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
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21
Cont
Figure 5.1
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Cont
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Cont
5.2
Substituting dA Bdy from figure 5.2 in which B
channel top width at free surface,continuity
becomes
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Cont
Ity in the flow direction x ,the ?Q/
? X in 5.3 can be written as
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Cont
5.4
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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
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Cont
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Cont
5.6
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Cont
5.7
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Cont
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Cont
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Cont
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Cont
34
Cont
7.9
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Cont
7.10
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Cont
5.11
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Cont
5.12
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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
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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

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Cont.
5.14
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Cont.
5.15
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Cont
5.15
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Cont
5.16
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Cont
5.2 and 5.13 with the time derivative terms set
to zero.
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Cont
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Cont
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Cont
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Momentum Equation
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Momentum equation
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Momentum equation
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Momentum equation
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Momentum Equation
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Momentum equation
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Momentum Equation
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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.
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Cont
Equation 5.15 with the for going simplications is
multiplied alaternatively by the quantity
Equation 7.5
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Cont
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Cont
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Cont
5.18 and 5.19
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Cont
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Cont..
5.20
Tion 5.18 and 5.19
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Cont
5.18 and
In 5.19
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Cont
5.21a
5.21b
5.21c
5.21d
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Cont
In 5.18 and 5.19
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Cont.
67
Cont.
68
Cont

• Change 7.2 to 5.2 as an example in the
  following statement .

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The end !

• Thank you !