Chapter 8 Flow Characteristics and Modelling of Head

Chapter 8

Flow Characteristics and Modelling ofHead-discharge Relationships for Weirs

8.1 Introduction

In Chapters 5 and 7, the formulations of the numerical models for the simulations of

flow surface and bed pressure profiles of steady rapidly varied open channel flows were

discussed. These models have been developed for the solutions of various types of

Boussinesq equations for the purpose of examining the nature of their solutions. In

this chapter, detailed descriptions of the free and submerged flow characteristics of a

family of broad-crested weirs will be presented. The main differences between the flow

behaviours of short- and broad-crested weirs will be briefly described first, followed by

the discussion of the hydraulics of a broad-crested weir. Also, the nature of the weir

flow problems and the procedure for the numerical simulation of these flow problems

for establishing head-discharge relationships will be described. A brief discussion of

the numerical results in comparison with experimental data will be presented at the end

of this chapter. All discussions in this chapter will be focused on broad-crested weirs

with finite upstream and downstream slopes.

8.2 Critical flow condition and flow control in open channel

8.2.1 Critical flow theory

Critical flow is an intermediate flow state between subcritical and supercritical flows

in which the energy per unit weight of the flow is minimum for a given discharge or

alternatively it is a flow state corresponding to maximum unit discharge for a constant

head (Jaeger, 1949, cited in Hager, 1985b). This state of flow is characterised by flow

possessing a velocity equal to the translational velocity of a small wave. The velocity of

propagation of such a wave on still water in a non-rectangular channel (Montes, 1998,

p35) reads as

c =

rgA

αT, (8.1)

221

222 Chapter 8. Flow Characteristics and Modelling of Head-discharge

where c is the velocity of the wave and T is the top width of the channel. Subcritical

flow is known by the mean flow velocity less than the velocity of the wave whereas in

supercritical flow the opposite feature is true. Thus, the changes in depth communicated

by the passage of such waves depend on the relative velocity of the waves with respect

to the local flow. In the vicinity of the critical section the flow possesses appreciable

streamline curvatures and slopes which influence the magnitude of the critical depth of

flow. A general expression, which accounts for the effects of the streamline curvature,

for predicting the critical depth (Jaeger, 1957, p140) for flow in any cross-sectional

shape of a channel is given by

αQ2T

KpgA3=

α

KpFr2 = 1, (8.2)

whereKp is the pressure correction coefficient. Assumption of constant values of α and

Kp at a section (∂α/∂H = ∂Kp/∂H = 0) is used to develop this equation. In a num-

ber of cases of practical importance, however, the critical depth may be calculated using

reasonable simplifying assumptions based on the condition of the flow problem consid-

ered. For instance, for transcritical flow over a broad-crested weir the deviations of the

correction coefficients, Kp and α, from unity are insignificant (α ∼= Kp∼= 1.0). For

such type of weir, the resulting simplified equation can be used to estimate the critical

depth of the flow. The critical flow condition described above allows determination of

the head-discharge relationships for long-based weirs under free flow conditions.

8.2.2 Flow control sections in open channel

At a critical control section, the relationship between the depth and the discharge is

unique, independent of the channel roughness and other uncontrolled circumstances.

Such a unique stage-discharge relationship offers a theoretical basis for the measure-

ment of discharge in open channels. In practice the section of a long prismatic channel

with mild slope serves as a control for establishing a relationship between head and dis-

charge (Fenton, 2001). In this case the control is due to friction in the channel giving

a unique relationship between the flow and the slope of the channel, the stage, chan-

nel geometry, and roughness. Montes (1998, p50) states that the unique features of a

critical control section compared to other sections, which also exhibit a single valued

stage-discharge relationship, is the fact that in a critical control section there is a pas-

sage from subcritical to supercritical flow so that the upstream region is isolated from

small perturbations generated downstream. The precise location of such control sec-

Chapter 8. Flow Characteristics and Modelling of Head-discharge 223

tion is dependent on the slope of the channel, and it occurs in a channel reach where the

slope changes from mild to steep.

Artificial control structures such as a weir can also be used as a flow control device in

the absence of a control section in open channels. Due to the existence of a critical

section on the crest of the structures, the upstream flow becomes independent of the

tailwater flow condition. This results in the establishment of a consistent relationship

between the overflow head and the discharge. Similar to the channel control section,

the discharge is computed from a single measurement of depth upstream of the control

structure. The present study focused on a control established by an embankment type

of flow control structure.

8.3 Classification of weirs and location of gauging stations

8.3.1 General discussion and classification of weirs

A weir with finite crest width in the direction of the flow is referred to as long-based weir

(Chadwick and Morfett, 1998, p403). The long-based weir may also be classified based

on the value of the overflow head to crest length ratio as a broad-crested or short-crested

weir.

A broad-crested weir is an overflow structure with a horizontal crest above which the

deviation from a hydrostatic pressure distribution because of vertical acceleration may

be neglected. In other words, the streamlines are nearly straight and parallel. The

criterion to obtain this situation as reported by Bos (1978, p15) is the length of the weir

crest in the direction of flow Lw, should be related to the total energy head over the weir

crest H1, as 0.08 ≤ H1/Lw ≤ 0.50. If H1/Lw is less than 0.08, then the energy losses

above the weir crest cannot be neglected, and undulations may occur on the crest. If

H1/Lw is less than or equal to 0.50, then only slight curvature of streamlines occurs

above the crest and a hydrostatic pressure distribution may be assumed. Experimental

studies of flow over a broad-crested weir indicate that the flow passes through the critical

state at some section on the crest, and the location of this section varies appreciably with

head and weir proportions.

The broad-crested weir is an intermediate case between a transition in which the flow

is wholly curvilinear and in which boundary resistance predominates, and hence both

accelerative and viscous effects must be considered in its analysis. However, the influ-


ence of the latter effect on the discharge capacity of the weir is limited to small overflow

head.

Short-crested weirs are those overflow structures in which the streamlines of the flow

above the weir crest have pronounced curvatures and slopes (see Figure 8.1). This char-

acter of the flow streamlines has a significant influence on the head-discharge relation-

ships of the structures. As discussed by Bos (1978, p27), the main difference between a

broad-crested weir and a short-crested weir is that nowhere above the short crest can the

curvature of the streamlines be neglected; there is thus no hydrostatic pressure distribu-

tion anywhere over the crest of the weir. Depending on the magnitude of the ratio of the

overflow head to the length of the crest of the weir, the same flow measuring structure

can act as a broad-crested weir or a short-crested weir. H1/Lw ∼= 0.33 is the delineating

value which separates these two types of flow control structures. The two-dimensional

flow patterns over a short-crested weir require the application of a higher-order flow

model for the complete description of the flow problem.

Figure 8.1: Flow over a short-crested weir (q = 462.1 cm2 / s)

8.3.2 Gauging stations for overflow head and tailwater depth

The head measurement station for free flow conditions should be located sufficiently

far upstream of the structure to avoid the influence of the curvature of the flow surface

on the head measurement. It should also be close enough to minimise the energy loss

between the head measurement station and the structure. Harrison (1967) suggested

that for a streamlined broad-crested weir the position of the upstream gauging station

should be at least 1.7 times the overflow head upstream from the face of the structure.

Bos (1978, p58) generalised the location of this station for different weir profiles and


recommended the position of the gauging station at a distance equal to between two

to four times the maximum overflow head from the structure. The accuracy of the

discharge measurement depends solely on the precision of the reading of the head over

the control structure. In this thesis, a procedure based on the established knowledge for

the location of this station will be incorporated in the numerical model, which simulates

transcritical flow over trapezoidal profile weirs, for the purpose of developing head-

discharge relationships for these weirs. Detailed discussion of the procedure will be

presented in Section 8.7.

