Favre averaged NavierStokes equations
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(14 intermediate revisions not shown)  
Line 1:  Line 1:  
  ==  +  == Instantaneous equations == 
The instantaneous continuity equation (1), momentum equation (2) and energy equation (3) for a compressible fluid can be written as:  The instantaneous continuity equation (1), momentum equation (2) and energy equation (3) for a compressible fluid can be written as:  
Line 99:  Line 99:  
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  +  == Favre averaged equations == 
  It is not possible to solve the instantaneous equations directly for  +  It is not possible to solve the instantaneous equations directly for most engineering applications. At the Reynolds numbers typically present in real cases these equations have very chaotic turbulent solutions, and it is necessary to model the influence of the smallest scales. Most turbulence models are based on onepoint averaging of the instantaneous equations. The averaging procedure will be described in the following sections. 
  == Averaging ==  +  === Averaging === 
Let <math>\Phi</math> be any dependent variable. It is convenient to define  Let <math>\Phi</math> be any dependent variable. It is convenient to define  
two different types of averaging of <math>\Phi</math>:  two different types of averaging of <math>\Phi</math>:  
  * Classical time  +  * Classical time averaging ([[Reynolds averaging]]): 
<table width="100%">  <table width="100%">  
<tr><td>  <tr><td>  
Line 119:  Line 119:  
</table>  </table>  
  * Density weighted time  +  * Density weighted time averaging ([[Favre averaging]]): 
<table width="100%">  <table width="100%">  
<tr><td>  <tr><td>  
Line 131:  Line 131:  
Note that with the above definitions <math>\overline{\Phi'} = 0</math>, but <math>\overline{\Phi''} \neq 0</math>.  Note that with the above definitions <math>\overline{\Phi'} = 0</math>, but <math>\overline{\Phi''} \neq 0</math>.  
  == Open  +  === Open turbulent equations === 
In order to obtain an averaged form of the governing equations, the instantaneous continuity equation (1), momentum  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 (  +  equation (2) and energy equation (3) are timeaveraged. Introducing a density weighted time average decomposition (11) of <math>u_i</math> and <math>e_0</math>, and a standard time average decomposition (10) of <math>\rho</math> and <math>p</math> gives the following exact open equations: 
<table width="100%"><tr><td>  <table width="100%"><tr><td>  
Line 174:  Line 174:  
</td><td width="5%">(15)</td></tr></table>  </td><td width="5%">(15)</td></tr></table>  
  Where the turbulent energy,  +  Where the turbulent energy, <math>k</math>, is defined by: 
<table width="100%"><tr><td>  <table width="100%"><tr><td>  
  :<math>k \equiv \  +  :<math>k \equiv \widetilde{\frac{u''_k u''_k}{2}}</math> 
</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 and modeling === 
  <math>  +  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 \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%">(21)</td></tr></table>  
  <math>  +  <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>  
+  
+  The terms marked with <math>(1^*)(8^*)</math> are unknown, and have to be modeled in some way.  
+  
+  Term <math>(1^*)</math> and <math>(4^*)</math> can be modeled using an eddyviscosity assumption for the Reynolds stresses, <math>\tau_{ij}^{turb}</math>:  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \tau_{ij}^{turb} \equiv  
+   \overline{\rho u''_i u''_j} \approx  
+  2 \mu_t \widetilde{S_{ij}^*}   
+  \frac{2}{3} \overline{\rho} k \delta_{ij}  
+  </math>  
+  </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.  
+  
+  Term <math>(2^*)</math> and <math>(8^*)</math> can be neglected if:  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  \left \widetilde{\tau_{ij}} \right >>  
+  \left \overline{\tau''_{ij}} \right  
+  </math>  
+  </td><td width="5%">(25)</td></tr></table>  
+  
+  This is true for virtually all flows.  
+  
+  Term <math>(3^*)</math>, corresponding to turbulent transport of heat, can be modeled using a gradient approximation for the turbulent heatflux:  
+  
+  <table width="100%"><tr><td>  
+  :<math>  
+  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}  
+  </math>  
+  </td><td width="5%">(26)</td></tr></table>  
+  
+  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>  
+  
+  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}{\partial x_i}\left[ \overline{\rho} \widetilde{u_i} \right] = 0  
+  </math>  
+  </td><td width="5%">(30)</td></tr></table>  
+  
+  <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 199:  Line 352:  
= 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  +  [[Category:Fluid dynamics]][[Category:Equations]] 
Latest revision as of 20:33, 24 November 2005
Contents 
Instantaneous 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 most engineering applications. At the Reynolds numbers typically present in real cases these equations have very chaotic turbulent solutions, and it is necessary to model the influence of the smallest scales. Most turbulence models are based on onepoint averaging of the instantaneous equations. The averaging procedure will be described in the following sections.
Averaging
Let be any dependent variable. It is convenient to define two different types of averaging of :
 Classical time averaging (Reynolds averaging):

(10) 

 Density weighted time averaging (Favre averaging):
 (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 (11) of and , and a standard time average decomposition (10) 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 and modeling
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.