ON THE INFLUENCE OF OUTER LARGE -SCALE STRUCTURES

ON NEAR -WALL TURBULENCE IN CHANNEL FLOW

L. Agostini and M.A. Leschziner

Aeronautics Department, Imperial College London, London SW7 2AZ

mike.leschziner@imperial.ac.uk

Abstract

DNS data for channel flow atReτ = 1025 are usedto analyse the interaction between large outer scalesin the log-law region – referred to assuper-streaks –and the small-scale, streaky, streamwise-velocity fluc-tuations in the viscosity-affected near-wall layer. Thestudy is inspired by extensive experimental investi-gations by Mathis, Marusic and Hutchins, culminat-ing in a predictive model that describes, in a suppos-edly universal manner, the “footprinting” and “mod-ulating” effects of the outer structures on the small-scale near-wall motions. The approach used herein isbased on the examination of joint PDFs for the small-scale fluctuations, conditioned on regions of large-scale footprints. The large and small scales are sep-arated by means of the Huang-Hilbert empirical-modedecomposition, the validity of which is demonstratedby way of pre-multiplied energy spectra, correlationmaps and energy profiles for both scales. Observationsderived from the PDFs then form the basis of assessingthe validity of the assumptions underlying the model.Although the present observations support some ele-ments of the model, the results imply that modulationby negative and positive large-scale fluctuations differgreatly – an asymmetric response that is not compati-ble with the model.

1 Introduction

Friction-drag reduction hinges on methods that reducenear-wall turbulence, the structure of which is char-acterized by streaks and associated small-scale, quasi-streamwise vortices. One effective method is to impartan oscillatory spanwise Stokes layer onto the turbulentboundary layer by means of in-plane wall oscillations.This is known to result in drag-reduction margins of upto 45% in optimal actuation conditions. However, thismargin declines approximately withRe−0.2, and thisrate of degradation is suspected to be partly rooted inthe influence of energetic outer structures on the near-wall layer. Touber and Leschziner (2012) and Agostiniet al. (2014) provide clear evidence, forReτ = 500and 1025, Respectively, of outer structures causingamplification and virtual annihilation of the streaks inpatches subjected to high- and low-velocity near-wall

“footprints” of the large outer scales during differentphases of the actuation cycle. This top-down effect ofthe outer motions on the streak intensity is referred toas “modulation”.

The fundamental processes involved in footprint-ing and modulation in canonical boundary layers havereceived much attention in recent years, boosted bythe discovery of an outer ‘secondary peak’ (or plateau)in turbulence energy and its pre-multiplied longitudi-nal energy spectrum at 0.1-0.2 of the boundary-layerthickness, which becomes increasingly pronouncedwith rising Reynolds number. In an especially re-markable series of studies, stretching over a period ofsome 10 years, Marusic, Mathis, Hutchins and theircollaborators (e.g. Mathis et al. (2009a,b); Marusicet al. (2010a); Hutchins et al. (2011); Mathis et al.(2013)) have investigated, experimentally, the phe-nomena in question, leading Mathis et al. (2011) and-Marusic et al. (2010b) to propose an empirical rela-tionship which ‘predicts’ the effects of the large-scaleouter fluctuations on the small-scale near-wall mo-tions:

u+p (y

+) = u∗(y+){1 + β(y+)u+

O,LS

(

y+O , θLS

)

}+ α(y+)u+

O,LS

(

y+O , θLS

)

(1)

In which the terms pre-multiplied by the empiricalcoefficientsα andβ express, respectively, the effectsof footprinting and modulation by the outer motionsu+

O,LS

(

y+O)

on the canonical fieldu∗(y+) that wouldexist if there were no large-scale structures, andθLS

is the angle linking the outer motions with their foot-prints (reflecting a lag). The coefficientsα andβ de-pend on the wall distancesy+ andy+O , but are of or-der0.7 and0.04, respectively, outside the viscous sub-layer, varying only modestly beyondy+ ≈ 20. Themain concept underpinning eq. 1 is, therefore, that thenear-wall intensity can be ‘predicted’, at any Reynoldsnumber and irrespective of the intensity of the outermotion, by imparting empirical corrections to the uni-versal fieldu∗(y+) that involve the Reynolds-number-dependent outer motions.

