chapter 9: relations for hydraulic resistance in rivers

1

1D SEDIMENT TRANSPORT MORPHODYNAMICSwith applications to

RIVERS AND TURBIDITY CURRENTS© Gary Parker November, 2004

CHAPTER 9:RELATIONS FOR HYDRAULIC RESISTANCE IN RIVERS

Dunes in the Mississippi River, New Orleans, USA

Image from LUMCON web page:http://

weather.lumcon.edu/weatherdata/audubon/map.html

Dunes on an exposed point bar in the meandering Fly River,

Papua New Guinea

Sediment transport often creates bedforms such as dunes. These bedforms are accompanied by form drag, and so reduce the ability of the flow to transport sediment.

http://weather.lumcon.edu/weatherdata/audubon/map.html

http://weather.lumcon.edu/weatherdata/audubon/map.html

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SKIN FRICTION AND FORM DRAG: THE CONCEPTSThe drag force acting on a body can be decomposed into skin friction and form drag. The former is generated by the viscous shear stress acting tangentially to the body. The latter is generated by the normal stress (mostly pressure) acting on a body. The Newtonian constitutive relation for water is

i

j

j

iij,vij,vijij x

uxu,p

Here ij denotes the stress acting in the jth direction on a face normal to the ith direction, p denotes the pressure, ij denotes the Kronecker delta ( = 1 if i = j and 0 if i j), ui = (u1, u2, u3) denotes the velocity vector and xi = (x1, x2, x3) denotes the position vector. The drag force Di on a body is given as

dSnD jjii

where ji is evaluated at the surface of the body, ni denotes a local unit vector outward normal to the surface of the body, and dS denotes an infinitesimal element of surface area.

3



SKIN FRICTION AND FORM DRAG: THE CONCEPTS contd.The drag force Di can be decomposed into a component due to skin friction Dsi and a component due to form drag Dfi as follows:

Drag due to skin friction consists of that part of the drag that pulls the surface of the body tangentially. Form drag consists of that part of the drag that pushes the body in normally. Only the former is thought to directly contribute to sediment transport. Now in the diagrams below let and denote the skin friction and form drag forces on the area element dS, denote a unit tangential vector to the surface in the x direction and denote a unit vector normal to the surface.

fisii DDD

dSpnD,dSnxu

xudSnD ifij

i

j

j

ijji,vsi

dSnzuDd tx

bodys

dSnpDd nbodyf

sD

fD

txn

nn

dS

pnn

dS

x

z

txn

u

4



SKIN FRICTION AND FORM DRAG: THE CONCEPTS contd.

Let D denote the drag force in the flow direction and nx denote the component of the unit outward normal vector to the surface in the flow direction. At sufficiently high Reynolds number, the drag on a streamlined body is mostly skin friction. The drag on a blunt body behind which flow separation occurs is mostly form drag. (The pressure in the separation bubble equilibrates with the low pressure at the point of separation.)

u

z

u

flow flow

separation bubble

p p

p

dSzuDD

bodys dSnpDD xbodyf

5



EINSTEIN DECOMPOSITIONEinstein (1950); Einstein and Barbarossa (1952)

When bedforms are not present, all of the drag on the bed is skin friction. This tangential drag force acts to pull the sediment along. When bedforms such as dunes are present, part of the drag is form drag associated with (most prominently) flow separation behind the dunes. Since this form drag is composed of stress that acts normal to the bed surface, it does not contribute directly to the motion of bed grains. As a result it is usually subtracted out in performing bedload calculations.

U

separation bubble

H

6



EINSTEIN DECOMPOSITION contd.

Consider an equilibrium (normal) flow over a bed with mean streamwise slope S that is covered with bedforms. The flow has average depth H and velocity U averaged over depth and the bedforms. The boundary shear stress averaged over the bedforms is given by the normal flow relation

U

separation bubble

H

gHSUC 2fb

7




Now smooth out the bedforms, “glue” the sediment to the bed so it remains flat but offers the same microscopic roughness as the case with bedforms, and run a flow over it with the same mean velocity U and bed slope S. In the absence of the bedforms, the resistance is skin friction only. Due to the absence of bedforms the skin friction coefficient Cfs and the flow depth Hs should be less than the corresponding values with bedforms.

U

separation bubble

H

SgHUC s2

fsbs

UHs

Skin friction + form drag Skin friction only

The difference between the two characterizes form drag.

