i n g c o e f f i c i e n t CA22 pi.o 0.0 N-S calc. A / r = 0 . 2 N-S calc. A / r = 0 . 4 T a s a i & al. e x p . Nr-0.2
T a s a i & al. e x p . Nr-OA
P a p a n i k o l a o u & al. calc.
0.5 F i g . 10. F i r s t - o r d e r force F ^ ' 0)320 D Kr=ci:^/g 260 240 h - N - S calc. A / r = 0 . 2 - N - S calc. A / r = 0 . 4 T a s a i & al. e x p . A / r = 0 . 2 T a s a i & al. e x p . A / r = 0 . 4 - P a p a n i k o l a o u & al. calc.
K r l c c ^ / g
F i g . 1 1 . F i r s t - o r d e r pliase
F i g . 12. Second-order force F f ) F i g - 13. Second-order phase ^3
CM21 CA22 ^i[deg] Fi'^ ^2 [deg] Jo
coarse mesh 0.643 0.350 0.555 210 0.449 -30 -0.115 0.225
fine mesh 0.625 0.348 0.567 209 0.428 -33 0.104 0.228
T h e agreement between the t w o meshes is good except f o r the v e r t i c a l d r i f t force. T h u s i t seems reasonable t o save c o m p u t i n g t i m e t o calculate a l l quantities derived f r o m pressure i n t e g r a t i o n w i t h the coarse mesh, as is done i n the f o l l o w i n g .
C o m p u t a t i o n s w i t h different numbers of iterations on nonhnearities show o n l y smah dif-ferences ( F i g . 6 ) . T h u s f u r t h e r c o m p u t a t i o n s f o r the circular cyhnder w i h be made w i t h o u t supplementary iterations on nonlinearities.
F i g . 7 shows t h e t i m e v a r i a t i o n of the non-dimensional pressure force a f t e r s u b t r a c t i n g Fb and Rhyit) f o r A/r = 0.4 and Kr = u'r/g = 0.64. Steady-state osciUations develop v e r y quickly. T h e agreement w i t h non-hnear calculations f o r a perfect fluid by Hwang et al. (1987) is g o o d ; t h e difference at the start of the s i m u l a t i o n is due t o d i f f e r e n t i n i t i a l m o t i o n .
T h e c o m p u t a t i o n s i n viscous fluid w i t h A/r = 0.2 and 0.4 and OJ between 1.1 a n d 4.43 r a d / s used a l e n g t h of the fluid d o m a i n between 15 t o 100 m (radius r = l m ) and a t i m e step of 10~^s. A s stated before, o n l y pressure forces have been taken i n t o account f o r c a l c u l a t i n g t h e h y d r o d y n a m i c coefficients. O n Figs. 8-15 various results are compared w i t h e x p e r i m e n t a l d a t a by Yamashita (1977), Tasai and Koterayama (1976) and w i t h perfect fiuid c o m p u t a t i o n s of Papanikolaou (1987) and Papanikolaou and Nowacki (1984). T h e radius o f circular cylinders used i n t h e experiments was 0.20 m , corresponding t o iZ„ of about 5 • 10^. T h u s the l a m i n a r flow assumption seems j u s t i f i e d .
E x p e r i m e n t a l variations of added mass and d a m p i n g coefficients are w e l l r e p r o d u c e d i n our calculations. However, added masses are overestimated f o r A/r = 0.4, and d a m p i n g coefficients are underestimated f o r A/r = 0.2. For low frequencies there are large differences i n the measured d a m p i n g coefficient between Y a m a s h i t a and Tasai et a l . , p r o b a b l y due t o the measuring difficulties at l o w frequencies. I n these cases, supplementary c o m p u t a t i o n s w i t h fine grids showed t h a t the viscous p a r t of the force is negligible. F i r s t - o r d e r forces a n d phases and second-order phases are quite g o o d f o r b o t h amphtudes. T h e v e r t i c a l d r i f t force is u n d e r p r e d i c t e d f o r A/r = 0.2. For A/r = 0.4 and h i g h frequencies there are n u m e r i c a l problems due t o strong m o t i o n s of the free surface near the body. T h i s m a y correspond t o t h e b e g i n n i n g of wave b r e a k i n g (see n e x t p a r t ) and should e x p l a i n the d i s c o n t i n u i t y of t h e curves f o r second- and t h i r d - o r d e r forces.
To conclude, our viscous fluid c o m p u t a t i o n s give a good p r e d i c t i o n of the v a r i a t i o n o f t h e h y d r o d y n a m i c coefficients w i t h frequency f o r b o t h amplitudes investigated.
