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sunfluidh:sunfluidh_link_equations_data_set [2016/11/30 12:38]
yann
sunfluidh:sunfluidh_link_equations_data_set [2018/12/17 14:58] (Version actuelle)
yann [Links equations & Data set]
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 ===== Links equations & Data set ===== ===== Links equations & Data set =====
 +
 [[:start|Click here to come back to the previous page]] [[:start|Click here to come back to the previous page]]
-[[sunfluidh:sunfluidh_data_setup|here to go to the data-set page]]+ 
 +[[sunfluidh:sunfluidh_data_setup|Click here to go to the data-set page]] 
 <WRAP info> <WRAP info>
 The code Sunfluidh solves the Navier-Stokes equations by means of an incremental projection method. The code Sunfluidh solves the Navier-Stokes equations by means of an incremental projection method.
    * In the prediction step, the Navier-Stokes equations are solved in order to estimate the velocity field $\vec{V}^*$ without ensuring the mass conservation ($\nabla \cdot \vec{V}=0$ for incompressible flows or $\frac{\partial \rho}{\partial t}+\nabla \cdot (\rho\vec{V})= 0$ for low Mach number flows).    * In the prediction step, the Navier-Stokes equations are solved in order to estimate the velocity field $\vec{V}^*$ without ensuring the mass conservation ($\nabla \cdot \vec{V}=0$ for incompressible flows or $\frac{\partial \rho}{\partial t}+\nabla \cdot (\rho\vec{V})= 0$ for low Mach number flows).
-   * In the projection step, The mass conservation is ensured by solving a Poisson's equation. The result $\phi$ corresponds to the time increment of the "dynamical" pressure (the part of the pressure associated to the dynamics) and its gradient is the velocity correction that ensures the mass conservation. \\+   * In the projection step, The mass conservation is ensured by solving a Poisson's equation. The result $\phi$ corresponds to the time increment of the "dynamical" pressure (the part of the static pressure associated to the dynamics) and its gradient is the velocity correction that ensures the mass conservation. For incompressible flows, $P_{dyn}$ is the static pressure defined up to a constant. \\
  
 $$ P_{dyn}^{n+1} = P_{dyn}^{n} + \phi$$ $$ P_{dyn}^{n+1} = P_{dyn}^{n} + \phi$$
 $$ \vec{V}^{n+1}= \vec{V}^* - \frac{\Delta t}{\rho} \nabla \phi$$ $$ \vec{V}^{n+1}= \vec{V}^* - \frac{\Delta t}{\rho} \nabla \phi$$
  
-For more details on the projection methods, see the document here.+**For more details on the projection methods, see the document on the numerical methods used in Sunfluidh [[sunfluidh:sunfluidh_edu_documents|(click here)]].**
  
 </WRAP> </WRAP>
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    * the different sets of governing equations that depend on the flow is either incompressible or dilatable (low Mach number hypothesis).     * the different sets of governing equations that depend on the flow is either incompressible or dilatable (low Mach number hypothesis). 
    * the different formulations of the Poisson's equation in respect with the problem treated.    * the different formulations of the Poisson's equation in respect with the problem treated.
-   * the links between the equations, physical quantities and [[sunfluidh:sunfluidh_data_setup|the data set]] +   * the links between the equations, physical quantities and **[[sunfluidh:sunfluidh_data_setup|the input data setup]] ** 
  
 </note> </note>
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 </note> </note>
  
-[[sunfluidh:sunfluidh_data_setup|here to come back to the data-set page]]+[[sunfluidh:sunfluidh_data_setup|Click here to come back to the data-set page]]
  
 ----- -----
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 <note important> <note important>
-We remind the user that the low Mach hypothesis is based on the hypothesis of the scale splitting between the thermodynamics and dynamics phenomena. As a consequence the pressure is read as $P= P_{th}+P_{dyn}$, where $P_{th}$ is the thermodynamical pressure and $P_{dyn}$ the dynamical pressure.  $P_{th}$ is supposed to be uniform over the domain and is defined by the equation of state. $P_dyn$ is solved from the Poisson's equation (see the projection methods).+We remind the user that the low Mach hypothesis is based on the hypothesis of the scale splitting between the thermodynamics and dynamics phenomena. As a consequence the pressure is read as $P= P_{th}+P_{dyn}$, where $P_{th}$ is the thermodynamical pressure and $P_{dyn}$ the "dynamicalpressure (part of the static pressure that contributes to the fluid motion).  $P_{th}$ is supposed to be uniform over the domain and is defined by the equation of state. $P_dyn$ is solved from the Poisson's equation (see the projection methods).
 </note> </note>
  
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 </note> </note>
  
-[[sunfluidh:sunfluidh_data_setup|here to come back to the data-set page]]+[[sunfluidh:sunfluidh_data_setup|Click here to come back to the data-set page]]
  
 ===== The different formulations of the Poisson's equation ===== ===== The different formulations of the Poisson's equation =====
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 $$ \nabla \frac{1}{\rho}\cdot \nabla \Phi= \frac{ \nabla \cdot \vec{V}^* - \nabla \cdot \vec{V}^{n+1} }{\alpha \Delta t} $$ $$ \nabla \frac{1}{\rho}\cdot \nabla \Phi= \frac{ \nabla \cdot \vec{V}^* - \nabla \cdot \vec{V}^{n+1} }{\alpha \Delta t} $$
  
-where $\nabla \cdot \vec{V}^{n+1}$ is estimated from the differential equation of state$+where $\nabla \cdot \vec{V}^{n+1}$ is estimated from the differential equation of state.
  
 [[sunfluidh:sunfluidh_data_setup|here to come back to the data-set page]] [[sunfluidh:sunfluidh_data_setup|here to come back to the data-set page]]
 +
 +<WRAP info>
 +**Information on numerical methods used for solving these equations is available in the pdf document present [[ :sunfluidh:sunfluidh_full_documents | here]].**
 +</WRAP>
 +
  
 ===== Link between the data set & the variables in equations ===== ===== Link between the data set & the variables in equations =====
 +
 +**__The knowledge of the data setup is recommanded in order to better understand the links.__** [[sunfluidh:sunfluidh_data_setup|Click here to go to the data-set page]]
 +
  
 ^ List of variables ^ Definition       ^ namelists where the physical quantity is defined^ ^ List of variables ^ Definition       ^ namelists where the physical quantity is defined^
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 [[:start|Click here to come back to the previous page]] [[:start|Click here to come back to the previous page]]
-[[sunfluidh:sunfluidh_data_setup|here to go to the data-set page]]+ 
 +[[sunfluidh:sunfluidh_data_setup|Click here to go to the data-set page]]
sunfluidh/sunfluidh_link_equations_data_set.1480505912.txt.gz · Dernière modification: 2016/11/30 12:38 de yann