Chapter 4: Non uniform flow in open channels

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Chapter 4Non uniform flow in open channels
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Learning outcomes

• By the end of this lesson, students should be
  able to
• Relate the concept of specific energy and
  momentum equations in the effect of change in bed
  level - Broad Crested Weir
• Relate the concept of specific energy and
  momentum equations in the effect of lateral
  contraction of channel ( Venturi Flume)

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Introduction

• Analysis of steady non uniform flow in open
  channels.
• Non uniform flow occurs in transitions where
  there is change in cross section or obstruction
  in channel.
• Analysis requires a different approach, requiring
  the use of the energy equation in a different
  form.

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Specific energy alternative depths of flow

• Specific energy, E,
• (16.1)
• For a wide rectangular channel, mean velocity is,
• While the volume rate of flow per unit width,

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• Substituting v q into E,
• (16.2)
• (16.3)
• This equation has 3 roots
• 1 root is negative unreal
• 2 roots are positive real, which give 2
  alternate depths
• Larger depth deep slow flow (subcritical/
  tranquil/ streaming flow).
• Smaller depth shallow fast flow (supercritical/
  shooting flow)

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• Critical depth, DC
• Depth at which the 2 roots coincide
• q qmax
• E Emin
• To find DC
• When dE/dD 0,
• (16.4)

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• Sub. from (16.4) to (16.2),
• Therefore, for rectangular channel,
• Differentiating (16.3) assuming E is constant
• (16.5)

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• When , (16.5) becomes,
• Or (16.6)

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• Critical velocity velocity of flow corresponding
  to critical depth.
• Sub. into (16.1),

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DC for non rectangular sections
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• For any shape and cross sectional area the E for
  any D,
• Since v Q/A,
• (16.7)

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• For flow at DC and vC, Emin, differentiating
  (16.7),
• (16.8)
• But a change in depth will produce a change in
  cross-sectional area, therefore dA/dDB.
• (16.9)

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• For critical flow,
• From (16.9)
• (16.10)
• where is the average depth.

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Froude Number

• Assume a surface wave of height dZ is propagated
  from left to right of observer.
• Wave is brought to rest relative to observer by
  imposing a velocity c equal to wave velocity on
  the observer, flow will appear steady.

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• Sub. du to (5.32),
• If wave height dZ is small,

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• Ratio of the stream velocity u to the propagation
  velocity c-u is known as Froude Number Fr.
• (5.34)
• Fr can be used to determine the type of flow for
  open channel.

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• For critical flow conditions, the Froude Number
  is,
• If v lt vc subcritical flow
• If v gt vc supercritical flow
• Important difference,
• Subcritical disturbances can travel upstream
  and downstream thus enabling downstream
  conditions to determine the behavior of the flow.
• Supercritical disturbances cannot travel
  upstream and thus downstream cannot control the
  behavior of the flow.

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Tranquil Critical Shooting
Subcritical Critical Supercritical
Depth D gt DC D DC D lt DC
Velocity v lt vC v vC v gt vC
Fr Fr lt 1 Fr 1 Fr gt 1
Channel slope Mild Critical Steep
Control Downstream - Upstream
Disturbance Wave can travel upstream Standing waves Waves cannot travel upstream
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• Figure 4.1 shows that when flow is in the region
  of
• Dc , small changes of E and q results in
  relatively large changes in D.
• Small surface waves are therefore easily formed
  but since velocity of propagation vp vc, these
  waves will be stationary or standing waves.
• Their presence therefore, an indication of
  critical flow conditions

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• Figure 4.1a Plot of D vs q for a constant E

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• Line OA from figure below can be drawn at 45
  through the origin.
• If scales for E and D are the same, horizontal
  distances from vertical axis to line OA will be
  equal to D, and the distance from OA to the
  specific energy curve will be v2/2g.
• If q is constant,
• Tranquil flow
• D increases, v increases, E curve is asymptotic
  to OA
• Shooting flow
• D decreases, v increases, E curve will be
  asymptotic to the E axis.

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• Which of the two alternate depths for a given E
  will occur at a cross section depends on the
  slope of the channel.
• Critical slope, sc, is defined as the slope of
  the channel which will maintain flow at critical
  depth, DC.
• Uniform tranquil s lt sc mild slope
• Uniform shooting flow s gt sc steep slope

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Example 4.1

• A rectangular channel 8 m wide conveys water at
  a rate of 15 m3/s. If the velocity in the channel
  is 1.5 m/s, determine
• a) E
• b) DC
• c) vc
• d) Emin
• e) Type of flow

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Example 4.2

• Determine the critical depth in the trapezoidal
  channel shown below if the discharge in the
  channel is 0.34 m3/s. The channel has side slopes
  with a vertical to horizontal ratio of 11.

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Example 4.3

• Determine the critical depth in a channel of
  triangular cross section conveying water at a
  velocity of 2.75 m/s and at a depth of 1.25 m.
  The channel has side slopes of 12.

