Low pressure de Laval simulation convergence problem
 

Hey!

I am interested in low pressure and high Knudsen number nozzle flows, trying to study the applicability of continuum approach. At the moment I have been trying to establish an identical case as shown in Chung's article http://arc.aiaa.org/doi/pdf/10.2514/6.1993-727.

There is a small nozzle with throat diameter 2.55mm and with expansion area ratio 66. Flow properties: Substance nitrogen, stagnation temperature 300K, stagnation pressure 474Pa, Re=270 and Knudsen number 2.3e-03.

I am aware that the Knudsen number is relatively high and proper approach should include molecular flow simulations. I am interested to study how and where the breakdown of continuum model arises so exact solution is not the goal at this point. I have already managed to achieve one benchmark for "straight" nozzle flow exiting to vacuum and results were similar as in literature in continuum region, but this was with much higher stagnation pressure ~Mpa and temperature T~2000K.

First I built a 2D-axisymmetric grid which included ~50000 cells in nozzle area with first cell size next to the wall being 0.005 mm (trying to reach y+~1). Exiting area was 20xnozzle exit diameter and hight was 10 times the exit diameter. The whole grid contained ~80000 cells. Maximum skewness was 0.5, max aspect ratio ~800, growing ratio ~1.1.

Boundary conditions:
Operating pressure 0
inlet: pressure inlet 474Mpa, T=300K
axis:axis
nozzle wall: wall
all the other boundaries: pressure outlet , p=0, T=300

Solver: implicit density based
I tried inviscid, laminar and k-e turbulence with courant number 0.01->2 (increasing along the simulation manually) and I didn't achieve convergence. x,y-velocities and turbulence parameters converge ~1e-03 but continuity and energy stays ~1e-01. Converging is really really slow (10000-30000 iteration) and usually while increasing the Courant number in the end the continuity explodes in some point.

I was sure that the problem would have been the grid because continuity is not converging. I created new better grids and the newest one has 400000 cells with skewness 0.25 and max aspect ratio 90. So quality should be ok. Still the problem exists.

The flow field reminds in a way the result I want to achieve but still converging problem is significant.

I am starting to believe that for this case the low stagnation pressure makes it that the solution is not converging.

I am interested to hear all kind of suggestions and ideas.

No successes yet. Simulation is tablewith residuals in 1e-02, but do not go smaller...

You can consider decreasing time-step. Try adaptive time step and also try adaptive mesh keeping the courant number is fixed.

It could help you determine how small your time-step should be.

Yeah I found out that the timestep had to be ~e-08 ->e-06 to get converging results. But then the calculus takes ages. Then I reduced the number of cells in my grid and suddenly I got convergence with steady state solver.

I think that my problem was Fluent's rounding errors, even though I ran with double precision...

Now it converges very well with 1st order accuracy but 2nd order is still problematic...