2.1. Global definition of the convection velocity

As mentioned in the Introduction, since Taylor’s hypothesis is not rigorously valid in wall-bounded turbulence, the convection velocity $C$ does not necessarily coincide with the mean velocity and should be defined as the value of $C$ for which the following convection equation best describes the fluctuations:

(2.1) $$\begin{eqnarray}\frac{1}{C}\partial _{t}u+\partial _{x}u=0.\end{eqnarray}$$

For the optimal value of $C$ , the square of the residual $(1/C)\partial _{t}u+\partial _{x}u$ of the convection equation is not expected to rigorously vanish, but it is as close to zero as possible. The present global convection velocity $C_{u}$ , which accounts for all scales, is the value of $C$ that provides the minimum value of $\mathscr{D}(C)$ , where

(2.2) $$\begin{eqnarray}\mathscr{D}(C)=\frac{\displaystyle E\left(\left(\frac{1}{C}\partial _{t}u+\partial _{x}u\right)^{2}\right)}{E((\partial _{x}u)^{2})}.\end{eqnarray}$$

It should be noted that in their case, del Álamo & Jiménez (Reference del Álamo and Jiménez2009) minimized the dual quantity $(E((\partial _{t}u+C\partial _{x}u)^{2})/E((\partial _{t}u)^{2}))$ . The roles of the streamwise and temporal derivatives are swapped consistently with the derivation of a dual method suited to spatially developing flows. Crucially, the expectation of the squared residual of the convection equation is normalized by a quantity that does not depend on the convection velocity $C$ . Minimizing either $\mathscr{D}(C)$ or $E(((1/C)\partial _{t}u+\partial _{x}u)^{2})$ over $C$ is equivalent, and leads to the following condition for $C_{u}$ :

(2.3) $$\begin{eqnarray}\frac{\partial \mathscr{D}}{\partial C}(C_{u})=0\Rightarrow C_{u}=-\frac{E((\partial _{t}u)^{2})}{E(\partial _{t}u\,\partial _{x}u)}.\end{eqnarray}$$

This differs from the dual global convection velocity defined in equation (2.4) of del Álamo & Jiménez (Reference del Álamo and Jiménez2009) as $-(E(\partial _{t}u\partial _{x}u)/E((\partial _{x}u)^{2}))$ . A discussion on how a different choice of the quantity to be minimized leads to other definitions of the convection velocity may be found in appendix A. It should be noted that (2.3) involves only local signals at the streamwise location $x$ , a strong advantage in a spatially developing flow such as the boundary layer since one of the motivations for assessing the convection velocity is to recover a fictitious spatial spectrum from time signals at the chosen Reynolds number station. Besides the least-squares definition of the global convection velocity, an interpretation based on turbulence physics in a moving frame of reference is given in § 6. Moreover, it will be demonstrated in § 5.2 that the present convection velocity is equivalent to a correlation-based definition commonly used in experiments.

The smaller $\mathscr{D}(C_{u})$ , the higher the level of validity of Taylor’s hypothesis. We define a suitable correlation coefficient by

(2.4) $$\begin{eqnarray}1-{\it\gamma}_{u}^{2}=\mathscr{D}(C_{u}),\quad {\it\gamma}_{u}\geqslant 0,\quad \text{so that }{\it\gamma}_{u}=\frac{|E(\partial _{t}u\,\partial _{x}u)|}{\sqrt{E((\partial _{t}u)^{2})E((\partial _{x}u)^{2})}}.\end{eqnarray}$$

Here ${\it\gamma}_{u}$ varies between 1 (perfect convection process, vanishing $\mathscr{D}(C_{u})$ ) and 0. Remarkably it coincides with the dual case given by equation (2.5) of del Álamo & Jiménez (Reference del Álamo and Jiménez2009), unlike the convection velocity $C_{u}$ . The closer to frozen are the turbulent fluctuations, the more correlated are their time and space derivatives, because the residual of (2.1) is smaller.

The mathematical expectations involved in the convection velocity $C_{u}$ (2.3) and in the associated correlation coefficient ${\it\gamma}_{u}$ (2.4) are assessed using the PSD of the time signals, which may be estimated using Welch’s method because the signals are statistically stationary. Recall that the cross-power spectral density $S_{ab}(\,f)$ of signals $a(t)$ and $b(t)$ is defined by

(2.5) $$\begin{eqnarray}R_{ab}({\it\tau})=E(a(t)b(t+{\it\tau})),\end{eqnarray}$$

(2.6a,b ) $$\begin{eqnarray}S_{ab}(\,f)=\int _{-\infty }^{\infty }R_{ab}({\it\tau})\exp (-\text{i}2{\rm\pi}f{\it\tau})\,\text{d}{\it\tau},\quad R_{ab}({\it\tau})=\int _{-\infty }^{\infty }S_{ab}(\,f)\exp (\text{i}2{\rm\pi}f{\it\tau})\,\text{d}f.\end{eqnarray}$$

From (2.3), (2.5) and (2.6), the equation for the convection velocity becomes

(2.7) $$\begin{eqnarray}C_{u}=-\frac{\displaystyle \int _{-\infty }^{\infty }S_{\partial _{t}u\partial _{t}u}(\,f)\,\text{d}f}{\displaystyle \int _{-\infty }^{\infty }S_{\partial _{t}u\,\partial _{x}u}(\,f)\,\text{d}f}.\end{eqnarray}$$

There is no need to estimate the time derivative of the signal, thanks to the two following properties, for statistically stationary time signals $a(t)$ and $b(t)$ :

(2.8a,b ) $$\begin{eqnarray}S_{a\,\partial _{t}b}(\,f)=2\text{i}{\rm\pi}fS_{ab}(\,f)\quad \text{and}\quad S_{\partial _{t}a\;b}(\,f)=-2\text{i}{\rm\pi}fS_{ab}(\,f),\end{eqnarray}$$

where the assumption of statistical stationarity is necessary for the latter equation (see appendix B). Equation (2.7) then becomes

(2.9) $$\begin{eqnarray}C_{u}=-\frac{\displaystyle \int _{-\infty }^{\infty }(2{\rm\pi}f)^{2}S_{uu}(\,f)\,\text{d}f}{\displaystyle \int _{-\infty }^{\infty }-2\text{i}{\rm\pi}fS_{u\,\partial _{x}u}(\,f)\,\text{d}f}.\end{eqnarray}$$

Since $u(t)$ is stationary, $S_{uu}(\,f)$ is real-valued and symmetrical ( $S_{uu}(-f)=S_{uu}(\,f)$ ). Moreover, $S_{u\,\partial _{x}u}(-f)=S_{u\,\partial _{x}u}(\,f)^{\ast }$ (Hermitian symmetry), so (2.7) finally becomes

(2.10) $$\begin{eqnarray}C_{u}=-\frac{\displaystyle \int _{0}^{\infty }(2{\rm\pi}f)^{2}S_{uu}(\,f)\,\text{d}f}{\displaystyle \int _{0}^{\infty }2{\rm\pi}f\,\text{Im}(S_{u\,\partial _{x}u}(\,f))\,\text{d}f}.\end{eqnarray}$$

In the same way, (2.4) may be rewritten as

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\gamma}_{u}=\frac{\displaystyle \left|\int _{-\infty }^{\infty }-2\text{i}{\rm\pi}fS_{u\,\partial _{x}u}(\,f)\,\text{d}f\right|}{\displaystyle \sqrt{\int _{-\infty }^{\infty }(2{\rm\pi}f)^{2}S_{uu}(\,f)\,\text{d}f}\sqrt{\int _{-\infty }^{\infty }S_{\partial _{x}u\,\partial _{x}u}(\,f)\,\text{d}f}}, & \displaystyle\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\gamma}_{u}=\frac{\displaystyle \left|\int _{0}^{\infty }2{\rm\pi}f\,\text{Im}(S_{u\,\partial _{x}u}(\,f))\,\text{d}f\right|}{\displaystyle \sqrt{\int _{0}^{\infty }(2{\rm\pi}f)^{2}S_{uu}(\,f)\,\text{d}f}\sqrt{\int _{0}^{\infty }S_{\partial _{x}u\,\partial _{x}u}(\,f)\,\text{d}f}}. & \displaystyle\end{eqnarray}$$

$u(t)$

$\partial _{x}u(t)$

2.2. Frequency-dependent convection velocity

A frequency-dependent convection velocity may be identified with the phase velocity of a single Fourier mode, as illustrated in the Introduction (1.16). While this is done by del Álamo & Jiménez (Reference del Álamo and Jiménez2009), where discrete Fourier modes were readily available from the spectral code, transposition to the present dual case is not obvious. Indeed, the time signals from numerical simulations have a finite length, so that the PSD estimation represents the average contribution of a frequency band of finite width to the energy of the signal. On the contrary, the physical problem is defined over an infinite time length, so that there exists an infinity of time Fourier modes $(\,f)$ continuously distributed over the frequency range, each mode having zero probability of being encountered because of its zero frequency support width. One singular Fourier mode $(\,f)$ of finite amplitude, whose phase velocity is given by (1.16), consequently has zero average contribution to the signal energy because it has zero probability of existing. Moreover, averaging (1.16) is problematic, as emphasized in the Introduction. Because of the nature of the time signals available from numerical simulations, the present frequency-dependent convection velocity is derived by restricting the global convection velocity (2.10) to a frequency band in the limit of a vanishing width. The resulting formula depends on the power spectral density instead of the amplitude of a singular Fourier mode as in del Álamo & Jiménez (Reference del Álamo and Jiménez2009) and (1.16).

