tolrol¿Õ i
TIDAL PROPAGATION
IN THE
GULF OF CARPENTARIA
by
Michele Marie Rienecker B.Sc. (ttons. ), University of Queensland
Thesis submitted for the degree of Doctor of PhilosoPhy
in the University of Adelaide Department of Applied Mathematics
December L978
tli" 1-qT0 [^t.ro{r.i 11ì 0,q J TABLE OF CONTENTS
SUMMARY (i)
SIGNED STATEMENT ( ii)
ACKNOIIILEDGEMENTS (íii)
CHAPTER 1 INTRODUCTION I
CHAPTER 2 THE PROBLEM: A RECTÆ{GULAR RESONATOR ON A SEMI-INFINITE CHANNEL 2.L The Tidal Equations 5 2.2 Frequency Response Analysis 6 2.3 Reformulation of the Equations for an Analytic Model 8
CHAPTER 3 THE ANA],YTIC SOLUTION 3.1 The Method of Solution and the Boundary Conditions 10 3.2 Solution for the Channel Region 11 3.3 Solution for the ResonaÈor Region 13 3.4 Solution folthe Junction Region 15 3.5 The Remaining Matchíng Conditions and the Galerkin Technique 20
CHAPTER 4 AN EXTENSION TO THE MODEL: AN ADJOINING CHANNEL 4.1 The Equatíons 35 4,2 The Solutions 37 4.3 Determination of ô 46
CHAPTER 5 TI4IO NUMERICAL MODELS 47 5.1 A Linear Fínite-oifference Numeríca1 Model 4B 5,2 The EVP Method 52 5.3 Stability, Consistency and Convergence 55 5.4 The Friction Parameter 59 5.5 A Non-Linear Model 6L 5.6 Consistency, Convergence and Stability 64 CHAPTER 6 APPLICATION TO TIIE GI]LF OF CARPENTARIA 6.1 The Gulf of CarPentaría 68 6.2 The AnalYtic Model of ChaPter 3 72 6.3 The AnalYtic Model of ChaPter 4 89 6.4 The Linear Numerical Model 95 6.5 The Non-linear Numerical Model 105 6.6 The Programs 110 6.7 The Response of the Gulf to Tidal Forcing 110
116 CHAPTER 7 CONCLUSION
1t_8 APPENDIX 1 The Representation of Bottom Friction
L27 APPENDIX 2 \te Galerkin and Collocation Methods
APPENDIX 3 Evaluation of the Integral Form for 6z(x,y) 130
APPENDIX 4 The Classes of Elements for the Non-linear Model and Their Associated Finite-Difference Equations 135
138 BIBLIOGRAPHY (i)
SUMMARY
This thesis considers tidal propagation in a rectangular resonator-channel system, with specific reference to the Gulf of
Carpentaria, situated to Èhe North of Australia. The linearízed form of the trdo-dimensional depth-averaged equations of continuity and momenEum conservation is used. An analytic solution is found by dividing the area into regions of constant deprh. In this manner, a solution is found for Èhe case of a síngle connecting channel and then for the exEended case of two connecting channels, associated with either neglecÈing or including the effect of tidal flux from Torres Strait into the Gulf. Results from the analytic model are used to provide tidal inputs for two numerical models, both of which use explicir finite- difference approximations. The first numerical model is linear and is developed to account for realistic boundary and bathynetry variations, with the emphasis on obtaining a model with small- compuÈer time and memory requirements. Since, in shallower coastal areas, the non- linear Lerms in Èhe continuity and momentum equations become more important, the second numerical model developed includes these terms to determíne their effect on the resonator as a whole. The two models show favourable agreement, thus verifying the usefulness of the linear mode 1 . ( ii)
SIGNED STATEMENT
I hereby declare that this thesis contains no material which has been accepted for the award of any other degree or diploma in any University and, to the best of my knowledge, it contains no material previously published by any other person, except where due reference ís made in the text of the thesie.
M.M. RIENECKER. (iii)
ACKNOI,üLEDGEMENTS
I would like to thank my st¡pervisor, Dr. B.J. Noye, for his advice and guidance in completíng this thesis.
Many thanks are also due Èo Dr. Michael Teubner for his help
and encouragement throughout, Èo Mrs. Angela McKay for her excellent, accurate typing, to Mr, G. de Vries for preparing the diagrams and to Mr. Phil Leppard for his advice on computing.
