Forces on a flexible pile

FORCES ON A FLEXIBLE PILE A. D. K. Laird

Professor of Mechanical Engineering University of California, Berkeley

ABSTRACT

This paper presents some background, hypotheses, and conclusions in connection "ith current research at the University of California, Berkeley, on flexibly mounted cylinders ,·,ith two degrees of freedom in waves and streams.

GENERAL CONSIDERATIONS

The maintenance of pole, stake, and pile structures in streams and vraves has been important to man from antiquity. Because of the common occur-rence of the flow of a fluid past a cylinder, and the simplicity of the boundaries, one might expect that long ago all related problems ,fQuld have been solved. However, if one experiments with forcing a pole rapidly through still water, he ,rill find that the pole oscillates laterally and that he can reduce these oscillations, and thus move the pole more easily, by grasping it firmly. These observations suggest a lack of simplicity of the system and an explanation for the present dependence on empirical information in the solution of related engineering problems. The art of building surfaces above vrater level has been augmented by engineering and scientific investigations so numerous that even citation of all the most important is prohibitive. Current practice produces successful offshore structures, but more design information will be needed to build them competitively in ever deeper water.

Marris (1964), Morkovin (1964), and others have summarized much of the pertinent data on the flow details around rigidly supported cylinders and described means of representing phenomena. This paper reviews progress

toward solution of problems basic to the design of structures in moving fluids, and toward explanation of differences bet"een force patterns imposed on actual structures under field conditions and on the idealized "infinitely long rigid circular cylinder in a steady stream".

TECHNICAL CONSIDERATIONS

The customary supports for platforms over vrater are various kinds of stiff-skinned cylinders. Design of the elements of the underpinning vrill include choices of cross-sectional shape, length to diameter ratio, the number to be used, their spacing, and the orientation of the structure to ,faves or currents. ,Thether the support is considered to be an island or piling is relative and involves the diameter to length ratio. For interactions vrith ,,-ind ,.,aves, a cylindrical fill ,lQuld be an island, but for tidal waves, might be a rigid pile.

LARGE CYLINDERS AND DIFFRACTION

~acCamy and Fuchs (1954) have shmm that vrave diffraction is im-portant vrhen the diameter of a cylindrical support element is comparable vrith

t h e wavelength. R e s u l t s o f i n v e s t i g a t i o n s o f t h e d i f f r a c t i o n o f sound waves, such as done hy Wiener ( 1 9 ^ 7 ) , are a p p l i c a b l e , as c o n f i r m e d by L a i r d (1955)> f o r c y l i n d e r diameters g r e a t e r t h a n about 20 p e r c e n t o f t h e wavelength-Since such c y l i n d e r s g e n e r a l l y have g r e a t r i g i d i t y , t h e y may be regards'^ as examples o f elements w i t h f l e x i b i l i t i e s approaching zero. The f l u i d f o r c e s on an I s l a n d are u s u a l l y u n i m p o r t a n t ; more common c o n s i d e r a t i o n s are

t o p p i n g o f t h e i s l a n d by waves and f i n d i n g s u i t a b l e l o c a t i o n s f o r docks. e weather s i d e I s f r e q u e n t l y s u b j e c t e d t o wave c r e s t s 50 percent h i g h e r t h a n t h e c r e s t s o f oncoming waves. Crests on t h e l e e s i d e may be 20 p e r c e n t lower than t h e approaching waves. Phase angles are a l s o v e r y s i m i l a r t o those f o r t h e d i f f r a c t i o n o f sound waves.

VIBRATION TRANSMISSION

I t i s c l e a r t h a t t h e s m a l l e r t h e diameter o f unsupported p i l ^ ^ ' t h e g r e a t e r w i l l be t h e d e f l e c t i o n o f a g i v e n number o f them. A l s o , t ^ e y may o f f e r l e s s r e s i s t a n c e t o f l u i d s moving t h r o u g h them, and may t r a n s m i t l e s s l a t e r a l seismic m o t i o n from t h e ocean f l o o r t o t h e p l a t f o r m . Cross ^ b r a c i n g , o r l a r g e r diameter f o r t h e p i l e s , i n c r e a s e s s t i f f n e s s , t r a n S f f l i ^ ^ ^ " " ' and I n d i v i d u a l l o a d c a r r y i n g c a p a c i t y . I n a c o n s i d e r a t i o n o f t h e f o r c e s on a s t r u c t u r e t h e f l e x i b i l i t y o f i t s u n d e r p i n n i n g s i s i m p o r t a n t .

FORCE COEFFICIENTS

C a l c u l a t i o n of t h e f o r c e s on a v e r y r i g i d l y supported cyli»'^*^''^ "'"'^ a steady stream i s r e l a t i v e l y simple. I t i s customary t o r e s o l v e t h ® f o r c e I n t o a drag f o r c e ( r e p r e s e n t e d by a drag c o e f f i c i e n t ) i n t h e d i r e c t i ^ * ^ ° . f l u i d m o t i o n , and a l i f t , o r t r a n s v e r s e f o r c e ( r e p r e s e n t e d by a l i f t c o e f f i -c i e n t ) normal t o t h e m o t i o n and t o t h e a x i s o f t h e p i l e . The drag fo^-ce -can be c l o s e l y approximated b y t h a t f o r a r i g i d c y l i n d e r g i v e n by WleseX'b^^S^-'^ and Betz (1923). Data on t h e l i f t c o e f f i c i e n t s , which are s m a l l e r *^^Q^a^^ drag c o e f f i c i e n t s , have been extended r e c e n t l y by Bishop and Hassan ( ^ ' ' WAVE FORCE COEFFICIENTS