As mentioned in the Introduction, the velocity signals may be filtered before using (2.3) to obtain the convection velocity of the pass-filtered scales. However, filtering may result in aliasing and phase alteration. Instead of filtering, it is preferable to take advantage of the frequency decomposition already displayed by (2.10). The time Fourier transform and the streamwise derivation commute, $\widehat{\partial _{x}u}(\,f)=\partial _{x}\hat{u} (\,f)$ , because $x$ and $t$ are independent coordinates. Consequently, both signals $u(t)$ and $\partial _{x}u(t)$ involved in (2.10) are affected in the same way by a frequency filtering, and the upper and lower integrals in (2.10) may be restricted to the same narrow frequency band. The convection velocity $C_{u}(f_{0},{\it\epsilon})$ associated with the energy contained in a frequency band $[\,f_{0}-{\it\epsilon},f_{0}+{\it\epsilon}]$ becomes

(2.13) $$\begin{eqnarray}C_{u}(f_{0},{\it\epsilon})=-\frac{\displaystyle \int _{f_{0}-{\it\epsilon}}^{f_{0}+{\it\epsilon}}(2{\rm\pi}f)^{2}S_{uu}(\,f)\,\text{d}f}{\displaystyle \int _{f_{0}-{\it\epsilon}}^{f_{0}+{\it\epsilon}}2{\rm\pi}f\,\text{Im}(S_{u\,\partial _{x}u}(\,f))\,\text{d}f}.\end{eqnarray}$$

The formula for the frequency-dependent convection velocity $U_{c}(f_{0})=\lim _{{\it\epsilon}\rightarrow 0}C_{u}(f_{0},{\it\epsilon})$ is obtained for a vanishing bandwidth $2{\it\epsilon}\rightarrow 0$ . As soon as the PSDs are continuous, it reads

(2.14) $$\begin{eqnarray}U_{c}(\,f)=-\frac{2{\rm\pi}fS_{uu}(\,f)}{\text{Im}(S_{u\,\partial _{x}u}(\,f))}.\end{eqnarray}$$

The same derivation from (2.12) leads to the associated frequency-dependent correlation coefficient

(2.15) $$\begin{eqnarray}{\it\gamma}_{u}(\,f)=\frac{|\text{Im}(S_{u\,\partial _{x}u}(\,f))|}{\sqrt{S_{uu}(\,f)}\sqrt{S_{\partial _{x}u\,\partial _{x}u}(\,f)}}.\end{eqnarray}$$

The resulting frequency-dependent convection velocity (2.14) is very similar to the phase velocity of a single Fourier mode (1.16) $U_{c}(x;f)=-(2{\rm\pi}f\hat{u} (x,f)^{\ast }\hat{u} (x,f)/\text{Im}(\hat{u} (x,f)^{\ast }\partial _{x}\hat{u} (x,f)))$ , but it is suited to averaged power spectral densities. The issue of averaging (1.16) over several realizations, which is tackled by del Álamo & Jiménez (Reference del Álamo and Jiménez2009) by considering the ratio of the expectations $-(2{\rm\pi}fE(\hat{u} (x,f)^{\ast }\hat{u} (x,f))/\text{Im}(E(\hat{u} (x,f)^{\ast }\partial _{x}\hat{u} (x,f))))$ (not necessarily matching the expectation of the ratio), is naturally addressed here if the PSD is seen as a kind of averaging. In the simple case of a monochromatic perturbation at a constant convection velocity, (2.14) may be interpreted as the phase velocity. Otherwise, the interpretation relies on the least-squares optimization and infinitely narrow frequency-band restriction presented above. Regarding the correlation coefficient (2.15), no interpretation similar to the phase velocity seems possible for a single Fourier mode (none is given by del Álamo & Jiménez Reference del Álamo and Jiménez2009), so that its interpretation always relies on the least-squares derivation and frequency restriction.

The evaluation of (2.14) and (2.15) requires accurate computation of the local streamwise derivative of the time signal, $\partial _{x}u(t)$ (see the Introduction). Compared with the phase of the cross-spectrum method, which may be seen as a first-order approximation relying on only two streamwise stations, the streamwise derivatives in (2.14) and (2.15) may be assessed with high-order centred finite difference schemes using more than two streamwise stations, for which the evaluation of the numerical error is straightforward. Data storage must consequently be chosen in order to procure satisfying accuracy in the wavenumber range of interest specific to the available database, as illustrated in the next section.

3.1. Turbulent boundary layer dataset used for the illustration

The compressible Navier–Stokes equations are solved on multiblock-structured grids using the FLU3M code developed by ONERA. Implicit time integration is carried out by the second-order backward scheme of Gear. The convective fluxes are discretized by a modified low-dissipative $\text{AUSM}+(\text{P})$ scheme (Mary & Sagaut Reference Mary and Sagaut2002). The solver accuracy has been assessed for direct numerical simulation and large eddy simulation by comparison with experimental data including spectra in many studies involving various types of flows (see e.g. Larchevêque et al. Reference Larchevêque, Sagaut, Lê and Comte2004; Dandois, Garnier & Sagaut Reference Dandois, Garnier and Sagaut2007; Deck & Thorigny Reference Deck and Thorigny2007; Simon et al. Reference Simon, Deck, Guillen, Sagaut and Merlen2007; Weiss et al. Reference Weiss, Deck, Robinet and Sagaut2009; Gand et al. Reference Gand, Deck, Brunet and Sagaut2010; Weiss & Deck Reference Weiss and Deck2011). Wall-bounded turbulent flows have been addressed using the third mode of zonal detached eddy simulation (ZDES) as a wall-modelled large eddy simulation (WMLES) in Deck et al. (Reference Deck, Weiss, Pamiès and Garnier2011), Laraufie, Deck & Sagaut (Reference Laraufie, Deck and Sagaut2011), Laraufie, Deck & Sagaut (Reference Laraufie, Deck, Sagaut, Fu, Haase, Peng and Schwamborn2012), Deck & Laraufie (Reference Deck and Laraufie2013), Laraufie & Deck (Reference Laraufie and Deck2013) and in Deck et al. (Reference Deck, Renard, Laraufie and Sagaut2014a ). The use of this technique to perform WRLES is demonstrated in Deck et al. (Reference Deck, Renard, Laraufie and Weiss2014b ) with the simulation used for the present study. Details of the formulation of the ZDES technique may be found in Deck (Reference Deck2012). The turbulent inflow is provided by the synthetic eddy method adapted from Jarrin et al. (Reference Jarrin, Benhamadouche, Laurence and Prosser2006) by Pamiès et al. (Reference Pamiès, Weiss, Garnier, Deck and Sagaut2009) and extended to ZDES by Deck et al. (Reference Deck, Weiss, Pamiès and Garnier2011).

A spatially developing smooth flat plate zero-pressure-gradient turbulent boundary layer has been simulated over a very large range of Reynolds numbers, $3060\leqslant \mathit{Re}_{{\it\theta}}\leqslant 13\,650~(1070\leqslant \mathit{Re}_{{\it\tau}}\leqslant 3800)$ , in Deck et al. (Reference Deck, Renard, Laraufie and Weiss2014b ). The low free-stream Mach number $M_{\infty }=0.21$ and the no-slip adiabatic wall condition imply that direct comparisons with incompressible simulations and very low Mach number experiments are possible. The WRLES mesh resolution is ${\rm\Delta}x^{+}=50$ and ${\rm\Delta}z^{+}=12$ in the streamwise and spanwise directions respectively, so that the inner site dynamics is not modelled, even though it is under-resolved compared with DNS. The first mesh cell is located within one wall unit from the wall. The outer layer is finely discretized so that the subgrid scales carry a negligible turbulent kinetic energy there (Deck et al. Reference Deck, Renard, Laraufie and Weiss2014b ). 125 points are clustered in the boundary layer thickness ${\it\delta}_{13\,000}$ at $\mathit{Re}_{{\it\theta}}=13\,000$ . The computational domain is $342\,{\it\delta}_{0}$ long, $8.6\,{\it\delta}_{0}$ wide and $41\,{\it\delta}_{0}$ high, where ${\it\delta}_{0}$ is the initial boundary layer thickness, leading to a total number of grid points $N_{xyz}=806\times 10^{6}$ . The inner-scaled time step is very fine, around ${\rm\Delta}t^{+}=0.26$ , meeting the criterion by Choi & Moin (Reference Choi and Moin1994). Time signals are collected at the $\mathit{Re}_{{\it\theta}}=13\,000$ station over $1155\,{\it\delta}_{0}/U_{\infty }$ inertial times, where $U_{\infty }$ is the free-stream velocity. Nearly $2\times 10^{6}$ CPU hours have been spent on 936 Nehalem processors of the CINES superscalar computer Jade-SGI Altix Ice 8200. Three terabytes of data from selected volumes of the unsteady field have been stored, which is much less than would be needed to compute frequency–wavenumber spectra.

The numerical dataset has been thoroughly validated, including spectral data, in Deck et al. (Reference Deck, Renard, Laraufie and Weiss2014b ). Two visualizations relevant to the present study are presented in figures 1 and 2. This provides an insight not only into the shape of the largest resolved coherent motions near $\mathit{Re}_{{\it\theta}}=13\,000$ but also into physical space when interpreting spectral data.

Figure 1.