Observation of the flow surface profile on the downstream side of a broad-crested weir

for free flow conditions reveals that the tailwater depth increases gradually with distance

in the direction of the flow. From the theoretical point of view, the downstream mea-

suring gauge should be located at a section free of surface drawdown effects. However,

it is difficult to generalise the location of this section due to the complex nature of the

submerged flow behaviours downstream of the weir. For the purpose of analysing flow

behaviour related to submerged flow conditions, the tailwater depth can be measured at

a well-prescribed point in which the influences of the curvature of the water surface and

the tailgate effect on the tailwater depth measurement are insignificant.

8.4 Submergence ratio and modular limit

The submergence ratio at a flow control structure may be defined as the ratio of the

downstream flow depth to the upstream depth of flow above the crest of the structure.

For low submergence ratios, critical flow occurs at some section on the crest of the

structure, and the tailwater conditions have no effect on the upstream flow depth. This

flow condition is referred to as modular flow. For such flow condition, the discharge is

computed from a single measurement of depth upstream of the control structure. At very

high submergence ratios, critical flow no longer exists at any section on the crest of the

structure. This non-modular flow condition requires that two flow depths – upstream

overflow depth and tailwater depth above the crest – be measured to approximate the

discharge of the control structures. The modular limit is the value of the submergence

ratio when the flow just begins to be affected by the downstream level, that is, when the

flow control structure begins to be drowned (see e.g., Bos et al., 1984, p66). From the

practical point of view, the modular limit is very important in defining the limit up to

which a flow control structure can be calibrated for discharge measurement. In terms of


flow depths above the crest, the modular limit is described as:

ML =hlh1, (8.3)

where:ML = modular limit,

hl = modular limit tailwater depth with reference to the crest,

h1 = sill-referenced upstream flow depth.

The roughness of the surface of the control structure and approach channel have signif-

icant influence on the values of the modular limit (Kindsvater, 1964). The effect is to

decrease its value. However, the height of the weir has little influence on the modular

limit of the weir. Bos (1985, #4.25) presented an analytical procedure for estimating the

modular limit of a long-throated flume (also valid for an hydraulically similar broad-

crested weir) based on the estimation of the total energy losses between the upstream

and downstream gauging stations.

In this study, the limiting tailwater depth for a given discharge was determined experi-

mentally by gradually raising the tailwater and observing the tailwater level at which the

upstream water level began to rise. The result of these experiments to define the mod-

ular limit of the trapezoidal profile weirs is shown in Figure 8.2. This figure indicates

that the modular limit of the weirs decreases with increasing of the ratio of the crest ref-

erenced head to the length of the weir crest, H1/Lw. This implies that the effect of the

curvature of the streamline of the flow over the crest is to decrease the modular limit of

the weir (see e.g., Bos, 1978, p86). For the case of flow over a broad-crested trapezoidal

profile weir, the curvature of the streamline at the control section weakly affects the

modular limit of the flow. As a result such type of control structure has a higher modu-

lar limit compared to a short-crested trapezoidal profile weir (see Figure 8.2). This fact

suggests that flow over a broad-crested weir takes place with minimum loss of head as

compared to a similar short-crested trapezoidal profile weir. For a long broad-crested

weir (H1/Lw → 0), an upper limit value ofML = 0.85 is reached (Kindsvater, 1964).

8.5 Description of weir flow

8.5.1 Nature of the flow problem

Theoretical analyses of flows over trapezoidal profile weirs are complicated by a com-

bination of effects related to the approach channel condition, geometry of the control


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

H1/Lw

ML

Kindsvater (1964)Lw = 40 cmLw = 10 cm

Submerged flow

Free flow

Figure 8.2: Modular limit for trapezoidal profile weirs

structure, property of the fluid and flow patterns. Experimental investigations are also

complicated by the occurrence of several significantly different flow patterns under free

and submerged flow conditions. The characteristic difficulties associated with the flow

phenomena of these weirs makes both the theoretical and experimental analyses more

complicated and difficult.

The flow patterns for trapezoidal profile weirs involve boundary-layer growth under

conditions of acceleration and separation, and non-hydrostatic pressure and nonuniform

velocity distributions due to the curvature of the streamlines over the crest of the weir. It

is evident that a general analytical expression for the discharge of the weir is impossible

due to the complicated nature of the flow. However, existing theoretical procedures can

be used to predict the discharge under specific conditions that are related to the overflow

head to crest length ratio.

On the other hand, a numerical model based on a higher-order governing equation may

be used to establish head-discharge relationships for practical solution of the flow prob-

lems. Examining the flow profile over a trapezoidal profile weir shows that the up-

stream far side flow surface profile approaches the normal flow depth asymptotically

for approach channel having mild or steep bed slope. On the downstream side of the

structure, for different tailwater depths below the modular limit tailwater depth the flow


changes from one flow regime to another without affecting the upstream flow situation.

These flow characteristics of a trapezoidal profile weir complicate the locations of the

inflow and outflow sections of the computational domain of the numerical model. How-

ever, systematic specification of the boundary values for a particular flow condition may

give better numerical solution of the problem and overcome the limitations associated

with the existing theory. The numerical solution of such model must be validated with

experimental data for assessing the drawbacks of the model.

8.5.2 Flow behaviour over short- and broad-crested weirs

The actual flow behaviour over short- and broad-crested control structures is quite com-

plex, involving a three-dimensional velocity pattern as well as viscous effects. The

viscous effects, which are more pronounced for the case of long broad-crested control

structure, modify the distribution of the velocity of the flow and cause a loss of energy.

For such structures, a correction factor should be introduced to account for viscous ef-

fects.

For flow over a control structure, the two significant parameters which describe the flow

characteristics are the ratios of the overflow head to the height of the control structure,

and to the crest length of the structure. The former ratio is a function of the Froude

number of the incoming flow in the channel and it indicates the significance of the

velocity head of the flow in the prediction of the total overflow head. At relatively

higher Froude number, the approach velocity cannot be neglected in determining the

upstream total overflow head. The ratio of the overflow head to the crest length of the

structure measures the shortness or breadth of the structure. The value of this ratio

directly determines the degree of the curvature of the streamline of the flow over the

crest of the control structure for free flow conditions.

Escande (1939, cited in Fritz and Hager, 1998) classifies various types of flows over

cylindrical-crested weirs. This classification is equally applicable to other geometrical

shape flow control structures such as trapezoidal profile weirs (see e.g., Kindsvater,

1964; Wu and Rajaratnam, 1996; Fritz and Hager, 1998). Four different types of flow

may occur depending on the height of the tailwater depth. These are the:

1. Free overflow;

2. Plunging flow;


3. Surface wave flow; and

4. Surface jet flow.

Free overflow

For free overflow conditions, the critical depth is located at the crest of the trapezoidal

profile weir somewhat downstream of the upper edge of the crest. The position of the

critical section varies along the weir crest depending on the magnitude of the discharge

as shown in Figures 8.3 and 8.4. For a supercritical flow state at the outflow section,

a pure transcritical flow without any shock wave can be observed under free flow con-

ditions. For larger tailwater depth, a hydraulic jump occurs with its toe located at or

downstream of the toe of the control structure (see Figure 8.5). Fritz and Hager (1998)

noted that the flow configuration of the jump at the downstream side of the trapezoidal

profile weir is identical to the classical hydraulic jump. The position of the jump varies

along the bed of the downstream channel and is entirely controlled by the level of the

downstream tailwater depth. For free flow conditions, when the surface tension effects

are negligible, the discharge over the weir depends on the head above the crest of the

weir and is independent of the tailwater depth. The common broad-crested weir equa-

tion can be applied to estimate the discharge of the structure provided that the curvature

of the streamlines of the flow over the crest is insignificant.

