1. CE 3305 Engineering FLUID MECHANICS
Lecture 5: flow Visualization; Euler's equation

2. Outline
Flow fields
Euler’s Equation
Application to Practical Cases

3. Flow patterns Flow patterns and visualization in real fluids
Use markers such as: dye; smoke; heat are used to “see” how fluid moves
The markers are “tracers,” and the tracer hypothesis is used to infer behavior of the flow field
Timeline: a line formed by marking adjacent particles at some instant
Pathline: the trajectory of a particular fluid particle
Streakline: the trajectories of many different particles that pass through a common point in space
Streamline: a line in a flow field that is tangent to velocity (at that point). Flow does not cross a streamline. (3D equivalent is called a streamtube)
<visualization video>

4. Flow patterns
Uniform flow: a flow field where velocity does not change along a streamline (velocity does not vary with position)
Non-uniform flow: a flow field where velocity does vary with position.

5. Velocity field LaGrangian: consider an individual fluid particle
Eulerian: consider a location in space
Dimensionalit y (pg 121)

6. Velocity field
Velocity of each particle expressed all at once is called the velocity field

7. Velocity field
Usual convention is to use u,v,w as the velocity functions in a cartesian system

8. Velocity field Typically represented in: Cartesian coordinates
Pathline coordinates.

9. Flow types Uniform flow: velocity constant along a pathline
Non-uniform: velocity varies with position (Niagara Falls in a Canoe)
Steady flow: velocity constant in time
Unsteady flow: velocity varies with time

10. Flow types Laminar flow: velocity constant along a pathline
Turbulent: velocity varies with position (Niagara Falls in a Canoe)
Viscous: shear stresses impact flow
Inviscid: shear stress small enough to ignore
<turbulent and laminar video>

11. Flow types Boundary layer, wake, potential (inviscid) flow regions
<flow around cylinder video>

12. Flow types
Boundary layer, wake, potential (inviscid) flow regions

13. acceleration Recall from Mechanics: F = m a
Acceleration is vital to relate forces to flow

14. acceleration Recall from Mechanics: F = m a
Acceleration is vital to relate forces to flow

15. Particle kinematics (lagrangian)
Individual particles

16. Fluid kinematics (eulerian)
Pick a point in space; how does fluid behave at that point?

17. Eulerian reference system
Pick a point in space

18. Eulerian reference system
Pick a point in space

19. Eulerian reference system
Pick a point in space

20. Euler’s equation
We have just created a way to examine the acceleration vector in the context of a Eulerian reference frame

21. Example (application of definitions)
Problem Statement

22. Example (application of definitions)
Known:
Geometry; Velocity field (functions); Linear variation along nozzle

23. Example (application of definitions)
Unknown:
Local acceleration at ½ distance along nozzle at time t=2 seconds. (either a numerical value, or a function)

24. Example (application of definitions)
Governing Equations:
Definition of position, velocity, and acceleration in Eulerian system

25. Example (application of definitions)
Solution:

26. Example (application of definitions)
Solution:

27. Example (application of definitions)
Solution:

28. Example (application of definitions)
Discussion:
This problem is really an application of definitions and calculus
The acceleration in this example is a function of time and position, but the local part is a constant at any location (not the same constant).

29. Example (application of euler’s equation)
Problem:

30. Example (application of euler’s equation)
Known:
Constant acceleration
Tank dimensions
Water surface linear variation at free surface
Euler’s equation applies (we will treat water as a rigid body)
Pressure at free surface is 0 gage (this will be really important!)

31. Example (application of euler’s equation)
Unknown:
Value of constant acceleration in x-direction

32. Example (application of euler’s equation)
Governing Equations:
Euler’s equation
Gravitational acceleration constant
Water density (I assume the liquid is water; but it could be JP4)

33. Example (application of euler’s equation)
Solution:

34. Example (application of euler’s equation)
Solution:

35. Example (application of euler’s equation)
Solution:

36. Example (application of euler’s equation)
Solution:

37. Example (application of euler’s equation)
Discussion:
Liquid density did not matter because the term cancelled in Euler’s equation;
So same result using water or mercury (energy required would be different)
Expressed result in “gravity” reference (e.g. acceleration at 0.13G); then convert into numbers.
What do you think the liquid would look like if accelerate at 1.0G?

