Avkhadiev F.G., Elizarov A.M., Maklakov D.V. Optimal hydrodynamic
shape  design by the methods of complex analysis//Finite elements
in fluids.  New trends and applications,  Proceedings of the Int.
Conf.,  Venece,  15-21 October 1995.  P.1.,  M.Cecchi,  K.Morgan,
J.Periaux,  B.A.Schrefler, O.C.Zienkiewicz (Eds.), 1995. Univ. di
Padova, 1995. -- P.1639--1649.

General approaches   and   methods  for  solving  scientific  and
engineering problems of optimal shape design have been  developed
intensively  at  least  for  last two decades.  Usually numerical
technique is a base of design procedures.  The aim of this report
is  on  the  one  hand  to  demonstrate  benefits of applying the
analytical methods of complex analysis to the optimal  design  of
hydrodynamic shapes and on the other hand to obtain the solutions
to  new  problems  of  practical  interest.  The  directions   of
investigations can be grouped as follows.

1. Optimal shape design of airfoils and hydrofoils

A choice  of  a  control  function  is  a  question  of principal
importance in the shape design of airfoils.  "The natural method"
by  means of variation of geometrical forms is not convenient due
to laborious computations.  A  more  practical  way  consists  in
employing  the  conformal  mapping of the flow region outside the
profile onto the region  outside  the  unit  circle.  A  velocity
distribution  $V(\gamma  )$ along the circle can be specified and
after that it is a rather simple matter to compute the  shape  of
airfoil, the boundary layer and all the aerodynamic coefficients.
Thus,  all main features of the flow  can  be  expressed  by  the
function $V(\gamma )$.

a) The method of optimal shape design of airfoils  based  on  the
control by means of $V(\gamma )$ has been developed.  The problem
of optimization of aerodynamic characteristics has  been  reduced
to  finding  a  minimum of a nonlinear functional with constrains
depending on angles of attack.  The existence and  uniqueness  of
the minimum have been investigated. The range of angles of attack
have been found in which the optimal profile is realistic and has
positive thickness along its entire length.

b) Profiles  for  maritime  applications  (hydrofoils)  must   be
designed  with  respect to onset of cavitation.  The inception of
cavitation is an undesirable phenomenon,  which will takes  place
if  the  pressure  minimum  on the hydrofoil surface becomes less
than the vapor  pressure.  The  value  of  the  pressure  minimum
certainly depends on the angle of attack $\alpha$. So, the design
of hydrofoils is always based upon the diagram  which  shows  the
coefficient  of  pressure  minimum $C_{pmin}$ via $\alpha$.  This
diagram is called the pressure envelope.

Since the pressure envelope is the main  characteristic  for  any
hydrofoil  it  is  very  attractive  to take this function as the
control function for shape design.  To do this one needs to solve
the  problem  of  determining  the  shape of hydrofoil by a given
pressure envelope  $C_{pmin}(\alpha)$.  It  is  shown  that  this
problem  has  an  exact  analytical  solution.  By analyzing this
solution exact  mathematical  criteria  have  been  found,  which
demonstrate  weather  or  not  a  given  pressure envelope can be
realized for a certain  closed  profile.  Numerical  computations
shows the efficiency of the technique.

2. Exact estimates of hydrodynamic characteristics

The criteria  of  solvability  mentioned above allow a comparison
theorem for pressure envelopes to be proved,  which  leads  to  a
possibility  of solving certain problems of optimal shape design.
For instance,  the exact lower bounds of pressure  envelopes  for
symmetric  profiles  have  been  obtained  and  the corresponding
optimal profile shapes have been found  in  the  sense  that  the
obtained  profiles  are  the  best  from  the  point  of  view of
inception of cavitation in a given range of angles of attack.

An important problem of the theory of compressible  flow  over  a
rigid  body  is  to  determine  the  range  of the Mach number at
infinity $M_{\infty }$,  which realizes a  subsonic  flow  regime
everywhere  in the stream.  The upper bound $M^{*}$ of this range
is called the critical Mach number.

By the use of the Chaplygin gas model of compressible fluid for a
certain  class of lifting airfoils we have reduced the problem on
finding the maximum of $M_{\infty }$  to  a  minimax  problem  of
special  type.  The exact analytical solution to this problem has
been  constructed  by  employing  the  Lindel\"of  principle  for
conform transformations and,  therefore, the exact upper bound of
$M^{*}$ has been found.

3. Extremal problems of  the theory of jets

In the theory of jets and cavities there  are  only  few  results
related  to  finding  optimal  shapes and the corresponding exact
bounds of hydrodynamic forces exerted  on  curved  obstacles.  In
certain  areas  of  engineering  it  needs to determine the shape
which has a maximum braking force,  for example  in  constructing
parachutes or reversal devices of jet aircrafts. So, the problems
of designing the shape of maximum drag are  of  a  new  class  of
optimal  hydrodynamic problems of practical significance.  In the
report the following results in this area are presented.

The shape of a symmetric curved plate that moves in a fluid  with
formation  of  a wake and creates the maximum drag force has been
found.  To calculate the drag the  Joukovsky-Roshko-Eppler  model
has been used.  The optimal shapes can be interpreted as those of
ideal impermeable parachutes.

The shape of a symmetric curved plate which deflects a  free  jet
of  fixed  width  has  been  determined  so  that  the  angle  of
deflection is maximal.  Thus the best deflectors for the reversal
jet is found.