Statistical evidence of anasymptotic geometric structure to the momentum transporting motions in turbulent boundary layers

Abstract

The turbulence contribution to the mean flow is reflected by the motions producing the Reynolds shear stress (〈−uv〉) and its gradient. Recent analyses of the mean dynamical equation, along with data, evidence that these motions asymptotically exhibit self-similar geometric properties. This study discerns additional properties associated with the uv signal, with an emphasis on the magnitudes and length scales of its negative contributions. The signals analysed derive from high-resolution multi-wire hot-wire sensor data acquired in flat-plate turbulent boundary layers. Space-filling properties of the present signals are shown to reinforce previous observations, while the skewness of uv suggests a connection between the size and magnitude of the negative excursions on the inertial domain. Here, the size and length scales of the negative uv motions are shown to increase with distance from the wall, whereas their occurrences decrease. A joint analysis of the signal magnitudes and their corresponding lengths reveals that the length scales that contribute most to 〈−uv〉 are distinctly larger than the average geometric size of the negative uv motions. Co-spectra of the streamwise and wall-normal velocities, however, are shown to exhibit invariance across the inertial region when their wavelengths are normalized by the width distribution, W(y), of the scaling layer hierarchy, which renders the mean momentum equation invariant on the inertial domain.

This article is part of the themed issue ‘Toward the development of high-fidelity models of wall turbulence at large Reynolds number’.

1. Introduction

Studies of turbulent boundary layer flows often associate turbulent transport with the action of organized vortical motions. Consequently, the identification and characterization of the geometric aspects of the turbulence are often synonymous with searching for and quantifying the vortical motions within the flow. Such motions include lambda/hairpin vortices, counter-rotating vortices, vortical fissures and streaks, to name a few. In the mean momentum equation, however, the net effect of the turbulent motions is manifest as the gradient of Reynolds shear stress, d〈−uv〉/dy, where u and v are the streamwise and wall-normal fluctuation velocities, respectively, and y is the wall-normal distance. Therefore, in contrast to the vortical motions, it is the uv motions, and, more specifically, their average variation in y, that are of primary significance. Here, we do not necessarily question the relevance of vortical motions to the dynamics (indeed, the Reynolds stress gradient can be related to the difference of velocity vorticity products), but rather emphasize that d〈−uv〉/dy plays the direct determining role in establishing the distribution of mean momentum. Given this, the purpose of this paper is to investigate geometric features of the uv motions, and to relate these features to the dynamics of turbulent boundary layers (TBLs).

At this point, we mention that the inclusion of geometric features in the study of wall-turbulence has a long history, particularly since the discovery of coherent motions in boundary layers (e.g. [1,2]). Early modelling efforts based on a self-similar eddy hierarchy, primarily for the log region of the TBLs (e.g. [3,4]), has met with considerable success in the recent past (e.g. [5]). Along a slightly different line of enquiry, tools from flow instability have been used to understand the recurring motions in turbulent flows, the most notable being the physical mechanism of the self-sustaining near-wall cycle (e.g. [6]). In the log region, there have been suggestions of a similar self-sustaining process (e.g. [7]). Furthermore, the tools from the ‘non-modal’ stability analysis (e.g. [8]) have been used to plausibly explain the large-scale structures observed in turbulent boundary layers (e.g. [9]), and also to model more detailed features of wall turbulence (e.g. [10]).

