Analysis of paradoxes arising from the Chezy formula with … · 2015-07-29 · roughness...

IJydrological Sciences -Journal- des Sciences Hjdrologiques,41(5) October 1996 659

Analysis of paradoxes arising from the Chezy formula with constant roughness: I. Depth-discharge curve

WITOLD G. STRUPCZEWSKI Institute of Geophysics, Polish Academy of Sciences, Ks. Janusza 64, 01-452 Warsaw, Poland

ROMUALD SZYMKIEWICZ Technical University of Gdansk, Majakowskiego 11/12, 80-952 Gdansk, Poland

Abstract The Chezy friction formula for steady flow in a uniform symmetrical channel with constant slope-friction factor is examined mathematically. Firstly, a wide rectangular channel and a semi-circular channel are compared in respect of the mean flow velocity using the Chezy formula with the Manning, Chezy and logarithmic laws for velocity. Then the inverse Chezy problem, i.e. the determination of the channel shape above the reference level for a given depth/discharge rating curve, is posed and the differential-integral equation for its solution is derived. The rating curves used for computation are the results of multiplying the discharge for a trapezoidal shape above the reference level by an exponential function. To facilitate interpretation of the numerical results, the relationship between side slope and discharge is analysed. It is shown by the inverse problem solution that an exponential reduction of channel flow capacity changes linear channel sides into convex sides (making the cross section shape wider) while an exponential increase of capacity causes changes into concave sides (reducing a section width) which is against common sense.

Analyse des paradoxes résultant de la formule de Chezy avec rugosité constante: I. Courbe hauteur-débit Résumé La formule de Chézy pour l'écoulement permanent dans un canal symétrique dont le facteur pente-frottement est constant a été analysée mathématiquement. On a tout d'abord comparé un canal rectangulaire et un canal semi-circulaire eu égard à la vitesse moyenne de l'écoulement à l'aide de la formule de Chézy avec les lois de Manning, de Chézy et logarithmique pour la vitesse. On a ensuite posé le problème de Chézy inverse, c'est à dire celui de la détermination de la section transversale du canal au dessus d'un niveau initial, pour une courbe hauteur-débit donnée. Une équation différentielle-intégrale a été proposée pour résoudre ce problème. Les courbes utilisées dans les calculs ont été déterminées en multipliant le débit pour une section trapézoïdale au dessus du niveau de débit nul par une fonction exponentielle. Pour faciliter l'interprétation des résultats numériques on a analysé la relation entre la pente et le débit. On a montré, à l'aide de la solution du problème inverse, que la réduction de l'écoulement rend la section plus large, tandis que son augmentation implique la réduction du canal, ce qui contredit le sens commun.

Open for discussion until I April 1997

660 W. G. Strupczewski & R. Szymkiewicz

INTRODUCTION

The approximate character of Chezy's equation for uniform flow in an open channel is widely acknowledged, as can be seen from the numerous formulae for the Chezy coefficient quoted in the literature (Leliavsky, 1959; Chow, 1959). Among them the Manning formula has become the most widely used for open channel turbulent flow computations. Jarrett (1994) stated: "Given the extensive work in theoretical fluid mechanics during this century, it is remarkable that the Manning formula has not been superseded by a theoretically-based formula developed from modern fluid mechanics". However, as shown in numerous studies, the assumption of a single value of Manning's roughness coefficient, n, rarely holds in practice. The blame for the high variability of n with flow depth is placed mainly on the nonuniformity of both channel shape and surface roughness along the wetted perimeter of the cross section. The other reason may be that the distribution of shear stress on the channel bed is not, in general, uniform as assumed in Chezy's equation, making the equation insufficiently accurate for natural shapes.

Since in reality the ideal channel of uniform shape and roughness does not exist, the potential influence of its shape on the roughness coefficient may be overshadowed by all kinds of irregularities of natural channels. Chow (1959) named several factors affecting the variability of n stating that "there is no evidence about the size and shape of a channel as important factors affecting the value of n". A relationship of n to the hydraulic radius (or depth) and the friction slope or the bed material size has been frequently investigated (Limerinos, 1970; Jarrett, 1992). According to Jarrett (1985) the basic n value for a uniform channel does not vary with depth of flow if the ratio of the mean depth to the median diameter of bed-material particles is greater than five and less than 276. The channel shape is not considered directly in the evaluation of roughness but indirectly, to a certain extent, by means of a cross section irregularities adjustment factor (Aldridge & Garrett, 1973; Jarrett, 1985).

In fact, a significant dependence of n on shape would act as a strong argument against unlimited use of the Chezy equation with Manning roughness, showing either that its two shape parameters describe the section shape unsatisfactorily from the hydraulic point of view or that the form of the equation is not sufficiently accurate for practical purposes. Leliavsky (1959) has reviewed some experimental findings concerning the universal use of the hydraulic radius (R) in discharge formulae, concluding that "there are still many reasons for believing that the estimate of the hydraulic efficiency of a canal profile may possibly be derived in future from a more sound basis than the hydraulic radius". Yen (1992) indicated that the longevity of the century-old Manning formula is a mixed blessing. He questioned if the formula is fundamentally sound and practical or if it merely reflects a lack of progress.

Since steady state flow in an open channel has been quite well

Paradoxes arising from the Chezy formula — depth-discharge curve 661

recognized during the many years of irrigation practice, it is possible to compare the Chezy formula with practical common sense gained by experience of life. Therefore, an attempt has been made in this paper to investigate whether the formula is rationally sound without going into underlying assumptions and without using any empirical data. However, to start with, one may try to deduce from a rational basis using any monomial formula: what is the relationship of a roughness coefficient to a channel shape. Next, the only tool used here will be a mathematical analysis of the Chezy function. Its result, i.e. Chezy function properties, will be compared with common sense.

It will be considered as axiomatic that the flow capacity of a channel increases with the channel size. The solution of the Chezy inverse problem will be used to check whether the Chezy equation satisfies that axiom. Standard applications of the Chezy equation with Manning friction are either to determine the depth of flow for a specified flow rate and a channel shape, or to determine the discharge for a specified depth of flow and channel shape taking into account the resistance to flow in the channel and the friction slope. Therefore, the determination of a channel shape for a given depth (or area) discharge relationship, given both the roughness coefficient and the bed slope, can be considered as an inverse problem to that usually solved.

There are two variables used in the Chezy formula to characterize the cross section shape, viz. the cross section area (̂ 4) and the wetted perimeter (P). To find the unique shape for the flow area-depth and the wetted perimeter-depth relationships of a given section, the assumption of a symmetrical section is necessary except for those particular cases in which the relationships describe elliptical or equilateral trapezoidal shapes. The reason for this is to avoid an infinite number of solutions of the inverse problem caused by insensibility of the lumped cross section parameters on the left-hand or right-hand side of any cross section segment.

The study herein is a continuation of previous work (Strupczewski & Szymkiewicz, 1989; Strupczewski, 1996). Since the inverse solution concerns a shape above a reference level, geometrically feasible reference conditions are defined (Strupczewski & Szymkiewicz, 1989) and the differential-integral equation of the inverse problem is derived for a depth-discharge rating curve for a channel with constant side slopes above the reference level. Strupczewski (1996) using various steps in a numerical solution showed that the Chezy formula does not meet the above mentioned axiom. The algorithm of the solution has subsequently been considerably developed and improved, creating many more opportunities for analysing the Chezy equation. Impressions so far are that while practising hydrologists show a high interest in the problem, some scientists are rather less interested and consider that its time has passed. Since about the mid-1960s, there has been limited hydraulic research, primarily because funding has been diverted to other needed water resources and environmental studies. In fact, the idea for this study came directly from a practical problem while the senior author was working in Liberia and facing a shortage of flow measurement data (Strupczewski & Sua, 1983).


UNIFORM FLOW FORMULA

For practical purposes, a turbulent uniform open channel flow formula is usually expressed as:

Q = KSm = aARr-lSm (D

where a denotes a resistance factor, Q and A are the normal flow discharge and flow area respectively, R is the hydraulic radius and S the bottom slope. The term K is known as the conveyance of the channel section, being a measure of the carrying capacity of the section, and was introduced by Bakhmeteff (1932). The related dimensionless factor K' was extensively tabulated by King (1954) and by the US Corps of Engineers (1959).

Since both the surface roughness of the channel boundary and the water surface slope are assumed here to be constant, it is convenient to express the Chezy equation as the product of the slope-friction factor (SF) and the geometrical factor (G):

Q =SFxG ( la)

where SF = aSm is constant under the above assumption, while

G = ArIPr~l (2)

where P denotes the wetted perimeter. The exponent r equals 3/2 and 5/3 for Chezy and Manning friction, respectively.

