Small-Scale Phase Organization Through Large-Scale Inputs in a Turbulent Boundary Layer

3A-2 9 June 30 - July 3, 2015 Melbourne, Australia

SMALL-SCALE PHASE ORGANIZATION THROUGH LARGE-SCALE INPUTS IN A TURBULENT BOUNDARY LAYER

Subrahmanyam Duvvuri: & Beverley J. McKeon Graduate Aerospace Laboratories California Institute of Technology Pasadena, CA 91125, USA :[email protected]

ABSTRACT & Orlu¨ (2010) suggested that the amplitude modulation co- A synthetic large-scale motion is excited in a flat plate efficient is to a large extent another representation of a cross turbulent boundary layer experiment and its influence on term in the scale-decomposed skewness factor. A clear con- small-scale turbulence is studied. The synthetic scale is nection was established empirically by Mathis et al. (2011) 2 seen to alter the average natural phase relationships in a between the cross term 3uLuS of the skewness, obtained by quasi-deterministic manner, and exhibit a phase-organizing a scale-decomposition of the velocity signal u into large- influence on the directly coupled small-scales. The results and small-scale components (u uL uS), and the am- “ ` and analysis presented here are of interest from a scientific plitude modulation coefficient across a range of Reynolds perspective, and also suggest the possibility of engineering numbers in a turbulent boundary layer. Bernardini & Piroz- schemes for favorable manipulation of energetic small-scale zoli (2011) studied two-point velocity correlations obtained turbulence through practical large-scale inputs. from DNS data of a compressible turbulent boundary layer to show clear evidence of top-down influence of large-scale outer events on the small-scales in the inner part of the boundary layer, and the same was interpreted as amplitude BACKGROUND AND INTRODUCTION modulation. Chung & McKeon (2010) note that the am- Phase relationships in turbulent shear flows have been plitude modulation coefficient can also be interpreted as a an area of research interest since the pioneering works of phase relationship between the large scales and the small- Brown & Thomas (1977) and Bandyopadhyay & Hussain scale envelope. It is clear from the correlation studies thus (1984) on the correlations between large and small scales. far that a definitive phase relationship exists between large- This area has seen a recent surge in activity with several in- and small-scale activity in wall-bounded turbulent flows. vestigations, experimental and numerical, over the past two decades confirming the presence of very-large-scale mo- A formal relationship between the amplitude modula- tions (VLSM) in wall-bounded turbulent flows (see Smits tion coefficient and the skewness for a general statistically et al. 2011 and reference therein). Of particular interest, stationary signal was established recently by the authors from a scientific and an engineering perspective, is the influ- (Duvvuri & McKeon 2015, henceforth referred to as DM15) ence of large scales on small-scale turbulence through non- through a simple multi-scale analysis. Both the quantities linear coupling. The experimental study of Rao et al. (1971) were shown to be fundamentally a measure of phase in in- showing the outer scaling of turbulent bursts in the inner re- teractions between triadically consistent scales, i.e. sets of wavenumbers k ,k ,k such that k k k . Note that gion of a boundary layer provided the first clear evidence t l m nu l “ n ´ m of such scale coupling, and emphasized the significance of triadic interactions assume physical significance given the inner-outer interactions in wall turbulence. quadratic nature of non-linearity that governs coupling between scales. More interestingly, DM15 describes an exper- More recently, through a careful analysis of high- iment in which a synthetic large-scale motion was excited Reynolds number boundary layer data, Hutchins & Maru- in a turbulent boundary layer to generalize and study the in- sic (2007) suggested an amplitude modulation influence fluence of large-scale motions on small-scale turbulence. It on near-wall small-scale turbulence by large-scale motions was shown that the naturally existing phase relationships in centered in the log region. The modulation effect was later the flow can be manipulated in a quasi-deterministic man- quantified by Mathis et al. (2009) through a demodulation ner. The influence of the synthetic scale is felt strongly by scheme in which a correlation coefficient (termed ampli- directly coupled pairs of small-scale wavenumbers; a clear tude modulation coefficient R) between the large-scale ve- phase-organization is seen among the triadically coupled locity signal and an envelope of the small-scale velocity sig- small-scales. nal from a turbulent boundary layer was taken to be a measure of amplitude modulation. Jacobi & McKeon (2013) us- In this paper we briefly recount details of the DM15 ex- ing a co-spectral technique demonstrated that the strongest periment and multi-scale analysis, and expand upon aspects modulating influence in the large-scale signal comes from a not dealt with at length previously. In particular, we show wavenumber that matches the VLSM. Mathis et al. (2009) here in detail the formulation of the amplitude weighted tri- also noted an interesting similarity between the behavior of adic small-scale envelope and demonstrate the small-scale the amplitude modulation coefficient and skewness (S) of phase-organization. A connection is then made between the turbulence signal with wall-normal distance. Schlatter the triadic envelope and the oscillatory component of the

