Although the cross-correlation and cospectral approaches provide a rich source of information about the relationship between large- and small-scale motions and appear to be better suited than the cor- relation coefficient for robust analysis, nevertheless, the correlation coefficient possesses a remarkable similarity to the streamwise velocity skewness which merits further attention. Mathis et al. [2009a] first noted the similarity between the correlation coefficient describing the scale interaction and the streamwise skewness; subsequently Mathis et al. [2011] showed how the two quantities are in- timately related through a decomposition of the streamwise skewness into large- and small-scale contributions. The connection between these quantities ultimately helps to explain the variation in the zero-crossing locations of the two functions, where the zero-crossing has been shown in the preceding sections to be important to the physical interpretation of the structure of the boundary layer.
The streamwise skewness, denoted Su, when plotted against wall-normal location, typically ex-
hibits a zero-crossing near the wall, as well as an apparent region of tangency (possibly containing one or two additional zero-crossings) in the logarithmic region. Between the near-wall crossing and the tangent region, the skewness is nominally negative and convex, although many measurements at higher Reynolds numbers report this region as marginally positive (Fernholz and Finley [1996] and
¨
Orl¨u [2009]). In any case, the Reynolds number dependence is significant to all of these features. Mathis et al. [2011] divided the instantaneous velocity signal into large (uL) and small (uR) scale
signals and then expanded the definition of the third-moment of the velocity fluctuation, yielding equation 8.6
u23/2S
u=u3= (uL)3+3(uL)2(uR) +3(uL)(uR)2+ (uR)3 (8.6)
where the over-bars denote time-averages with the means of each component subtracted prior to av- eraging. Examining the contributions of the four terms indicates that the only negative contribution is from the small-scale skewness,(uR)3.
Figure 8.14: Illustrations of the relative orientation of the large-scale motions and corresponding envelopes of small-scale fluctuations, in the streamwise direction. (Top, left) Unperturbed, compare Chung and McKeon [2010]. (Top, right) Static perturbation, where the large-scale inclination (in red) is reversed in the region between the two internal layers, which was speculated to be an effect of the roughness perturbation. (Bottom) Dynamic perturbation in isolation (i.e., phase-locked), where the small-scale envelope is that of the large scales from the static perturbation, and the artificial large scale is in red. Note that the cross-over point tends to shift up in the case of the static perturbation and shifts down in a dynamic case where the artificial, highly inclined large structure dominate. Note also that the artificial scale is more inclined than the natural large scale but less than the natural small scales.
10−2 10−1 100 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 y/δ Su
Figure 8.15: The decomposed skewness, Su, at x/δ ≈ 3.4 with: black △ Su; red ◻ u3L; purple ▽
u2
LuR; green ◯ uLu2R; blue × u3R. The first and second internal layers in the perturbed flows are
marked with dashed and dotted lines, respectively. The filter size isτ =2.5δ/U, to match that of Mathis et al. [2011].
Mathis et al. [2011] reported that terms (uR)3 and 3(uL)(uR)2 produced the dominant con-
tributions to the skewness. However, they did not emphasize the filter-cutoff dependence of this conclusion: although the overall skewness, Su, or correlation coefficient, R, is insensitive to the
choice of filter-cutoff, obviously the distribution of the skewness amongst the different large and small scale components will depend quite strongly on cutoff size. Mathis et al. [2011] employed a filter cutoff of λ+
x = 7000 orλ/δ = 2.5. But, if a smaller filter cutoff is employed (λ/δ = 1), then
the terms(uL)3and(uL)2(uR)are no longer insignificant, because more of the large scale motions
(greater than 1δ) are being included with the uL terms, as opposed to being grouped into the uR
signal. The question of the dependence on filter cutoff becomes even more important in the case of a perturbed flow in which large-scale motions are synthesized. Figure 8.16 shows the skewness for the three flow conditions at a single streamwise location, employing the same cutoff used in Mathis et al. [2011] for consistency, but again caution should be exercised in interpreting the meaning of each of the four terms in the decomposition, until a more careful study of the filter cutoff effect can be performed.
The static perturbation tends to increase the magnitude of the skewness across the boundary layer, while extending the convex region further from the wall. The dynamic perturbation increases the magnitude of the skewness only farther from the wall, thereby increasing the region of negative skew. In order to identify the sources of these effects, figure 8.17 shows the decomposition of the skewness in the perturbed flows. The magnitude increase of the statically perturbed flow can be
10−2 10−1 100 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 y/δ Su
Figure 8.16: The skewness, Su, at x/δ ≈3.4 with: red unperturbed; green ◻ statically perturbed;
blue◊dynamically perturbed. The filter size isτ=2.5δ/U, to match that of Mathis et al. [2011].
attributed to increases in the uLu2R and u3R terms, which are both expected from the increase in
turbulence intensity observed within the stress bore. In the dynamic case, the negative skewness is associated with an increase in the magnitude of theu3
L term, which is expected due to the artificial
large scales produced in the dynamically actuated flow.
The changes in the components of skewness can also be expressed in the context of shifts in the location of the zero-crossing. The static impulse tended to exaggerate the near-wall positive skew observed previously in the uniform roughness studies (Bandyopadhyay and Watson [1988] and Keirsbulck et al. [2002]), which is a consequence of the increase in small-scale fluctuations from the roughness. The results in the observed zero-crossing shift to a location farther from the wall. The dynamic impulse, however, tended to increase the region of negative skewness, shifting the zero-crossing closer to the wall.
As Reynolds number increases, the near-wall zero-crossing of skewness tends to move nearer to the wall (in outer units), monotonically, increasing the region of negative skewness, as seen in the results of ¨Orl¨u [2009]. Thus the dynamic actuation tends to produce a skewness profile which shares this attribute of higher Reynolds number flows, despite the smaller momentum thickness. However, the presence of a near-wall zero-crossing may be obscured by both Reynolds number and spatial resolution effects, and therefore the observation that the dynamically perturbed flow appears similar to higher Reynolds number flows must remain tentative. Nevertheless, the connection between the phase information derived from correlation techniques and the streamwise skewness offers a potential avenue for generalizing the phase observations considered in the current study to a much broader set of more general experiments.
10−2 10−1 100 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 y/δ Su 10−2 10−1 100 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 y/δ Su
Figure 8.17: The decomposed skewness,Su, atx/δ≈3.4 with symbols following figure 8.15, for the
statically perturbed flow (left) and dynamically perturbed flow (right)