RECENT RESULTS OBTAINED FROM A NUMERICAL WAVE THEORY FOR HIGHLY NONLINEAR
SHALLOW WATER WAVES
Robert 6. Dean
Department of Coastal and Oceanographic Engineering U n i v e r s i t y of Florida
Presented a t the SYMPOSIUM ON LONG WAVES U n i v e r s i t y of Delaware
Newark, Delaware September 1 0 - 1 1 , 1970
RECENT RESULTS OBTAINED FROM A NUMERICAL WAVE THEORY FOR HIGHLY NONLINEAR
SHALLOW WATER WAVES
INTRODUCTION
A n a l y t i c a l representation of shallow water wave phenomena i s complicated due, i n p a r t , to the f a c t t h a t nonlinear features are important in a predominant number, i f not a l l , problems of shallow water wave motion. I t t h e r e f o r e may not even be approximately v a l i d to u t i l i z e the A i r y wave theory and to assume t h a t various wave components behave independently of one another.
At present, even f o r p e r i o d i c wave motion, p r e d i c t i o n s of shallow water wave phenomena based on various a v a i l a b l e theories d i f f e r by d i s t u r b i n g amounts.
In t h i s paper, several features of a numerical wave theory (Stream f u n c t i o n ) are reviewed and explored to demonstrate: (a) the agreement between theory and l a b o r a t o r y measurements, and (b) the d i f f e r e n c e s between the numerical wave theory and the A i r y wave theory. In p a r t i c u l a r , t o t a l wave energy, momentum and momentum f l u x , pressure response f a c t o r s , maximum drag f o r c e s , shoaling c o e f f i c i e n t s , e t c . are examined. The purpose of t h i s paper i s to d i r e c t a t t e n t i o n to the very s i g n i f i c a n t d i f f e r e n c e s t h a t e x i s t between the two theories i n shallow water and the need f o r a d d i t i o n a l research to resolve the d i f f e r e n c e s noted.
STREAM FUNCTION WAVE THEORY
The Stream Function Wave Theory has been presented extensively
e l s e w h e r e ^ ^ ^ . ( 2 ) , ( 3 ) , ( 4 ) g ^ ^ j w ü i therefore only be described b r i e f l y i n t h i s paper The theory i s s t r i c t l y a p p l i c a b l e f o r a two dimensional wave propagating without
change of form i n water of uniform depth. Advantages of the theory i n c l u d e : (1) the form of the s o l u t i o n i s i n h e r e n t l y b e t t e r suited (say than the Stokes' representation) f o r s a t i s f y i n g the nonlinear f r e e surface boundary c o n d i t i o n s , (2) the theory can be r e a d i l y extended to reasonably high o r d e r s , (3) the theory has been shown to provide b e t t e r f i t s to the boundary conditions than other theories established f o r shallow water c o n d i t i o n s , and (4) f o r the l i m i t e d data a v a i l a b l e , the theory provides a s i g n i f i c a n t l y b e t t e r f i t to horizontal water p a r t i c l e v e l o c i t i e s measured in the l a b o r a t o r y .
Figure 1 i s the r e s u l t of a study to determine which theories provided best agreement to the two nonlinear free surface boundary c o n d i t i o n s . The Stream f u n c t i o n was c a l c u l a t e d to the f i f t h o r d e r , and the study demonstrated t h a t
higher orders of the theory would have extended the "best f i t " of the Stream f u n c t i o n theory s i g n i f i c a n t l y i n t o the shallow water r e g i o n .