4.3 F r e e s u r f a c e e l e v a t i o n f o r t h e c i r c u l a r c y l i n d e r
I n the previous simulations, the free surface elevations h{t) were i n t e r p o l a t e d at each t i m e step i n a p o i n t located i n the m i d d l e of fiuid d o m a i n . A Fourier analysis
-1-00
h{t) = 5 ] /i„ cos{nu>t + 6n). (30) n = l
gives t h e non-dimensional a m p h t u d e of order n by A „ = hn/A. T h e first-order a m p h t u d e A i is c o m p a r e d t o e x p e r i m e n t a l results of Tasai and Koterayama (1976) and Kyozuka (1982) i n F i g . 16. For A/r — 0.2 the a m p l i t u d e is slightly overestimated, whereas the agreement is g o o d f o r A/r = 0.4. T h e wave a m p h t u d e needs several periods t o become s t a t i o n a r y , so reflections o n e x t e r n a l b o u n d a r y can occur. Results w o u l d p r o b a b l y be i m p r o v e d w i t h a larger d o m a i n .
T h e free surface elevation near the cylinder at different t i m e steps f o r a f o r c e d h e a v i n g m o t i o n y^t) = Acosut w i t h A/r = 0.4 and oj'r/g = 2.0 is compared i n Figs. 17a-f w i t h
0.2 R -0.1 •0.2 •0.3 N-S calc. A/r=0.2 N-S calc. A/r=0.4 Y a m a s h i t a exp. A/r=0.2 Y a m a s h i t a exp. A/r=0.4 Papanil<olaou & al. calc.
, A-A. Kr=a:^/g F i g . 14. V e r t i c a l d r i f t force F„ (2) CO CÖ 0.6 ^ N-S calc. A/r=0.2 - N-S calc. A/r=0.4 Y a m a s h i t a exp. A/r=0.2 Y a m a s h i t a exp. A/r=0.4 Kr=tt?r/g F i g . 15. T h i r d - o r d e r f o r c e F f )
_A Navier-Stokes calculations A/r=0.2|
A Tasai & al. experiments A/r=0.2
V Kyozuka experiments A/r=0.2
• I - 1.0
<
(b)
• V
- • — N a v i e r - S t o k e s calculations A/r=0.4| • Tasai & al. experiments A/r=0.4
V Kyozuka experiments A/r=0.4 1.0 1.5 2 . 0 ^ ^ 0.0 0.5 1.0 1.
F i g . 16. A m p l i t u d e o f f i r s t - o r d e r wave f o r A/r 0.2 (a) a n d 0.4 ( b )
co^r/g
-extrapolated free surface for re-gridding
-calculated free surface ^ . 1 3 0
-extrapolated free surface for re-gridding -calculated free surface
F i g . 18. Free surface e l e v a t i o n near a cyhnder w i t h a f m e mesh; h e a v i n g m o t i o n A sin(a)t) w i t h
A/r = 0.2 and UJ = 3 . 1 6 r a d / s .
c o m p u t a t i o n s i n perfect f l u i d by Hwang et al. (1987) who use a semi-Lagrangian b o u n d a r y element m e t h o d s i m i l a r t o t h a t of Faltinsen (1977). T h e agreement between b o t h m e t h o d s is g o o d . I n F i g . 17f H w a n g and al. give a water j e t at the intersection between cylinder a n d free surface. T h i s phenomenon stops the s i m u l a t i o n i n perfect flow a n d i n viscous fluid a n d m a y represent t h e b e g i n n i n g of wave breaking i n reality.
4.4 P r o b l e m s a t t h e i n t e r s e c t i o n o f t h e b o d y w i t h t h e f r e e s u r f a c e
C o m p u t a t i o n s on fine meshes have shown n u m e r i c a l problems at the intersection between t h e b o d y a n d t h e free surface. I n t h i s region t h e wave elevation is o f t e n o s c i l l a t i n g , w i t h v e r y h i g h local slopes, r e s u l t i n g i n divergence of calculations. T h i s was described by Alessandrini and Delhommeau (1995). I t is due t o t h e singular f o r m t a k e n b y t h e k i n e m a t i c free surface equation (11) at t h e intersection: at this p o i n t , b o t h the no-shp c o n d i t i o n and the free surface k i n e m a t i c c o n d i t i o n should be satisfied. We have
dh/dt + u^dh/dx^ = u'. (31) As ti^ = 0 a n d u' = 0, w i t h o u t care we could conclude t h a t dh/dt = 0, i.e. the free surface
cannot move. I n r e a l i t y , dh/dx^ is i n f i n i t e at the b o d y intersection, so t h a t u^dh/dx^ can have a finite value. T h e free surface must be tangent t o the body, g i v i n g a viscous meniscus d i f f e r e n t f r o m t h e surface tension meniscus. T h i s is verified n u m e r i c a l l y i n F i g . 18, where the c u r v a t u r e of the free surface depends on the v a r i a t i o n of t h e free surface elevation.
