def resolvent_formulation(ff):
#===============================================================#
#### Initialize 3D array to store approximated velocity field ###
#===============================================================#
# Velocity field is stacked in the wall-normal direction.
tmp = np.zeros((ff.Nx, ff.Nd*ff.Nm, ff.Nz), dtype=complex)
# Note that the number of gridpoints in the wall-normal direction
# are equal to the number of interior Chebyshev nodes. This is
# becasue when reshaped, the velocity field is missing its
# wall-boundaries, i.e. top and bottom xz-planes.
#===============================================================#
#### Calculate differentiation matrices and mean flow derivatives
#===============================================================#
chebyshev_differentiation, mean_flow_derivatives = ps.calculate_derivatives(ff.Ny, [], ff.baseflow)
#===============================================================#
#### Loop through Wavepacket ####
#===============================================================#
for triplet in range(0, len(ff.kx)):
#===========================================================#
#### Get fundamental wavenumbers from Wavepacket ####
#===========================================================#
fund_alpha = ff.kx[triplet]
fund_beta = ff.kz[triplet]
amplitude = ff.chi_tilde[triplet]
#===========================================================#
#### Calculate first Fourier modes ####
#===========================================================#
kx_array = fund_alpha * ff.Mx
kz_array = fund_beta * ff.Mz
#===========================================================#
#### Loop through the Fourier modes ####
#===========================================================#
for mx in range(0, len(kx_array)):
kx = kx_array[mx] # Streamwise Fourier mode
for mz in range(0, len(kz_array)):
kz = kz_array[mz] # Spanwise Fourier mode
if kx == 0 or kz == 0:
continue
print('kx:\t'+ str(kx) + "\tkz:\t" + str(kz))
#----------------------------------------------------
#### Calculate the state vectors -
#----------------------------------------------------
omega = kx * ff.c
wegihted_transfer_function, w = ps.calculate_transfer_function(kx, kz, ff.Re, ff.Nm, omega, chebyshev_differentiation, mean_flow_derivatives)
#----------------------------------------------------
#### Perform SVD -
#----------------------------------------------------
weighted_resolvent_modes, singular_values, weighted_forcing_modes = svd(wegihted_transfer_function)
#----------------------------------------------------
#### Test: SVD -
#----------------------------------------------------
Tests.SVD(weighted_resolvent_modes, singular_values, weighted_forcing_modes, wegihted_transfer_function, False)
#----------------------------------------------------
#### Test: Orthogonality -
#----------------------------------------------------
Tests.orthogonality(weighted_resolvent_modes)
Tests.orthogonality(weighted_forcing_modes)
#----------------------------------------------------
#### Test: Invertibility -
#----------------------------------------------------
Tests.invertible(np.diag(singular_values))
#----------------------------------------------------
#### Retrieve non-grid-weighted (physical) modes -
#----------------------------------------------------
weighted_resolvent_modes = np.asmatrix(weighted_resolvent_modes)
weighted_forcing_modes = np.asmatrix(weighted_forcing_modes)
resolvent_modes = np.linalg.solve(w, weighted_resolvent_modes)
forcing_modes = np.linalg.solve(w, weighted_forcing_modes)
#----------------------------------------------------
#### Test: Continuity condition -
#----------------------------------------------------
Tests.continuity(resolvent_modes, np.diag(singular_values), kx, kz, ff.Nm, chebyshev_differentiation['D1'])
#----------------------------------------------------
#### Fix phase of resolvent modes -
# based on critical layer or centreline -
#----------------------------------------------------
phase_shift = np.zeros((resolvent_modes.shape[1], resolvent_modes.shape[1]), dtype=np.complex128)
ind0 = ff.Nm/2 + 1 # Use centreline, unless
if ff.c < 1.0:
inds = Tests.indices(mean_flow_derivatives['U'].diagonal(), lambda x: x > ff.c)
if len(inds) > 0:
ind0 = inds[0]
np.fill_diagonal(phase_shift, np.exp(-1j * np.angle(resolvent_modes[ind0,:])))
resolvent_modes *= phase_shift
