Reynolds number asymptotics of wall-turbulence fluctuations

Abstract In continuation of our earlier work (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, R3; Chen & Sreenivasan, J. Fluid Mech., vol. 933, 2022a, A20 – together referred to as CS hereafter), we present a self-consistent Reynolds number asymptotics for wall-normal profiles of variances of streamwise and spanwise velocity fluctuations as well as root-mean-square pressure, across the entire flow region of channel and pipe flows and flat-plate boundary layers. It is first shown that, when normalized by peak values, the Reynolds number dependence and wall-normal variation of all three profiles can be decoupled, in excellent agreement with available data, sharing the common inner expansion of the type $\phi (y^+)=f_0(y^+)+f_1(y^+)/Re^{1/4}_\tau$, where $\phi$ is one of the quantities just mentioned, the functions $f_0$ and $f_1$ depend only on $y^+$, and $Re_\tau$ is the friction Reynolds number. Here, the superscript $+$ indicates normalization by wall variables. We show that this result is completely consistent with CS. Secondly, by matching the above inner expansion and the outer flow similarity form, a bounded variation $\phi (y^\ast )=\alpha _\phi -\beta _{\phi }y^{{\ast {1}/{4}}}$ is derived for the outer region where, for each $\phi$, the constants $\alpha _\phi$ and $\beta _{\phi }$ are independent of $Re_\tau$ and $y^\ast$ $\equiv y^+/Re_\tau$ – also in excellent agreement with simulations and experimental data. One of the predictions of the analysis is that, for asymptotically high Reynolds numbers, a finite plateau $\phi \approx \alpha _\phi$ appears in the outer region. This result sheds light on the intriguing issue of the outer shoulder of the variance of the streamwise velocity fluctuation, which should be bounded by the asymptotic plateau of approximately 10.


