chapter 3 river hydraulics and the channel

Chapter 3

River Hydraulics and the Channel 1. Role of Flow Velocity in the Continuity Relation A Few Definitions 2. Velocity and Balancing the Impelling and Resisting Forces 3. Boundary Shear Stress 4. Flow Resistance

Sources of Flow Resistance 5. Flow Resistance Formulae

Chezy Formula Darcy-Weisbach Equation

6. The Nature of Channel Roughness Grain Roughness Form Roughness

7. Manning n estimating procedures Direct Measurement Descriptive tables and photographs Cowan Method Strickler Equation

8. Problem-Solving with Manning Equation 9. What Determines the Shape of a River Channel?

The Equilibrium Channel An Empirical Approach

10. Some Further Reading 11. What’s Next?

1. The Role of Flow Velocity in the Continuity Relation You will recall from the previous chapter that the continuity relationship can be expressed by

equation (3.1) as:

wd = A = Q/v (3.1)

where the cross-sectional area of the channel is a direct function of the discharge and the flow

velocity. Our task in this chapter is to learn what controls the mean velocity, v. Although most

of this discussion is concerned with the mean velocity and mean flow conditions, we know

already that velocity varies with position in the channel. It can also vary with time (that is, it can

fluctuate). Even though velocity does vary in space and time it is useful at a broad level of

generalization to assume that flow in a river channel can be characterized simply by the mean

flow. Nevertheless, it will be useful before we begin our journey here to acknowledge a few

definitions that will come in handy now but will be even more useful as we continue.

Hickin: River Geomorphology: Chapter 3

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A Few Definitions

• Open-channel flow is the name given in fluid mechanics to flow through a channel that

has a free water-surface. It contrasts with closed-conduit flow (flow in a pipe) in which

there is no free water-surface. The difference is important because, unlike flow in closed

conduits, pressure variation in open-channel flow is expressed as a change in flow depth

(the water surface can rise or fall) and forces at work at the water/air interface can give

rise to surface waves. Such waves can exert significant control on the mean flow. The

definitions to follow all relate to open-channel flow although most can also be adapted to

pipe-flow as well.

• Steady flow is flow in which flow velocity (v) and flow depth (d) do not change with

time. To use the language of the calculus we say that flow velocity and flow depth are

time invariant (dv/dt = 0 and dd/dt = 0). Of course, the continuity relationship dictates

that, for a given discharge through an open channel with a rigid boundary, if dv/dt = 0

then it must also be true that dd/dt = 0.

• Unsteady flow, on the other hand, is flow in which flow velocity (v) and flow depth (d)

do change with time. Again, to use the language of the calculus we say that flow velocity

and flow depth vary with time (dv/dt ≠ 0 and dd/dt ≠ 0). Once again, the continuity

relationship dictates that, for a given discharge through an open channel with a rigid

boundary, if dv/dt ≠ 0 then it must also be true that dd/dt ≠ 0.

• Uniform flow is flow in which velocity and depth of flow do not change in space (ie,

with distance downstream, s). Again, to use the language of the calculus we say that flow

depth (and therefore flow velocity) does not vary spatially (dv/ds = 0 and dd/ds = 0).

Note that this does not mean that the channel bed and the water surface have to be planar

(flat). Uniform flow can also be achieved over a wavy bed if the water-surface has the

same parallel waveform so that flow depth does not vary downstream. It turns out,

however, that this kind of in-phase condition of the bed and water surface is possible but

an unlikely outcome in most river channels.

• Non-uniform flow is the converse of uniform flow: flow depth varies downstream along

the channel so that the depth of flow varies spatially along with the mean-flow velocity

(or dv/ds ≠ 0 and dd/ds ≠ 0).


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Most flows in rivers are unsteady and non-uniform although it is sometimes convenient in fluid

mechanics to assume that they are in fact steady uniform flows. Flow can be steady and

uniform, unsteady and uniform, unsteady and non-uniform but never unsteady and uniform or

steady and non-uniform. Think about a rectangular channel carrying a constant discharge; can

you see why these last two conditions (unsteady uniform flow and steady non-uniform flow) are

not possible?

We also need to recognize two general kinds of flow that control the way in which velocity

varies within the cross section of a channel in two quite different ways. The first is laminar flow

and the second is turbulent flow (Figure 3.1):

Figure 3.1: Some definitions in open-channel flow (from Knighton, 1998, Figure 4.1)


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• Laminar flow is flow in which the rate of fluid deformation (for example, the change in

velocity with height above the channel bed) is controlled only by the viscosity of the

fluid. Viscosity is that internal molecular property that determines how easily a fluid will

flow. We say that water is less viscous (more runny) than higher viscosity honey or oil,

for example. Laminar flow is envisioned as occurring as stacked laminae or layers of

flow in which the water particles do not move vertically between enveloping layers of

water. Any force that tends to create these vertical motions is dampened and overcome

by the opposing viscous forces. Laminar flow is not common in nature and probably

never occurs in natural channels. We consider it here because it provides a basis of

comparison for examining the alternative more realistic “turbulent flow.”

