This is a fairly simple experiment for the Fluid Mechanics Laboratory course. The lab procedure can be found by scrolling down; other resources are linked to below.

Buff. Behind him is the sliding glass door where he met disaster.


Introduction:

The objective of this experiment is to determine the height and velocity fields for solid body cylindrical rotation of a fluid. From a theoretical standpoint, we introduce the use of Bernoulli’s Equation in two ways: the derivation of the surface of the fluid when rotated, and b) the use of the Pitot tube to determine the velocity of the fluid.

Theory:

Consider the cylindrical rotation of a fluid as shown in Figure 1 below.

In order to determine the height profile along the surface of the fluid, we utilize Bernoulli’s equation:

(1)

Where

  • γ = the unit or specific weight of the fluid
  • gc = the gravitational constant
  • p = pressure
  • u = velocity of fluid, cm/sec
  • z = height of fluid, cm

Basically, Bernoulli’s equation is a conservation of energy statement: it states that energy is conserved from state 0 to state 1. It assumes that there are no losses due to friction of viscosity, something we will explore further in later experiments.

When we rotate the drum, we change the energy state of the fluid particles in the drum. Since we are concerned with the surface of the fluid, at any point on the surface the pressure is atmospheric and thus a constant, or

(2)

So we can eliminate the first terms on the left- and right-hand sides of Equation (1).

Next, from dynamics, at any point in the rotating vessel,

(3)

Where

  • r = the radial distance from the center of the cylinder, cm
  • ω = is the angular speed in radians/sec. For this experiment we will bring the vessel to a constant angular speed, and thus this is a constant.

From Equations (2) and (3), we can simplify Equation (1) to:

(4)

Taking r0 to be the center of the cylinder (i.e. r0 = 0) and solving for z1, we have:

(5)

It is worth noting that, at r1 = 0 , z1 = z0 , as we would expect. For CGS units (which are highly recommended here,) gc = 981 cm/sec2.

This covers the portion of the experiment where we consider the height of the fluid in the bowl. The fluid velocity at any point is given by Equation (3). However, we want to compare that with an experimentally obtained result, and to accomplish that we use another application of Bernoulli’s Equation: the Pitot tube. The theory behind the Pitot tube is discussed in the instructor’s piece entitled “Computing Open Channel Flow using a Pitot Tube.” Basically, the Pitot tube takes kinetic energy from the fluid and converts it into potential energy of fluid going up the tube. In terms of Equation (1), on the left-hand side the second term is equated to the third term on the right-hand side, all other terms being zero. This yields

(6)

To reduce confusion, we substitute the variable l for z0, as l is the height of the fluid column in the Pitot tube above the free surface of the water. Rearranging,

(7)

Procedure:

  1. Adjust the angular velocity of the experimental apparatus until the fluid reaches an optimal, steady state “bowl” shape. Be sure that the bottom of the fluid “bowl” is not in contact with the bottom of the cylindrical vessel; this will interfere with the fluid flow.
  2. Determine the angular velocity ω of the vessel. There are red marks on the vessel and frame; you can time the passage of these two marks for one (or better more) rotations either with a stopwatch or with a camera phone video. You must convert the revolutions per second/minute into radians/sec for your calculations.
  3. You will then take measurements of the radius (r) and height (z) of the fluid surface at several points (at least four.) These experimental values will be used to compare to your theoretical profile based on Equation (5). Don’t forget to measure the distance z0 from the bottom of the bowl to the datum (red mark) on the frame.
  4. After this you use a Pitot tube to measure the velocity of the fluid, applying Equation (7) at several points (again at least four) from the radius.

It’s not necessary for the two sets of radii to be the same, but it helps. If you do use the same data point, the resulting table of data should look like the data spreadsheet.

If you used two different sets of radii, you’ll need two separate tables for the height and the velocity comparisons. Keep in mind that ω and z0 are your constants from which you compute most of your results.

What to Include in the Report

  • A title sheet
  • A main body containing the following sections:
    • Objective(s): What was the purpose of the experiment?
    • Theory: Include your interpretation of this section using the information provided in the handout and the lab lecture. How can you determine the height and velocity profiles for stable body cylindrical rotation of a fluid? What assumptions must be made? What is a Pitot tube, and how can it be used to determine the velocity of a fluid? Include all equations used and explain the derivation in your own words.
    • Procedure: Provide a stepwise list of your procedure for measuring the height and velocity profiles of the fluid surface. Be explicit.
    • Observed Data: Put the data into tabular form, with units. Also include a few sentences to orient the reader.
    • Results/Discussion: Present and discuss the results of your measurements. Show me two plots: a plot of the theoretical height profile vs. the experimentally determined height, and a similar plot comparing the theoretical and experimental velocity profiles. How do the theoretical and actual values compare? What do you think were likely sources of error? Consider the assumptions that you’ve made. (Note: some students have difficulty determining the rotational speed of the drum. If you’re one of them, use the alternate procedure below and report your results.)
    • Conclusions: What did you learn? Did you accomplish your objectives? How well did your theoretical and experimental values compare? Can you identify any limitations of this experiment? Do you have any recommendations?
    • Appendix: Include the raw data sheet here. Also, show sample calculations for your theoretical height and velocity profiles.
    • References: Include any references cited here. Make sure you only add the references you cite in the text.

Alternate Procedure by Computing Rotational Speed

If you have difficulty determining the rotational speed, proceed in the analysis as follows:

  1. For each point, solve for the rotational speed using Equation (5). When you are done, you should have a set of rotational speeds for the data points. Average this data set. You may want to use Chauvenet’s Criterion to eliminate outliers.
  2. Determine the velocity at each point using Equation (7).
  3. Determine the rotational speed at each point using Equation (3). When you are done, you should have a second set of rotational speeds for the data points. Average this second data set.

Compare the rotational speeds of the two data sets. They should be identical or very close. If they are not, you need to explain why this is so.