Page 110 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 95

Adj = —w; = —150 (r — 75)
dy; _ ’
dn lon — 0

that is, equivalently, to choose w, (r) = -lj;Gn (r, r;) , where Gy (r, r;)
is the Green’s function for the Neumann problem associated with the

Laplace operator (Laplacian) in the domain D.
Retaking again the Euler system in the form

Aw = —-W,uU= Oy, v = —Oz'p, sie = 0,

we can write

b=-s for r’) In|r —r'| dr’,

and then we set u = dyy, v = —Ozy. But these equations seem to be
just the equations established for a system of point vortices, the integral

representation for ~ being replaced by the formula y% = > w; (r), valid

in the conditions of a point vortices system analogously a as a Riemann
integral is approximated by a Riemann sum. This suggests that an
inviscid incompressible flow can be approximated by the flow induced
by a discrete system of vortices, The convergence of solutions of the
discrete vortex equations to solutions of Euler’s equations as N — oo is

studied in [38] and in [61].
Vortex systems provide both a useful tool in the study of general
properties of Euler’s equations and a good starting point for setting up
effective algorithms for solving these equations in specific situations.

8. Thin Profile Theory

The theory of a wing with an infinite span (i.e., the theory of profiles)
requires knowledge of the conformal mapping of the profile outside, from
the physical plane (z) onto the outside of a disk from the plane (Z).
However, for an arbitrary (wing) profile, it is difficult to get effectively

this mapping; that is why, many times, one prefers the reverse procedure,
that is to construct (wing) profiles as images of some circumferences
through given conformal mappings. The Joukovski, Karman—Trefftz,
von Mises, etc. profiles belong to this category [69].
In the particular case of the thin profiles with weak curvature, the
problem of a flow past such a profile can be directly solved in a quite
simple approximative manner. More precisely, this time it will not be

necessary to determine the above mentioned conformal mapping but
only the solving, in the physical plane, of a boundary value problem of