• Turbulent flow is flow in which the rate of fluid deformation is controlled, not by fluid

viscosity (because it is too weak a force), but overwhelmingly by the internal chaotic

fluid motions we call turbulence. In turbulent flow the orderly arrangement of flow

layers found in laminar flow are disrupted because, as vigour of the flow increases,

viscosity is no longer strong enough to overcome the large forces tending to move water

particles vertically, thus allowing mixing of the flow.

2. Velocity and Balancing the Impelling and Resisting Forces

The balance of forces applied to any body, including a body of water, determines how it will

move. Water moves through a channel because the impelling force (FI) of gravity is able to

overcome the frictional resisting force (FR) opposing to flow. This is a simple expression of

Newton’s second law of motion (F = Ma). Three force–balance conditions are possible:

• FI > FR and the flow accelerates

• FI = FR and flow velocity is constant

• FI < FR and flow velocity experiences negative acceleration (a deceleration)

In steady uniform flow the velocity is constant and we therefore know that the impelling and

resisting forces must be in balance (zero net force). Because it is the case that, over short reaches

along a channel (say a channel width or so), flow velocity in a natural channel is sensibly

constant, we can safely assume that it approximates steady uniform flow. This assumption

allows us to derive some relationships that lead us to some very useful tools for analyzing flows

in rivers. This task will be aided by reference to Figure 3.2.


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Figure 3.2: A control volume of fluid in a rectangular channel of length, L, width w, and depth, d. The

impelling gravitational force F1 is opposed by the force of boundary resistance, FR.

Figure 3.2 shows a control volume of fluid flowing in a rectangular channel (we want to keep

things simple for now but the principles we will derive can be applied to a cross-section of any

shape).

The impelling force, FI , is the force of gravity acting over unit area on the bed. This force is

simply the weight of the water (the product of the water volume (wdL), fluid density (ρ) and the

acceleration of gravity (g = 98.1 m/s2) resolved in the downstream direction. In other words we

can say that:

FI = sinαALρg

where α is the angle that the bed slopes downstream. FI, like any force in mechanics, is

expressed in units of mass x acceleration or kg x m/s2; these force dimensions are known as

Newtons (N). So we might say, for example that the water exerts a force of 5.0 N on the channel

boundary.

d

L

FI

W

α

α

w

A = Wd

W = volume.density.gravity W = ALρg P = 2d + w

FR


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The resisting force, FR, is taken to be the force of friction being exerted by the bed and vertical

banks on the flow moving in our rectangular channel. This is the friction force operating in the

opposite direction to the impelling force that the flow has to overcome to accelerate or at least

equal if the velocity is to remain constant. This frictional force acts in the plane of the channel

boundary so it is a shear stress (τo), a force per unit area, acting over the total area of the channel

boundary supporting our control volume of water. Stated mathematically:

FR = τo(2d+w)L Since FI = FR in our model we can say that, for our control volume of flow: FI = FR

sinαALρg = τo(2d+w)L

or

€

τ o = sinαρg A2d + w

(3.2)

3. Boundary Shear Stress

Equation (3.2) provides us with a means of measuring the shear stress exerted on the channel

boundary by the flowing water. Shear stress is a force per unit area and is expressed as so many

N/m2.

In practice we introduce two simplifications in equation (3.2) to make it easier to use:

1. Because river channel slopes are almost always less than 5o, sin α ≈ tan α ≈ water-surface

slope, s. So we substitute s for sin α ;

2. The ratio

€

A2d + w

is known as the hydraulic radius, Rh. For channels in which w/d>20

(most channels in fact), Rh ≈ d. So we substitute d for

€

A2d + w

in equation (3.2).

These simplifications give:

τo = ρgds = γds (3.3)

where γ = the specific weight of water. Equation (3.3) is sometimes referred to simply as the

depth-slope product.


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4. Flow Resistance As shear stress increases in a river, so does the flow velocity. The degree to which the mean

velocity responds to increasing shear stress depends on the resistance to flow in the channel. Put

another way, any process in a river that bleeds off energy from driving the mean flow, represents

resistance to flow. The principal source of flow resistance for most rivers is the frictional

resistance that the water encounters as it moves across the channel boundary. But this is not the

only source. Indeed, under certain conditions, boundary friction can be overwhelmed by other

sources of flow resistance.

