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DISTRIBUTION CHANNELS

By Willi H . Hager1 and P. U. Volkart2

ABSTRACT: The equations governing one-dimensional steady flow in distribution channels are presented. They consist of the dynamical relation for the surface profile and appropriate outflow laws for the lateral outflow types side-weir, side-opening and bottom-opening. Further investigations concern simplifications regarding the frictional slope and the separation loss gradient. The integrated form of the equation for the free surface profile is of Bernoulli-type, yet it is based on the balance of axial momentum. A classification of typical solutions and specifications for the boundary conditions in distribution channels of rectangular cross section are given. It is demonstrated that pseudouni-form flow conditions result in uniform lateral outflow. Experiments have been performed to check the validity of the one-dimensional model equations. The computation procedure is summarized and plots of the general solution are presented.

INTRODUCTION

This investigation concerns steady, gradually varied open channel flows with spatially decreasing discharge in the flow direction. Channels where these conditions occur are called distribution channels.

The first systematic studies have been performed by Favre (4) by presenting the axial momentum equation, thereby assuming uniform velocity and hydrostatic pressure distribution. De Marchi (16) has simplified the equation describing the free surface profile by considering the energy line parallel to the (straight) channel bottom. The lateral outflow intensity of a side-weir is modeled using the conventional weir formula. The surface profiles in prismatic channels are classified in terms of the flow state (i.e., sub- or supercritical). Experimental investigations by Favre and Braedele (5), Gentilini (8), and Ferroglio (6,7) indicate satisfactory agreement with de Marchi's theory for low Froude numbers F, but show serious disagreement for F > 0.5. The failure of the theory has been attributed to the neglect of the friction gradient (22,24), inexact consideration of the momentum correction factor 3 (1), and inappropriate lateral outflow relations (23).

Thirty-five years after Favre's work, Coulomb, et al. (2), derive new outflow formulae for weir flow types by including the effect of the approaching velocity. However, their analysis is not confirmed experimentally. Subramanya, et al. (23), and Nadesamoorthy, et al. (17), present outflow relations for a side-weir of zero weir height. It is found that the outflow rate for a side-weir is always smaller compared to a usual weir of corresponding flow depth. Using energy considerations Yen, et

'Sr. Research Engr., Chaire de Constructions Hydrauliques, CCH, Dep. de Genie Civil, Ecole Polytechnique Federate de Lausanne, EPFL, CH-1015 Lausanne, Switzerland.

2Sr. Research Engr., Head Dam Hydr. Section, Lab. of Hydr., Hydrology and Glaciology, VAW, Federal Inst, of Tech., ETHZ, CH-8092 Zurich, Switzerland.

Note.—Discussion open until March 1, 1987. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on January 9, 1986. This paper is part of the Journal of Hydraulic Engineering, Vol. 112, No. 10, October, 1986. ©ASCE, ISSN 0733-9429/86/0010-0935/$01.00. Paper No. 20966.

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al. (25), obtain a new equation for the surface profile. The axial momentum balance yields an extended relation for the surface profile which, however, contains some unspecified parameters. Further studies relating to the surface profile equation have been conducted by Yen (26), El-Kashab, et al. (3), and Ranga Raju (20).

The purpose of the present investigation is to consider in more detail distribution channels; considerations include the side-weir, the side-opening, and the bottom-opening as lateral outflow type. Expressions for the lateral outflow intensity are given for completeness, but details will be presented in another investigation (13).

Suitable simplifications regarding the energy loss gradients are considered in order to solve for generalized surface profiles in distribution channels. The results are compared with typical observation in prismatic and nonprismatic rectangular channels. Beyond comparisons regarding the one-dimensional flow quantities, the spatial flow configuration is discussed.

