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What does the Boussinesq eddy viscosity hypothesis really means?

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Old   July 3, 2022, 23:17
Question What does the Boussinesq eddy viscosity hypothesis really means?
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Songrui LI
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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
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Old   July 4, 2022, 11:55
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Filippo Maria Denaro
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Quote:
Originally Posted by Ryan LI View Post
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.
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Old   July 4, 2022, 15:56
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Take the trace of both sides and see what happens
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Old   July 4, 2022, 20:55
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Originally Posted by Ryan LI View Post
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.
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Old   July 5, 2022, 10:53
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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.
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Old   July 7, 2022, 02:53
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Originally Posted by FMDenaro View Post
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
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Old   July 7, 2022, 02:55
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Take the trace of both sides and see what happens
Yes indeed, I think I should look deeper into the linear algebra
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Old   July 7, 2022, 15:40
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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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