4.1. Taylor's frozen hypothesis with local mean velocity $U(y)$ (TH-$\boldsymbol{U}$)

$\textbf{TH}\textrm{-}{\boldsymbol{U}}$ is verified here by comparing the streamwise spectra ${E_{pp}}({k_x},y)$ (equation (3.6)) to the frozen $\omega $-spectra $E_{pp}^F(k_x^F,y)$ (equation (3.8)) with the local mean velocity $U(y)$ applied for ${U_T}$. Even though the local mean velocity is quite small close to the wall, we apply it in computing the frozen $\omega $-spectra in (3.8). This serves to determine the wall-normal locations where the local mean velocity is not appropriate for ${U_T}$. The contour plots of the premultiplied 1-D streamwise spectra $k_x^ + E_{pp}^ += {k_x}{E_{pp}}/({\rho ^2}u_\tau ^4)$ and frozen $\omega $-spectra $k_x^{F + }E_{pp}^{F + } = k_x^FE_{pp}^F/({\rho ^2}u_\tau ^4)$ for the pressure fluctuations for R500 and R2000 are shown in figures 1(a-i) and 1(b-i), respectively. The spectra are plotted versus the streamwise wavelength and the distance from the wall.

Figure 1.

Results of $\textbf{TH}\textrm{-}{\boldsymbol{U}}$. (i) Contour lines of the premultiplied streamwise spectra ${k_x}{E_{pp}}/({\rho ^2}u_\tau ^4)$ (filled, black contour lines) and frozen $\omega $-spectra $k_x^F\; E_{pp}^F/({\rho ^2}u_\tau ^4)$ (open, cyan contour lines) of the pressure fluctuations versus ${y^ + }$ and the wavelength for (a) R500 and (b) R2000. In (a-i), the contour lines correspond to the values $[0.2\,:\,0.2\,:\,2.8]$, and they indicate the values $[0.2\,:\,0.2\,:\,3.8]$ in (b-i). In (i), the black and blue cross marks denote the peaks of the streamwise and frozen $\omega $-spectra, respectively. (ii) Contour lines of the difference ${D_{pp}}({k_x},y)$ between the streamwise spectra and frozen $\omega $-spectra for (a) R500 and (b) R2000. Black and blue colours correspond to positive and negative ${D_{pp}}({k_x},y)$, respectively. In (i) and (ii), the dashed red line indicates the wall-normal height ${y^ + } = 20$ and the grey line indicates the wall-normal distances ${y^ + } = 100$ in (a) and 400 in (b). In (a,b), the yellow circles indicate the ridges of the streamwise spectra with a streamwise wavelength of ${\lambda _{{x_r}}}(y)$.

In addition to the contour plots of the streamwise and frozen $\omega $-spectra, we present the contour maps for the difference between them in figure 1(ii) at the same Reynolds numbers. The difference (relative error) between the two spectra ${D_{pp}}({k_x},y)$ is defined as (Del Álamo & Jiménez Reference Del Álamo and Jiménez2009)

(4.1)\begin{equation}{D_{pp}}({k_x},y) = {k_x}\frac{{E_{pp}^F({k_x},y) - {E_{pp}}({k_x},y)}}{{\max [{k_x}{E_{pp}}({k_x},y)]}}.\end{equation}

It is noted that when computing ${D_{pp}}({k_x},y)$, a cubic spline interpolation scheme is applied for the frozen $\omega $-spectra to enable matching with the streamwise spectra.

Like the velocity fluctuations discussed in previous studies (e.g. Monty & Chong Reference Monty and Chong2009; Wu et al. Reference Wu, Baltzer and Adrian2012; Squire et al. Reference Squire, Hutchins, Morrill-Winter, Schultz, Klewicki and Marusic2017), the contour lines of the streamwise and frozen $\omega $-spectra of the pressure field do not perfectly overlap each other for some regions normal to the wall. For the streamwise velocity fluctuations, Monty & Chong (Reference Monty and Chong2009) indicated differences between both spectra in the region below ${y^ + } = 50$ for streamwise wavelengths ${\lambda _x} > 4h$. But for the pressure field, the situation is different. From figures 1(a-i) and 1(b-i), we can discriminate three regions between the wall and channel centre based on the comparison between both spectra. The first region is the near-wall region below ${y^ + } \approx 20$. Within this region, the contour lines of the streamwise spectra are completely different from those of the frozen $\omega $-spectra. While the contour lines of the streamwise spectra tend to be vertical, indicating that the spectral energy of the pressure fluctuations resides in turbulent structures of nearly the same length scales (Jiménez & Hoyas Reference Jiménez and Hoyas2008), the contour lines of the frozen $\omega $-spectra are inclined to the wall. This behaviour of the frozen $\omega $-spectra comes from the very small values of the local mean velocities in the near-wall region, which cannot reflect the convective nature of the pressure field. Hence, $\textbf{TH}\textrm{-}{\boldsymbol{U}}$ cannot be applied within this region as the difference between both spectra is quite large $(O({\pm} 50\,{\%}))$, as shown in figures 1(a-ii) and 1(b-ii).