The work associated wíth this thesis was carried Òut from
February 1975 to November 1978, during which time I was financed by
a Commonwealth Postgtaduate Research Award'
M.M. Rietrecker - 1
CHAPTER 1
INTRODUCTION
The Gulf of Carpentaria, located in tl're North-Eastern part of Australia, is an area of difficult access by either land or water and hence remains relatively uncharted and unstudied. However, over the past few years, there has been an increase in the nuirber of research programs in the area as its potential for the lucrative rnining and fishing industries has been recognized. Lirnited sectors, such as at lrleipa, PorÈ McArthur, Groote Eylandt and Gove Peninsular, have been surveyed as port facilities l{ere required by mining companies. The C.S.I.R.O. has reported on the hydrology of the region (Rochford (1966), Newell (1973)) and has made a preliminary study of the circulation in the Gulf (Cresswell (1971)). Further work by this organízatíoa is currently under way. The Gulf r^ras also chosen by Teleki et al (1973), as the test site for assessing the usefulness of satellite irnagery to the rnapping of hydrological parameters in areas of difficult access. Tides and wind are the only two mechanisms which generate Èhe currenÈs in the atea. For several months of the year it is the influence of the strong, sÈeady trade winds which drive the circulation. However, overall, the circulation and mixing of waters ín the Gulf are governed by the clockwise motíon of tides and by density gradients resulting from the stratification of the water masses. Stratification is partly induced by differential evaporation rates betvreen the Northern and Southern halves of the bay (Teleki et al (1973)). According to Newell (1973), "the annual evaporatiorr/precipitation budget of Èhe Gulf forms a very sma1l part of ?-
the total water exchange but is of great importance in influencing water movement.tt
One examPle of the effect, in the Gulf, of winds and precipitation/ evaporation is the striking feature of the annual cycle of about .7m rn the tides, the lowesttídes being r:ecorded at the end of the drl' season and the highest during the wet season' It is during the wet season, which occurs in the summer months, that the activity of cyclones sometimes has a disastrous effect, causing large surges. some description of this is given by Easton (1970) who outlines the general tidal features of the Gulf. Realistic and accurate modelling of the waÈer movement in the Gulf of Carpentaria is obviously an intricate affair, it being necessary to incorporate the effects of tide, wind, stratification, precipitation/ evaporation and Pressure surges, not to mention ríver run-off and sediment transport. However, whereas the effects of the other mechanisms diminish at certain times of the year, Eidal forcing is always Present; and it is only the response of the Gulf to tidal forcing which is investigated in this thesis. Once the Èi-dal response is understood, it is easier to sËudy the coupling of the tide with other effsu[s' The tides are caused by the movement of the sun and the moon and their changing gravitational pull on the $Iater of the earth; however, in coastal areas, such as gulfs and estuaries, a\¡Iay from the deep oceans, astronomical tidal forcing can usually be neglected compared to the direct forcing from the motion of adjacent r¡raters. From the results of Hamblin (1976) for different size basins, the maximum amplitude of the resPonse of the Gulf of Carpentaría to direct astronomical forcing could be expected to be only about 3 cm, a very sma1l contribution to the response as a whole In this thesis, Eidal propagation in Gulf systems is investigated by finding the resPonse of Ëhe system Èo tidal forcing on an open boundary' 3
One of Èhe main problems in ascertaining the accuracy or limitations of any model of an area is that reliable input data across the open boundary is rarely available, especially if the boundary is wide. Data is usually available from coastal areas, but this may be disturbed by local effects and is not always representative for the open sea. Values may be inter- polated between coastal areas, but this does not normally take into account the disÈribution of depth (see Hansen (1962)). A more accurate input may be obtained from the results of an analytic model which produces the input from the solution to the model equations. This still does not account for the true distribution of depth, but avoids the need to interpolate over wide areas. Analytic models have their limiEations in that they provide solutions only for simplified situations; however, they can be very useful in providing an insight into the important feaÈures of a model area. As well as providing the tidal forcing daÈa along an open boundary for numerical models, they may also act as a guideline to the accuracy or validity of these more complicated models. lüith this in mind, two analytic models of the tidal propagation in the Gulf of Carpentaria are developed. They a, e essentially extensions of Taylorrs (I92Oa) problem of the reflection of a Kelvin wave by an end barrier in a semi-infinite channel.
tr{illiams (L972) modelled the Gulf as a rectangular resonator on a serni-infinite channel. His first model neglects the effect of the Coriolis force; his second considers the frequency of rotation of the earEh to be small compared to the frequency of the tidal motion.
The models in this thesis are extensions of his work, incorporating, amongst other things, the dissipation of energy by bottom friction. Although
Èhe Coriolis parameter is taken to be a constant, no assumption is made as 4
to its value relative to the forcing frequency and, hence, this model is applicable to more general situations. The solutions are found for the linearized form of the two- dimensional depth-integrated equations of continuity and momentum conservation. I^Ihen it is impossible to find a solution which satisfies a boundary condiÈion exactly, a Galerkin technique is used to find an approxímate solution. Ttre first model, in keeping with lüilliams (1972) , considers no flux through Torres Strait, while the second allows for the presence of tidal forci.ng through this Straít by incorporating a second semi-infinite channel in the model. Torres Strait is a shallow region with an intricate array of íslands, reefs and atolls. .tt is not considered that the tides in this area have been modelled accurately; only the effect of motion through the Strait on the tides in the Gulf is of interest. The results from the second rnodel are used as input for a linear numerical model which accounts for more complicated boundaries and bottom topography. This numerical model is a frequency-response scheme, based on the EVP method described by Roache (1972), rather than a time-stepping mode1, and has the ariset of requiring very little computer time and storage.