Plunging flow

The first detailed experimental analysis of submerged flow patterns downstream of the

trapezoidal profile weirs was given by Fritz and Hager (1998). Different flow character-

istics were observed for the downstream submerged flow cases. These observations are

very similar to the results of the experimental studies of submerged flow patterns down-

stream of a sharp-crested weir (except breaking wave) by Wu and Rajaratnam (1996).

For a given discharge as the tailwater depth increases, the submerged flow downstream

of the trapezoidal profile weir passes through several regimes. Figure 8.7 illustrates the

observed flow surface profiles for different flow regimes for flow over a trapezoidal pro-

file weir. For the plunging jet flow, the flow over the weir plunges into the tailwater with

a concentration of forward flow along the bottom and a backward flow (surface roller)

at the surface (see Figure 8.6). The main flow diffuses as a plane submerged jet along

the downstream face of the structure, and hits this face and bed of the channel. The


-0.1

0.0

0.1

0.2

0.3

-1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6

Horizontal distance (m)

Flow

sur

face

(m)

Bed profile

H1/Lw = 0.463

H1/Lw = 0.634

H1/Lw = 0.795

H1/Lw = 0.952

H1/Lw = 1.156

Critical depth

Figure 8.3: Free flow over a short-crested trapezoidal profile weir

-0.1

0.0

0.1

0.2

0.3

-1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6


Flow

sur

face

(m)

Bed profileH1/Lw = 0.144H1/Lw = 0.196H1/Lw = 0.250H1/Lw = 0.280H1/Lw = 0.317Critical depth

Figure 8.4: Free flow over a broad-crested trapezoidal profile weir


Figure 8.5: Transcritical flow over a weir with hydraulic jump (q = 462.1 cm2 / s)

position of the starting point of the plunging flow varies along the downstream face of

the structure depending on the tailwater depth. The concentration of eddies decreases

with increasing of tailwater depth of this flow regime. For tailwater level nearly equal

to the upper limit tailwater depth, the starting point of the plunging flow is just at the

downstream edge of the crest of the structure. Shifting of the plunging flow starting po-

sition by a small distance upstream of the downstream end section of the crest changes

the flow pattern into surface wave flow. Figure 8.8 shows the velocity distribution pro-

file for the plunging flow regime. It is this flow pattern which causes maximum erosive

velocities on the downstream face of the trapezoidal profile weirs (Kindsvater, 1964).

Figure 8.6: Flow pattern for plunging flow regime (q = 462.1 cm2 /s)


-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0Horizontal distance (m)

Flow

sur

face

(m)

Bed profileFree overflowPlunging flowSurface wave flowSurface jet flow1: x = 1.145 m2: x = 1.345 m

q = 222.4 cm2/s

1 2

Figure 8.7: Typical flow profiles for different flow regimes

0.0

0.2

0.4

0.6

0.8

-40 -20 0 20 40 60 80 100Horizontal velocity (cm/s)

h s/H

Measured (1, x = 1.145 m)


Figure 8.8: Velocity distribution in plunging flow regime (q = 222.4 cm2 / s)


Surface wave flow

Increasing the tailwater level to a point above the upper limit depth for plunging and

below the modular limit tailwater depth results in the surface wave flow which is char-

acterised by the presence of the first standing wave near the downstream end of the weir

crest followed by waves of decreasing amplitude (see Figures 8.7 and 8.9). The for-

ward flow for this flow regime is along the surface and backward flow along the bottom

and lower face of the weir. This fact is clearly indicated in the velocity profile which

is shown in Figure 8.10. The position and amplitude of the standing waves depend on

the level of the tailwater depth. At higher tailwater depth, the stationary waves near the

end section of the crest have considerable curvature. Further downstream, however, the

curvatures of the waves are insignificant.

For tailwater depth less than the modular limit tailwater level, the flow transition over

the control structure is from subcritical to supercritical and then to subcritical with min-

imum loss of energy. The interesting thing is that the latter transition is with the forma-

tion of undular surfaces which resemble in character the nearly two-dimensional (with-

out the appearance of any cross waves) undular hydraulic jump. Fawer (1937, cited in

Jaeger, 1957, p152) stated that the surface of the jump is undular if the supercritical

sequent depth is greater than 67% of the critical depth of flow. The analysis of the ex-

perimental data of this study confirms Fawer’s observation (see Table 8.1). When the

trough of the first standing wave passes through a flow section near the axis of sym-

metry of the weir, the flow over the structure completely changes to submerged flow.

Compared to the plunging flow regime, the erosive tendencies of this flow regime are a

minimum.

Table 8.1: Supercritical sequent depth to critical depth ratios for undular jump

Discharge (cm2 / s) Conjugate depth, h1 (cm) Critical depth,Hc (cm) h1/Hc (%)626.22 6.87 7.37 93.22563.14 6.44 6.86 93.88495.70 5.92 6.30 93.97422.17 5.45 5.66 96.23

Surface jet flow

In the case of surface jet flow in which the tailwater level is above the modular limit

tailwater depth, the approach flow is completely submerged. The configuration of the


Figure 8.9: Surface wave flow regime (q = 462.1 cm2 / s)

0.0

0.2

0.4

0.6

0.8

-20 -10 0 10 20 30 40 50 60 70 80

Horizontal velocity (cm/s)

h s/H



Figure 8.10: Velocity distribution in surface wave flow (q = 222.4 cm2 / s)


flow profile depends on the degree of submergence of the flow. For lower submergence

ratios, undular surfaces can be seen on the crest of the weir. At higher submergence

ratios, the flow profile becomes almost horizontal as indicated in Figures 8.7 and 8.11.

The forward flow is again along the surface with bottom recirculation in the region

near the downstream end of the structure (see Figure 8.12). In both the surface wave

and surface jet flow cases, the flow remains as a jet at the surface in the downstream

channel.

For flow control structures such as weirs, the discharge over the structure up to the mod-

ular limit tailwater depth depends on the upstream overflow head. Previous experimen-

tal investigations (see e.g., Kindsvater, 1964) for flow over broad-crested trapezoidal

profile weirs show that this depth corresponds to the surface wave flow regime of the

downstream submerged flow condition. This in turn implies that the free flow discharge

equation can be used to estimate the discharge over a broad-crested type of such weir

not only in the pure transcritical flow condition but also in the downstream submerged

flow condition up to the modular limit tailwater depth. If the tailwater depth is above

the modular limit depth, the flow control structure will submerge totally and moreover,

it no longer acts as a flow control structure. Because of submergence, the discharge ca-

pacity of the structure decreases markedly. This is the behaviour of the surface jet flow

regime.

8.5.3 Free flow transition ranges

For a definite range of tailwater levels, a given discharge produces either a plunging

flow or a surface wave flow on the downstream side of the trapezoidal profile weirs. The

transition from plunging flow to surface wave flow, and vice-versa, occurs within a well

defined range of tailwater levels (Kindsvater, 1957). The upper limit of the transition

range corresponds to the maximum tailwater depth in which the plunging flow remains

as a stable plunging flow. A small increment of the tailwater depth above the upper

limit value abruptly changes this flow pattern to surface wave flow. The lower limit

of the transition range is the minimum tailwater level in which the stable surface wave

flow pattern converts abruptly to plunging flow for tailwater depth slightly below this

minimum value. Between the upper and lower limits of the transition either flow may

occur. This particular switching mechanism adds to the complexity of the submerged

flow pattern downstream of the weir. Figures 8.13 and 8.14 show details of flow near


Figure 8.11: Submerged flow over a weir with 97% submergence ratio

0.0

0.2

0.4

0.6

0.8

-10 0 10 20 30 40 50 60 70 80Horizontal velocity (cm/s)

h s/H

Measured (1, x = 1.145 m)Measured (1, x = 1.345 m)

Figure 8.12: Velocity distribution in surface jet flow regime


the lower and upper limit of the transition range. In practice these sequences of events

would normally occur during a rising and falling of flood stages. The slope of the

downstream face of the weir and surface roughness influence the transition range limits.

