3.1. Introductory comment

In the following section, mainly the three incompressible DNS/Large-Eddy Simulation (LES) of Schlatter et al. (Reference Schlatter, Örlü, Li, Brethouwer, Fransson, Johansson, Alfredsson and Henningson2009), Schlatter & Örlü (Reference Schlatter and Örlü2010), Sillero et al. (Reference Sillero, Jiménez and Moser2013) and Eitel-Amor et al. (Reference Eitel-Amor, Örlü and Schlatter2014) are frequently referred to in order to support the conclusions drawn and to link them with existing data. However, since these data either do not cover the ‘blind spot’ in the range $4000 \lessapprox Re_\theta \lessapprox 5000$ , or inlet independence of the outer layer cannot be ensured within this region, an assessment of the respective data is necessary. To this end, figure 1 illustrates the $Re_\theta$ range covered by the simulations mentioned above. For all datasets depicted, the inflow lengths are represented as hatched lines or open markers until either $x \gtrapprox 200\delta _0$ or, if later, where the authors report confidence in shape factor, etc. Although it varies depending on the choice of inflow, $200\delta _0$ roughly represents a typical transit length for necessary turbulent wake recovery, compare with e.g. Sillero et al. (Reference Sillero, Jiménez and Moser2013), where ${\sim }250 \delta _0$ are required to span $1100\lt Re_\theta \lt 4800$ . For the present dataset, the inflow length has been extended to $300\delta _0$ to increase confidence of all following discussions. Note that such strictness is only necessary for highly sensitive quantities like the turbulent wake or shape factors.

Figure 1. The $Re$ ranges of numerical zero pressure gradient (ZPG) TBL studies. Hatched bars or open markers indicate either $\lessapprox 200\delta _0$ of development space or area upstream of where authors report confidence. The rationale behind the placement (in $Re_\theta$ ) of the logarithmic-layer ‘appearance’ and establishment of one decade of wall-normal extent can be found in Appendix A.

3.2. Friction coefficient

In figure 2, the effect of $Re$ on $c_f$ is examined. To compare compressible and incompressible data, both the present compressible DNS as well as the datasets of Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011, Reference Pirozzoli and Bernardini2013) are transformed into their equivalent incompressible representation (subscript $i$ ) by applying the ‘van Driest II’ transformation, which for an adiabatic wall reduces to

(3.1) \begin{equation} c_{f_i} = F_c \, c_{f_c}, \quad F_c = \frac {\big(\bar {T}_w/\bar {T}_e-1\big)}{\arcsin \left ( \sqrt {1-\bar {T}_e/\bar {T}_w} \right )^2} . \end{equation}

Here, $c_{f_i}$ is the transformed incompressible counterpart to $c_{f_c}$ , where $c_{f_c} = 2 \tau _w / (\bar {\rho }_e \bar {u}_e^2)$ for compressible cases. In addition to the DNS/LES data already introduced in figure 1, figure 2 shows experimental data from the OFI measurements of Österlund et al. (Reference Österlund, Johansson, Nagib and Hites2000) and Nagib et al. (Reference Nagib, Christophorou and Monkewitz2006), representative of the state of the art in direct measurement of $\tau _w$ . The low- $Re$ power-law correlation of Smits et al. (Reference Smits, Matheson and Joubert1983)

(3.2) \begin{equation} c_f \approx 0.024 \, Re_\theta ^{-0.25} \end{equation}

is plotted, as well as a curve of the form

(3.3) \begin{equation} c_f \approx a \, Re_\theta ^b + \frac {c}{Re_\theta }, \quad \text{ with }\; a = 0.01844,\ b = -0.2235,\ c = 0.2979 . \end{equation}

The latter simply represents an empirical best fit to the present data to highlight trends in panels ( $b$ ) and ( $c$ ). Note that the addition of a $c/Re$ term also better captures the $c_f$ trend for the low- $Re$ region of the dataset of Schlatter & Örlü (Reference Schlatter and Örlü2010) rather than a pure power-law relationship. For the high- $Re$ regime, the ‘Coles–Fernholz 2’ correlation

