474 Fundamentals of Fluid Mechanics and Transport Phenomena
8.6.2.5.3. Engineering formulae
We will designate under this category the input-output relations derived from
knowledge models and which are simple enough to be immediately useful for
example for quick dimensional assessment of components of a system in view of a
realization or for the representation of a sub-system in a more complex system.
These formulae are the result of reduced state representations whose pertinence has
been previously verified. They can also be obtained by simplification of an
analytical solution when one exists, for example by truncating a series (example of
section 8.6.2.4.2) or when using composite representations (section 8.3.2.2.3).
8.6.2.5.4.
Established regimes
The parametric expression of established regimes, obtained in section 8.5.2 in
the form of a series, can be simplified by truncation of this. Let us recall that the
representation in the form of a series is only of interest if it is possible to limit this to
a small number of terms. If this is not the case, this indicates that thermal boundary
layers exist in the domain, and it is therefore preferable to find a direct expression
which represents the boundary layer and to describe a composite solution by
matched asymptotic expansion (see section 8.3.2.2.3).
8.7. Application in fluid mechanics and transfer in flows
The evolution equations of a discrete or continuous system as a time function are
parabolic (or irreversible). In the presence of flow, the evolution variable is the time
only when Lagrangian variables are used. In Eulerian variables, the evolution speed
of a quantity
g is no longer represented by its temporal derivative tg ww but by the
material derivative
dtdg . This representation does not change the parabolic
character of the balance equations for extensive quantities with Euler variables
expressed along curvilinear abscissa of trajectories (or characteristic curves), which
is a parabolic variable equivalent to the time with Lagrange variables (see
interpretation of section 5.2.1) upstream then becoming the equivalent of the past. In
steady flow, the time variable disappears and the evolution variable becomes the
coordinate of the particle trajectories: systems studied thus appear as dynamic
systems along the trajectories. The same is true of boundary layer equations, or more
generally of the evolution of fluid properties along trajectories. These preceding
ideas are thus applicable to problems encountered in flows.
The methods evoked in this chapter are encountered when dealing with the
solution of flow and transfer problems in boundary layers ([SCH 99], [YIH 77]).
However, these are generally non-linear and computations cannot be effected in as
complete a manner as in this chapter. Writing balance equations, ultimately with
approximations which may be more or less global, leads to state representations