on studies of reflections in tidal rivers and their...

On studies of reflections in tidal rivers and their branches

O n studies of reflections in tidal rivers and their branches

D r . S. D a s Gupta , Professor of Electrical Engineering Judavpur University, Calcutta-32, India and M r . K . K . Bandyopadhyay , Senior Scientific Officer, Hydraulic Study Dept . Dalcutta Port Commissioners 20, G a r d e n Reach R o a d , Calcutta-43, India

S U M M A R Y : The determination of reflection and energy content in a branched tidal river due to the inception of a tidal wave is important in order to study the general variation of tidal level at various points of interest and discharge through various sections of the river and its branches. With a linearised version of the governing differential equations, the scatter matrix representation of the tidal channel handles the situation with ease; in fact, scattering takes care of the multiple reflections which occur in such a complicated system and gives a general expression of the reflecedt components, at each port, corresponding to a given set of incident waves at different ports.

In the present paper, a scatter matrix for an exponential tidal channel has been derived and also the general problem of any tidal channel with branching has been considered.

Examples have been given from River Hooghly (India) in conjunction with its tributary River Rupnarain.

Études de réflexions dans les rivières à marée et dans leurs bras.

R É S U M É : La détermination dans une rivière à marée ramifiée, de la réflexion et du contenu d'énergie provoquée par la propagation de l'onde de marée, est importante pour l'étude générale de la variation des niveaux aux différents points présentant de l'intérêt, et des débits dans les différentes sections de la rivière et de ses ramifications. La représentation par la matrice de dispersion des canaux de marées maîtrise facilement la situation, à l'aide de l'expression linéaire des équations différentielles qui gouvernent le phénomène. En fait, la dispersion s'occupe de multiples réflexions qui ont lieu dans ces systèmes compliqués et donne une expression générale des composantes réfléchies, dans chaque port, correspondant à une série de vagues incidentes dans des ports différents.

Dans cette communication la matrice de dispersion d'un canal à marée exponentiel a été dérivée et on a aussi examiné le problème général des canaux a marées avec des ramifications.

O n a donné des exemples de la rivière Hooghly (Inde) en conjonction avec sa rivière tributaire — Rupnarain.

1. INTRODUCTION

T h e relationship between the w a v e amplitude and discharge in a tidal river with or without branches m a y be analysed, as a first approximation based on a linearised mathematical model . Such a model is identical to that used in electrical network analysis, the wave amplitude H in the channel being analogous to the voltage E in the electrical network, and the discharge Q analogous to current / , respectively. While such analyses are not of use for any precise interpretation or for exact design data, nevertheless, for m a n y situations and to obtain a rough quantitative idea of m a n y problems such estimates are of considerable use as a first approximation. Studies of reflection in connection with the siltation at the tidal apex of the tidal channel (Bandyopadhyay and M a z u m d a r , 1968a), are important. T h e calculation of reflection on the assumption of non-linearity is extremely complicated (Schônfeld, 1951); in the case of branched channels it is convenient and feasible to appropriate using linear models.

273

S. Das Gupta and K. K Bandyopadhyay

T o describe the properties of a tidal channel by a linearised model, several formalisms such as the transfer matric formalism (Bandyopadhyay and D a s Cupta, 1969a) and open circuit impedance formalism (mainly in electrical circuits) are in use. For instance, in a transfer matrix formalism, the input and output quantities of H and Q respectively are interrelated. This formalism is relatively convenient when there is branching and when either the input on output quantities are given. If however the boundary conditions at the land-ward end are prescribed in terms of the hydraulic impedance at that end, scatering matrix formalism has a particular advantage. In this formalism, the question of energy transfer from a finite impedance tidal wave generator, with given boundary conditions at the land-ward end is handled, probably, in the best possible manner. In fact, scattering takes care of the multiple reflections which take place in a system and gives a general expression of the reflected components, at each port, corresponding to a given set of incident waves at different ports. Knowledge of reflection characteristics is of considerable importance in a tidal channel in view of corrective engineering works. Scatter matrix formalisation is ideally suited for determination of reflection and its energy content.

