Page 117 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

102. BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

9.2 The Determination of the Fluid Flow Induced

by the Motion of an Obstacle in the Fluid.
The Case of the Circular Cylinder

Let us consider an obstacle, bounded by the contour (C), which is
moving in the fluid mass supposed at rest at infinity. We know that the
circulation along the contour (C) is necessarily constant; in the sequel,

we limit ourselves to the case when this constant is zero.
Our aim, using the above notation, is to determine at any instant f,
afunction f(z) holomorphic outside (C), whose derivative ag is zero at
far distances and whose imaginary part along (C), fulfils the condition

v=a(ty—pwe- "4 (2? + y”) .

Suppose now, for sake of simplicity, that we solve first, the following
particular cases of the initially proposed problem, which are distinct by
the values characterizing the obstacle rototranslation:
Na=1,8=0,2=0;
2a=0,8=1,02=0;
3)a=0,6=0,2=1.

In all these cases we may assume that the corresponding complex
potential f is independent of time (the attached domains having a fixed
in time shape); denote by f(z), f(z), f(z), the complex potentials
which correspond to these three cases respectively.

It is obvious that, in general, a , 8 , 2k being supposed arbitrary
continuous functions of time, the function

f (z,t) = a(t) f (z) + B(t) fF (2) + 2H FO (2)

represents a solution of the initial proposed problem’®. One could prove
that the flow thus determined is unique, according to the uniqueness of
the respective Dirichlet problem. Concerning the effective determination
of the functions f(z), in the first two cases (when the displacement

of the obstacle is a uniform translation of unit velocity) the fluid flow
watched from Axy, can be identified with a steady flow of the type
already studied in the section devoted to the theory of profiles. The
third case is that of a uniform rotation. This case, as the previous two,
can be explicitly solved if we know the canonical conformal mapping of

the outside of (C) onto the exterior of a circular circumference.

'6The solution of the respective Dirichlet problem being a linear functional of the boundary
data.