Favre averaged NavierStokes equations
From CFDWiki
Line 272:  Line 272:  
\frac{2}{3} \overline{\rho} k \delta_{ij}  \frac{2}{3} \overline{\rho} k \delta_{ij}  
</math>  </math>  
  </td><td width="5%  +  </td><td width="5%">(24)</td></tr></table> 
Where <math>\mu_t</math> is a turbulent viscosity, which is estimated with a turbulence model. The last term is included in order to ensure that the trace of the Reynolds stress tensor is equal to <math>2 \rho k</math>, as it should be.  Where <math>\mu_t</math> is a turbulent viscosity, which is estimated with a turbulence model. The last term is included in order to ensure that the trace of the Reynolds stress tensor is equal to <math>2 \rho k</math>, as it should be.  
Term <math>(2^*)</math> and <math>(8^*)</math> can be neglected if:  Term <math>(2^*)</math> and <math>(8^*)</math> can be neglected if:  
  
<table width="100%"><tr><td>  <table width="100%"><tr><td>  
Line 284:  Line 283:  
\left \overline{\tau''_{ij}} \right  \left \overline{\tau''_{ij}} \right  
</math>  </math>  
  </td><td width="5%  +  </td><td width="5%">(25)</td></tr></table> 
This is true for virtually all flows.  This is true for virtually all flows.  
Line 292:  Line 291:  
<table width="100%"><tr><td>  <table width="100%"><tr><td>  
:<math>  :<math>  
  q_j^{turb} \  +  q_j^{turb} \equiv 
C_p \overline{\rho u''_j T} \approx  C_p \overline{\rho u''_j T} \approx  
 C_p \frac{\mu_t}{Pr_t} \frac{\partial \widetilde{T}}{\partial x_j}   C_p \frac{\mu_t}{Pr_t} \frac{\partial \widetilde{T}}{\partial x_j}  
</math>  </math>  
  </td><td width="5%"  +  </td><td width="5%"">(26)</td></tr></table> 
  Where <math>Pr_t</math> is a turbulent Prandtl number. Often a constant <math>Pr_t  +  Where <math>Pr_t</math> is a turbulent Prandtl number. Often a constant <math>Pr_t \approx 0.9</math> is used. 
  +  
+  Term <math>(5^*)</math> and <math>(7^*)</math>, corresponding to turbulent transport and molecular diffusion of turbulent energy, can be neglected if the turbulent energy is small compared to the enthalpy:  
+  <table width="100%"><tr><td>  
+  :<math>  
+  k << \widetilde{h} = C_p \widetilde{T}  
+  </math>  
+  </td><td width="5%">(27)</td></tr></table>  
+  This is a reasonable approximation for most flows below the hypersonic regime. A better approximation might be a gradient expression of the form:  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \overline{\frac{\rho u''_j u''_i u''_i}{2}}   
+  \overline{u''_i \tau_{ij}} \approx  
+   \left( \mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j}  
+  </math>  
+  </td><td width="5%">(28)</td></tr></table>  
  <math>  +  Where <math>\sigma_k</math> is a model constant. This approximation will not be included in the derived formulas below. Instead term <math>(5^*)</math> and <math>(7^*)</math> will be set to zero in the energy equation. 
+  
+  Term <math>(6^*)</math> is an artifact from the Favre averaging. It is related to heat conduction effects associated with temperature fluctuations.It can be be neglected if:  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \left \frac{\partial^2 \widetilde{T}}{\partial x_j^2} \right >>  
+  \left \frac{\partial^2 \overline{T''}}{\partial x_j^2} \right  
+  </math>  
+  </td><td width="5%">(29)</td></tr></table>  
+  
+  This is true for virtually all flows, and has been assumed in all follwing equations.  
