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ANSWER - Tide propagation in a sloping bottom converging banks estuary : Différence entre versions

De Wikhydro
(Expression of the simplified model)
(Analytical solution)
(13 révisions intermédiaires par un utilisateur sont masquées)
Ligne 1 : Ligne 1 :
 
== Context ==  
 
== Context ==  
Cette page fait partie de la démarche collaborative <big><span style="color:Green">'''[[ANSWER - disciplines|ANSWER]]''' </span></big>, dont l'objectif est de faire collaborer scientifiques et grand public autour du domaine de l'eau.<br />
 
  
Cette fiche fait suite à la fiche [[ANSWER - Propagation d'une onde dans un estuaire à pente du fond inclinée|"Propagation d'une onde dans un estuaire à pente du fond inclinée"]] mais traite de la propagation d'une onde dans un estuaire à gabarit rectangulaire dont la section se rétrécie linéairement vers l'amont et dont la pente du fond est également linéaire.
+
<big>'''[[ANSWER - Propagation de la marée dans un estuaire à fond pentu et aux berges convergentes linéaires|Version française]]'''</big>
  
 
This page is part of the collaborative approach<big><span style="color:Green">'''[[ANSWER - disciplines|ANSWER]]''' </span></big>, whose goal is to make scientists and general public collaborate in the field of water. <br />
 
This page is part of the collaborative approach<big><span style="color:Green">'''[[ANSWER - disciplines|ANSWER]]''' </span></big>, whose goal is to make scientists and general public collaborate in the field of water. <br />
Ligne 42 : Ligne 41 :
 
* <math>b(x)=b_0\dfrac{ x }{ x_0 }</math> canal width which varies linearly with the abscisse,  <math> b_0 </math> is the canal width at abscisse <math> x_0 </math> upstream of the estuary. <math> p_b=b_0/x_0 </math>
 
* <math>b(x)=b_0\dfrac{ x }{ x_0 }</math> canal width which varies linearly with the abscisse,  <math> b_0 </math> is the canal width at abscisse <math> x_0 </math> upstream of the estuary. <math> p_b=b_0/x_0 </math>
 
* <math>H(x)=H_0\dfrac{ x }{ x_0 }</math> canal depth which varies linearly with the abscisse. <math> p_f=H_0/x_0 </math>
 
* <math>H(x)=H_0\dfrac{ x }{ x_0 }</math> canal depth which varies linearly with the abscisse. <math> p_f=H_0/x_0 </math>
*  <math> f </math> linearized roughness coefficient.<br />
+
*  <math> f </math> linearized roughness coefficient
  
 
The equations governing the phenomenon are the linearized 1D Saint-Venant equations.
 
The equations governing the phenomenon are the linearized 1D Saint-Venant equations.
Ligne 53 : Ligne 52 :
 
\dfrac{ \partial Q }{ \partial t }+gS \dfrac{ \partial h }{ \partial x }+fQ=0  
 
\dfrac{ \partial Q }{ \partial t }+gS \dfrac{ \partial h }{ \partial x }+fQ=0  
 
\end{cases}
 
\end{cases}
</math><br />
+
</math>
  
 
Deriving the first equation with respect to time and the second with respect to space and eliminating the common term, we obtain:
 
Deriving the first equation with respect to time and the second with respect to space and eliminating the common term, we obtain:
Ligne 59 : Ligne 58 :
 
<math>
 
<math>
 
-b\dfrac{ \partial^2 h }{ \partial t^2 }+ g\dfrac { \partial }{ \partial x }\left[\Big S\dfrac{ \partial h }{ \partial x }\right]\Big +f\dfrac{ \partial Q }{ \partial x}=0  
 
-b\dfrac{ \partial^2 h }{ \partial t^2 }+ g\dfrac { \partial }{ \partial x }\left[\Big S\dfrac{ \partial h }{ \partial x }\right]\Big +f\dfrac{ \partial Q }{ \partial x}=0  
</math><br />
+
</math>
  
 
we are going to suppose that <math>h</math> is of the form <math>h=A(x)e^{-i\sigma t}</math>. Deriving, we obtain :
 
we are going to suppose that <math>h</math> is of the form <math>h=A(x)e^{-i\sigma t}</math>. Deriving, we obtain :
Ligne 69 : Ligne 68 :
 
<math>
 
<math>
 
x^2\dfrac{ \partial^2 h }{ \partial x ^2}+2x\dfrac{ \partial h }{ \partial x} + \left[\Big \sigma (\sigma+fi) \right] \dfrac{ 1}{gp_f}} xh=0
 
x^2\dfrac{ \partial^2 h }{ \partial x ^2}+2x\dfrac{ \partial h }{ \partial x} + \left[\Big \sigma (\sigma+fi) \right] \dfrac{ 1}{gp_f}} xh=0
</math><br />
+
</math>
Posons: <br />
+
 
 +
Posons:
 +
 
 
<math>
 
<math>
 
k^2=\dfrac{ \sigma^2}{ gH}  
 
k^2=\dfrac{ \sigma^2}{ gH}  
 
\text{ wave number  et }<br />
 
\text{ wave number  et }<br />
k_f^2=\sigma (\sigma+fi) \dfrac{ x_0}{ gH_0}\right]</math>
+
k_f^2=\sigma (\sigma+fi) \dfrac{ x_0}{ gH_0}\right]
 +
</math>
  
 
This term is a constant
 
This term is a constant
Ligne 86 : Ligne 88 :
 
* <math>U_m</math> the mean speed on a period
 
* <math>U_m</math> the mean speed on a period
  
'' 'Note' '': this equation shows us that the distribution of the wave height is independent of the linear convergence rate of banks <math> p_b </ math>, but depends on the height of the water average <math> H_0 </ math>.
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'' 'Note' '': this equation shows us that the distribution of the wave height is independent of the linear convergence rate of banks <math> p_b </math>, but depends on the height of the water average <math> H_0 </math>.
  