F o l l o w i n g Morison e t a l ( l 9 5 0 ) , i t i s common p r a c t i c e t o x - . data and p r e d i c t wave f o r c e s on p i l e s by s e p a r a t i n g t h e r e s i s t a n c e f o r c e , i n ^ t h e d i r e c t i o n o f m o t i o n , i n t o a drag f o r c e , p r o p o r t i o n a l t o t h e s q u a ^ ^ ° ^ f l u i d v e l o c i t y , and an I n e r t i a f o r c e ( r e p r e s e n t e d b y a mass c o e f f l e i ^-"^ ' '. p r o p o r t i o n a l t o t h e f l u i d a c c e l e r a t i o n . Drag and mass c o e f f i c i e n t s f °r c u l a r c y l i n d e r s have been g i v e n f o r ocean waves by VJlegel e t a l ( 1 9 5 ' ^ ^ o t h e r s , and from l a b o r a t o r y s t u d i e s by many, i n c l u d i n g Keulegan and C a r p e n e r (1958) and McNown and Keulegan ( 1 9 5 9 ) , who a l s o showed t h a t t h e StroTJ-^^^^ t +' or r a t i o o f t h e t i m e f o r f l o w t o t h e t i m e f o r v o r t e x f o r m a t i o n , was 3-™P°^ ^" • For S t r o u h a l times l e s s t h a n 0.1, i n e r t i a i s u n i m p o r t a n t and s t e a d y s t a t e rags are n o t s t r i c t l y a p p l i c a b l e . For S t r o u h a l times between about 1 a n c i

i n e r t i a i s i m p o r t a n t .

EDDY SHEDDING AHD CYLINDER RESPONSE

i n e r t i a l e f f e c t s may be n e g l i g i b l e , b u t such s m a l l p i l e s u s u a l l y have s i g n i f i -cant f l e x i b i l i t y . Over t h e range o f Reynolds numbers o f p r a c t i c a l i n t e r e s t , the shedding o f eddies causes t r a n s v e r s e f o r c e s t h a t have s i g n i f i c a n t frequency components a p p r o x i m a t e l y p r o p o r t i o n a l t o t h e v e l o c i t y o f t h e f l u i d . This frequency i s r e f e r r e d t o as t h e eddy shedding frequency. The S t r o u h a l number i s t h e p r o p o r t i o n a l i t y c o n s t a n t which takes d i f f e r e n t values depending on the c r o s s - s e c t i o n a l shape o f t h e p i l e , and v a r i e s some w i t h Reynolds number. The eddy-associated f o r c e s occur over a frequency range which tends t o be l a r g e r near t h e Reynolds number f o r t r a n s i t i o n from l a m i n a r t o t u r b u l e n t boundary l a y e r on t h e c y l i n d e r s . Since s t r u c t u r e s t e n d t o v i b r a t e a t t h e i r n a t u r a l f r e q u e n c i e s , f l u i d v e l o c i t y which causes an eddy-shedding frequency c l o s e t o t h e n a t u r a l frequency o f t h e c y l i n d e r w i l l cause v i b r a t i o n s . Even r e l a t i v e l y r i g i d c y l i n d e r s may s u s t a i n damage, as demonstrated by t h e f a i l u r e o f many smokestacks. Penzien ( 1 9 5 7 ) and o t h e r s have shown t h a t t h e c y l i n d e r responds at a frequency near t h e eddy shedding frequency when t h e v e l o c i t y i s below those which e x c i t e a n a t u r a l frequency ( s i m i l a r t o F i g u r e 9 B ) . For a c o n s i d e r a b l e range o f v e l o c i t i e s which e x c i t e t h i s n a t u r a l f r e q u e n c y , response i s a p p r o x i m a t e l y c o n s t a n t a t or below t h i s n a t u r a l frequency. At h i g h e r speeds, response r e v e r t s t o t h e eddy shedding frequency. Methods f o r p r e -v e n t i n g t h i s t y p e o f damage i n c l u d e t h e r e d u c t i o n o f t h e resonant a m p l i t u d e s of v i b r a t i o n s near t h e n a t u r a l f r e q u e n c y , by I n c r e a s i n g t h e s t r u c t u r a l damping, or by g u y i n g ; and t h e p r e v e n t i o n o f t h e occurrence o f t h e p e r i o d i c f o r c e s by suppressing p e r i o d i c eddy shedding w i t h devices such as s p i r a l or p e r f o r a t e d f i n s .

WAKE CAPTURE

I f a c y l i n d e r i s s u f f i c i e n t l y f l e x i b l e , i t s nearresonant v i b r a -t i o n a l ampli-tudes may I n f l u e n c e -t h e eddy shedding mechanism so s -t r o n g l y -t h a -t eddies are shed a t t h e v i b r a t i o n a l frequency. The v i b r a t i o n t h u s becomes s e l f - e x c i t e d . Wake c a p t u r e , o r l o c k i n g i n o f t h e eddy shedding w i t h t h e v i b r a t i o n s , has been t r e a t e d by Bishop and Hassan ( 1 9 6 3 ) f o r t h e one degree of freedom, c r o s s - c u r r e n t , m e c h a n i c a l l y d r i v e n o s c i l l a t i o n o f a c i r c u l a r c y l i n d e r . T h i s r e s e a r c h showed t h a t f o r a g i v e n stream v e l o c i t y , t h e o s c i l -l a t i o n s o f t h e wake depended o n -l y on t h e o s c i -l -l a t i o n s o f t h e c y -l i n d e r , and n o t on I t s n a t u r a l frequency. H y s t e r e s i s e f f e c t s and an I n c r e a s e o f mean drag w i t h o s c i l l a t i o n a m p l i t u d e were n o t i c e d .

Toebes and Eagleson ( 1 9 6 1 ) and Eagleson e t a l (1961+) gave accounts of r e s e a r c h on o s c i l l a t i o n s i n one degree o f freedom, w i t h t h i c k f l a t p l a t e s as c y l i n d e r s , which showed t h e resonant e f f e c t s and t h e s y n c h r o n i z a t i o n o f the wake mentioned above. At t h e h i g h e r Reynolds numbers, t h e wake m o t i o n was more c o h e r e n t , or b e t t e r o r g a n i z e d i n t o a v o r t e x s t r e e t when t h e wake was i n t h e c a p t u r e d s t a t e t h a n when I t was n o t .

