Page 115 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

100. BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

9. Unsteady Irrotational Flows Generated by the

Motion of a Body in an Inviscid
Incompressible Fluid

In what follows we will formulate the mathematical problem for deter-
mination of the fluid flow induced by a general displacement (motion) in
the fluid mass of a rigid body, this fluid flow being unsteady (in general).
Before considering separately either the 2-dimensional (plane) or the 3-
dimensional case, we remark that the problem of a uniform displacement
of a body with the velocity —v,, in a fluid at rest, is completely equiv-
alent with the problem of a uniform free-stream of velocity v, past the

same body but supposed fixed. This fact comes out at once, if one con-
siders also, besides the fixed system of axes, a mobile reference frame
rigidly linked to the body and we express the position vector (radius)
of the same point within these two systems, namely r = r’+ro; then, by
derivation, one deduces a similar relation between the velocity vectors

expressed in the two systems, that is v’ = v-+vo . Hence, the rest state
at infinity versus the fixed system (v = Q), will be the state of a uniform
motion with the velocity v.. within the mobile system where the body
could be seen fixed (being rigidly linked to it).

9.1 The 2-Dimensional (Plane) Case

In general, when we deal with the case of unsteady plane flows we
need first to introduce a fixed system of axes OXY. With respect to

this system, at any instant t, the flow will be determined by its complex
potential F(z, t), defined up to an additive function of time. The uniform
derivative of this complex potential will provide the components U and
V on the axes OX and OY.

The function F(Z, t) in the domain (D), where it is defined at any mo-
ment ¢, iseither a uniform function (which means a holomorphic function
of Z) or the sum of a holomorphic function and some logarithmic terms,
the critical points of these last ones being interior to the connected com-
ponents ( A, ) of the complement of (D). “A priori”, the coefficients

ratte of these logarithmic terms can depend on time but, under our
assumption, I’, are necessary constant. If this does not happen, the
circulation along a fluid contour encircling ( A, ), a contour which is

followed during the motion, will not be constant, in contradiction with
the Thompson theorem.
The determination of F should be done by using both the initial

conditions (a specific feature for the unsteady flows) and the boundary
conditions attached to the problem.