This formalism has been well-developed in network analysis (Carlin and Giordano, 1964a), and has frequently been used in micro-wave (Ghose, 1963) and other circuits (King, 1955). It appears to the authors that this concept is presented in tidal channel analysis for the first time in this paper. In section 2, the results of a general formulation (Carlin and Giordano, 1964¿) of the scatter matrix with complex port normalisation numbers has been summarised. Section 3 deals with scatter matrix formulation of a transmission channel having an exponentially tapered width. Section-4 deals with the problem of analysis of a branched tidal channel. Examples are given in section 5.

2. GENERAL FORMULATION OF SCATTER MATRIX

The complex port normalisation number Z¡ for the y'-th port in an «-port system is chosen, retaining the principal properties of a scatter matrix S = [sjk] listed below :

P P O P E R T I E S O F T H E S C A T T E R I N G M A T R I X

(1 ) The [sjk]2 at the operating frequency is the ratio of the power delivered to the complex

impedance at port / i.e. Z¡ to the available power at port k, when all ports are terminated in loads equal to their respective port normalisation numbers, and port k is fed with a generator having an internal impedance ZR.

(2) T h e reflection factor SJJ is related to the ratio of the delivered power P- to the available power PAj by the equation.

iJ- = l-\sjj\2 (2.1)

(3) The dissipation matrix Q = E—SS to be a positive hermitian matrix for a passive network, with S as the complex conjugate transpose of S, i.e. adjoint of S, and E is the identity matrix given by

~1 0 (T

0 0 1

274


(4) S is symmetric for a reciprocal «-port system.

(5) S is unitary for a loss-less «-port system.

The linear transformation then takes the form

2 U * V'rJ

2 Lv> t V> tJ

(2.3)

(2.4)

where:

o t is the incident wave and bk the reflected wave respectively; hk is the actual tidal elevation and qk the actual discharge; Zk the port normalisation number with R e Zk = rk> 0, at the &-th port, and Z*k is the conjugate of Zk.

The scattering matrix S of an «-port system represents a linear transformation matrix transforming a column «-vector incident wave amplitude a, to an «-vector reflected counterpart b.

Thus b = S-a (2.5)

If w e have a system with given open-circuit impedance matrix Z (Weinberg, 1962), then choosing a set of port normalization numbers Z 0 given by:

2o =

"Z( 0 0 0

Z2 0 0

Where

Re Z 0 = R =

0 0 Z3 0 ..

0 0 0 0 . .

>i 0

0 r2

0 0

(2.6)

0

0 (2.7)

Subject to the condition r¡ > 0 for all /'. Then it can be shown (Carlin and Giordano, 1964c) that the scattering matrix is given by

S = E-2Ri{Z0 +Zyl Ri: (2.8)

275

S. Das Gupta and K. K. Bandyopadhyay

3. SCATTER MATRIX FORMULATION OF A N EXPONENTIALLY TAPERED TIDAL CHANNEL

Consider a non-uniform tidal channel shown in figure 1 terminated by a hydraulic impedance ZR. Then the linearised differential equations of continuity and motion take the form:

dQ(x)

ex

8H(x)

dx

= Y(x) • H(x)

Z(x) • Q(x)

(3-1)

(3.2)

TIDAL

INPUT

NON-UNIFORM TIDAL CHANNEL

X=0

FIGURE 1.

Where Z(x) and Y(x) are the hydraulic impedance and admittance per unit length respectively at a point x of the channel; H(x) and Q(x) are wave amplitude and discharge respectively. Taking the origin of x at the landward end of the channel, let

and

Y(x) = Y,

Z(x) = Zy

(3.3)

(3.4)

assuming an exponentially tapered width/where Yx and Zj are the respective admittance and impedance per unit length at x = 0 and is the taper rate. Then the equation for Hy

takes the form:

where

d2H m . n

7 7 + n — - y H = o ox ox

y2 = Y(x)-Z(x)= YXZX

(3.5)

Noting that at x = 0, H = HR, Q = QR, the differential equation may be solved for H(x) which takes the form, after rearrangement and simplification:

H(x) = (coshy*x + -^-smhy*x)e-"xl2HR + ^±sinhy*xe-"xl2 Q'R (3.6) 2y* y*

276


where:

\ + ' '

Using equation (3.2) in equation (3.6) and simplifying w e have, for Q(x)

Q(X) = Z~ 6 ~ 4 ^ ) S i n h ^ £ ^ 2 H » + (Cosh?** ~ Y~i Sinhy*x\ + <*I2Q'R

(3.7)

In matrix terms, the value of Hs = H (L) and Qs = Q(L) at x = L.