8




bf = b - bs = mean bed shear stress due to form drag of bedformsCff = Cf – Cfs = friction coefficient associated with form dragHf = H – Hs = mean depth associated with form drag

U

separation bubble

H

SgHUC s2

fsbs

UHs



SgHUC f2

ffbf

SHHgUCC fs2

fffsbfbs

gHSUC 2fb

9



SKIN FRICTION

Skin friction can be computed using the techniques developed in Chapter 5; where = 0.4 and r = 8.1,

U

separation bubble

HU

Hs



SgHUC s2

fsbs

s

s2/1fs k

H11n1C 6/1

s

sr

2/1fs k

HC

or

2n,Dnk k90sks

10



FORM DRAG OF DUNES: EINSTEIN AND BARBAROSSA (1952)One of the first relations developed to predict the form drag in rivers in which dunes predominate is that of Einstein and Barbarossa (1952). They obtained an empirical form for Cff as a function of s*, where

35

2s

35

bs35s RgD

uRgD

bs

su0.0001

0.001

0.01

0.1

0.01 0.1 1 10

s35*

Cff

denotes the Shields number due to skin friction and D35 is the grain size such that 35 percent of a bed surface sample is finer. Note that

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The total shear velocity u*, shear velocity due to skin friction u*s and shear velocity due to bedforms u*f, and the associated Shields numbers are defined as

Engelund and Hansen (1967) determined the following empirical relation for lower-regime form drag due to dune resistance;

or thus

50s

2f

f50s

2s

s50s

2

RgDu,

RgDu,

RgDu

2s 4.006.0

Note that bedforms are absent (skin friction only) when s* = *; bedforms are present when s* < *. The relation is designed to be used with the following skin friction predictor:

Engelund and Hansen (1967) also present a form drag relation for upper-regime bedforms (antidunes).

FORM DRAG OF DUNES: ENGELUND AND HANSEN (1967)

bf

fbs

sb u,u,u

2sf 4.006.0

s

s2/1fs k

H11n1C 65ss D2k

12



Engelund-Hansen Bedform Resistance Predictor

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3x

x E-H RelationNo form drag

s

s

f

No form drag

Engelund-Hansen

FORM DRAG OF DUNES: ENGELUND AND HANSEN (1967) contd.

13



DEPTH-DISCHARGE PREDICTIONS WITH THE FORM DRAG PREDICTOR OF ENGELUND AND HANSEN (1967)

Form drag relations allow for a prediction of flow depth H and velocity U as a function of water discharge per unit width qw. In order to do this with the relation of Engelund and Hansen (1967) it is necessary to specify the stream slope S, bed material sizes Ds50 and Ds65, submerged specific gravity of the sediment R. The computation proceeds as follows for the case of normal flow, for which b = u*

2 = gHS.

Compute ks from Ds65.Assume a value (a series of values) of Hs.Assuming normal flow, compute u*s = (gHsS)1/2 and s* =u*s

2/(RgDs50).Compute * from s* according to Engelund-Hansen.Again assuming normal flow, * = (HS)/(RDs50) so that H = RDs50*/S.Compute Czs = Cfs

-1/2 from Hs/ks and the skin friction predictor.Compute the velocity U from the relation U/u*s = Czs.Compute the water discharge per unit width qw = UH.Plot H versus qw.

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The form drag predictor of Engelund and Hansen (1967) tends to work well for sand-bed streams at laboratory scale. It also works well at small to medium field scale, i.e. in streams in which dunes give way to upper-regime plane bed before bankfull flow is achieved. It works rather poorly for large, low-slope sand-bed rivers, in which dunes are usually never washed out even at or above bankfull flow. Wright and Parker (2004) have modified it to accurately cover the entire range.

This relation is designed to be used with the skin friction predictor

where strat is a correction for flow stratification which can be set equal to unity in the absence of other information (see original reference).

numberFroudegHU

Fr 8.07.0s 7.005.0 Fr

FORM DRAG OF DUNES: WRIGHT AND PARKER (2004)

61

s

s

strat

2/1fs k

H32.8C

90ss D3k

15



0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Middle Loup NiobraraRio Grande RedAtchafalaya MississippiEngelund-Hansen

sk*

*

*sk*

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Middle Loup NiobraraRio Grande RedAtchafalaya MississippiNew relation

sk*

7.0*Fr 7.0Fr

ss

COMPARISON OF FORM DRAG PREDICTORS AGAINST FIELD DATA

Engelund and Hansen (1967) Wright and Parker (2004)

The Niobrara and Middle Loup are small sand-bed streams. The Rio Grande is a middle-sized sand-bed stream. The Red, Atchafalaya and Mississippi Rivers are large sand-bed streams.

16



DEPTH-DISCHARGE PREDICTIONS WITH THE FORM DRAG PREDICTOR OF WRIGHT AND PARKER (2004)

The computation proceeds as follows for the case of normal flow, for which b = u*

2 = gHS. The stratification correction is not implemented here for simplicity.

Compute ks from Ds90.Assume a value (a series of values) of Hs.Assuming normal flow, compute u*s = (gHsS)1/2 and s* =u*s

2/(RgDs50).Compute the velocity U from the skin friction predictor.Compute from the indicated equation.Compute H from the indicated equation.Compute the water discharge per unit width qw = UH.Plot H versus qw.