OCEAM DATA

Wave f o r c e data from a t e s t s i t e on t h e h a l f m i l e l o n g dock a t Davenport, C a l i f o r n i a , on t h e exposed P a c i f i c c o a s t , were r e p o r t e d by W i e g e l et a l ( 1 9 5 7 ) . These d a t a , a l t h o u g h a t h i g h e r Reynolds numbers, are s i m i l a r t o d a t a t a k e n by d i f f e r e n t f o r c e r e c o r d i n g systems a t l e s s exposed ocean

COASTAL ENGINEERING

t e s t s i t e s . The Davenport t e s t c y l i n d e r s were l o c a t e d about two bays from t h e seaward end o f t h e dock and were surrounded by o t h e r p i l e s some f o u r t o t e n diameters away. The f o r c e sensor was a f o o t l o n g s e c t i o n a t t h e m i d d l e of t h e 12 f t . l e n g t h o f t h e t e s t c y l i n d e r . Consequently, a l t h o u g h t h e r e was some o s c i l l a t i o n o f t h e t e s t p i l e , t h e r e c o u l d have been no resonant e f f e c t s o f t h e a m p l i t u d e , so t h e amplitudes o f t h e f o r c e r e c o r d s r e p r e s e n t e d f o r c e s e x e r t e d d i r e c t l y by t h e w a t e r . The l a r g e d e v i a t i o n s o f t h e d a t a from l a b o r a t o r y d a t a had no ready e x p l a n a t i o n and prompted f u r t h e r r e s e a r c h a t t h e U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y .

There were known t o be alongshore and t i d a l c u r r e n t s a t t h e t e s t s i t e which c o u l d account f o r some 25 p e r c e n t o f t h e d e v i a t i o n s , on t h e average, b u t n o t between successive waves or between d i f f e r e n t phases o f t h e same wave.

There are a l s o s e v e r a l assumptions i n v o l v e d which might cause d e v i a t i o n s when data f o r f o r c e s on p i l e s i n waves and i n streams are compared. I t i s t a c i t l y assumed t h a t t h e d i s t a n c e t r a v e l l e d by t h e water as t h e wave passes (which i s a p p r o x i m a t e l y equal t o t h e wave h e i g h t ) i s l a r g e compared t o t h e c y l i n d e r d i a m e t e r , so t h a t t h e f l o w i s p r a c t i c a l l y f u l l y developed, o r t h a t a q u a s i -steady s t a t e e x i s t s . V/hen i n e r t i a e f f e c t s are i m p o r t a n t , t h e y are u s u a l l y separated from t h e drag f o r c e s on t h e assumption t h a t t h e two are a d d i t i v e . The mass c o e f f i c i e n t i s o f t e n assigned a c o n s t a n t v a l u e o r i s o t h e r w i s e account-ed f o r e m p i r i c a l l y . Although these assumptions t e n d t o improve f o r l o n g e r p e r i o d s , t h e y are apt t o be bad f o r wave h e i g h t s l e s s t h a n 2 o r 3 d i a m e t e r s . I t must be assumed a l s o t h a t t h e v e r t i c a l v e l o c i t y components o f t h e water m o t i o n caused by t h e waves do not a f f e c t t h e f o r c e s p u r p o r t e d l y caused by t h e h o r i z o n t a l components. I t i s i m p r o b a b l e , however, t h a t any c o m b i n a t i o n of t h e s e m i s c e l l a n e o u s e f f e c t s c o u l d r e s u l t i n t h e observed f i v e f o l d d e v i -a t i o n s .

ACCELERATION

Because t h e most obvious d i f f e r e n c e between wave-induced v e l o c i t i e s and stream v e l o c i t i e s was a c c e l e r a t i o n , and because o f t h e i n d i c a t i o n s by Lunnon (1928) t h a t i t s t r o n g l y i n f l u e n c e d t h e r e s i s t a n c e o f spheres under some c o n d i t i o n s , i t was decided t o t e s t a c c e l e r a t i o n as a s i g n i f i c a n t f a c t o r i n t h e apparent s c a t t e r o f s i m u l a t e d wave f o r c e data. I v e r s e n and B a l e n t ( l 9 5 l ) had p r e v i o u s l y experimented w i t h c o n s t a n t f o r c e a c c e l e r a t i o n e f f e c t s and o b t a i n e d c o r r e l a t i o n s , b u t showed l a r g e d e v i a t i o n s f o r p a r t o f t h e range o f e x p e r i m e n t a t i o n . Keim (1956) i n v e s t i g a t e d t h e e f f e c t s o f Reynolds number i n t h i s range o f d e v i a t i o n , b u t showed o n l y minor e f f e c t s . L a i r d e t a l (1959) r e p o r t e d t h e r e s i s t a n c e f o r c e on a r e l a t i v e l y r i g i d l y mounted c y l i n d e r p a r a l l e l t o t h e water s u r f a c e d u r i n g a c c e l e r a t i o n , steady m o t i o n , and d e c e l e r a t i o n as t h e c y l i n d e r was towed along 100 f t . o f channel. They showed t h a t r e s i s t a n c e f o r c e - a c c e l e r a t i o n modulus c o r r e l a t i o n s (Crooke 1955) were d e s t r o y e d by boun-dary l a y e r t r a n s i t i o n e f f e c t s . Since no s i g n i f i c a n t f o r c e s were found t h a t c o u l d not be accounted f o r by known drag and i n e r t i a e f f e c t s , a c c e l e r a t i o n was c o n s i d e r e d e l i m i n a t e d as a cause o f s e r i o u s d e v i a t i o n s o f t h e data. EDDY FORCES AND SHELTERING BY NEIGHBORS

They a l s o found t h a t t h e presence o f a p a r a l l e l n e i g h b o r i n g c y l i n d e r l e a d i n g t h e t e s t c y l i n d e r caused o s c i l l a t i o n s i n t h e r e s i s t a n c e f o r c e a t l e a s t

