Measurements of particle velocities in laboratory-scale random waves

Coastal Engineering, 6 (1982) 233--254 Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

283

MEASUREMENTS OF PARTICLE VELOCITIES IN LABORATORY- SCALE RANDOM WAVES

K. ANASTASIOU, R.G. TICKELL and J.R. CHAPLIN

Civil Engineering Department, The University of Liverpool, P.O. Box 147, Liverpool L69 3BX (Great Britain)

(Received March 17, 1981; revised and accepted January 11, 1982)

ABSTRACT

Anastasiou, K., Tickell, R.G. and Chaplin, J.R., 1982. Measurements of particle velocities in laboratory-scale random waves. Coastal Eng., 6: 233--254.

The paper describes measurements of water particle velocities in laboratory-scale irregular, non-periodic surface waves. The measurements were taken over a range of elevations using Laser Doppler anemometry and included observations of particle kinematics at two points separated in plan. The observed statistical and spectral properties were compared with those predicted by both traditional and intermittent linear random wave theory. For elevations which are always submerged, the measured properties were in good agreement with both theoretical approaches. This was not the case for points near mean water level, where the observed properties were approximated more closely by the intermittent approach. However, some departure between observations and the intermittent approach was evident for elevations above mean water level.

INTRODUCTION

A knowledge of the particle kinematics is essential for reliable estimates of wave action in a coastal or offshore environment. The numerical models currently in use calculate velocities and accelerations by either a deterministic wave theory with independent parameters of wave height, wave period and mean water depth, or linear random wave theory (Borgman, 1969, 1972; Holmes and Tickell, 1979), or second-order random wave theory (Hudspeth, 1975; Sharma and Dean, 1979), in which case the mean water depth and the energy spectrum of the free surface elevation are the only re- quirements. The deterministic approach has the advantage that accurate solutions are available for waves of finite amplitude, while the random approach attempts to describe the properties of an irregular sea environment. Previous studies (Tsuchiya and Yamagouchi, 1974; Grace, 1976; Lee et al., 1976; Chakrabarti, 1980) indicate that linear random wave theory (hereafter referred to as LRWT) provides a reasonable first-order solution for the flow at points which are continuously submerged, but fl~is is not the case

0378-8839/82/0000--0000/~02.75 © 1982 Ekevier Scientific Publishing Company

234

for elevations near or above the mean water level (MWL). Using LRWT as a starting point, Tmlg (Tung, 1975; Pajouhi and Tung, 1975) derived expressions for the statistical distributions and spectral properties of particle kinematics at such elevations.

Unfortunately, the lack of full-scale or laboratory.scale data for points in the vicinity of the free surface has previously prevented validation of this formulation. In this paper, we report measurements of particle kinematics near and above mean water level for laboratory-scale random waves, using a two-channel Laser Doppler anemometer, with a view to investigating the probabilistic properties of such in~.~'mittent flow fields, In addition to the measurements taken over a range of elevations but at one particular cross- section, observations of particle kinematics at two points separated in plan in the direction of wave propagation have been made in order to investigate the accuracy of LRWT in predicting the correlation in flow at such points. In this case, the points considered were submerged during all observations. The present study presents the data collected in the programme and compares the observed properties to those predicted by LRWT and to those pre dicted by the modified linear random wave theory (hereafter referred to as MLRWT), allowing for the effects of the free surface.

THEORY

Linear random wave theory applied to long.crested seas and in the al> sence of currents predicts that the free surface elevation q, horizontm particle velocity u, and the vertical particle velocity v, will be zero mean processes with Gaussian distributions. The statistical and spectral parameters for particle kinematics can be derived from the one-sided free surface elevation energy spectrum S , , (w) using the appropriate transfer functions. The variance of ~ is:

E[~ ~] = a . n' =- f S . n ( ~ ) d ~ (1) o

E[.] being the expectation operator; and ~ being the angular frequency in md/sec.

The energy spectra for u, v are:

S.u(~) = w 2 c°sh2 {h(z + d)} sinh ~ {k~) s , . ( , ~ ) (2)

sinh 2 {k(z + d)} s,,(~) = ~ .,, ,~h' {kd.} " S,~(~) (8)

where k is the wave number related to w by the exPression (o' = gk tanh (kd}; g is the acceleration due to gravity; ~

235

z is the elevation considered, measured from MWL; and d is the water depth.

The variances of u, p are given by integratio, of their respective spec~a in a similar manner to eq. 1. The energy spectra for u, v are real quantities.

For uni-directional waves, the cross-spectra between u, u, ~ for points separated in plan in the direction of wave propagation are given by the following expressions:

cosh {k(z + d)} 8nu(W ) = w ' s i nh{kd} e x p { i k A x } S ~ n ( ~ )

= H . u ( ~ ) S . . ( ~ )

(4)

shah (k(z + d)) 8~v(W) = iw sinh {kd} exp{ikAx}8~(w) (5)

= H.v(~) S .~(~)

Suv(CO) ffi iw cosh {k(z + d ) } s inh{k ( z + d)}

sinh 2 {kd} '

= Huv(~ )S~(~ )

exp { ik A x } Snn ( o~ ) (6)

where H~u(W), H~v(~O) and Huv(O~) are the transfer functions between (7, u) (7, v), and (u, v) respectively; Ax is the horizontal distance between the two points; and i is ~/-1.

