Open channels

April 18, 2007
Open channel hydraulics
John Fenton
Abstract
This course of 15 lectures provides an introduction to open channel hydraulics, the generic name for
the study of flows in rivers, canals, and sewers, where the distinguishing characteristic is that the
surface is unconfined. This means that the location of the surface is also part of the problem, and
allows for the existence of waves – generally making things more interesting!
At the conclusion of this subject students will understand the nature of flows and waves in open
channels and be capable of solving a wide range of commonly encountered problems.
Table of Contents
References . . . . . . . . . . . . . . . . . . . . . . . 2
1. Introduction . . . . . . . . . . . . . . . . . . . . . 3
1.1 Types of channel flowtobestudied . . . . . . . . . . . . 4
1.2 Properties of channel flow . . . . . . . . . . . . . . 5
2. Conservation of energy in open channel flow . . . . . . . . . . . 9
2.1 The head/elevation diagram and alternative depths of flow . . . . . 9
2.2 Critical flow . . . . . . . . . . . . . . . . . . . 11
2.3 TheFroudenumber . . . . . . . . . . . . . . . . 12
2.4 Waterlevelchangesatlocaltransitionsinchannels . . . . . . . 13
2.5 Somepracticalconsiderations . . . . . . . . . . . . . 15
2.6 Critical flowasacontrol-broad-crestedweirs . . . . . . . . 17
3. Conservation of momentum in open channel flow . . . . . . . . . 18
3.1 Integralmomentumtheorem . . . . . . . . . . . . . . 18
3.2 Flow under a sluice gate and the hydraulic jump . . . . . . . . 21
3.3 The effects of streams on obstacles and obstacles on streams . . . . 24
4. Uniform flowinprismaticchannels . . . . . . . . . . . . . . 29
4.1 Features of uniform flow and relationships for uniform flow . . . . 29
4.2 Computationofnormaldepth . . . . . . . . . . . . . 30
4.3 Conveyance . . . . . . . . . . . . . . . . . . . 31
5. Steady gradually-varied non-uniform flow . . . . . . . . . . . 32
5.1 Derivation of the gradually-varied flowequation . . . . . . . . 32
5.2 Properties of gradually-varied flow and the governing equation . . . 34
5.3 Classification system for gradually-varied flows . . . . . . . . 34
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Open channel hydraulics John Fenton
5.4 Somepracticalconsiderations . . . . . . . . . . . . . 35
5.5 Numerical solution of the gradually-varied flowequation . . . . . 35
5.6 Analyticalsolution . . . . . . . . . . . . . . . . . 40
6. Unsteady flow . . . . . . . . . . . . . . . . . . . . 42
6.1 Massconservationequation . . . . . . . . . . . . . . 42
6.2 Momentum conservation equation – the low inertia approximation . . 43
6.3 Diffusion routing and nature of wave propagation in waterways . . . 45
7. Structures in open channels and flowmeasurement . . . . . . . . . 47
7.1 Overshotgate-thesharp-crestedweir . . . . . . . . . . . 47
7.2 Triangularweir . . . . . . . . . . . . . . . . . . 48
7.3 Broad-crested weirs – critical flowasacontrol . . . . . . . . 48
7.4 Freeoverfall . . . . . . . . . . . . . . . . . . . 49
7.5 Undershotsluicegate . . . . . . . . . . . . . . . . 49
7.6 Drownedundershotgate . . . . . . . . . . . . . . . 50
7.7 DethridgeMeter . . . . . . . . . . . . . . . . . 50
8. The measurement of flowinriversandcanals . . . . . . . . . . 50
8.1 Methodswhichdonotusestructures . . . . . . . . . . . 50
8.2 The hydraulics of a gauging station . . . . . . . . . . . . 53
8.3 Ratingcurves . . . . . . . . . . . . . . . . . . 54
9. Loose-boundary hydraulics . . . . . . . . . . . . . . . . 56
9.1 Sedimenttransport . . . . . . . . . . . . . . . . . 56
9.2 Incipientmotion . . . . . . . . . . . . . . . . . 57
9.3 Turbulent flowinstreams . . . . . . . . . . . . . . . 58
9.4 Dimensionalsimilitude . . . . . . . . . . . . . . . 58
9.5 Bed-loadrateoftransport–Bagnold’sformula . . . . . . . . 59
9.6 Bedforms . . . . . . . . . . . . . . . . . . . 59
References
Ackers, P., White, W. R., Perkins, J. A. & Harrison, A. J. M. (1978) Weirs and Flumes for Flow
Measurement, Wiley.
Boiten, W. (2000) Hydrometry, Balkema.
Bos, M. G. (1978) Discharge Measurement Structures, Second Edn, International Institute for Land
Reclamation and Improvement, Wageningen.
Fenton, J. D. (2002) The application of numerical methods and mathematics to hydrography, in Proc.
11th Australasian Hydrographic Conference, Sydney, 3 July - 6 July 2002.
Fenton, J. D. & Abbott, J. E. (1977) Initial movement of grains on a stream bed: the effect of relative
protrusion, Proc. Roy. Soc. Lond. A 352, 523–537.
French, R. H. (1985) Open-Channel Hydraulics, McGraw-Hill, New York.
Henderson, F. M. (1966) Open Channel Flow, Macmillan, New York.
Herschy, R. W. (1995) Streamflow Measurement, Second Edn, Spon, London.
Jaeger, C. (1956) Engineering Fluid Mechanics, Blackie, London.
Montes, S. (1998) Hydraulics of Open Channel Flow, ASCE, New York.
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Novak, P., Moffat, A. I. B., Nalluri, C. & Narayanan, R. (2001) Hydraulic Structures, Third Edn, Spon,
London.
Yalin, M. S. & Ferreira da Silva, A. M. (2001) Fluvial Processes, IAHR, Delft.
Useful references
The following table shows some of the many references available, which the lecturer may refer to in
these notes, or which students might find useful for further reading. For most books in the list, The
University of Melbourne Engineering Library Reference Numbers are given.
Reference Comments
Bos, M. G. (1978), Discharge Measurement Structures, second edn, International Insti-tute
for Land Reclamation and Improvement, Wageningen.
Good encyclopaedic treatment
of structures
Bos, M. G., Replogle, J. A. & Clemmens, A. J. (1984), Flow Measuring Flumes for
Open Channel Systems, Wiley.
Good encyclopaedic treatment
of structures
Chanson, H. (1999), The Hydraulics of Open Channel Flow, Arnold, London. Good technical book, moderate
level, also sediment aspects
Chaudhry, M. H. (1993), Open-channel flow, Prentice-Hall. Good technical book
Chow, V. T. (1959), Open-channel Hydraulics, McGraw-Hill, New York. Classic, now dated, not so read-able
Dooge, J. C. I. (1992) , The Manning formula in context, in B. C. Yen, ed., Channel
Flow Resistance: Centennial of Manning’s Formula, Water Resources Publications,
Littleton, Colorado, pp. 136–185.
Interesting history of Man-ning’s
law
Fenton, J. D. & Keller, R. J. (2001), The calculation of streamflow from measurements
of stage, Technical Report 01/6, Co-operative Research Centre for Catchment Hydrol-ogy,
Monash University.
Two level treatment - practical
aspects plus high level review
of theory
Francis, J. & Minton, P. (1984), Civil Engineering Hydraulics, fifth edn, Arnold, Lon-don.
Good elementary introduction
French, R. H. (1985), Open-Channel Hydraulics, McGraw-Hill, New York. Wide general treatment
Henderson, F. M. (1966), Open Channel Flow, Macmillan, New York. Classic, high level, readable
Hicks, D. M. & Mason, P. D. (1991 ) , Roughness Characteristics of New Zealand
Rivers, DSIR Marine and Freshwater, Wellington.
Interesting presentation of
Manning’s n for different
streams
Jain, S. C. (2001), Open-Channel Flow, Wiley. High level, but terse and read-able
Montes, S. (1998), Hydraulics of Open Channel Flow, ASCE, New York. Encyclopaedic
Novak, P., Moffat, A. I. B., Nalluri, C. & Narayanan, R. (2001), Hydraulic Structures,
third edn, Spon, London.
Standard readable presentation
of structures
Townson, J. M. (1991), Free-surface Hydraulics, Unwin Hyman, London. Simple, readable, mathematical
1. Introduction
The flow of water with an unconfined free surface at atmospheric pressure presents some of the most
common problems of fluid mechanics to civil and environmental engineers. Rivers, canals, drainage
canals, floods, and sewers provide a number of important applications which have led to the theories and
methods of open channel hydraulics. The main distinguishing characteristic of such studies is that the
location of the surface is also part of the problem. This allows the existence of waves, both stationary
and travelling. In most cases, where the waterway is much longer than it is wide or deep, it is possible to
treat the problem as an essentially one-dimensional one, and a number of simple and powerful methods
have been developed.
In this course we attempt a slightly more general view than is customary, where we allow for real fluid
effects as much as possible by allowing for the variation of velocity over the waterway cross section. We
recognise that we can treat this approximately, but it remains an often-unknown aspect of each problem.
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This reminds us that we are obtaining approximate solutions to approximate problems, but it does allow
some simplifications to be made.
The basic approximation in open channel hydraulics, which is usually a very good one, is that variation
along the channel is gradual. One of the most important consequences of this is that the pressure in the
water is given by the hydrostatic approximation, that it is proportional to the depth of water above.
In Australia there is a slightly non-standard nomenclature which is often used, namely to use the word
”channel” for a canal, which is a waterway which is usually constructed, and with a uniform section.
We will use the more international English convention, that such a waterway is called a canal, and we
will use the words ”waterway”, ”stream”, or ”channel” as generic terms which can describe any type of
irregular river or regular canal or sewer with a free surface.
1.1 Types of channel flow to be studied
(a) Steady uniform flow
dn
(b) Steady gradually-varied flow
dn Normal depth
(c) Steady rapidly-varied flow
(d) Unsteady flow
Figure 1-1. Different types of flow in an open channel
Case (a) – Steady uniform flow: Steady flow is where there is no change with time, ∂/∂t ≡ 0.
Distant from control structures, gravity and friction are in balance, and if the cross-section is constant,
the flow is uniform, ∂/∂x ≡ 0. We will examine empirical laws which predict flow for given bed slope
and roughness and channel geometry.
Case (b) – Steady gradually-varied flow: Gravity and friction are in balance here too, but when a
control is introduced which imposes a water level at a certain point, the height of the surface varies along
the channel for some distance. For this case we will develop the differential equation which describes
how conditions vary along the waterway.
Case (c) – Steady rapidly-varied flow: Figure 1-1(c) shows three separate gradually-varied flow
states separated by two rapidly-varied regions: (1) flow under a sluice gate and (2) a hydraulic jump.
The complete problem as presented in the figure is too difficult for us to study, as the basic hydraulic
approximation that variation is gradual and that the pressure distribution is hydrostatic breaks down in the
rapid transitions between the different gradually-varied states. We can, however, analyse such problems
by considering each of the almost-uniform flow states and consider energy or momentum conservation
between them as appropriate. In these sorts of problems we will assume that the slope of the stream
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balances the friction losses and we treat such problems as frictionless flow over a generally-horizontal
bed, so that for the individual states between rapidly-varied regions we usually consider the flow to be
uniform and frictionless, so that the whole problem is modelled as a sequence of quasi-uniform flow
states.
Case (d) – Unsteady flow: Here conditions vary with time and position as a wave traverses the
waterway. We will obtain some results for this problem too.
1.2 Properties of channel flow
z = η
y
z
min z = z
Figure 1-2. Cross-section of flow, showing isovels, contours on which velocity normal to the section is constant.
Consider a section of a waterway of arbitrary section, as shown in Figure 1-2. The x co-ordinate is
horizontal along the direction of the waterway (normal to the page), y is transverse, and z is vertical. At
the section shown the free surface is z = η, which we have shown to be horizontal across the section,
which is a good approximation in many flows.
1.2.1 Discharge across a cross-section
The volume flux or discharge Q at any point is
Q =
Z
A
u dA = UA
where u is the velocity component in the x or downstream direction, and A is the cross-sectional area.
This equation defines the mean horizontal velocity over the section U . In most hydraulic applications
the discharge is a more important quantity than the velocity, as it is the volume of water and its rate of
propagation, the discharge, which are important.
1.2.2 A generalisation – net discharge across a control surface
Having obtained the expression for volume flux across a plane surface where the velocity vector is
normal to the surface, we introduce a generalisation to a control volume of arbitrary shape bounded by a
control surface CS. If u is the velocity vector at any point throughout the control volume and ˆn is a unit
vector with direction normal to and directed outwards from a point on the control surface, then u · ˆn on
the control surface is the component of velocity normal to the control surface. If dS is an elemental area
of the control surface, then the rate at which fluid volume is leaving across the control surface over that
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elemental area is u · ˆndS, and integrating gives
Total rate at which fluid volume is leaving across the control surface =
Z
CS
u · ˆndS. (1.1)
If we consider a finite length of channel as shown in Figure 1-3, with the control surface made up of
u1
nˆ1
u2
nˆ 2
Figure 1-3. Section of waterway and control surface with vertical ends
the bed of the channel, two vertical planes across the channel at stations 1 and 2, and an imaginary
enclosing surface somewhere above the water level, then if the channel bed is impermeable, u · n ˆ= 0
there; u = 0 on the upper surface; on the left (upstream) vertical plane u · n ˆ= −u1, where u1 is
the horizontal component of velocity (which varies across the section); and on the right (downstream)
vertical plane u · n ˆ= +u2. Substituting into equation (1.1) we have
Z
Total rate at which fluid volume is leaving across the control surface = −
A1
u1 dA +
Z
A2
u2 dA
= −Q1 + Q2.
If the flow is steady and there is no increase of volume inside the control surface, then the total rate of
volume leaving is zero and we have Q1 = Q2.
While that result is obvious, the results for more general situations are not so obvious, and we will
generalise this approach to rather more complicated situations – notably where the water surface in the
Control Surface is changing.
1.2.3 A further generalisation – transport of other quantities across the control surface
We saw that u · ˆndS is the volume flux through an elemental area – if wemultiply by fluid density ρ then
ρ u · ˆndS is the rate at which fluid mass is leaving across an elemental area of the control surface, with
a corresponding integral over the whole surface. Mass flux is actually more fundamental than volume
flux, for volume is not necessarily conserved in situations such as compressible flow where the density
varies. However in most hydraulic engineering applications we can consider volume to be conserved.
Similarly we can compute the rate at which almost any physical quantity, vector or scalar, is being
transported across the control surface. For example, multiplying the mass rate of transfer by the fluid
velocity u gives the rate at which fluid momentum is leaving across the control surface, ρuu · ˆndS.
1.2.4 The energy equation in integral form for steady flow
Bernoulli’s theorem states that:
In steady, frictionless, incompressible flow, the energy per unit mass p/ρ+gz +V 2/2 is constant
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along a streamline,
where V is the fluid speed, V 2 = u2+v2+w2, inwhich (u, v, w) are velocity components in a cartesian
co-ordinate system (x, y, z) with z vertically upwards, g is gravitational acceleration, p is pressure and
ρ is fluid density. In hydraulic engineering it is usually more convenient to divide by g such that we say
that the head p/ρg + z + V 2/2g is constant along a streamline.
In open channel flows (and pipes too, actually, but this seems never to be done) we have to consider
the situation where the energy per unit mass varies across the section (the velocity near pipe walls and
channel boundaries is smaller than in the middle while pressures and elevations are the same). In this
case we cannot apply Bernoulli’s theorem across streamlines. Instead, we use an integral form of the
energy equation, although almost universally textbooks then neglect variation across the flow and refer
to the governing theorem as ”Bernoulli”. Here we try not to do that.
The energy equation in integral form can be written for a control volume CV bounded by a control
surface CS, where there is no heat added or work done on the fluid in the control volume:
∂
∂t
Z
CV
ρ e dV
| {z }
Rate at which energy is increasing inside the CV
+
Z
(p + ρe) u.ˆndS
| {z }
CS
Rate at which energy is leaving the CS
= 0, (1.2)
where t is time, ρ is density, dV is an element of volume, e is the internal energy per unit mass of fluid,
which in hydraulics is the sum of potential and kinetic energies
e = gz +
1
2
¡
u2 + v2 + w2¢
,
where the velocity vector u = (u, v, w) in a cartesian coordinate system (x, y, z) with x horizontally
along the channel and z upwards, ˆn is a unit vector as above, p is pressure, and dS is an elemental area
of the control surface.
Here we consider steady flow so that the first term in equation (1.2) is zero. The equation becomes:
Z
CS
³
p + ρgz +
ρ
2
¡
u2 + v2 + w2¢´
u.ˆn dS = 0.
We intend to consider problems such as flows in open channels where there is usually no important
contribution from lateral flows so that we only need to consider flow entering across one transverse face
of the control surface across a pipe or channel and leaving by another. To do this we have the problem
of integrating the contribution over a cross-section denoted by A which we also use as the symbol for
the cross-sectional area. When we evaluate the integral over such a section we will take u to be the
velocity along the channel, perpendicular to the section, and v and w to be perpendicular to that. The
contribution over a section of area A is then ±E, where E is the integral over the cross-section:
E =
Z
A
³
p + ρgz +
ρ
2
¡
u2 + v2 + w2¢´
u dA, (1.3)
and we take the ± depending on whether the flow is leaving/entering the control surface, because u.ˆn =
±u. In the case of no losses, E is constant along the channel. The quantity ρQE is the total rate of
energy transmission across the section.
Now we consider the individual contributions:
(a) Velocity head term ρ
2
R
A
¡
u2 + v2 + w2
¢
u dA
If the flow is swirling, then the v and w components will contribute, and if the flow is turbulent there
will be extra contributions as well. It seems that the sensible thing to do is to recognise that all velocity
components and velocity fluctuations will be of a scale given by the mean flow velocity in the stream at
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that point,and so we simply write, for the moment ignoring the coefficient ρ/2:
Z
A
¡
u2 + v2 + w2¢
u dA = αU 3A = α
Q3
A2 , (1.4)
which defines α as a coefficient which will be somewhat greater than unity, given by
α =
R
A
¡
u2 + v2 + w2
¢
u dA
U 3A
. (1.5)
Conventional presentations define it as being merely due to the non-uniformity of velocity distribution
across the channel:
α =
R
A u3 dA
U 3A
,
however we suggest that is more properly written containing the other velocity components (and turbu-lent
contributions as well, ideally). This coefficient is known as a Coriolis coefficient, in honour of the
French engineer who introduced it.
Most presentations of open channel theory adopt the approximation that there is no variation of velocity
over the section, such that it is assumed that α = 1, however that is not accurate. Montes (1998, p27)
quotes laboratory measurements over a smooth concrete bed giving values of α of 1.035-1.064, while
for rougher boundaries such as earth channels larger values are found, such as 1.25 for irrigation canals
in southern Chile and 1.35 in the Rhine River. For compound channels very much larger values may be
encountered. It would seem desirable to include this parameter in our work, which we will do.
(b) Pressure and potential head terms
These are combined as Z
A
(p + ρgz) u dA. (1.6)
The approximation we now make, common throughout almost all open-channel hydraulics, is the ”hy-drostatic
approximation”, that pressure at a point of elevation z is given by
p ≈ ρg × height of water above = ρg (η − z) , (1.7)
where the free surface directly above has elevation η. This is the expression obtained in hydrostatics for
a fluid which is not moving. It is an excellent approximation in open channel hydraulics except where
the flow is strongly curved, such as where there are short waves on the flow, or near a structure which
disturbs the flow. Substituting equation (1.7) into equation (1.6) gives
ρg
Z
A
η u dA,
for the combination of the pressure and potential head terms. If we make the reasonable assumption that
η is constant across the channel the contribution becomes
ρgη
Z
A
u dA = ρgηQ,
from the definition of discharge Q.
(c) Combined terms
Substituting both that expression and equation (1.4) into (1.3) we obtain
E = ρgQ
μ
η +
α
2g
Q2
A2
¶
, (1.8)
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which, in the absence of losses, would be constant along a channel. This energy flux across entry and
exit faces is that which should be calculated, such that it is weighted with respect to the mass flow rate.
Most presentations pretend that one can just apply Bernoulli’s theorem, which is really only valid along
a streamline. However our results in the end are not much different. We can introduce the concept of the
Mean Total Head H such that
H =
Energy flux
g × Mass flux =
E
g × ρQ
= η +
α
2g
Q2
A2 , (1.9)
which has units of length and is easily related to elevation in many hydraulic engineering applications,
relative to an arbitrary datum. The integral version, equation (1.8), is more fundamental, although in
common applications it is simpler to use the mean total head H, which will simply be referred to as the
head of the flow. Although almost all presentations of open channel hydraulics assume α = 1, we will
retain the general value, as a better model of the fundamentals of the problem, which is more accurate,
but also is a reminder that although we are trying to model reality better, its value is uncertain to a degree,
and so are any results we obtain. In this way, it is hoped, we will maintain a sceptical attitude to the
application of theory and ensuing results.
(d) Application to a single length of channel – including energy losses
We will represent energy losses by ΔE. For a length of channel where there are no other entry or exit
points for fluid, we have
Eout = Ein − ΔE,
giving, from equation (1.8):
ρQout
μ
gη +
α
2
Q2
A2
¶
out
= ρQin
μ
gη +
α
2
Q2
A2
¶
in − ΔE,
and as there is no mass entering or leaving, Qout = Qin = Q, we can divide through by ρQ and by g, as
is common in hydraulics:
μ
η +
α
2g
Q2
A2
¶
out
=
μ
η +
α
2g
Q2
A2
¶
in − ΔH,
where we have written ΔE = ρgQ × ΔH, where ΔH is the head loss. In spite of our attempts to use
energy flux, as Q is constant and could be eliminated, in this head form the terms appear as they are used
in conventional applications appealing to Bernoulli’s theorem, but with the addition of the α coefficients.
2. Conservation of energy in open channel flow
In this section and the following one we examine the state of flow in a channel section by calculating the
energy and momentum flux at that section, while ignoring the fact that the flow at that section might be
slowly changing. We are essentially assuming that the flow is locally uniform – i.e. it is constant along
the channel, ∂/∂x ≡ 0. This enables us to solve some problems, at least to a first, approximate, order.
We can make useful deductions about the behaviour of flows in different sections, and the effects of
gates, hydraulic jumps, etc.. Often this sort of analysis is applied to parts of a rather more complicated
flow, such as that shown in Figure 1-1(c) above, where a gate converts a deep slow flow to a faster shallow
flow but with the same energy flux, and then via an hydraulic jump the flow can increase dramatically in
depth, losing energy through turbulence but with the same momentum flux.
2.1 The head/elevation diagram and alternative depths of flow
Consider a steady (∂/∂t ≡ 0) flow where any disturbances are long, such that the pressure is hydro-static.
We make a departure from other presentations. Conventionally (beginning with Bakhmeteff in
1912) they introduce a co-ordinate origin at the bed of the stream and introduce the concept of ”specific
energy”, which is actually the head relative to that special co-ordinate origin. We believe that the use of
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that datum somehow suggests that the treatment and the results obtained are special in some way. Also,
for irregular cross-sections such as in rivers, the ”bed” or lowest point of the section is poorly defined,
and we want to minimise our reliance on such a point. Instead, we will use an arbitrary datum for the
head, as it is in keeping with other areas of hydraulics and open channel theory.
Over an arbitrary section such as in Figure 1-2, from equation (1.9), the head relative to the datum can
be written
H = η +
αQ2
2g
1
A2(η)
, (2.1)
where we have emphasised that the cross-sectional area for a given section is a known function of surface
elevation, such that we write A(η). A typical graph showing the dependence of H upon η is shown in
Figure 2-1, which has been drawn for a particular cross-section and a constant value of discharge Q,
such that the coefficient αQ2/2g in equation (2.1) is constant.
η1
ηc
η2
zmin
Hc
Surface
elevation
η
H = η + αQ2
Head H = E/ρgQ
2g
1
A2(η)
H = η
1
2
Figure 2-1. Variation of head with surface elevation for a particular cross-section and discharge
The figure has a number of important features, due to the combination of the linear increasing function
η and the function 1/A2(η) which decreases with η.
• In the shallow flow limit as η → zmin (i.e. the depth of flow, and hence the cross-sectional area
A(η), both go to zero while holding discharge constant) the value of H ∼ αQ2/2gA2(η) becomes
very large, and goes to ∞ in the limit.
• In the other limit of deep water, as η becomes large, H ∼ η, as the velocity contribution becomes
negligible.
• In between these two limits there is a minimum value of head, at which the flow is called critical
flow, where the surface elevation is ηc and the head Hc.
• For all other H greater than Hc there are two values of depth possible, i.e. there are two different
flow states possible for the same head.
• The state with the larger depth is called tranquil, slow, or sub-critical flow, where the potential to
make waves is relatively small.
• The other state, with smaller depth, of course has faster flow velocity, and is called shooting, fast, or
super-critical flow. There is more wave-making potential here, but it is still theoretically possible
for the flow to be uniform.
• The two alternative depths for the same discharge and energy have been called alternate depths.
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That terminology seems to be not quite right – alternate means ”occur or cause to occur by turns,
go repeatedly from one to another”. Alternative seems better - ”available as another choice”, and
we will use that.
• In the vicinity of the critical point, where it is easier for flow to pass from one state to another, the
flow can very easily form waves (and our hydrostatic approximation would break down).
• Flows can pass from one state to the other. Consider the flow past a sluice gate in a channel as
shown in Figure 1-1(c). The relatively deep slow flow passes under the gate, suffering a large
reduction in momentum due to the force exerted by the gate and emerging as a shallower faster
flow, but with the same energy. These are, for example, the conditions at the points labelled 1 and
2 respectively in Figure 2-1. If we have a flow with head corresponding to that at the point 1 with
surface elevation η1 then the alternative depth is η2 as shown. It seems that it is not possible to
go in the other direction, from super-critical flow to sub-critical flow without some loss of energy,
but nevertheless sometimes it is necessary to calculate the corresponding sub-critical depth. The
mathematical process of solving either problem, equivalent to reading off the depths on the graph,
is one of solving the equation
αQ2
2gA2(η1)
+ η1
| {z }
H1
=
αQ2
2gA2(η2)
+ η2
| {z }
H2
(2.2)
for η2 if η1 is given, or vice versa. Even for a rectangular section this equation is a nonlinear tran-scendental
equation which has to be solved numerically by procedures such as Newton’s method.
2.2 Critical flow
δη
δA
B
Figure 2-2. Cross-section of waterway with increment of water level
We now need to find what the condition for critical flow is, where the head is a minimum. Equation (2.1)
is
H = η +
α
2g
Q2
A2(η)
,
and critical flow is when dH/dη = 0:
dH
dη
= 1−
αQ2
gA3(η) ×
dA
dη
= 0.
The problem now is to evaluate the derivative dA/dη. From Figure 2-2, in the limit as δη → 0 the
element of area δA = B δη,such that dA/dη = B, the width of the free surface. Substituting, we have
the condition for critical flow:
α
Q2B
gA3 = 1. (2.3)
11