Numerical schlieren (magnitude of density gradient in a streamwise/wall-normal plane) near the $\mathit{Re}_{{\it\theta}}=13\,000$ station.

Figure 2.

$u^{+}=20$ isosurface coloured by the wall distance near the $\mathit{Re}_{{\it\theta}}=13\,000$ station.

Figure 1 presents a numerical schlieren in a streamwise/wall-normal plane near the $\mathit{Re}_{{\it\theta}}=13\,000$ station, revealing the presence of inclined structures through the whole boundary layer thickness, which suggest packets of hairpins, consistent with the experimental findings leading to the hairpin packet paradigm of Adrian et al. (Reference Adrian, Meinhart and Tomkins2000). Some of these structures are indicated by dashed lines. The length of these packets is typically $1\,{\it\delta}$ to $3\,{\it\delta}$ , scaling with the local boundary layer thickness. Figure 2 presents, on the other hand, a horizontal plane projection of the isosurface of the instantaneous streamwise velocity $u^{+}=20$ , coloured by the wall distance. This reveals the presence of very long, meandering areas of low velocity (i.e. the given velocity level is reached much higher in the boundary layer, so that it appears in green to red instead of blue). This kind of coherent motion, which is more than $5\,{\it\delta}$ long, has been described in more detail in Deck et al. (Reference Deck, Renard, Laraufie and Weiss2014b ), suggesting that these are resolved superstructures. It should be noted that the superstructures seen in figure 2 are longer than the packets of hairpins identified in figure 1. The scale separation and its Reynolds number dependence are characterized in the next section by spectral analysis. The separation should be large enough for the scale-dependent convection velocity analysis to be relevant.

3.2. A brief spectral characterization of the resolved streamwise velocity fluctuations

The time PSD defined in (2.5) and (2.6) is estimated by Welch’s method (Welch Reference Welch1967) and further averaged in the spanwise direction. When both investigated signals coincide, the one-sided PSD $G_{aa}(\,f)\;(=2S_{aa}(\,f)$ for $f>0)$ of the statistically stationary time signal $a(t)$ is preferred, leading to

(3.1) $$\begin{eqnarray}\int _{0}^{\infty }G_{aa}(\,f)\,\text{d}f=E(a^{2})~\left(=\int _{0}^{\infty }G_{aa}({\it\omega})\,\text{d}{\it\omega}\right).\end{eqnarray}$$

In the following, the variable between brackets will implicitly denote the support of the PSD. For instance, $G_{aa}(\,f)$ and $G_{aa}({\it\omega})$ differ by a factor $2{\rm\pi}$ .

Here the streamwise PSD of the streamwise velocity fluctuations $G_{uu}(k_{x})$ is presented, and pre-multiplied so that the area below its semi-logarithmic plot is proportional to the variance:

(3.2) $$\begin{eqnarray}E(u^{2})=\int _{0}^{\infty }G_{uu}(k_{x})\,\text{d}k_{x}=\int _{-\infty }^{\infty }k_{x}G_{uu}(k_{x})\,\text{d}\ln (k_{x}).\end{eqnarray}$$

The streamwise PSD (depending on the streamwise wavenumber $k_{x}$ ) is reconstructed from the estimated time PSD (depending on the frequency $f$ ) using Taylor’s hypothesis with the local Reynolds-averaged streamwise velocity as the convection velocity ( $U_{c}=\langle U\rangle (y)$ ). This choice enables comparison with experimental spectra which are deduced from time signals in the same manner. The one-sided pre-multiplied streamwise PSD of streamwise velocity $k_{x}G_{uu}(k_{x})/u_{{\it\tau}}^{2}$ non-dimensionalized by the friction velocity at the station $\mathit{Re}_{{\it\theta}}=13\,000~(\mathit{Re}_{{\it\tau}}=3600)$ is plotted in figure 3 versus the wall distance and the streamwise wavelength together with experimental data at similar Reynolds numbers. Along with the streamwise mean velocity and turbulence intensity profiles, this reveals overall good agreement of the present simulation with the experiments, the outer layer being especially well resolved, as shown in more detail in Deck et al. (Reference Deck, Renard, Laraufie and Weiss2014b ). The outer energy site at large wavelengths (of the order of $6\,{\it\delta}$ ) is well captured by the simulation. This is consistent with the visualization (figure 2) of the superstructures, to which this site corresponds (see Marusic, Mathis & Hutchins Reference Marusic, Mathis and Hutchins2010). Consistent with the WRLES modelling strategy, the inner layer is somewhat under-resolved.

Figure 3.

Streamwise velocity profiles at $\mathit{Re}_{{\it\theta}}=13\,000~(\mathit{Re}_{{\it\tau}}=3600)$ . (a) Mean velocity (solid line) compared with experimental data by DeGraaff & Eaton (Reference DeGraaff and Eaton2000) (circles), and $u_{rms}^{+}$ (dash-dotted line) compared with the model by Marusic, Uddin & Perry (Reference Marusic, Uddin and Perry1997) and Marusic & Kunkel (Reference Marusic and Kunkel2003) (squares). (b) One-sided pre-multiplied streamwise PSD of streamwise velocity $k_{x}G_{uu}(k_{x})/u_{{\it\tau}}^{2}$ (dispersion relation $U_{c}=\langle U\rangle (y)$ ), compared with experimental data from Mathis et al. (Reference Mathis, Hutchins and Marusic2009) (solid isolines, $\mathit{Re}_{{\it\tau}}=2800$ ) and from Marusic et al. (Reference Marusic, Mathis and Hutchins2010) (dashed isolines, $\mathit{Re}_{{\it\tau}}=3900$ ). Outer scaling for all data, inner scaling for the present simulation only.

The Reynolds number dependence of the spectrum is evaluated in figure 4. In the absence of comparison with experiments, the choice of the convection velocity assessment is free, and a correlation-based definition $U_{c}=U_{corr}(y)$ globally representing all scales of motion is retained to reconstruct the spatial spectra; this method, used in Deck et al. (Reference Deck, Renard, Laraufie and Sagaut2014a ), will be discussed again in § 5.1. Strikingly, the outer energy site corresponding to the superstructures at $\mathit{Re}_{{\it\theta}}=13\,000$ , associated with wavelengths ${\it\lambda}_{x}=6\,{\it\delta}$ and larger, is missing at $\mathit{Re}_{{\it\theta}}=5200$ , which is consistent with the increase in scale separation and in the strength of superstructures with increasing Reynolds number (see e.g. Marusic et al. Reference Marusic, Mathis and Hutchins2010). On the contrary, the inner-scaled inner energy site does not evolve much from one Reynolds number to the other, consistent with the universality of the inner dynamics. It may also be noted that the outer layer contains some energy near ${\it\lambda}_{x}\approx 2{-}3\,{\it\delta}$ at both $\mathit{Re}_{{\it\theta}}=5200$ and $\mathit{Re}_{{\it\theta}}=13\,000$ . This possibly corresponds to the packets of hairpins visualized in figure 1, but seems to be somewhat overestimated compared with experiments, and deserves further investigation which is beyond the scope of the present study.

Figure 4.

Reynolds number impact on the pre-multiplied power spectral density of the streamwise velocity $k_{x}G_{uu}(k_{x})/u_{{\it\tau}}^{2}$ (colours and isolines): (a) $\mathit{Re}_{{\it\theta}}=5200$ ; (b) $\mathit{Re}_{{\it\theta}}=13\,000$ . Dispersion relation given by $U_{c}=U_{corr}$ (correlation-based convection velocity).

It is clear from figure 4 that the convection of the superstructures may only be studied at the highest Reynolds number station. Consequently, the new spectral method is illustrated at the $\mathit{Re}_{{\it\theta}}=13\,000$ station for the rest of the present study.

3.3. Spectral assessment of the convection velocity in the boundary layer at $\mathit{Re}_{{\it\theta}}=13\,000$

As motivated in the previous section, (2.14) and (2.15) are evaluated at the $\mathit{Re}_{{\it\theta}}=13\,000$ station. The signal $u(t)$ is stored at five successive nodes of the mesh, and a fourth-order-accurate centred finite difference scheme is used:

(3.3) $$\begin{eqnarray}\partial _{x}u_{i}=\frac{-u_{i+2}+8u_{i+1}-8u_{i-1}+u_{i-2}}{12{\rm\Delta}x}+O({\rm\Delta}x^{4}).\end{eqnarray}$$

It should be noted that a second-order scheme would not be accurate enough in the wavenumber range of interest, given that the spacing ${\rm\Delta}x^{+}=50$ is imposed by the mesh resolution and cannot be reduced, as preliminary tests pointed out for ${\it\lambda}_{x}\leqslant {\it\delta}$ . The possibility to resort to more than two nodes to evaluate the derivative is seen as a significant advantage in terms of numerical error control compared with methods such as the phase method described in the Introduction. Indeed, the order of accuracy of the streamwise derivative should intuitively be at least as high as the accuracy of the spatial discretization used in the flow solver, since both rely on the same spacing ${\rm\Delta}x$ . Classical accuracy evaluations show that with the present scheme, the error in the convection velocity (2.14) is $0.00017\,\%$ at ${\it\lambda}_{x}={\it\delta}_{13\,000}$ , $1.6\,\%$ at ${\it\lambda}_{x}=0.1\,{\it\delta}_{13\,000}$ , and is kept below $2\,\%$ for all wavenumbers presented in the figures of the present study. Being able to assess the numerical error is an important asset, since experimental data are lacking for a direct comparison at the same Reynolds number in the same flow (see table 1).