t w i c e as l a r g e as t h e steady s t a t e drag f o r c e , D^, i n t h e absence o f a n e i g h -b o r , and t h a t t h e mean drag f o r c e was reduced t o a h a l f o r a f i f t h o f ( F i g u r e l ) . Unmeasured l i f t f o r c e s appeared t o be l a r g e : f o r example, i n one case, the 1*50 l b . tow c a r r i a g e was l i f t e d o f f i t s r a i l s . L a i r d e t a l ( 1 9 6 0 ) , u s i n g v e r t i c a l c y l i n d e r s a t t a c h e d t o a frame suspended by veires and s w i n g i n g over s t i l l water as an i n v e r t e d wave model ( c y l i n d e r s p e r f o r m i n g n e a r l y simple harmonic m o t i o n t h r o u g h s t i l l w a t e r ) , showed t h e l a r g e o s c i l l a t o r y f o r c e s caused by a n e i g h b o r i n g c y l i n d e r vrere due t o t h e i n f l u e n c e on t h e t e s t c y l i n d e r of eddies shed by t h e l e a d i n g neighbor ( F i g u r e 2 ) . They showed a l s o t h a t t h e s h e l t e r i n g e f f e c t o f neighbors caused a r e d u c t i o n i n mean drag when t h e t e s t c y l i n d e r was i n i t s neighbor's wake. \-Then a neighbor o r n e i g h b o r s were t o t h e s i d e , t h e c y l i n d e r s u f f e r e d an i n c r e a s e o f drag due t o t h e b l o c k i n g e f f e c t . I t was a l s o shown by L a i r d et a l (1959, I 9 6 0 ) t h a t when t h e c y l i n d e r s were s t a r t e d from r e s t , t h e r e s u l t s w i t h and w i t h o u t n e i g h b o r s were t h e same up t o t h e t i m e the t e s t c y l i n d e r reached t h e wake o f i t s n e i g h b o r s ( F i g u r e 3 ) . From t h a t t i m e on t h e spacing was i m p o r t a n t . With a l e a d i n g neighbor one diameter ahead, a n e g a t i v e drag developed t h a t was almost v i b r a t i o n f r e e . W i t h one neighbor 2 and 1* diameters ahead, l i f t ( F i g u r e 1+) and drag o s c i l l a t i o n s o f t h e order of Dq developed. For l a r g e r s p a c i n g s , t h e o s c i l l a t i o n s and s h e l t e r i n g e f f e c t s decreased and were n e a r l y n e g l i g i b l e a t 12 diameters spacing.

FORCE CALCULATIOHS BY POTENTIAL THEORY

A t h e o r e t i c a l i n v e s t i g a t i o n based on p o t e n t i a l t h e o r y and observed eddy b e h a v i o r by L a i r d (1961) r e s u l t e d i n a method o f c a l c u l a t i o n which gave r e s u l t s ( F i g u r e 5) s i m i l a r t o t h e e x p e r i m e n t a l r e s u l t s o f L a i r d e t a l ( 1 9 6 0 ) , and showed t h a t cross c u r r e n t s c o u l d make i m p o r t a n t d i f f e r e n c e s i n t h e e f f e c t s of t h e eddies on t h e c y l i n d e r .

SHELTERING AUD ORIENTATION EFFECTS ON GROUPS

The e f f e c t s o f spacing and o f o r i e n t a t i o n t o t h e f l u i d m o t i o n o f a n e a r l y symmetric group o f 21+ c i r c u l a r c y l i n d e r s were r e p o r t e d by L a i r d and V/arren (1963). The group was r i g i d l y mounted v e r t i c a l l y between h o r i z o n t a l d i s k s , t h e upper o f which was a t t a c h e d by k s t i f f f o r c e bars t o a wheeled c a r r i a g e d r i v e n along r a i l s b y a crank and arm. I t was shown t h a t t h e drag c o e f f i c i e n t f o r a s i n g l e c y l i n d e r i n steady f l o w was approached f o r i n d i v i d u a l c y l i n d e r s as t h e spacing approached 8 diameters ( F i g u r e 6 ) . S h e l t e r i n g reduced the mean drag from about O.9 o f i n f i n i t e spacing values f o r 1* diameter s p a c i n g , to 0.35 f o r 1 diameter spacing ( c y l i n d e r s t o u c h i n g ) . Maximum t r a n s v e r s e

f o r c e s t o 1.5 t i m e s steady f l o w v a l u e s were r e c o r d e d f o r spacings from 2 t o k d i a m e t e r s . O r i e n t a t i o n had l i t t l e e f f e c t . I n r e a l waves t h e r e would be phase e f f e c t s ( m i s s i n g here) because n o t a l l c y l i n d e r s would be i n t h e same p a r t o f t h e wave c y c l e a t any g i v e n t i m e .

FORCE SYNCHRONIZATION

The experiment w i t h many (21+) c y l i n d e r s appeared t o show l e s s syn-c h r o n i z a t i o n o f l i f t and drag o s syn-c i l l a t i o n s t h a n t h e experiments w i t h few (2 t o 5) c y l i n d e r s I n i n v e r t e d wave model m o t i o n ( L a i r d I 9 6 U ) . Landweber (191*2) has shown t h e o r e t i c a l l y and e x p e r i m e n t a l l y t h a t t h e s y n c h r o n i z a t i o n f o r 2 p a r a l l e l c i r c u l a r c y l i n d e r s s i d e by s i d e i n a steady stream was e s s e n t i a l l y

254

complete over a wide range o f l a t e r a l spacings. These o b s e r v a t i o n s suggest e i t h e r t h a t s y n c h r o n i z a t i o n i s reduced as t h e number o f c y l i n d e r s i n c r e a s e s , or t h a t t o and f r o motion d i d not a l l o w t i m e f o r s y n c h r o n i z a t i o n t o set i n as

i n t h e steady stream experiments.

SELF EXCITED VIBRATIONS IN OSCILLATORY MOTION

E f f e c t s o f p i l e f l e x i b i l i t y a l s o are b e i n g i n v e s t i g a t e d . L a i r d (1962) gave some data from a s i n g l e v e r t i c a l c y l i n d e r , w i t h v a r i e d support f l e x i b i l i t y , and n a t u r a l f r e q u e n c y , moved t o and f r o t h r o u g h water by t h e c a r r i a g e used f o r t h e group experiments ( L a i r d and Warren I 9 6 3 ) . The c y l i n d e r supported by a s l e n d e r c a n t i l e v e r had two degrees o f freedom ( l i f t w i s e and dragwise) . \'Ihen t h e eddy shedding f r e q u e n c i e s were c l o s e t o t h e n a t u r a l f r e q u e n c i e s , apparent resonant e f f e c t s occircred i n t h e drag and l i f t f o r c e s w i t h c o e f f i c i e n t s from I4- t o 5 times t h o s e f o r r i g i d c y l i n d e r s i n u n i f o r m f l o w

( F i g u r e 7 ) . For a l l f l e x i b i l i t i e s , s p e c t r a l a n a l y s i s showed an i n c r e a s i n g amount o f energy i n t h e fundamental v i b r a t i o n a l mode as t h e eddy frequency i n c r e a s e d toward t h e n a t u r a l frequency. The p r i n c i p a l v i b r a t i o n s on t h e drag r e c o r d s tended t o be a t t w i c e t h e frequency o f those on t h e l i f t r e c o r d s . These f i n d i n g s are i n g e n e r a l agreement w i t h p r e v i o u s f i n d i n g s on s e l f e x c i t e d systems w i t h one degree o f freedom. There was evidence o f t h e c a r r i a g e f r e -quency i n f l u e n c i n g t h e response f r e q u e n c i e s as p r e d i c t e d by an elementary t h e o r e t i c a l a n a l y s i s .

