J. Fluid Mech. (2018), vol. 852, pp. 641–662. c© Cambridge University Press 2018doi:10.1017/jfm.2018.554

641

Dependence of small-scale energetics on largescales in turbulent flows

M. F. Howland1,2,† and X. I. A. Yang1,3

1Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA2Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA

3Mechanical and Nuclear Engineering, Pennsylvania State University, State College, PA 16802, USA

(Received 17 November 2017; revised 1 June 2018; accepted 6 July 2018)

In a turbulent flow, small- and large-scale fluid motions are coupled. In this work, weinvestigate the small-scale response to large-scale fluctuations in turbulent flows anddiscuss the implications on large eddy simulation (LES) wall modelling. The interscaleinteraction in wall-bounded flows was previously parameterized in the predictive inner–outer (PIO) model, where the amplitude of the small scales responds linearly to thelarge-scale fluctuations. While this assumed linearity is valid in the viscous sublayer,it is an insufficient approximation of the true interscale interaction in wall-normaldistances within the buffer layer and above. Within these regions, a piecewise linearresponse function (piecewise with respect to large-scale fluctuations being positive ornegative) appears to be more appropriate. In addition to proposing a new responsefunction, we relate the amplitude modulation process to the Townsend attached eddyhypothesis. This connection allows us to make theoretical predictions on the modelparameters within the PIO model. We use these parameters to apply the PIO modelto wall-modelled LES. Further, we present empirical evidence of amplitude modulationin isotropic turbulence. The evidence suggests that the existence of nonlinear interscaleinteractions in the form of amplitude modulation does not rely on the presence of anon-penetrating boundary, but on the presence of a range of viscosity-dominated scalesand a range of inertial-dominated scales.

Key words: turbulence modelling, turbulence theory, turbulent boundary layers

1. Introduction1.1. Background and motivation

Predicting small-scale turbulence using large-scale information remains a focus ofturbulence modelling. The underlying physical connections between large scalesand small scales in turbulent boundary layers have been studied in many differentcontexts. For example, in large eddy simulations (LES), large-scale structures areresolved by computational grids, while the effects of small scales are modelledthrough subgrid-scale models (Meneveau & Katz 2000). Here we focus on a specificinterscale interaction, termed ‘amplitude modulation’, which has received significant

† Email address for correspondence: mhowland@stanford.edu

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642 M. F. Howland and X. I. A. Yang

attention since the early studies by Marusic, Mathis & Hutchins (2010) and Mathis,Hutchins & Marusic (2011). While amplitude modulation may be a general physicalprocess in turbulence, in recent literature on wall-bounded flows it is often usedto refer to a specific interscale interaction whereby small scales become more orless energetic as a function of large scales (Mathis, Hutchins & Marusic 2009a).A good discussion of the amplitude modulation process in other turbulent flows(e.g. free shear flows) can be found in Fiscaletti, Ganapathisubramani & Elsinga(2015). Although the presence of large scales, e.g. streamwise elongated streaks, isa salient feature of boundary-layer flows, large scales are, in fact, less energeticthan small scales at low to moderate Reynolds numbers (Hutchins & Marusic 2007).It is only at high Reynolds numbers, when large scales are well separated fromsmall scales in Fourier space, that large scales have comparable energetic content tosmall scales (Hutchins et al. 2012). This scale separation at high Reynolds numbersalso allows for the study of interscale interactions between large and small scalesdirectly. Substantial evidence can already be found in the recent literature revealingthe presence of amplitude modulation, which was parameterized with the predictiveinner–outer (PIO) model (see, e.g. Marusic et al. 2010; Mathis et al. 2011)

u′i = αu′o,L + [1+ βu′o,L]u∗

i,S, (1.1)

where u′o,L is the large-scale streamwise velocity fluctuation at an outer location, andu′i is the streamwise velocity fluctuation at an inner location. We use subscriptsi and o to indicate quantities evaluated at an inner and an outer wall-normallocation, respectively. Subscripts L and S denote large- and small-scale quantities,respectively, and superscript ′ is used for fluctuating quantities (quantities with theirmean subtracted from them). The large scales in (1.1) may be loosely defined tobe the fluctuations whose characteristic length scales are greater than a pre-definedlength scale l, and the small scales are simply the complement (Mathis et al. 2009a).The superposition of the large- and small-scale fluctuations in the near-wall regionis parameterized using α, and the amplitude modulation is parameterized using β.The velocity signal u∗i,S is the small-scale fluctuation in the absence of inner–outerinteractions (see e.g. Mathis et al. 2011; Agostini & Leschziner 2014, for detaileddiscussion). In practice, both of the coefficients α and β need to be calibratedempirically. Unless otherwise noted, we will use wall units for normalization purposes.

In addition to having two parameters that need to be determined empirically,equation (1.1) contains a few other weaknesses. First, the large and small scales aredefined using a definitive, prescribed length scale, l. This is rather inconvenient forapplications including LES where the LES filter, i.e. the grid spacing, is unlikelyto be commensurate with the prescribed length scale l. Second, and probably moreimportantly, it was shown in Ganapathisubramani et al. (2012) and Agostini &Leschziner (2014) that the amplitude of small scales does not depend linearly onlarge scales, i.e. the parameter β is not only a function of the Reynolds numberand the wall-normal distance, but also a function of u′o,L itself. Last, due to theunavailability of u∗i,S, the use of (1.1) has been limited to data analyses, with fewsuccessful attempts using the PIO model in a predictive manner (Inoue et al. 2012;Sidebottom et al. 2014).

Defining large and small scales in a turbulent flow without an arbitrary pre-definedscale separation has been a topic of interest since the identification of the amplitudemodulation process. It was argued in Hutchins & Marusic (2007) and Mathiset al. (2009a) that the presence of amplitude modulation (and generally interscale

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Dependence of small-scale energetics on large scales in turbulent flows 643

interactions) depends on a scale separation between the inertia-dominated large scalesand the viscosity-affected small scales. In Hutchins & Marusic (2007), the largescales and the small scales are selected such that the inner peak in the premultipliedstreamwise energy spectrum is separated from the outer peak. Following this definition,a fixed length scale that separates the inner and the outer peaks was used in Hutchins& Marusic (2007) to define the large and small scales. Hutchins & Marusic (2007)and Mathis et al. (2009a) used l = δ. A number of subsequent studies followedHutchins & Marusic (2007) but used l+ = 7000, where + indicates normalizationby wall units (see e.g. Mathis et al. 2011, 2013; Ganapathisubramani et al. 2012;Talluru et al. 2014; Baars et al. 2015; Baars, Hutchins & Marusic 2017). Morerecently, Baars, Hutchins & Marusic (2016) used a spectral stochastic estimation todetermine the separation length scale. Further, moving away from one-dimensionalspatial filtration, Agostini & Leschziner (2016b) utilized a novel method, empiricalmode decomposition, to separate islands of large and small scales.

Despite the seemingly stringent definition, using an l that separates the inner andouter peaks admits certain arbitrariness in the modelling framework. This convenientlyallows us to define the large scales as the resolved eddies in a typical wall modelLES (WMLES), a potential application for the PIO model. Briefly, WMLES is asubset of LES where not only the inertial range scales, but also the near-wall eddies,are modelled (Piomelli & Balaras 2002; Bose & Park 2017). Consider, for example,using (1.1) in a typical WMLES of high Reynolds number wall turbulence, e.g. Reτ =19 000, which uses Ny = O(10) grid points across the boundary layer, e.g. Ny = 16.The filtering length scale l+ ≈ 7000 will in fact be commensurate with the resolvedvelocity at the first off-wall grid point at y+ ≈ 1200, assuming an eddy inclinationangle of approximately 10◦.