1 normal streamwise Reynolds stress, and finally, efforts to temporal mode for a relatively large streamwise extent. As model and explain the experimentally observed phase be- the forcing is spatially-impulsive and applied only at one havior of the same are very briefly discussed. location, the synthetic mode gradually decays with downstream distance (see figure 2 of DM14).

SYNTHETIC LARGE-SCALE MOTION Jacobi & McKeon (2011) performed a systematic study The Synthetic Large Scale of a spatially-impulsive k-type wall roughness element in a Following Hussain & Reynolds (1970), the turbulent fluctuations are further decomposed as u y,t u y turbulent boundary layer and demonstrated its effectiveness p q “ p q ` u y,t into coherent and turbulent parts, here u is the pe- in exciting a spatio-temporal mode (or a traveling wave) in 1p q the flow. The same technique is used here with modifica- riodic velocity component associated with the forcingr and u is rest of the turbulence. u y is obtained by subjecting tions to minimize static roughness effects and focus on the 1 p q r u y,t to a narrow-band-pass filter around the forcing fre- dynamic forcing of a single spatial (streamwise direction) p q and temporal scale. The experimental description below can quency and then phase-averagingr the filter output with re- be found in greater detail in DM15 and in Duvvuri & McK- spect to the input forcing signal (see DM14 for details); it eon (2014), henceforth referred to as DM14. represents the amplitude and shape of the synthetic mode. Figure 3 shows u y at station-1 (x 2.7δ) over one tempo- p q “ ral period; u y is also calculated at x 3.6δ and x 5.4δ p q “ “ Experimental Set-up (not shown here).r A flat plate zero-pressure-gradient boundary layer flow The streamwiser wavenumber ksls for the mode is esti- was perturbed (or forced) by a spatially-impulsive patch mated by tracking the change in phase of u with x (between of dynamic roughness; the flow has a free-stream velocity successive measurement stations) at its wall-normal peak U 22.1 m/s and is fully turbulent at the perturbation lo- amplitude location. The phase value is picked from the peak 8 “ r cation with momentum thickness based Reynolds number location as it suffers the least from de-correlating effects Re 2750 (see figure 1). The roughness element con- θ « from nearby spectral content during the phase-averaging sists of a thin straight rib at the wall aligned along spanwise procedure. From the change in phase of u between the first direction, and oscillated sinusoidally in the wall-normal di- two measurement stations (where the synthetic mode activ- rection (y) at a set frequency f = 50 Hz to force a single ity is the strongest), k is estimated to be 0.42δ 1 (λ sls r ´ sls “ spanwise constant (2D) spatio-temporal mode in the down- 15δ). The mode velocity is then c ω ksls 0.59U , and “ { “ 8 stream boundary layer. The temporal wavenumber ω (= it’s critical layer location y where c U y is estimated c “ p cq 2π f ) is set by the forcing frequency and the spatial (stream- using the mean velocity profile to be 0.072δ (y 68). c` « wise direction x) wavenumber is dependent on the rough- While this estimate of yc is fairly good, as judged by its ness geometry; it is determined a posteriori from exper- proximity to the wall-normal peak in u, it is important to imental data as explained shortly. Phase-locked hot wire note its sensitivity to ksls due to the sharp mean velocity measurements were made at three different downstream lo- gradient in the near-wall region. A slightr uncertainty in the cations of the perturbation as marked in figure 1; data pre- estimation of ksls (and hence c), possibly due to the finite sented in this paper is from the first measurement station number of phase-averaging cycles, can result in a signifi- (referred to as station-1) located at x 2.7δ downstream y “ cant change to c. In the following section we show the of the roughness perturbation (δ 16.55 mm is the local “ influence of the synthetic large scale on the natural triadic boundary layer thickness). At this location Re 2780 and θ “ phase relationships in a turbulent boundary layer. the friction Reynolds number is estimated using the Coles- Fernholz empirical relation to be Re 940. τ « TRIADIC SCALE INTERACTIONS Turbulence Spectrum An understanding of the triadic interactions between The time-resolved hot wire velocity signal U y,t , scales of turbulence provides valuable insight into the com- p q where t is time, is decomposed into mean U y and fluc- plex dynamics of turbulent flows. For instance, Sharma p q tuating components u y,t (Reynolds decomposition). Fig- & McKeon (2013) show through analysis of the Navier- p q ure 2 shows the power spectrum of the fluctuations Euu Stokes resolvent operator that a set of three triadically con- p q for the forced flow in a pre-multiplied form, and also the sistent spatio-temporal modes is able to produce complex difference spectra showing the fractional change in power structures such as modulating packets of hairpin vortices levels between forced and canonical flows EF EC rel- observed in wall-bounded turbulent flows. We summarize uu ´ uu ative to the canonical flow EC with no forcing. Note here uu ` ˘ here the analysis of DM15 showing skewness and ampli- that temporal data is projected onto the streamwise direc- tude modulation coefficient to be measures of phase in tri- ` ˘ tion at all wall-normal locations using the local mean ve- adic interactions. locity U y . The energetic narrow-band activity extending Consider a signal u of statistically stationary velocity p q in the wall-normal direction in the large-scale region of the fluctuations in streamwise direction x spectrum shows the presence of a synthetic scale, and this is confirmed by the difference spectrum. The cut-off filter to separate large and small scales for subsequent analysis is 8 u α sin k x φ , (1) set at λ 5δ, where the separation between the synthetic “ i p i ` iq γ “ i 1 large scale and energetic small scales is clear for all y. ÿ“ It is important to note that multiple scales are excited immediately downstream of the roughness pertur- with 0 k k for i j and k being the largest scale in ă i ă j ă 1 bation (x 1). However, these evanescent modes de- the signal. The skewness S u3 σ 3, where denotes À “ x y{ x y cay quickly and the flow is dominated by a single spatio- the mean operator and σ is the standard deviation of u, can

