National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

HYDRAULICS OF MOUNTAIN RIVERS

Gary Parker, University of Illinois River in Taiwan: Image courtesy C. Stark 1

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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 2

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 channel floodplain 100 floodplain 10 H B 1 0.0001

0.001

0.01

0.1

S

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 H bf /B bf << 1, where the subscript “bf” denotes “bankfull”. As a result the hydraulic radius R h is usually approximated reasonably accurately by the average depth. In terms of a rectangular channel, R h  HB B  2 H   1 H 2 H B  H 3 Alta Brit Ida Colo

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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  c bed area) [M/L/T 2 ] = Coulomb coefficient of resistance of a granule on a granular bed [1] D = characteristic grain size (e.g. surface median size D s50 ) Recalling that R = (  s /  ) – 1, the Shields Number  *      b R gD 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   ~  b D 2 3  c 4 3  R g D 2 The characterization of bed mobility thus requires a quantification of boundary shear stress at the bed.

4

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

THRESHOLD OF MOTION IN MOUNTAIN STREAMS

The threshold of motion is often expressed in terms of a Shields curve (Shields,   c  * at the threshold of motion, and

Re

grain Reynolds number, defined in the lecture on hydraulic geometry as p denote

Re

p 

RgD



D

Based Neill’s (1968) work on coarse sedimentg, Parker et al. (2003) amended the Brownlie (1981) fit of the original Shields curve to the form   c  0 .

5 [ 0 .

22

Re

 0 .

6 p  0 .

06  10 (  7 .

7

Re

 0 .

6 p ) ] The asymptotic value of for large

Re

p , i.e. coarse sediment, is 0.03. Consider the case of quartz (R = 1.65) in water at 20  C (  = 1x10 -6 m 2 /s). The smallest value of D s50 for the data set introduced in the lecture on hydraulic geometry is 27 mm, in which case

Re

p   c   c 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 5 morphodynamically insignificant, not precisely 0.

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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   bf 50  H bf S R D s 50 where all parameters were defined in the lecture on hydraulic geometry. A plot of   bf 50 versus

Bankfull Shields Number versus Dimensionless Discharge

 Q bf gD s 50 D 2 s 50 is given to the right. The data are those from the lecture on hydraulic geometry. The average value of is 0.0486, bf 50 i.e about 1.62 times the Shields number at the threshold of motion. Most alluvial gravel-bed streams move size D bankfull flow.

s50 at 1 0.1

0.01

0.001

1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 Alta Brit Ida Colo average threshold 6

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

THE EINSTEIN NUMBER: A DIMENSIONLESS QUANTIFICATION OF BEDLOAD TRANSPORT RATE

q b = volume bedload transport rate per unit width [L D = characteristic grain size [L] R = (  s /  ) – 1  1.65 for natural sediment [1] 2 /T] q   q b R gD D One standard approach to the quantification of bedload transport is the specification of the functional form q   f (   ) 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 M üller (1948) by Wong (2003): q    3   .

97 (    0 .

0495 ) 3 / 2 0 ,    ,   0 .

0495  0 .

0495 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).

7

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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 Cz  U u   C f  1 / 2 and the standard bed resistance coefficient C f D’arcy-Weisbach resistance coefficient); (= f/8, where f denotes the C f  u U  2    b U 2  ( Cz )  2 Note that as the bed shear stress increases, C f

decreases.

increases

and Cz 8

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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.

Cz  U u   C f  1 / 2  2 .

5  n   H 11 k s   where k s = a roughness height characterizing the bumpiness of the bed [L]. A close approximation is offered by the Manning-Strickler formulation: Cz  U u   C f  1 / 2   r   H k s   1 / 6 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 D s90 k s  n k D s 90 denotes the surface sediment size such that 90 percent of the surface material is finer, and n k is a dimensionless number between 1.5 and 3. For example, Kamphuis (1974) evaluated n k as equal to 2.

9

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

COMPARISION OF KEULEGAN AND MANNING-STRICKLER RELATIONS

 r = 8.1

100 Cz  8 .

1   H k s   1 / 6 10 1 1 10

H/k s

100 1000 Keulegan Parker Version of Manning Strickler Note that Cz does not vary strongly with depth. It is often approximated as a constant in broad brush calculations.

10

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

TEST OF RESISTANCE RELATION AGAINST MOBILE-BED DATA WITHOUT BEDFORMS FROM LABORATORY FLUMES

100.00

Cz  8 .

1   R b k s   1 / 6 The data in question are from all the experiments without bedforms used by Meyer-Peter and M üller (1948) to develop their bedload relation.