Physics-based modelling of the parameters in the PIO model (α and β) is moredifficult than defining the large and the small scales. The mechanism behindthe amplitude modulation process was considered in Chernyshenko, Marusic &Mathis (2012) and later in Zhang & Chernyshenko (2016). The authors proposed aquasi-steady–quasi-homogeneous (QSQH) model and made predictions of α and β.According to the QSQH model, large-scale motions modulate the local wall shearstresses, thereby modulating the local velocity and length scales. The modulated localvelocity and length scales then modulate the energetics of the small scales. Followingthe above argument to its logical conclusions, the authors predicted that

α(yi)=

U(yi)+ yodUdyi

U(yo)+ yodUdyo

, β(yi)=

1+yi

STD(ui)

dSTD(ui)

dyi

U(yo)+ yodUo

dyo

, (1.2a,b)

where quantities with · are evaluated at wall-normal locations defined based on localvelocity and length scales. Equation (1.2) was found to agree fairly well with the data.However, because the authors had to use a specifically chosen filter for defining thelarge and small scales, and because both the mean velocity and the Reynolds stressprofiles are required to compute α and β, the usefulness of (1.2) in predictive modelsis limited.

Zhang & Chernyshenko (2016) presumed that (1.1) is a good approximation of theinterscale interaction. The validity of (1.1) was challenged in Agostini & Leschziner(2014). Following previous works, including those by Agostini & Leschziner (2014)and Hwang et al. (2016), we will generalize the modelling framework by formally

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644 M. F. Howland and X. I. A. Yang

including a u′o,L dependence in β. We will also make predictions of α and β andassess the usefulness of the PIO model for LES wall modelling. In WMLES, wemodel the amplitude modulation of the wall shear stresses in the spatial domain.Earlier studies of amplitude modulation usually do not distinguish between thetemporal and the spatial amplitude modulations, which are two different processes(see Jacobi & McKeon 2013; Fiscaletti et al. 2015; Awasthi & Anderson 2018; Yang& Howland 2018).

The rest of the paper is organized as follows: in § 1.2, a generalized modellingframework is presented; in § 1.3, we briefly summarize the hierarchical randomadditive process (HRAP) model, which will be providing a few key physical insights.In § 2, we use HRAP to estimate the modelling parameters in the PIO model. It isworth noting that the analysis here shows that the HRAP model (and therefore theattached eddy model) admits interscale interactions. We then measure directly theresponse of the small-scale energetics to the large scales in § 3. We will also brieflydiscuss amplitude modulation in isotropic turbulence (HIT) in § 4 to provide someinsights on the physical mechanism of this phenomena. The usefulness of the PIOmodel in the context of WMLES is discussed in detail in § 5. A new LES wall modelformulation that is based on the PIO model is presented, and model performanceis compared with that of the commonly used equilibrium wall model. Concludingremarks are given in § 6.

1.2. Generalized amplitude modulation frameworkIn this subsection, we will generalize the modelling framework for amplitudemodulation in wall-bounded flows by accounting for the dependence of β upon thelarge-scale velocities. A generalization of the PIO model is imperative consideringthe nonlinearity of the small-scale response to large scales (Agostini & Leschziner2014; Agostini, Leschziner & Gaitonde 2016; Hwang et al. 2016), and a lack ofconsideration of the phase difference between the small and the large scales (Chung& McKeon 2010; Jacobi & McKeon 2013; Baars et al. 2017). We start by defininga response function f in replacement of β. The response function depends on twoarbitrary heights, y1 and y2, between which an interscale interaction occurs, andthe large-scale velocity u′L, i.e. f (y1, y2, u′L) (see figure 1 for details). Aside fromthe specific modelling details used by the series of studies following Marusic et al.(2010), the streamwise velocity fluctuation at an inner location may be modelled as

u′i(yi, t)= u′L(yi, t)+ u′S(yi, t)= αu′i,L,m + [1+ f (yi, yo, u′i,L,m)]u∗

i,S,m, (1.3)

where the velocity fluctuation at an inner location is decomposed into small and largescales, with small scales modelled as [1+ f (yi, yo,u′i,L,m)]u

∗

i,S,m. The subscript m denotesquantities that are modelled. Without loss of generality, the following discussion willfocus on (1.3).

Next, we determine the modelling parameters. The modelling parameter α is simplythe correlation between u′i,L and u′i,L,m, normalized by the ratio of their standarddeviations (Mathis et al. 2013; Agostini & Leschziner 2016b), defined as follows

α ≡〈u′i,L,mu′i,L〉

STD(u′i,L,m)STD(u′i,L)STD(u′i,L)

STD(u′i,L,m), (1.4)

where STD(·) is the standard deviation of the bracketed statistical quantity. Todetermine f (yi, yo, u′i,L,m) and u∗i,S,m, a characterization of the small-scale energetics is

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Dependence of small-scale energetics on large scales in turbulent flows 645

Flow direction

Wall

Modulation

Small-scaleLarge-scale

Modulation

Cor

rela

ted

Unco

rrelat

ed

FIGURE 1. (Colour online) A sketch of the interscale interaction in wall-bounded flows.The wall-normal direction is y. The outer and inner locations are yo and yi, respectively.The large scales at the two wall-normal locations are well correlated, with the large-scalefluctuations at the inner location slightly displaced in the streamwise direction becauseof the tilting of flow structures in the flow direction. The small scales at the two wall-normal locations are, generally, not correlated. The large scales at the two wall-normallocations modulate the amplitude of the small scales at the respective wall-normal heights,but because the large scales at the two wall-normal locations are correlated, the large-scalefluctuations at an outer location also modulate the small-scale fluctuations at the innerlocation.

required. A few characterizations of the small-scale energetics have been used in thecontext of interscale interaction modelling. For example, Ganapathisubramani et al.(2012), Agostini & Leschziner (2014) and Agostini & Leschziner (2016a) used u′2S ,and Mathis et al. (2009a) and Baars et al. (2015) used the envelope of the smallscales env(u′S). For many purposes, both characterizations lead to very similar results(Baars et al. 2017; Howland & Yang 2017). In this work, we follow Mathis et al.(2009b) and use env(u′S) for characterizing the small-scale energetics, which canbe obtained by conducting a Hilbert transformation of the small-scale signal. Theresponse function f (yi, yo, u′i,L,m) is such that the signal computed according to

u∗i,S,m =u′i − αu′i,L,m

1+ f (yi, yo, u′i,L,m)(1.5)

is not modulated by the large scales, i.e.

〈envL(u∗i,S,m)u′

i,L,m〉 = 0. (1.6)

Equation (1.6) provides one constraint for the response function f and allows onemodelling parameter to be determined (e.g. β).

Separately, the correlation 〈envL(u′i,S)u′

i,L,m〉 measures the response function at agiven wall-normal height f (y1= y, y2= y, u′L) directly. Because env(u′S)≈ 1+ f (y, y, u′L)and envL(u′S)≈ f (y, y, u′L), it follows from (1.3) that

〈envL(u′S)|u′

L〉 ≈ f (y, y, u′L). (1.7)

Here 〈A|B〉 is the ensemble average of A given condition B. If the response functionis a linear function of u′L, i.e. if f (y, y, u′L)=βu′L, the amplitude modulation coefficientβ is positively correlated with the single-point correlation R, defined as

R≡〈envL(u′S)u

′

L〉

STD(envL(u′S))STD(u′L). (1.8)

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646 M. F. Howland and X. I. A. Yang

This Pearson-type correlation coefficient R has been used to quantify amplitudemodulation in a number of previous studies (see, e.g. Mathis et al. 2009a; Baarset al. 2015, 2017). While R proves to be useful in a number of applications (e.g.Tang & Jiang 2018), relying solely on R for measuring amplitude modulation leadsto ambiguities. On one hand, R may be an overly optimistic measure of amplitudemodulation because it is also a measure of velocity skewness (Agostini et al. 2016).On the other hand, R may be too conservative because of a lack of consideration ofthe phase difference between the envelope of the small scales and the large scales inthe bulk region (Chung & McKeon 2010; Jacobi & McKeon 2013). More importantly,equation (1.8) is an integrated measure, and if the response function is not a linearfunction of u′L, physical interpretations relying on R (or β) become difficult. For theremainder of the present work, we will only use the coefficient R if f = βu′L, inwhich case R∼ β.