38. Example (application of euler’s equation)
Problem

39. Example (application of euler’s equation)
Known:
System at equilibrium
Constant angular velocity
Polar coordinates

40. Example (application of euler’s equation)
Unknown:
Equation of height change in two arms of the manometer

41. Example (application of euler’s equation)
Governing equations:
Euler’s equation

42. Example (application of euler’s equation)
Solution

43. Example (application of euler’s equation)
Solution

44. Example (application of euler’s equation)
Solution

45. Example (application of euler’s equation)
Solution

46. Example (application of euler’s equation)
Solution

47. Example (application of euler’s equation)
Discussion
Manometer fluid density cancel
Height change proportional to the radial component of acceleration
Could express as multiples of gravity
Centrifuges are used to generate huge accelerations (the angular velocity and radial arm length are squared)
Centrifugal pumps work about the same way

48. Outline
Bernoulli Equation
Application to some practical cases

49. Bernoulli’s equation Sort of a derivation Start with Euler’s equation
Textbook derives along a streamline (which saves a step).
Start with Euler’s equation

50. Bernoulli’s equation Select a useful coordinate system
Incompressible fluid

51. Bernoulli’s equation Write in differential form
Only showing x- component to the right; similar structure for y and z.
Z-component will have a weight term.
Require irrotational flow (vorticity vanishes)
<watch vorticity video>

52. Bernoulli’s equation
Y and Z acceleration terms

53. Bernoulli’s equation
Irrotational (zero vorticity) lets us refactor the cross- terms in the acceleration vector
Euler’s equation after substitutions (still ugly calculus)
Use chain rule

54. words

55. Bernoulli’s equation Rearrange the component equations
Group terms within the partial differential operation
Recall definition of the length of a vector in 3- space

56. Bernoulli’s equation Recall what a constant does when differentiated
Three derivatives, all equal to each other and all equal to zero and all with respect to a different variable
They must be the same function!

57. Bernoulli’s equation
The textbook derives along a streamline (which by definition means flow is irrotational)
Typically the equation is memorized as total head between two locations on the same streamline
Bernoulli’s equation is a special case of Euler’s equation of motion
It can be applied to compressible flow with minor modifications

58. Example using bernoulli’s equation
Problem Statement

59. Example using bernoulli’s equation
Known
Total head in system
Free surface and outlet pressure
Water is working fluid

60. Example using bernoulli’s equation
Unknown
Velocity at outlet

61. Example using bernoulli’s equation
Governing Equations
Bernoulli’s equation

62. example
Solution

63. example Discussion Water/oil same (specific weight cancels)
Steady flow requires important assumption about relative “areas”
Assumed pressure across “jet” is zero
No frictional losses (yet – that’s coming soon!)

64. Example bernoulli’s equation
Problem Statement

65. Example bernoulli’s equation
Known:
Working head
Outlet velocity
Working fluid (water)

66. Example bernoulli’s equation
Unknown:
Outlet pressure

67. Example bernoulli’s equation
Governing equation:
Bernoulli’s equation

68. example
Solution

69. example
Discussion
Almost same as prior example; but to find pressure, we need to know the working fluid
Outlet velocity specified, don’t know if it is a jet, so no assumption about pressure
No frictional losses (yet!)

70. Bernoulli example
Problem Statement

71. Bernoulli example Known: Speed of jet at fountain nozzle
Vertical speed of water at apogee (high point) in fountain
Working fluid (water)

72. Bernoulli example
Unknown:
Height of the jet (fountain)

73. Bernoulli example
Governing Equations:
Bernoulli’s equation

74. example
Solution

75. example
Solution

76. example
Discussion
Working fluid was irrelevant (it does matter when we introduce friction later on)
Height controlled by exit speed
Imagine the speed at the Bellagio Fountains in Las Vegas, Nevada (find a video)

77. Next Time
Reynold’s Transport Theorem