Title: Unsteady flow in open channel Flows
1 Chapter
Six
- 6. Unsteady flow in open channels
- 6.1 Introduction
- Unsteady flow changes with time
steady flow does not. - The difference is not an absolute one ,but may be
dependent on the observer. - Suppose for example that a land slide falls into
river and partially blocks it, sending wave
upstream. - surge wave ,often simply called a surge ,is a
moving wave front which brings about an abrupt
change din depth another example of this
phenomena is tidal bore by which the tide
invades certain rivers.
2Introduction
- Now an observer on the bank would see this as an
unsteady flow-phenomenon, since the flow changes
its velocity and depth as the sure passes him. - However,an observer who is moving along with the
surge sees the situation as one of steady
flow.,atleast in the first stage of movement the
before the surge begins to decay. - He is level with stationary wave front ,there is
flow of unchanging velocity and depth upstream of
him(assuming the river has a uniform slope and
cross section) and downstream of him.
3Introduction
- The distinction being made here is not academic
one, for the equation of motion is very much
easier to write down and manipulate for steady
flow than for unsteady flow. - Now an observer on the bank would see this as an
unsteady flow-phenomenon, since the flow changes
its velocity and depth as the sure passes him. - However, an observer who is moving along with the
surge sees the situation as one of steady flow.,
atleast in the first stage of movement the
before the surge begins to decay. - He is level with stationary wave front, there is
flow of unchanging velocity and depth upstream of
him (assuming the river has a uniform slope and
cross section) and downstream of him.
4Introduction
- The distinction being made here is not academic
one, for the equation of motion is very much
easier to write down and manipulate for steady
flow than for unsteady flow. - There are of course ,many cases in practice where
there is no such dependence on the view point
observer and flow would be classified as steady
or unsteady as the case may be by any observer.
5Introduction
- Such case is a progress of flood wave down a
river a man standing on the bank would clearly
see the phenomena as unsteady so would another
man moving down stream and keeping pace with peak
flood, since the magnitude of the peak discharge
itself tends to reduce as the flood moves
downstream. - In a problem such as this one can not take the
easy out by transposing to a steady flow case,
the problem treated as that of unsteady flow.
6Introduction
- Unsteady flow occurs where the flow parameters
vary with time at a fixed point. - Problems
- Oscillatory Sea waves.
- Predicting water level in river flood
- Dam break flood waves
- Surge due to gate operation e.g in irrigation
canal.
7Introduction
- Waves Definitions
- a wave is a temporal variation in the water
surface which is propagated through flood
medium. - The celerity of the wave is the speed of
propagation of the disturbance relative to the
fluid.
8 Waves classification
- CAPILLARY -due to surface tension
- Elastic- due to fluid compression
- Gravity waves
- Oscillatory wave e.g sea water
- Zero net mass transport
- Translator waves e.g flood waves
- net transport of fluid in direction of wave
- Solitary wave
- Rising limb single peak followed by and preceded
by steady flow - Â
9 Waves classification
- Wave train created by sequence of several waves
- Further definition
- Down stream wave - moves down channel slope
- Upstream wave - moves up channel slope
- Increase in level from steady flow positive wave
- Decrease in level from steady flow negative
wave - Monoclonal- single faced
- Two faced symmetrical or asymmetrical
10 Waves classification
- Deep water waves- only surface layers disturbed
-
Shallow water waves entire depth disturbed
bottom effect
11 Wave celerity
12 Wave celerity
13 Wave celerity
14 Wave celerity
15Development of St.Venant Equations
- There are five assumptions to derive the
equations(yevjevich and chaudry 1993). - The shallow water approximation apply so that
vertical accelerations are negligible,resulting
in a vertical pressure distribution that is
hydrostatic and the depth,y, is small compared
to the wave length so that the wave celerity c
(gy)1/2
16Development of St.Venant Equations
- 2. The channel bottom slope is small, so that
In the hydrostatic pressure force formulation is
approximately unity, and
Channel bed slope, where ? is angle of channel
bed relative to the horizontal.
3. The channel bed is stable ,so that the bed
elevations do not change with the time.
17Development of St.Venant Equations
- 4.0 The flow can be represented as
one-dimensional with - Horizontal water surface across any cross
section such that transverse velocities are
negligible and - An average boundary shear stress that can be
applied to the whole cross section. - 5.0 The frictional bed resistance is the same in
unsteady flow as in steady flow, so that manning
equation or chezs equation can be used to
evaluate the mean boundary shear stress.
18Development of St.Venant Equations
19Continuity Equation
5.1
20(No Transcript)
21Cont
Figure 5.1
22Cont
23Cont
5.2
Substituting dA Bdy from figure 5.2 in which B
channel top width at free surface,continuity
becomes
24Cont
Ity in the flow direction x ,the ?Q/
? X in 5.3 can be written as
25Cont
5.4
26Cont
- Where the first term on the right side of 5.4
represents the derivative of A with respect to x
while holding y constant .For prismatic channels
,this term goes to zero. Finally with thse
substition fo r ?Q/?X and then ?A/ ?X, and
dividing through by the top width,B, the
continuity equation reduces to
5.5
27Cont
28Cont
5.6
29Cont
5.7
30Cont
31Cont
32Cont
33Cont
34Cont
7.9
35Cont
7.10
36Cont
5.11
37Cont
5.12
38Cont
- Substituting equation 5.8 and 5.12 into equation
5.7 dividing by ??x ,and letting ?x go to zero
results in
5.13
39Cont
Of equation 5.13 come from 1 the time rate of
change of momentum inside the control volume.
2 the net momentum flux out of the control
volume and
40Cont.
- 3. The momentum flux of the lateral inflow all
in the x direction equation 5.13 represents the
momentum equation in conservation form for a
prismatic channel. - This is simply means that if the terms on the
right hand side of the equation are conserved
and this may be the most appropriate form in
which to apply some numerical solution sschemes. - Equation 5.13 sometimes is placed in reduced form
by applying the product rule of
differentiation,substituting for
41Cont.
5.14
42Cont.
5.15
43Cont
5.15
44Cont
5.16
45Cont
5.2 and 5.13 with the time derivative terms set
to zero.
46Cont
47Cont
48Cont
49Momentum Equation
50Momentum equation
51Momentum equation
52Momentum equation
53Momentum Equation
54Momentum equation
55Momentum Equation
566.4 method of characteristics
7.5 and 7.15 allows them to be replaced by four
ordinary differential equations in the x-t plane
x represents the flow direction and t is time .
Much simpler, ordinary differential equations
must be satisfied along two inherent
characteristics direction or paths in the x-t
plane in the characteristics has fallen out of
favor because of difficulties involved in the
supercritical case with the formation of
surges,it has the advantage of being more
accurate and lending a deeper understanding of
the physics of shallow water wave problems as
well as the mathematics is essential in some
explicit finite difference techniques,
specifically for Explaining kinematic wave
routing.
57Cont
Equation 5.15 with the for going simplications is
multiplied alaternatively by the quantity
Equation 7.5
58Cont
59Cont
60Cont
5.18 and 5.19
61Cont
62Cont..
5.20
Tion 5.18 and 5.19
63Cont
5.18 and
In 5.19
64Cont
5.21a
5.21b
5.21c
5.21d
65Cont
In 5.18 and 5.19
66Cont.
67Cont.
68Cont
- Change 7.2 to 5.2 as an example in the
following statement .
69The end !