H, ( Coshy*L + - ? - Sinhy*Ly *L / 2 ^ Sinhy*Le

V 2y J y*

*JC-»LI2

£ Y l - - ^ ) S i n h y *LE + " L / 2 (coshy*L - -!L Sinhy*L\£+"L/2 QL

(3

Thus in the transfer matrix formalism of relating the output quantities H R , Q"R with the input quantities H s , Q s (Bandyopadhyay and D a s Gupta, 19686) as follows

A

C

B

D

HR

LQ'R.

(3.9)

W e have, comparing equations (3.8) with (3.9)

A = ( Cosh/L + -5j Sinhy*LV nL/2

B = ^ S i n h y * L e - " L / 2

y

* / 2

Z , V 4y 1 - - ^ — i S i n h y ' L e "

,f . . * 2 /

D = /coshy*L - - ! L Sinhy*LyL/2

(3.10)

It is to be noted that AD — BC = 1, which is true in general for reciprocal systemsf.

"(•. Although, the linearised version of the schematised channel is reciprocal, the actual system is not.

277

S. Das Gupta and K . K . Bandyopadhyay

Rewritting equation (3.9) as a relation between the w a v e aplitude vector H T = [HSHR] and discharge vector Q T = [ G s ô R ] t , with the help of a transformation matrix Z of hydraulic impedance , w e have

H =

H.

HR

A

C

i

c

l~

c

D

cJ

—

V,

QR

= ZQ (3.11)

W h e r e , Q ' R = Q R i.e. Q R is the discharge entering the port, k n o w n as the receiving port. T h u s negative values of Q R will indicate actual discharge at the receiving end port.

T h e impedance matrix Z is given by

Z =

A C

1 - C

1 c

to

C

(3.12)

i.e.

2 = ^il_£-^/2

4

(cothy*L + - ^ V " L / 2 Cosech/L

Cosech y* L Cothy*L--^)£ ' ,li./2

(3.13)

It is to be noted that

A Z = IZI = zW"1 (3.14)

The image impedances (Kerchner and Corcoran, 1960a) Z 0 v and Z0R at the receiving and sending end are obtained by calculating the geometric mean of the open circuit and short circuit impedances at the sending and receiving end.

Thus considering equation

1 / C ( D / C - 1 / C )

DIC }- AB

DC

ZXE -i,L

y | C o s h y * L - - ^ S i n h r * L

Sinh2y*L + - 1 / 2

1 -If I -

(3.15)

\. T en the superscript stands for transpose of a vector.

278


The image impedance Z0R may be computed in a similar manner, and is given below:

Z - I™ ZoR ~ SAC

y* Í C o s h y * L + - ^ ; Sinn y * L

SinhVL + 1 - n

2y*J J

1/2

(3.16)

Then

Z 0 =

"Z„. 0

o zn

(3.17)

If the values of Z0, and Z as per equation (3.17) and (3.13) respectively are substituted in equation (2.8) the scatter matrix is obtained.

4. SCATTER-MATR1X FORMULATION OF A BRANCHED TIDAL CHANNEL

The wave amplitude and discharge data at different gauge points (stations) may be used to compute the scatter matrix for a branched tidal channel. Let us consider a branched tidal channel as shown in figure 2. Here S is the seaward end, R2 is the landward end of the branch 0R2 and R3 the landward end of the branch 0R3. Sj is an intermediate gauge station on the main channel OS (before branching). It is required that we find the scatter matrix for the 3-port arrangements shown in the dotted box with the port numbers

FIGURE 2.

