Active and inactive components of the streamwise velocity in wall-bounded turbulence
aa r X i v : . [ phy s i c s . f l u - dyn ] S e p This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Active and inactive components of thestreamwise velocity in wall-boundedturbulence
Rahul Deshpande , Jason P. Monty , and Ivan Marusic † Department of Mechanical Engineering, University of Melbourne, Parkville,VIC 3010, Australia (Received xx; revised xx; accepted xx)
Townsend (1961) introduced the concept of active and inactive motions for wall-bounded turbulent flows, where the active motions are solely responsible for producingthe Reynolds shear stress, the key momentum transport term in these flows. While thewall-normal component of velocity is associated exclusively with the active motions,the wall-parallel components of velocity are associated with both active and inactivemotions. In this paper, we propose a method to segregate the active and inactivecomponents of the 2-D energy spectrum of the streamwise velocity, thereby allowing usto test the self-similarity characteristics of the former which are central to theoreticalmodels for wall-turbulence. The approach is based on analyzing datasets comprisingtwo-point streamwise velocity signals coupled with a spectral linear stochastic estimation(SLSE) based procedure. The data considered span a friction Reynolds number range Re τ ∼ O (10 ) – O (10 ). The procedure linearly decomposes the full 2-D spectrum (Φ)into two components, Φ ia and Φ a , comprising contributions predominantly from theinactive and active motions, respectively. This is confirmed by Φ a exhibiting wall-scaling,for both streamwise and spanwise wavelengths, corresponding well with the Reynoldsshear stress cospectra reported in the literature. Both Φ a and Φ ia are found to depictprominent self-similar characteristics in the inertially dominated region close to the wall,suggestive of contributions from Townsend’s attached eddies. Inactive contributionsfrom the attached eddies reveal pure k − -scaling for the associated 1-D spectra (where k is the streamwise/spanwise wavenumber), lending empirical support to the attachededdy model of Perry & Chong (1982). Key words: boundary layer structure, turbulent boundary layers, turbulent flows
1. Introduction and motivation
The attached eddy model (Perry & Chong 1982; Marusic & Monty 2019), based onTownsend’s attached eddy hypothesis (Townsend 1976), is a conceptual model throughwhich the kinematics in a wall-bounded flow can be statistically represented by ahierarchy of geometrically self-similar attached eddies that are inertially dominated(inviscid), and randomly distributed in the flow field. Here, the term ‘attached’ refers toa flow structure whose geometric extent, i.e. the size of its velocity field, scales with itsdistance from the wall ( z ) and mean friction velocity ( U τ ), respectively. As per Townsend(1976), the attached eddies have a population density inversely proportional to theirheight ( H ), which ranges between O ( z min ) . H . O ( δ ), where z min corresponds to the † Email address for correspondence: [email protected]
R. Deshpande, J. P. Monty and I. Marusic start of the inertial region, while δ is the boundary layer thickness. At any z & z min ,the cumulative contribution from the range of attached eddies results in the streamwiseand spanwise turbulence intensities varying logarithmically as a function of z , while thewall-normal variance is a constant following: u = B − A ln( zδ ) ,v = B − A ln( zδ ) ,w = B , and uw + = B , (1.1)where A , A , B , B , B and B are constants. Here, u , v and w are the velocityfluctuations along the streamwise ( x ), spanwise ( y ) and wall-normal ( z ) directions,respectively, while superscript ‘+’ denotes normalization by U τ and kinematic viscosity( ν ). Recent literature (Jimenez & Hoyas 2008; Baidya et al. et al. z + ∼ u has been more convincing from high Re τ experimental datasets (Hultmark et al. et al. Re τ simulations (Jimenez & Hoyas 2008; Lee & Moser 2015), likely owing to the lackof scale separation resulting in the self-similar contributions becoming obscured by thenon-self-similar contributions at the same scale (Jimenez & Hoyas 2008; Rosenberg et al. b ). Recently, Baars & Marusic (2020 b ) were able to segregatethese two contributions, consequently revealing the near-wall logarithmic growth (of u )due to self-similar contributions down to z + ∼
80, with a slope of 0.98 (= A ; also knownas the Townsend-Perry constant).Given that the turbulence intensities in (1.1) equate to the integrated spectral energyin the respective velocity fluctuations (that is, u = R ∞ φ uu dk x , where φ uu is the one-dimensional (1-D) streamwise velocity spectrum and k x is the streamwise wavenumber),the contribution from the hierarchy of attached eddies also manifests itself in the energyspectra of the two wall-parallel velocity components; in the form of the so-called k − x -scaling (Perry & Chong 1982). This scaling has been predicted previously via dimensionalanalysis and other theoretical arguments (Perry & Abell 1977; Perry et al. et al. et al. (1986) further arguing that the respectivepremultiplied spectra ( k + x φ + uu , k + x φ + vv ) should plateau at a constant value equal to therate of logarithmic decay ( A and A ) for u and v . These predictions, however, arerarely observed at finite Re τ , likely due to the flow containing a mixture of self-similarattached eddies and other non-self-similar flow structures. The difficulty in separating thetwo contributions may explain the lack of convincing empirical evidence of the k − x -scalingfor φ uu , and its association with A , in the literature (Nickels et al. et al. a , b ).Noting that experiments show that u and v varies with Reynolds number in theinertial region while uw + does not (as per equations 1.1), Townsend (1961) commentedthat “ it is difficult to reconcile these observations without supposing that the motionat any point consists of two components, an active component responsible for turbulenttransfer and determined by the stress distribution and an inactive component which doesnot transfer momentum or interact with the universal component. ” He further elaborated“ that the inactive motion is a meandering or swirling motion made up from attachededdies of large size which contribute to the Reynolds stress much further from the wall thanthe point of observation. ” This definition of active and inactive motions, however, seems ctive and inactive components of the streamwise velocity in wall turbulence Active and inactive motions
In the simplest attached eddy model, attached eddies are the only eddying motionspresent in the boundary layer, and they lead to ‘active’ and ‘inactive’ contributions. Thekey reason for this is the nature of the velocity signature from individual attached eddiesin this inviscid model. The impermeability boundary condition at the wall enforces w = 0at the wall, but allows slip (and hence finite u and v at the wall). This is achieved byproducing attached eddy velocity fields using a vortex structure with image vortex pairsin the plane of the wall. The result is a spatially localised w -velocity signature from theattached eddies - this is well illustrated in figure 1 of Perry et al. (1986). Consequently,at any wall-normal location z in the inertial region, active motions are solely due to thevelocity fields of the attached eddies of height, H ∼ O ( z ), and these contribute to u ( z ), v ( z ), w ( z ) and hence uw ( z ). The inactive motions, however, are caused by the velocityfields from relatively large and taller attached eddies of height O ( z ) ≪ H . O ( δ ), andwhile these eddies contribute to u ( z ) and v ( z ), they make no significant contributionto w ( z ). Hence, the inactive motions do not contribute to uw ( z ) (or w ( z )). Therefore,while both active and inactive motions contribute to u ( z ) (and v ( z )), there are onlyactive contributions to uw ( z ) (or w ( z )). The consequence of this is that active motionsare the component of attached-eddy contributions that have pure wall-scaling ( z and U τ ). The remaining attached eddy contributions are the relatively large scale inactivemotions which, together with the inverse probability distribution of scales as per AEH,lead to the logarithmic decay of u and v (equation 1.1) with z .Given the above, the resulting attached eddy velocity fields can thus be decomposedfollowing Panton (2007): u = u active + u inactive ,v = v active + v inactive ,w = w active , (1.2)and as the active and inactive velocity fields are uncorrelated (Townsend 1961; Bradshaw1967), the Reynolds stresses in equation (1.1) can also be decomposed as: u = u + u ,v = v + v ,w = w ,uw = ( u active )( w active ) . (1.3)Here, the active and inactive motions can be deemed uncorrelated only if we ignorethe non-linear interactions across these motions, such as modulation, which havebeen shown to exist previously (Morrison 2007; Mathis et al. et al. et al. et al. a , b ;Deshpande et al. et al. R. Deshpande, J. P. Monty and I. Marusic contributions need to be recognized and appropriately accounted for while consideringthe decomposition in (1.3). They include the fine dissipative scales, as well as thosecorresponding to the inertial sub-range (Perry et al. et al. Re τ → ∞ , which the inviscid AEH models. Other contributions includethose from the very-large-scale-motions or superstructures (SS), which are associatedwith tall and large δ -scaled eddies spanning across the inertial region and contributingsubstantively to u and v (Baars & Marusic 2020 a , b ; Deshpande et al. et al. w , which is confirmed by the wall-scaling exhibited by the 1-D w -spectra(Bradshaw 1967; Morrison et al. et al. u and v in the inertial region. The total inactivecontributions can thus be segregated as: u = u , AE + u , SS , and v = v , AE + v , SS , (1.4)where u , SS and u , AE represent inactive contributions from the δ -scaledsuperstructures and self-similar attached eddies, respectively. It is the presence of theformer, which obscures the pure logarithmic decay of u with z , as well as the true k − x -scaling in the associated 1-D spectra (Jimenez & Hoyas 2008; Rosenberg et al. a , b ). 1.2. Present contributions
The present study first proposes a methodology to estimate u and u in theinertially-dominated region of a canonical wall-bounded flow. Developing this capabilityof segregating the active from the inactive component, especially for u , is of use tothe wall-turbulence modelling community, since it is u active which contributes to themomentum transfer (equation 1.3). The present methodology exploits the characteristicof the inactive motions (say at a given wall-normal distance z o in the inertial region) beingchiefly created by large eddies relative to the active motions at z o ; these inactive motionsare coherent across a significant wall-normal distance (Townsend 1976; Baars et al. z o as ‘swirling’motions that influence the velocity field at all wall heights below z o , including the wall-shear stress, via low frequency variations (see also § u -signals recorded at z o and below, downto the wall (say at a reference wall-normal location z r ). Recent work on the 1-D linearcoherence spectrum by Baars et al. (2017) and Deshpande et al. (2019) has shown thata scale-by-scale cross-correlation of the synchronously acquired u -signals, at z o and z r ,isolates the energetic motions coherent across z o and z r , which may be deemed as inactivefor the case of z r ≪ z o . Following (1.3), the isolated energy contribution from theinactive motions ( u ) can simply be subtracted from the total u -energy at z o ( u ) to yield contributions predominated by the active motions at z o . This makes thepresent approach different to previous analytical efforts, such as Panton (2007), whereinthe active contributions were simply assumed to be proportional to the Reynolds shearstress to estimate the inactive contributions.The methodology adopted here to segregate the active and inactive contributions, ctive and inactive components of the streamwise velocity in wall turbulence u , AE ) and from the δ -scaled superstructures ( u , SS ). While contributions from the latter are known tobe predominant across the inertial region, the self-similar attached eddy contributionsto the inactive motions reduce significantly beyond the δ -scaled upper bound of thelogarithmic (log) region (Baars & Marusic 2020 a , b ). By choosing the reference wall-normal location at this upper bound, say at a z r ≫ z o , the scale-by-scale cross-correlationof the synchronously acquired u -signals at these z o and z r would isolate u , SS , whichfollowing (1.4) can be used to estimate u , AE .To this end, two zero-pressure gradient turbulent boundary layer (ZPG TBL) datasets,comprising multi-point u -fluctuations measured synchronously across a wide range ofwall-normal (∆ z = | z o - z r | ) and spanwise (∆ y ) spacings, are considered. The datasetsinclude measurements across the inertially-dominated (log) region, and the TBLs spana decade of Re τ , permitting us to test for: (i) the universal wall-scaling of the u -spectra associated with the active motions at z o , and (ii) the k − x -scaling of the u -spectraassociated with the self-similar attached eddies inactive with respect to z o . These data arefirst used to directly compute the 2-D u -spectrum (Chandran et al. λ x = 2 π/k x )and spanwise ( λ y = 2 π/k y ) wavelengths coherent across z o and z r (Deshpande et al. et al. (2006), Baars et al. (2016)) based procedure, whichestimates the subset of the 2-D u -energy spectrum at z o , associated with specific coherentmotions coexisting at z o .
2. ZPG TBL datasets
Two ZPG TBL datasets, consisting of synchronous multi-point u -velocity fluctuations,are considered for analysis in the present study. One is the Re τ ≈ et al. (2014), while the other is the Re τ ≈
14 000 experimental dataset, a partof which has been reported previously in Deshpande et al. (2020). A brief description ofthe two datasets is presented below.2.1.
Multi-point measurements at Re τ ≈
14 000
The high Re τ dataset was acquired in the large Melbourne wind tunnel (HRNBLWT)under nominal ZPG conditions and low free-stream turbulence levels (Marusic et al. ≃ × ×
27 m. The very longlength (27 m), and capability to generate free-stream speeds of up to 45 ms − , permitZPG TBL measurements to the order of Re τ (= δU τ /ν ) ≈
26 000 in this facility.In the present study, all measurements were conducted at a location approximately20 m from the start of the working section, at a free-stream speed of U ∞ ≈
20 ms − ,resulting in a ZPG TBL at Re τ ≈
14 000. The TBL thickness δ here is estimated via themodified Coles law of the wake fit (Jones et al. et al. HW − ), the arrangement of which is depicted infigure 1(a). Wollaston hotwire probes of diameter, d ≈ µ m and exposed sensor length, l ≈ et al. l + (= lU τ /ν ) ≈
22 for the given measurements. This hotwire length is sufficientlysmall compared to the energetic spanwise wavelengths in the inertial region, which can be
R. Deshpande, J. P. Monty and I. Marusic Re τ ≈
14 000 (Deshpande et al. Re τ ≈ et al. Set − up z + o z + r T U ∞ /δ (∆ y ) max z + o z + r (∆ x ) max (∆ y ) max Φ , 100, 200,318, 477, 750,1025, 2250 ≈ z + o
19 500 2.7 δ , 120 –250 = z + o δ δ Φ cross
19 500 2.5 δ
120 – 250 δ δ Φ cross δ – – – – Table 1.
A summary of the ZPG TBL datasets comprising synchronized multi-point u -signalsat z + r and z + o used to compute two types of 2-D u -spectra, Φ and Φ cross . The terminology hasbeen described in § Re τ ; Marusic et al. (2013)), while the values in bold represent the near-wallreference location. Superscript ‘+’ denotes normalization in viscous units. λ + x λ + y Re τ ≈ filt. DNS ) Re τ ≈ filt. DNS ) Re τ ≈
14 000 (
Exp ) +r (z = 15) = 0.15 HW HW HW HW zyy y y y (a)(i) Setup for z o z = r (fixed)(fixed) wall HW HW HW HW y y z o z = r z o y y cross (fixed)(fixed)(ii) Setup for wall (b) Figure 1. (a) Schematic of the experimental set-up in HRNBLWT showing relative positioningand movement of the four hot-wire probes ( HW − ) for reconstructing the 2-D correlationcorresponding to (i) Φ and (ii) Φ cross . Mean flow direction is along x . In the case of (ii), HW − are positioned at either z r ≪ z o or z r ≫ z o , depending on the desired experiment (table 1). (b)Constant energy contours for Φ( z + o = z + r ≈
15) = 0.15, computed from the present experimentaland the converged DNS dataset of Sillero et al. (2014), plotted as a function of viscous-scaledwavelengths. Estimates from the DNS are box-filtered along y to mimic the spatial resolutionof the hotwire sensors. inferred from the spanwise spectra of the u -velocity component from any published DNSdataset (for instance, see figure 9 of Lee & Moser (2015)). The sensors were operatedin a constant temperature mode using an in-house Melbourne University ConstantTemperature Anemometer (MUCTA) at an overheat ratio of 1.8 and at a viscous-scaledsampling rate, ∆ T + ≡ U τ / ( νf s ) ≈ f s refers to sampling frequency.The experimental set-up, as depicted in figure 1, allows HW − to be traversed inthe spanwise direction at a consistent wall-normal distance of z o , while HW − remainstationary at a fixed spanwise and wall-normal ( z r ) location throughout the measurement.To calibrate the probes, the same procedure as that employed by Chandran et al. (2017) was implemented with HW , HW and HW simultaneously calibrated at a ctive and inactive components of the streamwise velocity in wall turbulence HW as a reference.Simultaneously acquired u -signals from the four hotwires are used to reconstruct thetwo-point correlation: R u o u r ( z o , z r ; ∆ x, ∆ y ) = u ( z r ; x, y ) u ( z o ; x + ∆ x, y + ∆ y ) (2.1)for the ∆ y range, 0 ≤ ∆ y ≤ (∆ y ) max and the total sampling duration ( T ) of the u -signals listed in table 1, with the overbar denoting ensemble time average. Taylor’s frozenturbulence hypothesis, which considers all the coherent structures coexisting at z o to beconvecting at the mean velocity at z o (i.e. U c = U ( z o )), is used to convert R u o u r froma function of time to that of ∆ x , with U c denoting the convection velocity assumed at z o . Following this, the 2-D Fourier transform of R u o u r is computed to obtain the 2-Dspectrum as: φ u o u r ( z o , z r ; k x , k y ) = Z Z ∞−∞ R u o u r ( z o , z r ; ∆ x, ∆ y ) e − j π ( k x ∆ x + k y ∆ y ) d (∆ x ) d (∆ y ) , (2.2)with j a unit imaginary number.For this study, we are only concerned with two types of 2-D spectra, Φ and Φ cross which are defined as: Φ( z + o ; λ x , λ y ) = | k + x k + y φ + u o u o ( z + o ; λ x , λ y ) | andΦ cross ( z + o , z + r ; λ x , λ y ) = | k + x k + y φ + u o u r ( z + o , z + r ; λ x , λ y ) | , (2.3)with the R u o u r corresponding to the former and latter, reconstructed via hotwire arrange-ments depicted in figure 1(a,i) and 1(a,ii), respectively. Here, z + o = z o U τ ν and k + x = k x νU τ (with similar definitions for other associated terms), where the superscript ‘+’ indicatesnormalization in viscous units. Table 1 details the exact wall-normal locations for which Φand Φ cross are computed, with ( || ) referring to the modulus operation. The present anal-ysis is focused in the inertially-dominated region, considered nominally to exist beyond z + o &
100 (Nickels et al. et al. a ), based on the empirical evidence discussed in §
1. While Φ represents contributionsfrom all coexisting motions at z o , Φ cross consists of contributions from only those motionsthat are coherent across z o and z r (Deshpande et al. §
3) to estimate subsets of Φ( z o ) representingcontributions from a specific family of coherent motions coexisting at z o . Φ cross hasbeen estimated for two different reference wall-normal positions ( z r ; table 1), eachtargeted at isolating specific contributions. The measurements to obtain Φ cross ( z + o , z + r ≈ Re τ ), however, were conducted following the same methodology as that adoptedfor Φ cross ( z + o , z + r ≈ et al. (2020)and may be consulted for further details.The present study also reports the first measurements of Φ in the near-wall region( z + o = z + r ≈ §
3) being adopted in thepresent study. Figure 1(b) compares the constant energy contour for this experimentallyestimated Φ against the same computed from the converged 2-D u -correlations availablefrom the DNS dataset of Sillero et al. (2014). While a reasonable overlap of contours isobserved in the small-scale range (figure 1(b)), when plotted as a function of viscous-scaled wavelengths, a prominent ‘footprint’ can be noted appearing for the large scaleswith increase in Re τ . This is representative of the increasing influence of the large scales inthe near-wall region with increase in Re τ , as discussed by Hutchins & Marusic (2007) andHutchins et al. (2009). Here, the spectra from the DNS are box-filtered for better one-to-one comparison with the experimental spectrum, wherein the energy in the small-scales R. Deshpande, J. P. Monty and I. Marusic is underestimated due to the spatial resolution of the hotwire sensor (Hutchins et al. y -direction, by following the samemethodology as outlined in Chin et al. (2009), taking into consideration the viscous-scaled hotwire sensor length corresponding to the measurements ( l + ≈ Re τ DNS estimates at large wavelengths. This is possiblydue to the failure of Taylor’s hypothesis for these large-scales in the near-wall region(del Álamo & Jiménez 2009; Monty & Chong 2009). This inconsistency, however, doesn’taffect any of the forthcoming analysis since all the calculations ( §
3) for the experimentaldataset are carried out in the frequency domain before converting to λ x via Taylor’shypothesis. 2.2. DNS dataset
A low Re τ dataset from the ZPG TBL DNS of Sillero et al. (2014) is also consideredin the present study. Thirteen raw DNS volumes, each of which is a subset of their fullcomputational domain between x ≈ δ and x ≈ δ , are selected to ensure a limited Re τ increase along x . Streamwise velocities u ( z + o ; x , y ) extracted from these fields are usedto compute Φ( z + o ) and Φ cross ( z + o , z + r ≈
15) following (2.1) – (2.3), at z + o and z + r consistentwith the experimental dataset (table 1). A similar analysis is also conducted using theinstantaneous wall-normal velocity fluctuations, w ( z + o ; x , y ) extracted from this dataset.It is used to establish the efficacy of the SLSE-based methodology being implementedhere to segregate active and inactive contributions, the results from which are discussedin appendix 1.
3. Energy decomposition into active and inactive contributions
As discussed in §
1, the inactive motions at z o are predominantly large motions (withrespect to z o ) that are coherent across a significant wall-normal distance. This forms thebasis for decomposing Φ( z o ). Classically, the size and scaling of the coherent structureshave been interpreted via two-point cross-correlations (Ganapathisubramani et al. et al. et al. z o ) intoits inactive and residual component (figure 2) by utilizing the scale-by-scale couplingbetween u -signals simultaneously measured at two wall-normal locations, z + o (in theinertially-dominated region) and z + r ≈
15, ensuring z + r ≪ z + o . A linear decompositionwas deemed sufficient for this purpose given the fact that the coupling has been computedbetween velocity signals at both ends (Guezennec 1989; Baars et al. § cross ( z + o , z + r ≈ z + o for both the experimental and DNS datasets,represents this scale-by-scale coupling. On comparing Φ cross ( z + o , z + r ≈
15) and Φ( z + o )contours from the two datasets at various z + o in figures 2(a,c), the former is representativeof energetic large-scales that can be associated with the motions inactive at z + o . It isevident that Φ cross ( z + o , z + r ≈
15) also inherently comprises energy contributions from the ctive and inactive components of the streamwise velocity in wall turbulence -3 -2 -1 λ y / δ ( a ) z + o ≈ , z + r ≈ Φ ( z + o ) = 0 . Φ ( z + r ) = 0 . Φ cross ( z + o , z + r ) = 0 . ( b ) z + o ≈ Φ ( z + o ) = 0 . Φ ia ( z + o ) = 0 . Φ a ( z + o ) = 0 . -1 -1 λ x / δ -3 -2 -1 λ y / δ ( c ) z + o ≈ . Re τ , z + r ≈ Φ ( z + o ) = 0 . Φ ( z + r ) = 0 . Φ cross ( z + o , z + r ) = 0 . -1 -1 λ x / δ ( d ) z + o ≈ . Re τ DNS Exp λ y ∼ λ x Φ ( z + o ) = 0 . Φ ia ( z + o ) = 0 . Φ a ( z + o ) = 0 . Figure 2. (a,c) Constant energy contours for Φ( z + o ), Φ cross ( z + o , z + r ≈
15) and Φ( z + r ≈ z + o ≈
100 and (c) z + o ≈ Re τ . (b,d) Constant energycontours for Φ ia ( z + o ) and Φ a ( z + o ), computed via (3.1) and (3.2), plotted at the same energy leveland z + o as in (a,c), respectively. In (a-d), contours on the left side correspond to those computedfrom the DNS data while those on the right are from the experimental data. Dashed green linesrepresent the linear relationship, λ y ∼ λ x . δ -scaled superstructures ( λ x & δ ), which are known to extend from the wall and spanacross the inertial region (Baars & Marusic 2020 a , b ; Deshpande et al. et al. cross in conjunction with the SLSE (Tinney et al. et al. ia ) associated with the inactivemotions at z o following:Φ ia ( z + o ; λ x , λ y ) = [Φ cross ( z + o , z + r ≈ λ x , λ y )] Φ( z + r ≈ λ x , λ y ) . (3.1)Interested readers may refer to appendix 1 to see the step-by-step procedure to arriveat the expression in (3.1). The mathematical operation in the above equation suggestsΦ ia ( z + o ) to be essentially a normalized version of Φ cross ( z + o , z + r ≈ z + r ≈ λ x byinvoking Taylor’s hypothesis, using U c = U ( z o ) (Baars et al. ia can be simply subtracted from Φ to leavea residual: Φ a ( z + o ; λ x , λ y ) = Φ( z + o ; λ x , λ y ) − Φ ia ( z + o ; λ x , λ y ) , (3.2)with Φ, Φ ia and Φ a representative of u , u and u , respectively. If the flowconsisted of only active and inactive inertial motions, Φ a and Φ ia would be the activeand inactive component, respectively. However, we refer to Φ a as the residual spectrum,given that it also comprises small contributions from the fine dissipative scales as well0 R. Deshpande, J. P. Monty and I. Marusic as those corresponding to the inertial sub-range ( § a , whichare associated predominantly with the inertial active motions.Figures 2(b,d) show the constant energy contours for the two components Φ ia ( z + o ) andΦ a ( z + o ), computed via (3.1) and (3.2), using the corresponding inputs plotted in figures2(a,c), respectively. While Φ ia takes up the large-scale portion of Φ, Φ a is restricted to thesmall-scale end of the spectrum. This is in spite of the fact that Φ cross ( z + o , z + r ≈
15) alsocomprises contributions from relatively small scales at z + o ≈
100 (figure 2(a)) and can beexplained by the linear transfer kernel (equations 6.4 and 6.5), which has been computedat various z + o for the DNS dataset and shown in figure 8(a) in appendix 1. Interestingly,at z + o ≈
100 (figure 2(b)), both Φ a ( z + o ) and Φ ia ( z + o ) can be seen to follow the λ y ∼ λ x relationship representative of geometric self-similarity, which is otherwise obscured for Φin the intermediate and large-scale range (Chandran et al. et al. ia and Φ a is consistent with the hypothesis ofTownsend (1961, 1976), who originally described both the active and inactive motions tobe associated purely with the attached eddy contributions, but conforming to a differentrange of scales: the active motions at z + o conform to the attached eddies with height, H∼ O ( z o ), while the inactive motions conform to relatively large eddies with O ( z o ) ≪ H . O ( δ ) (see § ia reduceswith increase in z + o , with energy contours at z + o ≈ Re τ (figure 2(d)) correspondingpredominantly to the tall δ -scaled superstructures coexisting across the inertial region.This likely explains why Φ ia contours do not align along λ y ∼ λ x at z + o farthest fromthe wall. It also forms the basis for choosing z + r ≈ Re τ as a reference wall heightwhile implementing the SLSE methodology to isolate the superstructure contributions,which will be discussed later in § a , on the other hand, comprises a significant rangeof scales irrespective of the change in z + o , with the contours simply shifting to relativelylarger scales, which is suggestive of its distance-from-the-wall ( z o ) scaling. Having definedthe procedure to obtain Φ a and Φ ia , next we test for z o - and δ -scaling to verify the extentto which the respective spectra can be associated with the active and inactive motions.
4. Active component of the streamwise velocity spectrum
Figures 3(a,b) show the constant energy contours of Φ a (= 0.15), computed from bothDNS and experimental datasets, plotted as a function of wavelengths scaled with z o and δ , respectively. The contours are plotted for Φ a across 100 . z + o . Re τ and are seento reasonably follow wall-scaling, that is, when the wavelengths are normalized by z o .In contrast, no such collapse is observed when the wavelengths are scaled with δ . It isnoted that this behaviour is only apparent after separating Φ a from Φ. For comparison,figure 4 in § z o - and δ -scalingin the intermediate and large-scale wavelength ranges, respectively (due to the wall-parallel velocity field associated with both the active and inactive motions (Bradshaw1967; Baidya et al. z o -scaling behaviour noted for Φ a is consistent with active motions. This can beseen by comparing the scaling behaviour of 1D w -spectra and 1D uw -cospectra, whichhave been shown to follow wall-scaling and exhibit a behaviour exclusively associatedwith active motions (Bradshaw 1967; Morrison et al. et al. a is integrated along λ y and λ x to obtain the corresponding premultiplied 1-Dspectra as a function of λ x (Φ a,x ; figure 3(c)) and λ y (Φ a,y ; figure 3(d)), respectively. Alsoplotted, are the pre-multiplied 1-D w -spectra (figure 3(f)) and uw -cospectra (figure 3(e))at 100 . z + o . Re τ from the Re τ ≈
10 000 dataset of Baidya et al. (2017), measured ctive and inactive components of the streamwise velocity in wall turbulence -1 ( d ) Φ a , y λ y /z o ∼ z o -1 ( c ) Φ a , x λ x /z o ∼ z o -2 -1 λ x /z o ( f ) k + x φ + ww z + o Baidya et al. (2017); Re τ ≈
10 000 -1 λ x /z o ( e ) - k + x φ + u w ∼ z o -2 -1 λ x / δ -2 -1 λ y / δ ( b ) z o / δ Φ a = 0 . DN S Φ a = 0 . Exp λ x /z o λ y / z o λ x = λ y ( a ) alongintegrate xy alongintegrate Figure 3. (a,b) Constant energy contours for Φ a ( z + o ) at energy level of 0.15 plotted for various z + o as a function of wavelengths scaled with (a) z o and (b) δ . Contours in red and blue correspondto Φ a estimated for the experimental and DNS datasets respectively (table 1), with dark to lightshading indicating an increase in z + o following 100 . z + o . Re τ for Re τ corresponding torespective datasets. Dashed green lines represent the linear relationship, λ x = 3 λ y . (c,d) Φ a ( z + o )integrated across λ y and λ x to obtain its corresponding 1-D version as a function of (c) λ x and(d) λ y respectively, each plotted with wavelengths scaled by z o . Same colour coding is followedas that described for (a,b). (e,f) Pre-multiplied streamwise 1-D cospectra/spectra for the (e)Reynolds shear stress and (f) wall-normal velocity plotted as a function of λ x scaled with z o .This data is from the Re τ ≈
10 000 dataset of Baidya et al. (2017) for various z + o . Dark to lightshading corresponds to the increase in z + o following 100 . z + o . Re τ , where Re τ ≈
10 000. R. Deshpande, J. P. Monty and I. Marusic at the same experimental facility as Deshpande et al. (2020). When the wavelengths arescaled with z o , the 1-D spectra in figures 3(c-f) are observed to collapse for λ & z o ,in-line with the characteristics of active motions. Further, both Φ a,x and k + x φ + uw peakat λ x ∼ z o , supporting the argument that the motions associated with Φ a contributeto the Reynolds shear stress and can hence be deemed active in the sense of Townsend(1961, 1976). The efficacy of the present SLSE-based methodology, to extract energeticcontributions from the active motions, can also be tested by implementing it on similartwo-point statistics computed for the w -velocity component. Given that the w -componentis associated exclusively with the active motions ( § w -component. Interested readers may refer to appendix 1 where the SLSE analysisconducted on the w -component has been discussed.Small scales ( λ ≪ z o ), which correspond to the viscous dissipative scales or thosefollowing the inertial sub-range scaling, do not scale with distance from the wall, ex-plaining the deviation from the collapse of the 1-D spectra in figures 3(c,d,f). A similardeviation, although at a much smaller magnitude, is also observed for the Reynolds shearstress cospectra, which eventually drops to zero at λ x /z o . a,x estimated from the DNS and experimental datasets also validatesthe use of the local mean velocity as the convection velocity ( U c = U ( z o )) for the activemotions, which seems intuitive given these are localized at z o .The present analysis, which is conducted along both the x and y directions, also revealsthe dominant spanwise wavelength corresponding to the active motions, i.e. λ y ∼ z o (figure 3(d)). This yields the dominant streamwise/spanwise aspect ratio of λ x / λ y ∼ Re τ (indicated by dashed linein figures 3(a,b)). A similar SLSE-based analysis, as implemented here for the u -velocityspectrum, was conducted on the DNS dataset to analyze the active component of the v - and w -velocity spectrum (not shown here for brevity). These components were alsofound to exhibit wall-scaling, across the inertially-dominated region, with the contours ofthe spectrum following the self-similar relationship, λ x / λ y ∼ λ x / λ y ∼ v and w -velocity spectrum, respectively. Interestingly, the aspect ratio found for the active u -spectrum matches that of the self-similar wall-coherent vortex clusters ( λ x ∼ λ y )investigated by del Álamo et al. (2006) and Hwang (2015), revealing information whichmay be useful for modelling the active motions in future works. The close agreement withHwang (2015) further suggests Φ a and Φ ia , both of which comprise of prominent self-similar contributions (figures 2,3,4), correspond well with the two component attachededdy structure proposed by Hwang (2015) for a wall-bounded turbulent flow. In theircase, Hwang (2015) defined motions at a given spanwise scale to be composed of twodistinct components: the first is the long streaky flow structure, attached to the wall andhaving significant turbulent kinetic energy, but inactive in the inner-region. The energycontributions from these motions are represented by Φ ia . While, the second componentcorresponds to the short and tall self-similar vortex packets which are active in the inner-region, and hence would contribute to Φ a .
5. Inactive component of the streamwise velocity spectrum
Figure 4 shows the constant energy contours of Φ (figures 4(a,b)) and Φ ia (figures4(c,d)), computed for the experimental dataset, plotted as a function of wavelengthsscaled with z o (figures 4(a,c)) and δ (figures 4(b,d)). These contours are plotted atthe same energy level and for the same z + o , as in figures 3(a,b). Consistent with the ctive and inactive components of the streamwise velocity in wall turbulence -2 -1 λ y / δ ( b ) z + o -2 -1 λ x / δ -2 -1 λ y / δ ( d ) z + o λ y / z o λ y ∼ λ x ( a ) Φ = 0 . Exp λ x /z o λ y / z o ( c ) Φ ia = 0 . Exp
Figure 4.
Constant energy contours for (a,b) Φ( z + o ) and (c,d) Φ ia ( z + o ) at energy level of 0.15plotted for various z + o as a function of wavelengths scaled with (a,c) z o and (b,d) δ , respectively.All data in (a-d) corresponds to the high Re τ experimental dataset reported in table 1, withdark to light shading indicating an increase in z + o following 100 . z + o . Re τ . Dashed greenlines represent the linear relationship, λ y ∼ λ x . -3 -2 -1 z o / δ contributions from Φ ia ; Expcontributions from Φ AEia ; Exp -3 -2 -1 z o / δ u ; ( Samie et al. contributions from Φ ; Expcontributions from Φ a ; Exp (a) (b)
Figure 5.
Comparison of the normalized streamwise turbulence intensities obtained byintegrating Φ (= u ), Φ a ( ≈ u ), Φ ia ( ≈ u ) and Φ AEia ( ≈ u , AE ) for the high Re τ experimental dataset described in table 1. Also plotted for comparison is the well-resolved u measured by Samie et al. (2018) across a ZPG TBL maintained at an Re τ comparable tothe present experimental dataset. The dashed green line in (b) represents the logarithmic decayof u described by (1.1) with A = 0.98 (Baars & Marusic 2020 b ), while the dash-dotted goldenline in (a) represents a constant u = 2.7. R. Deshpande, J. P. Monty and I. Marusic observations of Bradshaw (1967) and Baidya et al. (2017) for the 1-D u -spectra, Φcontours can be observed to be exhibiting z o -scaling in the intermediate scales ( O (1) . λ/z o . O (10)) and δ -scaling for the large scales ( λ & O ( δ )). This is due to contributionsfrom both the active as well as the inactive motions to Φ.Φ ia also exhibits both z o - and δ -scaling, with the scale range for z o -scaling, however,much narrower than that observed for Φ. This can be attributed to the fact that Φ ia ( z o )comprises contributions from the attached eddies of height, O ( z o ) ≪ H . O ( δ ), as wellas the δ -scaled superstructures ( § § z o close to the wall,due to which a clear λ y ∼ λ x trend is discernible in Φ ia ( z o ). Townsend (1976), however,described the classification of an eddy as ‘active’ or ‘inactive’ to be a relative concept,dependent on the wall-normal location under consideration. Hence, the tall attachededdies which are inactive relative to z + o ≈
100 may qualify as active at greater wall-heights. This explains the narrowing down of the Φ ia ( z o ) contours to the largest scaleswith increase in z + o (figure 4(d)), until only the superstructure contributions remain at z + o ≈ Re τ . The latter explains the deviation of the contours from the linear relationship,as z + o moves away from the wall.The reduction in the attached eddy contributions, with increase in z + o , translates intoa drop of the cumulative streamwise turbulence intensity, i.e. ! ∞ Φ ia d ( lnλ x ) d ( lnλ y ),plotted in figure 5(b) for the experimental dataset. Also shown alongside in figure 5(a)are cumulative contributions obtained by integrating Φ and Φ a for 100 . z + o . Re τ .Figure 5 also includes, for reference, the well-resolved u profile of Samie et al. (2018)across the entire boundary layer, as well as a log law with A = 0.98 proposed by(Baars & Marusic 2020 b ). As is evident from the plot, the contributions from both Φ andΦ ia decay with z/δ very similarly, however, they only approximately follow the A = 0.98log law. This disagreement can be associated with the δ -scaled superstructure contribu-tions existing in both Φ and Φ ia (Jimenez & Hoyas 2008; Baars & Marusic 2020 b ), giventhat the expressions in (1.1) are valid strictly for self-similar attached eddy contributionsalone ( § ia in the next sub-section, by following the same SLSE-based methodology discussedpreviously in §
3. Returning to figure 5, a similar variation for both the profiles obtainedon integrating Φ and Φ ia leads to the cumulative energy contributions from Φ a ( ≈ U τ and z (Townsend 1961; Bradshaw 1967).5.1. Inactive contributions from the self-similar attached eddies
Here, we consider isolating the inactive contributions from the self-similar attachededdies, by first estimating the δ -scaled superstructure contributions to Φ( z o ). As discussedpreviously in § z + ∼ Re τ ). This is supported by the scale-by-scale coupling (Φ cross )computed from the u -signals simultaneously measured at z + o ( ≈ z + r ≈ Re τ plotted in figure 6(a), where the energy contours can be seen to be restrictedonly to the very large scale end of Φ, indicative of the superstructure signature. The choiceof z + r ≈ Re τ is also consistent with Baars & Marusic (2020 a , b ), who also used it as areference location to extract the superstructure contribution. They recommended keeping z + o . z + r / z + r ≫ z + o , which explains the present Φ cross ( z + o , z + r ctive and inactive components of the streamwise velocity in wall turbulence -2 -1 -1 λ x / δ -3 -2 -1 λ y / δ ( a ) z + o ≈ ,z + r ≈ . Re τ z + o ≈ , z + r ≈ . Re τ Φ ( z + o ) = 0 . Φ ( z + r ) = 0 . Φ cross ( z + o , z + r ) = 0 . -2 -1 -1 λ x / δ -3 -2 -1 λ y / δ ( b ) z + o ≈ z + o ≈ λ y ∼ λ x Φ ( z + o ) = 0 . Φ SSia ( z + o ) = 0 . Φ AEia ( z + o ) = 0 . Figure 6. (a) Constant energy contours for Φ( z + o ), Φ cross ( z + o , z + r ≈ Re τ ) and Φ( z + r ≈ Re τ ) at energy level of 0.15 plotted for z + o ≈
100 and 318. (b) Constant energy contoursfor Φ
AEia ( z + o ) and Φ SSia ( z + o ), computed via (5.1) and (5.2), plotted at the same energy level and z + o as in (a). All contours in (a,b) are computed from the high Re τ experimental data. Dashedgreen lines represent the linear relationship, λ y ∼ λ x . ≈ Re τ ) measurements conducted at only three wall-normal locations ( z + o ) in theinertially dominated region. It is worth noting here that owing to this condition, thecross-spectrum analysis to isolate the superstructure contribution is only possible onthe high Re τ experimental dataset. On computing Φ cross ( z + o , z + r ≈ Re τ ) from theexperimental data, it is used in conjunction with the SLSE (appendix 1) to obtain alinear stochastic estimate of the spectrum (Φ SSia ) associated with the superstructurecontributions at z o following:Φ SSia ( z + o ; λ x , λ y ) = [Φ cross ( z + o , z + r ≈ . Re τ ; λ x , λ y )] Φ( z + r ≈ . Re τ ; λ x , λ y ) . (5.1)The above expression is similar to (3.1) discussed in §
3, with the calculations in (5.1)also carried out first in the frequency domain, followed by the conversion to λ x done byinvoking Taylor’s hypothesis using U c = U ( z + r ≈ Re τ ) (Baars & Marusic 2020 a , b ).The choice of U c in (5.1) is based on the ‘global’ nature and high convection speedsof the δ -scaled superstructures (Jimenez & Hoyas 2008; del Álamo & Jiménez 2009;Monty & Chong 2009).Contours associated with all the energy spectra in (5.1) have been plotted in figure6(b), with Φ SSia centred around a δ -scaled location of λ x ∼ δ , λ y ∼ δ , representativeof the superstructures. Following the linear superposition assumption in (1.4), Φ SSia ( z o )can be simply subtracted from Φ ia ( z o ) to estimate the inactive contributions from theattached eddies at z o (Φ AEia ):Φ
AEia ( z + o ; λ x , λ y ) = Φ ia ( z + o ; λ x , λ y ) − Φ SSia ( z + o ; λ x , λ y ) , (5.2)with Φ SSia and Φ
AEia representative of u , SS and u , AE , respectively. The claimis also supported by the Φ AEia contours plotted in figure 6(b), which are seen to followthe λ y ∼ λ x relationship representing geometric self-similarity. u , AE , obtained viaintegrating Φ AEia at the three z + o , has also been plotted in figure 5(b). While the trendlooks promising when compared with the u expression in (1.1), three data pointsare not sufficient to firmly establish the present claim (for this additional data at evenhigher Reynolds number would be required). However, to confirm the association ofΦ AEia with pure attached eddy contributions, we check for the constant energy plateau(representative of the k − x -scaling) in the corresponding premultiplied 1-D spectra. In thisrespect, the parameters in the present analysis align well with the necessary conditions6 R. Deshpande, J. P. Monty and I. Marusic -1 λ /z o ( a ) z + o ≈ A x ≈ A y ≈ . A ′ x A ′ y Φ x Φ y Φ AEia,x Φ AEia,y -1 λ /z o ( b ) z + o ≈ A x ≈ A y ≈ . A ′ x A ′ y -1 λ /z o ( c ) z + o ≈ A x ≈ A y A ′ x A ′ y ≈ . Figure 7. Φ( z + o ) and Φ AEia ( z + o ) integrated across λ y and λ x to obtain their correspondingpremultiplied 1-D version as a function of λ x (Φ x , Φ AEia,x ; in solid line) and λ y (Φ y , Φ AEia,y ; indash-dotted line), respectively, for z + o ≈ (a) 100, (a) 200 and (c) 318. Also highlighted are thepeaks/plateaus of Φ AEia,x and Φ
AEia,y ( A x , A y ), along with those of Φ x and Φ y ( A ′ x , A ′ y ). proposed by Perry & Chong (1982) and Nickels et al. (2005) to observe a clear k − x region,i.e. to measure sufficiently close to the wall in a high Re τ wall-bounded flow.To this end, Φ AEia is integrated along λ y and λ x to obtain the corresponding pre-multiplied 1-D spectra as a function of λ x (Φ AEia,x ) and λ y (Φ AEia,y ), respectively, whichhas been plotted for the three z + o in figure 7. Also plotted for reference in the samefigure are the premultiplied 1-D spectra (Φ x ,Φ y ) obtained by integrating Φ in the samemanner. Indeed, Φ AEia,x can be observed to be plateauing at A x ≈ z + o , which is consistent with the Townsend-Perry constant ( A ) estimated byBaars & Marusic (2020 b ) from the streamwise turbulence intensity profile. The span ofthe Φ AEia,x plateau, however, shrinks in size with the increase in z + o , likely due to decreasein the hierarchy of attached eddies inactive at z o (Perry & Chong 1982). To the best ofthe authors’ knowledge, the present result is the first empirical evidence that establishesconsistency between the logarithmic decay rate of the streamwise turbulence intensityand the constant energy plateau from the premultiplied 1-D u -spectrum, as argued byPerry et al. (1986) in the case of pure attached eddy contributions. This consistency,however, was not observed in the recent effort by Baars & Marusic (2020 a , b ) due tothe energy decomposition conducted directly for the 1-D u -spectra in their case. Thatanalysis neglected the scale-specific coherence over the spanwise direction, which hasbeen duly considered in the present study using the new experimental data.The present analysis also reveals the plateau, A y in the premultiplied spanwise 1-D spectra (Φ AEia,y ), which is found to be nominally equal to A x for all z + o . A y ≈ A x obtained here, thus satisfies the necessary condition proposed by Chandran et al. (2017)to associate the 2-D spectrum, Φ AEia with purely self-similar contributions. Φ y is alsoobserved to have a plateau at A ′ y ≈ z + o ≈ A ′ y = A ′ x , with both values changing asa function of z + o . This behaviour can be associated with the non-self-similar contributionsin Φ obscuring the pure self-similar characteristics, which have been successfully isolatedin the present study in the form of Φ AEia . ctive and inactive components of the streamwise velocity in wall turbulence
6. Concluding remarks
The present study proposes a methodology to extract the u -energy spectrum associatedwith the active and inactive motions (Townsend 1961, 1976) coexisting at any z o inthe inertially-dominated region of a wall-bounded flow. The methodology is based onisolating the streamwise turbulent energy associated with the inactive motions from thetotal energy, based on their known characteristic of being larger than the coexisting activemotions and coherent across a substantial wall-normal range (Townsend 1961, 1976). Thisis tested using ZPG TBL datasets comprising two-point u -signals, synchronously acquiredat z o and a near-wall location ( z r ), such that z r ≪ z o . The velocity-velocity coupling,constructed by cross-correlating these u -signals, is fed into an SLSE-based procedurewhich linearly decomposes the full 2-D spectrum Φ( z o ) into components representativeof the active (Φ a ) and inactive (Φ ia ) motions at z o .Φ a is found to exhibit z o -scaling across a decade of Re τ , and is also consistent with thecharacteristics depicted by the Reynolds shear stress cospectra, thereby confirming theassociation of Φ a with the active motions. Analysis conducted across both spatially-(DNS) and temporally-resolved (experimental) datasets also confirms the validity ofTaylor’s hypothesis for the active motions. Further, decomposition of Φ into Φ a andΦ ia brings out the self-similar characteristic of the two spectra, which is consistent withTownsend’s hypothesis on both active and inactive motions essentially being associatedwith contributions from the attached eddies, but of different sizes. In terms of usefulnessto reduced-order modelling, the present study highlights the close match between thegeometry of the active motions and the self-similar vortex clusters investigated previously(del Álamo et al. a ( z o ) is found to be associated predominantly with the self-similar attachededdies of height H ∼ O ( z o ), Φ ia ( z o ) is found to have contributions from both, therelatively tall self-similar attached eddies ( O ( z o ) ≪ H . O ( δ )) as well as the large δ -scalededdies associated with the superstructures. The latter is confirmed by the reduced self-similar contributions to Φ ia with increasing z o , due to the large attached eddies qualifyingas active in accordance to the original concept given by Townsend (1961, 1976). Thepresent study also segregates the inactive contributions from the attached eddies (Φ AEia ),from those coming from the δ -scaled superstructures (Φ SSia ), by utilizing the same SLSE-based methodology used earlier. The estimation of Φ
AEia reveals the constant energyplateau, representative of k − -scaling, in the corresponding premultiplied streamwiseand spanwise 1-D u -spectra. Both these spectra were found to plateau at A ≈ A obtained from the streamwise turbulence intensity profiles (Baars & Marusic 2020 b ), asargued by Perry & Chong (1982) for the case of pure attached eddy contributions. Acknowledgements
The authors wish to acknowledge the Australian Research Council for financial supportand are thankful to the authors of Sillero et al. (2014) and Baidya et al. (2017) formaking their respective data available. The authors also thank Dr. D. Chandran forassistance with the experiments, and Dr. W. J. Baars and Dr. A. Madhusudanan forhelpful discussions related to SLSE. The authors are also grateful to the anonymousreviewers for their helpful comments which significantly improved the quality of themanuscript.8
R. Deshpande, J. P. Monty and I. Marusic
Declaration of Interests
The authors report no conflict of interest.
Appendix 1: SLSE methodology adopted for energy decomposition
Here, we demonstrate the methodology to estimate a component of the full u -energyspectrum at z o , comprising contributions from specific coherent motions coexisting at z o ,via the spectral linear stochastic estimation (SLSE) approach. The procedure has beenadopted from previous studies in the literature employing SLSE (Tinney et al. et al. et al. z r ) to give a scale-specific conditional output (at z o ) following: e u E ( z o ; λ x , λ y ) = H L ( z o , z r ; λ x , λ y ) e u ( z r ; λ x , λ y ) , (6.1)where ˜ u ( z r ; λ x , λ y ) is the 2-D Fourier transform of u ( z r ) in x and y . Here, the superscript E represents the estimated quantity and H L represents the scale-specific linear transferkernel. It should be noted that the SLSE approach enables accurate estimation of onlythose scales (at z o ) that are coherent across z o and z r . Equation (6.1) can be further usedto estimate the 2-D energy spectrum, Φ E at z o (Madhusudanan et al. E ( z o ; λ x , λ y ) = | H L ( z o , z r ; λ x , λ y ) | Φ( z r ; λ x , λ y ) . (6.2)To obtain u E and Φ E at z o , the transfer kernel H L is required to be computed from anensemble of data following: H L ( z o , z r ; λ x , λ y ) = h e u ( z o ; λ x , λ y ) e u ∗ ( z r ; λ x , λ y ) ih e u ( z r ; λ x , λ y ) e u ∗ ( z r ; λ x , λ y ) i = | H L ( z o , z r ; λ x , λ y ) | e iψ ( z o ,z r ; λ x ,λ y ) , (6.3)with | H L | and ψ the scale-specific gain and phase respectively, and the asterisk ( ∗ ), anglebrackets ( hi ) and vertical bars ( || ) denoting the complex conjugate, ensemble averagingand modulus, respectively. Considering z + r as the reference wall-normal location used inthe present study, | H L | from equation (6.3) can be simply expressed as a function of thetwo types of 2-D spectra computed from the multi-point datasets (refer §
2) at various z + o in the inertial region following: | H L ( z + o , z + r ; λ x , λ y ) | = Φ cross ( z + o , z + r ; λ x , λ y )Φ( z + r ; λ x , λ y ) . (6.4)In the case of z + r ≪ z + o , Φ E ( z o ; λ x , λ y ) would be representative of the energy contribu-tions from all coexisting motions taller than z o , which as per our discussion in § E ( z o ; λ x , λ y ) | z + r ≈ → Φ ia ( z o ; λ x , λ y ). A simplified expression for Φ ia can be deducedfrom (6.2) and (6.4) as follows:Φ ia ( z + o ; λ x , λ y ) = | H L ( z + o , z + r ≈ λ x , λ y ) | Φ( z + r ≈ λ x , λ y )= [Φ cross ( z + o , z + r ≈ λ x , λ y )] Φ( z + r ≈ λ x , λ y ) . (6.5)Similarly, in case of z + r ≫ z + o , Φ E ( z o ; λ x , λ y ) would be representative of the energycontributions from all coexisting motions at z o that are taller than z r , which as perour discussion in § E ( z o ; λ x , λ y ) | z + r ≈ . Re τ → Φ SSia ( z o ; λ x , λ y ), and can be ctive and inactive components of the streamwise velocity in wall turbulence -1 λ x / δ -2 -1 λ y / δ ( a ) u − component z + o | H L | = 0 . | H L | max Φ ( z + r ≈
15) = 0 . -2 -1 λ x / δ -2 -1 λ y / δ ( b ) w − component z + o | G L | = 0 . | G L | max Ψ ( z + r ≈
15) = 0 . Figure 8.
Contours for the linear transfer kernel (black), for z + r ≈
15, and the near-wall 2-Dspectra (magenta) for the (a) u - and (b) w -velocity component. Here, the transfer kernels arecomputed at various z + o from the DNS dataset of Sillero et al. (2014), described in table 1. Darkto light shading indicates an increase in z + o following 100 . z + o . Re τ . The contour levelsfor the transfer kernels, | H L | and | G L | correspond to approximately 10% of the maximumvalue recorded for the kernel at the respective z + o , while that for Ψ( z + r ≈
15) has intentionallybeen kept very low to highlight no overlap with the associated transfer kernel, | G L | . estimated following:Φ SSia ( z + o ; λ x , λ y ) = [Φ cross ( z + o , z + r ≈ . Re τ ; λ x , λ y )] Φ( z + r ≈ . Re τ ; λ x , λ y ) . (6.6)Availability of both the numerator and denominator in the above expressions (table 1)allows direct computation of Φ ia and Φ SSia , without separately estimating | H L | , forboth the datasets. It should be noted here that Φ ia is computed in the frequency domainfor the experimental dataset, with the conversion to λ x obtained by invoking Taylor’shypothesis, using U c = U ( z o ) (Baars et al. SSia is also computed in thesimilar manner, however with the conversion to λ x achieved by using U c = U ( z + r ≈ Re τ ) (Baars & Marusic 2020 a , b ).As can be noted from (6.5), the essential information on energetic motions coherentacross z o and z r is embedded in | H L | which is translated into Φ ia (or Φ SSia ) viascale-by-scale amplification/attenuation provided by Φ( z + r ). For example, figure 8(a)shows the | H L ( z + o , z + r ≈ | computed from the DNS dataset at various z + o listedin table 1, along with Φ( z + r ≈
15) for the same dataset. It is evident from the plotthat | H L ( z + o ≈ , z + r ≈ | contours conform predominantly to the large scales of thespectrum, with the contours moving very gradually to even larger scales with increase in z + o . This explains the observation noted in § ia and Φ a respectively taking up thehigher and lower end of the full spectrum, Φ( z o ).We also use this opportunity to test the efficacy of the SLSE-based methodology, usedhere to estimate energy contributions from the active motions. To this end, the sameprocedure as that outlined in (6.1) – (6.5), is implemented on the wall-parallel w -velocityfields also retrieved from the DNS dataset, at the same z + o and z + r as that selected forthe u -velocity field (table 1). The w -velocity fields are used to compute the two types of2-D spectra, computed previously for the u -component, at the same z + o in the inertial0 R. Deshpande, J. P. Monty and I. Marusic region following: Ψ( z + o ; λ x , λ y ) = | k + x k + y φ + w o w o ( z + o ; λ x , λ y ) | andΨ cross ( z + o , z + r ≈ λ x , λ y ) = | k + x k + y φ + w o w r ( z + o , z + r ≈ λ x , λ y ) | , (6.7)where (6.7) is analogous to (2.3) expressed for the u -component. Similarly, the inactivecomponent Ψ ia of the w -velocity field can be computed via an expression similar to (6.5)given previously for the u -component:Ψ ia ( z + o ; λ x , λ y ) = | G L ( z + o , z + r ≈ λ x , λ y ) | Ψ( z + r ≈ λ x , λ y ) , where | G L ( z + o , z + r ≈ λ x , λ y ) | = Ψ cross ( z + o , z + r ≈ λ x , λ y )Ψ( z + r ≈ λ x , λ y ) . (6.8)Accordingly, Ψ a ( z o ) = Ψ( z o ) – Ψ ia ( z o ). Given the fact that the w -component is pre-dominantly associated with the active motions (Bradshaw 1967; Morrison et al. et al. a ( z o ) ≈ Ψ( z o ), we would expect the SLSE procedure to revealcumulative energy contributions from Ψ ia to be negligible. This is possible if | G L | andΨ( z + r ≈
15) do not overlap at common scales (as per equation 6.8). Figure 8(b) shows | G L | contours computed for various z + o from the DNS dataset alongside Ψ( z + r ≈ ia ( z o ) ≈ a ( z o ) ≈ Ψ( z o ), which proves the effectiveness of theSLSE-based methodology in extracting the energy spectrum associated with the activemotions. REFERENCESdel Álamo, J. C. & Jiménez, J.
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