Favre averaged NavierStokes equations
From CFDWiki
Line 180:  Line 180:  
</td><td width="5%">(16)</td></tr></table>  </td><td width="5%">(16)</td></tr></table>  
  Equation (12), (13) and (14) are referred to as the Favre averaged NavierStokes equations. <math>\overline{\rho}</math>, <math>\widetilde{u_i}</math> and <math>\widetilde{e_0}</math> are the primary solution variables.  +  Equation (12), (13) and (14) are referred to as the Favre averaged NavierStokes equations. <math>\overline{\rho}</math>, <math>\widetilde{u_i}</math> and <math>\widetilde{e_0}</math> are the primary solution variables. Note that this is an open set of partial differential equations that contains several unkown correlation terms. In order to obtain a closed form of equations that can be solver it is neccessary to model these unknown correlation terms. 
=== Approximations ===  === Approximations ===  
+  
+  To analyze equation (12), (13) and (14) it is convenient to rewrite the unknown terms in the following way:  
+  
+  <table width="100%"><tr><td>  
+  :<math>\overline{\tau_{ji}} = \widetilde{\tau_{ji}} + \overline{\tau''_{ji}}</math>  
+  </td><td width="5%">(17)</td></tr></table>  
+  
+  <table width="100%"><tr><td>  
+  :<math>\overline{u''_j p} + \overline{\rho u''_j e''_0} =  
+  C_p \overline{\rho u''_j T} +  
+  u_i \overline{\rho u''_i u''_j} + \overline{\frac{\rho u''_j u''_i u''_i}{2}}  
+  </math>  
+  </td><td width="5%">(18)</td></tr></table>  
+  
+  <table width="100%"><tr><td>  
+  :<math>\overline{q_j} =  
+   \overline{C_p \frac{\mu}{Pr} \frac{\partial T}{\partial x_j}} =  
+   C_p \frac{\mu}{Pr} \frac{\partial \widetilde{T}}{\partial x_j}   
+  C_p \frac{\mu}{Pr} \frac{\partial \overline{T''}}{\partial x_j}  
+  </math>  
+  </td><td width="5%">(19)</td></tr></table>  
+  
+  
+  <table width="100%"><tr><td>  
+  :<math>\overline{u_i \tau_{ij}} =  
+  \widetilde{u_i} \widetilde{\tau_{ij}} + \overline{u''_i \tau_{ij}} + \widetilde{u_i} \overline{\tau''_{ij}}  
+  </math>  
+  </td><td width="5%">(20)</td></tr></table>  
+  
+  Where the perfect gas relations (8) and Fourier's law (6) have been used. Note also that fluctuations in the molecular viscosity, <math>\mu</math>, have been neglected.  
+  
+  Inserting (17)(20) into (12), (13) and (14) gives:  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \frac{\partial \overline{\rho}}{\partial t} +  
+  \frac{\partial}{\partial x_i}\left[ \overline{\rho} \widetilde{u_i} \right] = 0  
+  </math>  
+  </td><td width="5%">(21)</td></tr></table>  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \frac{\partial}{\partial t}\left( \overline{\rho} \widetilde{u_i} \right) +  
+  \frac{\partial}{\partial x_j}  
+  \left[  
+  \overline{\rho} \widetilde{u_i} \widetilde{u_j} + \overline{p} \delta_{ij} +  
+  \underbrace{\overline{\rho u''_i u''_j}}_{(1^*)}  \widetilde{\tau_{ji}}   
+  \underbrace{\overline{\tau''_{ji}}}_{(2^*)}  
+  \right]  
+  = 0  
+  </math>  
+  </td><td width="5%">(22)</td></tr></table>  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \frac{\partial}{\partial t}  
+  \left(\overline{\rho} \widetilde{e_0} \right) +  
+  \frac{\partial}{\partial x_j}  
+  [  
+  \overline{\rho} \widetilde{u_j} \widetilde{e_0} +  
+  \widetilde{u_j} \overline{p} +  
+  \underbrace{C_p \overline{\rho u''_j T}}_{(3^*)} +  
+  \underbrace{\widetilde{u_i} \overline{\rho u''_i u''_j}}_{(4^*)} +  
+  \underbrace{\overline{\frac{\rho u''_j u''_i u''_i}{2}}}_{(5^*)}  
+  </math>  
