Help with tau constraints

[qucchia]

Hi,

I am new to Dedalus and spectral methods and don't quite get how to impose boundary conditions using Tau methods. For example, for a 2D viscous fluid with equations:

I want to impose that u is zero at the borders in the x axis.
Implementation so far:

problem = d3.IVP([v, rho], namespace=locals())

#Substitutions

dx = lambda A: d3.Differentiate(A, coords['x'])

dy = lambda A: d3.Differentiate(A, coords['y'])

lap =lambda A: d3.Laplacian(A)

idn = dist.TensorField((coords, coords), name='idn')

idn['g'][0,0] = 1

idn['g'][0,1] = 0

idn['g'][1,0] = 0

idn['g'][1,1] = 1

P = (u1**2)*(rho-rho0)

grad_v = d3.Gradient(v)

grad_vt = d3.TransposeComponents(grad_v)

Dn = 1/2 * (grad_v + grad_vt)

#Equations

problem.add_equation('dt(rho)  = - v@grad(rho) -rho*div(v) ')

problem.add_equation('dt(v) = -v@grad(v) -(1/rho)*div(rho*v*v + idn*P ) + 2*mu/rho*div(Dn)')

What I want to add:

ex, ey = coords.unit_vector_fields(dist)

problem.add_equation("dt(v(x='left')) @ ey = 0")

problem.add_equation("dt(v(x='right')) @ ey = 0")

I've tried imitating examples by using a lift_basis and putting them in the equations but

it hasn't worked out (I get Factor is exactly singular)

Any advice / help is greatly appreciated!

[Daniel Lecoanet]

Hi,

You need to write your equations in a way such that the diffusion term is linear and on the LHS of the equation. This can be done by changing the definition of your variables, e.g., here: https://ui.adsabs.harvard.edu/abs/2014ApJ...797...94L. There was a more recent example of this on this users list a few weeks ago.

Daniel

On Dec 7, 2024, at 12:17 PM, qucchia <yij...@gmail.com> wrote:

Hi,

I am new to Dedalus and spectral methods and don't quite get how to impose boundary conditions using Tau methods. For example, for a 2D viscous fluid with equations:

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