# Alternative Method:
# # Non-weighted forcing modes (physical forcing modes)
# unweighted_forcing_modes = np.linalg.solve(state_vecs['w'], forcing_modes_h.conjugate().T)
# unweighted_forcing_modes *= phase_shift
# unweighted_H = np.linalg.inv(state_vecs['w']) * state_vecs['H'] * state_vecs['w']
# unweighted_vel_modes = unweighted_H * unweighted_forcing_modes * np.diag(1.0/singular_values)
# delta_r = unweighted_vel_modes.real - resolvent_modes.real
# delta_i = unweighted_vel_modes.imag - resolvent_modes.imag
#----------------------------------------------------
#### Calculate spectral velocity vector -
# u_tilde = psi * sigma * xi -
# u_tilde = psi * chi -
#----------------------------------------------------
u_tilde = resolvent_modes[:, 0] * amplitude # Rank 1
#----------------------------------------------------
# Inverse fourier transform -
#----------------------------------------------------
u_tilde = np.asmatrix(u_tilde)
physical_ff = np.zeros((ff.Nx, ff.Nd*ff.Nm, ff.Nz), dtype=complex)
physical_ff = ps.my_ifft(u_tilde, kx, kz, ff)
tmp += physical_ff
return tmp
def deconstruct_field(original_ff_spectral,
kx_array,
kz_array,
Nm,
y,
c,
Re,
baseflow,
mean_profile,
sparse):
'''
Deconstruct given flow field and return the resolvent modes,
forcing modes, singular values and amplitude coefficients
needed to reconstruct it.
=================================================================
INPUTS:
=================================================================
original_ff_spectral: 3D array of flow field to
approximate in xz spectral form
with Chebyshev nodes in
wall-normal direction
(Chebyshev spacing).
Stacked in wall-normal direction:
dimensions: (Nx, Nd*Ny, Nz).
kx_array: 1D array of streamwise Fourier
modes.
kz_array: 1D array of spanwise Fourier
modes.
Nm: Number of interior Chebyshev nodes,
i.e. Ny - 2 (endpoints removed).
y: Grid points in the wall-normal
direction.
c: Wavespeed.
Re: Reynolds number based on laminar
baseflow centreline velocity, i.e.
Re = 1 / nu.
baseflow: Base flow type: [laminar, Couette],
laminar means Plane Poiseuille.
mean_profile: 1D array of streamwise velocity
profile of the turbulent mean in
wall-normal direction,
(endpoints included).
sparse: Boolean that determines whether to
use sparse or full SVD algorithm.
=================================================================
OUTPUTS:
=================================================================
A dictionary named 'deconstructed_dict' which contains:
resolvent_modes: 4D array of resolvent modes
at each Fourier mode combination:
resolvent_modes[mx, mz, :, :] gives
column vectors of given rank, i.e.
rank = resolvent_modes.shape[3].
forcing_modes: 4D array of forcing modes
at each Fourier mode combination:=
resolvent_modes[mx, mz, :, :] gives
column vectors of given rank, i.e.
rank = resolvent_modes.shape[3].
singular_values: 3D array of singular values
at each Fourier mode combination:
singular_values[mx, mz, :] gives
1D array of singular values, where
rank = len(singular_values[mx, mz, :]).
coefficients: 3D array of complex amplitude
coefficients at each
Fourier mode combination:
coefficients[mx, mz, :] gives
1D array of coefficients, where
rank = len(coefficients[mx, mz, :]).
'''
#===============================================================#
#### Store the resolvent modes and amplitude coefficients ####
# at each Fourier mode pair #
#===============================================================#
resolvent_modes_array = np.zeros((len(kx_array), len(kz_array), 3*Nm, 3*Nm), dtype=complex)
forcing_modes_array = np.zeros((len(kx_array), len(kz_array), 3*Nm, 3*Nm), dtype=complex)
coefficients_array = np.zeros((len(kx_array), len(kz_array), 3*Nm), dtype=complex)
sing_vals_array = np.zeros((len(kx_array), len(kz_array), 3*Nm), dtype=float)
#===============================================================#
#### Calculate differentiation matrices and mean flow derivatives
#===============================================================#
chebyshev_differentiation, mean_flow_derivatives = ps.calculate_derivatives(Nm+2, mean_profile, baseflow)
#===============================================================#
#### Loop through wavenumbers ####
#===============================================================#
startTime = datetime.now()
for mx in range(0, len(kx_array)):
kx = kx_array[mx]
# print('\nkx:'+ str(kx))
for mz in range(0, len(kz_array)):
kz = kz_array[mz]