The convection velocity and correlation coefficient, (2.14) and (2.15), are presented in figure 5 at $\mathit{Re}_{{\it\theta}}=13\,000$ using the dispersion relation $2{\rm\pi}f=-k_{x}\,U_{c}(\,f)$ . Consistent with theory (Lin Reference Lin1953), ${\it\gamma}_{u}$ suggests that the validity of Taylor’s hypothesis is all the better as smaller structures are considered further away from the wall. Indeed, the length scale of the structure and the mean shear (which increases nearer the wall) are sources of anisotropy, whereas Taylor’s hypothesis originates in homogeneous isotropic turbulence. However, even the footprint of the largest length scales does not feature a correlation coefficient as low as the reported minima of del Álamo & Jiménez (Reference del Álamo and Jiménez2009) (the correlation coefficient definitions coincide in the dual cases). In figure 6, ${\it\gamma}_{u}(y;f)$ is plotted instead of ${\it\gamma}_{u}(y;{\it\lambda}_{x})$ . The variations of ${\it\gamma}_{u}$ for constant $f$ with respect to $y$ suggest that the small values corresponding to the largest scales very near the wall do not result from low-frequency resolution limitations from the finite-length signals, because these limitations depend only on $f$ , not on $y$ . The small ${\it\gamma}_{u}$ of the largest scales seem to be related to the flow physics instead, in agreement with Lin (Reference Lin1953), more specifically to the intermittency of these scales. Indeed, Buxton et al. (Reference Buxton, de Kat and Ganapathisubramani2013) report in a plane shear layer that the tail of distribution of the probability density of the convection velocity is larger for the largest scales, i.e. the convection velocity fluctuates more towards unexpected values, which is attributed to intermittency.

Figure 5.

Convection velocity $U_{c}/U_{\infty }$ (coloured contours, a) and correlation coefficient ${\it\gamma}_{u}$ (coloured contours, b) at $\mathit{Re}_{{\it\theta}}=13\,000$ as a function of the wavelength and the wall distance. Solid lines are isolines of $k_{x}G_{uu}(k_{x})/u_{{\it\tau}}^{2}$ .

Figure 6.

Correlation coefficient ${\it\gamma}_{u}$ (coloured contours) at $\mathit{Re}_{{\it\theta}}=13\,000$ as a function of the frequency and the wall distance. Solid lines are isolines of $k_{x}G_{uu}(k_{x})/u_{{\it\tau}}^{2}$ .

The outer-scaled convection velocity is presented in figure 5. The largest scales (around ${\it\lambda}_{x}=10\,{\it\delta}$ , associated with superstructures) have a relatively uniform convection velocity across the whole boundary layer thickness. This uniformity is a necessary condition for the coherence of the motion, as recalled in the Introduction. Near-wall confinement implies that very large-scale motions must extend to higher distances above the wall than smaller scales, where the mean velocity is larger, so that the largest scales are convected faster than the smaller scales near the wall. In figure 5 the footprint of the largest scales is indeed faster (consistent with Hutchins et al. Reference Hutchins, Monty, Ganapathisubramani, Ng and Marusic2011) and their top is slower than the other structures.

The convection velocity is compared with the mean velocity in figure 7(a). Near the wall, it is significantly larger than the mean velocity at all length scales. It is slightly smaller than the mean in the outer part of the outer layer, in good qualitative agreement with Favre et al. (Reference Favre, Gaviglio and Dumas1967), del Álamo & Jiménez (Reference del Álamo and Jiménez2009), Chung & McKeon (Reference Chung and McKeon2010), Lee & Sung (Reference Lee and Sung2011) and Atkinson et al. (Reference Atkinson, Buchmann and Soria2015), among others. However, the lack of datasets matching the present Reynolds number and type of flow prevents a direct comparison. The three energy sites (inner, outer in the range of scales of the hairpin packets, and outer superstructure site) feature a common property: the convection velocity is higher than the mean velocity near the foot of the site, and lower than the mean near the top of the site. This is especially consistent with del Álamo & Jiménez (Reference del Álamo and Jiménez2009) assessing the convection velocity in channel flow as the mean velocity averaged over the height of the structure.

Figure 7.

Convection velocity divided by the mean streamwise velocity $U_{c}/\langle U\rangle$ (coloured contours, (a) and by the correlation-based convection velocity $U_{c}/U_{corr}$ (coloured contours, (b) at $\mathit{Re}_{{\it\theta}}=13\,000$ as a function of the wavelength and the wall distance. Black solid lines, isolines of $k_{x}G_{uu}(k_{x})/u_{{\it\tau}}^{2}$ ; white solid line, line of equal convection velocity and mean velocity $U_{c}=\langle U\rangle$ .

Near the wall, even the small structures are convected much faster than the mean velocity. It is possible that their core is located higher in the boundary layer, where the mean velocity is larger, if the wall distance considered is smaller than the wall-normal extent of the structure. Another possible explanation derives from the amplitude modulation of the small-scale near-wall activity by the largest scales, which also superimpose their large-scale contribution to the velocity fluctuations (see Mathis et al. Reference Mathis, Hutchins and Marusic2011). Most of the small-scale energy is generated in the presence of large-scale positive velocity fluctuations because of the associated higher near-wall shear. Since the convection velocity (2.14) is evaluated from PSDs, it relies on energy-carrying events, so that it may be close to the velocity in the presence of large-scale positive velocity fluctuations, which is larger than the mean.

Figure 7 also depicts (white solid line) the locus where $U_{c}(\,f)=\langle U\rangle$ . This is, to some extent, reminiscent of the so-called critical layer associated with hydrodynamic instabilities, which is the height where the phase speed of a linear perturbation mode is equal to the mean velocity, a singularity for inviscid parallel shear flows. However, the critical layer is no longer singular in viscous parallel flows, and the maximum amplitude of the perturbation mode is not necessarily in the vicinity of this layer (Charru Reference Charru2011). Moreover, the present flow is not exactly parallel.

Finally, figure 7(b) compares the frequency-dependent convection velocity with the global correlation-based convection velocity $U_{corr}(y)$ which does not depend on frequency and has already been used in figure 4. Globally speaking, both velocities are close to each other. Near the wall, the largest scales are convected much faster than the smaller scales. The lower convection velocity of the extremely large wavelengths very near the wall is not significant, because of the low energy content of the signal at the largest length scales which leads to poor signal-to-noise ratios. Figure 7 will be further discussed in § 5.

4.1. Dilatation factor

Reconstruction of the streamwise PSD $S_{uu}(k_{x})$ from the estimated time PSD $S_{uu}(\,f)$ and the frequency-dependent convection velocity $U_{c}(\,f)$ (2.14), where the frequency and the streamwise wavenumber are related by the dispersion relation $2{\rm\pi}f=-U_{c}(\,f)\,k_{x}$ , must preserve the signal energy:

(4.1) $$\begin{eqnarray}\int _{-\infty }^{\infty }S_{uu}(\,f)\,\text{d}f=E(u^{2})=\int _{-\infty }^{\infty }S_{uu}(k_{x})\,\text{d}k_{x}.\end{eqnarray}$$

Seeing the dispersion relation as a change of variables, the following condition must hold:

(4.2) $$\begin{eqnarray}S_{uu}\left(k_{x}=-\frac{2{\rm\pi}f}{U_{c}(\,f)}\right)=S_{uu}(\,f)\left|\frac{\text{d}f}{\text{d}k_{x}(\,f)}\right|,\end{eqnarray}$$

where $|\text{d}f/\text{d}k_{x}(\,f)|$ is the absolute value of the Jacobian of the frequency–wavenumber transformation resulting from the dispersion relation $2{\rm\pi}f=-U_{c}(f(k_{x}))\,k_{x}$ . Deriving the latter relation with respect to $k_{x}$ leads to

(4.3) $$\begin{eqnarray}\frac{\text{d}f}{\text{d}k_{x}(\,f)}=\frac{1}{2{\rm\pi}}\frac{-U_{c}(\,f)}{\displaystyle 1+k_{x}\frac{\text{d}U_{c}(\,f)}{\text{d}(2{\rm\pi}f)}}=\frac{1}{2{\rm\pi}}\frac{-U_{c}(\,f)}{\displaystyle 1-\frac{2{\rm\pi}f}{U_{c}(\,f)}\frac{\text{d}U_{c}(\,f)}{\text{d}(2{\rm\pi}f)}}\quad (U_{c}(\,f)\neq 0).\end{eqnarray}$$

$U_{c}(\,f)\neq 0$ is not a restrictive assumption, because the method is meant for time stationary spatially developing flows. Working in a frame of reference where the turbulent statistics are time stationary and where $U_{c}$ vanishes would contradict streamwise heterogeneity. Furthermore, reconstructing a streamwise fluctuation from its time signal is in fact impossible if it is convected at zero velocity. In the case of the boundary layer where the reference frame is the solid wall, all turbulent fluctuations are convected towards $x>0$ at $U_{c}(\,f)>0$ , but the general case is considered in the following.

