[Eifoehn4]

Is a two-dimensional simulation able to reproduce the law of wall, especially the log-law region?

I am interested in the comparison of different models:

1. 2D RANS, with wall-function
2. 2D RANS, wall-resolved
3. High resolved 2D simulation without turbulence model (either solved transient or steady)

How do commercial tools, e.g. Fluent, handle this? Thank you for the discussion.

[LuckyTran]

1 & 2 There's 0 difference between 2D RANS and 3D RANS
3 It gives a law of the wall, but a 2D one and not a 3D one. So yes and simultaneously, no.

[Eifoehn4]

Thank you LuckyTran. This was also my guess. This means (1), (2) and (3) are comparable in 3D however not in 2D.

Or in other words: A high resolved 2D simulation without turbulence model will not reproduce the log-law region.

[LuckyTran]

It produces a law of the wall but with a different Karman constant.

[arjun]

I do not think commercial codes even need to worry about it. 2D equation are special approximation on 3d equations.

The point is that because in our nature we have 3d world and navier stokes are written for it. This is what commecial codes provide for.

Originally Posted by Eifoehn4

Is a two-dimensional simulation able to reproduce the law of wall, especially the log-law region.

I am interested in the comparison of different models:

1. 2D RANS, with wall-function
2. 2D RANS, wall-resolved
3. High resolved 2D simulation without turbulence model (either solved transient or steady)

How do commercial tools, e.g. Fluent, handle this? Thank you for the discussion.

[Eifoehn4]

Originally Posted by arjun

I do not think commercial codes even need to worry about it. 2D equation are special approximation on 3d equations.

The point is that because in our nature we have 3d world and navier stokes are written for it. This is what commecial codes provide for.

Yes that's obviously true. There is not need to derive a pseudo-turbulence model to recover the exact solution for the 2D equations.

Originally Posted by LuckyTran

It produces a law of the wall but with a different Karman constant.

Do you have a reference for the different value in 2D? Currently I would expect a viscous and buffer layer, only.

[FMDenaro]

A simple example is a flow around a wing. If wing is finite, you have no homogeneous direction, the flow remains 3D even in statistical sense. Of course, also the profile of velocities will change.

But if you assume a homogeneous wing, that is a 3D wing extending periodically in spanwise direction, your 3D solution is only a 2D solution repeated in that direction.
The key is when you can afford the physical model with a 2D RANS ...

[andy_]

Originally Posted by Eifoehn4

Is a two-dimensional simulation able to reproduce the law of wall, especially the log-law region.

Turbulence cannot exist in 2D because the energy cascade due to vortex stretching requires 3 dimensions. What happens in 2D is the vortices generated by flow instabilities keep growing in size given they cannot roll up in the 3rd dimension and become smaller and more intense until dissipated by viscosity. Such behaviour can be seen in temperature inversions and jupiter's red spot. This 2D unsteady motion is nothing like turbulence and cannot be nudged towards turbulence because they are close to an inverse of each w.r.t. to the behaviour of the time and length scales.

If you project a 2D geometry to form a 3D one with periodic boundary conditions on the ends then you will be able to simulate a law of the wall behaviour if the periodic conditions are far enough apart for the motion on the planes to not influence each other through the solution domain. This will depend on the length scale of the largest turbulent motion which tends to vary from flow to flow.

[Eifoehn4]

Originally Posted by andy_

Turbulence cannot exist in 2D because the energy cascade due to vortex stretching requires 3 dimensions. What happens in 2D is the vortices generated by flow instabilities keep growing in size given they cannot roll up in the 3rd dimension and become smaller and more intense until dissipated by viscosity. Such behaviour can be seen in temperature inversions and jupiter's red spot. This 2D unsteady motion is nothing like turbulence and cannot be nudged towards turbulence because they are close to an inverse of each w.r.t. to the behaviour of the time and length scales.

If you project a 2D geometry to form a 3D one with periodic boundary conditions on the ends then you will be able to simulate a law of the wall behaviour if the periodic conditions are far enough apart for the motion on the planes to not influence each other through the solution domain. This will depend on the length scale of the largest turbulent motion which tends to vary from flow to flow.

It is quite clear for me that turbulence cannot exist in 2D due to the missing vortex stretching effect in z-direction.

My question is actually quite simple: You have a highly resolved 2D simulation of a flat plate without turbulence model and without wall function (steady or unsteady using the 2D equations). You take a line plot through the boundary.

Is a logarithmic region visible? Yes or no?

Edit: I meant "cannot exist in 2D"

[JBeilke]

Why don't you try it and tell us, what you find?

[FMDenaro]

Originally Posted by Eifoehn4

It is quite clear for me that turbulence cannot exist in 2D due to the missing vortex stretching effect in z-direction.

My question is actually quite simple: You have a highly resolved 2D simulation of a flat plate without turbulence model and without wall function (steady or unsteady using the 2D equations). You take a line plot through the boundary.