This can be rewritten as
α
(Q/A)2
g (A/B)
= 1,
and as Q/A = U , the mean velocity over the section, and A/B = D, the mean depth of flow, this means
that
Critical flow occurs when α
U 2
gD
= 1, that is, when α ×
(Mean velocity)2
g × Mean depth = 1. (2.4)
We write this as
αF 2 = 1 or √αF = 1, (2.5)
where the symbol F is the Froude number, defined by:
F =
Q/A p
gA/B
=
U
√gD
=
Mean velocity
√g × Mean depth.
The usual statement in textbooks is that ”critical flow occurs when the Froude number is 1”. We have
chosen to generalise this slightly by allowing for the coefficient α not necessarily being equal to 1, giving
αF 2 = 1at critical flow. Any form of the condition, equation (2.3), (2.4) or (2.5) can be used. The mean
depth at which flow is critical is the ”critical depth”:
Dc = α
U 2
g
= α
Q2
gA2 . (2.6)
2.3 The Froude number
The dimensionless Froude number is traditionally used in hydraulic engineering to express the relative
importance of inertia and gravity forces, and occurs throughout open channel hydraulics. It is relevant
where the water has a free surface. It almost always appears in the form of αF 2 rather than F . It might
be helpful here to define F by writing
F 2 =
Q2B
gA3 .
Consider a calculation where we attempt to quantify the relative importance of kinetic and potential
energies of a flow – and as the depth is the only vertical scale we have we will use that to express the
potential energy. We write
Mean kinetic energy per unit mass
Mean potential energy per unit mass =
12
αU 2
gD
2 αF 2,
= 1
which indicates something of the nature of the dimensionless number αF 2.
Flows which are fast and shallow have large Froude numbers, and those which are slow and deep have
small Froude numbers. For example, consider a river or canal which is 2m deep flowing at 0.5ms−1
(make some effort to imagine it - we can well believe that it would be able to flow with little surface
disturbance!). We have
F =
U
√gD ≈
0.5
√10 × 2
= 0.11 and F 2 = 0.012 ,
and we can imagine that the rough relative importance of the kinetic energy contribution to the potential
contribution really might be of the order of this 1%. Now consider flow in a street gutter after rain. The
velocity might also be 0.5ms−1, while the depth might be as little as 2 cm. The Froude number is
F =
U
√gD ≈
0.5
√10 × 0.02
= 1.1 and F 2 = 1.2 ,
12

which is just super-critical, and we can easily imagine it to have many waves and disturbances on it due
to irregularities in the gutter.
It is clear that αF 2 expresses the scale of the importance of kinetic energy to potential energy, even
if not in a 1 : 1 manner (the factor of 1/2). It seems that αF 2 is a better expression of the relative
importance than the traditional use of F . In fact, we suspect that as it always seems to appear in the form
αF 2 = αU 2/gD, we could define an improved Froude number, Fimproved = αU 2/gD, which explicitly
recognises (a) that U 2/gD is more fundamental than U/√gD, and (b) that it is the weighted value of u2
over the whole section, αU 2, which better expresses the importance of dynamic contributions. However,
we will use the traditional definition F = U/√gD. In tutorials, assignments and exams, unless advised
otherwise, you may assume α = 1, as has been almost universally done in textbooks and engineering
practice. However we will retain α as a parameter in these lecture notes, and we recommend it also in
professional practice. Retaining it will, in general, give more accurate results, but also, retaining it while
usually not being quite sure of its actual value reminds us that we should not take numerical results as
accurately or as seriously as we might. Note that, in the spirit of this, we might well use g ≈ 10 in
practical calculations!
Rectangular channel
There are some special simple features of rectangular channels. These are also applicable to wide chan-nels,
where the section properties do not vary much with depth, and they can be modelled by equivalent
rectangular channels, or more usually, purely in terms of a unit width. We now find the conditions for
critical flow in a rectangular section of breadth b and depth h. We have A = bh. From equation (2.3) the
condition for critical flow for this section is:
αQ2
gb2h3 = 1, (2.7)
but as Q = U bh, this is the condition
αU 2
gh
= 1. (2.8)
Some useful results follow if we consider the volume flow per unit width q:
q =
Q
b
=
U bh
b
= U h. (2.9)
Eliminating Q from (2.7) or U from (2.8) or simply using (2.6) with Dc = hc for the rectangular section
gives the critical depth, when H is a minimum:
hc =
μ
α
q2
g
¶1/3
. (2.10)
This shows that the critical depth hc for rectangular or wide channels depends only on the flow per unit
width, and not on any other section properties. As for a rectangular channel it is obvious and convenient
to place the origin on the bed, such that η = h. Then equation (2.1) for critical conditions when H is a
minimum, H = Hc becomes
Hc = hc +
α
2g
Q2
A2c
= hc +
α
2g
Q2
b2h2c
= hc +
αq2
2g
1
h2c
,
and using equation (2.10) to eliminate the q2 term:
Hc = hc +
h3c
2
1
h2c
=
3
2
hc or, hc =
2
3
Hc. (2.11)
2.4 Water level changes at local transitions in channels
Now we consider some simple transitions in open channels from one bed condition to another.
13

