Issues on modelling of isotropic part of Reynolds (SGS) stresses

Likun · February 4, 2015, 05:11

Hi Foamers,

Recently I am confused about the modelling of Reynolds or SGS stresses in openFoam. Specifically, I don't understand how the isotropic part the the SGS stresses have been considered. According to the turbulent viscosity assumption, the Reynolds or SGS stresses
$\tau_{ij} = -2 \mu_{sgs} \tilde S_{ij}^{D} + \frac{1}{3} \tau_{ij} \delta_{ij} = -2 \mu_{sgs} \tilde S_{ij}^{D} + \frac{2}{3}\bar \rho k \delta_{ij}$

where $\tilde S_{ij}^{D} = \frac{1}{2}\left( \frac{\partial \tilde u_{i}}{\partial x_{j}} + \frac{\partial \tilde u_{j}}{\partial x_{i}} \right) - \frac{1}{3} \frac{\partial \tilde u_{k}}{\partial x_{k}} \delta_{ij}$ is deviatoric part of the local strain rate tensor.

As discussed in http://www.cfd-online.com/Forums/ope...sonicfoam.html and http://www.cfd-online.com/Forums/openfoam-solving/58214-calculating-divdevreff.html#post388688, , the isotropic part of Reynolds (SGS) stresses, $\frac{1}{3} \tau_{ij} \delta_{ij}$ or $\frac{2}{3}\bar \rho k \delta_{ij}$ , are not included in the

term of UEqn

Code:

fvVectorMatrix UEqn 
(    
      fvm::ddt(rho, U)     
    + fvm::div(phi, U)   
    + turbulence->divDevRhoReff(U) 
);

And I can not find the source code, where this term has been considered. So I suppose that this term has been simply ignored in openFoam! Someone please correct me if I am wrong.

In many literature, including the book of Pope, it says that this term is absorbed into a modified pressure which leads to the following modelled momentum equation:

$\frac{\partial \bar \rho \tilde u_{i}}{\partial t} + \frac{\partial \left(\bar \rho \tilde u_{i} \tilde u_{j} \right)}{\partial x_{j}} = -\frac{\partial \bar p}{\partial x_{i}} \left( \bar p + \frac{2}{3} k \right) + \frac{\partial}{\partial j} \left(2 \left(\mu + \mu_{sgs} \right) \tilde S_{ij}^{D} \right) + \bar S_{M,i}$

It is clear that if the turbulent (SGS) kinetic energy

is much smaller than the thermodynamic pressure

, this term is indeed negligible.

So my questions are:

1. Is it true that the $\frac{1}{3} \tau_{ij} \delta_{ij}$ term has been neglected in OpenFoam? If this is true, the pressure (in compressible part) is the real pressure, right?

2. How this term can be included, since we already have the value for

?

Any idea will be appreciated.

Best,
Likun

tiam · June 25, 2015, 06:57

Realized that this issue is not really clear to me either

. Any answers since 2013?

Likun · January 25, 2016, 03:51

New discussions on this issue can find here: http://www.cfd-online.com/Forums/ope...tml#post582044

February 4, 2015, 05:11	Issues on modelling of isotropic part of Reynolds (SGS) stresses	#1
Likun Member Likun Join Date: Feb 2013 Posts: 52 Rep Power: 13	Hi Foamers, Recently I am confused about the modelling of Reynolds or SGS stresses in openFoam. Specifically, I don't understand how the isotropic part the the SGS stresses have been considered. According to the turbulent viscosity assumption, the Reynolds or SGS stresses $\tau_{ij} = -2 \mu_{sgs} \tilde S_{ij}^{D} + \frac{1}{3} \tau_{ij} \delta_{ij} = -2 \mu_{sgs} \tilde S_{ij}^{D} + \frac{2}{3}\bar \rho k \delta_{ij}$ where $\tilde S_{ij}^{D} = \frac{1}{2}\left( \frac{\partial \tilde u_{i}}{\partial x_{j}} + \frac{\partial \tilde u_{j}}{\partial x_{i}} \right) - \frac{1}{3} \frac{\partial \tilde u_{k}}{\partial x_{k}} \delta_{ij}$ is deviatoric part of the local strain rate tensor. As discussed in http://www.cfd-online.com/Forums/ope...sonicfoam.html and http://www.cfd-online.com/Forums/openfoam-solving/58214-calculating-divdevreff.html#post388688, , the isotropic part of Reynolds (SGS) stresses, $\frac{1}{3} \tau_{ij} \delta_{ij}$ or $\frac{2}{3}\bar \rho k \delta_{ij}$ , are not included in the term of UEqn Code: fvVectorMatrix UEqn ( fvm::ddt(rho, U) + fvm::div(phi, U) + turbulence->divDevRhoReff(U) ); And I can not find the source code, where this term has been considered. So I suppose that this term has been simply ignored in openFoam! Someone please correct me if I am wrong. In many literature, including the book of Pope, it says that this term is absorbed into a modified pressure which leads to the following modelled momentum equation: $\frac{\partial \bar \rho \tilde u_{i}}{\partial t} + \frac{\partial \left(\bar \rho \tilde u_{i} \tilde u_{j} \right)}{\partial x_{j}} = -\frac{\partial \bar p}{\partial x_{i}} \left( \bar p + \frac{2}{3} k \right) + \frac{\partial}{\partial j} \left(2 \left(\mu + \mu_{sgs} \right) \tilde S_{ij}^{D} \right) + \bar S_{M,i}$ It is clear that if the turbulent (SGS) kinetic energy is much smaller than the thermodynamic pressure , this term is indeed negligible. So my questions are: 1. Is it true that the $\frac{1}{3} \tau_{ij} \delta_{ij}$ term has been neglected in OpenFoam? If this is true, the pressure (in compressible part) is the real pressure, right? 2. How this term can be included, since we already have the value for ? Any idea will be appreciated. Best, Likun ztnuaa likes this.

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June 25, 2015, 06:57		#2
tiam Senior Member Timofey Mukha Join Date: Mar 2012 Location: Stockholm, Sweden Posts: 119 Rep Power: 14	Realized that this issue is not really clear to me either . Any answers since 2013?

January 25, 2016, 03:51		#3
Likun Member Likun Join Date: Feb 2013 Posts: 52 Rep Power: 13	New discussions on this issue can find here: http://www.cfd-online.com/Forums/ope...tml#post582044