 
== Analytical solution ==
 
== Analytical solution ==
On reconnait une équation de type Bessel:<br />
+
We recognize a Bessel type equation:<br />
  
 
<math>
 
<math>
Ligne 95 : Ligne 97 :
 
</math><br />
 
</math><br />
  
La solution de cette équation est donnée par:<br />
+
The solution of this equation is given by:<br />
  
 
<math>
 
<math>
Ligne 101 : Ligne 103 :
 
</math><br /><br />
 
</math><br /><br />
  
Les fonctions <math> J_{P/r}</math> et <math> Y_{P/r}</math> sont les [http://fr.wikipedia.org/wiki/Fonction_de_Bessel fonctions de Bessel] respectivement de première et de seconde espèce.<br /><br />
+
The fonctions <math> J_{P/r}</math> and <math> Y_{P/r}</math> are [http://fr.wikipedia.org/wiki/Fonction_de_Bessel fonctions de Bessel] first and second kind respectively.<br /><br />
Dans notre cas, nous avons : <math> p=1/2, \beta=0, \alpha^2=k_f^2 , r=1/2</math> avec <math> P=\sqrt{p^2-\beta^2}=0 </math> <br />
+
In our case, we have : <math> p=1/2, \beta=0, \alpha^2=k_f^2 , r=1/2</math> with <math> P=\sqrt{p^2-\beta^2}=0 </math> <br />
d'où:<br /><br />
+
where:<br /><br />
 
<math>
 
<math>
 
h(x)=\dfrac{ 1 } { \sqrt x} \left[ c_{1}  J_{1} (2k_f\sqrt x)+c_{2}  Y_{1} (2k_f\sqrt x) \right]e^{-i\sigma t}
 
h(x)=\dfrac{ 1 } { \sqrt x} \left[ c_{1}  J_{1} (2k_f\sqrt x)+c_{2}  Y_{1} (2k_f\sqrt x) \right]e^{-i\sigma t}
 
</math><br />
 
</math><br />
  
L'équation de quantité de mouvement nous permet de calculer la vitesse <math> u(x,t):</math> <br /><br />
+
The equation of momentum allows us to calculate the velocity <math> u(x,t):</math> <br /><br />
 
<math>
 
<math>
 
b\dfrac{ \partial h } { \partial t}+\dfrac{ \partial }{ \partial x} (bHu)=0<br />
 
b\dfrac{ \partial h } { \partial t}+\dfrac{ \partial }{ \partial x} (bHu)=0<br />
 
</math>
 
</math>
  
soit:<br />
+
also:<br />
  
 
<math>
 
<math>
Ligne 119 : Ligne 121 :
 
</math><br />
 
</math><br />
  
Posons: <math>z=2k_f \sqrt x </math>, il vient : <math>dz=k_f \dfrac{1}{\sqrt x} dx</math> et <math>dx= \dfrac{z}{2k_f^2}dz </math>
+
Given: <math>z=2k_f \sqrt x </math>, il vient : <math>dz=k_f \dfrac{1}{\sqrt x} dx</math> et <math>dx= \dfrac{z}{2k_f^2}dz </math>
  
En substituant ces expressions dans l'équation précédente, nous obtenons:
+
By substituting these expressions in the previous equation, we obtain:
  
 
<math>bHu= \dfrac{ b_0 } { x_0}  \dfrac{ i\sigma} { 4k_f^3} e^{-i\sigma t} \int_{x} z^2 \left[  c_{1}  J_{1} (z)+c_{2}  Y_{1} (z) \right]\, \mathrm dz
 
<math>bHu= \dfrac{ b_0 } { x_0}  \dfrac{ i\sigma} { 4k_f^3} e^{-i\sigma t} \int_{x} z^2 \left[  c_{1}  J_{1} (z)+c_{2}  Y_{1} (z) \right]\, \mathrm dz
 
</math><br />
 
</math><br />
Pour calculer cette intégrale, nous utilisons la relation de récurrence suivante :  
+
To calculate this integral, we use the following recurrence relation:
 
<math>
 
<math>
\dfrac{d}{dx} (x^n J_{n}(x))=x^n J_{n-1}(x) </math>, de même pour <math>Y_{n}(x)</math><br />
+
\dfrac{d}{dx} (x^n J_{n}(x))=x^n J_{n-1}(x) </math>, the same for <math>Y_{n}(x)</math><br />
Nous obtenons :<br />
+
We obtain :<br />
 
<math>
 
<math>
 
u=i\sigma \dfrac{ 1} { k_f}  \dfrac{x_0} { H_0}  \dfrac{1} { x} \left[  c_{1}  J_{2} (2k_f \sqrt x )+c_{2}  Y_{2} (2k_f \sqrt x ) \right]  e^{-i\sigma t}  
 
u=i\sigma \dfrac{ 1} { k_f}  \dfrac{x_0} { H_0}  \dfrac{1} { x} \left[  c_{1}  J_{2} (2k_f \sqrt x )+c_{2}  Y_{2} (2k_f \sqrt x ) \right]  e^{-i\sigma t}  
 
</math><br />
 
</math><br />
  
'''Synthèse : expression de la solution analytique finale'''<br />
+
'''Summary: expression of the final analytical solution'''<br />
  
En résumé, l'expression analytique exprimant la variation du niveau d'eau et de la vitesse moyenne en toute section du domaine sont données par les équations suivantes:<br /><br />
+
In summary, the analytical expression expressing the variation of the water level and the mean velocity in any section of the domain are given by the following equations:<br /><br />
  
 
<math>
 
<math>
Ligne 146 : Ligne 148 :
 
</math><br />
 
</math><br />
 
<br />
 
<br />
A partir des expressions générales de <math>h(x,t)</math> et de <math>u(x,t)</math>, nous pouvons déterminer la solution en fixant 2 conditions aux limites du domaine. Nous allons considérer les deux cas suivants:
+
Starting from the general expressions of  <math>h(x,t)</math> et de <math>u(x,t)</math>, we can determine the solution by setting 2 boundary conditions. We will consider the following two cases:
* propagation d'une onde incidente à partir de l'aval (droite) vers l'amont du domaine avec condition de sortie libre à l'extrémité amont (gauche)
+
* propagation of an incoming wave from downstream (right) to upstream of the domain with open boundary condition at the upstream end (left)
* réflexion de la même onde incidente à partir de l'aval (droite) vers l'amont avec condition de réflexion totale à l'extrémité amont (gauche)
+
* reflection of the same incoming wave from downstream (right) to upstream with total reflection at the upstream end (left)
  
== Configuration n°1 : estuaire sans frottement - sortie libre ==
+
== Configuration n°1 : estuary without roughness  - radiation boundary condition  ==
  
Nous allons imposer 2 conditions limites pour déterminer les deux constantes d'intégration.<br />
+
We impose 2 boundary conditions to determine the two integration constants.<br />
* une condition limite aval d'entrée de l'onde à l'intérieur du domaine par la droite <math> h(x_1,t)=A(x_1,t) e^{-i\sigma t}=a(x_1,t)e^{i(kx_1 -\sigma t)}=a(x_1,t)e^{i\phi} \quad\forall t</math><br />
+
* an entering boundary condition at the entrance from the right side  <math> h(x_1,t)=A(x_1,t) e^{-i\sigma t}=a(x_1,t)e^{i(kx_1 -\sigma t)}=a(x_1,t)e^{i\phi} \quad\forall t</math><br />
* une condition limite amont de sortie de l'onde sous forme de condition de sortie libre de type Sommerfeld:<br />
+
* an exit boundary condition of Sommerfeld radiation type :<br />
 
