. Avant, Quand il passe sous la vague, le traceur commencè a s'´ etirer, prenant une forme elliptique, la figure(5.6(e)) représente l'´ evolution temporelle du demi-grand axe et du demi-petit axe de cette ellipse ; on constante que le rapport entre le grand axe et le petit axe augmente dans unepremì ere phase puis commencè a diminuer, ` A la sortie de la vague, il a une forme elliptique, qu'il conservera dans la suite de l'´ ecoulement, La Figure (5.6(f)) représente l'´ evolution temporelle de l'angle ? entre le grand axe et l'axe Ox

M. Le-champ and M. Pour-calculer-le-champ, on prend un maillage cartésien régulier sur D avec 40 000 noeuds (200 fois 200) et on calcule la valeur de M-FTLE pour chaque noeud du maillage La Figure (5.7) présente deux exemples de champs M-FTLE obtenus pour deux temps d'intégration T = 0.5s et T = 1s. On observe que dans le champ de M- FTLE donne des valeurs maximales le long de la surface libre. On observe aussi que les valeurs de M-FTLE sont constantes sur le côté de la vague et augmentent quand les points s'approchent de la vague

L. Champs-de and M. Les-lignes-de-courant-ainsi-que-le, critère Q sont présentés sur la figure(5.12) pour di?érents instants au passage du front Avant l'arrivée du mascaret, on voit que les valeurs du critère Q sont presqué egalesàegales`egalesà zéro, sauf des valeursàvaleurs`valeursà proximité de la surface libre sur la figure A l'arrivée du mascaret, en haut de canal, une zone de Q négative indique qu'une zone de cisaillement a lieù a la rencontre entre l'´ ecoulement principal et le ressaut se propageant en sens inverse sur les figures, les lignes de courant matérialisent des tourbillons au fond de canal, p.1212

L. Figure, 13) représente quelques exemples de champ M-FTLE obtenus pour di?érents instants au passage du front et pour le temps d'intégration T = 2 s

L. Lignes-de-courant-ainsi-que-le-critère and Q. Sont-présentés-sur-la-figure, 23) pour di?érents instants au passage du front Avant l'arrivée du mascaret, on voit que les valeurs du critère Q sont presqué egalesàegales`egalesà zéro, sauf des valeursàvaleursà proximité de la surface libre sur la figure (5.12(a)). A l'arrivée du mascaret, en haut(zone de mélange) du canal et au fond du canal (dans la couche limite) les lignes de courant matérialisent des tourbillons sur les figures(f)) o` u les régions positives de Q mettent en exergue une zone o` u le taux de rotation est supérieur au taux de déformation et traduit la présence d'un tourbillon. La figure(5.24) présente quelques exemples de champs M-FTLE avec la méthode des moments d'ordre 2, pp.12-12

|. La-figure, 24(a)) présente le champ de M-FTLE avant l'arrivée du mascaret. ? On voit que les M-FTLE sont maximales près de la surface libre

@. Au, On remarque que les valeurs de M-FTLE sont faibles parce que l'´ ecoulement initial est uniforme. Si on prend un domaine de forme circulaire contenant le traceur de centre x 0 (150, 200) et de rayon r 0 = 1, on remarque lesélémentsleséléments suivants : ? La trajectoire s'incurve vers le haut

?. L-'´-evolution-du-moment and P. Xy, est divisé en deux phases Unepremì ere phase de diminution qui représente une rotation dans le sens horaire du traceur de particules. Une seconde phase d'augmentation avec un dépassement de valeur initiale, ce qui exprime une rotation dans le sens trigonométrique, mais l'angle de rotation est très petit (Figure 5, p.25

?. Sur and L. Figure, 25(e)) on observe que lesévolutionslesévolutions du demi-petit axe et du demi-grand axe de l'ellipse. On note que la déformation du domaine circulaire contenant le traceur est presque négligeable lors du transport (voir figure, p.25

@. Au, on note une augmentation de valeurs de M-FTLE parce que, les trajectoire sont plus courtes par rapport aux trajectoires qui se trouvent au centre de l'´ ecoulement : on constateçaconstate¸constateça sur la Figure (5.26) o` u on considère des positons initiales du domaine, pp.36-38

M. Adaptatif, exposants de Lyapunov conna??tconna??t des variations très violentes Un maillage régulier ne nous permet pas d'avoir des résultats précisprécisà moins d'appliquer la précision nécessairè a la capture des crêtescrêtesà tout le domaine, provoquant une explosion du temps de calcul. C'est l` a que l'adaptation de maillage entre en jeu. L'idée est de concentrer le nombre de mailles l` a o` u c'est réellement nécessaire, leprobì eme mérite notre attention et notre temps de calcul. il serait très intéressant de calculer le champ de M-FTLE sur un maillage adaptatif

/. Transport, Nous avons commencécommencéà aborder ce sujetàsujetà la fin du chapitre trois, mais sans le développer par la suite. Mais il serait très intéressant de calculer le champ de M-FTLE

´. Ecoulement-compressible, ´ etude desécoulementsdesécoulements compressibles mais cela nécessite de nouvellé equations de transport pour calculer le champ de M-FTLE

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