Sources of flow resistance

• Viscous resistance

• Turbulent resistance

• Internal distortion resistance

• Spill resistance

• Boundary (skin) resistance (grain roughness and form roughness)

Viscous resistance, as we noted earlier, refers to the internal “runniness” of a fluid. It is

sometimes termed molecular viscosity because it simply reflects the molecular structure of the

fluid and is essentially fixed for each kind of fluid although it does vary significantly with

temperature. Because it is a fixed property of each fluid, viscosity tends to be ignored or taken

as a given in discussions of flow resistance in rivers. But it is an important base-line source of

flow resistance that we should not overlook.

Turbulent resistance refers to the dampening of the mean flow by the presence of the small-

scale chaotic motions of water particles. These motions are three-dimensional and consist of

small parcels of water than spin and twist and tumble as the flow moves along. Some have

proposed that we might think of these small eddies in the flow as constituting an additional type

of viscosity: eddy viscosity. Eddies are generated at the bed of a river as the water flows over the

boundary but they are convected into the flow where they are carried along and slowly diffuse as

they lose their rotational energy to viscous resistance.


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The distinction between small-scale turbulence and the scale of larger disturbances to the flow

(such as large eddies shed from bank protuberances or eddy shedding from boulders on the bed,

for example) is entirely arbitrary. Some scientists have suggested that this larger scale of

discrete flow structure should be termed macroturbulence and this term has become quite widely

adopted in England and North America. For our purposes any flow structure scaled up to the

depth of flow we can call turbulence and is thus a source of turbulent resistance.

Internal distortion resistance refers to the energy lost to still larger flow structures such as the

energy lost to the mean flow that is bled off to drive the secondary circulation in bends (helical

flow). Other things being equal, there is greater resistance to flow through a meandering channel

than through a straight channel. Whenever there are sudden changes in bank alignment causing

abrupt changes in channel width or depth, the flow encounters internal distortion resistance.

Spill resistance is encountered where there are such severe changes in channel morphology

(from narrow to wide) that flow literally runs into the slack water ahead of it. The most extreme

example of spill resistance is a waterfall where the flow plunges into a pool. Spill resistance can

far exceed all other kinds of flow resistance in certain special circumstances. In mountain

streams, for example, it is not uncommon to encounter a high-energy kind of flow (called

supercritical flow) that is associated with severe water-surface deformation including breaking

surface waves and chutes that “collide” with downstream pools. Here spill resistance can be

extreme.

Boundary or skin resistance, however, is the principal source of flow resistance in most natural

channels. Boundary resistance depends on boundary roughness: smooth boundaries exert low

boundary resistance and rough boundaries give rise to high boundary resistance. It is difficult in

practice to decouple boundary resistance from turbulent resistance because both increase as

boundary roughness increases. Boundary roughness, as we shall soon see, is itself not an

uncomplicated concept. It is often thought of as consisting of two components: grain roughness

and form roughness. Grain roughness is the roughness that relates to the individual particles or

grains making up the sediments forming the channel boundary. Form roughness is the roughness

that relates to aggregates of particles forming bedforms and bars. These features are more


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difficult to analyze because, unlike the fixed grain size, they are transient and change as the flow

changes.

5. Flow Resistance Formulae It turns out that, by countless repeated observations in open-channel flow, shear stress has been

shown to be proportional to the velocity squared (also shown to be the case by dimensional

analysis):

τ ∝ v2 or

€

v ∝ τ

Since τ = γds, we can say that

€

v = C ds and

€

C =Vds

(3.4)

The coefficient of proportionality, C, is called Chezy C and

€

v = C ds is known as the Chezy

equation after the 18th-century French hydraulician who introduced it (http://chezy.sdsu.edu/).

Chezy C is a measure of flow conductance or efficiency. Other things being equal, flow velocity

increases as C increases (that is, it is the inverse of flow resistance).

Chezy C by itself was found difficult to assess and several attempts were made to evaluate it in

terms of variables that could be easily determined in the field. In 1891 Robert Manning, an Irish

engineer determined Chezy C empirically in terms of parameters that could be related to a river

channel in the field:

€

C =d 16

n

where n is the Manning roughness factor and ranges over an order of magnitude from 0.01-0.10

and d is flow depth.

Combining the Chezy equation and the Manning’s estimate of Chezy C yields


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€

C =v d s

d 16

n=

v

d 12s12

v = d 16d

12s12

n=

d 23s12

n

or

€

v = d 23s12

n (3.5)

Equation (3.5) was once known as the Chezy-Manning equation but now is known simply as the

Manning equation. The equation is in metric units but is not dimensionally balanced.

