Edward J. Hickin: River Hydraulics and Channel Form
Chapter 4
Mean flow and flow resistance in open channels
Mean boundary shear stress
Mean velocity and flow resistance
The Chezy and Darcy-Weisbach relations
The Manning equation
Sources of flow resistance
Estimating Manningʼs n
Here in Chapter 4 we begin to explore some of the ways in which river scientists have
found it useful to deal with the reality that water does not move as an ideal frictionless
fluid. Real flows resist motion in various ways and in doing so they consume energy
and do work. Again, it will be useful at the outset to take a simple approach to
understanding these resisting forces and we will here consider only the mean flow
properties. Variations in the forces acting within the flow is a topic we will put off until
later in Chapter 5.
Mean boundary shear stress
Water is impelled downstream by the force of gravity acting against the opposing
frictional force or shear stress exerted against it by the boundary. If the flow is uniform,
velocity does not change downstream and one may conclude from Newton's first law of
motion (a body will continue to move with constant velocity in a straight line unless
acted on by some net force) that the impelling and resisting forces must be in balance.
These conditions allow a formulation of the boundary shear stress, τo (the subscript o
Chapter 4: Mean flow and flow resistance in open channels
denotes 'at the boundary'). The relevant forces are (see Figure 4.1 for further definition
of terms):
L
v
θ
L
A
v
Ws
θ
bed
θ
p
profile
W
Ws = W sine θ
4.1: Definition diagram for derivation of the mean shear-stress formula
Impelling force (Ws) = downslope component of the total weight of water,
Ws = W sine θ = ρgAL sine θ...............................................(4.1)
Resisting force (Fo) = boundary shear stress x total bed area
Fo = τo PL ......................................................(4.2)
If Ws = Fo,
ρgAL sine θ = τo PL ...................................................(4.3)
A
or τo = ρg P sine θ ................................................(4.4)
The ratio A/P is known as the hydraulic radius, Rh (m). Making this substitution in
equation (4.4), and noting that sine θ = tan θ = slope, s, for small values of θ (< 5o),
and that ρg = γ, we can write:
τo = γRhs ≅ γds ..............................................(4.5)
Equation (4.5) defines the mean boundary shear stress and often is simply referred to as
the 'depth-slope product' because it turns out that this expression can be simplified
further in application because the hydraulic radius normally is approximated by the
mean depth (d) of the channel in most rivers (see Sample Problem 4.1).
4.2
Chapter 4: Mean flow and flow resistance in open channels
Sample Problem 4.1: A 2.00 m-deep uniform flow discharges through a 100.00 m-wide rectangular
channel at a slope of 0.001 (1.000 m drop for every horizontal km). If the water temperature is 10oC,
calculate the mean shear stress (a) based on the hydraulic radius, and (b) based on the mean flow
depth.
A
2.00 x 100.00
=
= 1.92 m. Figure 2.1 indicates
p
(2.00 +100.00 + 2.00)
that the specific weight of water at 10oC is 9.804 Nm-3; substituting the appropriate values in equation
(4.5) yields:
τo = γRhs = 9804 x 1.92 x 0.001 = 18.8 = 19 Nm-2.
Solution: (a) In this rectangular channel, Rh =
(b) The corresponding calculation using the mean depth y rather than Rh, yields:
τo = γ y s = 9804 x 2.00 x 0.001 = 19.6 = 20 Nm-2
Although it is useful to consider this classical derivation of boundary shear stress as the
downstream component of the weight of the water body within the channel, we should
note also that equation (4.5) can simply be regarded as a special case of the momentum
equation. Where flow is steady and uniform (y1 = y2 and v1 = v2), equation (3.7) reduces
to:
γVosin θ - τoAb = 0
so that
τo = γ
Vo
sinθ = γRhsinθ, or for low slopes where sinθ = tanθ = s, τo = γRhs.
Ab
It is important to remember that τo in equation (4.5) is a measure of the mean boundary
shear stress for a channel cross section and that it tells us nothing about the variation of
shear stress within the section. Furthermore we must not forget that equation (4.5) is
strictly valid only for uniform flow. Now as it turns out, this assumption can be relaxed to
some extent because τo also applies reasonably well to gradually varied flow in which,
at least for short reaches of channel, the flow approximates uniform conditions.
Nevertheless, the further the streamlines diverge from uniform flow conditions, the
greater the error in equation (4.5).
4.3
Chapter 4: Mean flow and flow resistance in open channels
Mean velocity and flow resistance
The Chezy and Darcy-Weisbach relations
Given a certain impelling force (shear stress) in a flow, the equilibrated mean velocity
will depend on the level of resistance to flow encountered as the water moves through
the channel. High levels of resistance mean that large amounts of energy are being bled
from the flow system leaving less to maintain motion; thus, velocity will be relatively low.
Similarly, low resistance means that only a small amount of the total energy is being
consumed per unit channel length and the larger available energy balance means that
velocity will be correspondingly higher. Although the general concept of energy
conservation and loss is a straightforward notion, the particular mechanisms of loss flow resistance - are quite complex.
Defining flow resistance, the relationship between the mean shear stress and the mean
flow velocity in rivers, has been a central problem in river studies for a very long time,
but it continues to defy a complete analytical solution. The French engineer A. de
Chezy perhaps was the first (in 1775) to address this problem for uniform flow
conditions and he reasoned on the basis of common observation that:
τo = kv2 ..........................................................................(4.6)
where k is a boundary roughness coefficient. Physical reasoning suggests that in
general the boundary shear stress must depend on the fluid mass transport rate so a
more complete version of equation (4.6) might be written:
τo = aρv2 ........................................................................(4.7)
where a may be any other dimensionless property of the fluid or the boundary (τo/ρv2 is
conveniently dimensionless). Actually, regardless of whether shear stress is assumed
to be a function of velocity, or of velocity and water density (or specific weight of water),
dimensional analysis (see Chapter 1) inevitably leads us to the conclusion that τo α v2.