From (4.3) and (4.2) and the dispersion relation, the pre-multiplied spectra are related as follows:

(4.4) $$\begin{eqnarray}k_{x}S_{uu}\left(k_{x}=-\frac{2{\rm\pi}f}{U_{c}(\,f)}\right)=fS_{uu}(\,f)\frac{-\text{sign}(U_{c}(\,f))}{\displaystyle \left|1-\frac{f}{U_{c}(\,f)}\frac{\text{d}U_{c}(\,f)}{\text{d}f}\right|}.\end{eqnarray}$$

For one-sided PSDs, only positive frequencies and wavenumbers are considered (e.g. $G_{uu}(\,f)$ represents both the contributions of $S_{uu}(\,f)$ and $S_{uu}(-f)=S_{uu}(\,f)$ ). The frequency–wavenumber relation becomes $2{\rm\pi}f=|U_{c}(\,f)|\,k_{x}$ , and the pre-multiplied one-sided PSDs are related by the dilatation factor

(4.5) $$\begin{eqnarray}k_{x}G_{uu}\left(k_{x}=\frac{2{\rm\pi}f}{|U_{c}(\,f)|}\right)=fG_{uu}(\,f)\underbrace{\frac{1}{\displaystyle \left|1-\frac{f}{U_{c}(\,f)}\frac{\text{d}U_{c}(\,f)}{\text{d}f}\right|}}_{dilatation~factor}\end{eqnarray}$$

where we recall that $\int _{0}^{\infty }G_{uu}(k_{x})\,\text{d}k_{x}=E(u^{2})$ . The dilatation factor is a local energy-preservation correction to the pre-multiplied PSD taking into account the frequency dependence of $U_{c}(\,f)$ . When $U_{c}(\,f)$ does not vary with $f$ , this factor reduces to 1 and time and streamwise pre-multiplied PSDs are equal. The factor coincides with the ratio of the group velocity to the phase velocity defined from the dispersion relation, as mentioned in del Álamo & Jiménez (Reference del Álamo and Jiménez2009). The group velocity is discussed in § 7.

The dilatation factor defined by (4.5) is shown in figure 8. It is smaller than 1 between the inner site and the footprint of the superstructures. Indeed, the foot of the superstructures is advected faster than the surrounding structures. Since a given wavenumber advected at a higher convection velocity has a higher frequency, the wavenumber gap between both sites is wider than the frequency gap, so the PSD must be reduced to preserve the integral over this area. As for the values near the very largest and smallest wavelengths, they are dominated by noise in these low-energy regions, which is not an issue since it is only a correction factor to the PSD, which is in any case small. The dilatation factor is used in the next section to reconstruct the streamwise spectrum.

Figure 8.

Dilatation factor (4.5) (coloured contours) at $\mathit{Re}_{{\it\theta}}=13\,000$ as a function of the wavelength and the wall distance. Solid lines are isolines of $k_{x}G_{uu}(k_{x})/u_{{\it\tau}}^{2}$ . The values at the largest wavelengths as well as at the smallest wavelengths should not be considered, because the energy in these regions is too low for a convenient signal-to-noise ratio to be achieved, leading to inaccuracies in the assessment of the dilatation factor.

4.2. Comparison of the reconstructed streamwise spectrum with classical strategies

The streamwise spectrum is reconstructed from the frequency spectrum using (4.5). The convection velocity $U_{c}(\,f)$ is involved in the frequency-to-wavenumber transformation and in the dilatation factor. This is compared in figure 9 with two classical reconstruction strategies differing by the convection velocity definition, which does not depend on frequency (the dilation factor is unity). The mean velocity $U_{c}=\langle U\rangle$ is used in the first strategy, frequently found in experiments. The second strategy uses the correlation-based global convection velocity $U_{c}=U_{corr}$ , already mentioned previously and further discussed in § 5. The most significant variation is near the wall, where the convection velocity is significantly larger than the mean velocity, so that the first strategy ( $U_{c}=\langle U\rangle$ ) is very different from the others ( $U_{c}=U_{corr}$ and $U_{c}(\,f)$ ). The differences between the latter strategies, although $U_{c}(\,f)$ is based on a global value equivalent to $U_{corr}$ , as is demonstrated in § 5.2, illustrate the advantage of taking into account frequency dependence. First, the wavelength of the footprint of the superstructures is underestimated if a uniform convection velocity is used, as pointed out by del Álamo & Jiménez (Reference del Álamo and Jiménez2009) and observed in figure 9. Indeed, the footprint is convected faster than the other fluctuations near the wall, requiring an accurate frequency-to-wavenumber transformation (whereas the dilatation factor has little impact). Only with the frequency-dependent assessment is the wavelength of the energy site associated with the superstructures and their footprint constant across the boundary layer. A second effect, even though smaller, is related to the dilatation factor being smaller than unity between the inner site and the footprint of superstructures (figure 8). This results in a better circularity of the inner site in figure 9 compared with the spurious elongation of its large-wavelength edge when no frequency dependence is considered.

Figure 9.

Pre-multiplied streamwise PSD $k_{x}G_{uu}(k_{x})/u_{{\it\tau}}^{2}$ estimated from the temporal spectrum at $\mathit{Re}_{{\it\theta}}=13\,000$ as a function of the wavelength and the wall distance: (a) $U_{c}=\langle U\rangle$ ; (b) $U_{c}=U_{corr}$ ; (c) $U_{c}=U_{c}(\,f)$ .

One may conclude that the main differences are located in the inner layer, where the correlation-based strategy strongly differs from the mean velocity. The improvements from the scale dependence are more subtle, so it is required only for a refined analysis of the spectrum, or to get physical insight into the turbulence dynamics through convection properties and the validity of Taylor’s hypothesis.

5.1. Global convection velocity and global correlation coefficient

The global convection velocity $C_{u}$ is expressed as a weighted harmonic average of its frequency-dependent counterpart $U_{c}(\,f)$ by inserting (2.14) into (2.10) and finally introducing one-sided power spectral densities (with factors 2 for $f>0$ cancelling out):

(5.1) $$\begin{eqnarray}C_{u}=\frac{\displaystyle \int _{0}^{\infty }f^{2}G_{uu}(\,f)\,\text{d}f}{\displaystyle \int _{0}^{\infty }\frac{1}{U_{c}(\,f)}f^{2}G_{uu}(\,f)\,\text{d}f},\end{eqnarray}$$

which is dual to equation (2.7) of del Álamo & Jiménez (Reference del Álamo and Jiménez2009).

From (2.12) and (2.14), one may rewrite the global correlation coefficient as

(5.2) $$\begin{eqnarray}{\it\gamma}_{u}=\frac{\displaystyle \left|\int _{0}^{\infty }\frac{1}{U_{c}(\,f)}(2{\rm\pi}f)^{2}G_{uu}(\,f)\,\text{d}f\right|}{\displaystyle \sqrt{\int _{0}^{\infty }(2{\rm\pi}f)^{2}G_{uu}(\,f)\,\text{d}f}\sqrt{\int _{0}^{\infty }G_{\partial _{x}u\,\partial _{x}u}(\,f)\,\text{d}f}}\end{eqnarray}$$

and, using (5.1),

(5.3) $$\begin{eqnarray}{\it\gamma}_{u}=\frac{1}{|C_{u}|}\sqrt{\frac{\displaystyle \int _{0}^{\infty }(2{\rm\pi}f)^{2}G_{uu}(\,f)\,\text{d}f}{\displaystyle \int _{0}^{\infty }G_{\partial _{x}u\,\partial _{x}u}(\,f)\,\text{d}f}},\end{eqnarray}$$

which is dual to equation (2.8) of del Álamo & Jiménez (Reference del Álamo and Jiménez2009).

Alternatively, ${\it\gamma}_{u}$ is written as a function of the frequency-dependent ${\it\gamma}_{u}(\,f)$ , inserting (2.15) into (2.12),

(5.4) $$\begin{eqnarray}{\it\gamma}_{u}=\frac{\displaystyle \left|\int _{0}^{\infty }\text{sign}(\text{Im}(S_{u\partial _{x}u}(\,f)))\,f\sqrt{G_{uu}(\,f)}\,\sqrt{G_{\partial _{x}u\partial _{x}u}(\,f)}\,{\it\gamma}_{u}(\,f)\,\text{d}f\right|}{\displaystyle \sqrt{\int _{0}^{\infty }f^{2}G_{uu}(\,f)\,\text{d}f}\sqrt{\int _{0}^{\infty }G_{\partial _{x}u\,\partial _{x}u}(\,f)\,\text{d}f}},\end{eqnarray}$$

which becomes

(5.5) $$\begin{eqnarray}{\it\gamma}_{u}=\frac{\displaystyle \left|\int _{0}^{\infty }\text{sign}(U_{c}(\,f))\,f\sqrt{G_{uu}(\,f)}\,\sqrt{G_{\partial _{x}u\partial _{x}u}(\,f)}\,{\it\gamma}_{u}(\,f)\,\text{d}f\right|}{\displaystyle \sqrt{\int _{0}^{\infty }f^{2}G_{uu}(\,f)\,\text{d}f}\sqrt{\int _{0}^{\infty }G_{\partial _{x}u\,\partial _{x}u}(\,f)\,\text{d}f}},\end{eqnarray}$$

noting in (2.14) that $\text{sign}(\text{Im}(S_{u\partial _{x}u}(\,f)))=-\text{sign}(fU_{c}(\,f))$ since $S_{uu}(\,f)\geqslant 0$ ( $u(t)$ being statistically stationary). As discussed in § 4.1, one may assume that $U_{c}(\,f)$ does not vanish. The PSDs have been assumed to be continuous functions of $f$ in § 2.2, so $U_{c}(\,f)$ is also continuous. Consequently, $U_{c}(\,f)$ has a constant sign, and (5.5) becomes