Introduction
The Reynolds number dependencies of the variances of streamwise and spanwise velocity fluctuations as well as pressure fluctuations are thought to present exceptional challenges for the classical notion of wall scaling (Marusic et al. 2010;Smits et al. 2011).A salient example is that, when scaled in wall units, the peak values of these quantities near the wall grow with increasing Reynolds number (though the peak locations are remarkably invariant; see, e.g., Sreenivasan (1989)).In CS, the growth of these peaks was cast as a finite Reynolds number effect, and it was shown that a bounded growth model (discussed below) fits the data better.In this paper, we turn attention to wall-normal profiles of the variances of these fluctuations.This work is an alternative to the attachededdy hypothesis by Townsend (1956), which ascribes a logarithmic decay for fluctuations in the outer flow as φ + (y * ) = B φ − A φ ln y * . (1.1) Here, φ + represents the variance of uu + or ww + ; the superscript + indicates normalization by u τ and ν, and u, v, w for fluctuation velocities in the streamwise (x), wall-normal (y) and spanwise or azimuthal (z) directions; y * = y/δ where δ is the flow thickness; the slope A φ and intercept B φ are constants independent of y * and the friction Reynoldss number Re τ = u τ δ/ν, where u τ ≡ (τ w /ρ) 1/2 is the friction velocity, but may depend on φ.
The rationale behind (1.1), as discussed by Marusic & Monty (2019), is that the number density of the attached eddies that contribute to turbulent fluctuations varies inversely with y * , and an integration with respect to y * leads to the total fluctuation intensity given by (1.1).Some consequences of this idea have been explored in laboratory measurements (EXP) (Metzger & Klewicki 2001;Hultmark et al. 2012;Vincenti et al. 2013;Willert et al. 2017;Samie et al. 2018;Ono et al. 2022) as well as direct numerical simulations (DNS) (Wu & Moin 2009;Schlatter & Örlü 2010;Jimenez et al. 2010;Lee & Moser 2015;Pirozzoli et al. 2021;Hoyas et al. 2022;Yao et al. 2023).The resulting findings have been discussed in terms of mixed scaling (DeGraaff & Eaton 2000), multi-regime of the power-law spectrum (Vassilicos et al. 2015;Diaz-Daniel et al. 2017), inner-outer interactions (Marusic et al. 2017) and random addictive process (Yang & Lozano-Durán 2017).The notion of attached eddies has been extended to study high-order moments of single point velocity fluctuations (Meneveau & Marusic 2013) as well as to velocity structure functions (de Silva et al. 2015).
Pressure fluctuations have also received attention in the past sixty years (Bradshaw 1967;Klewicki et al. 2008;Panton et al. 2017), in part because of their importance for aircraft cabin noise.By extending Townsend's attached-eddy hypothesis, Bradshaw (1967) obtained a k −1 spectrum by an inner-outer matching in wavenumber space and hence an ln Re τ growth of wall pressure fluctuation.The k −1 spectrum so deduced is marginally detected in laboratory boundary layers at Re τ ≈ 6000 (Tsuji et al. 2007), but not in the DNS data so far.This unsatisfactory situation prompted Panton et al. (2017) to develop alternative matching analysis in the spatial domain, also yielding the log-profile of (1.1).This is reminiscent of Hultmark (2012) who derived the ln y * variation in pipes by matching uu + between the inner and outer regions.It is worth noting that the Re τ effects included in these models are not part of Townsend's original attached-eddy hypothesis.
While the above works suggest a boundless growth of turbulence peaks as Re τ → ∞, CS argued that the observed variations are bounded at very high Reynolds numbers and follow a defect law of the type Here, φ + ∞ the asymptotically bounded value of φ + p (Re τ ) and c φ,∞ are the fixed coefficients.The underlying physics depends on the slight imbalance that exists between wall dissipation and maximum production in the turbulent energy budget at any finite Reynolds number, and on their tendency to eventually balance each other.Subsequently, Monkewitz (2022) showed that an asymptotic expansion of uu + profiles with the Re −1/4 τ gauge function from CS reproduced data better than ln Re τ (Smits et al. 2021).Recent measurements of Ono et al. (2022) in pipes for Re τ ranging from 990 to 20750 are also supportive of the bounded behavior.The results of CS have been checked against DNS data in the open channel (Yao et al. 2022) and compressible channel (Gerolymos & Vallet 2023), indicating the universality of the bounded behavior for different flow conditions.Indeed, Hoyas et al. (2022) reported that the wall pressure fluctuation in their DNS channel data for Re τ up to 10 4 might also be bounded.
Yet, to differentiate between the two sets of results beyond doubt, one clearly requires much higher Reynolds numbers than currently covered (or likely to be covered for the foreseeable future) in laboratory experiments or DNS (Nagib et al. 2022).Measurements in the atmospheric boundary layer (Metzger & Klewicki 2001;Metzger et al. 2007) might be thought of as helpful but various uncertainties characteristic of field measurements prevent a decisive conclusion there also.At the current stage, new theoretical ideas are highly desired to provide additional insights.As noted by Klewicki (2022), the bounded growth, once accepted, would necessitate a reassessment of a number of earlier empirical findings.In this spirit, we obtain an alternative to (1.1) by using the bounded behavior of (1.2), providing a more complete description of the asymptotic behavior of wall turbulence (including pressure).
Specifically, we first decouple the Re τ dependence from the wall-normal variation for the root-mean-square (rms) profiles by using peak values for normalization.Then we develop a matching procedure between the inner viscous and outer inviscid regions, yielding in the outer flow a defect law of the type (1.3) Here, φ + represents not only uu + and ww + but also the rms of pressure fluctuation p ′+ (superscript prime denotes the rms).
To verify (1.3), DNS data sets are collected for those with clear Re τ trend for uu + , ww + and p ′+ , all publicly available.In particular, we use the DNS on channels by Lee & Moser (2015) for Re τ from 550 to 5200, on pipes by Pirozzoli et al. (2021) for Re τ from 500 to 6000, and on TBLs by Schlatter et al. (2009Schlatter et al. ( , 2010) ) for Re τ from 490 to 1270.Higher Re τ data in the literature (Sillero et al. 2013;Hoyas et al. 2022) are also included for comparison.For experiments, we select uu + data from the Princeton pipe by Hultmark et al. (2012) for Re τ from 5411 to 98187, from the Princeton TBLs by Vallikivi et al. (2015) for Re τ from 4635 to 25062, and from the Melbourne TBLs by Samie et al. (2018) for Re τ from 6000 to 20000.Channel experiments are not collected here because their (limited) Re τ variation has been covered by the DNS of Lee & Moser (2015).Note that the data uncertainty, especially concerning the probe resolution in experiments and grid resolution in the DNS, are not addressed in this paper (see CS for a brief discussion).We do wish to state, however, that there is much need for betterresolved data.
The paper is organized as follows.Section 2 presents data collapse for the inner flow region, which leads to the uniform expansion scheme presented there.Section 3 begins with the verification of inviscid similarity in the outer region, followed by the derivation of the defect decay, using comprehensive data comparisons.Section 4 is devoted to a discussion of the geometry effect.A perspective and summary of the results are given in section 5.