I, 2 and 3 corresponding to Sly R2 and R3 respectively. It has been assumed that the transfer matrix parameters are available for smaller sub-sections of SS¡, St0, 0R3, The transfer matrix of the overall section between S and S¡, S , and 0, 0 and R2 also between 0 and R3, may then be computed (Bandyopadhyay and Das Gupta, 1969a). Methods for obtaining the transfer matrix of a single sub-section between two subsequent gauge stations from wave amplitude and discharge data (Bandyopadhyay and Das Gupta, 1969a) or from schematized scaled models are available (Bandyopadhyay and Mazumdar, 19696).

279


W h e n a channel has roughly an exponential taper, it is, as will be evident from section 3 of this paper, not necessary to synthesise the transfer matrices of the sub-sections into a single transfer matrix. The values of the taper rate r\, the apparent propagation constant 7*, the impedance rate Z t at some point on the branch, and the length of the line L, are required for obtaining the transfer matrix.

Once the transfer matrices are available, the impedance matrix Z is obtained as per equation (3.12).

The tidal input at St m a y be considered generated from an equivalent Thevenin generator having H T h as the wave amplitude and an internal hydraulic impedance ZTh. These m a y be determined as shown in Appendix-I.

For well-known reasons, iSe port normalising number at S1 has been assumed as

•¿V/i-The tidal channel illustrated in figure-3, then can be schematically shown as:

FIGURE 3.

For convenience, the port quantities have been relabelled in this figure. Thus Sl has a wave amplitude Hi and a discharge Q\ into the tidal channel, while at its junction point end, the wave amplitude is feeding it at a value H \ and discharge Q\. The impedance matrix involved is Z 1 . Thus the governing equation for this section is:

7/

H\

7 1 7 1 "~' ¿11 ¿12

_Z2i ¿22_IL(¿2_

"21" (4.1)

as these are linearised equations, the system is reciprocal, hence Z 2 1 = Zxl in general Similarly for branches 0/J2 and OR3 are respectively.

and

"HÏ -

_«L

-HT

_«L

V71 ¿il

72

F73 ¿ u

73

—¿21

7 2 "1 ¿12

7 2

¿22_

7 3 1 ¿12

7 3

¿22_

-QT

_ôL

-QT

_GL

(4.2)

(4.3)

280


Also at the junction point,

H\ = H] = H\ = H (Say)

and

Ql + Ql + Ql = o Equations (4.1), (4.2) and (4.3) m a y be rewritten in the form:

(4.4)

(4.5)

\-H\-

Hl Hi

H\

Hi

_*L

=

0

0

71

i,2\ 0

_0

0

72

Z , 2 2

0

0

72

•¿21

0

0

0

73

0

0

7 3

•¿21

7 n z,1 2

0

0

7 1

•¿22

0

0

0

72

¿12 0

0

z2

•¿11

0

0 ~

0

73

•¿12

0

0

7 3

• ¿ 1 1 —

reí e2 el

e2

Ql Lôï

(4.6)

The matrix equation may be partitioned as above and as we have essentially three more relations in equations (4.4) and (4.5), the unknown Q\, Q\ and Q\ m a y be eliminated, thus resulting in a matrix relation between the set of wave amplitude column vector, consisting of H\, H2, 7 / | and the discharge vector consisting of Q\, Q\ and Q\ as given below (assuming Z , 2 = Z2 ¡ in general).

H\

Hi

m

Where,

"Z1

¿ n y1 712 + J 22¿12 '

+ Yl2712

1 22-¿12

Y2 V 1 7 1 7 2

I H I 22-¿12-¿12

y2 y1 7 1 7 2

J 11 J 22z"I2z'l;

7 2 _ Y2 72* Z-22 J 1 1 ¿"12 '

y3 yl 7I 7 3 7 1 1 ' 22¿12¿i:

+ y2 72 2

1 11^12

y3 y1 7 1 7 3

i u / 22¿12¿12 y2 y3 72 7 3

x 11 * I 1 ^12^12

y2 V 3 7 2 7 3

1 1 I ' 1 1 •£'12'^12

YUZH +

Vil 7 3 2 1 1 1 ^ 1 2

GÍ

e

Ci

(4.7)

Y=Yln+ Y2ti + y3,

and in general Y = 0 If w e get

tfr = [Hj Hi Hi]

QT = lQ\ Ql Ql]

