Loading...

PPT – HYDRAULICS OF MOUNTAIN RIVERS PowerPoint presentation | free to download - id: 86fbc-MGVlZ

The Adobe Flash plugin is needed to view this content

HYDRAULICS OF MOUNTAIN RIVERS Gary Parker,

University of Illinois

River in Taiwan Image courtesy C. Stark

TOPICS COVERED

- This lecture is not intended to provide a full

treatment of open channel flow. Nearly all

undergraduate texts in fluid mechanics for civil

engineers have sections on open channel flow

(e.g. Crowe et al., 2001). Three texts that

specifically focus on open channel flow are those

by Henderson (1966), Chaudhry (1993) and Jain

(2000). - Topics treated here include
- Approximations for the channel
- Shields number, Einstein number, generic

bedload equation - Boundary resistance in mountain streams Chezy

and Manning-Strickler forms - Skin friction and form drag in mountain rivers
- Backwater and the backwater length
- Normal (steady, uniform) flow
- Calculations of flow and sediment transport

using the normal flow assumption

SIMPLIFICATION OF CHANNEL CROSS-SECTIONAL SHAPE

River channel cross sections have complicated

shapes. In a 1D analysis, it is appropriate to

approximate the shape as a rectangle, so that B

denotes channel width and H denotes channel depth

(reflecting the cross-sectionally averaged depth

of the actual cross-section). As was seen the

lecture on hydraulic geometry, natural channels

are generally wide in the sense that Hbf/Bbf ltlt

1, where the subscript bf denotes bankfull.

As a result the hydraulic radius Rh is usually

approximated reasonably accurately by the average

depth. In terms of a rectangular channel,

THE SHIELDS NUMBER A KEY DIMENSIONLESS PARAMETER

QUANTIFYING SEDIMENT MOBILITY

?b boundary shear stress at the bed ( bed drag

force acting on the flow per unit bed area)

M/L/T2 ?c Coulomb coefficient of resistance

of a granule on a granular bed 1 D

characteristic grain size (e.g. surface median

size Ds50) Recalling that R (?s/?) 1, the

Shields Number ? is defined as

It can be interpreted as scaling the ratio

impelling force of flow drag acting on a bed

particle to the Coulomb force resisting motion

acting on the same particle, so that

The characterization of bed mobility thus

requires a quantification of boundary shear

stress at the bed.

THRESHOLD OF MOTION IN MOUNTAIN STREAMS

The threshold of motion is often expressed in

terms of a Shields curve (Shields, 1936). Let

denote the value of ? at the threshold of

motion, and Rep denote grain Reynolds number,

defined in the lecture on hydraulic geometry

as Based Neills (1968) work on coarse

sedimentg, Parker et al. (2003) amended the

Brownlie (1981) fit of the original Shields curve

to the form The asymptotic value of for

large Rep, i.e. coarse sediment, is 0.03.

Consider the case of quartz (R 1.65) in water

at 20?C (? 1x10-6 m2/s). The smallest value of

Ds50 for the data set introduced in the lecture

on hydraulic geometry is 27 mm, in which case Rep

17,850 and 0.0289. So an approximate

value of of 0.03 is appropriate for most

coarse-bedded mountain streams. In point of

fact, there is no sharply-defined threshold of

motion. The value of 0.03 should be interpreted

to be a value below which the bedload transport

rate is morphodynamically insignificant, not

precisely 0.

SHIELDS NUMBER AT BANKFULL FLOW IN MOUNTAIN

STREAMS

It will be shown in Slide 26 that for the case of

mountain streams the Shields number at bankfull

flow can be estimated as where all parameters

were defined in the lecture on hydraulic

geometry. A plot of

versus is given to the right. The

data are those from the lecture on hydraulic

geometry. The average value of is

0.0486, i.e about 1.62 times the Shields number

at the threshold of motion. Most alluvial

gravel-bed streams move size Ds50 at bankfull

flow.

THE EINSTEIN NUMBER A DIMENSIONLESS

QUANTIFICATION OF BEDLOAD TRANSPORT RATE

qb volume bedload transport rate per unit width

L2/T D characteristic grain size L R

(?s/?) 1 ? 1.65 for natural sediment 1

One standard approach to the quantification of

bedload transport is the specification of the

functional form

An example appropriate for the bedload transport

of gravel of uniform size is the modified form

of the bedload transport equation of Meyer-Peter

and Müller (1948) by Wong (2003) In field

gravel-bed rivers, however, a) the gravel is

rarely uniform and b) gravel is commonly

transported at Shields numbers below 0.0495 (see

previous slide).

BOUNDARY RESISTANCE IN MOUNTAIN STREAMS

Let Q flow discharge L/T B water surface

width L H cross-sectionally averaged depth

L U Q/(BH) cross-sectionally averaged flow

velocity L u (?b/?)1/2 shear velocity

L/T Two dimensionless bed resistance

coefficients are defined here the Chezy

resistance coefficient Cz given as and the

standard bed resistance coefficient Cf ( f/8,

where f denotes the Darcy-Weisbach resistance

coefficient) Note that as the bed shear

stress increases, Cf increases and Cz decreases.

RESISTANCE RELATIONS FOR HYDRAULICALLY ROUGH FLOW

Mountain streams are almost invariably in the

range of hydraulically rough flow, for which

resistance becomes independent of kinematic

viscosity ?. Keulegan (1938) offered the

following relation for hydraulically rough flow.

where ks a roughness height characterizing the

bumpiness of the bed L. A close approximation

is offered by the Manning-Strickler formulation

Parker (1991) suggested a value of ?r of 8.1 for

gravel-bed streams.

The roughness height over a flat bed of coarse

grains (no bedforms) is given as

where Ds90 denotes the surface sediment size such

that 90 percent of the surface material is finer,

and nk is a dimensionless number between 1.5 and

3. For example, Kamphuis (1974) evaluated nk as

equal to 2.

COMPARISION OF KEULEGAN AND MANNING-STRICKLER

RELATIONS ?r 8.1

Note that Cz does not vary strongly with depth.

It is often approximated as a constant in

broad-brush calculations.

TEST OF RESISTANCE RELATION AGAINST MOBILE-BED

DATA WITHOUT BEDFORMS FROM LABORATORY FLUMES

The data in question are from all the experiments

without bedforms used by Meyer-Peter and Müller

(1948) to develop their bedload relation.

Here Rb denotes the bed component of hydraulic

radius Rh rather than depth the flumes were too

narrow to allow the approximation Rh ? H.

FORM DRAG AND SKIN FRICTION

The formulas of the previous slide hold only for

flat, coarse granular beds. River beds are

rarely flat. Lowland sand-bed streams typically

contain dunes. Dunes are not common in mountain

streams, but such bedforms as bars, pool-riffle

sequences and step-pool sequences are common.

All such features, as well as planform

irregularity, contributed added resistance, so Cz

is usually lower, and Cf is usually higher, than

predicted by the equation of the previous slide.

Step-pool pattern in the Hiyamizudani river,

Japan. Cour. K. Hasegawa

Bars in the Rhine River, Switzerland. Cour. M.

Jaeggi

FORM DRAG AND SKIN FRICTION contd.

The component of the resistance coefficient Cfs

due to the flow acting on the grains themselves

is known as skin friction, and the extra

component Cff is denoted as form drag (due to

bedforms), to that the total resistance

coefficient Cf is given as

Cfs can be computed from the relations of Slides

10 and 11. In bar-dominated mountain streams,

form-drag is prominent at lower flows, but is

muted at flood flows (see images to the right).

In steeper streams with pool-riffle patterns, and

in particular step-pool patterns, form drag is

likely significant even at flood flows.

Elbow River, Alberta, Canada at low flow and

100-year flood. Cour. Alberta Research Council

BACKWATER

If (most) alluvial streams are disturbed at a

point, the effect of that disturbance tends to

propagate upstream. For example, the effect of a

lake (slowing the flow down) or a waterfall

(speeding the flow up) is felt upstream, as

illustrated below.

Backwater from a lake

Backwater from a waterfall

Backwater effects are mediated by a dimensionless

number known as the Froude number Fr, where

BACKWATER contd.

For most alluvial mountain streams, the Froude

number at bankfull flow Frbf satisfies the

condition As seen from the lecture on hydraulic

geometry. For the case Fr lt 1 backwater effects

propagate upstream, so that the effect of a

disturbance is felt upstream of it.

BACKWATER contd.

Supercritical flow (Fr gt 1) does occur in very

steep mountain streams, and in particular streams

with step-pool patterns and bedrock streams. In

the case of a supercritical flow the effect of a

disturbance propagates downstream rather than

upstream.

Stream in the interior of British Columbia,

Canada. Cour. B. Eaton.

Sustained supercritical flow over an alluvial or

bedrock bed is unstable, and usually devolves

into a series of steps punctuated by hydraulic

jumps at formative flow.

