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\AtBeginDocument{{\noindent\small
2004-Fez conference on Differential Equations and Mechanics \newline
{\em Electronic Journal of Differential Equations},
Conference 11, 2004, pp. 103--108.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{103}

\begin{document}

\title[\hfilneg EJDE/Conf/11 \hfil Quasi spectral data construction]
{Quasi spectral data construction in two
points in Partially nonhomogeneous turbulence}

\author[M. O. Cherkaoui Malki \& S. Sayouri\hfil EJDE/Conf/11 \hfilneg]
{Mohammed ou\c{c}amah Cherkaoui Malki, Salaheddine Sayouri} % in alphabetical order

\address{M. O. Cherkaoui Malki \hfill\break
Laboratoire d'informatique \\
D\'{e}partement de Math\'{e}matiques et Informatique \\
Facult\'{e} des Sciences Dhar Mehraz \\
B.P. 1796 F\`{e}s-Atlas Morocco}
 \email{cherkaouimmo@hotmail.com}

\address{Salaheddine Sayouri \hfill\break
LPTA, D\'{e}partement de Physique \\
Facult\'{e} des Sciences Dhar Mehraz \\
B.P. 1796 F\`{e}s-Atlas Morocco}
\email{ssayouri@yahoo.com}

\date{}
\thanks{Published October 15, 2004.}
\subjclass[2000]{65Z05, 65R10, 76F55, 42B99}
\keywords{Navier-Stokes; nonlinearity; Green's function; \hfill\break\indent
mathematical formalism; quasi spectral analysis; data construction}

\begin{abstract}
 In this work, we study the numerical solution of the equations
 of correlations -or moment of order two - associated with the
 Navier-Stokes equations. We treat the spectral transformations of these
 equations by admitting directions of homogeneities; the problem of finding
 suitable initial conditions within the framework of the numerical resolution
 and the writing in two points is dealt with. We provide a new method of
 construction of these initial conditions in an intermediate space between
 physical space and spectral space (quasi spectral space). This original method
 departs from the formalism known in the homogeneous case and takes into
 account the presence of the walls. It is all the more interesting as the
 experimental data never give enough point of calculation making it possible to
 obtain these quantities in a quasi spectral space.
\end{abstract}

\maketitle
\numberwithin{equation}{section}

\section{Introduction}

When one is interested in nonhomogeneous turbulence, the
Navier-Stokes equations \eqref{e1.1} and the models based on a
description in only one point such as turbulent viscosity \cite{b1},
``the $k-\varepsilon$'' \cite{l1} or other models, provide only one
partial description of all of the phenomena associated with these
flows. One thus expects the development of new writings in two (or
more) points (i.e. descriptions taking into account the
interaction between the various structures of turbulence).
 \begin{equation}
\frac{\partial}{\partial t}V_{i}+V_{j}\frac{\partial}{\partial
x_{j}} V_{i}-\eta\Delta V_{i}+\frac{\partial}{\partial
x_{i}}P=0\label{e1.1}
\end{equation}
where $V_{i}$ is the velocity component along $x_{i}$,
$P$ is the pressure, and $\Delta$ is the Laplacian

The models in two points are based on the resolution of the
equations of the correlations \cite{c1}, $Q_{ij}(x,x')$, given by
\begin{equation}
Q_{ij}(x,x')=\langle v_{i}(x)v_{j}(x')\rangle \label{e1.2}
\end{equation}

where $\langle\cdot \rangle$ stands for the statistical average
and where $v_i$ ($i=1,2,3$) are the velocity fluctuations
and are defined by
$V_{i}=\langle V_{i}\rangle +v_{i}$
One must then solve the tensor of Reynolds \eqref{e1.3} in which each
term is described by a nonlinear equation containing other terms of the tensor.
\begin{equation}
\begin{pmatrix}
Q_{11} & Q_{12} & Q_{13}\\ Q_{21} & Q_{22} & Q_{23}\\
Q_{31} & Q_{32} & Q_{33}
\end{pmatrix} \label{e1.3}
\end{equation}
Moreover, to be able to carry out a complete analysis, we must
simultaneously determine turbulence in physical and spectral
spaces. Obviously, development of such models is not easy and the
numerical resolution of the new equations obtained starting from
the equations of Navier-stokes is even more complicated; we must
deal with problems in terms of mathematical formalism, numerical
processing and physical modelling \cite{c2,c3,c4,k1}.

A priority, for solving correlations equation, is fixed in the
search for formulations being able to lower the constraints of
calculation and storage.

Besides, The complexity of the double correlation's tensor and the
non linearity make very difficult the control of the evolution of
any inaccuracy allowed at the beginning of calculations.