1.3. Hierarchical random additive process (HRAP)In anticipation of the results in later sections, we briefly summarize a recentlydeveloped modelling framework for wall-bounded turbulence, namely the HRAP(Yang, Marusic & Meneveau 2016a,b). The HRAP is a reinterpretation of theTownsend attached eddy hypothesis (Townsend 1976), and it models high Reynoldsnumber boundary-layer flows as collections of self-similar, wall-attached eddies.Following Townsend (1976), the eddy population density is P(y) ∼ 1/y. Theinstantaneous velocity at a wall-normal distance y is modelled as a superpositionof all the eddy-induced velocities there

u′y = a1 + a2 + · · · + aNy, Ny ∼ log(δ/y), (1.9)

where an addend ai represents the contribution from an attached eddy of size δ/2i

(therefore a small index i corresponds to a large-scale eddy). In the context ofWMLES, large-scale eddies (a1, a2, etc.) are resolved by the computational grids, butsmall-scale eddies (aNy , aNy−1, etc.) will need to be modelled. The exact discriminationbetween the large and small scales depends on the grid resolution. Comparing (1.3)and (1.9), we note first that both models account for the superposition of large scalesand small scales at a near-wall location. Second, we note that (1.3) groups the largescales in one term u′i,L,m, while (1.9) explicitly details the contents within the largeand small scales, and relates these contents to the near-wall eddies.

2. Amplitude modulation and the HRAP model2.1. Amplitude modulation of the velocity fluctuations

Following the discussion in § 1.3, according to the HRAP formalism, the near-wallvelocity may be modelled as

u′i = a1 + a2 + · · · + aNi . (2.1)

We group the addends to large scales

u′i,L = a1 + · · · + aNo (2.2)

and small scalesu′i,S = aNo+1 + · · · + aNi, (2.3)

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Dependence of small-scale energetics on large scales in turbulent flows 647

where u′i = u′i,L + u′i,S. According to the HRAP model, u′i,L is also the velocityfluctuation at a wall-normal height yo∼ δ/2No and therefore, the large-scale fluctuationat a near-wall location may be modelled using velocity fluctuations at a wall-normallocation further away from the wall. This is an important insight, which will be usedin the later sections. The distinction between the small- and large-scale componentsis arbitrary as long as 1<No <Ni.

We model u′S given a large-scale fluctuation u′L. The subscript i is dropped here forbrevity. We will not start with (1.1), but instead, we will show (1.1) is the end pointof the following analysis. First, because u′S is a fluctuating quantity whose mean isremoved, 〈u′S|u

′

L〉 = 0. Second, 〈u′S2|u′L〉 is

〈u′2S |u′

L〉= 〈(aNo+1+· · ·+ aNi)2|u′L〉∼ 〈(Ni−No− 1)a2

|u′L〉∼ 〈u′2τ |u′

L〉 · log(

yo

yi

)D, (2.4)

where yi ∼ δ/2Ni , and D is a damping function (its detailed functional form will notbe relevant here). It follows from (2.4) that the large scales affect the energetics ofthe small scales through the local velocity scale uτ , a mechanism that was discussedearlier in Zhang & Chernyshenko (2016) and Baars et al. (2017). The variance 〈u′2τ |u

′

L〉

may be estimated by invoking the law of the wall

〈u′2τ |u′

L〉 =

[κ(U(yo)+ u′L)

log(yo/δν)

]2

, (2.5)

where U(yo) is the mean velocity at the wall-normal height yo, and δν is an inner,viscous length scale defined as δν = ν/〈uτ 〉 · exp(−κB) for smooth walls. B isthe additive constant in the law of the wall. Equation (2.5) is often used in awall-modelled LES context, where wall shear stress is modelled as a function ofthe velocity away from the wall (Piomelli & Balaras 2002; Bose & Park 2017).Equations (2.4) and (2.5) provide estimates of the variance of u′S. The same proceduremay be followed for estimating the higher-order statistics, but for now we model u′Sas

u′S|u′

L = uτ ·G(µ, σ 2)=κ(U(y0)+ u′L)

log(yo/δν)G(µ, σ 2), (2.6)

where G(µ, σ 2) is a stochastic quantity whose mean µ and variance σ 2 are known.We rewrite (2.6) as

u′+S |u′+

L =

(1+

1U+(yo)

u′+L

)G(µ, σ 2), (2.7)

which conforms to (1.1), and provides us with a prediction of the modulationcoefficient β,

βHRAP = 1/U+(yo). (2.8)

Both (2.7) and (2.8) rely on the response function being f = βu′L. The detailedfunctional form of the response function will be determined in § 3.1. For now, wecompare (2.8) and (1.2) to the measurements from a Reτ = 2000 channel flow directnumerical simulation (DNS) (Hoyas & Jiménez 2006). The modelled large-scalevelocity fluctuation at the inner location is u′i,L,m = u′o. The outer velocity is directlyused as the modelled inner large-scale velocity without filtration at a pre-specifiedlength scale. The modulation coefficient β is obtained using (1.5) and (1.6).

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648 M. F. Howland and X. I. A. Yang

The reader is directed to Mathis et al. (2011) for detailed procedures for determiningβ from data. Additionally, the reader is directed to Zhang & Chernyshenko (2016)for procedures for obtaining the theoretical prediction of β. Figure 3 shows themeasured and model predicted modulation coefficient β as functions of both theinner and the outer location. Within the viscous sublayer, β is independent of theinner location, and the measurements agree reasonably well with (2.8). For a givenyi, both (2.8) and (1.2) predict that β decreases as a function of yo, which bearsout in the measurements. Without using the optimal filtered defined in Zhang &Chernyshenko (2016), equation (1.2) is slightly less accurate than (2.8).

2.2. Amplitude modulation of wall shear stressesTo use (2.8) in WMLES, we will need to consider the amplitude modulation ofthe wall shear stresses. Amplitude modulation of wall shear stresses was previouslyconsidered in Mathis et al. (2013) as:

τ ′w = fu′o,L + [1+ f ]τ ∗w,S,m. (2.9)

The amplitude modulation response function is again denoted as f and followingMathis et al. (2013), the modelling parameter α equals f . Without loss of generality,f in (2.9) is a nonlinear function of u′o,L.

3. Small-scale response to large-scale fluctuationsIn this section, we directly measure the response function according to (1.7) to

assess the degree to which an approximation of linearity is a sufficient estimation. Theresults in this section are confined to single point measurements.

3.1. Empirical response functionFor the present measurements, we use boundary-layer flow data at friction Reynoldsnumbers Reτ = 6500, 10 000 and 13 000. Details of the data sets can be found inHutchins et al. (2009) and Talluru et al. (2014). We focus on spatial amplitudemodulation using the local velocity as the convective velocity (Yang & Howland2018). Following Mathis et al. (2009a), we define the large and small scales using apre-specified length scale l+=4000. We will also use a filtering length scale l+=7000to show the robustness of the measurements. The large and the small scales in (1.7)are at the same wall-normal height. We will not take into consideration the phasedifference between the large and the small scales considering the high Reτ of thepresent data (Baars et al. 2017).

Figure 4(a–c) shows the response function f = 〈envL(u′S)|u′

L〉 as functions of thelarge-scale velocity fluctuation at three wall-normal locations, y+ ≈ 5, y+ ≈ 75 andy+ ≈ 200. Results of channel flow DNS at Reτ = 2000 and 5200 (Hoyas & Jiménez2006; Lee & Moser 2015) are included for comparison. The DNS results agreereasonably well with the results obtained using the hot-wire data at similar Reynoldsnumbers. We make a few observations. First, the response function depends on thewall-normal distance but not Reτ when the Reynolds number is sufficiently high(e.g. Reτ & 2000). Second, the response function is linear, f = βu′L, only withinthe viscous sublayer (y+ < 5). Third, beyond the viscous sublayer, the small-scalefluctuation responds differently to the large-scale fluctuation depending on whetheru′L is positive or negative, as also noted in Hwang et al. (2016). In the logarithmic

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Dependence of small-scale energetics on large scales in turbulent flows 649

region (y+ ≈ 100–200), 〈envL(u′S)|u′

L〉 and u′L are positively correlated for u′L < 0,and are negatively correlated for u′L > 0, leading to positive and negative amplitudemodulation, respectively. Last, the response function f (u′L) is approximately a linearfunction of u′L for u′L < 0 and u′L > 0 at all wall-normal locations. Figure 5(a–c)shows 〈envL(u′S)|u

′

L〉 at wall-normal locations y/δ ≈ 0.1, 0.2 and 0.4, and the sameobservations can be made. In addition, the response function depends only weaklyon the wall-normal location between y/δ≈ 0.1 and 0.4. Results are similar when theanalysis is repeated using u′2S (or |u′S|) as the metric of the small scale energetics (notshown for brevity).

