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tkespec.py
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tkespec.py
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# -*- coding: utf-8 -*-
"""
Created on Fri May 9 10:14:44 2014
@author: tsaad
"""
import numpy as np
from numpy.fft import fftn
from numpy import sqrt, zeros, conj, pi, arange, ones, convolve
# ------------------------------------------------------------------------------
def movingaverage(interval, window_size):
window = ones(int(window_size)) / float(window_size)
return convolve(interval, window, 'same')
# ------------------------------------------------------------------------------
def compute_tke_spectrum_1d(u, lx, ly, lz, smooth):
"""
Given a velocity field u this function computes the kinetic energy
spectrum of that velocity field in spectral space. This procedure consists of the
following steps:
1. Compute the spectral representation of u using a fast Fourier transform.
This returns uf (the f stands for Fourier)
2. Compute the point-wise kinetic energy Ef (kx, ky, kz) = 1/2 * (uf)* conjugate(uf)
3. For every wave number triplet (kx, ky, kz) we have a corresponding spectral kinetic energy
Ef(kx, ky, kz). To extract a one dimensional spectrum, E(k), we integrate Ef(kx,ky,kz) over
the surface of a sphere of radius k = sqrt(kx^2 + ky^2 + kz^2). In other words
E(k) = sum( E(kx,ky,kz), for all (kx,ky,kz) such that k = sqrt(kx^2 + ky^2 + kz^2) ).
Parameters:
-----------
u: 3D array
The x-velocity component.
v: 3D array
The y-velocity component.
w: 3D array
The z-velocity component.
lx: float
The domain size in the x-direction.
ly: float
The domain size in the y-direction.
lz: float
The domain size in the z-direction.
smooth: boolean
A boolean to smooth the computed spectrum for nice visualization.
"""
nx = len(u[:, 0, 0])
ny = len(u[0, :, 0])
nz = len(u[0, 0, :])
nt = nx * ny * nz
n = max(nx, ny, nz) # int(np.round(np.power(nt,1.0/3.0)))
uh = fftn(u) / nt
# tkeh = zeros((nx, ny, nz))
tkeh = 0.5 * (uh * conj(uh)).real
length = max(lx, ly, lz)
knorm = 2.0 * pi / length
kxmax = nx / 2
kymax = ny / 2
kzmax = nz / 2
wave_numbers = knorm * arange(0, n)
tke_spectrum = zeros(len(wave_numbers))
for kx in range(nx):
rkx = kx
if kx > kxmax:
rkx = rkx - nx
for ky in range(ny):
rky = ky
if ky > kymax:
rky = rky - ny
for kz in range(nz):
rkz = kz
if kz > kzmax:
rkz = rkz - nz
rk = sqrt(rkx * rkx + rky * rky + rkz * rkz)
k = int(np.round(rk))
print('k = ', k)
tke_spectrum[k] = tke_spectrum[k] + tkeh[kx, ky, kz]
tke_spectrum = tke_spectrum / knorm
if smooth:
tkespecsmooth = movingaverage(tke_spectrum, 5) # smooth the spectrum
tkespecsmooth[0:4] = tke_spectrum[0:4] # get the first 4 values from the original data
tke_spectrum = tkespecsmooth
knyquist = knorm * min(nx, ny, nz) / 2
return knyquist, wave_numbers, tke_spectrum
# ------------------------------------------------------------------------------
def compute_tke_spectrum(u, v, w, lx, ly, lz, smooth):
"""
Given a velocity field u, v, w, this function computes the kinetic energy
spectrum of that velocity field in spectral space. This procedure consists of the
following steps:
1. Compute the spectral representation of u, v, and w using a fast Fourier transform.
This returns uf, vf, and wf (the f stands for Fourier)
2. Compute the point-wise kinetic energy Ef (kx, ky, kz) = 1/2 * (uf, vf, wf)* conjugate(uf, vf, wf)
3. For every wave number triplet (kx, ky, kz) we have a corresponding spectral kinetic energy
Ef(kx, ky, kz). To extract a one dimensional spectrum, E(k), we integrate Ef(kx,ky,kz) over
the surface of a sphere of radius k = sqrt(kx^2 + ky^2 + kz^2). In other words
E(k) = sum( E(kx,ky,kz), for all (kx,ky,kz) such that k = sqrt(kx^2 + ky^2 + kz^2) ).
Parameters:
-----------
u: 3D array
The x-velocity component.
v: 3D array
The y-velocity component.
w: 3D array
The z-velocity component.
lx: float
The domain size in the x-direction.
ly: float
The domain size in the y-direction.
lz: float
The domain size in the z-direction.
smooth: boolean
A boolean to smooth the computed spectrum for nice visualization.