The second region extends from ${y^ + } \approx 20$ to ${y^ + } \approx 100\textrm{ }(y/h \approx 0.2)$ for R500 in figure 1(a-i) and to ${y^ + } \approx 400\textrm{ }(y/h \approx 0.2)$ for R2000 in figure 1(b-i). Generally, the streamwise and frozen $\omega $-spectra appear qualitatively similar. Within this range of Reynolds numbers, both spectra indicate the existence of only the inner peak reported in previous studies (Luhar et al. Reference Luhar, Sharma and Mckeon2014; Tsuji, Marusic & Johansson Reference Tsuji, Marusic and Johansson2016; Mehrez et al. Reference Mehrez, Philip, Yamamoto and Tsuji2019a). For both spectra, the streamwise wavelength associated with the inner peak is $\lambda _{{x_{peak}}}^ +\approx 250$. However, its location varies slightly among the spectra. For the streamwise spectra, it is located at approximately $y_{pea{k_T}}^ += 25$, while the frozen $\omega $-spectra suggest its location around $y_{pea{k_F}}^ += 21$. In addition, it is noted that the contour lines of the streamwise and frozen $\omega $-spectra match each other in the short-wavelength regime. In the long-wavelength regime, a slight difference between the two contour plots can be discerned. The short- and long-wavelength regimes are roughly discriminated in the figures based on the ridges of the streamwise spectra that indicate the peaks of the spectra at the various wall-normal locations (highlighted using yellow circles). The streamwise wavelength associated with the ridge of the streamwise spectra at each wall-normal location is denoted by ${\lambda _{{x_r}}}(y)$. It is worth mentioning that the relative error ${D_{pp}}({k_x},y)$ in this second region is of $O({\pm} 5\,{\%})$. However, it seems to increase with Reynolds number, as observed in figures 1(a-ii) and 1(b-ii).

The third region starts at ${y^ + } \approx 100\textrm{ }(y/h \approx 0.2)$ and $400\textrm{ }(y/h \approx 0.2)$ for R500 and R2000, respectively, and ends at the centre of the channel. The relative error between the two spectra almost vanishes, even for the higher Reynolds number R2000 (figure 1b-ii). The contour lines of the streamwise and frozen $\omega $-spectra overlap each other perfectly. Accordingly, $\textbf{TH}\textrm{-}{\boldsymbol{U}}$ is applicable in this region.

The behaviour indicated in the previous paragraphs may be perceived more clearly in figure 2, which shows the premultiplied streamwise spectra and frozen $\omega $-spectra versus the wavelength at three locations normal to the wall for R2000. The three locations shown in the figure represent the three regions described in the previous paragraphs. At $y/h \approx 0.2$, as shown in figure 2(c), a perfect consistency between the two spectra can be observed as they collapse quite well. However, this perfect overlap between the two spectra is only observed for short wavelengths at ${y^ + } \approx 70$ in figure 2(b). The spectra are different for long wavelengths. In addition, it is noted that the peak of the streamwise spectrum is attenuated relative to that of the frozen one.

Figure 2.

Results of $\textbf{TH}\textrm{-}{\boldsymbol{U}}$. Profiles of the premultiplied streamwise spectra ${k_x}{E_{pp}}/({\rho ^2}u_\tau ^4)$ (solid curve) and frozen $\omega $-spectra $k_x^F\; E_{pp}^F/({\rho ^2}u_\tau ^4)$ (dotted curve) of the pressure fluctuations versus the wavelength for R2000 at (a–c) different wall-normal locations.

In figure 2(a), the frozen $\omega $-spectrum in the viscous sublayer at ${y^ + } \approx 5$ is shifted to shorter wavelengths than the streamwise spectrum. For the case of the velocity field, Squire et al. (Reference Squire, Hutchins, Morrill-Winter, Schultz, Klewicki and Marusic2017) pointed out that this shift is attributed to a violation of $\textbf{TH}\textrm{-}{\boldsymbol{U}}$ where the velocity fluctuation intensity is high.

