# 1-D turbulent energy spectra in homogeneous direction (non-isotropic)

I am trying to compute the one-dimensional energy spectra for my channel-flow simulation. I have already written a post-processing script to achieve this; however, I need to validate my code before proceeding.

To do so, I am taking the two-point cross-correlation (per given plane) vector from a DNS database. Then, I am trying to apply my code and plot against the 1D energy spectra provided by the same database and in the same (homogeneous) direction.

I am always getting the wrong output, but I cannot figure out what went wrong. I have looked into several reference books and forum posts here and there with no tangible result as of yet.

The steps I am following are:

Given a two-point cross-correlation vector in the streamwise (i.e. x-direction) and homogeneous direction, I am applying the following (e.g. in Matlab form):

% where Ruu is the correlation vector from the DNS database
N = length(Ruu);
Nk = 2^nextpow2(N);

% Fourier transform data
Bx1 = zeros(Nk, 1);

for k=1:Nk
for n=1:N
Bx1(k) = Bx1(k) + (1/N)*Ruu(n)*exp(-2i*pi*(k-1)*(n-1)/N);
end
end

% wavenumbers initialization
kx = zeros(Nk, 1);
% total distance between correlations used
Lx = (max(x) - min(x));
for n=1:Nk
% streawise coordinates to wavenumber
kx(n) = pi*(n-1)/Lx;
end
% calculate 1D streamwise energy spectra
Eu = Bx1.*conj(Bx1);

% show only first half due to symmetry
loglog(kx(1:end/2), 2*Eu(1:end/2))


The resulting figure, in case you were wondering, is depicted below.

The output of such a figure is nowhere near what I am looking for. Can someone shed some insight on the issue?

Reference DNS Data

Moser, Robert D.; Kim, John; Mansour, Nagi N., Direct numerical simulation of turbulent channel flow up to $$Re_{\tau} = 590$$, Phys. Fluids 11, No. 4, 943-945 (1999). ZBL1147.76463.

• I am also trying to plot Euu. Your code is really helpful for me. Do you solve this problem? Thanks in advance! Kind regards, Guan
– 崔冠哲
Jun 1 '21 at 11:40

First, the fft(u).*conjg(fft(u)) is equal to fft(R), R is 1 dimension the correlation function.
Bx1(k) = Bx1(k) + (1/N)*Ruu(n)*exp(-2i*pi*(k-1)*(n-1)/N).