Figures 2 , 3, and 4 represent comparisons of the maximum h o r i z o n t a l water p a r t i c l e v e l o c i t i e s as c a l c u l a t e d using a number of wave theories and as measured by Le Méhaute', e t . a l . ^ ^ ^ ; the wave conditions f a l l w i t h i n the intermediate and shallow water ranges. Computation of the standard deviations between each of the theories and the measurements (8 d i f f e r e n t waves) i n d i c a t e d t h a t the Stream f u n c t i o n provided the l e a s t standard deviation,^.with the second lowest standard d e v i a t i o n approximately 40% higher than t h a t provided by the Stream f u n c t i o n theory. Figures 2 , 3, and 4 span the range of r e l a t i v e agreement between the Stream
f u n c t i o n theory and the data (8 cases) w i t h the best f i t shown i n Figure 2 , an average f i t i n Figure 3, and the worst f i t i n Figure 4. Figures 5 and 6 are comparisons of the p r e d i c t i o n s of various wave theories w i t h a measured maximum v e r t i c a l water p a r t i c l e v e l o c i t y d i s t r i b u t i o n and w i t h a measured wave p r o f i l e , r e s p e c t i v e l y . In both cases, the f i t provided by the Stream f u n c t i o n wave theory
10^ 10-O 10-2 10-5 B R E A K I N G L I M I T A I R Y WATER WAVES 10-2
F I G U R E I.
I 0 - ' 100 W A T E R W A V E S 10' h / T 2 ( f t . / s e c ? )P E m O D I C WAVE T H E O R I E S PROVIDING B E S T F I T T O
DYNAMIC F R E E S U R F A C E BOUNDARY C O N D I T I O N
( A N A L Y T I C A L AND S T R E A M F U N C T I O N TT T H E O R I E S )
3Fifiir ® 5 . Vsrtleo ! W ®f ® r Psrtici ® Veleclt ^ , Cos e S
CM O
is considered as good or b e t t e r than t h a t provided by the other t h e o r i e s . Of p a r t i c u l a r i n t e r e s t , i s the r e l a t i v e l y poor f i t associated w i t h some of the theories developed p a r t i c u l a r l y f o r shallow water c o n d i t i o n s .
RESULTS OBTAINED FROM STREAM FUNCTION THEORY
I f i t i s accepted t h a t previous i n v e s t i g a t i o n s i n d i c a t e t h a t the Stream f u n c t i o n wave theory does provide some s i g n i f i c a n t advantage over other a v a i l a b l e t h e o r i e s , then i t i s i n t e r e s t i n g to compare d i f f e r e n c e s t h a t would r e s u l t by use of the Stream f u n c t i o n wave theory and the A i r y wave theory. The A i r y wave theory was chosen f o r comparison, because i t i s the most widely employed theory f o r a l l r e l a t i v e depths, i n c l u d i n g the shallow water r e g i o n . For example, important shallow water c a l c u l a t i o n s employing the A i r y wave theory i n c l u d e : s h o é l i n g , energy, momentum f l u x , r e f r a c t i o n , e t c . Examples of the d i f f e r e n c e s i n several v a r i a b l e s f o l l o w .
Maximum Drag Forces and Moments
The maximum drag force i s a v a r i a b l e of considerable engineering importance Figure 7 presents the percentage d i f f e r e n c e s i n maximum drag forces t h a t are obtained by use of the Stream f u n c t i o n and A i r y wave theorys. I t i s seen t h a t
in deep water, the maximum percentage d i f f e r e n c e i s approximately 5% whereas i n shallow water the corresponding d i f f e r e n c e i s greater than 60%. I t i s reasonable to suppose t h a t comparison w i t h the A i r y wave theory w i l l unduly exaggerate the disagreement between theories i n the shallow water r e g i o n ; t h i s i s not necessaril the case. For example, c a l c u l a t i o n s presented i n an e a r l i e r paper^"^^have shown t h a t the Cnoidal wave theory as presented by Laitone^^^would p r e d i c t maximum drag forces t h a t are 105% greater than the Stream f u n c t i o n wave theory f o r a near-breaking wave i n shallow water.
h / T ^ ( f t . / s e c ? )
F I G U R E 7.
PERCENTAGE DIFFERENCES IN MAXIMUM DRAG
F O R C E S ; S T R E A M FUNCTION V S . LINEAR
WAVE T H E O R Y .
Figure 8 presents s i m i l a r information f o r the maximum drag moment. Pressure Response Factor
The pressure response f a c t o r , Kp, based on the wave height i s defined as
P D ( 0 ° ) - PD(180°) K p =
where Pp denotes the dynamic pressure which the pressure sensor "sees" and Y i s the s p e c i f i c weight of water. The pressure response f a c t o r i s important, because the pressure sensor i s s t i l l the most convenient type of wave gage to deploy, e s p e c i a l l y i f there are no s t r u c t u r e s a v a i l a b l e as wave gage supports at the s i t e of i n t e r e s t .
