Page 110 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 93

Fig. 2.29

pressure. (For incompressible flow, at least, only pressure difference is of significance
and not the absolute value of the pressure.) This allows us to introduce the following
non-dimensional flow variables:
U = u/Um, V = v/UX, W = w/U,, and P =p/(pUL) (2.98)

If, by writing x = XL etc. the non-dimensional variables given in Eqns (2.97) and
(2.98) are substituted into Eqns (2.94) and (2.95) with the body-force terms omitted,
we obtain the Navier-Stokes equations in the form:
au av aw
-+-+-=o (2.99)
ax aY az

(2.100a)

(2.100b)
DW dP 1 8W @‘W @‘W
--
DT --E+%(=+dr2+=) (2.1 OOC)
where the short-hand notation (2.96) for the material derivative has been used.
A feature of Eqns (2.100) is the appearance of the dimensionless quantity known
as the Reynolds number:
PUmL
Re=- (2.101)
P
From the manner in which it has emerged from making the Navier-Stokes equations
dimensionless, it is evident that the Reynolds number (see also Section 1.4) represents
the ratio of the inertial to the viscous terms (i.e. the ratio of rate of change of
momentum to the viscous force). It would be difficult to overstate the significance
of Reynolds number for aerodynamics.
It should now be clear from Eqns (2.99) and (2.100) that if one were to calculate
the non-dimensional flow field for a given shape - a circular cylinder, for example -
the overall flow pattern obtained would depend on the Reynolds number and, in the
case of unsteady flows, on the dimensionless time T. The flow around a circular
cylinder is a good example for illustrating just how much the flow pattern can change
over a wide range of Reynolds number. See Section 7.5 and Fig. 7.14 in particular.
Incidentally, the simple dimensional analysis carried out above shows that it is not
always necessary to solve equations in order to extract useful information from them.
For high-speed flows where compressibility becomes important the absolute value
of pressure becomes significant. As explained in Section 2.3.4 (see also Section 1.4),
this leads to the appearance of the Mach number, M (the ratio of the flow speed to the
speed of sound), in the stagnation pressure coefficient. Thus, when compressibility