Page 73 - Aerodynamics for Engineering Students

56 Aerodynamics for Engineering Students

direction as the local velocity vector’. Since this is identical to the condition at a
solid boundary it follows that:

(a) any streamline may be replaced by a solid boundary without modifying the
flow. (This only strictly true if viscous effects are ignored.)
(b) any solid boundary is itself a streamline of the flow around it.

(ii) A filament (or streak) line - the line taken up by successive particles of fluid
passing through some given point. A fine filament of smoke injected into the
flow through a nozzle traces out a filament line. The lines shown in Fig. 2.2 are
examples of this.
(iii) A path line or particle path - the path traced out by any one particle of the fluid
in motion.
In unsteady flow, these three are in general different, while in steady flow all three are
identical. Also in steady flow it is convenient to define a stream tube as an imaginary
bundle of adjacent streamlines.

2.2 One-dimensional flow: the basic equations

In all real flow situations the physical laws of conservation apply. These refer to the
conservation respectively of mass, momentum and energy. The equation of state
completes the set that needs to be solved if some or all of the parameters controlling
the flow are unknown. If a real flow can be ‘modelled’ by a similar but simplified
system then the degree of complexity in handling the resulting equations may be
considerably reduced.
Historically, the lack of mathematical tools available to the engineer required that
considerable simplifying assumptions should be made. The simplifications used
depend on the particular problem but are not arbitrary. In fact, judgement is required
to decide which parameters in a flow process may be reasonably ignored, at least to
a first approximation. For example, in much of aerodynamics the gas (air) is con-
sidered to behave as an incompressible fluid (see Section 2.3.4), and an even wider
assumption is that the air flow is unaffected by its viscosity. This last assumption
would appear at first to be utterly inappropriate since viscosity plays an important
role in the mechanism by which aerodynamic force is transmitted from the air flow to
the body and vice versa. Nevertheless the science of aerodynamics progressed far on
this assumption, and much of the aeronautical technology available followed from
theories based on it.
Other examples will be invoked from time to time and it is salutory, and good
engineering practice, to acknowledge those ‘simplifying’ assumptions made in order
to arrive at an understanding of, or a solution to, a physical problem.

2.2.1 One-dimensional flow: the basic equations
of conservation
A prime simplification of the algebra involved without any loss of physical signifi-
cance may be made by examining the changes in the flow properties along a stream
tube that is essentially straight or for which the cross-section changes slowly (i.e.
so-called quasi-one-dimensional flow).