Sub-critical flow over a step in a channel or a narrowing of the channel section: Consider the
1 2
Δ
Figure 2-3. Subcritical flow passing over a rise in the bed
Surface
elevation
η
¾
Head H = E/ρgQ
Critical constriction
Constriction
Upstream section
1
2
4
3
2’
Figure 2-4. Head/Surface-elevation relationships for three cross-sections
flow as shown in Figure 2-3. At the upstream section the (H, η) diagram can be drawn as indicated in
Figure 2-4. Now consider another section at an elevation and possible constriction of the channel. The
corresponding curve on Figure 2-4 goes to infinity at the higher value of zmin and the curve can be shown
to be pushed to the right by this raising of the bed and/or a narrowing of the section. At this stage it is not
obvious that the water surface does drop down as shown in Figure 2-3, but it is immediately explained
if we consider the point 1 on Figure 2-4 corresponding to the initial conditions. As we assume that no
energy is lost in travelling over the channel constriction, the surface level must be as shown at point 2
on Figure 2-4, directly below 1 with the same value of H, and we see how, possibly against expectation,
the surface really must drop down if subcritical flow passes through a constriction.
Sub-critical flow over a step or a narrowing of the channel section causing critical flow: Consider
14

now the case where the step Δ is high enough and/or the constriction narrow enough that the previously
sub-critical flow is brought to critical, going from point 1 as before, but this time going to point 2’ on
Figure 2-4. This shows that for the given discharge, the section cannot be constricted more than this
amount which would just take it to critical. Otherwise, the (H, η) curve for this section would be moved
further to the right and there would be no real depth solutions and no flow possible. In this case the
flow in the constriction would remain critical but the upstream depth would have to increase so as to
make the flow possible. The step is then acting as a weir, controlling the flow such that there is a unique
relationship between flow and depth.
Super-critical flow over a step in a channel or a narrowing of the channel section: Now consider
super-critical flow over the same constriction as shown in Figure 2-5. In this case the depth actually
increases as the water passes over the step, going from 3 to 4, as the construction in Figure 2-4 shows.
Δ
3
4
Figure 2-5. Supercritical flow passing over a hump in the bed.
Themathematical problem in each of these cases is to solve an equation similar to (2.2) for η2, expressing
the fact that the head is the same at the two sections:
αQ2
2gA21
(η1)
+ η1
| {z }
H1
=
αQ2
2gA22
(η2)
+ η2
| {z }
H2
. (2.12)
As the relationship between area and elevation at 2 is different from that at 1, we have shown two
different functions for area as a function of elevation, A1(η1) and A2(η2).
Example: A rectangular channel of width b1 carries a flow of Q, with a depth h1. The channel
section is narrowed to a width b2 and the bed raised by Δ, such that the flow depth above the bed
is now h2. Set up the equation which must be solved for h2.
Equation (2.12) can be used. If we place the datum on the bed at 1, then η1 = h1 and A1(η1) =
b1η1 = b1h1. Also, η2 = Δ + h2 and A2(η2) = b2 (η2 − Δ) = b2h2. The equation becomes
αQ2
2gb21
h21
+ h1 =
αQ2
2gb22
h22
+ Δ + h2, to be solved for h2, OR,
αQ2
2gb21
h21
+ h1 =
αQ2
2gb22
(η2 − Δ)2 + η2, to be solved for η2.
In either case the equation, after multiplying through by h2 or η2 respectively, becomes a cubic,
which has no simple analytical solution and generally has to be solved numerically. Below we
will present methods for this.
2.5 Some practical considerations
2.5.1 Trapezoidal sections
15

γ
1
B
h
W
Figure 2-6. Trapezoidal section showing important quantities
Most canals are excavated to a trapezoidal section, and this is often used as a convenient approximation
to river cross-sections too. In many of the problems in this course we will consider the case of trapezoidal
sections. We will introduce the terms defined in Figure 2-6: the bottom width is W , the depth is h, the
top width is B, and the batter slope, defined to be the ratio of H:V dimensions is γ. From these the
following important section properties are easily obtained:
Top width : B = W +2γh
Area : A = h (W + γh)
p
1 + γ2h,
Wetted perimeter : P = W + 2
where we will see that the wetted perimeter is an important quantity when we consider friction in chan-nels.
(Ex. Obtain these relations).
2.5.2 Solution methods for alternative depths
Here we consider the problem of solving equation (2.12) numerically:
αQ2
2gA21
(η1)
+ η1 =
αQ2
2gA22
(η2)
+ η2,
where we assume that we know the upstream conditions at point 1 and we have to find η2. The right side
shows sufficiently complicated dependence on η2 that even for rectangular sections we have to solve this
problem numerically. Reference can be made to any book on numerical methods for solving nonlinear
equations, but here we briefly describe some techniques and then develop a simplified version of a robust
method
1. Trial and error - evaluate the right side of the equation with various values of η2 until it agreeswith
the left side. This is simple, but slow to converge and not suitable for machine computation.
2. Direct iteration - re-arrange the equation in the form
η2 = H1 −
αQ2
2gA22
(η2)
and successively evaluate the right side and substitute for η2. We can show that this converges only
if the flow at 2 is subcritical (αF 2 < 1), the more common case. Provided one is aware of that
limitation, the method is simple to apply.
3. Bisection - choose an initial interval in which it is known a solution lies (the value of the function
changes sign), then successively halve the interval and determine in which half the solution lies each
time until the interval is small enough. Robust, not quite as simply programmed, but will always
converge to a solution.
4. Newton’s method - make an estimate and then make successively better ones by travelling down the
local tangent. This is fast, and reliable if a solution exists. We write the equation to be solved as
f (η2) = η2 +
αQ2
2gA22
(η2) − H1 (= 0 when the solution η2 is found). (2.13)
16

Then, if η(n)
2 is the nth estimate of the solution, Newton’s method gives a better estimate:
η(n+1)
2 = η(n)
2 −
f (η(n)
2 )
f 0(η(n)
2 )
, (2.14)
where f 0(h2) = ∂f /∂η2. In our case, from (2.13):
f 0(η2) =
∂f (η2)
∂η2
= 1−
αQ2
gA32
(η2)
∂A2
∂η2
= 1−
αQ2B2(η2)
gA(η2)
32
= 1− αF 2(η2),
which is a simple result - obtained using the procedure we used for finding critical flow in an
arbitrary section. Hence, the procedure (2.14) is
η(n+1)
2 = η(n)
2 −
η(n)
2 + αQ2
2gA22
2 ) −H1
(η(n)
1−αF (n)2
2
. (2.15)
Note that this will not converge as quickly if the flow at 2 is critical, where both numerator and
denominator go to zero as the solution is approached, but the quotient is still finite. This expression
looks complicated, but it is simple to implement on a computer, although is too complicated to
appear on an examination paper in this course.
These methods will be examined in tutorials.
2.6 Critical flow as a control - broad-crested weirs
For a given discharge, the (H, η) diagram showed that the bed cannot be raised or the section narrowed
more than the amount which would just take it to critical. Otherwise there would be no real depth
solutions and no flow possible. If the channel were constricted even more, then the depth of flow over
the raised bed would remain constant at the critical depth, and the upstream depth would have to increase
so as to make the flow possible. The step is then acting as a weir, controlling the flow.
hc
hc
hc
Figure 2-7. A broad-crested weir spillway, showing the critical depth over it providing a control.
Consider the situation shown in Figure 2-7 where the bed falls away after the horizontal section, such as
on a spillway. The flow upstream is subcritical, but the flow downstream is fast (supercritical). Some-where
between the two, the flow depth must become critical - the flow reaches its critical depth at some
point on top of the weir, and the weir provides a control for the flow, such that a relationship between
flow and depth exists. In this case, the head upstream (the height of the upstream water surface above the
sill) uniquely determines the discharge, and it is enough to measure the upstream surface elevation where
the flow is slow and the kinetic part of the head negligible to provide a point on a unique relationship
between that head over the weir and the discharge. No other surface elevation need be measured.
Figure 2-8 shows a horizontal flow control, a broad-crested weir, in a channel. In recent years there has
been a widespread development (but not in Australia, unusually) of such broad-crested weirs placed in
streams where the flow is subcritical both before and after the weir, but passes through critical on the
17

weir. There is a small energy loss after the flume. The advantage is that it is only necessary to measure
the upstream head over the weir.
Small energy loss
hc
hc hc
Figure 2-8. A broad-crested weir
3. Conservation of momentum in open channel flow
3.1 Integral momentum theorem
P
Control volume
M1
1 2
M2
Figure 3-1. Obstacle in stream reducing the momentum flux
We have applied energy conservation principles. Now we will apply momentum. We will consider, like
several problems above, relatively short reaches and channels of prismatic (constant) cross-section such
that the small contributions due to friction and the component of gravity down the channel are roughly in
balance. Figure 3-1 shows the important horizontal contributions to force and momentum in the channel,
where there is a structure applying a force P to the fluid in the control volume we have drawn.
The momentum theorem applied to the control volume shown can be stated: the net momentum flux
leaving the control volume is equal to the net force applied to the fluid in the control volume. The
momentum flux is defined to be the surface integral over the control surface CS:
Z
CS
(p ˆn + u ρu.ˆn) dS,
where ˆn is a unit vector normal to the surface, such that the pressure contribution on an element of area
dS is the force p dS times the unit normal vector ˆn giving its direction; u is the velocity vector such
that u.ˆn is the component of velocity normal to the surface, u.ˆn dS is the volume rate of flow across the
surface, multiplying by density gives the mass rate of flow across the surface ρu.ˆn dS, and multiplying
by velocity gives uρu.ˆn dS, the momentum rate of flow across the surface.
We introduce i, a unit vector in the x direction. On the face 1 of the control surface in Figure 3-1, as the
outwards normal is in the upstream direction, we have ˆn = −i, and u = u1i, giving u.ˆn = −u1and the
18

vector momentum flux across face 1 is
M1 = −i
R
A1
¡
p1 + ρu21
¢
dA = −i M1,
where the scalar quantity
M1 =
R
A1
¡
p1 + ρu21
¢
dA.
Similarly, on face 2 of the control surface, as the outwards normal is in the downstream direction, we
have ˆn = i and u = u2i, giving u.ˆn = +u2 and the vector momentum flux across face 2 is
M2 = +i
R
A2
¡
p2 + ρu22
¢
dA = +i M2
with scalar quantity
M2 =
R
A2
¡
p2 + ρu22
¢
dA.
Using the momentum theorem, and recognising that the horizontal component of the force of the body
on the fluid is −P i, then we have, writing it as a vector equation but including only x (i) components:
M1 +M2 = −P i
AsM1 = −M1i andM2 = +M2i, we can write it as a scalar equation giving:
P = M1 − M2, (3.1)
where P is the force of the water on the body (or bodies).
3.1.1 Momentum flux across a section of channel
From the above, it can be seen how useful is the concept of the horizontal momentum flux at a section of
the flow in a waterway:
M =
Z
A
¡
p + ρu2¢
dA.
We attach different signs to the contributions depending on whether the fluid is leaving (+ve) or en-tering
(-ve) the control volume. As elsewhere in these lectures on open channel hydraulics we use the
hydrostatic approximation for the pressure: p = ρg(η − z), which gives
M = ρ
Z
A
¡
g(η − z) + u2¢
dA.
Now we evaluate this in terms of the quantities at the section.
Pressure and elevation contribution ρ
R
A g(η − z) dA : The integral
R
A (η − z) dA is simply the
first moment of area about a transverse horizontal axis at the surface, we can write it as
R
A (η − z) dA = A¯h, (3.2)
where ¯h
is the depth of the centroid of the section below the surface.
Velocity contribution ρ
R
A u2 dA : Now we have the task of evaluating the square of the horizontal
velocity over the section. As with the kinetic energy integral, it seems that the sensible thing to do is
to recognise that all velocity components and velocity fluctuations will be of a scale given by the mean
flow velocity in the stream at that point, and so we simply write
Z
A
u2 dA = βU 2A = β
Q2
A
, (3.3)
19

which defines β as a coefficient which will be somewhat greater than unity, given by
β =
R
A u2 dA
U 2A
. (3.4)
This coefficient is known as a Boussinesq coefficient, in honour of the French engineer who introduced
it, who did much important work in the area of the non-uniformity of velocity and the non-hydrostatic
nature of the pressure distribution. Most presentations of open channel theory adopt the approximation
that there is no variation of velocity over the section, such that it is assumed that β = 1. Typical real
values are β = 1.05 − 1.15, somewhat less than the Coriolis energy coefficient α.
Combining: We can substitute to give the expression we will use for the Momentum Flux:
M = ρ
¡
gA¯h
+ βU 2A
¢
= ρ
μ
gA¯h
+ β
Q2
A
¶
= ρg
μ
A(h)¯ h(h) +
βQ2
g
1
A(h)
¶
(3.5)
where we have shown the dependence on depth in each term. This expression can be compared with that
for the head as defined in equation (2.1) but here expressed relative to the bottom of the channel:
H = h +
αQ2
2g
1
A2(h)
.
The variation with h is different between this and equation (3.5). For large h, H ∼ h, while M ∼ A(h) h(h), which for a rectangular section goes like h2. For small h, H ∼ 1/A2(h), and M ∼ 1/A(h).
Note that we can re-write equation (3.5) in terms of Froude number (actually appearing as F 2 – yet
again) to indicate the relative importance of the two parts, which we could think of as ”static” and
”dynamic” contributions:
M = ρgA¯h
μ
1+βF 2 A/B
¯h
¶
.
The ratio (A/B) /¯ h, mean depth to centroid depth, will have a value typically of about 2.
Example: Calculate (a) Head (using the channel bottom as datum) and (b) Momentum flux, for a
rectangular section of breadth b and depth h.
We have A = bh, h = h/2. Substituting into equations (2.1) and (3.5) we obtain
H = h +
αQ2
2gb2 ×
1
h2 and,
M = ρ
μ
gb
2 × h2+
βQ2
b ×
1
h
¶
.
Note the quite different variation with h between the two quantities.
3.1.2 Minimum momentum flux and critical depth
We calculate the condition for minimum M:
∂M
∂h
=
∂
∂h
(A(h)¯ h(h)) −
βQ2
g
1
A2(h)
∂A
∂h
= 0. (3.6)
The derivative of the first moment of area about the surface is obtained by considering the surface
increased by an amount h + δh
∂(A¯ h)
∂h
= lim
δh→0
(A(h)¯ h(h))h+δh − A(h)¯ h(h))
δh
. (3.7)
The situation is as shown in Figure 3-2. The first moment of area about an axis transverse to the channel
20

Depth to centroid of hatched area: δh/2
B
h
Depth to centroid of white area: h +δh
δh
Figure 3-2. Geometrical interpretation of calculation of position of centroid
at the new surface is:
(A(h)¯ h(h))h+δh = A(h) × (¯h
+ δh) + B × δh ×
δh
2
,
so that, substituting into equation (3.7), in the limit δh → 0,
∂(A¯ h)
∂h
= lim
δh→0
A(h) × (¯h
+ δh) + B × δh × δh/2 − A(h)¯ h(h))
δh
= A(h) = A, (3.8)
which is surprisingly simple. Substituting both this and ∂A/∂h = B in equation (3.6), we get the
condition for minimum M:
βQ2B
gA3 = βF 2 = 1, (3.9)
which is a similar condition for the minimum energy, but as in general α6= β, the condition for minimum
momentum is not the same as that for minimum energy.
3.1.3 Momentum flux -depth diagram
If the cross-section changes or there are other obstacles to the flow, the sides of the channel and/or the
obstacles will also exert a force along the channel on the fluid. We can solve for the total force exerted
between two sections if we know the depth at each. In the same way as we could draw an (H, η) diagram
for a given channel section, we can draw an (M, η) diagram. It is more convenient here to choose the
datum on the bed of the channel so that we can interpret the surface elevation η as the depth h. Figure
3-3 shows a momentum flux – depth (M, h) diagram. Note that it shows some of the main features of
the (H, h) diagram, with two possible depths for the same momentum flux – called conjugate depths.
However the limiting behaviours for small and large depths are different for momentum, compared with
energy.
3.2 Flow under a sluice gate and the hydraulic jump
Consider the flow problem shown at the top of Figure 3-4, with sub-critical flow (section 0) controlled
by a sluice gate. The flow emerges from under the gate flowing fast (super-critically, section 1). There
has been little energy loss in the short interval 0-1, but the force of the gate on the flow has substantially
reduced its momentum flux. It could remain in this state, however here we suppose that the downstream
level is high enough such that a hydraulic jump occurs, where there is a violent turbulent motion and in
a short distance the water changes to sub-critical flow again. In the jump there has been little momentum
loss, but the turbulence has caused a significant loss of energy between 1-2. After the jump, at stage 2,
the flow is sub-critical again. We refer to this depth as being sequent to the original depth.
In the bottom part of Figure 3-4 we combine the (H, h) and (M, h) diagrams, so that the vertical axis is
21