Eq.1 has two important implications that are opento question:

(i) positive and negative large-scale fluctuationscause equally-weighted modifications to the

small-scale fluctuations - i.e. the unperturbedfield u∗ is “symmetrically” altered;

(ii) scaling of all quantities (and thus also coefficientsα andβ) with the mean-friction velocity implieslinearity, in the sense that the Reynolds-numberuniversality expressed by eq. 1 is not conditionalon large-scale variations in the friction velocityinduced by the LS motions or the energy of theouter structures.

The present paper examines the validity of eq. 1,focusing on implication (i) as part of a broader studythat also encompasses implication (ii). The route takenentails a statistical analysis of DNS data for channelflow atReτ = 1025, reported in Agostini et al. (2014).Although this Reynolds number is relatively low, incomparison with experimental configurations, the flowis appropriate to the aims pursued in this study, as willtranspire. The analysis has led the present authors topropose an alternative to the model 1, but this is re-ported elsewhere.

2 The Flow and its Analysis

Results presented herein arise from a DNS for a canon-ical channel flow atReτ = 1025, performed over abox of length, height and depth4πh × 2h × 2πh, re-spectively, covered by1056×528×1056(= 589×106)nodes. The corresponding cell dimensions were∆x+,∆y+min, ∆y+max, ∆z+ = 12.2, 0.4, 7.2, 6.1. Thedetails of the simulation and its accuracy are discussedin Agostini et al. (2014).

The present statistical analysis is based on process-ing 60 spatial snapshots of the solution over a timeinterval of t+ = tu2

τ/ν = 1000. This approach isin contrast to that of Marusic and his collaborators,which is based on the processing of temporal signalsover large periods of time, recorded at specific spa-tial wall-normal locations, over a range of wall-normaldistances.

A representative (raw)x − z snapshot aty+ =13.5 is given in figure 1(a). This shows contours ofstreamwise-velocity fluctuations. The plot conveys aclear view of both the small-scale streaks, which are atmaximum strength at the wall-normal location chosen,and of the footprints of large-scale outer structures,which typically have a length of 5-10 channel half-heights orx+ = O

(

5− 10× 103)

for the presentReynolds number.

Given snapshots of the form shown in figure 1(a),the footprints need to be separated from the small-scale motions. This can be done here using a methodcalled “Huang-Hilbert Empirical Mode Decomposi-tion” (EMD) (Huang et al., 1998). The EMD splits anysignal into a set ofIntrinsic Mode Functions (IMFs)based purely on the local characteristic time/spacescales of the signal. The method requires no pre-determined functional elements, such as Fourier or

(a) (b)

Figure 1: Snapshot of the streamwise-velocity fluctuationsat y+ = 13.5: (a) complete signal; (b) large-scale veloc-ity fluctuations; islands with red/blue boundaries identifypositive/negative fluctuations within the extreme 10% bands(tails) of the PDF of the large-scale fluctuations (see figure5(b) and related description).

wavelet functions. Rather, the IMFs are the EMD-generated basis functions, which arise purely from thegiven signal itself. Unlike Fourier methods, the EMDdoes not require filters to separate the scales, and doesinvolve filter-induced loss of energy. The applicationof 2-d extended version of the standard 1-d EMD tosnapshots of the form of figure 1(a) leads to the rep-resentation shown in figure 1(b) for the large-scalestreamwise-velocity fluctuations, in which the islandssurrounded by the line contours are areas within whichthe large-scale motions exceed a certain limit definedand discussed below. Typically, the large-scale ve-locity fluctuations within these islands are around 15-20% of the mean velocity (aty+ = 13.5). Analogouslarge-scale fields may be obtained for the spanwise-and wall-normal-velocity components. Snapshots ofthe form shown in figure 1(a) were decomposed intofour modes, the first three representing the small scalesand the last the large scales. This choice is justified onthe basis of a preliminary analysis, involving an ex-amination of pre-multiplied energy spectra, two-pointcorrelation maps, and energy profiles for the separatedlarge- and small-scale motions; only a small part ofthis study can be included herein.

