In this paper, experimental utilization of the proper orthogonal decomposition for the high Reynolds number flat plate turbulent boundary layer is studied. The data was obtained in the large LML wind tunnel, The boundary layer thickness was approximately 30 cm in the end of 21.6 m long test section. The streamwise component of the turbulent boundary layer velocity field was measured using a hot-wire rake of 143 probes. The array of probes enabled
simultaneous sampling and therefore computing the cross-spectral tensor which formed the kernel of the Lumley integral equation.
The normalized eigenvalue distribution shows that the first POD mode integrated over fre-quency and azimuthal mode number has more than 40% of the turbulence kinetic energy, while the second one has about 20% of the total turbulence kinetic energy. It is possible to recover about 90% of the kinetic energy only using the first four POD modes. Similar figures are ob-tained at both Reynolds number with no significant indication of Reynolds number dependence.
The eigenspectra always peak near zero frequency and most of the large scale features are found below 100 Hz and 50 Hz for the high and low Reynolds number cases studied here respectively. This could be the effect of aliasing from the missing dimension as suggested by George2. The results indicate that it might be possible to scale the eigenspectra especially in the inertial range of turbulence similar to the similarity of the eigenfunctions and eigenspectra within the outer layer of turbulent wall flow. The kinetic energy distribution is maximum at spanwise Fourier mode-1, while there is a slight difference between spanwise Fourier mode-0 and-1. The normalized eigenvalue distribution obtained from two different Reynolds numbers have almost the same distribution and features.
The pre-multiplied eigenspectra are produced in a similar fashion as the pre-multiplied ve-locity spectra. Instead of wave-number spectra, frequency domain spectra were used because of difficulty in assigning a true convective velocity. The pre-multiplied eigenspectra have only one peak whereas the pre-multiplied velocity spectra have two peaks; one in the near-wall region and another one in the log-layer. The comparison between the pre-multiplied velocity spectra and the pre-multiplied eigenspectra shows that frequency of the peak of pre-multiplied eigenspectra coincides with frequency of the secondary peak observed in the pre-multiplied velocity spectra.
This means that the large scale modes of turbulent boundary layer are indeed connected to the physical mechanisms causing the secondary peak of the pre-multiplied velocity spectrum. In support of this, tests of inner and outer scaling of pre-multiplied eigenspectra at both Reynolds numbers suggested the outer scaling parameters outperform the inner scaling parameters. One should always note that the near-wall peak in turbulence kinetic energy is confined to a very small area, only a few percent of δ, as we consider the entire turbulent boundary layer. In fact, 90% of the boundary layer is outer flow. Even though the magnitude of local turbulence
kinetic energy is large near the wall, its net contribution to the total turbulence kinetic energy is small. It is therefore not surprising to see only one peak in pre-multiplied eigenspectra away from the near-wall region.
Reconstructed velocities were studied to see the effect of energetic and non-energetic modes of the POD on the turbulence statistics. At both Reynolds numbers, the rms profiles based on energetic reconstruction (with modes carrying significant amount of energy) and that based on non-energetic reconstruction (with modes carrying very small amount energy, and essentially the modes which are not considered energetic) indicate three distinct behaviors of the fluctu-ating velocities across the boundary layer. The higher Reynolds number case produces better defined layers due to absence of low Reynolds number and viscosity effects. The reconstructed rms based only on the energetic part of the modes decays exponentially between the log-layer and the wall, stays approximately constant through the log-layer, then decays exponentially after the log-layer. On the other hand, the non-energetic reconstruction shows that urmsdecays exponentially from wall to the top of the log-layer and stays flat at its minimum once the log-layer is over. Similar pictures are also seen in lower Reynolds number test, but this time the effect of viscosity and Reynolds number effects are observed. This is because the probe locations are fixed in both cases, while the wall unit is doubled in the low Reynolds number case. The outer scaling seems performs much better for both energetic and non-energetic re-constructed statistics which suggests that small scale motions right outside the inner sublayer are strongly influenced by the large scale outer layer motions.
These findings supports earlier findings of Tutkun et al30, in which authors showed the strong connection between the near-wall region and outer layer by studying the two-point correlations of the same data. Amplitude modulation of small-scale motions near the wall by the large scale motions of outer layer has previously been documented for both turbulent boundary layer flow and turbulent channel flow50. The statistics of reconstructed turbulence based on our POD results indicate that large energy carrying POD modes strongly influence the small-scale, non-energetic, fluctuations, not only in near-wall region but also throughout the boundary layer.
Appendix A: POD Analysis Procedure and Numerical Implementation
The POD integral equation given by Eq. (6) can be solved numerically by approximating the integral using the trapezoidal rule:
Z
wherefiand ∆yirepresent the values of the functionf at the grid points (or measurement points in this case) and value of the integrand around these grid points respectively. Justification of using the trapezoidal rule in computing the POD integral equation is detailed by Moin and Moser16. The numerically discretized form of the POD integral equation, Eq. (6), results in an eigenvalue problem, which can be written as follows:
Aφ(n) =λ(n)φ(n) (A2) Solution to the eigenvalue problem given by Eq. (A2) is easy to obtain using a standard numerical eigenvalue solver if the kernel, A, is Hermitian symmetric. Because of the logarith-mic stretching of the probes along the wall-normal direction, the A in this experiment is not Hermitian symmetric as shown below:
A=
where the first matrix is the two-point cross-spectral tensor and the second matrix is the diagonal matrix with the integrands. The remedy to make the kernel Hermitian symmetric is to rearrange Eq. (A2) and multiply both sides of it with the diagonal matrix of square-roots of the integrands:
where w,S and φ(n) are:
The solution to Eq. (A4) does not produce the correct eigenfunctions because of multiplica-tion of both sides of Eq. (A2) by w. Therefore, the eigenfunctions, ψ, which will be obtained from the solutions are:
The correct eigenfunctions can be found by multiplying the solutions with the inverse of the diagonal matrix formed by the square-root of the integrand elements:
φ(n)=
The number of eigenfunctions and eigenvalues depends on the size of the kernel which is the two-point cross-spectral tensor. The kernel is a square matrix of (M×N)×(M×N), where
M is the number of velocity components included in the analysis and N is the number of grids, or measurement locations within the field of interest10,17. In our experiments, we had 11 probes in the wall-normal direction,y, spaced logarithmically and only the streamwise velocity component of turbulent velocities was measured. Therefore, the size of the the kernel is 11×11 for each pair of spanwise Fourier mode, k, and frequency, f. Thus, the maximum number of POD modes which can be extracted from the current setting is 11 for each k and f.
ACKNOWLEDGMENTS
The authors would like to thank Peter B. V. Johansson, Faraz Mehdi, Michel Stanislas, Jean-Marc Foucaut, Sebastien Coudert, Jim Kostas, J¨oel Delville and Carine Fourment for their collaboration during the experiments. Unfortunately, the last two members of the team (J. Delville and C. Fourment) passed away in July 2014 because of a traffic accident while returning back to Poitiers from a measurement campaign in Modane. They are greatly missed by the turbulence and flow control community.
The original experimental data were generated by the WALLTURB project. WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the CEC under the 6th framework program (CONTRACT No: AST4-CT-2005-516008). Tutkun’s work is partially financed by the research project DOMT – Developments in Optical Measurement Technologies funded by the Research Council of Norway with project number 231491 under the Frinatek program.
∗ murat.tutkun@ife.no
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