Is a logarithmic region visible? Yes or no?

Edit: I meant "cannot exist in 2D"

are you thinking a sort of DNS or LES in 2D? or a high resolved RANS?
The problem you stated has zero mean spanwise flow but in LES/DNS your 2D solution would be unsteady, even without the vortex stretching. Then you have to do the ensemble averaging of data. If the effect of the vortex stretching is disregardable in average, you could get some acceptable approximation in the averaged. Note that I am thinking to a region of fully developed turbulence. If you think to develop turbulence from a laminar BL, the transition regions is not well represented.

But such a case is well suited and would be resolved very well by a 2D RANS.

[Eifoehn4]

Originally Posted by JBeilke

Why don't you try it and tell us, what you find?

I did a test yesterday (a fast one) and could not reproduce a logarithmic region.

[Eifoehn4]

Originally Posted by FMDenaro

are you thinking a sort of DNS or LES in 2D? or a high resolved RANS?
The problem you stated has zero mean spanwise flow but in LES/DNS your 2D solution would be unsteady, even without the vortex stretching. Then you have to do the ensemble averaging of data. If the effect of the vortex stretching is disregardable in average, you could get some acceptable approximation in the averaged. Note that I am thinking to a region of fully developed turbulence. If you think to develop turbulence from a laminar BL, the transition regions is not well represented.

But such a case is well suited and would be resolved very well by a 2D RANS.

I think that's the actual confusion. My solution was time-resolved, however never unsteady. Would an unsteady solution in 2D in the limit of and ever occur? I think not, or?

[FMDenaro]

Originally Posted by Eifoehn4

I think that's the actual confusion. My solution was time-resolved, however never unsteady. Would an unsteady solution in 2D in the limit of and ever occur? I think not, or?

If you have a resolved grid, a high accurate method, the 2d solution will develop instability and produces a time-dependent flow even in 2D.
Be sure you grid is really so fine, ensure that the lenght is chosen to be long enough and use a scheme without numerical dissipation.

[LuckyTran]

I bet you have done a transient laminar calc and that's why you only see a viscous region, because that's exactly what a laminar flow is, it's viscous everywhere.

[sbaffini]

Don't take me wrong, you all know what I'm going to write, but I feel like a summary is needed.

A 2D unsteady Navier-Stokes simulation with no additional term (e.g. turbulence model) is laminar. Laminar in the special way of a system that can only be laminar. Laminar meaning that none of the defining features of turbulence are present.

This 2D simulation, as such, can't reproduce the log-law as the underlying dynamics, <u'v'>, is different between the two.

Analogously, using a proper model for <u'v'> in a steady 2D simulation is sufficient to get a log law, as any 2D RANS test will show.

[Eifoehn4]

Originally Posted by sbaffini

Don't take me wrong, you all know what I'm going to write, but I feel like a summary is needed.

A 2D unsteady Navier-Stokes simulation with no additional term (e.g. turbulence model) is laminar. Laminar in the special way of a system that can only be laminar. Laminar meaning that none of the defining features of turbulence are present.

This 2D simulation, as such, can't reproduce the log-law as the underlying dynamics, <u'v'>, is different between the two.

Analogously, using a proper model for <u'v'> in a steady 2D simulation is sufficient to get a log law, as any 2D RANS test will show.

Thank's for your reply @sbaffini. Your explanations are in line with my current understanding. However, if I have some time and enough resources I will rerun the case with my high order spectral code. But I doubt that the transient simulation will ever get unsteady.

[andy_]

Originally Posted by Eifoehn4

Your explanations are in line with my current understanding. However, if I have some time and enough resources I will rerun the case with my high order spectral code. But I doubt that the transient simulation will ever get unsteady.

Performing that simulation looks like a good idea to me if you doubt flow becomes unstable at high Reynolds numbers. Laminar 2D flow instabilities are often what initiates turbulence for example vortex streets, developing free or boundary shear layers,... What happens when motion in the 3rd direction is suppressed like in temperature inversions or strong gravity fields is interesting but of limited practical interest due to rarity.

[FMDenaro]

Originally Posted by Eifoehn4

Thank's for your reply @sbaffini. Your explanations are in line with my current understanding. However, if I have some time and enough resources I will rerun the case with my high order spectral code. But I doubt that the transient simulation will ever get unsteady.

How do you define the extension of your computational domain? Are you sure you extended it at least up to Re_x=O(10^5) and the BL is resolved (apart the region near x=0)?

[LuckyTran]

It's not clear to me what steps were taken to even ensure that the 2D simulation is a proper DNS. Btw, people have done 2D DNS before. I want to point out that even a 3D high resolution mesh can produce a fully laminar solution if you do not take all the proper steps to include perturbations in the initial condition and the boundary conditions. The challenges you are facing are not caused by "2D turbulence"