OPEN CHANNEL FLOW: uniform motion

M. Pilotti - lectures of Environmental Hydraulics

In order to have a uniform flow, a prismatic channel is a necessary

condition.

This channel, of trapezoidal cross section (b= 6m, B=17 m), is

used to convey Q = 51 mc/s of drinkable water to a large

american town. Its length is 300 kms.

However, this is not a sufficient condition because many man-made

structures can interact with the flow causing departure from

uniform flow (e.g., the gate on the left)

In these situations uniform motion still holds but one has to be

sufficiently far away from the disturbance

How much far away is far ? We have to compute the profiles…

OPEN CHANNEL FLOW: Specific Energy


(G.2b) Specific Energy with respect to the thalweg, with Q constant2

22

)(22 hgA

Qh

g

UhE

αα +=+=

kkk

k

EkEkEk

kkE

gB

Qk

kBkgA

Q

hB

hAh

Frhg

U

dh

dA

hgA

Q

dh

dE

5

3;

5

4;

3

2

2

)()(

1

)(

)(

011)(

1

32

2

3

2

22

3

2

===

+=

=

=

=

=−=−=−=

α

α

ααα(E.1) Minimum of E(h)

(E.2) Equivalent (average)

Hydraulic Depth

(E.3) General expression

for critical depth

(E.4) Critical depth in

rectangular channel

General espression for

Specific Energy in critical

Condition

E(k) in Rectangular, Triangular and Parabolic cross-section;



For a given channel

section and a given

discharge the critical

depth yc depends only

on the geometry of the

section, while the

normal depth h = h0

depends on the slope

of the channel and the

roughness coefficient.

Depending on the relative position between h0 and k, the bottom slope is defined as

Mild slope: h0 > k (see figure above); Steep slope: h0 < k; Critical slope: h0 = k



E(Y) for different cross sections

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Y [m]

E [

m]

Circular; R=2 m

Triang.; θ θ θ θ = 90°

R; B=4 m

Rect.; B= 8 mTrap; B= 4m, side slope 1/1

Q= 5 mc/s

Triang.; θ θ θ θ = 135°

OPEN CHANNEL FLOW: Froude number


Let us underline the meaning of the Froude number.

Let us consider an infinitely wide channel where water flows in uniform motion

with depth h and velocity U.

If perturbation affects the whole water column (tsunami like), we have a

wave of positive height dh that may travel upstream and downstream with absolute

celerity ±a. Due to its passage U is modified, as U-dU.

Given that the motion is an unsteady one, it is convenient to study the process

as seen from astride the wave. This is a inertial frame of reference so that

both energy and mass balance can be written in terms of relative velocity.

We can write

( ) ( ) ( ) ( )

( ) ( )( ) ( )

)1(

0

220

;22

mmm

rrr

FrghcUghUa

h

aUdhdUdhhadUUhaU

ds

dQ

g

aUdUdh

g

adUUdhh

g

aUh

ds

dE

aUUaUU

vvv

rr

tra

===

−=+−−=−=

−=−−++=−+=

−=+=+=

energy

balance

mass

balance

this is the wave relative celerity with respect to the flow U

OPEN CHANNEL FLOW: Froude number


Let us observe our final result

)1( mmm FrghcUghUa ===

Where c is the wave relative celerity according to Lagrange (1788)

(see following part of the course on Unsteady motion).

Accordingly: if Fr < 1 then there is a positive value of a and

a negative one and every perturbation can move

both upstream and downstream.

if Fr > 1 both values of a are positive, so that

the wave cannot propagate upstream.

if Fr = 1, c=U (see previous slides) and a = 0

Note that a is generally different from U. The infinitely small

wave propagates with a celerity that is different from the

average mass velocity, U.

OPEN CHANNEL FLOW: overall significance of the Froude number;


( )

Re),Fr,,x(fp

Re);,Fr,,x(fv

vRe

pzFrD

vD

Frgh

Uhg

U

pg

U

FrgL

U

g

L

U

gV

vVv

FrgL

U

gL

U

gV

vVv

**

**

****

d

ττ

τ

γγ

γ

ρρρ

ρρ

ρ

ρρρ

ρ

ρ

ρρ

==

∆+∇−∇=

===

=∆=∆∝−

∇

==∝∇

r

rr

11

2

1

2

122

2

22

22

22

2

1

22

2Froude number can be introduced when studying

jets, as the ratio between inertial and gravitational

Forces

In general terms it should take into

account the density of the fluid

where the jet is taking place

(densimetric Froude number)

But also in open channel flow as the semi-ratio

between the kinetic energy per unit weight over

the energy related to pressure (after Bakhmeteff,

1912).