The impetus for the present effort to consider the geometric features in the uv motions is provided by the findings in the study by Klewicki et al. [11]. Building upon the analysis framework of Fife and co-workers (e.g. [12–15]), this study exploits the dynamic self-similarities admitted by the mean momentum equation to expose a complementary geometric structure. The theoretical framework leverages the leading balances of terms in the mean dynamical equation and, via variable transformations, the exchange of leading balance as a function of the increasing scale of the momentum transporting motions with wall-normal distance. These analyses evidence the existence of an underlying hierarchy (distribution) of scaling layers, W(y), whose widths physically represent the characteristic size of the turbulent motions responsible for momentum transport, which is dominated in the wallward direction, i.e. by the negative uv motions, or uv_<, herein. The y variation of W, dW/dy, is of central importance, as it is analytically shown to approach a constant on the domain where the leading-order balance solely involves inertial terms. In the context of the mean momentum equation, this is the origin of distance-from-the-wall scaling that underlies a logarithmic mean velocity profile, and recent empirical studies at high Reynolds number support the onset of logarithmic dependence on the inertial domain as specified by the present theory (e.g. [5,16]).

The features of the theory most relevant to this study relate to the fact that the coordinate stretching function, ϕ, that yields an invariant form of the mean momentum equation across the entire hierarchy of scaling layers is analytically given by , and thus this scaling parameter approaches constancy, , as the Reynolds number tends to infinity. Indeed, the theory specifies that , where κ is the leading coefficient in the logarithmic mean profile equation (i.e. the von Kármán constant). By employing a valid but discrete construction of the layer hierarchy, Klewicki et al. [11] show that the quantity approaches a constant α on the inertial domain, and through a number of empirical measures show that α equals unity to within a couple of per cent, i.e. within the measurement uncertainty. Remarkably, the condition α = 1 indicates that , where is the golden ratio. Geometrically, α = 1 corresponds to an extended version of distance-from-the-wall scaling. It requires both a proportionality between each layer width and its distance from the wall (derived from the theory), and the same proportionality between adjacent layers; this is rationally expected as the number of hierarchy layers, and thus Reynolds number, tends to infinity. The analysis then extends these findings to surmise that the fraction of time that the uv signal is negative on the inertial domain should equal , while direct measurements revealed an approximately constant value close to 0.62 at the highest Reynolds number explored. Overall, these findings suggest some kind of geometrically self-similar arrangement of uv motions on the inertial domain, and the aim of this work is to further clarify the geometric structure of these negative uv motions.

2. Experimental database

For the purposes of this paper, we employ the experimental database of u and v signals acquired at the High Reynolds Number Boundary Layer Wind Tunnel (HRNBLWT) and at the Flow Physics Facility (FPF) located at the University of Melbourne and at the University of New Hampshire, respectively, using hot-wire anemometry with four sensing elements. The details of the experimental procedure are presented in Morrill-Winter et al. [17, 18], and table 1 shows the relevant experimental parameters, where the friction Reynolds number ranges from 2400 to 16 400. Here, the friction velocity is constructed from the wall shear (with U the total streamwise velocity) and the kinematic viscosity ν, and δ is the boundary layer thickness based on the composite velocity profile [19]. Note that the superscript + will represent normalization by viscous scales; for example, , and . Also, since we intend to evaluate the lengths of the zero-crossings in the uv signal, which is susceptible to noise, we have filtered all the uv signals using a second-order Butterworth filter with a cut-off frequency of .

Table 1.

Experimental parameters. is the free stream velocity and l is the representative length of the sensor.

3. Geometric features of negative uv motions

The time series of the uv signal is converted into a binary signal, with −1 for negative uv magnitudes and +1 for the positives. The length (or the time period) of the −1 portion of the signal divided by the total length of the signal is the negative time fraction. Figure 1a shows this time fraction for different Reynolds numbers (presented by different symbols) as a function of the normalized wall distance . The normalization (by , the intermediate length scale) is to highlight the fact that in boundary layers the beginning of the inertial region starts at . The averaged time fraction within the inertial region (ending near 0.15δ⁺) is presented in figure 1b for increasing δ⁺, where it is evident that the time fraction approaches a constant with increasing Reynolds number. Here, we note that the start of the inertial layer may be different for boundary layers than in pipes and channels. That is, earlier results that are largely based on channel flow analyses put the onset of the inertial domain at while our more recent measurements (in [18]) for the boundary layer (used herein) indicate that this onset is near . Furthermore, we point out that the precise values of the constants 2.6 and 3.6 are obtained empirically from the experimental data, even though the theory predicts these constants to be . Although a slight difference in the beginning of the inertial region is noted for the two flows, the analysis of the uv motions, however, is plausibly similar in both flows, especially in the inertial regions, where we expect to see self-similar behaviours.