Later we will see that the variable a in equation (4.7) is a dimensionless number (not
4.4
Chapter 4: Mean flow and flow resistance in open channels
necessarily constant) which variously depends on the boundary roughness, the
Reynolds number, and on the cross-sectional shape of the channel.
Combining equations (4.5) and (4.7) yields the relationship:
τo = γRhs = aρv2 ......................................................(4.8)
from which it follows that
v=
g
aRhs
(noting again that γ =ρg) ……....(4.9)
Replacing the radical g/a by one constant, C, in equation (4.9) we have the formula
v = C Rhs ....................................................................(4.10)
which is known as the Chezy equation after its originator. Chezy C is a measure of flow
efficiency or conductance; as C increases for a given depth-slope product (shear stress)
so does the velocity. In other words, Chezy C is inversely related to the resistance to
flow. Evaluating Chezy C for natural channels has been a major engineering and
scientific enterprise during the last two centuries.
Many of our ideas about flow resistance in open channels were developed directly from,
or in parallel with, studies of flow through pipes. By the mid-19th century several
experimentalists, including d'Aubuisson (1840), Weisbach (1845), and Darcy (1854),
had showed that, in a run of cylindrical pipe, head loss (ΔH) varies directly with the
velocity head (v2/2g) and pipe length (L), and inversely with pipe diameter (D). They
proposed empirical head-loss equations of the form:
L v2
ΔH = ƒ D 2g ...............................................................(4.11)
where ƒ is a dimensionless coefficient of proportionality, called the friction factor or more
commonly, the Darcy-Weisbach resistance coefficient. Equation (4.11) now called the
Darcy pipe flow equation, is essentially similar to the Chezy equation for open channel
flow.
4.5
Chapter 4: Mean flow and flow resistance in open channels
Adaption of equation (4.11) to steady uniform flow in an open channel involves
recognizing that ΔH/L is equivalent to the water-surface slope (s) and that D can be
converted to an equivalent depth through the hydraulic radius. For a circular pipe of
diameter D, Rh = A/P =
π(D/2)2
= D/4 so that D = 4Rh. Rearranging equation (4.11)
πD
and making these substitutions for ΔH/L and D give:
ΔH 2g
ƒ =D L
v2
=
8gRhs
..........................................(4.12)
v2
The general structure of equation (4.12) and its similarity to the Chezy equation, perhaps
becomes more apparent if it is written in the form:
v2 =
8gRhs
or v =
∂
8gRhs
or
∂
From equation (4.13) it now is clearly apparent that C =
v=
8g
∂
8g
∂
Rhs ...............(4.13)
.................…….............(4.14)
Of course, both the Chezy and Darcy-Weisbach equations can also be written in terms
of shear stress, giving:
v2
C2 = gρ τo
and
or
C=
v2
gρ τo .........................................(4.15)
8 τo
ƒ = ρ 2 ....................................................................................(4.16)
v
Here it might be useful to introduce the concept of shear velocity, v* = τ/ρ ; v* is not a
velocity in the sense of a time rate of displacement but it does have the dimensions of
velocity (LT-1, ms-1) and is a very useful term in dimensional analysis. We will have
considerably more use for this term later when we consider the nature of velocity
distributions. Meanwhile, we can rewrite equation (4.16) in terms of the shear velocity to
give:
8
ƒ = 2 (v*)2
v
4.6
Chapter 4: Mean flow and flow resistance in open channels
and by taking roots,
v
8
=
.............................................…………...............(4.17)
v*
∂
If it is assumed that shear stress (τo) at a pipe wall is a function only of the following
variables, τo = f(v, D, ρ, µ, k), dimensional analysis suggests the dependency
ƒ = f (Re, D/k) .................................................................(4.18)
where k is the representative height of the roughness elements on the pipe wall and D/k
therefore is an inverse measure of relative roughness ('relative smoothness').
The term Re in equation (4.18), known as the Reynolds number, is a dimensionless ratio
which expresses the relative importance of inertial and viscous forces in the flow:
inertial forces
vL
Re = viscous forces = ν
It is a subject to which we will return and consider more fully in Chapter 5 but for now we
must at least consider the nature of the conceptual basis of the Reynolds number
because it bears directly on the present discussion.
Flow is conceptualized as occurring in two modes with fundamentally different
constraints: laminar and turbulent flow. In laminar flow water is thought to move as a
stack of individual laminae of fluid in which vertical mixing of fluid among laminae is
prevented by the forces of viscosity. As we noted in Chapter 2, viscosity is the internal
molecular property of fluids which constrains the rate at which they deform (flow) when
subjected to some stress. Laminar flow can only occur, however, if the viscous forces
are large compared with the inertial forces represented by flow velocity and some
characteristic length specifying the scale of the flow. In other words, laminar flow is the
domain of relatively low Reynolds number. In cases of flow where the inertial forces
become large with respect to the viscous forces, the viscosity is no longer the property
which limits the rate of fluid deformation and the nature of flow is considered to be quite
different. In this turbulent flow domain, fluid motion no longer occurs in distinct laminae
4.7
Chapter 4: Mean flow and flow resistance in open channels
without mixing but rather is a much more chaotic motion in which water particles break
the bounds of viscosity and move significant vertical distances in the flow. In this flow
dominated by chaotic motion, called turbulent flow, , turbulence itself is the
characteristic that limits the rate of fluid deformation. In other words, turbulent flow is
the domain of relatively high Reynolds number. As you might imagine, the resistance to
flow encountered by water moving though a pipe or channel is not independent of the
type of motion, laminar or turbulent, exhibited by the flow.