Momentum flux M
Depth h
h1
h2
3
2
1
4
Rectangular: M ~ h2
Rectangular: M ~ 1/ h
P
h4 P
h3
Figure 3-3. Momentum flux – depth diagram, showing effects of a momentum loss P for subcritical and supercrit-ical
flow.
0 h 0
P
Head H, Momentum flux M
Depth
h
Head H
Momentum flux M
0
2 2
1 1
H 0 = H1 M 1 = M 2 M 0
2 h
1 h
P
0 1 2
Figure 3-4. Combined Head and Momentum diagrams for the sluice gate and hydraulic jump problem
22

depth h and the two horizontal axes are head H and momentum flux M, with different scales. We now
outline the procedure we follow to analyse the problem of flow under a sluice gate, with upstream force
P , and a subsequent hydraulic jump.
• We are given the discharge Q and the upstream depth h0, and we know the cross-sectional details
of the channel.
• We can compute the energy and momentum at 0, H0 and M0 (see points 0 on the M −H −h plot).
• As energy is conserved between 0 and 1, the depth h1 can be calculated by solving the energy
equation with H1 = H0, possibly using Newton’s method.
• In fact, this depth may not always be realisable, if the gate is not set at about the right position. The
flow at the lip of the gate leaves it vertically, and turns around to horizontal, so that the gate opening
must be larger than h1. A rough guide is that the gate opening must be such that h1 ≈ 0.6×Gate
opening.
• With this h1 we can calculate the momentum flux M1.
• The force on the gate P (assuming that the channel is prismatic) can be calculated from:
P = M0 − M1 = ρ
μ
gAh+β
Q2
A
¶
0 − ρ
μ
gAh + β
Q2
A
¶
1
• Across the hydraulic jump momentum is conserved, such that M2 = M1:
μ
gAh+β
Q2
A
¶
2
=
μ
gAh+β
Q2
A
¶
1
• This gives a nonlinear equation for h2 to be solved numerically (note that A and¯h
are both functions
of h). In the case of a rectangular channel the equation can be written
1
2
gh21
+
βq2
h1
=
1
2
gh22
+
βq2
h2
,
where q = Q/b, the discharge per unit width. In fact it can be solved analytically. Grouping like
terms on each side and factorising:
(h2 − h1)(h2 + h1) =
2βq2
g
μ
1
h1 −
1
h2
¶
,
h22
h1 + h21h2 −
2βq2
g
= 0,
which is a quadratic in h2, with solutions
h2 = −
h1
2 ±
s
h21
4
+
2βq2
gh1
,
but we cannot have a negative depth, and so only the positive sign is taken. Dividing through by
h1:
h2
h1
= −
1
2
+
s
1
4
+
2βq2
gh31
=
1
2
μq
1 + 8βF 2
¶
1 −1
• Sometimes the actual depth of the downstream flow is determined by the boundary condition further
downstream. If it is not deep enough the actual jump may be an undular hydraulic jump, which
does not dissipate as much energy, with periodic waves downstream.
• The pair of depths (h1, h2) for which the flow has the same momentum are traditionally called the
conjugate depths.
23

• The loss in energy H2 − H1 can be calculated. For a rectangular channel it can be shown that
ΔH = H1 − H2 =
(h2 − h1)3
4h1h2
.
3.3 The effects of streams on obstacles and obstacles on streams
3.3.1 Interpretation of the effects of obstacles in a flow
Slow (sub-critical) approach flow Figure 3-5 shows that the effect of a drag force is to lower the
P
P
h
1 h
2 h
1 2 M
c h
Figure 3-5. Effect of obstacles on a subcritical flow
water surface (counter-intuitive!?) if the flow is slow (sub-critical).
Fast (super-critical) approach flow Figure 3-6 shows that the effect of a drag force on a super-critical
P
h
2 P h
1 h
1 2 M
c h
Figure 3-6. Effect of obstacles on a supercritical flow
flow is to raise the water surface. In fact, the effect of the local force only spreads gradually through
the stream by turbulent diffusion, and the predicted change in cross-section will apply some distance
downstream where the flow has become uniform (rather further than in the diagrams here).
A practical example is the fast flow downstream of a spillway, shown in Figure 3-7, where the flow
becomes subcritical via a hydraulic jump. If spillway blocks are used, the water level downstream need
not be as high, possibly with large savings in channel construction.
3.3.2 Bridge piers - slow approach flow
Consider flow past bridge piers as shown in Figure 3-8. As the bridge piers extend throughout the flow,
for the velocity on the pier we will take the mean upstream velocity V = Q/A1, and equation (3.14) can
24

P
2 P h
2*
1
2
*
2 h = depth without blocks
1 h
1 2 M
Figure 3-7. Effect of spillway blocks on lowering the water level in a spillway pool
Plan
Side elevation
c h
1
2 P
c h
1 2 M
Figure 3-8. Flow past bridge piers and their effect on the flow
be used.
3.3.3 Flow in a narrowing channel - choked flow
We consider cases where the width reduction is more than in a typical bridge pier problem, such that the
flow in the throat may become critical, the throat becomes a control, and the flow is said to be choked. If
so, the upstream depth is increased, to produce a larger momentum flux there so that the imposed force
due to the convergence now just produces critical flow in the throat. In problems such as these, it is very
helpful to remember that for a rectangular section, equation (2.10):
hc =
¡
αq2/g
¢1/3
, or, re-written, q =
p
gh3c
/α,
where q = Q/b, the flow per unit width, and also to observe that at critical depth, equation (2.11):
H = hc +
αQ2
2gb2h2c
= hc +
αq2
2gh2c
=
3
2
hc, so that hc =
2
3
H.
It is clear that by reducing b, q = Q/b is increased, until in this case, criticality is reached. While
25

c h
Elevation
c h
Plan
Figure 3-9. Flow through contraction sufficiently narrow that it becomes critical
generally this is not a good thing, as the bridge would then become a control, where there is a relationship
between flow and depth, this becomes an advantage in flow measurement applications. In critical flow
flumes only an upstream head is needed to calculate the flow, and the structure is deliberately designed
to bring about critical depth at the throat. One way of ensuring this is by putting in a rise in the bed at
the throat. Note that in the diagram the critical depth on the hump is greater than that upstream because
the width has been narrowed.
3.3.4 Drag force on an obstacle
As well as sluices and weirs, many different types of obstacles can be placed in a stream, such as the piers
of a bridge, blocks on the bed, Iowa vanes, the bars of a trash-rack etc. or possibly more importantly,
the effects of trees placed in rivers (”Large Woody Debris”), used in their environmental rehabilitation.
It might be important to know what the forces on the obstacles are, or in flood studies, what effects the
obstacles have on the river.
Substituting equation (3.5) into equation (3.1) (P = M1 − M2) gives the expression:
P = ρ
μ
gA¯h
+ β
Q2
A
¶
1 − ρ
μ
gA¯h
+ β
Q2
A
¶
2
, (3.10)
so that if we know the depth upstream and downstream of an obstacle, the force on it can be calculated.
Usually, however, the calculation does not proceed in that direction, as one wants to calculate the effect
of the obstacle on water levels. The effects of drag can be estimated by knowing the area of the object
measured transverse to the flow, a, the drag coefficient Cd, and V , themean fluid speed past the object:
P =
1
2
ρCdV 2a, (3.11)
and so, substituting into equation (3.10) gives, after dividing by density,
1
2
CdV 2a =
μ
gA¯h
+ β
Q2
A
¶
1 −
μ
gA¯h
+ β
Q2
A
¶
2
. (3.12)
We will write the velocity V on the obstacle as being proportional to the upstream velocity, such that we
write
V 2 = γd
μ
Q
A1
¶2
, (3.13)
26

where γd is a coefficient which recognises that the velocity which impinges on the object is generally
not equal to the mean velocity in the flow. For a small object near the bed, γd could be quite small; for
an object near the surface it will be slightly greater than 1; for objects of a vertical scale that of the whole
depth, γd ≈ 1. Equation (3.12) becomes
1
2
γd Cd
Q2
A21
a =
μ
gA¯h
+ β
Q2
A
¶
1 −
μ
gA¯h
+ β
Q2
A
¶
2
(3.14)
A typical problem is where the downstream water level is given (sub-critical flow, so that the control is
downstream), and we want to know by how much the water level will be raised upstream if an obstacle
is installed. As both A1 and h1 are functions of h1, the solution is given by solving this transcendental
equation for h1. In the spirit of approximation which can be used in open channel hydraulics, and in the
interest of simplicity and insight, we now obtain an approximate solution.
3.3.5 An approximate method for estimating the effect of channel obstructions on flooding
Momentum flux M
Depth h
1 h
3
2
1
4
4 h P
3 h
Tangent to (M,h) curve
Approximate h1
Exact h1
2 h
Figure 3-10. Momentum flux – depth diagram showing the approximate value of d1 calculated by approximating
the curve by its tangent at 2.
Now an approximation to equation (3.14) will be obtained which enables a direct calculation of the
change in water level due to an obstacle, without solving the transcendental equation. We consider a
linearised version of the equation, which means that locally we assume a straight-line approximation to
the momentum diagram, for a small reduction in momentum, as shown in Figure 3-10.
Consider a small change of surface elevation δh going from section 1 to section 2, and write the expres-sion
for the downstream area
A2 = A1 + B1δh.
It has been shown above (equation 3.8) that
∂(A¯ h)
∂h
= A,
and so we can write an expression for A2¯h
2 in terms of A1¯h
1 and the small change in surface elevation:
A2¯h
2 = A1¯h
1 + δh
∂(A¯ h)
∂h
¯¯¯¯
1
= A1¯h
1 + δh A1,
27

and so equation (3.14) gives us, after dividing through by g:
1
2
γd Cd
Q2
gA21
a = −δh A1 + β
Q2
gA1 − β
Q2
g (A1 + B1δh)
= −δh A1 + β
Q2
gA1
Ã
1 −
μ
1 +
B1
A1
¶
−1
δh
!
.
Now we use a power series expansion in δh to simplify the last term, neglecting terms like (δh)2. For ε
small, (1 + ε)−1 ≈ 1 − ε, and so
1
2
γd Cd
Q2
gA21
a ≈ −δh A1 + β
Q2B1
gA21
δh.
We can now solve this to give an explicit approximation for δh:
δh ≈
12
γd Cd
Q2
gA31
a
β Q2B1
gA31
− 1
.
It is simpler to divide both sides by the mean depth A1/B1 to give:
δh
A1/B1
=
1
2 γd Cd F 2
1
a
A1
βF 2
1 − 1
.
We do not have to worry here that for subcritical flow we do not necessarily know the conditions at
point 1, but instead we know them at the downstream point 2. Within our linearising approximation, we
can use either the values at 1 or 2 in this expression, and so we generalise by dropping the subscripts
altogether, so that we write
δh
A/B
=
1
2 γd Cd F 2 a
A
βF 2 − 1
= 1
2 γd Cd
a
A ×
F 2
βF 2 − 1
. (3.15)
Thus we see that the relative change of depth (change of depth divided by mean depth) is directly
proportional to the coefficient of drag and the fractional area of the blockage, as we might expect. The
result is modified by a term which is a function of the square of the Froude number. For subcritical
flow the denominator is negative, and so is δh, so that the surface drops, as we expect, and as can be
seen when we solve the problem exactly using the momentum diagram. If upstream is supercritical,
the surface rises. Clearly, if the flow is near critical (βF 2
1 ≈ 1) the change in depth will be large (the
gradient on the momentum diagram is vertical), when the theory will have limited validity.
Example: In a proposal for the rehabilitation of a river it is proposed to install a number of logs
(”Large Woody Debris” or ”Engineered Log Jam”). If a single log of diameter 500mm and 10m
long were placed transverse to the flow, calculate the effect on river height. The stream is roughly
100m wide, say 10m deep in a severe flood, with a drag coefficient Cd ≈ 1. The all-important
velocities are a bit uncertain. We might assume a mean velocity of say 6ms−1, and velocity on
the log of 2ms−1. Assume β = 1.1.
We have the values
A = 100 × 10 = 1000m2, a = 0.5 × 10 = 5m2
F 2 = U 2/gD = 62/10/10 = 0.36, γd = 22/62 ≈ 0.1
and substituting into equation (3.15) gives
δh
A/B ≈
1
2 γdCd
a
A1
β − 1
F 2
1
=
1
2 × 0.1 × 1 × 5
1000
1.1 − 1
0.36
= −1.5 × 10−4,
so that multiplying by the mean depth, δη = −1.5 × 10−4 × 10 = −1.5mm. The negative value is the
28

change as we go downstream, thus we see that the flow upstream is raised by 1.5mm.
4. Uniform flow in prismatic channels
Uniform flow is where the depth does not change along the waterway. For this to occur the channel
properties also must not change along the stream, such that the channel is prismatic, and this occurs only
in constructed canals. However in rivers if we need to calculate a flow or depth, it is common to use
a cross-section which is representative of the reach being considered, and to assume it constant for the
application of this theory.
4.1 Features of uniform flow and relationships for uniform flow
• There are two forces in balance in steady flow:
– The component of gravity downstream along the channel, and
– the shear stress at the sides which offers resistance to the flow, which increases with flow veloc-ity.
• If a channel is long and prismatic (slope and section do not change) then far from the effects of
controls the two can be in balance, and if the flow is steady, the mean flow velocity and flow depth
remain constant along the channel, giving uniform flow, at normal depth.
A
L
P
τ0
τ0
θ
g sin θ
g
Figure 4-1. Slice of uniform channel flow showing shear forces and body forces per unit mass acting
Consider a slice of uniform flow in a channel of length L and cross-sectional area A, as shown in
Figure 4-1. The component of gravity force along the channel is ρ × AL × g sin θ, where θ is the
angle of inclination of the channel, assumed positive downwards. The shear force is τ 0 × L × P ,
where τ 0 is the shear stress, and P is the wetted perimeter of the cross-section. As the two are in
balance for uniform flow, we obtain
τ 0
ρ
= g
A
P
sin θ.
Now, τ 0/ρ has units of velocity squared; we combine g and the coefficient relating the mean
29

velocity U at a section to that velocity, giving Chézy’s law (1768):
U = C
p
RS0,
where C is the Chézy coefficient (with units L1/2T−1), R = A/P is the hydraulic radius (L), and
S0 = sinθ is the slope of the bed, positive downwards. The tradition in engineering is that we use
the tangent of the slope angle, so this is valid for small slopes such that sin θ ≈ tan θ.
• However there is experimental evidence that C depends on the hydraulic radius in the form C ∼ R1/6 (Gauckler, Manning), and the law widely used is Manning’s Law:
U =
1
n
R2/3S1/2
0 ,
where n is the Manning coefficient (units of L−1/3T), which increases with increasing roughness.
Typical values are: concrete - 0.013, irrigation channels - 0.025, clean natural streams - 0.03,
streams with large boulders - 0.05, streams with many trees - 0.07. Usually the units are not shown.
• Multiplying by the area, Manning’s formula gives the discharge:
Q = U A =
1
n
A5/3
P 2/3
p
S0, (4.1)
in which both A and P are functions of the flow depth. Similarly, Chézy’s law gives
Q = C
A3/2
P 1/2
p
S0. (4.2)
Both equations show how flow increases with cross-sectional area and slope and decreases with
wetted perimeter.
4.2 Computation of normal depth
If the discharge, slope, and the appropriate roughness coefficient are known, either of equations (4.1)
and (4.2) is a transcendental equation for the normal depth hn, which can be solved by the methods
described earlier. We can gain some insight and develop a simple scheme by considering a trapezoidal
cross-section, where the bottom width is W , the depth is h, and the batter slopes are (H:V) γ : 1 (see
Figure 2-6). The following properties are easily shown to hold (the results have already been presented
above):
Top width B W +2γh
Area A h(W + γh)
Wetted perimeter P W +2
p
1 + γ2h
In the case of wide channels, (i.e. channels rather wider than they are deep, h ¿ W , which is a common
case) the wetted perimeter does not show a lot of variation with depth h. Similarly in the expression for
the area, the second factor W +γh (the mean width) does not show a lot of variation with h either – most
of the variation is in the first part h. Hence, if we assume that these properties hold for cross-sections of
a more general nature, we can rewrite Manning’s law:
Q =
1
n
A5/3(h)
P 2/3(h)
p
S0 =
√S0
n
(A(h)/h)5/3
P 2/3(h) × h5/3,
where most of the variation with h is contained in the last term h5/3, and by solving for that term we can
re-write the equation in a form suitable for direct iteration
h =
μ
Qn
√S0
¶3/5
×
P 2/5(h)
A(h)/h
,
30