<math> h(x,t)=a(x_0) e^{i\phi }</math><br />
 
<math> h(x,t)=a(x_0) e^{i\phi }</math><br />
  
En dérivant cette expression, nous obtenons:<br />
+
Deriving this expression, we get:<br />
  
 
<math> \dfrac{  dh }{ dx }=e^{i\phi}\dfrac{ da }{ dx }+iae^{i\phi}\dfrac{  d\phi }{ d x }</math><br />
 
<math> \dfrac{  dh }{ dx }=e^{i\phi}\dfrac{ da }{ dx }+iae^{i\phi}\dfrac{  d\phi }{ d x }</math><br />
  
Chaque terme exprime un processus:
+
Each term expresses a process:
* prise en compte du shoaling : <math> \dfrac { d a }{ d x }=- \dfrac { 1 }{ 4 } \dfrac { p_0 }{ H } a</math>. La pente étant nulle, le terme de shoaling s'annule.
+
* taking into account shoaling : <math> \dfrac { d a }{ d x }=- \dfrac { 1 }{ 4 } \dfrac { p_0 }{ H } a</math>. The slope being zero, the shoaling term is canceled.
* équation eikonale : <math> \dfrac{ d\phi }{ d x }=k </math>(<math> k </math> est le nombre d'onde)
+
* eikonale equation : <math> \dfrac{ d\phi }{ d x }=k </math>(<math> k </math> is the wave number)
  
Nous obtenons:<br />
+
We get:<br />
  
 
<math> \dfrac{ d h }{ d x }= ae^{i\phi}(-\dfrac{1}{4}\dfrac{p_0}{H}+ik) </math><br />
 
<math> \dfrac{ d h }{ d x }= ae^{i\phi}(-\dfrac{1}{4}\dfrac{p_0}{H}+ik) </math><br />
  
ou encore:<br />
+
or:<br />
  
 
<math> \dfrac{ d h }{ d x }= A(x,t)e^{-i\sigma t}(-\dfrac{1}{4}\dfrac{p_0}{H}+ik) </math><br />
 
<math> \dfrac{ d h }{ d x }= A(x,t)e^{-i\sigma t}(-\dfrac{1}{4}\dfrac{p_0}{H}+ik) </math><br />
  
soit:<br />
+
sand:<br />
  
 
<math> (1) \qquad \dfrac{ d A }{ d x }= A(x,t)(-\dfrac{1}{4}\dfrac{p_0}{H}+ik) </math>
 
<math> (1) \qquad \dfrac{ d A }{ d x }= A(x,t)(-\dfrac{1}{4}\dfrac{p_0}{H}+ik) </math>
  
Or
+
and
 
<math>
 
<math>
 
A=\dfrac{1}{\sqrt x}\Big( c_{1} J_{1} (2k_f \sqrt x )+c_{2} Y_{1} (2k_f \sqrt x ) \Big)
 
A=\dfrac{1}{\sqrt x}\Big( c_{1} J_{1} (2k_f \sqrt x )+c_{2} Y_{1} (2k_f \sqrt x ) \Big)
 
</math><br />
 
</math><br />
  
En dérivant cette expression, nous obtenons:<br />
+
Deriving this expression, we get.:<br />
  
 
<math>(2)  \qquad \dfrac{dA}{dx}=-\dfrac{1}{2}\dfrac{1}{x\sqrt x }\Big[c_{1} J_{1} (2k_f \sqrt x )+c_{2} Y_{1} (2k_f \sqrt x ) \Big] - \dfrac{k_f}{x} \Big[ c_{1} J_{2} (2k_f \sqrt x )+c_{2} Y_{2} (2k_f \sqrt x ) \Big]</math><br />
 
<math>(2)  \qquad \dfrac{dA}{dx}=-\dfrac{1}{2}\dfrac{1}{x\sqrt x }\Big[c_{1} J_{1} (2k_f \sqrt x )+c_{2} Y_{1} (2k_f \sqrt x ) \Big] - \dfrac{k_f}{x} \Big[ c_{1} J_{2} (2k_f \sqrt x )+c_{2} Y_{2} (2k_f \sqrt x ) \Big]</math><br />
  
sachant que :<br />
+
knowing that :<br />
 
<math> \dfrac {d } {dz}  J_{1}(z) =- J_{2} (z)\quad \text{et}\quad \dfrac {d } {dz}  Y_{1}(z) =- Y_{2} (z)</math><br />
 
<math> \dfrac {d } {dz}  J_{1}(z) =- J_{2} (z)\quad \text{et}\quad \dfrac {d } {dz}  Y_{1}(z) =- Y_{2} (z)</math><br />
  
En applicant la relation (2) en <math> x=x_0  </math> et en posant :<br />
+
Applying relation (2) at <math> x=x_0  </math> and stating :<br />
 
<math>\gamma =-\dfrac{1}{2 }-ikx_0 \qquad\text {et} \qquad \delta=-k_f \sqrt x_0 </math> <br />
 
<math>\gamma =-\dfrac{1}{2 }-ikx_0 \qquad\text {et} \qquad \delta=-k_f \sqrt x_0 </math> <br />
Nous obtenons le système suivant à résoudre pour calculer les coefficients :<math> c_{1} \text {et} c_{2} </math>
+
We get the following system to solve to calculate the coefficients :<math> c_{1} \text {et} c_{2} </math>
  
 
<math>
 
<math>
Ligne 202 : Ligne 204 :
 
</math><br />
 
</math><br />
  
Ceci conduit à la solution suivante :<br />
+
This leads to the following solution :<br />
  
 
<math>
 
<math>
Ligne 221 : Ligne 223 :
 
</math><br />
 
</math><br />
  
'''Cas d'application : translation d'une onde sinusoïdale'''
+
'' 'Application case: translation of a sinusoidal wave' ''
  
Les caractéristiques de cet exemple sont les suivantes:
+
The characteristics of this example are as follows:
* longueur du canal : 10 000 km
+
* length of the canal: 10,000 km
* période de la marée : 900s
+
* tide period: 900s
* amplitude : 1 m à l'amont
+
* amplitude: 1 m upstream
* pente du fond : 0,0001 m/m
+
* bottom slope: 0.0001 m / m
* profondeur à l'entrée du canal à l'aval : 100 m et à sa sortie amont : 20 m <br />
+
* depth at the entrance to the downstream channel: 100 m and at its outlet upstream: 20 m <br />
L'animation suivante représente la propagation d'une onde de marée de 12 heures dans un estuaire dont le fond remonte linéairement et les berges convergent linéairement.
+
The following animation represents the propagation of a 12-hour tidal wave in an estuary whose bottom rises linearly and the banks converge linearly.
  
