Mechanics of Fluids

168 Physical similarity and dimensional analysis
from fluid motion. These can be broadly classified as pressure coefficients,
force coefficients and friction coefficients, respectively. Amongst the force
coefficients are the lift and drag coefficients, of fundamental importance
in aircraft aerodynamics. The friction coefficients include amongst their
number the friction factor associated with pipe flow, and the skin friction
coefficient used to describe in a dimensionless form the surface shear stresses
of external fluid flow. Examples of dimensionless groups which fall outside
these named categories are the Strouhal number, which arises in the treatment
of cyclical phenomena, and the mass flow parameter, which is a useful
concept when dealing with compressible flows in pipes. Apart from the pressure
coefficients, which will be discussed explicitly, albeit briefly, all other
dimensionless groups will be considered as they arise.
5.5.1 The pressure coefficient and related coefficients
Pressure forces are always present and are therefore represented in any
complete description of fluid flow. When expressed in dimensionless form,
the ratio of pressure forces to other types of forces appears. For example,
the ratio
|Pressure force|
|Inertia force|
is proportional to �p∗l2 ϱl2 �p∗
=
u2 ϱu2 where �p∗ represents the difference in piezometric pressure between two
points in the flow. In fluid dynamics, it has become normal practice to use
the ratio �p∗ / 1
2ϱu2 , the 1
2 being inserted so that the denominator represents
kinetic energy divided by volume or, for an incompressible fluid, the dynamic
pressure of the stream (see Section 3.7.1). This latter form is usually known
as the pressure coefficient, denoted by the symbol Cp.
Several other similar coefficients are, in essence, variants of the pressure
coefficient. Amongst these are the static and total pressure loss coefficients,
widely used to describe the dissipation of mechanical energy that occurs in
internal fluid flows, and the pressure recovery coefficient more specifically
used to describe the properties of diffusers. Also in this category are the
corresponding head loss coefficients.
5.5.2 The discharge coefficient
The discharge coefficient is an important dimensionless parameter which
relates the flow rate through a differential-pressure flow-metering device,
such as an orifice plate, nozzle or venturi tube, to the pressure distribution
the flow generates. It was first introduced in Chapter 3, where it was
used to adjust theoretical values of mass flow rate (or volumetric flow rate),
derived from simplistic mathematical models of fluid motion which ignored
the effects of viscosity, to yield improved comparisons with the behaviour
of real flows. The method of dimensional analysis provides a more rigorous
justification for the use of the discharge coefficient, since, from the outset,
the analysis takes full account of the viscous nature of real fluids.