U n f o r t u n a t e l y t h i s free surface behaviour leads t o a s i n g u l a r i t y i n the c o m p u t a t i o n of t h e m e t r i c . I n t h i s case t h e mass conservation cannot be used t o o b t a i n t h e v e r t i c a l v e l o c i t y at t h e free surface due t o a severe divergence of c o m p u t a t i o n s . These problems have been solved by using a D i r i c h l e t c o n d i t i o n f o r the v e r t i c a l velocity at the free surface, a n d by a change i n c o m p u t i n g t h e free surface elevation near t o the body. T o avoid t h e s i n g u l a r i t y of t h e m e t r i c , t h e free surface elevation is e x t r a p o l a t e d i n the v i c i n i t y of the b o d y as shown i n F i g . 18. T h i s e x t r a p o l a t i o n is used o n l y f o r the r e - g r i d d i n g and f o r the c a l c u l a t i o n o f t h e m e t r i c (Alessandrini and Delhommeau 1994). I n spite of these m o d i f i c a t i o n s , convergence on fine meshes is d i f f i c u l t t o o b t a i n near t h e body. I t e r a t i o n s on nonlinearities seem t o a t t e n u a t e t h i s p r o b l e m .
4.5 C o m p u t a t i o n s o f f o r c e s for a r e c t a n g u l a r c y l i n d e r
T h e influence of viscosity is weak f o r a circular cyhnder i n forced heaving. T h e r e f o r e we made c o m p u t a t i o n s f o r a rectangle where viscous effects are more i m p o r t a n t . A first c a l c u l a t i o n was f o r f o r c e d heaving m o t i o n yc{t) = Asinut and LOB'/V = 1000, uj'B/2g = 2, A/B = 0.3, B/d = 1.0 (here i/ = 0.018m^/s and B = 2 m ) f o l l o w i n g c o m p u t a t i o n s of Yeung and Ananthakrishnan (1992). B and d are b r e a d t h and d r a f t at rest, respectively. For t h e rectangle t h e meshes at t h e b o d y were refined sufficiently t o resolve b o u n d a r y layer effects f o r t h e a c t u a l
value. We used a mesh of 6000 nodes, and the t i m e step was 2 • 10~^s.
For comparison w i t h Yeung a n d A n a n t h a k r i s h n a n , R f y i f ) was d i v i d e d i n t o t h e shear c o m -ponent
a n d the n o r m a l component
Rfyn{t) = 2pv—b\de\ (33) These components a n d the pressure force are nondimensionalised b y pB'^uj'.
C o m p u t a t i o n s w i t h 1 or 3 i t e r a t i o n s on nonhnearities show o n l y s m a l l differences i n Rpy{t) ( F i g . 19). A l s o t h e viscous forces Rfys{t) and Rjyn{t) are s i m i l a r f o r d i f f e r e n t numbers of nonlinear i t e r a t i o n s . So one nonlinear i t e r a t i o n appears enough. A m p l i t u d e s of the shear a n d n o r m a l components are a p p r o x i m a t e l y 10% and 2% of the v e r t i c a l pressure force ( F i g . 19).
F i g . 17. Free surface elevation o b t a i n e d by f o r c e d heaving m o t i o n Acos{u>t) o f a circular cylinder; A/r = 0.4, u)^r/g = 2.0 — Navier-Stokes c o m p u t a t i o n s ; • Perfect f l o w c o m p u t a t i o n s .
(M
F i g . 19. Pressttre force and viscous force f o r a rectangular cyhnder i n heave Asin{ut). C o m -p u t a t i o n s w i t h o u t ( — ) , w i t h one ( • ) and w i t h three ( A ) nonhnear i t e r a t i o n s . u>B^/u = 1000 (u = O . l S m V s ) , u>'-B/2g = 2.0, A/B = 0.30, B/d = 1.0.
T h i s agrees weh w i t h d a t a given by Yeung and A n a n t h a k r i s h n a n (about 10% f o r the shear and 1% f o r the n o r m a l c o m p o n e n t ) .
F u r t h e r c o m p u t a t i o n s used the parameter uB' jv = 1.8 • lO'' and the same values as before f o r oj'B/2g, A/B and B/d. (Here u = l O - ^ m V s and B = 2 m ) . T h e mesh h a d 6000 nodes t o o , a n d the t i m e step was 5 • 10~^s. F i g . 20 shows nondimensional forces c o m p u t e d w i t h d i f f e r e n t numbers of i t e r a t i o n s on nonhnearities. C o m p a r i n g Figs. 19 and 20 shows the i n f l u e n c e of viscosity: the pressure force is the same, b u t the force due t o the shear stress is i m p o r t a n t o n l y f o r the s m a l l uB'/v. I n F i g . 20 the n o r m a l viscous force is n o t shown because of i t s smaUness.