where the first term on the right is a constant for any particular problem, and the second term is expected
to be a relatively slowly-varying function of depth, so that the whole right side varies slowly with depth –
a primary requirement that the direct iteration scheme be convergent and indeed be quickly convergent.
Experience with typical trapezoidal sections shows that this works well and is quickly convergent. How-ever,
it also works well for flow in circular sections such as sewers, where over a wide range of depths
the mean width does not vary much with depth either. For small flows and depths in sewers this is not
so, and a more complicated method might have to be used.
Example: Calculate the normal depth in a trapezoidal channel of slope 0.001, Manning’s coef-ficient
n = 0.04, width 10m, with batter slopes 2 : 1, carrying a flow of 20m3 s−1. We have
A = h (10 + 2 h), P = 10+4.472 h, giving the scheme
h =
μ
Qn
√S0
¶3/5
×
(10 + 4.472 h)2/5
10 + 2 h
= 6.948 ×
(10 + 4.472 h)2/5
10 + 2 h
and starting with h = 2 we have the sequence of approximations: 2.000, 1.609, 1.639, 1.637 -
quite satisfactory in its simplicity and speed.
4.3 Conveyance
It is often convenient to use the conveyance K which contains all the roughness and cross-section prop-erties,
such that for steady uniform flow
Q = K
p
S0,
such that, using an electrical analogy, the flow (current) is given by a ”conductance” (here conveyance)
multiplied by a driving potential, which, here in this nonlinear case, is the square root of the bed slope.
In more general non-uniform flows below we will see that we use the square root of the head gradient.
With this definition, if we use Manning’s law for the flow, K is defined by
K =
1
n × A
μ
A
P
¶2/3
=
1
n ×
A5/3
P 2/3
, (4.3)
where K is a function of the roughness and the local depth and cross-section properties. Textbooks often
use conveyance to provide methods for computing the equivalent conveyance of compound sections such
as that shown in Figure 4-2. However, for such cases where a river has overflowed its banks, the flow
situation is much more likely to be more two-dimensional than one-dimensional. The extent of the
various elemental areas and the Manning’s roughnesses of the different parts are all such as to often
render a detailed ”rational” calculation unjustified.
2 3
1
Figure 4-2.
In the compound channel in the figure, even though the surface might actually be curved as shown and
31

the downstream slope and/or bed slope might be different across the channel, the tradition is that we
assume it to be the same. The velocities in the individual sections are, in general, different. We write
Manning’s law for each section based on the mean bed slope:
Q1 = K1S1/2
0 , Q2 = K2S1/2
0 , Q3 = K3S1/2
0
In a general case with n sub-sections, the total discharge is
Q =
Xn
i=1
Qi =
Xn
i=1
KiS1/2
0 = S1/2
0
Xn
i=1
Ki = S1/2
0 K
where we use the symbol K for the total conveyance:
K =
Xn
i=1
Ki =
Xn
i=1
A5/3
i
niP 2/3
i
.
5. Steady gradually-varied non-uniform flow
Steady gradually-varied flow is where the conditions (possibly the cross-section, but often just the sur-face
elevation) vary slowly along the channel but do not change with time. The most common situation
where this arises is in the vicinity of a control in a channel, where there may be a structure such as a
weir, which has a particular discharge relationship between the water surface level and the discharge.
Far away from the control, the flow may be uniform, and there the relationship between surface elevation
and discharge is in general a different one, typically being given by Manning’s law, (4.1). The transition
between conditions at the control and where there is uniform flow is described by the gradually-varied
flow equation, which is an ordinary differential equation for the water surface height. The solution will
approach uniform flow if the channel is prismatic, but in general we can treat non-prismatic waterways
also.
In sub-critical flow the flow is relatively slow, and the effects of any control can propagate back up the
channel, and so it is that the numerical solution of the gradually-varied flow equation also proceeds in
that direction. On the other hand, in super-critical flow, all disturbances are swept downstream, so that
the effects of a control cannot be felt upstream, and numerical solution also proceeds downstream from
the control.
Solution of the gradually-varied flow equation is a commonly-encountered problem in open channel
hydraulics, as it is used to determine, for example, how far upstream water levels might be increased,
and hence flooding enhanced, due to downstream works, such as the installation of a bridge.
5.1 Derivation of the gradually-varied flow equation
Consider the elemental section of waterway of length Δx shown in Figure 5-1. We have shown stations
1 and 2 in what might be considered the reverse order, but we will see that for the more common sub-critical
flow, numerical solution of the governing equation will proceed back up the stream. Considering
stations 1 and 2:
Total head at 2 = H2
Total head at 1 = H1 = H2 − HL,
and we introduce the concept of the friction slope Sf which is the gradient of the total energy line such
that HL = Sf × Δx. This gives
H1 = H2 − Sf Δx,
32

U 2 / 2 g Total energy line
α 2
U 2 / 2 g
α 1
2 h
S f Δx
S0 Δx
2 1
1 h
Sub-critical flow
Δx
Figure 5-1. Elemental section of waterway
and if we introduce the Taylor series expansion for H1:
H1 = H2 + Δx
dH
dx
+ . . . ,
substituting and taking the limit Δx → 0 gives
dH
dx
= −Sf , (5.1)
an ordinary differential equation for the head as a function of x.
To obtain the frictional slope, we use either of the frictional laws of Chézy orManning (or a smooth-wall
formula), where we make the assumption that the equation may be extended from uniform flow (where
the friction slope equals the constant bed slope) to this non-uniform case, such that the discharge at any
point is given by, for the case of Manning:
Q =
1
n
A5/3
P 2/3
p
Sf ,
but where we have used the friction slope Sf rather than bed slope S0, as in uniform flow. Solving for
Sf : the friction slope is given by
Sf =
Q2
K2(h)
, (5.2)
where we have used the conveyance K, which was defined in equation (4.3), but we repeat here,
K (h) =
1
n
A5/3
P 2/3
,
showing the section properties to be a function of the local depth, where we have restricted our attention
to prismatic channels on constant slope. This now means that for a given constant discharge we can
write the differential equation (5.1) as
dH
dx
= −Sf (h). (5.3)
As we have had to use local depth on the right side, we have to show the head to be a function of depth
h, so that we write
H = h + zmin +
α
2g
Q2
A2(h)
. (5.4)
33

Differentiating:
dH
dx
=
dh
dx
+
dzmin
dx −
α
g
Q2
A3(h)
dA(h)
dx
. (5.5)
The derivative dzmin/dx = −S0, where S0 is the bed slope,which we have defined to be positive for the
usual case of a downwards-sloping channel. Now we have to express the dA(h)/dx in terms of other
quantities. In our earlier work we saw that if the surface changed by an amount Δh, then the change in
area due to this was ΔA = B Δh, and so we can write dA(h)/dx = B dh/dx, and substituting these
results into equation (5.5) gives
dH
dx
= −S0 +
μ
1 −
α
g
Q2B(h)
A3(h)
¶
dh
dx
= −S0 +
¡
1 − αF 2(h)
¢ dh
dx
,
where the Froude number has entered, shown here as a function of depth. Finally, substituting into (5.3)
we obtain
dh
dx
=
S0 − Sf (h)
1 − αF 2(h)
=
S0 − Q2/K2(h)
1 − αF 2(h)
, (5.6)
a differential equation for depth h as a function of x, where on the right we have shown the functional
dependence of the various terms. This, or the less-explicit form (5.3), are forms of the gradually-varied
flow equation, from which a number of properties can be inferred.
5.2 Properties of gradually-varied flow and the governing equation
• The equation and its solutions are important, in that they tell us how far the effects of a structure or
works in or on a stream extend upstream or downstream.
• It is an ordinary differential equation of first order, hence one boundary condition must be supplied
to obtain the solution. In sub-critical flow, this is the depth at a downstream control; in super-critical
flow it is the depth at an upstream control.
• In general that boundary depth is not equal to the normal depth, and the differential equation de-scribes
the transition from the boundary depth to normal depth – upstream for sub-critical flow,
downstream for supercritical flow. The solutions look like exponential decay curves, and below we
will show that they are, to a first approximation.
• If that approximation is made, the resulting analytical solution is useful in providing us with some
insight into the quantities which govern the extent of the upstream or downstream influence.
• The differential equation is nonlinear, and the dependence on h is complicated, such that analytical
solution is not possible without an approximation, and we will usually use numerical methods.
• The uniform flow limit satisfies the differential equation, for when Sf = S0, dh/dx = 0, and the
depth does not change.
• As the flow approaches critical flow, when αF 2 → 1, then dh/dx → ∞, and the surface becomes
vertical. This violates the assumption we made that the flow is gradually varied and the pressure
distribution is hydrostatic. This is the one great failure of our open channel hydraulics at this level,
that it cannot describe the transition between sub- and super-critical flow.
5.3 Classification system for gradually-varied flows
The differential equation can be used as the basis for a dual classification system of gradually-varied
flows:
• one based on 5 conditions for slope, essentially as to how the normal depth compares with critical
depth, and 3 conditions for the actual depth, and how it compares with both normal, and critical
depths, as shown in the Table:
34

Slope classification
Steep slope: hn < hc
Critical slope: hn = hc
Mild slope: hn > hc
Horizontal slope: hn = ∞
Adverse slope: hn does not exist
Depth classification
Zone 1: h > hn and hc
Zone 2: h between hn and hc
Zone 3: h < hn and hc
Figure 5-2 shows the behaviour of the various solutions. In practice, the most commonly encountered
are the M1, the backwater curve on a mild slope; M2, the drop-down curve on a mild slope, and S2, the
drop-down curve on a steep slope.
5.4 Some practical considerations
5.4.1 Flood inundation studies
Figure 5-3 shows a typical subdivision of a river and its flood plain for a flood inundation study, where
solution of the gradually-varied flow equation would be required. It might be wondered how the present
methods can be used for problems which are unsteady, such as the passage of a substantial flood, where
on the front face of the flood wave the water surface is steeper and on the back face it is less steep. In
many situations, however, the variation of the water slope about the steady slope is relatively small, and
the wavelength of the flood is long, so that the steady model can be used as a convenient approxima-tion.
The inaccuracies of knowledge of the geometry and roughness are probably such as to mask the
numerical inaccuracies of the solution. Below we will present some possible methods and compare their
accuracy.
5.4.2 Incorporation of losses
It is possible to incorporate the losses due, say, to a sudden expansion or contraction of the channel, such
as shown in Figure 5-4. After an expansion the excess velocity head is destroyed through turbulence.
Before an expansion the losses will not be so large, but there will be some extra losses due to the
convergence and enhanced friction. We assume that the expansion/contraction head loss can be written
ΔHe = C
μ
Q2
2gA22
−
Q2
2gA21
¶
,
where C ≈ 0.3 for expansions and 0.1 for a contraction.
5.5 Numerical solution of the gradually-varied flow equation
Consider the gradually-varied flow equation (5.6)
dh
dx
=
S0 − Sf (h)
1 − αF 2(h)
,
where both Sf (h) = Q2/K2(h) and F 2(h) = Q2B(h)/gA3(h) are functions of Q as well as the depth
h. However as Q is constant for a particular problem we do not show the functional dependence on
it. The equation is a differential equation of first order, and to obtain solutions it is necessary to have a
boundary condition h = h0 at a certain x = x0, which will be provided by a control. The problem may
be solved using any of a number of methods available for solving ordinary differential equations which
35

Figure 5-2. Typical gradually-varied flow surface profiles, drawn by Dr I. C. O’Neill.
36

Typical cross-section used for 1-D analysis
Edge of flood plain / Extent of 1% flood
River banks
Figure 5-3. Practical river problem with subdivision
2 1
Figure 5-4. Flow separation and head loss due to a contraction
are described in books on numerical methods. These methods are usually accurate and can be found
in many standard software packages. It is surprising that books on open channels do not recognise that
the problem of numerical solution of the gradually-varied flow equation is actually a standard numerical
problem, although practical details may make it more complicated. Instead, such texts use methods
such as the ”Direct step method” and the ”Standard step method”. There are several software packages
such as HEC-RAS which use such methods, but solution of the gradually-varied flow equation is not a
difficult problem to solve for specific problems in practice if one knows that it is merely the solution of
a differential equation, and here we briefly set out the nature of such schemes.
The direction of solution is very important. If the different conventional cases in Figure 5-2 are exam-ined,
it can be seen that for the mild slope (sub-critical flow) cases that the surface decays somewhat
exponentially to normal depth upstream from a downstream control, whereas for steep slope (super-critical
flow) cases the surface decays exponentially to normal depth downstream from an upstream
control. This means that to obtain numerical solutions we will always solve (a) for sub-critical flow:
from the control upstream, and (b) for super-critical flow: from the control downstream.
5.5.1 Euler’s method
The simplest (Euler) scheme to advance the solution from (xi, hi) to (xi + Δxi, hi+1) is
xi+1 ≈ xi + Δxi, where Δxi is negative for subcritical flow, (5.7)
dh
hi+1 ≈ hi + Δxi
dx
¯¯¯¯
i
= hi + Δxi
S0 − Sf (hi)
1 − αF 2(hi)
. (5.8)
This is the simplest but least accurate of all methods – yet it might be appropriate for open channel
37

problems where quantities may only be known approximately. One can use simple modifications such
as Heun’s method to gain better accuracy – or even more simply, just take smaller steps Δxi.
5.5.2 Heun’s method
In this case the value of hi+1 calculated from equation (5.8) is used as a first estimate h∗i+1, then the
value of the right hand side of the differential equation is also calculated there, and the mean of the two
values taken. That is,
xi+1 ≈ xi + Δxi, again where Δxi is negative for subcritical flow, (5.9)
h∗i+1 = hi + Δxi
S0 − Sf (hi)
1 − αF 2(hi)
, (5.10)
hi+1 = hi +
Δxi
2
μ
S0 − Sf (hi)
1 − αF 2(hi)
+
S0 − Sf (h∗i+1)
1 − αF 2(h∗i+1)
¶
. (5.11)
Neither of these two methods are presented in hydraulics textbooks as alternatives. Although they are
simple and flexible, they are not as accurate as other less-convenient methods described further below.
The step Δxi can be varied at will, to suit possible irregularly spaced cross-sectional data.
5.5.3 Predictor-corrector method – Trapezoidal method
This is simply an iteration of the last method, whereby the step in equation (5.11) is repeated several
times, at each stage setting h∗i+1 equal to the updated value of hi+1. This gives an accurate and conve-nient
method, and it is surprising that it has not been used.
5.5.4 Direct step method
Textbooks do present the Direct Step method, which is applied by taking steps in the height and calcu-lating
the corresponding step in x. It is only applicable to problems where the channel is prismatic. The
reciprocal of equation (5.1) is
dx
dH
= −
1
Sf
,
which is then approximated by a version of Heun’s method, but which is not a correct rational approxi-mation:
Δx = −
ΔH
¯ Sf
, (5.12)
where a mean value of the friction slope is used. The procedure is: for the control point x0 and h0,
calculate H0 from equation (5.4), then assume a finite value of depth change Δh to compute h1 =
h0 + Δh, from which H1 is calculated from equation (5.4), giving ΔH = H1 − H0. Then with ¯ Sf =
(Sf (h0)+Sf (h1))/2, equation (5.12) is used to calculate the corresponding Δx, giving x1 = x0 +Δx.
The process is then repeated to give x2 and h2 and so on. It is important to choose the correct sign
of Δh such that computations proceed in the right direction such that, for example, Δx is negative for
sub-critical flow, and computations proceed upstream.
The method has the theoretical disadvantage that it is an inconsistent approximation, in that it should
actually be computing the mean of 1/Sf, namely 1/Sf , rather than 1/ ¯ Sf . More importantly it has
practical disadvantages, such that it is applicable only to prismatic sections, results are not obtained at
specified points in x, and as uniform flow is approached the Δx become infinitely large. However it is a
surprisingly accurate method.
5.5.5 Standard step method
This is an implicit method, requiring numerical solution of a transcendental equation at each step. It
can be used for irregular channels, and is rather more general. In this case, the distance interval Δx is
specified and the corresponding depth change calculated. In the Standard step method the procedure is
38

