1. FLOWING FLUIDS AND PRESSURE VARIATION

2. Pressure differences are (often) the forces that move fluids
e.g., pressure is low at the center of a hurricane.
“For your culture”: The force balance for hurricanes is the pressure force vs. the Coriolis force. That’s why hurricanes on the northern hemisphere always spin counter-clockwise. In a tornado, the balance is between pressure force and centrifugal acceleration; a tornado can spin either way…

3. Lagrangian and Eulerian Descriptions of Fluid Motion
Eulerian: Observer stays at a fixed point in space.
Lagrangian: Observer moves along with a given fluid particle.
We consider first the Lagrangian case. The position of a fluid particle at a given time t can be written as a Cartesian vector
For a complete description of a flow, we need to know at every point. Usually one references each particle to its initial position:

4. For fluid mechanics, the more convenient description is usually the Eulerian one:

5. “For your Culture”: In solid mechanics, we often use Lagrangian methods. We don’t need to follow “every particle”, but just some small volumes.
When displacements get large (e.g. in fluid flow), the deforming grid gets problematic. But we sometimes use mixed approaches, e.g. Lagrangian tracers in a Eulerian frame.

6. Streamlines and Flow Patterns
Figure 4.3 (p. 84)
Streamlines are used for visualizing the flow. Several streamlines make up a flow pattern.
A streamline is a line drawn through the flow field such that the flow vector is tangent to it at every point at a given instant in time.

7. Uniform vs. Non-Uniform Flow
Using s as the spatial variable along the path (i.e., along a streamline):
Flow is uniform if
Examples of uniform flow:
Note that the velocity along different streamlines need not be the same! (in these cases it probably isn’t).

8. Examples of non-uniform flow:
Converging flow: speed increases along each streamline.
Vortex flow: Speed is constant along each streamline, but the direction of the velocity vector changes.

9. Steady vs. Unsteady Flow
For steady flow, the velocity at a point or along a streamline does not change with time:
Any of the previous examples can be steady or unsteady, depending on whether or not the flow is accelerating:

10. Turbulent flow in a jet
Turbulence is associated with intense mixing and unsteady flow.

11. Flow around an airfoil:
Partly laminar, i.e., flowing past the object in “layers” (laminae).
Turbulence forms mostly downstream from the airfoil.
(Flow becomes more turbulent with increased angle of attack.)

12. Flow inside a pipe:
Laminar Turbulent
Turbulent flow is nearly constant across a pipe.
Flow in a pipe becomes turbulent either because of high velocity, because of large pipe diameter, or because of low viscosity.

13. Methods for Developing Flow Patterns
(i.e., finding the velocity field)
Analytical Methods: The governing equations (mostly the “Navier-Stokes” equation) are non-linear. Closed-form solutions to these equations only exist for special, strongly simplified cases.
Computational Methods: The authors of the book seem to feel that this is not overly useful. As a numerical modeler I disagree. For experimental methods, it is often difficult to find materials that scale properly to large scales (e.g., entire oceans or the Earth’s mantle).
Experimental Methods: Very useful for complicated flows, especially flows that involve turbulence. While the corrects physical laws are known, time and space resolution make turbulence tricky in numerical models.

14. Pathline, Streakline, and Streamline
Important concepts in flow visualization:
In steady flows, all 3 are the same.
Pathline: The line (or path) that a given fluid particle takes.
Streakline: The line formed by all fluid particles that have passed through a given fixed point.
Streamline: A continuous line that is tangent to the velocity vectors everywhere along its path (at a given moment in time).

15. xxxxx
xxxxx
Streakline at t = t0
Streakline at t > t0

16. Acceleration: Normal and Tangential Components
Velocity can be written as:
where V(s,t) is the speed and
is a unit vector tangential to the velocity.
The derivative of the speed is (since ds/dt = V):

17. The time derivative of the unit vector is non-zero because the direction of the unit vector changes. The centripetal acceleration is
so that the total acceleration becomes

18. Acceleration in Cartesian Coordinates
This is probably one of the most fundamental concepts of the course, but it is not very intuitive! or

19. From last page, we had
This derivative is called the full derivative or material derivative.
It is often written D/Dt instead of d/dt.
It can apply to other quantities as well.