Since the inverse solution of the Chezy formula raises the question of a relationship between the roughness coefficient and the section shape, it is worth looking first at what can be deduced about the relationship from the rational basis of any monomial formula for uniform flow in an open channel. Four aspects are relevant:

(a) for uniform flow there is a balance between boundary resistance and gravity, so that:

T0P = y AS (3)

where r0 is the average boundary shear stress, y the specific gravity of water and S the slope (i.e. the sine of the angle between the direction of flow and the horizontal);

(b) in the case of rough turbulent flow in a uniform channel of a regular shape, the simplest assumption for the velocity v at a given distance, y, from the rough boundary is to express it as a ratio to the so-called shear velocity defined as:

v* = fji or, substituting the value of r0 from equation (3)

V* = yJgRS

(4)

(4a)

Paradoxes arising from the Chezy formula - depth-discharge curve 663

where p and g are the fluid density and the gravitational acceleration, respectively. The ratio may then be expressed as a function of the relative distance from the boundary, i.e.:

y_

y0

(5)

where y0 is a representative roughness height. In the case of the Chezy and Manning formulae, this universal

velocity relationship is written as:

y_

y0

(5a)

with/? = 0 (i.e. v constant which in fact unreal) for Chezy and/? = 1/6 for Manning. It is to be noted that from theoretical concepts of open channel flow mechanics equation (5) takes the form of the Prandtl-von Kârmân universal velocity distribution law:

1 In y_ ya

(5b)

where k is von Kârmân's constant (~0.4);

in the two-dimensional case of a wide rectangular channel or the axi-symmetrical case of a semi-circular channel flowing full, the boundary shear will be the same at all points on the wetted perimeter and the length of the normals from the boundary to the surface will all have the same value, ymax. Under these conditions, the velocity distribution formula of equation (5a) can be integrated to give the mean velocity, V, in a wide rectangular section:

V =

vdA av*B yPdy av*Bh{P+l) (6)

A Ay/ V ' (p + DAyf

where B is the width of the channel, while in the semi-circular section:

V =

véA

irav

^y0f

yP+ldy = irav h(p+2) (7)

(p+2)Ay/

Substituting equation (4a) for v* and expressing the depth h by the hydraulic radius for the wide rectangular channel yields:

W. G. Strupczewski & R. Szymkiewicz

V = aRP -4gRS

and for the semi-circular section:

V ( P + 2 ) y / '

y/gRS

(6a)

(7a)

For the semi-circular form, equation (7a) can be rewritten in the form of equation (6a) as:

V = F-aRP

(p^)y0p

-yfgRS

where F is the shape factor:

p _ 2(P+1> <P + D

(7b)

(8) (p+2)

and has the value 1.2088 for the Chezy equation with Manning friction (p = 1/6) and 1 for Chezy friction (p = 0). The difference of 21 % between the values of the shape factor using Manning friction for two model shapes for uniform flow in an open channel shows that the influence of channel shape on Manning roughness may be quite significant. It should be noted that the above result has been obtained from a rational basis and the Chezy-Manning equation, i.e., if the underlying assumptions are correct the result is correct as well. However, it is the only simple conclusive way to confront properties of the Chezy equation (with constant Manning or Chezy roughness) with reality. This will be done further in the next sections and continued in the second part of this study (Strupczewski & Szymkiewicz, 1996).

Assuming that the logarithmic law of velocity distribution (equation (5b)) is correct and proceeding in a similar way the shape factor can be shown to be:

In

F = y0 R

0.5

In R_

To + Ï1-X

R

(9)

In this case the shape factor tends very slowly to unity for the hydraulic radius/representative roughness height ratio, goes to infinity, i.e. for frictionless channels, as shown in Fig. 1.

for other shapes, the distribution of shear stress along the wetted perimeter is not uniform and the isovels (lines of equal velocity) will not be similar to those in the two special cases above.


MATHEMATICAL FORMULATION

Let a cross section of a prismatic channel contain water up to a certain level, h0, called the initial or reference level. Its geometry at the reference level is defined by area, A0, wetted perimeter, P0 and the width of the channel T0; alternatively by A0, the hydraulic radius, R0 and T0. For simplicity of notation, the origin of the /z-axis will be at the reference level, i.e. h0 = 0.

F

R/y0 Fig. 1 Shape factor as a function of the ratio of the hydraulic radius to the roughness height for the logarithmic law of velocity distribution.

