# Fluid Mechanics-61341

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1 An-Najah National University College of Engineering Fluid Mechanics Chapter [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed 1 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

2 Euler s Equation 2 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

3 Euler s Equation 3 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

4 Euler s Equation 4 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

5 Bernoulli s Equation 5 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

6 Bernoulli s Equation Pressure head Velocity head Elevation head Constant 6 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

7 The Energy Line (EL) and the Hydraulic Grade Line (HGL) Each term in the Bernoulli s equation is a type of head P/g = Pressure Head V 2 /2g n = Velocity Head Z = Elevation head EL is the sum of these three heads HGL is the sum of the elevation and the pressure heads 7 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

8 The Energy Line (EL) and the Hydraulic Grade Line (HGL) 8 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

9 The Energy Line (EL) and the Hydraulic Grade Line (HGL) Understanding the graphical approach of EL and HGL is key to understanding what forces are supplying the energy that water holds 1 P/ g V 2 /2g Z EL HGL Q 2 V 2 /2g P/g Z Point 1: Majority of energy stored in the water is in the Pressure Head Point 2: Majority of energy stored in the water is in the elevation head If the tube was symmetrical, then the velocity would be constant, and the HGL would be level 9 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

10 Bernoulli s Equation (Uniform Cross Section) For uniform cross sections streamtubes, the velocity a cross the entire section is uniform as a result Bernoulli s equation becomes: 10 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

11 Example 1 11 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

12 Example 1 (Solution) 12 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

13 Application of Bernoulli s Equation Torricelli s theorem 13 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

14 Torricelli s Theorem An ideal fluid is one that is incompressible and has no resistance to shear stress. Ideal fluids do non actually exist, but sometimes it is useful to consider what happen to an ideal fluid in a particular fluid flow problem in order to simplify the problem 14 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

15 Torricelli s Theorem Taking the datum at the center of the nozzle and choosing the center streamline give h = z + p/g in the reservoir where velocities are negligible Writing Bernoulli s equation for a streamline between the reservoir and the tip of the nozzle shown as in Fig. 5.4 p1 z 1 h p2 V 2g 2 n, Torricelli's equation resultsif p 2 0 h V 2g 2 n V 2g n h 15 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

16 For freely falling body Torricelli s Theorem as u V h g V g V h h g V n n n Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid 16 Dr. Sameer Shadeed

17 Torricelli s Theorem (Free Jets) The velocity of a jet of water is clearly related to the depth of water above the hole The greater the depth, the higher the velocity 17 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

18 Example 2 18 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

19 Example 2 (Solution) 19 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

20 Example 2 (Solution) 20 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

21 Example 2 (Solution) 21 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

22 Example 2 (Solution) 22 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

23 Example 3 23 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

24 Example 3 (Solution) 24 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

25 Example 3 (Solution) 25 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

26 Application of Bernoulli s Equation 26 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

27 Application of Bernoulli s Equation stagnation point 27 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

28 Stagnation Points On any body in a flowing fluid, there is a stagnation point. Some fluid flows over and some under the body. The dividing line (the stagnation streamline) terminates at the stagnation point. The velocity decreases as the fluid approaches the stagnation point. The pressure at the stagnation point is the pressure obtained when a flowing fluid is decelerated to zero speed stagnation point 28 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

29 Example 4 29 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

30 Example 4 (Solution) 30 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

31 Example 5 Determine the difference in pressure between points 1 and 2. Hint: Point 1 is called a stagnation point, because the air particle along that streamline, when it hits the biker s face, has a zero velocity 31 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

32 Example 5 (Solution) Assume a coordinate system fixed to the bike (from this system, the bike is stationary, and the world moves past it). Therefore, the air is moving at the speed of the bike. Thus, V 2 = Velocity of the Biker Apply Bernoulli s equation from 1 to 2 Point 1 = Point 2 P 1 /g air + V 12 /2g + z 1 = P 2 /g air + V 22 /2g + z 2 Knowing the z 1 = z 2 and that V 1 = 0, we can simplify the equation P 1 /g air = P 2 /g air + V 22 /2g P 1 P 2 = ( V 22 /2g ) g air 32 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

33 Example 5 (Solution) If the Biker is traveling at 5 m/s, what pressure does he feel on his face if the g air = N/m 3? We can assume P 2 = 0, because it is only atmospheric pressure P 1 = ( V 22 /2g )(g air ) P 1 = ((5) 2 /(2(9.81)) x P 1 = 15.3 N/m 2 (gage pressure) If the biker s face has a surface area of 300 cm 2 He feels a force of 15.3 x 300x10-4 = 0.46 N 33 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

34 Application of Bernoulli s Equation 34 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

35 Application of Bernoulli s Equation 35 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

36 Example 6 36 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

37 Example 6 (Solution) 37 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

38 Example 6 (Solution) 38 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

39 Example 6 (Solution) 39 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

40 Application of Bernoulli s Equation 40 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

41 Application of Bernoulli s Equation 41 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

42 Example 7 42 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

43 Example 7 (Solution) 43 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

44 Example 7 (Solution) 44 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

45 Example 7 (Solution) 45 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

46 Example 7 (Solution) 46 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

47 Example 8 47 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

48 Example 8 (Solution) 48 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

49 Example 8 (Solution) 49 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

50 The Work Energy Equation 50 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

51 The Work Energy Equation 51 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

52 The Work Energy Equation 52 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

53 Example 9 53 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

54 Example 9 (Solution) 54 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

55 Example 9 (Solution) 55 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

56 Example 10 Calculate the power output of this turbine 56 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

57 Example 10 (Solution) 57 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

58 Example 10 (Solution) 58 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

59 Example 11 Water is pumped from a large lake into an irrigation canal of rectangular cross section 3 m wide, producing the flow situation shown in the figure. Calculate the required pump power assuming ideal flow. 59 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

60 Example 11 (Solution) 60 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

61 Example 11 (Solution) 61 Fluid Mechanics-2nd Semester [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed

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