GOVERNING SYSTEM OF EQUATIONS

Equation of the Surface Profile.—Consider steady, one-dimensional, and gradually varied flow of a homogeneous and incompressible medium. Its average cross-sectional velocity is V = Q/A, in which Q = discharge and A = the wetted cross section. Assuming essentially hydrostatic pressure distribution, the longitudinal momentum theorem yields (4)

h' = So- S/ + 7^--(1 + P -

Q2P'

PQ_23A

gA3dx

PQ_2M

gA3 dh

li-cos<t>\ QQ1

V gA*

1 - ^ — (1)

in which primes denote ordinary differentiation with respect to the longitudinal coordinate x; S0 = the bottom slope; Sf = the friction slope; (3 = the momentum correction coefficient; g = gravitational acceleration; and U = lateral outflow velocity, deflected by the angle <j> from the channel axis (see Fig. 1). Yen, et al. (25), present an equation more general than Eq. 1, yet it contains some unspecified parameters.

The denominator in Eq. 1 corresponds to (1 - F2), in which

-» -x b)

FIG. 1.—Definition of Parameters in a Side-Weir

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(2) \gA3 dhj

denotes the Froude number; F = 1 corresponds to the critical flow condition. The numerator of Eq. 1 consists of the difference of the bottom slope and the friction gradient (S0 - Sf); further terms account for the nonprismatic channel geometry and the outflow characteiistics; the last term includes the longitudinal variation of p. Eq. 1 allows determination of the surface profile h(x) and the discharge characteristics Q(x), once the geometry of the distribution channel (S0, A, 8A/dx, dA/dh) and the hydraulic parameters (Sf, (3, JJ'Cos <J>, Q') are specified.

It can be demonstrated that the effect of p on h(x) and Q(x) is limited to subcritical flows (12). However, compared to the influence of other parameters described, it can be neglected to the lowest order of approximation. Therefore, ensuing considerations assume P = 1 and p ' = 0.

Lateral Outflow.—The outflow types side-weir, side-opening and bottom-opening, as shown in Fig. 2, are considered. The weir height is denoted by w, and the opening width by a. The quantities w and a may vary with x. Only free lateral outflow conditions will be studied.

For strictly plane flow, the usual discharge characteristics of weirs and orifices are relevant. Discharge then depends on the outflow geometry, the discharge coefficient, and the head on the weir.

In contrast to the plane flow condition, flow in distribution channels exhibits considerable spatial variation. This has been taken into account by Coulomb, et al. (2), Subramanya, et al. (23), and Nadesamoorthy, et al. (17), for side-weirs, Schattulat (21) for side-openings, and Nasser, et al. (18), for bottom-openings. However, all of the formulations require significant restrictions regarding their general application.

A rigorous analysis of the lateral outflow characteristics indicates that the outflow intensity, Q' = dQ/dx, and the average cross-sectional outflow angle, 4>, depend on a typical outflow height and the outflow velocity. Based upon a quasi two-dimensional approach, the results (12,13)

Q' = —n 5

3(1 + 9) y)

1/2-v

W side-weir (3a)

Q' = -n*D 2gH3

L3(4 - 3y)J

1/2

1 - (S + 9) (i - y) 1/2

side-opening (3b)

(a)

FIG. 2.—(a) Side-Weir; (ft) Side-Opening; and (c) Bottom-Opening; Free Outflow Conditions


Q' = -0 .62D <2gH H I 1/2

; side-opening (3c) ( 2 - y )

are obtained, y = h/H is the relative flow depth; H is the local energy head; W = w/H is the relative weir height; S is the head-loss gradient; 9 = (l/h)(8A/dx) is the channel convergence; c is a weir shape factor being c = 1 for a sharp-crested weir; n* reflects whether one (n* = 1) or both (n * = 2) sides of the channel have side-weirs or side-openings; and D = a/H is the relative opening width. For plane flow conditions (y -* 1) Eqs. 3a-c become identical with the usual discharge relations for a sharp-crested weir and orifice. Eqs. 3a-c consist of the plane outflow law, extended by terms accounting for the approaching velocity (0 < y < 1), the lateral outflow angle, and the channel geometry (terms in { } of Eqs. 3a-b). These latter become insignificant for flows in prismatic, nearly horizontal channels.