Following the above discussion, we may re-write the response function as follows

f (u′L)= β<0u′L, for u′L < 0; f (u′L)= β>0u′L, for u′L > 0. (3.1a,b)

Because the small-scale fluctuation responds differently to the large-scale fluctuationdepending on the sign of u′L, and because R∼ β if f ∼ u′L, it is useful to consider aconditional amplitude modulation metric

R>0 ≡〈envL(u′S)u

′

L|u′

L > 0〉STD(envL(u′S)|u

′L > 0)STD(u′L|u′L > 0)

(3.2)

and similarly R<0. The conditional R defined in (3.2) is shown in figure 4(d) asfunctions of y+ for boundary-layer and channel flows at various Reynolds numbers.Figure 5(d) shows the same quantities as a function of y/δ. A few observations canbe made. First, within the viscous sublayer, R<0 ≈ R>0. Second, R>0 is 0 at y+ ≈ 20,and R<0 is 0 at y+ ≈ 500.

If the true response function follows (3.1), we may compute β< and β> similarlyas β in (1.1), except that, here, the constraint (1.6) will be applied for u′L < 0 andu′L > 0, respectively:

〈envL(u∗i,S,m)u′

i,L,m|u′

i,L,m ≶ 0〉 = 0. (3.3)

The response function f (u′i,L,m), as defined in (3.1), is such that the universal signal,computed according to (1.5), satisfies (3.3).

To account for the nonlinearity of the response function f as a function of u′L, wecan re-cast (2.8) to

β≶0 = 1/〈U+(yo)+ u′o|u′

o ≶ 0〉. (3.4)

A consequence of (3.4) is that the predicted amplitude modulation coefficient is largerfor u′L < 0 than for u′L > 0, which is consistent with figure 4. Figure 6 shows themeasured β≶0 as functions of the inner location yi for two outer locations (y+o = 215and 400). According to figure 6, β<0 ≈ β>0 in the viscous sublayer. Equation (3.4),however, is still successful only within the viscous sublayer.

For WMLES, the quantity of interest is the wall shear stress, which is a quantity inthe viscous sublayer, where f ∼ u′L. Since the response function f ∼ u′L, the usefulnessof (1.3) in the context of WMLES depends only on whether the large scales canbe commensurate with the LES grids. To test the sensitivity of the results on theseparating length scale, we repeat the above analysis for a different filtering lengthscale, l+ = 7000. Figure 7 shows the measured response function as functions of u′Land very similar results are obtained compared to the l+ = 4000 case.

Finally, we comment on the phase difference between the large scales and thesmall scales, and its connections to the amplitude modulation. Chung & McKeon(2010) and Jacobi & McKeon (2013) argued that amplitude modulation (measuredusing the correlation coefficient R) is but a measure of the phase difference between

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650 M. F. Howland and X. I. A. Yang

Flow direction

Attached eddies

FIGURE 2. Schematic of the modelled boundary layer. The attached eddies are spacefilling. On a vertical plane cut, as shown here, the number of eddies doubles as the sizehalves. An eddy affects the shaded region below it. The velocity at a generic point inthe flow field is a result of the additive superposition of all of the eddy-induced velocityfields there.

the large-scale fluctuations and the envelope of the small scales. A counter argumentwas proposed in Baars et al. (2017) where the authors argued that the phase shift isa finite Reynolds number effect. It is worth noting that the phase difference betweensmall and large scales is a potential explanation for the deviation from linearity off . While a full discussion of this point is beyond of the scope of this work, wenote that the phase difference between the large and small scales is not relevant toWMLES, where one typically uses the instantaneous LES velocity directly above thewall to predict the local wall shear stress (Bose & Park 2017).

3.2. Positive and negative amplitude modulationThe small scales in the near-wall region are subjected to changes in the local Reynoldsnumbers, which are modulated by the large-scale fluctuations. Equations (2.8)and (3.4) are models for such effects. According to HRAP and the sketch in figure 2,a large-scale eddy affects the flow beneath it (see e.g. Yang et al. 2016c; Yang &Lozano-Durán 2017; Yang et al. 2017), and therefore it is unlikely that amplitudemodulation due to large-scale modulation of the local Reynolds numbers woulddepend on the inner location. This expectation bears out in both (2.8) and (3.4),where the predicted response function is not a function of yi. However, in additionto the above mentioned mechanism that leads to positive amplitude modulation, thereare also mechanisms that could lead to negative amplitude modulation. Negativeamplitude modulation is defined as the decrease of small-scale energetics withincreasing large-scale velocity. Negative modulation manifests as a negative slopein f . Jacobi & McKeon (2013) argued that negative amplitude modulation arises asa result of the phase difference between the envelope of the small scales and thelarge-scale fluctuations. Zhang & Chernyshenko (2016), on the other hand, arguedthat the negative amplitude modulation is a combined effect of the modulation of thewall-normal coordinate and the decrease of STD(u′) as a function of the wall-normaldistance. Agostini & Leschziner (2016b) argued that negative modulation is theresult of the splatting motions which are the result of sweeps towards the wall. Last,Baars et al. (2015, 2017) argued that the negative modulation is due to intrusions ofnon-/less-turbulent free stream flow. These mechanisms compete with the mechanismthat is responsible for positive amplitude modulation, and lead to the overall behaviourof the response function in figures 4, 5 and 7.

Since β≷0 is a quantity that measures the overall effects of both the positiveamplitude modulation and the negative amplitude modulation, β≷0 itself is not

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Dependence of small-scale energetics on large scales in turbulent flows 651

0.08

0.10

0.06

0.04

0.02

0100 102101

0.08

0.10

0.12

0.06

0.04

0.02

0100 200 300 400

QSQH, (1.12)HRAP, (2.8)

QSQH, (1.12)HRAP, (2.8)

(a) (b)

FIGURE 3. (Colour online) (a) Symbols: measured amplitude modulation coefficients β asfunctions of y+i for outer locations, y+o = 200, 400 and 500 (which correspond to yo/δ =0.1, 0.2 and 0.25) for Reτ = 2000 channel flow DNS (Hoyas & Jiménez 2006). Differentcolours are used for different yo. Dashed lines: predictions from (1.2). Thin solid lines:(2.8). (b) Same as (a) but for β as functions of y+o for y+i = 1, 5 and 10.

0.9

0.3

–0.3

–0.6

–0.9

0

0.60.225

–0.225

–0.450

0

0.450

–2–4 0 2 4 –2–4 0 2 4

500

0.225

–0.225

–0.450

0

0.450

–2–4 0 2 4

0

–0.2

–0.4

0.2

0.4

0.6

0.8

1.0

102101

R

(a) (b)

(c) (d )

FIGURE 4. (Colour online) The response function 〈envL(u′S)|u′

L〉 as a function of the large-scale fluctuation u′L at wall-normal locations y+ ≈ 5 (a), y+ ≈ 75 (b) and y+ ≈ 200 (c).Solid lines are linear fits of the data for u′L > 0 and u′L < 0, respectively. The slopes are0.32, −0.01 and −0.04 for u′L > 0 in (a–c), and are 0.32, 0.06 and 0.04 for u′L < 0. Thedashed black line indicates u′L=0. The y scale is magnified in (b,c) for better visualization.(d) Conditional correlation coefficient R≶ as functions of y+. Solid lines represent R>0 anddashed lines represent R<0. Solid vertical lines represent the buffer layer of 30< y+< 100.