Surface roughness tends to lower the upper and lower limit values of the transition range

(Kindsvater, 1964).

Figure 8.13: Flow near the lower limit transition range (q = 512.2 cm2 / s)

In this study, the upper and lower limits of the transition range for flow over trapezoidal

profile weirs were determined experimentally by gradually raising and lowering the

tailwater and observing the levels at which the transition took place. Figure 8.15 shows

the results of the tests made to define the lower and upper transition stage using the mean

curves of the experimental data. In this figure the transition submergence, ξt = htc/h1(htc and h1 = crest-referenced tailwater depth and overflow depth respectively) is shown

versus the ratio of the overflow head to weir crest length, H1/Lw. The transition range

separates the upper and lower limit curve, starting at 0 < ξt < 0.60 for H1/Lw = 0,

decreasing to 0.768 < ξt < 0.874 for H1/Lw = 1.045. As the overflow head to weir

crest length ratio increases, the upper and lower limit curves of the transition range rise

gradually particularly forH1/Lw ≥ 0.20 (see Figure 8.15). This shows that the effect of

the curvature of the streamlines of the flow over the crest of the weir is to decrease the

transition range of the free flow. It clear from this figure that flows are always plunging

for tailwater level below the weir crest but above the subcritical sequent depth of the free

jump (i.e., ξt < 0). It also indicates that surface wave flows always occur for transition

submergence value greater than 0.874 (i.e., ξt > 0.874).


Figure 8.14: Flow near the upper limit transition range

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

H1/Lw

ξ t

Upper limitLower limitKindsvater (1964)

Transition range

Figure 8.15: Free flow transition range for trapezoidal profile weir


8.5.4 Nature of bed pressure distributions in different flow regimes

The level of the tailwater depth affects the distribution of the pressure on the surface of

the trapezoidal profile weir. Figure 8.16 illustrates the observed bed pressure profiles

in different flow regimes (the case of rising tailwater depth). In the regions around the

corners of the broad-crested trapezoidal profile weir, the streamline curvatures for the

free flow regime are very sharp (see Figure 8.4). Consequently, the pressure distribu-

tions strongly deviate from hydrostatic in these regions. For tailwater depth below the

upper limit transition depth, the submerged flow on the downstream side of the weir be-

comes plunging flow. In this flow regime, the bed pressure on the downstream face of

the trapezoidal profile weir is higher than the corresponding pressure for free flow situa-

tion due to the increasing of the level of the tailwater on this face of the weir. Examining

the flow profile of this flow regime for tailwater depth very close to the upper limit tran-

sition depth (see Figure 8.14) reveals that the pressure distribution is nearly hydrostatic.

For lower tailwater depth in the plunging flow regime, the curvature of the streamline

influences the bed pressure distributions on the downstream face of the structure.

-0.1

0.0

0.1

0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2


Bed

pre

ssur

e (m

)

Free flowPlunging flowSurface wave flowSurface jet flow

q = 222.4 cm2/s

Weir crest

Figure 8.16: Typical bed pressure distributions for different flow regimes (Lw = 400 mm)

The surface wave flow regime is characterised by the presence of waves of decreasing

amplitude in the direction of the flow as shown in Figure 8.7. This implies that the


pressure distribution within this flow regime is non-hydrostatic. As the tailwater level

approaches the modular limit tailwater depth value, the influence of the curvature of

the flow surface on the pressure distribution increases with increasing of the curvature

of the waves. Compared to the plunging flow bed pressure values, the bed pressures

on the downstream face of the structure are relatively large due to the increasing of

tailwater level and flow surface curvature (this is true for positive curvature effects).

At a higher degree of submergence, the flow profile in the surface jet flow regime is

nearly horizontal. For this condition, the pressure distribution within the flow region is

purely hydrostatic. The shape of the bed pressure distribution curve along the length of

the structure is similar to the inverted shape of the trapezoidal profile weir (see Figure

8.16). The bed pressure corresponding to this flow regime is the maximum of all the

cases of the flow regimes related to the weir flow situations.

8.5.5 Hydraulics of broad-crested weirs

Free flow discharge equation

Because of the complex nature of the flow over weirs which is influenced by roughness,

turbulence levels, geometry of the structure and several other parameters, it is difficult to

develop a free flow discharge equation precisely. The discharge formulas, independent

of the shape of the weir, relate the discharge to the upstream overflow depth via the

discharge coefficient. Hence, the discharge coefficient represents the combined effects

of all parameters that influence the free flow pattern. The accuracy of determining the

discharge over the weir under free flow conditions depends on the reliability and validity

of the discharge coefficient, and the sensitivity to the measurement of head over the flow

control structure.

For a broad-crested weir with rectangular control section, the head discharge relation

that takes into account the approach velocity head is given by the following equation:

Q = CdBp2g

µh1 +

αV 20

2g

¶3/2, (8.4)

where:Cd = discharge coefficient,

V0 = approach velocity,

B = width of weir perpendicular to the flow.


In the above equation, equation (8.4), an energy head correction coefficient, α, is in-

troduced to take into account non-uniformity of the approach velocity arising from

the effects of the shape of the channel cross-section, boundary irregularities and cur-

vature of the streamlines. For low overflow energy head with respect to weir height

(H1/Hw < 1/6), the magnitude of the approach velocity is very small. However, the

non-uniformity of the approach velocity increases with increasing of the overflow en-

ergy head relative to the weir height. According to Bazin (1898, cited in Fritz and Hager,

1998), the energy head correction factor for large approach velocity is 5/3.

French (1985, #8.3) based on the work of Bos (1978), provides the following criteria to

analyse flow over a broad-crested weir:

• H1/Lw < 0.08, flow over the crest is subcritical, and the weir cannot be used to

determine the discharge.

• 0.08 ≤ H1/Lw ≤ 0.33, the discharge equation used in this range will estimate the

flow rate accurately.

• 0.33 ≤ H1/Lw ≤ 1.50 to 1.80, the weir is no longer broad crested but it should be

classified as a short-crested weir.

• 1.50 ≤ H1/Lw, the flow pattern over the weir crest is unstable and the nappe may

separate completely from the crest. The weir characteristics approach a sharp-

crested weir.

Bos (1978, p28) suggests that for flow over a short-crested weir with rectangular control

section, a head-discharge equation similar in structure to equation (8.4) can be used for

discharge computation. The discharge coefficient takes into account the effect of the

streamline curvature besides other factors. Such type of weir has a higher discharge

coefficient compared to a broad-crested weir due to the substantial curvature of the

streamline over the crest of the weir.

Empirical discharge equation for submerged flow

Equations of discharge for free flow have been derived on the basis of a simple energy

analysis. The analysis was made possible because critical-flow control occurs on the

crest of the control structure when the flow is under free flow conditions. For submerged

overflow conditions, the flow passes over the structure in a subcritical state so that the

discharge depends on both the upstream and downstream water levels. Several investi-


gators proposed different empirical formulas to predict this non-modular discharge. Du

Buat (1816) presented an equation for the computation of submerged discharge over a

trapezoidal shaped weir. He considered the submerged flow as a flow consisting of free

flow over the weir and flow through a submerged orifice under the tailwater referenced

head. This simplifying assumption cannot be justified but gives a framework in which a

constant can be attached to the equation. This equation has the form (Ellis, 1947, p79):

Qs = CsBph1 − htc

¡h1 +

12htc¢, (8.5)

where:Qs = discharge passing under submerged condition,

Cs = coefficient of non-modular discharge.