The outflow intensity Q' of a distribution channel (y < 1) is smaller than the corresponding discharge per unit width of the plane flow configuration. Eqs. 3a-c compare well with observations (±10%), provided flow conditions are gradually varied (12,13).

Friction Gradient.—The frictional gradient Sf accounts for the turbulent energy dissipation and can be approximated by the Manning-Strick-ler formula for flows in the turbulent rough flow regime. According to (12) its magnitude remains small for F < 1 while, for F > 1, its longitudinal variation is insignificant. This is mainly because of the finite extension of the distribution channel. Consequently, Sf = Sf(x) can be replaced by the average Sfa — (Sfu + Sfd)/2, in which indices u and d denote the limiting up- and downstream cross sections of the distribution channel, respectively.

Head Loss Gradient Due to Bifurcation.—The term

/ U- cos <b\QQ'

"'--(*—ir*)i& <4» in the numerator of Eq. 1 corresponds to the energy loss gradient due to the lateral outflow. H's vanishes for Q = 0, Q' = 0 (particular cases), or V = U-cos 4> O = 1); the latter case corresponds to poten-flow conditions (4). For V > (<)• U-cos <j>, H's is positive (negative). This mechanical energy increase and decrease of the straight-through branch is, of course, compensated for by a lateral energy transfer (9).

The longitudinal variation of H's can be demonstrated to be also of minor importance in terms of effect on the surface profile (12). The integrated energy variation, AHS, along the outflow length AL, may be written as AHS = % • V2J2g, £ being the head loss coefficient. The averaged head loss gradient Sb due to the lateral outflow is then

AHS £ • Vi Sb = = (5)

AL 2#AL v '

in which Vu > 0 is the upstream velocity; and AL the lateral outflow length. £ has been evaluated as (9)

4AQ/AQ 1\ { _ i o : V Q T S J •• (6)


The Integrated Dynamical Relation.—For constant bottom slope S0, and with Sf„ and Sb as given previously, the total gradient

S = S0 - Sfa - Sb (7)

is independent of both x and h. Eq. 1 then becomes (with (3 = 1)

QQ' | Q2 dA)(1 Q2 M V 1

#A2 gA3dx)\ gA3 dh, h' = (S-^fl + ~ n - 1 - - T 1 - ) (8)

whose solution is

Q2

H = H0 + Sx = h + -r—2 (9) 2gA2

with H0 as constant of integration. Eq. 9 is the well-known Bernoulli equation, but, in fact, it corresponds to the transformed axial momentum relation. Nevertheless, H will be referred to as energy head. It should be noted that, for the usual bottom slopes, S is much smaller than unity.

TYPES OF SURFACE PROFILES

Consider a rectangular, nonprismatic channel. With linearly varied channel width B, the cross-sectional area is

A = Bh = (b + Ox) h (10)

in which b is the (constant) channel width at the origin x = 0 and 9 = dB/dx. Differentiating Eq. 9 with respect to x gives

Q2e QQ')!, Q2

gB3h2 gB2h2){ gB2h3

or, by using the transformation t = h — Sx

ft'=lS+^-^ H - ^ I i (")

V 2 , „ . M Q'h F2 S + — ^

B Q (1 - F2)"1 (12)

in which F2 = Q2/(gB2h3) is the Froude number. In contrast to h(x), t(x) describes the surface profile with respect to the slope S. Let a = Q'h/ 0; $ = Qh/B; and

<r = S + * - a (13)

then Eq. 12 reduces to

aF2

• ' - r ^ <"> This transformed dynamical equation allows a simple description of all possible surface profiles. The sign of t' in Eq. 14 depends on the sign of a and F. CT = 0 corresponds to a local extremum of t(x); except for critical flow conditions, F = 1 (singularity discussed in Ref. 12).