Re τ -scaling for near wall region
When scaled by wall variables, an asymptotic expansion for uu + , ww + , and p ′ + , represented by φ + , can be written as where g is the gauge function of Re τ ; f 0 , f 1 and f 2 (as well as f introduced below in (2.2)) are general functions depending merely on the wall-normal distance y + , and h.o.t.indicates high order terms.For the streamwise mean velocity φ + = U + , a first order truncation of (2.1) is fairly accurate near the wall.as already remarked, for turbulent fluctuations, the wall scaling has received challenges, reflected in the notable Re τ -dependence for the near wall uu + and ww + as well as pressure fluctuation p ′+ .Note that uv + and vv + are thought as 'active' motions by Townsend (1956), with bounded maximum values, with the latest evidence being given, for example, in Smits et al. (2021) and Yao et al. (2022).In the rest of this section, we first show data collapse of uu + , ww + and p ′+ after normalization by their corresponding peak values, and then summarize a common expansion for these quantities, which is actually a second-order truncation of (2.1).

Data collapse for the inner flow region
Figure 1 shows the profiles of uu + , top panels for the channel, middle for the pipe and bottom for the TBL.While the left column displays marked Re τ variations, the right column illustrates excellent data collapse after normalization by peak values.That is, Note that according to CS, the peak location is an invariant at y + p ≈ 15.This is generally accepted as correct (at least since Sreenivasan (1989)); see Smits et al. (2021).On the other hand, Pirozzoli et al. (2021) commented that the invariant peak location in CS is violated by their pipe data, which shows that y + p slightly increases from 14.28 at Re τ ≈ 500 to 15.14 at Re τ = 6000.Nevertheless, using a finer near-wall resolutions than in Pirozzoli et al. (2021), a later study by Yao et al. (2023) 180, 2000, 5000, respectively).This range of variation is typically seen by others as well (Moser et al. 1999;Jimenez et al. 2010;Chin et al. 2014), but is regarded as small here, owing possibly to secondary reasons such as the grid and probe resolution.
Similar to (2.2), data collapse for the spanwise velocity fluctuation is achieved via where ww + p is the peak value, and h is a y + -dependent function.As shown in figure 2, different Re τ curves are in close agreement with each other, with the self-preserving range from the wall to the peak (at y + ≈ 45) or beyond.We note a marginal Re τ dependence on this peak location; it is not clear whether they arise from numerical uncertainty or physical modulation by outer flow structures.
Coming now to pressure fluctuations, the left column of figures 3 shows Re τ dependence of p ′+ .The best collapse is obtained by plotting p ′+ p − p ′+ , as shown in the right column.On this basis, we may write where j is (in general) a y + -dependent function.The collapse extends from wall to the peak (at y + ≈ 30), with p ′+ p − p ′+ w a constant around 0.4.This constancy inspires us to postulate (2.4).From (2.4) one has p ′+ /p ′+ p = 1 − j(y + )/p ′+ p (Re τ ), in which an increasing p ′+ p with Re τ would eventually spoil the data collapse if one plotted p ′+ /p ′+ p .That is the reason why (2.2) or (2.3) is not applied to p ′+ .Note that Panton et al. (2017) attempted another data collapse by using pp + w − pp + , but it is not as satisfactory as (2.4) in figure 3, as discussed later in section 3.

Summary for the near wall scaling
The above comparisons demonstrate that the Re τ and y + dependencies could be decoupled after a proper normalization by peak values.Recalling (1.2) for the Re τ -scaling of the peak values and substituting it for uu + p , ww + p and p ′+ p into (2.2),(2.3) and (2.4), respectively, one has a uniform expansion  Lee & Moser (2015), for pipes by Pirozzoli et al. (2021), and for TBLs by Schlatter et al. (2009Schlatter et al. ( , 2010)).where φ 0 (y Note that (2.5) is a specific case of which is a second-order truncation of (2.1) that was initiated first by Spalart & Abe (2021) and Monkewitz (2022).If f 0 = 0, the above (2.6)reduces to which is the scaling proposed by Smits et al. (2021) for uu + with g(Re τ ) = ǫ + x−w (streamwise wall dissipation).However, Smits et al. (2021) found that their proposal did not work as well for ww + , which they speculated was due to different superposition and modulation enforced by outer flow structures.Here, we show that replacing wall dissipation by peak value, i.e. g = φ + p , (2.7) applies for both uu + and ww + .Even so, (2.7) is not proper for pressure due to a constancy p ′+ p − p ′+ w as explained earlier.Thus, a nonzero f 0 is needed in (2.6) when taking pressure into consideration, which is missed in Smits et al. (2021).
Finally, we recall that from (2.6), Monkewitz (2022) developed a composite model for uu + , which shows that g = Re −1/4 τ yields a better data description than the alternative g = ln Re τ by Smits et al. (2021).The gauge function g = Re −1/4 τ in (2.6) restores wall scaling for asymptotically high Re τ , and we will use it below to derive an outer decay profile, which has not not achieved before.
3. Re τ -scaling for outer region and the defect law for fluctuations Similar to (2.1) for the inner region, an asymptotic expansion for the outer flow reads where y * = y/δ = y + /Re τ is the outer unit; F 0 , F 1 and F 2 are general functions depending on y * , and G(Re τ ) is the gauge function.In analogy to law of the wall, the inviscid outer flow similarity corresponds to the first order truncation in (3.1), i.e.
When we take φ + = U + e − U + , the resulting equation for the mean velocity is known as the velocity defect law.Here, the subscript e indicates the value at y = δ or y * = 1, i.e., the centerline for channel and pipe flows, and the boundary layer edge for TBL.This law has been tested extensively in the literature-see Nagib et al. (2007) and She et al. (2017) for recent efforts.We shall now develop the equation for the fluctuating quantities.
An assessment of outer similarity for the three profiles is shown in figure 4, with the top panels for the channel, middle for the pipe and bottom for the TBL.It is remarkable that the pressure fluctuation displays an excellent data collapse from y * = 0.1 to y * = 1.In the same flow region, the spanwise velocity variance data also collapse well, with a deviation of the order of 0.1 (scaled on u 2 τ ).For streamwise velocity, the outer similarity holds better towards the outer edge though discernible Re τ dependence exists for small Re τ profiles.For Re τ > 1000, the streamwise variance profiles also collapse together closely (figures 5-6), thus supporting the inviscid similarity of (3.2) at high Re τ .