 
{|
 
{|
Ligne 238 : Ligne 240 :
 
|}
 
|}
  
Cet exemple montre que l'onde générée à l'aval (droite sur le schéma) avec une amplitude de 1m se propage vers l'amont (gauche) en se déformant : diminution de la longueur d'onde, augmentation de l'amplitude.<br />
+
This example shows that the wave generated downstream (right on the diagram) with an amplitude of 1m propagates upstream (left) while deforming: decrease of the wavelength, increase of the amplitude. <br />
A l'extrémité gauche, cette onde sort du domaine sans générer d'onde de réflexion, ce qui valide la condition de sortie libre utilisée ici.<br />
+
At the left end, this wave leaves the domain without generating a reflection wave, which validates the free output condition used here. <br />
Le terme de shoaling introduit est d'un ordre de grandeur inférieur au terme eikonale et ne produit aucune modification sensible à la sortie de l'onde.
+
The term shoaling introduced is of an order of magnitude less than the term eikonal and produces no significant change in the exit of the wave.
Par ailleurs, la courbe enveloppe montre clairement une amplification du signal vers l'amont.
+
Moreover, the envelope curve clearly shows an amplification of the signal in the upstream direction.
'''
+
Comparaison avec le code de calcul TELEMAC2D'''
+
== Configuration n°2 : estuaire sans frottement - réflexion d'une onde à l'intérieur du domaine / berges et fond linéaires ==
+
  
Introduisons les conditions limites suivantes, qui correspondent à :<br />
+
== Configuration n°2 : estuary without roughness - reflection of a wave inside the domain : linearly converging banks and sloping bottom  ==
* une condition limite aval d'entrée de l'onde à l'intérieur du domaine par la droite <math> h(x_1,t)=A_1(x_1,t) e^{-i\sigma t}\quad\forall t</math><br />
+
 
* une condition limité amont de réflexion totale de l'onde <math> u(x)=0 \quad\forall t</math><br />
+
Let's introduce the following boundary conditions, which correspond to :<br />
Nous aboutissons au système suivant :
+
* a downstream boundary condition which generates an entering wave inside the domain from the right  <math> h(x_1,t)=A_1(x_1,t) e^{-i\sigma t}\quad\forall t</math><br />
 +
* an upstream total reflective boundary condition <math> u(x)=0 \quad\forall t</math><br />
 +
We get the following system :
  
 
<math>
 
<math>
Ligne 259 : Ligne 260 :
 
\end{cases}
 
\end{cases}
 
</math><br />
 
</math><br />
Dont la résolution conduit à:<br />
+
Whose resolution leads to:<br />
 
<math>
 
<math>
 
\begin{cases}  
 
\begin{cases}  
Ligne 276 : Ligne 277 :
 
\end{cases}
 
\end{cases}
 
</math><br />
 
</math><br />
On vérifie bien que <math> h(x_1,t) = A_1 e^{-i\sigma t}</math><br />
+
We verify that <math> h(x_1,t) = A_1 e^{-i\sigma t}</math><br />
  
'''Cas d'application : réflexion d'une onde sinusoïdale'''
+
'' 'Application case: reflection of a sinusoidal wave' ''
  
Le même cas que ci-dessus est utilisé, mais en imposant une condition limite de réflexion à l'amont. Dans ce cas, une onde réfléchie de même période, de même amplitude mais de sens de propagation opposée  vient se superposer à l'onde incidente. Ceci conduit donc au niveau de la frontière "mer" (de droite), à imposer 2 fois l'amplitude de l'onde incidente à la frontière "mer".
+
The same case as above is used, but imposing a total reflection boundary condition. In this case, a reflected wave of the same period, the same amplitude but opposite direction of propagation is superimposed on the incident wave. This is equivalent as to impose twice the amplitude of the incident wave at the "sea" boundary.
  
 
{|
 
{|
Ligne 289 : Ligne 290 :
 
|}
 
|}
  
 +
This animation shows the formation of regular crest and troughs which represents a total reflection.
  
 +
The wavelength decreases progressively upstream as a function of the height of water. There is also an increase in the amplitude upstream.
  
Cette animation montre la formation de ventres et de creux réguliers ce qui représente une réflexion totale.  
+
Thus, in the absence of any friction, the wave propagates without loss of energy under the effect of the contraction of the banks and the slope of the bottom (to be compared with the similar case without slope where no deformation is observe).
  
La longueur d'onde diminue de manière progressive vers l'amont en fonction de la hauteur d'eau. On note également une augmentation de l'amplitude des battements vers l'amont.
+
'''Comparaison with numerical model TELEMAC2D'''
 
+
Ainsi, en l'absence de tout frottement, l'onde se propage sans perte d'énergie sous l'effet de la contraction des berges et de la pente du fond (à comparer avec le cas similaire sans pente où l'on n'observe aucune déformation).
+
 
+
'''Comparaison avec le code de calcul TELEMAC2D'''
+
  
 
<html><iframe frameborder="0" width="480" height="270" src="//www.dailymotion.com/embed/video/x4a7xub" allowfullscreen></iframe><br /><a href="http://www.dailymotion.com/video/x4a7xub_propagation-d-une-onde-dans-un-estuaire-a-pente-lineaire-et-convergence-de-berges-lineaire_school" target="_blank">Propagation d&#039;une onde dans un estuaire &agrave; pente...</a> <i>par <a href="http://www.dailymotion.com/Wikhydro" target="_blank">Wikhydro</a></i></html>
 
<html><iframe frameborder="0" width="480" height="270" src="//www.dailymotion.com/embed/video/x4a7xub" allowfullscreen></iframe><br /><a href="http://www.dailymotion.com/video/x4a7xub_propagation-d-une-onde-dans-un-estuaire-a-pente-lineaire-et-convergence-de-berges-lineaire_school" target="_blank">Propagation d&#039;une onde dans un estuaire &agrave; pente...</a> <i>par <a href="http://www.dailymotion.com/Wikhydro" target="_blank">Wikhydro</a></i></html>
  
== Configuration n°3 : Estuaire HYPERSYNCHRONE avec frottement / K=80 ==
+
== Configuration n°3 : HYPERSYNCHRONE ESTUARY with roughness / K=80 ==
Nous avons repris le cas précédent de la propagation de l'onde avec une sortie libre, mais nous lui avons rajouté un peu de frottement sous la forme d'un coefficient de Strickler de 50.
+
We take the previous case of the wave propagation with a radiation condition, but we add a little roughness in the form of a Strickler coefficient of 50.
  