Its results also compare very well with a more complicaÈed non-linear numerical scheme which is developed to determine the effects of the non- linear terms on the tidal motion in the Gulf and to assess Lhe usefulness of the more simple linear model. Both numerical models use a finite- difference approximation to the two-dimensional depth-integrated form of the equations of continuity and momentum conservation which govern fluid motion. 5
CHAPTER 2
THE PROBLEM: A RECTANGULAR RESONATOR ON A SEMI-INFINI.TE CHANNEL
2.L The Tidal Equations
The general two-dimensional depth-averaged equations governing fluid motion have been derived by such authors as Dronkers (1964) and Nihoul (1975). These equations, as given by Nihoul, may be written in vector form as
$tt*zl + V.{(rr+z¡01 =0 2.L,IA
ò g + q.Vq + = v(ez) + 2.I.lb ðt I b^_e - - ;L S S [
where g = (U,V) is the depth-averaged horízonLal velocity,
h is the depÈh at mean-sea-leve1,
Z is the surface elevation above mean-sea-leve1, t is the time coordinate,
f is the Coriolis Parameter ' vector in the vertical direction, 5 is the unit
e is the acceleration due to gravitY,
Y is a bottom friction Parameter, V= tãI,¡T' xry are orthogonaL Cartesian coordinates, positively increasing to the East and North resPecÈivelY,
and E represents the contribution from other factors such as external forces, atmospheric pressure gradients, wind stress effects and t.urbulence and shear effects. 6
In this study of the response of some systems to tidal forcing on an open boundary, the effects of the t.erm { are neglected. The equations 2.1.L can be further simplified to yield linear equations r¿hich have the advantage of superposition of solutions. The assumptions (and a discussion of their validity) implicit in such an approximation may be found, for example, inHendershottand Munk (1970) and Noye and
Tronson (1978).
The linearízed shallow r¡rater r¡rave equations may then be written
2.I.2a 3cnul*3(r,vlâx "'" ðy ".' ' =-Yðt
AU *;u v = 2.1.2b ðt - f - c #
#.;u+fu= -tK 2 .I.2c in which the Coriolis parameter is considered to be constant and the friction parametèr, r, is some linear approximaEion to yllqll A discussion of this línearízation of the quadratic friction term is given in Appendi-x 1 along with different forms which may be used to model Y
2.2 Frequency Res Ana 1ys is
Equations 2,L2 are used to model the tidal propagation in a channel- resonator system, as shown in Figure 2.1, where the motion is produced in response to an input ne{60(xry).t* }, of period T = 2rfu, travelling along the channel in the positive x-direction. The equations are solved subject to the input eoeÅ-t and the real Parts of the soluÈions for Z, U and V will give the elevation and velocity fields at any insËant in time. (i = ,/-l¡. 7
b CHANNEL JUNCTION (Region 3) (Region 2) v
x
d RESONATOR (Region 1)
a
Figure 2. 1 A rectangular resonator on a semi-infinite channel.
The area is divided into three regions with a
depËh-step at the common boundaries of each region. 8
Since the equations are linear, it follows that
Z(x,y,t) = 6(x,y)"-lt''tt U(x,y,t) = u(x,y)e-iu,t V(x,y, t) = v(x,y).-i<'rt so that the explicit time dependence in 2.1.2 may be removed. The equations governing the spatial variation of the fluid motion are therefore
(-it¡.¡+r/h)u - f ., = -, t 2.2.1a
(-ir¡+r/h)v+f,r=-*Foây 2.2.rb
* it¡6 2.2.Ic 3rn'¡dx 3(n*,)dy subject to the boundary condiÈions
u(O,y) = o, -d v(x,O) = 0, x(-a 2 .2.2e and a radiation condition that the input wave does not excite other $/aves travelling in Èhe same direction.
