As suggested in the last section, if there are more than 4 variables in the problem, and only 3 dimensional quantities (M, L, T), then we cannot find a unique relation between the variables. The best we can hope for is to find dimensionless groups of variables, usually just referred to as dimensionless groups, on which the problem depends. In fact, this is quite a good approach, for reasons that will be discussed in the next section.
Buckingham Pi is a procedure for determining dimensionless groups from the variables in the problem.
Let us assume that there are n = 3 dimensional quantities to consider - mass, length and time. The problem
involves m = 6 variables, denoted A . . . F. In general we can derive m - n dimensionless groups, often denoted
1,
2..., using the following procedure
This is probably best illustrated by a worked example. The head loss in a horizontal pipe in turbulent
flow is related to the pressure drop p, and is a measure of the resistance to flow in the pipe. It
depends on the diameter of the pipe D, the viscosity
and density
, the length of the pipe l, the
velocity of the flow v and the surface roughness
. We start by listing the dimensions of these
parameters
D | [L] |
v | [LT-1] |
![]() | [ML-3] |
![]() | [ML-1T-2] |
![]() | [ML-1T-1] |
l | [L] |
![]() | [ ] |
We will choose D, v and as repeating variables. Our first dimensionless group involves
p, in the
form
0 | = c + d | ||
0 | = -b - 2d | ||
0 | = a + b - 3c - d |
c | = -d | ||
b | = -2d | ||
0 | = a - 2d + 3d - d ![]() |
We can repeat the process with to get the second dimensionless group :
0 | = c + d | ||
0 | = -b - d | ||
0 | = a + b - 3c - d |
The next dimensionless group will involve l with dimension L. However one of the repeating variables is the diameter D, and so the ratio of the two is already dimensionless. So the next dimensionless group is
From this analysis we have successfully determined that the turbulent flow in a roughened pipe depends on a head loss parameter
This is as far as we can go using dimensional analysis. Experiment however shows that the pressure drop depends linearly on the length of the pipe, so we can make this relation explicit :
There are a number of helpful short cuts that can simplify matters :
All of these short cuts were used in the previous example.