Figure 1.

(a) The time fraction that uv < 0 for versus wall position normalized by the intermediate length scale. (b) The average time fraction uv < 0 over the inertial subdomain. Symbols are tabulated in table 1. (Online version in colour.)

Insights regarding the behaviour of the time fraction are gained by considering the lengths of the negative uv motions, say, L_<. Here, the negative time segments (t_<) are converted into spatial lengths by employing the local mean velocity at the corresponding wall location, i.e. L_< = t_<U(y). Note that the time fraction is the same as the length fraction and is independent of the convection velocity used. Additionally, the relevant L_< in the present case is observed to be less than δ/10, and hence is hardly affected by the use of Taylor's frozen turbulence hypothesis [20]—even if it were an issue. The average of these lengths in inner units 〈L_<⁺〉 is presented in figure 2 on log–log axes, where the lighter (grey) symbols are outside the inertial domain. Clearly, 〈L_<⁺〉 increases with distance from the wall. In the inertial region, the average length increases approximately as . In fact, the power-law exponent is curiously close to 1/Φ², as indicated by the solid line in the figure. We note that the power law has a range that seems to extend beyond y/δ = 0.15.

Figure 2.

Distribution of the average lengths of negative uv motions on log–log axes at and 16 400. The darker symbols (cf. table 1) are inside the inertial region (). (Online version in colour.)

As an interesting aside, we note that the average lengths of u signal, both positive and negative, have been investigated by Sreenivasan and co-workers (e.g. [21,22]). It turns out that the average length is proportional to the Taylor microscale, and this is true for any random signal that obeys a Gaussian probability density function. In the inertial region or log region of the turbulent boundary layer, the Taylor microscale is proportional to (y⁺)^1/2. Figure 2 also shows a line of slope (dashed) for comparison. Evidently, 〈L_<⁺〉 follows a different scaling. In fact, it will be made clear later (in figure 3) that the probability density function (PDF) of the uv signal is far from being Gaussian, ruling out the expectation of a (y⁺)^1/2 scaling. Furthermore, we observe that the signal spends a large portion of its time at values close to zero. This is likely to cause some uncertainty in determining 〈L_<⁺〉; however, the signals were obtained from two different experimental facilities and both show highly similar behaviour.

Figure 3.

PDF and CDF of −uv⁺ at δ⁺=10 100. (a) At . (b) At all y⁺ locations measured, and presented as a contour plot. The hatched areas on the CDF correspond to confidence bounds about the median, specifically (or 95% confidence), (or 50% confidence), and (or 20% confidence). The PDF magnitudes are given by the colour bar. Note the colour increments are logarithmically spaced. (Online version in colour.)

The above discussion shows that, with close approximation, the negative time fraction in the inertial region, where is the number of negative uv segments within the total length of the signal . Taking , where C is possibly a weak function of δ⁺,

3.1

This suggests that, with increasing wall height, the average number of segments decreases, which of course is a consequence of the increasing lengths with a constant time fraction.

4. Magnitude of uv motions

Discerning a self-similar structure that connects the geometry and dynamics of the momentum transporting motions naturally leads to consideration of the signal magnitudes. Features of the magnitudes of uv motions are represented by the PDF, and an example is given in figure 3. Figure 3a shows the PDF of the −uv⁺ signal for y⁺ near the start of the inertial domain at δ⁺=10 100 with a dark (coloured) solid line, whereas the light (grey) line is the cumulative density function (CDF). The hatched regions show the percentage of data about the median. The highly non-Gaussian nature of the PDF and the high concentration of data around zero magnitude are clear. Figure 3b presents the PDFs as a function of y⁺. Interestingly, the shape of the PDF is nearly invariant over most of the boundary layer, with an obvious reduction in width in the outer region. This reduction in the PDF size in the outer part is a consequence of the non-turbulent region interspersed with the turbulent uv-containing region. In fact, the (externally) intermittent region starts at a y⁺ of about [23], which is roughly the location where the width of the PDF size starts to decrease.

