Page 101 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

86 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

Hence the normal component of the total fluid flow at the point (xj, y;)
is

Voo COS Bi Le | on ~ ds 5
mee
—

which, vanishing for every 2 (the slip-condition), will generate a linear
algebraic system to determine the unknowns 7), 72, ... , Yj, + » %n- But
this time, in contrast with the source panel method, the system should be

completed with the Joukovski condition yz = 0. In fact, the fulfilment
of this last condition could be performed by considering two small panels
(panels 7 and 1—1), in the neighborhood of the sharp trailing edge, such
that the control points 7 and 7 — 1 are close enough to the trailing edge,
and imposing that ~; = —+y,-,. This leads to the “a priori’ fulfilment
of the Joukovski condition. At the same time, to avoid the approach of
an over-determined system of nm unknowns with n + 1 equations we will

ignore the slip-condition at one of the control points and so we get again
a system of n linear algebraic equations with munknowns, which can be
solved by conventional techniques.
Obviously, the obtained solution, besides the slip-condition, will sat-
isfy the Joukovski condition too. More, the tangential velocities to the

boundary are equal to y which could be seen clearly supposing that, at
every point inside the body (on the “lower” part of the vortex sheet
too) the velocity ug = 0. Hence, the velocity outside the vortex sheet is
y = Uy — Ug = U, — 0 = uy, So that the local velocities tangential to the
surface (boundary) are equal to the local values of +.
Concerning the circulation, if S; is the length of the j-th panel, then
the circulation due to the j-th panel is y;S; and the total circulation is
nm n
l= $3 7;5; and, correspondingly, the lift L is pogVoo > Sj
i i
Finally, we remark that the accuracy problems have encouraged the
development of some higher-order panel techniques. Thus a “second-

order” panel method assumes a linear variation of yy over a given panel
such that, once the values of -y are matched at the edges to its neighbors,
the values of y at the boundary points become the unknowns to be
solved. Yet the slip-condition, in terms of the normal velocity at the
control points, is still applied.
There is also a trend to develop panel techniques using a combination
of source panels and vortex panels (source panels to accurately represent
“the thickness” of the profile while vortex panels to effectively provide

the circulation). At the same time, there are many discussions on the
control point to be ignored for “closing” the algebraic system in the case