The Manning equation is very important because, in spite of its antiquity, it is widely used in

river engineering and applied work in fluvial geomorphology today. Manning n is thought of as

an index of channel roughness when it is based on velocity, depth and slope measurements in the

field, but as such it is really a coefficient of proportionality that reflects all sources of flow

resistance and not just simply channel roughness.

The Darcy-Weisbach resistance equation

Although Manning n is widely used in engineering, an alternative coefficient, the Darcy-

Weisbach resistance coefficient, is preferred by scientists because it is dimensionally

homogeneous. The conceptual basis of the Darcy-Weisbach resistance coefficient, ff, is that it is

simply the ratio of shear stress/velocity2, or

€

ff = f shear stressvelocity 2

= f 1

C2

(3.6)

The precise form of equation (3.6), the Darcy-Weisbach resistance equation, is

€

ff =8gdsv 2

(3.7)

When Darcy-Weisbach ff is calculated from measurements of d, s and v in the field, like

Manning n, it too is a coefficient of proportionality that therefore reflects all types of flow

resistance present in the river.


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6. The Nature of Channel Roughness

Throughout our discussion of flow resistance and channel roughness to this point we have

avoided a very difficult problem: what precisely is channel roughness? The conventional

wisdom is that it consists of two parts: grain roughness and form roughness.

Grain roughness is taken to be that component of roughness that relates directly to the size (grain

diameter) of particles constituting the boundary materials. For example, at a qualitative level we

can say that a boundary made of sand is less rough than one made of fine gravel because the

grains of sand are smaller than the particles of gravel. Making a more precise quantitative

statement about grain roughness requires us to digress for a bit because grain diameter is itself

not a straightforward concept.

Grain diameter in natural sediment varies considerably and in order to characterize the

size of grains in a sample of sediment we must have some basis for generalizing such as using an

average or some other measure of central tendency. It turns out that many geomorphologists

think that the most important grain-size influencing channel roughness for the purpose discussing

flow resistance is not the average size but rather the size of larger particles in any given sample.

In order to bring some objectivity and consistency to measures of these coarser particles the

grain-size distribution has become the basis of defining effective grain size. In general

geomorphologists use the Wentworth grade scale (borrowed from sedimentary geologists) when

discussing grain size (Figure 3.3):

SEDIMENT TYPE GRAIN-SIZE RANGE

Clay <1/256 mm

Silt 1/256 – 1/16 mm

Sand 1/16 – 2 mm

Gravel 2 – 64 mm

Cobbles 64 – 256 mm

Boulders >256 mm

Figure 3.3: The Wentworth grade scale used to describe grain size.


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The Wentworth grade scale is based on the median size of a sample of sediment grains. We

designate the median size as D50 because it is the grain size in the cumulative distribution of

grain size, half of which (50%) is coarser and half therefore is finer. An example of how we

construct a cumulative grain-size distribution graph is shown in Figure 3.4.

Grain

Diameter D, cm Ranked

D Sample %

(N=18) Cumulative %

Finer than 1.0 0.2 1/18*100 = 5.56 5.56 0.4 0.3 5.56 11.12 0.2 0.4 5.56 16.68 0.8 0.6 5.56 22.24 1.5 0.7 5.56 27.80 2.0 0.8 5.56 33.36 0.9 0.9 5.56 38.92 0.3 1.0 5.56 44.48 0.7 1.2 5.56 50.04 1.2 1.5 5.56 55.60 3.2 1.9 5.56 61.16 2.5 2.0 5.56 66.72 2.0 2.0 5.56 72.28 1.9 2.3 5.56 77.84 0.6 2.5 5.56 83.40 2.6 2.6 5.56 88.96 3.0 3.0 5.56 94.52 2.3 3.2 5.56 100.00

Figure 3.4: A cumulative grain-size analysis and graph for determining grain-size percentiles

D50 = 1.25 cm

Cumulative grain-size distribution

0

20

40

60

80

100

120

0 0.5 1 1.5 2 2.5 3 3.5 Diameter, cm

Percent finer

D84 = 2.50 cm cm


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Here the median grain size, D50, is 1.25 cm and D84 is 2.50 cm.

In research on flow resistance a number of studies have shown that resistance is more sensitive to

the higher grain-size percentiles than D50. For example, it is common for D84 to be used as a

measure of effective grain size. Remember, D84 is simply mathematical shorthand for saying

“that grain size in a sediment sample in which 84 per cent of the grains are finer (16 per cent are

coarser). In the case of the example in Figure 3.4 we say that “84 per cent of the grains in the

sample are finer than 2.50 cm”. For the statistically-inclined reader, D84 represents two standard

deviations above the mean if the grain-size distribution is normal.

