...Eq(11.6) The energy loss in the jump is dependent on the two depths y 1 and y 2 3 = E =...Eq(11.7)

Transcription

1 . Open Channel Flow Contd.5 Hydraulic Jump A hydraulic jump occurs when water in an open channel is flowing supercritical and is slowed by a deepening of the channel or obstruction in the channel. The slowing causes the water to suddenly jump to the other specific energy state (slow, deep flow). One example of the formation of a hydraulic jump is at the downstream of a dam spillway (See figure below). Figure.4: Hydraulic jump at the bottom of a dam spillway For a hydraulic jump to occur the flow before the jump must be super-critical (fast and thin). That is, in the figure above y must be less than the critical depth y c and the Froude Number must be greater than (Fr > ). The downstream depth after the jump can be calculated by the following equation: y y = [ (+ 8Fr ) ].....Eq(.6) The energy loss in the jump is dependent on the two depths y and y (y y) E E = E = Eq(.7) 4yy The figure below illustrates what happens in a hydraulic jump using a Specific Energy curve. The flow enters the jump at the super-critical depth of y and specific energy E. In the jump the depth increases abruptly. If there is no energy loss the depth after the jump would have been y as this is the alternate depth for the Energy E. However, there is energy dissipation at the jump. Because of this energy loss E, the actual depth corresponds to y with an energy level of E which is less than E. F:\ENGG5009_0\Word\Notes\L4 Open Channel Flow - Part.docx 58

2 This surge/jump can move up or downstream of the channel at a velocity given by gy y Speed of the surge = (+ ) Eq(.8) y If the speed of this surge = Upstream velocity, the jump is stationary. Conversely it may travel up- or down-stream if the surge speed is less than or greater than the upstream velocity. y and y are known as the conjugate depths for the hydraulic jump. Example.: Water is discharged under a sluice gate from a reservoir into a horizontal rectangular channel at a rate of 8m³/s. The channel is m wide and is made of unfinished formed concrete. A hydraulic jump is formed at a point where the water depth is m. Determine (a) The velocity before the jump (b) the depth after the jump (c) the velocity after the jump (d) the energy dissipated in the jump. F:\ENGG5009_0\Word\Notes\L4 Open Channel Flow - Part.docx 59

3 .6 Open Channel flow measurement Two widely used devices used to measure flow in open channels are Weirs and Flumes. In each case the cross section of the flow is changed at the structure which in turn causes a change in the water level. This water level relative to a point on the structure is then used to relate to the flow rate in the channel..6. Weirs A weir is a barrier that is placed across the open channel so that the water flows as a free jet into the channel beyond the barrier. The typical design of a weir has a sharp edge on the upstream side and allows the flow to spring away as a free jet called nappe with good aeration below the nappe. A well-designed weir flow is shown in the figure below. The head H above the weir crest drives the flow and the flow rate is expressed as a function of this head. This relationship is known as the rating equation and plot of Head Vs. Discharge is known as a rating curve. By measuring the head for any given flow, one is able to estimate the flow rate over the weir using the rating curve/equation. Conversely, the head required to provide a given flow rate too could be read off the rating curve. Figure.5: Flow over a well-designed weir Various notch shapes are used in weirs. Some of the common cross sections are shown in the figure below. Rectangular weir Contracted weir Cipoletti weir V notch weir Rectangular weir: This is also known as a suppressed weir. The crest width L extends to full width of the channel in which it is installed. F:\ENGG5009_0\Word\Notes\L4 Open Channel Flow - Part.docx 60

4 Contracted weir: This has sides extended inward from the sides of the channel. The water must contract as it flows around the sides of the weir thereby decreasing the effective length of the weir. Weir length is thus adjusted in the equation (see table below). Cipoletti weir: Although this is also contracted from the sides of the channel, the opening edges slope outward thus negating the contraction seen in the rectangular weir. Weir length in the equation need not be adjusted. V notch weir: Angles from 5 to 0 are acceptable. The table below gives the Rating Equations for the various weir shapes. Table.: Rating equations for weirs Weir Equation Notes Rectangular C = Coefficient of discharge Q = CDL g H Rectangular n=0 fully suppressed, n= one side Q =.84( L = 0. nh Contracted )H contraction, n= fully contracted. Cipoletti Q =.84L H V notch θ = 4.8C tan H 5 Q C = Coefficient of discharge.6. Flumes Flumes make use of the relationship between the flow rate and critical flow depth to estimate the flow in a channel. Flumes are contractions placed in the flow channels to achieve its critical depth within this structure. For some pictures of flumes see the power point presentation material for this lecture. Critical Flow flumes use the unique relationship between discharge and flow depth when critical flow occurs. The flumes are designed to cause the flow to achieve its critical depth within the flume structure. Flumes and orifices are seldom used these days to measure flow rates. Instead, current meters, ultrasonic flow meters, magnetic flow meters are the more commonly used advanced approaches for measuring flow. F:\ENGG5009_0\Word\Notes\L4 Open Channel Flow - Part.docx 6

5 .7 Slope classification An important derivation from the specific energy equation (refer to Eqn.4 and Fig.) is that at minimum Specific energy E c, the flow depth is the critical depth y c and the relationship between flow rate and depth at this state can be expressed as (for a rectangular cross section only) y c Q q = = B g g.....eq(.9) Where Q :Discharge (m³/s); B: Width of channel (m); q: Discharge per unit width of channel(m²/s). Proof: The associated Specific energy is E min = y c Steep slope: When the normal depth < Critical depth Y n < y c Steep slope Mild slope: When the normal depth > Critical depth Y n > y c Mild slope Critical slope: When the normal depth=critical depth Y n = y c Critical slope Example.4: A rectangular channel 4m wide and at a slope of :500 carries water at a flow rate of 4.9m³/s. The Manning s n for the channel bed is 0.0. Determine the following: (i) Normal depth of flow (Hint: Use Manning s Equation) (ii) Critical depth for this flow rate (Hint : Use Eqn.9) (iii) The type of slope of this channel (Hint: Use criteria above) (iv) Froude Number of the flow (Hint : Fr = V/ (gh)) (v) What type of flow is this? (Hint: Table.) F:\ENGG5009_0\Word\Notes\L4 Open Channel Flow - Part.docx 6