The value of Cs in this equation must be determined by experiment. The accuracy of

the prediction of the submerged discharge using equation (8.5) depends not only on the

measurement of the upstream and downstream depths but also on the predetermined

value of the submerged discharge coefficient. However, it is difficult to formulate a sub-

merged flow equation independent of the tailwater depth. The most convenient alterna-

tive is an empirical solution based on experimental data analysis and free flow discharge

equation (see e.g., Hager, 1994; Wu and Rajaratnam, 1996; Fritz and Hager, 1998).

A simple functional relationship, which describes the effect of submergence on the dis-

charge capacity of the weir, in terms of the free flow discharge, qfree can be expressed

as

qsub = ϕ0qfree, (8.6a)

ϕ0 = G

µhtlhll

¶, (8.6b)

where:qsub = the submerged flow rate of the control structure,

G = function for the submerged flow reduction factor,

htl = tailwater depth with reference to the modular limit depth (ht − hl),ht = the tailwater depth above the channel bed at the downstream

gauging station,

hll = overflow depth with reference to the modular limit depth (H − hl),H = channel bed referenced upstream depth of flow at the

corresponding gauging station,

ϕ0 = the submerged flow reduction factor.


It has to be noted that the ratio, htl/hll, is not the same as the submergence ratio of a

flow control structure which was defined in Section 8.4. The above functional relation-

ship can be determined from the plot of experimentally determined discharges (free and

submerged) and submergence ratio with reference to the modular limit tailwater depth

using data modelling techniques. Since the free flow discharge is constant over the full

range of the conditions considered, the empirical solutions based on equation (8.6b) are

adequate to predict the submerged discharge. This method will be applied in this study

to establish an empirical relationship for the submerged discharge of the trapezoidal

profile weirs. The results will be presented in Section 8.9.

Submerged flow control structures are not recommended for the practical measurement

of discharge. This is because of the following reasons (Bos, 1985, p66):

• the submerged flow reduction factor for a given control structure is not only a func-

tion of submergence ratio but also a function of the free overflow discharge. How-

ever, this discharge is to be measured.

• for a control structure with higher modular limit, the submerged flow reduction

factor cannot be determined at the required accuracy. Any errors related to the

measurement of both the upstream and downstream heads with respect to the crest

of the weir directly influence the value of the submergence ratio.

• estimation of the non-modular flow rate of a structure requires measurement of both

the upstream and downstream crest-referenced heads. From the practical point of

view, however, measurement of two heads is time consuming and expensive.

• the submerged flow reduction factor for a given flow control structure is basically

determined based on available experimental data. This requires the construction

of different sizes of physical models to conduct the experiments in a laboratory for

the complete range of discharges. In general, this process is relatively expensive.

8.6 Theoretical weir discharge coefficients

The common head-discharge relationship for a flow control structure is formulated

based on a number of idealised assumptions such as absence of energy losses between

the gauging and control sections; uniform velocity distribution in both sections; and

negligible streamline curvatures at the gauging and control sections. However, these

simplifying assumptions are not often correct. A discharge coefficient must be intro-


duced to take into account the effects of these assumptions in the estimation of dis-

charge. This coefficient is theoretically determined by assuming hydrostatic pressure

distribution at the crest and face of the flow control structure. However, the curve for

the actual pressure distribution at these sections lies below the curve for the hydrostatic

pressure distribution due to the effect of the negative curvature of the streamlines. It was

shown by Matthew (1963) that the discharge coefficient for flow over a circular-crested

weir is affected by both the streamlines curvatures and the absolute scale of the flow.

Free flow over a weir is characterised by flow transition from subcritical to supercriti-

cal states. In the vicinity of this transition the streamlines of the flow have considerable

curvature and slope. Owing to this fact, the application of the conventional method of

analysis to such kind of flow problem results in underestimating the discharge capac-

ity of the weir. Based on the BTMU equation, a general expression for weir discharge

coefficients will be developed here. The theoretical discharge coefficient expression in-

cludes terms which account for the impact of the curvature of the streamlines. This

implies that the effect of the non-hydrostatic pressure distribution is implicitly incorpo-

rated in the resulting discharge coefficient equation.

Rewriting equation (8.4) for the head-discharge relationship under free flow conditions

as

q = Cdp2gH

3/21 , (8.7)

where:q = discharge per unit width,

H1 = upstream total energy head above the weir crest (h1 + αV 20/2g).

Using the flow equation, equation (3.50a), the discharge capacity of the control structure

as a function of the flow depth and other hydraulic parameters at a section can be written

as

q2 =gH (Hx + Z

0b + Sf)

−β ω12ϕ

³Hxxx + ξ0Hxx + 2

ω0ω1

³Z000b2+

Z00b Z0b

H

´´+ βHx

H2

, (8.8)

where:H = flow depth above the bed,

ϕ = 1 + Z 02b .


For a trapezoidal-shaped flow control structure, the contribution of the bed curvature is

zero, i.e. Z 00b = Z 000b = 0. Using this fact, equation (8.8) reduces to

q2 =gH (Hx + Z

0b + Sf)

−β ω12ϕ(Hxxx + ξ0Hxx) + βHx

H2

. (8.9)

Equation (8.9) includes terms which reflect the effect of the curvature of the streamlines.

However, for flow over a broad-crested weir with H1/Lw ≤ 0.50, the flow surface over

the crest of the weir has almost a constant slope with negligible curvature (Hxxx =

Hxx = 0). For this kind of flow phenomenon, equation (8.9) becomes

q2 =gH3 (Hx + Z

0b + Sf)

βHx. (8.10)

Using equation (8.7) in equation (8.10) and further simplifying, one obtains

Cd =

Ã1

√2H

3/21

!µH3 (Hx + Z

0b + Sf)

βHx

¶1/2. (8.11)

Equation (8.11) relates the coefficient of discharge with the hydraulic parameters of the

flow and the total head at the gauging stations. This equation is valid for flow over the

weir with hydrostatic pressure distribution or insignificant curvature of streamline.

In contrast to a broad-crested weir, the flow pattern over a short-crested weir is charac-

terised by pronounced curvatures of streamlines. This behaviour influences the head-

discharge relationship as well as the modular limit of the flow control structure. Insert-

ing equation (8.7) into equation (8.9) and simplifying the resulting expression yields the

following equation:

Cd =

Ã1

√2H

3/21

!ÃH (Hx + Z

0b + Sf)

−β ω12ϕ(Hxxx + ξ0Hxx) + βHx

H2

!1/2. (8.12)

If the geometric characteristics of the surface streamline at a particular section (for in-

stance, crest section near the axis of symmetry of the weir) over the crest of the weir is

known in addition to the flow parameters, one can use equation (8.12) to estimate the

coefficient of discharge for flow over a short-crested weir with H1/Lw between 0.50

and 1.50. The values ofHxxx, Hxx andHx can be determined numerically using exper-

imental data for a given trapezoidal weir geometry to obtain the discharge coefficient,

Cd.

Using measured values of discharge and overflow head, the experimental discharge co-

efficients were computed from equation (8.7). These values are compared with the the-

oretical discharge coefficients estimated from equation (8.12) for free flow conditions


in Figure 8.17. The figure also shows the mean trend curve for the experimental dis-

charge coefficients and the corresponding equation. It can be seen from this figure that

the agreement between the predicted and the experimental result is fairly good.