Figure 2 shows contour maps of pre-multiplied en-ergy spectra,kxΦ+

uu, of the streamwise fluctuationsacrossy+, presented in three different ways. Figures2(a) and (b) were both derived from the raw fields.The difference between the two is that the spectra infigure 2(b) are normalized forms of the those in fig-ure 2(a), such that the area under any spectrum at anylocationy+ is 1. This normalisation accentuates thedominant modes in different parts of the wall-normaldomain. The implication is that the near-wall layer

y+

λ+ x

101

102

102

103

104

0

0.5

1

1.5

2

(a)

y+

λ+ x

101

102

102

103

104

0

0.1

0.2

0.3

0.4

0.5

(b)

y+

λ+ x

101

102

102

103

104

(c)

Figure 2: Contours of pre-multiplied energy spectrakxΦ+uu

for streamwise-velocity fluctuations : (a) standard (un-scaled) representation; (b) scaled field, subject to normal-isation (kxΦ

+uu(y

+))/(u′u′+(y+)); (c) contours of pre-

multiplied (unscaled) spectra for the LS mode (gray lines)and SS mode (black lines). Contours are separated by theconstant interval 0.13, with lowest value being 0.13.

is dominated by small-scale (“SS”, henceforth) modesof streamwise length scaleλ+

x = O(1000), while theouter region, aty+ ≈ 150, is dominated by largescales (“LS”, henceforth),λ+

x = O(5000 − 10000).When the raw signal is decomposed into four modesand pre-multiplied energy spectra are obtained, sepa-rately, for the sum of the first three modes and mode 4,the resulting two sets of contour maps are those givenin figure 2(c). This figure thus illustrates that the LSmodes are clearly delineated and closely related to thecontour maps in figure 2(b). Moreover, the contoursof the mode 4 are seen to penetrate deeply into theviscous sublayer, reflecting the pronounced footprint-ing of the LS structures observed in this layer. If, in-stead of applying the EMD to yield figure 2(c), en-ergy spectra are derived from the raw fluctuation field,subject to low-pass and high-pass filters with cut-offat λ+

x = 3000, the resulting spectra feature respec-tive small-scale and large-scale contour fields whichare close to those shown in figure 2(c).

Next, they+-wise energy distributions of the SSand LS modes (modes 1-3 and mode 4, respec-tively) are determined and compared in figure 3 todistributions reported by Marusic et al. (2010b) forfriction-Reynolds-number values of 3900, 7300 and39000. The present SS energy is seen to agree wellwith the “universal”, Reynolds-number-independentdistributions of Marusic et al. (2010b). However,the energy of the LS motions is strongly Reynolds-

101

102

103

104

0

1

2

3

4

5

6

7

8

9

y+

<(u

+ SS)2

>x,z,t

(a)

101

102

103

104

0

1

2

3

4

5

6

7

y+

<(u

+ LS)2

>x,z,t

(b)

Figure 3: Profiles of streamwise turbulence intensity pro-file: (a) SS scales, (b) LS scales; present results forReτ =1025: dashed lines and full circles; other profiles: experi-mental results of Marusic et al. (2010b) forReτ = 3900,7300, 39000; stars identify the predicted wall-normal loca-tions at which the maximum LS energy is expected, given byy+

O= 3.9

√Reτ .

number-dependent, and the present level is thus signif-icantly lower than those corresponding to the higherReynolds-number values. However, the level isbroadly consistent with the trend suggested by theexperimental data. Furthermore, empirical relationsgiven by Marusic et al. (2010b) for the variations of thetotal energy aty+ ≈ 15 andy+O ≈ 3.9

√Reτ (the lat-

ter the position of maximum large-scale energy), canbe used to evaluate the expected LS energy at thesetwo locations at the present Reynolds number. Theseare shown in figure 3 by circles and stars, respectively,with the lowest variation arising from the present data.

