Page 100 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 83

Eqn (2.46) to simplify Eqn (2.58a,b), Eqns (2.62) and (2.63) apply equally well to
compressible flow. In order to show this to be true, it is necessary to allow density to
vary in the derivation of Term (i) and to simplify it using the compressible form of the
continuity Eqn (2.45).

2.6.1 The Euler equations

For some applications in aerodynamics it can be an acceptable approximation to
neglect the viscous stresses. In this case Eqns (2.66) simplify to
(Z ;: ;;)
p -++-+v- =pg,-- aP (2.68a)
ax
dv dv dP
-pg
p -++-+v- dv)- Y ay (2.68 b)
--
( at ax ay
These equations are known as the Euler equations. In principle, Eqns (2.68a,b),
together with the continuity Eqn (2.46), can be solved to give the velocity components
u and v and pressurep. However, in general, this is difficult because Eqns (2.68a,b) can
be regarded as the governing equations for u and v, but p does not appear explicitly in
the continuity equation. Except for special cases, solution of the Euler equations can
only be achieved numerically using a computer. A very special and comparatively
simple case is irrotational flow (see Section 2.7.6). For ths case the Euler equations
reduce to a single simpler equation - the Laplace equation. This equation is much more
amenable to analytical solution and this is the subject of Chapter 3.

2.7 Rates of strain, rotational flow and vorticity

As they stand, the momentum Eqns (2.66) (or 2.67), together with the continuity Eqn
(2.46) (or 2.47) cannot be solved, even in principle, for the flow velocity and pressure.
Before this is possible it is necessary to link the viscous stresses to the velocity field
through a constitutive equation. Air, and all other homogeneous gases and liquids,
are closely approximated by the Newtonian fluid model. This means that the viscous
stress is proportional to the rate of strain. Below we consider the distortion experi-
enced by an infinitesimal fluid element as it travels through the flow field. In this way
we can derive the rate of strain in terms of velocity gradients. The important flow
properties, vorticity and circulation will also emerge as part of this process.

2.7.1 Distortion of fluid element in flow field
Figure 2.24 shows how a fluid element is transformed as it moves through a flow
field. In general the transformation comprises the following operations:
(i) Translation - movement from one position to another.
(ii) DilationlCompression ~ the shape remains invariant, but volume reduces or increases.
For incompressible flow the volume remains invariant from one position to another.
(iii) Distortion - change of shape keeping the volume invariant.
Distortion can be decomposed into anticlockwise rotation through angle
(a - p)/2 and a shear of angle (a + /3)/2.

The angles a and p are the shear strains.