The correlation coefficient between 7, u f~,r zero time lag T is given by:

o o

f S ,~u(w)dw 0 r~u(0) ffi (7)

O~Ou

where 07, Ou are the standard deviations of 7, u, respectively. Similar expressions to eq. 7 can be derived for rqv and ruv. The coherence function between 7, u is given by:

IS~u(¢~)l 2 ~ " ( ~ ) : s ~ ( ~ ) a . . ( ~ ) (8)

with similar expressions to eq. 8 for 7~v(~O ) and 72uv(W). The probability density functions for 7, u, v are Gaussian, that is:

p(,7) = 4 2 - 2 (9)

with similar expressions to eq. 9 for u and v. For a Gaussian process, the skewness (A) and kurtosis (B) are zero where

these two important parameters are as defined below:

236

A = .us/~ B = ( ~ / ~ ) - 3

~j = E[ (~ - E[~I)J]

(io)

In the above formulae for the statistical distributions and energy spectra of the particle kinematics, no provision has been made for the fact that the measuring point is not necessarily submerged at all times. For example, a point on MWL remains submerged for only 50% of the total observation time, a point at an elevation of o~ above MWL remains submerged for only 15.87% of the total observation t~nae. The instantaneous particle velocity u(z, t) taking into account the submergence or not of the point at elevation z can be defined as:

"u(z, t) = u(z, t) for t such that ~(t) t> z

ffi 0 for t such that 7/(t) < z (11)

where u(z, t) is derived from LRWT. A similar expression to eq. 11 may be derived for v. The random vari-

ables u (or v) and ~/ are not statistically independent. The events [~(t) >i z] and [7/(t) < z] are, however, mutually exclusive and by the theorem of Total Probability (Papoulis, 1965, Chapter 4):

P6(u) = H(u) Prob[~ < z] + Pu,~>z(u) Prob[~/I> z]

= H(u)[1 - P~(z)] + Pu~.~,(u~P~(z) (12)

where H(u) is the Heaviside unit step function; Pul - (u) is the conditional cumulative distribution function, or cdf ~ Z . of u given ~/1> z; and P~(z) is the unconditional cdf of ~.

The corresponding probability density function for u(or v) is obtained by differentiating P~(u) with respect to u. Thus:

P 6 ( u ) = 6 ( u ) [ 1 - P ( z ) ] + pul~z(U)P~(z)

Where

(~3)

8(u) = dH(u)/du

(u) is the delta function. Also:

(14)

pulq>z(u) = f Pu~ (u, ~)d~/P~(z) (15)

where pu~(U, ~) is the joint probability density function for u and ~, which formean zero Gaussian random variables is:

287

1 pu,(U. ~)-- 2~o~o~ " " '- ̀- r ~ x

( _,:., . [.7 +__., ..,,.<o,.,,])_ × exp 2[1 - r~,~0)] O~u %o u (16)

Substitution of expressions 15 and 16 into eq. 13 yields, after some mani- pulation, the expression for the probability density function of the intermittent ~rocess, as given below:

I z ( " ) r',- r,,,u<Ol,,,<,. 1 p~(u) --. [1 - O(b)]8(u) + - - O o. ~ L ,Vi~-r,~.(O)'J

(17)

where b = z/%, Z(b) ffi (1/%/2~)exp(-0.5bl), O(b) ffi f~'Z(~,)d~, A similar expression to eq. 17 may be derived for v. Tung showed that

while p~(u) describes a non-zero mean, skewed process, pv(v), describes a zero mean unskewed process.

The energy spectrum of the intermittent process may be derived from the Fourier transform of the co-variance function:

Sffff(~) = 2 ; Rffff(Qexp(i~or)dr (18) 0

where:

a ~ ( r ) - E[~(z. t )~(z , t + 7)] - E2[~(z. t)] (19)

and

E[~(z, t)] ffi Our~ , (O)Z(b ) (20)

The first quantity in the RHS of eq. 19 may be written as:

El~(z. t)~(z, t÷~)] ffi

E[Hfn(t)- z)H(n(t+T) - z )Elu(z, t)u(z, t+ril~(t)~(t+r].] (21)

See Papoulis, 1965, Chapter 8. The conditional expected value in the above expression nay be expressed

as a function of Ou, %, r . . (Q, ru.(~) and ruu(~). Substitut i.on of this expression into the RHS of eq. 19 yields for the co-variance f~: action:

r b [ 1 - r.,,,,~'lil + R ~ ( ~ ) ffi o2ufruu(.)L(b, b, r,n(Q) + 2r.u(O)rnu(.)bZ(b).t l. 7-1 r~,(.)'3

1 + , v ~ , i i ~,,(,)]" = [r,~.(o) ÷ r,~,,(~)-

238

2%u(0) r~(Q %u(7)] Z ( ~/1- +x/2b r-~n (Q'/ - o2ur~u(O)Z2(b) (22)

in which:

L[b, b, %~(Q] : J d~/, J p~,~2(~/l,~h)d~/2 (23) b b

where p~, ~2 (~,~2) has a form similar to that of eq. 16. A similar expression to eq. 22 may be derived for u.