(3.4) \begin{equation} c_f \approx 2 \big[ \kappa ^{-1} \ln (Re_\theta ) + C \big]^{-2} \quad \text{ with }\; \kappa = 0.384,\ C = 4.127 \end{equation}

is plotted (Fernholz & Finley Reference Fernholz and Finley1996; Nagib et al. Reference Nagib, Chauhan and Monkewitz2007). Note that $\kappa = 0.384$ is used in accordance with its high- $Re$ calibration in Nagib et al. (Reference Nagib, Chauhan and Monkewitz2007), whereas later $\kappa = 0.41$ is consistently utilised as a fixed constant for cross-comparing primarily low- and moderate- $Re$ datasets.

Figure 2. Friction coefficient $c_f$ vs $Re_\theta$ with compressible scaling (a,b) and compensated representation (c,d). Panels (b,d) are zooms of (a,c), respectively.

As was previously shown in a variety of studies, e.g. Schlatter & Örlü (Reference Schlatter and Örlü2010), the low- $Re$ numerical data align generally well with the low- $Re$ power-law behaviour in panels ( $a$ ) and ( $b$ ), if inlet development regions with more rapidly decreasing $c_f$ are ignored. For $Re_\theta \gtrapprox 5000$ , both existing numerical and experimental data clearly confirm the high- $Re$ correlation. As indicated by the present newly computed data, represented by the solid black line, the changeover from the low- to the high- $Re$ regime occurs rather abruptly, at least in logarithmic representation. A ‘bend’ is distinctly visible in the $c_f$ distribution, especially in the zoomed-in representation of panel ( $b$ ). To more closely investigate the abrupt bend in $c_f$ , panels (c,d) normalise all data of ( $a$ ) by the fitted low- $Re$ correlation. The DNS data of Schlatter & Örlü (Reference Schlatter and Örlü2010) terminate precisely at $Re_\theta$ where the bend occurs, however, agreement with the newly generated data up to that point supports the plausibility of a sharp bend. Sillero et al. (Reference Sillero, Jiménez and Moser2013) report confidence in their data for $Re_\theta \gtrapprox 4800$ , precisely the region where their data overlap with the present data. Perhaps the most important evidence for the sharpness of the low-to-moderate $Re$ cross-over comes from LES data of Eitel-Amor et al. (Reference Eitel-Amor, Örlü and Schlatter2014), which are reported to be independent of initial conditions by $Re_\theta \approx 1500$ , but are subject to the uncertainty of a slightly under-resolving numerical grid. Analogous to the present simulation, these data show a notably sharp bend at a comparable $Re_{\theta }$ , although showing a consistent ${\sim }2\,\%$ offset with respect to the fully resolved incompressible DNS of Schlatter & Örlü (Reference Schlatter and Örlü2010).

3.3. Outer layer

As been shown in a variety of different studies, e.g. Fernholz & Finley (Reference Fernholz and Finley1980) and Nagib & Chauhan (Reference Nagib and Chauhan2008); Smits (Reference Smits2024), the turbulent wake appears to develop to a visually $Re$ -independent state at a possible value of approximately $4000\lessapprox Re_\theta \lessapprox 5000$ , which coincides remarkably well with the $Re$ of the present data at which $c_f$ shows a distinct bend. It is explicitly noted, however, that these trends are subject to significant scatter, see e.g. Smits (Reference Smits2024), thus, existing boundary-layer data at large $Re$ indicate this supposed $Re$ -independence is not firmly established. The present section evaluates the wake development as well as its relation to the Reynolds-number range where $c_f$ has been shown to diverge from typical low- $Re$ correlations. To this end, figure 3(a,b) shows the streamwise velocity $\bar {u}_i$ profiles at several equally spaced $Re_{\theta _i}$ stations in inner and outer scaling, where $\bar {u}_i$ is the incompressible-transformed mean streamwise velocity according to van Driest (Reference van Driest1951)

(3.5) \begin{equation} \bar {u}_i^+ = \int _0^{\bar {u}^+}{ \sqrt {\frac {\bar {\rho }}{\bar {\rho }_w}} } {\rm d} \bar {u}^+ . \end{equation}