Form Roughness is that component of roughness that relates to aggregates of grains in the form

of bedforms and bars. Form roughness can be considerably greater than grain roughness,

especially in steep rugged mountain channels. Examples of bedforms in sandbed rivers are

shown in Figures 3.5 – 3.7:

Figure 3.5: Gravel and sand have accumulated around young willows to create rib-like sedimentary forms that

have form roughness larger than the sediment grains that form them.


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Figure 3.6: Ripple fields formed on bar surfaces of a sand-bed river.

Figure 3.7: Dunes and ripples on dunes formed on a point-bar surface of a sand-bed river.


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The case of bedforms that occur in sand-bed rivers has been much studied although we still have

a great deal to learn about the role that bedforms play in accounting for the resistance to flow in

channels. In Figure 3.8 is reproduced David Knighton’s Figure 4.3 from his fluvial

Figure 3.8: Form roughness elements (from Knighton, 1998: Figure 4.3).

geomorphology textbook. It summarizes the findings of scientists who conducted a series of

experiments in the 1960s in a large engineering flume at Colorado State University at Fort

Collins designed to allow bedforms to develop in sandy sediment. The experiments showed that,

as discharge (and shear stress and flow velocity) is increased in the flow over a sand bed, a

sequence of bedforms develops: ripples ⇒ ripples on dunes ⇒ dunes ⇒ washed-out dunes ⇒

plane bed ⇒ antidunes ⇒ chutes and pools. The overall roughness of the channel (measured by

Darcy-Weisbach ff) varied according to the bedforms as shown above in Figure 3.5.

The flow-resistance effects of changing bedforms in sandbed rivers are sometimes evident in

discharge rating curves where measured discharge is related to the flow stage. In Figure 3.9


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Figure 3.9: This synthetic discharge rating curve shows typical responses to changing bedform regime.

the discontinuities or “kinks” in the graph relate to the shift in the bedform domains as discharge

increases. For example, in Figure 3.9A, discharge increases as an orderly direct function of

gauge height while ripples dominate the bedforms but as they are replaced by dunes in the

transition zone, discharge increases with very little change in gauge height because the lower

resistance allows the velocity to increase and accommodate most of the change in discharge

(refer to Figure 3.8). Once ripples begin to form again on the surface of newly developed dunes

on the bed, however, flow resistance increases yet again and discharge and gauge-height increase

together once more. Figure 3.9B shows the same set of transitions with gauge height as the

dependent variable. The rate of change in the stage flattens off in the transition zone as changes

in velocity accommodate the increasing discharge and then increases once again as flow

resistance increases in the zone of ripples on dunes.

7. Estimating flow resistance

Depending on the purpose of the measurements, flow resistance can be measured directly or it

can be estimated by a variety of indirect methods. These are outlined below.

Transition zone (reduced resistance)

Disc

harg

e

Gauge height (stage)

Ripples zone (high flow resistance)

Transition zone (reduced flow resistance)

Zone of dunes with ripples (resumption of higher flow resistance)

Gau

ge h

eigh

t (st

age)

Discharge

Ripples zone (high flow resistance)

Zone of dunes with ripples (resumption of higher flow resistance)

A B


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Direct measurement of flow resistance is usually done when the flow resistance is the

variable of interest (rather than the prediction of velocity). For example, a geomorphologist

might be interested in comparing the flow resistance in a river as she moves along the channel

from the headwaters to the mouth. Measured data required to compute Darcy-Weisbach ff and

Manning n are flow depth, water-surface-slope, and flow velocity at the downstream sequence of

measurement stations; substitution in equations (3.5) and (3.7) yields the desired result.

It is important to be reminded, however, that calculating Manning n from direct measurements

yields a roughness factor that reflects all sources of flow resistance and not simply channel

roughness. In this case Manning n is simply a coefficient of proportionality in equation (3.7); in

a sense it is an equivalent roughness corresponding to the resistance conditions in the channel.

For example, it is entirely possible that a channel with a very smooth boundary (low roughness

factor, n) but which is meandering tortuously across its floodplain has a higher Manning n than a

channel with a rougher boundary but a straight channel. The high flow resistance caused by

meandering translates into a higher value for the roughness factor than the smooth channel

boundary by itself would suggest.

Estimating Darcy-Weisbach ff from grain size is a procedure commonly adopted where

there is no possibility of obtaining direct measurements and also where we can be confident that

most of the flow resistance is related to the roughness of the channel boundary. Most river

scientists regard skin resistance as the dominant source of flow resistance in medium to large-

sized rivers. There is less certainty about the more complex small steep headwater channels.