Figs. 21 and 22 show the velocity and v o r t i c i t y fields i n t h e v i c i n i t y of t h e b o d y f o r t = 2.bbT and / = 3.18T respectively, T being the heave p e r i o d . T w o i t e r a t i o n s on nonlinearities were used, r e s u l t i n g i n free surface elevations i n the v i c i n i t y of the b o d y w h i c h are smoother t h a n those c o m p u t e d w i t h o u t supplementary nonlinear i t e r a t i o n s . O u r results have been compared w i t h those o f Yeung and Ananthakrishnan (1991) w h o used also Navier-Stokes c o m p u t a t i o n s . T h e agreement between the t w o calculations is quite good; however, because Yeung a n d A n a n -t h a k r i s h n a n ' s c o m p u -t a -t i o n s comprised -the whole body, -t h e i r v o r -t i c i -t y resul-ts are shgh-tly asy-m e t r i c ( F i g . 22 b ) .
5 C o n c l u s i o n
W e solve the two-dimensional unsteady l a m i n a r Navier-Stokes equations f o r f o r c e d m o t i o n of a b o d y at a free surface. A comparison of our results f o r n - t h harmonic force coefficients w i t h those given b y perfect fiuid computations and w i t h experimental results prove the accuracy of t h e m e t h o d . T h e m a i n interest of the C P U - t i m e intensive viscous fluid m o d e l is t h e s t u d y of flows a r o u n d sharp-edged bodies where the influence of viscosity is i m p o r t a n t , a n d o n l y Navier-Stokes c o m p u t a t i o n s are expected t o predict the h y d r o d y n a m i c coefficients accurately. However, the t r e a t m e n t of the intersection between b o d y and free surface needs s t i l l some i m p r o v e m e n t s . T h e next step planned is t o use a f u l l y coupled v e l o c i t y / p r e s s u r e / f r e e surface elevation f o r m u l a t i o n solved by a direct m e t h o d (Alessandrini and Delhommeau 1995).
6 R e f e r e n c e s
A L E S S A N D R I N I , B . and D E L H O M M E A U , G. (1994), Simulaüon of three-dimensional unsteady viscous free surface flow around a ship model, Int. Jour, for Num. Meth. in Fluids 19, 321-342
A L E S S A N D R I N I B . and D E L H O M M E A U , G. (1995), Simulation numérique de Vécoulement turbulent incompressible auiour d'une carène de navire: vers une prise en compte rigoureuse des conditions de surface libre en fluide visqneux, 5èmes Journées de I'Hydrodynamique, Rouen
ERIKSSON, L . E . (1982), Generation of boundary-conforming grids around wing-body configurations using transfinite interpolation, A I A A Journal 20(10), 1313-1320
FALTINSEN, O . M . (1977), Numerical solutions of transient nonlinear free-surface motion outside or inside moving bodies, Proc. 2nd I n t . Conf. on Num. Ship Hydrodyn., Berkeley, 347-357
H W A N G , J.H., K I M , Y . J . and K I M , S.Y. (1987), Nonlinear forces due to two-dimensional forced oscil-lation, Proc. l U T A M Symp. on Nonlinear Water Waves, Tokyo, Springer-Verlag, 231-238
K Y O Z U K A Y . (1982), Experimental study on second-order forces acting on cylindrical body in waves, Proc. 14th Symp. on Naval Hydrodynamics, A n n Arbor, 319-382
LEE, C . M . (1968), The second-order theory of heaving cylinders in a free surface, J. of Ship Research 12, 313-327
NICHOLS, B . D . and H I R T , C W . (1977), Nonlinear hydrodynamic forces on floating bodies, Proc. 2nd Int. Conf. on N u m . Ship Hydrodyn., Berkeley, 382-394
P A P A N I K O L A O U , A . (1987), On calculations of nonlinear wave-body interaction effects, Proc. l U T A M Symp. on Nonlinear Water Waves, Tokyo, Springer-Verlag, 247-258
P A P A N I K O L A O U , A . and N O W A C K I , H . (1984), Second-order theory of oscillating cylinders in a regular steep wave, Proc. 13th Symp. on Naval Hydrod., 303-331
PARISSIS, G. (1966), Second-order potentials and forces for oscillating cylinders on a free surface, MIT-Report 66-10, Dept. of Ocean Eng.
