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z5I>a}$7~KHmMPu*R6yHfPwO^43MxYdvZghe_5~rM?lGlRl)5nA*ajOwuCg>nPDh1{ z8kg_Pi-aKSxyeVX4c$pxXFsUjiV*L-YLeXFZq>{Fz6=r<(cOugBxi`)Jmn1-A*f)q z;WyG-t4-zPiAwke-NK(DFX8^5!a4)&&&UM&MDD%e%k&pGYdS5&|4FHL0|8t?Uapov T6i!|aZhk=&T3Q)3S(N_*x#;c8 diff --git a/Docs/velocity_correction.tex b/Docs/velocity_correction.tex index a19616e22..0343e7b70 100644 --- a/Docs/velocity_correction.tex +++ b/Docs/velocity_correction.tex @@ -1,6 +1,45 @@ -\subsection{Correction de vitesse pour la modélisation d'un écoulement incompressible dans une boîte} - -\maketitle +% \documentclass[a4paper,10pt]{article} +% \usepackage[utf8x]{inputenc} +% \usepackage{amsmath} +% % \usepackage{showkeys} % affiche les labels et references +% \usepackage[T1]{fontenc} +% \usepackage{multirow} +% \usepackage{amsthm,amssymb} +% \setlength{\parskip}{12pt} %taille du saut entre deux paragraphes +% \usepackage{graphicx} +% \graphicspath{{./images/}} +% \usepackage{pdfsync} % permet de cliquer sur le pdf pour avoir l endroit ou il y a le code latex +% \usepackage{cancel} +% \usepackage{xcolor} +% \usepackage[left=3cm,right=3cm,top=2.5cm,bottom=2.5cm]{geometry} +% \overfullrule 5pt %Pour d\'esigner les listes couples d'un *full carre noir la ou l on sort +% +% \usepackage[pdftex]{hyperref} +% \hypersetup{ +% pdftitle={Flow past 3D sphere}, +% pdfauthor={Mimeau Chloe}, +% pdfcreator={PARMES}, +% pdfkeywords={sphere, potential flow, velocity correction, flow rate}, +% pdfsubject={Flow past 3D sphere} +% % pagebackref=false, +% %colorlinks=false,citecolor=red, +% %linkcolor=blue, +% %plainpages=false, +% %breaklinks=true, +% % pdfpagelabels=true +% } +% +% +% \theoremstyle{plain} +% \newtheorem{Theoreme}{Th\'eor\`eme}[section] +% %opening +% \title{Correction de vitesse pour la modélisation d'un écoulement incompressible dans une boîte} +% \author{} +% \date{} +% +% \begin{document} +% +% \maketitle Lors de la résolution de l'équation de Poisson $\Delta \mathbf{u} = - \nabla \boldsymbol{\omega} $, le mode 0 des transformées de Fourier rapides de la vitesse est imposé à 0, ce qui entraîne une modification des valeurs moyennes des composantes de la vitesse et donc des composantes de la vitesse elles-mêmes lorsque la transformée de Fourier inverse est appliquée. Une correction de la vitesse est donc nécessaire après la résolution de l'équation de Poisson, à chaque itération. On introduit la notation suivante pour exprimer la vitesse corrigée : @@ -9,7 +48,7 @@ On introduit la notation suivante pour exprimer la vitesse corrigée : \bf{u}= \widetilde{u} + \overline{u}\label{ux} \end{align} -où $\widetilde{\bf{u}} $ désigne la vitesse en sortie de FFT et où $\overline{\bf{u}} $ dénote la moyenne en espace de la vitesse, que l'on cherche ici à