mean boundary shear stress mean velocity and flow …hickin/fluids/chapt4-mean flow.pdfedward j....

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

4.2

denotes 'at the boundary'). The relevant forces are (see Figure 4.1 for further definition

of terms):

A

L

p

θ

v

Wθ

θ

v

profile

bed

L

Ws

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)

or τo = ρg A

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).


4.3

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.

Solution: (a) In this rectangular channel, Rh = Ap =

2.00 x 100.00(2.00 +100.00 + 2.00) = 1.92 m. Figure 2.1 indicates

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.

(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 VoAb

sinθ = γRhsinθ, or for low slopes where sinθ = tanθ = s, τo = γRhs.

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).


4.4

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


4.5

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 = gaRhs (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:

ΔH = ƒ LD v2

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.


4.6

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)2πD = D/4 so that D = 4Rh. Rearranging equation (4.11)

and making these substitutions for ΔH/L and D give:

ƒ = D ΔHL 2g

v2 = 8gRhs

v2 ..........................................(4.12)

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 v = 8g

∂ Rhs ...............(4.13)

From equation (4.13) it now is clearly apparent that C = 8g∂ .................…….............(4.14)

Of course, both the Chezy and Darcy-Weisbach equations can also be written in terms of shear stress, giving:

C2 = gρ v2τo or C = gρv2

τo .........................................(4.15)

and ƒ = 8ρ τov2 ....................................................................................(4.16)

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: ƒ = 8v2 (v*)2


4.7

and by taking roots, vv* = 8

∂ .............................................…………...............(4.17)

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:

Re = inertial forcesviscous forces = vL

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


4.8

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

roughness element height and the thickness of the viscosity-dominated flow. In laminar 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


4.9

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)

2 000

1 000 10 000 100 000 1 000 000Reynold s number, Re

Eq. (4.20)

0.01

0.02

0.03

0.040.05

0.10

0.07 D/k

1014504252120

61.0

30.0

rela

tive

roug

hnes

s in

crea

sing

turbulent flowlaminar flow

Res

ista

nce

coef

ficie

nt,

ƒ

D

k

internal pipe geometry showing roughness elements

Figure 4.2: The Stanton diagram showing ƒ = (Re, D/k); after Rouse (1946).


4.10

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:

C = kRh1/6

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

n ............................................................(4.22)

A corresponding equation for discharge can be written:

Q = A Rh2/3s1/2

n = wd Rh2/3s1/2

n ...................................(4.23)

where A, w, and d, are respectively cross-sectional area, width, and mean depth, of the channel.


4.11

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.

Solution: From equation (3.88), Q = wd Rh2/3s1/2

n or 100 = 20d Rh2/3(0.0001)1/2

0.017 or 8.5 = dRh2/3

Since Rh = AP =

20d2d+20 , substituting for Rh yields 8.5 = d

20d

2d+20 2/3. Raising both sides of

the

equation by the power 3/2 yields: 24.782 = d3/2

20d

2d+20 or 49.563d + 495.631 = 20d5/2. Solving by

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


4.12

€

1f

= k8g

Rh1/6

n .....................................................(4.24)

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


4.13

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


4.14

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


4.15

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.


4.16

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 cross-sectional 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.


4.17

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.


4.18

Type and condition of channel Typical magnitude range of Manning's 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

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).


4.19

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


4.20

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

€

1f

= 18g

Rh1/6

n , equation (4.25) can

be written 18g

Rh1/6

n = 1.0 + 2.0 log (y/D84) .................................(4.27a)

or if Rh ≈ y, simplification leads to: n = y1/6

8.859(1.0+2.0log yD84

) ..........……............(4.27b)

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,

n = 0.0151D501/6 (for D50 in millimetres) ..................... (4.28a)

or 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.


4.21

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 cross-sectional condition; 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 and area shape changes 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

4.4: Determination of Manning's roughness coefficient by the Cowan procedure (after Chow, 1959).


4.22

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.

.01

.02

.03

.04

.05

.06

.07

.08

.09

0Flow depth, y (metres)

Man

ning

's n

Eqn 4.28; D = 0.00550Eqn 4.27; D = 0.0184

Eqn 4.28; D = 0.0550

Eqn 4.27; D = 0.1084Eqn 4.28; D = 0.2550Eqn 4.27; D = 0.0584

0.5 1.0 1.5 2.0 2.5 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.


4.23

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:

Q = wd y2/3s1/2

n = w y5/3s1/2

n or 300 = 100 y5/3(0.0005)1/2

n = 2.236 y5/3

n

Introducing the Strickler equation (n = 0.0478D501/6) here yields:

300 = 2.236 y5/3

0.0478D501/6 = 2.236 y5/3

0.0478(0.1)1/6

which simplifies to 300 = 2.236 y5/3

0.033 and further to 300 = 67.758 y5/3

So, y5/3 = 4.428 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

that v* = τ/ρ = gys = 0.14 so water surface slope must have been s = 0.0589.81x3 = 0.00067.

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

8.859(1.0+2.0logy

D84) ; log

yD84

= 1.3829 and D84 = 0.125 m or 125 mm.

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!


4.24

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) 473-475. Einstein, H.A. and Barbarossa, N.L., 1952, River channel roughness: American Society of Civil Engineers Transactions, 117, 1121-1146.


4.25

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.

mean boundary shear stress mean velocity and flow …hickin/fluids/chapt4-mean flow.pdfedward j....

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