3.2 Methodology
3.2.2 DNS Velocity Fields
The DNS velocity fields were kindly provided by X. Wu who used the second-order finite difference code described in Wu & Moin (2008) at Re = 24,580 (Reτ = 685), and with a domain length
of 30R (kmin = 0.21). The number of grid points in the streamwise, wall-normal, and azimuthal
directions was respectively 2048×256 ×1024. The DNS data was subsampled by a factor of 4 in the streamwise and azimuthal directions to decrease the size of the data files. The flow domain is large enough for the spatial-averaged velocity profile and streamwise turbulence intensity profile to be converged, even with the subsampling.
Two sets of velocity fields were available. For the first set, the flow was periodically sampled in time at a rate of 1 sample every 7.2 dimensionless time units, based on the pipe radius and bulk velocity. A total of 21 samples was taken over 150 dimensionless time units. The number of samples and sampling duration for the first data set could not be chosen by the authors and constrained the number of Fourier modes that could be resolved with the data. The second set was randomly sampled based on the parameters derived in the present study in order to apply compressive sampling.
The 21 periodically sampled DNS velocity fields were used to check whether the frequency content of the Fourier modes was indeed sparse. Each velocity field was Fourier transformed in the streamwise and azimuthal directions by applying a 2D FFT,
u(x, r, θ, t) =X
k,n
ck,n(r, t)ei(kx+nθ), (3.7)
and normalized such that
255kmin X k=0 127 X n=−128 ck,n(r)c∗k,n(r) =u0 2 (r), (3.8)
where ∗ denotes the complex conjugate and k
min = 0.21 for a domain length of 30R. The 2D
spatial Fourier modes are referred to by their wavenumber pair (k, n). Only the positive k half plane was kept since the spectrum of a real-valued signal is symmetric. The 2D Fourier spectrum was integrated in the wall-normal direction and averaged in time to identify the 2D Fourier modes that contribute most to the streamwise turbulence intensity, effectively a crude POD over a limited resolvable parameter space.
The range of 2D Fourier modes that were free from aliasing effects was estimated based on their streamwise wavenumber by comparing the frequency range that can be resolved with the available DNS data to the empirical bounds on the frequency content of the flow. The dynamically signifi- cant frequency content in wall-bounded turbulence was estimated as a function of the streamwise wavenumber by assuming that the highest and lowest convection velocities correspond respectively to the centerline velocity and 10 times the friction velocity. The upper bound comes from the anal- ysis of the Orr-Sommerfeld equation by Joseph (1968) showing that, for laminar channel flow, the
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Figure 3.2: Frequency range resolved by the periodically sampled DNS data (delimited by the two horizontal dashed lines) compared to the empirical upper and lower bounds on the DNS frequency content (solid lines) as a function of the streamwise wavenumber k. The shaded area shows the time-resolved streamwise wavenumber range for the available data.
real part of the eigenvalues, which corresponds to the convection velocity of the disturbances, is restricted to the range set by the mean flow. The lower bound was estimated based on the results of Morrison et al. (1971), who used experimental data to decompose the buffer layer as a sum of propagating waves, and found that the lowest convection velocity is about 10 times the friction velocity,u= 10uτ, or 0.44 ¯U atRe= 24,580. The value for the lowest convection velocity is broadly
supported in the literature, as summarized by LeHewet al.(2011).
The upper and lower bounds on the frequency content as a function of the streamwise wavenum- ber are depicted on figure 3.2. The highest frequency that could be resolved with the available DNS data is given by the inverse of the sampling rate fmax = 71.2 = 0.14 and is reached at k = 0.69
for modes convecting at the centerline velocity. All the modes with k ≥ 0.69 have part of their frequency content aliased to lower frequencies, such that only the frequency content of the lowest three streamwise wavenumbersk= 0.21, 0.42, and 0.63, for various azimuthal wavenumbers, could accurately be extracted from the available DNS data by applying a FFT in time, because the Nyquist criterion is satisfied for these three modes.
Similarly to the periodically sampled DNS velocity fields, the randomly sampled fields were decomposed as a Fourier series in the streamwise and azimuthal directions using a 2D FFT, and the 2D Fourier spectrum was integrated in the wall-normal direction and averaged in time, to identify the most energetic spatial modes. Compressive sampling was used to extract the frequency content of these most energetic modes at each wall-normal location, following the same methodology as for the synthetic velocity fields. The frequency spectrum was integrated in the wall-normal direction to identity the sparse frequencies, defined as the frequencies containing not less than 10% of the energy
in the peak frequency.
As an alternative approach, compressive sampling was also applied globally on the `2-norm of
the temporal Fourier coefficients in the wall-normal direction, i.e., to minimize
Nopti X j=1 Z 1 0 ck,n,2πjdf(r)c∗k,n,2πjdf(r)rdr, (3.9)
where∗denotes the complex conjugate. The 2D Fourier modes were interpolated on a uniform grid
in the radial direction and premultiplied by √r, such that the`2-norm corresponds to the energy
norm in cylindrical coordinates. The latter method only needs to be applied once, instead of at each wall-normal location separately, and tends to select frequencies that are energetic over a wide range of wall-normal locations. However, the optimization algorithm converges significantly slower due to the significant increase in the number of constraints fromNs (the number of samples) toNs×Nr
(the number of samples times the number of grid points in the radial direction). The results obtained with the two methods were compared to identify the best method to use in wall-bounded turbulence.