Page 120 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 105

function y is determined, the pressure can be calculated by applying the

Bernoulli theorem.

9.4 General Method for Determining of the Fluid
Flow Induced by the Displacement of an
Arbitrary System of Profiles Embedded in

the Fluid in the Presence of an *A Priori’
Given Basic Flow

In what follows we intend to give a brief survey on a new method which
allows us the solving of any direct problem of plane hydrodynamics, 1.e.,
to determine the fluid flow induced by a general displacement in the
inviscid fluid mass, of an arbitrary system of profiles, possibly in the

presence of unlimited walls, in the conditions of the pre-existence of an
already given “basic” flow which could present even a (finite) number of
singularities.
The great advantage of this method consists, not only in its general-
ity but also in the fact that it can be easily adapted to the numerical
calculations. A CVBM joined to this general method will be presented

later in this book.
From the mathematical point of view, by avoiding the conformal map-
ping technique, the method solves the proposed problem by using some
appropriate singular integral equations which, under our assumptions,
lead to a system of regular integral Fredholm equations. By imposing
some additional hypotheses on both the profiles and the “a priori’ ex-

isting basic flow, one establishes also, together with the solving of the
involved algebraic system, the existence and uniqueness theorems for the
respective integral equations.

9.4.1 The Mathematical Considerations and the
Presentation of the Method in the Case of Only One

Profile Moving in an Unlimited Fluid
Let us consider'’, as being given, a plane potential inviscid fluid flow
called the basic flow. Let wg(z) be the complex velocity of this basic
fluid flow.

Let us now imagine the fluid flow induced by a general displacement
(roto-translation) of an arbitrary profile in the fluid mass. Of course this
flow will superpose on that basic fluid flow. In what follows, we want

'7 Ror more details and even for the consideration of a general case of ”n” profiles, one could
read the paper of T. Petrila [103]. An extension of this method to the case of profiles with
sharp trailing edge and of the influence of some unlimited walls on the flow can also be found
in the papers of T. Petrila [102], [101].