2 Figure 1. Schematic of the experimental set-up (not to scale). The turbulent boundary layer is forced by a spatially impulsive single rib dynamic roughness at a streamwise location (x 0) where the momentum thickness Reynolds number Re 2750. “ θ « The rib has a height of 0.76 mm (oscillation amplitude) and a width (streamwise extent) of 1.5 mm. Phase-locked hot wire measurements are made at three downstream stations as marked.

Figure 2. Pre-multiplied power spectrum (left) of streamwise velocity fluctuations for the forced flow and the difference spectrum (right) between the forced and canonical flows at station-1 (x 2.7δ). The difference spectrum shows the fractional “ change in power levels in presence of the synthetic mode EF EC relative to the canonical flow EC . In both plots the uu ´ uu uu vertical line at λ 5δ indicates the scale filter cut-off location, and the horizontal dashed line at y 0.072δ indicates the γ “ ` ˘ c`“ ˘ estimated critical layer location for the synthetic mode.

be reduced to the form ponents using a spatial Fourier ﬁlter a set wavenumber kγ

γ 1 6 ´ 8 S αlαmαn sin φl φm φn u α sin k x φ , u α sin k x φ . (3) “ 4σ 3 p ` ´ q L “ i p i ` iq S “ i p i ` iq l,m,n i 1 i γ @ ÿ | “ “ kl km kn ÿ ÿ k ăk ăk l ` m“ n (2) Note that for λγ 2π kγ is set at 5δ for experimental data 3 8 2 “ { α αn sin 2φ φn . analysis. An envelope E of the small-scale signal is ob- ` 4σ 3 l p l ´ q l 1 tained using a Hilbert transform procedure (see DM15 for k ÿ“2k p n“ l q details), and R is defined to be the correlation coefficient between the large-scale signal and the large-scale component From the above equation it is seen that skewness is nothing of E , written as EL. From equation 3, the expression for R but a weighted (by mode amplitudes) and normalized (by can be reduced to the form 3 σ ) sum of the quantity sin φ φm φn over all sets of p l ` ´ q u E 1 triads k ,k ,k and wavenumber pairs k ,k such that L L l m n l n R x 1 y 1 αlαmαn sin φl φm φn , t u t u “ 2 2 2 2 “ Ω p ` ´ q kl kn km and 2kl kn respectively. uL EL l,m,n “ ´ “ k@ ÿk |k We now consider the amplitude modulation coefficient n´ m“ l @ D @ D 0 kl kγ R following the procedure outlined by Mathis et al. (2009). kă,k ăk m ną γ The velocity signal is split into large- and small-scale com- (4)

