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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.

2
Introduction
  • 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.

3
Introduction
  • 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.

4
Introduction
  • 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.

5
Introduction
  • 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.

6
Introduction
  • 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.

7
Introduction
  • 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
  •  

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
15
Development 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

16
Development 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.
17
Development 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.

18
Development of St.Venant Equations
19
Continuity Equation
  • k

5.1
20
(No Transcript)
21
Cont
Figure 5.1
22
Cont
23
Cont
5.2
Substituting dA Bdy from figure 5.2 in which B
channel top width at free surface,continuity
becomes
24
Cont
Ity in the flow direction x ,the ?Q/
? X in 5.3 can be written as
25
Cont
5.4
26
Cont
  • 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
27
Cont
28
Cont
5.6
29
Cont
5.7
30
Cont
31
Cont
32
Cont
33
Cont
34
Cont
7.9
35
Cont
7.10
36
Cont
5.11
37
Cont
5.12
38
Cont
  • Substituting equation 5.8 and 5.12 into equation
    5.7 dividing by ??x ,and letting ?x go to zero
    results in

5.13
39
Cont
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
40
Cont.
  • 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

41
Cont.
5.14
42
Cont.
5.15
43
Cont
5.15
44
Cont
5.16
45
Cont
5.2 and 5.13 with the time derivative terms set
to zero.
46
Cont
47
Cont
48
Cont
49
Momentum Equation
50
Momentum equation
51
Momentum equation
52
Momentum equation
53
Momentum Equation
54
Momentum equation
55
Momentum Equation
56
6.4 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.
57
Cont
Equation 5.15 with the for going simplications is
multiplied alaternatively by the quantity
Equation 7.5
58
Cont
59
Cont
60
Cont
5.18 and 5.19
61
Cont
62
Cont..
5.20
Tion 5.18 and 5.19
63
Cont
5.18 and
In 5.19
64
Cont
5.21a
5.21b
5.21c
5.21d
65
Cont
In 5.18 and 5.19
66
Cont.
67
Cont.
68
Cont
  • Change 7.2 to 5.2 as an example in the
    following statement .

69
The end !
  • Thank you !
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