# sys.stdout.write(".")
# sys.stdout.flush()
if kx == 0 and kz == 0: # Zeroth Fourier modes
resolvent_modes_array[mx, mz, :, 0] = original_ff_spectral[mx, :, mz]
continue # Start the loop again
#--------------------------------------------------------
#### Calculate the state vectors -
#--------------------------------------------------------
omega = kx * c
wegihted_transfer_function, w = ps.calculate_transfer_function(kx, kz, Re, Nm, omega, chebyshev_differentiation, mean_flow_derivatives)
#--------------------------------------------------------
#### Perform SVD -
#--------------------------------------------------------
if sparse:
weighted_resolvent_modes, singular_values, weighted_forcing_modes = svd(wegihted_transfer_function, full_matrices=False)
else:
weighted_resolvent_modes, singular_values, weighted_forcing_modes = svd(wegihted_transfer_function)
#--------------------------------------------------------
#### Test: SVD -
#--------------------------------------------------------
Tests.SVD(weighted_resolvent_modes, singular_values, weighted_forcing_modes, wegihted_transfer_function, sparse)
#--------------------------------------------------------
#### Test: Orthogonality -
#--------------------------------------------------------
Tests.orthogonality(weighted_resolvent_modes)
Tests.orthogonality(weighted_forcing_modes)
#--------------------------------------------------------
#### Test: Invertibility -
#--------------------------------------------------------
Tests.invertible(np.diag(singular_values))
#--------------------------------------------------------
#### Retrieve non-grid-weighted (physical) modes -
#--------------------------------------------------------
weighted_resolvent_modes = np.asmatrix(weighted_resolvent_modes)
weighted_forcing_modes = np.asmatrix(weighted_forcing_modes)
resolvent_modes = np.linalg.solve(w, weighted_resolvent_modes)
forcing_modes = np.linalg.solve(w, weighted_forcing_modes)
#--------------------------------------------------------
#### Tests: Continuity condition -
#--------------------------------------------------------
Tests.continuity(resolvent_modes, np.diag(singular_values), kx, kz, Nm, chebyshev_differentiation['D1'])
#--------------------------------------------------------
#### Fix phase of resolvent modes -
# based on critical layer or centreline -
#--------------------------------------------------------
phase_shift = np.zeros((resolvent_modes.shape[1], resolvent_modes.shape[1]), dtype=np.complex128)
ind0 = Nm/2 + 1 # Use centreline, unless
if c < 1.0:
inds = Tests.indices(mean_flow_derivatives['U'].diagonal(), lambda x: x > c)
if len(inds) > 0:
ind0 = inds[0]
np.fill_diagonal(phase_shift, np.exp(-1j * np.angle(resolvent_modes[ind0,:])))
resolvent_modes *= phase_shift
#--------------------------------------------------------
#### Project resolvent modes to get coefficients -
#--------------------------------------------------------
# Initialize amplitude coefficients vector
xi = np.zeros((3*Nm, 1), dtype=complex)
# Projection
xi = inv(np.diag(singular_values)) * resolvent_modes.H * w.H * w * np.asmatrix(original_ff_spectral[mx, :, mz]).T
Tests.projection(xi, np.diag(singular_values), resolvent_modes, np.asmatrix(original_ff_spectral[mx, :, mz]).T, 1e-10)
#--------------------------------------------------------
#### Store resolvent modes, singular values and coeffs. -
#--------------------------------------------------------
resolvent_modes_array[mx, mz, :, :] = resolvent_modes
forcing_modes_array[mx, mz, :, :] = forcing_modes
coefficients_array[mx, mz, :] = np.squeeze(np.asarray(xi))
sing_vals_array[mx, mz, :] = singular_values
calcTime = datetime.now() - startTime
# print("\n\n\n")
# print(calcTime)
# print("\n\n\n")
deconstructed_dict = {}
deconstructed_dict['resolvent_modes'] = resolvent_modes_array