The results in figures 1 and 2 indicate that $\textbf{TH}\textrm{-}{\boldsymbol{U}}$ can be applied to the pressure field in channel flows above $y/h \approx 0.2$. Between ${y^ + } \approx 20$ and $y/h \approx 0.2$, $\textbf{TH}\textrm{-}{\boldsymbol{U}}$ is invalid for structures with wavelengths larger than approximately the ridges of the premultiplied spectra. Such structures seem to propagate with velocities that differ from the local mean. Generally, this deviation from $\textbf{TH}\textrm{-}{\boldsymbol{U}}$ does not exceed $5\,{\%}$ within this region, but it increases with Reynolds number. Finally, in the near-wall region up to ${y^ + } \approx 20$, $\textbf{TH}\textrm{-}{\boldsymbol{U}}$ is invalid where the maximum error of $O(50\,{\%})$. Therefore, the average pressure convection velocity ${C_p}(y)$ is estimated.

4.2. Taylor's frozen hypothesis with average (overall) convection velocity ${{C}_p}(y)$ (TH-${C_{\boldsymbol{p}}}$)

The results for the pressure average convection velocity ${C_p}(y)$ (computed using (3.4)) are shown in figure 3 for R180, R500 and R2000. In the figure, ${C_p}(y)$ is plotted versus the distance from the wall in inner and outer scaling in figures 3(a) and 3(b), respectively. Also shown in the figure is the local mean velocity at the same Reynolds numbers. The results of ${{C}_p}(y)$ are generally consistent with those of Kim & Hussain (Reference Kim and Hussain1993) at $R{e_\tau } = 180$. From the wall up to ${y^ + } \approx 10$, the average convection velocity is larger than the mean velocity, being nearly constant with values of approximately $11.40{u_\tau }$–$12.05{u_\tau }$ $(0.55{U_{cl}}$–$0.66{U_{cl}})$. Geng et al. (Reference Geng, He, Wang, Xu, Lozano-Durán and Wallace2015) reported the Reynolds number dependence trend close to the wall upon estimating the convection velocities of the velocity fluctuations in channels. In their study, the convection velocities of the velocity components at $R{e_\tau } = 205$ overestimated those at $R{e_\tau } = 932$.

Figure 3.

The average convection velocity of the pressure fluctuations ${C}_p^ + (y)$ for R180 (black), R500 (blue) and R2000 (brown) in (a) inner and (b) outer scaling for the wall distance. In the two panels, the local mean velocity ${U^ + }(y)$ is plotted versus the distance from the wall for R180 (triangles), R500 (squares) and R2000 (circles). (c) The difference between the average convection velocity of the pressure fluctuations and the local mean velocity ${C}_p^ + (y) - {U^ + }(y)$ for R180 (black), R500 (blue) and R2000 (brown) in outer scaling for the wall distance. The inset shows the difference ${C}_p^ + (y) - {U^ + }(y)$ in inner scaling for the wall distance. (d) The average convection velocities of the pressure fluctuations ${C}_p^ + (y)$ (brown), streamwise velocity ${C}_u^ + (y)$ (red), wall-normal velocity ${C}_v^ + (y)$ (magenta) and spanwise velocity ${C}_w^ + (y)$ (green), and the local mean velocity ${U^ + }(y)$ (open circles) for R2000 are plotted versus the distance from the wall in logarithmic scale. The average convection velocity of the streamwise velocity ${C}_u^ + (y)$ (filled circles) at $R{e_\tau } = 950$ from Del Álamo & Jiménez (Reference Del Álamo and Jiménez2009) is plotted versus ${y^ + }$. The subscript $\phi $ in the $y$-axis label ${C}_\phi ^ + $ stands for p, u, v and w.

In addition, our result $(0.66{U_{cl}})$ for R180 is smaller than the value reported by Kim & Hussain (Reference Kim and Hussain1993), which was $0.75{U_{cl}}$ at the same Reynolds number. This difference between the two values may be assigned to the difference between the method used here to compute the convection velocity and the method used in the aforementioned study. Del Álamo & Jiménez (Reference Del Álamo and Jiménez2009) indicated that (3.4) weights ${{C}_p}(y)$ towards the convection velocities of high wavenumbers (containing higher spectral energy). However, Kim & Hussain (Reference Kim and Hussain1993) computed the convection velocity from the space–time correlation using a time delay of $\mathrm{\Delta }{t^ + } = 18$, a value which they indicated to be too large for the high wavenumbers. Kim & Hussain (Reference Kim and Hussain1993) examined the dependence of the convection velocity on the time delay $\mathrm{\Delta }{t^ + } = 3$, 18 and 27. The convection velocity in their study was found to change by about $18\,{\%}$ between $\mathrm{\Delta }{t^ + } = 3$ and 27 close to the wall (at ${y^ + } \approx 5$). Accordingly, it is inferred that ${{C}_p}(y)$ computed in this study is smaller than that provided by Kim & Hussain (Reference Kim and Hussain1993) for locations close to the wall. For the same reasons, the reported average convection velocity of the wall pressure is smaller than that indicated by Choi & Moin (Reference Choi and Moin1990) and Jeon et al. (Reference Jeon, Choi, Yoo and Moin1999) ($0.72{U_{cl}}$ at $R{e_\tau } = 180$). From ${y^ + } \approx 10$ to ${y^ + } \approx 20$, the present convection velocity ${{C}_p}(y)$ increases very slightly until it becomes equal to ${U^ + }(y)$ at ${y^ + } \approx 20$, as shown in figure 3(a) for R180 and R2000. Beyond this location, the average convection velocity is consistent with the local mean velocity, being slightly smaller than ${U^ + }(y)$.