P I Q U E T , J. and VISONNEAU, M . (1991), Computation ofthe flow past shiplike hull, Comp. & Fluids 19(2), 183-215
POTASH, R.L. (1971), Second-order theory of oscillating cylinders, J. of Ship Res. 15(4), 295-324
SHANKS, S.P. and T H O M P S O N , J.F. (1977), Numencal solution ofthe Navier-Stokes equations for 2D hydrofoils m or below a free surface, Proc. 2nd Int. ConL on Num. Ship Hydrodyn., Berkeley, 202-220 T A S A I , F. and K O T E R A Y A M A , W. (1976), Nonlinear hydrodynamic forces acting on cylinders heaving
on the surface of a fluid. Rep. of Res. Inst, for A p p l . Mech., Kyushu Univ., vol. 23 no. 77
T E L S T E , J.G. (1985), Calculation of fluid motion resulting from large-amplitude forced heave motion of a two-dimensional cylinder in a free surface, Proc. 4th Int. Conf. Num. Ship Hydrodyn., Washington D . C , 82-93
URSELL, F. (1949), On the heaving motion ofa circular cylinder on the free surface of a fluid, Quartely J. of Mech. and A p p l . Math. 2, 218-231
V A N DER VORST, H.A. (1992), Bi-CGSTAB: a fast and smoothly converging variant of bi-CG for the solution of nonsymetric linear systems, J. Sci. Stat. Comp. 13
V I S O N N E A U , M . (1993), Simulation numérique des equations de Navier-Stokes pour un fluide visqueux incompress ible, Ecole de printemps de M E N , Carcans-Maubuisson, May
Y A M A S H I T A , S. (1977), Calculations ofthe hydrodynamic forces acting upon thin cylinders oscillating vertically with large amplitude, J. Soc. Naval Arch, of Japan 141, 61-69
Y E U N G , R.W. and A N A N T H A K R I S H N A N , P. (1991), Large-amplitude oscillation of two-dimensional bodies in a viscous fluid with a free surface, 6th Int. Workshop on Water Waves and Floating Bodies, Woods Hole
Y E U N G , R.W. and A N A N T H A K R I S H N A N , P. (1992), Oscillation of a floating body in a viscous fluid, J. of Eng. Math. 26, 211-230
Y E U N G , R.W. and Y U , C F . (1991), Viscosity effects on the radiation hydrodynamics of horizontal cylinders, J. of Offshore Mech. and Artie Eng. 1(A), 309-316
I . . . . I . . . . I , . 1 . 1 . J
' ' , — 10 15 20 rat
F i g . 20. Pressirre force and viscous force for a rectangrdar cylinder i n lieave Asm.{ut). C o m -p u t a t i o n s w i t h o u t ( — ) , w i t h one ( • ) and three ( A ) nonhnear i t e r a t i o n s . OJB^^IV = 1.8 • 10^ {v = l O - ' ^ m V s ) , u>'^B/2g = 2.0, A/B = 0.30, B/d = 1.0.
F i g . 2 1 . Velocity and v o r t i c i t y fields at t = 2.bbT [T =lieave p e r i o d ) . Present m e t h o d (c and d ) , c o m p u t a t i o n s o f Yeung and Ananthakrishnan (a and b ) . Heaving m o t i o n Asmiut) w i t h A/B = 0.30, B/d ^ 1.0, uj'^Bj2g ^ 2.0, uB'^jv = 1.8 • 10^ [P = l O ^ W / s ) .
F i g . 22. Velocity and v o r t i c i t y fields at f = 3.18T ( T =heave p e r i o d ) . Present m e t h o d (c and d ) , c o m p u t a t i o n s of Yeung and Ananthakrishnan (a and b ) . Heaving m o t i o n Asiii{u>t) w i t h A/B - 0.30, B/d = 1.0, u;''B/2g = 2.0, uBy^ = 1.8 • 1 0 ^
Fourier Representation of Near-Field Free-Surface Flows ^
F r a n c i s N o b l e s s e , D a v i d Taylor M o d e l Basin^ C h i Y a n g , George Mason University^
1. I n t r o d u c t i o n
W i t h i n the classical frequency-domain analysis of ship motions based on the Green f u n c t i o n s a t i s f y i n g the usual hnear free-surface b o u n d a r y c o n d i t i o n , free-surface effects are defined by a Fourier superposition of elementary waves exp[k(- i {a(-\-l3r])], where a and /? are Fourier variables, k = ^a^^^"^ is the wavenumber, and (.^ , ' ' ? ) ( < 0) are nondimensional coordinates (the C axis is v e r t i c a l and points u p w a r d , and the plane C = 0 is the mean free surface). Specifically, the t e r m representing free-surface eff'ects i n the Green f u n c t i o n G ' ( f ; x), where I = (^, ??, C < 0) is the flow-observation p o i n t and x = (,T, y, ^ < 0) is the singular p o i n t , and the free-surface p o t e n t i a l ^•'^in the F o u r i e r - K o c h i n approach are defined b y
4 . ^ { ^ ; } = ^hm f d f J f j a {^^Pt ^ - + ' ^ ( - - + / ^ ^ ) ] } exp[ ( a ^ + ^ , ) ] / [ i ^ + i . s i g n ( i ? , ) ] .
Here S = S{a,(3) is a spectrum function w h i c h is defined by a d i s t r i b u t i o n of the elementary wave s o l u t i o n ex-p[kz + i(ax + Py)] of the Laplace equation over the mean w e t t e d huU and the mean waterhne of the ship. Specific spectrum f u n c t i o n s are defined w i t h i n the Fourier-K o c h i n f o r m u l a t i o n developed i n Noblesse and Yang (1995). However, we are n o t concerned w i t h specific spectrum f u n c t i o n s here. Indeed, we consider the Fourier representation (1) f o r a generic s p e c t r u m f u n c t i o n <S , i.e. f o r a,n a r b i t r a r y d i s t r i b u t i o n of sources a n d / o r dipoles.