deconstructed_dict['forcing_modes'] = forcing_modes_array
deconstructed_dict['singular_values'] = sing_vals_array
deconstructed_dict['coefficients'] = coefficients_array
return deconstructed_dict
def test_deconstruct_field(original_ff_spectral,
kx_array,
kz_array,
Nm,
y,
c,
Re,
baseflow,
mean_profile,
sparse):
#===============================================================#
#### Store the resolvent modes and amplitude coefficients ####
# at each Fourier mode pair #
#===============================================================#
resolvent_modes_array = np.zeros((len(kx_array), len(kz_array), 3*Nm, 3*Nm), dtype=np.complex128)
forcing_modes_array = np.zeros((len(kx_array), len(kz_array), 3*Nm, 3*Nm), dtype=np.complex128)
coefficients_array = np.zeros((len(kx_array), len(kz_array), 3*Nm), dtype=np.complex128)
sing_vals_array = np.zeros((len(kx_array), len(kz_array), 3*Nm), dtype=np.float64)
#================================================================
#### Loop through wavenumbers
#================================================================
chebyshev_differentiation, mean_flow_derivatives = ps.calculate_derivatives(Nm+2, mean_profile, baseflow)
#================================================================
#### FFT mean profile
#================================================================
spectral_deviation_profile = np.zeros((3*Nm),dtype=complex)
if len(mean_profile) == 0:
spectral_deviation_profile = original_ff_spectral[0,:,0]
else:
# Remove baseflow from turbulent mean profile
baseflow_profile = []
if baseflow == "lam": # Laminary base flow
baseflow_profile = 1.0 - y**2.0
elif baseflow == "cou": # Couette base flow
baseflow_profile = y
# Remove baseflow from mean to get turbulent deviation profile
vel_profile = mean_profile - np.asarray(baseflow_profile)
deviation_profile_sp = np.fft.fft(vel_profile)
spectral_deviation_profile[1:Nm] = deviation_profile_sp[1:Nm]
#================================================================
#### Loop through wavenumbers
#================================================================
startTime = datetime.now()
for mx in range(0, len(kx_array)):
kx = kx_array[mx]
print('\n\nkx:'+ str(kx))
for mz in range(0, len(kz_array)):
kz = kz_array[mz]
sys.stdout.write(".")
sys.stdout.flush()
if kx == 0 and kz == 0: # Zeroth Fourier modes
resolvent_modes_array[mx, mz, :, 0] = spectral_deviation_profile
continue # Start the loop again
#--------------------------------------------------------
#### Calculate the state vectors
#--------------------------------------------------------
omega = kx * c
wegihted_transfer_function, w = ps.calculate_transfer_function(kx, kz, Re, Nm, omega, chebyshev_differentiation, mean_flow_derivatives)
#--------------------------------------------------------
#### Perform SVD
#--------------------------------------------------------
if sparse:
weighted_resolvent_modes, singular_values, weighted_forcing_modes = svd(wegihted_transfer_function, full_matrices=False)
else:
weighted_resolvent_modes, singular_values, weighted_forcing_modes = svd(wegihted_transfer_function)
#--------------------------------------------------------
#### Test: SVD
#--------------------------------------------------------
Tests.SVD(weighted_resolvent_modes, singular_values, weighted_forcing_modes, wegihted_transfer_function, sparse)
#--------------------------------------------------------
#### Test: Orthogonality
#--------------------------------------------------------
Tests.orthogonality(weighted_resolvent_modes)
Tests.orthogonality(weighted_forcing_modes)
#--------------------------------------------------------
#### Test: Invertibility
#--------------------------------------------------------
Tests.invertible(np.diag(singular_values))