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652 M. F. Howland and X. I. A. Yang

0.225

–0.225

–0.450

0

0.450

1–1–2–3–4 0 2 3 4

0.225

–0.225

–0.450

0

0.450

1–1–2–3–4 0 2 3 4

0.225

–0.225

–0.450

0

0.450

1–1–2–3–4 0 2 3 4

0

–0.2

–0.4

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2 0.3 0.4 0.5

R

(a) (b)

(c) (d )

FIGURE 5. (Colour online) The response function 〈envL(u′S)|u′

L〉 as functions of the large-scale fluctuation u′L at wall-normal locations y/δ = 0.1 (a), 0.2 (b) 0.4 (c). The slope is−0.05 for u′L > 0 in (a–c) and is −0.04 for u′L < 0. (d) R≶0 as functions of y/δ. Solidlines represent R>0 and dashed lines represent R<0.

0.020.01

0 HRAP, (3.4)

0.030.040.050.060.070.08

–0.01100 102101

0.020.01

0

0.030.040.050.060.070.08

–0.01100 102101

(a) (b)

FIGURE 6. (Colour online) The measured β≶0 as functions of y+i for channel flowat Reτ = 2000 (+ symbols) at (a) y+o = 215 and (b) 400. The solid lines are the predictionsusing (3.4).

independent of the inner location. If positive amplitude modulation is fully accountedfor in (3.4), we could remove this form of interscale interaction from the responsefunction using (1.5), (3.1) and (3.4), i.e. u′i,d,S≡ u′i,S− (1+β≷0u′o)u

∗

i,S is only negativelymodulated by the large scales. This expectation is confirmed in figure 8, where〈envL(u′i,d,S)|u

′

L〉 are shown as functions of u′L for a few inner locations and a fixedouter location y+o = 270 (in a Reτ = 5200 channel (Lee & Moser 2015)).

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Dependence of small-scale energetics on large scales in turbulent flows 653

0.9

0.3

–0.3

–0.6

–0.9

0

0.6

–2–4 0 2 4

(a)

0.225

–0.225

–0.450

0

0.450

–2–4 0 2 4

(b)

0.225

–0.225

–0.450

0

0.450

–2–4 0 2 4

0.225

–0.225

–0.450

0

0.450

–2–4 0 2 4

(c) (d )

FIGURE 7. (Colour online) (a–c) Same as in figure 4 except with l+ = 7000. (d) Sameas in (a) except with y/δ = 0.2. The linear fits are the same as in figures 4 and 5.

1–1–2–3–4 0 2 3 4 1–1–2–3–4 0 2 3 4 1–1–2–3–4 0 2 3 4

0.9

0.3

–0.3–0.6–0.9

0

0.60.225

–0.225

0.450

–0.450

0

0.225

–0.225

0.450

–0.450

0

(a) (b) (c)

FIGURE 8. (Colour online) The response functions 〈envL(u′i,d,S)|u′

L〉 and 〈envL(u′S)|u′

L〉

measured using Reτ = 5200 channel flow DNS. The vertical axis for (a–c) is 〈envL(·)|u′L〉where (·) denotes either u′i,S or u′i,d,S. The outer location is at y+o = 270. The inner locationis at yi = 10 for (a), y+i = 70 for (b) and y+i = 140 for (c).

3.3. Spatial and temporal amplitude modulationThe discussion to this point has been limited to spatial amplitude modulation. Inthis subsection, we will repeat the analysis in § 3.1 for the temporal amplitudemodulation process. Hot-wire measurements are used directly for this purpose.Temporal amplitude modulation is the measure of amplitude modulation using Taylor’shypothesis and a mean convective velocity for the conversion of a temporal signal tospace (Yang & Howland 2018). For brevity we only show results of the Reτ = 13 000boundary layer. Figure 9 shows the measured response function 〈envL(u′S)|u

′

L〉 asfunctions of the large-scale velocity fluctuation at wall-normal locations y+ ≈ 10, 75and 200 (results at other wall-normal locations are similar and are not shown here forbrevity). At y+≈ 10, spatial and temporal modulations are not distinguishably differentaccording to this metric. Beyond y+ ≈ 10, the two processes lead to quantitatively

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654 M. F. Howland and X. I. A. Yang

1–1–2–3–4 0 2 3 4 1–1–2–3–4 0 2 3 4 1–1–2–3–4 0 2 3 4

0.9

0.3

–0.3–0.6–0.9

0

0.6 0.225

–0.225

0.450

–0.450

0

0.225

–0.225

0.450

–0.450

0

(a) (b) (c)

SpatialTemporal

FIGURE 9. (Colour online) 〈envL(u′S)|u′

L〉 as functions of u′L at y+ ≈ 10 (a), 75 (b) and200 (c) for a boundary-layer flow at Reτ = 13 000. Orange: temporal data; blue: spatialdata (see Yang & Howland (2018) for a detailed discussion on the conversion of temporalhot-wire data to spatial coordinates).

102 102500 500

0

–0.2

–0.4

0.2

0.4

0

–0.2

–0.4

0.2

0.4

SpatialTemporal

(a) (b)

R<0

FIGURE 10. (Colour online) Conditional R≶0 as a function of y+ for Reτ = 13 000boundary-layer flow. (a) R>0 (b) R<0. Orange: temporal data; blue: spatial data.

different results. Temporal amplitude modulation is stronger than the same process inspatial coordinates, i.e. for u′L < 0, the small scales are more subdued, and for u′L > 0,the small scales are more energetic. Nevertheless, the response function can still beapproximated using a piecewise linear function (3.1), and therefore, the discussion in§ 3.1 is equally useful for temporal amplitude modulation.

Figure 10 shows the R≶0 measured using the temporal data as functions ofthe wall-normal distance. Spatial amplitude modulation and temporal amplitudemodulation are only quantitatively different. As R≶0 are measures of β≶0, figures 9and 10 and suggest that the temporally measured R≶0 are generally greater than theirspatial counterparts in the buffer layer and in the logarithmic region.

4. Homogeneous isotropic turbulenceThe amplitude modulation process has mainly been considered for wall-bounded

flows. However, interscale interactions were also found in free shear flows includingmixing layers, wakes and jets (Bandyopadhyay & Hussain 1984; Buxton &Ganapathisubramani 2014). In a recent theoretical investigation, Johnson & Meneveau(2017) suggested that amplitude modulation might also be present in isotropicturbulence. In this section, we examine the presence of amplitude modulation inHIT by measuring directly the response function (as in § 3.1). In particular, we aimto garner HIT in an investigation of the source of negative amplitude modulationwhich has been observed in boundary-layer flows and discussed in detail in § 3.2.

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Dependence of small-scale energetics on large scales in turbulent flows 655

1 10 100 300

10–2

10–4

10–6

10–8

100

E(k)

k

FIGURE 11. (Colour online) Time-averaged radial kinetic energy spectrum for the forcedHIT of Li et al. (2008). The dashed black line indicates the inertial subrange with aconstant slope of −5/3.

For this purpose, we use DNS data of HIT at Taylor micro-scale Reynolds numberReλ = 433 (details of these data can be found in Li et al. 2008). The computationaldomain is of size 10243, and therefore there are k = 512 Fourier modes in eachdirection. Figure 11 shows the energy spectrum as a function of the wavenumber.The classic inertial scaling exist between k= 1 and k= 100 (see figure 11).

To study the interscale interaction, we compute again the response function〈envL(u′S)||u

′

L|〉, where u′S and u′L are velocity fluctuations whose characteristic lengthscales are smaller and larger than a pre-specified length scale (in Fourier space). Thevelocity field is high-pass filtered at k = 1 to remove velocity fluctuations at thewavelengths at which the forcing is applied (Li et al. 2008). Given the flow isotropy,〈envL(u′S)||u

′

L|〉 = 〈envL(u′S)| − |u′

L|〉, therefore only 〈envL(u′S)||u′

L|〉 will be shown.Reasonable statistical convergence is attained by averaging in the three homogeneousdirections, for three velocity components and among two statistically independentrealizations. The mean of the envelope is removed following convention. Positiveslope in the response function is positive amplitude modulation (R> 0) and negativeslope is negative amplitude modulation (R < 0). Figure 12 shows 〈envL(u′S)||u

′

L|〉 asfunctions of the large-scale fluctuation. The small scales are normalized by theirenergy content TKES for better visualization.