Finally it arises when Navier Stokes equations are made dimensionless.

This result is particularly important because it dictates the Froude similarity

criterion.

OPEN CHANNEL FLOW: Physical models and the Froude similarity criterion


When Re is sufficiently large, dynamic similitude for fixed bed models is obtained by imposing Froude similarity.

This is shown taking as an example the Cancano Test case.

In 1943, in the middle of War World II, under the effect of the beginning of the bombing of Milan and

the precedent of the attacks on 16 May 1943 to the Ruhr dams, where mines, factories and houses

where flooded for 80 Km and over 1000 people drowned, it was considered that Cancano dam could

be regarded as a military target. The arched gravity Cancano dam had been built in 1921 in val Fraele,

Valtellina, north Italy, to the purpose of the hydropower supply of Milan. Accordingly, it was asked to

Prof. De Marchi, one of the leading hydraulicians of the time, to study the effects of a possible

bombing of Cancano I dam.

Let us define the ratios between model (m) and prototype (p)

If Froude similarity is imposed, then the constraints

hold

And eventually

lvp

m

p

m

m

m

p

p ;h

h

U

U;

gh

U

gh

Uλλ ===

p

mv

p

m

p

ml U

U;

l

l

h

h === λλ

;t

tl

p

mt λλ ==

OPEN CHANNEL FLOW: Physical models and the Froude similarity criterion


A physical model of the first 16 km stretch of the alpine valley; total and partial collapses of the dam were considered

and discharge hydrographs were measured in three locations

500

1==p

ml l

lλ

6

25

1065

1422

1

⋅==

===

.Q

Q.

/l

p

m

ltv

λ

λλλ

See: De Marchi, G. Sull’onda di piena che seguirebbe al crollo della diga di Cancano. L’Energia Elettrica, 1945, 22, 319-340.

Pilotti M., Maranzoni A., Milanesi L., Tomirotti M., Valerio G., Hydraulic hazard mapping in alpine dam break prone areas: the Cancano dam case,

IAHR Congress, 2013, Chengdu, China, Tsinghua University Press, Beijing, ISBN 978-7-89414-588-8, (on USB), 7 pp.

OPEN CHANNEL FLOW: steady flow profiles in prismatic channels


Let us consider a gradually varied flow, i.e. one in which vertical acceleration on the cross section are negligible,

and, accordingly, an hydrostatic pressure distribution is present. This happens if the slope of the channel is small

and the geometry of the boundary is such that the streamlines are practically parallel. Let us consider Q=constant.

Under the above hypotheses,starting from energy equation of gradually varied flow.

dx

dA

A

E

dx

dy

y

ES

gA

Qyz

dx

d

dx

dHb ∂

∂+∂∂+−=

++=

2

2

2

fSdx

dH −=

23

2

2

2

112

, FrgA

bQ

gA

Qy

dy

d

dy

dE

dydE

SS

dx

dy fb −=−=

+=

−=

2

20

2

2

22

1

1

1

)(1

Fr

QQS

Fr

SKQS

dx

dyb

bb −

−=−

−=

This equation provides y(x) for a general channel.

Let us now consider a prismatic channels, so that

A=A(y(x)) and Sb=constanty

Edx

dA

A

ESS

dx

dy fb

∂∂

∂∂−−

=



From the equation of gradually varied flow in a prismatic channel, the following general properties of the flow profile

y(x) are easily obtained:

.

(asymptotic to a horizontal line)

220

2

1,1,);(

);(FrD

Q

QSN

QyD

QyN

dx

dyb −=

−==

bSdx

dy

D

N

Fr

Qy →⇒

→→

⇒

→∞→

⇒∞→ 1

1

0 0

0 0

−∞→−∞→

⇒

∞→∞→

⇒→D

N

Fr

Qy

00

0 00 →⇒

≠→

⇒→⇒→dx

dy

D

NQQyy

∞→⇒

→≠

⇒→⇒→dx

dy

D

NFryy c

0

0 1

(asymptotic to normal-depth line)

(asymptotic to a vertical line)

(dy/dx→ ∞ if Manning eq. Is used for Q0)