déterminer. L'écoulement s'effectue dans la direction $x$ et la vitesse est initialisée en posant ${\bf{u}}(t=0)=(u_x, u_y, u_z)=(u_{\infty}, 0, 0)$. +où $\widetilde{\bf{u}} $ désigne la vitesse en sortie de FFT et où $\overline{\bf{u}} $ dénote la moyenne en espace de la vitesse, que l'on cherche ici à déterminer. L'écoulement s'effectue dans la direction $x$ et la vitesse est initialisée en posant ${\bf{u}}(t=0)=(u_{x \infty}, u_{y \infty}, u_{z \infty})$. $\overline{u_x}$ est un flux constant en espace, par conséquent $\dfrac{\partial \overline{u_x}}{\partial x} = \dfrac{\partial \overline{u_x}}{\partial y} = \dfrac{\partial \overline{u_x}}{\partial z} = 0$. Ainsi le champ de vorticité moyen en espace est donné en 2D par : @@ -20,7 +59,7 @@ et en 3D par : \begin{align} \overline{\omega_x} & = \dfrac{\partial \overline{u_z}}{\partial y} - \dfrac{\partial \overline{u_y}}{\partial z} \\ \overline{\omega_y} & = -\dfrac{\partial \overline{u_z}}{\partial x} \label{wy} \\ -\overline{\omega_z} & = -\dfrac{\partial \overline{u_y}}{\partial x} \label{wz} +\overline{\omega_z} & = \dfrac{\partial \overline{u_y}}{\partial x} \label{wz} \end{align} La correction de la vitesse se fait par réajustement du débit à l'entrée du domaine de calcul. Cette correction est développée ci-dessous composante par composante tout d'abord en 2D puis sera étendue au cas 3D. @@ -39,9 +78,9 @@ On note $B$ le coté correspondant au bord d'entrée du domaine de calcul. En pa \Longleftrightarrow \ \ \overline{u_x} = \dfrac{\text{débit souhaité}}{L_y} - \dfrac{\text{débit calculé}}{L_y} \end{align} -où $L_y$ dénote la longueur du coté $B$. Or le débit que l'on souhaite imposer à l'entrée du domaine est égal à $u_{\infty} L_y$ et donc : +où $L_y$ dénote la longueur du coté $B$. Or le débit que l'on souhaite imposer à l'entrée du domaine est égal à $u_{x \infty} \ L_y$ et donc : \begin{equation} -\boxed{u_x = \widetilde{u_x} + u_{\infty} - \dfrac{\text{débit calculé}}{L_y}} +\boxed{u_x = u_{x \infty} + \widetilde{u_x} - \dfrac{\int\limits_B \widetilde{u_x} \ dy}{L_y}} \end{equation} @@ -60,12 +99,12 @@ Or ici le débit souhaité est égal a 0. Donc : \begin{align} 0 = \underbrace{\int\limits_B \widetilde{u_y} \ dy}_{\text{débit calculé}} + \overline{\omega} x_0 L_y + L_y c \\ ~ -\Longleftrightarrow c = -\dfrac{\text{débit calculé}}{L_y} - \overline{\omega} x_0 +\Longleftrightarrow c = -\dfrac{\int\limits_B \widetilde{u_y} \ dy}{L_y} - \overline{\omega} x_0 \end{align} et donc : \begin{equation} -\boxed{u_y = \widetilde{u_y} + \overline{\omega} x - \overline{\omega} x_0 - \dfrac{\text{débit calculé}}{L_y}} +\boxed{u_y = u_{y \infty} + \widetilde{u_y} + \overline{\omega}(x-x_0) - \dfrac{\int\limits_B \widetilde{u_y} \ dy}{L_y}} \end{equation} où $x_0$ dénote la coordonnée