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Unit III Rapidly Varied Flows

Unit III Rapidly Varied Flows. The hydraulic jump is the phenomenon that occurs where there is an abrupt transition from super critical flow to sub critical flow. The most important factor that affects the hydraulic jump is the Froude number.

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Unit III Rapidly Varied Flows

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  1. Unit IIIRapidly Varied Flows

  2. The hydraulic jump is the phenomenon that occurs where there is an abrupt transition from super critical flow to sub critical flow. The most important factor that affects the hydraulic jump is the Froude number. The most typical cases for the location of hydraulic jump are: Jump below a sluice gate. Jump at the toe of a spillway. Jump at a glacis. (glacis is the name given to sloping floors provided in hydraulic structures.) Hydraulic Jump- Define

  3. Example Jump at a glacis Jump below a sluice gate Jump at the toe of a spillway

  4. General Expression for Hydraulic Jump: In the analysis of hydraulic jumps, the following assumptions are made: The length of hydraulic jump is small. Consequently, the loss of head due to friction is negligible. The flow is uniform and pressure distribution is due to hydrostatic before and after the jump. The slope of the bed of the channel is very small, so that the component of the weight of the fluid in the direction of the flow is neglected. Hydraulic Jump- Assumption

  5. Comments: • This is the general equation governing the hydraulic jump for any shape of channel. • The sum of two terms is called specific force (M). So, the equation can be written as: • M1 = M2 • This equation shows that the specific forcebefore the hydraulic jump is equal to that after the jump. 6

  6. Classification of the Jump • The hydraulic jump can be classified based on initial Froude number as • Undular (F1 = 1.0 − 1.7) • Weak (F1 = 1.7 − 2.5) • Oscillating (F1 = 2.5 − 4.5) • Steady (F1 = 4.5 − 9.0), and • Strong (F1 > 9.0)

  7. Classification of the Jump

  8. Classification of the Jump

  9. Applications of the Hydraulic Jump

  10. Define: Surge A surge is a moving wave front which results in an abrupt change of the depth of flow. It is a rapidly varied unsteady flow condition Two Types Positive – which results in an increase depth of flow Negative – Which results in decrease depth of flow

  11. Positive surge Type B – Positive surge (Advancing Upstream) Ex: Tail gate closed suddenly. Type A – Positive surge (Advancing Downstream) Ex: Head Gate is opened suddenly.

  12. Negative Surge Type C – Negative Surge (Retreating Downstream) Ex: Head Gate is closed suddenly. Type D – Negative Surge (Retreating Upstream) Ex: Tail gate opened suddenly.

  13. Assumptions • Channel is horizontal and frictionless; • Pressure distribution is hydrostatic at locations away from the front; • Velocity is uniform within the cross section, at location away from the front; • Change in the flow depth at the front occurs over a very short distance; • Water surfaces behind and ahead of the wave front are parallel to the bed.

  14. Case A: Surge due to sudden in crease of flow For example, consider the movement of a positive surge wave in x-direction in an open channel having an irregular cross section. Here, as the surge moves with an absolute velocity, Vw, flow depth becomes equal to y2 behind the surge. Undistributed flow depth ahead of the surge is y1. The corresponding flow velocities behind and ahead of the slope front are V2 and V1 respectively. The surge has been created due to a sudden change of flow rate from Q1 to Q2.

  15. Surge due to sudden in crease of flow Absolute Velocity of Surge Wave To make it to steady flow , apply Vw in opposite direction to V1 and V2 and the surge.

  16. (1) (2) (3) (4)

  17. Applying momentum equation to the control volume of Fig (5) (6) Sub Eq. 2 in Eq. 6 (7)

  18. Sub Eq. 3 in Eq. 7 and subsequent simplification leads to (8) (9) (10)

  19. Now, substitution of Eq. (4) in Eq. (7) and subsequent simplification leads to (11) Equations (10) and (11) can be used to determine the surge wave velocity and the surge height, if we know the values of undisturbed flow depth, y1, flow rate before the surge, Q1, and the flow rate after the surge, Q2. Equations (10) and (11) are non-linear equations. They can be solved by an appropriate numerical technique. For rectangular channels, Eqs. (10) and (11) simplify to the following. (12) (13)

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