1. Elementary Fluid Dynamics: The Bernoulli Equation CEE 331 June 25, 2015 CEE 331 June 25, 2015 

2. Bernoulli Along a Streamline zy x Separate acceleration due to gravity. Coordinate system may be in any orientation! k is vertical, s is in direction of flow, n is normal. Component of g in s direction Note: No shear forces! Therefore flow must be frictionless. Steady state (no change in p wrt time)

3. Bernoulli Along a Streamline 0 (n is constant along streamline, dn=0) Write acceleration as derivative wrt s Chain rule and Can we eliminate the partial derivative?

4. Integrate F=ma Along a Streamline If density is constant… Along a streamline But density is a function of ________. pressure Eliminate ds Now let’s integrate…

5. Bernoulli Equation ä Assumptions needed for Bernoulli Equation ä Eliminate the constant in the Bernoulli equation? _______________________________________ ä Bernoulli equation does not include ä ___________________________ ä Assumptions needed for Bernoulli Equation ä Eliminate the constant in the Bernoulli equation? _______________________________________ ä Bernoulli equation does not include ä ___________________________ Apply at two points along a streamline. Mechanical energy to thermal energy Heat transfer, Shaft Work äFrictionless äSteady äConstant density (incompressible) äAlong a streamline

6. Bernoulli Equation The Bernoulli Equation is a statement of the conservation of ____________________ Mechanical Energy p.e.k.e. Pressure head Elevation head Velocity head Piezometric head Total head Energy Grade Line Hydraulic Grade Line

7. Bernoulli Equation: Simple Case (V = 0) ä Reservoir (V = 0) ä Put one point on the surface, one point anywhere else ä Reservoir (V = 0) ä Put one point on the surface, one point anywhere else z Elevation datum Pressure datum 1 2 Same as we found using statics We didn’t cross any streamlines so this analysis is okay!

8. Mechanical Energy Conserved Hydraulic and Energy Grade Lines (neglecting losses for now) The 2 cm diameter jet is 5 m lower than the surface of the reservoir. What is the flow rate (Q)? z Elevation datum Pressure datum? __________________ Atmospheric pressure Teams z

9. Bernoulli Equation: Simple Case (p = 0 or constant) ä What is an example of a fluid experiencing a change in elevation, but remaining at a constant pressure? ________ Free jet

10. Bernoulli Equation Application: Stagnation Tube ä What happens when the water starts flowing in the channel? ä Does the orientation of the tube matter? _______ ä How high does the water rise in the stagnation tube? ä How do we choose the points on the streamline? ä What happens when the water starts flowing in the channel? ä Does the orientation of the tube matter? _______ ä How high does the water rise in the stagnation tube? ä How do we choose the points on the streamline? Stagnation point Yes!

11. 2a1a 2b 1b Bernoulli Equation Application: Stagnation Tube ä 1a-2a ä _______________ ä 1b-2a ä _______________ ä 1a-2b ä _______________ _____________ ä 1a-2a ä _______________ ä 1b-2a ä _______________ ä 1a-2b ä _______________ _____________ Same streamline Crosses || streamlines Doesn’t cross streamlines z V = f(  p) V = f(z 2 ) V= f(p 2, z 2 )

12. Stagnation Tube ä Great for measuring __________________ ä How could you measure Q? ä Could you use a stagnation tube in a pipeline? ä What problem might you encounter? ä How could you modify the stagnation tube to solve the problem? ä Great for measuring __________________ ä How could you measure Q? ä Could you use a stagnation tube in a pipeline? ä What problem might you encounter? ä How could you modify the stagnation tube to solve the problem? EGL ( defined for a point )

13. Pitot Tubes ä Used to measure air speed on airplanes ä Can connect a differential pressure transducer to directly measure V 2 /2g ä Can be used to measure the flow of water in pipelines ä Used to measure air speed on airplanes ä Can connect a differential pressure transducer to directly measure V 2 /2g ä Can be used to measure the flow of water in pipelines Point measurement!

14. Pitot Tube V V1 =V1 = 1 2 Connect two ports to differential pressure transducer. Make sure Pitot tube is completely filled with the fluid that is being measured. Solve for velocity as function of pressure difference z 1 = z 2 Static pressure tap Stagnation pressure tap 0

15. Bernoulli Normal to the Streamlines Separate acceleration due to gravity. Coordinate system may be in any orientation! Component of g in n direction

16. Bernoulli Normal to the Streamlines 0 (s is constant normal to streamline) R is local radius of curvature and n is toward the center of the radius of curvature

17. Integrate F=ma Normal to the Streamlines If density is constant… Normal to streamline Multiply by dn Integrate

18. n Pressure Change Across Streamlines If you cross streamlines that are straight and parallel, then ___________ and the pressure is ____________. hydrostatic r As r decreases p ______________decreases

19. Relaxed Assumptions for Bernoulli Equation ä Frictionless ä Steady ä Constant density (incompressible) ä Along a streamline ä Frictionless ä Steady ä Constant density (incompressible) ä Along a streamline Small energy loss (accelerating flow, short distances) Or gradually varying Small changes in density Don’t cross curved streamlines

20. Bernoulli Equation Applications ä Stagnation tube ä Pitot tube ä Free Jets ä Orifice ä Venturi ä Sluice gate ä Sharp-crested weir ä Stagnation tube ä Pitot tube ä Free Jets ä Orifice ä Venturi ä Sluice gate ä Sharp-crested weir Applicable to contracting streamlines (accelerating flow). Teams

21. Ping Pong Ball Why does the ping pong ball try to return to the center of the jet? What forces are acting on the ball when it is not centered on the jet? How does the ball choose the distance above the source of the jet? n r Teams

22. Summary ä By integrating F=ma along a streamline we found… ä That energy can be converted between pressure, elevation, and velocity ä That we can understand many simple flows by applying the Bernoulli equation ä However, the Bernoulli equation can not be applied to flows where viscosity is large or where mechanical energy is converted into thermal energy. ä By integrating F=ma along a streamline we found… ä That energy can be converted between pressure, elevation, and velocity ä That we can understand many simple flows by applying the Bernoulli equation ä However, the Bernoulli equation can not be applied to flows where viscosity is large or where mechanical energy is converted into thermal energy.

23. Jet Problem ä How could you choose your elevation datum to help simplify the problem? ä How can you pick 2 locations where you know enough of the parameters to actually find the velocity? ä You have one equation (so one unknown!) ä How could you choose your elevation datum to help simplify the problem? ä How can you pick 2 locations where you know enough of the parameters to actually find the velocity? ä You have one equation (so one unknown!)

24. Jet Solution The 2 cm diameter jet is 5 m lower than the surface of the reservoir. What is the flow rate (Q)? z Elevation datum Are the 2 points on the same streamline?

25. Example: Venturi

26. How would you find the flow (Q) given the pressure drop between point 1 and 2 and the diameters of the two sections? You may assume the head loss is negligible. Draw the EGL and the HGL over the contracting section of the Venturi. 1 2 hhhh How many unknowns? What equations will you use?

27. Example Venturi