2.3 Reformulation of the EquaËions for an Analytic Model
The equations 2.2.1 can be solved analytically if it is assumed that the depth is a constant; the equations being forrnulated in terms of 6(xry) only and the solution found by a separation of variables technique Manipulation of equations 2.2.|a and b yields the relations 9
u(x,y) = - g{(-io+r/h)2 +f'}-t{(-ir¡+r/h)ff.f fft 2.3.La
2 v(x,y) = - g{(-io+r/h) * f'}-t {- f * + (-ir¡+r/h)#} 2.3.lb and substitution of these expressions into 2.2.\c yields the differential equation governing Ç :-
v'Ç*#f'{ffiffi.l6=o 2.3.2
Defining
q = r/htrt 2.3.3a
0=flw 2.3.3b
2.3.3c these equations rnay be rewritten as
u(x,y) = - fit(r+i6¡'-o'\-'{i(r+iq>Nf, - t #} 2.3 .4a
v(x,y) = - fit(l+io)'-a'j-'{u t + i(r+i4)ff} 2 "3.4b
Y'e*x'e=o 2.3.5 and solved subject to the condiÈions 2.2.2. 10
CHAPTER 3
THE ANALYTIC SOLUTION
3.1 The MeÈhod of SoluËion and the Boundary Conditions
To find Èhe solution to equations 2:3.4 and 2.3.5 in the area depicted in Figure 2.1, the system is divided inÈo three constant depth regions as shown - the channel, the junction and the resonator. Ihe equations are solved (as far as possible) in each separate region and then the elevations and volume transports are matched at the colnmon boundaries of adjacent regions. Thus, the solutions are required to Ehe equations
lvz+{)e, =0, j=r,2,3 subject Ëo the conditions (i) va(x,O) = vs(x,b) = 0, x < -a
( ii) v2(xrb)=0, -a(x<0
( iii) u2(0,y)=0, 0( iv) v1(x,-d)=0, -a(v) ur(O,y) = ur(-a,y) = 0, -d < y < 0 (vi) 6r(x,o) = Ez(x,O)' -a < x < 0
(vii) h¡v1(x,0) = h2v2(x,0), -a < x < 0 (vl11) ÇzGa,y) = ÇtGa,Y), 0 < y < b ( ix) h2u2(-ary) = hgug(-a,y), 0 { y < b and (x) a radiation condition in the channel region. (rn the above, j= 1 refers Ëo theresonator, j = 2 to the junction and j=3 to the channel, and each of the u. (xry), v, (x,y) may be found from the appropriate forms of 2,3.4.) 11
The condition at the corner (-a,0) is given as vs(-a,0)
= ,rr(-a,0) = 0, while the restrictions v2(-a,0) = hlv1(-a,O) /hz and uz(-a,0) = hsus(-a,0)/hz do not necessarily drive these latter velocities to zero and hence could provide a shear effect. lrlhereas it is recognised that this may be erroneous, the exact condition on the velocity at such corners is not known and the use of the above condition is not considered to be too detrimental to the solution as a whole. The inclusion of the Coriolis force (0 I 0) prohibits the finding of an exact solution to this system. Even for Ehe simpler problem of a constant depth channel with a barrier aE one end (Taylor (I92Oa), Defant
(1961)) one has to resort to an approximate method to find a solution which satisfies Ehe zero normal velocity condition at the closed end of the channel. Here, for those boundary or matching conditions which cannot be satisfied exactly, a Galerkin method, which is an approximaÈion technique, ís used. This technique is discussed in Appendix 2.
3.2 Solution for the Channel Resion
In the channel region, the equation which governs the surface elevation is
YzÇt + xlÇ, = o, 3.2.I where
^.2 -_ or2 {(r+iqù2-e2} 3.2.2 ^s cr\ --( l.Io'f- subject to
vg(x,O) = vs(x,b) = 0, for x d -a 3.2.3 and the radiation condition. L2
The general solution, found by separation of variables, is
( o) x+Ks y 4r(x,y)=60(x,y)+Aoe -ik 3
æ k 1ß 1.rv ( e) b -| 3 + e '-{ co 6--L k 3 .2.4 I -b s .0n I & where Eo(x,y) is the input described in Section 2.2 and
Yz K3= 3.2.5 '{*
(o) (u2 Y2 ì. 3 k 3 (r+iqr¡ .2.6a ttn ' I
. (r) + K3 {r:- , ur/Ð'f Lez 3 .2.6b
with n.{t![) ] > O so Ehar rhe radiation condition is satisfied. It inrnediately follows that f*{tlf) } > O so that the waves with coefficient as x + -æ' fo, whích travel- back up the channel, have a finite amplitude The input ldave must also satisfy 3.2"1and 3'2'3, so thaÈ
( o) ik x; K¡ Y 6o (x,y) = doê
@ o' ( [) k ,. b + e : - k3 .-s1 3 .2.7 h y-Tt e I {"."&'ry "+)
Each component in 3.2.7 decays exPonentially as x increases. If the rate of decay of the zetoth mode is less than that of the higher modes and the channel is sufficienEly long, the contribution to 4o(x,y) from the higher modes (1, > 0) can be neglected relative Ëo the contríbution from the ltave correspondíng to [ = 0. Hence, the inpuÈ wave is aPProx-
ímated as
3.2.8 Ço(x,y) = "tnlo)*'K'" 13
and Lhe surface elevation in the channel region may be written
(o) ( o) ik x- Kry* -ik x+K5y 6r(x,y) = e 3 Ao e
æ ( [) -ik ¡X .Q,nv .([) b .Q,nvì + I & e co "bêJ + K3 I; 3 .2.9 l= t { "t"-E.-i
The coefficient 0,s is omitted in 3.2.8 and 3.2.9 since the system is linear and os serves only as a scaling factor. Relative amplitudes and phases at different locations may be obtained by using 3.2.8 as input, and the actual amplitude or phase at any location may be found if the results are scaled according to some reference point. Using the relations 2.3.4, the velocities are found to be
ulo)*+K'Y ,=19, - [- n!o)*' aot(ro) ua(x,y) = t¡(1+i0g) [ "'*Ío)x'K¡Y* "-t
æ F -ik 1o'.1n,0, .Q.nv +L & e .e É no¿ 3. 2. 10 2=t "o"ff # "i )l
( [) -ik x vg(x,y) noe *[ Kz + (tJ']"'"\' 3.2.tL
3.3 Solution for the Resonator Reeion
To determine the surface elevation in the resonator, the equation
Yzer*xl6r=0, 3.3.1 where
2 62 {(r+i4 t)2-02} 3.3.2 Bhr ( r+iOt ¡
must be solved, subject to the condiÈions
ur (-a ry) =u¡(Qry)=0, -dThe separation of variables technique is used once again to obtain a solution which satisfies 3.3.3, resulting in the expression
(x,y) Bo cosh Krx cos po(y+d) + i sinh Ktx sin E, = { Po(y+d)
. l¡rx (Y+d) . uo{"."T cos p'(y+a) . o, s1n- sln Pq oÏ, 1,.fu h a
¡Co cosh K1x sin ps(y+d) i sinh K1x cos po(y+d) { - I L¡rx i0 a . [nx sin po(y+¿) - pl s l-n- cos tp (y+d) .ico I1.TõJ G a [= r fcos- 3.3.5
ut-^( u2 | )v" where K = -lehr (1+i0r)J 3.3.6
3.3 .7 a Po fficl+io,))
Le z+ 3.3.7b Pp {'t -(Ð'}',
The condition 3.3.4, which cannot be satisfied exactly, is treated, using the Galerkin process, in Section 3.5.
