\documentclass[reqno]{amsart} 

\AtBeginDocument{{\noindent\small 
2004-Fez conference on Differential Equations and Mechanics \newline
{\em Electronic Journal of Differential Equations},
Conference 11, 2004, pp. 95--102.\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}{95}

\begin{document} 

\title[\hfilneg EJDE/Conf/11 \hfil Green's functions]
{Green's functions and closing in pressure in partially nonhomogeneous
 turbulence} 

\author[M. O. Cherkaoui M. \& E. Omari A. \hfil EJDE/Conf/11 \hfilneg]
{Mohammed ou\c{c}amah Cherkaoui Malki, Omari Alaoui Elkebir}  % 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{E. Omari Alaoui\hfill\break
Laboratoire de M\'{e}canique des fluides\\
D\'{e}partement de Physique \\
Facult\'{e} des Sciences Dhar Mehraz \\
B.P. 1796 F\`{e}s-Atlas Morocco}
\email{cherkaouimmo@hotmail.com}

\date{}
\thanks{Published October 15, 2004.}
\subjclass[2000]{65Z05, 65R10, 65R32, 76F55}
\keywords{Navier-Stokes; nonlinearity; Green's function; quasi spectral analysis;
\hfill\break\indent mathematical formalism;  data construction}


\begin{abstract}
 In this work, we are interested with the numerical solution of the equations
 of the correlations -or moment of order two - associated with the
 Navier-Stokes equations. The method of closing in pressure which is based on
 the elimination of the terms of pressure present in these equations by using
 the functions of Green, is completely re-examined. We underline the Green's
 functions divergence problems which occurred with the traditional method of
 closing. Then we establish a new formalism which makes it possible to
 circumvent these problems. We present and confront in the course of our
 presentation two methods of construction of Green's functions according to the
 choice of the boundary conditions, namely the method of images and the
 spectral method.
\end{abstract}

\maketitle
\numberwithin{equation}{section}

\section{Introduction}

 When one is interested in non homogeneous turbulence, the
Navier-Stokes equations 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, which are
defined by
$V_{i}=\langle V_{i}\rangle +v_{i}$
one must then solve the tensor of Reynolds in which each term is
described by a nonlinear equation which contains other terms of the tensor.
\[
\begin{pmatrix}
Q_{11} & Q_{12} & Q_{13}\\
Q_{21} & Q_{22} & Q_{23}\\
Q_{31} & Q_{32} & Q_{33}
\end{pmatrix}
\]
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,e1,j1}.
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 or at the representation of pressure
as a function of velocities.

\section{Conventional closing in pressure of the Navier-Stokes equation}

To simplify the search for a model of turbulence and the corresponding
algorithm, it is adequate from our point of view to restrict the
equations to the velocity terms. We thus understand by ``closing in pressure''
the representation of the terms of pressure present in the Navier-Stokes
equations as functions of the velocity components.

The general approach of closing consists in introducing the Green's kernel
during a formal calculation and to use the equations relative to the
incompressible fluids: $\sum\frac{\partial}{\partial xi}Vi=0$.

In general, Green's functions are presented in the form of series of
functions; their introduction systematically generates truncation errors in
the algorithm of the numerical resolution. Considering the complexity of the
tensorial writing of the double correlations and the non linearity, the
control of the evolution of these errors is then very difficult.

The conventional procedure consists of a first step, in writing the pressure
in an integral form such as
\begin{equation}
P(x)=\!\int_{\Omega}G(x,x")\Delta P(x")dx"+\int_{\Gamma}\frac{\partial
}{\partial n}G(x,x")P(x")dx"-\int_{\Gamma}G(x,x")
\frac{\partial}{\partial n}P(x")dx"\label{e2.1}
\end{equation}
where $\Omega$ is the flow domain, $\Gamma$ its border and
$\frac{\partial}{\partial n}$  the normal derivative to $\Gamma$.
$G(x,x')$ is the Green's function which  on $\Omega$ satisfies
\begin{equation}
\Delta _xG(x,x')=\delta(x-x^{'})\,. \label{e2.2}
\end{equation}
Note that here no more boundary conditions are imposed for $G(x,x')$ on
$\Gamma$.
One obtains \eqref{e2.1} by two formal applications of Gauss theorem from:
\begin{equation}
P(x)=\int_{\Omega}\Delta G(x,x")P(x")dx"\label{e2.3}
\end{equation}
which is the starting point of all the attempts of closing of the 
conventional procedure.