The wake strength is defined as the local deviation from an assumed ‘standard’ logarithmic law

(3.6) \begin{equation} \bar {u}_{\rm log}^+ = \frac {1}{\kappa } \ln \left ( y^{+} \right ) + B, \end{equation}

with $\kappa = 0.41,B = 5.2$ . As indicated in figure 3(a,b), the wake strength $\Delta \bar {u}^+ = \bar {u}_i^+ - \bar {u}_{\log}^+$ saturates for $Re_{\theta } \gtrapprox 4100$ . The collapse, annotated by $\bigcirc\kern-8pt\small\textrm{II}$ , is particularly visible in panel ( $b$ ). We note that use of the van Driest transformation increases the velocity profile in the wake region with respect to incompressible data, therefore the wake strength to increases by a Mach-number-dependent, but Reynolds-number-independent offset (Wenzel et al. Reference Wenzel, Selent, Kloker and Rist2018). Thus, conclusions obtained for the present supersonic case reliably transfer to the incompressible regime although the absolute value of the wake strength is increased; implications will be detailed where appropriate. Furthermore, it is emphasised that the tight collapse of the data in the range $4100 \lessapprox Re_{\theta _i} \lessapprox 5660$ spans a respectable $Re$ range of $\Delta Re_{\theta _i} \gtrapprox 1500$ and thus extends notably into the moderate- $Re$ regime. As depicted in figure 3( $c$ ), the $Re$ dependence of this saturating trend is often quantified by the wake parameter $\Pi$ , which is defined as (Coles Reference Coles1956)

(3.7) \begin{equation} \Pi = \frac {\kappa }{2} \left ( \bar {u}_{99}^+ - \frac {1}{\kappa } \ln {\left (\delta _{99}^+\right )} - B \right ) , \end{equation}

hence proportional to the wake strength at $y=\delta _{99}$ . For all data considered, the same $\kappa = 0.41, B = 5.2$ are utilised, which generally shifts high- $Re$ data points to an artificially high $\Pi$ compared with if $\kappa ,B,\Pi$ were fit simultaneously; the assimilated data from Österlund (Reference Österlund1999) are most significantly affected by this simplification. It is further noted that some of the scatter observed regarding the wake may likely be associated with the fact that all of the constants in the profile equation are slowly settling into asymptotic values. Furthermore, as addressed in the $c_f$ discussion, the definition of $Re_{\theta _i}$ renders the occurrence of the $c_f$ bend notably independent of the Mach number. Thus, differences in panel ( $c$ ) between transformed and incompressible data are only on the ordinate, where transformed compressible data maintain a higher value of $\Pi$ as a consequence of the van Driest transformation. For the present transformed data, $\Pi$ increases ${\propto }\ln (Re_{\theta _i})$ for $Re_{\theta _i} \lessapprox 4100$ after which it abruptly saturates and stays approximately constant. Note, however, that some caution is warranted about the generalisation of this collapse at higher $Re$ , seeing that most boundary-layer properties evolve logarithmically over decades of $Re$ . Naturally, this change in slope occurs at the same $Re_{\theta _i}$ as the $c_f$ bend, which supports the notion that wake convergence – or more specifically the completion of the relative lift of the wake out of the inner layer – is likely the driver for the change in the $c_f$ slope. However, the observed abruptly saturating progression of $\Pi$ seems unfamiliar, as all empirical trend lines known by the authors to date have been formulated by assimilating scattered data to an assumed smooth function (Coles Reference Coles1962, Reference Coles1987; Cebeci & Smith Reference Cebeci and Smith1974; Chauhan et al. Reference Chauhan, Monkewitz and Nagib2009). Interestingly, partial evidence supporting abrupt wake-convergence behaviour can be gathered from existing data. The dataset of Schlatter & Örlü (Reference Schlatter and Örlü2010), for instance, which ends slightly before the expected $Re$ of the bend feature, follows a semi-logarithmic trend up to $Re_{\theta } \lessapprox 4100$ , without displaying any tendency towards smoothly rounding. Data from Sillero et al. (Reference Sillero, Jiménez and Moser2013) are estimated to have an inlet-independent wake – thus high confidence in $\Pi$ – only for $Re_\theta \gtrapprox 4800$ , beyond which approximate wake saturation is seen. Thus, if the systematic shift in $\Pi$ for the present transformed data is manually compensated for (grey offset curve in panel ( $c$ ) indicated by $\bigcirc\kern-7pt\small\textrm{I}$ ), it can be interpreted as a plausible bridge between both datasets, approximated by