T h e f u n c t i o n D = D{a,/3) i n (1) is the dispersion function, and Df = dD/df where ƒ is a wave frequency. Specifically, the dispersion f u n c t i o n f o r wave d i f f r a c t i o n - r a d i a t i o n b y a ship advancing at constant speed along a.straiglit p a t h , chosen as the ^ axis ( t h i s axis points t o w a r d t h e ship b o w ) , t h r o u g h regular waves i n water of i n f i n i t e d e p t h and l a t e r a l extent, is
D = { f - F a f - k w i t h k = V o H ^ s i g n ( P / ) = s i g n ( / - F a ) . (2) Here ƒ = a; ^/L/g is the nondimensional wave frequency and F='U/\fgL is t h e Froude number,
w i t h w = encounter frequency of i n c o m i n g waves, L = ship l e n g t h and U = ship speed.
A c c u r a t e evaluation of the singular double Fourier i n t e g r a l (1) is c r i t i c a l f o r using
free-surface Green f u n c t i o n s , be i t f o r d e t e r m i n i n g near-field fiows v i a a. free-free-surface Green-function method, f o r couphng an inner near-field calculation method which accounts f o r viscous a n d / o r nonhnear effects w i t h an outer linear potential-flow representation, or f o r evaluating the f a r - f i e l d waves corresponding t o a given near-field flow (predicted b y any calculation m e t h o d , i n c l u d i n g p o t e n t i a l - f l o w methods based on R a n k i n e singularities and viscous-flow m e t h o d s ) .
T h e free-surface t e r m G ^ defined b y (1) is analyzed i n immerous studies of free-surface Green f u n c t i o n s using contour integration i n the complex plane. T h i s classical m e t h o d of analysis o f t h e Green f u n c t i o n and the subsequent huh-panel and waterhne-segment i n t e g r a t i o n involve s u b s t a n t i a l d i f f i c u l t i e s , w h i c h have hindered the development of reliable and p r a c t i c a l c a l c u l a t i o n methods based o n the free-surface Green function ( 1 ) . T h e double Fourier i n t e g r a l
/
oo roo
df5 I daAexp[-i{ai+(3vi)]l[D + iesign{Df)] (3)
-oo J-oo
•'This study was supported by the Independent Research program of the David Taylor Model Basin, with additional support from the Carrier Channel Guidance program.
^Code 542, Bethesda, MD 20084-5000, USA. 'Fairfax, VA, USA.
is analyzed i n Noblesse and Chen (1996) using an alternative m e t h o d t o the classical contour-i n t e g r a t contour-i o n approach. T h contour-i s alternatcontour-ive m e t h o d , based on a dcontour-irect analyscontour-is ( w contour-i t h o u t contour-i n t r o d u c contour-i n g an i n t e g r a t i o n c o n t o u r ) i n the real Fourier space ( a , / 3 ) , shows t h a t the generic free-surface p o t e n t i a l defined by the Fourier representation (3) can be expressed as
4 > ^ = r ( 4 )
where cf)^ and cj)^ correspond t o a wave component and a non-oscihatory near-field ( l o c a l ) flow component. T h e wave component ^ ' ^ i s given by an integral along the curve(s) defined i n the Fourier plane by the dispersion r e l a t i o n D{a,f3) = 0:
<t>^=-iT^y2 ds[sign(Df) + sign{^Da + vDp)]Aexp[-i{aC+/3r2)]/\\VD\\ (5) D=o -^^=0
where {D^,Dp) = (OD/da ,dD/d(3), \\VD\\ = {Dl + D j y / ' , ^ ^ ^ ^ means s u m m a t i o n over all the dispersion curves D = 0, and ds stands f o r the d i f f e r e n t i a l element of arc l e n g t h of the dispersion curves. T h e near-field component (j)^ i n (4) is given by
/
CO roo
dp daA<ixp[-i{a^+(iy)]l[D-ies\gn{iD„ + yDp)] (6)
•oo J-oo
where D — i£s\gix{^Da-]-r]Dp) may be replaced by D outside a dispersion curve, and we have
sign(e-D„
+ vDp) =
QïïeW =
Expression (6) f o r the near-field component (j)^ \s adequate f o r a n a l y t i c a l purposes, as is i l l u s t r a t e d by the analysis of the Green f u n c t i o n of wave d i f f r a c t i o n - r a d i a t i o n at l o w f o r w a r d speed given i n Noblesse and Chen (1996). Expression (6) can also be used f o r purposes of n u m e r i c a l evaluation i f e i n the expression D~i e sign{^Da+r]Di3) is sufficiently smah. However, t h i s m e t h o d of n u m e r i c a l l y evaluating the near-field component ^ ^ i s n o t p r a c t i c a l because the i n t e g r a n d of (6) corresponding t o the smah values of e required t o o b t a i n sufficient accuracy is sharply peaked at a dispersion curve, as is shown i n this study. A Fourier representation of t h e near-field component (f)^ w h i c h can be evaluated numerically i n an accurate and p r a c t i c a l manner (an essential requirement f o r a free-surface Green-function calculation method a n d f o r couphng an inner near-field calculation method w i t h an outer linear potential-flow represen-tation) is obtained here. T h i s Fourier representation of the near-field component (p^ and the Fourier representation (5) o f t h e wave component (f)^ in the decomposition (4) i n t o wave and near-field components y i e l d a m a t h e m a t i c a l representation o f the generic free-surface p o t e n t i a l (f)^ t h a t is weU suited f o r a n a l y t i c a l and n u m e r i c a l purposes i n b o t h t h e near field and t h e f a r field (where (p^ fti ( f ) ^ ) , as is shown i n this study f o r steady ship waves and t i m e - h a r m o n i c offshore-structure waves.
