highlights of open channel hydraulics and sediment transport

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1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to

RIVERS AND TURBIDITY CURRENTS© Gary Parker November, 2004

HIGHLIGHTS OF OPEN CHANNEL HYDRAULICS AND SEDIMENT TRANSPORT

Dam at Hiram Falls on the Saco River near Hiram, Maine, USA

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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 in Chapter 3, natural channels are generally wide in the sense that Hbf/Bbf << 1, where the subscript “bf” denotes “bankfull”. As a result the hydraulic radius Rh is usually approximated accurately by the average depth. In terms of a rectangular channel,

H

B

channel floodplainfloodplain

H

BH

21

H

H2B

HBRh

3



THE SHIELDS NUMBER:A KEY DIMENSIONLESS PARAMETER QUANTIFYING SEDIMENT MOBILITY

gDRb

3

c

2b

2g

34

~

DR

D

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]

Recalling that R = (s/) – 1, the Shields Number * is defined as

It can be interpreted as a ratio scaling the ratio impelling force of flow drag acting on a 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.

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QUANTIFICATION OF BOUNDARY SHEAR STRESS AT THE BED

2/1fC

u

UCz

U = cross-sectionally averaged flow velocity ( depth-averaged flow velocity in the wide channels studied here) [L/T]

u* = shear velocity [L/T]

Cf = dimensionless bed resistance coefficient [1]

Cz = dimensionless Chezy resistance coefficient [1]

2b

f UC

BH

QU

bu

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RESISTANCE RELATIONS FOR HYDRAULICALLY ROUGH FLOW

6/1

sr

2/1f k

HC

u

UCz

Keulegan (1938) formulation:

90sks Dnk

s

2/1f k

H11n

1C

u

UCz

where = 0.4 denotes the dimensionless Karman constant and ks = a roughnessheight characterizing the bumpiness of the bed [L].

Manning-Strickler formulation:

where r is a dimensionless constant between 8 and 9. Parker (1991) suggested a value of r of 8.1 for gravel-bed streams.

Roughness height over a flat bed (no bedforms):

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.

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COMPARISION OF KEULEGAN AND MANNING-STRICKLER RELATIONSr = 8.1

Note that Cz does not vary strongly with depth. It is often approximated as a constant in broad-brush calculations.

1

10

100

1 10 100 1000

H/ks

Cz

Keulegan

Parker Version of Manning-Strickler

6/1

sk

H1.8Cz

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1.00

10.00

100.00

1.00 10.00 100.00

Rb/ks

Cz

ETH 52

Gilbert 116

Parker Version of Manning-Strickler

6/1

s

b

k

R1.8Cz

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

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NORMAL FLOW

Normal flow is an equilibrium state defined by a perfect balance between the downstream gravitational impelling force and resistive bed force. The resulting flow is constant in time and in the downstream, or x direction.

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]b = 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 Chapter 3, the bed slope angle of the great majority of alluvial rivers is sufficiently small to allow the approximations

1cos,Stansin

xB

x

gHxBS

bBx

H

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UHBBqQUHq www Conservation of downstream momentum:

Impelling force (downstream component of weight of water) = resistive force

xBxSgHBsinxgHB b

gHSb

Reduce to obtain depth-slope product rule for normal flow:

NORMAL FLOW contd.

Conservation of water mass (= conservation of water volume as water can be treated as incompressible):

xB

x

gHxBS

bBx

H

gHSu

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ESTIMATED CHEZY RESISTANCE COEFFICIENTS FOR BANKFULL FLOWBASED ON NORMAL FLOW ASSUMPTION FOR u*

The plot below is from Chapter 3

1

10

100

1 10 100 1000 10000 100000

Grav BritGrav AltaGrav IdaSand MultSand Sing

bfCz

H

50

bf

bfbfbf

bf

bankfull

bf D

HH,

SgHHB

Q

u

UCz

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Relation for Shields stress at normal equilibrium:(for sediment mobility calculations)

gHSUC 2f

RELATION BETWEEN qw, S and H AT NORMAL EQUILIBRIUM

DR

HS

gDRb

DR

S

g

qC 3/23/12wf

3/12wf

gS

qCH

2/12/1z

2/12/1

f

SHgCSHC

gU or

Reduce the relation for momentum conservation b = gHS with the resistance form b = CfU2:

Generalized Chezy velocity relation

Further eliminating U with the relation for water mass conservation qw = UH and solving for flow depth:

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ESTIMATED SHIELDS NUMBERS FOR BANKFULL FLOWBASED ON NORMAL FLOW ASSUMPTION FOR b

The plot below is from Chapter 3

25050

bf

50

bf

50

b50bf

DgD

QQ,

DR

SH

gDR

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02 1.E+04 1.E+06 1.E+08 1.E+10 1.E+12 1.E+14