The cases Q' > 0 (locally increasing discharge), 9 > 0, and S > > 0 (producing hydraulic jumps and/or extremely nonuniform outflow intensities) are not further considered. Fig. 3 classifies the remaining types


e=o

S - 0 s«o

0<O

S - 0 s-o —

a>0 a<0 a>0

FIG. 3.—Types of Surface Profiles in Rectangular, Nonprlsmatic Distribution Channels; (• • •) a = 0, ( ) Possible Development of Hydraulic Jump; Flow Direction from Left to Right

of surface profiles. Distinction is made between prismatic (6 = 0) and nonprismatic converging channels (6 < 0) with S-0 and S « 0. The classification according to de Marchi (16) is significantly widened by including the effects of S and 6.

PSEUDOUNIFORM FLOW

It can be shown (12) that \S - S0\ < 0.01 for arbitrary subcritical flows, whence t(x) — h(x). For a distribution channel of locally constant outflow geometry, fixed energy head, and outflow length AL, the maximum lateral outflow AQ results from a horizontal flow surface. The difference H - h = V2/2g then remains constant, and so does the average velocity V, although the discharge varies with x. This peculiarity is referred to as the pseudouniform flow condition. It is achieved by proper modeling of the longitudinal channel geometry. Fig. 4 shows two typical examples of linear decrease of cross-sectional area A.

The purpose of the following computations, restricted to rectangular cross section, is to determine the channel geometry and corresponding

B,

1 a) "AQ\

m

b)

AQ 5V :AZ

FIG. 4.—Generation of Pseudouniform Flow State by: (a) Linear Cross-Section Contraction; or (b) Linear Channel Bottom increase


n*D

,iy.

0.5

PN (c)

tffl-^01

1L -̂0.4

0 0.25 0.5

FIG. 5.—Relative Flow Depth yPN for Pseudouniform Flow Condition According to Eq. 18 as a Function of Contraction Ratio 6 and Outflow Geometry: (a) Side-Weir; (b) Side-Opening; and (c) Bottom-Opening; (•••) Critical Flow Condition ym = 2/3

flow depth such that maximum lateral outflow results. Assuming: (1) A gradually varied channel geometry; (2) a horizontal flow profile; (3) a constant lateral outflow intensity; and (4) compensation of the head losses by bottom slope (S = 0) yields between the up- and downstream cross sections

H = K +Vl hd + V\

(15) 2g 2g + Az

Az being the difference of the respective bottom elevations. For hu = ha

+ Az (horizontal flow surface), Eq. 15 reduces to Vu = Vd. From continuity considerations V = Qu/{Buhu) = Qd/(Bdhd) = constant, thus

AQ_Az AB_Az AB

Qu hu B„ hu Bu

(16)

Ramamurthy, et al. (19), have performed tests in side-weirs by varying either B = B(x) or z = z(x) according to Eq. 16. The present analysis accounts for the combined variation of the channel width and the bottom profile for arbitrary distribution channels. A good agreement has been found when comparing predictions with observations (12).

The pseudouniform flow condition (subscript PN) implies a constant flow depth h = hPN, provided F # 1, whence

( T = S + • Qhp Q'h PN

B Q = 0. (17)

according to Eq. 14. Eliminating Q with Eq. 9, and inserting Eq. 10 yields

Q'PN = dhPNV2g(H-hPN) (18)

This is evaluated in Fig. 5 for the considered outflow types. The pseudouniform flow condition in side- and bottom-openings has

physical relevance only for the subcritical flow state. Consequently, dashed curves have been drawn in Fig. 5 for yPN < 2/3. Application of Eq. 16 enables design of a distribution channel with maximum lateral outflow, constant cross-sectional velocity, and uniform lateral discharge distribution.

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BOUNDARY CONDITIONS

The preceding analyzed flow condition corresponds to a particular case and is often applied for design discharge. For all other discharges, flow conditions must be determined using Eqs. 8 and 9, or, upon eliminating Q

h'= V2g(H-h) | 2(H h)dA

8* dx

2{H - h) dA

~A dh (19)

in which H = H0 + Sx. The general solution of Eq. 19 must start in a control section, which coincides with the conditions as given in Table 1 (12). Control points for all surface profiles with a > 0 are given by F = 0 (dead end) for subcritical flow conditions, and by F = 1 for supercritical flow conditions.