3 Figure 3. Synthetic mode u (left), triadic small-scale envelope E (center), and Reynolds stress component Rxx (right) over one temporal period at station-1. The top and bottom panels are scaled linearly and logarithmically respectively in y. Data in 1 2 2 this ﬁgure are in raw units, msr ´ for u, and m s´ for E and Rxx.r r

r r r 2 1 2 1 where Ω u 2 E 2 is the normalization factor for the “ x Ly x L y covariance u E . Only triadic wavenumbers contribute to x L Ly R in a manner similar to S. However, the filtering process places a restriction on the class of triadic sets k ,k ,k ; t l m nu assuming without a loss of generality that k k k , l ă m ă n the large scale kl and the small scales km,kn necessarily lie on either side of the filter cut-off kγ to make a non-zero contribution R. Following the phase interpretation of Chung & McK- eon (2010), it is seen that φ φ φ ∆φ in equation 4 l ` m ´ n “ is the phase difference (with a π 2 radians offset) between { the large-scale kl and the envelope of the triadic small- scales km,kn. Hence R can be interpreted as an amplitude- weighted (and normalized) phase measure sin ∆φ of all p q large- and small-scale triadic interactions. Notice that skewness (equation 2) is also a measure of the same phase quan- F C tity, but over all triadic interactions with no scale restric- Figure 4. Changes in the skewness ∆S S S ( ) and “ F´ C‚ tions. By considering the quantities u3 , u3 , u2 u in the amplitude modulation coefficient ∆R R R (‚) in x Sy x Ly x L Sy “ ´ a manner similar to u3 and u E , the following exact presence of the synthetic large scale. x y x L Ly relationship between S and R can be written (see DM15 for details) critical layer location. The results in figure 4 suggest that the synthetic large-scale drives the envelope of all small- 3 3 3 2 σ S 1.5ΩR uS uL 3 uLuS . (5) scales towards being in (∆S, ∆R 0) and out (∆S, ∆R 0) “ ` x y ` x y ` x y ą ă of phase with it below and above its critical layer location The influence of the synthetic large scale on the nat- respectively, thereby altering the natural large- and small- urally existing triadic phase relationship can be seen in scale phase relationships (or the degree of amplitude mod- changes to values of S and R. Figure 4 shows the differ- ulation) in the flow. ence ∆S and ∆R between the forced and canonical flows in the wall-normal region around the critical layer. There is a marked increase in the values of S and R in the region SMALL-SCALE ORGANIZATION 0.02δ y 0.1δ and decrease in the region 0.1δ y The direct influence of the synthetic large-scale is seen ă ă ă ă 0.4δ in presence of the synthetic large-scale. The cross-over on the small-scales that couple in a triadic manner. Consider location y δ 0.1 corresponds well to the synthetic mode two small-scale triadic wavenumbers k ,k ( k ) that cou- { “ m n ą γ peak location (see figure 5), and is close to the estimated ple directly to the synthetic large-scale, i.e. k k k ; n ´ m “ sls 4 the envelope of km,kn has a wavenumber of ksls due to triadic small-scales are in phase with the synthetic large- the triadic condition. An average triadic envelope of all scale (Ψ 1) near the wall and out of phase (Ψ 1) “ “ ´ such small-scale wavenumbers km,kn can be obtained in away from the wall; a sharp phase jump of π radians occurs the following simple manner. The small-scale velocity uS at y 0.04δ, close to the estimated critical layer location. 2 “ (equation 3) is squared (uS) then phase-averaged with re- The phase behavior of the directly coupled small-scales is spect to the forcing signal and mean removed (denoted by consistent with the altered large and small-scale phase rela- 2 2 tionship suggested by skewness and amplitude modulation uS uS E ). The phase-averaging procedure picks out ´ x y “ 2 coefficient. the component corresponding to ksls from uS, r r r The quantity E can also be interpreted in a more tradi- tional manner in terms of the normal streamwise component 1 2 E α α cos k x φ φ . (6) Rxx u of ther Reynolds stress tensor R. The oscilla- “ 2 m n p sls ` n ´ mq ” x y m,n tory` component˘ of Rxx due to the forcing (denoted by Rxx) k @kÿ |k r n´ m“ sls can be obtained by the phase-averaging procedure described km,kn kγ ą earlier; we write R u2 u2 . Although E and Rr xx “ ´ x y xx are similar quantities, the scale restriction imposed on the For a specific set of small-scale triadic wavenumbers km,kn velocity fluctuationsr in ther calculationr of E meansr thatr it such that u α sin k x φ α sin k x φ , an mn “ m p m ` mq ` n p n ` nq represents only the contributions from small-scale velocity envelope A x can be written as the square of the ana- p q fluctuations to the oscillating Reynolds stress.r lytic function modulus following DM2015, i.e. A x p q “ The phase variation of Rxx y , seen in figure 3 (right u2 x 2 x u p q mn H , where H is the Hilbert transform of mn. panel), is very similar to that of E y . The phase rela- p q` p q p q After removing the mean terms A reduces to the following tionship between synthetic larger scale and the oscillating form Reynolds stress can be quantified inr terms of correlation coefficient Φ (similar to Ψ) A x 2αmαn cos kn km x φn φm p q “ rp ´ q ` ´ s (7) 2αmαn cos kslsx φn φm . “ p ` ´ q u Rxx Φ . (9) A1 2 E 1 2 “ 2 { 2 { A straightforward comparison between equations 6 and 7 u r rRxx reveals that, in essence, E captures the amplitude-weighted A E A E average phase of the triadic envelope across all sets of r r small-scale wavenumbersr that couple directly with the syn- As expected from a visual inspection of figure 3, a phase thetic scale. jump of π radians between u and Rxx is seen from the be- The center panels in figure 3 show E from the exper- havior of Φ near the critical layer. This phase jump can iment. A certain correlation between the synthetic large be predicted from a simplifiedr transferr function formula- scale (u) and the average small-scale triadicr envelope (E ) tion between the isolated synthetic scale and the oscillatory can be seen by just a visual inspection of the figure. The Reynolds stress obtained from the governing Navier-Stokes two quantitiesr are in phase close to the wall, and outr of equations, this is a subject of ongoing research (Chung et phase away from it following an abrupt jump of π radians al., in preparation). The role of the critical layer in anchor- in the phase of E close to the critical layer location. The ing the wall-normal location of phase reversal between large phase relationship can be analyzed in a quantitative manner and small scales, i.e. the zero-crossing locations of R and Ψ by defining a correlationr coefficient Ψ between u and E , is also under investigation. akin to the amplitude modulation coefficient R r r u E CONCLUDING REMARKS Ψ . (8) The natural triadic phase relationships in a turbulent A1 2 E 1 2 “ 2 { 2 { boundary were manipulated by synthetically introducing a u r rE large-scale motion in the flow. By interpreting the skewness A E A E and the amplitude modulation coefficient of the streamwise r r Note that R gives the average phase between all large- and velocity signal as a measure of the average phase in tri- small-scales in the flow (including the synthetic scale when adically consistent wavenumber interactions, the synthetic present), whereas Ψ is a measure of the phase between the large scale is shown to alter such phase relationships in synthetic scale and the average triadic envelope of small- a quasi-deterministic manner. Note that the phase rela- scales that are in direct coupling. Thus Ψ represents the tionships are discussed here in a time-averaged sense, the coupling due to one wavenumber within the broader uL. instantaneous flow field is still turbulent, and hence the Ψ y at station-1 is shown in figure 5 along with the normal- term quasi-deterministic. The influence of the synthetic p q ized energy of the synthetic large-scale. The wall-normal large-scale on directly coupled small scales is understood locations where the synthetic large-scale energy drops to by defining an average triadic small-scale envelope E ; a 15% of its peak value are shown by dash-dot lines for ref- clear phase-locking or organization effect is seen on the erence; the region between these lines is where effects of small scales. The quantity E is related to the tradition-r the synthetic large-scale are expected to dominate. As the ally studied normal streamwise component of the oscilla- energy in the synthetic mode drops close to zero, the cor- tory Reynolds stress Rxx. Effortsr are presently being made relation coefficient Ψ becomes noisy (outside the 15% en- to model the observed phase relationships between the iso- ergy reference lines). A clear organization in phase of di- lated synthetic scale andr the Reynolds stress directly from rectly coupled small-scales by the synthetic large-scale is the Navier-Stokes equations, and also understand the role of seen in the region where the synthetic scale is active. The the critical layer in the same context.