dans la direction $x$ du bord d'entrée du domaine de calcul. @@ -77,42 +116,152 @@ où $x_0$ dénote la coordonnée dans la direction $x$ du bord d'entrée du doma On note $S$ la surface correspondant au bord d'entrée du domaine de calcul. En partant de la décomposition \ref{ux} on a alors : \begin{align} -\underbrace{\iint\limits_S u_x \ dydz}_{\text{débit souhaité}} = \underbrace{\iint\limits_S \widetilde{u_x} \ dydz}_{\text{débit calculé}} + \iint\limits_S \overline{u_x} \ dydz \\ -~ -\Longleftrightarrow \ \ \overline{u_x} = \dfrac{\text{débit souhaité}}{L_yL_z} - \dfrac{\text{débit calculé}}{L_yL_z} \label{debSouhaiteUx3D} +\underbrace{\iint\limits_S u_x \ dydz}_{\text{débit souhaité}} &= \underbrace{\iint\limits_S \widetilde{u_x} \ dydz}_{\text{débit calculé}} + \iint\limits_S \overline{u_x} \ dydz \\ +\Longleftrightarrow \ \ \overline{u_x} &= \dfrac{\text{débit souhaité}}{L_yL_z} - \dfrac{\text{débit calculé}}{L_yL_z} \label{debSouhaiteUx3D} \nonumber \\ +\Longleftrightarrow \ \ \overline{u_x} &= u_{x \infty} - \dfrac{\iint\limits_S \widetilde{u_x} \ dydz}{L_yL_z} \label{debSouhaiteUx3D} \nonumber \end{align} et donc : \begin{equation} -\boxed{u_x = \widetilde{u_x} + u_{\infty} - \dfrac{\text{débit calculé}}{L_yL_z}} +\boxed{u_x = u_{x \infty} + \widetilde{u_x} - \dfrac{\iint\limits_S \widetilde{u_x} \ dydz}{L_yL_z}} \end{equation} \underline{Correction de $u_y$}:\\ -D'après la relation \ref{wz} on a $\overline{u_y} = \overline{\omega_z}x +c$ avec $c$ une constante. D'après \ref{ux} on obtient donc que $u_y = \widetilde{u_y} + \overline{\omega_z} x +c$. Ainsi, +D'après la relation \ref{wz} on a $\overline{u_y} = \overline{\omega_z}x +c_1$ avec $c_1$ une constante. D'après \ref{ux} on obtient donc que $u_y = \widetilde{u_y} + \overline{\omega_z} x +c_1$. Ainsi, \begin{align} -\iint\limits_S u_y \ dydz = \iint\limits_S \widetilde{u_y} \ dydz + \iint\limits_S \overline{\omega_z}x \ dydz + \iint\limits_S c \ dydz \\ +\iint\limits_S u_y \ dydz = \iint\limits_S \widetilde{u_y} \ dydz + \iint\limits_S \overline{\omega_z}x \ dydz + \iint\limits_S c_1 \ dydz \\ ~ -\Longleftrightarrow \ \underbrace{\iint\limits_S u_y \ dydz}_{\text{débit souhaité}} = \underbrace{\iint\limits_S \widetilde{u_y} \ dydz}_{\text{débit calculé}} + \ \overline{\omega_z}\ x_0L_yLz + cL_yL_z +\Longleftrightarrow \ \underbrace{\iint\limits_S u_y \ dydz}_{\text{débit souhaité}} = \underbrace{\iint\limits_S \widetilde{u_y} \ dydz}_{\text{débit calculé}} + \ \overline{\omega_z}\ x_0L_yLz + c_1 L_yL_z \end{align} où $x_0$ désigne la coordonnée dans la direction $x$ de la surface d'entrée $S$ du domaine de calcul. -Donc $\ \ \ c= \dfrac{\text{débit souhaité}}{L_yL_z} - \dfrac{\text{débit calculé}}{L_yL_z} - \overline{\omega_z}\ x_0$ +Donc +\begin{align} +c_1 &= \dfrac{\text{débit souhaité}}{L_yL_z} - \dfrac{\text{débit calculé}}{L_yL_z} - \overline{\omega_z}\ x_0 \nonumber \\ +c_1 &= u_{y \infty} - \dfrac{\iint\limits_S \widetilde{u_y} \ dydz}{L_yL_z} - \overline{\omega_z}\ x_0 \nonumber +\end{align} -Or le débit souhaité est égal a 0, par conséquent : +Et ainsi : \begin{equation} -\boxed{u_y = \widetilde{u_y} + \overline{\omega_z}x - \dfrac{\text{débit calculé}}{L_yL_z} - \overline{\omega_z}\ x_0} +\boxed{u_y = u_{y \infty} + \widetilde{u_y} + \overline{\omega_z}(x-x_0) - \dfrac{\iint\limits_S \widetilde{u_y} \ dydz}{L_yL_z}}\label{correc_Y} \end{equation} \underline{Correction de $u_z$}:\\ -Sur le même principe on a d'après la relation \ref{wy} que $\overline{u_z} = -\overline{\omega_y}x +c$ avec $c$ une constante. D'après \ref{ux} on obtient donc que $u_z = \widetilde{u_z} - \overline{\omega_y} x +c$. Ainsi en procédant de même que pour $u_y$ on obtient : +Sur le même principe on a d'après la relation \ref{wy} que $\overline{u_z} = -\overline{\omega_y}x +c_2$ avec $c_2$ une constante. D'après \ref{ux} on obtient donc que $u_z = \widetilde{u_z} - \overline{\omega_y} x +c_2$. Ainsi en procédant de même que pour $u_y$ on obtient : +\begin{equation} +\boxed{u_z = u_{z \infty} + \widetilde{u_z} - \overline{\omega_y}(x-x_0) - \dfrac{\iint\limits_S \widetilde{u_z} \ dydz}{L_yL_z}}\label{correc_Z} +\end{equation} + +%%%%%%%% Remarque 1 pour la constance des terme sde correction en Y et en Z %%%%%%%%%% + +{\bf{Remarque 1:}} + +Nous allons justifier ici la constance des termes $c_1$ et $c_2$ utilisés dans les corrections de $u_y$ et $u_z$. + +Tout d'abord, revenons sur la constante $\overline{\omega_z}$. Cette constante, de part les conditions de bord choisies dans notre problème, est nulle. En effet : +\begin{align} +\overline{\omega_z} &= \dfrac{\partial \overline{u_z}}{\partial y} - \dfrac{\partial \overline{u_y}}{\partial z}\\ +~ +\Longleftrightarrow \ \iiint\limits_{\Omega} \overline{\omega_z} &= \iiint\limits_{\Omega} \dfrac{\partial \overline{u_z}}{\partial y} - \dfrac{\partial \overline{u_y}}{\partial z} =0 \label{wz_zero} +\end{align} +car, en effet, des conditions périodiques sont imposées sur les bords dans les directions $Y$ et $Z$. Par conséquent, $\overline{\omega_z}$ est nul. + +De façon générale, à partir des relations \ref{wy} et \ref{wz}, on peut exprimer les flux $\overline{u_x}$, $\overline{u_y}$ et $\overline{u_z}$ comme suit : +\begin{align} +\overline{u_x} &= c_0\\ +\overline{u_y} &= \overline{\omega_z} x + f(y,z)\\ +\overline{u_z} &= -\overline{\omega_y} x + g(y,z) +\end{align} +% +où $c_0$ est une constante et où $f$ et $g$ sont des fonctions dépendant de $y$ et de $z$. +Ainsi, on a : +\begin{align} +\dfrac{\partial \overline{u_x}}{\partial x}&= 0 \ ; \quad \dfrac{\partial \overline{u_x}}{\partial y} = 0 \ ; \quad \dfrac{\partial \overline{u_x}}{\partial z}= 0\\ +\dfrac{\partial \overline{u_y}}{\partial x}&= \overline{\omega_z} \ ; \quad \dfrac{\partial \overline{u_y}}{\partial y} =\dfrac{\partial f}{\partial y} \ ; \quad \dfrac{\partial \overline{u_y}}{\partial z}= \dfrac{\partial f}{\partial z}\\ +\dfrac{\partial \overline{u_z}}{\partial x}&= -\overline{\omega_y} \ ; \quad \dfrac{\partial \overline{u_z}}{\partial y} =\dfrac{\partial g}{\partial y} \ ; \quad \dfrac{\partial \overline{u_z}}{\partial z}= \dfrac{\partial g}{\partial z} +\end{align} + +Par conséquent, à partir de la condition d'incompressibilité, on a : +\begin{equation} +\text{div}\ {\bf{u}} = 0 \ \Longleftrightarrow \ \dfrac{\partial \overline{u_x}}{\partial x} + \dfrac{\partial \overline{u_y}}{\partial y} + \dfrac{\partial \overline{u_z}}{\partial z} = 0 \ \Longleftrightarrow \ \boxed{\dfrac{\partial f}{\partial y} + \dfrac{\partial g}{\partial z} = 0}\label{div_f_g} +\end{equation} + +D'autre part, on sait d'après \ref{wz_zero} que $\dfrac{\partial \overline{u_z}}{\partial y} - \dfrac{\partial \overline{u_y}}{\partial z}=0$, donc : +\begin{equation} +\boxed{\dfrac{\partial g}{\partial y} - \dfrac{\partial f}{\partial z} = 0}\label{rot_f_g} +\end{equation} + +A partir de ces deux équations, on obtient les conditions suivantes sur $f$ et $g$ : +\begin{center} + $\partial_y$(\ref{div_f_g}) - $\partial_z$(\ref{rot_f_g}) $\Longleftrightarrow$ $\Delta f =0$ \\ +$\partial_z$(\ref{div_f_g}) + $\partial_y$(\ref{rot_f_g}) $\Longleftrightarrow$ $\Delta g =0$ +\end{center} + + +Par conséquent, les solutions du système composé des équations \ref{div_f_g} et \ref{rot_f_g} sont les fonctions $f$ et $g$ s'exprimant comme les combinaisons linéaires suivantes de $y$ et $z$ : +\begin{equation} + \left\{ + \begin{align} +f(y,z) &= ay + bz + c_1\\ +g(y,z) &= by - az + c_2, \qquad \text{avec} \ a,b, c_1, c_2 \in \mathbb{R} + \end{align} + \right. +\end{equation} + +En intégrant l'équation de correction $\bf{u}= \widetilde{u} + \overline{u}$ sur le bord d'entrée du domaine de calcul, on obtient le système d'équations suivant : +\begin{equation} + \left\{ + \begin{align} +\iint\limits_S u_y \ dydz = \iint\limits_S \widetilde{u_y} \ dydz + \underbrace{\iint\limits_S \overline{\omega_z}x \ dydz + \iint\limits_S f(y,z) \ dydz}_{\iint\limits_S \overline{u_y} \ dydz}\\ +\iint\limits_S u_z \ dydz = \iint\limits_S \widetilde{u_z} \ dydz \underbrace{- \iint\limits_S \overline{\omega_y}x \ dydz + \iint\limits_S g(y,z) \ dydz}_{\iint\limits_S \overline{u_z} \ dydz} + \end{align} + \right. +\end{equation} + +\begin{equation} + \Longleftrightarrow\left\{ + \begin{align} +\iint\limits_S u_y \ dydz = \iint\limits_S \widetilde{u_y} \ dydz + \iint\limits_S \overline{\omega_z}x \ dydz + \iint\limits_S ay + bz + c_1\ dydz\\ +\iint\limits_S u_z \ dydz = \iint\limits_S \widetilde{u_z} \ dydz - \iint\limits_S \overline{\omega_y}x \ dydz + \iint\limits_S by - az + c_2\ dydz + \end{align} + \right. +\end{equation} + +Pour des raisons de lisibilité, on pose les notations suivantes : +\begin{eqnarray} +\alpha_y &=& \iint\limits_S u_y \ dydz \quad ; \quad \alpha_z = \iint\limits_S u_z \ dydz\\ +\beta_y &=& \iint\limits_S \widetilde{u_y} \ dydz \quad ; \quad \beta_z = \iint\limits_S \widetilde{u_z} \ dydz\\ +\gamma_y &=& \iint\limits_S \overline{\omega_z}x \ dydz \quad ; \quad \gamma_z = - \iint\limits_S \overline{\omega_y}x \ dydz +\end{eqnarray} +Ainsi, on a : +% \begin{equation} -\boxed{u_z = \widetilde{u_z} - \overline{\omega_y}x - \dfrac{\iint\limits_S \widetilde{u_z} \ dydz}{L_yL_z} + \overline{\omega_y}\ x_0} + \left\{ + \begin{align} +\alpha_y &=& \beta_y + \gamma_y + aL_z \left[ \dfrac{1}{2} y^2\right]^{Ly_{+}}_{Ly_{-}} + bL_y \left[ \dfrac{1}{2} z^2\right]^{Lz_{+}}_{Lz_{-}} + c_1 L_yL_z\\ +\alpha_z &=& \beta_z + \gamma_z + bL_z \left[ \dfrac{1}{2} y^2\right]^{Ly_{+}}_{Ly_{-}} - aL_y \left[ \dfrac{1}{2} z^2\right]^{Lz_{+}}_{Lz_{-}} + c_2 L_yL_z\\ + \end{align} + \right. \end{equation} -{\bf{Remarque :}} +\begin{equation} + \Longleftrightarrow \left\{ + \begin{align} +\alpha_y &=& \beta_y + \gamma_y + a\dfrac{L_z}{2}\left[ Ly_{+}^2 - Ly_{-}^2\right] + b\dfrac{L_y}{2}\left[ Lz_{+}^2 - Lz_{-}^2\right] + c_1 L_yL_z\\ +\alpha_z &=& \beta_z + \gamma_z + b\dfrac{L_z}{2}\left[ Ly_{+}^2 - Ly_{-}^2\right] - a\dfrac{L_y}{2}\left[ Lz_{+}^2 - Lz_{-}^2\right] + c_2 L_yL_z\\ + \end{align} + \right. +\end{equation} + +Or, lorsque le domaine est centré en 0 dans les directions $Y$ et $Z$ (comme c'est le cas dans notre étude), les termes $\left[ Ly_{+}^2 - Ly_{-}^2\right]$ et $\left[ Lz_{+}^2 - Lz_{-}^2\right]$ sont nuls, et dans ce cas, les formules de correction des composantes $u_y$ et $u_z$ de la vitesse sont bien celles données par les équations \ref{correc_Y} et \ref{correc_Z} . + + +%%%%%%%% Remarque 2 pour écoulements potentiels autour cylindre/sphère %%%%%%%%%%% + +{\bf{Remarque 2:}} Dans le cadre d'un écoulement autour d'un cylindre (2D) ou d'une sphère (3D), on peut également choisir l'écoulement potentiel comme condition initiale de l'écoulement. @@ -180,8 +329,5 @@ u_z &= -u_{\infty} \dfrac{3R^3 xz}{2 r^5} % \begin{equation} % \underbrace{\iint\limits_S u_x \ dydz}_{\text{débit souhaité}} = \iint\limits_S \ \dfrac{-u_{\infty} x}{r} \left ( 1 - \dfrac{R^3}{r^3} \right ) \ dydz = -u_{\infty} x_0 \iint\limits_S \ \dfrac{1}{r} \left ( 1 - \dfrac{R^3}{r^3} \right ) \ dydz % \end{equation} -%%% Local Variables: -%%% mode: latex -%%% TeX-master: "manual" -%%% End: +% \end{document} -- GitLab