[FluidKo]

Hello everyone.
I have very many questions of Reynolds Stress Model.
I've studied k- SST Model before but I can't understand review paper of Reynolds Stress model.

First I'm going to write down link of this paper.
https://ntrs.nasa.gov/api/citations/...9950018160.pdf

Cause of the many questions, it can be irritating.
Sorry.
But I need help.



1. Equation (37) in page 8

A. Why laplacian of fluctuating pressure is emerged?

I guess it is emergied to derive Equation (39) and Equation (40).
Right?

B. Why it is derived like that?

Why fluctuating pressure is related to fluctuating velocity gradient and mean velocity gradient?
Where is it derived from?

2. Equation (25) in page 7

A. Why we call this 'pressure-strain correlation'?

I think equation (25) is related to fluctuating pressure and fluctuating velocity gradient.
So I think 'fluctuating pressure-fluctuating strain correlation' is more clear nomenclature.
But why they call it pressure-strain?
Is there any special reason?

3. Equation (39) in page 8 and Equation (40) in page 9

A. Meaning of 'slow pressure-strain' and 'rapid pressure-strain'

In this paper author said these mean 'slow pressure-strain' and 'rapid pressure-strain'.
But I can't understand the meaning of slow and rapid.
What is matched with slow and rapid?
Is it meaning 'Pressure is changed slowly or rapidly.'?
Or what?
Actually I don't know Green's function but I've known these two equations are derived from Equation (37) in page 8.
But what is the meaning of 'slow pressure-strain' and 'rapid pressure-strain'?
And also where is the meaning of 'slow' in Equation (39) and 'rapid' in Equation (40)?

4. Equation (43) in Page 9

A. Meaning of

I guess dot means (Material differentiation). Is it right?

5. Equation (44) in page 9

A. Meaning of

I guess this also have similar meaing with upper.
Material differentiation.
Is it right?

6. In page 9, "The fundamental assumptions underlying two-equation models are that the turbulence is locally homogeneous and an equilibrium state is reached where attain constant values that are largely independent of the initial conditions."

A. Indepedent Constant

What it the exact meaing of independent constant about initial condition?
And also why it should be independent constant about initial condition?

Thanks for reading this long questions.
I'm not good at English so please excuse my bad English.
Also I've studied k- SST Model before but I'm not enough.
I'm not professional about turbulence.
But I'm interested in that field.
So help me please
Thanks

[LuckyTran]

1. That's the Poisson equation and for an incompressible flow, the pressure satisfies the Poisson equation. The gradient of pressure appears in the momentum equation. You can take the divergence of the momentum equation (and then substitute the continuity equation into it) to derive this Poisson equation.

2. It doesn't matter whether or not you call it fluctuating pressure and fluctuating strain because the correlation between a constant (i.e. a mean) and anything is zero. There's no ambiguity.

3. Pressure can be decomposed into two parts (one slow and one rapid). Both satisfy the Poisson equation. The rapid correlation has a source term that contains the mean velocity gradient. The slow correlation has a source term that does not and the slow correlation is primarily non-linearities that indirectly interact with the mean flow. Rapid/slow is with regard to the rate at which they interact and distort one another (pressure and strain). You don't have to call it rapid and slow if you don't want to because rapid/slow comes from distortion theory.

4&5. Yes (almost always) but (sometimes) no. Overdot is time rate of change which for the purposes of this conversation means material derivative (without going in-depth into why the material and total derivative would not be the same).

6. Take for example a turbulent viscosity which depends only on the mean velocity and mean velocity gradient (in other words, every Boussinesq-like turbulence model). The turbulent viscosity (i.e. the turbulence) depends only on the current state of the flow. It has no hysterisis and doesn't know where it has been. This is equivalent to assuming that the constants are independent of the initial conditions. The turbulence is treated as a state variable, which is a big conceptual assumption. If your turbulent viscosity doesn't contain a path integral in it, your turbulence model falls into this category. If paths are involved then we need to be really careful about that material derivative in 4&5.

[FluidKo]

Originally Posted by LuckyTran

1. That's the Poisson equation and for an incompressible flow, the pressure satisfies the Poisson equation. The gradient of pressure appears in the momentum equation. You can take the divergence of the momentum equation (and then substitute the continuity equation into it) to derive this Poisson equation.

2. It doesn't matter whether or not you call it fluctuating pressure and fluctuating strain because the correlation between a constant (i.e. a mean) and anything is zero. There's no ambiguity.

