Page 85 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

70 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

5. Principles of the (Wing) Profiles Theory
5.1 Flow Past a (Wing) Profile for an Incidence

and a Circulation ‘“‘a priori’? Given

Let (c) be a contour — the right section, in the working plane, of an
arbitrary cylinder; in aerohydrodynamics such a cylinder could be seen
as an airfoil or a wing of a very large (“infinite”) span (to ensure the
plane feature of the flow) and the respective right section (c) is called
wing profile or shorter profile’.
The main problem of the theory of profiles is to study the steady flow
of a fluid past a profile (obstacle), a flow which behaves at infinity (that

means for | z | very large) as a uniform flow of complex velocity

Voe ** = Vocosa — iVosina.

By incidence of the profile with respect to Oz, we will understand

the angle @ made by the velocity vector at far field (infinity) with the «
- axis. Besides the incidence of the profile let us also establish precisely
(“a priori’) the circulation T of the flow around the profile.
The determination of the complex potential comes then to the search
for an analytic function f(z) such that:
z
1) f(z) — # Jog is an analytic and uniform function in (d);
2) its imaginary part is constant along (c);

3) lim ff = Voe™*.
|z|+00 dz
Let (D) be the domain of the plane (Z) defined by |Z| > R and let
z = H(Z) or Z = h(z) be the canonical conformal mapping® which maps
(D) onto the domain (d), the exterior of the given profile (c).

The complex potential F(Z) of the associated (transformed) flow will
satisfy the properties 1), 2) and 3) provided that f and z are replaced by
F and Z, while (d) and (c) are replaced, respectively, by (D) and (C).
More precisely, the fulfilment of the conditions 1) and 2) comes from the
already studied parallelism between f(z) and F(Z), while the condition

‘With regard to the geometry of profiles, some additional considerations can be found, for
instance, in the Caius Iacob book “Introduction mathématique 4 la mécanique des fluides”’,
pp. 652-654 [69]. In this book, starting with p. 435, some special classes of profiles are
envisaged too.
’We recall the following basic theorem: “ There is a unique conformal mapping, called canon-
ical, of the domain (d) — the outside of the closed contour (c) — onto the outside of a circular
circumference (C) of radius R, centered at the origin, a mapping which in V(oo) admits a
co
development in the form z=Z+ >> <The radius R of the circumference (C) is an “a
n=0 2
priori” unknown length which depends only on the given contour (c).