The velocities may be found from 2.3.4 and are
oo a( i rnl*î . (Ð'){* cos pr(y+d) Q=t -t . lnx * sin pO (y+d) s1n- 3.3.8 Ct a vr(x,y) =ffil - nopo{sinh K1x cos p'(y+d) + i cosh K1x sin no(r+a)}
[r* pt (v+d) + ' sin (v.d)] i %{ * cos Pt "o"'l'rr Po Q=t *; "i.r + Copo{- sint K¡x sin po(y+d) + i cosh K¡x cos noly*a)} [r* - .r{Ë sin eo(r+d) - ' pQ cos pr(y+d) oi, fi "i' "o"'tr* ) 3. 3.9 15
3.4 Solution for the Junction Region
The surface elevation in the junction must satisfy the equation
v"Ç" * x|e, = 0, 3.4.1 where
,,2 = 62 {(r+iOz)2-o2i 3.4.2 L2 Et'z ( 1+i0z ) and the conditions
vz(x,b) = 0, -a < x < 0 3 .4.3
uz(0,y) = o, o < Y < b 3.4.4
The solution to 3.4.1, which also satisfies 3.4.3¡ mâY be written as
1r(x,y) ß(À¡"{)t*,À0 sinh s(y-b) + s(1+i0z)cosh s(y-b)}aÀ r 3.4.5
where À2 - s, = X| 3.4.6
and ß(À) is an unknown complex function of À Only a discrete seÈ of À-values satisfies the matching condition (vii) given in Section 3.1,
hrvr(x,0) = hzvz(xrO), -a < x < 0. 3 .4.7
From 3.4.5 ít is found thaË
o frt",0) + i(l+iQrlfff",ol
-I {x2ez - sb dÀ 3.4.8 r ß(À).{À* "r(1+iQ2)2} "int -æ so that 3.4.7 may be wriÈten as T6
-iÀx l. 1 I xzoz - sinh sb dÀ J-u, " I "2{t+íqr¡'I .4 ^,
=o urnr{"inh K1x cos ped + i cosh r1x sin psd} [- oo u2a * sin pou i Br 0 G cos Pl d sin T t no "'" T) e 1 { trrt I + Copo - sinh Krx sin Pod + i cosh K1x cos P¡d I
- .Q,nx pqd 1 cos dcos- 3 .4.9 I .r{t sin st" - Pg Pp a L *fi T in which
Ilence
-lttx .'iÀx ß(u)e - sb du dx + I" r {u'u' "'(1+iQ2)'z}"i't, -a -oo
CI iÀx e gs cosh K¡x + f s sinh K1x 2tr r
æ oo e Il'lTX N'ITX + t sln 3 .4. 10 Þ"fte) e cos-+ I n a n= I n= I - )u" where
go i lo [- no sin pod + Co cos Pod] fo = - po[Bo cos pod + Co sin po-d1 + gn = i Pn [- B' sin P,rd + Cn cos P.d], ¡ e Z' a tiL f 0 [B' cos pnd + c sin pnd] , ,, € z* 3,4"11 n nn gh1 ,,
To evaluate the left-hand-side of the equaliÈy sign in 3.4.10, the definition of vz(x,O) is extended to Èhe infinite domain by
, X(-â vz(x,0) {32(xr0), -â { x < 0 t0 , x)0 t7
The definition for x ( -a is essentially an analytic continuaÈion into region 3. The extension for x > 0 is quite arbitrary, but, as long as the results are not used outside the original domain of definition of vz(x,y), this device causes no loss ín generality' Thus, equation 3.4.10 reduces to
í{x202 ß(À) - "2(1+iQ2)2}"intr "b
* .-iÀn cosh K¡a + K1 sinh rra}l * Fret- iÀ {iÀ
K1a + Kl cosh K1a}] + #[xr - "-^' {iÀ sinh
æ íÀ grr [1 - (-l)""-i1\a , 2 ¡ÍLT( n i À d
nn/?. L + i t , tl _ (_l)n"-iÀ" l 3.4.12 ,,=", " À'-(a!!)- I a
The expression for ß(À) which is obtained from 3.4.I2 may be substituted now satisfies into 3 .4.5 to yield the new expression for Çz(x,y) which 3.4.7, the condition which ensures the continuity of volume flux from region 1 to regio¡ 2:-