Above the reference level (h0) the geometrical factor (equation (2)) is:

G=fh{h) for h > h„ (10)

Given the initial conditions (A0, P0, T0) the task is to find the symmetrical cross section profile above the initial level, i.e.:

T = 4>(h) for h > h„ (11)

The restriction of a symmetrical section is applied in order to get the unique shape for the surface width-depth profile given by equation (11).

By means of the inverse problem solution it will be possible to check whether the Chezy formula is conformable with common sense with respect to the channel capacity/size axiom and, if not, to assess whether such nonconforming shapes can be frequently met in natural or trained river channels.

Because equation (11) contains only T and h variables, A and P in equation (2) and their derivatives with respect to depth shall be expressed in terms of these two variables by means of the following geometrical relationships, the last two of which are valid for a symmetrical section only:

A =A0 + Tdh (12a)


— = T (12b) dh

P = P0 + 2 f [l +(d772d/z)2f5d/z (12c)

— = 2[l+(d772d/z)2f5 (12d)

A basic shape (1) will be assumed with its stage-discharge rating curve derived from the Chezy equation with Manning or Chezy friction. Modifying the stage-discharge equation over the reference level, the inverse problem solution will be used to generate secondary shapes (2) which will be analysed from the point of view of the capacity/size axiom.

For simplicity assume that the basic channel (1) has constant sides of slope z above an initial level h0 = 0. Then the relationships (12a-d) take the form:

Ax = A0 + T0h+zh2 (14a)

i l l = T0 + 2zh (14b) dh

(14c) Px = P0 + 2hyi+z

dP I = 2V1 +Z2 <14d)

dh ¥

so that equation (2) becomes:

(A0+T0h+Zh2r G1 = r .1 (14)

(P0 + 2hil+z2 )

Therefore the basic section shape (1) of the prismatic channel is symmetrical and composed of a reference shape defined at a reference level h0 by A0, P0 and T0, with an upper part with sides constant slope z above the level

K A secondary shape (2) related to the basic shape (1) may be found from:

G2(h) = CGx(h) (15) where:

C = i 1 for h < 0 ( 1 6 )

C{h) for h > 0


in which C(h) is a positive, differentiable and monotonically increasing or decreasing function.

EQUATION FOR A SPECIFIED DEPTH-DISCHARGE CURVE

The solution of the inverse Chezy problem may be obtained numerically from the equivalence of terms:

d/z d/z d/z 1 (17)

with the initial conditions Ax = A2 = A0, Px = P2 = P0, Tx = T2 = T0 for h = 0. G2 and Gx are given by equations (2) and (14), respectively, while their first derivatives with respect to stage h are given by:

dG2 CG,

d/L - n dP, ( r - 1 ) A2 d/z d/z

and

(18)

ÙGX = G,

r(T0+2zh) _ 2 ( r - l ) V l + z

An + Tnh+zh2 f ^ 0 ° Pn+2hyl+z

Substituting equations (18), (19) and (2) into equation (17) yields:

r c L 4 2 _ ( r _ 1 ) d P 2

A2 d/z P2 d/z r <Wj_ ( r - l ) d ^ i

^ j d/z d/z J_dC Cd/T

(19)

(17a)

Then substituting equations (12b,d) and (14a,d) into equation (17a) gives T2 in terms of h and P2:

0.5

^2 °2 1 +

d ^

2d/z (17b)

r(T0 + 2zh) _ 2 ( r - l ) V l + z

4,+^+z*2 V 2 */n5

+ 1 dC + Cd/z"

It remains to eliminate P2 from equation (17b) by expressing it in terms of h. To do so substitute G2 from equations (15) and (14) into equation (2):


4 G,

l/(r-l)

CG,

l/(r-l)

A2r/{r-l)(P0+2hil+z2 ) (20)

Substituting equation (20) into equation (17b) and introducing the auxiliary functions:

^(h) = 2(A0 + T0h +zh2)rl{r~l)/(P0+2hjl +z2 )

<j)2(A) = (r-lM" r / ( r_1)

(21a)

(21b)

Uh)

one arrives at:

r(T0+2zh) _ 2(r-l) \ / l+z2

c w * > (po+2hir^F') (21c)

.Lji-^.c"™ f 1 2

1 + dr2

àh

0.5

3 Cd/z

(17c)

Reordering and denoting A2 = A, T2 = T gives the differential-integral equation:

dF = 2C1/(1"r)

d^ ~~ 0 r 0 2 ^ A 3 C d / z 1 2 (22)

