fluid_mechanics

372 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling
Governing equations
expressed in
terms of dimensionless
variables lead
to the appropriate
dimensionless
groups.
the Euler number, g/V 2 the reciprocal of the square of the Froude number, and mrV/ the reciprocal
of the Reynolds number. From this analysis it is now clear how each of the dimensionless
groups can be interpreted as the ratio of two forces, and how these groups arise naturally from the
governing equations.
Although we really have not helped ourselves with regard to obtaining an analytical solution
to these equations 1they are still complicated and not amenable to an analytical solution2, the dimensionless
forms of the equations, Eqs. 7.31, 7.34, and 7.35, can be used to establish similarity
requirements. From these equations it follows that if two systems are governed by these equations,
then the solutions 1in terms of u*, v*, p*, x*, y*, and t*2 will be the same if the four parameters
/tV, p 0rV 2 , V 2 g/, and rV/m are equal for the two systems. The two systems will be dynamically
similar. Of course, boundary and initial conditions expressed in dimensionless form must
also be equal for the two systems, and this will require complete geometric similarity. These are
the same similarity requirements that would be determined by a dimensional analysis if the same
variables were considered. However, the advantage of working with the governing equations is that
the variables appear naturally in the equations, and we do not have to worry about omitting an important
one, provided the governing equations are correctly specified. We can thus use this method
to deduce the conditions under which two solutions will be similar even though one of the solutions
will most likely be obtained experimentally.
In the foregoing analysis we have considered a general case in which the flow may be
unsteady, and both the actual pressure level, p 0 , and the effect of gravity are important. A
reduction in the number of similarity requirements can be achieved if one or more of these conditions
is removed. For example, if the flow is steady the dimensionless group, /tV, can be
eliminated.
The actual pressure level will only be of importance if we are concerned with cavitation. If not,
the flow patterns and the pressure differences will not depend on the pressure level. In this case, p 0
can be taken as rV 2 1or 1 2rV 2 2, and the Euler number can be eliminated as a similarity requirement.
However, if we are concerned about cavitation 1which will occur in the flow field if the pressure at
certain points reaches the vapor pressure, p v 2, then the actual pressure level is important. Usually, in
this case, the characteristic pressure, p 0 , is defined relative to the vapor pressure such that p 0 p r p v
where p r is some reference pressure within the flow field. With p 0 defined in this manner, the similarity
parameter p 0rV 2 becomes 1 p r p v 2rV 2 . This parameter is frequently written as
1 p r p v 2 1 2rV 2 , and in this form, as was noted previously in Section 7.6, is called the cavitation number.
Thus we can conclude that if cavitation is not of concern we do not need a similarity parameter
involving p 0 , but if cavitation is to be modeled, then the cavitation number becomes an important
similarity parameter.
The Froude number, which arises because of the inclusion of gravity, is important for problems
in which there is a free surface. Examples of these types of problems include the study of
rivers, flow through hydraulic structures such as spillways, and the drag on ships. In these situations
the shape of the free surface is influenced by gravity, and therefore the Froude number becomes
an important similarity parameter. However, if there are no free surfaces, the only effect of
gravity is to superimpose a hydrostatic pressure distribution on the pressure distribution created by
the fluid motion. The hydrostatic distribution can be eliminated from the governing equation 1Eq.
7.302 by defining a new pressure, p¿ p rgy, and with this change the Froude number does not
appear in the nondimensional governing equations.
We conclude from this discussion that for the steady flow of an incompressible fluid without
free surfaces, dynamic and kinematic similarity will be achieved if 1for geometrically similar
systems2 Reynolds number similarity exists. If free surfaces are involved, Froude number similarity
must also be maintained. For free-surface flows we have tacitly assumed that surface tension
is not important. We would find, however, that if surface tension is included, its effect would appear
in the free-surface boundary condition, and the Weber number, rV 2 /s, would become an additional
similarity parameter. In addition, if the governing equations for compressible fluids are
considered, the Mach number, Vc, would appear as an additional similarity parameter.
It is clear that all the common dimensionless groups that we previously developed by using
dimensional analysis appear in the governing equations that describe fluid motion when these equations
are expressed in terms of dimensionless variables. Thus, the use of the governing equations
to obtain similarity laws provides an alternative to dimensional analysis. This approach has the