To highlight the non-Gaussian character of the PDFs, we show the skewness of the profiles in figure 4. Not only is the skewness non-zero, it also attains a rather convincingly constant value of about 1.62 on the inertial domain. This value is intriguingly (and surprisingly) close to Φ, as shown by the horizontal solid black like in figure 4.

Figure 4.

The skewness of −uv versus wall-normal position normalized by the intermediate length scale. Symbols are tabulated in table 1. (Online version in colour.)

5. Connection between the length scales and the magnitude of uv motions

The apparent appearance of in the magnitude and the geometry of uv motions is intriguing, and, at present, the structural properties leading to these observations are not clear. It seems quite likely, however, that the magnitude of the uv motions and the length scale of these motions are inherently related, and this motivates further enquiry. To this end, we use two techniques: (i) joint PDF of negative area of the uv signal and the corresponding lengths, and (ii)spectral analysis.

(a). Joint PDF of negative areas and lengths

In an attempt to understand any dependence between the uv magnitude and the corresponding length scale, we construct a box signal that has the same zero-crossing positions and the same areas of the positive and negative excursions as the uv signal. An example is shown in figure 5a, where the grey region has the same area in both the top and bottom panels. This construction therefore assigns an area (positive or negative) to every corresponding zero-crossing length. Here, we consider only the negative areas and the corresponding lengths L_<. A joint PDF of A⁺_< and L⁺_< is presented in figure 5b, the diagonal shape of which indicates that larger lengths correlate with larger areas or uv magnitudes. We note that the mean value of A⁺_<, , is related to the negative contribution to , 〈uv⁺_<〉, because

5.1

Furthermore, to understand the dependence of L⁺_< on A⁺_<, we evaluate A⁺_< conditioned on L⁺_<, using

5.2

which follows from the usual conditional probability equation, (e.g. [24]). The expression is plotted in figure 5c as a solid line, which shows the contribution that L⁺_< makes to . Here is the area under the curve, as seen by integrating (5.2) over all L⁺_<. From (5.1), it is clear that the L⁺_< that makes the maximum contribution to also contributes maximally to 〈uv⁺_<〉. This maximum L⁺_< is also shown in figure 5c with a horizontal grey solid line. If, however, A⁺_< is independent of L⁺_<, then , and (5.2) is equal to , which is also plotted in figure 5c using a dashed line. It is evident that the dashed and the solid lines are markedly different, representing the degree of dependence of the magnitudes of the negative uv motion on their associated length scales. Thehorizontal dashed line in figure 5c also shows 〈L⁺_<〉, which indeed differs from the maximum L⁺_< location of . This suggests that, while 〈L⁺_<〉 might connect to an underlying geometric structure, a length scale larger than 〈L⁺_<〉 is dynamically more relevant. A preliminary analysis shows that the maximum L⁺_< location increases with y according to a power-law dependence that seems greater than 1/Φ². Further analysis is, however, required before more firm conclusions can be drawn.

Figure 5.

Magnitude and length-scale analysis of the negative part of the uv signal for and at (which is the measurement point closest to the middle of the inertial region). (a)(i) Part of the uv signal versus . (ii) Box signal with the same areas (positive and negative) and L⁺_< as that in (i). (b) Joint PDF of the L⁺_< and A⁺_<. The contour levels are log₁₀ spaced, from to in increments of . (c) Conditional mean of A⁺_< with respect to L⁺_< (solid curve) and the case if A⁺_< was independent of L⁺_< (dashed curve). The maximum of L⁺_< is given by the solid horizontal line, and the horizontal dashed line denotes the value of 〈L⁺_<〉. (Online version in colour.)