DRAG ANOMALY IN OSCILLATORY MOTION

A p p a r e n t l y , l a r g e mean drag f o r c e s are a s s o c i a t e d w i t h s m a l l t r a n s v e r s e a m p l i t u d e s : f o r example, a p p r o x i m a t e l y 3 t i m e s t h e steady drag a t a m p l i -tudes o f 2 p e r c e n t o f t h e diameter ( F i g u r e 8 ) . For c y l i n d e r s o f f i n i t e l e n g t h , the steady s t a t e drag c o e f f i c i e n t s t e n d t o be l o w e r , as shown by VJleselberger and Betz ( 1 9 2 3 ) . However, i n a l l o f t h e a u t h o r ' s r e s u l t s w i t h v e r t i c a l c y l i n -ders p r o t r u d i n g t h r o u g h t h e water s u r f a c e , t h e r e d u c t i o n o f drag appeared t o be l e s s . V?hether these two e f f e c t s e x i s t and i f t h e y are r e l a t e d should be i n v e s t i g a t e d . I t i s d o u b t f u l i f t h e t h r e e d i m e n s i o n a l e f f e c t s mentioned by Macovsky (1958) c o u l d e x p l a i n t h e i n c r e a s e o f drag. I t i s a l s o d o u b t f u l t h a t v e l o c i t y d i s t r i b u t i o n e f f e c t s d e s c r i b e d by Masch and Moore (1960) were i n v o l v e d . FLEXIBLE CYLINDER IN AN OSCILLATED GROUP

R e s u l t s o f an experiment w i t h a f l e x i b l y supported c y l i n d e r a t t h e c e n t e r o f a group o f 5 p a r a l l e l v e r t i c a l c y l i n d e r s i n a c r u c i f o r m p a t t e r n were g i v e n by L a i r d (196U). The group was a t t a c h e d t o t h e o s c i l l a t e d c a r r i a g e mentioned above. The h o u t l y i n g c y l i n d e r s were r i g i d l y supported. This two degree o f freedom f l e x i b l e c y l i n d e r a l s o showed t h e same k i n d o f a m p l i t u d e and f r e q u e n c y responses as found f o r one degree o f freedom s e l f e x c i t e d c y l i n -d e r s , b u t t h e o r i e n t a t i o n o f t h e group t o t h e m o t i o n was i m p o r t a n t . As might be p r e d i c t e d from p r e v i o u s t e s t s , when one c y l i n d e r was l e a d i n g , s h e l t e r i n g and eddy e f f e c t s were l a r g e r t h a n when two c y l i n d e r s were ahead and t o t h e s i d e s . SINGLE FLEXIBLE CYLINDER IN A STREAM

s u p p o r t e d by a f l e x i b l e c a n t i l e v e r o f square c r o s s - s e c t i o n and v a r i a b l e l e n g t h experienced t r a n s v e r s e o s c i l l a t i o n s s i m i l a r t o those f o r one degree o f freedom systems d e s c r i b e d by Toebes and Eagleson (1961) and o t h e r s . For c o n d i t i o n s o f wake c a p t u r e , t h e o s c i l l a t i o n s i n t h e d i r e c t i o n o f f l o w were a t t w i c e t h e frequency o f t h e l i f t w i s e o s c i l l a t i o n s so t h a t t h e c y l i n d e r f o l l o w e d a c r e s c e n t shaped p a t h . Motion p i c t u r e s showed t h a t , f o r c o n d i t i o n s o f wake c a p t u r e , t h e wake was about t w i c e as wide as f o r c o n d i t i o n s o f non-capture.

The t y p e o f response t o be expected from a f l e x i b l y supported c y l i n -der w i t h two l a t e r a l degrees o f freedom o f equal f l e x i b i l i t y i s i l l u s t r a t e d i n F i g u r e 9. The Reynolds number f o r which t h e eddy shedding f r e q u e n c y , f g , equals t h e n a t u r a l v i b r a t i o n a l f r e q u e n c y , f n , o f t h e c y l i n d e r i s r e p r e s e n t e d by t h e v e r t i c a l l i n e . The l i f t f r e q u e n c y , f L , tends t o have two values which are l e s s t h a n f g , o r J u s t l e s s t h a n f ^ , up t o t h e v e l o c i t y o f wake c a p t u r e , at which p o i n t f L assumes one v a l u e and t h e f r e q u e n c y , f ^ g , o f t h e o s c i l l a t i o n s i n t h e drag d i r e c t i o n t e n d r a p i d l y t o 2f-^. At a s u b s t a n t i a l l y l a r g e r v e l o c i t y , fx, r i s e s above tj^. For s t i l l l a r g e r v e l o c i t i e s , b a r r i n g t h e c l o s e approach t o o t h e r n a t u r a l f r e q u e n c i e s o f t h e system, f L i n c r e a s e s r a p i d l y t o a p p r o x i m a t e l y f g . These changes i n frequency c o i n c i d e w i t h apparent resonant e f f e c t s i n C l r , t h e c o e f f i c i e n t o f l i f t response based on t h e am plitude o f t r a n s v e r s e o s c i l -l a t i o n s . The dragwise o s c i -l -l a t i o n c o e f f i c i e n t , Cd2, a l s o goes t h r o u g h an

apparent resonance. The mean drag c o e f f i c i e n t , C-q, a p p a r e n t l y s t a r t s l a r g e a t s m a l l v i b r a t i o n a l amplitudes as shoim a l s o i n F i g u r e 8. This u n e x p l a i n e d

l a r g e apparent drag c o e f f i c i e n t s h o u l d be i n v e s t i g a t e d f u r t h e r . decreases r a p i d l y w i t h i n c r e a s i n g v e l o c i t y t o a normal v a l u e j u s t b e f o r e wake c a p t u r e . I t s a b r u p t i n c r e a s e w i t h Rg i s a s s o c i a t e d w i t h t h e abrupt i n c r e a s e i n t r a n s -v e r s e a m p l i t u d e s . This i n c r e a s e o f Cd i s r e l a t e d t o t h a t r e p o r t e d b y Bishop and Hassan ( 1 9 6 3 ) . Beyond resonance, r e t u r n s t o normal.