(5.6) $$\begin{eqnarray}{\it\gamma}_{u}=\frac{\displaystyle \int _{0}^{\infty }\sqrt{f^{2}G_{uu}(\,f)}\,\sqrt{G_{\partial _{x}u\partial _{x}u}(\,f)}\,{\it\gamma}_{u}(\,f)\,\text{d}f}{\displaystyle \sqrt{\int _{0}^{\infty }f^{2}G_{uu}(\,f)\,\text{d}f}\sqrt{\int _{0}^{\infty }G_{\partial _{x}u\,\partial _{x}u}(\,f)\,\text{d}f}}\quad (U_{c}(\,f)\neq 0\;\forall f).\end{eqnarray}$$

This expression is new and was not derived, even in the dual case, by del Álamo & Jiménez (Reference del Álamo and Jiménez2009); ${\it\gamma}_{u}$ may be seen as a weighted average of the correlation coefficient ${\it\gamma}_{u}(\,f)$ over $f$ , similar to (5.1) for $U_{c}(\,f)$ , but this average is biased since $\int _{0}^{\infty }\!\sqrt{f^{2}G_{uu}(\,f)}\,\sqrt{G_{\partial _{x}u\partial _{x}u}(\,f)}\,\text{d}f\neq \sqrt{\int _{0}^{\infty }f^{2}G_{uu}(\,f)\,\text{d}f}\sqrt{\int _{0}^{\infty }G_{\partial _{x}u\,\partial _{x}u}(\,f)\,\text{d}f}$ a priori (unless $U_{c}(\,f)$ is constant over all $f$ ).

The present global convection velocity $C_{u}$ (5.1) and the associated correlation coefficient (5.3) are shown in figure 10, together with the mean velocity and the correlation-based convection velocity $U_{corr}$ mentioned previously (and discussed in more detail in the next section). Consistent with the findings in the literature, the global convection velocity is close to the mean velocity, except near the wall where it is significantly larger and has a nearly constant value. In the outer layer, it is slightly smaller than the mean velocity. The global correlation coefficient has high values, especially in the outer layer, and decreases near the wall as predicted by Lin (Reference Lin1953).

Figure 10.

Outer-scaled streamwise velocities $U/U_{\infty }$ at $\mathit{Re}_{{\it\theta}}=13\,000$ : mean velocity $\langle U\rangle$ , correlation-based convection velocity $U_{corr}$ , global convection velocity $C_{u}$ , and partially integrated global convection velocity ( $C_{u}({\it\lambda}_{x}>4{\it\delta})$ and $C_{u}({\it\lambda}_{x}<4{\it\delta})$ ). The associated correlation coefficients ${\it\gamma}_{u}$ are plotted on the right-hand axis.

The convection of the superstructures is isolated from the smaller structures in figure 10 considering only ${\it\lambda}_{x}/{\it\delta}\geqslant 4$ by partially integrating (5.1) and (5.3). The frequency range is determined from the wavenumber range using the dispersion relation $2{\rm\pi}f=-U_{c}(\,f)\,k_{x}$ and assuming that $k_{x}(\,f)$ is a monotonous function, which is the case because the variations of $U_{c}(\,f)$ are slow enough. The largest length scales ( ${\it\lambda}_{x}/{\it\delta}\geqslant 4$ ) are convected faster than the other scales near the wall, but the associated correlation coefficient is relatively low (consistent with Lin Reference Lin1953). There is very little difference between the convection velocity and correlation coefficient accounting for either all scales or those scales such that ${\it\lambda}_{x}/{\it\delta}\leqslant 4$ , suggesting that the contribution of the superstructures is not taken much into account through the weighted average process described by (5.1) and (5.6). This will be easily explained via the physical interpretation of $C_{u}$ given in § 6.2, revealing that the dissipative scales dominate its value, and incidentally emphasizing that the global profiles of $C_{u}$ and ${\it\gamma}_{u}$ (all scales) presented in figure 10 may depend on the resolution of the numerical simulation. In contrast, the values describing the largest scales only ( ${\it\lambda}_{x}/{\it\delta}>4$ ) do not directly depend on the resolution.

Furthermore, $C_{u}$ and $U_{corr}$ are very close to each other. A theoretical equivalence between both definitions in the limit of vanishing correlation separation for $U_{corr}$ is demonstrated in the next section. This suggests that the evaluation of $C_{u}$ , which could be altered by numerical errors from PSD estimations and discrete integration, is reliable because it coincides with $U_{corr}$ in agreement with theory (the correlation-based method features fewer sources of numerical errors). In turn, this validates the numerical assessment of $U_{c}(\,f)$ (because it determines $C_{u}$ ), which cannot be done experimentally because no dataset is available at the same Reynolds number in the same flow. The theoretical equality between $U_{corr}$ and $C_{u}$ , which is demonstrated in the next section, suggests the interpretation of figure 7(b) as depicting the variations of $U_{c}(\,f)$ around its weighted frequency harmonic average $C_{u}$ .

5.2. Equivalence with the correlation-based convection velocity

The correlation-based convection velocity $U_{corr}$ mentioned in previous sections results from a linear fit to the relation between ${\rm\Delta}x$ and ${\it\tau}$ obtained by maximizing the correlation coefficient $(E(u(x,t)u(x+{\rm\Delta}x,t+{\it\tau}))/\sqrt{E(u(x,t)^{2})E(u(x+{\rm\Delta}x,t+{\it\tau})^{2})})$ over the time delay ${\it\tau}$ for five very small streamwise separations ${\rm\Delta}x$ (matching the mesh spacing) from the time signals stored at five successive mesh nodes. Wills (Reference Wills1964) recommends resorting to the correlation coefficient rather than to the correlation $E(u(x,t)u(x+{\rm\Delta}x,t+{\it\tau}))$ for streamwise heterogeneous flows. The smallest possible value ${\rm\Delta}x$ is used so that even the smallest shortest-lived structures are quasi-frozen over ${\it\tau}$ and $U_{corr}$ accounts for the convection of all scales. A dual method maximizing the coefficient over ${\rm\Delta}x$ for a given ${\it\tau}$ would not have been possible because the numerical constraints of the simulation lead to very small time step values compared with the streamwise mesh spacing and the convection velocity.

Figure 10 shows that $U_{corr}$ and the present global convection velocity $C_{u}$ (2.3) are very close to each other. This is explained here by demonstrating that minimizing $\mathscr{D}(C)=E(((1/C)\partial _{t}u+\partial _{x}u)^{2})/E((\partial _{x}u)^{2})$ over $C$ , which defines $C_{u}$ , is equivalent to maximizing the correlation coefficient $E(u(x,t)u(x+{\rm\Delta}x,t+{\it\tau}))/\sqrt{E(u(x,t)^{2})E(u(x+{\rm\Delta}x,t+{\it\tau})^{2})}$ over ${\it\tau}$ for a vanishing value of ${\rm\Delta}x$ , which defines $U_{corr}$ because the mesh spacing is very small. Indeed, the latter method is equivalent to maximizing $E(u(x,t)u(x+{\rm\Delta}x,t+{\rm\Delta}x/C))/\sqrt{E(u(x,t)^{2})E(u(x+{\rm\Delta}x,t+({\rm\Delta}x/C))^{2})}$ over the convection velocity $C$ for a given vanishing ${\rm\Delta}x$ (the time delay ${\it\tau}$ is ${\it\tau}={\rm\Delta}x/C$ ). To second order in ${\rm\Delta}x$ , one may write

(5.7) $$\begin{eqnarray}\displaystyle u\left(x+{\rm\Delta}x,t+\frac{{\rm\Delta}x}{C}\right) & \mathop{=}_{{\rm\Delta}x\rightarrow 0} & \displaystyle u(x,t)+\left(\partial _{x}u+\frac{1}{C}\partial _{t}u\right){\rm\Delta}x\nonumber\\ \displaystyle & & \displaystyle +\left(\partial _{xx}u+\frac{2}{C}\partial _{xt}u+\frac{1}{C^{2}}\partial _{tt}u\right)\frac{{\rm\Delta}x^{2}}{2}+o({\rm\Delta}x^{2}),\end{eqnarray}$$

so it may be shown that the correlation coefficient becomes

(5.8) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\displaystyle E\left(u(x,t)u\left(x+{\rm\Delta}x,t+\frac{{\rm\Delta}x}{C}\right)\right)}{\displaystyle \sqrt{E(u(x,t)^{2})E\left(u\left(x+{\rm\Delta}x,t+\frac{{\rm\Delta}x}{C}\right)^{2}\right)}}\nonumber\\ \displaystyle & & \displaystyle \quad \mathop{=}_{{\rm\Delta}x\rightarrow 0}1+\left(\frac{\displaystyle \left(E\left(u\left(\partial _{x}u+\frac{1}{C}\partial _{t}u\right)\right)\right)^{2}}{2(E(u^{2}))^{2}}-\frac{\displaystyle E\left(\left(\partial _{x}u+\frac{1}{C}\partial _{t}u\right)^{2}\right)}{2E(u^{2})}\right)(x,t)\nonumber\\ \displaystyle & & \displaystyle \qquad \quad \times \,{\rm\Delta}x^{2}+o({\rm\Delta}x^{2}).\end{eqnarray}$$

