def test_that_is_incompressible_projection():
N = 128
L = 2 * np.pi
vt = VectorTool(N, L)
u = np.random.random((2, N, N))
u = vt.dealias(vt.div_free_proj(u))
assert vt.is_incompressible(u)
def test_dealias_of_velocity_with_components_equal_to_sinkx_where_k_is_above_two_thirds_dealias_boundary_is_zero(
):
L = 2.0 * np.pi
N = 128
vt = VectorTool(N, L)
kabove = np.ceil(vt.kmax_dealias) + 1
v = np.zeros((2, N, N))
v[0] = np.sin(2. * np.pi / L * kabove * vt.X[0])
v[0] = np.sin(2. * np.pi / L * kabove * vt.X[1])
z = np.zeros(v.shape)
assert np.allclose(vt.dealias(v), z)
def test_dealias_of_velocity_with_components_equal_to_sinkx_where_k_is_below_two_thirds_dealias_boundary_is_zero(
):
L = 2.0 * np.pi
N = 128
vt = VectorTool(N, L)
kabove = np.floor(vt.kmax_dealias) - 1
v = np.zeros((2, N, N))
v[0] = np.sin(2. * np.pi / L * kabove * vt.X[0])
v[0] = np.sin(2. * np.pi / L * kabove * vt.X[1])
z = np.zeros(v.shape)
assert np.allclose(vt.dealias(v), z)
def test_h1norm_of_sin_flow():
N = 128
L = 2.0
vt = VectorTool(N, L)
X = np.mgrid[:N, :N].astype(float) * (L / N)
u = np.zeros((2, N, N))
u[0] = np.sin(2. * np.pi / L * X[1])
assert np.isclose(vt.h1norm(u)**2., 2. * np.pi**2.)
def test_that_l2norm_of_vector_with_siny_for_xcomponent_equals_sqrt_of_half_of_Lsq(
):
L = 2.0 * np.pi
N = 128
vt = VectorTool(N, L)
X = np.mgrid[:N, :N].astype(float) * (L / N)
u = np.zeros((2, N, N))
u[0] = np.sin(2.0 * np.pi * X[1] / L)
assert np.isclose(vt.l2norm(u), (0.5 * L**2)**0.5)
def test_subtract_mean_of_velocity():
L = 2.0
N = 128
vt = VectorTool(N, L)
u = np.zeros((2, N, N))
sinkx = np.sin((2 * np.pi / L) * vt.X[0])
u[0] = sinkx + 2.0
u[1] = sinkx + 1.0
assert np.allclose(vt.subtract_mean(u), np.array([sinkx, sinkx]))
def test_that_is_incompressible_function_returns_true_for_cellular_flow():
N = 128
L = 2 * np.pi
vt = VectorTool(N, L)
u = np.zeros((2, N, N))
u[0] = np.sin((2.0 * np.pi / L) * vt.X[0]) * \
np.sin((2.0 * np.pi / L) * vt.X[1])
u[1] = np.cos((2.0 * np.pi / L) * vt.X[0]) * \
np.cos((2.0 * np.pi / L) * vt.X[1])
assert vt.is_incompressible(u)
def test_ifft_of_fft_equals_original_vector_function():
L = 10.5
N = 128
k = 2.0 * np.pi / L
X = np.mgrid[:N, :N].astype(float) * (L / N)
u = np.zeros((2, N, N))
u[0] = np.sin(k * X[1])
u[1] = np.cos(k * X[0])
vt = VectorTool(N, L)
assert np.allclose(vt.ifft(vt.fft(u)), u)
def test_lap_invlap_of_u_is_u_minus_mean_value():
L = 2.0 * np.pi
N = 128
vt = VectorTool(N, L)
u = np.zeros((2, N, N))
# Compressible flow
u[0] = np.sin((2.0 * np.pi / L) * vt.X[0]) * \
