Cong re s s
ON, 1963
ARCHIEF
-
Lab.
y. Scheepsbouwkun
14
Technhe Hog
eschool
Deift
LABORATORY WAVE GENETÀTIONA COMPARISON OF THEORETICAL AND EXPERIÌIENTkL PERFORMANCE
by
Alan A. Smith
Lecturer in Civil Engineering
The Royal College of Science and Technolor, Glasgow,
Scotland.
[n certain problems of wzve research it is desirable that the generator may be pre-set bo produce a wave of specified characteristics without the necessity of trial and
error djustment. In this papor, following a brief outline of the methods and theory of
ve generation, a relationship is derived between wave amplitude and paddle
ccentricity for any relative depth
(h/L).
A description of a wave generator for two menzicnil studies is then given, together with the derivation of the necessarylibration curves. Ex-perinients carried out to test the validity of the theory are escribed and the results used to evaluate an efficiency factor for the wave generator.
'¼
/
j')ans certains prcblemes de houle, c'est desirable que le generateur soit regle par Lvance pour produire une orde avec caractristiques specifiées, sans avoir besoin de
rocder a' l'ajustage par essais et erreurs. DnR ce papier, suivant un courte Lescription des mthodes et theories de generation de houle, une équation est dérivée !ntre l'amplitude de houle et l'eccentricite de volet pour une profondeur relatif (h,'L).
ne description d'un genérateur de houle pour les 'etudes de deux dirnemsious est dormie .vec la derivation des courbes de calibrage nécessaires.
Des expériences faites pour ester la validité de le théorie et les resultats employas pour évaluer Un facteur
'efficience pour le gnérateur de houle.
-(1) H. Lamb. 1iydrodynamic&'.
- -. -_
;-- ,,
-LBORATOPY Wt.VE GENERATION
A COMPARISON OF THEORFICL kND EXPERIMENTAL PERFORMANCE
By Alan A. Smith
The Royal College of Science and Technolor, Glasgow, Scotland
INTPCLUCTION:
1.1 In most studies of wave phenomena in hydraulic models it is acceptable to adjust the generator, more or less by trial ana error, until an incident wave of the desired characteristics is obtained. In one particular investigation at the Royal Collcge, however, a study is being made of the effect of waves on the mixing of a stratified system, the investigation being part of a brcader programme of research into demsity difference phenomena. In this case, therefore, it is essential that a wave train of specified characteristics be established with the minimum of delay following the start of a test run. This in turn demands that the various controls of the wave gene'ator should be pre-set to predicted values thus eliminating the necessity for trial and error
adjustment. In this paper a description is given of the solution adopted and a comparison made between theoretical and experimental performance.
BASIC P UIRTS CF A WAVE GENERATOR
2..1 Irrespective of whether waves are to be studied in two or in three dimensions, the
basic requirements of the generator will be the same.
2.2 The machine should be capable of producing a series of regular waves of uniform characteristics and period and free from surface irregularities and sub-harmonics. Furthermore the machine should be adjustable to produce a wide range of period and amplitude. An important exception to this general requirenent arises in the study of wave impact, run-up and other phenomena which require waves with a spectrum of wave
height mad lenth such as is found in nature.
2.3 The machine should be mechanically sound, requiring the minimum of maintenance, having few or preferably no bearings or joints in water and incorporating a convenient and adequate source of mechs,ical power.
-2. As might be expected, the solution for any particular investigation will usunlly
represent a compromise between these two sets cf conditions in order that the machine is economically justifiable.
HODS OF GNERATION:
3.1 One of the simplest methods of wsve generation consists of varying the displacement of a p'ìrtially submerged body. This may be effected, by means of a mechanical plunger or by using a pneumatic device to withdraw and inject alternately a suitable volume of water. Such machines, however, are seldom suitable for waves of any appreciable steepness since
the disturbance at the generator prevents regular waves from being formed within a distance of several wave lengths from the source.
3.2 A more popular solution consists of an inflexible diaphra or paddle which is made to oscillate horizontally either as a piston, as a paddle hinRed at the bed, or with a combination of these two modes of movement. The relative merits cf these various alternatives may be compared in the light of the theoretical particle movement which exists in a regular wave.