The integration of eq. 18 is very difficult to achieve and must be carried out numerically. Without showing the detailed derivations (Paouhi and Tung, 1975), eq. 22 may be expanded as a Taylor's series around ruu(7) = r~u(Q = %~(r) = 0, yielding the following approximate expression for the energy spectrum of the intermittent process:

8~(~o) =( °-~)2r~u(O)b'Z'(b)$nn(o~)+2

+ Q2 (b)Suu(~O)

Ou % r~u(O)bZ(b)Q(b)Snu(~) +

(24)

with a similar expression to eq. 24 for o. Equations !7, 18, 22 and 24 show that the statistical and spectral prop-

erties for the particle kinematics in an intermittent situation can be de- duced from the energy spectrum of the free surface elevation.

EXPERIMENTAL PROCEDURE

Measurements were taken in the random wave flume in the Department of Civil Engineering at Liverpool University. The flume is 18 m long and 0. 75 m wide with a maximum working water depth of 0.90 m. The wave generator is of the piston type and it is servo~ontrolled. The spending beach has a slope of i • 5.3 and consists of a thick layer of jute on galva- nised steel sheets. Tests carried out with regular waves indicated a reflection coefficient (HR/HI) of approximately 6%.

Three, mechanically generated, surface elevation spectra were used for the tests ($1, $2, $3) all being of the Pierson-Moskowitz type with an equation:

0.0081g 2 r,o.74 o 4 ] ~o s exp L ~4

= g / U

U is the wind speed

(25)

The three mechanically generated surface elevation spectra corresponded

239

to values of U of 2.0, 2.5 and 2.8 m/s, respectively. The control signals for the paddle were generated digitally on a mini-computer and recorded, via the digital to analogue converter onto a F.M. tape deck. The signals were re- played into the wave generator control console while the time histories of surface elevation and particle velocitie.~ were sampled on-line and store# on a magnetic disk, for subsequent off-line analysis.

During" all the experimental conditions (S1, $2, $3), horizontal and vertical particle velocities were measured in one cross-section at elevations o f - 0 . 2 0 m, -0 .10 m, -0.02 m, 0.0 m, +0.02 m relative to MWL by using the twin-channel laser. In another cross-section horizontal particle velocities were measured by the smgle~hannel laser only at z = -0.20 m (all three in- puts). The distance between the two cross-sections was 0.42 m. The water depth was 0.70 m, throughout the test programme.

During a typical run, two channels were used for two wave gauges, four channels for u, v and their corresponding drop-out signals at a point in the same cross-section as one of the wave gauges, and two channels for the horizontal particle velocity and drop-out signal at a point in the cross- section of the other wave gauge. The drop-out signals indicated when the laser equipment failed to trace the velocity, for example, when the beam intersection was above the water surface.

Surface elevation time histories were measured with resistance-type wave gauges which proved very stable transducers throughout the test programme.

Both laser Doppler anemometers were operated in forward scatter mode. The first anemometer was a DISA Mk. I system which was capable of measuring particle velocities in only one direction, that being the horizontal direction for the present tests, and the second was a DISA Mk. II system with much improved specification and capable of measuring simultaneously velocities in two directions on the same vertical plane. The Mk. II system was used for the velocity measurements at points near and above the MWL because of its capability to track the Doppler signal with impressive con- s~stency. At these points, the axis of the laser system was tilted upwards, in an attempt to ensure that the intersection point of the beam~ was the first point along the beam path from the laser to become immersed in or emerge from the water. The photomultiplier was also angled upwards to avoid inter- ference to its line of sight due to the movement of the free surface

DATA ANALYSIS

Digitisation of the free surface elevation and velocity time se "es was carried at a rate of 20 Hz and samples of 10,000 points were collected. The time series were subjected to statistical and spectral analysis. The statistical analysis involved calculation of the first four statistical moments, coefficients of skewness and kurtosis and probability density functions (pdf.s) for each time series. In estimating pdf.s 2.5 class intervals per standard deviation were used, e.g., ~ = %/2.5 (Bendat and P~ersol,.-1971, Chapter 4).

240

For the spectral calculations both segmental and frequency smoothing were employed. Each record was divided in to 19 segments of 512 points each. A cosine taper d(t) was applied on each segment of length T, to smooth dis- continuities at the two ends of the segment (Newland, 1978, Chapter 11), Le.:

hi(t) = d(t)n(t)

The cosine taper had the form:

1 ( 1 1 0 , t ) d(t) =~- - cos - - ~ for 0 ~ t ~ T/IO

(26)

T 9T d(t) = 1.0 f o r - - - ~ t ~ - - -

10 10

10~ t - 1 10 9 T

d(t) ffi ~ + cos r for ~-~ ~ t ~ T

(27)

Spectral estimates were obtained for each semment and corresponding estimates for each frequency were summed and averaged. On the resulting values a 5-point frequency smoothing was applied. Thus the degrees of free- dom.per independent spectral estimate were 19 × 5 × 2 ffi 190, with an ef- fective bandwidth of 1.19 rads/sec.

