VARIATIONAL APPROACH TO THE DETERMINATION
OF MEANDERING FLOWS

M.S. Yalin,  A.M. Ferreira and Da Silva

Department of Civil Engineering, Queen’s University

Kingston, Ontario, Canada K7L 3N6

E-mail: amsilva@civil.queensu.ca

Abstract: This paper concerns the determination of the vertically-averaged flow in a sine-generated meandering channel; the geometry of the bed surface is specified. In contrast to previous theoretical works, the local resistance factor of the meandering flow is treated as an unknown function. Accordingly, it is suggested that in addition to the two (longitudinal and radial) equations of motion and the equation of continuity, a fourth equation is needed to reveal the vertically-averaged meandering flow. In this paper, a fourth (variational) equation is derived based on the assumption that the free surface is shaped by nature in such a way as to render its variation (in any direction) to occur in the “smoothest” possible manner. Finally, a comparison between results of the present variational equation and experimental measurements is provided.

  Keywords: sine-generated channels, vertically-averaged flow, variational approach

1  INTRODUCTION

In agreement with the conditions prevailing in natural rivers, the present considerations are confined to rough turbulent flows, “large” values of the width-to-depth ratio ( , say), and small values of the Fr-number ( ). It is assumed that the flow rate Q is constant; the meandering channel centreline is a sine-generated curve [1], [6].

A meandering flow is determined by solving the vertically-averaged (longitudinal and radial) equations of motion and continuity which, in the channel-fitted coordinates  and n (see Fig. 1a), can be expressed as follows

           (1)

               (2)

                         (3)

(see List of Symbols). The three unknown functions , ,  can be computed from the system of Eqs. (1) to (3) only if the resistance factor , where , is known. In most of the previous theoretical works ([2], [4], [5], etc.) the resistance factor  of a meandering flow was invariably identified with the (known) friction factor of a straight uniform flow, i.e. . In fact, however, since the characteristics  and  vary from one location  in the flow plan to another, it is only natural to expect their ratio, viz , also to vary depending on location. But this means that  is not a known quantity. Rather, it is just another unknown function which (in analogy to ,  and ) must be computed.

Fig. 1

It follows that we have actually not three but four unknown functions, viz , ,  and , and thus that four equations are needed to reveal a vertically averaged sine-generated meandering flow past the bed of any given geometry. In this paper an attempt is made to reveal the additional (to Eqs. (1) to (3)) fourth equation.

The bed surface is assumed to be the deformed surface of an alluvial meandering channel at any time t: the channel banks are rigid, the initial movable bed at  is flat, equilibrium is achieved at  (see Fig. 1b). The specified geometry of the bed surface is determined by the (known) function ; alternatively the function  may be used. It is further assumed that at any cross-section and at any t, the cross-sectional average value of , viz  , is equal to  (i.e. the areas of the cross-sectional erosion and deposition are treated as identical (at any cross-section and at any t)).

2  VARIATIONAL EQUATION

[1] No matter what  might be, the elevation  of the free surface always increases when the outer bank is approached. The rate of this increment is the largest at the flow sections around the apex , and it is approximately zero at the crossovers  and  (i.e.  (for any )). The increment of  with r is an inevitable consequence of the channel curvature, and the only query on the score is how such an increment takes place. It would be reasonable to postulate that the free surface is shaped by nature in such a way as to render its variations (in every direction) to occur in the “smoothest” possible manner; for only in this case the pressure within the fluid will vary from one location to another in the least perceptible form and fluid will flow most “comfortably”. In mathematical terms this would mean that the free surface should be such, that the integral of its -values, or rather of its -values, over the plan area of a meander loop, should be the smallest possible:

                                 (4)

(The involvement of the longitudinal free surface slope , in addition to , is redundant, for  and  are interrelated ( )).

Fig. 2

The differential  of the plan area  (see Fig. 2) is given by

                  (5)

Since the channel is sine-generated, B/R  is  a known function of x_c (see e.g. [6]), viz

  (= )                      (6)

Using the notations

,                           (7)

we determine

                           (8)

while the consideration of (5), (6) and (8) makes it possible to express (4) as

              (9)

Since  appears only in the known function , the minimization above implies

                      (10)

In this expression, which is assumed to be valid for all -sections, the integrand F (= ) is to be treated as a function of and  only. Substituting this F into the Euler equation

      i.e.                (11)

one obtains

                             (12)

i.e.

        or                           (13)

which is the additional (fourth) equation sought.