Dry Meadow Creek, Calif., USA. Cour. M. Neumann

THE BACKWATER LENGTH

The characteristic distance Lb upstream (in the

case of the more usual subcritical flow) or

downstream (in the case of supercritical flow) to

which the effect of a disturbance is felt is

known as the backwater length. Let Hd the flow

depth at the disturbance L S down-channel

slope of the river 1. The backwater length is

then given as Taking Hb to scale with Hbf,

some estimates of the backwater length are given

below. Estimates of the backwater length

obtained in this way average to 2.1 km, 1.1 km,

0.2 km, and 5.2 km for the data sets for Alberta,

Britain, Idaho and Colorado introduced in the

lecture on hydraulic geometry. That is,

backwater lengths tend to be very short in

mountain streams.

NORMAL FLOW

Normal flow is an equilibrium state defined by a

perfect balance between the downstream

gravitational impelling force and resistive bed

force, in the absence of any perturbation due to

backwater. The resulting flow is constant in

time and in the downstream, or x direction. The

approximation of normal flow is often a very good

one in mountain streams.

- Parameters
- x downstream coordinate L
- H flow depth L
- U flow velocity L/T
- qw water discharge per unit width L2T-1
- B width L
- Qw qwB water discharge L3/T
- g acceleration of gravity L/T2
- bed angle 1
- tb bed boundary shear stress M/L/T2
- S tan? streamwise bed slope 1
- (cos ? ? 1 sin ? ? tan ? ? S)
- water density M/L3

As can be seen from the lecture on hydraulic

geometry, the bed slope S of most river, even

most mountain rivers, is sufficiently small to

allow the approximations

NORMAL FLOW contd.

Conservation of water mass ( conservation of

water volume as water can be treated as

incompressible)

Conservation of downstream momentum Impelling

force (downstream component of weight of water)

resistive force

Reduce to obtain depth-slope product rule for

normal flow

CHEZY RESISTANCE COEFFICIENT AT BANKFULL FLOW

Using the normal flow approximation, it is found

between the relations

that Cz can be estimated as

Regression of all four data sets

This is how Czbf was estimated in the lecture on

hydraulic geometry, as shown to the right. If Cz

can be estimaged, the flow velocity U is then

given as This relation is known as Chezys law.

MANNING-STRICKLER RESISTANCE RELATION FOR PLANE

BED

Using the normal flow approximation and the form

for Cz for a plane bed in the absence of bedforms

given in Slide 11, it is found that or solving

for U, This is known as a Manning-Strickler

resistance relation, where Mannings n (a

parameter that should be relegated to the dustbin

due to its perverse dimensions) is given

as (But you must remember to use MKS units

for n, whereas the equation for U works for any

consistent set of units).

MANNING-STRICKLER RESISTANCE RELATION FOR

MOUNTAIN RIVERS AT BANKFULL FLOW

The regression of the data of Slide 21 for

mountain rivers at bankfull flow yields the

relation where Ubf Qbf/(HbfBbf), or

thus This represents a generalized

Manning-Strickler relation for mountain streams.

FORM DRAG VERSUS SKIN FRICTION AT BANKFULL FLOW

In order to compare form drag versus skin

friction at bankfull flow, it is necessary to

estimate the roughness height ks. Here the

Kamphuis (1974) relation Is used in conjunction

with the reasonable estimate Then defining

Cf,bf, Cfs,bf and Cff,bf as the values of the

total resistance coefficient, the resistance

coefficient due to skin friction and the

resistance coefficient due to form drag,

respectively at bankfull flow, it is found

that The fraction of resistance that is form

drag Fform is thus given as

FORM DRAG VERSUS SKIN FRICTION AT BANKFULL FLOW

contd.

The fraction of resistance that is form drag at

bankfull flow in mountain streams is less than

0.5. The fraction is 0.2 0.3 in relatively deep

mountain streams (Hbf/Ds50) gt 20, but can be

above 0.3 in relatively shallow mountain streams.

Deeper streams tend to have lower slopes, and

shallower streams tend to have higher slopes, as

shown in the next slide.

SHIELDS NUMBER AT BANKFULL FLOW USING THE NORMAL

FLOW ASSUMPTION

Using the definition of the Shields number ? and

the estimate for bed shear stress ?b from the

normal flow approximation,

the following estimate is obtained for the

Shields number at bankfull flow This is the

origin of the estimate of Shields number used in

the chapter on hydraulic geometry and in Slide 7

of this lecture. A crude approximation of the

plot to the right yields so that S decreases as

Hbf/Ds50 increases.

CALCULATING THE FLOW AT NORMAL EQUILIBRIUM

CHEZY FORMULATION

Between the relations

it can be shown that

Thus if the water discharge per unit width qw,

down-channel bed slope S, characteristic bed

grain size D and submerged specific gravity R are

known, and if the Chezy resistance coefficient Cz

can be estimated, the flow depth H, flow velocity

U, bed shear stress ?b and Shields number ? can

be computed as indicated above.