(3.8) \begin{equation} \Pi (Re_{\theta }) \approx \begin{cases} \begin{array}{cc} -0.8769 + 0.1816\, \ln (Re_\theta ) & \text{for } Re_{\theta } \lessapprox 4125\\[5pt] -0.8769 + 0.1816\, \ln (4125) \approx 0.635 & \text{for } Re_{\theta } \gtrapprox 4125 \end{array}\end{cases}\kern-15pt, \end{equation}

indicated by the black dotted line in figure 3( $c$ ). The LES data of Eitel-Amor et al. (Reference Eitel-Amor, Örlü and Schlatter2014) are slightly lower compared with the trend curve (corresponding to the higher $c_f$ ) but converge towards a consistent $\Pi$ value at higher $Re$ where the effective viscous grid resolution successively increases.

Figure 3. Mean-velocity profiles and wake parameter. The black dotted line in panel ( $c$ ) indicates the fit given in (3.8).

3.4. Remark

The previous discussion has confirmed the utility of $\Pi$ as an indicator of the outer-layer similarity, which can be visually observed in figure 3(a,b). However, the somewhat ‘artificial’ nature of $\Pi$ as pointed out in Chauhan et al. (Reference Chauhan, Monkewitz and Nagib2009), motivates a further remark. One difficulty (among others) relating to the accurate evaluation of $\Pi$ is its dependence on the log-law constant $\kappa$ and offset $B$ , which cannot be assumed constant in the lower- $Re$ regime, as scale separation is not sufficient to cultivate a true, unambiguous logarithmic region. At low $Re$ , $\kappa ,B$ must be approximated either by assumption or through fitting the mean profile, neither of which is robust, which itself can lead to scatter among the data. Thus, to support the above discussions, a related metric, as discussed in e.g. Nagib et al. (Reference Nagib, Chauhan and Monkewitz2007) and Sanmiguel Vila et al. (Reference Sanmiguel Vila, Vinuesa, Discetti, Ianiro, Schlatter and Örlü2017), is depicted in figure 6 of Appendix B, showing the streamwise evolution of the Rotta–Clauser length scale $\varDelta =\int _0^\infty (\bar {u}_e^+ - \bar {u}^+) \textrm{d}y$ in relation to $\delta _{99}$ . According to classical theory, this wake metric is assumed to become approximately constant as a well-developed outer layer becomes self-similar (like $\Pi$ ), however, with the benefit of being independent of the pre-supposed existence of a logarithmic layer. Evaluation of $\varDelta /\delta _{99}$ in figure 6 shows the same qualitative trend as $\Pi$ in figure 3( $c$ ), including sudden saturation at a plateau value. Thus, the resemblance between both metrics strongly supports the discussion above. Additionally, the streamwise development of the wake has been evaluated in Appendix C within the framework of the physically consistent profile form suggested in Monkewitz, Chauhan & Nagib (Reference Monkewitz, Chauhan and Nagib2007). As with $\varDelta /\delta _{99}$ , the general observation regarding the onset of outer-layer velocity profile self-similarity is qualitatively consistent.