Flows with a < 0 have a second control point. A rigorous analysis yields Taylor polynomials valid in the neighborhood of the specified control points, by which the numerical integration of Eq. 19 can be initiated (for results see below).

EXPERIMENTS

The objectives pursued with the experiments were: (1) To check the validity of the major assumptions on which the presented theory is based; (2) to compare experimental results with the predictions; and (3) to investigate typical spatial flow configurations. These objectives have been realized with two different channels, both having rectangular cross sections of the width 0.30 m. The first, having an approximate length of 5 m, will be referred to here as Single; the second, of which the length is 50 m approximately, will be referred to as Manifold. Single has an iron base and therefore could only be used for side-weirs and side-openings. The lateral outflow length is AL = 1.0 m. Manifold, however, has a total possible outflow length of approximately 3 m. Up to eight single openings of any of the three lateral outflow types are able to be used. The main purpose of single is to investigate the spatial flow characteristics, i.e., detailed measurement of the velocity field and the water surface. Manifold, on the other hand, facilitates examination of a series of single openings, and observations are mainly restricted to the one-dimensional flow characteristics, i.e., the average development of cross-sectional velocity, the surface profile; and the lateral outflow angle. The present investigation considers only results obtained with single; manifold is described in (11,12,15). Table 2 specifies the model parameters.

TABLE 1.—Conditions at Control Points for Sub- and Supercritical Flow Conditions as a Function of a (See also Fig. 3)

d) o- > 0 <x < 0

F < 1 (2)

F = 0 F = 1

• F > 1 (3)

F = 1 y = W or y = D

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TABLE 2.—Variation of Basic Parameters of Model SINGLE

Parameter (1)

Inflow discharge Bottom inclination Contraction angle Bottom rise Weir height Side-opening

Variation (2)

0 < Q„ < 45 L/s -4% < S „ < 5% -17" < 9 £ 0 0 < A z < 0.20 m 0 < w s 0.20 m 0 < a < 0.05 m

The lateral outflow from Single is directed into a side-channel spillway. Each specific outflow geometry has been obtained using elements of height 0.05 m. These are inserted into two vertical slots on either side of the outflow section. Positioning the element(s) in the bottom of the section provides side-weirs of variable height, while positioning them into the top of the section leads to side-openings of variable aperture. Combinations are also used.

A simple instrumentation is chosen, being reasonably accurate and time-efficient to use. These requirements are attained with a gauge of precision ±0.1 mm for depth reading, a velocity meter (Mini-Air-Water, Schiltknecht AG, Switzerland) with an 8 mm propeller diameter and three variable domains of velocity determination (0.05 < V < 5.0 m/s), and an angle meter. The last instrument resembles a flag and transmits the local flow angle directly to an angle scale. 21 extensive tests (spatial flow configuration) have been performed in Single, and 50 additional tests in Single and Manifold to measure the one-dimensional flow properties. Test results are given in Ref. 12; some typical findings will be discussed.

SPATIAL FLOW BEHAVIOR

According to the one-dimensional flow theory, the surface profile in distribution channels is influenced by the local Froude number F, the channel and lateral outflow geometries, and the up- and downstream flow conditions. Fig. 6 shows the simplified, spatial flow mechanism for subcritical flow conditions in a side-weir. The most typical flow situations are obtained for Qd = 0, corresponding to AQ = Qu. All other

FIG. 6.—Different Flow Domains in a Side-Weir with Dead End for Subcritical Flow Conditions; Surface and Bottom Flow Directions Are Indicated as Solid and Dashed Lines, Respectively


subcritical flow configurations are contained in this limit case (12). Four flow zones, schematically shown in Fig. 6, can be distinguished from each other.