Values for the gain factor and coherence function for the observed time series were obtained from cross.spectral estimates (Bendat and Piersol, 1971, Chapter 5), e.g., between 7, u:

+ =

where: 8nu(~) ffi Cnu(~)- iQ~u(~); =

C.u(~) is the coincident spectrum, Q ,u (~ ) is the Quadrature spectrum:, IH, u(~)] is the gain factor; and 7~u(~0) is the coherence function.

The 95% confidence limits for the spectral, coherence function and gain factor estimates w~re calculated as suggested by Bendat a n d Piersol (1971, C h a p t e r 6 ) . . .... " . ~ .... ,. . . . .

The measured spectra of surface elevation ~ were u s e d t o calculate theoret~ ical spectra and statistical, parameters for ~the p ~ e kinematics. Spectral values for the linear theory were obtained by employing transfer func t ions

241

derived from eqs. 2--6. Spectral values for the intermittent case were o1> tained from eq. 24. In all calculations of theoretical spectra for tl~e particle velocities, an upper freq,Jency cut-off of 3Wpeak was applied, i.e., three times the frequency of the peak value of S,,(c0).

RESULTS AND DISCUSSION

Pree surface elevation

All three measured spectra of free surface elevation were in close agreement with the target spectra. Figure 1 shows the observed and target spectra for n corresponding to a value of U ffi 2.0 m/s. It is noted that the measured spectrum shows no obvious second-order peak.

Table I shows some statistical properties of the observed time series for together with the target standard deviations for the same records. All the

observed time series for ~ possessed non-zero coefficients of skewness and

• I x l O ' ~ -

° ; IO

v 8

3 3

c

v - ! x l o - S . o L 9 "

8 "

Q

S -

v l i i I I v

Measured Sl~ctrum (including 95% confidence I;mits o,q • 2 0 - 4 0 r a m ) .

Theoretical spectrum - - O - - (c~ I , 2 1 . 2 5 m m ) .

! x 10"6 L ~ I . ~ 2 4 6 8 10 12 14 16

Angular frequency W ( r a d s l x c )

Fig. 1. Energy spectrum of free surface elevation for input $1.

242

TABLE I

Statistical properties for observed time series for

eetrum $1 $2 u = 2.0 m/s

Properties ,

% (target) (mm) 21.25 % (measured) (ram) 20.43 A (skewness, measured) 0.114 B (kmtx)ds, measured) 0.283

uffi 2.5 m/s $ 3 u = 2 .8 m / s

33.21 41.65 30.00 39.10

0 .186 0.237 0 . 5 8 0 0 .764

kurtosis, which would not be the case for a truly Gaussian process. The values of the coefficients of skewness and kurtosis increased with % which suggests that the chaz~teristic steepness of the records increased with %. Values for the significant wave height Hs and average zero crossing period Tz obtained from the observed spectra for ~ were used to calculate values of Hs/L z where Lz is the wave length corresponding to Tz for the water depth of 0.70 m. The values of Hs/Lz so obtained for the test conditions ($1, $2, $3) were 0.0478, 0.0503 and 0.0556, respectively, thus possessing an up- ward trend similar to the one shown by the values of the coefficients of skewness and kurtosis. The measured standard deviations for ~ were consistently lower than the target values. In the past, tests performed on the same experimental facility with periodic waves revealed that the wavemaker showed a tendency to attenuate the high-frequency components (w > 4~ rads/ sec), which offers an explanation for the small discrepancies between measured and target values for %.

5 0

4 O

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o o 3

2

-31o

-4o!

- 5 0 ;

! I l I l I ! I' ! I i I f I ! I I I I I

J , G o u s s i a n d i s t r i b u t i o n 1o'11 = 2 0 " 4 m i n i .

O b s Q r v e d d i s t r i b u t i o n .

m

m I

Fig. 2. Cumulative distribution Of f~ee surfa~ elevation for input $ 1 .

i I I i I I i i l, I I I " I I I " i i | I I

04 0.2 0,5 t 2 5 'tO 20 30 40:50 'a0 ~ rio 9 0 : 9 5 9S 9 9 99"S Cumulative d is t r ibu t ion (%, less thon).

243

Figure 2 shows the observed cumulative probability distribution (cdf.) together with a Gaussian cdf ft.ted in terms of the measured values of o~, over a range -2 ,2o~ < v < 2.2o~. The chi-'square goodness-of-fit test for a = 0.05 level of significance between the observed and Gaussian cdf.s was passed, indicating that the observed cdf.s for ~ were in good agreement with the corresponding Gaussian cdf.s. It is not unusual to find that the Chi. square test is passed although the observed distribution shows small deviations from the theoretical values of skewness and kurtosis.

CI u C

CI j.,

8

ti t 9zl0"!l l- ~ 9 " ' " ~ t |t ! ' ! t C o h a ~ n c , function J ~ , u2(O)i

7 - ( i nc lud ing 9 5 % confidonco~ ~, 6 l i m i t s ) " " ,

IxlO-I 9zlO' l

0

'i 6 S,, IO-t

u & c

o 3

o U

2

Ix I 0 ~ g x l ~ I

8 ? I

5.10-1

" i

I x I0"!

t !

!

t t]

Spoctrum $3

t Spoctrum $2

J S l ~ c t r u m $1

2 . 4 6 0 10 12 14 16 Angulor frequency W ( rods lsoc)

Fig. 3. Coherence function for horizontal velocities for each input (z, = z= = - 0 . 2 0 m, A x = 0 . 4 2 m, z l o n - -9.80).