3.5. Logarithmic overlap region

The previous sections have demonstrated the high sensitivity of the mean-velocity profile to largely unavoidable uncertainties at the inlet boundary, which could potentially obscure fine features of the streamwise evolution of e.g. $c_f$ or the wake strength if sufficient relaxation length is not provided. Following these considerations, attention is turned to the region between the buffer layer and the wake region and how it behaves in conjunction with the wake saturation. For convenience, such will be referred to as the logarithmic overlap layer (‘log layer’), although clearly no significant region of precise logarithmic proportionality yet exists at this $Re$ . To this end, the velocity-defect profile $\bar {u}_{i,e}^+ - \bar {u}_i^+ \approx f(y/\delta )$ is illustrated in figure 4(a,c,e) for several $Re_{\theta _i}$ , where panel $(c)$ is simply a normalisation of panel ( $a$ ) by $f() = 3.752 - 0.41^{-1} \ln (y/\delta )$ for magnification purposes; the constant is fitted after constraining $\kappa = 0.41$ . For reference, panel $(e)$ focuses once again on the wake region to link the discussion of log layer with the aforementioned lifting process of the wake, where it is clear that, for $Re_{\theta _i} \gtrapprox 4000$ , the wake defect profile is highly self-similar with respect to $\delta _{{99}_i}$ . The indicator function $\Xi _i = y^+ (d \bar {u}_i^+ / d y^+)$ is shown in figure 4(b,d) along with incompressible reference functions from Monkewitz et al. (Reference Monkewitz, Chauhan and Nagib2007); panel ( $d$ ) is a zoom of panel ( $b$ ).

Figure 4. Mean velocity-defect profiles (a,c,d) and diagnostic function (b,d). For reference, the Padé approximant functions of Monkewitz et al. (Reference Monkewitz, Chauhan and Nagib2007) are plotted with their original parameter values.

Figure 5. Visualisation of the instantaneous flow. Panel ( $a$ ) is a side view of the full domain geometry. Panels (b,c) show the magnitude of the density gradient. Panels (d,e) show mean-removed shear stress $\tau _{uy}^\prime$ at the wall. Panels (f,g) show isosurfaces of $\lambda _2$ criterion coloured by mean-removed streamwise velocity $u^\prime$ . All plots are provided at two different $Re$ , namely upstream (left) and downstream (right) of the point of wake saturation.

Most prominently, both panels ( $a$ ) and ( $c$ ) illustrate the development of the defect profile log layer (blue shades) followed by a rather abrupt arrival upon a visually identical slope/offset for $Re_{\theta _i} \gtrapprox 4000$ (white and red shades). Such convergence becomes more visually apparent through examination of $\Xi _i$ in panels (b,d). Here, a solidification of a local minimum occurs around $y^+ \approx 70$ for the same $Re_{\theta _i}$ at which: (i) the bend in $c_f$ occurs, (ii) $\Pi$ becomes $Re$ -independent and (iii) the velocity-defect log-layer slope and offset become $Re$ -independent. Therefore, present data clearly support that the point of wake convergence at $Re_{\theta _i} \approx 4100$ also marks a state at which a unique characteristic feature in the development of the precursor to the log layer emerges, namely a local minimum in $\Xi _i$ at roughly $y^+\approx 70$ . Note that this conclusion is in line with the LES study by Eitel-Amor et al. (Reference Eitel-Amor, Örlü and Schlatter2014), where comparable trends have been shown. Further increase in $Re$ would see the $\Xi _i$ curve envelope continually approach a characteristic similar to the $y^+ (P_{23}+P_{25})$ curve, eventually flattening out at $\kappa \approx 0.384$ for $y^+ \gtrapprox 300$ in the high- $Re$ regime for incompressible flows. It is worth noting, however, that recent studies such as Hoyas et al. (Reference Hoyas, Vinuesa, Schmid and Nagib2024) and Monkewitz (Reference Monkewitz2024), focusing on channel and pipe flows, have indicated that the indicator function exhibits a slight sensitivity to grid spacing and distribution. This sensitivity, combined with the influence of time averaging, suggests that a more precise discussion of the scaling properties of the overlap layer may not be meaningful at this stage. Nevertheless, the overall trends are largely consistent with the trends suggested by the (incompressible) composite model of Monkewitz et al. (Reference Monkewitz, Chauhan and Nagib2007), with the implications being threefold: first, the transformed compressible and incompressible trends are highly comparable, again adding confidence to previous conclusions. Second, the bend in $c_f$ corresponds to a Reynolds number at which no clear log layer has yet formed, thus it cannot be attributed to the high- $Re$ regime. Third, lack of a log layer in the moderate- $Re$ regime is not attributable to insufficient wake development, although the absence of a $Re$ -independent minimum $\Xi$ might be an indicator.