The extent of the inflow zone (1) depends on the magnitude of the downstream submergence. Its domain is characterized by nearly uniform velocity distributions and almost horizontal transverse surface profiles. The flow accelerates at the entrance of the distribution channel, and potential energy is transformed into kinetic energy along the weir plane, which forms the lateral outflow zone (2). Besides the increase in the magnitude of the velocities, the direction of the accelerating flow deviates considerably from the channel axis. Streamlines are sloped and curved, but lateral outflow may be approximated as potential flow. Particles situated opposite to the outflow zone, however, are not able to follow the lateral outflow because of downstream submergence. Their movement is slowed and, if downstream submergence is sufficient, a separation zone (3) is developed. Note that some streamlines do not leave the distribution channel, but form the backflow zone (4) at the dead end. While the average flow velocity becomes negligible, significant eddy formation is created by the two zones.

It seems ambitious to approach these complex processes by a one-

FIG. 7.—Velocity Profiles in Distribution Channels for Subcritical Flow State at Various Positions y (in cm) over Channel Bottom: (a) Side-Weir (Run C, F„ = 0.38, w = 0.15 m, S0 = 0.3%); Flow Surface with Lines of Constant Depth (in em) [Bottom of (a)]; (h) Side-Opening (Run M, F„ = 0.52, a = 1.9 cm, S„ = 0.2%); Flow Direction is from Left to Right


dimensional flow theory. Theoretical considerations have shown, however, that while the simplified procedure drastically reduces the information concerning the spatial flow characteristics, it (more importantly) describes fairly the overall phenomena (12). For practical purposes the aim is generally to specify the lateral outflow rate and the surface profile, while ignoring the actual distribution mechanism. The first two quantities, among others, are sufficiently reproduced by the one-dimensional theory just presented. It seems that using the simplified procedure is reasonable even though details of the spatial process are not elucidated.

Fig. 7 shows typical spatial flow configurations for subcritical flow conditions for a side-weir with AQ/Q„ = 1, and a second for a side-opening with AQ/Q„ = 1/2 (AQ being trie lateral outflow discharge). The inflow zones exhibit almost rectangular velocity profiles, which change to nearly triangular profiles along the lateral outflow length. This process implies mechanical energy dissipation due to separation and fric-tional losses.

A typical evaluation of an experiment for supercritical conditions is shown in Fig. 8. The lateral outflow consists of a side-weir of zero weir height (3 s j s 4 m). Velocity profiles have been plotted for y = 1.5, 3 and 4.5 cm above the channel bottom as well as the surface h(x,z). An oblique hydraulic jump forms at x — 4.5 m because of the side-wall discontinuity at x = 4.0 m. Compared to subcritical flow, this example reveals almost uniform velocity distributions and decreasing flow depth in the flow direction (see Fig. 3). Since flow is accelerated, additional mechanical losses almost vanish. Wall friction now dominates because

FIG. 8.—Velocity Profiles for: (a) Supercritical Flow State at Various Positions y (In cm) over Bottom; (h) Corresponding Water Surface (Run J, F„ = 2.06, Fd = 3.16, w = 0, S0 = 2%)


TABLE 3.—Longitudinal Development of One-Dimensional Flow Parameter In Terms of Flow State F in Prismatic Channel

Development in flow direction

(1) h(x) V(x)

Q'(x)

m -H's(x) Sf(x)

Flow State

Subcritlcal (2)

Rising Decelerating Nonuniform Rising Increasing Decreasing —> 0 Decreasing

Supercritical (3)

Falling Accelerating Uniform Falling Decreasing Increasing -» 0 Increasing

of the relatively high velocities. As has been demonstrated (12), however, the longitudinal variation of the friction slope Sf tends to zero for large Froude numbers; Sf therefore is nearly a constant. The longitudinal development of different parameters is summarized in Table 3. Experimental findings about the behavior of the parameters <|>, H's, and Sf make possible the simplified one-dimensional analysis already presented.