Assuming that the flow is statistically stationary, and since the mathematical expectation $E(\cdot )$ commutes with the time and space derivation (one of Reynolds’ axioms), we conclude that $E(u\,(1/C)\partial _{t}u)=(1/C)\partial _{t}E(u^{2}/2)=0$ , and that $E(u\partial _{x}u)=\partial _{x}E(u^{2}/2)$ . This leads to the final result, valid for a flow that is statistically stationary in time:

(5.9) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\displaystyle E\left(u(x,t)u\left(x+{\rm\Delta}x,t+\frac{{\rm\Delta}x}{C}\right)\right)}{\displaystyle \sqrt{E(u(x,t)^{2})E\left(u\left(x+{\rm\Delta}x,t+\frac{{\rm\Delta}x}{C}\right)^{2}\right)}}\nonumber\\ \displaystyle & & \displaystyle \quad \mathop{=}_{{\rm\Delta}x\rightarrow 0}1+\left(\left(\frac{\partial _{x}E(u^{2})}{2E(u^{2})}\right)^{2}-\frac{\displaystyle E\left(\left(\partial _{x}u+\frac{1}{C}\partial _{t}u\right)^{2}\right)}{E(u^{2})}\right)(x)\times \frac{{\rm\Delta}x^{2}}{2}+o({\rm\Delta}x^{2}),\qquad\end{eqnarray}$$

from which it is concluded that maximizing the correlation coefficient $(E(u(x,t)u(x+{\rm\Delta}x,t+{\rm\Delta}x/C)))/\sqrt{E(u(x,t)^{2})E(u(x+{\rm\Delta}x,t+{\rm\Delta}x/C)^{2})}$ over $C$ for a given very small value of ${\rm\Delta}x$ , in the limit of vanishing ${\rm\Delta}x$ , is equivalent to the minimization of $E((\partial _{x}u+(1/C)\partial _{t}u)^{2})$ over $C$ , which is itself equivalent to the minimization of $\mathscr{D}(C)=E(((1/C)\partial _{t}u+\partial _{x}u)^{2})/E((\partial _{x}u)^{2})$ over $C$ . This shows that for a vanishing ${\rm\Delta}x$ , $U_{corr}$ and $C_{u}$ are equivalent, consistent with the observations in figure 10. The coexistence of the correlation-based and least-squares-minimized-residual definitions leads to two different physical interpretations given in the next section.

6.1. Initial velocity of a turbulence-freezing frame of reference

The convection velocity $U_{corr}$ described previously is related to a broad category of velocities defined by maximizing the correlation $R_{uu}({\rm\Delta}x,{\it\tau})=E(u(x,t)u(x+{\rm\Delta}x,t+{\it\tau}))$ . While $U_{corr}$ relies on a maximization over ${\it\tau}$ for a given ${\rm\Delta}x$ , the most intuitive physical interpretation of these methods is obtained in the dual case of a maximization over ${\rm\Delta}x$ for a given ${\it\tau}$ , which is treated first.

Defining the value ${\rm\Delta}x_{max}({\it\tau})$ of ${\rm\Delta}x$ that maximizes $R_{uu}({\rm\Delta}x,{\it\tau})$ for a given ${\it\tau}$ , one notices that $R_{uu}({\rm\Delta}x_{max}({\it\tau}),{\it\tau})$ coincides with the time correlation in a frame of reference that is moving such that its origin at time ${\it\tau}$ has the streamwise coordinate ${\rm\Delta}x_{max}({\it\tau})$ . Fisher & Davies (Reference Fisher and Davies1963) and Wills (Reference Wills1964) point out that in this frame of reference, the correlation decays as slowly as possible in time, by definition of ${\rm\Delta}x_{max}({\it\tau})$ . The classical convection velocity defined as the ratio ${\rm\Delta}x_{max}({\it\tau})/{\it\tau}$ and which depends on ${\it\tau}$ may be seen as the mean velocity of this frame of reference during the time ${\it\tau}$ . The frame is not necessarily inertial, because its instantaneous velocity $\text{d}({\rm\Delta}x_{max}({\it\tau}))/\text{d}{\it\tau}$ may vary in time. Favre et al. (Reference Favre, Gaviglio and Dumas1967) consider the value of $\text{d}({\rm\Delta}x_{max}({\it\tau}))/\text{d}{\it\tau}$ at a given time ${\it\tau}$ as another possible definition of the convection velocity (but this is done in a mathematical way, with no reference to the instantaneous velocity of a moving frame). Furthermore, Favre et al. (Reference Favre, Gaviglio and Dumas1967) show that if the flow is homogeneous along $x$ , both mean and instantaneous velocities coincide when ${\it\tau}$ vanishes, and are given by

(6.1) $$\begin{eqnarray}\lim _{{\it\tau}\rightarrow 0}\frac{{\rm\Delta}x_{max}({\it\tau})}{{\it\tau}}({\it\tau})=\lim _{{\it\tau}\rightarrow 0}\frac{\text{d}({\rm\Delta}x_{max}({\it\tau}))}{\text{d}{\it\tau}}({\it\tau})=-\frac{E(\partial _{t}u\partial _{x}u)}{E(\partial _{x}u\partial _{x}u)}.\end{eqnarray}$$

The latter expression from Favre et al. (Reference Favre, Gaviglio and Dumas1967) coincides with equation (2.4) of del Álamo & Jiménez (Reference del Álamo and Jiménez2009), even though the derivations are different. The global convection velocity of del Álamo & Jiménez (Reference del Álamo and Jiménez2009) may be interpreted, based on Favre et al. (Reference Favre, Gaviglio and Dumas1967), as the initial velocity of a moving frame in which the correlation decays as slowly as possible in time (turbulence is frozen), assuming streamwise homogeneity.

As mentioned in § 5.2, the present dataset requires dual definitions, all the more as the flow is spatially developing. With the value ${\it\tau}_{max}({\rm\Delta}x)$ of ${\it\tau}$ that maximizes $R_{uu}({\rm\Delta}x,{\it\tau})$ for a given ${\rm\Delta}x$ , $R_{uu}({\rm\Delta}x,{\it\tau}_{max}({\rm\Delta}x))$ coincides with the time correlation in a frame of reference that is moving such that its origin reaches the streamwise position ${\rm\Delta}x$ at time ${\it\tau}_{max}({\rm\Delta}x)$ . By definition of ${\it\tau}_{max}({\rm\Delta}x)$ , this ensures the slowest possible spatial decay of the correlation. The classically defined convection velocity ${\rm\Delta}x/{\it\tau}_{max}({\rm\Delta}x)$ is seen as the (time) average velocity of this frame of reference over ${\rm\Delta}x$ (on which it depends). Similarly to the dual case, the instantaneous velocity of the frame $\text{d}({\rm\Delta}x)/\text{d}({\it\tau}_{max}({\rm\Delta}x))$ depends on ${\rm\Delta}x$ and may be used as a convection velocity (as did Favre et al. Reference Favre, Gaviglio and Dumas1967 mathematically). Favre et al. (Reference Favre, Gaviglio and Dumas1967) show that as soon as the flow is statistically stationary (in time, since no spatial homogeneity is required for their derivation), both mean and instantaneous velocities coincide when ${\rm\Delta}x$ vanishes, and are given by

(6.2) $$\begin{eqnarray}\lim _{{\rm\Delta}x\rightarrow 0}\frac{{\rm\Delta}x}{{\it\tau}_{max}({\rm\Delta}x)}({\rm\Delta}x)=\lim _{{\rm\Delta}x\rightarrow 0}\frac{\text{d}({\rm\Delta}x)}{\text{d}({\it\tau}_{max}({\rm\Delta}x))}({\rm\Delta}x)=-\frac{E(\partial _{t}u\partial _{t}u)}{E(\partial _{t}u\partial _{x}u)}(x).\end{eqnarray}$$

The latter expression from Favre et al. (Reference Favre, Gaviglio and Dumas1967) coincides with the global convection velocity $C_{u}$ defined in (2.3). This relation between the velocity $C_{u}$ (2.3) minimizing the squared residual expectation and the limit of the convection velocity maximizing the correlation for a given ${\rm\Delta}x$ is consistent with the previous direct demonstration that $C_{u}$ also coincides with $U_{corr}$ defined by maximizing the correlation coefficient (instead of the correlation) for a vanishing ${\rm\Delta}x$ . $C_{u}$ and $U_{corr}$ for a vanishing ${\rm\Delta}x$ are interpreted as the initial velocity of the moving frame of reference that provides the slowest spatial decay of the correlation, i.e. turbulence is frozen over the longest possible distance, with no other assumption than statistical time stationarity.

6.2. Turbulence micro scale maximization in a convected frame of reference

The present global convection velocity $C_{u}$ (2.3) minimizes the expectation of the squared residual of the convection equation $E(((1/C)\partial _{t}u+\partial _{x}u)^{2})$ . This mathematical definition coincides with the phase velocity in the non-fluctuating convection of a single Fourier mode (see the Introduction, (1.16)). A physical interpretation for a broadband turbulence signal with fluctuating convection velocity, which is lacking in the literature known to the authors, is proposed here.