ETH 52 10.00

Gilbert 116 Parker Version of Manning Strickler 1.00

1.00

10.00

R b /k s

100.00

Here R b denotes the bed component of hydraulic radius R h rather than depth: the flumes were too narrow to allow the approximation R h  H.

11

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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 C z is usually lower, and C f usually higher, than predicted by the equation of the previous slide.

is Bars in the Rhine River, Switzerland. Cour. M. Jaeggi Step-pool pattern in the Hiyamizudani river, Japan. Cour. K. Hasegawa 12

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

FORM DRAG AND SKIN FRICTION contd.

The component of the resistance coefficient C fs due to the flow acting on the grains themselves is known as skin friction, and the extra component C ff is denoted as form drag (due to bedforms), to that the total resistance coefficient C f is given as

C

f 

C

f s 

C

f f C fs 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 13

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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

Fr

 U gH 14

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

BACKWATER contd.

For most alluvial mountain streams, the Froude number at bankfull flow

Fr

bf satisfies the condition

Fr

bf 

1 ( subcritica l flow )

As seen from the lecture on hydraulic geometry. For the case

Fr

< 1 backwater effects propagate

upstream

, so that the effect of a disturbance is felt upstream of it.

10 1 Alta Brit Ida Colo 0.1

0.0001

0.001

S

0.01

0.1

15

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

BACKWATER contd.

Supercritical flow (

Fr

> 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.

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

16

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

THE BACKWATER LENGTH

The characteristic distance L b 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 H d = the flow depth at the disturbance [L] S = down-channel slope of the river [1].

The backwater length is then given as L b  H d S Taking H b below.

River

to scale with H bf , some estimates of the backwater length are given South Fork Clearwater River, Idaho, USA Minnesota River, Wilmarth, USA Fly River, Kuambit, PNG

Bed

Gravel Sand Sand H bf (m) 1.06 4.6 9.45 S 0.0055 0.00019 0.000051 L b (m) 0.2 km 24.2 km 185.3 km 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

.

17

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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:  b B  x  x x = downstream coordinate [L] H = flow depth [L] U = flow velocity [L/T] q w = water discharge per unit width [L B = width [L] 2 T -1 ] Q w = q w B = water discharge [L 3 /T] g = acceleration of gravity [L/T 2 ]   S = tan   b = bed angle [1] = bed boundary shear stress [M/L/T 2 ] = streamwise bed slope [1] (cos   1; sin   = water density [M/L 3 ] tan   S) x  H B  gH  xBS 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 18 sin   tan   S , cos   1

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

NORMAL FLOW contd.

Conservation of water mass (= conservation of water volume as water can be treated as incompressible): q w  UH Q w  q w B  UHB Conservation of downstream momentum: Impelling force (downstream component of weight of water) = resistive force  gHB  x sin    gHB  xS   b B  x Reduce to obtain depth-slope product rule for normal flow:  b B  x  x  b   gHS x  H u   gHS B  gH  xBS 19

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

CHEZY RESISTANCE COEFFICIENT AT BANKFULL FLOW

Using the normal flow approximation, it is found between the relations Cz  U u  that Cz can be estimated as Cz  U gHS  BH Q gHS 100 u   gHS

Bankfull Chezy Number versus H bf /D s50

Regression of all four data sets: Cz bf  4 .

39   H bf D s 50   0 .

210 This is how Cz bf 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 10 U  Cz gHS This relation is known as

Chezy’s law

.

1 1 10

H bf /D s50

100 20 Alta Brit Ida Colo

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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 Cz  U gHS  8 .

1   H k s   1 / 6 or solving for U, U  8 .

1 k 1 / s 6 g H 2 / 3 S 1 / 2 This is known as a Manning Strickler resistance relation, where Manning’s “n” (a parameter that should be relegated to the dustbin due to its perverse dimensions) is given as 1  8 .

1 g n k 1 s / 6 (But you must remember to use MKS units for n, whereas the equation for U works for any consistent set of units).

21

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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 Cz bf  U bf gH bf S  4 .

39   H bf D s 50   0 .

210 where U bf = Q bf /(H bf B bf ), or thus U bf  4 .

39 D s 0 .

210 50 g H 0 bf .

710 S 1 / 2 This represents a generalized Manning-Strickler relation for mountain streams.

22

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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 k s . Here the Kamphuis (1974) relation k s  2 D s 90 Is used in conjunction with the reasonable estimate D s 90  3 D s 50 Then defining C f,bf , C fs,bf and C ff,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 C f , bf  Cz  bf 2  ( 4 .

39 )  2   H D s bf 50    0 .

420 Cf s , bf  ( ( 6 ) 1 / 8 .

1 ) 3 2   H D s 50    ( 1 / 3 ) Cf f , bf  C f , bf  C f s , bf The fraction of resistance that is form drag F form is thus given as F f orm  C f , bf  C f , bf C f s , bf 23

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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 (H bf /D s50 ) > 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.