The aim is now to determine $\Delta P$ in the domain $P(x")$, and
$\frac{\partial}{\partial n}P(x")$ on $\Gamma$ as functions of the
velocities; see equation \eqref{e2.1}.

- In general $\Delta P$ is obtained from the Poisson's equation \eqref{e2.4}; 
in the case of incompressible fluids, the writing of the Laplacian of 
pressure in terms of velocities is simplified and takes the form:
\begin{equation}
\bigskip\Delta P(x)=f(\partial Vi/\partial xk)i,k=\frac{\partial}{\partial
xi}(Vj\frac{\partial}{\partial xj}Vi)\label{e2.4}
\end{equation}

- Moreover$\frac{\partial P}{\partial n}$ is deduced by making degenerate the
Navier-Stokes equations at $\Gamma$ which only needs to know the geometry of
the flow.

- $P(x)$: for the calculation of the remaining integral at $\Gamma$, one
introduces the following two limit problems:
Dirichlet type problem:
\begin{equation}
\begin{gathered}
\Delta_{x}G(x,x^{'})=\delta(x-x^{'}) \quad\mbox{in } \Omega\,,\\
G(X,X')=0\quad\mbox{on } \Gamma\,;
\end{gathered}\label{e2.5}
\end{equation} 
Neumann type problem:
\begin{equation}
\begin{gathered} 
\Delta_{x}G(x,x^{'})=\delta(x-x^{'}) \quad\mbox{in } \Omega\,,\\
\frac{\partial G}{\partial n}(x,x^{'})=0\quad\mbox{on } \Gamma
\end{gathered} \label{e2.6}
\end{equation}

Now the assumptions can be taken on the Green's function at $\Gamma$.
 In \cite{h1} it has been established a judicious procedure which consists in 
choosing boundary conditions on the Green's kernel in such a manner 
to eliminate the quantities
which one does not manage to translate. So, he imposes on the Green's kernel
boundary conditions of the Neumann type \eqref{e2.6} of such kind to 
eliminate the
integral on $\Gamma$ containing the terms in $P(x)$ in \eqref{e2.1}. 
One obtains then:
\begin{equation}
P(x)=\int_{\Omega}G(x,x")f(\frac{\partial Vi}{\partial x_{k}})_{_{i,k}%
}dx"-\int_{\Gamma}G(x,x")\frac{\partial P(x")}{dn}dx\label{e2.7}
\end{equation}
Thus, this simplified equation is of great interest and does not
introduce any additional assumption on the pressure.

However, the problem which remains is the convergence of $G(x,x')$ given
by \eqref{e2.6} and which we are dealing with in the next section.

\section{Green's functions and their convergence}

We present here two techniques of construction of the Green's functions. We
will try by the same occasion to clear up the origin of the divergence of some
of these functions.

\subsection*{Method of images}

The method of images has the advantage of providing an analytical solution.
This method takes as a starting point that known in electrostatics and more
precisely the method of coverings. It consists in constructing a charge
distribution which would produce the equipotential fixed beforehand.

More generally, when in electrostatics one imposes equipotentials and that one
wants to know the potential in an other given point of space, one adopts the
process known as the process of covering.
This process consists in determining the charge distribution, which would
produce these equipotentials. Then, one identifies the desired potential with
that which would produce the charge distribution thus determined.

To understand the concept of charge and image let us point out the following
solutions of the problems of Dirichlet and Neumann for the common cases:

\subsubsection*{The entire $\mathbb{R}^{3}$ space}
When the domain represents the entire $\mathbb{R}^{3}$ space, the two systems
\eqref{e2.5} and \eqref{e2.6} have the same solution $G\infty$ which is written as:
\begin{equation}
G_{\infty}(x,x')=\frac{-1}{4\pi\| x-x^{'}\|}\label{e3.1}
\end{equation}
This function is identified (except for a constant) with the potential
associated with a charge $q=-1$ placed in x. One will thus speak about
``turbulence charge'' by associating the potential $G\infty$ to it (In
electrostatics, one associate a potential to each charge).

\subsubsection*{Semi infinite domain}
When the domain is semi infinite (or semi closed) the Green's functions are 
respectively:
\begin{equation}
G_{1}(x,x')=\frac{-1}{4\pi\| x-x^{'}\|}+\frac
{1}{4\pi\| x^{\ast}-x^{'}\|}\label{e3.2}
\end{equation}
For system \eqref{e2.5}, and
\begin{equation}
G_{2}(x,x')= \frac{-1}{4\pi\| x-x^{'}\|
}+\frac{-1}{4\pi\| x^{\ast}-x^{'}\|}\label{e3.3}
\end{equation}
for system \eqref{e2.6}.
where $x\ast$ is the symmetric of $x$ with respect to the wall.