iÀx À +iK f Çz(x"y) e 0( À) dÀ I À K I + r .-iÀ(x+a) {go []. cosh Kra - it<, sinh Kla] K1a] - f o [À sính K1a - iKr cosh ]ffiul I {le'+iTr"} I [- --,^* \v\^,'o(À) u/\dÀ n= I J " )r, 1*rl -æA
æ -tÀ( x+a) + (-1)" e 3.4.L3 I J- n= I a 18
where
À0 sinh s( -b +s ( 1+i )cosh s( -b) 3.4.14 0(À) = a 1+i0, sinh sb
The detaí1s of the contour integration used to evaluate these integrals are given in Appendix 3, the result being
* Go Çr(x,y) = Eo "inlo)*'K'(v-b) "{nlo)**K'(v'b) oo ( r) i k ,X cos w_ À r.!e) ryl + I Eg e b Jl,r' "i" bl l=t
æ ( [) ( Q) tlry. -tk ,X eb . + Gg e cos &!r + K2 Sr-n I b ln b l=t It-Io,I
+ oofcosh K,x cosh so(l-b) . ;#øsinhK,x sinh sr{v-u)] î t s[r*. i0 .^'tr -..- sinh so(y-b)] * Do^ cosh so(y-b)so (v-b) - ;{ "1" + ,=1, [cos "o"h + ro[sinh K,x cosh so(v-b) .;ffi; cosh Krx sinh sr{v-u)] [r* in [r* . ro["ir, so(v-b) . ' oi, "o"¡ G# "o" "i"¡ "o(r-u)] 3 .4. 15 r¿ith
so(1+iôz) (oo,ro) iCI (Bo,fo) 3 .4.L6a 2l tozr!+sfi ( 1+iQr) sính sob ifi (1+iQ2)sO (o0,tr'o) = - (go,fo), Le z+ , 3.4.16b sot rr$l' - "'o(1+i4r)2lsintr
where T9
2 lu' so Xz I
+ sg -*;\, Lez,
r/- . ( o) kz ffirr+io'z))
. ( r) + K2 = {*i - (Ð'r, 9"e2, l.' 3.4.17 K2 Í and the fn and gn are as defined in 3'4'LL' 2.3.4, are The velocities, which are obtained using the relations
, *!o) x- K,( v- b) ,t2 ( x, y) ,db {- uour,o' e nlo) *+K'( v'b) + cs k{zo) "-
æ + u!0, * onng EI .,*Ír).[- "o, {r * "t" T] e ¡
æ *Í o'. (e) + I Gr 2 cos I "-' [u +.'#É"'"ry]
. cosh K1x sính s't'-O)] +Do iKr sinh K1x cosh so(y-b) * É I oo l,'t¡x 0 r¡2 l.tx . 9'trs].n cosh sO (y-b) + cos sinh sn 1y-b) I + I Dr AA -rlgltzaLl l=t - - - . sinh K1x sinh s'f'-o)] a Fo iKl cosh K¡x cosh se(v-b) * #
æ @ cosh so(y-b) . sinh sort-o)l] 3.4. 18 + I F,[, T "o" "ir,$ 9.= t ** 20
6) < rl ik .* 2 + Ur vz(x,fi = Ere [^; (Ð']"'" b rffr e. I I
@ \" -ik oI ¡rv Gg e 1o',. -; . +L .e,nL (+)']"'" b Q=t
2 * lo rl *' . (uf," cosh K1x sinh so(Y-b) "o I ) 2 o sinh so(v-b) .oï,"i[(-t ) (T)']"'" T
. r<¡x sinh se(v-b) * ,o* [*î (nt" )']"t"n
q) 2 1 ( o) sinh socr-o)) + Fg (k 2 ) (Ð']"t" T e I 1 "Q 3.4.19
llilliams (Lg72) has omitte and the Galerkin Te chnioue 3 5 The Remaini ne Ma tchins Conditions
been satisfied Not all the conditions listed in SecÈion 3.1 have involve, as yet by the exPressions found for Çt,Çz and Çs which still ' which remain to be satisfied unknown comPlex coefficients ' The conditions
ate d < b (1) continuit.y of elevation at x = -a' 0 y ( < 3.5.1 that is, Çz(-a,Y) = 6g(-a,Y), o Y b ( < (2) continuity of volume flux at x = -a' 0 y b 3.5.2 that ís, in2u2(-arY) = h3ua(-ary), 0 { y < b ( < 0 (3) continuity of elevation at y = 0, -ê x 3. 5.3 that is, 6r(x,o) = 4z(x,0), -a < x < 0 2l
(4) zero normal velocity at the boundary x=0 that is, u2(0,y) = 0' 0 < y < b 3.5.4 (5) zero normal velocity at the boundary y=-d that is, Vl (xr-d) = 0, -a < x < 0' 3.5.5
Since the expressions for l-he elevaËions (3.2.9,3.3.5,3.4.15)