281


and

z =

Zl1 '22^12 +

y 1 2 7 1 2

1 22^12

y2 y1 71 7 2

y2 y1 71 7 2

7 u ' 22^12^1;

z2

•^22 ^ . 1 ^ 2 +

y2 2 7 2 2

y3 y1 71 73 y2 y3 72 73

'11 ' 2 2 ¿ 1 2 A 1 2 / 1 1 ' 11 z ' 1 2 z ' 1 2

y3 y1 71 73

' 11 * 22^12^12

y2 y3 72 73

1 11 * 11 ^ 1 2 ^ 1 2

Z 2 2 ' 1 1 Z 1 2 +

1 11^12

We have H = ZQ

Then as shown in section 3, choosing the port normalising numbers

2o =

"Z 0 1 0 0

0 z02 0

_o 0 z0 3 .

The scatter matrix may be obtained as per equations (2.8)

S = E-2R*(Z 0+Z)~ 1R i

(4.9)

(4.10)

(4.11)

5. EXAMPLES

The River Hooghly (India), which is the tidal waterway for the Port of Calcutta, has been used as an example, from already published data of 4.5.62. This tidal channel has an important tributary joining it at a distance of about 45 national miles from the sea.

For the purpose of scatter matrix computation, this complex channel has been divided into the following sectors as shown in figure 4.

(a) Sector-0 is the portion between the sea face and Diamond Harbour. This is roughly an exponential channel (Bandyopadhyay, Mazumdar and Das Gupta, 1968c) and is being replaced by an equivalent tidal wave (Thevenin) generator in conjunction with an equivalent hydraulic impedance.

(¿>) Sectors 1, 2 and 3 of the system constitute the tidal channel network.

Sector 1—stretches between Hooghly Point and Diamond Harbour (8 r . m ) . Sector 2—stretches between Hooghly Point and Dainan (along the main tributary)

(32 n . m ) . Sector 3—stretches between Hooghly Point and Bansbaria (along the main channel)

(66 n . m ) .

282


BANSBARIA

CALCUTTA

FIGURE 4.

It m a y be mentioned that Sector 3 also has an exponential taper (Bandyopadhyay, Mazumdar and Das Gupta, 1968c). Sectors 1 and 2 have been considered from usual four terminal network concepts (general ABCD constants). The data are as follows:

Sector-0

Zj (at diamond Harbour)

n

L

Sector-2

A = 0.7442/21.6» B = 248.08/52.5° C = 4.3105x10" D = 0.682/59.4»

3/99.6°

= 0.292/39.3°ohms = 0.05/n.m. = 0.0357/53° = 38.5 n . m .

Sector-3

Zl (at Bransbaria)

n

L

Sector-1

A = 0.978/1.2° B = 30.9/47.4° C = 1.9278 x l 0 - 3 / - 8 9 . 5 ' D = 0.978/1.2°

= 6.34/18.7°ohms. = 0.0225/n.m. = 0.042/60.4» = 65 n . m .

283:


6. RESULTS

(a) The Thevenin equivalent tide generator and the series equivalent hydraulic impedance may be computed from the relationships 3.10 and (1-3) and (1-4).

ZTh = 5.807/20.3° o h m .

(¿») The impedances Zfi, for the K-Xh Sector (K = 1, 2, 3) as detailed in equations may be obtained using equations (3.11) and (4.1) and the data of respective sectors as

For K = 1

For K = 2

Z K =

-7K 7K

2 2 I

z1 =

z2 =

"507.31 / -89 .3° 518.72 / -90 .5°

518.72 / - 9 0 . 5 o 507.31 / -89 .3°

"172.65/ -78 .0 o 231.99 / - 9 9 . 6 o "

_ 2 3 1 . 9 9 / - 9 9 . 6 o 158.22 / - 4 0 . 2 o

For K = 3

Z3 =

"42 .485 / -50 .4 o 46 .96/ -156 .3 o "

.46.96 / -156.3° 120.22 / -18 .3° .

ZTh for Z 0 1 , Z0R for Z 0 2 and Z0R for Z03 (the impedances looking from the receiving end at Dainan (Sector-2) and at Bansbaria (Sector-3) has been chosen as the normalising impedances.