For v e l o c i t i e s t o o low t o produce s i g n i f i c a n t t r a n s v e r s e o s c i l l a t i o n s , Mandini (1965) showed t h a t t h e r e i s a range o f v e l o c i t i e s over w h i c h t h e c y l i n -der may o s c i l l a t e i n t h e d i r e c t i o n o f f l o w and shed a p a i r o f eddies each t i m e i t s t a r t s i n t h e upstream d i r e c t i o n . D u r i n g t r a n s v e r s e o s c i l l a t i o n , as t h e c y l i n d e r s t a r t s t o move away from i t s maximum a m p l i t u d e , i t sheds t h e eddy i t i s l e a v i n g .

FLUID FORCE MODEL

Maclean {l96h) a n a l y t i c a l l y a p p l i e d a m o d i f i e d Z - t r a n s f o r m t o p u l s e t r a i n s a c t i n g on a h y p o t h e t i c a l springmassdamped system chosen as a l i n e a r -i z e d a p p r o x -i m a t -i o n t o a c y l -i n d e r d e s c r -i b e d by F r -i t z l e r (1964). The p u l s e t r a -i n r o u g h l y s i m u l a t e d f o r c e s o f t h e water on t h e c y l i n d e r . The c a l c u l a t e d steady s t a t e response curves showed o s c i l l a t i o n s a t f r e q u e n c i e s a s s o c i a t e d w i t h t h e d u r a t i o n o f t h e p u l s e and w i t h t h e n a t u r a l frequency o f t h e c y l i n d e r . Under near resonant c o n d i t i o n s , t h e o s c i l l a t i o n s near t h e n a t u r a l f r e q u e n c y were o f the o r d e r o f 5 times those a t much lower f r e q u e n c i e s .

NOM-CIRCULAR CYLINDER BEHAVIOR

Hoerner (1958) gave f o r c e c o e f f i c i e n t s f o r a wide v a r i e t y o f c y l i n d e r s . L i n d s e y ( 1 9 3 8 ) , Roshko (l95't) and o t h e r s have g i v e n data on s t r u c t u r a l shapes used as p i l e s and cross b r a c i n g .

P a r k i n s o n and Brooks ( I 9 6 I ) measured p r e s s u r e d i s t r i b u t i o n s around r e l a t i v e l y r i g i d l y supported c y l i n d e r s w i t h D, square and r e c t a n g u l a r c r o s s -s e c t i o n -s and c a l c u l a t e d l i f t c o e f f i c i e n t -s a-s f u n c t i o n -s o f yaw angle ( F i g u r e 10). The absence o f l i f t a t any angle t o 20 degrees f o r a D, and a 1 t o 2 r e c t a n g l e w i t h l o n g s i d e across t h e f l o w , i s n o t e w o r t h y . Mhen these c y l i n d e r s were f l e x i b l y mounted, t h e ones f r e e o f yaw e f f e c t s o s c i l l a t e d i n a steady stream l i k e c i r c u l a r c y l i n d e r s i n s e l f - e x c i t e d m o t i o n produced by eddy shedding. They a l s o v i b r a t e d i n t h e i r t o r s i o n a l modes. Those r e c t a n g u l a r c y l i n d e r s w i t h r a t i o s o f dimension across t h e f l o w t o dimension w i t h t h e f l o w s m a l l e r t h a n h t o 3 were l e s s s u b j e c t t o eddy-induced v i b r a t i o n s b u t plunged o r g a l l o p e d l a t e r a l l y w i t h o s c i l l a t i o n amplitudes i n c r e a s i n g w i t h f l o w v e l o c i t y . There was no s i g n i f i c a n t r o t a t i o n o f t h e c y l i n d e r s ; t h e l i f t f o r c e arose because t h e t r a n s v e r s e v e l o c i t y o f t h e c y l i n d e r c r e a t e d an e f f e c t i v e yaw angle. The r a t e o f grovrth o f a m p l i t u d e from r e s t a l s o was measured and c a l c u l a t e d .

P a r k i n s o n and Smith (196U) r e - a n a l y z e d t h e s t a b i l i t y o f t h e above square c y l i n d e r by u s i n g t h e r i g i d c y l i n d e r r e s u l t s and assuming s m a l l nonl i n e a r damping. They p r e d i c t e d o s c i nonl nonl a t i o n h y s t e r e s i s r e s u nonl t i n g from o v e r nonl a p -p i n g o f two s t a b l e l i m i t c y c l e s , t h e magnitudes o f t h e steady s t a t e o s c i l l a t i o n s as a f u n c t i o n o f v e l o c i t y , and t h e number o f c y c l e s from r e s t t o t h e steady s t a t e a m p l i t u d e . From t h e form o f t h e i r e q u a t i o n s t h e y deduced a parameter which c o r r e l a t e d a wide range o f d a t a w i t h remarkable accuracy. This paper does much t o e x p l a i n t h e t r a n s v e r s e g a l l o p i n g o f r e c t a n g u l a r c y l i n d e r s which i s n o t a s s o c i a t e d w i t h t h e resonance between eddy shedding and t h e n a t u r a l frequency o f t h e c y l i n d e r s .

REFEEEHCES

Bishop, R. E. D., and Hassan, A. Y. (1963). The l i f t and drag f o r c e s on a c i r c u l a r c y l i n d e r i n a f l o w i n g f l u i d : Proc. Roy. Soc. Lond. , V o l . 277, pp. 32-50.

