2D Rayleigh-Benard

[Jason Lee]

Dear all,

I'm still working on RB convection these days.
A question here is that I found even though the Rayleigh number is 5000, the water flow is static.
I post the two temperature perturbation maps at t = 500 and 1157.
The no-slip boundary condition is used.
This is kind of weird since the critical Rayleigh number should be around 1700 according to these curves above.

So I would like to ask that whether you guys have tested this before?

My setup is:

# Parameters
Lx, Lz = (2.*math.sqrt(2), 1.)
Prandtl = 6.8
Rayleigh = 5e3

# Create bases and domain
x_basis = de.Fourier('x', 128, interval=(0, Lx))
z_basis = de.Chebyshev('z', 128, interval=(0, Lz))
domain = de.Domain([x_basis, z_basis], grid_dtype=np.float64)

# 2D Boussinesq hydrodynamics
problem = de.IVP(domain, variables=['p','b','u','w','bz','uz','wz','vort'])
problem.parameters['P'] = (Rayleigh * Prandtl)**(-1/2)
problem.parameters['R'] = (Rayleigh / Prandtl)**(-1/2)
problem.parameters['F'] = F = 1
problem.add_equation("dx(u) + wz = 0")
problem.add_equation("dt(b) - P*(dx(dx(b)) + dz(bz)) - w         = -(u*dx(b) + w*bz)")
problem.add_equation("dt(u) - R*(dx(dx(u)) + dz(uz)) + dx(p)     = -(u*dx(u) + w*uz)")
problem.add_equation("dt(w) - R*(dx(dx(w)) + dz(wz)) + dz(p) - b = -(u*dx(w) + w*wz)")
problem.add_equation("bz - dz(b) = 0")
problem.add_equation("uz - dz(u) = 0")
problem.add_equation("wz - dz(w) = 0")
problem.add_equation("uz - dx(w) + vort = 0")
problem.add_bc("left(b) = 0")
problem.add_bc("left(u) = 0")
problem.add_bc("left(w) = 0")
problem.add_bc("right(b) = 0")
problem.add_bc("right(u) = 0")
problem.add_bc("right(w) = 0", condition="(nx != 0)")
problem.add_bc("integ(p, 'z') = 0", condition="(nx == 0)")

# Build solver
solver = problem.build_solver(de.timesteppers.RK443)
logger.info('Solver built')

# Initial conditions
x = domain.grid(0)
z = domain.grid(1)
b = solver.state['b']
bz = solver.state['bz']
u = solver.state['u']
w = solver.state['w']
vort = solver.state['vort']

# Random perturbations, initialized globally for same results in parallel
gshape = domain.dist.grid_layout.global_shape(scales=1)
slices = domain.dist.grid_layout.slices(scales=1)
rand = np.random.RandomState(seed=42)
noise = rand.standard_normal(gshape)[slices]

# Linear background + perturbations damped at walls
#zb, zt = z_basis.interval
bpert = noise
#bpert = rd.array_gen_from_file('tmpr_init.dat')
b['g'] = bpert
u['g'] = rd.array_gen_from_file('uvel_init.dat')
w['g'] = rd.array_gen_from_file('wvel_init.dat')
# b.differentiate('z', out=bz)

# Integration parameters
solver.stop_sim_time = 2000
solver.stop_wall_time = 20000 * 60.
solver.stop_iteration = np.inf

# CFL
CFL = flow_tools.CFL(solver, initial_dt=2.5e-3, cadence=5, safety=0.1,
                     max_change=1.5, min_change=0.5, max_dt=0.0025)
CFL.add_velocities(('u', 'w'))

Thank you guys

Regards

Jason

[Jason Lee]

Hey guys,

I'm really confused. Hope someone could help.

Best

Jason

[Daniel Lecoanet]

Hi Jason,

You might be interested in looking at the following paper where we calculated 2D convective steady states for Rayleigh numbers up to 4x10^5:

http://adsabs.harvard.edu/abs/2014ApJ...797...94L

So it is completely normal to find steady states at Rayleigh numbers substantially larger than the critical Rayleigh number.  These steady states are often unstable to 3D perturbations.

Daniel

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[Ben Brown]

Jason,

     At moderately large aspect ratios (e.g. A=4), we've been seeing stable flywheel states establish even at very high Ra (10^8 and higher) in 2-D Rayleigh-Benard.  They're very robust.

--Ben

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[Jason Lee]

Thank you Daniel! I will check it out.

[Jason Lee]

Thanks Ben. That would be interesting.

[Jason Lee]

Hi, Daniel,

There must be some misleading. The problem in my post is that the water is static which means there is almost no motion in the flow rather than the steady state. Actually that is the thing confusing me.
Hope you guys know why.

Thanks

Jason

[Keaton Burns]

Hi Jason,

Can you send along all the scripts and initial condition data files you’re using, so we can test it out locally?

Thanks,

-Keaton

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[Jason Lee]

Hi Keaton,

Sure.
Please note that the python script is quite verbose after I added some output lines and it cannot be parallel computed since I didn't use related syntax.

Thank you.

Jason

[Keaton Burns]

Hi Jason,

I ran it until about t=50 on my laptop and made the attached video using the dedalus analysis hdf5 outputs.  From this it looks like the instability is saturating into steady rolls.  Are you seeing actually static (zero) velocities in your text-based outputs?  If so, maybe there’s an issue with how you’re writing out to the text files?

-Keaton

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