The measured dependence of ƒ on the character of pipe flow was established in several
classical experiments conducted during the first few decades of this century, notably by
Stanton (1914), Nikuradse (1932-35), and Colebrook and White (1937-39); some of
these early observations are reviewed in A.S.C.E. (1963). These studies confirm the
validity of equation (4.12); typical results are summarized in the Stanton diagram shown
in Figure 4.2.
The Stanton diagram shows measured values of ƒ plotted against Reynolds number for
a wide range of pipe roughness. The data show that, in laminar flow (where Re<2000),
the resistance to flow is entirely dependent on Re and is quite independent of the
relative roughness. That is, resistance is a single-valued function of Reynolds number
[equation (4.19): ƒ = 64/Re] regardless of the pipe roughness. Beyond a narrow
transition zone (Re = 2000-3000), however, the flow becomes fully turbulent and
resistance to flow becomes independent of Re (ƒ versus Re is a horizontal plot for given
roughness) and entirely dependent on D/k. In this fully turbulent domain, resistance to
flow is described by
1/
f = 1.14 + 2.0 log (D/k) ..................................(4.20)
The key to interpreting the behaviour of ƒ in Figure 4.2 is the relationship between the
€ the thickness of the viscosity-dominated flow. In laminar
roughness element height and
flow all the roughness elements are enclosed in viscous fluid in which there is no lateral
mixing. In consequence, the roughness geometry has no influence on the flow and
4.8
Chapter 4: Mean flow and flow resistance in open channels
resistance is a simple function of Reynolds number. In the fully turbulent domain,
laminar flow is thought to persist in a very thin film or viscous sublayer next to the pipe
wall which becomes even thinner as Reynolds number increases. In cases where the
roughness elements are relatively large and protrude through the surface of the viscous
sublayer, they exert a profound influence on the entire flow to the extent that
resistance is dependent only on the relative roughness. Here the flow encounters
hydrodynamically rough boundaries. The lower envelope of the family of resistance
curves in the fully turbulent domain of figure 4.2 represents a convergence
corresponding to hydrodynamically smooth boundaries in which the roughness
elements are fully enclosed by the viscous sublayer. Here the effects of wall roughness
are isolated from the main flow and the dependency on Reynolds number is once again
apparent.
Eq. (4.19)
0.07
Eq. (4.20)
0.05
61.0
0.04
120
0.03
252
504
1014
0.02
2 000
laminar flow
0.01
D/k
30.0
internal pipe geometry showing
roughness elements
turbulent flow
1 000
10 000
relative roughness
increasing
Resistance coefficient, ƒ
0.10
100 000
1 000 000
Reynold s number, Re
Figure 4.2: The Stanton diagram showing ƒ = (Re , D/k); after Rouse (1946).
4.9
k
D
Chapter 4: Mean flow and flow resistance in open channels
Rivers move through channels as fully turbulent flow over hydrodynamically rough
boundaries and resistance to flow appears to be determined largely by relative
roughness of the boundary. Nevertheless, the viscous sublayer may be important for
certain processes operating in the vicinity of the bed and we will have to revisit this
notion later when we consider the distribution of velocity close to the boundary.
Graphs similar to those in Figure 4.2 can be derived for Chezy C (called Moody
diagrams) using the transform of equation (4.9) although to my knowledge no primary
data have ever been collected for the purpose of an independent characterization so
obviously they show the same results with the same interpretation.
The Manning Equation
The most well known and widely applied assessment of Chezy C was provided in 1891
by the Irish engineer, R. Manning. He used experimental data from his own studies
and from the results of others to derive the empirical relationship:
kRh1/6
C= n
................................................................(4.21)
where n is a measure of channel roughness and k = 1.49 (for Imperial units) or 1.0 (for
SI units). Combining equations (4.10) and (4.21) yields the Chezy-Manning equation (or
simply, the Manning equation as it is now known), the SI-unit version of which states
that:
v=
Rh2/3s1/2
............................................................(4.22)
n
A corresponding equation for discharge can be written:
Q =A
Rh2/3s1/2
Rh2/3s1/2
=
wd
...................................(4.23)
n
n
where A, w, and d, are respectively cross-sectional area, width, and mean depth, of the
channel.
4.10
Chapter 4: Mean flow and flow resistance in open channels
Equations (4.22) and (4.23) are widely used today by river engineers to predict the mean
velocity and discharge through open channels from measured values of hydraulic
radius (or mean depth in wide channels), channel width, and water-surface slope, and
an estimate of the roughness coefficient, n. Manning's n computed from measured
velocities typically varies between 0.01 and 0.10 in natural channels. Sample Problem
4.2 illustrates a typical application of the Manning equation.
Sample Problem 4.2
Problem: An open channel of rectangular section 20 m wide has a slope of 0.0001. Calculate the depth
of uniform flow and the mean velocity in this channel if the discharge is 100 m3s-1.
Rh2/3(0.0001)1/2
or 8.5 = dRh2/3
0.017
 20d  2/3
20d
=
, substituting for Rh yields 8.5 = d
.
