FluidMech_2014_5.ppt

OPEN CHANNEL FLOW
Basic terms in study of open channel flow are: discharge or
cross-sectional area A , wetted perimeter O, hydraulic radius
Rh = A/O and volume discharge Q.
For the rectangular cross-section:
(for “real” river h/B0 ; Rh = h).
Irrigation channels are often built with trapezoidal cross-
section. Sewage channels are mostly made with circular cross-
section.
h
A
h
B h
R
O
h
B h
B

 
 
/
2 1
2

Partially regulated channels often occur as composite
channels.
Natural river beds are nonuniform.
The depth is defined as vertical distance between the free
surface and the lowest
point on the line of
cross section.
If the bed is made of resistant material the profile will remain
stable and unchanged over time. Otherwise, erosion and
deposition will lead to a meandering process.
OPEN CHANNEL FLOW
Nature main river bad
Man-made regulation
(inundation + dike)

Classification for open channel flow: uniform and nonuniform
Classification for open channel flow: steady - unsteady
OPEN CHANNEL FLOW

laminar and turbulent
Critical Reynolds number for open channel flow
D = 4Rh  Rekrit  500
subcritical, critical and supercritical
h
h
V
4
R
R
e=
υ
OPEN CHANNEL FLOW

OPEN CHANNEL FLOW – resistance and turbulen flow
For the practical purposes one can adopt the logarithmic velocity
profile along the vertical:
It is a non-dimensional equation that relates velocity u in particular
point of vertical axes, mean velocity V and “shear velocity” u*.
The channel bed with higher roughness induces more intense
turbulence. That will cause a decrease in near-bottom velocities, as
well as an increase in free surface velocity.
Using the logarithmic velocity profile, Koriolis kinetic energy
coefficient  has the value 1,04.
 
 
 
*
u
u y
=
1
+
2
,
5 1
+
l
n
V V h

The distribution of shear stresses along the contour of trapezoidal
cross-section is given in the figure:
The consequence of non-uniform stress distribution is the onset of
„weak“ secondary flow in cross section and shift of maximum
velocity position from the surface to the deeper layer.
OPEN CHANNEL FLOW – resistance and turbulen flow

The first task in engineering practice is to calculate the cross-
section average velocity V and volume discharge Q for an
arbitrary cross-section A. In the case of uniform flow Chezy
equation is commonly used:
C - Chezy roughness coefficient
I0 – slope of channel bottom
For the uniform flow condition: I0 = IPL= IEL
IPL – slope of piezometric line (slope of free surface)
IEL – slope of energy line IEL
To define Chezy‘s roughness coefficient in practice is frequently
used Manning's coefficient of roughness n:
h
0
V
=
C
R
I
6
h
1
C= R
n
OPEN CHANNEL FLOW – uniform flow

OPEN CHANNEL FLOW – uniform flow
For uniform and stationary flow conditions Chezy equation would
define the relation Q(h) = V(h)*A(h) or so-called “consumption
curve”.

OPEN CHANNEL FLOW – local changes in cross-section geometry
The occurrence of “sudden” changes in channel geometry causes
gradually changes in the flow geometry. The disturbances of free
surface are present “upstream” and “downstream” from the
position of change.
Intensified vorticity in the vicinity of geometry disturbance is
responsible for the “local” losses of mechanical energy:
V – average velocity in flow cross-section
2
V
V
h =ξ
2
g

OPEN CHANNEL FLOW – cross-section specific energy
The flow through an arbitrary cross-section can be also divided
according to the gravity participation in the overall inertia force.
The appropriate regimes are termed as: subcritical, critical or
supercritical. We need to introduce the idea of specific energy E of
cross-section that is defined by equation:
In order to yield the extrema of specific energy function E we apply
the first derivate:
 dA = Bdh  Froude
number (**2)
2
2
α
Q
E=
h
+
2
g
A
2
3
d
E α
Q
d
A
=
1
-
d
h g
Ad
h
2
3
α
Q
B=1
g
A