Matching for the defect law for fluctuations
Based on the above inner viscous expansion and outer inviscid similarity, we now develop a matching procedure to derive the analytical form in the intermediate zone.To start with, one may invoke the derivation of Millikan (1938) for the log-law, which has been extended (for example) for uu + by Hultmark (2012) and for pp + by Panton et al. (2017).Nevertheless, different orders of matching would lead to different scaling proposals, and here we present a short account of matching to obtain the defect law for fluctuations.Later we will address the question of how the current defect law is consistent with Millikan's matching analysis.
It is readily verified that (1.3) matches (3.2) in the outer region and (2.6) in the inner, hence offering a common description for the overlap region.It should be mentioned that a logarithmic decay of (1.1) can also be obtained if the gauge function Re  Before turning to the data in evidence for (1.3), we discuss how the defect law conforms with the classical matching analysis.In fact, by following the matching of Millikan (1938), as discussed for example in Pope (2000), one has where A 0 and A 1 are coefficients independent of y + or y * .Note that A 1 involving the Re τdependence is a higher-order modification compared to the leading order coefficient A 0 , which has been introduced for the streamwise mean velocity by Wosnik et al. (2000) and Monkewitz & Nagib (2023) .Here, particularly for A 0 < 0, (3.5) yields the logarithmic decay, e.g. for uu + as in Hultmark (2012), and for pp + as in Panton et al. (2017).
(3.6) Further, with A 1 (Re τ ) = O(1/ ln Re τ ), one has which means that φ + is bounded at any specific y + s location.Conversely, a bounded φ + would exclude a constant logarithmic slope but require the slope to decrease with Re τ according to O(1/ ln Re τ ).
One can easily check that (1.3) falls into this category of (3.7) as follows.Calculate the local logarithmic slope of (1.3) yielding