Our study led us to the quasi spectral formulation. The
one-dimensional Fourier transforms are employed according only to
the homogeneity directions.The partial equations with the
derivative are the simplest and allow exploiting symmetries which
are presented by the flow geometry (non homogeneous partial
turbulence). The space quasi spectral contains at the same time
components of the wave number - according to one or several
homogeneity directions - and of the Cartesian co-ordinates (on a
nonhomogeneous directions).

 Besides the physical interest,
 calculations in the quasi spectral domain present two other advantages:
\\
- It makes it possible to write Green's functions in an exact
analytical form.
\\
- It allows optimization of the problem of the partial derivative
while making it possible to take into account various degrees of
homogeneity of the nonhomogenous flow (partially nonhomogenous
turbulence).

 As mentioned previously, the production of the data
in a quasi-spectral space for the physical quantities of
references constitutes a problem because of the lack of points of
calculation: for reasons of wake of probe, for example, the
experiments never provide enough points of calculation to evaluate
these quantities in the quasi spectral space.

It is thus important, due to the complexity of the equations and
the non linearity, to determine suitable initial conditions, and
to set up processes which, on the basis of homogeneous quantities,
take into account the presence of walls and produce physically
acceptable nonhomogeneous quantities. We propose in what follows
the processes that we built for this purpose and discuss their
advantages and disadvantages.

\section{Nonhomogeneous data construction}

The problem of the influence of the starting data always arises when one wants
to judge the quality of a physical model. A suitable representation of these
data is of primary importance. In this paragraph we provide techniques which
make it possible to build these data in a quasi-spectral space.

Let $x_{2}$ be a direction of non homogeneity transverse to the wall.
$N(n_{1},n_{2},n_{3})$ and $M(m_{1},m_{2},m_{3})$ be two points
of the wall by which the transversal line passes,
$X(x_{1},x_{2},x_{3})$ an internal point
belonging to $[N,M]$ (see figure 1)


\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(90,53)(0,0)
\put(38,32){\line(-1,4){3.4}}
\put(38,32){\line(4,1){15}}
\put(38,32){\line(1,1){10}}
\put(30,44){$x_1$}
\put(52,33){$x_2$}
\put(43,42){$x_3$}
%
\put(10,15){\line(4,1){71}}
\put(30,19.29){$\bullet$}
\put(30,16){$x$}
\put(60,26.74){$\bullet$}
\put(60,23){$x'$}
\qbezier(6,40)(6,20)(16,0)
\put(6.5,32){\line(-4,-1){7}}
\put(8,22){\line(-4,-1){7}}
\put(11,12){\line(-4,-1){7}}
\put(15,2){\line(-4,-1){7}}
\qbezier(78,54)(78,34)(88,14)
\put(78.5,47){\line(4,1){7}}
\put(80,37){\line(4,1){7}}
\put(82.5,27){\line(4,1){7}}
\put(86.8,17){\line(4,1){7}}
\put(13,11.5){$M$}
\put(77,27){$N$}
\end{picture}
\end{center}
\caption{One direction of non homogeneity ($x_2$)}
\end{figure}

Let us consider the characteristic scales of the flow
where $\epsilon$ is dissipation,
 $u$'s are velocities,  $l$ is the length,
and $\vec{K}(k_{1},k_{2},k_{3})$ is the wave number with $K$ its module.

Let $E^{h}(K)$ be the homogeneous and isotropic energy spectrum associated on
the scales$u,\epsilon$ and $l$.
$E^{h}(K)$ can come from a modelling of the type $K$-$\epsilon$ or
turbulent viscosity (or different as we will take it further).