Fr < 1  subcritical
Fr > 1  supercritical
Fr = 1  critical
The depth at which Fr = 1 one terms as critical depth hkr and
associated cross-section average velocity as critical velocity vkr.
If discharge Q and specific energy E are fixed (Q, E = const.), two
solutions for h arise from above equation (h1 and h2).
Those two depths approach each other when decreasing the
specific energy E. At minimum specific energy E = Emin only one
depth can exist and that is the critical depth hkr (Fr2 = Fr =1)
Critical depth hkr for rectangular cross-section is obtained explicitly:
(q = hkr*Vkr )
q
=
Q
/
B  /
V qh
2
2
q
E=h
+
2
g
h
2
3 2 q
h-
E
h+ =
0
2
g
1
/
3
2
k
r
q
h =
g
 
 
 
m
i
n k
r k
r
1 3
E=
h
+
h
=
h
2 2

Specific energy diagram (specific energy curve for cross-section)
(specific energy E is the function of depth h under the constant
specific discharge q)
h>hkr (Fr < 1) subcritical
condition
h<hkr (Fr > 1) supercritical
condition
IMPORTANT:
One curve is obtained
by fixed Q (or q) and
varying the channel
bed slope I

For any available specific energy E exists the corresponding
maximum discharge qmax that is “transported” at critical condition
(hkr and Vkr).
Example: the flow over a submerged hump .

Flows in open channels will overcome the “obstacles” (weirs)
on “economic” way, achieving the critical depth in the vicinity
of the structure crest (highest datum-level).
Flow regime upstream of the obstacle is subcritical, on the
weir profile or over the submerged wide weir is critical, and
downstream of obstacle depends on channel bottom slope
(I0<Ikrit , h>hkr, Fr < 1  subcritical
I0>Ikrit , h<hkr, Fr > 1  supercritical).

Let us apply the idea of specific energy to the example of
steady flow under the plate with the neglected lines and local
losses.
The depths before and after the plate (h1, h2) are related to the
same specific discharge q and specific energy E.

Raising the plate above the critical depth hkr results in
maximum possible discharge qmax for the available specific
energy E.

Flow above the hump with
relatively “low”
denivelation h
(we assume neglected
energy losses).
subcritical
supercritical
MIRNO (Fr < 1) SILOVITO (Fr >1)
bottom rise dz/dx > 0
bottom lowering dz/dx < 0
dh/dx < 0
dh/dx > 0
dh/dx > 0
dh/dx < 0
width widening dB/dx > 0
constriction dB/dx < 0
dh/dx > 0
dh/dx < 0
dh/dx < 0
dh/dx > 0

OPEN CHANNEL FLOW – overflow and underflow
Flow over sharp-crested weir
P is the weir height, B is channel width, h0 is incoming depth
and V0 is average cross section incoming velocity in subcritical
regime of flow. Energy losses are to be neglected.
Above the highest weir datum one would find the critical
depth :
hp - overflowing depth (vertical distance from weir crest to the
free surface at distance of 4hp from the weir cross section).
 
2
0
k
r 0 p
V
2 2
h
=E
-
P
=h
+
3 3 2
g
 
 
 

Considering the critical condition over the weir crest the
overflowing discharge can be calculated applying the equation:
After a few steps of editing the above equation one gets:
CQ - nondimensional discharge coefficient
In case of “high” weir hp/P0 ; V00:
The above equation is also used for the other types of weirs.
The value of discharge coefficient CQ is obtained experimentally.
Generally, discharge coefficient CQ depends on:
 
 
 
 
 
 
3
/
2
2
3 0
k
r p
V
2
Q
=
q
B
=
g
h
B
=
gh
+ B
3 2
g
3
/
2
Q p
Q
=
C
2
g
h
B
3
/
2
Q
12
C
= =
0
,
3
8
5
3
2



p
p 0 0
0 0
0
0 σ
h
ρ
h V V
h V
,
F
r
=, ,
R
e
=
,
W
e
=
P υ
g
h
f
o
r
m
,
r
o
u
g
h
n
e
s
s

More common used form is the ogee weir or ogee spillway
with the rounded crest :

Underflow
At a sufficient upstream distance from the gate (plate) the
streamlines are parallel and the pressure distribution along the
water column is hydrostatic.