Comparison with data
Figures 5-8 show the data comparisons with (1.1) and (1.3) for uu + , ww + and p ′+ , respectively.Table 1 collects all the parameters for the three profiles, arising from these fits to the data, which will now be discussed in greater detail.Similar to figures 1-3, top panels of figures 5-8 are for channel, middle for pipe and bottom for boundary layer flows; the left column is for abscissa that are in linear units while right are for logarithmic units.A difference from figures 1-3 is that the data in figures 5-8 are denoted by symbols with lines (black), so that (1.1) and (1.3) are better marked to guide the eye.
Particularly for uu + , to avoid distractions by data scatter at small Re τ , we collect in figure 5 only high Re τ profiles from DNS, namely, Re τ from 2000 to 10 4 for channel; 2000 to 6000 for pipe; and 1270 to 1990 for TBL.Compared to figures 4 a,d,g, it is clear in figures 5 a,c,e that all high Re τ profiles collapse on each other in the flow range for 0.1 y * 1, thus bearing out the inviscid similarity.This is confirmed again by experimental data in figure 6 corresponding to higher Re τ .
Note that the logarithmic behavior advanced in the literature (Hultmark et al. 2012;Marusic et al. 2013;Samie et al. 2018) is indicated by green dotted line.Although it characterizes data in the region from 0.1 y * 0.3, the value of the intercept B φ needs to be adjusted for the three flows from 1.61 to 2.2, while A φ holds constant around 1.26.In contrast, the red dashed line represents (1.3) with the same α φ = 10 and β φ = 9.3, which reproduces data well for channel, pipe and TBL flows, covering not only the logarithmic range but also the so-called wake region, almost all the way to the centerline of channel and pipe flows.
One may imagine that the data in figure 5 have not reached the asymptotic state and that (1.1) might agree with data better for higher Re τ .But experimental data from Princeton pipes (Hultmark et al. 2012) with Re τ covering one more decade, e.g.Re τ from about 5000 to 10 5 , do not show any improvement of the fit to (1.1); see figures 6a-b.A similar observation is also true for the TBL, as shown in figures 6c-f.Moreover, for ww + and p ′+ , (1.3) extends almost all the way to the centerline of channel and pipe flows.Here, data in figures 7 and 8 are the same DNS groups as in figures 1 and contain those low Re τ profiles.The agreement with (1.3) is excellent at the smallest Re τ ≈ 500 for channel and pipe, in contrast to the log variation that agrees with data only for Re τ 2000.Therefore, in both y * and Re τ ranges, (1.3) covers a wider range than (1.1).
Three further points will -rednow be discussed.First, the difference between (1.3) and (1.1) is more vital for asymptotically high Re τ .For (1.1), an infinitely large of φ + ∝ ln Re τ would arise as y * → 0 and Re τ → ∞.In contrast, (1.3) assigns a plateau of φ + ≈ α φ in the same limit.Such an asymptotic plateau implies that turbulent eddies in the bulk would be in a quasi-equilibrium state in the sense that their contribution to φ + is invariant when y * changes.Mimicking Townsend's terminology, the decreasing influence of attached eddies away from the wall would be compensated by the increasing influence of detached eddies, so that their total contribution to φ + remains invariant.Note also that according to (1.3), the outer peak of uu + , if it exists, should be bounded by uu + ≈ 10.Clarification of such differences of perspectives in the asymptotic state require future work.
Second, while (1.3) adheres closely with TBL data of Vallikivi et al. (2015) up to y * = 1 (figure 6c), it is slightly and uniformly higher than TBL data of Samie et al. (2018) for y * > 0.6 (figure 6e).This is due to the fact that former set of data are obtained for flow over a flat plate mounted in the same pipe in which data of Hultmark et al. (2012) are measured.Therefore, the data of Vallikivi et al. (2015) in its outer wake region resemble the center behavior of pipe data by Hultmark et al. (2012), both in agreement with (1.3) up to y * = 1.In contrast, TBL data of Samie et al. (2018) are measured in the Melbourne wind tunnel with the normal free stream boundary condition, so that a vanishing uu + ≈ 0 towards boundary layer edge is observed (figures 6 e&f), which is   the bounded variation of (3.12) according to CS. Symbols are DNS data, squares for channel (Lee & Moser 2015), circles for pipe (Pirozzoli et al. 2021) and diamonds for TBL (Schlatter et al. 2009).Note that DNS channel data at Re τ = 150, 300, 400 of Iwamoto et al. (2002), and Re τ = 180, 550, 944, 2000 of Hoyas & Jimenez (2006) are also included here.lower than (1.3).Such a difference reflects the wake influence on uu + in TBL.In fact, the wake influence is much sharper for ww + and p ′+ in TBL (figures 7e & 8e).This issue will be addressed in the section 4.2.
The third and final point is that for p ′+ , the green dotted line in figure 8  3) in figure 8. Nevertheless, as shown in figures 9a-c, pp + w − pp + versus y + produces no data collapse for y + > 5. Particularly for the trough located at about y + = 30, the data are markedly lower for increasing Re τ , thus creating a challenge for the inner-outer matching analysis.To reconcile this challenge, Panton et al. (2017) introduced two logarithmic slopes, i.e.A CP = 2.56 for the common part of presumed log profile in the overlap layer, and another A w = 2.24 for the Re τ variation of the wall pressure.Following this fix, one can estimate

Discussion on flow geometry influence
4.1.Near wall universality We focus here on the geometry effects.First, a universal data collapse is summarized by unifying (2.2), (2.3) and (2.4) together, i.e.
Moreover, as y + moves from the wall to the peak location, it is interesting to check whether f φ (y + ) is universal for channel, pipe and TBL flows.This is indeed verified in figures 10a, b & c, for uu + , ww + and p ′+ , respectively.Profiles from these three wall flows collapsed together from the wall to the peak location, which means that Re τ -dependence and geometry influence are canceled by the ratio of relative variations composed of φ + w and φ + p .This is conceivable if the near wall region is viewed as a self-organized entity, so that superposition and modulation effects enforced by the outer flow structures are characterized to the first order by wall and peak values.