"""
nx = len(u[:, 0, 0])
ny = len(v[0, :, 0])
nz = len(w[0, 0, :])
nt = nx * ny * nz
n = nx # int(np.round(np.power(nt,1.0/3.0)))
uh = fftn(u) / nt
vh = fftn(v) / nt
wh = fftn(w) / nt
tkeh = 0.5 * (uh * conj(uh) + vh * conj(vh) + wh * conj(wh)).real
k0x = 2.0 * pi / lx
k0y = 2.0 * pi / ly
k0z = 2.0 * pi / lz
knorm = (k0x + k0y + k0z) / 3.0
print('knorm = ', knorm)
kxmax = nx / 2
kymax = ny / 2
kzmax = nz / 2
# dk = (knorm - kmax)/n
# wn = knorm + 0.5 * dk + arange(0, nmodes) * dk
wave_numbers = knorm * arange(0, n)
tke_spectrum = zeros(len(wave_numbers))
for kx in range(-nx//2, nx//2-1):
for ky in range(-ny//2, ny//2-1):
for kz in range(-nz//2, nz//2-1):
rk = sqrt(kx**2 + ky**2 + kz**2)
k = int(np.round(rk))
tke_spectrum[k] += tkeh[kx, ky, kz]
# for kx in range(nx):
# rkx = kx
# if kx > kxmax:
# rkx = rkx - nx
# for ky in range(ny):
# rky = ky
# if ky > kymax:
# rky = rky - ny
# for kz in range(nz):
# rkz = kz
# if kz > kzmax:
# rkz = rkz - nz
# rk = sqrt(rkx * rkx + rky * rky + rkz * rkz)
# k = int(np.round(rk))
# tke_spectrum[k] = tke_spectrum[k] + tkeh[kx, ky, kz]
tke_spectrum = tke_spectrum / knorm
# tke_spectrum = tke_spectrum[1:]
# wave_numbers = wave_numbers[1:]
if smooth:
tkespecsmooth = movingaverage(tke_spectrum, 5) # smooth the spectrum
tkespecsmooth[0:4] = tke_spectrum[0:4] # get the first 4 values from the original data
tke_spectrum = tkespecsmooth
knyquist = knorm * min(nx, ny, nz) / 2
return knyquist, wave_numbers, tke_spectrum
# ------------------------------------------------------------------------------
def compute_tke_spectrum2d(u, v, lx, ly, smooth):
"""
Given a velocity field u, v, w, this function computes the kinetic energy
spectrum of that velocity field in spectral space. This procedure consists of the
following steps:
1. Compute the spectral representation of u, v, and w using a fast Fourier transform.
This returns uf, vf, and wf (the f stands for Fourier)
2. Compute the point-wise kinetic energy Ef (kx, ky, kz) = 1/2 * (uf, vf, wf)* conjugate(uf, vf, wf)
3. For every wave number triplet (kx, ky, kz) we have a corresponding spectral kinetic energy
Ef(kx, ky, kz). To extract a one dimensional spectrum, E(k), we integrate Ef(kx,ky,kz) over
the surface of a sphere of radius k = sqrt(kx^2 + ky^2 + kz^2). In other words
E(k) = sum( E(kx,ky,kz), for all (kx,ky,kz) such that k = sqrt(kx^2 + ky^2 + kz^2) ).
Parameters:
-----------
u: 3D array
The x-velocity component.
v: 3D array
The y-velocity component.
w: 3D array
The z-velocity component.
lx: float
The domain size in the x-direction.
ly: float
The domain size in the y-direction.
lz: float
The domain size in the z-direction.
smooth: boolean
A boolean to smooth the computed spectrum for nice visualization.
"""
nx = len(u[:, 0])
ny = len(v[0, :])
nt = nx * ny
n = nx # int(np.round(np.power(nt,1.0/3.0)))
uh = fftn(u) / nt
vh = fftn(v) / nt
tkeh = 0.5 * (uh * conj(uh) + vh * conj(vh))
k0x = 2.0 * pi / lx
k0y = 2.0 * pi / ly
knorm = (k0x + k0y) / 2.0
print('knorm = ', knorm)
kxmax = nx / 2
kymax = ny / 2
# dk = (knorm - kmax)/n
# wn = knorm + 0.5 * dk + arange(0, nmodes) * dk
wave_numbers = knorm * arange(0, n)
tke_spectrum = zeros(len(wave_numbers))
for kx in range(-nx//2, nx//2-1):
for ky in range(-ny//2, ny//2-1):
rk = sqrt(kx**2 + ky**2)
k = int(np.round(rk))
tke_spectrum[k] += tkeh[kx, ky]
tke_spectrum = tke_spectrum / knorm
# tke_spectrum = tke_spectrum[1:]
# wave_numbers = wave_numbers[1:]
if smooth:
tkespecsmooth = movingaverage(tke_spectrum, 5) # smooth the spectrum
tkespecsmooth[0:4] = tke_spectrum[0:4] # get the first 4 values from the original data
tke_spectrum = tkespecsmooth
knyquist = knorm * min(nx, ny) / 2
return knyquist, wave_numbers, tke_spectrum
# ------------------------------------------------------------------------------
def compute_tke_spectrum_flatarrays(u, v, w, nx, ny, nz, lx, ly, lz, smooth):
unew = u.reshape([nx, ny, nz])
vnew = v.reshape([nx, ny, nz])
wnew = w.reshape([nx, ny, nz])
k, w, espec = compute_tke_spectrum(unew, vnew, wnew, lx, ly, lz, smooth)
return k, w, espec