# Simple turbulence model appropriate for buoyancy-driven cavity like problem

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.

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)

• 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

• 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 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
• @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 1:44
• 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 3:12
• @JanBlechta. Maybe you could post a labeled diagram of the geometry? – Bill Barth May 15 '13 at 12:15