244

Particle kinematics

Results from measurements at two different positions along the line of wave propagation Cross-spectral analysis between the horizontal velocity records for z =

-0 .20 m and ~ x = 0.42 m was used to produce the coherence function values (eq. 28) between the velocities (see Fig. 3). While values of coherence at frequencies close to the primary peak frequency are over 0.9, they fall below 0.5 at frequencies above twice the primary peak frequency. The same is obserced for frequencies below the peak frequency of each spectrum.

Since the values of the coherence function may be viewed as a measure of the linearity of a system and the absence or otherwise of noise, it is evi-

I x l O

,,,,,, O

'o ? o t. 6

E

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2 :N 4, ; . Q ¢n C ci 'o

"~ I x l 0 "4 ( .

o ~ 7

m 6

Ix lO-$

I I i , , ! 'I t Iqeosurad S l ~ c t r u m ( i n c l u d i n g 95 % c o n f i d e n c e l i m i t s ) .

- - -o - - - M.L.R.W.T. s p e c t r u m . [ ~ q n 2 4

- - -O-- - L.R.W.T. s p e c t r u m [oqn2

iH l iU I m e a s u r e d ( inc lud ing 95 "l . c o n f i d a n t e l i m i t s ) .

- - - I H l i u l f r o m L.R.W.T. [eqn.4

!

S -

& -

3 -

I I i | ,. , , • i I , 2 4 6 8 10 12 14 16

A n g u l a r f r e q u e n c y b) ( r o d s l s e c )

- I 0 a g •

- ? 3 - 6 =

- S X

- 4 c 0

c q~

- 2 . _ c 0 t9

I

Fig. 4. Spectrum o f horizontal velocity and gain factor for input 81 (z = -0 .20 m, Ax = 0.42 m, z/% = -9.80).

245

dent that for points which are continuously submerged, LRWT is satis- factory for frequencies up to twice the primary peak frequency. The coherence function values are used in the estimation of the confidence bands on the transfer functions and, therefore, they provide guidance on the region over which the latter may be reliably estimated. Figure 4 compares the measured spectrum of horizontal velocity at z = -0 .20 m to the corresponding velocity spectrum derived from the surface elevation spectrum measured at a distance of 0.42 m from the laser position. In the same figure the experimental (eq. 28) and theoretical values (eq. 4) of the gain factor are plotted against frequency. Taking into account the 95% confidence limits, the experimental values are consistently lower than the theoret~ ical values, contrary to the corresponding autospectra which agree very well. It is thought that this may be due, in part, tc reflections from the spending beach which produce different time histories of ~ at different positions in the direction of wave propagation, thus influencing the particle kinematic values.

Results from measurements at seoeral eleoations on the same position along the line o f waue propagation Figures 5 and 6 show the observed and theoretical spectra for u and 0 at

z = -0 .10 m. In the same figures the observed values of the gain factors between ~ and the velocities (eq. 28) as well as the theoretical values derived from LRWT (eqs. 4 and 5) are plotted against the frequency ~. The 95% confidence limits for the observed values are also shown. The agreement between theoretical and observed values is very good up to frequencies near twice the primary peak frequency. However, there is a tendency for the observed values of the gain ~actor to lie slightly below the theoretical values. Generally, (see previous discussion of Fig. 4), agreement between theoretical and experimental results was better when velocities were considered at a point in the same cross-section as the wave gauge. The disagreement between measured and theoretical results derived from LRWT at frequencies near and above twice the primary peak frequency (Figs. 5 and 6) could be a~ributed to non-linear terms becoming important at these frequencies, but no ifivestigation of this factor w~s carried out during the course of the present work.

The observed spectra for ~ were used to produce velocity spectra for points near and above MWL, employing both LRWT (eqs. 2 and 3) and MLRWT (eq. 24). These spectra were compared to the observed particle kinematic spectra (see Figs. 7 to 12).

Although the points in the observed spectra do not follow a regular pat~ tern, MLRWT based on eq. 24 provides consistently closer agreement with the observed values than LRWT. The latter always gives larger spectral values at all frequencies. The degree of over-estimation increases with the in- crease in the relative elevation (b = z/o~) of the measurement point above MWL. For example, with input spec~nnn S1 and at z = +0.02 .m, LRWT

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t M

ea

sure

d s

pe

ctru

m

(in

clu

din

g

95

% c

on

fid

en

ce

_ li

mit

s).

--o

---

M.L

,R.W

.T.

spic

tru

m.[

ilq

n24

L,R

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, sp

ect

rum

[e

qn 3

]

IHllY

I

me

asu

red

(i

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ud

ing

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% c

on

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en

ce

lim

its)

.

----

--

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vl

fro

m

L.R

.W.T

. [e

qn.~

I 12

14

(ro

ds

I see

).

!:

S Z

- 2

.:-

0

Gl

I ~

Mea

sura

d sp

ectr

um

# M

.L.R

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. sP

"ctr

um'|e

qn'2

4 #

L R

W T

$1

:~ct

rum

te

qn. 2

1 x

tO"

i

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~

!

