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RIVERS & STREAMSHYDRAULICS

Benoit Cushman-Roisin

Dartmouth College

River flow is 3D and unsteady (turbulent).

As a result, the downstream velocity, aligned with the channel, dominates the flow.

The 1D assumption may be made, withh(x,t) = water depthu(x,t) = water velocity

with x = downstream distance and t = time.

Another assumption may be made: Incompressibility

density = constant = 1000 kg/m3 (freshwater)

But, length of river >> width & depth

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The reduction to two flow variables [h(x,t) and u(x,t)] necessitates only two physical statements.

These are:

1. Conservation of mass (what goes in, goes out)

2. Downstream momentum budget (with 3 forces: pressure, gravity and friction).

volume

Budget for a stretch dx of river

S = Slope = sin P = Wetted perimeter

A = Area W = Width h = max depth

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Conservation of mass:

Amount stored in stretch dx = what goes in – what goes out.

0, 0

0

constant

0

t dt

x x dxtAdx Au dt Au dt

dx dt

A Aut x

A Aut x

This equation is attributed to Leonardo da Vinci (1452-1519), which he wrote in an algebraic way instead of using derivatives as we do now.

where A(h) is a known function of the water depth h (channel profile given)

Then

Case of a rectangular channel with constant width:

A = Wh, with W = constant

P = h + W + h

00

hux

ht

Aux

At

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Momentum Budget:

Momentum inside stretch Momentum entering atd

xdt

Momentum exiting at

Pushing pressure force in rear

Braking pressure force ahead

Downslope gravitational force

Braking frictional force along bottom

x dx

0Pressure force ( ) ( )

height from deepest point at bottom

( ) hydrostatic (gage) pressure ( )

( ) channel width at level (0 )

h

pF pdA p z w z dz

z

p z g h z

w z z z h

x

hgA

x

h

dh

dF

x

F

gAdzzwg

dzzgwzwzhgdh

dF

dzzwzhgF

pp

h

h

hzp

h

p

)(

)()]()([

)()(

0

0

0

For later:

a function of h only

5

2

2

( )sin

( )

bottom stress bottom area

( )

( )

with bottom stress

and wetted perimeter

b

D

b D

mg

Adx gS

gASdx

Pdx

C u Pdx

C u

P

Gravitational force

Frictional force

Putting it altogether:

dxPuC

gASdx

FF

AuAudt

AudxAudx

D

dxxpxp

dxxx

tdttt

2

at at

at

2

at

2at a

||

][][

or, in differential form:

22 uPCgASx

FAu

xAu

t Dp

and after some simplifications and use of volume conservation:

2uA

PCgS

x

hg

x

uu

t

uD

Momentum in and out

Pressure force, rear and front

Downslope gravity

Bottom friction

inertia gravityfriction

pF hgA

x x

from earlier

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For convenience, we define the hydraulic radius:

so that the momentum equation becomes:

For a broad flat channel, which is a good approximation for most rivers:

2

D

u u h uu g gS C

t x x R

h

uCgS

x

hg

x

uu

t

uh

W

Wh

hW

WhR D

2

2

This equation is attributed to Adhemar de Saint-Venant (1797-1886).

Together, this momentum equation and the mass-conservation equation form a 2 x 2 nonlinear system for the flow variables h and u.

perimeter wetted

area sectional-cross

P

AR

1. Uniform frictional flow:

0 and 0

xt

Only the momentum equation remains and it becomes:

DD C

ghSu

h

uCgS 0

2

The balance is between the forward force of gravity and the retarding force of bottom friction.

The formula is due to Antoine de Chézy (1718-1798).

The Chézy formula specifies one relation between the velocity u and the water depth h. How can these quantities be determined separately?

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Answer: We need to know the volumetric flow (discharge) of the river!

3

1

3

1

2

2

thengiven, is When

WC

gSQu

gSW

QCh

WhuAuQ

D

D

Note that both water depth and velocity increase with the discharge. This explains why the water level rises and the current increases simultaneously when the river discharge rises.