FIG. 9.—(a) Velocity Profiles In Slde-Welrs with Linear Channel Width Contraction (Run H); {b) Linear Channel Bottom Rise (Run L); y and z (In cm) Are Distances from Channel Bottom and Side-Waif Opposite to Outflow Plane, Respectively


Typical experimental results regarding distribution channels with converging width and rising channel bottom are shown in Fig. 9. The first example concerns a channel with Bu/Bd = 2, and AQ/Q„ = 1/2 which exhibits pseudouniform flow conditions. As is shown by the plot, these are not only given by the one-dimensional theory but are also demonstrated by the spatial velocity profiles. In contrast to Fig. 7, no separation and backflow zones are visible because these are now "filled" with the contracting channel profile. The analogous tendencies are represented by the second example with a local linear rise of channel bottom, in which Az/hu = 3/4 and AQ/Q„ = 1. Consequently, selected channel geometry according to Eq. 16 yields simple velocity and pressure fields, uniform lateral outflow behavior, and low additional energy losses. The importance of pseudouniform flow, therefore, is not only theoretical, but also practical.

COMPUTATION PROCEDURE

The aim of flow computations in distribution channels is either determination of the channel geometry for prescribed inflow and outflow, or evaluation of the surface profile and the lateral outflow for fixed channel geometry and inflow or outflow. The first problem corresponds to the design of the channel geometry, whereas the second yields information for the remaining cases. Both problems involve the surface profile; consequently, it must be computed even if not sought.

The solutions of Eq. 19 have been found by an explicit numerical integration scheme, thereby accounting for the appropriate outflow law according to Eqs. 3a-c. For rectangular distribtion channels the nondi-mensional parameters

Kx h - 6 - / w a X — ; y = - ; 6 = - ; ; = - ; W = ~; D = - (20)

b y H K ' K H H v ' are introduced in which; = Sb/H and K = n*c. The surface profile y(X) then depends on 0,;', and W or D. Typical evaluations are shown in Fig. 10 for sub- and supercritical flow in a side-weir. Control points of the first plot are y = 1 (F = 0) and y = 2/3 (F = 1), while y = 2/3 (F = 1) and y = W are the respective locations for the second plot according to Table 1.

Consider the curve W = 0.2 in Fig. 10 (top); since F < 1, the curve starts at (0,1) and moves in the negative X-direction. Critical flow is reached at X = -2.45, indicating the maximum length of possible sub-critical flow. For X < -2.45 a hydraulic jump must occur. Note that the direction of computation is reversed to the flow direction. Consider the curve W = 0.8 on the same plot. Evidently, since W > 2/3, only sub-critical flow state is possible with Q' < 0. The first possible curve has its origin at (0,1) with a positive inclination in the flow direction; a minimum appears at approximately y = 0.975. The second curve starts at (0,0.8) with a negative inclination, and tends also to the pseudouniform flow depth (<r = 0).

The surface profile for supercritical flow conditions commences at (0, 2/3), and the computation proceeds in the positive direction (see Fig. 10, bottom). For 6 = -0.2, the maximum lateral outflow length is X =


FIG. 10.—Nondimensional Surface Profiles y(X) in a Side-Weir with / = 0.02 and 6 = -0.20 for: (a) Subcrltlcal; and (b) Supercritical Flow Conditions

-1/6 = 5, and y(X = 5) = 0.505. The control point for 0.2 < y < 0.5 is y = 0.2. The surface profile has positive inclination in the flow direction until a = 0 (X = 4.2, y = 0.51) and then becomes identical with the first corresponding curve.

Sixty-seven diagrams as presented in Fig. 10 are prepared in Ref. 12, including the surface profiles for side-weirs, side-openings, and bottom-openings in rectangular channels. Parameter combinations are / = -0.02, 0, +0.02, and 9 = -0.4, -0.2, - 0 . 1 , 0 for F < 1 and F > 1. These allow computation of nearly all relevant problems involving rectangular distribution channels. In particular, the spatial discharge distribution Q(x) is obtained by computing V2/(2g) = H(l - y), and Q = BHyV.

The computation procedure must be modified for flows with hydraulic jumps along the lateral outflow length. Recently, the effect of transitional flow state (F = 1) has been investigated by accounting for the


effective transverse pressure distribution (10,15). It has been found that a standing surface wave pattern appears for subcritical flows, while a local modification of the surface profile at the inflow zone must be accounted for in supercritical flows. For the remaining cases predictions have been compared with observations (12,13), and deviations of less than 5% regarding the lateral outflow discharge and the average surface profile are obtained. Consequently, the present hydraulic model may serve to predict accurately the governing flow behavior.

