Page 109 - Aerodynamics for Engineering Students

92 Aerodynamics for Engineering Students

For example, the left-hand side of Eqn (2.95a) represents the total rate of change of
the x component of momentum per unit volume. Indeed it is often written as:
D d d d d
-
Du where --- +u-+ v-+ w- (2.96)
Dt- at ax ay az
is called the total or material derivative. It represents the total rate of change with time
following the fluid motion. The left-hand sides of Eqns (2.95bYc) can be written in a
similar form. The three terms on the right-hand side represent the x components of body
force, pressure force and viscous force respectively acting on a unit volume of fluid.
The compressible versions of the Navier-Stokes equations plus the continuity
equation encompass almost the whole of aerodynamics. To be sure, applications
involving combustion or rarified flow would require additional chemical and phys-
ical principles, but most of aerodynamics is contained within the Navier-Stokes
equations. Why, then, do we need the rest of the book, not to mention the remaining
vast, ever-growing, literature devoted to aerodynamics? Given the power of modern
computers, could we not merely solve the Navier-Stokes equations numerically for
any aerodynamics application of interest? The short answer is no! Moreover, there is
no prospect of it ever being possible. To explain fully why this is so is rather difficult.
We will, nevertheless, attempt to give a brief indication of the nature of the problem.
Let us begin by noting that the Navier-Stokes equations are a set of partial
differential equations. Few analytical solutions exist that are useful in aerodynamics.
(The most useful examples will be described in Section 2.10.) Accordingly, it is
essential to seek approximate solutions. Nowadays, it is often possible to obtain very
accurate numerical solutions by using computers. In many respects these can be
regarded almost as exact solutions, although one must never forget that computer-
generated solutions are subject to error. It is by no means simple to obtain such
numerical solutions of the Navier-Stokes equations. There are two main sources of
difficulty. First, the equations are nonlinear. The nonlinearity arises from the left-
hand sides, i.e. the terms representing the rate of change of momentum - the so-called
inertial terms. To appreciate why these terms are nonlinear, simply note that when
you take a term on the right-hand side of the equations, e.g. the pressure terms, when
the flow variable (e.g. pressure) is doubled the term is also doubled in magnitude.
This is also true for the viscous terms. Thus these terms are proportional to the
unknown flow variables, i.e. they are linear. Now consider a typical inertial term, say
uduldx. This term is plainly proportional to u2 and not u, and is therefore nonlinear.
The second source of difficulty is more subtle. It involves the complex effects of
viscosity.
In order to understand this second point better, it is necessary to make the Navier-
Stokes equations non-dimensional. The motivation for working with non-dimen-
sional variables and equations is that it helps to make the theory scale-invariant and
accordingly more universal (see Section 1.4). In order to fix ideas, let us consider the air
flowing at speed U, towards a body, a circular cylinder or wing say, of length L. See
Fig. 2.29. The space variables x, y, and z can be made nondimensional by dividing by
L. L/U, can be used as the reference time to make time non-dimensional. Thus we
introduce the non-dimensional coordinates

X=x/L, Y=y/L, Z=z/L, and T=tU/L (2.97)
U, can be used as the reference flow speed to make the velocity components
dimensionless and pU& (c.f. Bernoulli equation Eqn (2.16)) used as the reference