Grain roughness is mainly a function of relative roughness (d/D or Rh/D). A widely used version

applicable to open-channels is the Wolman equation. The Wolman equation is a semi-empirical

correlation between ff and the relative roughness, y/D84 (y = flow depth, d, or hydraulic radius,

Rh):

€

1ff

= 2log yD84

+1.0 (3.8)

I say semi-empirical because the structure of equation (3.8) does have some theoretical basis in

early studies of pipe flow although the details need not concern us here. Figure 3.10 shows a

graph of a very similar updated function derived by David Knighton and presented in his 1998


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textbook (the published equation below actually is incorrect and here the correct version appears

above the original in bold).

Figure 3.10: Relationship between friction factor (ff) and relative roughness (from Knighton, 1998).

For Rh/D84>1.0 the Wolman and Knighton equations yield sensibly identical results.

Estimating Manning n from grain size is also based on an empirical correlation, in this

case known as the Strickler equation:

n = 0.0151D501/6 where D50 is in mm (3.9A)

and n = 0.0478D501/6 where D50 is in metres (3.9B)

The Strickler equation was developed in the European Alps and applies ideally to gravel-bed

rivers where grain roughness is the primary source of channel roughness. In any case, it yields a

minimum estimate of Manning n.

€

1ff

= 0.82ln(4.35RD84)


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So, for D50 = 2.5 cm (or 0.025 m) equation (3.9B) yields: n = 0.0478(0.025)1/6 = 0.026.

Combining equation (3.9B) with the Manning equation yields:

€

v =d2 / 3s1/ 2

0.0478D501/ 6 (3.10)

Equation (3.10) is commonly employed today by river engineers in need of flow estimates in

ungauged gravel-bed rivers.

Estimating Manning n from descriptive tables and photographs

Descriptive tables of typical ranges of Manning n, like those shown in Figure 3.11, are

commonly used by river scientists and engineers. Figure 3.11 lists typical values of Manning n

for three general types of channel: artificial channels and canals, small streams where the

bankfull width is less than 35 m, and large rivers where the bankfull width exceeds 35 m. You

will note that sandbed rivers with their complex response to bedforms get special treatment here.

A fourth category of channel condition, floodplain surfaces, provides guidance to Manning n in

overbank flows. Although such tables are a useful guide to Manning n, especially to the novice

assessor, their effective use requires considerable judgment. Their use is usually combined with

photographs of river channels for which Manning n has been established by measurement. One

such very useful publication developed for agencies in the U.S. is Barnes (1967). Two sets of

plates from that publication are shown in Figure 3.12.

The Cowan Method of estimating Manning n is named for its American originator

(Cowan, 1956) and is designed to allow an integration of estimates of the resistance components.

Cowan’s operational formula is:

n = (n0+n1+n2+n3+n4)m (3.11)

where each term is defined in Figure 3.14. The term n0 assigns a roughness based on boundary

materials alone. The term n1 is an estimate of the effect of within cross-sectional irregularity,

negligible for a very regular shape (trapezoidal or semi-circular, for example) and up to an

additional 0.20 roughness units for channels that have severe boundary irregularities such as

those caused by bank slumping. The term n2 also refers to channel irregularity but this time for

changes along the channel. If the changes along the channel are minor or gradual no adjustment


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Condition of channel Typical magnitude range of Manning n

Minimum Normal Maximum A: Artificial channels and canals 1. Smooth concrete: 0.012 2. Ordinary concrete lining: 0.013 3. Shot concrete, untroweled, and earth channels in best condition: 0.015 4. Straight unlined earth canals in good condition: 0.020

B: Small streams (bankfull width <35 m) (a) Low-slope streams (on plains)

1. Clean, straight, bankfull stage, no deep pools; 0.025 0.030 0.033 2. Same as above but more gravel and weeds: 0.030 0.035 0.040 3. Clean, winding, some pools and shoals: 0.033 0.040 0.045 4. Same as above, but some weed and gravel: 0.035 0.045 0.050 5. Same as above but at less than bankfull stage: 0.040 0.048 0.055 6. Same as above but with more gravel present: 0.045 0.050 0.060 7. Sluggish reaches, weedy, deep pools: 0.050 0.070 0.080 8. Very weedy reaches, deep pools; or floodways with a heavy stand of timber: 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: 0.017 0.028 (b) a bed of dunes: 0.018 0.035 2. Near critical or transitional flow over washed-out dunes: 0.014 0.024 3. Upper regime flow (F>1.0) with (a) a plane bed: 0.011 0.015 (b) standing waves: 0.012 0.016 (c) antidunes: 0.012 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: 0.030 0.040 0.050 2. Bed of cobbles with large boulders: 0.040 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: 0.025 0.060 2. Irregular and rough boundary: 0.035 0.100