Integrating (12), we determine

                        (14)

where  reflects the variation of  along the centreline  (where ):  Integrating now (14) we obtain

                            (15)

where reflects the variation of  along : .

3  DETERMINATION OF C1 AND C2

The cross-sectional average value of can be expressed in the (well known) form

   i.e.                        (16)

where  is close to unity. On the other hand,  can be determined by averaging the expression (14) of  along the flow width :

                  (17)

where is always smaller than unity, and the logarithmic multiplier in (17) is thus always positive.

Equating the two expressions of  above, we determine

                     (18)

where

   ;      ;                  (19)

The substitution of (18) in (15) gives

             (20)

Since  the expression (20) of  can be converted into the expression of , viz

                      (21)

where

  ;                      (22)

The term  reflects the variation of along the centreline .

Averaging the expression (21) of over the flow width, we obtain

                    (23)

But  where  varies (in a known manner) with , and  with  and . Hence

                      (24)

Substituting (24) in (23) and subtracting the resulting equation from (21), we obtain

             (25)

where

     i.e                           (26)

Note that if (as mentioned in the Introduction) the cross-sectional and depositional areas are equal, then . Note also that if the bed is flat, then .

4  COMPARISON WITH EXPERIMENT

Fig. 3 shows how the -values computed from the relation (25) compare with the measured values of  reported in Silva 1995 [3]. The measurements were carried out in a sine-generated laboratory channel having  (sinuosity ). The flow cross-section was rectangular, the flow width was B=0.40m; the meander wavelength was m. The channel walls were made of plexiglass; the (rigid) channel bed was formed by a cohesionless granular material having D₅₀=2.2mm. The following experimental conditions were used: flow rate l/s, channel average flow depth h_av=3.0cm, centreline bed slope . The measurement region of the channel consisted of three consecutive meander loops (henceforward referred to as , , ): in each loop measurements were carried out in several r-points of eight equally spaced cross-sections ( -sections). In Fig. 3 only the data measured at the crossover- and apex-sections are plotted. The solid lines in this Figure are the computed distributions of across the channel width for the aforementioned cross-sections: the computations were carried out using  (flat bed) and . The agreement of the computed lines in Fig. 3 with the data is favourable.

 Fig. 3

5  FINAL REMARKS

The four unknown functions , ,  and  of a vertically-averaged sine-generated meandering flow having a specified geometry of the bed surface can be revealed by solving numerically the system of equations formed by Eqs. (1) to (3) and the present variational equation (Eq. (25)). Noting that for a given bed geometry (i.e. for given ), Eq. (25) is a condition which must be satisfied only by the unknown function , the  aforementioned system can be shown symbolically as

        (Eq. Motion (1) along )                        (27)

        (Eq. Motion (2) along )                        (28)

              (Eq. Continuity (3))                           (29)

                    (Variational Eq. (25))                        (30)

The present research efforts of the authors are directed towards the simultaneous solution of the system of equations above.

References

[1]    Leopold, L.B., Langbein, W.B. (1966): River meanders. Sci. Am., 214.

[2]    Shimizu, Y. (1991): A study on the prediction of flows and bed deformation in alluvial streams. (In Japanese) Civil Engrg. Research Inst. Rept., Hokkaido Development Bureau, Sapporo, Japan.

[3]    Silva, A.M.F. (1995): Turbulent flow in sine-generated meandering channels. Ph.D. Thesis, Department of Civil Engineering, Queen’s University, Kingston, Canada.

[4]    Smith, J.D., McLean, S.R. (1984): A model for flow in meandering streams. Water Resources Research, Vol. 20, No. 9.

[5]   Struiksma, N., Crosato, A. (1989): Analysis of a 2-D bed topography model for rivers. in “River Meandering”, S. Ikeda and G. Parker eds., American Geophysical Union, Water Resources Monograph, 12.

[6]    Yalin, M.S., Silva, A.M.F.: Fluvial processes. IAHR Monograph, The Netherlands (In Press).

List of relevant symbols