In order to extract statistical data pertinent to theinteractions between the large outer and small innerscales, and thus to eq. 1, 1-d and 2-d (joint) PDFsof SS velocity fluctuations have been assembled, con-ditional on regions of high-velocity, low-velocity andnear-zero LS footprints the first two being identifiedby the red/blue islands in 1. Regions of low-velocityand high-velocity and LS motions are defined hereas those which fall, respectively, into the lowest andhighest10% bands (tails) of the PDF of the entireLS field, while regions which are, essentially, devoidof LS footprints are defined as those which fall intothe central10% of the PDF. This is illustrated in fig-ure 5. The PDF in figure 5(a) gives the distributionof all LS motions contained in figure 1(b), while thetwo PDFs in figure 5(b) relate, respectively, to the top10% (in terms of area) of positive (red) and negative(blue) LS fluctuations in the complete PDF of figure5(a). These correspond, respectively, to the red andblue islands in figure 1(b), and LS fluctuations thereinare of order 20% and 15%, respectively, of the meanvelocity in the same plane. It needs to be emphasizedhere that there is no fundamentally profound reasonfor the present focus on 10% bands within the LS PDF.This is a choice that reflect the wish to bring out asclearly as is possible differences in the effects of posi-tive and negative LS fluctuations on the SS motion. An

−10 −5 0 5 10

0

0.005

0.01

0.015

0.02

u+

pdf(u+)

(a)

−10 −5 0 5 10

0

0.005

0.01

0.015

0.02

u+SS

pdf(u+ SS)

(b)

−10 −5 0 5 10

0

0.005

0.01

0.015

0.02

u+SS/LS

pdf(u+ SS/LS)

(c)

Figure 4: PDFs ofu-velocity fluctuations aty+ = 13.5pertaining to the bands of 10% extreme positive LS events(solid line), 10% minimum LS events (chain line) and 10%extreme negative events (dashed line); (a) total fluctuations;(b) SS fluctuations only; (c) SS fluctuations normalized withthe LS velocity, eq. 2.

analysis based on extended bands (up to 40%) in noway changes, qualitatively, the conclusions presentedherein.

With the extreme 10% regions so identified, PDFsare then constructed of the SS motions within the blueand red regions, so to enable an examination of theeffects of the footprints on the SS motions. A fea-ture of the PDF in figure 5(a), which will be rel-evant to the discussion to follow, is that it is onlyweakly asymmetric, with extreme positive fluctuationsslightly more prevalent than extreme negative ones.This weak asymmetry (relative to the principal axes)also applies to the joint(u− v) PDF at the same wall-normal location, shown in figure 5(c) and assembledwith v+LS determined from the application of the EMDto the wall-normal component. The near-symmetry ofthe PDF in figure 5(c) will later serve as a backgroundagainst which to contrast corresponding PDFs for theSS motions, which display a much higher degree ofasymmetry.

3 Small-scale response

3.1 Interactions with positive vs.negative large-scale fluctuations

Most results presented below are in the form of PDFsat the wall-normal locationy+ = 13.5, with mean-friction scaling of the velocity fluctuations. As such,

−5 0 5

0

0.1

0.2

0.3

0.4

u+LS

pdf(u+ LS)

(a)

−5 0 5

0

0.1

0.2

0.3

0.4

u+LS

pdf(u+ LS)

(b)

u+

LS

v+

LS

−2 0 2

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

(c)

u+

LS

v+

LS

−2 0 2

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

(d)

Figure 5: PDFs of large-scaleu-velocity fluctuations (EMDmode 4) aty+ = 13.5: (a) the whole 1-d PDF; (b) partialPDFs for the 10% lowest and highest velocity fluctuationsfor which the statistics of SS motions are studied; (c) joint2-d (u−v) PDF of large-scale velocity fluctuations equivalentto (a); (d) partial joint 2-d(u − v) PDFs equivalent to (b).PDF contours identify 0.1-0.9 of the PDF height at constantincrement 0.1.

they pertain to the “linear” model eq.1, in whichmean-friction scaling is also used.

Some basic arguments are first conveyed by meansof 1-d PDFs of the formpdf(u+). Figures 4(a) - (c)give, respectively, PDFs of the totalu-fluctuation field,of the SS-fluctuation field – i.e. with the LS motionssubtracted from those of figure 4(a), and of the SS fluc-tuations normalized, for reasons explained below, withthe absolute LS velocity as follows:

u+

SS/LS = u+

SS × < U1,LS >x,z,t

u1,LS+ < U1,LS >x,z,t(2)

where< U1,LS >x,z,t is the average velocity aty+ = 13.5. Each plot contains three PDFs, relatingto the lowest, middle and highest 10% bands, respec-tively, in the PDF of the large-scale motions shown infigure 5(a). Unless stated explicitly otherwise, normal-isation is performed with the mean-friction velocity.