 
{|
 
{|
Ligne 312 : Ligne 311 :
  
  
L'animation montre un effet certain mais de faible amplitude sur l'évolution du niveau d'eau. Celui-ci est tout de même légèrement amorti par rapport à la solution sans frottement (configuration n°1).
+
The animation shows a effective but small effect on the evolution of the water level. This one is still slightly damped compared to the solution without friction (configuration n ° 1).
La courbe enveloppe montre l'effet d'amplification du signal vers l'amont (gauche)
+
The envelope curve shows the signal amplification effect upstream (left)
  
'''Comparaison avec TELEMAC 2D'''
+
== Configuration n°4 : HYPOSYNCHRONE ESTUARY with roughness  / K=50 ==
  
== Configuration n°4 : Estuaire HYPOSYNCHRONE avec frottement / K=50 ==
+
We have taken the previous case of the propagation of the wave with a radiation boundary condition exit, but we added a little more friction in the form of a Strickler coefficient of 50.
 
+
Nous avons repris le cas précédent de la propagation de l'onde avec une sortie libre, mais nous lui avons rajouté un peu plus de frottement sous la forme d'un coefficient de Strickler de 50.
+
  
  
Ligne 329 : Ligne 326 :
 
|}
 
|}
  
L'animation montre un effet certain, plus important que dans le cas précédent sur l'évolution du niveau d'eau. Celui-ci est beaucoup plus amorti par rapport à la solution sans frottement (configuration n°1).
+
The animation shows a effective effect, more important than in the previous case on the evolution of the water level. This one is much more damped compared to the solution without friction (configuration n ° 1).
 
+
La courbe enveloppe montre une nette atténuation du signal vers l'amont, ce qui rend l'estuaire hyposynchrone.
+
  
'''Comparaison avec le code de calcul TELEMAC2D'''
+
The envelope curve shows a clear attenuation of the signal upstream, which makes the estuary hyposynchronous.
  
== Conclusion générale sur la simulation de la propagation d'une onde dans un estuaire à pente du fond constante et à section constante ==
+
== General conclusion on the simulation of the propagation of a wave in an estuary with linearly sloping bottom and linearly decreasing section ==
  
Les travaux qui ont été à l'origine de ce papier ont permis de dériver les équations susceptibles de représenter l'entrée d'une onde dans un estuaire caractérisé par un fond en pente linéarisée ascendant vers l'amont et un gabarit dont les berges convergent linéairement également vers l'amont.
+
The works consist in the derivation of the equations that represent the entry of a wave in an estuary characterized by a linearly sloping bottom and whose banks converge linearly in the upstream direction.
  
Plusieurs enseignements peuvent être tirées de ces résultats:
+
Several lessons can be drawn from these results:
* les conditions limites de sortie libre de Sommerfeld donnent toutes satisfactions : l'on sort sans générer en interne du domaine des ondes de réflexion
+
* the boundary condition of radiation of Sommerfeld give all satisfactions: one quit the upstream boundary without generating reflection waves inside the domain
* la condition de réflexion totale imposée en fond d'estuaire représente donc bien le phénomène par la solution analytique. Par contre, l'utilisation du code numérique génère également une onde de réflexion à l'intérieur du domaine qui vient se superposer à la condition limite amont et nuit à la restitution correcte, allant même jusqu'à faire exploser le modèle. Un remarque est ici nécessaire sur ce que l'on représente exactement avec la solution limite. Si l'on veut représenter la superposition de l'onde incidente et de l'onde réfléchie, il faut doubler l'onde à la condition limite. Ainsi, on voit ici que l'enrichissement des 2 méthodes est réciproque.
+
* the condition of total reflection imposed at the upstream entrance of the estuary (river side) is well represented by the analytical solution. On the other hand, the use of the numerical code also generates a reflection wave inside the domain which is superimposed on the upstream boundary condition and harms to the correct restitution, even going as far as exploding the model. A remark is necessary here on what exactly is represented with the boundary condition. If one wants to represent the superposition of the incident wave and the reflected wave, it is necessary to double the wave amplitude at the boundary condition. Thus, we see here that the enrichment of the two methods is reciprocal.
* la prise en compte d'un frottement est également fort instructif. Tout d'abord, il faut rappeler que sans frottement, l'estuaire est HYPERSYNCHRONE, du fait de sa pente ascendante et de son gabarit rétréci. Si l'on rajoute un peut de frottement (K=80), l'onde diminue, ce qui est bien conforme à la théorie.Par contre, si le frottement augmente, le train d'onde est beaucoup plus écrasé en amplitude, amenant même l'estuaire à devenir HYPOSYNCHRONE.
+
* taking into account the friction term, even linearized, is also very instructive. First of all, it must be remembered that without friction, the estuary is HYPERSYNCHRONE, because of its upward slope and its converging banks. If we add a little friction (K = 80), the wave decreases, which is in line with the theory. On the other hand, if the friction increases, the wave train is much more damped in amplitude, causing even the estuary to become HYPOSYNCHRONE.
  
 
== Bibliographie ==
 
== Bibliographie ==
Ligne 354 : Ligne 349 :
  
 
== Code Scilab ==
 
== Code Scilab ==
Les animations précédentes ont été réalisées à l'aide le l'application SCILAB. <br />
+
The previous animations were made using the SCILAB application. <br />
Elles peuvent être utilisées pour reproduire le graphique. Il suffit de sélectionner l'ensemble du texte dans le fichier *.pdf et de le copier dans l'éditeur du logiciel puis d'exécuter le programme.
+
They can be used to reproduce the graph. Just select all the text in the * .pdf file and copy it to the software editor and run the program.
Le fichier est disponible ici :  [[File:Estuaire avec correction.pdf]] fournit le code SCILAB du programme qui fournit les animations précédentes.
+
The file is available here :  [[File:Estuaire avec correction.pdf]] provides the SCILAB code of the program of the previous animations.
  
  

Version du 22 mars 2018 à 11:47

Sommaire

Context

Version française 

This page is part of the collaborative approachANSWER , whose goal is to make scientists and general public collaborate in the field of water.

This sheet is a continuation of the "Propagation d'une onde dans un estuaire à pente du fond inclinée" but deals with the propagation of a wave in an estuary with a rectangular section whose section narrows linearly upstream and whose slope of the bottom is also linear.

Mathematical modeling of wave propagation phenomenon in an estuary

We will start from the one-dimensional Saint-Venant equations system, which will have been linearized to study the deformation of a wave during its propagation along a schematic estuary.
The linearization of this system will lead to an equation that will be solved directly, thus providing the temporal variation of the water level and the average celerity per section.
We will study 4 configurations:

  1. frictionless propagation with upstream boundary condition of open exit
  2. total reflection at the downstream boundary of the estuary and propagation without friction
  3. propagation with slight friction and upstream boundary condition of open exit
  4. propagation with strong friction and upstream boundary condition of open exit

We will take into account in this example a channel of rectangular section which narrows from downstream (the sea) to upstream (the river), a constant linear slope ascending towards the bottom and a convergence of the banks.

We will compare the results obtained by these analytical solutions with those obtained with TELEMAC2D two-dimensional finite element hydrodynamic computation code.

Hierarchy of simplifying hypotheses

The simplifications that follow allow us to develop a linear model of propagation of a wave within a simplified geometry domain, corresponding to a bottom slope canal, which can approach a schematic inside which spreads a surge wave.

'Navier-Stokes'

→ incompressible fluid
→ integration in a calculation section (rectangular channel) ==> Saint-Venant 1D
→ negligible acceleration
→ linearized friction

Expression of the simplified model

From the above assumptions, consider a rectangular cross-sectional channel with a constant bottom slope and a constant bank convergence rate.

Dessin pente nulle et berges linéaires.png

Soit :

  • $ h(x,t) $ water level
  • $ u(x,t) $ flow mean velocity of the flow in the section $ S(x)=b(x)H(x) $
  • $ Q(x,t)=S(x,t)u(x,t) $ discharge at section $ x $
  • $ b(x)=b_0\dfrac{ x }{ x_0 } $ canal width which varies linearly with the abscisse, $ b_0 $ is the canal width at abscisse $ x_0 $ upstream of the estuary. $ p_b=b_0/x_0 $
  • $ H(x)=H_0\dfrac{ x }{ x_0 } $ canal depth which varies linearly with the abscisse. $ p_f=H_0/x_0 $
  • $ f $ linearized roughness coefficient

The equations governing the phenomenon are the linearized 1D Saint-Venant equations.

$ \begin{cases} b\dfrac{ \partial h }{ \partial t }+ \dfrac{ \partial Q }{ \partial x }=0 } \\ \\ \dfrac{ \partial Q }{ \partial t }+gS \dfrac{ \partial h }{ \partial x }+fQ=0 \end{cases} $

Deriving the first equation with respect to time and the second with respect to space and eliminating the common term, we obtain:

$ -b\dfrac{ \partial^2 h }{ \partial t^2 }+ g\dfrac { \partial }{ \partial x }\left[\Big S\dfrac{ \partial h }{ \partial x }\right]\Big +f\dfrac{ \partial Q }{ \partial x}=0 $

we are going to suppose that $ h $ is of the form $ h=A(x)e^{-i\sigma t} $. Deriving, we obtain :

$ \dfrac{ \partial h }{ \partial t}=(-i\sigma) A(x) e^{-i\sigma t} = -i\sigma h \qquad et \qquad \dfrac{ \partial^2 h }{ \partial t^2}=-\sigma^2 h $

By reporting these values in the equation above :

$ x^2\dfrac{ \partial^2 h }{ \partial x ^2}+2x\dfrac{ \partial h }{ \partial x} + \left[\Big \sigma (\sigma+fi) \right] \dfrac{ 1}{gp_f}} xh=0 $

Posons:

$ k^2=\dfrac{ \sigma^2}{ gH} \text{ wave number et }<br /> k_f^2=\sigma (\sigma+fi) \dfrac{ x_0}{ gH_0}\right] $

This term is a constant

Moreover, the friction can be expressed as follows :

$ f=\dfrac{8}{3\pi}\dfrac{g}{K^{2}{H_m^{4/3}}}\left| U_m \right| $ avec :

  • $ K $ is the Strickler coefficient close to 0.002 - 0.003
  • $ H_m $ the mean depth
  • $ U_m $ the mean speed on a period

'Note' : this equation shows us that the distribution of the wave height is independent of the linear convergence rate of banks $ p_b $, but depends on the height of the water average $ H_0 $.

Analytical solution

We recognize a Bessel type equation:

$ x^2y''+(2p+1) x y'+(\alpha^2x^{2r}+\beta^2)y=0 $

The solution of this equation is given by:

$ y=x^{-p} \left[ c_{1} J_{P/r} (\alpha x^{r}/r) + c_{2} Y_{P/r} (\alpha x^{r}/r) \right] $

The fonctions $ J_{P/r} $ and $ Y_{P/r} $ are fonctions de Bessel first and second kind respectively.

In our case, we have : $ p=1/2, \beta=0, \alpha^2=k_f^2 , r=1/2 $ with $ P=\sqrt{p^2-\beta^2}=0 $
where:

$ h(x)=\dfrac{ 1 } { \sqrt x} \left[ c_{1} J_{1} (2k_f\sqrt x)+c_{2} Y_{1} (2k_f\sqrt x) \right]e^{-i\sigma t} $

The equation of momentum allows us to calculate the velocity $ u(x,t): $

$ b\dfrac{ \partial h } { \partial t}+\dfrac{ \partial }{ \partial x} (bHu)=0<br /> $

also:

$ bHu=i\sigma \dfrac{ b_0 } { x_0}e^{-i\sigma t} \int_{x} \sqrt x \left[ c_{1} J_{1} (2k_f \sqrt x)+c_{2} Y_{1} (2k_f \sqrt x) \right]\, \mathrm dx $

Given: $ z=2k_f \sqrt x $, il vient : $ dz=k_f \dfrac{1}{\sqrt x} dx $ et $ dx= \dfrac{z}{2k_f^2}dz $

By substituting these expressions in the previous equation, we obtain:

$ bHu= \dfrac{ b_0 } { x_0} \dfrac{ i\sigma} { 4k_f^3} e^{-i\sigma t} \int_{x} z^2 \left[ c_{1} J_{1} (z)+c_{2} Y_{1} (z) \right]\, \mathrm dz $
To calculate this integral, we use the following recurrence relation: $ \dfrac{d}{dx} (x^n J_{n}(x))=x^n J_{n-1}(x) $, the same for $ Y_{n}(x) $
We obtain :
$ u=i\sigma \dfrac{ 1} { k_f} \dfrac{x_0} { H_0} \dfrac{1} { x} \left[ c_{1} J_{2} (2k_f \sqrt x )+c_{2} Y_{2} (2k_f \sqrt x ) \right] e^{-i\sigma t} $

