fluid_mechanics

348 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling
V7.3 Reynolds
number
No separation
Re ≈ 0.2
Laminar boundary layer,
wide turbulent wake
Re ≈ 20,000
V7.4 Froude
number
The magnitude of the weight of the particle, F G , is F G gm, so the ratio of the inertia to the gravitational
force is
Thus, the force ratio F IF is proportional to V 2 G
g/, and the square root of this ratio, V1g/, is called
the Froude number. We see that a physical interpretation of the Froude number is that it is a measure
of, or an index of, the relative importance of inertial forces acting on fluid particles to the weight
of the particle. Note that the Froude number is not really equal to this force ratio, but is simply
some type of average measure of the influence of these two forces. In a problem in which gravity
1or weight2 is not important, the Froude number would not appear as an important pi term. A similar
interpretation in terms of indices of force ratios can be given to the other dimensionless groups,
as indicated in Table 7.1, and a further discussion of the basis for this type of interpretation is given
in the last section in this chapter. Some additional details about these important dimensionless groups
are given below, and the types of application or problem in which they arise are briefly noted in the
last column of Table 7.1.
Reynolds Number. The Reynolds number is undoubtedly the most famous dimensionless
parameter in fluid mechanics. It is named in honor of Osborne Reynolds 11842–19122, a
British engineer who first demonstrated that this combination of variables could be used as a criterion
to distinguish between laminar and turbulent flow. In most fluid flow problems there will
be a characteristic length, /, and a velocity, V, as well as the fluid properties of density, r, and
viscosity, m, which are relevant variables in the problem. Thus, with these variables the Reynolds
number
arises naturally from the dimensional analysis. The Reynolds number is a measure of the ratio of
the inertia force on an element of fluid to the viscous force on an element. When these two types
of forces are important in a given problem, the Reynolds number will play an important role. However,
if the Reynolds number is very small 1Re 12, this is an indication that the viscous forces
are dominant in the problem, and it may be possible to neglect the inertial effects; that is, the density
of the fluid will not be an important variable. Flows at very small Reynolds numbers are commonly
referred to as “creeping flows” as discussed in Section 6.10. Conversely, for large Reynolds
number flows, viscous effects are small relative to inertial effects and for these cases it may be
possible to neglect the effect of viscosity and consider the problem as one involving a “nonviscous”
fluid. This type of problem is considered in detail in Sections 6.4 through 6.7. An example
of the importance of the Reynolds number in determining the flow physics is shown in the figure
in the margin for flow past a circular cylinder at two different Re values. This flow is discussed
further in Chapter 9.
Froude Number.
The Froude number
F I
V 2
F G g/ V * dV * s
s
ds*
Re rV/
m
Fr
V
1g/
is distinguished from the other dimensionless groups in Table 7.1 in that it contains the acceleration
of gravity, g. The acceleration of gravity becomes an important variable in a fluid dynamics
problem in which the fluid weight is an important force. As discussed, the Froude number is
a measure of the ratio of the inertia force on an element of fluid to the weight of the element. It
will generally be important in problems involving flows with free surfaces since gravity principally
affects this type of flow. Typical problems would include the study of the flow of water
around ships 1with the resulting wave action2 or flow through rivers or open conduits. The Froude
number is named in honor of William Froude 11810–18792, a British civil engineer, mathematician,
and naval architect who pioneered the use of towing tanks for the study of ship design. It
is to be noted that the Froude number is also commonly defined as the square of the Froude number
listed in Table 7.1.