This first comparison brings to light substantial dif-ferences in the manner in which the positive and neg-ative LS motions affect the SS fluctuations. Impor-tantly, thisasymmetric response is not associated withthe superposition of the LS motions onto the SS fluc-tuations - an effect that is included in figure 4(a), but isexcluded from (b). When the LS motion is removed,the SS PDF associated with negative LS fluctuationsis close to Gaussian, while that associated with pos-itive LS fluctuations is skewed, the latter character-ized by a predominance of relatively weak positive

∆x+

y+

−3000 −2000 −1000 0 1000 2000 3000

101

102

0

0.2

0.4

0.6

0.8

1

Figure 6: Correlation of LS fluctuations at anyy+ locationwith those aty+

O≈ 150.

fluctuations and relatively strong negative fluctuations.The removal of the LS motions corresponds, essen-tially, to the shift implied by theα-related term ineq. 1, which reflects the assumption of a superpo-sition process. Here, however,α = 1 (by implica-tion), because the LS information is available, and soused, on the same plane as the SS field, rather than aty+O . The rationale of normalizing the SS fluctuationswith the absolute LS velocity, as done in figure 4(c),is rooted in the observation that the velocity ratio ineq. 2 agrees closely with the normalized friction ve-locity uτ/uτ,LS, the latter evaluated by applying theEMD scale-decomposition to the planey+ = 3 and re-stricting attention to the 10% extreme bands of the LSfluctuations. The implication of figure 4(c) is, there-fore, that the SS fluctuations neither scale universallywith uτ nor withuτ,LS when these fluctuations are de-termined or considered at a fixedy+ location. Thefact that the modulation of the SS fluctuations differsmarkedly for positive and negative LS fluctuations, incontrast to the implication of eq. 1, will be argued be-low to reflect the impact of splatting associated withsweeps. As a consequence, the modulation is not sim-ply representable in terms of the streamwise LS fluc-tuations alone.

The use by Mathis et al. (2011) ofθL ≈ 12.5◦

andα = O(0.7) in eq. 1 to correlate the LS motionsnear the wall to those aty+O = 3.9

√Reτ is examined

in Figs. 6 and 7, respectively, the former containingthe correlation map for the LS motions, and the lattershowing they+-wise variations of the standard devi-ation of the PDF for the streamwise LS motions, thusreflecting their intensity.

They+-wise locus of the maximum correlation co-efficient in figure 6 suggestsθLS ≈ 12.5◦ for thestraight line connecting the maxima at locationsy+ =15 andy+O = 3.9

√Reτ , while figure 7 shows that the

intensity of the LS motions is fairly constant, downto the viscous sublayer within which the standard de-viation drops progressively. The implication is thatα = O(1) is more appropriate than0.7, at least fory+ above the viscous sublayer. Hoewever, this appliesonly to the present Reynolds number. As figure 3(b)suggests (by the gradient of the lines connecting re-spective circles and stars),α declines with increasingReynolds number.

100

101

102

0

0.5

1

1.5

2

y+

std(u

+ LS)

Figure 7: y-wise variation of the standard deviation of thePDFs of the LS motions.

In eq. 1,u∗ is assumed to be the canonical sig-nal that would have been recorded in the absence ofLS effects. In the present context of spatial statistics,it is reasonable to suppose that equivalent conditionsprevail in areas to which the central portion of the LSPDF in figure 5(a) relates. Figure 4 shows, by chainlines, the 1-d PDFs of the SS motions that relate tothe central10% band in the LS PDF. As might be ex-pected, the SS fluctuation field in this central band hasfeatures intermediate between the those within the twoextreme areas of the LS PDF of figure 5. However, thisdoes not suffice, on its own, to judge this field as beingfree fromany LS influences. The interpretation of theu∗ field as either being or not being “universal” is oneaspect of an examination of the validity of Mathis etal’s model (Mathis et al., 2011) by reference to joint(u − v) PDFs, considered next.