and velocities (3.2.10, 3.2.IL, 3.3.8, 3.3.9, 3.4.18, 3.4.19) involve infinite sums, no explicit expression or value can be found for each coefficient, and it is obvious that no finite combination of terms will satisfy the above conditions exactly. Hence, some approximation technique must be used. Techniques widely used in such circumstances belong to the
Method of !üeighted Residuals (Finlayson, L972), from r¿hich class, the
most cormnonly used are probably Collocation and the Galerkin method' discussed in Appendíx 2' These, with particular emphasis on the latter ' are The Galerkin technique is used here' Each of the series in the expressions for elevations or velocitíes is truncated after l, = N; and, for each of the five conditions above' 1¡+1) weighting functions are used" The resulting equations, together with the appropriate 2(N+1) equations from 3.4.16, yield a system of 71¡+1) simulÈaneous linear equations in the 7(t't+t) unknown coefficients
Gl [ 0'1'""N' s, Bl , Cg., Dl , EQ, F[, , = The five conditions are no\'/ treated in turn' 22
(1) Continuity of elevation at x = -ê, 0Substirution of 3.2.9 and 3.4.15 into equation 3.5.1 yields
( o) æ ( [) .tk a-Kr(v-b) -ik . lnvl + Er e ,ã cos Ar ß1 s r-n Eo e [ b b4l Q= r ( 1) ,*:o)a+Kr(v-b) t k tTtf_ +Goe + Gr e "à cos &y + ße sln I i b b
ioK r +Do cosh K¡a cosh so(y-b) (t+i4r¡"0 sinh Kla sinh so(y-b)
. i0K, K1a sinh so(Y-b) + Fo - sinh K1a cosh ss(y-b) n#O}.; cosh
I i0 [n 9 (-1) so (y-b) + F (-r) sinh sO (y-b) .i Dr cosh I r ( 1+i0z [=t [=r ""I ( o) ik a+K3 y + Aoe
æ ( r) i k + Ar e ao cos Ut + p1 sln W 3.5.6 I¡ b b where
0b ( r) + ge = k Lez 3.5.7 a Gîot l,rr 2t 0b ( r) + 3.5.7b 9g k !"ez ÎT+ïõ--J [1T 3
The Galerkin equations are produced using
u) (-a,y)dy, R = 0,1,...,N, 3. 5.8 trG",y)dy = r m Ë ? o
u) are chosen to be where the m
K_y t^) =e, o
mfiy + p sin mlTy ,m)0 3.5.9 W=cosm b m b
and the ËrGary), tr(-ary) are the truncated-series forms of the expressions for Çr(-a,y), 6r(-a,l) given in 3"5'6 (this notational
convention is carríed Èhroughout without further explanation) ' 23
The resulting equations are
( 0) ( 0) k 1 tk a o 1 t8 K_b -K^b e a a*t -"*'o +Go e -e E o K3-K2 Ks+Kz ([) r -tk a 1 a Ln - 2 Ka+ tl ( -1) -r] *)Ete b ""'o x2*(4)'3Þ ( r) - ik ,r -r] +)Gte [*, - ß, f][t-'r' ""'o '.-Frf . o, K," r,o - sinh K1a t, ["o"r, ] **30
iOK . - Kra rro . ' cosh Kta t,] t, ;ft [ "l"n Tffi
( -1)r + Dr(-r)r T,.n * I Fq Trn I _r*, dJ # *1-"i (o) o) a nl o -ik *Ao 1 .t " ", -r] =be 2Kt [", ([) ik a 1 - 3 3.5.10 *)Ale [*. - oo {][r-'ro "*' '-1] x'*(Æ)'3D ( 0) -ik a+K.b, (Note thaÈ, if hg = hz, the term with coefficient Es is Esb e
where
1, 0r1r... rN T Kr("K'b - cosh sob) - so sinh sob , = ,9. =
0r1r...,N, T K, sinh sob - - cosh s'b), [ = r9. = "ß("*'b
and 24
