fluid_mechanics

7.6 Common Dimensionless Groups in Fluid Mechanics 349
The Mach number
is a commonly used
dimensionless parameter
in compressible
flow
problems.
V7.5 Strouhal
number
Euler Number.
can be interpreted as a measure of the ratio of pressure forces to inertial forces, where p is some
characteristic pressure in the flow field. Very often the Euler number is written in terms of a pressure
difference, so that Also, this combination expressed as ¢p 1 2
¢p, Eu ¢prV 2 .
2 rV is called
the pressure coefficient. Some form of the Euler number would normally be used in problems in
which pressure or the pressure difference between two points is an important variable. The Euler
number is named in honor of Leonhard Euler 11707–17832, a famous Swiss mathematician who
pioneered work on the relationship between pressure and flow. For problems in which cavitation
is of concern, the dimensionless group 1 p r p v 2 1 2
2 rV is commonly used, where p v is the vapor
pressure and p r is some reference pressure. Although this dimensionless group has the same form
as the Euler number, it is generally referred to as the cavitation number.
Cauchy Number and Mach Number.
and the Mach number
The Cauchy number
are important dimensionless groups in problems in which fluid compressibility is a significant factor.
Since the speed of sound, c, in a fluid is equal to c 1E v r 1see Section 1.7.32, it follows
that
r
Ma V A E v
and the square of the Mach number
is equal to the Cauchy number. Thus, either number 1but not both2 may be used in problems in
which fluid compressibility is important. Both numbers can be interpreted as representing an index
of the ratio of inertial forces to compressibility forces. When the Mach number is relatively
small 1say, less than 0.32, the inertial forces induced by the fluid motion are not sufficiently large
to cause a significant change in the fluid density, and in this case the compressibility of the fluid
can be neglected. The Mach number is the more commonly used parameter in compressible flow
problems, particularly in the fields of gas dynamics and aerodynamics. The Cauchy number is
named in honor of Augustin Louis de Cauchy 11789–18572, a French engineer, mathematician, and
hydrodynamicist. The Mach number is named in honor of Ernst Mach 11838–19162, an Austrian
physicist and philosopher.
Strouhal Number.
The Euler number
Eu
p
rV 2
Ca rV 2
Ma V c
Ma 2 rV 2
Ca
E v
The Strouhal number
St v/
V
is a dimensionless parameter that is likely to be important in unsteady, oscillating flow problems
in which the frequency of the oscillation is v. It represents a measure of the ratio of inertial forces
due to the unsteadiness of the flow 1local acceleration2 to the inertial forces due to changes in velocity
from point to point in the flow field 1convective acceleration2. This type of unsteady flow
may develop when a fluid flows past a solid body 1such as a wire or cable2 placed in the moving
stream. For example, in a certain Reynolds number range, a periodic flow will develop downstream
from a cylinder placed in a moving fluid due to a regular pattern of vortices that are shed from the
body. 1See the photograph at the beginning of this chapter and Fig. 9.21.2 This system of vortices,
called a Kármán vortex trail [named after Theodor von Kármán 11881–19632, a famous fluid
E v