The interpretation is set in a convected inertial frame of reference with streamwise and time coordinates $(x_{c}=x-Ct,t)$ , moving at a constant velocity $C$ along $x$ with respect to the usual frame of reference with coordinates $(x,t)$ where the wall is still. The expectation $E$ is seen as a spanwise and ensemble average so that it is a Reynolds average $\langle \cdot \rangle$ in both frames, commuting with derivatives. The streamwise velocity $U$ in the usual frame becomes $U-C$ in the convected frame. The velocity fluctuation $u=U-\langle U\rangle$ and its variance $E(u^{2})$ are the same in both frames because $C$ is constant. The partial time derivative in the convected frame $(\partial u/\partial t)_{x_{c}}$ is a function of the partial time and streamwise derivatives in the usual frame,

(6.3) $$\begin{eqnarray}\left(\frac{\partial u}{\partial t}\right)_{x_{c}}=\left(\frac{\partial u}{\partial t}\right)_{x}\left(\frac{\partial t}{\partial t}\right)_{x_{c}}+\left(\frac{\partial u}{\partial x}\right)_{t}\left(\frac{\partial x}{\partial t}\right)_{x_{c}}=\left(\frac{\partial u}{\partial t}\right)_{x}+\left(\frac{\partial u}{\partial x}\right)_{t}C\end{eqnarray}$$

so that the expectation of the squared residual becomes

(6.4) $$\begin{eqnarray}E\left(\left(\frac{1}{C}\partial _{t}u+\partial _{x}u\right)^{2}\right)=\frac{1}{C^{2}}E\left(\left(\frac{\partial u}{\partial t}\right)_{x_{c}}^{2}\right),\end{eqnarray}$$

which will now be related to the temporal auto-correlation of $u$ in the convected frame, defined as

(6.5) $$\begin{eqnarray}\displaystyle R_{uu}^{c}(x_{c},t;{\it\tau}) & = & \displaystyle E(u(x_{c},t)u(x_{c},t+{\it\tau}))\nonumber\\ \displaystyle & = & \displaystyle E(u(x=x_{c}+Ct,t)u(x+{\rm\Delta}x=x_{c}+C(t+{\it\tau}),t+{\it\tau})).\end{eqnarray}$$

Since ${\it\tau}$ is independent from the space and time variables, one obtains

(6.6) $$\begin{eqnarray}\displaystyle \frac{\text{d}^{2}R_{uu}^{c}(x_{c},t;{\it\tau})}{\text{d}{\it\tau}^{2}} & = & \displaystyle E(u(x_{c},t)(\partial _{tt}u(x_{c},t+{\it\tau}))_{x_{c}})\nonumber\\ \displaystyle & \Rightarrow & \displaystyle \frac{\text{d}^{2}R_{uu}^{c}(x_{c},t;0)}{\text{d}{\it\tau}^{2}}=E(u(x_{c},t)(\partial _{tt}u(x_{c},t))_{x_{c}}),\end{eqnarray}$$

(6.7) $$\begin{eqnarray}\displaystyle \frac{\text{d}^{2}R_{uu}^{c}(x_{c},t;0)}{\text{d}{\it\tau}^{2}} & = & \displaystyle (\partial _{tt}E(u(x_{c},t)^{2}/2))_{x_{c}}-E((\partial _{t}u(x_{c},t))_{x_{c}}^{2}).\end{eqnarray}$$

$(\partial _{tt}E(u(x_{c},t)^{2}/2))_{x_{c}}=C^{2}\,(\partial _{xx}E(u(x_{c},t)^{2}/2))_{t}$

(6.8) $$\begin{eqnarray}E\left(\left(\frac{1}{C}\partial _{t}u+\partial _{x}u\right)^{2}\right)=(\partial _{xx}E(u^{2}/2))_{t}-\frac{1}{C^{2}}\frac{\text{d}^{2}R_{uu}^{c}(x_{c},t;0)}{\text{d}{\it\tau}^{2}}.\end{eqnarray}$$

The global convection velocity is defined by the minimization over $C$ of $\mathscr{D}(C)=(E(((1/C)\partial _{t}u+\partial _{x}u)^{2}))/E((\partial _{x}u)^{2})$ , or equivalently of $\mathscr{E}(C)=(E(((1/C)\partial _{t}u+\partial _{x}u)^{2}))/2E(u^{2})$ , rewritten from (6.8):

(6.9) $$\begin{eqnarray}\mathscr{E}(C)=\frac{(\partial _{xx}E(u^{2}/2))_{t}}{2E(u^{2})}-\frac{\displaystyle \frac{\text{d}^{2}R_{uu}^{c}(x_{c},t;0)}{\text{d}{\it\tau}^{2}}}{2C^{2}E(u^{2})}=\underbrace{\frac{(\partial _{xx}E(u^{2}/2))_{t}}{2E(u^{2})}}_{independent\,of\,C}+\underbrace{\frac{\displaystyle -\frac{\text{d}^{2}R_{uu}^{c}}{\text{d}{\it\tau}^{2}}(x_{c},t;0)}{2C^{2}R_{uu}^{c}(x_{c},t;0)}}_{\mathscr{F}(C)}.\end{eqnarray}$$

Minimizing $\mathscr{D}(C)$ is equivalent to minimizing $\mathscr{F}(C)=1/(C^{2}{\it\tau}_{c}^{2})$ over $C$ , where the Eulerian micro time scale in the convected frame of reference ${\it\tau}_{c}$ is defined by (see Hinze Reference Hinze1959)

(6.10) $$\begin{eqnarray}\frac{\displaystyle -\frac{\text{d}^{2}R_{uu}^{c}}{\text{d}{\it\tau}^{2}}(x_{c},t;0)}{2R_{uu}^{c}(x_{c},t;0)}=\frac{1}{{\it\tau}_{c}^{2}}.\end{eqnarray}$$

Note that in the particular case when streamwise homogeneity is assumed in addition to statistical time stationarity, the flow is also time stationary in the convected frame (and homogeneous along $x_{c}$ ). In this case, ${\it\tau}_{c}$ is the time delay ${\it\tau}$ when the osculating parabola to $R_{uu}^{c}(x_{c},t;{\it\tau})$ at ${\it\tau}=0$ reaches a zero correlation.

Coming back to the general case, the global convection velocity $C_{u}$ (2.3) is interpreted as the value of $C$ maximizing the length scale ${\it\lambda}_{c}=|C|{\it\tau}_{c}$ , corresponding to the distance travelled by the convected frame during a time equal to the Eulerian micro time scale of turbulence ${\it\tau}_{c}$ in this convected frame, i.e. the distance over which the coherence of turbulent fluctuations in the micro scale sense is preserved by following their convection. This interpretation applies to the equivalent correlation-based velocity $U_{corr}$ (see § 5.2), which is frequently used experimentally with a vanishing streamwise separation between two probes.

In the dual case of del Álamo & Jiménez (Reference del Álamo and Jiménez2009), $(E((\partial _{t}u+C\partial _{x}u)^{2})/E((\partial _{t}u)^{2}))=C^{2}\mathscr{D}(C)(E((\partial _{x}u)^{2})/E((\partial _{t}u)^{2}))$ is minimized instead of $\mathscr{D}(C)=(E(((1/C)\partial _{t}u+\partial _{x}u)^{2})/E((\partial _{x}u)^{2}))$ . Equivalently $C^{2}\mathscr{E}(C)$ is minimized over $C$ . From (6.9) multiplied by $C^{2}$ , one may conclude that a physical interpretation is also possible, but only assuming streamwise homogeneity of the flow (consistent with the requirements of the dual method, and similar to the time stationarity required for the present method). With this assumption, the method of del Álamo & Jiménez (Reference del Álamo and Jiménez2009) minimizes $C^{2}\mathscr{F}(C)$ , i.e. maximizes the micro time scale ${\it\tau}_{c}$ in the convected frame, which is more intuitive than maximizing ${\it\lambda}_{c}$ in the present method but requires streamwise homogeneity.

The maximization of the micro scale is reminiscent of another method to assess the convection velocity maximizing the integral scale, in a convected inertial frame of reference, using correlations computed for all possible values of both streamwise and time separations (see Fisher & Davies Reference Fisher and Davies1963; Wills Reference Wills1964). This is not suited to the present case especially because spatial growth of the boundary layer requires a local (in $x$ ) formulation. Putting the emphasis on the large scales through the integral scale may also be irrelevant considering the validity of Taylor’s hypothesis for the largest near-wall scales (Lin Reference Lin1953), whereas $C_{u}$ , in contrast, is dominated by the small dissipative scales related to the micro scale. As demonstrated in § 5.1, the convection of the largest scales may still be isolated since the method is spectral.

The relation between the present global convection velocity $C_{u}$ and the micro scale is emphasized by the weight $f^{2}G_{uu}(\,f)$ in (5.1) for the weighted harmonic average giving $C_{u}$ from $U_{c}(\,f)$ , confirming that $C_{u}$ and $U_{corr}$ are dominated by the dissipative scales and explaining the results of figure 10 for the smaller-scale-restricted convection velocity. This, however, implies that $C_{u}$ and $U_{corr}$ (massively used in experiments) are dominated by the smallest resolved scales, which are not necessarily the dissipative scales if an under-resolved simulation or experiment is considered, possibly explaining some discrepancies between datasets of different resolution (see e.g. a comparison in LeHew et al. Reference LeHew, Guala and McKeon2011) if the convection velocity of the dissipative scales and of larger scales is different. This suggests that a safe way of comparing convection velocities from different datasets, other than DNS or very well-resolved experiments, is to resort to the partial integration of $C_{u}$ (5.1) limited to frequencies suitably resolved by all datasets. This emphasizes the advantage of the spectral convection velocity compared with the correlation-based definition, all the more as their global values coincide (§ 5.2).