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Which turbulence model is suitable for resolving incompressible buoyancy-driven flow of a fluid within an cylindrical ampoule?

I prefer turbulence model which is sufficiently simple so that fully coupled (UFL) variational form of Navier-Stokes-Fourier is changed in way that each added term

  • is treated by Newton solver, or

  • is precomputed prior to assembling tensors at every Newton iteration

so that no additional fixed-point iteration scheme is needed.

You can answer immediately or read description of problem below if useful.

Timedependent solution

This is coupled problem of flow of melt in the ampoule, flow of air in the furnace (outside of the ampoule) and heat conduction in rest parts of the system (including crystal on bottom of the ampoule).

For purposes of this question is relevant flow of the melt in the ampoule which is similiar to buoyancy-driven cavity (especially in top part) but

  • problem is cylindrically symmetric, symmetry axis is on the left

  • boundary conditions for temperature are more complicated than in classical benchmark; furnace wall (right boundary) has non-monotonic temperature profile (see image below - note that interior of the ampoule occupies cca $0.3<x<0.4$, $x$ being vertical coordinate) - increasing with altitude on most of the wall but decreasing near top; top and bottom boundaries of fluid have temperature continuous with outter wall (it is coupled to the rest of the system; bottom boundary is not planar)

Temperature profile

  • almost all material coefficients are temperature dependent

Flow in the ampoule resembles buoyancy-driven cavity in top part where temperature profile induces unstable stratification of fluid. This cavity-like flow is bounded by stably-stratified fluid from below. This was achieved with two orders of magnitude higher viscosity than target one and is non-stationary solution.

It was tested that stationary solution with implicit Euler in time and Newton solver accounting for non-linearity can be computed with three orders higher viscosity than target one. For only two orders higher viscosity stationary solution seems to not beiing stable but time-marching with Crank-Nicolson scheme with small timesteps is possible. I'm currently going to switch to implicit Euler timesteping, lower timestep more, try adding SUPG/PSPG stabilization and check how small viscosity is manageable. But I guess that three orders lower viscosity than that enabling stationary solution will induce turbulence.

For the target viscosity Grashof anfd Rayleigh numbers are $$ \mathrm{Gr = 3.6E8},$$ $$ \mathrm{Ra = 1.1E9},$$ taking whole length of ampoule as chracteristic length and $850-775 = 75\;\mathrm{K}$ as temperature difference scale; $775\;\mathrm{K}$ is temperature at bottom of the ampoule and $850\;\mathrm{K}$ is peak temperature of temperature profile. Note that this numbers could few orders of magnitude lower taking into account that:

  • one could take only depth of unstably-stratified region as a characteristic lenght

  • one could take smaller temperature difference corresponding to unstably-stratified region

  • peak furnace temperature $850\;\mathrm{K}$ is not reached inside the ampoule

Jan Blechta
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  • Do you have a feel for the size of the Rayleigh number for the real problem? That should tell you pretty quickly whether it would be turbulent. – Bill Barth May 14 '13 at 21:39
  • I've posted some buoyancy-Navier-Stokes results earlier today in the g+ FEniCS community https://plus.google.com/110475963061639463862/posts/5NWw8dzrtst. Viscosity is high, Re=1.0e7, hence you see a bunch of turbulence. – Nico Schlömer May 14 '13 at 21:53
  • @BillBarth: Gr = 3.6E8, Ra = 1.1E9 but note that they could one order of magnitude lower taking for characteristic length only depth of unstably stratified region. Also temperature difference is not so large in this region so numbers can finally be two or three orders of magnitude lower. – Jan Blechta May 14 '13 at 22:19
  • @Nico: Nice. I would say it is a light turbulence, is it? And you resolved it with fine grid and SUPG, did you? That casts doubt on answer http://scicomp.stackexchange.com/a/7159/4254 – Jan Blechta May 14 '13 at 22:39
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    @Nico 1. Viscosity would be low for high Re. 2. If your parameters are set at $Re=10^7$, then that solution is drowning in numerical viscosity (imperfectly---see the artifacts). – Jed Brown May 15 '13 at 01:44
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    2D turbulence doesn't make a lot of sense to compute in the first place. It has different phenomenology to 3D turbulence, and may lead you to the wrong conclusions. It sounds like you have a pretty large range of parameters of interest. At $Ra \sim 10^6$ or $10^7$, I wouldn't expect much in the way of turbulence, so you should be able to get away with a fine mesh and/or some sort of stabilization. – Bill Barth May 15 '13 at 03:12
  • @BillBarth: I don't have large range of parameters. $Gr$ and $Ra$ are uncertain because it is not obvious which length and temperature scale applies to problem. – Jan Blechta May 15 '13 at 10:27
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    @JanBlechta. Maybe you could post a labeled diagram of the geometry? – Bill Barth May 15 '13 at 12:15
  • Assuming your parameters would give you turbulence in the 3D case for the scenario you describe, adding a turbulence model for the 2D case makes sense if you decide that what you're simulating is an ensemble average of the behavior for a 3D problem taken along the symmetry present in/out of the screen. – Rhys Ulerich May 15 '13 at 14:20
  • @RhysUlerich: Of course, that's same like for 2D Cartesian (i.e. translational symmetry) simulation of turbulence. This is also average along direction perpendicular to computational domain. – Jan Blechta May 15 '13 at 18:55
  • @BillBarth: Figures added, problem explained more deeply. – Jan Blechta May 15 '13 at 22:01
  • @JanBlechta: I think you mean Grashof number, not Grassmann number. – Geoff Oxberry May 15 '13 at 22:29
  • @GeoffOxberry: Yes, funny mistake. – Jan Blechta May 15 '13 at 22:35

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