61

s

s

s kH32.8

SgHU

8.07.0

50ss gH

URD

HS7.005.0

61

s

ss k

HSgH32.8U

13/207.0

50s

Ug

SRDH

4/5

s

7.005.0

The relations can be written as:

or alternatively as:

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PREDICTION OF BEDLOAD TRANSPORT IN A STREAM IN WHICH DUNES MAY BE PRESENT

If dunes are not present, the calculation of bedload transport may proceed using the techniques of Chapter 7.

If dunes are present, the calculation is based not on the total boundary shear stress b, but rather just that component due to skin friction bs. Thus in the case of relations for uniform sediment D, the following transformation must be made

so that the bedload relation of e.g. Ashida and Michiue (1972) is recast as

In the case of the normal flow

and the calculation can proceed from the calculation of thedepth-discharge relation.

RgDu

RgD

2sbs

s

05.0,17q ccscsb

RDDHs

s

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Depth-Discharge and Bedload Calculator Uses a) Wright-Parker formulation for flow resistance (without stratification correction)b) Ashida-Michiue formulation for bedload transport,

Input ParametersS 4.00E-05 bed slopeD50 0.3 mm median sediment sizeD90 0.6 mm size such that 90% of the sediment is finernk 3 factor such that ks = nk Ds90R 1.65 submerged specific gravity of sediment

InputHs (m)

s* U (m/s) H (m) qw (m2/s)* s*/* Fr u* (m/s) u*s (m/s) qb (m

2/s)0.800 0.0646 0.4072 0.008 0.254 0.1036 0.0206 3.145 0.258 0.00999 0.01772 1.60E-07 Discard results whenever s*/* > 10.900 0.0727 0.4405 0.014 0.544 0.2397 0.0440 1.654 0.191 0.01461 0.01879 3.72E-071.000 0.0808 0.4725 0.02 0.906 0.4279 0.0732 1.104 0.159 0.01885 0.01981 6.64E-071.100 0.0889 0.5035 0.027 1.323 0.6664 0.1069 0.831 0.14 0.02279 0.02078 1.03E-061.200 0.0970 0.5336 0.034 1.788 0.9539 0.1444 0.671 0.127 0.02648 0.0217 1.47E-061.300 0.1051 0.5629 0.042 2.29 1.2891 0.1851 0.568 0.119 0.02998 0.02259 1.97E-061.400 0.1131 0.5914 0.049 2.826 1.6712 0.2284 0.495 0.112 0.0333 0.02344 2.53E-061.500 0.1212 0.6192 0.057 3.39 2.0994 0.2740 0.442 0.107 0.03647 0.02426 3.15E-061.600 0.1293 0.6464 0.066 3.98 2.5729 0.3216 0.402 0.103 0.03952 0.02506 3.83E-061.700 0.1374 0.6731 0.074 4.592 3.0912 0.3711 0.37 0.1 0.04245 0.02583 4.57E-061.800 0.1455 0.6992 0.083 5.225 3.6536 0.4222 0.344 0.098 0.04528 0.02658 5.35E-061.900 0.1535 0.7249 0.092 5.876 4.2598 0.4749 0.323 0.095 0.04802 0.0273 6.19E-062.000 0.1616 0.7501 0.101 6.545 4.9092 0.5289 0.306 0.094 0.05068 0.02801 7.08E-062.100 0.1697 0.7749 0.11 7.229 5.6015 0.5841 0.291 0.092 0.05326 0.02871 8.01E-062.200 0.1778 0.7993 0.119 7.927 6.3362 0.6406 0.278 0.091 0.05577 0.02938 8.99E-062.300 0.1859 0.8234 0.129 8.639 7.1130 0.6981 0.266 0.089 0.05822 0.03004 1.00E-052.400 0.1939 0.8471 0.138 9.364 7.9316 0.7567 0.256 0.088 0.06062 0.03069 1.11E-052.500 0.2020 0.8704 0.148 10.1 8.7917 0.8162 0.248 0.087 0.06296 0.03132 1.22E-05

50

ss RD

SH

61

s

ss k

HSgH32.8U

4/5

s

7.005.0

13/207.0

50s

Ug

SRD

H

UHqw

50RDHS

gHU

Fr

gHSu

SgHu ss

)05.0)(05.0(DRgDq ss5050b

SAMPLE PREDICTION OF FLOW AND BEDLOAD TRANSPORT

The basis for the calculation is a large sand-bed stream. The calculation uses Wright-Parker (without stratification correction) and Ashida-Michiue.