• :)

3 2

t. g

11

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3 21

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"5 .

2

t~tO

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2 4

Ang

ular

fr

eque

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~) (

rodS

lSeC

)

Fig

. 7.

S

pe

ctru

m

of

ho

~o

nta

l ve

loci

ty

fo~

in

pu

t $

1

(z

= -0

.02

m

, z/

o~

=

--0

.98

).

Fig

. 8.

Sp

ectr

um

of

vert

ical

vel

ocit

y fo

r in

pu

t 51

(z

= -0

.02

m,

z lo

~ =

-0.9

8).

.I sl

O "~

tn u~

' M

easu

red

spe

ctru

m

(inc

ludi

ng 9

5"/,

con

fidon

ce

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mit

s).

• ---

O-"

M

.L.R

.W.T

- sp

octru

m.~

cKIn

.24~

' --

-O--

- L.

R.W

.T"

spe

ctru

m,

tecl

n. 3]

6 A

ngul

ar

freq

uenc

y tO

(ro

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15

b:

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-,I

• i

i ~

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I I

w I

T M

easu

red

spe

ctru

m

/ ~

(inc

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ng

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nce.

2

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li

mit

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/ --

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- M

.L.R

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. sp

ect

rum

[eq

r~2~

~

L.R

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. sp

ect

rum

[e

qn.2

~ 81

f -

~ 7

- ~

6 _

~,"°,,

'E-

/ \

t

u s

i .

..

= .;F

i

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,

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I I

i t

!x10

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(5

8 10

12

14

16

An

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lar

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qu

en

cy

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rod

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Fig

. 9.

Sp

ectr

um

of

hori

zont

al v

eloc

ity

for

inpu

t S1

(z

= 0.

0 m

, zlo

~ F

ig.

10.

Sp

ectr

um

of

vert

ical

vel

ocit

y fo

r in

pu

t S1

(z

- 0.

0 m

, z/

o, i

!I' ! II ? ,

o 6

~. ,.

~ 3

. in

s-

&-

3-

2- 1

,.,o.,L!

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.

- 0.

0).

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rum

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ng 9

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R W

T

spe

ctru

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|iron

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i • -

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spec

trum

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3 |1

tt

6 8

10

12

t4

Ang

ular

fr

eq

ue

ncy

tU

(ra

0s!

sec)

0o

!f 6 ('1) } M e o s u r e d s p e c t r u m ( inc luding ]

95% c o n f i d e n c e l i m i t s ) . -I ( 2 ) - - o - - M.L.R.W.T. spec t rum. [eqn.24] (3) --I:)-- L.R.W.T. spec t rum. [eqn.2 ] (4) - -x- - As (2)ca lcu lated at z ,~O.O18m (5) - -6-- As ( 2 ) c a l c u l a t e d a t z , t 0.O22m.

249

"~ lx lO-3 'o g ~ e

E 6 "-" S

3 4

O7

C 0 'O

6

S

%

I xl0 "51 I, I t I I I I I I 2 4 6 8 10 12 14 16

A n g u l a r f r e q u e n c y (O ( r o d s l s e c )

Fig. 11. Spectrum of horizontal velocity for input S1 (z = +0.02 m, z /oq - 0.98).

gives a peak spectral value of the hi rizontal velocity of 3.0 × 10 -3 m 2/fads sec, while MLRWT gives 0.66 × 10 -3 m 2/fads sec. The observed value is 0.38 ×~ 10 -3. m2/rads sec. Also, MLRWT tends to give higher spectral values than the observed ones for frequencies up to twice fl~e primary peak frequency. For frequencies above twice the primary peak frequency the o1~ served .values are ~ higher. These differences between the spectral values become less pronounced as the standard deviation of the input spectrum increases, thus reducing the value of the ~relative elevation of zlo,~. For z = + 0 , 0 2 m and~for input S1 (see Fig.~ 12) the observed spectral values for the vertical~velocity~, arehigher a t frequencies about.twice the primary peak fre- quenCythanat-frequencies~close.to the primary peak frequency. This is not the case for the spectrum of the horizontal velocity. In other words, for

250

w I ! " '.,.. ~ 11 I (;) | Measured spectrum (including 95% conf idence l im i t s ) . -

(2) ~ M.L.R.W.T. spec t rum. [eqn.24].

• • (~(~2m:

--~ 1 . 1 0 - ~

o

'~ 1=10"

• ". S l -

(3~ --O-- L.R.W.T. s l ~ c t r u m . [eqno 3 ] (4) - -x- - As(2) calculated a t z -÷ 0.018m

m s 7 / \\'I" '2 i -. 3 %%

2

I~10. S J I 2 4 6 8 10 12 14 16

Angular f raquoncy l0 ( rads lsec ) .

Fig. 12. Spectrum of vertical velocity for input 81 ( z = +0.09- m , z/oq = 0 . 9 8 ) .

high posi t ive .relative elevat ions, t h e . s p e c t r u m for o seems to possess a wider b a n d w i d t h - t h a n the corresponding, specUmm, f o r u, T h e .results p resen ted for such e l eva t ions indicate .that t he re is a -cons iderable scope, fo r i m p r o v e m e n t u p o n t h e pred ic t ions o f ~ R W T , .-. ,~. . .