20. For a simpler example, let’s look at the material derivative of temperature T in one dimension: T(x,t)
At a given point x0, a change in temperature can be caused by two different mechanisms:
The temperature of the local fluid particle changes (e.g., due to heat conduction, radioactive heating, etc…):
All fluid particles keep their temperature, but the velocity u brings a new particle to x0 which has a different temperature:

21. The changes are called “local change” and “convective change” (the convective change is also called “advective change”)
= local temperature change
= local acceleration in x
= convective temperature change
= convective acceleration in x

22. Example 4.1 (p. 94): Find the acceleration half-way through the nozzle
Velocity is given as
Taking the x-derivative of u, and multiplying it times u gives:
Just plug in the values and x = 0.5L to get the answer…

23. Lagrangian reference frame:
In the Lagrangian frame (moving along with a fluid particle), there is no convective acceleration. Because you’re staying with a given particle, no other particle can come in and bring with it a different velocity.
Eulerian:
The convective terms may be seen as a correction due to the fact that new particles with different properties are moving into our observation volume.

24. Note that conservation laws naturally apply in the Lagrangian frame:
A conserved quantity such as total energy E remains constant in a given material volume.
An Eulerian observer sees different material volumes flow past, each of them possibly with different E.

25. Euler’s Equation
Euler’s equation is Newton’s 2nd Law applied to a continuous fluid.
Recall Newton’s 2nd law for a particle (balance in l-direction):
This law is fundamental and thus also applies to fluid particles:
(There may be additional forces…)

27. Now we shrink the fluid element to
(partial derivative since p may be a function of other coord.s and time)
and
so that
Euler’s equation
(force balance in a moving fluid)

28. Uniform acceleration of a tank of liquid (Fig. 4.13)
Horizontal balance:
Vertical balance: (hydrostatic!)

29. Derivation of the Bernoulli Equation:
Start with Euler’s equation applied along a pathline:
Assume steady flow ( ); all s-derivatives are now full derivatives.
Integrating with respect to s, we get
Bernoulli’s eqn.

30. Recall what we just did to get the Bernoulli equation:
Assume steady flow (don’t apply this to anything else!)
Integrate forces (per volume) along a pathline.
The integral of force along a distance gives us energy:
Note that the velocity term is the kinetic energy per unit volume.
Also note that energy / volume has same units as pressure.
The kinetic energy / volume is also known as kinetic pressure.
Along a streamline in steady, inviscid flow, the sum of piezometric pressure plus kinetic pressure is constant.

31. Application of the Bernoulli Equation: Stagnation Tube
Apply to points 1 and 2
(same depth z):
Point 2 is a stagnation point (velocity is zero)

32. From the geometry of the flow we know (we observe or else we just assume; we haven’t really derived this) that at both points there is no vertical acceleration. The vertical balance is thus hydrostatic:
The stagnation tube is a simple device for measuring velocity.

33. Notice that we had to have a free surface at the top of the fluid.
If the flow is in a pressurized pipe, we don’t know whether the piezometric head measured by l is due to static pressure in the pipe or due to flow.
Hence the Pitot tube…

34. Pitot tube:
Look carefully; the Pitot tube is really 2 tubes in 1.
If we know the static pressure at point 1 and the dynamic pressure at point 2, we have all we need to find the velocity.
For more details see the book, p. 102.

35. Rotation and Vorticity
Rotation of a fluid element in a rotating tank of fluid
(solid body rotation).

36. Rotation of fluid element in flow between moving and
stationary parallel plates
You can think of the “cruciforms” as small paddle wheels that are free to rotate about their center.
If the paddle wheel rotates, the flow is rotational at that point.

37. As
And similarly

38. The net rate of rotation of the bisector is

39. The rotation rate we just found was that about the z-axis; hence, we may call it
and similarly
The rate-of-rotation vector is
Irrotational flow requires (i.e., for all 3 components)

40. The property more frequently used is the vorticity

41. Vortices
A vortex is the motion of many fluid particles around a common center. The streamlines are concentric circles.
Choose coordinates such that z is perpendicular to flow.
In polar coordinates, the vorticity is (see p. 112 for details)
(V is function of r, only)
Solid body rotation (forced vortex): or

42. Vortex with irrotational flow (free vortex):
A paddle wheel does not rotate in a free vortex!

43. Forced vortex (interior) and
free vortex (outside):
Good approximation to naturally occurring
vortices such as
tornadoes.
Euler’s equation for any vortex:

44. We can find the pressure variation in different vortices
(let’s assume constant height z):
In general:
Solid body rotation:
Free vortex (irrotational):

45. Application to forced vortex (solid body rotation):
with
Pressure as function of
z and r
p = 0 gives free surface