fluid_mechanics

370 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling
F I G U R E 7.9 Instrumented,
small-waterplane-area, twin
hull (SWATH) model suspended from
a towing carriage. (Photograph courtesy
of the U.S. Navy’s David W.
Taylor Research Center.)
on the ship. Ship models are widely used to study new designs, but the tests require extensive facilities
1see Fig. 7.92.
It is clear from this brief discussion of various types of models involving free-surface flows
that the design and use of such models requires considerable ingenuity, as well as a good understanding
of the physical phenomena involved. This is generally true for most model studies. Modeling
is both an art and a science. Motion picture producers make extensive use of model ships,
fires, explosions, and the like. It is interesting to attempt to observe the flow differences between
these distorted model flows and the real thing.
7.10 Similitude Based on Governing Differential Equations
Similarity laws can
be directly developed
from the equations
governing the
phenomenon of
interest.
In the preceding sections of this chapter, dimensional analysis has been used to obtain similarity
laws. This is a simple, straightforward approach to modeling, which is widely used. The use of dimensional
analysis requires only a knowledge of the variables that influence the phenomenon of
interest. Although the simplicity of this approach is attractive, it must be recognized that omission
of one or more important variables may lead to serious errors in the model design. An alternative
approach is available if the equations 1usually differential equations2 governing the phenomenon
are known. In this situation similarity laws can be developed from the governing equations, even
though it may not be possible to obtain analytic solutions to the equations.
To illustrate the procedure, consider the flow of an incompressible Newtonian fluid. For simplicity
we will restrict our attention to two-dimensional flow, although the results are applicable
to the general three-dimensional case. From Chapter 6 we know that the governing equations are
the continuity equation
and the Navier–Stokes equations
0u
0x 0v
0y 0
r a 0u
0t u 0u
0x v 0u
0y b 0p 0x m a 02 u
0x 2 02 u
0 2 y b
r a 0v
0t u 0v
0x v 0v
0y b 0p 0y rg m a 02 v
0x 2 02 v
0y 2 b
(7.28)
(7.29)
(7.30)
where the y axis is vertical, so that the gravitational body force, rg, only appears in the “y equation.”
To continue the mathematical description of the problem, boundary conditions are required.
For example, velocities on all boundaries may be specified; that is, u u B and v v B at all boundary
points x x B and y y B . In some types of problems it may be necessary to specify the pressure
over some part of the boundary. For time-dependent problems, initial conditions would also
have to be provided, which means that the values of all dependent variables would be given at
some time 1usually taken at t 02.
Once the governing equations, including boundary and initial conditions, are known, we are
ready to proceed to develop similarity requirements. The next step is to define a new set of