OPEN CHANNEL FLOW

1
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.
2
OPEN CHANNEL FLOW
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.
Man-made regulation (inundation dike)
Nature main river bad
3
OPEN CHANNEL FLOW
Classification for open channel flow uniform and
nonuniform Classification for open
channel flow steady - unsteady
4
OPEN CHANNEL FLOW
laminar and turbulent
Critical Reynolds number for open channel
flow D 4Rh ? Rekrit
? 500 subcritical, critical and supercritical
5
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.
6
OPEN CHANNEL FLOW resistance and turbulen flow
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.
7
OPEN CHANNEL FLOW uniform 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 Chezys roughness coefficient in
practice is frequently used Manning's coefficient
of roughness n
8
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.
9
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
10
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)
11
OPEN CHANNEL FLOW cross-section specific energy
Fr lt 1 ? subcritical Fr gt 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 hkrVkr )
12
OPEN CHANNEL FLOW cross-section specific energy
Specific energy diagram (specific energy curve
for cross-section) (specific energy E is the
function of depth h under the constant specific
discharge q) hgthkr (Fr lt 1) subcritical
condition hlthkr (Fr gt 1)
supercritical condition
IMPORTANT One curve is obtained by fixed Q (or
q) and varying the channel bed slope I
13
OPEN CHANNEL FLOW cross-section specific energy
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 .
14
OPEN CHANNEL FLOW cross-section specific energy
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 (I0ltIkrit ,
hgthkr, Fr lt 1 ? subcritical I0gtIkrit , hlthkr,
Fr gt 1 ? supercritical).
15
OPEN CHANNEL FLOW cross-section specific energy
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.
16
OPEN CHANNEL FLOW cross-section specific energy
Raising the plate above the critical depth hkr
results in maximum possible discharge qmax for
the available specific energy E.
17
OPEN CHANNEL FLOW cross-section specific energy
Flow above the hump with relatively low
denivelation ?h (we assume neglected energy
losses).
subcritical
supercritical
MIRNO (Fr lt 1) SILOVITO (Fr gt1)
bottom rise dz/dx gt 0 bottom lowering dz/dx lt 0 dh/dx lt 0 dh/dx gt 0 dh/dx gt 0 dh/dx lt 0
width widening dB/dx gt 0 constriction dB/dx lt 0 dh/dx gt 0 dh/dx lt 0 dh/dx lt 0 dh/dx gt 0
18
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).
19
OPEN CHANNEL FLOW overflow and underflow
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
20
OPEN CHANNEL FLOW overflow and underflow
More common used form is the ogee weir or ogee
spillway with the rounded crest
21
OPEN CHANNEL FLOW overflow and underflow
Underflow At a sufficient upstream distance from
the gate (plate) the streamlines are parallel and
the pressure distribution along the water column
is hydrostatic.
22
OPEN CHANNEL FLOW overflow and underflow
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).
23
OPEN CHANNEL FLOW overflow and underflow

• For the flow under the gate of width B following
  equality are valid
• - contraction cross-section area A1CC s B
• continuity equation Q0Bh0V0 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.

24
OPEN CHANNEL FLOW hydraulic jump
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
25
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
V1h1V2h2 h1,h2 - first and second
conjugate depth - averaged tangential
stresses at the bottom Lj - length of
hydraulic jump Member?Lj is negligible in
comparison to the pressure force.
Normal hydraulic jump
26
OPEN CHANNEL FLOW 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
27
OPEN CHANNEL FLOW hydraulic jump
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.
28
OPEN CHANNEL FLOW hydraulic jump
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.
29
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.
30
GROUNDWATER FLOW IN POROUS MEDIA
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 ).
31
GROUNDWATER FLOW IN POROUS MEDIA
In general 3D case k is the tensor dependent on
geological strata and flowing liquid. Groundwater
flow can be observed as potential flow.
32
GROUNDWATER FLOW IN POROUS MEDIA
In potential flow the flow field is defined with
flow mesh that consists of equipotentials and
streamlines.
33
GROUNDWATER FLOW IN POROUS MEDIA
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).
34
GROUNDWATER FLOW IN POROUS MEDIA
Aquifers appear in two characteristic forms,
confined (pressurized) and unconfined (with the
free surface).
35
GROUNDWATER FLOW IN POROUS MEDIA
In the case of unconfined aquifers it is useful
to introduce the concept of Girinsky potential (?
specific discharge potential).
36
GROUNDWATER FLOW IN POROUS MEDIA
We analyze only the simpler cases of wells that
are constructed continuously from surface up to
the impermeable floor.
Unconfined aquifer
37
GROUNDWATER FLOW IN POROUS MEDIA
Confined aquifer
38
GROUNDWATER FLOW IN POROUS MEDIA
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.
39
GROUNDWATER FLOW IN POROUS MEDIA
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.
40
GROUNDWATER FLOW IN POROUS MEDIA
Unconfined aquifer
Confined aquifer
41
GROUNDWATER FLOW IN POROUS MEDIA
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.
42
GROUNDWATER FLOW IN POROUS MEDIA
Confined aquifer
WATERCOURSE
43
GROUNDWATER FLOW IN POROUS MEDIA
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.
44
GROUNDWATER FLOW IN POROUS MEDIA
Confined aquifer
VERTICAL NON-PERMEABLE BARRIER
45
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
46
FORCES ON IMMERSED SOLID BODY
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
47
FORCES ON IMMERSED SOLID BODY

48
FORCES ON IMMERSED SOLID BODY

• 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 lt 0), the boundary layer
  remains taped on the contour. The onset of
  inverse pressure gradient (dp/ds gt 0) triggers
  the boundary layer separation, accompanied with
  the eddy formation.

49
FORCES ON IMMERSED SOLID BODY
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 Fx.
50
FORCES ON IMMERSED SOLID BODY
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).
51
FORCES ON IMMERSED SOLID BODY
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 lt 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)
52
FORCES ON IMMERSED SOLID BODY
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 )
2D forms
3D forms