TORETICAL PARTI CLE MOVE' 'T DIS TRIBUIION:
1+.l It has been shown (1) that regular waves of small anpllude produce a particle
motion which takes the form cf small closed orbits of elliptical form, the proportions of which depend
on:-the relative depth, i.e. depth/wavelength.
the height above the bed of the mean position of the particle. The axes of these ellipses
are:-Horizontal cr,shi2n'r (i)
sin-h
2rtri,/isinh 2mrv/L
where 2.a wave height
L wavelength
h
= heht of still water surface above bed
y = height of mean position of the particle above the bed.
4.2 Obviously when y = h - i.e. for particles on the free surface - the vertical axie of the ellipse is equal to the wave height 2a. In very deep water (h L) the
e pressions sinh 2wh/L and cosh 2rrh,/L are practically equal co that the surface orbits t're circles. With increasing dcpth bcicw the surface, these circular orbIts rapidly decrease in size until at pth below the surface equal approximately to L/2, the particle motion is almost negligible. This accounts for the fact that waves do not "feel" the bottom until they are in wter of a depth approximately equoi. to half their wavelength. For example ripples approach a shoreline practically free from distortion, whereas seiche waves are refracted and distorted even by the topography of the ocean bed. 4.3 At the other extreme, when the wavelength is very long in relation to the depth, the particle orbits become pronouïced ellipses, the horizontal axes of which reduce only
slihtiy fron the surface to the bed. The vertical axes vary from a maxin'ni of 2a at the free surface, to zero at the bed, thus producing on the bed a straight line
oscillatory motion.
4.4 Evidently the articulated paddle g-ives a fairly good approximation of particle movement distribution for deep water waves (L - 2h) while the piston generator is suitable for waves in sh1 low water (L 8h). Where intermediate values of (b/L) are involved a strong argument may be made in favour of some form of linkage which enables finite but unequal horizontal movements to be produced at y = O and y h. This is particularly true if reasonably pure waves are to be produced as close as possible to the paddle.
4.5 Various possibilities exist for such a linkage but irrespective cf the method employed, a first m'proximation to the desired horizontal particle movement distribution will be obtained
if:-Horiznntal tixis at y = h cosh.2-h/L
o h /T (
)
horizontal axis at y = O - cosh o - C S
w
3
If the paddle eccentricity at any height y is denoted by e then:
ey = e0 + (eh - e0).y/h (4)
and from equation (3)
eh
r
i
ey
cosh + h L1 - coch 2o ] eh
(5)
Althouch the distribution of horizontal eccentricity defined by (5) represents a
considerable improvement over the piston and flap type generators for intermediate values of (h/L) ib will be appreciated that the particle movement at intermediate depths will be distorted, particularly with deep water waves with values of (/L) in excess of 0.5 say.
TORY OF WAVE GENATION:
5.1 Based on the assumptions of an incornpresalble, non-viscous
fluid and
irrotati-rnr1 flow Biesci (2) has developed expressìons for the particle motion resulting from anyspecified paddle movement in a two-diensicnal system. Neglecting the perturbations which exist near the paddle the wave half anplitude may be ex-cressed
as:-in which nl Q =
2(2/T).
Ley.coh.my.dy
a -(2x/T)L
cosh'.my.dy 103-(2) F. Ejesel: study of a certain type of wave machine".
La Houille Blanche No. 4, l95.
'i
'21.
(6)
(7)
and m = (2c,ÌL)
(8)
In equation (7) the function e(y) describes the distribution of horizontal eccentricity at the paddle as a function of height above the bed y.
5.2 The author bas used this theory, substituting for e(y) the expression already developed in equation (5). In this way a relationship between wove amplitude and paddle eccentricity which is a function of the relative depth (h/L) is obtained.