We remark that $G_{1}$ can be obtained in placing a charge $q'=-q$ in a
symmetrical point $x\ast$ of $x$ with respect to the plane$(x_{2}=0)$.
In the same way, if we changes the sign of q' then we find the analytical
solution of \eqref{e2.6}:

The determination of the Green's function is thus equivalent to that of a
``turbulence charges'' distribution whose equipotential zero coincides with
the frontier of the domain.
One will retain that in the first case, one has an alternation of sign
whereas, in the second, one cumulates quantities of the same sign.

\subsubsection*{Case of the plane channel}
Let us now consider the plane channel represented below:

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(100,61)(0,-6)
\put(0,35){\line(1,0){100}}
\put(0,35){\line(3,2){28.2}}
\put(45,53){${}^0q^1$}
\put(70,40){$\Gamma_2$}
%
\put(0,23){($\Omega$)}
\put(80,10){\line(1,0){15}}
\put(80,10){\line(0,1){15}}
\put(80,10){\line(3,2){14.1}}
\put(75,22){$x_2$}
\put(93,6){$x_1$}
\put(95,16){$x_3$}
%
\put(0,0){\line(1,0){100}}
\put(0,0){\line(3,2){28.2}}
\put(45,18){${}^0q$}
\put(45,-8){${}^0q^0$}
\put(15,5){$\Gamma_1$}
\end{picture}
\end{center}
\caption{Case of plane channel}
\end{figure}

According to what we have just observed in the case of half spaces and, by
considering initially only the plane $\Pi_{1}$, we start by placing a
symmetrical charge $q'$ of $q$ with respect to $\Pi_{1}$ in order to
satisfy the condition at the edge of this plane. We will place according to
this same point of view another charge $q"$ symmetrical of $q$ with respect to
$\Pi_{2}$ satisfying the condition at the edge of this second plane.
Obviously, these two new charges produce secondary effects which we must
neutralize thereafter. These effects are due to the influence of $q"$on the
$\Pi_{1}$ plane and that of $q'$ on$\Pi_{2}$.

We then place two new charges, the first one symmetrical of $q"$ with respect
to $\Pi_{1}$ and the second one symmetrical of $q'$ with respect to
$\Pi_{2}$ and so on.
Gradually, we set up an (infinite) charge distribution which satisfies at the
same time the two conditions.

It is then verified easily that the contribution to the potential of each
charge at the frontiers is neutralized respectively on each wall by that of
the two charges which are respectively symmetrical for them.

The solution that we then obtain for the problem of Neumann is an ensemble of
contributions resulting from charges of the same sign placed at distances in
``$1/r$'':
\begin{equation}
G_{N}=\frac{-1}{4\pi}\sum_{n=-\infty}^{n=+\infty}\frac{1}{\|
x-x^{'}-4ane_{2}\|}+\frac{1}{\| x-x^{'}-2a(2n+1)e_{2}%
+2x_{2}^{'}e_{2}\|}\label{e3.4}
\end{equation}
and that relative to the problem of Dirichlet is presented in the form of
alternate series (resulting from charges of opposite signs):
\begin{equation}
G_{D}=\frac{-1}{4\pi}\sum_{n=-\infty}^{n=+\infty}\frac{(-1)^{n}}{\|
x-x^{'}-4ane_{2}\|}+\frac{(-1)^{n}}{\| x-x^{'
}-2a(2n+1)e_{2}+2x_{2}^{'}e_{2}\|}\label{e3.5}
\end{equation}
Note that the function $G_{N}$ diverges for any couple of points $(x,x')$
of $\Omega\times\Omega$ and that the function$G_{D}$ is convergent in any
point of $\Omega\times\Omega$.
Thus when one uses the method of images, the study of the coherence of the
final solution obtained constitutes the final step. So nothing prevents from
pushing a little further the analogy and of saying that, as in electrostatics,
where one cannot conceive infinite potential in a point, one cannot preserve,
as for as we are concerned , only the Green's functions which take finished
values inside the domain.

\subsubsection*{More complex geometry}
When the geometry of the field is more complex the method of the images can
still be applied. It is enough for that to realize that the process that we
applied for two planes can be done for a polyhedric field delimited by N
planes. Moreover, in the case of curvilinear frontiers, we can always approach
the desired Green's function by that associated to a close polyhedric geometry.
The problem which remains is the divergence of the functions associated to
Neumann's system: the solution is provided by cumulating contributions of the
same sign.