TKES =

∫∞

kfilt

E(k) dk, (4.1)

where kfilt is the cutoff filtration wavenumber.We make a few observations. First, 〈envL(u′S)||u

′

L|〉 6= 0, therefore interscaleinteraction is present in HIT. Similar observations are made if the small-scaleenergetics are measured using u′2s . This positive amplitude modulation process mayresult from similar mechanisms that lead to a positive amplitude modulation in wallturbulence. For example, large |u′L| leads to a high local Reynolds number, which inturn leads to more energetic small scales (Zhang & Chernyshenko 2016). Second, theresponse of the small scales to large-scale velocity fluctuation increases as a functionof the filtration wavenumber k until k ≈ 100, at which wavenumber u′S is dominatedby viscous effects and u′L is dominated by inertial effects. Third, negative amplitudemodulation is not found at any filtration wavenumber kfilt, suggesting that mechanisms

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656 M. F. Howland and X. I. A. Yang

0 0.5 1.0 1.5–0.1

0.1

0.2

0.3

0

¯env

L(u� S)

||u� L|˘

/�TK

E S

FIGURE 12. (Colour online) The response function 〈envL(u′S)|u′

L〉 computed using forcedHIT data. Scale separation is made at kfilt = 10, 100 and 300.

that lead to negative amplitude modulation may be specific to wall-bounded flows.Considering the physical mechanisms of negative amplitude modulation as discussedin Baars et al. (2017) and § 3.2, we may note that these effects likely will notmanifest in HIT. As a result, these effects and negative amplitude modulation maybe unique to wall-bounded flows.

5. Wall-modelled large eddy simulation

In this section, a new LES wall model formulation is developed based on theamplitude modulation of the wall shear stresses, i.e. (2.9). A connection is madebetween the new amplitude modulation based model and the slip wall model (Bose &Moin 2014). The performance of this wall model is then compared to the commonlyused equilibrium wall model (Piomelli & Balaras 2002).

5.1. Model formulationFollowing the discussion in § 3, we can approximate the response function withinthe viscous sublayer as a linear function of the large-scale velocity fluctuationf (yi, yo, u′o,L)= βu′o,L. As a result, equation (2.9) reduces to

τ ′w = βu′o,L + (1+ βu′o,L)τ∗

w,S,m. (5.1)

WMLES requires the filtered wall shear stresses to integrate the LES equations in thebulk region (Piomelli & Balaras 2002; Choi & Moin 2012). If the large scales arecommensurate with the LES grids, filtering both sides of (5.1) leads to

τ ′w = τ′

w,L = βu′o,L, (5.2)

and there is no need to specify the universal small-scale wall shear stress fluctuation.Here · is the LES filtering operation. The modelling parameter β is specified accordingto (2.8), β = 1/U+(yo). It then follows from (5.2) that

τw = 〈τw〉 + τ ′w = 〈τw〉 +1

U(y1)u′(y= y1), (5.3)

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Dependence of small-scale energetics on large scales in turbulent flows 657

where y1 is the distance between the first off-wall grid point and the wall. If the firstoff-wall grid point is in the logarithmic range, equation (5.3) reduces to

τw = 〈τw〉 +κ u′

log(y1/δν), (5.4)

where κ ≈ 0.4 is the von Kármán constant, δν = ν/uτ · exp(−κB) for smooth walls,and B ≈ 5 is the additive constant in the law of the wall. Different from (5.4), theequilibrium wall model reads

τw =

[κu

log(y1/δν)

]2

≈ 〈τw〉 +2κ u′

log(y1/δν), (5.5)

leading to a slightly different model for the wall shear stress fluctuations. Before weproceed and compare the two models in WMLES, we make a connection between(5.4) and the recently developed slip wall model (Bose & Moin 2014), where a slipvelocity is applied at the wall in replacement of the no-slip condition. The slip velocityis specified according to the following Robin-type boundary condition,

us − lp∂ us

∂n= 0, (5.6)

where us is the slip velocity, n is the wall-normal direction and lp is the slip length.Bose & Moin (2014) used a dynamic procedure for determining the slip length, here,we turn to well-established Reynolds Averaged Navier–Stokes (RANS) closures for anestimate of lp. We start by considering the equilibrium momentum equation at the firstoff-wall grid point

u2τ − νT

∂〈u〉∂n= 0, (5.7)

where uτ is the friction velocity, uτ = κ〈u〉/ log(y1/δν) according to the law of the wall.The eddy viscosity is νT = κuτy1. Dividing both sides of (5.7) by uτ and replacing theremaining uτ with κ〈u〉/ log(y1/δν), we obtain

κ〈u〉log(y1/δν)

− κy1∂〈u〉∂n= 0. (5.8)

By re-arranging the terms, equation (5.8) leads to

〈u〉 − y1 log(y1/δν)∂〈u〉∂n= 0. (5.9)

While 〈·〉 is not equivalent to ·, equation (5.9) provides us with an estimate of the sliplength

lp = y1 log(y1/δν). (5.10)

A different, physics-based interpretation of the slip wall model formulation may befound in Lozano-Durán et al. (2017). The arguments in Lozano-Durán et al. (2017)are not detailed here as providing a physical interpretation to the slip model is notthe focus of this paper. Here, we use (5.6) and (5.10) to estimate the wall shear stresspredicted by the slip model,

τw = νT∂ us

∂n=νT

lpus =

κuτlog(y1/δν)

us. (5.11)

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658 M. F. Howland and X. I. A. Yang

Normalizing the quantities in (5.11) with wall units and dropping the + superscriptfor brevity, equation (5.11) leads to

τw =κy1

log(y1/δν)us = 〈τw〉 +

κ u′

log(y1/δν), (5.12)

which happens to be the same form as (5.4). While detailed discussion falls out of thescope of this paper, this result may be the reason why the slip wall model outperformsthe equilibrium model in channel flow (Bae et al. 2018).

5.2. LES set-upWe test the new wall model (5.4) in WMLES and compare its performance withthat of the equilibrium wall model. The in-house pseudo-spectral code LESGO isused. Details of this code may be found in Bou-Zeid, Meneveau & Parlange (2005)and Anderson & Meneveau (2011). Here, we briefly summarize the main featuresof the code. The code solves the filtered Navier–Stokes equation in a half-channelwith periodic boundary conditions in both the streamwise and the spanwise directions.The half-channel is driven by an imposed pressure gradient. A zero-stress conditionis imposed at the top boundary, and the bottom wall is modelled using a wallmodel. A pseudo-spectral scheme is used in both the streamwise and the spanwisedirection, and a second-order finite difference scheme is used for spatial discretizationin the wall-normal direction. The subgrid-scale stresses are modelled using thescale-dependent Lagrangian-averaged Smagorinsky model (Bou-Zeid et al. 2005). Forthe calculations here, the computational domain is 2πδ× 1δ× 2πδ in the streamwise,wall-normal and spanwise directions respectively. Two grids are used: 643 and 1283.All the statistics are averaged for 10 flow throughs after a statistically stationary stateis reached. The flow through time is tf = Lx/uo with Lx and uo being the extent of thecomputational domain in the streamwise direction and the volume-averaged velocity.Here, we compare the performance of the new wall model in (5.3) to that of theequilibrium wall model given by (5.5). For the particular flow here, the mean wallshear stress is known from the mean momentum balance; therefore, the wall shearstress fluctuation is the quantity of interest. The wall model input yo = 3.22× 10−5δ,which corresponds to a Reτ = 4200 boundary layer.

