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CE 261 Open Channel Flow uvm/~imohamme/ce261/

CE 261 Open Channel Flow http://www.uvm.edu/~imohamme/ce261/. Ibrahim Mohammed http://www.uvm.edu/~imohamme imohamme@uvm.edu. Who Am I?. I am a Post Doctorate Associate at the EPSCoR Vermont

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CE 261 Open Channel Flow uvm/~imohamme/ce261/

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  1. CE 261Open Channel Flowhttp://www.uvm.edu/~imohamme/ce261/ Ibrahim Mohammed http://www.uvm.edu/~imohamme imohamme@uvm.edu

  2. Who Am I? • I am a Post Doctorate Associate at the EPSCoR Vermont • (EPSCoR) is a program designed to fulfill the National Science Foundation's (NSF) mandate to promote scientific progress nationwide • My research is focused on physical and statistical hydrological modeling and their relations with climate and land cover

  3. Why we are here? • Open Channel flow is needed for planning, design and operation of water resources projects Mel Martin joins a crowd watching the raging Whetstone Brook surge over the falls in downtown Brattleboro, Vt. on Sunday, Aug. 28, 2011. / AP Photo/The Brattleboro Reformer, Chris Bertelsen

  4. Concepts • The flow of water in a conduit may be either open channel flow or pipe flow • Tunnels, pipes, and aqueducts are closed conduits, whereas rivers, streams, estuaries are open channels.

  5. Unit Conversion • Handy Online Reference (http://www.onlineconversion.com/) Excerpt from (Crowe, 2009)

  6. Comparison between pipe flow and open channel flow (Chow, 1959)

  7. Classifications of free surface flows Steady and Unsteady flow: Time as the Criterion Uniform and Varied flow: Space as the Criterion (Chaudhry, 2008)

  8. (Chow, 1959)

  9. Reynolds Number • Laminar flow is when liquid particles appear to move in definite smooth paths and the flow appears to be as a movement of thin layers on top of each other (free surface laminar flow is extremely rare in real life applications) • Turbulent flow is when liquid particles move in irregular paths which are not fixed with respect to either time or space

  10. Reynolds Number (2) • The relative magnitude of viscous and inertial forces determines whether the flow is laminar or turbulent • Viscous forces dominate, flow is laminar • Inertial forces dominate, flow is turbulent • Transition from laminar to turbulent occurs for Re = 600 mean flow velocity a characteristic length kinematic viscosity of the liquid

  11. Reynolds Number (3) • Characteristic length may be hydraulic depth or hydraulic radius for free surface flows • Hydraulic Depth: flow area divided the water-surface width • Hydraulic Radius: flow area divided by the wetted perimeter

  12. mean flow velocity [L/T] a characteristic length [L] kinematic viscosity of the liquid = μ / ρ = [M/LT] / [M/L3] = [L2/T] = [1] Reynolds number is a dimensionless!!!

  13. Froude Number • When the flow velocity is equal to the velocity of a gravity wave having small amplitude, flow is said to be critical • Flow velocity is less than critical velocity = subcritical flow • Flow velocity is greater than critical velocity = supercritical flow

  14. Froude Number (2) • The Froude number is equal to the ratio of inertial and gravitational forces. For rectangular channel, it is defined as: Subcritical Flow (Fr < 1); critical Flow (Fr = 1); Supercritical Flow (Fr > 1)

  15. Properties of channel cross sections y depth of flow at a section (vertical) d depth of flow section (normal to the flow direction) Z stage B top width A flow area P wetted perimeter R = A/P hydraulic radius D = A/B hydraulic depth (Chaudhry, 2008)

  16. Velocity Distribution • Flow velocity varies due to shear stress at the bottom and at the sides of the channel • Flow velocity may have components in all three Cartesian coordinate directions (vertical and transverse are small) (Chaudhry, 2008)

  17. Velocity Distribution Example • Q. The water velocity in the channel shown in the figure has a distribution across the vertical section equal to . What is the discharge in the channel if the water is 2 m deep (d = 2 m), the channel is 5 m deep, and the maximum velocity is 3 m/s? Example 5.3 from (Crowe, 2009)

  18. Example Solution • Discharge equation is Channel is 5 m wide, so differential area is . Using given velocity distribution,

  19. Energy Coefficient • Since flow velocity in a channel section usually varies from one point to another then: • Flow velocity varies from one point to another due to shear stress at the bottom and at the sides of the channel and due to the pressence of free surface.

  20. Energy coefficient • Coriolis coefficient αis obtained from the Kinetic energy transfer through area ΔA per unit time

  21. Momentum Coefficient • Momentum coefficient accounts for non-uniform velocity distribution. • Boussinesq coefficient

  22. turbulent flow in a straight channel having a rectangular, trapezoidal, or circular cross section, alpha is less than 1.15

  23. Example • The flow velocities measured at various flow depth in a wide rectangular flume are shown in graph. Determine the values of α and β? Flow depth (m) Velocity (m/s) Flow Velocity measured at various flow depths in a wide rectangular flume

  24. Example Solution • Let us consider a unit width of the flume. Then we can replace A in the equations for energy and momentum coefficients (Chaudhry equations 1-7 & 1-14) by the flow depth. Mean flow velocity is:

  25. Example Solution (2) • Using Simpson’s rule for the numerical integration,

  26. Pressure Distribution(Stationary, Vertical) • Pressure distribution in a channel section depends on the flow condition • Pressure intensity is directly proportional to the depth below free surface (Chaudhry, 2008)

  27. Pressure Distribution(Stationary, Horizontal) (Chaudhry, 2008)

  28. Parallel Flow in a Sloping Channel (Chaudhry, 2008)

  29. Curvilinear Flow r = radius of curvature of the streamline, V = flow velocity at the point under consideration (Chaudhry, 2008)

  30. Reynolds Transport Theorem • The Reynolds transport theorem relates the flow variables for a specified fluid mass to that of a specific flow region. • It is utilized to derive the governing equations for steady and unsteady flow conditions. System, control surface, and control volume in a flow field Figure 5.6 (Crowe, 2009)

  31. Reynolds Transport Theorem (2) • Derived by considering the rate of change of an extensive property of a system as it passes through a control volume Figure 5.9 (Crowe, 2009) Extensive property is any property that depends on the amount of matter present. (mass, momentum, energy) Intensive property is any property that is independent of the amount of matter present. (pressure , temperature)

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