Figure 8 shows PDFs of the entire(u − v)-fluctuation field, of the total fluctuations in the blueand red areas of figure 1, of the SS fluctuations in theblue and red areas, and of the SS fluctuations scaled bythe LS velocity, in accordance with eq. 2, withv+SS/LS

(and later alsow+

SS/LS) obtained upon replacingu+

SS

by v+SS (or w+

SS). The circular and diamond-shapedsymbols represent the centres of gravity of the PDFsin question. The collapse of the centres of gravity infigure 8(c) points to the validity of the superpositioncomponent associated withα in eq. 1. There is, fur-thermore, clear evidence that positive LS perturbationscause considerably stronger modulation of the SS mo-tions than negative perturbations. Once scaled by theLS velocity, figure 8(d), the SS fluctuations displayfair universality in quadrant 2, associated with ejec-tions, but substantial departures from universality inquadrant 4, associated with sweeps. That these drasticdifferences are not linked to a bias in the LS motionsis demonstrated in figure 5(c), which shows the near-symmetric PDF of the complete field of LS motions.

Figure 9 gives joint(u − w) PDFs, which corre-spond to the(u − v) PDFs in figure 8. These suggestthat the skewed shape of the PDF in figure 4, asso-ciated with positive LS motions, is driven by sweepswhich transport relatively weak SS fluctuations from

u+

v+

−5 0 5−1.5

−1

−0.5

0

0.5

1

1.5

(a)

u+

v+

−5 0 5−1.5

−1

−0.5

0

0.5

1

1.5

(b)

u+

SS

v+

SS

−5 0 5−1.5

−1

−0.5

0

0.5

1

1.5

(c)

u+

SS/LS

v+SS/LS

−5 0 5−1.5

−1

−0.5

0

0.5

1

1.5

(d)

Figure 8: Joint(u− v) PDFs aty+ = 13.5 of (a) total fluc-tuation field; (b) total fluctuations in blue and red regions infigure 1(b), respectively; (c) SS fluctuations; (d) SS fluctua-tions normalized by LS fluctuations. Red contours : +10%LS fluctuations, blue contours -10% LS fluctuations. PDFcontours identify 0.1-0.9 of the PDF height at constant in-crement 0.1, subject to total PDF volume normalized to 1.

beyond the buffer layer downwards. In contrast, SSejections are weaker, more numerous and more nor-mally distributed, thus unaffected by the bias causedby sweeps. In common with the(u − v) PDFs, thecentres of gravity of the(u − w) PDFs collapse uponthe removal of the LS motion. Similarly, the SS mo-tions scaled by the LS velocity collapse for negativeLS motions, but not for positive ones.

The conclusion thus emerging, so far, from theabove discussion is that modulation is not a “ symmet-ric” process, in the sense of positive and negative LSmotions having the same weight on the SS field, andthat the lack of symmetry is caused by major differ-ences in the effects of sweeps and ejections on the SSstructure. In particular, the(u−w) PDFs bring to lightthat sweeping motions go hand in hand with strongspanwise fluctuations in quadrants 1 and 4, which arecharacteristic of splatting. This process, and its effectson the SS motions, is not accounted for in eq. 1 andcannot be captured by the model.

3.2 Universal SS motions - theu∗-field

If, despite the above incompatibility, the model, eq. 1,is to be retained, it is possible to determine (or ratherestimate) the values forα andβ that secure the bestpossible compliance with the present data. To this end,eq. 1 is re-cast as follows:

u∗ =u+ − αu+

O,LS

1 + βu+

O,LS

(3)

u+

w+

−5 0 5−4

−2

0

2

4

(a)

u+

w+

−5 0 5−4

−2

0

2

4

(b)

u+

SS

w+

SS

−5 0 5−4

−2

0

2

4

(c)

u+

SS/LS

w+

SS/LS

−5 0 5−4

−2

0

2

4

(d)

Figure 9: Joint(u−w) PDFs aty+ = 13.5 of (a) total fluc-tuation field; (b) total fluctuations in blue and red regions,respectively in figure 1(b); (c) SS fluctuations; (d) SS fluc-tuations normalized by LS fluctuations. Contours levels: seecaption of Fig. 8.