Thu8 : -a
-
= 2x
Ch 104 -2coh.mh
- +
51n11.rnh-inh inh tnh.cosh.inh
cosh.mh +
-sinh.rnh
(9)
TS
function is tabulated below for variou3 values of (h/L) and 13 shown graphically in Figure 5.D3IGN AND CON3TtJCTION:
6.1 In section 4 it was shown that a reasonably close aproxiriation to the theorctical pirticle ttioveinent distribution is attainable over a range of relative depth (1/L) value by the use of a paddle the motion of which inay be made to vary between the extremes of
pure translation and rotation about an ads at bed level. Th15 advantage is of partialar value in any fli,rre generator with which long waves are to be produced in a f]use of limited length, since by this inearis the length available as a tect region free
from end effects can be siii1icantly increased. A wave generator of this type in use at t}'e Royal College uses the principle of the suspended paddle, the linkage being arranged as shown diagramiaticaily in figure 3. The padrile consists of a rid inred L supported by two rairs of links, one pair of which is adjustable. This adjatcent is achieved by suprorting the utper ends of the lickri in a frame which itself rotates about an axis coincidcnt with the pins connecting th pdc1le to the lower end of the odjustable links. The arrangement is shown in Figure 1, which illustrates the screwed rod by which the links may be positioned. A simple scale measures the chord distance and facilitates calibration as described in section 7.
62
Variation of the eccentricity is of coarse essential in a facility intended to stu1y a range of wave characteristics, but a further refinement incorporated in some generators, is the ability to alter the eccentricity while the machine is running. Thedevice ud in this instance is illustrated diagrammatically in Figure 2, and cc-ists
of a crank bar with a pin working on a screwed rod. The latter may be rotated through a pair of bevel gears by means of a shaft concentric with the main drive shaft, any rota' ion of one sht relative to the other thus producing a change in the eccentricity of the crank pin. When rivariing normally both shafts are made to rotate in the serie
direction and at the same speed by men of a pair of differential gear boxes. One of the differential casings may be rotated by means of a handwheel, thus producing a
rotation cf the inner shaft relative to the outer drive shaft irrespective of the speed of the latter. For convenience the handwheel also operates a digital counter through sitii-.able gearing so that the eccentricity may be read directly to 0.01 inch. The eccentricity may be reduced to zero to enable the speed adjustment to be made and checked if necessary without generating waves.
6. Control of the wave period and thus wavelength is achieved using a
varible speed gear box mounted bet"een the electric motor and the reduction gear box. Speed variations of 9 1 are obtainable and may be pre-set accurately by means of a digital counter on the control ksob. An long as adequate power is provided the npeed-torque characteristics of the unit are reasonably flat; even when wave amplitudes are increased by 2 or 3 times, the change in wavelength is generally within 2%.
CALIBRATION 0F Ïfft SUSPENSION LINKAGE:
7.1 With the system of linkage described in. section 6 and illutrated in F'igure 3, it is evident that the position of the adjustable link uniquely determines the ratio of the horizontal displacement of the paddle face at bed level and at eny height
h above bed
level. Considering the geometry of the linkage, a first approximation may be made by computing the position of the instanteneous centre of rotation of the paddle for any
position of the moveable link, and therefter proportioning horizontal displacement to
height above thia centre of rotation. At high values of (h/L), however, where the centre is only a small distance below the bed, the distortion of the linkage under the action of even moderate eccentricities renders thio method inaccurate. A more exact
h/L 1.00 0.80
O.0
0.50OiO
O. 0.25 0.20 0.15 0.125 0.10Also eh e0 -BC - (CD - h) tan Arc BC - CD.tan R (13) eh Xh
-
= cash (2rh/L) (3') co XoFrom eqll-&tions (12), (13) and (3') the typical curves of Figure (2) are obtained by which for any desired values of h arid
L:-(a) the required position of the adjustable link is determined
and (b) the ratio eh/e and hence a/e are obtained using the curve of equation (9) or some modified form of the equation.
7.3
Prdiction of the required period or frequency for the required va1ue ofh and
L is eariily effected using a recognised expression for wave celerity, such as equation (i'+).
= =
/-*.tauh
[i2 ]
If a large number of such calculations are envisaged a series of curves say be constructed to facilitate the solution. A simplified example of such a diagram is given in Figure (6).
EERThETAL TECHNIQUE:
B.i
The tests were carried out in a glass-sided rectangular flume about ft. long by 1.25 ft. wide. No convergence or smoothing filters were incorporated in front of the paddle but absorbing beach filters were installed at both extremities of the flume toinimise reflections.