CONCLUSIONS

This investigation summarizes analytical and experimental investigations of flows in distribution channels. It is found that:

1. The axial momentum equation, coupled with appropriate outflow relations accounting for the spatial flow behavior, defines the solution of the surface profile and the local discharge distribution. The usual weir and orifice formulas, valid for plane flow conditions, are replaced by expressions including effects of the lateral outflow geometry, the outflow angle, and the velocity of approach. Their influence is significant for supercritical flows.

2. The optimal outflow characteristics are obtained for pseudouniform flow conditions, which are achieved by geometrical modeling of the distribution channel. Either linear longitudinal width contraction, or negative bottom inclination, or both, yield uniform lateral outflow intensity and constant channel velocity.

3. The general solution of the surface profile is found by accounting for appropriate boundary conditions at properly chosen control points. The computation of particular problems is facilitated by a classification of the surface profiles and the application of diagrams.

4. The spatial lateral outflow mechanism is explained using velocity profiles and the flow surface. The differences between sub- and supercritical flow conditions are not only reflected by the one-dimensional equations, but also are elucidated by the discussion of the pertinent spatial flow parameters.

ACKNOWLEDGMENTS

This project has been executed at the Laboratory of Hydraulics, Hydrology, and Glaciology, VAW, Institute of Technology, ETH, Zurich, under the supervision of D. Vischer, Director, to whom the writers would like to express their gratitude. At the same time, the first writer would like to thank Kuster & Hager, Hydraulics and Wastewater Engineering for having supported him. The manuscript has been improved by J. Whittaker.

APPENDIX I.—REFERENCES

1. Ackers, P., "A Theoretical Consideration of Side Weirs as Stormwater Overflows," Proceedings of the Institution of Civil Engineers, Vol. 6, 1957, pp. 250-269.


2. Coulomb, R., de St.-Martin, J. M., and Nougaro, J., "Etude Th6orique de la Repartition du D6bit le Long d'un DeVersoir Lateral," Le Ginie Civil, Tome 144, No. 4, 1967, pp. 311-316.

3. El-Kashab, A., and Smith, K. V. H., "Experimental Investigation of Flow over Side Weirs," Journal of the Hydraulics Division, ASCE, Vol. 102, No. HY9, Sept., 1976, pp. 1255-1268.

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APPENDIX II.—NOTATION

The following symbols are used in this paper:

A a B b c

D F g H

AH H's

Ho h J T

; K

n* Q

AQ Q' s

st Sf So

t u V w w X X

y y z z a

= = = = = = = = = = =

= = = = = = = = = = = = = = = = = = = = = = = = =

channel area; opening width; channel width; constant channel width; weir shape coefficient; a/H, relative opening height; Froude number; gravitational acceleration; energy head; head loss; energy loss gradient (primes indicate ordinary differentiation with respect to x); energy head at control point; flow depth; relative slope; reduced relative slope; weir shape coefficient; number of outflow sides; discharge; lateral outflow discharge; outflow intensity; total energy gradient; average branch loss gradient; friction gradient; channel inclination; h - Sx, transformed flow depth; average outflow velocity; average channel velocity; w/H, relative weir height; weir height; nondimensional length coordinate; length coordinate; h/H, relative flow depth; transverse coordinate; vertical bottom coordinate; vertical coordinate; Q'h/Q, relative outflow rate;


0 = momentum correction coefficient; AL = length of distribution channel;

6 = tangent of diverging angle; 6 = e/K, reduced diverging angle; £ = head-loss coefficient;

<J = S + $ - a, pseudouniform flow parameter; $ = relative channel divergence; and <S? = outflow angle.

Subscripts d = downstream;

ex = experimental; PN = pseudouniform;

th = theoretical; and M = upstream.


distribution channels - uliege.be · nonprismatic channel geometry and the outflow characteiistics;...

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