Page 114 - Bird R.B. Transport phenomena

§3.7 Dimensional Analysis of the Equations of Change 99

Fluid
. approaches
from * = -<*> У'
with uniform
• velocity v (

Fig. 3.7-1. Transverse flow around a cylinder.

Now we rewrite the problem in terms of variables made dimensionless with the characteristic
>
<
length D, velocity у , and modified pressure 3 . The resulting dimensionless equations of
ж x
change are
2
J
(V • v) = 0, and Ц + [v • Vv] = - VSf> + - V v (3.7-17,18)
dt R e
in which Re = Dv p/fx. The corresponding initial and boundary conditions are:
x
I.C iix 2 + y >\orii\z\>\(L/D), v = 8 x (3.7-19)
2
2
B.C.1 as x 2 + j/ + z -+oo, v^6 x (3.7-20)
2
2
B.C. 2 ifx 2 + y <|and|z|<^(L/D), v = 0 (3.7-21)
B.C.3 as*->-°o ti/ = 0, 0>-* 0 (3.7-22)
a
If we were bright enough to be able to solve the dimensionless equations of change along with
the dimensionless boundary conditions, the solutions would have to be of the following form:
v = v(x, y, z, t, Re, L/D) and Ф = Ф(х, у, z, r, Re, L/D) (3.7-23,24)
That is, the dimensionless velocity and dimensionless modified pressure can depend only
on the dimensionless parameters Re and L/D and the dimensionless independent variables
x, y, z, and t.
This completes the dimensional analysis of the problem. We have not solved the flow
problem, but have decided on a convenient set of dimensionless variables to restate the prob-
lem and suggest the form of the solution. The analysis shows that if we wish to catalog the
flow patterns for flow past a cylinder, it will suffice to record them (e.g., photographically) for
a series of Reynolds numbers Re = Dv^p/fi and L/D values; thus, separate investigations
into the roles of L, D, v , p, and /л are unnecessary. Such a simplification saves a lot of time
x
and expense. Similar comments apply to the tabulation of numerical results, in the event that
7 8
one decides to make a numerical assault on the problem. '

7 Analytical solutions of this problem at very small Re and infinite L/D are reviewed in L.
Rosenhead (ed.), Laminar Boundary Layers, Oxford University Press (1963), Chapter IV. An important
feature of this two-dimensional problem is the absence of a "creeping flow" solution. Thus the [v • Vv]-
term in the equation of motion has to be included, even in the limit as Re —> 0 (see Problem 3B.9). This is
in sharp contrast to the situation for slow flow around a sphere (see §2.6 and §4.2) and around other
finite, three-dimensional objects.
8
For computer studies of the flow around a long cylinder, see F. H. Harlow and J. E. From, Scientific
American, 212,104-110 (1965), and S. J. Sherwin and G. E. Karniadakis, Comput. Math., 123,189-229 (1995).