2. P r a c t i c a l F o u r i e r r e p r e s e n t a t i o n o f n e a r - f i e l d c o m p o n e n t
Noblesse and Chen (1996) show t h a t the near-field component <?!)^ can be expressed as
(f)^ = (/)2-'^i w i t h (7a)
/
OO poo
d(3 da Aexp[-i{aC+f3r])]/D and ( 7 b )
-oo J —CO
Ti = -iT^Y, / dss\gn{iD„^r]Dii) Aexp[-i{ai+(iri)]l\\SJD\\. (7c) D=o - ^ ^ = 0
T h e representation (7) is used, instead of expression ( 6 ) , as a s t a r t i n g p o i n t f o r the analysis of the near-field component (f)^ developed below.
2.1. D i s p e r s i o n s t r i p s
T l i e representation (7) defines the near-field component (p^ as the difference between t h e double i n t e g r a l (t>2 and the single i n t e g r a l (f>i . We seek a representation of the n e a r - f i e l d component cj)^ w h i c h involves only a double integral by expressing t h e single i n t e g r a l 4>i a l o n g t h e dispersion curve(s) D = 0 as a double i n t e g r a l . To t h i s end, we define t h e dispersion strip(s) -4:(JW <D< AaW, where t h e positive real constant a and f u n c t i o n W{a, (5) c o n t r o l t h e w i d t h o f t h e dispersion s t r i p ( s ) . Specifically, t h e w i d t h of the dispersion s t r i p -^aW <D<AaW, given by 2d7i = 2dD{dn/dD) = 8aW/ldD/dn), is equal t o 8crT4^/||Vi?|| since dD/dn = \\VD\\ , see (13) i n Noblesse and Chen (1996). Dispersion strip(s) of nearly constant w i d t h can be defined i f t h e width-function W{a,j3) is chosen as W - WVDW . F u r t h e r m o r e , the w i d t h of a dispersion s t r i p is equal t o 8cr i f V F = l l V i ^ l l • Other choices f o r the f u n c t i o n W can be used, as is shown for steady flows f u r t h e r on i n t h i s study. We define A as
A = D/W. ( 8 ) T h e analysis developed i n t h i s study uses t w o even real f u n c t i o n s of A/a w h i c h are n e g l i g i b l y
small outside a dispersion s t r i p - 4 ( T < A < 4 ( T . These t w o /oca/mng f u n c t i o n s , denoted Er a n d Ei, are chosen i n section 2.4 as the p r o d u c t of the exponential f u n c t i o n e x p [ - ( A / c r ) 7 2 ] b y p o l y n o m i a l s i n ( A / c r ) ^ . T h e f u n c t i o n e x p [ - ( A / c 7 ) 2 / 2 ] is equal t o e x p ( - 8 ) 3 X 10"'* at t h e
edges D = ±4aW of a dispersion s t r i p .
2 . 2 . D o u b l e - i n t e g r a l a p p r o x i m a t i o n of s i n g l e i n t e g r a l T h e single i n t e g r a l (7c) can be expressed i n t h e f o r m
<j>i = - i { C / c 7 ) y I dssign(^Dc>-{-rjDp)oAoexv[-i{ao^+(3or])]\\VD\\^'[ E,{A/a)dA
D=0
where Ei is an even f u n c t i o n s a t i s f y i n g t h e c o n d i t i o n
/
oo roo
Ei{A/a)dA^2 Ei{A/a)dA = aTr/C, (9)
-oo Jo
C is a real constant ( d e t e r m i n e d f u r t h e r o n ) , a n d the subscript 0 means t h a t the corresponding f u n c t i o n is evaluated at the dispersion curve D = 0. T h e i n t e g r a l (/>! can be expressed as
^ = - i ( C / a ) V / dssign{CD„-\-riDp)o r dAEi{A/cj) Aex]>[-iia^+P7j)]/\\VD\\
JD=0 J-OO
+i{C/a)Y, I dssign{^D^ + TiDp)o r dAE,iA/a)A{A,s) (10)
D=0 J J - o o
w i t h A = A e x p [ - i ( a e + / 3 ? ? ) ] / l | V £ ' | | - A o e x p [ - i ( « o ^ + M ] / | | V i ? | | o • ( H )
T h e f u n c t i o n A ( A , s ) , w h i c h vanishes at a dispersion curve A = 0 , m a y be a p p r o x i m a t e d b y t h e T a y l o r series A ( A , s) = A i ( s ) A -(- A 2 ( s ) A ^ - l - A 3 ( s ) A ^ - l - • • • i n t h e v i c i n i t y of a dispersion curve. T e r m b y t e r m i n t e g r a t i o n of t h i s expansion yields
/
oo roo roo
E,{A/a)A{A,s)dA = Ai{s) E,{A/a) A dA + A2{s) E,{A/(7) A'dA + • • • .