The flow is pronouncedly nonuniform in gate cross-section
(streamlines are not parallel).
At a certain downstream distance from the gate appears
another cross-section with parallel streamlines, so-called
contraction cross-section.
The ratio between the gate opening height “s” and contracted
depth h1 is termed as contraction coefficient CC (obtained
experimentally ; h1 = CC s).

For the flow under the gate of width B following equality are valid:
- contraction cross-section area A1=CC s B
- continuity equation Q0=Bh0V0 = Q1= BCC sV1
- specific energy is equal for the both of cross-sections E0 = E1
(energy losses are neglected)
In case of horizontal bottom and great upstream depth s/h0 0 ;
Fr0  0 the discharge coefficient reads CQ = 0,611.
2 2
0 1
0 C
V V
h
+ =
C
s
+
2
g 2
g
 2
1 0C 0
V
=
2
g
h
-
C
s
+
V
C C 0
0
s
Q
=
C1
-
C 2
g
h
s
B
h
Q 0
Q
=
C
2
g
h
s
B

The transition from supercritical to subcritical flow regime is
related to the hydraulic phenomena called hydraulic jump.
Rapid decrease in average velocity and increase of depth,
including the high degree of mechanic energy loss takes place
in the hydraulic jump.
Because of energy loss hv , the concept of specific energy is no
longer valid and applicable (energy loss in hydraulic jump is not
apriori known).
Normal
hydraulic
jump
OPEN CHANNEL FLOW – hydraulic jump

We can use the law of momentum conservation and apply it
on the control volume that include normal hydraulic jump.
q =V1h1=V2h2
h1,h2 - first and second conjugate depth
- averaged tangential stresses at the bottom
Lj - length of hydraulic jump
MemberL is negligible in comparison to the pressure force.
 
2 2
1 2
0
j 21
h h
ρ
g
-
ρ
g
-
τ
L
=
ρ
q
V
-
V
2 2
0
Normal
hydraulic
jump

Applying the momentum and continuity equations one
defines the relationship between the conjugate depths h1 i h2:
Fr1 - Froude number in cross section where h1
After the calculation of h2 and V2 , one can find the intensity of
energy loss (dissipation) within the hydraulic jump by :
 
2
1
2 1
h
h
= 1
+
8
F
r-
1
2
 
2
2
1 2
h
h
= 1
+
8
F
r-
1
2
1
1
v
F
r = >
1
g
h
 
3
2 1
v
1 2
h-h
h=
4
h
h

The length of hydraulic jump is to be determined experimentally.
For the practical purpose one can use relation Lj  6,1h2.
Increase of Fr1 causes the decrease of h2/h1 and hv/E1.
If downstream normal depth h is greater then second conjugate
depth h2, hydraulic jump will occur in so-called submerged form.
Conversely, hydraulic jump will occur in so-called thrown form that
can threat the bottom stability.

If submerged condition is not assured, one has to carry out the
so-called stilling basin.
Stilling basin provides the stabilization (localization) of
hydraulic jump within its geometry.
Stilling basin is carried out after the spillway, excavating the
cave below the level of natural river bottom.

GROUNDWATER FLOW IN POROUS MEDIA
Darcy velocity v = Q/A applies primarily in the analysis of
groundwater flow (flow in porous underground aquifers).
It relies on the assumption of the continuum (v = Q/A ) where
the presence of the solid phase within the flow cross-section
A is not taken into account.
“Real” velocity is higher then Darcy velocity, what is especially
important in the analysis of pollution transport in aquifers.