.2 Altho'igh the use of equation (14) enables the period for any desired wavelength
to be predicted quite accurately, it ws
felt
that direct measurement of wavelength was essential for the purpose of reliable comarison with the theory. The arrangement used for wavelength measurement consisted of two point gauges with a D.C. potential of 15olts, set to the still water level. The signal from each of these probes is a regular square wave and hen both signals are exactly in phase the points must be an exact multiple cf the wavelength apart. A similar arrangement has been used by
arLsford (3) which depends on the cperator gauging by eye when two neon flashes occur inu1tcnecua1y. In the present icots the signals were displayed on tuo beam DSCillCscore with a persistent image screen. Tuning of the time base of the
oscilloscope produces a rt,ationary trace so that one moveable probe may be accurately positioned to give
signals
exactly in phase. The method is speedy, reliable and yields results of high accuracy - usually well within 0.5%.(3)
G. Raniford 1'A method of measuring wave length by synchronisation of visual signals". La Houille Blanche No.5, 1951.
r
105-uialysis in outlined below with reference to Figure
3.
7.2 If point A on the paddle element is subjected to some horizontal eccentricity + e, it will e-çperence a vertical displacement which is dependent on 8 , R and e
Accurately this displacement is given
by:-r
-lRsinB-e
-lRsine+e]
True rise = R Lc03(smn
R
- cos(sin R)j
(io)
In the machine under consideration the approximation
Approximate rise = 2.e.tan
e
- -
(il)ay be shown to be accurate to within 0.5% with the ¡raximum values of O and e/R likely to be encountered. Since CB is constant the angle
e
may be found and used tocalculate the deficiency of horizontal displacement of the paddle face at any depth, compared with the maximum eccentricity e. Hence the following relationships are
obtained:-(CD - h) Arc
- =
i -.tan
(12)106
-8.3 Wave amplitude was measured by a resistivity gauge ap displayed on a qulek
remponse pen recorder. Although the absorbing beach at the end of the flume was quite efficient the usual procedure was followed of measuring the amplitude ov'er a distance of at least half a wavelength in order to detect any reflected wave. Even with the lowest
values of wave steepness tested, the coefficient of reflection was not apprechle and
it was estimated that it was unnecessary to allow for second order effects as des?ribed b7 Carry (4) in his analysis of partial clapotis. The incident wave amplitude 2a was tiken, therefore, as the average of the minimum and maximum amplitudes measured at the node and antinode points.
CONCLUSIONS:
9.1 Figure (5) shows the theoretical relationship between (a/eh) qnd (h/L) together
v-ith experimental points within the range 0.1 - (b/L) < 0.7. Wavelengths used varied
from
1.6
ft to 8 ft. in depths of0.6
ft.,o.8
ft. and 1.0 ft. Care vas taken to avoid wavelengths of a value close to (n - ) times the flume breadth or multiples of this value, in order to prevont the build up of cross-oscillations. The e,cperimental pointsrhow a fairly consistent trend, and tend to be grouped about a line displaced by %
from the theoretical law for (a/eh). This would infer an efficiency factor of the ord'r of 0.95. A possible reason for this discrepancy might be leakage beneath the raddle which although small, cannot be completely eliminated by virtue of the geometry of the suspension linkage.
9.2 Preliminary observation irdicates tbt good orbital movements are obtained very close to the padile but as yet these have not been photographed and analysed
quantitatively.
9.3 The results of this initial study indicate that a suitable technique can be developed whereby a wave machine may be pre-set to produce a wave of desired
characteristics immed5ately the generator is started. This is not only valuable in
mnimising mixing at the intrface of a stratified miscible system before the regular
train of waves is established hut is convenient and time saving when the facility is used for undergraduate teaching and demonstration.
AC}GOWEDG{TS:
The experiments described were carried out in the Civil neering laboratory of The Poyo]. College, Glasgow, with the permission of Professor W. Framer to whom the writer is indebted for encouragement, assistance and constructive criticism in the preparation
of this paper.
r
Fig. 1.I
t-
107 -\C
!
FR2.
Fig. 2.Mochanisni for eccentricity variation.
e D D1
VEL
F1_
COU7C Fig. 3. AC7UAL tsAssuMo
ISEView of wave generator and linkage. Geometry of suspension linkage.
5h-4 A -r,
cc»JcE-r! 'C
z,
4.
14
-
108-Fig. 4. Typical calibration curves.
OP 1 o, ,g,
hf.
Fig. 5. Theoretical and experirzental values of (a/eh) as a function of (h/Is).
Fig.