-co J — oo J— oo
T h e integrals i n v o l v i n g o d d powers of A are zero f o r any even f u n c t i o n Ei{A/a). I t follows t h a t t h e second i n t e g r a l i n (10) is negligibly small i f
roo
/ E,{A/a) Jo
A 2 ™ d A = 0 f o r l < m < M (12a)
roo
( I / C T ) ! / P ; ( A / C T ) A^^^C^A I < 1 f o r m > M + l . (12b)
Functions Ei w l i i c l i satisfy conditions (9) and (12a) are defined i n section 2.4 . Expression (10) tiierefore becomes
~
- ^ ^ E
J j , ^ / ' J _ J ^ s i g n ( e i ? „ + ??i^^)o e x p [ - i ( « e + / ? 7 / ) ] since (8) yields dA = dD {l-AdW/dD)/W. T h e r e l a t i o ndW/dD = \W-VD/\\VD\\\ (13) w h i c h follows f r o m (24) i n Noblesse and Chen (1996), yields dA = dDfl/\\VD\\ w i t h
Ü = [\\WD\\-DVW-VD/{W\\VD\\)]/W. (14) T h e t r a n s f o r m a t i o n of coordinates (s,D) ( a , / ? ) f i n a l l y yields
foo roo
4>i^-i{Cla) dp dasign{CD^ + rjDp)onEiAexp[-i{a^+p7])]/\\VD\\, (15) J—oo J — oo
w h i c h expresses the single i n t e g r a l (7c) as a double i n t e g r a l . T h i s double-integral representa-t i o n of (f)i can be combined w i representa-t h representa-the double i n representa-t e g r a l ^2 i n (7a). T h e r e s u l representa-t i n g double-inrepresenta-tegral representation of the near-field component (f)^ is given i n section 2.5 , after the double i n t e g r a l ^2 defined b y (7b) is m o d i f i e d i n t o a f o r m better suited f o r n u m e r i c a l evaluation.
2 . 3 . A c l a s s o f s i n g u l a r d o u b l e i n t e g r a l s
T h e double i n t e g r a l (7b) is of the f o r m J^^dp J^^daN/D where the d e n o m i n a t o r D vanishes along one (or several) curve(s) and the n u m e r a t o r N is finite at the curve(s) D = 0 . T h i s i n t e g r a l can be expressed as
/
oo roo roo rco roo roo
dp daN/D= dp daN{l-Er)/D+ dp daNE^/D. (16)
•00 J — oo J — oo J—oo J — oo J — oo
T h e f u n c t i o n Er = Er{A/a), where A = D/W, is an even real f u n c t i o n t h a t is neghgibly smaU outside the s t r i p ( s ) defined by -AaW <D <AaW, and the i n t e g r a n d N{l-Er)/D vanishes i f D = 0 because the even f u n c t i o n Ej.(A/a) is assumed t o satisfy the c o n d i t i o n
Er{0) = 1 . (17) T h e o n l y significant c o n t r i b u t i o n t o the second i n t e g r a l on the r i g h t of (16) stems f r o m the
strips -4aW <D<4aW. T h i s i n t e g r a l can be rendered negligibly smah, so t h a t (16) becomes
/
CO rco POO POO
dp daN/D^ dp daN{l-Er)/D, (18)
-00 J — oo J— CO J — oo
i f the locahzing f u n c t i o n Er{A/a) is chosen i n the manner explained f u r t h e r on. I f the f u n c t i o n D also vanishes at an isolated p o i n t (ao,/3o), this singular point must he outside the s t r i p ( s ) -4aW < D < 4aW associated w i t h the curve(s) D = 0 , so t h a t the f u n c t i o n E,. is n e g h g i b l y smah at the p o i n t (an ,/9o) and the i n t e g r a n d on the r.h.s. of (18) is i d e n t i c a l t o the i n t e g r a n d on the l e f t i n the v i c i n i t y of this p o i n t . T h u s , the s t r i p ( s ) associated w i t h the singular curve(s) D = 0 are assumed s u f f i c i e n t l y t h i n to exclude any isolated singular p o i n t .
T h e second i n t e g r a l on the r.h.s. of (16) may be expressed as
E / ds [ dD \\VD\\-^NEr/D
D=0 J^=^ J-oo