Cd = -5.4259(H1/Lw)3 + 4.6894(H1/Lw)2 - 1.1456(H1/Lw ) + 0.4422

0.20

0.25

0.30

0.35

0.40

0.45

0.10 0.15 0.20 0.25 0.30 0.35

H1/Lw

Cd

Experimentally determined

Predicted

Figure 8.17: Comparison of experimentally determined and predicted discharge coefficients

8.7 Model development for establishing head-dischargerelationships

8.7.1 Formulation of the boundary value problem

As described before, a trapezoidal shaped weir over which water is flowing may be

treated as a short- or broad-crested weir depending on the magnitude of the overflow

head to weir crest length ratio. When free flow exists over the trapezoidal profile weirs,

the common free flow discharge equation, equation (8.4), may be used to determine

the quantity of water flowing over the weir with reasonable accuracy. However, this

equation does not give satisfactory results for the case of flow over a short-crested weir

due to the pronounced curvatures of streamlines over the crest of the weir. This indicates

that a general model, which includes the effects of the curvature of the streamlines

implicitly or explicitly, is essential for establishing discharge rating curves for such

types of weirs. Therefore, the main objective of this part of the thesis is to examine the


feasibility of the BTMU equation for the development of head-discharge relationships

for short- and broad-crested trapezoidal profile weirs. Hence, the research questions to

be answered in this simulation study are:

1. As a one-dimensional model typically does not have the ability to define a coef-

ficient of discharge, how can flow over a trapezoidal profile weir be accurately

portrayed?

2. Does the numerical model give similar overflow heads to those determined exper-

imentally?

The solution procedure of the numerical model to estimate the crest-referenced head

corresponding to the discharge over the trapezoidal profile weirs under free flow condi-

tions using the BTMU equation requires the specification of three boundary values. It

was mentioned in the preceding section that different flow regimes exist on the down-

stream side of this weir corresponding to different tailwater depths under the same free

flow conditions. These flow situations complicate the precise measurement of the tail-

water depth for free flow conditions. From the practical point of view, it is advantageous

to measure one flow depth in the subcritical flow region for the purpose of calibrating

the head-discharge relationships of the weirs. Because of these, a numerical procedure

based on the specification of boundary values at the inflow section only is employed

here for the solution of the BTMU equation to provide head-discharge relationships for

such types of weirs. This approach applies a solution procedure based on the Newton-

Raphson iterative scheme (similar to the method discussed in Chapter 5), which is en-

tirely different from a solution procedure based on the conventional shooting method.

The flow profile at the upstream far section is asymptotic to the normal depth of flow.

Depending on the height of the flow control structure and the slope of the approaching

channel, there are two possible flow surface profiles. If the height of the flow control

structure is greater than or equal to the normal depth of the approach flow , the flow pro-

file isM1 type or a backwater curve. Otherwise the profile isM2 or a drawdown curve.

However, in most practical cases the flow control structure is such that the approach

flow profile is a backwater curve.

For this flow simulation problem, the unknowns are a set of flow depths along the length

of the computational domain for a given discharge of the weir. Two additional equations

that relate the flow parameters with the slope and curvature of the flow surface at the


inflow section are necessary for a unique numerical solution of the flow problem. As

discussed in Chapter 3, the gradually varied flow equation is employed here for this

purpose. Based on the assumption of quasi-uniform flow condition upstream of the

inflow section, the conveyance factor of the flow at this section can be expressed as

a function of the section properties using the well-known Manning’s equation for a

rough bed channel. Therefore, the numerical model is to be developed based on the

known values of the parameters related to the geometry of the control structure and

the upstream boundary conditions. Assuming that the approach channel is wide, the

hydraulic radius of the flow at any section is approximated by the corresponding depth

of flow. This indicates that the effect of wall friction is neglected in this flow simulation

procedure as in the case of the common two-dimensional flow computational models.

Similar to the upstream inflow section of the computational domain, the gauging station

should be situated sufficiently far upstream of the trapezoidal profile weir to avoid the

influence of the curvature of the water surface on the magnitude of the estimated over-

flow depth. According to Bos et al. (1984, p36) this section is located at a distance of

the larger of the following two values:

• between two and three times the maximum crest-referenced head from the up-

stream end of the crest;

• the maximum crest-referenced head from the heel of the trapezoidal profile weir.

From the computational point of view, however, the maximum overflow head is not

known a priori to fix the position of the gauging station. In this numerical simulation

procedure, the overflow head corresponding to the maximum discharge at the heel of

the trapezoidal profile weir will be used to locate the gauging station approximately.

The computational domain for the numerical solution of the weir flow problem consid-

ered is shown in Figure 8.18. In this figure AB is the inflow section, CD is the outflow

section and BM is the approach channel bed. Also, MUVN is the trapezoidal pro-

file weir. The inflow section of the computational domain is located in a region where

the flow is assumed to be nearly horizontal with hydrostatic pressure distribution. This

quasi-uniform flow condition before the inflow section of the solution domain simpli-

fies the evaluation of the boundary values at this section using the gradually varied

flow equation. For the given boundary values and discharge at the inflow section, it is

required to determine the upstream depth of flow above the crest of the weir at the gaug-


ing stationGS. For this purpose, the computational domain bounded by the free surface,

inflow and outflow sections, and the solid flow boundary is discretised into equal size

steps in x as shown.

Nodes

Z

X

G

SGauging station-GS

A

B

CD

j = 0 j = J

U V

M N

h

Figure 8.18: Computational domain for transcritical flow over a weir

Boundary conditions

The complete numerical solution of the problem using the third-order governing equa-

tion requires specifying three boundary conditions. At the inflow section, the flow depth

is specified as an inflow boundary condition. For the subcritical approach flow, this

value corresponds to the quasi-uniform flow condition. The additional two boundary

conditions at the inflow section – the slope and curvature of the flow surface – are eval-

uated based on the specified flow depth and discharge at this section using equations

(3.27) and (3.33).

8.7.2 Computational model development

For the purpose of discretisation, rewriting the general form of the flow equation, equa-

tion (5.4), for flow in prismatic channel (after multiplying both sides by h3) as

h3d3H

dx3+ h3ξ0

d2H

dx2+ h3ξ1

dH

dx+ h3ξ2 = 0. (8.13)


The five-point upwind finite difference approximations are used here to replace the

derivative terms of the BTMU equation. These finite difference representations (Bick-

ley, 1941) for derivatives at node j areµdH

dx

¶j

=1

24h(−2Hj−3 + 12Hj−2 − 36Hj−1 + 20Hj + 6Hj+1) +O

¡h5¢, (8.14)

µd2H

dx2

¶j

=1

12h2(−Hj−3 + 4Hj−2 + 6Hj−1 − 20Hj + 11Hj+1) +O

¡h5¢, (8.15)

µd3H

dx3

¶j

=1

4h3(2Hj−3 − 12Hj−2 + 24Hj−1 − 20Hj + 6Hj+1) +O

¡h5¢, (8.16)

where O(h5) is the fifth-order neglected terms in each approximation.

Substituting equations (8.14), (8.15) and (8.16) into the general equation, equation

(8.13), gives

14(2Hj−3 − 12Hj−2 + 24Hj−1 − 20Hj + 6Hj+1)

+ 112hξ0,j (−Hj−3 + 4Hj−2 + 6Hj−1 − 20Hj + 11Hj+1)

+ 124h2ξ1,j (−2Hj−3 + 12Hj−2 − 36Hj−1 + 20Hj + 6Hj+1) + h3ξ2,j = 0. (8.17)

Simplifying the above expression and assembling similar terms together gives the gen-

eral finite difference equation for the derivative at node j

Hj+1¡6h2ξ1,j + 22hξ0,j + 36

¢+Hj

¡20h2ξ1,j − 40hξ0,j − 120

¢+Hj−1

¡−36h2ξ1,j + 12hξ0,j + 144

¢+Hj−2

¡12h2ξ1,j + 8hξ0,j − 72

¢+Hj−3

¡−2h2ξ1,j − 2hξ0,j + 12

¢+ 24h3ξ2,j = 0. (8.18)

In the solution domain, equation (8.18) is applied to evaluate nodal values between 1

and J − 1 inclusive. However, the use of equation (8.18) at the nodal point j = 1

introduces two unknowns which are external to the solution domain. Since the value of

the nodal point at j = 0 is known, we can use the additional two boundary conditions

at j = 0 to evaluate the nodal values at j = −1 and j = −2. Thus,

h

µdH

dx

¶0

= hSH =124(−2H−3 + 12H−2 − 36H−1 + 20H0 + 6H1) , (8.19)

h2µd2H

dx2

¶0

= h2κH =112(−H−3 + 4H−2 + 6H−1 − 20H0 + 11H1) , (8.20)

where SH and κH are the slope and curvature of the water surface at j = 0 respectively.