5 Figure 5. Correlation coefﬁcients Ψ ( ), Φ (‚), and normalized synthetic mode energy (–). The estimated critical layer ‚ location yc is marked by a dashed line, and the locations where the synthetic mode energy drops to 15% of it’s peak value are marked by dash-dot lines.

The experimental technique presented here helps us in a turbulent boundary layer. Journal of Fluid Mechan- better understand the scale coupling in turbulent wall- ics 767, R4. bounded flows through deterministic perturbations. In con- Hussain, A. K. M. F. & Reynolds, W. C. 1970 The mechan- junction with the resolvent based model of the Navier- ics of an organized wave in turbulent shear flow. Journal Stokes equations (e.g. Sharma & McKeon, 2013), the re- of Fluid Mechanics 41, 241–258. sults presented here open the possibility of setting up a Hutchins, N. & Marusic, I. 2007 Large-scale influences in practical framework for favorable manipulation of energetic near-wall turbulence. Philosophical Transactions of the near-wall small-scale turbulence through large-scale inputs. Royal Society A 365, 647–664. The authors gratefully acknowledge AFOSR (grant FA Jacobi, I. & McKeon, B. J. 2011 Dynamic roughness per- 9550-12-1-0469, program manager D. Smith) for the finan- turbation of a turbulent boundary layer. Journal of Fluid cial support of this work, and a graduate fellowship (SD) Mechanics 688, 258–296. from the Resnick Institute at Caltech. Jacobi, I. & McKeon, B. J. 2013 Phase relationships between large and small scales in the turbulent boundary layer. Experiments in Fluids 54 (3), 1481. REFERENCES Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale Bandyopadhyay, P. R. & Hussain, A. K. M. F. 1984 The amplitude modulation of the small-scale structures in tur- coupling between scales in shear flows. Physics of Fluids bulent boundary layers. Journal of Fluid Mechanics 628, 27 (9), 2221–2228. 311–337. Bernardini, M. & Pirozzoli, S. 2011 Inner/outer layer in- Mathis, R., Marusic, I., Hutchins, N. & Sreenivasan, K. R. teractions in turbulent boundary layers: A refined mea- 2011 The relationship between the velocity skewness sure for the large-scale amplitude modulation mecha- and the amplitude modulation of the small scale by the nism. Physics of Fluids 23 (6), 061701. large scale in turbulent boundary layers. Physics of Flu- Brown, G. L. & Thomas, A. S. W. 1977 Large structure ids 23 (12), 121702. in a turbulent boundary layer. Physics of Fluids 20 (10), Rao, K. N., Narasimha, R. & Narayanan, M. A. B. 1971 The S243–S252. ‘bursting’ phenomenon in a turbulent boundary layer. Chung, D., Duvvuri, S., Jacobi, I. & McKeon, B. J. (in Journal of Fluid Mechanics 48, 339–352. preparation) Interactions between scales in wall turbu- Schlatter, P. & Orlu,¨ R. 2010 Quantifying the interaction lence: phase relationships, amplitude modulation and the between large and small scales in wall-bounded turbu- importance of critical layers. lent flows: A note of caution. Physics of Fluids 22 (5), Chung, D. & McKeon, B. J. 2010 Large-eddy simulation 051704. of large-scale structures in long channel flow. Journal of Sharma, A. S. & McKeon, B. J. 2013 On coherent structure Fluid Mechanics 661, 341–364. in wall turbulence. Journal of Fluid Mechanics 728, 196– Duvvuri, S. & McKeon, B. J. 2014 Phase relationships in 238. presence of a synthetic large-scale in a turbulent bound- Smits, A. J., McKeon, B. J. & Marusic, I. 2011 High- ary layer. In 44th AIAA Fluid Dynamics Conference, At- Reynolds number wall turbulence. Annual Review of lanta GA, June 2014. AIAA paper 2014-2883. Fluid Mechanics 43, 353–375. Duvvuri, S. & McKeon, B. J. 2015 Triadic scale interactions