Z 0 1 = 5.807 /20.3° o h m s

Using equation (3.16)

and

Hence

Z 0 2 (Sector - 2) = 250.4 / -6 .65° ohms .

Zo =

Z 0 3 (Sector -

5.807 /20.3°

0

0

-3) = 98.6/-27.3°

0

250.4/-6.65 o

0

ohms.

0

0

98.6 /-27.3°_

From equations (4.9) and (4.11).

284


T h e scatter matrix has been computed from the relationship:

S = E-2Ri(Z + Z0)~1Ri

1.43 / - 1 6 2 . 2 o - 1 2 . 3 9 / - 9 2 . 0 o - 1 4 . 9 0 / + 5 5 . 5 o "

- 1 2 . 3 9 / - 9 2 . 0 o 67.56 / + 1 5 6 . 0 ° - 1 3 . 4 0 / - 9 0 . 8 °

_ - 1 4 . 9 0 / + 55.5 0 - 1 3 . 4 0 / - 9 0 . 8 o 76.16 / - 1 5 0 . 0 ° .

7. DISCUSSION

(o) T h e mathematical tool used here is already in existence in electrical networks.

(b) A m o n g the other formalisms of representing the performances of tidal channel or electrical networks, the scatter matrix formalism is particularly suitable, under k n o w n boundary conditions in an «-port channel or network system.

(c) T h e nonlinear aspects of the system leave further scope of analysis.

8. CONCLUSION

A generalised treatment of reflection and multiple reflection has been exposed with the help of scatter matrix formalism.

For the first order approximation of the estimates of reflection, the method is adequate. T h e scatter matrix lends itself a physical interpretation of the tidal channel reflection

and helps to solve the tidal amplitude and discharge problems for k n o w n boundary conditions.

9. A C K N O W L E D G E M E N T

T h e authors are indebted to D r . S . K . Bhattacharya, Chief Hydraulic Engineer, Port Commissioners, Calcutta and Prof. H . C . G u h a , Vice-Chancellor, Jadavpur University for their help.

REFERENCES

1. B A N D Y O P A D H Y A Y , K . K . and M A Z U M D A R , N . C . (1968a): "Studyof Reflection Coefficient of a Tidal Channel under different boundary conditions", XIHth Congress of Theoretical and Applied Mechanics (Durgapur-lndia).

2. S C H O N F E L D , J .C . (1951): "Propagation of Tides and Similar Waves" , Author's Thesis, Delft (Netherlands).

3. B A N D Y O P A D H Y A Y , K . K . and D A S G U P T A , S. (1969a): "Identification of the parameters of a Tidal Channel from simulation and other approaches", IVth Congress of the Internationa Federation of Automatic Control, Warsaw (Poland).

4. C A R L I N , H . J . and G I O R D A N O , A . D . (1964a, b, c): "Network Theory", Englewood Cliffs, N . J . , Prentice Hall Inc., pp. 222-230, pp. 326-327 and p. 328.

5. G H O S E , R . N . (1963): "Microwave Circuit Theory and Analysis", N e w York, McGraw-Hill Book C o . Inc., pp. 201-211.

285


6. K I N G , R . W . P . (1955): " Transmission Line Theory", N e w York, McGraw-Hill Book C o . Inc., pp. 304-314.

7. W E I N B E R G , L . (1962): "Network Analysis and Synthesis", N e w York, McGraw-Hill Book C o . Inc., p. 104.

8. B A N D Y O P A D H Y A Y , K . K . and D A S G U P T A , S. (1968n): "Analysis of a Tidal Channel with the help of Four Terminal Network concept", Journal of the Institution of Engineers (India), vol. 49, N o . 2 (Part EL-1) , p. 4.

9. K E R C H N E R , M . R . and C O R C O R A N , G . F . (1960a): "Alternating Current Circuits", (4th edition). N e w York, John Wiley & Sons Inc., p. 440.

10. B A N D Y O P A D H Y A Y , K . K . and M A Z U M D A R , N . C . (1969b): "Scaling of Tidal Channel Parameters for the construction of Electronic Analogue Model", International Journal of Electronics (London). (Accepted for publication in March 1969).