I t is~-most i m p o r t a n t that,the.~ t rue .. m e a s u r e m e n t elevation, is e s ~ b l i s h e d when. ~ g : v e l o c i t y . m e a s u r e m e n t s ~ a ~ v e : M W L , ! . F i l ~ r e s .11. and 12 ~ show u a n d o:-. s ~ t r a ~ c a I - c u l ~ - ~ m i ~ t h e ~ e / ~ i ~ ~ ! e l ~ a ~ o n ~ i . ~ ~ b y ~:us~g~.

f rom t h e base line s p e c ~ ; . T h e d i f f e r e n c e s - . a r e , . f a r m O m i p r o n o u n c e d in • the case .o f - ; thever t i ca I velocity.~..~.~i.~ .~-~.~., : ...i :~ .~ :-..~ _.. • . . . . . . .~ ... ...

2 5 1

n (u) ( m ' l s e c ) S 'O-

S.0-

4 .0 -

3 .0-

2 .0 -

1.0-

p~q<z) • 0 . 8 3 6 z • O . 0 2 m

zion1 , 0 - 9 8

-3 -~ -I 0 ! 2 3

. . . . XO I----!'-- I

p (~<z) • 0 .5 z - O ' O ~

zl~q ,0 .0

0

x o ) x

x I ! I

x

man - - 1 ° I

- 3 - 2 - I 0 I 2 3 - 3 - 2 - I

u ( m s e c ' l )

0 p(l~<z) • 0.164

z • - O . O ~ m z l~q - - 0 .98

0.16,; ~x

I I • ! I 0 I 2 3 -2 -I

L.R.W.T.p.d. t M.L.R.W.T.p.d.f. Maosu rcd p.d.f.

p(~l<z) • 0.0 z • - O.1Orn

Z t ~ • - 4 - 9 0

I I ! 0 I 2 (x 10"11

Fig. 13 . Probab i l i ty d e n s i t y f u n c t i o n o f h o r i z o n t a l v e l o c i t y for i n p u t $ 1 (z/o n = 0 . 9 8 , 0 .0 , -0.98, -4.90).

..- LR.W.T.p.d. f . p (v ) (m-1$¢c) o M.L.R.W.T.p.d.f.

6.0 ] ~ x M e o s u r , d p.d.f.

] p ( , , z : . 0 . 8 3 6 ] p i l~<z) . 0 . 5 p ( , < z ) • 0 .164 p ( 1 } < z ) . 0 . 0 • O.02m J z .O .Om | z , .- O . O 2 m l x z . - 0 , 1 O r e

zlGq • 0 .98 z lo~ • 0.0 z l o ~ • - 0 .98 z loT i • - 4 9 0

S'0

~-0

.o .. l:-l f' 1 . 0

0 T-~I , , , T I - I I I l I ~ . ~

- 3 . 2 - i 0 , 2 3 - ) - 2 - i 0 i 2 3 -2 -i o i 2 -2 -i o i 2(41o) v ( m s i c "t)

Fig. 14 . Probab i l i ty d e n s i t y f u n c t i o n o f vert ica l v e l o c i t y for i n p u t S l ( z /o n = 0 . 9 8 , 0 . 0 ,

-0.98, -4.90).

252

Figures 13 and 14 show the observed probability distribution for u and v for input S1 and for four elevations. The same figures include the pdf. for u, v derived from LRWT and MLRWT. For the point z = -0 .10 m, both theories give identical results which are in close agreement with the observed values. AS the measurement position approaches and then emerges above MWL, there is an increasing difference between the pdf. derived from LRWT and the observed pdf. While for z = -0.02 m and z = 0.0 m observed pdf. values agree well with those derived from MLRWT, the agreement is not good for z = +0.02 m. For this elevagon, MLRWT gives higher probability levels for the same positive value of the independent variable u (or v). This is more evident in the case of the horizontal velocity. The o!~ served pdf. values for the vertical velocity v show a high degree of agreement with the pdf. values derived from MLRWT. The pdf.s for u are generally skewed while the pdf.s for u are unskewed. It is hoped that use of a second-order random wave theory (Hudspeth, 1975; Sharma and Dean, 1979) will improve upon the pdf. estimates for the particle kinematics at these positions.

CONCLUSIONS AND RECOMMENDATIONS

As noted previously, the results of this study are, necessarily, specific to the generation of random wave fields in the particular facility used. However, in two senses, the results are of more general interest. Firstly, the interpretation of model tests in such an experimental faciliw will be depen- dent on the agreement or otherwise between the theory and the observed behaviour. Secondly, these studies serve a useful purpose in highlighting areas which should be given particular consideration in planning and carry- ing out other data collection programmes, including those at full scale.

The statistical and spectral analysis of the data collected during the test programme revealed the following points:

(1) The measured surface elevation spectra were in close agreement with the target spectra without obvious second-order peaks.

(2) The records of surface elevation possessed non-zero coefficients of skewness and kurtosis which increased with the characteristic steepness of the records. This constituted a departure from the Gaussian assumption.

(3) Notwithstanding the above comment, the observed probability density functions of surface elevation agreed very well with the Gaussian densities in the range -2 .2% < ~ < 2.2 %.