3. Pressure can be decomposed into two parts (one slow and one rapid). Both satisfy the Poisson equation. The rapid correlation has a source term that contains the mean velocity gradient. The slow correlation has a source term that does not and the slow correlation is primarily non-linearities that indirectly interact with the mean flow. Rapid/slow is with regard to the rate at which they interact and distort one another (pressure and strain). You don't have to call it rapid and slow if you don't want to because rapid/slow comes from distortion theory.

4&5. Yes (almost always) but (sometimes) no. Overdot is time rate of change which for the purposes of this conversation means material derivative (without going in-depth into why the material and total derivative would not be the same).

6. Take for example a turbulent viscosity which depends only on the mean velocity and mean velocity gradient (in other words, every Boussinesq-like turbulence model). The turbulent viscosity (i.e. the turbulence) depends only on the current state of the flow. It has no hysterisis and doesn't know where it has been. This is equivalent to assuming that the constants are independent of the initial conditions. The turbulence is treated as a state variable, which is a big conceptual assumption. If your turbulent viscosity doesn't contain a path integral in it, your turbulence model falls into this category. If paths are involved then we need to be really careful about that material derivative in 4&5.

I have two questions.

First, in third question, you've told me 'rapid pressure-strain ' interacts with mean velocity gradient directly and 'slow pressure-strain ' interacts with mean velocity gradient indirectly.
But I don't know about that. I think this is related with distortion theory that you've told me. Right?
I want to get recommendation of paper to learn what is distortion theory.

Edit: Oh I've found content related to this from the text book by S.B.Pope.

Second, in sixth question, you've told me path.
But I can't understand the meaning of the path.
In my opinion, I guess path that you're meaning is pathline(trajectory)
because you've talked
'There is no hysterisis in turbulence. There is no function(Doesn't matter with initial condition) to describe accurate turbulent state that varies over each time. It is random.',

If we know all of informations about pathlines for whole particles, then it is possible to consider every situation at every each times.
It means we can find change of turbulence viscosity and any other change about turbulence at every each time along the pathline.(We know history and where turbulence has been)
Do I understand it well?

Thanks

[LuckyTran]

Real eddies and turbulence have some memory of where they've been. If you know the history, you can maybe build models to take that into account. But that is not the way we do this today and that is the fundamental assumption being mentioned. We don't need a complete history of every fluid particle dating back to the Big Bang to solve this. We know that that temporal correlation time-scale of turbulence is small. At the same-time, we know that time-scale is not zero. We only need the history up to this time-scale and maybe a little bit more.



Let's say you are doing RANS/URANS using your favorite turbulence model today). Your mut is at most a function of x,y,z, and u,v,w at the current time t. That is, given that u=u(x,y,z,t), v=v(x,y,z,t), and w=w(x,y,z,t) you have mut=mut(x,y,z,u(x,y,z,t),v(x,y,z,t),w(x,y,z,t)). Only the current time t exists in this dependency (and not any previous time t'). And if only the current t is allowed, then you have assumed that all your turbulence constants depend only on the current t and not the initial conditions. This issue also applies to LES but in a limited scope in the SGS model. A generalized turbulence model would need to know where every eddy has been at last up to t' going back the temporal time-scale. That's what you would need to do to no longer make this fundamental assumption.

[FluidKo]

Originally Posted by LuckyTran

Real eddies and turbulence have some memory of where they've been. If you know the history, you can maybe build models to take that into account. But that is not the way we do this today and that is the fundamental assumption being mentioned. We don't need a complete history of every fluid particle dating back to the Big Bang to solve this. We know that that temporal correlation time-scale of turbulence is small. At the same-time, we know that time-scale is not zero. We only need the history up to this time-scale and maybe a little bit more.



Let's say you are doing RANS/URANS using your favorite turbulence model today). Your mut is at most a function of x,y,z, and u,v,w at the current time t. That is, given that u=u(x,y,z,t), v=v(x,y,z,t), and w=w(x,y,z,t) you have mut=mut(x,y,z,u(x,y,z,t),v(x,y,z,t),w(x,y,z,t)). Only the current time t exists in this dependency (and not any previous time t'). And if only the current t is allowed, then you have assumed that all your turbulence constants depend only on the current t and not the initial conditions. This issue also applies to LES but in a limited scope in the SGS model. A generalized turbulence model would need to know where every eddy has been at last up to t' going back the temporal time-scale. That's what you would need to do to no longer make this fundamental assumption.

Thank you very much for always answering what I don't know.