to write
ΔH = −Sf Δx,
and then write it as
H2(h2) − H1(h1) = −
Δx
2
(Sf1 + Sf2) ,
for sections 1 and 2, where the mean value of the friction slope is used. This gives
α
Q2
2gA22
+ Z2 + h2 = α
Q2
2gA21
+ Z1 + h1 −
Δx
2
(Sf1 + Sf2) ,
where Z1 and Z2 are the elevations of the bed. This is a transcendental equation for h2, as this determines
A2, P2, and Sf2. Solution could be by any of the methods we have had for solving transcendental
equations, such as direct iteration, bisection, or Newton’s method.
Although the Standard step method is an accurate and stable approximation, the lecturer considers it
unnecessarily complicated, as it requires solution of a transcendental equation at each step. It would be
much simpler to use a simple explicit Euler or Heun’s method as described above.
Example: Consider a simple backwater problem to test the accuracy of the various methods. A
trapezoidal channel with bottom width W = 10m, side batter slopes of 2:1, is laid on a slope of
S0 = 10−4, and carries a flow of Q = 15m3
s−1. Manning’s coefficient is n = 0.025. At the
downstream control the depth is 2.5m. Calculate the surface profile (and how far the effect of the
control extends upstream). Use 10 computational steps over a length of 30km.
2.5
2.4
2.3
2.2
2.1
2
-45 -40 -35 -30 -25 -20 -15 -10 -5 0
Depth (m)
x (km)
Accurate solution
Analytical approximation
Trapezoidal
Standard step
Direct step
Figure 5-5. Comparison of different solution methods – depth plotted.
Figure 5-5 shows the results of the computations, where depth is plotted, while Figure 5-6 shows the
same results, but where surface elevation is plotted, to show what the surface profile actually looks
like. For relatively few computational points Euler’s method was not accurate, and neither was Heun’s
method, and have not been plotted. The basis of accuracy is shown by the solid line, from a highly-accurate
Runge-Kutta 4th order method. This is not recommended as a method, however, as it makes use
of information from three intermediate points at each step, information which in non-prismatic channels
is not available. It can be seen that the relatively simple Trapezoidal method is sufficiently accurate,
certainly of acceptable practical accuracy. The Direct Step method was slightly more accurate, but the
results show one of its disadvantages, that the distance between computational points becomes large as
uniform flow is approached, and the points are at awkward distances. The last plotted point is at about
−25km; using points closer to normal depth gave inaccurate results. The Standard Step method was
very accurate, but is not plotted as it is complicated to apply. Of course, if more computational points
39

6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
Accurate solution and normal depth
Analytical approximation
-45 -40 -35 -30 -25 -20 -15 -10 -5 0
Surface elevation (m)
x (km)
Trapezoidal
Standard step
Direct step
Figure 5-6. Comparison of different solution methods – elevation plotted.
were taken, more accurate results could be obtained. In this example we deliberately chose relatively
few steps (10) so that the numerical accuracies of the methods could be compared.
Also plotted on the figures is a dotted line corresponding to the analytical solution which will be de-veloped
below. Although this was not as accurate as the numerical solutions, it does give a simple
approximate result for the rate of decay and how far upstream the effects of the control extend. For
many practical problems, this accuracy and simplicity may be enough.
The channel dimensions are typical of a large irrigation canal in the Murray Valley - it is interesting that
the effects of the control extend for some 30km!
To conclude with a recommendation: the trapezoidal method, Heun’s method iterated several times
is simple, accurate, and convenient. If, however, a simple approximate solution is enough, then the
following analytical solution can be used.
5.6 Analytical solution
Whereas the numerical solutions give us numbers to analyse, sometimes very few actual numbers are
required, such as merely requiring how far upstream water levels are raised to a certain level, the effect
of downstream works on flooding, for example. Here we introduce a different way of looking at a
physical problem in hydraulics, where we obtain an approximate mathematical solution so that we can
provide equations which reveal to us more of the nature of the problem than do numbers. Sometimes an
understanding of what is important is more useful than numbers.
Consider the water surface depth to be written
h(x) = h0 + h1(x),
where we use the symbol h0 for the constant normal depth, and h1(x) is a relatively small departure
of the surface from the uniform normal depth. We use the governing differential equation (5.6) but we
assume that the Froude number squared is sufficiently small that it can be ignored. This is not essential,
but it makes the equations simpler to write and read. (As an example, consider a typical stream flowing
at 0.5 m/s with a depth of 2m, giving F 2 = 0.0125 - there are many cases where F 2 can be neglected).
40

The simplified differential equation can be written
dh
dx
= S0 − S(h),
where for purposes of simplicity we have dropped the subscript f on the friction slope, now represented
by S. Substituting our expansion, we obtain
dh1
dx
= S0 − S(h0 + h1(x)). (5.13)
Now we introduce the approximation that the h1 term is relatively small such that we can write for the
friction term its Taylor expansion about normal flow:
S(h0 + h1(x)) = S(h0) + h1(x) ×
dS
dh
(h0) + Terms proportional to h21
.
We ignore the quadratic terms, write dS/dh(h0) as S/
0 , and substituting into equation (5.13), we obtain
dh1
dx
= −S/
0 h1
where we have used S(h0) = S0. This is an ordinary differential equation which we can solve analyt-ically.
We have achieved this by ”linearising” about the uniform flow. Now, by separation of variables
we can obtain the solution
h1 = Ge−S/
0 x,
and the full solution is
h = h0 + Ge−S/
0 x, (5.14)
where G is a constant which would be evaluated by satisfying the boundary condition at the control.
This shows that the water surface is actually approximated by an exponential curve passing from the
value of depth at the control to normal depth. In fact, we will see that as S/
0 is negative, far upstream as
x → −∞,
the water surface approaches normal depth.
Now we obtain an expression for S/
0 in terms of the channel dimensions. From Manning’s law,
S = n2Q2 P 4/3
A10/3 ,
and differentiating gives
S/ = n2Q2
Ã
4
3
P 1/3
A10/3
dP
dh −
10
3
P 4/3
A13/3
dA
dh
!
,
which we can factorise, substitute dA/dh = B, and recognising the term outside the brackets, we obtain
an analytical expression for the coefficient of x in the exponential function:
0 = n2Q2 P 4/3
−S/
0
A10/3
0
μ
10
3
B
A −
4
3
dP/dh
P
¶¯¯¯¯
0
= S0
μ
10
3
B0
A0 −
4
3
dP0/dh0
P0
¶
.
The larger this number, the more rapid is the decay with x. The formula shows that more rapid decay
occurs with steeper slopes (large S0), smaller depths (B0/A0 = 1/D0, where D0 is the mean depth -
if it decreases the overall coefficient increases), and smaller widths (P0 is closely related to width, the
term involving it can be written d(log P0)/dh0: if P0 decreases the term decreases - relatively slowly -
but the negative sign means that the effect is to increase the magnitude of the overall coefficient). Hence,
generally the water surface approaches normal depth more quickly for steeper, shallower and narrower
(i.e. steeper and smaller) streams. The free surface will decay to 10% of its original departure from
41

normal in a distance X0.1, where
e−S/
0 X0.1 = 0.1.
In the example above, we obtain −S/
0 = 0.00015,giving X0.1 = −15400m, which seems roughly right
when compared with the figure above. In a further interval of the same distance the surface will decay
to 10% of this, 1% of its original magnitude, and so on. This theory has shown us something of the
power of obtaining approximate solutions where no other analytical solutions exist, to provide us with a
simpler understanding of the nature of the problem.
6. Unsteady flow
Top of Control Volume
Water surface
y
z
x
q
Q+ΔQ
Q
Δx
Figure 6-1. Element of non-prismatic waterway showing control volume extended into the air
We now consider the full equations for unsteady non-uniform flow. The fundamental assumption we
make is that the flow is slowly varying along the channel. The mathematics uses a number of concepts
from vector calculus, however we find that we can obtain general equations very powerfully, and the
assumptions and approximations (actually very few!) are clear.
6.1 Mass conservation equation
Consider the elemental section of thickness Δx of non-uniform waterway shown in Figure 6-1, bounded
by two vertical planes parallel to the y − z plane. Consider also the control volume made up of this
elemental section, but continued into the air such that the bottom and lateral boundaries are the river
banks, and the upper boundary is arbitrary but never intersected by the water.
The Mass Conservation equation in integral form is, written for a control volume CV bounded by a
control surface CS,
∂
∂t
Z
CV
ρ dV
| {z }
Total mass in CV
+
Z
CS
ρ u.ˆn dS
| {z }
Rate of flow of mass across boundary
= 0,
where t is time, dV is an element of volume, u is the velocity vector, ˆn is a unit vector with direction
normal to and directed outwards from the control surface such that u.ˆn is the component of velocity
normal to the surface at any point, and dS is the elemental area of the control surface.
As the density of the air is negligible compared with the water, the domain of integration in the first
integral reduces to the volume of water in the control volume, and considering the elemental slice,
42

dV = Δx dA , where dA is an element of cross-sectional area, the term becomes
ρΔx
∂
∂t
Z
A
dA = ρΔx
∂A
∂t
Now considering the second integral, on the upstream face of the control surface, u.ˆn = −u , where u
is the x component of velocity, so that the contribution due to flow entering the control volume is
−ρ
Z
A
u dA = −ρQ.
Similarly the downstream face contribution is
+ρ (Q + ΔQ) = +ρ
μ
Q +
∂Q
∂x
¶
.
Δx
On the boundaries which are the banks of the stream, the velocity component normal to the boundary
is very small and poorly-known. We will include it in a suitably approximate manner. We lump this
contribution from groundwater, inflow from rainfall, and tributaries entering the waterway, as a volume
rate of q per unit length entering the stream. The rate at which mass enters the control volume is ρqΔx
(i.e. an outflow of −ρqΔx). Combining the contributions from the rate of change of mass in the CV and
the net contribution across the two faces, and dividing by ρΔx we have the unsteady mass conservation
equation
∂A
∂t
+
∂Q
∂x
= q. (6.1)
Remarkably for hydraulics, this is an almost-exact equation - the only significant approximation we have
made is that the waterway is straight! If we want to use surface elevation as a variable in terms of surface
area, it is easily shown that in an increment of time δt if the surface changes by an amount δη, then the
area changes by an amount δA = B × δη, from which we obtain ∂A/∂t = B × ∂η/∂t, and the mass
conservation equation can be written
B
∂η
∂t
+
∂Q
∂x
= q. (6.2)
The assumption that the waterway is straight has almost universally been made. Fenton&Nalder (1995)1
have considered waterways curved in plan (i.e. most rivers!) and obtained the result (cf. equation 6.1):
³
1 −
nm
r
´ ∂A
∂t
+
∂Q
∂s
= q,
where nm is the transverse offset of the centre of the river surface from the curved streamwise reference
axis s, and r is the radius of curvature of that axis. Usually nm is small compared with r, and the
curvature term is a relatively small one. It can be seen that if it is possible to choose the reference axis
to coincide with the centre of the river viewed in plan, then nm = 0 and curvature has no effect on this
equation. This choice of axis is not always possible, however, as the geometry of the river changes with
surface height.
6.2 Momentum conservation equation – the low inertia approximation
We can repeat the procedure above, but this time considering momentum conservation. It can be shown
that we obtain
∂Q
∂t
+ β
∂
∂x
μ
Q2
A
¶
+ gA
∂η
∂x
= −gASf . (6.3)
Equations (6.2) and (6.3) are the long wave or Saint-Venant equations. They are used to simulate wave
1 Fenton, J. D. & Nalder, G. V. (1995), Long wave equations for waterways curved in plan, in Proc. 26th
Congress IAHR, London, Vol. 1, pp. 573–578.
43

motions in rivers and canals, notably the propagation of flood waves and the routine simulation of irri-gation
channel operations. There is a software industry which specialises in numerical solutions. It can
be shown mathematically that solutions of these equations look like two families of waves, propagat-ing
at two different velocities, one upstream, the other downstream, with differing amounts of diffusive
dissipation. The upstream propagating waves show rather more dissipation. The propagation in both
directions is important in situations where transients are rapid, such as in hydro-electric supply canals.
In most situations, however, the flow velocity is relatively small, such that a large simplification is pos-sible.
We can show that the first two terms in equation (6.3), relative to the others, are of the order of
magnitude of F 2, and can be ignored.
Firstly consider ∂Q/∂t. The discharge Q is of an order of magnitude V ×W ×D, where V is a typical
velocity, W is a typical width, and D is a typical mean depth. The time scale of motion is given by L/V ,
where L is a typical length scale of the wave motion down the waterway, and our velocity scale gives a
measure of how quickly it is swept past. Hence,
∂Q
∂t
is of a scale V × W × D
L/V
=
V 2W D
L
.
Now we examine the scale of the term gA∂η/∂x, and here we assume that the vertical scale of our
disturbances is the vertical scale of the channel D, giving
gA
∂η
∂x
is of a scale g × W × D ×
D
L
=
gW D2
L
.
Now we compare the magnitudes of the two terms ∂Q/∂t : gA∂η/∂x and we find that the ratio of the
two terms is of a scale
V 2W D
L ×
L
gW D2 =
V 2
gD
, the scale of the Froude number squared.
Hence, possibly to our surprise, we find that the relative magnitude of the term ∂Q/∂t is roughly F 2,
and in many flows in rivers and canals this is a small quantity and terms of this size can be ignored.
Examining the second term in equation (6.3) it might be more obvious that it too is also of order F 2.
If we neglect both such terms of order F 2, making the ”low-inertia” approximation, we find that the
momentum equation (6.3) can simply be approximated by
∂η
∂x
+ Sf = 0, (6.4)
which expresses the fact that, even in a generally unsteady situation, the surface slope and the friction
slope are the same magnitude. Now we use an empirical friction law for the friction slope Sf in terms
of conveyance K, so that we write
Sf =
Q2
K2 ,
where the dependence of K on depth at a section would be given by Manning’s law. Substituting this
into (6.4) gives us an accurate approximation for the discharge in terms of the slope:
Q = K
r
−
∂η
∂x
, (6.5)
even in a generally unsteady flow situation, provided the Froude number is sufficiently small. (Note that
∂η/∂x is always negative in situations where this theory applies!). This provides us with a good method
of measuring the discharge - if we can calibrate a gauging station to give the conveyance as a function
of surface height, then by measuring the surface slope we can get the discharge.
At this point it is easier to introduce the local depth h such that if Z is the local elevation of the bottom,
η = Z + h and ∂η
∂x
=
∂Z
∂x
+
∂h
∂x
=
∂h
∂x − S0,
44