Summary: expression of the final analytical solution

In summary, the analytical expression expressing the variation of the water level and the mean velocity in any section of the domain are given by the following equations:

$ h(x,t)=\Re \Bigl(\left \dfrac{ 1} { \sqrt x} [ c_{1} J_{1} (2k_f \sqrt x)+c_{2} Y_{1} (2k_f \sqrt x) \right] e^{-i\sigma t}\Bigr) $

$ u(x,t)=\Re \Bigl( i\sigma \dfrac{ 1} { k_f} \dfrac{x_0} { H_0} \dfrac{1} { x} \left[ c_{1} J_{2} (2k_f \sqrt x) + c_{2} Y_{2} (2k_f \sqrt x) \right] e^{-i\sigma t}}\Bigr) $

Starting from the general expressions of $ h(x,t) $ et de $ u(x,t) $, we can determine the solution by setting 2 boundary conditions. We will consider the following two cases:

  • propagation of an incoming wave from downstream (right) to upstream of the domain with open boundary condition at the upstream end (left)
  • reflection of the same incoming wave from downstream (right) to upstream with total reflection at the upstream end (left)

Configuration n°1 : estuary without roughness - radiation boundary condition

We impose 2 boundary conditions to determine the two integration constants.

  • an entering boundary condition at the entrance from the right side $ h(x_1,t)=A(x_1,t) e^{-i\sigma t}=a(x_1,t)e^{i(kx_1 -\sigma t)}=a(x_1,t)e^{i\phi} \quad\forall t $
  • an exit boundary condition of Sommerfeld radiation type :

$ h(x,t)=a(x_0) e^{i\phi } $

Deriving this expression, we get:

$ \dfrac{ dh }{ dx }=e^{i\phi}\dfrac{ da }{ dx }+iae^{i\phi}\dfrac{ d\phi }{ d x } $

Each term expresses a process:

  • taking into account shoaling : $ \dfrac { d a }{ d x }=- \dfrac { 1 }{ 4 } \dfrac { p_0 }{ H } a $. The slope being zero, the shoaling term is canceled.
  • eikonale equation : $ \dfrac{ d\phi }{ d x }=k $($ k $ is the wave number)

We get:

$ \dfrac{ d h }{ d x }= ae^{i\phi}(-\dfrac{1}{4}\dfrac{p_0}{H}+ik) $

or:

$ \dfrac{ d h }{ d x }= A(x,t)e^{-i\sigma t}(-\dfrac{1}{4}\dfrac{p_0}{H}+ik) $

sand:

$ (1) \qquad \dfrac{ d A }{ d x }= A(x,t)(-\dfrac{1}{4}\dfrac{p_0}{H}+ik) $

and $ A=\dfrac{1}{\sqrt x}\Big( c_{1} J_{1} (2k_f \sqrt x )+c_{2} Y_{1} (2k_f \sqrt x ) \Big) $

Deriving this expression, we get.:

$ (2) \qquad \dfrac{dA}{dx}=-\dfrac{1}{2}\dfrac{1}{x\sqrt x }\Big[c_{1} J_{1} (2k_f \sqrt x )+c_{2} Y_{1} (2k_f \sqrt x ) \Big] - \dfrac{k_f}{x} \Big[ c_{1} J_{2} (2k_f \sqrt x )+c_{2} Y_{2} (2k_f \sqrt x ) \Big] $

knowing that :
$ \dfrac {d } {dz} J_{1}(z) =- J_{2} (z)\quad \text{et}\quad \dfrac {d } {dz} Y_{1}(z) =- Y_{2} (z) $

Applying relation (2) at $ x=x_0 $ and stating :
$ \gamma =-\dfrac{1}{2 }-ikx_0 \qquad\text {et} \qquad \delta=-k_f \sqrt x_0 $
We get the following system to solve to calculate the coefficients :$ c_{1} \text {et} c_{2} $

$ \begin{cases} c_{1} ( \gamma J_{1}^0 + \delta J_{2}^0 )+ c_{2}( \gamma Y_{1}^0 + \delta Y_{2}^0 )=0 \\ \\ c_{1}J_{1}^1 +c_{2}Y_{1}^1 =A_1 \sqrt x_1 \end{cases} $

This leads to the following solution :

$ \begin{cases} D=( \gamma J_{1}^0 + \delta J_{2}^0 ) Y_{1}^1 - ( \gamma Y_{1}^0 + \delta Y_{2}^0 )J_{1}^1 \\ D_{c_1}=-A_1\sqrt {x_1}( \gamma Y_{1}^0 + \delta Y_{2}^0 ) \\ D_{c_2}=A_1\sqrt {x_1} ( \gamma J_{1}^0 + \delta J_{2}^0 ) \\ c_1=D_{c_1}/D \qquad c_2=D_{c_2}/D \\ h(x,t)=\Re \Bigl(\left \dfrac{ 1} { \sqrt x} [ c_{1} J_{1} (2k_f \sqrt x)+c_{2} Y_{1} (2k_f \sqrt x) ] e^{-i\sigma t}\Bigr) \\ u(x,t)=\Re \Bigl( i\sigma \dfrac{ 1} { k_f} \dfrac{x_0} { H_0} \dfrac{1} { x} \left[ c_{1} J_{2} (2k_f \sqrt x) + c_{2} Y_{2} (2k_f \sqrt x) \right] e^{-i\sigma t}}\Bigr) \end{cases} $

'Application case: translation of a sinusoidal wave'

The characteristics of this example are as follows:

  • length of the canal: 10,000 km
  • tide period: 900s
  • amplitude: 1 m upstream
  • bottom slope: 0.0001 m / m
  • depth at the entrance to the downstream channel: 100 m and at its outlet upstream: 20 m

The following animation represents the propagation of a 12-hour tidal wave in an estuary whose bottom rises linearly and the banks converge linearly.

Sortie libre ss frottement.gif
Enveloppe Sortie libre pb=0.001 - K=0 - pf=0.00112.gif

This example shows that the wave generated downstream (right on the diagram) with an amplitude of 1m propagates upstream (left) while deforming: decrease of the wavelength, increase of the amplitude.
At the left end, this wave leaves the domain without generating a reflection wave, which validates the free output condition used here.
The term shoaling introduced is of an order of magnitude less than the term eikonal and produces no significant change in the exit of the wave. Moreover, the envelope curve clearly shows an amplification of the signal in the upstream direction.