Eliminating the nodal value at j = −3 from equation (8.20) using equation (8.19); the

explicit expression for the nodal value at j = −2 in terms of the nodal point values at


j = −1, 0 and 1 becomes

H−2 = 6hSH + 12H−1 − 15H0 + 4H1 − 6h2κH . (8.21)

Application of the finite difference equation, equation (8.18), at node j = 0 results in

the following expression:

H1¡6h2ξ1,0 + 22hξ0,0 + 36

¢+H0

¡20h2ξ1,0 − 40hξ0,0 − 120

¢+H−1

¡−36h2ξ1,0 + 12hξ0,0 + 144

¢+H−2

¡12h2ξ1,0 + 8hξ0,0 − 72

¢+H−3

¡−2h2ξ1,0 − 2hξ0,0 + 12

¢+ 24h3ξ2,0 = 0. (8.22)

Solving the nodal value at j = −1 in terms of the values of the nodal point 0 and 1 using

equations (8.19), (8.21) and (8.22) gives

H−1 =

µ1

Φ0 − C0

¶⎛⎝ A0(24hSH − 80H0 + 27H1 − 36h2κH)+B0(6hSH − 15H0 + 4H1 − 6h2κH)+H1E0 +H0D0 + F0

⎞⎠ , (8.23)

where:A0 = 6h2ξ1,0 + 22hξ0,0 + 36,

B0 = 20h2ξ1,0 − 40hξ0,0 − 120,C0 = −36h2ξ1,0 + 12hξ0,0 + 144,D0 = 12h2ξ1,0 + 8hξ0,0 − 72,E0 = −2h2ξ1,0 − 2hξ0,0 + 12,F0 = 24h3ξ2,0,

Φ0 = −54A0 − 12B0.

Similarly, applying equation (8.18) at the last nodal point J will introduce unknowns

external to the computational domain. The best way of solving this problem is to discre-

tise the differential flow equation using the backward finite difference approximations.

These approximations for nodal point j in terms of nodal values at j, j−1, j−2... areµdH

dx

¶j

=1

24h(6Hj−4 − 32Hj−3 + 72Hj−2 − 96Hj−1 + 50Hj) +O

¡h5¢, (8.24)

µd2H

dx2

¶j

=1

12h2(11Hj−4 − 56Hj−3 + 114Hj−2 − 104Hj−1 + 35Hj) +O

¡h5¢,

(8.25)

µd3H

dx3

¶j

=1

4h3(6Hj−4 − 28Hj−3 + 48Hj−2 − 36Hj−1 + 10Hj) +O

¡h5¢. (8.26)

Inserting the above three equations (equations (8.24), (8.25) and (8.26)) into the general

equation, equation (8.13), and simplifying the resulting expression gives the following


finite difference equation for derivative at node j:

Hj¡50h2ξ1,j + 70hξ0,j + 60

¢+Hj−1

¡−96h2ξ1,j − 208hξ0,j − 216

¢+Hj−2

¡72h2ξ1,j + 228hξ0,j + 288

¢+Hj−3

¡−32h2ξ1,j − 112hξ0,j − 168

¢+Hj−4

¡6h2ξ1,j + 22hξ0,j + 36

¢+ 24h3ξ2,j = 0. (8.27)

Equation (8.27) is employed at the outflow section (j = J) only.

Equations (8.18), (8.21), (8.23) and (8.27) are solved numerically to simulate the flow

surface profile for the given discharge over the control structure. The overflow head at

the gauging station will be determined from the predicted flow depth at this station and

known height of the trapezoidal profile weir above the upstream floor level.

As mentioned before, the incoming flow to the inflow boundary section is either gradu-

ally varied or nearly parallel flow. The specified depth of flow, and the estimated slope

and curvature of the flow surface using the appropriate equations are used as the inflow

boundary values to solve the problem numerically.

8.7.3 Solution procedure

A simple iterative technique is employed here based on the fact that the estimated po-

sition of the free surface profile for the given discharge is systematically improved by

successively eliminating the errors resulting from the assumed initial position of the

free surface, thus finding stable convergent solutions. The essence of this technique is

as follows:

1. The slope and curvature of the flow surface at the inflow section are computed

using the appropriate equations for the specified flow depth and corresponding

discharge.

2. Using the Bernoulli and continuity equations, the initial position of the free surface

is estimated.

3. Similar procedures and solving technique as for the case of the two-point boundary

value method for flow over curved beds are adopted in this numerical simulation

procedure. The only difference is that the boundary values, specified at the up-

stream inflow section, are used in the iteration process to get the final free surface

profile. For this numerical model, the Jacobian matrix elements in each iteration

process are computed numerically.


4. The location of the upstream gauging station is fixed approximately using a simple

iteration procedure that uses the existing criterion from the literature for this pur-

pose. The required overflow depth is then calculated based on the known height of

the weir for this theoretical position of the gauging station.

5. The procedure (steps 1-4) is repeated with different discharge values to obtain a

head-discharge curve.

8.8 Model results for discharge rating curves

As discussed before, the numerical model simulates flow profiles for flow over trape-

zoidal profile weirs under free flow conditions for the purpose of developing head-

discharge relationships. The flow profile predictions of this model are verified with

measurements for free flow with different overflow head to crest length ratios. Some of

the simulation results are shown in Figures 8.19 - 8.21. For all cases of the simulation,

the agreement between the predictions and measurements is good.

-0.2

0.0

0.2

0.4

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0


Flow

sur

face

(m)

Bed profile Predicted (H1/Lw = 0.20)

Predicted (H1/Lw = 0.25) Measured

Weir crest length = 40 cm

Figure 8.19: Flow surface profiles for free flow over a broad-crested weir (smooth bed)

The computed head-discharge curves for flow over short- and broad-crested trapezoidal

profile weirs are compared with the experimental results in Figure 8.22. For discharge

less than 6L/s, the predicted values are extended using a data modelling technique and


-0.2

0.0

0.2

0.4

-1.0 -0.5 0.0 0.5 1.0 1.5


Flow

sur

face

(m)


Predicted (H1/Lw = 0.80) Measured


Figure 8.20: Flow surface profiles for free flow over a short-crested weir (smooth bed)

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

-1.0 -0.5 0.0 0.5 1.0 1.5


Flow

sur

face

(m)


Predicted (H1/Lw = 0.72) Predicted (H1/Lw = 0.86)

Measured


Figure 8.21: Comparison of predicted and measured flow profiles (rough bed)


are shown in this figure by dashed lines. The numerical solutions demonstrate good

agreement with the experimental data for both weirs. This figure also compares the

simulated discharge rating curves of the short- and broad-crested types of these weirs.