Crooke, R. C. ( 1 9 5 5 ) . Re-analysis o f e x i s t i n g wave f o r c e data on model p i l e s : U. S. Army Corps. Eng., BEB, Tech. Memo No. 71 ( A p r i l ) .

Eagleson, P. S., Nontsopoulas, G. K., and D a i l y , J . W. ( 1 9 6 4 ) . The n a t u r e of s e l f - e x c i t a t i o n i n t h e f l o w - i n d u c e d v i b r a t i o n o f f l a t p l a t e s : ASME, Trans., J o u r . Bas. Eng., V o l . 86, pp. 599-6o6 (September).

F r i t z l e r , G. L. (1961+). H y d r o e l a s t i c v i b r a t i o n s o f c i r c u l a r c y l i n d e r s . M.S. T h e s i s , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y .

Hoerner, S. F. (1958). Fluid-dynamic d r a g ; p r a c t i c a l i n f o r m a t i o n on a e r o d y n a m i c drag and hydrodynamic r e s i s t a n c e : 2nd Ed., M i d l a n d Park, W.J.

I v e r s e n , H. Vf. , and B a l e n t , R. A. (1951). A c o r r e l a t i n g modulus f o r f l u i d r e s i s t a n c e i n a c c e l e r a t e d m o t i o n : Jour. A p p l . Phys., V o l . 22, Ho. 3, pp. 32I+-328.

Keim, S. R. ( 1 9 5 6 ) . F l u i d r e s i s t a n c e t o c y l i n d e r s i n a c c e l e r a t e d m o t i o n : ASCE, Proc. Jour. Hyd. D i v . , V o l . 82, paper 1113, l l * pp. (December).

Keulegan, G. H., and Carpenter, L. H. ( 1 9 5 8 ) . Forces on c y l i n d e r s and p l a t e s i n an o s c i l l a t o r y f l u i d : Jour. Res. H a t l . Bur. Stds. , 6 0 , pp. 1+23-1*0 (May). L a i r d , A. D. K. ( l 9 5 5 ) . A model study o f wave a c t i o n on a c y l i n d r i c a l i s l a n d :

Amer. Geo. Union, Trans., V o l . 3 6 , Ho. 2 , pp. 2 7 9 - 2 8 $ .

L a i r d , A. D. K. , Johnson, C. A., and VJalker, R. W. ( 1 9 5 9 ) . Water f o r c e s on a c c e l e r a t e d c y l i n d e r s : ASCE, Waterways and Harbors D i v . Proc. V o l . 85 \-nil

pp. 9 9 - 1 1 9 ; ASCE, Trans., V o l . 1 2 5 , P t . 1 , ( 1 9 6 O ) , pp. 6 5 2 - 6 6 6 .

L a i r d , A. D. K. , Johnson, C. A., and Vfalker, R. W. ( 1 9 6 0 ) . Vfater eddy f o r c e s on o s c i l l a t i n g c y l i n d e r s : ASCE J o u r . Hyd. D i v . P r o c , V o l . 8 6 , No. HY9,

Paper Wo. 2 6 5 2 , pp. 1+3-51+, and ASCE, Trans., V o l . 1 2 7 , P a r t 1 ( I 9 6 2 ) ,

pp. 3 3 5 - 3 5 1 .

L a i r d , A. D. K. ( 1 9 6 1 ) . Eddy f o r c e s on r i g i d c y l i n d e r s : ASCE, P r o c , J o u r . Waterways and Harbors D i v . , V o l . 8 7 , Wo. \mh, pp. 5 3 - 6 7 .

L a i r d , A. D. K. ( 1 9 6 2 ) . Water f o r c e s on f l e x i b l e o x c i l l a t i n g c y l i n d e r s : ASCE, P r o c , Jour. Waterways and Harbors D i v . , V o l . 8 8 , Wo. \rvI3, pp.

1 2 5 - 1 3 7 .

L a i r d , A. D. K., and Warren, R. P. ( 1 9 6 3 ) . Groups o f v e r t i c a l c y l i n d e r s o s c i l l a t i n g i n w a t e r : ASCE, P r o c , Jour, o f Eng. Mech. D i v . , V o l . 9 8 ,

Wo. EMI, pp. 2 5 - 3 5 .

L a i r d , A. D. K. (196I+). Wave f o r c e s on p i l i n g : Univ. o f C a l i f . , I n s t . Eng. Res., Report Bo. HPS-61t-l ( J u n e ) .

Landweber, L. ( 1 9 I + 2 ) . Flow about a p a i r o f a d j a c e n t , p a r a l l e l c y l i n d e r s normal t o a stream: t h e o r e t i c a l a n a l y s i s : Report 1+85, U.S. Wavy, The David Vf. T a y l o r Model B a s i n , Vfashington, D.C.

L i n d s e y , W. F. ( 1 9 3 8 ) . Drag o f c y l i n d e r s o f simple shapes: WACA, TR. 619

pp. 1 6 9 - 1 7 6 .

Lunnon, R. G. ( 1 9 2 8 ) . F l u i d r e s i s t a n c e t o moving spheres: Proc. Roy. Soc. Lond. ( A ) , V o l . I I 8 , p. 68O.

MacCamy, R. C , and Fuchs, R. A. (195*+). Wave f o r c e s on p i l e s ; a d i f f r a t i o n t h e o r y : Univ. o f C a l i f . , I n s t . Eng. Res., and BEB, Tech. Memo. Wo. 69

(December).

MacWoira, J . S., and Keulegan, G. H. ( 1 9 5 9 ) . V o r t e x f o r m a t i o n and r e s i s t a n c e i n p e r i o d i c m o t i o n : ASCE, P r o c , V o l . 8 5 , Wo. EMI, Paper I 8 9 I + , 6 pp. ( J a n u a r y ) .

McLean, V/. J . (1961+). Behavior o f a f l e x i b l y supported c y l i n d e r i n a f l u i d stream: M.S. T h e s i s , Univ. o f C a l i f . , B e r k e l e y .

258

1190, ;s: Report i " -David T a y l o r Model B a s i n , Washington, D.C. ( J u l y ) .