in which case equation (6.5) can be written
Q = K
r
S0 −
∂h
∂x
, (6.6)
6.3 Diffusion routing and nature of wave propagation in waterways
Now we eliminate the discharge Q from the equations by simply substituting equation (6.6) into the
mass conservation equation (6.2), noting that as the bed does not move, ∂η/∂t = ∂h/∂t, to give the
single partial differential equation in the single variable h: ,
∂h
∂t
+
1
B
∂
∂x
Ã
K
r
S0 −
∂h
∂x
!
= 0 (6.7)
The conveyance is usually expressed as a function of roughness and of the local depth h, so that we
can perform the differentiation in equation (6.7) and we assume that if the variation of the local depth is
small compared with the overall slope so that |∂h/∂x| ¿ S0 we can write
∂h
∂t
+
√S0
B
dK
| {zdh}
Propagation velocity
∂h
∂x
=
K
|2B{√zS0}
Diffusion coefficient
∂2h
∂x2 (6.8)
We now have a rather simpler single equation in a single unknown. This is an advection-diffusion
equation, and the nature of it is rather clearer than our original pair of equations. It has solutions which
propagate at the propagation velocity shown. We write the equation as
∂h
∂t
+ c
∂h
∂x
= ν
∂2h
∂x2 , (6.9)
where c is a propagation speed, the kinematic wave speed, and ν is a diffusion coefficient (with units of
L2T−1), given by
c =
√S0
B
dK
dh
and ν =
K
2B√S0
. (6.10)
As K = 1/n × A5/3(h)/P 2/3(h), we can differentiate to give the kinematic wave speed c for an
arbitrary section. However for the purposes of this course we can consider a wide rectangular channel
(h ¿ B) such that A ≈ bh and P ≈ B = b such that K = 1/n × Bh5/3. Differentiating, dK/dh =
1/n × 5/3 × Bh2/3, and substituting into (6.10) we obtain
c =
5
3 ×
√S0
n
h2/3.
Now, the velocity of flow in this waterway is U = 1/n × (A/P )2/3 √S0, which for our wide-channel
approximation is U ≈ √S0/n × h2/3, and so we obtain the approximate relationship for the speed of
propagation of disturbances in a wide channel:
c ≈
5
3
U, (6.11)
which is a very simple expression: the speed of propagation of disturbances is approximately 123
times
the mean speed of the water.
In all text books another wave velocity is often presented, which is c =
p
gA/B, where A/B is the
mean depth of the water. This is obtained from dynamical considerations of long waves in still water. It
is indeed the order of magnitude of the speed at which waves do move over essentially still water, and
is widely used, incorrectly, to estimate the speed of disturbances in rivers and canals. It is called the
dynamic wave speed, and part of waves do travel at this speed. However, provided the Froude number
is small, such that F 2 ¿ 1, equation (6.11) is a good approximation to the speed at which the bulk of
45

disturbances propagate.
1000
950
900
850
0 1 2 3 4 5 6 7 8
Discharge
(Ml/d)
Time (Hours)
Inlet
Outlet
Figure 6-2. Inflow and outflow hydrographs for short pool on a shallow slope showing the effects of friction
Now to consider the effects of diffusion, if we examine the real or simulated propagation of waves in
streams, the apparent motion is of waves propagating at the kinematic wave speed, but showing marked
diminution in size as they propagate. We consider as a test case, a pool 7km long, bottom width of
7m, batter slopes of 1.5:1, a longitudinal slope of 0.0001, a depth of 2.1m at the downstream gate,
and Manning’s n = 0.02. We consider a base flow of 10 m3s−1 increased linearly by 25% up to a
maximum of 12.5 m3s−1 and back down to the base flow over a period of two hours. A computer
program simulated conditions in the canal. The results presented in Figure 6-2 show that over the length
of only 7km the peak discharge has decreased by about 50% and has spread out considerably in time -
the wave propagation is not just a simple translation. In fact, the peak takes about 55 minutes to traverse
the pool, whereas using the dynamic wave speed that time would be 30 minutes, while simply using
the kinematic wave speed it would be 140 minutes. The wave has diffused considerably, showing that
simple deductions based on a wave speed are only part of the picture. This difference might be important
for flood warning operations.
It is important to find out more about the real nature of wave propagation in waterways. Here we provide
a simple tool for estimating the relative importance of diffusion. If we were to scale the advection-diffusion
equation (6.9) such that it was in terms of a dimensionless variable x/L which would be of
order of magnitude 1, then the ratio of the importance of the diffusion term to the advection term can be
shown to be ν/cL. This looks like the inverse of a Reynolds number (which is correct – the Reynolds
number is the inverse of a dimensionless viscosity or diffusion number). Now we substitute in the
approximations for a wide channel, giving:
A measure of the importance of diffusion =
ν
cL
=
K
2B√S0 ×
B
√S0K0(h) L
=
K/K0(h)
2 S0L
, and as K ∝ h5/3 this gives
≈
3
10 ×
h
S0L
.
This is a useful result, for it shows us the effects of diffusion very simply, as h is the depth of the stream,
and S0L is the amount by which a stream drops over the reach of interest, we have
A measure of the importance of diffusion ≈
3
10 ×
Depth of stream
Drop of stream .
46

For the example above, this is about 0.8, showing that diffusion is as important as advection, and re-minding
us that the problem is not the simple translation of a wave.
This result is also interesting in considering different types of streams – a steep shallow mountain stream
will show little diffusion, whereas a deep gently sloping stream will have marked diffusion!
7. Structures in open channels and flow measurement
The main texts to which reference can be made are Ackers, White, Perkins & Harrison (1978), French
(1985), Henderson (1966), Novak (2001). Another useful scholarly reference is that by Jaeger (1956).
7.1 Overshot gate - the sharp-crested weir
6
- h
?
6
H
?
6 C
P
?
y
D
6
-
x
Figure 7-1. Side view of sharp-crested weir. The dotted line shows the notional pressure distribution above the
crest.
The calculation of the discharge over a sharp-crested weir is one of the great misleading results in
hydraulic engineering, which is widely known to be so, yet every textbook reproduces it. Instead of the
actual pressure distribution shown dotted in Figure 7-1, the pressure is assumed to be zero over the crest,
then Bernoulli’s law applied, and then it is assumed that the fluid velocity is horizontal only (whereas it
is actually vertical at the crest). It is necessary to introduce a coefficient of discharge which has a value
of 0.6 − 0.7 to make the theory work.
A more correct way to proceed is to use dimensional analysis – firstly to consider a weir of infinite
length and in water of infinite depth. The head over the weir is H (as shown in Figure 7-1), gravitational
acceleration is g, the discharge is q per unit length. We have three quantities, two dimensions (L and
T), and so by the Buckingham π theorem, we have only a single quantity affecting results, which must
therefore be a constant. Collecting the terms in the only possible dimensionless combination, we have
q p
gH3
= C = constant,
hence we obtain the well-known 3/2 law:
q = C√gH3/2.
If we now consider a finite width of channel B, finite width of weir b, and a finite apron height P , with
47

now a total discharge Q we obtain
Q
p
gH3
b
= f (b/B, H/P, b/P ).
In fact, an approximate expression can be written
Q = C√g b H3/2, (7.1)
where C is a dimensionless coefficient. There is relatively little variation about a constant value of C,
and a useful approximate expression is
Q = 0.6√gb (ηu − zc)3/2
where ηu is the upstream surface elevation and zc is the elevation of the crest.
7.2 Triangular weir
An analysis as misguided as the traditional one produces the result quoted by French (1985) on p352,
where some results of Bos (1978) are quoted. The expression is
Q = C
8
15
p
2g tan
θ
2
H2.5,
where C is the coefficient of discharge, and where θ is the angle between the sides of the triangle. A
typical result from French (1985) is that C is roughly 0.58, which has been found to agree well with
experiment. We prefer to write the expression as
Q = 0.44√g tan
θ
2
H2.5.
7.3 Broad-crested weirs – critical flow as a control
In this case theory is rather more applicable, and we have already considered that in this course. The
flow upstream is subcritical, but the flow in the structure is made to reach critical - the flow reaches
its critical depth at some point on top of the weir, and the weir provides a control for the flow. In this
case, the head upstream (the height of the upstream water surface above the sill) uniquely determines
the discharge, and it is enough to measure the upstream surface elevation where the flow is slow and the
kinetic part of the head negligible to provide a point on a unique relationship between that head over the
weir and the discharge. No other surface elevation need be measured. Such a horizontal flow control is
called a broad-crested weir. We can apply simply theory of critical flows to give the discharge per unit
Small energy loss
hc
hc hc
Figure 7-2. Broad-crested weir in a canal showing substantial recovery of energy
width q:
q =
p
gh3c
=
q
g
¡2
3Hc
¢3
=
q
8
gH3/2
27 c
48

where
Hc = H1 − Δ = h1 + α
U 2
1
2g − Δ = h1 +
αQ2
2gA21
− Δ.
As A1, the upstream area, is a function of h1, the relationship for discharge is not simply a 3/2 power
law, but for sufficiently low upstream Froude number the deviation will not be large. In practice the
discharge for the weir as a whole is written, introducing various discharge coefficients Montes (1998,
p250),
Q = 0.54CsCeCv b√gh3/2
1 .
7.4 Free overfall
b h
Figure 7-3. Free overfall
Consider the free overfall shown in Figure 7-3. It can be shown that the discharge per unit width is
q = 1.65√gh3/2
b .
7.5 Undershot sluice gate
1 2
Figure 7-4. Sluice gate
Consider the case where the gate is not drowned, but a stream of supercritical flow can exist for some
distance downstream. Applying the energy theorem between a point at section 1 and one at 2:
h1 +
α1
2g
μ
q
h1
¶2
= h2 +
α2
2g
μ
q
h2
¶2
.
49

Solving for q (and assuming α1 = α2 = α) we obtain:
q =
1
√α
√2gh1h2 √h1 + h2
,
which it is easier to write in dimensionless form:
q p
2gh31
=
1
√α
p h2/h1
1 + h2/h1
. (7.2)
In practice we know the upstream depth h1 and the gate opening h, such that
h2 = Cch,
where Cc is the coefficient of contraction. A number of theoretical and experimental studies have been
made of this, but the variation is not large. Henderson (1966, p205) shows some of these but ends up
recommending a constant value of 0.61.
Equation (7.2) is a convenient way of representation, as h1 is given, and so we find that the dimensionless
discharge is simply a function of h2/h1, the depth ratio, or h/h1. We can approximate the result by
noting that h2/h1 is small, and using α = 1,
q p
2gh31
=
p h2/h1
1 + h2/h1 ≈
h2
h1
,
and we find that to first order, for small gate openings,
q ≈
p
2gh1 × h2,
and the result is almost obvious, that the discharge per unit width is given by a velocity corresponding
to the full upstream head of water multiplied by the downstream depth of flow.
7.6 Drowned undershot gate
In this case the analysis is rather more complicated, but we apply energy and momentum considerations
where appropriate and find that we can extract a useful result.
7.7 Dethridge Meter
Developed in Australia, these have been almost universally used in our irrigation industry. They are the
familiar water-wheels one sees in irrigation areas. They are now being replaced by more sophisticated
methods.
8. The measurement of flow in rivers and canals
We have considered each of the major type of structures separately, as they are used for control of flows
in channels, as well as measurement. Now we briefly describe a variety of methods which are used to
measure flow without the use of structures, such that there is no obstacle to the flow. Then we consider
how the movement of waves in rivers might affect the results, how the results are analysed and used.
8.1 Methods which do not use structures
1. Velocity area method (”current meter method”): The area of cross-section is determined from
soundings, and flow velocities are measured using propeller current meters, electromagnetic sen-sors,
or floats. The mean flow velocity is deduced from points distributed systematically over the
river cross-section. In fact, what this usually means is that two or more velocity measurements are
made on each of a number of vertical lines, and any one of several empirical expressions used to
50

calculate the mean velocity on each vertical, the lot then being integrated across the channel. One
of the most common numerical methods used by hydrographers (the ”Mean-Section Method”) is
wrong and should not be used. Rather, the Trapezoidal rule should be used, which is just as easily
implemented. In a gauging in which the lecturer participated, a flow of 1693 Ml/d was calculated
using the Mean-Section Method. Using the Trapezoidal rule, the flow calculated was 1721 Ml/d,
a difference of 1.6%. Although the difference was not great, practitioners should be discouraged
from using a formula which is wrong. The lecturer has developed a family of new methods for more
accurate implementation of this method (see Fenton 2002).
2. Stage-Discharge Method: Here a stage-discharge relationship or rating curve is built up, typically
using the velocity-area method, so that the functional relationship Qr(η) can be determined, where
Qr is the rated discharge. Subsequent measurements of the surface elevation at some time t, such as
a daily measurement, are then used to give the discharge:
Q(t) = Qr(η(t)).
This is very widely used and is the routine method of flow measurement. It will be described below.
3. Slope-Stage-Discharge Method: The method is presented in some books and in International and
Australian Standards, however, especially in the latter, the presentation is confusing and at a low
level, where no reference is made to the fact that underlying it the slope is being measured. Instead,
the fall is described, which is the change in surface elevation between two surface elevation gauges
and is simply the slope multiplied by the distance between them. No theoretical justification is
provided and it is presented in a phenomenological sense (see, for example, Herschy 1995). An
exception is Boiten (2000), however even that presentation loses sight of the pragmatic nature of
determining a stage-conveyance relationship, and instead uses Manning’s law in its classical form
Q =
1
n
A5/3
P 2/3
p
Sη ,
where Sη is the free surface slope, Sη = −∂η/∂x, and it is assumed that the discharge must be
given using these precise geometrical quantities of A and P . The lecturer suggests a rather more
empirical approach, developing a stage-conveyance relationship by measuring the stage and the
slope and using the relationship Q = K(η)
p
Sη ,where K(η) is the conveyance, to determine a
particular value of K for measured values of stage and slope. Then, routinely, if the two were
measured, discharge would be calculated in the form
Q(t) = K(η(t))
q
Sη(t).
4. Ultrasonic flow measurement: This is a method used in the irrigation industry in Australia, but is
Figure 8-1. Array of four ultrasonic beams in a channel
also being used in rivers in the USA. Consider the situation shown in Figure 8-1, where some three
51

or four beams of ultrasonic sound are propagated diagonally across a stream at different levels. The
time of travel of sound in one direction is measured, as is the time in the other. The difference
can be used to compute the mean velocity along that path, i.e. at that level. These values then
have to be integrated in the vertical. Commercial implementations of the latter process are woefully
inadequate, and installations of this type of meter do not gain the accuracy of which they are easily
capable.
5. Electromagnetic methods: The motion of water flowing in an open channel cuts a vertical mag-
Signal probes Coil for producing magnetic field
Figure 8-2. Electromagnetic installation, showing coil and signal probes
netic field which is generated using a large coil buried beneath the river bed, through which an
electric current is driven. An electromotive force is induced in the water and measured by signal
probes at each side of the channel. This very small voltage is directly proportional to the average
velocity of flow in the cross-section. This is particularly suited to measurement of effluent, water in
treatment works, and in power stations, where the channel is rectangular and made of concrete; as
well as in situations where there is much weed growth, or high sediment concentrations, unstable
bed conditions, backwater effects, or reverse flow. This has the advantage that it is an integrat-ing
method, however in the end recourse has to be made to empirical relationships between the
measured electrical quantities and the flow.
6. Acoustic-Doppler Current Profiling methods: In these, a beam of sound of a known frequency
is transmitted into the fluid, often from a boat. When the sound strikes moving particles or regions
of density difference moving at a certain speed, the sound is reflected back and received by a sen-sor
mounted beside the transmitter. According to the Doppler effect, the difference in frequency
between the transmitted and received waves is a direct measurement of velocity. In practice there
are many particles in the fluid and the greater the area of flow moving at a particular velocity, the
greater the number of reflections with that frequency shift. Potentially this method is very accu-rate,
as it purports to be able to obtain the velocity over quite small regions and integrate them up.
However, this method does not measure in the top 15% of the depth or near the boundaries, and
the assumption that it is possible to extract detailed velocity profile data from a signal seems to be
optimistic. The lecturer remains unconvinced that this method is as accurate as is claimed.
7. Slope area method: This is used to calculate peak discharges after the passage of a flood. An ideal
site is a reach of uniform channel in which the flood peak profile is defined on both banks by high
water marks. From this information the slope, the cross-sectional area and wetted perimeter can be
obtained, and the discharge computed with the Manning formula or the Chézy formula. To do this
however, roughness coefficients must be known, such as Manning’s n in the formula
Q =
1
n
A5/3
P 2/3
p
¯ S,
where A is the area, P the perimeter, and ¯ S the slope. It is approximate at best.
8. Dilution methods In channels where cross-sectional areas are difficult to determine (e.g. steep
52