Equation (22) has been solved numerically by the Runge-Kutta fourth order method. An exponential function may be taken for C(h):

C = i 1 for h < 0 (16a)

exp(e/z) for h > 0 i.e. negative values of e, according to the Chezy-Manning formula, result in the shape (2) carrying a lower discharge than shape (1) for any h over the reference level, i.e.:

G2(h) < Gx(h) for h > 0 and e < 0

The computer program used solved equation (22) step by step in finite difference form, starting from the initial conditions (A0, P0, T0, G0) at h0 = 0 up to an arbitrary upper limit, /zmax taken here as equal to the initial depth. It produced values of T, P, A and G for both shapes (1) and (2) and


0 < h < /zmax. The step (Ah) of the computation was each time selected to fulfil two accuracy conditions:

1. there shall be no deviation of the solution from a straight line for shape (1), i.e. for e = 0; and

2. equation (15) shall be preserved within the whole range of computation, i.e. for 0 < h < /zmax.

Obviously it is sufficient to check both conditions at the /zmax stage. In general, a shorter step Ah is applied for small values of the side slope z than for large values.

FLOW DISCHARGE VS SIDE SLOPE

In order to make the results of computation fully understandable, it is useful to analyse the relationship of the discharge (or equivalently of the geometrical factor, G) to the side slope via the Chezy formula. To do so, expand the depth-geometrical factor relationship of equation (10) into a Taylor series around h, assuming its differentiability:

G(h+Ah) = G(h) + AhG(h) + ^-G"(h) + ... (23)

For a small depth increment, ôh, all terms higher than the linear one can be neglected and equation (23) reduces to:

GQi + ôh) = G(h) + Oh G'(h) (24)

Taking the basic shape (1) and expansion around the reference level, i.e. h = 0 + , one finds:

Gx(ôh,z) s G^+ôhGy'iO^z) for ôh > 0 (24a)

where G[(h, z) is given by equation (19). The derivative of G|(0+,z) with respect to z is:

i-G^O+.z) - - 2 ( ^ i G l ( Q ) ( 2 5 )

Pjl +z2

Therefore,

' r.T 2(r-l) maxGjXCnz) = G^O'.O)

z K P0

Gj(0) (26)

and consequently from equation (24a):

Gx(àh,z} < Gt(ôh,Zp for zt > Zj > 0 and Ôh > 0 (27)

which contradicts the capacity/size axiom.


One can deduce from equation (26) the existence of an upper bound to e in equation (16a). Confronting equations (17) and (26):

< 2 ( r - l ) 1+z2 - 1 (28)

Lack of a finite lower bound to e points out the possibility of getting from equation (22) a shape for G being a monotonically decreasing function of h.

For a pair (z, ,zj) where z-t > Zj and the initial conditions, one can derive the upper limit of stage interval h* for which:

G{h,z,) < G(h,zp

as the greater of two roots of the equation:

Gih*,Zi) = Gih*,Zj)

(29)

(30)

where G{h*,z) is given by equation (14). Example results are illustrated in Fig. 2 for a trapezoidal reference shape with A0 = 9, P0 = 12.324 and Tg - 12, and for unit initial depth, side slopes z0 — 3, and Manning friction. For Zj = -5, 1, 2 and 3 vs z{ = 0 (vertical wall) one gets h* = .16, .33, .56 and .68 respectively, while h* = .97 is obtained for zt = 2.5 and Zj = 3.0, which are quite considerable values if compared with the unit reference depth. There is a rapid rise of every curve at its origin, except for the one corresponding to a vertical side slope. This feature is not dependent on reference conditions.

1.0

0.8

0.2

0.0

z j

.0

=

f .5

1.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.8 1.8 2.0 2.2 2.4 2.6 2.8 3.0

Fig. 2 Stage h* of two shapes' equivalence in respect to flow discharge. Reference condition: trapezoidal channel A0 = 9, T0 = 12, P0 = 12.324, i.e. z„ = 3.

One may argue that due to the discontinuity of the G^h) derivatives at the reference level, such a shape should not be considered as a simple one, i.e.


suitable for direct application of the Manning formula. In fact, whatever a natural shape is, its representation got from measurements is composed of several straight line segments, i.e. it has several vertices. Therefore, in the above case it is rather the difference between G{(0~) and G{(0+) which is decisive in distinguishing between a simple and a complex shape.