The homogeneous and isotropic spectral correlation $\Phi_{ij}^{h}$ associated
to $E^{h}(K)$ is calculated from
\begin{equation}
\Phi_{ij}^{h}(k_{1},k_{2},k_{3})=\frac{E_{{}}^{h}(K)}{4\pi K^{2}}(\delta
_{ij}-\frac{k_{i}k_{j}}{K^{2}})\label{e2.1}
\end{equation}
The passage to a quasi spectral space according to $x_2$ is done by applying the
traditional Fourier transforms:
\[
\hat{Q}_{ij}^{h}(k_{1},r,k_{3})=\hat{Q}_{ij}^{h}(k_{1},x_{2},x_{2}+r,k_{3})
=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\Phi_{ij}
^{h}(k_{1},k_{2},k_{3})\exp(ik_{2}r)dk_{2} %\label{e2.2} \end{equation}
\]
The problem now is to produce non homogeneous correlations $\hat{Q}_{ij}$.
For this purpose, it is initially necessary to notice the
following differences between the functions $\hat{Q}_{ij}^{h}$ and
the nonhomogeneous correlations $\hat{Q}_{ij}$ according to $x_{2}$:
\\
(i)  The homogeneous correlations verify the condition:
\[
\hat{Q}_{ij}^{h}(k_{1},x_{2},x_{2}+r,k_{3})=\hat{Q}_{ij}^{h}(k_{1}
,x_{2},x_{2}-r,k_{3})\,.
\]
Which is significantly false in the nonhomogenous case especially
close to the walls where the double correlations have a broad
asymmetry.
\\
(ii) Boundary condition at the frontiers
\[
Q_{ij}^{h}(k_{1},x_{2},n_{2},k_{3})=Q_{ij}^{h}(k_{1},x_{2},m_{2},k_{3})=0
\]
($n_{2}$ and $m_{2}$ coordinates of the transverse frontier points)
are not respected.

To estimate the non homogeneous correlations Hamadiche \cite{h1} has
proposed to take
\begin{equation}
\hat{Q}_{ij}(k_{1},x_{2},x_{2}+r,k_{3})=\frac{1}{2\pi}\exp(ik_{2}r)\frac
{1}{2\pi}\int_{-\infty}^{+\infty}\Phi_{ij}^{h}(k_{1},k_{2},k_{3})dk_{2}
\label{e2.3}
\end{equation}
and to cancel these quantities at the wall.
However this writing is symmetrical and does not generate contributions for
the imaginary parts although these last-mentioned contain information about
the signal obtained after Fourier transformation.

\subsection*{First model}
To improve the preceding writing the first idea \cite{c3} consists in using the
functions of cap of Lebesgue:
$Lx_{2}()$ is linear on $[x_{2},m_{2}]$ and $[n_{2},x_{2}]$ respectively,
$Lx_{2}(n_{2})=0$,
$Lx_{2}(m_{2})=0$,
$Lx_{2}(x_{2})=1$ (See figure2).

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1.2mm}
\begin{picture}(71,30)(4,5)
\put(15,10){\line(1,0){60}}
\put(15,5){\line(0,1){30}}
\dashline{1}(15,30)(50,30)
\dashline{1}(15,20)(50,20)
\dashline{1}(15,16.66)(30,16.66)
\dashline{1}(30,10)(30,30)
\dashline{1}(50,10)(50,30)
%
\put(20,10){\line(1,2){10}}
\put(20,10){\line(3,2){30}}
\put(30,30){\line(2,-1){40}}
\put(50,30){\line(1,-1){20}}
%
\put(12,29.5){1}
\put(12,9.5){0}
\put(4,16){$Lx'_2(x_2)$}
\put(4,20.5){$Lx_2(x'_2)$}
\put(19,7){$m_2$}
\put(29,7){$x_2$}
\put(49,7){$x'_2$}
\put(69,7){$n_2$}
\put(22.5,25){$Lx_2$}
\put(55.5,25){$Lx'_2$}
\end{picture}
\end{center}
\caption{Lebesgue's functions}
\end{figure}

One defines then the non-homogenous correlation as follows:
$$
\hat{Q}_{ij}(k_{1},x_{2},x'_{2},k_{3})=\hat{Q}_{ij}^{h}(k_{1}
,x_{2},x'_{2},k_{3})\ L_{x_{2}}(x_{2}')
$$
This interpretation has the advantage of making it possible to gradually take
into account the presence of the walls.
One can however check that it does not establish the fundamental equality
\begin{equation}
\hat{Q}_{ij}(k_{1},x_{2},x'_{2},k_{3})=\hat{Q}_{ji}(k_{1},x'
_{2},x_{2},k_{3})\label{e2.4}
\end{equation}
because $Lx_{2}(x'_{2})\neq Lx'_{2}(x_{2}$ (see figure 2)
which lead us to try to improve it.

\subsection*{Second model: Information by the First Most Distant Point (IFMDP)}

We develop another technique which has the advantage of observing all the
conditions quoted previously.
It consists in informing the points of the most distant wall initially and
proceeding by intervals in the following way:
To construct $\hat{Q}_{ij}(k_{1},x_{2},.,k_{3})$ (for fixed values of
$x_{2},k_{1}$ and $k_{3}$)
\\
1- One is interested initially in the interval $[x,M]$ for all the values of
$x_{2}$, we inform only the half interval starting from $x_{2}$ and moving
towards the most distant wall from the point of calculation (interval
$[x_{2},m_{2}]$) for all the values of $i$ and $j$.
See two distinct examples of functions
$\hat{Q}_{ij}(k_{1},x_{2},.,k_{3})$ and $\hat{Q}_{ji}(k_{1},x'_{2},.,k_{3})$
in figure 3.