#--------------------------------------------------------
#### Retrieve non-grid-weighted (physical) modes
#--------------------------------------------------------
weighted_resolvent_modes = np.asmatrix(weighted_resolvent_modes)
weighted_forcing_modes = np.asmatrix(weighted_forcing_modes)
resolvent_modes = np.linalg.solve(w, weighted_resolvent_modes)
forcing_modes = np.linalg.solve(w, weighted_forcing_modes)
#--------------------------------------------------------
#### Check that the continuity condition is satisfied
#--------------------------------------------------------
Tests.continuity(resolvent_modes, np.diag(singular_values), kx, kz, Nm, chebyshev_differentiation['D1'])
#--------------------------------------------------------
#### Fix phase of resolvent modes based on critical layer or centreline
#--------------------------------------------------------
phase_shift = np.zeros((resolvent_modes.shape[1], resolvent_modes.shape[1]), dtype=np.complex128)
ind0 = Nm/2 + 1 # Use centreline, unless
if c < 1.0:
inds = Tests.indices(mean_flow_derivatives['U'].diagonal(), lambda x: x > c)
if len(inds) > 0:
ind0 = inds[0]
np.fill_diagonal(phase_shift, np.exp(-1j * np.angle(resolvent_modes[ind0,:])))
resolvent_modes *= phase_shift
#--------------------------------------------------------
#### Project resolvent modes to get amplitude coefficients
#--------------------------------------------------------
# Initialize amplitude coefficients vector
ampl_coeffs = np.zeros((3*Nm, 1), dtype=np.complex128)
# Projection
ampl_coeffs = inv(np.diag(singular_values)) * resolvent_modes.H * w.H * w * np.asmatrix(original_ff_spectral[mx, :, mz]).T
Tests.projection(ampl_coeffs, np.diag(singular_values), resolvent_modes, np.asmatrix(original_ff_spectral[mx, :, mz]).T, 1e-10)
# phase_test = False
# norm_test = False
# xi_norm = np.linalg.norm(xi)
# if kz == 2.0 or kz == -2.0:
# #xi_norm >= 1e-10:
# # Phase test
# if phase_test:
# print("\nApplying phase shift at: " + str(kz) + "\n ")
# # shift the field by pi/3
# xi += 1.0j*(np.pi/3.0)
#
# # Norm test
# if norm_test:
# print("\nApplying norm doubling.")
# # Double the energy of the field
# xi *= np.sqrt(2.0) + 1.0j*np.sqrt(2.0)
#
# # Fix xi to see if same coefficients are retrieved from projection
# if fixXi:
# xi[0] = 1.0
# xi[1] = 0.0
# print("\nXI:")
# print("norm: " + str(xi_norm))
# print(xi)
# print("")
# print(xi)
#--------------------------------------------------------
#### Store resolvent modes, singular values and amplitude coeffs.
#--------------------------------------------------------
resolvent_modes_array[mx, mz, :, :] = resolvent_modes
forcing_modes_array[mx, mz, :, :] = forcing_modes
coefficients_array[mx, mz, :] = np.squeeze(np.asarray(ampl_coeffs))
sing_vals_array[mx, mz, :] = singular_values
calcTime = datetime.now() - startTime
print("\n\n\n")
print(calcTime)
print("\n\n\n")
deconstructed_dict = {}
deconstructed_dict['resolvent_modes'] = resolvent_modes_array
deconstructed_dict['forcing_modes'] = forcing_modes_array
deconstructed_dict['singular_values'] = sing_vals_array
deconstructed_dict['coefficients'] = coefficients_array
return deconstructed_dict
import PseudoSpectralMethods as ps
from numpy.linalg import svd
from numpy import diag
alpha = 5.0 # Streamwise Fourier mode
beta = 1.0 # Spanwise Fourier mode
c = 0.5 # Wavespeed
omega = c * alpha # Temporal frequency
Re = 1000.0 # Reynolds number
N = 31 # Wall-normal resolution
Nm = N - 2 # Wall-normal resolution (endpoints removed)
bf = "lam" # Baseflow
vel_profile = [] # Mean velocity profile (if known)
chebyshev_differentiation, mean_flow_derivatives = ps.calculate_derivatives(N, vel_profile, bf)
H, w = ps.calculate_transfer_function(alpha, beta, Re, Nm, omega, chebyshev_differentiation, mean_flow_derivatives)
U, S, V = svd(H)
S = diag(S)
print("Fin")