Darcy experimental device is used for determine the Darcy
filtration coefficient (hydraulic conductivity) k of particular
filter material.
It is obtained by measuring the Darcy velocity as v = Q/A and
pezometric slope I = h/l (ratio of piezometric drop h on
path length l ).
Δ
h
v=k
Δ
l
Q
Δ
l
k=
A
Δ
h

In general 3D case k is the tensor dependent on geological
strata and flowing liquid.
Groundwater flow can be observed as potential flow.
     
 
h
v k I I
l
h h h
v u
i vj w
k u k v k w k
x y z
k
h k
h k
h
v i j k v g
r
a
d k
h
x y z
k
h
p
v g
r
a
d h z
g
 



  

  
     
  
  
    
  

  

; ;
F
L
O
W
P
O
T
E
N
T
I
A
L
p
h
y
s
ic
a
ly
:

In potential flow the flow field is defined with flow mesh that
consists of equipotentials and streamlines.

Hydraulic (Dupuit) flow theory neglects the vertical component
of flow velocity (equipotentials are vertical lines).
Vertical component of the flow can not be ignored in the
vicinity of the well (strong deviation from Dipuit assumption).

Aquifers appear in two characteristic forms, confined
(pressurized) and unconfined (with the free surface).

 
q q grad
l
h
h
q k h k
l l
h
k
l l
h
k





 



      
 
  
 
 
 
2
2
2
2
2
2
In the case of unconfined aquifers it is useful to introduce the
concept of Girinsky potential ( specific discharge potential).

We analyze only the simpler cases of wells that are constructed
continuously from surface up to the impermeable floor.
n
o
n
l
i
n
e
a
rQ
-sr
e
l
a
t
i
o
n
 
2 2
0
H-
h
Q
=
πk
R
l
n
r
Unconfined aquifer

l
i
n
e
a
rQ
-sr
e
l
a
t
i
o
n
   0
H-
h
Q
=2
πk M
R
ln
r
Confined aquifer

Applying the Girinsky potential  (in case of pumping from an
unconfined aquifer) enables the linearization of the problem,
because of the linear relationship between pumping rate Q and
Girinsky potential drop .
This allows the application of superposition principle in the
case of well group in unconfined aquifer.
x
x x x x x
x
x
x x
k
H k
h
; ; ;
k
h k
H
; h H ; s H h
;
k
Q R
l
n L
I
N
E
A
R
R
E
L
A
T
I
O
N a
n
d
Q
r

       


  


      


 
    


  



2 2
0
0 0 0
2 2
2
0
0 0
0
2 2
2
2 2
2

Application of the superposition principle for a group of wells
(two or more wells) in the confined or unconfined aquifer
enables the calculation of piezometric height (confined
aquifers) and free surface (unconfined aquifer) at arbitrary
point in the horizontal plane.

  

 
 

  
   
  
   
 
    
 
 
 
1 2
1 2
1 2
1 1
2 2
0
0
2 2
2
2 2
x
x
m m
i
x i
i
x
x x
Q R Q R
ln ln
r r
Q R
ln
r
kH kh
 

 
1 2
1 2
1 2
2 2
x
x
s s s
Q R Q R
s l
n l
n
k
M
r k
M
r

 


1 1
2
m m
i
x i
i
Q
R
s s l
n
k
M
r
Unconfined aquifer
Confined aquifer

The idea of superposition can be used in analyzing the impact
of the open watercourses on the current field in the aquifer.
The watercourse is replaced by a fictive recharge-well that has
the same intensity and opposite sign (-Q means water inflow).
The fictive recharge-well is placed on the opposite side of the
watercourse and on the same distance L from the watercourse.

 




 
 
 
 
 

 

1 2
1 2
2
1
2 1 0
0
0
2 2
2
2
2
2
2
x
x
x
Q R Q R
s ln ln
kM r kM r
Q R R
s ln ln
kM r r
Q r
s ln
kM r
SNIŽENJE U ZDENCU:
r L r r
Q L
s ln
kM r
Confined aquifer
WATERCOURSE

The idea of superposition can be also used in analyzing the
impact of non-permeable vertical boundary (barrier) on the
current field in the aquifer.
The vertical barrier is replaced by a fictive extraction-well that
has the same intensity and sign (Q means water outflow). The
fictive extraction-well is placed on the opposite side of the
vertical barrier and on the same distance L from the vertical
barrier.