Macovsky, M. S. (1958). V o r t e x induced v i b r a t i o n s t u d i e s : Report 1*°

v i b r a t i n g M a n d i n i , R. V. (1965). Kinematics o f v o r t i c e s i n t h e e a r l y wake °

c i r c u l a r c y l i n d e r s : M.S. T h e s i s , Univ. o f C a l i f . , B e r k e l e y .

^ i n d u c e d M a r r i s , A. W. (1961+) . Review on v o r t e x s t r e e t s , p e r i o d i c wakes, 185-9°

v i b r a t i o n phenomena: ASME, Trans., Jour. Bas. Eng., V o l . 86, P ( J u n e ) .

=nt Masch, F. D., and Moore, W. L. ( I 9 6 0 ) . Drag f o r c e s i n v e l o c i t y

f l o w : ASCE. P r o c , Jour. Hyd. D i v . , V o l . 86, pp. 1-11 ( J u i y ^ •

, 1 9 5 0 ) .

M o r i s o n , J . R. , O'Brien, M. P., Johnson, J. W. , and Schaaf, S. A- ^ I I * * E , The f o r c e e x e r t e d by s u r f a c e waves on p i l e s : Jour Pet. Tecïi' ' ^ ] _ 9 5 ' - ' ) "

V o l . 2, Wo. 5, and Pet. Trans., AIMME, V o l . I 8 9 , pp. ihg-l"?^

^Xow i n -M o r k o v i n , N. V. (1961*). Flow around c i r c u l a r c y l i n d e r s—i n c l u d i J ^ ^ g g p a r a t e d

s t a b i l i t i e s and t r a n s i t i o n t o t u r b u l e n c e ; Symposium on F u l ^ y

Flows: ASME, NewYork, pp. 102-118 (May). .. 3 _ r i s t a b i l i t y

P a r k i n s o n , G. V., and Brooks, H. P. H. (1961). On t h e a e r o e l a s t - i ^ ^ 2 5 8 . o f b l u f f c y l i n d e r s : ASME, Jour. A p p l . Mech., V o l . 28, pp. 2 ?

g ^ e r o e l a s t i c P a r k i n s o n , G. V., and Smith, J. D. ( l 9 6 1 t ) . The square p r i s m a s ^ • -^"^'

n o n - l i n e a r o s c i l l a t o r : Quar. Jour. Mech. and A p p l . Math., 2, pp. 225-239.

^ ^ ^ c t u r e s : Penzien, J. ( 1 9 5 7 ) . The wind induced v i b r a t i o n o f c y l i n d r i c a l ^ ^y) •

ASCE, P r o c , V o l . 83, Ho. EMI, Paper Wo. l l l * l , 17 pp. ( J a n - ^ i ^ Roshko, A. (195!*). On t h e drag and shedding frequency o f two-tS-^*

b l u f f b o d i e s : NACA, TW, 3 l 6 9 , 29 pp.

Toebes, G. H., and Eagleson, P. S. ( 1 9 6 1 ) . H y d r o e l a s t i c vibra-fc 3-_^ . Bas. Eng., p l a t e s r e l a t e d t o t r a i l i n g edge geometry: ASME, Trans., J o ^

S e r i e s D. V o l . 83, p. 671.

on c i r c u l a r W i e g e l , R. L. Beebe, K. E. , and Moon, J. ( l 9 5 7 ) . Ocean wave fo^ -1;=''^^

c y l i n d r i c a l p i l e s : ASCE, P r o c , J o u r . Hyd. D i v . V o l . 83, and Trans., V o l . 121+, pp. 8 9 - I I 6 ( 1 9 5 9 ) .

^ c u l a r c y l m -Wiener, F. M. ( l 9 l ( 7 ) . Sound d i f f r a c t i o n by r i g i d spheres and <^

d e r s : Jour. Acoust. Soc. Amer., V o l . 19, pp. Itltlt-lt51 ( M a y ? ' ,„ 3_sehen Versucns

W i e s e l b e r g e r , C. , and B e t z , C. ( 1 9 2 3 ) . Ergebnisse der a e r o d y n ^ r * ^ a n s t a l t zu Göttingen: R. Aldenbourg, B e r l i n , V o l . 2, p. 2 6 : ^ "

6 0

4 0

2 0

0

" T 1 r " 1 1 1 1 r _{- 1 — r}

-No neighbor: Neighbor 6 in. Neighbor

9"^-i v v M 9"^-i l

^ M » ^

ahead, in line: ahead, in ür^^^

1

^

O

P

a. 2 . 3 3 - inch cylinder

No neighbor:

Neighbor l ^ " ^ ;

ahead, in

4 0

a / V A / W A Neighbor 9 in.

ahead, in line:

One Division =0.2 secon ^

b. 4.47- inch cylinder

Figure 1. Uniform motion drag forces for cylinders. Redrawn from L a i (1959).

COASTAL E N G I N E E R I N G

L i j O

-0.5 1.0 1.5 2.0 2.5

TIME FROM START (IN SECONDS)

t:zziroi Tzi'iT'^eir'''''''^^^^^^^ ^'^^ °^

a 2-in. neighbor at numbers of diameters shown on curves. Redrawn from Laird et al (1960).

O 0.5 1.0 1.5 2.0 2.5 3.0 3.5

TIME (seconds)

Figure 4. Lift faces on 2-in. test cylinder for the runs used in Fig. 3 having spacings shown. Redrawn from Laird et al (1960).

0.8

0 0.2 0.4 0.6 0.8

TIME (sec.)

Figure 5. Calculated forces and positions for a cylinder passing between two eddies previously shed by it. Redrawn from Laird (1961)

Dr

0

0

O

•

o

oO

o w

f e / f ,

.0

Figure 7. Drag and lift force ratios as functions of eddy frequency to cylinder natural frequency ratios for 2-in. diameter cylinder. From Laird (1962).

5

DO

r

1.0 1.5 2.0 2.5

h/ho

Figure 8. Drag force ratios as functions of walce width, h, to uniform walie width, h^, ratios at various natural frequencies, fj^.

Figure 9. Typical variations of force coefficients and response frequencies as functions of Reynolds number for a single half inch cylinder with a cantilever spring constant of 0.4 lb. p e r ft.