Favre averaged NavierStokes equations
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Equations (1)(9), supplemented with gas data for <math>\gamma</math>, <math>Pr</math>, <math>\mu</math> and perhaps <math>R</math>, form a closed set of partial differential equations, and need only be complemented with boundary conditions.  Equations (1)(9), supplemented with gas data for <math>\gamma</math>, <math>Pr</math>, <math>\mu</math> and perhaps <math>R</math>, 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 <math>\Phi</math> be any dependent variable. It is convenient to define  
+  two different types of averaging of <math>\Phi</math>:  
+  
+  * Classical time average (Reynolds average):  
+  <table width="100%">  
+  <tr><td>  
+  :<math>\overline{\Phi} \equiv \frac{1}{T} \int_T \Phi(t) dt</math>  
+  </td>  
+  <td rowspan="2" width="5%">(10)</td></tr>  
+  <tr><td>  
+  :<math>\Phi' \equiv \Phi  \overline{\Phi}</math>  
+  </td></tr>  
+  </table>  
+  
+  * Density weighted time average (Favre average):  
+  <table width="100%">  
+  <tr><td>  
+  :<math>\widetilde{\Phi} \equiv \frac{\overline{\rho \Phi}}{\overline{\rho}}</math>  
+  </td><td rowspan="2" width="5%">(11)</td></tr>  
+  <tr><td>  
+  :<math>\Phi'' \equiv \Phi  \widetilde{\Phi}</math>  
+  </td></tr>  
+  </table>  
+  
+  Note that with the above definitions <math>\overline{\Phi'} = 0</math>, but <math>\overline{\Phi''} \neq 0</math>.  
<math>  <math>  
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</math>  </math>  
  
[[Category:Fluid Dynamics]][[Category:Equations]]  [[Category:Fluid Dynamics]][[Category:Equations]] 
Revision as of 09:02, 5 September 2005
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) 
Insert Reference 
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 .