mountain streams) or where flow velocities are too high to be measured by current meters dilution
or tracer methods can be used, where continuity of the tracer material is used with steady flow.
The rate of input of tracer is measured, and downstream, after total mixing, the concentration is
measured. The discharge in the stream immediately follows.
9. Integrating float methods There is another rather charming and wonderful method which has been
very little exploited. At the moment it has the status of a single measurement method, however the
lecturer can foresee it being developed as a continuing method. Consider a single buoyant particle
(a float, an orange, an air bubble), which is released from a point on the bed. We assume that it
has a constant rise velocity w. As it rises it passes through a variable horizontal velocity field u(z),
where z is the vertical co-ordinate, it samples the horizontal velocity equally at each point. The
point where the particle reaches the surface downstream of the point at which it was released on
the bed can be shown to be directly proportional to the mean horizontal velocity experienced on its
vertical traverse. Now, if we were to release bubbles from a pipe across the bed of the stream, on
the bed, then the expression
Q = Bubble rise velocity × area on surface between bubble path pattern and line of release
is possibly the most direct and potentially the most accurate of all flow measurement methods!
8.2 The hydraulics of a gauging station
Almost universally the routine measurement of the state of a river is that of the stage, the surface eleva-tion
at a gauging station, usually specified relative to an arbitrary local datum. While surface elevation
is an important quantity in determining the danger of flooding, another important quantity is the actual
flow rate past the gauging station. Accurate knowledge of this instantaneous discharge - and its time in-tegral,
the total volume of flow - is crucial to many hydrologic investigations and to practical operations
of a river and its chief environmental and commercial resource, its water. Examples include decisions
on the allocation of water resources, the design of reservoirs and their associated spillways, the calibra-tion
of models, and the interaction with other computational components of a network. A typical set-up
High water
Low water
Gauging station Local control
Distant control
Flood
Channel control
⊗
Figure 8-3. Section of river showing different controls at different water levels and a flood moving downstream
of a gauging station where the water level is regularly measured is given in Figure 8-3 which shows a
longitudinal section of a stream. Downstream of the gauging station is usually some sort of fixed con-trol
which may be some local topography such as a rock ledge which means that for relatively small
flows there is a relationship between the head over the control and the discharge which passes. This will
control the flow for small flows. For larger flows the effect of the fixed control is to ”drown out”, to
become unimportant, and for some other part of the stream to control the flow, such as the larger river
downstream shown as a distant control in the figure, or even, if the downstream channel length is long
enough before encountering another local control, the section of channel downstream will itself become
the control, where the control is due to friction in the channel, giving a relationship between the slope
in the channel, the channel geometry and roughness and the flow. There may be more controls too, but
53

however many there are, if the channel were stable, and the flow steady (i.e. not changing with time
anywhere in the system) there would be a unique relationship between stage and discharge, however
complicated this might be due to various controls. In practice, the natures of the controls are usually
unknown. Previously we have exploited the fact that a good approximation to the momentum equation
for low Froude number flows is simply equation (6.5) or the equivalent (6.6), where it is likely that we
can express conveyance K as a function of the local elevation at a gauging station:
r
−
Q = K(η)
∂η
∂x
.
We have seen that this is quite a faithful reflection of the actual mechanics of the situation, and shows that
were we to measure the surface elevation η and its gradient ∂η/∂x at a point on a river, then provided we
knew K(η) we would have a quite accurate expression for the discharge. This implies that at a particular
gauging station the computed discharge is a function of both η and ∂η/∂x. However, the tradition in
river engineering has been to assume that the surface slope varies little from mean bed slope. That is, to
assume that
p
S0,
Q ≈ K(η)
so that discharge Q is inferred to be a unique function of the stage, a term for the surface elevation,
whether relative to a national datum or a local datum, and to ignore effects of surface gradient. Effec-tively,
this has been to assume that the surface gradient is a constant for a given stage, whether at the
front of a flood wave, or at the rear, or any other time.
8.3 Rating curves
In practice, at a gauging station the relationship between stage η and discharge Q has been continuously
measured and approximated for a long time, as it can change with time due to downstream condi-tions
changing or the bed moving, for example. At irregular intervals, such as quarterly, hydrographers
measure the stage and measure the flow accurately by measuring the water velocity at closely-spaced
intervals over the cross-section and using numerical integration to obtain the discharge. This provides a
single point on the Rating Curve, a plot of discharge horizontally against stage vertically. From a num-ber
of such points, the rating curve is built up. Subsequently it is assumed that for any stage reading,
the routine periodic measurement, the corresponding discharge, can be read off. This is what happens
then, at intervals such as hourly or daily, the stage is read and telemetered to a central data management
authority. From the rating curve for that stage, the corresponding discharge can be calculated. The figure
below shows the current rating curve for the Ovens River atWangaratta, where flow measurements have
been made since 1891. There are a couple of difficulties with such a curve, including reading results
off for small flows, where the curve is locally vertical, and for high flows where it is almost horizontal.
A traditional way of overcoming the difficulty of representing rating curves over a large range has been
to use log-log axes. However, this has no physical basis and has a number of practical difficulties, al-though
it has been recommended by International and Australian Standards.Hydraulic theory can help
here, for it can be used to show that the stage-discharge relationship will tend to show stage varying
approximately like η ∼ Q1/2, for both cases:
1. Flow across a U-shaped (parabolic) weir, the approximate situation for low flow at a gauging station,
when a local control such as a rock ledge controls the flow, and
2. Uniform flow down a U-shaped (parabolic) waterway for large flows, when the local control is
washed out and the waterway acts more like a uniform flow governed by Manning’s law.
In these cases, both parts of the relationship would plot as (different) straight lines on
¡√Q,η
¢
axes.
Here we plot the results from the above figure on such a square root scale for the discharge, and we
see that indeed at both small and large flows the rating curve is a straight line. This means that simpler
procedures of numerical approximation and interpolation could be used. Sometimes results have to be
taken by extrapolating the curve. If this has to be done, then linear extrapolation on the
¡√Q,η
¢
axes
might be reasonable, but it is still a procedure to be followed with great caution, as the actual geometry
54

14
13
12
11
10
9
8
7
6
0 50000 100000 150000 200000 250000
Discharge (Ml/d)
Stage
Figure 8-4. Rating curve using natural (Q, η) axes
for above-bank flows can vary a lot.In all of this we have not considered what actually happens to a
14
13
12
11
10
9
8
7
6
0 100 200 300 400 500
Sqrt(Discharge (Ml/d))
Stage
Figure 8-5. A rating curve using (√Q, η) axes
rating curve if the depth gradient ∂h/∂x is significant. Usually it is not important, but for a sufficiently
rapidly rising and falling flood we can imagine that it might be. Consider the front of a flood wave where
the depth increases towards the top of the flood. In this case ∂h/∂x will be negative, and equation (6.6)
Q = K
r
S0 −
∂h
∂x
,
shows that the discharge will be larger than if we had neglected this slope. Similarly, at the rear of
the flood wave the depth gradient will be positive, and the flow will be rather less. In this situation
the trajectory of the flood actually forms a loop on the Stage-discharge diagram. There are some ways
around this. One would be to recognise that in general, the discharge is not just a function of stage but
also depends on the surface slope. Hydraulics has told us that it might be better always to measure slope
as well as stage by measuring the stage at two points, and to develop a Stage-Conveyance relationship
rather than a Stage-Discharge relationship. This, however, lies in the future. Some recent research at
the Co-operative Research Centre for Catchment Hydrology has investigated some of these aspects and
produced some solutions.
55

9. Loose-boundary hydraulics
Most rivers and canals have beds of soil, of a more-or-less erodible nature. Water flowing in such streams
has the ability to scour the bed, to carry particles which are heavier than the water, and to deposit
material, hence changing the bed topography. This phenomenon is of great economic and ecological
importance, for example in predicting the scouring around and potential collapse of bridges, weirs,
channel banks etc., estimating the rate of siltation of reservoirs, predicting the possible form changes of
rivers with a threat to aquatic life etc..
9.1 Sediment transport
The grains forming the boundary of an alluvial stream have a finite weight and finite resisting ability,
including cohesion and coefficient of friction. They can be brought into motion if the forces due to
fluid motion acting on a sediment particle are greater than the resisting forces. Often this is expressed
in terms of disturbing and resisting stresses on the bed of the stream. If the shear stress τ acting at a
point on the flow boundary is greater than a certain critical value τ cr then grains will be removed from
that region, and the bed is said to scour there. We introduce the concept of a relative tractive force at
a point, τ /τ cr. If this is slightly greater than 1, only the grains forming the uppermost layer of the flow
boundary can be detached and transported. If τ /τ cr is greater than 1, but less than a certain amount,
then grains are transported by deterministic jumps in the neighbourhood of the bed. This mode of grain
transport is referred to as bed-load. If the ratio is large, then grains will be entrained into the flow and
will be carried downstream by turbulence. This transport mechanism is known as suspended-load. The
total transport rate is the sum of the two. The simultaneous motion of the transporting fluid and the
transported sediment is a form of two-phase flow. We can write all the variables which should dominate
the problem of the removal and transport of particles:
ρ Density of water ML−3
ρs Density of solid particles ML−3
ν Kinematic viscosity of water L2T−1
φ Diameter of grain L
g Gravitational acceleration LT−2
h Depth of flow L
τ Shear stress of water on bed ML−1T−2
As we have 7 such quantities and 3 fundamental dimensions involved, there are 4 dimensionless numbers
which can characterise the problem. In fact, it is convenient to replace g by g0 = g (ρs/ρ − 1), the
apparent p
submerged gravitational acceleration of the particles, and to replace τ by the shear velocity
u∗ =
τ /ρ. Convenient dimensionless variables, partly found from physical considerations, which
occur are
Θ =
u2∗
g0φ
=
τ
(ρs − ρ) gφ
, roughly the ratio of the shear force on a particle to its submerged weight
R∗ =
u∗φ
ν
, roughly the ratio of fluid inertia forces to viscous forces on the grain
G =
ρs
ρ
, the specific gravity of the bedmaterial, and
φ
h
, the ratio of grain size to water depth
Two important quantities here are R∗ which is the grain Reynolds number, and Θ the Shields parameter,
which can be thought of as a dimensionless stress.
56

9.2 Incipient motion
In the 1930s Shields conducted a number of experiments in Berlin and found that there was a narrow
band of demarcation between motion and no motion of bed particles, corresponding to incipient motion.
He represented these on a figure of Θ versus R∗. A slight problem with this is that the fluid velocity (in
the form of shear velocity) occurs in both quantities. It is more reasonable to introduce the dimensionless
grain size (see p7 of Yalin & Ferreira da Silva 2001):
δ =
μ
R2
∗
Θ
¶1/3
= φ
μ
g0
ν2
¶1/3
.
Here we consider what δ means. If we take a common value of G = 2.65, plus g = 9.8ms−2,
ν = 10−6m2 s−1(for 20◦C), then we obtain δ ≈ φ × 25000 in units of metres. If φ is specified in
terms of millimetres then we have δ ≈ 25 φ, and so for a range of particle sizes we have
δ 0.1 1 10 100 1000
φ (mm) 0.004 0.04 0.4 4 40
1
0.1
0.01
Yalin’s approximation to Shields’ data, eqn (9.1)
Bagnold’s conjecture for random beds
Motion
All beds random on the
scale of the particles
No motion
Beds flat in laboratories
0.1 1 10 100 1000
Dimensionless
shear stress
Θ
Dimensionless grain size δ
Beds random in nature
Figure 9-1. Incipient motion diagram
Figure 9-1 shows a representation of Shields’ results, using δ for the abcissa. Instead of the experimental
results we use a formula by Yalin which is an approximation to the results for incipient motion, giving
the critical value Θcr:
Θcr = 0.13 δ−0.392 e−0.015 δ2
+ 0.045
³
1 − e−0.068 δ
´
. (9.1)
Above the line, for larger values of Θ (and hence larger velocities or smaller and lighter grains), particles
will be entrained into the flow. Below the line, particles should be stable. For small particles there
appears to be a linear relationship (on these log-log axes), while for large particles the critical shear
stress, based on Shields’ laboratory experiments, approaches a constant value of about 0.045. In between
there is a dip in the curve, with a minimum at about φ = 14, corresponding to a grain size of 0.35mm,
about a fine sand.
Bagnold (personal communication) suggested that there was probably no fluid mechanical reason for
that, but that there is an implicit scale effect in the diagram, an artificial geometric effect, and suggested
that the Shields diagram has been widely misinterpreted. He suggested that for experiments with small
57

particles, while the overall bed may have been flattened, individual small grains may sit on top of others
and may project into the flow, so that the assemblage is random on a small scale. For large particles
(gravel, boulders, etc.) in nature, they too are free to project into the flow, however in the experiments
which determined the Shields diagram, the bed was made flat by levelling the tops of the large particles.
Hence, there is an artificial scale effect, and if one were only to consider random beds of particles which
are free to project into the flow above their immediate neighbours, while the bed might level on a scale
much larger than the particles. Fenton & Abbott (1977) followed Bagnold’s suggestion and examined
the effect of protrusion of particles into the stream. Although they did not obtain definitive results, they
were able to recommend that for large particles the value of Θcr was more like 0.01 than 0.045, which
seems to be an important difference, the factor of 1/4 requiring a fluid velocity for entrainment into
the flow of randomly-placed particles to be about half that of the sheltered case. A curve representing
Bagnold’s hypothesis, as partly borne out by the experiments, is shown on Figure 9-1.
9.3 Turbulent flow in streams
Now we consider some simple relations to relate this to physical quantities. The shear velocity u is a very convenient quantity indeed. If we consider the steady uniform flow in a channel, then the∗
component of gravity force down the channel on a slice of length Δx is ρgA Δx S0. However the shear
force resisting the gravity force is τ × P × Δx. Equating the two we obtain
τ = ρg
A
P
S0.
In this work it is sensible only to consider the wide channel case, such that A/P = h, the depth, giving
τ = ρgh S0,
or in terms of the shear velocity:
u∗ =
r
τ
ρ
=
p
gh S0,
and so in terms of the dimensionless stress:
Θ =
u2∗
g0φ
=
gh S0
g0φ
=
S0 h
(G − 1) φ
.
9.4 Dimensional similitude
In experiments with sediment transport, as in other areas of fluid mechanics, it is desirable to have the
same dimensionless numbers governing both experimental and full-scale situations. In this case we
would like the dimensionless particle size AND the dimensionless shear stress each to have the same
values in both model and full scale. Using the subscript m for model and no subscript for the full scale
situation, we then should have
δm = δ such that φm
μ
g0m
ν2
¶1/3
= φ
μ
g0
ν2
¶1/3
,
but as gravitational acceleration g and viscosity ν are the same in each, we can write
φm (Gm − 1)1/3 = φ (G − 1)1/3 .
Also we require the same dimensionless shear stress:
S0m hm
(Gm − 1) φm
=
S0 h
(G − 1) φ
.
In practice it is difficult to be able to satisfy all the dimensionless numbers.
58

9.5 Bed-load rate of transport – Bagnold’s formula
The volumetric rate of transport qsb per unit width is given by
qsb =
βub(τ − τ cr)
(ρs − ρ) g
,
where β is a function of δ, and ub is the flow velocity in the vicinity of the bed. In the case of a rough
turbulent flow, β ≈ 0.5. This formula is preferred, as it is simple, as accurate as any, and reflects the
meaning of the bed-load rate.
9.6 Bedforms
In most practical situations, sediments behave as non-cohesive materials, and the fluid flow can distort
the bed into various shapes. The interaction process is complex. At low velocities the bed does not move.
With increasing flow velocity the inception of movement occurs. The basic bed forms encountered are
ripples (usually of heights less than 0.1m), dunes, flat bed, standing waves, and antidunes. At high
velocities chutes and step-pools may form. Typical bed forms are summarised in Figure 9-2 below.
Figure 9-2. Bedforms and the Froude numbers at which they occur (after Richardson and Simons)
59

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