\subsection*{Quasi spectral method}
This method which serves here to validate the method of images consists in
transporting the differential systems \eqref{e2.5} and \eqref{e2.6} in a spectral space by a
Fourier transform (on one or more space variables) and to solve the new system
thus obtained.

It is hoped whereas that the solution of the spectral problem admits an
opposite Fourier transform and that this transform is also solution of the
initial problem.
In the case of the entire space or of the half space, one can see easily that
this process leads to the same solutions referred to above.

With regard to the plane channel and taking into account the geometry of the
domain, it is necessary to take the spectral transform:
\begin{equation}
\widehat{f}(k_{1},x_{2},k_{3})=\int\int f(x_{1},x_{2},x_{3})\exp
(-i(k_{1}x_{1}+k_{3}x_{3}))dx_{1}dx_{3}\label{e3.6}
\end{equation}
The differential system associated with the boundary conditions of the
Dirichlet type can be written
\begin{equation}
\begin{gathered}
(\frac{\partial^{2}}{\partial x_{2}^{2}}-q^{2})W^{13}(x_{2},x"_{2})=\frac
{1}{4\pi^{2}}\delta(x_{2}-x"_{2}),\\
W^{13}=0\quad\mbox{in }  \Gamma\times\Gamma
\end{gathered} \label{e3.7}
\end{equation}
where $q^{2}=k_{1}^{2}+k_{3}^{2}$
and
\begin{equation}
W^{13}(x_{2},x"_{2})=4\pi^{2}\hat{G}(k_{1},x_{2},x"_{2},k_{3}%
)\exp(i(k_{1}x_{1}+k_{3}x_{3}))\label{e3.8}
\end{equation}
The solution of this differential system can be obtained by taking into
account the following remarks:
\\
(1)  On any interval not containing $x'_{2}$, $W$ is the solution of
\begin{equation}
\begin{gathered}
(\frac{\partial^{2}}{\partial x_{2}}-q^{2})W^{13}(x_{2},x'_{2})=0,\\
W^{13}(x_{2},x'_{2})=0 \quad\mbox{for } x_{2}=a \mbox{ and } x_{2}=-a 
\end{gathered} \label{Po}
\end{equation}
(2)  $W$ is continuous on $[-a;a]$. Indeed, a discontinuity in $x_{2}=x'_{2}$
would lead to $\delta$ (undesirable in the second derivative).

(3) The jump: $w^{13}(x'_{2}+)-w^{13}(x'_{2}-)=1$ (Theorem of
derivation of the distributions represented by discontinuous functions).

The procedure then consists in finding $W+$, solution of \eqref{Po} on the
right of $x'_{2}$, and $W$, solution of \eqref{Po} on the left of
$x'_{2}$, and to connect them using conditions 2 and 3.

One finds that 
\begin{gather*}
W_{+}^{13}(x_{2},x"_{2})=\frac{C_{1}(x"_{2})}{q}sh[q(x_{2}-a)]
\quad\mbox{for } x_{2}>x"_{2} \\
W_{-}^{13}(x_{2},x"_{2})=\frac{C_{2}(x"_{2})}{q}sh[q(x_{2}+a)]
\quad\mbox{for } x_{2}<x"_{2}
\end{gather*}
with 
$$
C_{1}(x"_{2})=-\frac{sh[q(x"_{2}+a)]}{sh(2qa)}, \quad
C_{2}(x"_{2})=-\frac{sh[q(x"_{2}-a)]}{sh(2qa)}
$$
One can then transform this writing in the following condensed form:
$$
W^{13}(x_{2},x"_{2})=-\frac{e^{-q\mid x_{2}-x"_{2}\mid}}{2q}+\frac
{ch[q(x_{2}+x"_{2})]-e^{-2qa}ch[(q(x_{2}-x"_{2})]}{2qsh(2qa)}
$$
In the same way, the solution of the problem with condition of the Neumann
type is written:
$$
Z^{13}(x_{2},x"_{2}))=-\frac{e^{-q\mid x_{2}-x"_{2}\mid}}{2q}+\frac
{-ch[q(x_{2}+x"_{2})]-e^{-2qa}ch[(q(x_{2}-x"_{2})]}{2qsh(2qa)}
$$
The fractional terms of $W$ and $Z$ behave respectively in: - a continuous
function in$W$ in the vicinity of zero ($sh(q)$ is equivalent to q in the
vicinity of zero) - a function in [$1/q2$] in the vicinity of zero in the
expression of $Z$.

