An-Najah National University College of Engineering Fluid Mechanics-61341 Chapter [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed 1 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Euler s Equation 2 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Euler s Equation 3 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Euler s Equation 4 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Bernoulli s Equation 5 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Bernoulli s Equation Pressure head Velocity head Elevation head Constant 6 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
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 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
The Energy Line (EL) and the Hydraulic Grade Line (HGL) 8 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
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 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
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 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 1 11 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 1 (Solution) 12 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Application of Bernoulli s Equation Torricelli s theorem 13 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
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 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
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 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
For freely falling body Torricelli s Theorem as u V 2 2 2 2 h g V g V h h g V n n n 2 2 2 0 2 2 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid 16 Dr. Sameer Shadeed
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 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 2 18 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 2 (Solution) 19 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 2 (Solution) 20 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 2 (Solution) 21 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 2 (Solution) 22 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 3 23 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 3 (Solution) 24 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 3 (Solution) 25 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Application of Bernoulli s Equation 26 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Application of Bernoulli s Equation stagnation point 27 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
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 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 4 29 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 4 (Solution) 30 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
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 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
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 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 5 (Solution) If the Biker is traveling at 5 m/s, what pressure does he feel on his face if the g air = 12.01 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 12.01 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 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Application of Bernoulli s Equation 34 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Application of Bernoulli s Equation 35 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 6 36 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 6 (Solution) 37 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 6 (Solution) 38 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 6 (Solution) 39 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Application of Bernoulli s Equation 40 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Application of Bernoulli s Equation 41 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 7 42 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 7 (Solution) 43 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 7 (Solution) 44 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 7 (Solution) 45 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 7 (Solution) 46 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 8 47 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 8 (Solution) 48 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 8 (Solution) 49 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
The Work Energy Equation 50 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
The Work Energy Equation 51 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
The Work Energy Equation 52 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 9 53 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 9 (Solution) 54 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 9 (Solution) 55 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 10 Calculate the power output of this turbine 56 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 10 (Solution) 57 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 10 (Solution) 58 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
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 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 11 (Solution) 60 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed
Example 11 (Solution) 61 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed