def viscous_stress_tensor(state, grad_cv):
r"""Compute the viscous stress tensor.
The viscous stress tensor $\tau$ is defined by:
.. math::
\mathbf{\tau} = \mu\left(\nabla{\mathbf{v}}
+\left(\nabla{\mathbf{v}}\right)^T\right) + (\mu_B - \frac{2\mu}{3})
\left(\nabla\cdot\mathbf{v}\right)
Parameters
----------
state: :class:`~mirgecom.gas_model.FluidState`
Full conserved and thermal state of fluid
grad_cv: :class:`~mirgecom.fluid.ConservedVars`
Gradient of the fluid state
Returns
-------
numpy.ndarray
The viscous stress tensor
"""
return _compute_viscous_stress_tensor(dim=state.dim,
mu=state.viscosity,
mu_b=state.bulk_viscosity,
grad_v=velocity_gradient(
state.cv, grad_cv))
def test_velocity_gradient_eoc(actx_factory, dim):
"""Test that the velocity gradient converges at the proper rate."""
from mirgecom.fluid import velocity_gradient
actx = actx_factory()
order = 3
from pytools.convergence import EOCRecorder
eoc = EOCRecorder()
nel_1d_0 = 4
for hn1 in [1, 2, 3, 4]:
nel_1d = hn1 * nel_1d_0
h = 1/nel_1d
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
)
discr = EagerDGDiscretization(actx, mesh, order=order)
nodes = thaw(actx, discr.nodes())
zeros = discr.zeros(actx)
energy = zeros + 2.5
mass = nodes[dim-1]*nodes[dim-1]
velocity = make_obj_array([actx.np.cos(nodes[i]) for i in range(dim)])
mom = mass*velocity
q = join_conserved(dim, mass=mass, energy=energy, momentum=mom)
cv = split_conserved(dim, q)
grad_q = obj_array_vectorize(discr.grad, q)
grad_cv = split_conserved(dim, grad_q)
grad_v = velocity_gradient(discr, cv, grad_cv)
def exact_grad_row(xdata, gdim, dim):
exact_grad_row = make_obj_array([zeros for _ in range(dim)])
exact_grad_row[gdim] = -actx.np.sin(xdata)
return exact_grad_row
comp_err = make_obj_array([
discr.norm(grad_v[i] - exact_grad_row(nodes[i], i, dim), np.inf)
for i in range(dim)])
err_max = comp_err.max()
eoc.add_data_point(h, err_max)
logger.info(eoc)
assert (
eoc.order_estimate() >= order - 0.5
or eoc.max_error() < 1e-9
)
def test_velocity_gradient_structure(actx_factory):
"""Test gradv data structure, verifying usability with other helper routines."""
from mirgecom.fluid import velocity_gradient
actx = actx_factory()
dim = 3
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 + 1.0
mass = 2*ones
energy = zeros + 2.5
velocity_x = nodes[0] + 2*nodes[1] + 3*nodes[2]
velocity_y = 4*nodes[0] + 5*nodes[1] + 6*nodes[2]
velocity_z = 7*nodes[0] + 8*nodes[1] + 9*nodes[2]
velocity = make_obj_array([velocity_x, velocity_y, velocity_z])
mom = mass * velocity
cv = make_conserved(dim, mass=mass, energy=energy, momentum=mom)
from grudge.op import local_grad
grad_cv = local_grad(discr, cv)
grad_v = velocity_gradient(cv, grad_cv)
tol = 1e-11
exp_result = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
exp_trans = [[1, 4, 7], [2, 5, 8], [3, 6, 9]]
exp_trace = 15
assert grad_v.shape == (dim, dim)
from meshmode.dof_array import DOFArray
assert type(grad_v[0, 0]) == DOFArray
def inf_norm(x):
return actx.to_numpy(discr.norm(x, np.inf))
assert inf_norm(grad_v - exp_result) < tol
assert inf_norm(grad_v.T - exp_trans) < tol
assert inf_norm(np.trace(grad_v) - exp_trace) < tol
def test_velocity_gradient_sanity(actx_factory, dim, mass_exp, vel_fac):
"""Test that the grad(v) returns {0, I} for v={constant, r_xyz}."""
from mirgecom.fluid import velocity_gradient
actx = actx_factory()
nel_1d = 16
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 = 3
discr = EagerDGDiscretization(actx, mesh, order=order)
nodes = thaw(actx, discr.nodes())
zeros = discr.zeros(actx)
ones = zeros + 1.0
mass = 1 * ones
for i in range(mass_exp):
mass *= (mass + i)
energy = zeros + 2.5
velocity = vel_fac * nodes
mom = mass * velocity
cv = make_conserved(dim, mass=mass, energy=energy, momentum=mom)
from grudge.op import local_grad
grad_cv = make_conserved(dim, q=local_grad(discr, cv.join()))
grad_v = velocity_gradient(discr, cv, grad_cv)
tol = 1e-11
exp_result = vel_fac * np.eye(dim) * ones
grad_v_err = [
discr.norm(grad_v[i] - exp_result[i], np.inf) for i in range(dim)
]
assert max(grad_v_err) < tol
def viscous_stress_tensor(discr, eos, cv, grad_cv):
r"""Compute the viscous stress tensor.
The viscous stress tensor $\tau$ is defined by:
.. math::
\mathbf{\tau} = \mu\left(\nabla{\mathbf{v}}
+\left(\nabla{\mathbf{v}}\right)^T\right) + (\mu_B - \frac{2\mu}{3})
\left(\nabla\cdot\mathbf{v}\right)
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 viscous stress tensor
"""
dim = cv.dim
transport = eos.transport_model()
mu_b = transport.bulk_viscosity(eos, cv)
mu = transport.viscosity(eos, cv)
grad_v = velocity_gradient(discr, cv, grad_cv)
div_v = np.trace(grad_v)
return mu * (grad_v + grad_v.T) + (mu_b - 2 * mu / 3) * div_v * np.eye(dim)