D: Floodplain surfaces (a) Pasture, no brush Short to high grass: 0.025 0.035 0.050 (b) Cultivated areas 1. No crop 0.020 0.030 0.040 2. Mature row and field crops: 0.025 0.035 0.050 (c) Brush 1. Scattered brush and heavy weeds: 0.035 0.050 0.070 2. Light brush and trees in summer (full foliage): 0.040 0.060 0.080 3. Medium to dense brush in summer (full foliage): 0.070 0.100 0.160 (d) Trees 1. Dense willows in summer: 0.110 0.150 0.200 2. Heavy stand of timber, a few down trees, undergrowth: 0.080 0.100 0.120

Figure 3.11: 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).


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Figure 3.12: Photographs from a field manual designed to assist a field technician in estimating Manning n (from Barnes, 1967)

is necessary but severe alternations of narrow and wide sections, for example, might add up to

0.015 roughness units to the total estimate of Manning n. Term n3 recognizes the effects of

individual obstructions (a large boulder, a fallen tree, grounded logs, etc). Obstructions assessed

here must not have been included in n1 or n2 (that is, no double counting of effects). Severe

obstructions can add as much as 0.06 roughness units to the total roughness. The term n4

recognizes the effects of vegetation in the channel and as Figure 3.14 illustrates, vegetation type

Manning n = 0.026

Manning n = 0.059


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no Smooth alluvial boundary.............................................................................................. 0.020 Rock-cut boundary.................................................................................................... 0.025 Boundary Fine gravel ..................................................................................................................... 0.024 materials Coarse gravel boundary................................................................................................... 0.028

n1 Smooth: best attainable for the given materials;.............................................................. 0.000 Degree of slightly eroded banks.................................................................................................. 0.005 channel Moderate: Comparable with dredged channel in fair to poor condition; cross-sectional some minor bank slumping and erosion..................................................................... 0.010 irregularity Severe: Extensive bank slumps and moderate bank erosion; jagged irregular rock-cut materials............................................................................. 0.020

n2 Gradual: Changes in size and shape are gradual............................................................. 0.000 Occasional alternation: Large and small sections alternate Variation in occasionally or shape changes to cause occasional channel shifting of flow from side to side................................................................................. 0.005 cross-section Frequent alternation: Large & small sections alternate or shape changes shape and area cause frequent shifting of flow from side to side………............................... 0.010 to 0.015

Negligible........... 0.00 n3 Determination of n3 is based on the presence and characteristics of obstructions such as debris, slumps, 0.010 stumps, exposed roots, boulders and fallen and lodged Minor ............to 0.015 Relative logs. Conditions considered in other steps must not be effect of reevaluated (double counted) in this determination. In 0.020 obstructions judging the relative effect of obstructions, consider the Appreciable….to 0.030 extent to which the obstructions occupy or reduce average water area; the shape (sharp or smooth) and 0.040 position and spacing of the obstructions. 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 n4 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, Vegetation 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 Minor: Sinuosity index = 1.0 to1.2............................................................................... 1.00 Degree of Appreciable: Sinuosity index = 1.2 to 1.5.................................................................... 1.15 Meandering Severe: Sinuosity index >1.5........................................................................................ 1.30

Figure 3.13: The Cowan procedure for estimating Manning n [(n = n0+n1+n2+n3+n4)m].


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can greatly influence the overall roughness. Grasses and supple seedlings (which likely lay

down in streamlined fashion in the flow) can add up to about 0.02 roughness units to the total

roughness but bushes and brush (that more likely will stand up in the flow) can add as much as

1.00 roughness units to the total roughness.

An adjustment is also made in the Cowan procedure for the degree of channel meandering,

increasing the total bracketed terms in equation (3.11) by as much as 30% where the sinuosity

index exceeds 1.5. The sinuosity index is simply the ratio of the actual channel length between

two points along the channel and the straight-line distance between them.

Like learning to play the piano or the flute, practice makes perfect here as well! River scientists

and engineers who routinely assess Manning n on rivers quickly achieve a remarkable degree of

accuracy in assigning roughness factors in their work. Experience clearly counts for a lot in this

endeavour and you too will find that your estimates of Manning n will greatly improve with

repetition.

7. Problem-solving with basic hydraulic equations

Several problems are presented below to illustrate solutions based on relationships developed in

this chapter.

Problem 1: What is the average shear stress on the bed of Fraser River with its slope of

1 m in 10 km and a flow depth of 12.0 m?