Configuration n°2 : estuary without roughness - reflection of a wave inside the domain : linearly converging banks and sloping bottom

Let's introduce the following boundary conditions, which correspond to :

  • a downstream boundary condition which generates an entering wave inside the domain from the right $ h(x_1,t)=A_1(x_1,t) e^{-i\sigma t}\quad\forall t $
  • an upstream total reflective boundary condition $ u(x)=0 \quad\forall t $

We get the following system :

$ \begin{cases} c_1 J_{2}^0+c_2 Y_{2}^0 =0 \\ \\ c_1 J_{1}^1+c_2 Y_{1}^1 =A_1 \sqrt {x_1} \end{cases} $
Whose resolution leads to:
$ \begin{cases} D= J_{2}^0 Y_{1}^1 - J_{1}^1 Y_{2}^0 \\ D_{c_1}=-A_1\sqrt x_1 Y_{2}^0 \\ D_{c_2}=A_1\sqrt x_1 J_{2}^0 \\ c_1=D_{c_1}/D \qquad c_2=D_{c_2}/D \\ h(x,t)=\Re \Bigl(\left \dfrac{1} { \sqrt x} [ c_{1} J_{1} (2k_f \sqrt x)+c_{2} Y_{1} (2k_f \sqrt x) \right] e^{-i\sigma t}\Bigr) \\ u(x,t)=\Re \Bigl( i\sigma \dfrac{ 1} { k_f} \dfrac{x_0} { H_0} \dfrac{1} { x} \left[ c_{1} J_{2} (2k_f \sqrt x) + c_{2} Y_{2} (2k_f \sqrt x) \right] e^{-i\sigma t}}\Bigr) \end{cases} $
We verify that $ h(x_1,t) = A_1 e^{-i\sigma t} $

'Application case: reflection of a sinusoidal wave'

The same case as above is used, but imposing a total reflection boundary condition. In this case, a reflected wave of the same period, the same amplitude but opposite direction of propagation is superimposed on the incident wave. This is equivalent as to impose twice the amplitude of the incident wave at the "sea" boundary.

[[Documents.gif]]
Enveloppe Réflexion totale pb=0.001 - K=0 - pf=0.001 V224.gif

This animation shows the formation of regular crest and troughs which represents a total reflection.

The wavelength decreases progressively upstream as a function of the height of water. There is also an increase in the amplitude upstream.

Thus, in the absence of any friction, the wave propagates without loss of energy under the effect of the contraction of the banks and the slope of the bottom (to be compared with the similar case without slope where no deformation is observe).

Comparaison with numerical model TELEMAC2D


Propagation d'une onde dans un estuaire à pente... par Wikhydro

Configuration n°3 : HYPERSYNCHRONE ESTUARY with roughness / K=80

We take the previous case of the wave propagation with a radiation condition, but we add a little roughness in the form of a Strickler coefficient of 50.

K80.gif
Eveloppe Sortie libre pb=0.001 - K=80 - pf=0.00112.gif


The animation shows a effective but small effect on the evolution of the water level. This one is still slightly damped compared to the solution without friction (configuration n ° 1). The envelope curve shows the signal amplification effect upstream (left)

Configuration n°4 : HYPOSYNCHRONE ESTUARY with roughness / K=50

We have taken the previous case of the propagation of the wave with a radiation boundary condition exit, but we added a little more friction in the form of a Strickler coefficient of 50.


K50.gif
Eveloppe Sortie libre pb=0.001 - K=50 - pf=0.00112.gif

The animation shows a effective effect, more important than in the previous case on the evolution of the water level. This one is much more damped compared to the solution without friction (configuration n ° 1).

The envelope curve shows a clear attenuation of the signal upstream, which makes the estuary hyposynchronous.

General conclusion on the simulation of the propagation of a wave in an estuary with linearly sloping bottom and linearly decreasing section

The works consist in the derivation of the equations that represent the entry of a wave in an estuary characterized by a linearly sloping bottom and whose banks converge linearly in the upstream direction.

Several lessons can be drawn from these results:

  • the boundary condition of radiation of Sommerfeld give all satisfactions: one quit the upstream boundary without generating reflection waves inside the domain
  • the condition of total reflection imposed at the upstream entrance of the estuary (river side) is well represented by the analytical solution. On the other hand, the use of the numerical code also generates a reflection wave inside the domain which is superimposed on the upstream boundary condition and harms to the correct restitution, even going as far as exploding the model. A remark is necessary here on what exactly is represented with the boundary condition. If one wants to represent the superposition of the incident wave and the reflected wave, it is necessary to double the wave amplitude at the boundary condition. Thus, we see here that the enrichment of the two methods is reciprocal.
  • taking into account the friction term, even linearized, is also very instructive. First of all, it must be remembered that without friction, the estuary is HYPERSYNCHRONE, because of its upward slope and its converging banks. If we add a little friction (K = 80), the wave decreases, which is in line with the theory. On the other hand, if the friction increases, the wave train is much more damped in amplitude, causing even the estuary to become HYPOSYNCHRONE.

Bibliographie

  • Le Méhauté B., "An Introduction to Hydrodynamics & Waterways" Springer-Verlag, 1976, 315 p.
  • Thual O., "Hydrodynamique de l'environnement", Les éditions de l'Ecole Polytechnique, Oct. 2011, 314 p.
  • Dinguemans M. W., "Water Wave Propagation Over Uneven Bottoms - Part 1 - Linear Wave Propagation", Advanced Series on Ocean Engineering - Volume 13, World Scientific, 471 p.
  • Dinguemans M. W., "Water Wave Propagation Over Uneven Bottoms - Part 2 - Non-Linear Wave Propagation", Advanced Series on Ocean Engineering - Volume 13, World Scientific, 494 p.
  • Tanguy J.M. "Traité d'hydraulique environnementale", vol.2, "processus maritimes", reps. publi. JM Tanguy, Ed. Hermès - Lavoisier, 2010.
  • site du SHOM et plus particulièrement une page dédiée aux courants de marée
  • D. Prandle and M. Rahman, "Tidal Respo,se in Estuaries" Journal of the Physical Oceanography, Vol. 10, pp 1552 - 1573, 1980

Code Scilab

The previous animations were made using the SCILAB application.
They can be used to reproduce the graph. Just select all the text in the * .pdf file and copy it to the software editor and run the program. The file is available here : Fichier:Estuaire avec correction.pdf provides the SCILAB code of the program of the previous animations.



Le créateur de cet article est Jean-Michel Tanguy
Note : d'autres personnes peuvent avoir contribué au contenu de cet article, [Consultez l'historique].

  • Pour d'autres articles de cet auteur, voir ici.
  • Pour un aperçu des contributions de cet auteur, voir ici.
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