Wake modification in TBL
Note that in figures 7 and 8, ww + and p ′+ depart from (1.3) for y * 0.5 in TBL, but the agreement persists all the way to the centerline of channel and pipe flows.Recall the findings by Chen et al. (2019) that the total shear stress τ + and the Reynolds shear stress −uv + in the wake of the TBL differ notably from that in channel and pipe flows.The reason for this difference, according to Chen et al. (2019), is the nonzero mean momentum transport in the wall-normal direction of TBL, i.e.V /V e ∝ (y * ) 3/2 (the subscript e indicates the value at the boundary layer edge, as noted earlier).The latter leads to −uv + ≈ τ + ≈ 1 − (y * ) 3/2 in TBL, differing from −uv + = 1 − y * in channel and pipe flows for which V ≡ 0. Once we accept the difference in the −uv + behavior, we may expect a similar wake modification on ww + and p ′+ by the nonzero V in TBL.That is, where φ + represents ww + or p ′+ , and φ + uv represents −uv + .If so, substituting φ where the proportionality coefficient c φ is independent of y * but may depend on φ.
Verification of (4.3) for TBL is provided in figure 11.The agreement with data is quite satisfactory for y * > 0.2, and the fitting parameters are φ + e = 0 and c φ = 2.02 for ww + , while φ + e = 0.8 and c φ = 1.8 for p ′+ .This model for the wake flow could also be applied to describe uu + towards the free stream of TBL, but the deviation is fairly small as shown in figures 5e and 6e, and will not be considered further here.

Perspective and conclusions
New methods of analysis and generations of new experiments and simulations have revealed deeper layers of interesting questions on wall flow dynamics.Previously unthinkable questions, such as the universality of the Kármán constant in the mean flow description and the scaling of fluctuations in these flows, as well as the implications of the behavior of fluctuations for the mean velocity itself, can now be asked, and reasonable answers for them can be attempted.In contrast to the mean velocity, concerted effort to understand the scaling of fluctuations is relatively new.This paper, when taken together with our earlier work (Chen & Sreenivasan 2021, 2022), provides a self-consistent description of fluctuations in streamwise and spanwise velocity, as well as pressure fluctuations.One of the main qualitative conclusions of this work is that wall-normalized fluctuations are bounded even when Re τ → ∞, thus restoring the validity of the standard law of the wall.The alternative scenario of attached eddy hypothesis and its consequences lead to a different conclusion.
Aiming for an asymptotic description of fluctuations in canonical wall flows, we have obtained several new results, summarized as follows.First, excellent data collapse is achieved for the near-wall rms profiles of streamwise and spanwise velocity fluctuations ( uu + and ww + ) as well as pressure fluctuations (p ′+ ).Their spatial variations and the Reynolds number dependence are decoupled via the normalization through peak values.With the defect law for the peaks given in CS, a universal near wall expansion (2.6) is obtained with the specific gauge function g = Re −1/4 τ , consistent with that developed by Monkewitz (2022).
Moreover, a defect decay (1.3) is derived by matching (2.6) with the outer inviscid similarity (3.2).Compared to the log-profile by Townsend's attached eddy hypothesis, it is shown that (1.3) reproduces the data better, not only over a wider Re τ domain but also in a larger flow region.As indicated by (1.3), there would appear an asymptotic plateau as y * → 0 and Re τ → ∞, which implies a quasi-equilibrium state with contributions to fluctuations coming from all associated eddies that are invariant as wall-normal position changes.If so, the intriguing outer peak of streamwise fluctuation, if one exists, would be bounded by uu + ≈ 10.
Finally, a near wall universality (4.1) is obtained independent of both Reynolds number and flow geometries.In addition, a wake flow modification in TBL is introduced for ww + and p ′+ , which shows close agreement with data towards the boundary layer edge.
There is no gainsaying that more and better data are required to put all these results on a firmer foundation.It is exciting to await cleaner data at higher Reynolds numbers with improved resolution.