CHEZY ? MANNING-STRICKLER

For the case of plane-bed rough flow, the

following formulation for resistance was given in

Slide 10

The corresponding relation based on data for

mountain rivers at bankfull flow is (Slide

20) Assuming that ks 2 Ds90 and Ds90 3

Ds50, the above relation can be cast into the form

Both relations can be cast in terms of a

generalized Manning-Strickler formulation, such

that

CALCULATING THE FLOW AT NORMAL EQUILIBRIUM

MANNING-STRICKLER FORMULATION

Consider a generalized Manning-Strickler

resistance relation of the form where for

example ?g can be estimated as 5.92, ng can be

estimated as 0.210 and ks can be estimated as

2Ds90 for mountain gravel-bed streams at flood

flows (Slide 27). The relations for H, U, ?b and

? now become where

CALCULATING BEDLOAD TRANSPORT AT NORMAL

EQUILIBRIUM

For no particularly good reason, most

formulations of bedload transport in gravel-bed

streams have ignored form drag. For the sake of

illustration, we do so here. Consider a

flume-like river with no form drag and

containing uniform gravel of size D, roughness

height ks ( 2D) and submerged specific gravity

R. The reach has bed slope S, and is conveying

water discharge per unit width qw. For this case

it is reasonable to assume ng 1/6 and ?g 8.1,

i.e. the relation of Slide 10. The Shields

number can be computed from the previous slide as

and the volume bedload transport rate per

unit width q can be estimated from Slide 7 as

MANNING-STRICKLER STANDARD CASE OF ng 1/6

In the case of an exponent ng of 1/6 (the

standard Manning-Strickler exponent of Slide 10),

the relevant relations reduce to

REFERENCES

Brownlie, W. R., 1981, Prediction of flow depth

and sediment discharge in open channels, Report

No. KH-R-43A, W. M. Keck Laboratory of Hydraulics

and Water Resources, California Institute of

Technology, Pasadena, California, USA, 232

p. Chaudhry, M. H., 1993, Open-Channel Flow,

Prentice-Hall, Englewood Cliffs, 483 p. Crowe, C.

T., Elger, D. F. and Robertson, J. A., 2001,

Engineering Fluid Mechanics, John Wiley and sons,

New York, 7th Edition, 714 p. Gilbert, G.K.,

1914, Transportation of Debris by Running Water,

Professional Paper 86, U.S. Geological

Survey. Jain, S. C., 2000, Open-Channel Flow,

John Wiley and Sons, New York, 344 p. Kamphuis,

J. W., 1974, Determination of sand roughness for

fixed beds, Journal of Hydraulic Research, 12(2)

193-202. Keulegan, G. H., 1938, Laws of turbulent

flow in open channels, National Bureau of

Standards Research Paper RP 1151, USA. Henderson,

F. M., 1966, Open Channel Flow, Macmillan, New

York, 522 p. Meyer-Peter, E., Favre, H. and

Einstein, H.A., 1934, Neuere Versuchsresultate

über den Geschiebetrieb, Schweizerische

Bauzeitung, E.T.H., 103(13), Zurich,

Switzerland. Meyer-Peter, E. and Müller, R.,

1948, Formulas for Bed-Load Transport,

Proceedings, 2nd Congress, International

Association of Hydraulic Research, Stockholm

39-64. Neill, C. R., 1968, A reexamination of the

beginning of movement for coarse granular bed

materials, Report INT 68, Hydraulics Research

Station, Wallingford, England. Parker, G., 1991,

Selective sorting and abrasion of river gravel.

II Applications, Journal of Hydraulic

Engineering, 117(2) 150-171.

REFERENCES

Parker, G., Toro-Escobar, C. M., Ramey, M. and S.

Beck, 2003, The effect of floodwater extraction

on the morphology of mountain streams, Journal of

Hydraulic Engineering, 129(11), 885-895. Shields,

I. A., 1936, Anwendung der ahnlichkeitmechanik

und der turbulenzforschung auf die

gescheibebewegung, Mitt. Preuss Ver.-Anst., 26,

Berlin, Germany. Vanoni, V.A., 1975,

Sedimentation Engineering, ASCE Manuals and

Reports on Engineering Practice No. 54, American

Society of Civil Engineers (ASCE), New York.

Wong, M., 2003, Does the bedload equation of

Meyer-Peter and Müller fit its own data?,

Proceedings, 30th Congress, International

Association of Hydraulic Research, Thessaloniki,

J.F.K. Competition Volume 73-80.

For more information see Gary Parkers e-book 1D

Morphodynamics of Rivers and Turbidity Currents

http//cee.uiuc.edu/people/parkerg/morphodynamics

_e-book.htm