Depending on the magnitude ofH1/Lw, the same type of weir can act as a broad-crested

or a short-crested weir. At low flow rates the rating curves for the short- and broad-

crested types of trapezoidal profile weirs are identical; indicating the insignificance of

the influence of the curvature of the streamlines on the discharge characteristics of these

weirs at such flow rates (see Figure 8.22). As the discharge increases, clear differences

between the head-discharge curves of the broad-crested and short-crested weirs are ob-

served. This difference is due to the large increase of curvature of the streamline of the

flow over the crest of the short-crested trapezoidal profile weir. It can be observed from

Figure 8.22 that in the region of relatively high flow rate, the overflow depth required

to pass the given discharge over the broad-crested weir is larger than the corresponding

overflow depth for the short-crested weir. This implies that less energy is required to

pass a given flow over the short-crested types than the broad-crested weirs. The com-

parison result suggests that the curvature of the streamlines of the flow over the crest of

the trapezoidal profile weir has significant impact on the discharge capacity of this weir.

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30

Discharge (L/s)

Ove

rflo

w d

epth

(cm

)

Predicted (Lw = 10 cm)Measured (Lw = 10 cm)Predicted (Lw = 40 cm)Measured (Lw = 40 cm)Extension

Figure 8.22: Comparison between discharge rating curves for short- and broad-crested weirs(smooth bed)


Figure 8.23 illustrates the rating curve simulated by the model for a rough bed flow

situation. The figure also compares the model result with experimental data. The model

reproduces the trend of the rating curve accurately and the result is in good agreement

with experimental data.

4.3

6.3

8.3

10.3

12.3

14.3

0 5 10 15 20 25 30

Discharge (L/s)

Ove

rflo

w d

epth

(cm

)

Predicted (Lw = 15 cm)

Measured (Lw = 15 cm)

Figure 8.23: Comparison of predicted and measured discharge rating curve result (rough bed)

The mean and standard deviation of the relative percentage errors of the predicted over-

flow depths was computed for the considered weir flow situations. The relative percent-

age error, at a given discharge, is defined as (hc − hm) /hm× 100, where hc and hm are

the computed and measured overflow depths respectively. The mean and standard devi-

ation analysis results showed that the model slightly underestimated the overflow depths

for the short- and broad-crested weir flow situations. However, the magnitudes of the

relative errors for all cases of the flow are very small. The overall mean and standard

deviation of the errors for all flows are −0.52% and 2.49% respectively.

8.9 Submerged flow discharge

The functional relationship for the reduced discharge, defined by equation (8.6b), is

determined approximately from the analysis of the experimental data. Observations


on the variation of qsub/qfree with htl/hll are shown in Figures 8.24 and 8.25. As

expected, the figures show decreasing of the submerged discharge with increasing of

htl/hll. The value of qsub/qfree varies between 0 and 1, with qsub/qfree = 0 for com-

pletely submerged condition (htl/hll = 1) and qsub/qfree = 1 for free flow conditions

(htl/hll = 0) . Mean curves for the normalised representation of the experimental data

are fitted and shown in these figures. These mean curves can be described by the fol-

lowing general equation:

qsubqfree

=

µ1− htl

hll

¶υ

. (8.28)

Scattered measured data can be seen in the lower part of Figure 8.25. Surface waves

and turbulences especially on the downstream side of the weir model increased the dif-

ficulties of accurate measurement of tailwater depths due to the insufficient length of the

test section of the flume for this broad-crested type of weir. For flow over trapezoidal

profile weirs with different crest lengths, the ratio of the overflow head to the length

of the weir crest significantly affects the value of the exponent υ. In this study, the ex-

perimental data for submerged flows over short- and broad-crested trapezoidal profile

weirs were analysed. The values of the exponent were determined for these two trape-

zoidal shaped weirs. For broad-crested type of trapezoidal profile weir (Lw = 400 mm,

H1/Lw < 0.30), 1/υ = 7.0. Similarly for the short-crested trapezoidal shaped weir of

Lw = 100 mm (0.463 ≤H1/Lw ≤ 1.16), 1/υ = 10.8.

Comparison of the average curves of the submerged flow reduction factor for short-

and broad-crested trapezoidal profile weirs is shown in Figure 8.26. The result demon-

strates that for a given value of submergence ratio with respect to the modular limit

tailwater depth, the value of qsub/qfree is larger for short-crested than for broad-crested

trapezoidal shaped weirs. This is due to the lower modular limit and higher free flow

discharge of the short-crested weir which are the result of the considerable curvature

of the streamlines of the flow over the crest of the weir. The comparison result sug-

gests that short-crested trapezoidal profile weirs are more prone to submergence than

broad-crested types of trapezoidal shaped weirs.

If the flow depths at the upstream and downstream gauging stations are known for a

typical trapezoidal shaped weir, one can use the modular limit curve, the curve for the

submerged discharge (or the mean curve equation) and the appropriate rating curve for

free flow to predict the submerged discharge of the weir. As described in the previous


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

htl/hll

q sub

/qfr

ee H1/Lw = 0.463H1/Lw = 0.544H1/Lw = 0.634H1/Lw = 0.764H1/Lw = 0.795H1/Lw = 0.952H1/Lw = 0.965H1/Lw = 1.045H1/Lw = 1.156

qsub/qfree = (1-htl/hll)1/10.77

Figure 8.24: Variation of submerged flow reduction factor (Lw = 100 mm)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0

htl/hll

q sub

/qfr

ee

H1/Lw = 0.144H1/Lw = 0.177H1/Lw = 0.196H1/Lw = 0.220H1/Lw = 0.250H1/Lw = 0.267

qsub/qfree = (1-htl/hll)1/7

Figure 8.25: Variation of submerged flow reduction factor (Lw = 400 mm)


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

htl/hll

q sub

/qfr

ee

Lw = 100 mmLw = 400 mm

Figure 8.26: Comparison of submerged flow reduction factors for short- and broad-crestedweirs

section, estimation of non-modular discharge using any types of weirs is time consum-

ing and expensive processes.

8.10 Summary and conclusions

In this chapter, the free and submerged flow characteristics of trapezoidal profile weirs

as well as the main differences between the flow behaviours of short- and broad-crested

types of these weirs were discussed. The nature of the weir flow problem was addressed

in relation to the theoretical and experimental methods of analyses. A one-dimensional

numerical model based on the specification of boundary values at the inflow section for

establishing discharge-rating curves was developed. The predictions of this model were

compared with available experimental data. A good agreement was observed between

the predicted and measured values for smooth and rough bed flow situations.

The results of the analyses of the experimental data and the predictions of the numerical

model demonstrate the detailed dependence of the global flow characteristics of the

trapezoidal profile weirs on the curvature of the streamlines. Also, the existing model

for developing discharge rating curves under free flow conditions, which is valid only

for broad-crested weirs, was extended using the Boussinesq-type momentum equation


model (BTMU model). In the following chapter, the main conclusions of the current

study and recommendation for areas of further research will be presented.

Minerva Access is the Institutional Repository of The University of Melbourne

Author/s:Zerihun, Yebegaeshet Tsegaye

Title:A one-dimensional Boussinesq-type momentum model for steady rapidly varied openchannel flows

Date:2004-11

Citation:Zerihun, Y. T. (2004). A one-dimensional Boussinesq-type momentum model for steadyrapidly varied open channel flows. PhD thesis, Department of Civil and EnvironmentalEngineering, The University of Melbourne.

Publication Status:Unpublished

Persistent Link:http://hdl.handle.net/11343/39476

Terms and Conditions:Terms and Conditions: Copyright in works deposited in Minerva Access is retained by thecopyright owner. The work may not be altered without permission from the copyright owner.Readers may only download, print and save electronic copies of whole works for their ownpersonal non-commercial use. Any use that exceeds these limits requires permission fromthe copyright owner. Attribution is essential when quoting or paraphrasing from these works.

http://hdl.handle.net/11343/39476

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Chapter 8 Flow Characteristics and Modelling of Head