The naive version of the demodulation procedure employed on the unperturbed flow in section 7.6 can be applied to both perturbed regimes, in order to track the ratio of the peak frequency in the spectrum of the modulating signal to the frequency of the chosen carrier, shown in figure 8.18. As in the demodulation for the unperturbed flow, the marginal indication of a modulating relationship appeared only for a range of very large scale motions, with no obvious indication of smaller-scale involvement. The statically perturbed case appears almost identical to the unperturbed flow; the dynamic case shows that the range of large scale is limited in the region of the internal layers. The proper interpretation of these trends must wait for a more robust demodulation approach.
8.9
Summary
The relationship between large-scale and small-scale motions in the perturbed turbulent boundary layer was investigated using the correlation and cospectral techniques previously applied to the unperturbed boundary layer, as well as phase-locked analysis for the dynamically forced flow.
Phase-locked velocity maps of the streamwise and wall-normal fluctuating velocity components and Reynolds stresses allowed for identification of well-defined mode shapes in the envelopes of fluctuating quantities, much like the large-scale fluctuations were isolated previously. These mode shapes were then employed to consider the extent to which the phase relationships between large and small scales in the dynamically perturbed flow could be treated as a linear superposition of those
fδ/U ∞ y/ δ 10−2 10−1 100 101 102 10−2 10−1 100 f* m/f 0 0.2 0.4 0.6 0.8 1 fδ/U ∞ y/ δ 10−2 10−1 100 101 102 10−2 10−1 100 f* m/f 0 0.2 0.4 0.6 0.8 1
Figure 8.18: The ratio of the peak frequency of the information signal m(t) denoted f∗ m to the
corresponding frequency of the instantaneous velocity signal spectrum, f, is shown as a contour map, over the range of the instantaneous velocity signal frequencies and wall-normal locations, normalized in outer units. The dashed black lines mark the spectral limits set by the product detector; the circles are the ridgeline of the dominant interacting scales from figure 8.10. (Left) static (Right) dynamic
relationships in the statically perturbed flow and the synthetic large scales, treated in isolation. By employing the cross-correlation isocontour maps, it was shown that this linear superposition could capture a significant amount of phase relationship in the dynamically perturbed case, which is likely a consequence of the very long wavelength of the synthetic large scale.
The correlation coefficient and the closely related streamwise skewness were also employed to compare the perturbed flows; in particular, the decomposition of the streamwise skewness pro- posed by Mathis et al. [2011] was used to explore how the balance of contributions from large- and small-scale motions tended to modify the shape and zero-crossings of the correlation and skewness functions. The correlation coefficient was also reconstructed from the cross-correlation function in order to test the validity of Taylor’s hypothesis, revealing tentative support for the observation that the synthetic large scale tends to convect slower than the local mean velocity in the flow, across most of the boundary layer, beyond the location of the critical layer.
Finally, the cospectral density maps and accompanying phase maps were used to compare the relative phase lag profile (over the boundary layer thickness) between large- and small-scale motions, as a function of the scale size. The suppression of the lead of small scales over large scales in the static flow was observed to be consistent with the picture of the second internal layer as a newly ‘born’ boundary layer, indicated also in the phase-locked velocity maps of section 8.2; the more extreme trend in the dynamically perturbed case was consistent with the shallower inclination of
the synthetic large scales, which therefore tend to align more closely with the inclination of natural small-scale motions in the flow field. Moreover, the separability of the dynamic and static effects in the cross-correlation maps must be seen as a consequence of the phase-locked small-scale envelope capturing a significant amount of large-scale content in order to explain the phase trends from the cospectrum. This conclusion magnifies the overall message of the cospectral density maps, which is the profound importance of the use of an envelope procedure on the results of all of the correlation analyses considered.
Chapter 9
Conclusions
9.1
Summary of Results
The investigation of the perturbed boundary layer considered both a static roughness perturbation and a dynamically actuated roughness perturbation, and the relationship between these perturbed flows and the unperturbed zero-pressure gradient boundary layer. The static perturbation distorted the boundary layer layer in a mean sense, producing a stress bore and temporarily disrupting the near-wall processes associated with the equilibrium boundary layer. The stress bore provided an important means by which to consider the relaxation of the boundary layer back towards equilib- rium. A new scaling of the streamwise velocity gradient was proposed and was used in conjunction with statistical and spectral maps of the developing flow field downstream of the perturbation to better understand the structural effects of a roughness perturbation on the boundary layer. This characterization of the static perturbation was essential to interpreting the dynamic perturbation which was subsequently considered. The dynamic perturbation involved actuating the roughness patch temporally in order to target large-scale motions in the boundary layer. The extent to which the dynamic wave aspect of the dynamic perturbation could be considered a linear addition to the underlying roughness perturbation was a constant theme of this experiment, and was explored by spectral and statistical means, and later in the context of large- to small-scale phase relationships in the flow field. The large-scale motion produced in the boundary layer by the dynamic forcing was shown to manifest as a coherent and persistent very long wavelength motion which could be isolated and studied by phase-locked measurements. A resolvent-based technique was used to pre- dict the shape of the synthetic large-scale motion from its experimentally determined parameters, and the predictions were interpreted in the context of the stress bore associated with the roughness actuation. The relationship of the synthetic large-scale motion to smaller scales in the flow was then explored using a collection of correlation based techniques. In particular, the cospectral density be- tween large scales and the envelope of small-scale motions was used to explore both the unperturbed and perturbed boundary layers, and indicated the importance of the envelope technique to previ-
ous observations of an amplitude modulation relationship between scales in the boundary layer. A phase interpretation was developed to provide physical intuition to the relationships between large and small scales, and it was shown that the very large scale synthetic motion generated in the flow could be considered, to a significant extent, a linear superposition on the nonequilibrium base flow, due to the significant difference in its wavelength.