np.sin((2.0 * np.pi / L) * vt.X[1]) + 2.0
u[1] = -np.cos((2.0 * np.pi / L) * vt.X[0]) * \
np.cos((2.0 * np.pi / L) * vt.X[1]) + 1.0
assert np.allclose(vt.lap(vt.invlap(u)), vt.subtract_mean(u))
def test_curl_is_equal_to_curl_computed_with_grad_function():
N = 128
L = 2.0
kappa = 0.0
gamma = 1.0
v = np.random.random((2, N, N))
# print(np.shape(v))
st = ScalarTool(N, L)
vt = VectorTool(N, L)
v = vt.div_free_proj(v)
v = vt.dealias(v)
c = st.grad(v[1])[0] - st.grad(v[0])[1]
assert np.allclose(c, vt.curl(v))
def test_that_div_free_projector_converts_a_compressible_flow_to_incompressible(
):
N = 128
L = 2 * np.pi
vt = VectorTool(N, L)
u = np.zeros((2, N, N))
# Compressible flow
u[0] = np.sin((2.0 * np.pi / L) * vt.X[0]) * \
np.sin((2.0 * np.pi / L) * vt.X[1])
u[1] = -np.cos((2.0 * np.pi / L) * vt.X[0]) * \
np.cos((2.0 * np.pi / L) * vt.X[1])
w = np.zeros((2, N, N))
w[0] = np.sin((2.0 * np.pi / L) * vt.X[0]) * \
np.sin((2.0 * np.pi / L) * vt.X[1])
w[1] = np.cos((2.0 * np.pi / L) * vt.X[0]) * \
np.cos((2.0 * np.pi / L) * vt.X[1])
u = u + 0.1 * w
assert vt.is_incompressible(vt.div_free_proj(u))
def test_validate_div_free_projector():
L = 2.0 * np.pi
N = 128
st = ScalarTool(N, L)
vt = VectorTool(N, L)
u = np.zeros((2, N, N))
# Compressible flow
u[0] = np.sin((2.0 * np.pi / L) * vt.X[0]) * \
np.sin((2.0 * np.pi / L) * vt.X[1])
u[1] = -np.cos((2.0 * np.pi / L) * vt.X[0]) * \
np.cos((2.0 * np.pi / L) * vt.X[1])
w = np.zeros((2, N, N))
w[0] = np.sin((2.0 * np.pi / L) * vt.X[0]) * \
np.sin((2.0 * np.pi / L) * vt.X[1])
w[1] = np.cos((2.0 * np.pi / L) * vt.X[0]) * \
np.cos((2.0 * np.pi / L) * vt.X[1])
u = u + 0.1 * w
projection_alt = u - st.grad_invlap(vt.div(u))
print(np.amax(abs(projection_alt - vt.div_free_proj(u))))
assert np.allclose(projection_alt, vt.div_free_proj(u))
def test_h1normsq_of_vector_is_spatial_integral_neg_lapvec_times_vec():
N = 128
L = 2.0
kappa = 0.0
gamma = 1.0
v = np.random.random((2, N, N))
st = ScalarTool(N, L)
vt = VectorTool(N, L)
v = vt.div_free_proj(v)
v = vt.dealias(v)
a = vt.h1norm(v)**2
b = st.sint(np.sum(-vt.lap(v) * v, 0))
assert np.allclose(a, b)
def test_l2norm_squared_of_curl_of_vector_is_spatial_integral_of_neg_vector_times_lap_vector(
):
N = 128
L = 2.0
kappa = 0.0
gamma = 1.0
v = np.random.random((2, N, N))
# print(np.shape(v))
st = ScalarTool(N, L)
vt = VectorTool(N, L)
v = vt.div_free_proj(v)
v = vt.dealias(v)
curl = vt.curl(v)
a = st.l2norm(curl)**2.
b = st.sint(np.sum(-vt.lap(v) * v, 0))
assert np.allclose(a, b)