Grav BritGrav AltaSand MultSand SingGrav Ida

Q

50bf

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RELATIONS AT NORMAL EQUILIBRIUM WITH MANNING-STRICKLER RESISTANCE FORMULATION

6/1

sr

2/1f

3/12wf

k

HC

gS

qCH

RD

S

g

qk 10/710/3

2r

2w

3/1s

Relation for Shields stress at normal equilibrium:(for sediment mobility calculations)

10/3

2r

2w

3/1s

gS

qkH

2/13/26/1

sr

2/12/1

f

SHk

gSH

C

gU

6/1s

r2/13/2

k

g

n

1,SH

n

1U

Manning-Strickler velocity relation(n = Manning’s “n”)

Solve for H to find

Solve for U to find

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BUT NOT ALL OPEN-CHANNEL FLOWS ARE AT OR CLOSE TO EQUILIBRIUM!

Flow into standing water (lake or reservoir) usually takes the form of an M1 curve.

Flow over a free overfall (waterfall) usually takes the form of an M2 curve.

A key dimensionless parameter describing the way in which open-channel flow can deviate from normal equilibrium is the Froude number Fr: gH

UFr

And therefore the calculation of bed shear stress as b = gHS is not always accurate. In such cases it is necessary to compute the disquilibrium (e.g. gradually varied) flow and calculate the bed shear stress from the relation

2fb UC

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NON-STEADY, NON-UNIFORM 1D OPEN CHANNEL FLOWS:St. Venant Shallow Water Equations

0x

UH

t

H

Relation for water mass conservation (continuity):

Relation for momentum conservation:

2f

22

UCx

gHx

Hg

2

1

x

HU

t

UH

x = boundary (bed) attached nearly horizontal coordinate [L]y = upward normal coordinate [L] = bed elevation [L]S = tan - /x [1]H = normal (nearly vertical) flow depth [L]Here “normal” means “perpendicular to the bed” and has nothing to do with normal flow in the sense of equilibrium.

xy

H

Bed and water surface slopes exaggerated below for clarity.

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HYDRAULIC JUMP

subcritical

flow

supercritical

Supercritical (Fr >1) to subcritical (Fr < 1) flow.

1Fr1Fr

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ILLUSTRATION OF BEDLOAD TRANSPORT

Double-click on the image to see a video clip of bedload transport of 7 mm gravel in a flume (model river) at St. Anthony Falls Laboratory, University of Minnesota. (Wait a bit for the channel to fill with water.) Video clip from the experiments of Miguel Wong.

rte-bookbedload.mpg: to run without relinking, download to same folder as PowerPoint presentations.

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ILLUSTRATION OF MIXED TRANSPORT OF SUSPENDED LOAD AND BEDLOADDouble-click on the image to see the transport of sand and pea gravel by a turbidity current

(sediment underflow driven by suspended sediment) in a tank at St. Anthony Falls Laboratory. Suspended load is dominant, but bedload transport can also be seen. Video clip from

experiments of Alessandro Cantelli and Bin Yu.

rte-bookturbcurr.mpg: to run without relinking, download to same folder as PowerPoint presentations.

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PARAMETERS CHARACTERIZING SEDIMENT TRANSPORT

qb = Volume bedload transport rate per unit width [L2/T]qs = Volume suspended load transport rate per unit width [L2/T]qt = qb + qs = volume total bed material transport rate per unit width

[L2/T]qw = Volume wash load transport rate per unit width [L2/T] = water density [M/L3]s = sediment density [M/L3]R = (s/) – 1 = sediment submerged specific gravity [1]D = characteristic sediment size (e.g. Ds50) [L]* = dimensionless Shields number, = (HS)/(RD) for normal flow [1]

Dimensionless Einstein number for bedload transport

Dimensionless Einstein number for total bed material transport

DRgD

qq b

b

DRgD

qq t

t

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SOME GENERIC RELATIONS FOR SEDIMENT TRANSPORT

BEDLOAD TRANSPORT RELATIONS (e.g. gravel-bed stream)

Wong’s modified version of the relation of Meyer-Peter and Müller (1948)

Parker’s (1979) approximation of the Einstein (1950) relation

TOTAL BED MATERIAL LOAD TRANSPORT RELATION (e.g. sand-bed stream)

Engelund-Hansen relation (1967)

0495.0,97.3q c

5.1

cb

03.0,1)(2.11q c

5.4

c5.1b

2/5

ft )(

C

05.0q

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REFERENCESChaudhry, 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.Parker, G., 1991, Selective sorting and abrasion of river gravel. II: Applications, Journal of

Hydraulic Engineering, 117(2): 150-171. 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.

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