NUMERICAL RESULTS

The previous section led to the conclusion that, according to the Chezy formula, making side walls steeper results in an increase of discharge over a finite stage interval, which is against common sense. Furthermore, to avoid the argument about the discontinuity of the geometrical factor derivatives, one can replace equation (16a) by:

exp(e/za) with a > 1 for h > 0 Then for a basic shape (1) with both Gx{h) and G[(h) continuous one finds a secondary shape (2) with the same properties. It can be easily achieved here by assuming a constant side slope for the whole basic shape (1), i.e. taking z0 = z. Values of a = 2 and e = —1, —.1, -.01 , .01, .1, 1 were used in the computation of the secondary shapes (2) via equation (22). As a basic shape, a trapezoid was selected with the reference shape as in Fig. 2 and the side slopes z = z0 = 3, and Manning friction assumed. The results presented in Fig. 3 indicate high sensitivity of channel shape to any deviations from the basic stage-discharge rating curve, e.g., for e = .01 and —.01 and h = 1.5 the

(T - T0)/2 Fig. 3 Secondary shapes related by equations (15) and (16b) to the basic one shown above the reference level. The basic shape: trapezoidal with the reference values A0 = 9, T0 = 12 and P0 = 12.324 (z = z0 = 3). Manning friction, i.e. r = 5/3.


discharge ratio Q2/Q\ equals 1.0228 and .9777, respectively ,while the surface width ratio T2ITX equals .8959 and 1.1077, respectively. However, most importantly, note that secondary shapes wider than the basic shape correspond to lower flow capacity and vice versa. A third observation is that, while for negative values of the parameter e the solution is unbounded, there is an upper bound for positive value of e. The upper bound corresponds to the vertical slope (z = 0), which is in conformity with equation (26). Then the maximum possible flow increment (26), where subscript "0" denotes the bound, becomes insufficient to fulfil equation (15). The value of the bound decreases as the side slope of the basic shape decreases.

It is seen in Fig. 3 that the discharge decreases with a growing channel width in both the cases of convex and concave shapes. Bringing the capacity/ size axiom to mind, if the rating curve of the basic trapezoidal section shape got from the Chezy formula with a constant value of Manning roughness were correct, discharge would be underestimated for all secondary convex shapes and overestimated for concave shapes.

Quite similar results can be obtained repeating the computations for Chezy friction as well as using equation (16a) instead of (16b). Therefore, the blame for the disagreement of the results with common sense should be put first on the Chezy formula but not on the friction law.

CONCLUSIONS

The dependence of Manning's n on channel shape proceeds from the rationale of the Manning formula. The comparison drawn of the uniform flow formula based on its rationale for a wide rectangular channel and a semi-circular channel has pointed out a dependence of Manning's n on channel shape amounting to a difference of 21% between the values of n. One can expect differences for natural river channels to be greater than for the two model shapes. Therefore, the conviction that the shape of a channel is not an important factor affecting the value of n must have come from applications.

To deal with any shape, mathematical analysis of the Chezy function with constant roughness has been applied and its results have been confronted with common sense. It was shown (Fig. 2) that the maximum increase of discharge on the depth-discharge curve above a reference level corresponds to vertical side slopes (z = 0) of the channel cross section independently of a reference shape and then it decreases monotonically with increasing z. The flow capacity of a channel with vertical side slopes above a reference level remains the highest possible one over quite a long interval of depth.

The inverse solution has proved very suitable to study properties of the Chezy formula with constant roughness. Taking a trapezoid as a basic shape, it was found via the inverse solution that any reduction of flow capacity (made according to equations (16a) or (16b) changes the basic shape into one with convex sides. A greater reduction results in a greater convexity and conse-


quently in a wider channel (Fig. 3). The solution has no upper bound of depth. On the other hand, an increase of flow capacity changes constant slope sides into concave sides, decreasing the channel width (Fig. 3). The solution has an upper bound of depth where the flow increment demanded becomes greater than the maximum possible.

H. Minkowski (the co-author of the special theory of relativity and Hilbert's great friend) expressed his admiration of the profound relationships of convex sets in the famous statement: " Everything that is convex is interesting for me". Indeed to get meaningful results from the Chezy equation more attention should be put on channel shapes both convex and concave.

The properties of the Chezy function discussed above are contrary to common sense about the open channel flow process.

Acknowledgements The authors would like to thank the anonymous reviewers for their valuable comments and in particular to thank J. C. I. Dooge, who significantly enhanced the quality of the paper.

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Publ., Littleton, Colorado, USA.

Received 23 November 1995; accepted 1 December 1995

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