Dimensional Analysis

Reynolds Number

The Reynolds Number  is regularly cited as the most important parameter in fluid mechanics.  You have seen that this number can be used to predict whether flows will be laminar or turbulent, but the full significance of this and other non-dimensional combinations of parameters may not yet be completely understood.  It is the purpose of this document to discuss the following aspects of these parameters:

1.     Why is it useful to describe problems in terms of non-dimensional parameters?

2.     How does one determine which non-dimensional parameters to use?

3.     What is the relationship between such non-dimensional parameters and the governing differential equations?

4.     How can one apply these parameters to solutions of differential equations?

1.     Why is it useful to describe problems in terms of non-dimensional parameters?

Consider first the case of the simple RC electrical circuit.  The ratio of output voltate () to input voltage () is given by the following relationship:

.

Assume that you did not know this relationship and that you needed to measure the response of all possible RC circuits.  One way to do this would be to measure the output while varying each of , , , and .  Assume that you can get away with looking at 10 values of each parameter.  Since you need to vary 4 different parameters, you must make 10,000 measurements, which means that you will be in the instrumentation laboratory for a long time, and you will probably miss the homecoming game.  However, a dimensional analysis of the system will indicate that there are only two parameters that are independent of one another.  One of these is the voltage ratio, , and the other is the non-dimensional parameter .  Recall that  has the dimensions of time and that  has the dimensions of 1/time, indicating that the product  is non-dimensional.  Thus, it is necessary to vary only one parameter and measure the other ().  You could take 3 times as many meaningful measurements and still be out of the lab in a time that is orders of magnitude shorter.  Not only will you be able to make it to the homecoming game, but you will also have time to stop by the Huddle House and have a cup of coffee beforehand.  The analysis tells you that it is not even necessary to change the resistance and capacitance values of the circuit.  You can change  simply by changine .

2.     How does one determine which non-dimensional parameters to use?

The student should refer to section 7.7.1 in Middendorf and Engelmann for a discussion of the Buckingham p theorem.  The theorem describes a method to determine how many dimensionless parameters will be important in a problem given the dimensional parameters.  Because this type of analysis can be applied to any physical problem without the need to understand the governing differential equations, the student must be capable of performing it.

Exercise 1: Use the Buckingham Pi theorem to show that the two non-dimensional parameters of importance to the RC circuit described above are  and .  You will need to express ohms as volts/amp and capacitance as amp-s/volt.

Exercise 2: Use the Buckingham Pi theorem to determine the important non-dimensional parameters for an LRC circuit.

Exercise 3: Assume that you are given an RC circuit, but that the resistance is a function of current, i.e. .  Determine the important non-dimensional parameters governing the circuit.

3.     What is the relationship between such non-dimensional parameters and the governing differential equations?

Non-dimensionalization of the Navier Stokes equations

4.     How can one apply these parameters to solutions of differential equations?

Derivation of Poiseuille flow.