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Exercise

• A channel has a trapezoidal cross-section with a
  base width of 0.6 m and sides sloping at 450.
  When the flow along the channel is 20 m3 min-1,
  determine the critical depth. (0.27 m)

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Control sections

• Control sections cross sections at which the
  flow passes through the critical depth.
• Such sections are limiting factor in the design
  of channel. and some of the cases in which they
  occur are
• Transition from tranquil to shooting flow
• Entrance to a channel of steep slope from a
  reservoir
• Free outfall from a channel with a mild slope
• Change in bed level or change in width of channel

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Transition from tranquil to shooting flow
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• May occur where there is a change of bed slope s.
• Upstream slope is mild and s is less than the
  critical slope sc.
• Over s considerable distance the depth will
  change smoothly from D1 to D2.
• At the break in the slope, the depth will pass
  through DC forming a control section which
  regulates the upstream depth.
• At the tail end, the reverse transition from
  shooting to tranquil flow occurs suddenly by
  means of a hydraulic jump.

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Entrance to a channel of steep slope from a
reservoir

• If depth of flow in the channel is less than DC
  for the channel, water surface must pass through
  DC in the vicinity of the entrance, since
  conditions in the reservoir correspond to
  tranquil flow.

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Free Outfall from a Channel with a Mild Slope
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• If slope s of the channel is less than sc the
  upstream flow will be tranquil.
• At the outfall, theoretically, the depth will be
  critical, DC.
• In practice, gravitational acceleration will
  cause an increase of velocity at the brink so
  that D lt DC.
• While experiments indicate that depending on the
  slope upstream
• DC occurs at distance of between 3DC to 10DC from
  the brink.
• D at the brink is approximately 0.7DC.

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Flow over a broad-crested weir

• Broad-crested weir is an obstruction in the form
  of a raised portion of the bed extending across
  the full width of the channel with a flat upper
  surface or crest sufficiently broad in the
  direction of flow for the surface of the liquid
  to become parallel to the crest.
• Upstream edge is rounded to avoid separation
  losses that occur at a sharp edged.
• The flow upstream of the weir is tranquil and the
  conditions downstream of the weir allow a free
  fall over the weir.

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• The discharge over the weir will be, therefore,
  be the maximum possible and flow over the weir
  will take place at DC.
• For a rectangular channel,
• (16.11)

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• The specific energy, E, measured above the crest
  of the weir will be (assuming no losses),
• H is the height of the upstream water level
  above the crest and v is the mean velocity at a
  point upstream where flow is uniform.
• If the upstream depth is large compared with the
  depth over the weir, (v2/2g) is negligible,
  therefore,
• Rewriting (16.11),
• (16.12)

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• A single measurement of the head, H above the
  crest of the weir would then be sufficient to
  determine Q.
• Since , the depth over the crest
  of the weir is fixed, irrespective of its height.
  
• Any increase in the weir height will not change
  DC but will cause an increase in the depth of the
  flow upstream.
• Therefore, maximum height of the weir,

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• If the level of the flow downstream is raised,
  the surface level will be drawn down over the
  hump, but the depth may not fall to the critical
  depth.
• The rate of flow can be calculated by applying
  Bernoullis s Equation and continuity equation
  and depends on the difference in surface level
  upstream and over the weir.

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Example 4.4

• A broad crested weir 500 mm high is used to
  measure the discharge in a rectangular channel.
  The width of the channel is 20 m and the height
  of the channel upstream of the weir is 1.25 m.
  What is the discharge in the channel if water
  falls freely over the weir? Assume that the
  velocity upstream is very small. Determine the
  difference in water level between upstream and
  over the top of the weir.

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Effect of lateral contraction of a channel

• When width of a channel is reduced while bed
  remains flat, q increases.
• As channel narrows - neglecting losses, E remains
  constant for tranquil flow, depth will decrease
  while for shooting flow depth will increases.

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Free surface does not pass through DC
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• Arrangement forms a venturi flume (venturi
  meter).
• Applying energy equation between upstream
  throat and ignoring losses,

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• Actual discharge,
• Cd is a coefficient of discharge 0.95 to 0.99

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Free surface passes through DC
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• The flowrate is given by,
• Assuming that the upstream velocity head is
  negligible,
• (16.15)
• where H is the of the upstream free surface
  above bed level at the throat.

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Lateral contraction with hump
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• Height of upstream water level above the hump, H
  D1 Z
• When upstream conditions are tranquil and the bed
  slope is the same downstream as upstream,
  impossible for shooting flow to be maintained for
  any great distance from the throat.
• Revert to tranquil flow downstream by means of a
  hydraulic jump or standing wave.
• Venturi flume operating in this mode is known as
  standing wave flume.

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Example 4.5

• A venturi flume is constructed in a channel
  which is 3.5 m wide. If the throat width in the
  flume is 1.2 m and the depth upstream from the
  constriction is 1.25 m , calculate the discharge
  in the channel when the depth at the throat is
  1.2 m. If the conditions are such that a standing
  wave is formed, what is the discharge?

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Example 4.6

• A Venturi flume is 2.5m wide and 1.4m deep
  upstream with a throat width of 1.3m. Assuming
  that a standing wave form downstream, calculate
  the rate of flow of water if the discharge
  coefficient is 0.94. Do not ignore the velocity
  of approach.

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Review of past semesters questions
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APR 2010

• A 10 m wide channel conveys 25 m3/s of water at a
  depth of 1.6 m. Determine
• i) specific energy of the flowing water
• ii) critical depth, critical velocity and
  minimum specific energy

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APR 2010

• A venturi flume is 1.40 m wide at the entrance
  and 0.7 m wide at the throat. Determine the flow
  if the depths at the entrance and at the throat
  is 0.8 m and 0.6 m respectively. Neglect all
  losses.