def diffusive_flux(state, grad_cv):
r"""Compute the species diffusive flux vector, ($\mathbf{J}_{\alpha}$).
The species diffusive flux is defined by:
.. math::
\mathbf{J}_{\alpha} = -\rho{d}_{(\alpha)}\nabla{Y_{\alpha}}~~
(\mathtt{no~implied~sum}),
with species diffusivities ${d}_{\alpha}$, and species mass
fractions ${Y}_{\alpha}$.
Parameters
----------
state: :class:`~mirgecom.gas_model.FluidState`
Full fluid conserved and thermal state
grad_cv: :class:`~mirgecom.fluid.ConservedVars`
Gradient of the fluid state
Returns
-------
numpy.ndarray
The species diffusive flux vector, $\mathbf{J}_{\alpha}$
"""
return _compute_diffusive_flux(
state.mass_density, state.species_diffusivity,
species_mass_fraction_gradient(state.cv, grad_cv))
def diffusive_flux(discr, eos, cv, grad_cv):
r"""Compute the species diffusive flux vector, ($\mathbf{J}_{\alpha}$).
The species diffusive flux is defined by:
.. math::
\mathbf{J}_{\alpha} = -\rho{d}_{(\alpha)}\nabla{Y_{\alpha}}~~
(\mathtt{no~implied~sum}),
with species diffusivities ${d}_{\alpha}$, and species mass
fractions ${Y}_{\alpha}$.
Parameters
----------
discr: :class:`grudge.eager.EagerDGDiscretization`
The discretization to use
eos: :class:`~mirgecom.eos.GasEOS`
A gas equation of state with a non-empty
:class:`~mirgecom.transport.TransportModel`
cv: :class:`~mirgecom.fluid.ConservedVars`
Fluid state
grad_cv: :class:`~mirgecom.fluid.ConservedVars`
Gradient of the fluid state
Returns
-------
numpy.ndarray
The species diffusive flux vector, $\mathbf{J}_{\alpha}$
"""
transport = eos.transport_model()
grad_y = species_mass_fraction_gradient(discr, cv, grad_cv)
d = transport.species_diffusivity(eos, cv)
return -cv.mass * d.reshape(-1, 1) * grad_y
def test_species_mass_gradient(actx_factory, dim):
"""Test gradY structure and values against exact."""
actx = actx_factory()
nel_1d = 4
from meshmode.mesh.generation import generate_regular_rect_mesh
mesh = generate_regular_rect_mesh(
a=(1.0,) * dim, b=(2.0,) * dim, nelements_per_axis=(nel_1d,) * dim
)
order = 1
discr = EagerDGDiscretization(actx, mesh, order=order)
nodes = thaw(actx, discr.nodes())
zeros = discr.zeros(actx)
ones = zeros + 1
nspecies = 2*dim
mass = 2*ones # make mass != 1
energy = zeros + 2.5
velocity = make_obj_array([ones for _ in range(dim)])
mom = mass * velocity
# assemble y so that each one has simple, but unique grad components
y = make_obj_array([ones for _ in range(nspecies)])
for idim in range(dim):
ispec = 2*idim
y[ispec] = ispec*(idim*dim+1)*sum([(iidim+1)*nodes[iidim]
for iidim in range(dim)])
y[ispec+1] = -y[ispec]
species_mass = mass*y
cv = make_conserved(dim, mass=mass, energy=energy, momentum=mom,
species_mass=species_mass)
from grudge.op import local_grad
grad_cv = local_grad(discr, cv)
from mirgecom.fluid import species_mass_fraction_gradient
grad_y = species_mass_fraction_gradient(cv, grad_cv)
assert grad_y.shape == (nspecies, dim)
from meshmode.dof_array import DOFArray
assert type(grad_y[0, 0]) == DOFArray
def inf_norm(x):
return actx.to_numpy(discr.norm(x, np.inf))
tol = 1e-11
for idim in range(dim):
ispec = 2*idim
exact_grad = np.array([(ispec*(idim*dim+1))*(iidim+1)
for iidim in range(dim)])
assert inf_norm(grad_y[ispec] - exact_grad) < tol
assert inf_norm(grad_y[ispec+1] + exact_grad) < tol