The percentage d i f f e r e n c e s i n pressure response f a c t o r f o r a pressure sensor located at mid-depth are shown i n Figure 9. As an example, i f a pressure sensor indicates a t o t a l pressure of 607 psf between c r e s t and trough and h = 22.5 f t . and T = 15 s e c . , then the corresponding wave heights as calculated by the 1inear
and Stream f u n c t i o n wave theory would be: H l i n e a r - 9 . 9 f t . H, = 13.3 f t .
I t i s seen t h a t the nonlinear e f f e c t s can r e s u l t i n a considerable discrepancy i n i n t e r p r e t e d wave h e i g h t s .
Total Average Wave Energy
Figure 10 presents the percentage differences between Stream f u n c t i o n t o t a l wave energy and t h a t obtained through use of the A i r y theory. The negative signs i n d i c a t e t h a t , f o r a given wave h e i g h t , the energies c a l c u l a t e d by the A i r y
wave theory are the g r e a t e r . This r e s u l t may, at f i r s t , seem s u r p r i s i n g , however, i t i s c l e a r t h a t the 1 i m i t i n g p o t e n t i a l energy would be zero as the wave troughs
h / T ^ ( f t . / s e c . 2 )
F I G U R E 8 . P E R C E N T A G E D I F F E R E N C E S IN MAXIMUM DRAG
M O M E N T S ; S T R E A M F U N C T I O N V S . L I N E A R
10 2 5 h / T 2 ( f t . / 8 e c 2 )
FIGURE 9 . PERCENTAGE DIFFERENCES IN PRESSURE RESPONSE
F A C T O R S T R E A M FUNCTION V S . LINEAR WAVE
THEORY
IO' 2 5
h/T^(ft/8@e2)
F I G U R E 1 0 . I S O L I N E S O F P E R C E N T A G E D I F F E R E N C E S I N T O T A L E N E R G Y ;
become longer and shallower and the wave crest becomes more narrow. For example, the t o t a l average energy of the stream f u n c t i o n shallow water wave shown i n Figure 11 i s only 46% of t h a t f o r the A i r y theory shown.
Total Momentum
The d i f f e r e n c e s f o r the t o t a l momentum are presented in Figure 12. The maximum percentage d i f f e r e n c e , as defined i n the f i g u r e , i s greater than 300%. The reason t h a t the stream f u n c t i o n p r e d i c t s such markedly smaller momentum values i s due, i n large p a r t , to the peaked crests and long low troughs shown i n Figure 1 1 .
Total Momentum Flux
The momentum f l u x d i f f e r e n c e i s presented in Figure 13. Again i t i s noted t h a t the momentum f l u x c a l c u l a t e d from the Stream f u n c t i o n wave theory is generally less than c a l c u l a t e d by the A i r y wave theory. Since the momentum f l u x i s an
important agent i n several s u r f zone mechanisms ( e . g . set-up and longshore c u r r e n t ) d i f f e r e n c e s of the magnitude noted here could be of considerable importance.
Shoaling C o e f f i c i e n t
The percentage d i f f e r e n c e s i n shoaling c o e f f i c i e n t s f o r waves advancing normal to the shoreline are presented in Figure 14. As an example, assume the f o l l o w i n g deep water wave conditions
T = 10 sec. h = 1000 f t .