Raising both sides of
2d+20
2d+20
Solution: From equation (3.88), Q = wd
Since Rh =
the
A
P
Rh2/3s1/2
n
or 100 = 20d
 20d 
or 49.563d + 495.631 = 20d5/2. Solving by
2d+20
equation by the power 3/2 yields: 24.782 = d3/2
iteration yields d = 4.15 m. From continuity, v = 100/(20 x 4.15) = 1.21 ms-1. Note that this problem is
algebraically much simpler if Rh can be approximated by the mean depth, d.
Obviously the error in predicting the mean velocity from equation (4.22) is directly
proportional to the error in estimating Manning's n. For example, if the true value of n is
0.04 and was erroneously judged to be 0.03, the velocity and discharge would be
overestimated by 25 per cent. It is not surprising, therefore, that considerable attention
has been given to the problem of estimating accurately the magnitude of n in natural
channels. Before we examine some of these estimating procedures, however, we
should note two important properties of Manning's n and consider the factors which
contribute to most of its variation in natural channels.
First, equation (4.22) is not a dimensionally balanced physical statement and it can be
shown readily that n is not simply a length. From equation (4.14) we know that C
= 8g / ∂ , which on substitution in equation (3.86), gives
4.11
Chapter 4: Mean flow and flow resistance in open channels
k Rh1/6
1
=
.....................................................(4.24)
8g n
f
Since ƒ is dimensionless and n is dependent only on roughness, the factor k must have
€
the dimensions of g . Furthermore,
since Rh is a length, n must have the dimensions
L1/6; thus Rh/n6 is a measure of inverse relative roughness analogous to D/k in
equation (4.19). So equation (4.22) is not a 'rational' or physically deduced relationship
but rather an empirical relationship in which n is thought to express resistance to flow
related to 'roughness' of the boundary.
Second, when evaluating n by measurement (of Rh, s, and v), Manning's n simply
becomes a coefficient of proportionality which reflects all sources of flow resistance, not
just that related to boundary roughness. Consequently, two channels with identical
boundary materials may have quite different values of Manning's n if they differ
markedly with respect to other sources of flow resistance.
Sources of flow resistance
The more important of these sources of flow resistance are as follows:
1. Boundary roughness
2. Stage and discharge
3. Vegetation
4. Obstructions
5. Channel irregularity and alignment
6. Sediment load
Boundary roughness actually is a rather more difficult concept to define than you might
imagine. Although conventionally it is defined operationally as a roughness length (see
Figure 4.2) 'roughness' also depends on the spacing and shape of the roughness
elements. Widely spaced elements produce less roughness than those more closely
spaced, at least up to a certain close packing. If elements are shaped to allow close
4.12
Chapter 4: Mean flow and flow resistance in open channels
fitting, their spacing can be reduced to zero (all elements touch adjacent elements) and
boundary roughness will decline.
These difficulties aside, in the commonly assumed case of close-packed spherical
grains, roughness is taken as the mean grain size or as some fixed percentile of the
grain-size distribution. It has been shown (for example, see Wolman, 1955, Leopold,
Wolman and Miller, 1964 and Limerinos, 1970) that the relationship between flow
resistance and boundary roughness in open channels essentially is similar to the
uniformly distributed skin resistance for turbulent flow through rough pipes described in
Figure 4.2 and by equation (4.20):
1/
f = 1.0 + 2.0 log (y/D84) .............................................(4.25)
where D84 refers to the 84th percentile of the grain-size distribution measured in the
€
same units as the depth,
y.
But equation (4.25) applies only to a regular and smooth boundary in which grain
roughness is the only source of skin or boundary resistance. The boundaries of rivers
invariably have other scales of roughness expressed on them and these constitute
additional 'roughness'. For example, sediment transport creates bedforms such as
ripples, dunes, and various kinds of bars, and these may be even more important than
grain size in controlling flow velocity (see Einstein and Barbarossa, 1952 and Simons
and Richardson, 1966). Generally, the larger and more closely spaced the bedforms,
the greater the skin resistance and the magnitude of Manning's friction factor.
It is important not to think of boundary roughness as simply a local effect at the channel
perimeter. Turbulence and macroturbulence generated along the rough boundary is
transmitted throughout the entire flow. The rougher the boundary the greater the flow
disturbance there and the greater the intensity of turbulence (and flow resistance)
experienced by the entire flow.
Stage and discharge control on flow resistance in natural channels is implicit in
equation (4.25). As relative smoothness y/D84 increases, the Darcy-Weisbach
resistance coefficient declines. In other words, flow resistance obviously is stage
4.13
Chapter 4: Mean flow and flow resistance in open channels
dependent for a given absolute roughness. In general we might conclude that
Manning's n will decline as discharge and stage increases, a deduction that is widely
supported by observation. At low flows the larger roughness elements which are
'drowned out' at high discharges become increasingly more important contributors to
flow resistance and may even contribute differently in kind by deflecting the mean flow
lines and causing other additional resistance effects (see below under Obstructions).
Another factor which usually tends to reinforce the inverse relation between flow
resistance and stage is the typical difference between the roughness of the bed and
banks. Generally the banks are relatively smooth, often consisting of finer cohesive
material which presents smooth near-vertical faces to the flow. Therefore, the average
boundary roughness of the wetted perimeter of the channel declines as stage
increases. We also need to make a cautionary note here, however, because we can all
conceive of particular cases where the banks are actually rougher than the bed of the
channel. In such cases these two effects - drowning of bed roughness and
encountering greater bank roughness - will be opposed and the net effect on Manning's
n clearly will depend on the relative importance of each factor.