( 0) -ik a 1 mT Eo e 2 Kz+P - (-t)' m b "*,0 x'*(9)'zÞ . ( o) lR- e 1 +Goe 2 *, . - (-1)'"] [- o- i][e-K'b r'*(9)'ZD
* * Er - ßr I u,' å "-'*Í''"[, - o-ß-] I' "-t*Ío'"[o,.r0.. rnl
m) r) k a nÍ +c l. 1+pß * cc .' " * ßr I nt¿ mm I, [o-îo- "tf
iOK + (l+rQzJso,-:i;j:1þ' sinh K1a
+ cosh K¡a t* ]
- 1)e io Î'rr ( -1)r + Dg r, -T' ,9. Tqg I mTT 2 tg. L ( 1+i0z ) s 2 + ( ) "rl e. b "r'*(Tl' (o) -ik L 1 3 . 0," - ,-r)^ .-*'o =e t [", i][t ] rt*(S)3Þ ( o) k I + A e ,e - o- .K" - t] o [-, i][,-1)' r'*(4r)'3b
* o^r."'*l*) " * Ar * Pp I ,r = 1r...,N 3.5.11 [r.oå] I' "'*10'" [o*î0," ntf
where
TrL sQ sinh soo * Pm ("ottt b- (-1)'),L= 0,1,...,N S "Q (-1)-) * sinh sob, [ = 0r1r...rN Toç. = ,p(cosh sob - 0,,,,, S
N means I I I
N f'*"ans I 9=t lf-
^b m and l^ [(-1)r'--1] Ilm 3.5.t2 Xnr TT Fç 25
(2) ContinuitY of volume flux at x = -a, 0 { y < b
Using the expressions 3.2'10 and 3'4'l-8, equation 3'5'2 may be written as
o) o) *' ( t) kl a+Kr(v-b) nl '' v- + c¡ tlo) ei - eo t!o) "-' ([) -ik a/[) trJ El e t cos vr s1n w I 2 b b r 1 ( [) k A r) trr 2 r.( [" os + vQ s1n Ut .icr e 2L b b Q= t
K1a sinh * oo[- i*, sinh Kla cosh so(v-b). *# cosh 'o{v-u)]
0 t¡2 sinh Kla sinh ss(Y-b) +Fo iK¡ cosh K1a cosh ss(Y-b) - so ghz
&I (-r)l so (y-b) . t, c-tlo sinh so (v-b) * FQ i cosh ni, å $ rl (o) ( 0) lk a*Ky -ik â' K ... 3 3 hg e a 3v ¡ ¡(o) e hz ffi{-n'," 03 (r) a ß) k t( ["o" &v +1l0 sin Uv 3. 5. 13 .iAe e 3[ b b ) 9.= t
where tJ2 0 b vr =ñ tu-o=ot a 3.5.14a .Fîñ;
ts2 e b 9"eZ + 3.5.14b uq =;ñ;;( ç '
The Galerkin equations, resulting from
3. 5. 15 tú hz iz(-a,Y)dY = t) hs üs(-a,Y)dY I m I m o 0 K.Y wíth l,oo = ê
mTy lil = cos + 1rn, sin , m = 1,...,N, 3.5.16 m b ry
ate 26
( o) a 1 K b Kb -ik 2 I - no k!o) e t. e e' Ks-Kz L I (0) tk a1 K-b -K-b + co kto) e e' -e I K3+K2 J
( l) -ik . ( [) 1 -I Er e R2 . -r] *3*( [*, "- f]fr-rlo "*,0 b )' (l) r ik r (e) I +IGte , k 2 *?*t Lt¡\, [*,-"of][{-r)r.",o-.] b
. o, ir, sinh Kra ï,0 . cosh rra Tro [- * É ] ;fu30 t r o *ro li Kr cosh K1a T - 9'sinhK,ar' I L-ro so ghz 20 J K2-sz30
+lDr+## r,p +lFr r+ rl
(o) ik a 2Kb l, !1.19rì J - r(rol¡!r b e-inÍo)n + ¿ ¡(o) 1 3 [e t -t] hz (1+i0s) I o 3 F" 3 (r) - ik . ( r) 1 .à K3 (-1)r t]) +TfuE [*, - f][ "*.0 x'*(F)'3Þ " 3.5.t7 o) - ( o) -t nl a + K, b (Note that if h3 = l:2, the term wiEh coefficíent Ele rs -g þlt2 e ), 0 where
T Ku(eK'b - sob) - so sinh s'b, l, = 0,1,"',N P "o"h
TrQ K, sinh sob - ("*ob - cosh spb), [ = 0r1,. ,N "l " and, defining * tgb - (-1)-)' Trp = "l sinh soo um S(cosrt .0 = 0r1, . .. ,N (-1)-) * sinh s'b, Toç" = sg(cosh "tb - U- + 27
( 0) -ik 1 Eo e ,e . (-1)-] - k!o) mfi ,,", - K2+ ( [*, l]["",o 2 b f
I oì'a + co t!o) 1 -Kz+U MTT - (-t,*] IIIT.2 m b "tuì x2+( ]["-.,0 2 b,l ,o( tn) - E r.!'r[r - v u Þ et ^ n!'' 1+vu nt¿LmnìlI "-tnl'', l*. m 2 I nlm ([) -i k a E e 7 u!o'[u,"îo- - VQ I T L ¡nl
. . < lt + GO et*t a k