According to l i n e a r wave t h e o r y , the wave height i n 10 f t . of water depth would have increased to 4.9 f t . due to s h o a l i n g . The Stream f u n c t i o n wave theory would p r e d i c t a wave height of 5.9. As discussed p r e v i o u s l y , the primary
reason f o r the d i f f e r e n c e i s t h a t , f o r a given wave h e i g h t , the amount of wave
FIGURE II. COMPARISON OF IIth ORDER STREAM FUNCTION AND
B R E A K I N G I N D E X C H / H B » 1 . 0 H / H B « 0 . 7 5 —
H / H Q = 0.50 --^.yCv^
H / H B » 0.25 -y^y/^^'
/ / / / / \ U R V E ^ — ^ / / / / / / / /^ y^
/ / / / / / / / / , / / / / / / «/o D I F F/
/ / / S H A L L D W H Q - B R E A K I N G W A V E H E I G H T I N T E R M E D I A T E 1 ^ D E E P 4. W A T E R W A V E S ' • ' W A T E R W A V E S L _ 1 ^ W A T E R W A V E S ~ .-1,, , • , ,1 .• — • - i — ^ — — — — _ • • I — 1 • L 10 2 5 10 2 5 10° 2 5h / T ^ ( f t . / s e c ? )
F I G U R E 1 2 . P E R C E N T A G E D I F F E R E N C E S IN M O M E N T U M ;
S T R E A M F U N C T I O N V S . LINEAR WAVE THEORY.
O 10 5 2 5 2 5 2 10
BREAKING INDEX C l
H / H B = 1 . 0 — . H / H B - 0 . 7 5 / / H / H B * 0 . 5 0~//Jc7 y
H / H B = 0.25Ayy>^j
/ R V E - ^ / ' ' / / /
/ / / / / / • • / / / / / ^ / /4/ 7/ / /
/ / / ' 4"" / % D I F F ./ /
/ / / / / / / STREAM F N . — AIRY S T R E A M F N ./ / / /
/ SHALLOW 1 H g > BREAKING WAVE HEIGHT I N T E R M E D I A T E DEEP ^ WATER WAVES ' 1 1 W A T E R WAVE 1 1 WATER WAVES -1 -1 10'* 2 5 l O " 2 5 10° 2 5 10h / T ^ ( f t / s e a ^ )
F I G U R E 13. I S O U N E S O F P E R C E N T A G E D I F F E R E N C E S
IN M O M E N T U M F L U X j AIRY V S . S T R E A M
F U N C T I O N W A V E T H E O R I E S .
10" 5 2 5 2 2 1 B R E A K I N G I N D E X H / H B ' 1 , 0 - - ^ H / H B = 0 . 7 5 - ^ 1 ^ 1 ^ / ^
H / H B « 0 5 0 ^ ^ ^ v
^ y "
H / H B = 0 . 2 5/ / /
C U R V E / // / / / / /
//
/ / / //
/ / / / / / /X
-//
// J
//// .
/ / / /
/ // / / /
^ / / / / % D I F F . =/ A
S T R E A M F N . - A I R Y S T R E A M F N ./ / l ^ * / / /
/ C5 / / / //
/ / S H A L L O W ^ H B = B R E A K I N G W A V E H E I G H T I N T E R M E D I A T E J D E E P ^ W A T E R W A V E S - i ' W A T E R W A V E S I 1 • * r W A T E R W A V E S -1 — _ . 1 «0"' 2 5 10"' 2 5 10° 2 5h / T ^ ( f t . / s e c ^ )
FIGURE 14. ISOLINES OF P E R C E N T A G E D I F F E R E N C E S IN
SHOALING C O E F F I C I E N T ; AIRY VS. S T R E A M
FUNCTION W A V E T H E O R I E S .
19energy is less f o r a nonlinear wave than as c a l c u l a t e d by the A i r y wave theory. For a comparison of shoaling e f f e c t s as predicted by the Stream f u n c t i o n and A i r y wave theories and measurements, one set of laboratory data published by Bowen, e t . a l . ^''^is used. Figure 15a i l l u s t r a t e s the plane l a b o r a t o r y beach w i t h a slope of 1:12. Figures 15b and 15c i l l u s t r a t e the wave shoaling and the r e l a t i v e c r e s t and trough displacement, r e s p e c t i v e l y .
I t i s seen i n Figure 15b t h a t as the wave commences s h o a l i n g , the A i r y p r e d i c t i o n s are i n b e t t e r agreement than the Stream f u n c t i o n . As the wave
approaches breaking, the measured wave height deviates sharply from the A i r y theory and tends to be i n somewhat b e t t e r agreement w i t h the Stream f u n c t i o n p r e d i c t i o n s . I t should be noted t h a t a slope of 1:12 i s f a i r l y steep, and i t i s c l e a r t h a t there w i l 1 be a time (or distance) " l a g " between the time of passage of a wave over a p a r t i c u l a r depth and the response of the wave to t h a t depth.