Given appropriately chosen values forα andβ, ex-pected to be close to those proposed by Mathis et al.(2011), the question is whether the SS PDFs, condi-tional on the±10% extrema of the LS motion, can bemade to collapse, such a collapse being interpretableas representing the fieldu∗. An ambiguity that ariseswith this process relates to the interpretation ofv∗ andw∗. In the absence of a credible alternative, the as-sumption is made here thatv∗ = v+SS andw∗ = w+

SS ,respectively . A possible variation is to use eq. 3, withthe numerator replaced byv+SS andw+

SS to obtainv∗

andw∗, respectively. However, this variation has onlymarginal effects on the results to follow.

As shown in figure 10, use ofβ = 0.04, α = 0.7,with uO,LS taken fromy+O ≈ 150, subject toθLS =12.5◦, results in a fair correspondence in theu-wisewidths of the PDFs, but in significant differences intheir shape, especially in quadrant 4. Thus, althoughfigure 10 suggests that the amplitude, or intensity, oftheu∗-fluctuations is fairly insensitive to whether theLS motions are positive or negative, supporting Mathiset al’s model (with the particular empirical constantsused), the PDFs in figure 10 show thatu∗-field is not,in fact, universal, because the PDF of the SS fluctu-ations subjected to positive LS fluctuations is skewedand distorted (see also figure 4(b) ). Moreover, if thejoint (u∗− v∗) PDFs are projected onto theu∗ axis, toyield corresponding 1-d PDFs, the shape of the latterare quit similar to that shown in figure 4(c). This dis-tortion is, again, a consequence of the sweeping mo-tions – an effect that does not fall neatly under theheading “footprinting” and amplitude “modulation”,

u∗

v∗

−5 0 5−1.5

−1

−0.5

0

0.5

1

1.5

(a)

u∗

w∗

−5 0 5−4

−2

0

2

4

(b)

Figure 10: Joint PDFs: (a)u∗−v∗; (b)u∗−w∗, both derivedfrom eq. 3 aty+ = 13.5, with β = 0.04 andα = 0.7.Contours levels: see caption of Fig. 8.

and which cannot be represented purely by referenceto the outer LS motionsu+

O,LS.A first extension of the work reported herein ad-

dresses the question of whether scaling by the localLS-modified friction velocity, including the scaling iny+, improved the universality of eq. 1, and thusu∗ ineq. 3. The answer is conditionally affirmative, subjectto uncertainties arising from data-related limitations.In a second extension, an alternative phenomenologi-cal model to eq. 1 has been derived, but this is reportedelsewhere.

4 Summary and Conclusions

This study set out to examine, in general, the effects oflarge-scale motions in the log-law region on the small-scale streaks in the viscosity-affected near-wall layer,and to investigate, in particular, the validity of theconcepts underpinning the predictive model of Mathiset al. (2011), which expresses these effects by way ofsuperposition- and modulation-related terms acting ona “universal” small-scale field that is held to be unaf-fected by the large-scale motions.

With scaling based on the mean-friction velocity,as done by Mathis et al. (2011), the results show thatthere are significant differences in the response of thesmall-scale motions to negative and positive large-scale outer fluctuations. In other words, the responseof the small-scale motions is not “symmetric”, in thesense of the use theβ−term in the model of Mathiset al, which represents modulation. In reality, thesmall-scale streaks are modified in three ways, ratherthan two: by superposition, by modulation and by dis-tortions of the small-scale field caused the differen-tial influence of sweeps (splatting) and ejections (anti-splatting). The third process is brought to light by dis-tinctive distortions in both the(u − v) and (u − w)PDFs. The PDFs included herein, as well as othersat lowery+ values not included, suggest that the dif-ferential influence of the sweeps and ejections of thelarge-scale motions is felt down to the lower levels ofthe viscous sublayer (y+ ≈ 3). In particular, the pear-shaped(u−w) PDFs point to the presence of splatting

that is associated with sweeps, which is argued to bethe cause of the asymmetry in the modulation of thestreamwise small-scale fluctuations.

While the model of Mathis et al. (2011) cor-rectly represents the superposition effect exerted bythe large-scale structures on the near-wall structures,it does not appear to capture correctly the asymmet-ric modulation, due to the splatting-induced distortionsnoted above. Furthermore, use of the model equationsto extract the “universal” small-scale motions fromthe DNS data shows that these motions are not, infact, “universal”, insofar as these motions depend onwhether they originate from (or associated with) re-gions of positive or from negative large-scale fluctua-tions. These observations are not, therefore, consistentwith the model.

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