Note: As the discharge increases, the water depth (÷Q2/3) increases faster than the velocity (÷Q1/3).

Manning’s formula

River data show that the drag coefficient CD is not a constant but depends on depth.

If we use the logarithmic velocity profile of wall turbulence, we obtain:

20

2

]1)/[ln(

zhCD

and the Chezy formula becomes

1ln

02 z

hghS

C

ghSu

D

Using abundant data, Robert Manning (1816-1897) determined that a power of h was adequate, with the 2/3 power giving the best fit, and he wrote:

2/13/21SR

nu

in which the coefficient n is now called the Manning Coefficient.

Note that this expression is not dimensionally correct. So, care must be taken to use metric units.

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Examples of Manning Coefficients:

See also: https://wwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/index.htmFor details, consult: https://pubs.usgs.gov/wsp/2339/report.pdf

Clark Fork at St. Regis, Montanan = 0.028

Clark Fork above Missoula, Montanan = 0.030

Middle Fork Flathead Rivernear Essex, Montana

n = 0.041

South Fork Clearwater Rivernear Grangeville, Idaho

n = 0.051

Rock Creek near Darby, Montana

n = 0.075

Smooth cement canaln = 0.012

Numerical values of the Manning Coefficient:

CHANNEL TYPE n

Artificial channels finished cement 0.012

unfinished cement 0.014

brick work 0.015

rubble masonry 0.025

smooth dirt 0.022

gravel 0.025

with weeds 0.030

cobbles 0.035

Natural channels mountain streams 0.045

clean and straight 0.030

clean and winding 0.040

with weeds and stones 0.045

most rivers 0.035

with deep pools 0.040

irregular sides 0.045

dense side growth 0.080

Flood plains farmland 0.035

small brushes 0.125

with trees 0.150

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The enigma of Roman water engineers

Segovia, Spain

Roman engineers had no conception of time at the scale of the minute and second. So, they could not quantify the water velocity and dealt only with water depths.

So, how were they able to build properly designed aqueducts and sewage drains?

The Romans were lucky because velocity is directly related to water depth, and water depth could then be used as the only variable.

It also helped that water depth happens to be the more sensitive function of discharge among the two variables.

Answer:

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A nice exercise:

Subject the Chezy solution to small, time-dependent fluctuations,to find that it is stable only as long as

DCSFr 42

If this condition is not met,waves grow to finite amplitude.

These are so-called “roll waves”

Photo credit: Alden Adolph, Thayer School student, January 2013

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2. Steady frictionless flow:

02

:ascast becan equation momentum theand then

level) (sea datum reference a above )(elevation bottom thedefine weIf

:Momentum

constant 0)( :Mass

:become equations governing ofpair The

friction) (no 0 and s)(steadines 0 takeNow,

2

gbghu

dx

d

dx

dbS

xb

gSdx

dhg

dx

duu

W

Qhuhu

dx

d

Ct D

energy. potential )( energy, kinetic 2

:energy ofon conservati is principle Bernoulli theof essence The

principle. Bernoulli theasknown is and

1782),-(1700 Bernoulli Daniel todue isrelation is This

flow. thealong conserved is

2

expression theTherefore,

2

2

hbgu

gbghu

B

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3. Linear waves:

Now, ignore friction (CD = 0) and bottom slope (S = 0).

Then, linearize the equation around a basic state of no flow and a uniform water depth H:

( , ) ( , )h x t H a x t

Governing equations reduce to:

0 ,a u u a

H gt x t x

A solution is the shallow-water gravity wave:

( , ) sin ( )

( , ) sin ( )

a x t A k x ct

cAu x t k x ct

H

c gH

with

Summary of particular solutionsand cases to be considered later:

Shallow-water wave

Bernoulli

Chézy

Steady non-uniform flow

Roll wavesFlooding

t

Sx

DC