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Ryan LI July 3, 2022 23:17
What does the Boussinesq eddy viscosity hypothesis really means?
 
Hi CFDers

I just stepped into the mathematics of the RANS model and I got stuck. When I read the Boussinesq hypothesis, also called the eddy viscosity hypothesis, I saw this equation and the name of every term. Such as

\tau_{ij} = 2 \, \mu_t \, S_{ij}^* - \frac{2}{3} \rho k \delta_{ij}

where \mu_t denotes the eddy viscosity, S_{ij}^* the strain rate tensor and k the turbulent kinetic energy.

But I still have concerns,

After some research, here are what I know so far:
the strain rate tensor describes the rate of change of the deformation of the fluid in the neighborhood of a certain point. The turbulent momentum transfer results in the deformation of the fluid, and the deformation gives rise to viscous forces in its interior against the deformation, due to friction between adjacent fluid elements, and due to the internal stress tensor.

And the turbulent kinetic energy defines the energy of motion carried by fluid flow. And it dissipates down the turbulence energy cascade.

Here are what I still don't know：
Why the kinetic energy term in the above equation? I don't know how the turbulent kinetic energy is related to the stress tensor.
Why 2 and - \frac{2}{3} in the above equation? I don't know whether this is pure empirical or if there is a theory behind these factors.

Thank you

FMDenaro July 4, 2022 11:55
Quote:
Originally Posted by Ryan LI (Post 830929)
Hi CFDers

I just stepped into the mathematics of the RANS model and I got stuck. When I read the Boussinesq hypothesis, also called the eddy viscosity hypothesis, I saw this equation and the name of every term. Such as

\tau_{ij} = 2 \, \mu_t \, S_{ij}^* - \frac{2}{3} \rho k \delta_{ij}

where \mu_t denotes the eddy viscosity, S_{ij}^* the strain rate tensor and k the turbulent kinetic energy.

But I still have concerns,

After some research, here are what I know so far:
the strain rate tensor describes the rate of change of the deformation of the fluid in the neighborhood of a certain point. The turbulent momentum transfer results in the deformation of the fluid, and the deformation gives rise to viscous forces in its interior against the deformation, due to friction between adjacent fluid elements, and due to the internal stress tensor.

And the turbulent kinetic energy defines the energy of motion carried by fluid flow. And it dissipates down the turbulence energy cascade.

Here are what I still don't know：
Why the kinetic energy term in the above equation? I don't know how the turbulent kinetic energy is related to the stress tensor.
Why 2 and - \frac{2}{3} in the above equation? I don't know whether this is pure empirical or if there is a theory behind these factors.

Thank you



I suggest you to think about the decomposition of the tensor in isotropic and deviatoric parts. You will see the 1/3 appearing.

The model is for the deviatoric part of the stress, thus the isotropic part will be added to the pressure. The 2 appears exactly as in in the molecular diffusion term.

sbaffini July 4, 2022 15:56
Take the trace of both sides and see what happens

watermelon July 4, 2022 20:55
Quote:
Originally Posted by Ryan LI (Post 830929)
Hi CFDers

I just stepped into the mathematics of the RANS model and I got stuck. When I read the Boussinesq hypothesis, also called the eddy viscosity hypothesis, I saw this equation and the name of every term. Such as

\tau_{ij} = 2 \, \mu_t \, S_{ij}^* - \frac{2}{3} \rho k \delta_{ij}

where \mu_t denotes the eddy viscosity, S_{ij}^* the strain rate tensor and k the turbulent kinetic energy.

But I still have concerns,

After some research, here are what I know so far:
the strain rate tensor describes the rate of change of the deformation of the fluid in the neighborhood of a certain point. The turbulent momentum transfer results in the deformation of the fluid, and the deformation gives rise to viscous forces in its interior against the deformation, due to friction between adjacent fluid elements, and due to the internal stress tensor.

And the turbulent kinetic energy defines the energy of motion carried by fluid flow. And it dissipates down the turbulence energy cascade.

Here are what I still don't know：
Why the kinetic energy term in the above equation? I don't know how the turbulent kinetic energy is related to the stress tensor.
Why 2 and - \frac{2}{3} in the above equation? I don't know whether this is pure empirical or if there is a theory behind these factors.

Thank you
Hello, Songrui, Boussinesq eddy viscosity hypothesis is totally analogized from the N-S constitutive relationship. So, if you were familiar with the N-S relationship, the Boussinesq eddy viscosity will be understood too. This will interpret the coefficient 2.

About the coefficient 2/3, this is from the average of turbulent kinematic energy. Actually here is Pt, a "pressure" due to fluctuating velocity.

Best Regards.

LuckyTran July 5, 2022 10:53
Really it's about understanding where the 3 in 1/3 comes from. If you're asking why 2, then look at the definition of S and tell me why there is a 1/2 in it.

Ryan LI July 7, 2022 02:53
Quote:
Originally Posted by FMDenaro (Post 830974)
I suggest you to think about the decomposition of the tensor in isotropic and deviatoric parts. You will see the 1/3 appearing.

The model is for the deviatoric part of the stress, thus the isotropic part will be added to the pressure. The 2 appears exactly as in in the molecular diffusion term.
Thx Filippo for your kindly response

I followed your lead and I found this section demonstrates what you say and it really helps me.

https://en.wikipedia.org/wiki/Viscou...viscous_stress

Ryan LI July 7, 2022 02:55
Quote:
Originally Posted by sbaffini (Post 830982)
Take the trace of both sides and see what happens
Yes indeed, I think I should look deeper into the linear algebra

sbaffini July 7, 2022 15:40
Quote:
Originally Posted by Ryan LI (Post 831147)
Yes indeed, I think I should look deeper into the linear algebra
No, please, don't go into linear algebra, just, literally, take the trace of both sides, which are 3x3 tensors, so just take the sum of their 3 diagonal terms and take note of the definitions of all the terms


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