+  </td></tr><tr><td>  
+  :<math>  
+   C_p \frac{\mu}{Pr} \frac{\partial \widetilde{T}}{\partial x_j}  
+   \underbrace{C_p \frac{\mu}{Pr} \frac{\partial \overline{T''}}  
+  {\partial x_j}}_{(6^*)}  
+  \widetilde{u_i} \widetilde{\tau_{ij}}   
+  \underbrace{\overline{u''_i \tau_{ij}}}_{(7^*)}   
+  \underbrace{\widetilde{u_i} \overline{\tau''_{ij}}}_{(8^*)}  
+  ]  
+  = 0  
+  </math>  
+  </td><td width="5%" rowspan="2">(23)</td></tr></table>  
+  
+  
+  
+  
+  
+  
<math>  <math> 
Revision as of 09:58, 5 September 2005
Contents 
Instantaneuos Equations
The instantaneous continuity equation (1), momentum equation (2) and energy equation (3) for a compressible fluid can be written as:
 (1) 
 (2) 
 (3) 
For a Newtonian fluid, assuming Stokes Law for monoatomic gases, the viscous stress is given by:
 (4) 
Where the traceless viscous strainrate is defined by:
 (5) 
The heatflux, , is given by Fourier's law:
 (6) 
Where the laminar Prandtl number is defined by:
 (7) 
To close these equations it is also necessary to specify an equation of state. Assuming a calorically perfect gas the following relations are valid:
 (8) 
Where , , and are constant.
The total energy is defined by:
 (9) 
Note that the corresponding expression (15) for Favre averaged turbulent flows contains an extra term related to the turbulent energy.
Equations (1)(9), supplemented with gas data for , , and perhaps , form a closed set of partial differential equations, and need only be complemented with boundary conditions.
Favre Averaged Equations
It is not possible to solve the instantaneous equations directly for the applications of interest here. At the Reynolds numbers typically present in a turbine these equations have very chaotic turbulent solutions, and it is necessary to model the influence of the smallest scales. All turbulence models used in this work are based on onepoint averaging of the instantaneous equations. The averaging procedure used is described in the next sections.
Averaging
Let be any dependent variable. It is convenient to define two different types of averaging of :
 Classical time average (Reynolds average):

(10) 

 Density weighted time average (Favre average):
 (11) 

Note that with the above definitions , but .
Open Turbulent Equations
In order to obtain an averaged form of the governing equations, the instantaneous continuity equation (1), momentum equation (2) and energy equation (3) are timeaveraged. Introducing a density weighted time average decomposition (10) of and , and a standard time average decomposition (11) of and gives the following exact open equations:
 (12) 
 (13) 
 (14) 
The density averaged total energy is given by:
 (15) 
Where the turbulent energy, , is defined by:
 (16) 
Equation (12), (13) and (14) are referred to as the Favre averaged NavierStokes equations. , and are the primary solution variables. Note that this is an open set of partial differential equations that contains several unkown correlation terms. In order to obtain a closed form of equations that can be solver it is neccessary to model these unknown correlation terms.
Approximations
To analyze equation (12), (13) and (14) it is convenient to rewrite the unknown terms in the following way:
 (17) 
 (18) 
 (19) 
 (20) 
Where the perfect gas relations (8) and Fourier's law (6) have been used. Note also that fluctuations in the molecular viscosity, , have been neglected.
Inserting (17)(20) into (12), (13) and (14) gives:
 (21) 
 (22) 
 
 (23) 