Nevertheless, generally Manning's n will decline as stage and discharge increases up
to about bankfull level. Discharges beyond bankfull flow will spread out over the flood
plain of the river and the flow will encounter much greater relative roughness, resulting
in an overall increase in flow resistance.
Vegetation plays an important but complex role in controlling flow resistance in open
channels (Hickin, 1984). Vegetation commonly grows on the banks of rivers so it is an
important element of bank roughness. Although isolated trees and short grasses may
offer little resistance to flow, dense growths of bushes and vines may represent an
important example of the effects referred to in the cautionary note above.
Within-channel vegetation such as weeds and lilies may be important sources of flow
resistance in some low-velocity streams. The direct effect of vegetation as roughness
often is enhanced by the trapping of other organic debris being transported by the flow.
Here it may be useful to distinguish between this type of small organic debris and the
4.14
Chapter 4: Mean flow and flow resistance in open channels
effects of treefall and logs incorporated into the channel. The latter class of vegetative
material, termed large organic debris, will be considered below as an Obstruction.
The importance of vegetation as a roughness element also depends on stage and on
the physical structure of the plants involved. In the typical case, as stage increases,
flow resistance attributable to vegetation declines because many plants (such as young
willow and alder, for example) are flattened as they are submerged. Thus, at high flows
they present a much more streamlined shape to the flow than they do at low flows.
Certain other plants (such as blackberry bushes and cottonwood saplings), are not so
pliable and their form remains a source of great flow resistance as discharge increases.
In general, physical reasoning and considerable anecdotal evidence lead us to
conclude that, as the density and size of boundary vegetation increases in river
channels, so will Manning's n and flow resistance increase. Nevertheless, it must be
acknowledged that very little systematic study of the relation between flow resistance
(or Manning's n) and the character of riparian vegetation has been undertaken.
Obstructions in a river channel include fallen trees, log jams, large boulders, slumped
banks, bridge piers, and the like. All such occurrences contribute to increases in flow
resistance and the magnitude of Manning's n. The degree of increase clearly will
depend on the nature of the obstructions, their size, shape, number, and distribution
within the channel.
Large obstructions may result in the local acceleration of flow into the supercritical
domain, resulting in the formation of hydraulic jumps. Here, as we noted earlier, rapid
flow literally impacts on the more slowly moving downstream water mass, resulting in
the telescoping of stream lines and extreme energy loss through turbulence. This type
of flow resistance has been termed impact or spill resistance. Perhaps the most
extreme case of spill resistance occurs at the foot of a waterfall where a free-falling
stream impacts on a plunge pool before resuming channeled flow.
Early experimental studies (Leopold et al, 1960) showed that channel obstructions may
greatly increase flow resistance well beyond that attributable to boundary roughness.
4.15
Chapter 4: Mean flow and flow resistance in open channels
Indeed, spill resistance seems to be associated with substantial upward departures
from the 'square law' resistance of equation (4.12) even in flows for which the mean
Froude number is considerably less than unity.
Channel irregularity and alignment refers to major changes in the mean boundary
geometry such as downstream variations in wetted perimeter and cross-sectional size
and shape of the channel. Such large-scale irregularity may be introduced by sand and
gravel bars, ridges, depressions, pools and riffles on the channel bed, and by the
presence of very large boulders. Although a gradual and uniform change in crosssectional size and shape will not appreciably effect the magnitude of Manning's n,
abrupt changes or alternations of small to large sections may increase the magnitude of
Manning's n by 0.005 or more.
Of particular importance in this context is the additional flow resistance introduced by
the periodic alternation of flow and form that occurs in meandering channels. In river
bends with large radius of curvature, the resistance increment attributable to
meandering may be relatively low but in bends of tight curvature flow resistance and
Manning's n will be increased measurably. Wherever flowing water is forced to change
its direction of flow at channel bends the deflection creates internal distortion resistance
and energy dissipation by eddying, secondary circulation, and by increased shear rate.
Internal distortion resistance has been shown to be twice the magnitude of skin
resistance in very tightly curved channel bends in a laboratory flume (Leopold et al,
1960) and there is no reason to suppose that flow resistance and Manning's n does not
also increase with decreasing bend radius in natural rivers.
Sediment load probably is not an important factor influencing flow resistance in most
rivers but it does become a significant control in rivers which carry unusually high
concentrations of suspended sediment. In sufficient concentration suspended
sediment can actually dampen the turbulence in the flow and reduce the overall level of
flow resistance; important early contributors to these ideas were Vanoni and Nomicos
(1960), and Bagnold (1954), among others.
4.16
Chapter 4: Mean flow and flow resistance in open channels
A major world river carrying such extremely sediment-laden flows is the Yellow River in
China. This river is so heavily laden with wind-blown loess eroded from the Interior
Plateau that it commonly transports more sediment than water! The flow is so
dominated by the suspended particulates that its special character is becoming the
basis of a new sub-branch of fluid mechanics - that of hyperconcentrated flow.
It also has been argued (Chow, 1959) that, because energy is used to maintain
bedload transport, Manning's n must increase as the rate of bedload transport
increases. It is not likely, however, that this effect is measurable because it is
accompanied by other confounding changes in the flow. In any case, general
observation suggests that this factor exerts a relatively unimportant influence on the
resistance to flow in open channels.
Estimating Manning's n
Manning's n is most commonly estimated for a river channel by employing either (a)
descriptive rating tables and reference photographs; (b) the Cowan procedure; or (c)
empirical relations directly linking n to the size of the boundary material.