11. K E R C H N E R , M . R . and C O R C O R A N , G . F . (1960b): " Alternating Current Circuits", (4th edition). N e w York, John Wiley & Sons Inc., p. 197.

12. B A N D Y O P A D H Y A Y , K . K . , M A Z U M D A R , N . C . and D A S G U P T A , S. (1968c): " A method of estimation of Chezy's coefficient in a Tidal River", Journal of the Institution of Engineers (India), vol. 48, N o . 6 (Part EL-3), p. 406.

APPEND1X-I

Determination of the Thevenin Equivalent Generator at any point in a tidal channel

Considering a tidal wave with an input wave amplitude Hs at the seaward end as shown in the dotted box (fig. 5). If Hs and Qs are the wave amplitude and discharge respectively into the channel at the points S. Let Hs

l and Qsl be the same quantities respectively

at the points S1 with, of course, Qsx considered positive for a discharge out of S1. Then

w e have the following relations:

Hs = AH\ + BQl (1.1)

Q, = CHl + DQl (1.2)

A B C D

ÏV*-*** s ,

I I

J F I G U R E 5.

By definition (Kerchner and Corcoran, 19606), the Thevenin Generator consists of a wave amplitude HTh equal to that at the point S 1 with no output discharge i.e. with Qs

l = 0 with a hydraulic impedance ZTh in series. This is the ratio of H^ and Q* at Sl

when the end S is short circuited (in the electrical sense; in the hydraulic sense, this means opening the end of the channel) i.e. Hs = 0 in equation (1-1). Thus clearly as per definition, we set Q s 1 = 0 and Hs

l = HTh in equation (1-1) for determining HTh.

Hn = -H, (1.3)

286

Données sur les caractéristiques des courants fluviaux

and for determining ZTh, w e set Hs = 0 and noting that the direction of discharge will

be reversed, w e have from equation (1-1)

Données sur les caractéristiques des courants fluviaux de surface à l'embouchure du canal de Sulina

C . Bondar , V . Rovenja, I. State

A B S T R A C T : Intensive researches effected by surface floats (1 m draught) launched from the Sulina mouth have permitted to study the trajectories of the river currents and the limits of the river waters spreading into the sea.

The surface currents have been observed to disperse in two distinct forms, function of the hydrological Danube river regime and of the sea water penetration:

— the river jet shape, when the sea waters do not penetrate into the channel mouth ; — the fan shape, when the sea waters penetrate into the river bed.

The river water jet diverges into the sea at angles of 7 to 9° up to the zone where the river jet masses lose touch with the bottom.

The river water fan diverges in angles larger than 30°, the limits and the plane development of the fan being invariable with respect to the liquid discharges.

R É S U M É : Les nombreuses recherches effectuées à l'aide de flotteurs de surface (tirant d'eau de 1 m ) lancés depuis l'embouchure du canal de Sulina ont permis de déterminer le trajet des courants et les limites de dispersion dans la mer des masses fluviales.

E n fonction du régime hydrologique du Danube et de la pénétration des eaux marines dans le lit fluvial, on distingue deux situations de spectres des courants de surface :

— spectre du jet fluvial quand les eaux marines ne pénètrent pas dans le lit fluvial à l'embouchure ; — spectre en éventail quand les eaux marines pénètrent dans le lit fluvial.

La divergence du jet fluvial dans la mer se produit suivant des angles de 7 à 9° jusque dans la zone où les masses d'eau se détachent du fond marin.

La divergence de l'éventail fluvial se produit suivant des angles de plus de 30°, et les limites et le développement en plan restent invariables par rapport aux débits liquides.

1. GENERALITES

Les courants fluviaux à l'embouchure d u canal de Sulina peuvent être considérés c o m m e un prolongement de l'écoulement des eaux fluviales dans les eaux marines.

D u point de vue hydraulique cet écoulement a la forme d 'un jet liquide noyé à niveau libre, limité latéralement par les eaux de la m e r qui constituent de vraies parois liquides.

À rencontre des jets liquides habituels, les jets fluviaux qui se dispersent dans les mers et océans sont noyés dans une masse d 'eau de grande densité. À ce point de vue le contact

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on studies of reflections in tidal rivers and their...

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