5.3. WMLES resultsBoth the equilibrium wall model and the amplitude modulation based model capturethe mean velocity profile correctly (see figure 13). The two models only differ inhigh-order statistics. WMLES often leads to deficiencies in its predictions of thevariance of streamwise velocity (Park & Moin 2016; Park et al. 2016). Figure 14shows the variance of the streamwise velocity fluctuation for the WMLES usingboth the presently developed modulation-based wall model and the commonly usedequilibrium wall model. While both models are fairly accurate when the resolutionis high (1283), the modulation-based wall model is slightly more accurate forlow-resolution calculations. In WMLES of channel flow, one can typically expectthe grid convergence of the streamwise velocity variance from the centre of thechannel to the wall (Yang et al. 2016a,c). In figure 14, this expectation only bearsout for WMLES with the modulation-based model. Therefore, we may conclude thatthe modulation-based wall model leads to more accurately resolved bulk structures.

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Dependence of small-scale energetics on large scales in turbulent flows 659

10010–110–210–3 10010–110–210–30

5

10

15

25

20

30

0

5

10

15

25

20

30Log law

(a) (b)

FIGURE 13. (Colour online) (a) Mean velocity profiles as functions of the wall-normaldistance for WMLES with the modulation-based model. The grid sizes are 643 and 1283.(b) Same as (a) but for WMLES that use the equilibrium wall model.

10010–110–210–3 10010–110–210–3

10

0

2

4

6

8

10

0

2

4

6

8

DNS

(a) (b)

FIGURE 14. (Colour online) (a) Variance of the streamwise velocity fluctuations asfunctions of the wall-normal distance. The DNS results are obtained from Lozano-Durán& Jiménez (2014) and the two WMLES use the model in (5.3). The grid sizes are 643

and 1283. (b) Same as (a) but for WMLES that use the equilibrium wall model.

6. ConclusionsInterscale interaction in turbulent flows is investigated in this study through the lens

of amplitude modulation (Mathis et al. 2009a). We directly measured the responsefunction in wall-bounded flows as a function of the wall-normal distance and thelarge-scale velocity fluctuation. Although a linear response function as suggested byMarusic et al. (2010) is generally not accurate above the viscous sublayer, a piecewiselinear function (3.1) is found to be a good working approximation of the real responsefunction. The usefulness of the predictive inner–outer model (1.3) in the context ofWMLES is carefully assessed. We argue that it is possible to correspond the largeand small scales in the PIO model to the resolved and unresolved turbulence in thenear-wall region for a typical WMLES. Therefore, the model (1.3) can be useful toLES wall modelling if the modelling parameters can be specified in a non-empiricalmanner using (2.8). One specification is ventured in § 2, and the prediction agreesqualitatively with the measurements in the viscous sublayer (which is the region ofinterest if the wall shear stress is the quantity to be modelled).

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660 M. F. Howland and X. I. A. Yang

Further, the evidence suggests that positive amplitude modulation is invariant to thewall-normal height of the inner location and to the Reynolds number. The responsefunction f is a superposition of positive amplitude modulation and negative amplitudemodulation. We have related the positive amplitude modulation to the interscaleinteraction between the inertial-dominated motions and viscosity-dominated motions.Since the two ranges of scales are present in all turbulent flows, positive amplitudemodulation is expected to be present in HIT as well. This is confirmed in § 4.Although we have limited our discussion in this paper to the streamwise velocity andwall shear stress fluctuations, this discussion is likely to be relevant to the amplitudemodulation of spanwise and vertical velocity components as well (Mathis et al. 2013;Talluru et al. 2014).

Finally, a novel amplitude modulation-based LES wall model has been proposedand tested in turbulent channel flow at Reτ = 4200. The novel wall model showssmall improvements over the common equilibrium wall model for coarsely resolvedsimulations. The application of this wall model to more complex flows requires futureinvestigation and validation.

Acknowledgements

M.F.H. is funded through a National Science Foundation Graduate ResearchFellowship under grant no. DGE-1656518 and a Stanford Graduate Fellowship.X.I.A.Y. is funded by the US AFOSR (grant no. 1194592-1-TAAHO). The authorswould like to thank I. Marusic and M. Lee for making their data available andP. Johnson, A. Lozano-Durán and E. L. Holzbaur for their helpful comments on themanuscript.

REFERENCES

AGOSTINI, L. & LESCHZINER, M. A. 2014 On the influence of outer large-scale structures onnear-wall turbulence in channel flow. Phys. Fluids 26 (7), 075107.

AGOSTINI, L. & LESCHZINER, M. 2016a On the validity of the quasi-steady-turbulence hypothesisin representing the effects of large scales on small scales in boundary layers. Phys. Fluids28 (4), 045102.

AGOSTINI, L. & LESCHZINER, M. 2016b Predicting the response of small-scale near-wall turbulenceto large-scale outer motions. Phys. Fluids 28 (1), 015107.

AGOSTINI, L., LESCHZINER, M. & GAITONDE, D. 2016 Skewness-induced asymmetric modulationof small-scale turbulence by large-scale structures. Phys. Fluids 28 (1), 015110.

ANDERSON, W. & MENEVEAU, C. 2011 Dynamic roughness model for large-eddy simulation ofturbulent flow over multiscale, fractal-like rough surfaces. J. Fluid Mech. 679, 288–314.

AWASTHI, A. & ANDERSON, W. 2018 Numerical study of turbulent channel flow perturbed byspanwise topographic heterogeneity: amplitude and frequency modulation within low- andhigh-momentum pathways. Phys. Rev. Fluids 3 (4), 044602.

BAARS, W. J., HUTCHINS, N. & MARUSIC, I. 2016 Spectral stochastic estimation of high-Reynolds-number wall-bounded turbulence for a refined inner–outer interaction model. Phys. Rev. Fluids1 (5), 054406.

BAARS, W. J., HUTCHINS, N. & MARUSIC, I. 2017 Reynolds number trend of hierarchies and scaleinteractions in turbulent boundary layers. Phil. Trans. R. Soc. Lond. A 375 (2089), 20160077.

BAARS, W. J., TALLURU, K. M., HUTCHINS, N. & MARUSIC, I. 2015 Wavelet analysis of wallturbulence to study large-scale modulation of small scales. Exp. Fluids 56 (10), 188.

BAE, H. J., LOZANO-DURAN, A., BOSE, S. & MOIN, P. 2018 Turbulence intensities in large-eddysimulation of wall-bounded flows. Phys. Rev. Fluids 3 (1), 014610.

Dow

nloa

ded

from

htt

ps://

ww

w.c

ambr

idge

.org

/cor

e. S

tanf

ord

Libr

arie

s, o

n 13

Aug

201

8 at

16:

45:1

6, s

ubje

ct to

the

Cam

brid

ge C

ore

term

s of

use

, ava

ilabl

e at

htt

ps://

ww

w.c

ambr

idge

.org

/cor

e/te

rms.

htt

ps://

doi.o

rg/1

0.10

17/jf

m.2

018.

554

Dependence of small-scale energetics on large scales in turbulent flows 661

BANDYOPADHYAY, P. R. & HUSSAIN, A. 1984 The coupling between scales in shear flows. Phys.Fluids 27 (9), 2221–2228.

BOSE, S. & MOIN, P. 2014 A dynamic slip boundary condition for wall-modeled large-eddy simulation.Phys. Fluids 26 (1), 015104.

BOSE, S. T. & PARK, G. I. 2017 Wall-modeled LES for complex turbulent flows. Annu. Rev. FluidMech. 50 (1), 535–561.

BOU-ZEID, E., MENEVEAU, C. & PARLANGE, M. 2005 A scale-dependent lagrangian dynamic modelfor large eddy simulation of complex turbulent flows. Phys. Fluids 17 (2), 025105.

BUXTON, O. & GANAPATHISUBRAMANI, B. 2014 Concurrent scale interactions in the far-field of aturbulent mixing layer. Phys. Fluids 26 (12), 125106.

CHERNYSHENKO, S. I., MARUSIC, I. & MATHIS, R. 2012 Quasi-steady description of modulationeffects in wall turbulence. arXiv:1203.3714v1.

CHOI, H. & MOIN, P. 2012 Grid-point requirements for large eddy simulation: Chapman’s estimatesrevisited. Phys. Fluids 24 (1), 011702.

CHUNG, D. & MCKEON, B. J. 2010 Large-eddy simulation of large-scale structures in long channelflow. J. Fluid Mech. 661, 341–364.