An example of a simple descriptive rating table for Manning's n is shown in Figure 4.3.
Matching the field conditions to the nearest description facilitates the estimate of the
actual roughness factor. Although this estimating procedure is often quite a challenge
to the uninitiated, river engineers and scientists who work routinely at estimating
Manning's n quickly become adept at its quite accurate assessment based on a field
inspection.
4.17
Chapter 4: Mean flow and flow resistance in open channels
Type and condition of channel
Typical magnitude range of Manning's n
Minimum
A: Artificial channels and canals
1. Smooth concrete:
2. Ordinary concrete lining:
3. Shot concrete, untroweled, and earth channels in best condition:
4. Straight unlined earth canals in good condition:
Normal
Maximum
0.012
0.013
0.015
0.020
B: Small streams (bankfull width <35 m)
(a) Low-slope streams (on plains)
1.
2.
3.
4.
5.
6.
7.
8.
Clean, straight, bankfull stage, no deep pools:
Same as above but more gravel and weeds:
Clean, winding, some pools and shoals:
Same as above, but some weed and gravel:
Same as above but at less than bankfull stage:
Same as above but with more gravel present:
Sluggish reaches, weedy, deep pools:
Very weedy reaches, deep pools; or floodways
with a heavy stand of timber:
0.025
0.030
0.033
0.035
0.040
0.045
0.050
0.030
0.035
0.040
0.045
0.048
0.050
0.070
0.033
0.040
0.045
0.050
0.055
0.060
0.080
0.075
0.100
0.150
(b) Sand bed channels with no vegetation
(typical of small streams but applies to rivers of all scales)
1. Lower-regime flow (F<1.0) with
(a) a bed of ripples:
(b) a bed of dunes:
2. Near critical or transitional flow over washed-out dunes:
3. Upper regime flow (F>1.0) with
(a) a plane bed:
(b) standing waves:
(c) antidunes:
0.017
0.018
0.014
0.011
0.012
0.012
0.028
0.035
0.024
0.015
0.016
0.020
(c) Steep mountain streams (steep banks, trees
and brush along banks submerged at high stages
1. Bed of gravels, cobbles and a few small boulders:
2. Bed of cobbles with large boulders:
0.030
0.040
0.040
0.050
0.050
0.070
C: Large rivers (bankfull width> 35m); the value of n is less than
that for small streams of similar description because relative roughness
typically is lower and banks offer less effective resistance to flow.
1. Regular section with no boulders or brush:
2. Irregular and rough boundary:
0.025
0.035
D: Floodplain surfaces
(a) Pasture, no brush
Short to high grass:
(b) Cultivated areas
1. No crop
2. Mature row and field crops:
(c) Brush
1. Scattered brush and heavy weeds:
2. Light brush and trees in summer (full foliage):
3. Medium to dense brush in summer (full foliage):
(d) Trees
1. Dense willows in summer:
2. Heavy stand of timber, a few down trees, undergrowth:
0.060
0.100
0.025
0.035
0.050
0.020
0.025
0.030
0.035
0.040
0.050
0.035
0.040
0.070
0.050
0.060
0.100
0.070
0.080
0.160
0.110
0.080
0.150
0.100
0.200
0.120
4.3: A rating table for Manning's friction factor, n, based on the type and condition of the channel
boundary and flood plain and the nature of riparian vegetation (based on data from the U.S.
Department of Agriculture and Simons and Richardson, 1966).
4.18
Chapter 4: Mean flow and flow resistance in open channels
For those unpracticed at the task, estimating Manning's n also can be facilitated by
comparing the field site in question with photographs of river channels for which
Manning's n has been measured. One widely used set of reference photographs is
published by the United States Geological Survey (Barnes, 1967). The field operator
simply refers to the reference channel which most closely resembles the field
conditions in order to form an estimate of Manning's n.
Estimating Manning's n from a general rating table often involves the mental integration
of a number of quite different contributions to the overall roughness factor and the
Cowan procedure is one attempt to break the assessment down into component
estimates (Cowan, 1954).
The magnitude of Manning's n may be computed by:
n = (no + n1 + n2 + n3 + n4) m .............................................(4.26)
where
no = the minimum value of n for a straight uniform channel of given
boundary materials;
n1 = a surface irregularity correction;
n2 = a channel shape/size correction;
n3 = a correction for the influence of obstructions;
n4 = a vegetation correction factor;
and
m = a correction factor to account for the degree of meandering.The
appropriate values for the components in equation (4.26) are shown in Figure 4.4.
For example, a meandering channel (sinuosity index = 1.3) with a smooth unvegetated
alluvial boundary and slightly irregular banks, a pool and riffle sequence producing
frequently alternating but otherwise unobstructed flow, would be characterized by
equation (4.26) as follows:
n = (0.020 + 0.005 + 0.005 + 0.00 + 0.00 )1.15 = 0.035
4.19
Chapter 4: Mean flow and flow resistance in open channels
Such a channel would correspond with the channel type B(a)3 in Figure 4.3.