As regards the observed particle kinematics properties the following com- ments could be made.

For points more than 2% below M ' ~ e statistical and spectral properties of particle kinematics were described satisfactorily by linear random wave theory in the following way:

(a) The statistical distributions for u and u followed a Gaussian form. (b) The spectral values were in close agreement with those predicted by

253

linear random wave theory from a given surface elevation spectrum in the same vertical cross-section, as it can be seen from the 95% confidence limits.

(c) At frequencies above twice the primary peak frequency, the observed spectral values for u and v deviated from the values predicted by LRWT. It is poa~ible that this was due to second-order non-linear effects but the present programme has not attempted to investigate this point further.

(d) The observed transfer functions between surfac~ elevations and velocities were in close agreement to those given by LRWT. The agreement was better in the case when the velocity measurement points were in the same cross-section as the wave gauges, than in the case when a horizontal separa- tion was introduced. In the second case, the measured values were below the expected ones. I t is possible that reflections from the spending beach might have influenced these values.

(e) The measured values of the coherence function between two horizontal velocity records at points on z = -0 .20 m, spaced 0.42 m apart, showed a strong linear relationship at frequencies close to the primary peak frequency. At frequencies close to twice the primary peak frequency the values of the coherence function were below 0.5.

For measurements taken close to MWL, observed spectral values and prol> ability density functions of the velocities were in reasonable agreement with the appropriate values derived from the Modified Linear Random wave- theory. Corresponding values derived from Linear Random wave-theory were significantly higher than the observed values. The following points could be made:

(1) At frequencies close to the plY.mary peak frequency, the observed spectral values were below those given by MLRWT. However, at frequencies above twice the primary peak frequency the observed values were greater than those given by MLRWT, perhaps an indication that non-linear terms become important at these frequencies. This behaviour was more evident in the vertical velocity spectra.

(2) The probability distributions for the horizontal velocity were skewed. This was not the case for the vertical velocity. Closer agreement was achieved between the observed pdf.s for v and MLRWT, than between the observed pdf.s for u and MLRWT. This point was more evident for z/o,~ ~ 1, where MLRWT overestimated the observed pdf.

A similar but extended programme of work is currently in progress, employing different spectral shapes and water depths, in order to determine the influence of these factors upon the observed spectral and probabilistic properties of particle kinematics. Second-order random wave theory will be employed in parallel to LRWT and MLRWT in an at tempt to improve the description ~f particle kinematics at positions above MWL and provide deeper insight into the wave on members of offshore structures in the splash zone.

254

ACKNOWLEDGEMENT

This work forms par t of the Project "Particle Dynamics in Non-Linear Water Waves" funded bY the Science & Engineering Research Council through the North Western Universities Consort ium for Marine Technology.

REFERENCES

Bendat, J.S. and Piersol, A.G., 1971. Random Data: Analysis and Measurement Procedures. Wiley-Interscience, New York, N.Y., 407 pp.

Borgman, L.E., 1969. Ocean wave simulation for engineering design. Proc. ASCE, W~W4, 95: 557--583.

Borgman, L.E., 1972. Statistical models for ocean waves and wave forces. In: Ven Te Chow (Editor), Advances in Hydroscience, 8: 139--181.

Chaluabarti, S.I~, 1980. Laboratory generated waves and wave theories. Proc. ASCE, WW3, 106: 349--368.

Grace, R.A~, 1976. Near bottom water motion under ocean waves. Proc. 15th Coutal Eng. Conf., Honolulu, pp. 2371--2386.

Hansen~ J.B. and Svendsen, I.A., 1974. Laboratory generation of waves of constant form. Proc. 14th Coastal Eng. Conf., Copenhagen, pp. 321--339.

Holmes, P. and Tickell, R.G., 1979. Full-scale wave loading on cylinders. 2nd Int. Conf. on Behaviour of Offshore Structures, BOSS '79, London, Paper 79, pp. 1--16.

Hudspeth, R.G., 1975. Wave force predictions from non-linear random sea simulations. Offshore Technology Conf., Houston, OTC Paper 2193.

Lee, A., Greated, C.A. and Durrani, T.S., 1976. Velocities under periodic and random waves. Proc. 15th Coastal Eng. Conf., Honolulu, pp. 558--5"/4.

Newland, D.E., 19"/8. Random Vibrations and Specttzl Analysis. Longman, London, 285 pp. Pajouhi, K. and Tung, C.C., 1975. Statistics of random wave field, Proc. ASCE, WW4,

101: 435--449. Papoulis, A., 1965. Probability Random Variables and Stochastic Processes. McGraw-Hill,

New York, N.Y., 583 pp. Sharma, J.N. and Dean, R.G., 1979. Second-order directional seas and associated wave

forces. Offshore Technology Conf., Houston, OTC Paper 3645. Tsuchiya, Y. and Yamagouchi, M., 19"/4. Horizontal and vertical water particle velocities

induced by waves. 14th Coastal Eng. Conf., Copenhagen, pp. 555--568. Tung, C.C., 19"/5. Statistical properties of the kinematics and dyB~mics of a random

gravity wave field. J. Fluid Mech., 70(2): 951--255.

Documents

Measurements of particle velocities in laboratory-scale random waves