FISCALETTI, D., GANAPATHISUBRAMANI, B. & ELSINGA, G. E. 2015 Amplitude and frequencymodulation of the small scales in a jet. J. Fluid Mech. 772, 756–783.

GANAPATHISUBRAMANI, B., HUTCHINS, N., MONTY, J. P., CHUNG, D. & MARUSIC, I. 2012Amplitude and frequency modulation in wall turbulence. J. Fluid Mech. 712, 61–91.

HOWLAND, M. F. & YANG, X. I. A. 2017 Dependence of small-scale energetics on large scales inwall-bounded flows. In CTR Annual Research Briefs, pp. 215–228. Stanford University, CenterTurbulence Research.

HOYAS, S. & JIMÉNEZ, J. 2006 Scaling of the velocity fluctuations in turbulent channels up toReτ = 2003. Phys. Fluids 18 (1), 011702.

HUTCHINS, N., CHAUHAN, K., MARUSIC, I., MONTY, J. & KLEWICKI, J. 2012 Towards reconcilingthe large-scale structure of turbulent boundary layers in the atmosphere and laboratory.Boundary-Layer Meteorol. 145 (2), 273–306.

HUTCHINS, N. & MARUSIC, I. 2007 Large-scale influences in near-wall turbulence. Phil. Trans. R.Soc. Lond. A 365 (1852), 647–664.

HUTCHINS, N., NICKELS, T. B., MARUSIC, I. & CHONG, M. 2009 Hot-wire spatial resolution issuesin wall-bounded turbulence. J. Fluid Mech. 635, 103–136.

HWANG, J., LEE, J., SUNG, H. J. & ZAKI, T. A. 2016 Inner–outer interactions of large-scalestructures in turbulent channel flow. J. Fluid Mech. 790, 128–157.

INOUE, M., MATHIS, R., MARUSIC, I. & PULLIN, D. I. 2012 Inner-layer intensities for the flat-plateturbulent boundary layer combining a predictive wall-model with large-eddy simulations. Phys.Fluids 24 (7), 075102.

JACOBI, I. & MCKEON, B. J. 2013 Phase relationships between large and small scales in the turbulentboundary layer. Exp. Fluids 54 (3), 1481.

JOHNSON, P. L. & MENEVEAU, C. 2017 Turbulence intermittency in a multiple-time-scale Navier–Stokes-based reduced model. Phys. Rev. Fluids 2 (7), 072601.

LEE, M. & MOSER, R. D. 2015 Direct numerical simulation of turbulent channel flow up to Reτ ≈5200. J. Fluid Mech. 774, 395–415.

LI, Y., PERLMAN, E., WAN, M., YANG, Y., MENEVEAU, C., BURNS, R., CHEN, S., SZALAY, A. &EYINK, G. 2008 A public turbulence database cluster and applications to study lagrangianevolution of velocity increments in turbulence. J. Turbul. 9, N31.

LOZANO-DURÁN, A., BAE, H., BOSE, S. & MOIN, P. 2017 Dynamic wall models for the slipboundary condition. Annual Research Briefs 2017. Stanford University, Center TurbulenceResearch.

LOZANO-DURÁN, A. & JIMÉNEZ, J. 2014 Effect of the computational domain on direct simulationsof turbulent channels up to Reτ = 4200. Phys. Fluids 26 (1), 011702.

MARUSIC, I., MATHIS, R. & HUTCHINS, N. 2010 Predictive model for wall-bounded turbulent flow.Science 329 (5988), 193–196.

Dow

nloa

ded

from

htt

ps://

ww

w.c

ambr

idge

.org

/cor

e. S

tanf

ord

Libr

arie

s, o

n 13

Aug

201

8 at

16:

45:1

6, s

ubje

ct to

the

Cam

brid

ge C

ore

term

s of

use

, ava

ilabl

e at

htt

ps://

ww

w.c

ambr

idge

.org

/cor

e/te

rms.

htt

ps://

doi.o

rg/1

0.10

17/jf

m.2

018.

554

662 M. F. Howland and X. I. A. Yang

MATHIS, R., HUTCHINS, N. & MARUSIC, I. 2009a Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J. Fluid Mech. 628, 311–337.

MATHIS, R., HUTCHINS, N. & MARUSIC, I. 2011 A predictive inner–outer model for streamwiseturbulence statistics in wall-bounded flows. J. Fluid Mech. 681, 537–566.

MATHIS, R., MARUSIC, I., CHERNYSHENKO, S. I. & HUTCHINS, N. 2013 Estimating wall-shear-stressfluctuations given an outer region input. J. Fluid Mech. 715, 163.

MATHIS, R., MONTY, J. P., HUTCHINS, N. & MARUSIC, I. 2009b Comparison of large-scaleamplitude modulation in turbulent boundary layers, pipes, and channel flows. Phys. Fluids 21(11), 111703.

MENEVEAU, C. & KATZ, J. 2000 Scale-invariance and turbulence models for large-eddy simulation.Annu. Rev. Fluid Mech. 32 (1), 1–32.

PARK, G., HOWLAND, M. F., LOZANO-DURAN, A. & MOIN, P. 2016 Prediction of wall shear-stressfluctuations in wall-modeled large-eddy simulation. In APS Meeting Abstracts.

PARK, G. I. & MOIN, P. 2016 Space-time characteristics of wall-pressure and wall shear-stressfluctuations in wall-modeled large eddy simulation. Phys. Rev. Fluids 1 (2), 024404.

PIOMELLI, U. & BALARAS, E. 2002 Wall-layer models for large-eddy simulations. Annu. Rev. FluidMech. 34 (1), 349–374.

SIDEBOTTOM, W., CABRIT, O., MARUSIC, I., MENEVEAU, C., OOI, A. & JONES, D. 2014 Modellingof wall shear-stress fluctuations for large-eddy simulation. In Proc. 19th Australasian FluidMech. Conf., Melbourne, Australia.

TALLURU, K. M., BAIDYA, R., HUTCHINS, N. & MARUSIC, I. 2014 Amplitude modulation of allthree velocity components in turbulent boundary layers. J. Fluid Mech. 746, R1.

TANG, Z. & JIANG, N. 2018 Scale interaction and arrangement in a turbulent boundary layer perturbedby a wall-mounted cylindrical element. Phys. Fluids 30 (5), 055103.

TOWNSEND, A. A. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.YANG, X. I. A., BAIDYA, R., JOHNSON, P., MARUSIC, I. & MENEVEAU, C. 2017 Structure function

tensor scaling in the logarithmic region derived from the attached eddy model of wall-boundedturbulent flows. Phys. Rev. Fluids 2 (6), 064602.

YANG, X. I. A. & HOWLAND, M. F. 2018 Implication of Taylor’s hypothesis on measuring flowmodulation. J. Fluid Mech. 836, 222–237.

YANG, X. I. A. & LOZANO-DURÁN, A. 2017 A multifractal model for the momentum transferprocess in wall-bounded flows. J. Fluid Mech. 824, R2.

YANG, X. I. A., MARUSIC, I. & MENEVEAU, C. 2016a Hierarchical random additive process andlogarithmic scaling of generalized high order, two-point correlations in turbulent boundarylayer flow. Phys. Rev. Fluids 1 (2), 024402.

YANG, X. I. A., MARUSIC, I. & MENEVEAU, C. 2016b Moment generating functions and scalinglaws in the inertial layer of turbulent wall-bounded flows. J. Fluid Mech. 791, R2.

YANG, X. I. A., MENEVEAU, C., MARUSIC, I. & BIFERALE, L. 2016c Extended self-similarity inmoment-generating-functions in wall-bounded turbulence at high Reynolds number. Phys. Rev.Fluids 1 (4), 044405.

ZHANG, C. & CHERNYSHENKO, S. I. 2016 Quasisteady quasihomogeneous description of the scaleinteractions in near-wall turbulence. Phys. Rev. Fluids 1 (1), 014401.

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e. S

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arie

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8 at

16:

45:1

6, s

ubje

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w.c

ambr

idge

.org

/cor

e/te

rms.

htt

ps://

doi.o

rg/1

0.10

17/jf

m.2

018.

554