Manning's n may also be estimated directly from the roughness of the boundary
measured as a representative particle diameter. For example, implicit in equation (4.25)
is a relationship linking Manning's n and the size of the bed material. Noting from the
SI-unit version of equation (4.24) that
be written
1 Rh1/6
1
=
, equation (4.25) can
8g n
f
1 Rh1/6
= 1.0 + 2.0 log (y/D84) .................................(4.27a)
8g n
€
or if Rh ≈ y, simplification leads to:
n=
y1/6
y
8.859(1.0+2.0log D )
..........……............(4.27b)
84
A more commonly employed empirical relationship is that developed in 1923 by the
Swiss engineer, A. Strickler. He found that, for straight uniform reaches of gravel-bed
rivers in the Swiss Alps,
or
n = 0.0151D501/6
(for D50 in millimetres) ..................... (4.28a)
n = 0.0478D501/6
(for D50 in metres) ............................(4.28b)
It is important to note here that this Strickler relation, as it is known, and that expressed
by equation (4.27), only should be applied to gravel bed rivers in which total
resistance to flow is simply the result of skin resistance alone. As other sources of
resistance begin to contribute significantly to the total, these equations will increasingly
underestimate the actual magnitude of n. In this sense equations (4.27) and (4.28)
estimate the minimum value of Manning's n.
4.20
Chapter 4: Mean flow and flow resistance in open channels
no
Boundary
materials
n1
Degree of
channel
cross-sectional
irregularity
n2
Variation in
channel
cross-section
shape and area
Smooth alluvial boundary..........................................................................................0.020
Rock-cut boundary.....................................................................................................0.025
Fine gravel ................................................................................................................ 0.024
Coarse gravel boundary..............................................................................................0.028
Smooth: best attainable for the given materials..........................................................0.000
slightly eroded banks....................................................................................0.005
Moderate: Comparable with dredged channel in fair to poor
condition; some minor bank slumping and erosion......................................0.010
Severe: Extensive bank slumps and moderate bank erosion;
jagged irregular rock-cut materials...............................................................0.020
Gradual: Changes in size and shape are gradual........................................................0.000
Occasional alternation: Large and small sections alternate
occasionally or shape changes to cause occasional
shifting of flow from side to side..................................................................0.005
Frequent alternation: Large & small sections alternate or
shape changes cause frequent shifting of flow from
side to side.......................................................................................0.010 to 0.015
Negligible.................0.00
n3
Relative
effect of
obstructions
n4
Vegetation
Determination of n3 is based on the presence and
characteristics of obstructions such as debris, slumps,
stumps, exposed roots, boulders and fallen and lodged
logs. Conditions considered in other steps must not be
reevaluated (double counted) in this determination. In
judging the relative effect of obstructions, consider the
extent to which the obstructions occupy or reduce
average water area; the shape (sharp or smooth) and
position and spacing of the obstructions.
0.010
Minor ................to 0.015
0.020
Appreciable... .....to 0.030
0.040
Severe................to 0.060
Low: Dense but flexible grasses where flow depth is 2-3 x the
height of vegetation or supple tree seedlings (willow, poplar)
0.005
where flow depth is 3-4 x vegetation height...........................................................to 0.010
Medium: Turf grasses in flow 1-2 times vegetation height; stemmy grasses
where flow is 2-3 x vegetation height; moderately dense brush on
0.010
banks where Rh>0.7 m............................................................................................to 0.025
High: Turf grasses in flow of same height; foliage-free willow or poplar,
8-10 years old and intergrown with brush on channel banks
0.025
where Rh>0.7m; bushy willows, 1 year old, Rh>0.7m...........................................to 0.050
Very High: Turf grasses in flow half as deep; bushy willows (1 year old)
with weeds on banks; some vegetation on the bed; trees with
0.050
weeds and brush in full foliage where Rh>5m.........................................................to 0.100
m
Degree of
Meandering
Minor: Sinuosity index = 1.0 to1.2................................................................................1.00
Appreciable: Sinuosity index = 1.2 to 1.5.....................................................................1.15
Severe: Sinuosity index >1.5.........................................................................................1.30
4.4: Determination of Manning's roughness coefficient by the Cowan procedure (after Chow,
1959).
4.21
Chapter 4: Mean flow and flow resistance in open channels
The relative performance of equations (4.27) and (4.28) is illustrated in Figure 4.5. It has
been assumed here that D84=2D50 (on the basis of typical grain-size distributions in
gravel-bed rivers; see Shaw and Kellerhals (1982). For the finer gravels (D50 = 5 mm;
D84 = 10 mm), equations (4.27) and (4.28) essentially yield the same results (n = 0.02)
although there clearly is considerable divergence in the coarser gravels at relatively
shallow depths of flow (<1.0 m). In general it seems likely that equation (4.27) is the
more reliable estimator simply because it includes the effects of depth variation in the
'relative smoothness term', y/D84. Nevertheless, for bed material up to about 10 cm in
diameter and depths greater than about 0.5 m, the simpler Strickler relation appears
to perform adequately. Examples of the application of equations (4.27) and (4.28) are
shown in Sample Problems 4.3 and 4.4.
Eqn 4.27; D 84 = 0.05
Eqn 4.27; D 84 = 0.10
Eqn 4.27; D 84 = 0.01
Eqn 4.28; D 50 = 0.25
Eqn 4.28; D 50 = 0.05
Eqn 4.28; D 50 = 0.005
.09
Manning's n
.08
.07
.06
.05
.04
.03
.02
.01
0
0.5
1.0
1.5
2.0
2.5
Flow depth, y (metres)
3.0
3.5
4.5: Manning's n computed from equations (4.27) and (4.28) for a range of
particle size and depth of flow.
4.22
Chapter 4: Mean flow and flow resistance in open channels
Sample Problem 4.3
Problem: An open channel of rectangular section 100 m wide with a slope of 0.0005 must carry a
discharge of 300 m3s-1. If the bed consists of uniformly packed and roughly spherical pebbles of 10 cm
median diameter, what will be the depth of flow?
