1250 
The first term is a dependent parameter sometimes 
known as the Euler number E = V/+/2Ap/p, consisting 
primarily of the flow characteristics. The next two (or 
more) are length ratios. The last three are groups in- 
volving the fluid properties, known respectively as the 
Froude number F = V/+/aAy/p (gravity), the Rey- 
nolds number R = pVa/y (viscosity), and the Mach 
number M = V/~/e/p (compressibility). Evidently, 
even though the actual form of the function is unknown, 
the dependent parameter will be the same at model and 
prototype scale if the corresponding independent pa- 
rameters have the same numerical values—in other 
words, the model and the prototype flows will then 
(but only then) be mechanically similar. 
Since all problems in mechanics are expressible in 
terms of three-dimensional categories, the Il-theorem 
will be seen to reduce by three the number of terms in 
the function, each of the dimensionless parameters 
differing from the others by one variable only. It is 
thus possible in such problems to distribute the various 
fluid properties singly (except for the density) among 
the three independent parameters F, R, and M. A 
problem which combines mechanics and thermodynam- 
ics, on the contrary, requires one more dimensional 
category—temperature—for its expression, as well as 
such additional flow-variables as temperature rise and 
heat flux, and such fluid variables as thermal capacity 
and conductivity [20]. The functional relationships— 
and hence the similitude requirements—obviously be- 
come far more complex under these conditions. Finally, 
if electrical phenomena are also involved—as is con- 
ceivable in certain meteorological problems—any possi- 
bility of complete similitude is removed from practical 
consideration. 
In view of the great difficulty of reproducing large- 
scale phenomena in accurate detail at a convenient 
model scale, recourse must generally be had to various 
laboratory expedients. Some occurrences, of course, are 
relatively simple in their basic aspects, and hence may 
be simulated—and even investigated functionally—with 
relative ease. Typical of these is the matter of the 
velocity distribution in turbulent flow near very rough 
boundaries without regard to gravitational, viscous, 
or thermal effects. In many phenomena, however, the 
secondary effects can be ignored only as a first approxi- 
mation, and then must be restored to consideration by 
analytical or experimental means. Conditions illustra- 
tive of this situation are encountered in ship testing, in 
which the model hull is towed at such a rate as to pro- 
duce similarity of the gravity-wave pattern only, that 
portion of the resistance due to viscous action being 
evaluated from supplementary tests on thin plates [9]. 
Likewise, in some problems of heat transfer the tem- 
perature differences are too small to influence the flow 
pattern appreciably—or at the worst can be assumed to 
cause gravitational effects but not the more complex 
thermodynamic effects. However, many phases of fluid 
motion are completely impossible to simulate in even 
approximate detail, and efforts must be directed 
toward reproduction of the over-all occurrence regard- 
less of the discrepancies in minor effects. River models 
LABORATORY INVESTIGATIONS 
with movable beds [6] are outstanding examples of 
willful distortion of slope, roughness, and vertical and 
horizontal proportions to achieve this end; the model 
channels are exaggerated in depth to maintain turbu- 
lent flow, and the proper time and discharge scales for 
over-all similarity are then determined in preliminary 
tests through reproduction of known past occurrences * 
before the desired future occurrences are investigated. 
Laboratory Methods of Prototype Simulation 
Since air is the fluid medium involved in nearly all 
meteorological phenomena, the familiar wind tunnel of 
aeronautical research [19] is the type of pertinent equip- 
ment which first suggests itself. Such equipment has, 
in fact, already been used, without modification, for 
the study of flow over particular boundary configura- 
tions. The fact remains, however, that aeronautical 
wind-tunnels are designed for specific purposes which 
are generally quite different from those of research in 
meteorology. Whereas aeronautical studies require high 
speeds, test sections only long enough to accommodate 
model planes, and dynamometers capable of measuring 
complex force systems, meteorological needs usually 
involve high capacities rather than speeds, test sections 
which can be adapted to a wide variety of boundary 
forms, and means of measuring the characteristics of 
the flow pattern rather than the forces exerted. In fact, 
various meteorological studies have been made in an 
open hall without any tunnel whatever, the air flow 
being produced by means of a series of blowers directed 
as desired. 
Although a technique such as that last cited is often 
satisfactory for qualitative studies, the fact remains 
that the air stream in quantitative investigations must 
generally be uniform and of moderately low turbulence, 
a condition which cannot be achieved by unconfined 
jets from blowers except in their immediate vicinity. 
It is therefore preferable to house the model in a uniform 
duct with a properly formed bell inlet, and to draw the 
air through this duct by means of an exhaust fan or 
fans at the downstream end. Because of the large 
capacities rather than speeds (and hence the small 
pressure changes) which are involved, such ducts may 
be erected quite cheaply for the particular project in 
question, and then readily modified for the next prob- 
lem to be studied. The cost of a closed circuit on a large 
scale is generally prohibitive, and only smaller tunnels 
need be made recirculating. However, complete en- 
closure of the duct in a large hall is desirable in order to 
eliminate the effect of atmospheric disturbances upon 
the inflow, provided the hall is of sufficient size to permit 
stilling of the exhaust from the blowers before the air 
re-enters the inlet. On the other hand, tunnel systems 
utilizing smoke, gas, or heated air sometimes draw 
directly from and discharge into the out-of-doors; such 
tunnels require a relatively large stilling room just 
upstream from the test section to eliminate the variable 
effect of winds. While blowers may be obtained com- 
mercially for a wide variety of flow conditions, because 
of the small pressure changes involved in such studies 
the use of surplus airplane propellers driven at relatively 
