Surface tension is sensitive to temperature or concentrate. Unbalance surface tension due to temperature or concentrate drives the flow. This flow become turbulence for large enough temperature or concentrate gradient alone free surface.

Question:

How do we choose a suitable turbulence model for this flow ?

Thanks in advance.

What is the relative size of the important length scales? (i.e. the size of the solution region, the size of the motion in the surface, the size of the largest eddies and the size of the smallest).

Is the problem evolving in time or steady?

Are you fully resolving all the important scales in the the free surface or representing it an approximate manner? Is this what you want the "turbulence" model to represent?

Extrapolating based on "large enough" suggests the fluid is only just entering the turbulent regime. The only reliable approach here is to solve the governing equations directly (DNS in popular jargon) which does not use an explicit turbulence model.

I realize the above has not answered your question but I hope it may be of some help in starting to answer it.

Thanks your comments and suggestions.

The model is that a cylindrical liquid bridge of length L held by surface tension forces between two parallel, coaxial solid rods of equal radii R. A temperature difference DT is imposed over the liquid bridge at the disks.

L=100 mm, R=20 mm, DT=100 k, Prandtl number Pr=74 ...

According to experiment and numerical results in literatures and also our recent numerical results, we can sure this flow with above given condition is under turbulent regime. (Before entering turbulent regime for smaller DT, periodic oscillation is fund).

The size of the largest eddies is about L, but we do not know the smallest eddy size now.

This specified flow is driven by surface tension and has strong curvature of free surface. This simulation is time consuming according to our experience on simulation under periodic oscillory regime, it seems impossible to apply DNS method. For this reason, your suggestions and comments on choice of turbulent model will be helpful.

It looks like an interesting problem to study.

If you have a simulation code for the time varying laminar regime then this is also the DNS code. What prevents you using it at the higher flows? Aliasing? If so, there are numerical approaches and "turbulence modelling" you can follow.

What do you want to achieve with the numerical simulation? For example, an accurate understanding of the physical processes involved or, perhaps, a quick parametric study?

If it is the former, I see no alternative to directly simulating the physical processes given the flow has only just gone turbulent. There is no accurate Reynolds averaged general turbulence model for very low Reynolds number turbulent motion.

If it is the latter, then I would suggest considering almost anything that claims to be a low Reynolds number turbulence model that is simple (i.e. 0, 1 or 2 equation model) and defendable in applying it in the vicinity of a moving free surface (it is very unlikely that the empiricism in the model will have been "setup" for this flow regime). If tweaking the model has relatively little influence on the predicted flowfield then you might have a believable set of results. However, if tweaking the turbulence model significantly changes the predicted flowfield then you can be sure you are not reliably simulating the dominant physical processes for your problem and should view the results accordingly.

It is my great happiness to read your suggestions. Thanks.

This is the first time for me to consider flow involving turbulence, therefore please forgive my poor knowledge on it.

I can not understand the means of "tweaking the turbulence model" in your suggestion. Could you please explain this way to check the validation of turbulent model in more detail.

Your comments are that up to now, there is not any preferred turbulence mode for the flow driven by surface tension, is there ?

By "tweaking the turbulence model" I was not suggesting an attempt to validate the turbulence model but to determine if the physics of your problem was sensitive to the turbulence model. I think we can be confident that in the low Reynolds number turbulent regime a Reynolds averaged based turbulence model (i.e. a computationally cheap one) is not going to be reliable except, possibly, for the narrow range of flows it was "tuned up" to reproduce. The best we can hope for is that the modelled turbulent quantities are not greatly influencing the evolution of your flow (my ignorance of free surface flows is profuse so that one is down to you or someone else). If changing the value of the empirical constants (they usually have a range of sensible values but normally have to be adjusted altogether as a set) significantly alters the results you can be sure you are not reliably simulating the physics of your problem and must try another approach. If the predictions are relatively insensitive then you may have a believable simulation subject to the other modelling assumptions.

The bald answer to the question about a "preferred model" is that I do not know. There might be a model that everyone in the field has adopted. A literature survey would reveal what others had used in related fields, for example, wave studies for generating electrical power. However, adopting models based on what others had used rather than based on an understanding of the dominant physical processes in the problem under study is not something I would recommend for an academic study of any duration. It is, of course, defendable as a first go or if time is very pressing. I intend no criticism since I know almost nothing about you project.

The relevant parameter in your experiment is the Marangoni number Ma, which is proportional to the temperature difference and to the surface tension coefficient, and inversely prop. to the dynamic viscosity but you should know that. If the Marangoni number exceeds a certain value, it is well known that the flow becomes unstationnary (with dependance to the Rayleigh number, to the Prandtl number, and to the geometry). I guess that even with the rather high viscosity of your oil, your system reaches some turbulent motion since the surface tension coefficient is high enough. If you want to model this flow, the simplest way is in 2D to take a classical k-epsilon model with boundary conditions of no energy through the free surface, but still considering the classical Marangoni condition for the tangential stress. The surface will be considered as flat since the deviations are small. This will probably produce some good results, but only if the flow is really fully turbulent (which is not obvious in your case). Why this modelization? because nobody (to my knowledge) knows exactly how to model such turbulent flow near the free surface. Turbulence of interface is itself a research topic. see Zebib et al, Physics of fluids 28 (85) Favre et al, Physics of fluids 9 (97) Favre Eric, thesis of the Institut National Polytechnique de Grenoble (1997, in french). May I ask you why you study those kinds of flow? Sincerely, Eric.

Dear Dr. Favre,

Thanks your suggestions. Our study about thermocapillary flow by numerical simulating is financial supported by IHI company in Japan, and they plan to do this experiment. The computational conditions is also supported by them.

This thermocapillary is axis-symmery 2D flow for Ma<Mac, and it becomes 3D periodic and qusi-periodic flow (Pr>1, Pr Prandtl number), and finally turbulent flow with increasing of Ma(Marangoni number). We have simulated the flow under 3D periodic oscillatory regime. We think the flow with this giving conditions is turbulent flow but not fully turbulence. It means the some 3D periodic oscillatory characters are probablely still retained. Whether do we have to study this flow by using 3D model in this situation ?