+  
+  === Closed Approximated Equations ===  
+  
+  To summarize, the governing equations (21)(23), with assumptions (24), (25), (26), (27) and (29) can be written as in (30)(39). These equations are valid for a perfect gas. Note also that all fluctuations in the molecular viscosity have been neglected.  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
\frac{\partial \overline{\rho}}{\partial t} +  \frac{\partial \overline{\rho}}{\partial t} +  
\frac{\partial}{\partial x_i}\left[ \overline{\rho} \widetilde{u_i} \right] = 0  \frac{\partial}{\partial x_i}\left[ \overline{\rho} \widetilde{u_i} \right] = 0  
</math>  </math>  
+  </td><td width="5%">(30)</td></tr></table>  
  <math>  +  <table width="100%"><tr><td> 
+  :<math>  
\frac{\partial}{\partial t}\left( \overline{\rho} \widetilde{u_i} \right) +  \frac{\partial}{\partial t}\left( \overline{\rho} \widetilde{u_i} \right) +  
\frac{\partial}{\partial x_j}  \frac{\partial}{\partial x_j}  
Line 321:  Line 353:  
= 0  = 0  
</math>  </math>  
+  </td><td width="5%">(31)</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}  
+  \left[  
+  \overline{\rho} \widetilde{u_j} \widetilde{e_0} +  
+  \widetilde{u_j} \overline{p} +  
+  \widetilde{q_j^{tot}}   
+  \widetilde{u_i} \widetilde{\tau_{ij}^{tot}}  
+  \right] = 0  
+  </math>  
+  </td><td width="5%">(32)</td></tr></table>  
+  
+  Where  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \widetilde{\tau_{ij}^{tot}} \equiv \widetilde{\tau_{ij}^{lam}} + \widetilde{\tau_{ij}^{turb}}  
+  </math>  
+  </td><td width="5%">(33)</td></tr></table>  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \widetilde{\tau_{ij}^{lam}} \equiv  
+  \widetilde{\tau_{ij}} =  
+  \mu  
+  \left(  
+  \frac{\partial \widetilde{u_i} }{\partial x_j} +  
+  \frac{\partial \widetilde{u_j} }{\partial x_i}   
+  \frac{2}{3} \frac{\partial \widetilde{u_k} }{\partial x_k} \delta_{ij}  
+  \right)  
+  </math></td><td width="5%">(34)</td></tr></table>  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \widetilde{\tau_{ij}^{turb}} \equiv  
+   \overline{\rho u''_i u''_j} \approx  
+  \mu_t  
+  \left(  
+  \frac{\partial \widetilde{u_i} }{\partial x_j} +  
+  \frac{\partial \widetilde{u_j} }{\partial x_i}   
+  \frac{2}{3} \frac{\partial \widetilde{u_k} }{\partial x_k} \delta_{ij}  
+  \right)   
+  \frac{2}{3} \overline{\rho} k \delta_{ij}  
+  </math></td><td width="5%">(35)</td></tr></table>  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \widetilde{q_j^{tot}} \equiv \widetilde{q_j^{lam}} + \widetilde{q_j^{turb}}  
+  </math></td><td width="5%">(36)</td></tr></table>  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \widetilde{q_j^{lam}} \equiv  
+  \widetilde{q_j} \approx  
+   C_p \frac{\mu}{Pr} \frac{\partial \widetilde{T}}{\partial x_j} =  
+   \frac{\gamma}{\gamma1} \frac{\mu}{Pr} \frac{\partial}{\partial x_j}  
+  \left[ \frac{\overline{p}}{\overline{\rho}} \right]  
+  </math></td><td width="5%">(37)</td></tr></table>  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \widetilde{q_j^{turb}} \equiv  
+  C_p \overline{\rho u''_j T} \approx  
+   C_p \frac{\mu_t}{Pr_t} \frac{\partial \widetilde{T}}{\partial x_j} =  
+   \frac{\gamma}{\gamma1} \frac{\mu_t}{Pr_t} \frac{\partial}{\partial x_j}  
+  \left[ \frac{\overline{p}}{\overline{\rho}} \right]  
+  </math></td><td width="5%">(38)</td></tr></table>  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \overline{p} = \left( \gamma  1 \right) \overline{\rho}  
+  \left( \widetilde{e_0}  \frac{\widetilde{u_k} \widetilde{u_k}}{2}  k \right)  
+  </math></td><td width="5%">(39)</td></tr></table>  
+  
+  If a separate turbulence model is used to calculate <math>\mu_t</math>, <math>k</math> and <math>Pr_t</math>, and gas data is given for <math>\mu</math>, <math>\gamma</math> and <math>Pr</math> these equations form a closed set of partial differential equations, which can be solved numerically.  
[[Category:Fluid Dynamics]][[Category:Equations]]  [[Category:Fluid Dynamics]][[Category:Equations]] 
Revision as of 11:49, 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) 
The terms marked with are unknown, and have to be modeled in some way.
Term and can be modeled using an eddyviscosity assumption for the Reynolds stresses, :
 (24) 
Where is a turbulent viscosity, which is estimated with a turbulence model. The last term is included in order to ensure that the trace of the Reynolds stress tensor is equal to , as it should be.
Term and can be neglected if:
 (25) 
This is true for virtually all flows.
Term , corresponding to turbulent transport of heat, can be modeled using a gradient approximation for the turbulent heatflux:
 (26) 
Where is a turbulent Prandtl number. Often a constant is used.
Term and , corresponding to turbulent transport and molecular diffusion of turbulent energy, can be neglected if the turbulent energy is small compared to the enthalpy:
 (27) 
This is a reasonable approximation for most flows below the hypersonic regime. A better approximation might be a gradient expression of the form:
 (28) 
Where is a model constant. This approximation will not be included in the derived formulas below. Instead term and will be set to zero in the energy equation.
Term is an artifact from the Favre averaging. It is related to heat conduction effects associated with temperature fluctuations.It can be be neglected if:
 (29) 
This is true for virtually all flows, and has been assumed in all follwing equations.
Closed Approximated Equations
To summarize, the governing equations (21)(23), with assumptions (24), (25), (26), (27) and (29) can be written as in (30)(39). These equations are valid for a perfect gas. Note also that all fluctuations in the molecular viscosity have been neglected.
 (30) 
 (31) 
 (32) 
Where
 (33) 
 (34) 
 (35) 
 (36) 
 (37) 
 (38) 
 (39) 
If a separate turbulence model is used to calculate , and , and gas data is given for , and these equations form a closed set of partial differential equations, which can be solved numerically.