We remark that contrary to $W$ (the Dirichlet solution), the function $Z$
(Neumann solution) does not admit a transform of opposite Fourier in the
physical space.

In addition, the Green's function which we obtained in the physical space by
the method of images can be easily obtained from the functions $Z$ and $W.$ It
is enough, for that, to use the developments into exponential functions sh and
ch, and to apply then the inverse Fourier transforms.

\subsection*{Conclusion}
When constructing the solution relative to the plane channel, the recourse to
infinite charge distributions, caused by the principle of compensation, is due
to the presence of two parallel plans located on both sides of the charge $q$.
In other words it is the confinement of the charge (of the flow) in its domain
which gives solutions in the form of infinite series of functions. As the
contribution of each charge to the potential is presented in the form of a
general term of a series in $1/n$, the convergence of this series of function
cannot be carried out only if one utilizes charges of contrary signs.
Consequently, the choice of boundary condition of the Neumann type must be ignored.

Let us notice that the method described above can be extended to more complex
geometries even if it means to check the validity of the solution that it provides.

Table 1 gathers some results obtained for different geometries concerning the
two classes of limit problems which interest us.%

\begin{table}[ht]
\caption{Green's functions nature for different geometries}
\begin{center}
\begin{tabular}
[c]{|l|l|l|}\hline
& Neumann & Dirichlet\\\hline
$\mathbb{R}^{3}$ space & Convergent & Convergent\\\hline
Half plane & Convergent & Convergent\\\hline
Channel flow & Divergent & Convergent\\\hline
Polyhedric section & Divergent & Convergent\\\hline
\end{tabular}
\end{center}
\end{table}

\section{New closing}

To summarize what proceeds we can say that the problem is to choose a model of
closing which satisfies two fundamental conditions:

- The series of Green's function should not be divergent.

- The resulting integral should not lead to introducing additional assumptions
on the pressure.

And we can say that neither the condition of Dirichlet nor that of Neumann
applied to the pressure are appropriate. The former necessitates additional
physical conditions, the latter lead to a divergence of the Green's function.
Our governing idea consists in finding models of closing for each gradient
components of pressure

With the help of an identical procedure to that which we provided for the
pressure (see section 2.), we write each component of the gradient of pressure
in an integral form and establish the relations:
\begin{equation}
\begin{aligned}
\frac{\partial P}{\partial x_{i}}(x)
&=\int_{\Omega}G(x,x")\Delta_{x}%
\frac{\partial P}{\partial x"_{i}}(x")dx"+\int_{\Gamma}\frac{\partial
}{\partial n}G(x,x")\frac{\partial}{\partial x"_{i}}P(x")dx"\\
&\quad 
-\int_{\Gamma}G(x,x")\frac{\partial}{\partial n}(\frac{\partial}{\partial
x"_{i}}P(x"))dx"
\end{aligned}\label{e4.1}
\end{equation}
starting from
\begin{equation}
\frac{\partial P}{\partial x_{i}}(x)=\int_{\Omega}\Delta G(x,x")\frac
{\partial}{\partial x"_{i}}P(x")dx" \label{e4.2}
\end{equation}

Now, the choice of the limit conditions is then determined in a natural
 way on the basis of the constraints referred to above.
The conditions of Dirichlet type \eqref{e2.5} imposed on the limits of the domain on
the Green's function eliminate the second integral at G of the equation \eqref{e4.1}.
And it remains:
\begin{equation}
\frac{\partial P}{\partial x_{i}}(x)=\int_{\Omega}G(x,x")\Delta_{x}
\frac{\partial P}{\partial x"_{i}}(x")dx"+\int_{\Gamma}\frac{\partial
}{\partial n}G(x,x")\frac{\partial}{\partial x"_{i}}P(x")dx"\label{e4.3}
\end{equation}
To calculate the remaining integral at $\Gamma$, we must know the gradients of
pressure on it. However these quantities can directly be written starting from
the Navier-Stokes equations as we explained previously in section 2.

In summary, we can say that this new formulation has a double advantage, On
the one hand, concerning the determination of the values at the edge, we find
the same physical considerations as in the judicious choice of the
conventional procedure and, on the other hand, we utilize a new Green's
function which converges.
Finally, formal calculation described by the new procedure of closing can be
exploited numerically.

\subsection*{Conclusion}

We have solved the
problem of closing in pressure of the Navier -Stokes equations and we have
provided a new mathematical formalism which extends the field of application
of the models of closing using Green's function. This formalism makes it
possible to circumvent the problems of divergence of the Green's functions and
offers an alternative to the conventional formalisms.

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\end{document}