Page 112 - Bird R.B. Transport phenomena

§3.7 Dimensional Analysis of the Equations of Change 97

§3.7 DIMENSIONAL ANALYSIS OF THE EQUATIONS OF CHANGE
Suppose that we have taken experimental data on, or made photographs of, the flow
through some system that cannot be analyzed by solving the equations of change analyt-
ically. An example of such a system is the flow of a fluid through an orifice meter in a
pipe (this consists of a disk with a centered hole in it, placed in the tube, with a pressure-
sensing device upstream and downstream of the disk). Suppose now that we want to
scale up (or down) the experimental system, in order to build a new one in which exactly
the same flow patterns occur [but appropriately scaled up (or down)]. First of all, we
need to have geometric similarity: that is, the ratios of all dimensions of the pipe and ori-
fice plate in the original system and in the scaled-up (or scaled-down) system must be
the same. In addition, we must have dynamic similarity: that is, the dimensionless groups
(such as the Reynolds number) in the differential equations and boundary conditions
must be the same. The study of dynamic similarity is best understood by writing the
equations of change, along with boundary and initial conditions, in dimensionless
1 2
form. '
For simplicity we restrict the discussion here to fluids of constant density and vis-
cosity, for which the equations of change are Eqs. 3.1-5 and 3.5-7
(V • v) - 0 (3.7-1)
2
P§~v= -V3> + /xVv (3.7-2)
f
In most flow systems one can identify the following "scale factors": a characteristic
length / , a characteristic velocity v , and a characteristic modified pressure 2P 0 = p {) +
0
0
pgh 0 (for example, these might be a tube diameter, the average flow velocity, and the
modified pressure at the tube exit). Then we can define dimensionless variables and dif-
ferential operators as follows:
V
i = f y = l 2 = f U -f (3.7-3)
'o 'o *o 'o
v = ^ Ф = ^Р or * = ^ £ (3.7-4)
^ l /l
V = Z 0 V = Ь (д/дх) + Ъ {д/ду) + b (d/dz) (3.7-5)
х у z
2
2
V 2 = {д /дх ) + {S /dy ) 2 + (<? /r?z ) (3.7-6)
2
2
2
D/Dt = (l /v )(D/Dt) (3.7-7)
Q
Q
We have suggested two choices for the dimensionless pressure, the first one being con-
venient for high Reynolds numbers and the second for low Reynolds numbers. When
the equations of change in Eqs. 3.7-1 and 3.7-2 are rewritten in terms of the dimension-
less quantities, they become
(V • v) = 0 (3.7-8)
2
Ц, v = - V9> + T ^ - V v (3.7-9a)
Dt 1'o^oPj
2
or Ц; v = -[т^-lv* + [^Mv v (3.7-9b)
Dt

1
G. Birkhoff, Hydrodynamics, Dover, New York (1955), Chapter IV. Our dimensional analysis
procedure corresponds to Birkhoff s "complete inspectional analysis."
2
R. W. Powell, An Elementary Text in Hydraulics and Fluid Mechanics, Macmillan, New York (1951),
Chapter VIII; and H. Rouse and S. Ince, History of Hydraulics, Dover, New York (1963) have interesting
historical material regarding the dimensionless groups and the persons for whom they were named.