Figure 15c i l l u s t r a t e s t h a t the r e l a t i v e crest and trough elevations are i n good agreement w i t h the Stream f u n c t i o n p r e d i c t i o n s , although there i s some
i n d i c a t i o n of a " l a g " f o r near breaking c o n d i t i o n s .
CONCLUSIONS
Comparisons have shown t h a t the Stream f u n c t i o n wave theory i s i n reasonable agreement w i t h the equations describing water wave motion and w i t h maximum h o r i z o n t a l water p a r t i c l e v e l o c i t y data c o l l e c t e d by Le Mehaute', e t . a l . f o r shallow water waves. A number of the theories derived f o r shallow water conditions do not agree s a t i s f a c t o r i l y w i t h the measured data.
Although the l i n e a r wave theory i s used i n engineering p r a c t i c e f o r a number of important shallow water a p p l i c a t i o n s , i n c l u d i n g : s h o a l i n g , energy, pressure response f a c t o r , momentum f l u x , e t c . , i t has been shown t h a t s i g n i f i c a n t d i f f e r e n c e s
X (cm.)
- H ^ 50 100 150 250
« ^ ' ' Q p h s béc )
© Data
— - Stream Function Theory Airy Theory Conventional Breaking Height = 0.78 Note: H o = 6 . 4 5 c m . T " 1 . 1 4 sec. 100 b) Wav@ Shoaling 2 0 0 X (cmj
J
X (cm.)c) Relotivt C r e s t and Trough Elevations
FIGURE 15. COMPARISON OF STREAM FUNCTION AND
AIRY PREDICTIONS WITH MEASUREMENTS
BY BOWEN ET A L .
can e x i s t between shallow water values as calculated by the 1inear and Stream f u n c t i o n wave t h e o r i e s .
, Much a d d i t i o n a l high q u a l i t y laboratory and f i e l d shallow water data are required to f u r t h e r develop our understanding of the c a p a b i l i t i e s and 1 i m i t a t i o n s of the various a v a i l a b l e wave t h e o r i e s . The e x i s t i n g shallow water data are l i m i t e d in number and i n the variables and phase angles t h a t they represent.
REFERBNCES
Dean, R. G.,."Stri'earfl Function Repr'eSentation of Nonlinear' Ocean Waves," Journal o f Geophysical Research,^ 70(18), pp. 4561-4572, September, 1965. Dean, R, G., "Strjeam Function Wave Theory; V a l i d i t y and A p p l i c a t i o n , "
Proceedings, AStE Specialty Conference on Coastal Engineering. Chapter 12, Vp. 269-300, 19J65. ; ~
Dean, R.| i/Rel^tive V a l i d i t y of Wa^^er^ Wave Theories," Proceedings, ASCE Specialty Conference oh C i v i l Engineering i n the Oceans. San Francisco, pp. '1-30, 1968. (Also published in'Waterwayi~and Harbors J o u r n a l , American Society of C i v i l Enginfeers.; p. 105-119;, February, 1970)
Dean, R. G., and B: Le Mehaute', "Experimental V a l i d i t y b f Water Wave Theories, Paper presented at the 1970 ASCE S t r u c t u r a l Engineering Conference,
P o r t l a n d , Oregon, A p r i l 8 , 1970.
Le Méhaute', B . , D^ Divoky and A. L i n , "Shallow Water Waves: A Comparison of Theory and Experiment," Proceedings Eleventh I n t e r n a t i o n a l Conference on Coastal Engineering, ChapT.;7, pp. 86^TÜ7TT968.
Laitone, E. V , , "The Second Approximation to Cnoidal and S o l i t a r y Waves," Journal o f F l u i d Mechanics, V. 9, Part 3, pp. 430-444, November, 1960. Bowen, A. J . , D. L. Inman and V. P. Simmons, "Wave 'Set-Down' and Set-Up,"
Journal of Geophysical Research. V. 73, No, 8 , pp. 2569 - 2577, A p r i l , 1968.