(b). Spectral analysis

Typical analyses that use the co-spectra of u and v take into account the contributions to the magnitude of at particular length scales. Such analyses do not relate directly to the uv signal, and cannot distinguish between the negative and positive regions in the uv time series.

Nevertheless, spectral analysis provides useful information, and as an example the (pre-multiplied) co-spectrogram of −u and v, G, is presented in figure 6 for . The time (t) signal has been converted to space using the local mean velocity () with the corresponding wavenumber , and the ordinate in the figure is presented as the wavelength () normalized by viscous scales. We note that the pre-multiplied or weighted co-spectrum is independent of U. Figure 6 presents contour shades (colour) of the pre-multiplied co-spectra magnitudes. The dark solid line, which is , seems to pass roughly through the middle of the contours.

Figure 6.

The frequency pre-multiplied co-spectrogram of −u and v, G, for . The solid line indicates . (Online version in colour.)

In fact, the proportionality of the length scale with wall distance is a well-known empirical result in wall turbulence, especially in the inertial region [3,4]. More recently, and as discussed at the outset, the analysis that starts from the mean momentum equation of the wall-bounded turbulent flow has placed this distance-from-the-wall scaling on a more rigorous footing (e.g. [12–15]), and most recently for the zero-pressure-gradient turbulent boundary layer as well [18]. As described previously, a central element of the theory is the finding of a length-scale distribution, W(y), that allows the mean momentum equation to be written in a self-similar from. Here W is defined as

5.3

and on the inertial region .

The efficacy of W in scaling the data is evidenced in a slightly different manner in figure 7 using the same data as presented in figure 6. The different lines correspond to varying wall locations, and the darker (black) lines are in the inertial region. The abscissa is the wavelength normalized by W, where the constant is arbitrarily chosen so as to allow the peaks in the co-spectra to be nominally centred around one. The important feature of figure 7 is the near collapse of all the dark lines within the inertial region. This provides further evidence of the distance-from-the-wall scaling of the uv motions that directly relate to the wall-normal distribution of mean streamwise velocity and its logarithmic dependence.

Figure 7.

Frequency pre-multiplied co-spectra of −u and v versus wavelength normalized by for (). The black lines represent over the domain , and (lighter) blue lines represent co-spectra at every other wall location measured. (Online version in colour.)

A more direct (however, experimentally more challenging) way to show the same scaling is to differentiate the co-spectrum G with y. The differentiated co-spectrogram is presented in figure 8. Here, we also show the distribution of W obtained using relation (5.3) as white circular symbols. We note that (e.g. [25]), which makes the differentiated co-spectrogram particularly relevant for the mean equation. It is clear from figure 8 that the location where the differentiated co-spectrogram passes through zero coincides remarkably well with the slope of the W(y) profile, that is, the theoretically reasoned scaling. Figure 8 also shows the mean vorticity as , which in the inertial region should be directly proportional to y at sufficiently high Reynolds number.

Figure 8.

Wall distance derivative of the frequency pre-multiplied co-spectrogram of −u and v for . The circles denote , and squares represent (). The black filled symbols denote the wall location where reaches its maximum, the onset of the inertial domain, and the end of the inertial subdomain, respectively. (Online version incolour.)

(c). Qualitative example of the wall scaling in uv motions

Finally, we present a qualitative example of W scaling of uv motions from the direct numerical simulation (DNS) database of turbulent boundary layers by Sillero et al. [26] at . Figure 9 shows streamwise–spanwise planes of uv contours at four different wall locations presented as four columns. The blue colour denotes negative uv regions, whereas red denotes the positive uv regions, with grey regions close to zero. Figure 9a shows the actual planes from the DNS, while figure 9b is a zoomed-in view of the white boxed regions of panel (a). The sizes of the grey boxes are scaled with the local W, which increases linearly with increasing y. The boxes have been selected to have the same proportion of positive and negative uv areas. Figure 9a qualitatively shows that the length scales become larger with increasing y, whereas the number of regions decreases. The scaled panel (b), however, shows roughly the same structure across the y locations due to its scaling with W.

