1. Ch 4 Fluids in Motion

2. Introduction
In the previous chapters we have defined some basic properties of fluids and have considered various situations involving fluids that are at rest.
In general, fluids have a well-known tendency to move or flow. The slightest of shear stresses will cause the fluid to move. Similarly, an appropriate imbalance of normal stresses (pressure) will cause fluid motion.
In this chapter we will discuss various aspects of fluid motion without being concerned with the actual forces necessary to produce the motion. That is, we will consider the kinematics of the motion—(the velocity and acceleration of the fluid, and the description and visualization of its motion. No forces)
The analysis of the specific forces necessary to produce the motion (the dynamics of the motion) will be discussed in detail in the following chapters.
A wide variety of useful information can be gained from a thorough understanding of fluid kinematics. Such an understanding of how to describe and observe fluid motion is an essential step to the complete understanding of fluid dynamics.

3. Flow Patterns: Pathline
The pathline is the line traced out by a fluid particle.

4. Flow Patterns: Streamline
The streamline is a curve that is everywhere tangent to the local velocity vector.

5. Flow Patterns: Streakline
It is the instantaneous locus of all fluid particles that have passed through a given point.
If at point A in a flow field, a dye is injected, then the photograph of the dye streak would be a streakline. In other words, if fluid particles 1 through 4 have passed successively through point A, the shown dotted line (joining all these particles at time t) would be the streakline.

6. Dividing Streamline and Stagnation Point
When an object divides the flow, then the streamline that follows the flow division is called “dividing streamline”.
The point of division is called the stagnation point (since the flow is stagnant there).

7. Laminar and Turbulent Flow
In laminar flow, the fluid flows in layers parallel to each other. No mixing.
In turbulent flow, the velocity is fluctuating with time and a strong mixing occurs between fluid layers.

8. 1 Dimensional Flow
Not very good assumption but many practical flows could be modeled as 1 D flow
Pipe flow:
Function of radial position, r, only !
Duct flow:
Function of axial distance, x, only !

9. 2 Dimensional Flow

10. 3 Dimensional Flow

11. Methods of Predicting Velocity Field
Analytical methods:
Solving a set of equations to get the velocity field.
Numerical methods:
Solving the same set of equations using numerical methods.
Predicted streamline pattern over the Volvo ECC prototype.

12. Pathlines of floating particles
Experimental methods
Pathlines of floating particles
We inject fluid markers (ink or dust) and study the streamlines, pathlines and streaklines.
Smoke traces about an airfoil with a large angle of attack.

13. Volume Flow Rate
Flow rate (or discharge, Q) is defined as the rate at which a certain fluid volume passes through a given section in the flow stream.
V is assumed to be constant.

14. Volume Flow Rate, Q
If the velocity is constant over the cross section,
But if V is not normal to dA

15. Average Velocity
We define the average velocity as
But what is dA?

16. Average Velocity Continued ..

17. Mass Flow Rate

18. Acceleration in Cartesian and Streamline Coordinates
(a) Cartesian coordinates
(b) Streamline coordinates

19. Acceleration in Cartesian Coordinates
This is a vector result whose scalar components can be written as

20. 1D acceleration
2D acceleration
3D acceleration

21. Convective Acceleration
Local Acceleration

22. Streamline Coordinates
In the streamline coordinate system the flow is described in terms of one coordinate along the streamlines, denoted s, and the second coordinate normal to the streamlines, denoted n.

23. Acceleration in Streamline Coordinates
Tangential acceleration, at
Normal acceleration, an
Local
acceleration
Convective acceleration.

24. Uniform Flow Patterns
In the uniform flow, the velocity vector (magnitude + direction) does not change a long a streamline.The streamlines should be straight and parallel to each other.

25. Non-uniform Flow Patterns
In figure a, the streamlines are straight but not parallel. So, a change in the velocity magnitude will occur as we move along the streamline.
In figure b, the streamlines are parallel but they are not straight. So, a change in the velocity direction occurs.

26. System and Control Volume
A fluid system is a given quantity of matter consisting always of the same matter.
A control volume (CV) is a geometric volume defined in space and enclosed by a control surface.

27. Lagrangian Method
There are two approaches to describe the velocity of a flowing field.
The position of a specific fluid particle traveling along a pathline is recorded with time.

28. Eulerian Method
The properties of fluid particle passing a given point in space are recorded with time.
The Eulerian approach is generally used to analyze fluid motion.

29. Control Volume Equation (Reynolds Transport Equation)
This equation relates the time rate of change of a property of a system to the time rate of change of the property in a control volume plus the net efflux of the property across the control surface.

30. Intensive and Extensive properties of a System
Intensive properties are those that are independent of the mass of the system.
Extensive properties are those that are dependent on the system mass.
The amount of an extensive property that a system possesses at a given instant, can be determined by adding
up the amount associated with each fluid particle in the system.

31. Control Volume Equation (Reynolds Transport Equation)
Derivation of the
Control Volume Equation (Reynolds Transport Equation)
See also handouts

34. If the flow is steady, this term is zero
Uniform Velocity distribution
If the velocity is not uniform over the cross section
If the flow is steady, this term is zero

35. General form of the Integral continuity equation
Application of Reynolds Transport Equation to Conservation of Mass Principle (Integral Form of the Continuity Equation)
General form of the Integral continuity equation

36. Continuity at a Point Differential Form of the Continuity Equation
If the flow is steady
If the flow is also incompressible

37. Applications on Continuity Equation

38. Rotation
The rotational rate of a fluid element is the average rotational rate of two initially perpendicular sides of a fluid particle.

39. Vorticity
The Vorticity of a fluid particle is a vector equal to twice the rotational rate of the particle.
For irrotational flow:

40. Vortices
A forced vortex is a rotational flow with concentric circular streamlines in which the fluid rotates as a solid body.
A free (potential) vortex is an irrotational flow in which the velocity varies inversely as the distance from the center.

41. Separation
Separation in a flow occurs when the streamlines move a way from the body boundaries and a local re-circulation region occurs.