fluid_mechanics

358 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling
COMMENT This same model could also be used to predict
the drag per unit length, d / 1lbft or Nm2, on the prototype, since
the drag would depend on the same variables as those used for the
frequency. Thus, the similarity requirements would be the same
and with these requirements satisfied it follows that the drag per
unit length expressed in dimensionless form, such as d /DrV 2 ,
would be equal in model and prototype. The measured drag per
unit length on the model could then be related to the corresponding
drag per unit length on the prototype through the relationship
d / a D b a r b a V 2
b d
D m r m V /m
m
7.8.2 Model Scales
It is clear from the preceding section that the ratio of like quantities for the model and prototype
naturally arises from the similarity requirements. For example, if in a given problem there are two
length variables / 1 and / 2 , the resulting similarity requirement based on a pi term obtained from
these two variables is
so that
/ 1
/ 2
/ 1m
/ 2m
The ratio of a
model variable to
the corresponding
prototype variable
is called the scale
for that variable.
/ 1m
/ 2m
/ 1 / 2
We define the ratio / 1m/ 1 or / 2m/ 2 as the length scale. For true models there will be only one length
scale, and all lengths are fixed in accordance with this scale. There are, however, other scales such
as the velocity scale, V mV, density scale, r mr, viscosity scale, m mm, and so on. In fact, we can define
a scale for each of the variables in the problem. Thus, it is actually meaningless to talk about
a “scale” of a model without specifying which scale.
We will designate the length scale as l / , and other scales as l V , l r , l m , and so on, where the
subscript indicates the particular scale. Also, we will take the ratio of the model value to the prototype
value as the scale 1rather than the inverse2. Length scales are often specified, for example,
1
as 1 : 10 or as a 10 scale model. The meaning of this specification is that the model is one-tenth
the size of the prototype, and the tacit assumption is that all relevant lengths are scaled accordingly
so the model is geometrically similar to the prototype.
F l u i d s i n t h e N e w s
“Galloping Gertie” One of the most dramatic bridge collapses
occurred in 1940 when the Tacoma Narrows bridge, located near
Tacoma, Washington, failed due to aerodynamic instability. The
bridge had been nicknamed “Galloping Gertie” due to its tendency
to sway and move in high winds. On the fateful day of the collapse
the wind speed was 65 kmhr. This particular combination of a
high wind and the aeroelastic properties of the bridge created large
oscillations leading to its failure. The bridge was replaced in 1950,
and plans are underway to add a second bridge parallel to the ex-
isting structure. To determine possible wind interference effects
due to two bridges in close proximity, wind tunnel tests were run
in a 9 m 9 m wind tunnel operated by the National Research
Council of Canada. Models of the two side-by-side bridges, each
having a length scale of 1 : 211, were tested under various wind
conditions. Since the failure of the original Tacoma Narrows
bridge, it is now common practice to use wind tunnel model studies
during the design process to evaluate any bridge that is to be
subjected to wind-induced vibrations. (See Problem 7.72.)
7.8.3 Practical Aspects of Using Models
Validation of Model Design. Most model studies involve simplifying assumptions with
regard to the variables to be considered. Although the number of assumptions is frequently less
stringent than that required for mathematical models, they nevertheless introduce some uncertainty
in the model design. It is, therefore, desirable to check the design experimentally whenever
possible. In some situations the purpose of the model is to predict the effects of certain proposed
changes in a given prototype, and in this instance some actual prototype data may be available.
The model can be designed, constructed, and tested, and the model prediction can be compared