Discard first three rows

This calculation is implemented in:Rte-bookWPHydResAMBL.xls

19



DEPTH-DISCHARGE AND BEDLOAD RELATION FOR SAMPLE CALCULATIONWright-Parker depth-discharge predictor: Ashida-Michiue bedload transport relation

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0.0 2.0 4.0 6.0 8.0 10.0

qw (m2/s)

H (m

), q b

(m2 /s

)

0.0E+00

2.0E-06

4.0E-06

6.0E-06

8.0E-06

1.0E-05

1.2E-05

1.4E-05

Hqb

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A BULK PREDICTOR FOR DEPTH-DISCHARGE RELATIONSThe Brownlie (1982) empirical depth-discharge predictor has been demonstrated to be accurate for both laboratory and field sand-bed streams. It takes the lower-regime form

and the upper-regime form

where

Once H is known U = qw/H can be computed. It is then possible to back-calculate Hs from any appropriate relation for skin friction and the normal flow assumption, e.g.

Once Hs is known, s* = (HsS)/(RDs50) and thus the bedload transport rate can be computed. A discriminator between lower regime and upper regime canbe found in the original reference.

08013.0g

08750.06248.0w

1 S)Sq(S2836.0H

1050.0g

09188.06539.0w

1 S)Sq(S3724.0H

50s50s

ww

50s DgDqq,

DHH

6/1

s

sr

s kH

SgHU

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ANOTHER BULK PREDICTOR FOR FLOW RESISTANCEThe flow predictor of Karim and Kennedy (1981) takes the following form:

where qt denotes the total volume bed material load per unit width. Karim and Kennedy’s predictor for qt is presented in Chapter 11.

s10

310

5010

5010

s10

5050

t10

5010

s10

5050

t10

5010

vuog10Sog

DHog0411.0

Duogvuog2166.0

DRgDqogDuog

vuog0831.0

DRgDqog1665.09045.0

RgDUog

22



GENERALIZATION TO GRADUALLY VARIED FLOWS

The preceding calculations are predicated on the assumption of normal flow. In the case of gradually varied flow, the equation to be solved is

2f

1SS

dxdH

Fr

In the calculation of gradually varied flow the actual slope S should be replaced by the friction slope Sf in the relations for skin friction and form drag:

For example, the relations of Wright and Parker (without stratification correction) become

61

s

s

fs

w

kH32.8

SgHHq

8.07.0

2/3w

50s

f

50

fs

Hgq

RDHS7.005.0

RDSH

HgSgHgH

UCS fbb

2

ff

23



GENERALIZATION TO GRADUALLY VARIED FLOWS contd.

The flow is assumed to be subcritical. The depth H is assumed to be known at the downstream point H2; it is to be computed at the upstream point H1. The formulation can be discretized as

32

2w

31

2w

gHq

gHq 1

Sx

1

Sxx

21HH

2f21

1f21

21

flow

x

x1 x2

H1

H2

Now since qw and H2 are known, Hs1 and Sf1 can be computed (iteratively) from the two relations

61

s

1s

1f1s1

w

kH32.8

SgHHq

8.07.0

2/31

w

50s

1f1

50

1f1s

Hgq

RDSH7.005.0

RDSH

24



GENERALIZATION TO GRADUALLY VARIED FLOWS contd.

Once all quantities at x2 are computed, H1, Hs1 and Sf1 can be computed iteratively from the following three equations.

32

2w

31

2w

gHq

gHq 1

Sx

1

Sxx

21HH

2f21

1f21

21

flow

x

x1 x2

H1

H2

61

s

1s

1f1s1

w

kH32.8

SgHHq

8.07.0

2/31

w

50s

1f1

50

1f1s

Hgq

RDSH7.005.0

RDSH

25



REFERENCES FOR CHAPTER 9Ashida, K. and M. Michiue, 1972, Study on hydraulic resistance and bedload transport rate in

alluvial streams, Transactions, Japan Society of Civil Engineering, 206: 59-69 (in Japanese).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.

Einstein, H. A., 1950, The Bed-load Function for Sediment Transportation in Open Channel Flows, Technical Bulletin 1026, U.S. Dept. of the Army, Soil Conservation Service.

Einstein, H. A and Barbarossa, N. L., 1952, River Channel Roughness, Journal of Hydraulic Engineering, 117.

Engelund, F. and E. Hansen, 1967, A Monograph on Sediment Transport in Alluvial Streams, Technisk Vorlag, Copenhagen, Denmark.

Karim, F., and J. F. Kennedy, 1981, Computer-based predictors for sediment discharge and friction factor of alluvial streams, Report No. 242, Iowa Institute of Hydraulic Research, University of Iowa, Iowa City, Iowa.

Wright, S. and Parker, G., 2004, Flow resistance and suspended load in sand-bed rivers: simplified stratification model, Journal of Hydraulic Engineering, 130(8), 796-805.

chapter 9: relations for hydraulic resistance in rivers

Documents

drag force di

tangential drag force

normal vector

skin friction dsi

bed surface

normal stress

flow direction

unit vector normal