Figure 9.

Streamwise–spanwise plane of for the DNS of Sillero et al. [26], . (a) The full extent of the domain. (b) The zoomed-in views of the corresponding (a) panels shown by the respective white squares. Area coloured blue is , coloured grey is , and coloured red is ; in print is grey scale, the blue, red and grey colours appear darkest to lightest. (Online version in colour.)

6. Summary and concluding remarks

Motivated by previous observations, herein we have sought to focus directly on the uv motions (rather than the usual vortical motions) due to its presence in the mean momentum equation of turbulent boundary layers. Analyses of the data over a decade variation in Reynolds number show that the negative uv (uv_<) motions constitute about 62% of the overall time. The time fraction is almost a constant within the inertial region, and, consistent with the extended version of distance-from-the-wall scaling noted at the outset, seems to approach a value near with increasing Reynolds number. On the other hand, the average length (or time) associated with uv_<, 〈L⁺_<〉, increases with wall-normal distance, showing a power-law dependence of the form . Consequently, within the inertial region, the number of occurrences of uv_< decreases with increasing wall distance, whereas their length scales increase. The fact that the power-law exponent of 1/Φ² is different from is a consequence of a highly non-Gaussian −uv PDF with a non-zero skewness. Remarkably, this skewness attains a value very close to Φ in the inertial region across all Reynolds numbers, leading us to surmise an underlying relationship between the magnitude and length of the negative uv excursions.

The apparent occurrence of quantities close to Φ in various statistics of the uv motions is intriguing, and our understanding of this behaviour remains incomplete. In general, however, there is strong empirical and analytical evidence for a self-similar structure across the inertial region. As just noted, we suspect a link between the magnitude of uv and a length scale associated with it. We observe that the 〈L⁺_<〉 profile follows a power-law exponent , whereas from a straightforward distance-from-the-wall-based argument one would expect this value to be unity. In search of a link between magnitude and length, we then constructed a measure of negative uv magnitude, A_<, and examined the joint PDF between A_< and the corresponding L_<. The conditional A_< showed that the dynamically significant length, which has a maximum contribution to the magnitude of , is notably larger than 〈L⁺_<〉. Here, it is relevant to note that the net Reynolds shear stress is given by

6.1

where the positive counterparts A_> and N_> have analogous meanings to the negative ones. Also, d〈−uv〉/dy is now simply related to the positive and negative wall-normal gradients of the magnitude and number densities of the uv motions. In this paper, we have only considered the first term in the last expression of (6.1), and perhaps the behaviour of the second term needs to be clarified. A broader analysis of the uv PDFs across Reynolds numbers and wall-normal positions also seems warranted.

Notwithstanding the limitations of the above described analyses, the uv co-spectra shows a clear distance-from-the-wall scaling in the inertial region. This scaling follows from multiscale analysis of the mean momentum equation (which underlies and motivates the present research), and thus is distinct from other theoretical developments that simply assume wall scaling. The scaling is evidenced by a convincing coalescence of all of the uv co-spectra onto a single curve when the wavelength is normalized by the scaling layer width distribution, W(y), which yields an invariant form of the mean momentum equation. What remains unresolved is whether the extended version of wall scaling (analytically yielding , as described in the Introduction) becomes operative at high Reynolds number, and, if so, what are the geometric and dynamical relationships that explain how the self-similar structure of the mean flow is reflected (as empirically suggested) in the motions responsible for the wallward flux of momentum.?

Authors' contributions

All the authors contributed equally, they also all read and approved the manuscript.

Competing interests

The authors declare that they have no competing interests.

Funding

This study was financially supported by Australian Research Council and National Science Foundation.