Solution 1: From equation (3.3) we can say that

τo = ρgds = γds

Since ρ = 1000 kg/m3, g = 9.81 ms-2, and s = 1/10 000 = 0.0001

τo = (1000)(9.81)(12.0)(0.0001) = 11.8 Nm-2

Problem 2: A sensibly straight and rectangular channel has a smooth bed of fine to

medium gravel. Calculate the discharge if the bed slope is 0.0005, flow depth is 2.00 m

and the channel width is 150 m.

Solution 2: From continuity we know that Q = Av = (150)(2.00)v = 300v


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From the Manning equation,

€

v =d2 / 3s1/ 2

n=22/ 3(0.0005)1/ 2

n

From the table of Manning n we note that, for fine to medium gravel,

n = 0.03. So

€

Q =300(1.59)(0.0224)

0.03= 356m3s−1

Note that we could also have used the Strickler equation, n = 0.0151D501/6 (mm)

to estimate Manning n: n = 0.0151(25)1/6 = 0.03

Problem 3: A river channel with a rectangular cross-section has 3.00 m-high banks, is

200.00 m wide, and has a slope of 0.0001. Will a discharge of 1000.0 m3s-1 cause

flooding if Mannings n = 0.03?

Solution 3: From continuity (Q = wdv) we know that 1000.0 = 200dv and

€

v =1000200d

=5d

From the Manning equation,

€

v =d2 / 3s1/ 2

n=d2 / 3(0.0001)1/ 2

0.03= 0.33d2 / 3

Substituting for v we can say that

€

5d

= 0.33d2 / 3

5 = 0.33d5 / 3

d =50.33

3 / 5

d = 5.11m

It follows that the 1000 m3s-1 discharge cannot be contained by a channel of this size and

that overbank flooding will occur.

8. What Determines the Shape of a River Channel? Although we don’t have quite enough theory yet to deal with this question analytically here in

Chapter 3, our discussion of equilibrium channels in Chapter 2 provides a hint about how we

might go about this kind of analysis. If we could define the critical conditions for moving the

material forming the bed of a river channel we could combine this condition with the hydraulics

we have learned in this chapter to resolve the problem. We will return to this issue in Chapter 4.


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Meanwhile, there are some empirical studies that point clearly to the role of bank strength in

determining the form ratio of an alluvial channel, the most famous of which is the work of

Professor Stanley Schumm at the State University of Colorado at Fort Collins. He reasoned that,

because very sandy channels have very low banks (because sandy sediment is not strong enough

to form high vertical bank sections), sandbed channels are always very wide relative to their

small depths. Conversely, streams with mud-rich banks have strong cohesive sediments and can

be much deeper (and therefore narrower) than their sandy counterparts carrying the same

discharge. Schumm formalized these general relationships in an empirical model based on

sampling channel sediments in rivers in the American Midwest (Figure 3.14).

Figure 3.14: Relationship between channel form ratio (w/d) and

the silt-clay content of the channel boundary based on the work of Professor Stan Schumm (1960).

Figure 3.14 shows the strong inverse relationship between the shape of a river channel expressed

as the form ratio (w/d) and the strength of the boundary materials expressed as the percentage of

silt/clay that they contain. Channels consisting of sediments with about 50% or more silt-clay

have form ratios less than 10 (deep and narrow) while those formed in sediments with less than

about 5% silt clay have form ratios greater than 50 (wide and shallow).


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9. Some Further Reading

I suggest that you limit your further reading (beyond these notes) of more advanced material to

the following section of Knighton (1998): Knighton, D. 1998: Fluvial Forms and Processes: A New Perspective, Hodder Arnold Publication: pages 96-107.

Other cited references: Cowan, W. L., 1956: Estimating hydraulic roughness coefficients, Agricultural Engineering 37(7), 473-475. Barnes,H.H., 1967: Roughness characteristics of natural channels. Water Supply Paper 1894, US Geological Survey, Washington, DC, 213 pp. Schumm, S.A., 1960: The shape of alluvial channels in relation to sediment type. United States Geological Survey Professional Paper 352B, 17-30. Simons, D.B. and Richardson, E.V., 1966: Resistance to flow in alluvial channels. US Geological Survey Paper 422J.

10. What’s Next?

Now that we have mastered some basic concepts of open-channel fluid mechanics and how they

can be used to determine the size of a river channel, we need to turn our attention to the role that

sediment transport plays in river geomorphology. Once we have added ideas relating to

sediment entrainment to our toolbag we can take a more analytical approach to answering the

question: what controls the shape of a river channel? These important topics are the subject of

Chapter 4.

chapter 3 river hydraulics and the channel

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