Is this understanding of turbulence models correct?
 

I’m new to modeling with CFD. I wrote up this description of what a model “sees” when we try to model the flow of a plume of smoke arising from a cigarette. I’d greatly appreciate it if you could please read this, and tell me if there is anything I need to correct. This is for my own understanding. (and for anybody who may ask me about this in the future :-) ). Please be very finicky and pick out any errors either in the set-up or the description.

Consider 3 points in the flow field of the smoke plume of a cigarette:

http://i.min.us/ie1nQ8.jpg

1. Completely laminar flow : Just at the bottom of the plume. Nice, smooth flow.
2. Laminar-turbulent transition : A little distance higher when the flow starts to get disrupted.
3. Turbulent flow : A distance where the flow is completely turbulent.

I have 4 models I want to understand and compare:

1. Steady-state, laminar (i.e. constant viscosity) : This I understand to be simply the Navier-Stokes equations with all the time derivatives set to 0.
2. Transient laminar
3. Steady-state, turbulent: Let me assume a RANS-like model here, i.e one which adds an extra viscosity to the turbulent regions of the flow. Again, all time derivatives are identically set to 0.
4. Transient, turbulent.

Point 1: Completely laminar flow

The SS Laminar model simply solves for a space DE and all model assumptions are pretty much valid. There is good convergence through out.
The SS transient model steps in time, and at each time step it is a space DE with initial conditions. Again, all model assumptions are quite valid, and there is good convergence both in space and in time.
The two turbulent models are quite similar to the corresponding laminar ones, because it turns out that the calculated extra viscosity is very small.

We get similar velocity fields for all 4.

Point 2: Laminar-Turbulent transition

The SS Laminar model’s assumptions are starting to go wrong. It is becoming meaningless to say time derivatives are 0. The velocity field calculated at this stage simply does not make sense, and may lead to divergence.

The SS transient model also has the same problem, but the effects here are different. Time-stepping through the DEs correctly produces vortices and instabilities (real ones), but these instabilities start growing because the constant viscosity assumptions are wrong. The model breaks down and diverges. If it does converge, the results are meaningless.

The extra viscosity in the turbulent models start kicking in. In the SS turbulent model, the extra viscosity term helps to converge, and the value of the velocities in the field are in some sense time-averaged (I’m tempted to say ensemble-averaged, but I don’t see an ensemble here. What is the ensemble over?) because that is what the eqns with zero time derivatives mean.

The transient turbulent model is still chugging on, time stepping. The velocity fields at every time instant are “real”, except for the “smaller scale” disturbances that are clubbed into an extra viscosity. For example, if a parcel of fluid is rotating about its center and going up, the velocity field may show the parcel only going up. The rotation is happening at a smaller scale, and is not resolved. But the effects of the rotation are to, say, increase the heat transfer, and this is reflected by the increased extra viscosity.

Point 3: Fully turbulent flow

The SS laminar and transient laminar models are completely useless, and have most likely long diverged. They are more dangerous if they have converged, because now the solution is meaningless, even though it may have some plausible features.

The SS turbulent model is losing out on a lot of detail, and none of the transient features like vortices, eddies, etc are seen. But the general flow is still broadly OK, if you look at gross parameters like heat transfer, average velocities, etc.

The transinet turbulent model does show the transient features. It is correct up to the level where the flow effects are clubbed into a viscosity. The velocity fields aren’t exact, but when the flow is taken as a whole they give a “low pass filtered” version of the field.

A couple of things:

- Cigarette plumes will be driven by temperature (buoyancy). The constant thermodynamic properties assumption (viscosity, cp) would need to be checked.

- Mesh and time resolution has an impact: if you solve for the transient laminar equations on a very fine grid, you might be able to resolve turbulent structure (think intrinsic LES and pseudo-DNS). The numerical algorithms will have a role to play.

- There is an aspect that your interpretation of transient turbulent model that is incorrect (on the basis of RANS turbulence model). The RANS turbulence model aims to model the whole turbulence spectra, not the <<"smaller scale" disturbance>>. The turbulence structure that the model presents are the coherent turbulent ones, which should correspond to natural instabilities frequency and should have higher energy than the modeled turbulence. Your interpretation would be correct for a LES model.

Thank you Julien for your patience and help :)

1. Yes, I see what you mean. I was implicitly assuming a Boussinesq-approximation like case, but I understand that in a general case all material properties may vary.

2. Yes, this was something I missed. 'Numerical viscosity' would play an important role (anisotropically at that!) in messing up the solution. I am intrigued at your suggestion of looking at a fine grid transient constant viscosity method as a pseudo DNS. I will read up further on this.

3. This is critical: What would happen if the scale of the grid wasn't small enough to represent the higher frequencies of the spectrum? Is it that the field will be simply "numerically chopped" ?

If that is so, what exactly changes when we change the parameters of a RANS model? Suppose we have a very fine grid, so that numerical filtering of the field is not of concern. We change the values of a k and epsilon in such a model. What do we expect will change? (Since the full spectrum is modeled anyway)

Once again, thanks for your patience!!

I will try to clarify a couple of points:

1) DNS vs transient laminar:
- The equations are the same (the Navier-Stokes equations without averaging);
- DNS would need to be 3D, whereas 2D laminar is possible;
- DNS would minimise numerical diffusion using high order time and space discretisation;
- DNS would use a fine mesh to resolve turbulent structure up to the Kolmogorov scales.

2) RANS: modeled vs resolved turbulence
In very short, RANS models aim to resolve the turbulence. In theory, mesh resolution should not impact your solution - at least in a steady-state fashion - and definitively not the turbulent energy distribution.

Playing with the turbulence model coefficients are going to affect the behavior of the model:

Take the k-epsilon model (http://www.cfd-online.com/Wiki/Stand...psilon_model):
- Increase C_mu will increase turbulent dispersion (artificially?);
- Changing C_1epsilon and C_2epsilon will affect the balance of turbulence dissipation (which would be equivalent to changing the shape of the spectrum IMO).

Usually, one would change the coefficient to match the behavior of the turbulence model to a specific configuration (which has been investigated experimentally). This may relates to different turbulence behavior: mixing layer versus jet for example, or it may not.

Hope it helps more than it confuses.

In practice, I would personally recommends to use the default turbulence coefficients and try to get convergence (tough enough). And do occasional mesh sensitivity analysis.