Many problems benefit from transformation to dimensionless form. Doing so makes it easy to assess how systems of equations can be simplified for very large or very small values of key dimensionless groups, for example, the "creeping flow" assumptions in transport phenomena. Dimensionless forms are also widely used in reactor design.
The procedure to be followed can be outlined:
The scale factors are dimensional constants with the same dimensions as the variable they modify. Typical practice is to use some quantity "characteristic" of the system. For instance, the "characteristic length" in a pipe flow problem is usually defined to be the pipe diameter; in a channel flow problem, the channel width; and for the falling sphere problem, the sphere diameter.
In familiar problems, it is possible to assign values to the scale factors at the beginning -- and this is certainly more convenient. The advantage to determining scale factors at a later point is that it allows the equation to be adapted to a particular aspect of a problem.
The characteristic values used to make up the scale factors should be kept to a small set. For instance, consider a problem involving position, velocity, and time. These three variables involve only two dimensions, length and time, so only two characteristic values should be chosen. In pipe flow, it is hard to decide on a value of time that is "characteristic", so common practice is to select the pipe diameter and the average velocity as the scaling factors for length and for velocity, and then use the ratio (diameter)/(average velocity) to scale time. However, in a free convection problem there is no bulk velocity, so it would make sense to define a characteristic length and time and scale the velocity using (length)/(time).
To illustrate the procedure, it will be applied to the Navier-Stokes equation:
Examining the problem, it is clear that the independent variables are position and time, and that velocity is the independent variable. Scale factors will be needed for each of these. The equation also includes terms representing pressure and gravity. These are both needed to describe the system and are dimensional, so reference factors will be needed to make them dimensionless. Density and viscosity are properties of the fluid, not of the flow, so they can be left alone.
The scale and reference factors can then be used to define dimensionless quantities (identified by an asterisk), like so:
It is probably worthwhile to first look at how this substitution will work on the gradient operator:
Practice a little algebra to group all the terms, then divide through to drive one or more of the coefficients to one. Our focus this time is turbulent flow, which means the convective terms are expected to be important, so divide through to se the coefficient on the convection terms to one. We'll also divide through to clean up the boundary conditions:
The next step is to assign the values for the scale functions.
If you have the boundary conditions, they are very good indicators. One of the main uses of dimensionless equations is to put everything on a normalized scale from zero to one. Thus, it would be ideal if the dimensionless variables can be set up to run 0 to 1. This can be easily done by setting SL equal to R and SV equal to U.
It is also helpful to think about the physical system represented. What quantities characteristic of the system are easily identified? For pipe flow, clearly the diameter of the tube is important and is fixed, hence it would make tremendous sense to define SL = D or R.
The equation is then:
That leaves the gravity reference factor. Since gravity has a "universal" constant value, that makes a perfect reference factor, and RG=g.
Plug everything back in, and simplify:
Now, take a look at the groups of variables that result. First, convince yourself that they are dimensionless. Then, see if any look familiar. At least one should.
The groups that appear are the Reynolds Number and a second dimensionless group called the Froude Number.
If we want, we can rewrite the Navier-Stokes equation as:
Examining this equation provides useful physical insight into the problem. On the left hand side of the equation are the accumulation and convective transport terms. On the right hand side are the pressure, gravitational, and molecular transport terms. The equation has been normalized so that the convective transport term has a coefficient of one. Knowing this, we can understand the physical significance of the dimensionless groups.
The Reynolds number can be seen to represent the ratio of the convective (or inertial) transport to the molecular (or viscous) transport. Small Reynolds numbers thus mean that molecular transport is dominant, hence the flow is laminar. Large Reynolds numbers, as occur in turbulent flow, mean that convective transport is dominant.
Similarly, the Froude number represents the ratio of inertial forces (convective transport) to gravitational forces.
We can also use this form of the equation to consider limiting cases. First consider the case where the Reynolds number becomes very small (approaches zero). In this case, the molecular transport term becomes much much larger than the others meaning that they become negligible. Such a situation is called creeping flow. In the opposite case, the Reynolds number becomes so large that the molecular transport term becomes negligible. This case is called ideal flow.
The equation was normalized for flow dominated by convection. Had we chosen to focus on viscous flow, or gravity driven flow, or some other aspect, it might have led to a different set of dimensionless groups, and different physical insights.
All of the equations of change can be analyzed in similar fashion. This is the source of many of the common dimensionless numbers (Peclet, Euler, Brinkman, etc.)
References:
R.M. Price
Original: 3/97; 5/21/99
Revised: 9/27/2000, 10/12/2000
Copyright 2000 by R.M. Price -- All Rights Reserved