Solution : Since the channel is quite wide we will assume that Rh = y so that we can recast equation
(4.23) thus:
y2/3s1/2
y5/3s1/2
y5/3(0.0005)1/2
y5/3
Q = wd
= w
or 300 = 100
= 2.236
n
n
n
n
Introducing the Strickler equation (n = 0.0478D501/6) here yields:
300 = 2.236
y5/3
0.0478D501/6
which simplifies to
300 = 2.236
y5/3
0.033
So,
y5/3 = 4.428
= 2.236
y5/3
0.0478(0.1)1/6
and further to 300 = 67.758 y5/3
and y = 4.4283/5 = 2.44 m
Thus the flow depth will be 2.44 m and the mean velocity will be 1.23 ms-1.
Sample Problem 4.4
Problem: A field operator surveys a 200 m-wide rectangular channel carrying a 3 m-deep flow for which
the shear velocity and flow velocity were respectively measured at 0.14 ms-1 and 1.5 ms-1. Because her
detailed field notes were subsequently lost, you must estimate the water-surface slope and the median
size of the bed material at the time of the survey.
Solution: To solve this problem we first need to determine the water-surface slope from the known shear
velocity so that we can solve the Manning equation for n; n can then be used to estimate the size of the
bed material.
Since the channel has a high w/d ratio (200/3 = 67), we can safely assume that Rh = y. We know
0.058
that v* = τ/ρ = gys = 0.14 so water surface slope must have been s =
= 0.00067.
9.81x3
Thus the Manning equation for this channel can be written:
n =
y2/3s1/2
v
=
(3)2/3(0.00067)1/2
1.5
= 0.036
The link between Manning's n and D50 is provided by the Strickler relation [equation (4.28a)]:
n = 0.0151 D501/6 (D50 in mm)
or 0.06 = 0.0151 D501/6
Solving for D50 yields D50 = (0.036/0.0151)6 = 180 mm.
Alternatively, we might have used equation (3.92b) to estimate D84:
n = 0.036 =
31/6
y
8.859(1.0+2.0log
)
D84
; log
y
= 1.3829 and D84 = 0.125 m or 125 mm.
D84
Although these two solutions for particle size are of the same order of magnitude, the difference
(remember that D84/D50 ≈ 2.0 and that the equivalent D50 from equation (4.27) therefore is about 63 mm)
should remind us that these empirical equations constitute rather imprecise science!
4.23
Chapter 4: Mean flow and flow resistance in open channels
Some concluding remarks
Although the Strickler equation is used widely in gravel-bed river engineering, there are
other empirical relationships that might also be used (see Bray, 1982). All such
empirical relationships are similar in form and only work well when applied in
environments similar to those in which they were developed. Furthermore, we must not
forget that these relationships specify Manning's n for the given particle size or relative
roughness only. As such they are estimates of minimum n; actual values will be
higher to the extent that sources of flow resistance other than boundary roughness are
influencing flow in the channel.
In the discussion in this and earlier chapters we have been concerned only with the
mean velocity of flow and the factors controlling it. But of course we are all aware that,
in a natural channel, the velocity varies considerably within the flow, faster in the deeper
water in the middle of the stream and more slowly in the shallower water near the
banks. We now need to explore the nature of this within-flow variation in velocity and
the attempts that have been made to explain why it occurs.
References
A.S.C.E., 1963, Task force on friction factors in open channels: Proceedings of the American Society of
Civil Engineers, 89, HY2, 97.
Bagnold, R.A., 1954, Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid
under shear: The Royal Society of London Proceedings, Series A, 225, 49-63.
Barnes,H.H., 1967, Roughness characteristics of natural channels. Water Supply Paper 1894, US
Geological Survey, Washington, DC, 213 pp.
Bray, D.I., 1982, Flow resistance in gravel-bed rivers. In Gravel-bed Rivers, Hey, R.D., Bathurst, C. and
Thorne, C.R. (Editors), John Wiley and Sons, 109-132.
Chow, V.T., 1959, Open Channel Hydraulics. McGraw Hill, New York, 680p.
Cowan, W.L., 1954, Estimating hydraulic roughness coefficients. Agricultural Engineering, Vol 37 (7) 473475.
Einstein, H.A. and Barbarossa, N.L., 1952, River channel roughness: American Society of Civil Engineers
Transactions, 117, 1121-1146.
4.24
Chapter 4: Mean flow and flow resistance in open channels
Henderson, F.M., 1966, Open Channel Flow, Macmillan, New York.
Hickin, E.J., 1984, Vegetation and river channel dynamics: Canadian Geographer, 28 (2) 111-126.
Leopold, L.B., Bagnold, R.A., Wolman, M.G. and Brush, L.M., 1960, Flow resistance in sinuous or
irregular channels: U.S. Geological Survey Professional Paper 282D.
Leopold, L.B., Wolman, M.G. and Miller, J.P., 1964, Fluvial Processes in Geomorphology, Freeman San
Francisco, 522 p.
Limerinos, J.T., 1970, Determination of the Manning coefficient from measured bed roughness in natural
channels. Water Supply Paper 1898-B, US geological Survey, 47 pp.
Shaw, J. and Kellerhals, R., 1982, The composition of Recent alluvial gravels: Alberta Research Council,
Bulletin 41, 151p.
Simons, D.B. and Richardson, E.V., 1966, Resistance to flow in alluvial channels: United States
Geological Survey Professional Paper 422-J, 61p.
Vanoni, V.A. and Nomicos, G.N., 1960, Resistance properties of sediment laden streams: American
Society of Civil Engineers Transactions, 125, 1140-1175.
4.25