Figure 1 :
Figure 1: Wall-normal dependence of streamwise velocity fluctuation scaled in viscous units (abscissa in logarithmic scale) for a series of Re τ 's in channels (top panels), pipes (middle panels) and TBL flows (bottom panels).Left column: uu + versus y + .Right column: uu + normalized by its (inner) peak value uu + p , showing very good collapse.Colored lines indicate DNS data at different Re τ 's marked in the figure legends, for channels byLee & Moser (2015), for pipes byPirozzoli et al. (2021), and for TBLs bySchlatter et al. (2009Schlatter et al. ( , 2010)).

Figure 2 :
Figure2: The same plots as in figure1but for the spanwise velocity fluctuations.That is, top panels are for channels, middle for pipes and bottom for TBLs; left column for ww + while right for ww + / ww + p versus y + .Lines represent the same DNS data as in figure1.

Figure 3 :
Figure 3: Wall-normal dependence for the rms of pressure fluctuation p ′+ = pp +1/2 in channels (top panels), pipes (middle panels) and TBL flows (bottom panels).Left column for p ′+ while right for p ′+ p − p ′+ versus y + .Lines are the same DNS data as in figure 1.

Figure 4 :
Figure4: Wall-normal dependence of turbulence fluctuations in outer length unit y * = y/δ (the abscissa in linear scale).Top panels for channels, middle for pipes and bottom for TBLs.Left column for uu + ; middle for ww + and right for p ′+ .Lines are the same DNS data as in figure1.
ln Re τ ) at any given position y + s .Moreover, as y * increases, A local in (3.8) becomes larger, indicating a steeper logarithmic slope towards the outer flow.This picture is indeed supported by data in the comparison below, in contrast to (1.1) with a constant slope valid in a narrower flow domain.

Figure 8 :
Figure 8: R.M.S. of pressure fluctuation in channel (top), pipe (middle) and TBL (bottom) flows scaled in the outer unit.Dashed (red) lines indicate the bounded decay (1.3) with parameters in table 1. Dotted (green) lines indicate the logarithmic decay in (3.9), i.e. 0.27 − 2.56 ln(y * ) for channel, 0.6 − 2.45 ln(y * ) for pipe and 2.5 − 2.45 ln(y * ) for TBL.Symbols with lines are the same DNS data as in figure7.

Figure 9 :
Figure 9: Wall-normal dependence of the variance of pressure fluctuations is shown in the plots of pp +w − pp + versus y + , where pp + w is the wall-value of pp + .Lines are the same DNS data as in figure3, (a) for channels, (b) for pipes and (c) for TBLs.Note that lines depart markedly from each other with increasing Re τ in the region y + > 5. (d) Difference between the peak and wall values of pressure variance, i.e. pp + p − pp + w , for channel, pipe and TBL flows for a series of Re τ values.Dotted line (green) indicates the logarithmic growth by (3.11), i.e. 0.32 ln Re τ + 0.33.Dashed line (red) indicates the bounded variation of (3.12) according to CS. Symbols are DNS data, squares for channel(Lee & Moser 2015), circles for pipe(Pirozzoli et al. 2021) and diamonds for TBL(Schlatter et al. 2009).Note that DNS channel data at Re τ = 150, 300, 400 ofIwamoto et al. (2002), and Re τ = 180, 550, 944, 2000 ofHoyas & Jimenez (2006) are also included here.
found no such variation of y +

Table 1 :
Parameters in (1.3), for different fluctuations.Superscripts 'CH', 'Pipe' and 'TBL' represent channel, pipe and boundary layer flows, respectively.Note that both α φ and β φ vary only modestly among different flows, implying that essentially the same mechanisms applies for all flows.Moreover, β φ is quite close to α φ , as φ at the boundary layer edge y * = 1 is fairly small.
ln Re τ ≈ 0.32 ln Re τ , (3.11) which would break the wall scaling completely.As a comparison, figures 3b,d,f show that data of p ′+ w − p ′+ collapsed -redwell up to the trough, better than pp + w − pp + in figures 9a-c.Moreover, via the bounding relation p ′+ w ≈ 4.4 − 10.5/Re which depicts data satisfactorily in a wider Re τ range than (3.11) in figure9d.