00

0

10

0 >⇒

>>

⇒

<>

⇒>>dx

dy

D

N

Fr

QQyyy c

00

0

10

0 <⇒

><

⇒

<<

⇒<<dx

dy

D

N

Fr

QQyyyc

00

0

10

0 >⇒

<<

⇒

><

⇒<<dx

dy

D

N

Fr

QQyyy c

( ) 220

2 1,1, FrDQQSNDNdxdy b −=−==

Mild slope prismatic channel

M1 profile

M2 profile

M3 profile

OPEN CHANNEL FLOW: steady flow profiles in mild slope prismatic channels


From W. H. Graf and

M. S. Altinakar, 1998



Steep slope prismatic channel

S1 profile

S2 profile

S3 profile

00

0

10

0 >⇒

>>

⇒

<>

⇒>>dx

dy

D

N

Fr

QQyyy c

00

0

10

0 >⇒

<<

⇒

><

⇒<<dx

dy

D

N

Fr

QQyyy c

00

0

10

0 <⇒

<>

⇒

>>

⇒<<dx

dy

D

N

Fr

QQyyy c

( ) 220

20 1,1, FrDQQSNDNdxdy −=−==



Horizontal slope prismatic channel

Adverse slope prismatic channel

Critical slope prismatic channel

OPEN CHANNEL FLOW: hydraulic jump


Supercritical flow in mild slope prismatic channels (M3 profile) and subcritical flow in steep slope prismatic channels

(S1 profile) are limited downstream and upstream respectively at the critical depth. In these cases it may happen that

supercritical flow has to be followed by subcritical flow to cover the whole channel length.

The change from supercritical to subcritical flow takes place abruptly through a vortex known as the hydraulic jump,

characterized by considerable turbulence and energy loss.

The flow depths upstream and downsteam of the jump are called sequent depths or conjugate depths.



Due to the loss of linearity and to the unknown energy loss we have to revert to the Momentum balance

where

Π: pressure force acting on the given section, computed as

Π=γ ygA where yg is the depth of the centroid of flow area A

W: weight of the water enclosed between the sections;

Tf: total external force of friction acting along the boundary.

If the tractive force on the boundary and the component of the

weight compensate each other, we can write

According to which Specific Force S in conserved across the hydraulic jump.

This function has some interesting properties. For instance

fTgA

QW

gA

Q +Π+=+Π+ 22

2

11

2

sinγβθγβ

222

2

111

2

AygA

QAy

gA

QS gg γγγγ +=+=

( )222

2

2

2

2

11)(

FrAbAg

AQAA

gA

bQ

y

Ay

y

A

gA

Q

y

S g −=

−=+−=

∂∂

+∂∂−=

∂∂

Is zero when Fr=1, I.e., in critical conditions

∫∫∫∫∫

==+−==−=−

=YYYY

g

Y

g AdbdYfdY

dYfYYfdYf

dY

ddbY

dY

d

dY

Ayd

A

dbY

y0000

0 )(),(0)0,(1),(),()()()(

;

)()(

ξξξξξξξξξξξξ



From the momentum

equation

in the form S1=S2 it turn out

that since E2<E1 an energy

loss ∆E=E2-E1 takes place

across the jump

y1 is the initial depth (depth

before the jump) and y2 the

sequent depth.

Both are coniugate depths



For rectangular sections the condition of momentum conservation between sections 1 and 2 can be written as

Whose solution, due to the symmetry of the equation, can be put in one of the followig forms that can be used to

calculate downstream (or upstream) depth once upstream (or downstream) conditions are known:

The energy loss across the jump can be calculated as:

( )18121 2

11

2 −+= Fry

y ( )18121 2

22

1 −+= Fry

y

21

312

22

2

2

221

2

2

121 4)(

22 yy

yy

ybg

Qy

ybg

QyEE

−=−−+=−

2

2

2121

22

2

221

1

2 2)(

22 bg

Qyyyy

yb

ybg

Qyb

ybg

Q =+⇒+=+

OPEN CHANNEL FLOW: steady flow profiles in steep slope prismatic channels


Boundary conditions:

Q known;

If Fr<1, Y downstream and the

computation proceeds in the

upstream direction along the

channel.

If Fr>1, Y upstream and the

computation proceeds in the

downstream direction along the

channel.

From W. H. Graf and M. S. Altinakar, 1998

OPEN CHANNEL FLOW: qualitative profiles in complex channels


Real cases can be obtained by combining the simple

profiles seen before.

As a first step control section must be identified, where

the depth is known as a function of Q. From there one

starts computing the profile moving in the direction

dictated by the Froude number, as far as the critical

depth is reached.

At this stage, in some stretch of the channel, more than

a single profile is potentially present. The final choice

will be the one whose Specific Force prevails.

OPEN CHANNEL FLOW: steady flow profiles in various slope prismatic channels


From W. H. Graf and M. S. Altinakar, 1998

Horizontal slope prismatic channel

Adverse slope prismatic channel

Critical slope prismatic channel

NOTES ON OPEN CHANNEL FLOW - UniBs Water...

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