- 1. The equations which describe the flow of fluid are derived from three fundamental laws of physics (r/ship of fluid motion): Conservation of matter (or mass) Conservation of energy Conservation of momentum Assumptions have been done with in the control volume in each principles. 1.4. Energy & Momentum Principles in Open Channel Flow
- 2. A. Continuity equation For any control volume during the small time interval dt the principle of conservation of mass implies that the mass of flow entering the control volume minus the mass of flow leaving the control volume equals the change of mass within the control volume. If the flow is steady and the fluid incompressible the mass entering is equal to the mass leaving, so there is no change of mass within the control volume.
- 3. B. Energy Equation oConsider the forms of energy available for the above control volume. If the fluid moves from the upstream face 1, to the downstream face 2 in time dt over the length L. Similarly the total energy per unit weight of section two also computed and consider no energy is supplied between the inlet and outlet of the control volume, energy leaving equal to energy entering.
- 5. C. Momentum Equation The law of conservation of momentum says that a moving body cannot gain or lose momentum unless acted upon by an external force. The resultant force acting on a free body of fluid in any direction is equal to the time rate of change of momentum in that direction The flow may be compressible or incompressible, real (with friction) or ideal (frictionless), steady or unsteady moreover, the equation is not only valid along a streamline
- 6. Energy Equation Momentum Equation Relates the change in energy within the control volume Relates the overall forces on the boundary of the control volume Applicable to only steady flows in which the energy changes are negligible Applicable to steady and unsteady flows The fluid is ideal, incompressible, one dimensional Conditions within the control volume is not taken in to consideration. The theorem is useful when energy changes are known and velocity and pressure distribution are required The theorem is useful when energy changes are unknown and only overall knowledge of the flow is required Used to determine velocity distribution or pressure distribution Useful to determine the resultant force acting on the boundary of flow passage To determine the characteristics of flow when there is abrupt change of flow section ( sudden enlargement in pipe, hydraulic jump..) Useful when detailed information of the flow condition inside the control volume is not known
- 7. Application of Bernoulli's Equation for Uniform Flow interrupted by raised humbs velocity and depth of flow over the raised hump.
- 8. Specific Energy Equation We have to understand the flow condition
- 9. In order to adjust the water level in open channel “Channel Transitions” are incorporated Rise in bed elevation Drop in bed elevation Sudden enlargement in width Sudden Contraction in width The objective here in designing is to minimize the energy loss due to such channel transitions Concept of momentum , continuity and specific energy is used to solve such flow problems
- 10. Specific Energy Considering the energy correction factor= 1 and slope is insignificant The head attained by a fluid element per unit weight wrt channel bed as a datum:-
- 11. The two alternate depths represent two totally different flow regimes: slow & deep on the upper limb of the curve (sub- critical flow) & fast & shallow on the lower limb of the curve, (Super critical flow) Check this
- 13. Salient feature of critical flow Specific energy for a given discharge is minimum. The discharge for a given specific energy is maximum. The Froude number is equal to unity The velocity head is equal to one half of the hydraulic depth
- 14. Example A channel of a rectangular section, 7 m wide, discharges water at a rate of 18 m3/s with an average velocity of 3 m/s. Find: (10 points) A. Specific-energy head of the flowing water, B. Depth of water, when specific energy is minimum, C. Velocity of water, when specific energy is minimum, D. Minimum specific-energy head of the flowing water, E. Type of flow.
- 17. Home Study & Assignment Work Case 2…….When Specific energy is constant Y= f (Q) Critical depth in rectangular channels Critical depth for Non rectangular channels Computation of critical flow
- 18. 1.5. Hydraulic Jump Topics • Definition • Impulse momentum equation • Advantage of Hydraulic jump • Assumptions made for analysis of hydraulic jump
- 19. The hydraulic jump is an important feature in open channel flow and is an example of rapidly varied flow. A hydraulic jump occurs when a super-critical flow and a sub-critical flow meet. The jump is the mechanism for the two surfaces to join. They join in an extremely turbulent manner which causes large energy losses. Because of the large energy losses the energy or specific energy equation cannot be use in analysis, the momentum equation is used instead. If energy actually leaks from the system via frictional head loss the Bernoulli equation will overstate the energy available to the flow and the related predictions of velocity and depth will proportionately be in error. To recall our earlier strategy, we minimize this error by considering only short reaches of channel and only gradual transitions. In certain flow phenomena, however, we simply can no longer ignore the energy losses and we must look to alternative ways of describing the flow.
- 20. When subcritical flow accelerates into the supercritical state the transition often is smooth with gradually increasing velocity and decreasing depth bringing about a smooth drop in the water surface until the alternate depth is achieved. Any disturbance to the water surface is smoothed out by the surface or gravity wave propagation mechanism discussed earlier. In these circumstances energy losses are not great and the Bernoulli equation does a credible job of describing the changes to the flow. When supercritical flow changes to subcritical flow, however, there is no smoothing of the water surface upstream of the transition because the high downstream velocity prevents upstream diffusion of the water-surface deformation. As a result the transition to subcritical flow is sudden and marked by an abrupt discontinuity, or hydraulic jump, in the water.
- 22. Flow over weir ( Look The Linked Video)
- 23. Purposes of hydraulic jump To increase the water level on the d/s of the hydraulic structures To reduce the net up lift force by increasing the downward force due to the increased depth of water, To increase the discharge from a sluice gate by increasing the effective head causing flow For removing air pockets in a pipe line.
- 24. Analysis of Hydraulic Jump Assumptions The length of the hydraulic jump is small, consequently, the loss of head due to friction is negligible, The channel is horizontal as it has a very small longitudinal slope. The weight component in the direction of flow is negligible. The portion of channel in which the hydraulic jump occurs is taken as a control volume & it is assumed the just before & after the control volume, the flow is uniform & pressure distribution is hydrostatic. Objective: To describe (drive) geometry of channel undertaking hydraulic jump Let us consider a small reach of a channel in which the hydraulic jump occurs. The momentum of water passing through section (1) per unit time is given as:
- 25. The momentum of water passing through section (1) per unit time is given as: