Data-driven decomposition of the streamwise turbulence kinetic energy in boundary layers. Part 2: Integrated energy and
In Part 1, the scaling of the streamwise velocity energy spectra in turbulent boundary layers was considered. A spectral decomposition analysis provided a means to separate out attached and non-attached eddy contributions and was used to generate three spectral sub-components, one of which is a close representation of the spectral signature induced by self-similar, wall-attached turbulence. Since sub-components of the streamwise turbulence intensity follow directly from the integrated components of the velocity energy spectra, we here focus on the scaling of the former. Specific attention is given to the potential behaviour in spectra, at ultra-high , and its relation with the turbulence intensity adhering to a wall-normal logarithmic decay per Townsend’s attached-eddy hypothesis. This decay with a Townsend–Perry constant of is suggested to be universal across all Reynolds numbers considered. It is also demonstrated how the logarithmic-region results are consistent with the Reynolds-number increase of the streamwise turbulence intensity in the near-wall region.
Key words: wall-bounded turbulence, turbulence kinetic energy, spectral coherence
1 Introduction and context
Wall-normal trends of the streamwise turbulence intensity (TI), denoted as , are a prerequisite to modelling efforts of wall-bounded turbulence. Several previous models for have been hypothesis-based. For instance, the model of Marusic & Kunkel (2003) was inspired by the attached-eddy hypothesis (AEH, Townsend 1976), while Monkewitz & Nagib (2015) constructed a model via asymptotic expansions and Chen et al. (2018) via a dilation symmetry approach. Such models require validation and calibration (Monkewitz et al. 2017) and assumptions are inevitable for extrapolated conditions. Even with available wall-normal profiles of , from both numerical computations and experiments (e.g. Marusic et al. 2010a), definitive scalings remain elusive. This is mainly due to the weak dependence of on the Reynolds number, the limited Reynolds-number range over which direct numerical simulations are feasible and the practical challenges associated with experimental acquisition of fully-resolved data.
This introduction addresses Townsend’s AEH in § 1.1 and the widely researched logarithmic decay of the streamwise TI within the outer region of turbulent boundary layers (TBLs). Related to this is the contentious issue of the scaling in the streamwise velocity spectra , with being the streamwise wavenumber (and is the streamwise wavelength). Recall that energy spectra inform how the streamwise TI is distributed across wavenumbers, as the streamwise TI (strictly the velocity variance or normal stress) equates to the integrated spectral energy via Parselval’s theorem (e.g. ). After motivating the need to consider the streamwise TI scaling in the context of scalings, we briefly review Part 1 (Baars & Marusic 2019) in § 1.2, which considered a data-driven spectral decomposition.
Notation in this paper is identical to that used in Part 1. Recall that coordinates , and denote the streamwise, spanwise and wall-normal directions of the flow, whereas the friction Reynolds number is the ratio of (the boundary layer thickness) to the viscous length-scale . Here is the kinematic viscosity and is the friction velocity, with and being the wall-shear stress and fluid’s density, respectively. When a dimension of length is presented in outer-scaling, it is normalized with scale , while a viscous-scaling with is signified with superscript ‘+’. Lower-case represents the Reynolds decomposed fluctuations, while capital is used for the absolute mean.
1.1 Townsend–Perry constant in the context of the turbulence intensity and spectra
Townsend (1976) hypothesized that the energy-containing motions in TBLs are comprised of a hierarchy of geometrically self-similar eddying motions, that are inertially dominated (inviscid), attached to the wall and scalable with their distance to the wall (Marusic & Monty 2019). According to the classical model of attached eddies (Perry & Chong 1982), the wall-normal extents of the smallest attached eddies scale with inner variables, e.g., , while the largest scale on . Consequently, is a direct measure of the attached-eddy range of scales. Following the attached-eddy modelling framework, the streamwise TI within the logarithmic region adheres to
where and are constants; was dubbed the Townsend–Perry constant. A scaling (or a plateau in the premultiplied spectrum ) is consistent with the presence of a sufficient range of attached-eddy scales. Such a spectral scaling for the energy-containing, inertial range of anisotropic scales can be predicted with the aid of dimensional analysis, a spectral overlap argument and an assumed type of eddy similarity (e.g. Perry & Abell 1975; Davidson & Krogstad 2009). Perry et al. (1986) related the plateau magnitude of the premultiplied spectrum back to (1.1) and found that
An underlying assumption of (1.1), in combination with (1.1), is that all energy is induced by Townsend’s attached-eddy motions. And so, from detailed studies on the streamwise turbulence kinetic energy, from which profiles of and streamwise spectra are available, the Townsend–Perry constant inferred via either (1.1) or (1.1) should be equal, provided that attached-eddy turbulence dictates the scaling.
Thus far, evidence for (1.1) has remained inconclusive, mainly due to the limited spectral range over which this region may exist. It is instructive to present an energy spectrogram: premultiplied spectra at 40 logarithmically-spaced positions within the range are presented with iso-contours of in Figure 1(a). These spectra were obtained from hot-wire measurements at in Melbourne’s TBL facility (Baars et al. 2017a). Near-wall streaks (Kline et al. 1967) dominate the inner-spectral peak in the TBL spectrogram (identified with the marker at and ), while large-scale organized motions cause a broad spectral peak in the log-region, indicated with a marker at and (Mathis et al. 2009). Nickels et al. (2005) determined a region as , (wall-scaling) and (outer-scaling) at (triangular region ‘N’ in Figure 1a). This region satisfied (1.1) with . In § 2.1 of Part 1 it was suggested that this plateau in the spectrogram may be caused by a transitioning from the imprint signature of the inner-spectral peak, to the broad outer-spectral peak. It was furthermore determined from a coherence analysis (Baars et al. 2017b), relative to a wall-based reference, that wall-attached self similar motions only become spectrally energetic at (but could plateau at much larger , see Part 1). Based on this, it was suggested that a scaling in measured spectra is unlikely for . This is consistent with the study of Chandran et al. (2017), where experimentally acquired streamwise–spanwise 2D spectra of at were examined for a . They concluded that an appreciable scaling region can only appear for . Moreover, even for the highest laboratory data, the presence of a has been inconclusive (Morrison et al. 2002; Rosenberg et al. 2013; Vallikivi et al. 2015), while this region should grow with .
We now switch our attention to evidence for (1.1). A caveat in determining from profiles is that, generally, all turbulent scales are considered (integral of the entire spectrum). This approach inherently assumes that the attached-eddy structures dominate . Now, when accepting that the attached-eddy contribution to the overall turbulence intensity grows with (see Part 1), this assumption should become more valid. For this reason, Marusic et al. (2013) considered high data in the range (Winkel et al. 2012; Hultmark et al. 2012; Hutchins et al. 2012; Marusic et al. 2015), and inferred that (see Figure 1b). It is worth noting that the value for has changed significantly over time. For instance, values for have been quoted as 1.03 (Perry & Li 1990), 1.26 (Hultmark et al. 2012; Marusic et al. 2013; Örlü et al. 2017) and 1.65 (Yamamoto & Tsuji 2018). These variations in are largely due to the varying TI slope with and the different fitting regions for (1.1).
The aforementioned has illustrated that values found from profiles vary, while the AEH envisions a constant in (1.1): one that is invariant with . Moreover, values found from profiles do not agree with values for inferred from spectra via (1.1), despite that this is expected per the attached-eddy model (Perry et al. 1986). A central facet of this mismatch is the fact that (1.1) and (1.1) are restricted to attached-eddy turbulence only. For a quantitative insight into what portion of the turbulence kinetic energy is representative of attached-eddy turbulence, a spectral decomposition method was introduced in Part 1 and is summarized next.
1.2 Streamwise energy spectra and the triple decomposition
Data-driven spectral filters and were obtained with the aid of two-point measurements and spectral coherence analysis (confined to wall-normal separations only). It was verified that the two spectral filters were universal for . Filter function was formulated as
Subscript signifies the wall-based reference, on which this filter is based, and the three constants are: , and (Table 1, Part 1). A smooth filter was generated by convoluting (1.2) with a log-normal distribution, , spanning six standard deviations, corresponding to 1.2 decades in (details are provided in Part 1). Filter and equals a scale-dependent fraction of energy that is stochastically coherent with the near-wall region. Consequently, is the incoherent energy fraction.
As opposed to the wall-based filter , filter employs a reference position in the logarithmic region:
Filter constants are , and . A smooth filter was formed in a similar way as . Of the fraction of energy that is stochastically coherent with the near-wall region (via ), a sub-fraction of that energy is also coherent with in the logarithmic region (and this fraction is prescribed by ).
A triple decomposition for was formed from and in Part 1, following
Consequently, and Figure 2 illustrates this decomposition for (duplicate of Figure 14, Part 1). The three energy spectrograms of (1.2)–(1.2) are overlaid on the premultiplied energy spectrogram . Here, and the triple-decomposition is performed for . In the near-wall region, here taken as (nominally is used, roughly the wall-normal position at which the near-wall spectral peak becomes indistinguishable from the spectrogram), is -invariant and taken as . Throughout this work, the exact value of is of secondary importance, since small variations in this location do not affect conclusions, given a lower bound of the logarithmic region in viscous scaling, .
Component (Figure 2a) comprises the energy that is coherent via : large-scale wall-attached energy that is coherent with . This component includes spectral imprints of self-similar, wall-attached structures reaching beyond and non-self-similar wall-attached structures that are coherent with (e.g. VLSMs). Component (Figure 2c) is formed from the -based incoherent energy. This small-scale energy is wall-detached and includes detached (non)-self-similar motions, such as phase-inconsistent attached eddies, incoherent VLSMs, etc. The remaining component, , equals the wall-coherent energy that resides below and consists of self-similar and non-self-similar contributions. However, the non-self-similar contributions are likely to reside at large (reflecting global modes, Bullock et al. 1978; del Álamo & Jiménez 2003) and those of height are expected to be energetically insignificant.
1.3 Present contribution and outline
Coming back to § 1.1, we can now argue that can be inferred from profiles via (1.1), as long as the global-mode (VLSMs/superstructures) and Kolmogorov-type energy contributions are removed. This step is crucial, since both of these contributions constitute a clear -dependence (Part 1). And, the Reynolds-number dependent outer-spectral peak seems to interfere with a significant spectral range at which a may be expected (see spectra in Morrison et al. 2002; Nickels et al. 2005; Marusic et al. 2010b; Baidya et al. 2017; Samie et al. 2018). Thus, when re-assessing in this paper, both the spectral view and are considered simultaneously, while recognizing that is solely associated with the portion of the turbulence that obeys the AEH.
Next, in §§ 2.1-2.2, scalings of the streamwise TI are presented for a range of . Data used are the same as in Part 1 (Baars & Marusic 2019, § 3.2). Findings on the Townsend–Perry constant are reconciled in § 2.3, after which its relation to the near-wall TI growth, with , is presented in § 3. Finally, implications of our results for the development of a new empirical model for the streamwise TI are presented in § 4.
2 Scaling of the streamwise turbulence intensity
2.1 Methodology and logarithmic scalings
Figure 3(a) shows the three sub-components , and for the spectrum at (slice through Figure 2). When integrated, these sub-components form three contributions to the streamwise TI, being , and , respectively. In summary:
Figure 3(b) presents these three sub-components of the TI at , together with (open diamonds). Wall-normal profiles of the three sub-components are obtained when the energy is integrated for all (Figure 3c). Again, contributions are shown in a cumulative format: the bottom profile (squares) represents , the intermediate profile (circles) encompasses and the final profile (diamonds), , equals (by construction). Regarding the full profile, it is well-known that the near-wall streamwise TI is attenuated due to spatial resolution effects of hot-wires (Hutchins et al. 2009). Here the spanwise width of the hot-wire sensing length was . A corrected profile for the streamwise TI is superposed in Figure 3(b) with filled diamonds, following the method of Smits et al. (2011). Samie et al. (2018) confirmed that this correction scheme is valid for Reynolds numbers up to . Because the TI above the near-wall region (say ) is unaffected by spatial resolution issues, we can proceed our current analysis without hot-wire corrections.
The wall-incoherent component, , exhibits an increase in its energy-magnitude with increasing throughout the logarithmic region. This was anticipated as the spectral distribution of in Figure 2(c) indicates a clear broadening of its spectral energy band around (the hypotenuse of the triangle). Section 4 addresses the wall-normal trend of this TI component in more detail.
Components and have to be considered simultaneously. In Part 1 it was addressed how their spectral-equivalents and varied with . Figure 4(a) illustrate the dependence of the two TI sub-components on , by presenting (squares) and (lines) for a range of (indicated with the vertical lines). At low , the wall-attached self-similar motions not extending beyond contribute to , but its wall-normal range is limited (per definition, is non-existent above ). With increasing , the range of wall-attached self-similar motions increases, but global modes (or imprints of non-self-similar VLSMs/superstructures) that are restricted to also contribute to (due to the difficulty in spectrally decomposing the two, see § 5.2 in Part 1). Hence, does not exclusively contain energy imposed by wall-attached self-similar motions. When resides in the intermittent region, all global modes are being assigned to (and thus to ). This is reflected by the highest profile in Figure 4(a): in the process of increasing , a hump has appeared in the streamwise TI (approaching for ).
We now focus exclusively on as this sub-component is directly related to Townsend’s attached-eddy turbulence. Figure 4(b) shows for (the near-wall TI is irrelevant here). Although profiles do comprise a signature of wall-attached non-self-similar motions, two trends of its statistics are believed to be reflective of wall-attached self-similar motions:
At first, the magnitude of at is displayed in Figure 5(a), with forming the abscissa (with a finer -discretization than used in Figure 4). When assuming that the non-self-similar, large-scale motions have a negligible influence on the TI-trend at , and that obeys Townsend’s AEH, we arrive at
That is, an increase of mimics an increase in Reynolds number through the inclusion of more wall-attached scales in (see also Appendix B of Part 1). Data in Figure 5(a) adheres to (2.1) for approximately one decade in and fitting of the data residing at results in .
A second measure quantifying the trend in considers the decay of following (1.1). It is impossible to perform a direct fit of a logarithmic decay to the data of in Figure 4(b), because of the aforementioned issues (for large , the profiles are influenced by non-self-similar, global-mode turbulence). Instead, we define a logarithmic slope from the two profile end-points: and , via
Figure 4(b) displays the logarithmic slope for one profile (discrete point measurements were interpolated to exactly and the position at which becomes zero). Data in Figure 5(b), and its mean value , are in close agreement to from Figure 5(a). This is expected when obeys an attached-eddy scaling.
2.2 Reynolds number variation
We now assess how the identified logarithmic scalings via (2.1) and (2.1) depend on the Reynolds number. Single-point hot-wire measurements at a range of Reynolds numbers were employed in § 6 of Part 1, to address the Reynolds number variation of the triple-decomposed energy spectrograms. These same single-point hot-wire data are here processed via the procedure described previously (§ 2.1). At first, the profiles for these data are shown in Figure 6(a). For the three lowest Reynolds numbers (, 3 900 and 7 300: Hutchins et al. 2009), data were corrected for spatial attenuation effects (Smits et al. 2011), whereas the two other profiles ( and 19 300: Samie et al. 2018) comprise fully-resolved measurements. An energy-growth in the outer region presents itself through the emergence of a local maximum in (Samie et al. 2018), whereas at the same time, the near-wall TI grows with (Marusic et al. 2017).
Data of each Reynolds-number case are spectrally decomposed, to generate a similar output as was presented in Figure 4(a). For each of the five profiles in Figure 6(a), the result is shown in Figures 6(a–f), respectively. With the aid of (2.1) and (2.1), Figures 5(a,b) can now also be constructed for each of the five Reynolds numbers, as shown in Figures 7(a,b).
Especially at the two largest Reynolds numbers ( and 19 300), there is a consistent agreement between and . At the two lowest Reynolds numbers ( and 3 900), the slope extracted from the two profile end-points of exhibits a decreasing trend (top two profiles in Figure 7b). This is ascribed to the fact that the upward trend of (square symbols in Figures 6b,c) changes rapidly near the upper edge of the logarithmic region: its magnitude starts to decrease around in order to merge with the TI profiles in the wake region. Because of this decrease, there is a less rapid decay of the profiles near . When slope is determined from the two profile end-points, it causes a decreased slope. Generally, the limited scale separation in the triple-decomposed spectrograms at low Reynolds numbers exacerbates this issue (see also the spectrograms in Figure 18 of Part 1). Nevertheless, the clear logarithmic trends in Figure 7(a), reinforced by the consistent trends in Figure 7(b) at large , are indicative of our identified slopes being a reflection of attached-eddy type turbulence conforming with Townsend’s AEH.
2.3 Reconciling from trends in the turbulence intensity and spectra
Having re-assessed the wall-normal decay of the TI sub-component that is associated with Townsend’s attached-eddies (§§,2.1-2.2), we can now proceed with reconciling the status quo. Recall that (1.1) is restricted to the streamwise TI that is generated by inviscid, geometrically self-similar and wall-attached eddies only. Both and were inferred by only considering the sub-component of the TI that complies with Townsend’s assumptions. Therefore, those slopes are interpreted as . Figure 8 displays , for all Reynolds numbers, with the open square symbols. Uncertainty estimates are shown with the error bars and are based on 95 % confidence bounds from the fitting procedure of (2.1). Alongside, with the solid square symbols, values of are shown with the uncertainty estimates based on the 95 % confidence interval of the data points in Figures 5(b) and 7(b) residing at . Numerical values are summarized in Table 1. To complete quantification of (1.1) by considering energy only, offset can be determined. For this we have to introduce a new quantity , being the TI decay with a pure logarithmic decay:
One case of is considered for determining . Although offset depends on (see Figure 4b), we only have to consider the scenario for one specific to infer its Reynolds-number trend. Values for are shown on the bottom of Figure 8. Mean values for both and are found from the mean values of and in Table 1, resulting in
To indicate the effect of the variation in and with , six lines according to formulation (2.3), with the six values of and (down to ) are shown in the inset in Figure 8, together with the TI profiles of Figure 6(a). Variations in and (as well as the uncertainty estimates from the fitting procedure, listed in Table 1), result in indistinguishable logarithmic trends in relation to typical experimental scatter in the TI profiles (e.g. Winkel et al. 2012; Vincenti et al. 2013; Marusic et al. 2017; Örlü et al. 2017). Both and are thus considered to be Reynolds number invariant for .
A last set of data points in Figure 8 comprises the peak values of at , duplicated from Figure 20 in Part 1, to also consider in the context of the energy spectra. Because the scale separation in spectral space is still relatively limited at these Reynolds numbers, the peak value in the associated spectra () keeps maturing with (detailed in § 6.2, Part 1). At there is a consistency between the value for found from the TI trend and the peak/plateau in the associated spectrum. However, it is important to recall that no complete similarity has been observed in the associated spectra (e.g. Figure 19, Part 1). Only higher Reynolds number, fully-resolved data at the start of the logarithmic region, , can provide a definite answer on whether a plateau in the spectral component, associated with attached-eddies, truly converges towards a Reynolds number-invariant that is consistent with the TI trends. Thus far, Figure 8 does not exclude that possibility: a rough extrapolation of the peak values approaches , consistent with (2.3). Moreover, peak values are affected by the choice of when separating the wall-coherent energy into and . A refined spectral separation procedure would be required to yield an unambiguous result for a type of plateau in the premultiplied spectrum. Nevertheless, to the authors’ knowledge, our current work shows for the first time that a Reynolds number-invariant could be consistent with a potential plateau region at ultra high . Previously, (Perry & Li 1990; Marusic & Kunkel 2003) found from profiles was in close agreement with the spectral-based value of by Nickels et al. (2005), but this was strictly coincidental. The former was obtained at significantly lower Reynolds numbers (highest ) than the latter ().
3 in relation to the turbulence intensity in the near-wall region
Consistent scaling laws have recently emerged for the inner-peak of the streamwise TI. Samie et al. (2018) considered the maximum in the TI profiles, denoted as , from DNS and fully-resolved measurement data, to conclude that
with and . Nominally, the maxima reside at . Lee & Moser (2015) observed (3) through DNS channel flow up to and an increase of with is consistent with earlier studies (DeGraaff & Eaton 2000; Hutchins et al. 2009; Klewicki 2010). Peak values of the streamwise TI at , as a function of , are shown in Figure 9. DNS data include the channel flow of Lee & Moser (2015) and TBL flow of Sillero et al. (2013). Experimental data of TBLs are from studies performed in Melbourne’s boundary layer facility (Marusic et al. 2015; Samie et al. 2018), UNH’s Flow Physics Facility (Vincenti et al. 2013) and at Utah SLTEST (Metzger et al. 2001). Current data (employed in Figure 7). All these data (aside from Samie et al. 2018) were corrected for spatial resolution effects via Smits et al. (2011). ASL data of Metzger et al. (2001), with a relatively small hotwire length of , are uncorrected (Hutchins et al. 2009). High Reynolds number experimental pipe flow data are also included from the CICLoPE facility (Örlü et al. 2017; Willert et al. 2017), reaching up to . Hotwire data of Örlü et al. (2017) were again corrected for spatial resolution effects, whereas the PIV data of Willert et al. (2017) were nearly fully-resolved. Given the measurement uncertainty, (3) appears to represent the trend well for all the data (solid line).
Figure 9 also presents at , except for the unavailable Utah SLTEST data at this location. When the data at adhere to an attached-eddy scaling, the Reynolds-number growth of the streamwise TI can be described by , since (1.1) or (2.3) can be reformulated as
When fitting (3) to the data in Figure 9 with (2.3), the new offset-constant is determined as . Figure 9 shows that (3) represents the data well, meaning that the Reynolds-number behaviour of the streamwise TI, at a lower bound of the logarithmic region fixed in viscous scaling, e.g., , is predicted well through the attached-eddy scaling. This furthermore implies that the contribution of large-scale, global-mode VLSMs and small-scale wall-incoherent turbulence (reflected in and , respectively) do not, or negligibly, contribute to the Reynolds-number trend.
The question now remains how (3) and (3) are compatible (or how is consistent with ). Marusic & Kunkel (2003) proposed that the near-wall viscous region is influenced by the Reynolds number dependent, outer-layer streamwise TI. The validity of this proposition was strengthened by the superposition framework detailed in the literature (Hutchins & Marusic 2007; Marusic et al. 2010a; Mathis et al. 2011; Baars et al. 2016) and studies focusing on a near-wall component that is free of motions not scaling in inner units (e.g. Hu & Zheng 2018). Note however that a complex scale interaction and spectral energy transfer is present (de Giovanetti et al. 2017; Cho et al. 2018), in combination with an outer motion wall-shear-stress footprint (Abe et al. 2004; de Giovanetti et al. 2016). In summary, we move forward with the near-wall TI being composed of two contributions:
A universal function that is Reynolds-number invariant when scaled in inner units, denoted as . It mainly encompasses the inner-peak in the spectrogram induced by the near-wall cycle (NWC).
An additive component that accounts for the Reynolds-number dependent superposition of the outer region TI onto the near-wall viscous region. It is hypothesized that this Reynolds-number dependence is solely the result of the attached-eddy turbulence at . In simplest form, it can be hypothesized that the near-wall footprint drops off linearly in , to zero at , so that
where is found from (2.3), which is reformulated as
As is Reynolds number-invariant, grows with via (3) with . Refitting of (3) yields ; Figure 9 indicates that these constants represent the scattered data equally well as with and (adopted earlier from Samie et al. 2018). In order to extend the scaling validation to the entire near-wall region (not just ), reference DNS data of a ZPG TBL are utilized (Sillero et al. 2013). Figure 10(a) displays from the DNS at . Following (3), part of this near-wall TI is envisioned as the attached-eddy component, labelled as . The remaining TI forms . For any , data should now collapse when the near-wall attached-eddy contribution is subtracted from the near-wall TI profile. Figure 10(b) visualizes this assessment: the dashed line corresponds to from the DNS in Figure 10(a), the symbols correspond to the five Reynolds number profiles of Figure 6(a) and the 10 blue-coloured profiles span (taken from Marusic et al. 2017, where data were also corrected for the hotwire’s spatial attenuation effects). The excellent collapse of all data agrees with the two-part model, . In conclusion, is consistent with the Reynolds-number increase of the near-wall TI.
4 Towards an empirical model for the turbulence intensity
Evidence for a portion of the streamwise TI in the logarithmic region adhering to an attached-eddy scaling, was provided in § 2. Subsequently, § 3 highlighted its consistency with the near-wall region scaling trends. In addition, the data-driven spectral filters in Part 1 serve as direct evidence for the existence of attached-eddy turbulence in the spectral sense. Following the data-driven spectral decomposition, the streamwise TI was earlier analysed in terms of its three additive components, via (2.1):
Wall-coherent components and , computed by the data-driven approach, are not separable in the sense that one of these consists of attached-eddy turbulence only (the inherent difficulty of decomposing energy wall-coherent self-similar attached-eddies from that of wall-coherent, large-scale non-self-similar motions was discussed in § 5.2 of Part 1). However, was shown to closely represent the energy content associated with attached-eddies, and, was used to support (2.3)-(2.3). As such, for a future empirical model for the streamwise TI in the logarithmic region, we proceed by replacing with (e.g., obeys pure attached-eddy scaling). Consequently, needs to be replaced by a component encompassing all remaining energy (that was envisioned as the global/VLSM-type energy), denoted as , where subscript G stands for global. Wall-incoherent component remains unchanged, hence, the addition of the three sub-components still equals :
For the data considered in Figure 7, with , the three additive contributions of (4) are displayed in Figure 11(a-c) and Figures 11(b-f) for inner- and outer-scalings, respectively. Each component is now discussed.
The simplest approach for obtaining a scaling formulation for is to use Kolmogorov-type modelling as used in previous works (see Spalart 1988; Marusic et al. 1997). Spectral scaling of comprises a -scaling at the low wavenumber-end, while the higher wavenumber-end adheres to a scaling up to a wavenumber fixed in Kolmogorov scale , see Figures 15(f) and 21 in Part 1. A scaling can therefore be inferred from integrating a model spectrum from to , where and are constants. The latter boundary equals , with being a constant and . From the production-dissipation balance, and thus . Accordingly,
where , and are constants. When , will tend towards for large . At our practical values (Figure 11c), increases up to , after which a wake deviation occurs (Marusic et al. 1997). Fitting of (4) to the data in Figure 11(c), for , results largely in a Reynolds number-invariant contribution as shown by the profiles in Figure 12(a); it was visually verified that (4) described each experimental profile in Figure 11(c) well. Experimental uncertainties in can be a cause for the slight variations in observed in Figure 12(a). Average values for the constants, from the five profiles, are and (thick light blue line in Figure 12a).
For the attached-eddy energy , (2.3) was adopted. The near-wall decay trend following (3) is also drawn in Figures 11(b,e). The vertical offset of the AE component, being in (2.3), is dependent on the chosen . Further research should provide solutions on what offset (at what outer-scaled location should become zero) describes stochastic observations of attached structures. Conceivably, studies extracting instantaneous attached-eddy structures from full velocity fields can be instrumental to this (recent studies focusing on the instantaneous attached-eddy structures from DNS data include del Álamo et al. 2006; Lozano-Durán et al. 2012; Hwang & Sung 2018; Solak & Laval 2018).
Finally, appears as a broad hump throughout the logarithmic region (Figures 11a,d) and is mainly comprised of VLSM-type energy. No physical models exist for this component, other than merging of self-similar LSMs may be one of the mechanisms generating VLSMs/superstructures (Adrian et al. 2000). Future studies have to reveal Reynolds number scalings, and the physical mechanisms that generate and sustain the VLSMs; resolving their spatial and temporal dynamics (Kerhervé et al. 2017), as well as performing variational mode decompositions (Wang et al. 2018), are promising. For now, a full-empirical formulation would have to be constructed for a future streamwise TI model. Judiciously, is well-represented by a parabolic relation with logarithmic argument :
When (4) is fitted to the profiles in Figure 11(a), for , the data is well-represented by the fits shown in Figure 12(b). Generally, its energy content increases with , but a consistent monotonic trend is absent, owing to the experimental difficulties in acquiring repeatable and converged data at very large wavelengths (Samie 2017). Nevertheless, this global/VLSM component is responsible for the secondary peak (or hump) in (Hultmark et al. 2012; Vallikivi et al. 2015; Willert et al. 2017; Samie et al. 2018). Marusic et al. (2013) observed that a lower bound of the logarithmic region resembled the dependence (Sreenivasan & Sahay 1997; Wei et al. 2005; Klewicki et al. 2009; Morrill-Winter et al. 2017), which is in agreement with the peak locations of (see Figure 12b). This explains that a steeper logarithmic decay—than one with —has been observed in profiles ( in Marusic et al. 2013): the decay included, on top of the attached-eddy decay, the energy decay of global/VLSM-type energy.
5 Concluding remarks
Through the use of data-driven spectral filters for the streamwise velocity fluctuations (derived and applied in Part 1), a breakdown of the streamwise TI was assessed. Within the logarithmic region, here from up to , the TI is formed from three additive contributions, with a summary presented in Figure 13. The main outcomes of this work are listed as follows.
Scaling trends of the TI reflecting wall-attached, self-similar eddying motions, revealed evidence for a logarithmic region-scaling following the AEH with in (2.3). Over the range of Reynolds numbers investigated (), was found to be constant. A logarithmic decay of the attached-eddy turbulence, via , had to be explicitly assumed, because the wall-attached turbulence does comprise a signature of global/VLSM-type energy. It was hypothesized that this energy masks a true logarithmic region in the TI profiles, due to a shoulder of energy (Figure 13b