1 0.1

1 10

H/D s50

100 24

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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,      R b gD  b   gHS the following estimate is obtained for the Shields number at bankfull flow:   bf 50  H bf S R D s 50 This is the origin of the estimate of Shields number used in the chapter on hydraulic geometry and in Slide 7 of this lecture.

1 0.1

0.01

Bankfull Shields Number versus Dimensionless Discharge

Alta Brit Ida Colo average threshold A crude approximation of the plot to the right yields   bf 50  0 .

049 0.001

1.0E+02 so that S decreases as H bf /D s50 increases.

1.0E+03 1.0E+04 1.0E+05

Qhat

1.0E+06 1.0E+07 25

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

CALCULATING THE FLOW AT NORMAL EQUILIBRIUM: CHEZY FORMULATION

Between the relations q w  UH , U  g C f H 1 / 2 S 1 / 2 it can be shown that  C z 2 / 3 H  q w Cz gS g H 1 / 2 S 1 / 2 ,  b   ( Cz ) 2 U 2 ,     b  R gD  HS R D U  ( Cz ) 2 / 3  q w gS  1 / 3  b   ( Cz )  2 / 3  q w gS  2 / 3    1 ( Cz ) 2 / 3   q 2 w g   1 / 3 S 2 / 3 R D Thus if the water discharge per unit width q w , 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.

26

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

CHEZY



MANNING-STRICKLER

For the case of plane-bed rough flow, the following formulation for resistance was given in Slide 10: Cz  8 .

 1  H k s   1 / 6 The corresponding relation based on data for mountain rivers at bankfull flow is (Slide 20) Cz  4 .

39   H D s 50   0 .

210 Assuming that k s form = 2 D s90 and D s90 = 3 D s50 , the above relation can be cast into the Cz  5 .

92   H k s   0 .

210 Both relations can be cast in terms of a generalized Manning-Strickler formulation, such that Cz   g   H k s   n g 27

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

CALCULATING THE FLOW AT NORMAL EQUILIBRIUM: MANNING-STRICKLER FORMULATION

Consider a generalized Manning-Strickler resistance relation of the form Cz   g   H k s   n g  b where for example  g can be estimated as 5.92, n g can be estimated as 0.210 and k s can be estimated as 2D s90 The relations for H, U,  b for mountain gravel-bed streams at flood flows (Slide 27). and  * now become H   k g n g s q w gS n ms U    w ( 1  n ms ) g k n g s gS n ms n ms   g [ 1  ( 1 / 2 ) n ms ] k n g s q w  g n ms S [ 1  ( 1 / 2 ) n ms ]    g  ( 1 / 2 ) n ms k s ( n g n ms ) q w  g S [ 1  ( 1 / 2 ) n ms ] RD where n ms  ( 2 / 3 ) [( 1  ( 2 / 3 ) n g ] 28

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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 k s (~ 2D) and submerged specific gravity R. The reach has bed slope S, and is conveying water discharge per unit width q w . For this case it is reasonable to assume n g = 1/6 and  g = 8.1, i.e. the relation of Slide 10. The Shields number can be computed from the previous slide as    g  ( 1 / 2 ) n ms k s ( n g n ms ) q w  g n ms S [ 1  ( 1 / 2 ) n ms ] RD n ms  ( 2 / 3 ) [( 1  ( 2 / 3 ) n g ] and the volume bedload transport rate per unit width q can be estimated from Slide 7 as q b  3 .

97 R gD D (    0 .

0495 ) 3 / 2 29

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

MANNING-STRICKLER: STANDARD CASE OF n g = 1/6:

In the case of an exponent n g of 1/6 (the standard Manning-Strickler exponent of Slide 10), the relevant relations reduce to: n ms  3 5 H  k 1 s / 6 q w  g gS 3 / 5 U    w 2 / 5  g k 1 / s gS 6 3 / 5  b   g 7 / 10 k 1 / 6 s  g q w 3 / 5 S 7 / 10    g  3 / 10 k 1 / s 10 q w  g 3 / 5 S 7 / 10 RD 30

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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, 7 th 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

, 2 nd 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.

Hydraulic Engineering

, 117(2): 150-171.

II: Applications,

National Center for Earth-surface Dynamics

Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005

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,

30 th Congress, International Association of Hydraulic Research, Thessaloniki, J.F.K. Competition Volume: 73-80.

For more information see Gary Parker’s e-book:

1D Morphodynamics of Rivers and Turbidity Currents

http://cee.uiuc.edu/people/parkerg/morphodynamics_e-book.htm

32