 


 


 


1 2
2
1 2
2 1 0
2
0
0
2 2
2
2
2 2
x
x
Q R Q R
s ln ln
kM r kM r
Q R
s ln
kM r r
SNIŽENJE U ZDENCU:
r L; r r
Q R
s ln
kM L r
VERTICAL NON-PERMEABLE
BARRIER
Confined aquifer

FORCES ON IMMERSED SOLID BODY
A solid body experiences hydrodynamic forces when it moves
through fluid at rest (resistance). The same intensity of
hydrodynamic force will be present in situation where the
body is at rest and fluid flows around it.
Forces and their intensity are directly related to the viscosity.
At “very” low speed, viscosity is the major contributor in the
overall resistant force.
At “high” speed, viscosity has a noticeable effect only very
close to the solid body contour.
Occurrence of boundary layer separation from the solid body
surface (contour) depends on the body form

We analyze the two-dimensional body of arbitrary shape in
Cartesian plane.
Body surface area is A, and the infinitesimal surface element
dA is defined with an inclination angle  of against the positive
x axis.
Stresses along the body contour are divided into pressure
(normal) and shear (tangential).
Integrating the stresses along x direction gives drag force:
Integrating the stresses along x direction gives hydrodynamic
lift force:
   
 

 
 
X
A A
F
-
p
c
o
s
d
A
s
i
n
d
A
   
 

 
 
y
A A
F
-
p
s
i
n
d
A
c
o
s
d
A

Drag force Fx consists of two parts:
- form resistance (first member on the right hand side)
- friction resistance (second member on the right hand side)
In the case of flow around the body with symmetry contours lift
force Fy is equal to 0.
The primary engineering interest is related to the high-Reynolds
number flows where conditions in boundary layer are highly
dependent on the form of the solid body.
If the pressure gradient is negative along the body contour,
(dp/ds < 0), the boundary layer remains “taped” on the contour.
The onset of inverse pressure gradient (dp/ds > 0) triggers the
boundary layer separation, accompanied with the eddy formation.

Boundary layer separation takes place at the point of separation A.
In the case of oval body forms (e.g.. pier with circular cross section),
the separation point may not be fixed in time.
The intensity of drag force FX is highly dependent on the position A.
(A “more on the left” downstream region occupied with the large
eddies is larger  higher intensity of drag force)
In most engineering problems the form resistance FO has a major
contribution in drag force F .

IMPORTANT:
Created eddies extract the mechanical energy from the main
stream, so the integral of pressures acting on the body „second“
half is lower then the integral on “first” half (in x-direction).
In case of ideal fluid (rotation-free and inviscid) there is no
boundary layer. Consequently, separation and eddy production do
not exist, and the drag force is zero.
For the practical use one has defined the simple equations for the
calculation of drag force FX and form resistance FO:
CX , CO – non-dimensional coefficients of drag and form resistance
 - density of fluid
AP - orthogonal projection of body surface area on vertical
plane perpendicular to the flow direction (x direction).
2 2
0 0
X
X
P O
O
P X
O
V V
F
=
C
ρ
AF
=
C
ρ
AC
=
C
2 2

Coefficients CX and CO are drawn out from the experimentally
obtained results:
Generally, drag coefficient CX is the function of body form, Reynolds
number, roughness and Mach number (Ma - neglected influence in
most of the engineering problems, e.g. for Vair < 200km/h).
If the body is short and has sharp
edges, the viscous forces have
negligible influence
(fixed position of separation,
Re has no influence on FX and CX
so FX  FO and CX  CO)
O
O 2
0
P
F
C =
V
ρA
2

If the body has conspicuous extent in the direction of flow (thin
horizontal plate) viscous force dominates in drag force
(drag force FX  friction resistance force FT ):
 
 


X
A
F s
i
nd
A
2D forms 3D forms

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