notes on open channel flow - unibs water...
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NOTES ON OPEN CHANNEL FLOW
Profili di moto
permanente in un
canale e in una serie
di due canali -
Boudine, 1861
Prof. Marco Pilotti
DICATAM, Università degli Studi
di Brescia
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
M. Pilotti - lectures of Environmental Hydraulics
(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;
OPEN CHANNEL FLOW: Specific Energy
M. Pilotti - lectures of Environmental Hydraulics
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
OPEN CHANNEL FLOW: Specific Energy
M. Pilotti - lectures of Environmental Hydraulics
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
M. Pilotti - lectures of Environmental Hydraulics
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
M. Pilotti - lectures of Environmental Hydraulics
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;
M. Pilotti - lectures of Environmental Hydraulics
( )
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
M. Pilotti - lectures of Environmental Hydraulics
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
M. Pilotti - lectures of Environmental Hydraulics
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
M. Pilotti - lectures of Environmental Hydraulics
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
∂∂
∂∂−−
=
OPEN CHANNEL FLOW: steady flow profiles in prismatic channels
M. Pilotti - lectures of Environmental Hydraulics
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)
OPEN CHANNEL FLOW: steady flow profiles in prismatic channels
M. Pilotti - lectures of Environmental Hydraulics
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
M. Pilotti - lectures of Environmental Hydraulics
From W. H. Graf and
M. S. Altinakar, 1998
OPEN CHANNEL FLOW: steady flow profiles in prismatic channels
M. Pilotti - lectures of Environmental Hydraulics
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 −=−==
OPEN CHANNEL FLOW: steady flow profiles in prismatic channels
M. Pilotti - lectures of Environmental Hydraulics
Horizontal slope prismatic channel
Adverse slope prismatic channel
Critical slope prismatic channel
OPEN CHANNEL FLOW: hydraulic jump
M. Pilotti - lectures of Environmental Hydraulics
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.
OPEN CHANNEL FLOW: hydraulic jump
M. Pilotti - lectures of Environmental Hydraulics
OPEN CHANNEL FLOW: hydraulic jump
M. Pilotti - lectures of Environmental Hydraulics
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
OPEN CHANNEL FLOW: hydraulic jump
M. Pilotti - lectures of Environmental Hydraulics
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
OPEN CHANNEL FLOW: hydraulic jump
M. Pilotti - lectures of Environmental Hydraulics
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: hydraulic jump
M. Pilotti - lectures of Environmental Hydraulics
OPEN CHANNEL FLOW: steady flow profiles in steep slope prismatic channels
M. Pilotti - lectures of Environmental Hydraulics
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
M. Pilotti - lectures of Environmental Hydraulics
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
M. Pilotti - lectures of Environmental Hydraulics
From W. H. Graf and M. S. Altinakar, 1998
Horizontal slope prismatic channel
Adverse slope prismatic channel
Critical slope prismatic channel