\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(87,47)(-9,0)
\put(0,10){\line(1,0){75}}
\put(74.6,9.15){$\rightarrow$}
\put(0,0){\line(0,1){20}}
\put(0,26){\line(0,1){20}}
\put(-.9,46){$\uparrow$}
\put(5,8){\line(0,1){4}}
\put(70,8){\line(0,1){4}}
\dashline{1}(10,10)(10,45) 
\dashline{1}(20,10)(20,45)
\dashline{1}(10,23.1)(20,23.1)
\qbezier(10,45)(12,10)(70,10)
\qbezier(20,45)(25,-18)(70,10)
\put(9,6){$x'_2$}
\put(19,6){$x_2$}
\put(22,38){$Q_{ij}(x_2,\cdot)$}
\put(45,13){$Q_{ij}(x'_2,\cdot)$}
\put(-8,22){$Q_{ij}(x_2,x'_2$)}
\end{picture}
\end{center}
\caption{Construction of $\hat{Q}_{ij}(k_{1},x_{2},.,k_{3})$ by IFMDP}
\end{figure}

\noindent
2- Then, the rest of the points in the interval $]N, X[$ (for
example $\hat {Q}_{ij}(k_{1},x_{2},x'_{2},k_{3})$) are
precisely indicated with the help of equation \eqref{e2.4}.
Figure 4 represents the manner of informing $\hat{Q}_{ij}(k_{1},x_{2}
,x'_{2},k_{3})$.

The IFMDP technique is thus more precise and more adequate for the partial non
homogenous case. It has the advantage of making it possible to gradually take
into account the presence of the walls, it verifies both the two conditions
(i) and (ii) and fundamental equation \eqref{e2.4}.
We will further use IFMDP during numerical tests of validation of the new
mathematical closing of the pressure problem.

\section{Conclusion}

The spectral nature of turbulence has been taking into account by
using the translation of the equations in the quasi spectral
space. This space, intermediary between physical and spectral
spaces, is an additional field of study of turbulence and its
evolution. The associated formulation especially makes it possible
to divide, in stages, the consideration of the degrees of non
homogeneity. It also changes the nature of the equations to be
solved by decreasing considerably the problems of numerical
``congestion'' which generally characterize this type of
treatments.Among the methods of production of data in a
quasi-spectral space for the double correlations IFMDP technique
is more precise and more adequate for the partial non homogenous
case. It has the advantage of making it possible to gradually take
into account the presence of the walls, it verifies both the two
conditions (i) and (ii) and the fundamental equation \eqref{e2.4}.

\begin{thebibliography}{0}


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Gautier-Villard, Paris (1897).

\bibitem{c1} R. Comolet,
\emph{M\'{e}canique exp\'{e}rimentale des fluides Dynamique des fluides
r\'{e}els},  Masson 1994.


\bibitem{c2} M. M. O. Cherkaoui,
\emph{Contribution \`{a} l'\'{e}tude de la turbulence non homog\`{e}ne par
m\'{e}thode int\'{e}grale et spectrale},
Th\`{e}se de doctorat de l'Ecole Centrale de Lyon (1992).

\bibitem{c3} M. M. O. Cherkaoui, Omari Alaoui El K\'{e}bir, \& M. Hamadiche;
\emph{Mod\'{e}lisation des \'{e}quations de Navier-stokes pour un \'{e}coulement
non homog\`{e}ne'' 5\`{e}me Congr\`{e}s international de m\'{e}canique},
 Mohamedia (Avril 1999).

\bibitem{c4} Comte-Bellot,
\emph{Notes prises au cours de Melle Comte Bellot},
Support de cours Ecole centrale de Lyon 3ème année Ecully France.

\bibitem{h1} M. Hamadiche,
\emph{Analyse spectrale des m\'{e}canismes lin\'{e}aires des \'{e}coulements
turbulent s non homog\`{e}nes},
 Th\`{e}se de doctorat d'\'{e}tat Universit\'{e} Claude Bernard Lyon I (1985).

\bibitem{k1} J. Kim, P. Moin\& R. Moser,
\emph{Turbulence statistics in fully developed channel flow at low Reynolds number},
 J. F?. M?. vol, 177, pp. 133 - 166 (1987).

\bibitem{l1} B. E. Launder, D. B. Spalding,
\emph{Mathematical models of turbulence}, Academic Presse London (1972).

\end{thebibliography}

\end{document}