The difference between the pressure average convection velocity ${C}_p^ + (y)$ and the local men velocity ${U^ + }(y)$ is shown in figure 3(c) for R180, R500 and R2000. The difference is presented across the channel. It is clear in the figure that the difference is significant close to the wall which emphasizes that the local mean velocity is not applicable for representing Taylor convection velocity. Above the position ${y^ + } \approx 20$, the difference between ${C}_p^ + (y)$ and ${U^ + }(y)$ is small where it approaches a value of around $- 0.7$ in wall units for the different Reynolds numbers.

Superimposing the pressure average convection velocity on the local mean velocity helps to estimate the locations of the effective pressure field sources (Bull Reference Bull1967; Blake Reference Blake1970; Schewe Reference Schewe1983). For the wall pressure, Schewe (Reference Schewe1983) indicated that the pressure convection velocity is equal to the local mean velocity at ${y^ + } = 21$, whereas Kim (Reference Kim1989) and Kim & Hussain (Reference Kim and Hussain1993) estimated a higher location of ${y^ + } \approx 23$. Our results suggest the wall-normal location of ${y^ + } \approx 19.5$ for R180 and R2000 in close agreement with the aforementioned estimate by Schewe (Reference Schewe1983). As pointed out by Kim (Reference Kim1989), it is inferred that the main contribution to the wall pressure comes from vortex-like structures, which are centred at ${y^ + } \approx 20$ in the near-wall region (Kim et al. Reference Kim, Moin and Moser1987).

The average convection velocity of the pressure fluctuations ${C}_p^ + (y)$ is now compared with the convection velocities of the three velocity components at the same Reynolds number. The results are displayed in figure 3(d) for R2000. The average convection velocities of the pressure and velocity fields ${C}_\phi ^ + (y)$, where $\phi $ represents $p,u,v$ and w, are plotted versus the distance from the wall. The average convection velocities of the three velocity components are computed using the scheme presented by Del Álamo & Jiménez (Reference Del Álamo and Jiménez2009), which we apply here to the pressure field (equation (3.4)). However, the scale-dependent convection velocities of the three velocity components are computed from the momentum equations derived by Del Álamo & Jiménez (Reference Del Álamo and Jiménez2009) (their (2-11), (2-12), (2-13)) rather than the time series DNS dataset. Saving the time series data for the three velocity components in addition to the pressure field for our higher Reynolds number R2000 using the large computational domain $({L_x} \times {L_z} = (25.6 \times 9.6)h)$ requires a massive storage capacity (approximately 250 TB/turbulent field). Thus, for convenience, the time series dataset was saved only for the pressure field. For the velocity fluctuations, time realizations were saved at unequal time intervals.

The average convection velocity of the streamwise velocity in figure 3(d) matches that presented by Del Álamo & Jiménez (Reference Del Álamo and Jiménez2009) at a lower Reynolds number of $R{e_\tau } = 950$ which is also included in figure 3(d) for comparison. In addition, the current investigation of the average convection velocities of the three velocity components agrees with the previous results presented by Kim & Hussain (Reference Kim and Hussain1993) and Geng et al. (Reference Geng, He, Wang, Xu, Lozano-Durán and Wallace2015) at their lower Reynolds numbers of $R{e_\tau } = 180$ and up to 932, respectively. The average convection velocities of the three velocity components agree with each other remarkably well in the viscous sublayer below ${y^ + } \approx 5$. They are almost constant with a value of around $9{u_\tau }$. Beyond ${y^ + } \approx 5$, the convection velocities also increase slightly with the distance from the wall. Thus, the average convection velocity of the streamwise velocity seems to match the local mean velocity beyond ${y^ + } \approx 15$, whereas the other two velocity components have convection velocities that match the local mean velocity beyond ${y^ + } \approx 20$.

The larger values of the average convection velocities of the velocity fluctuations than the local mean velocity in the near-wall region originate from the footprint of large-scale structures, especially in the viscous sublayer. For u fluctuations, eddies with ${\lambda _x} \ge 2h$ and ${\lambda _z} \ge 0.4h$ contribute to the convection velocity in the near-wall region and propagate with a uniform velocity scaled with the bulk velocity (Del Álamo & Jiménez Reference Del Álamo and Jiménez2009). From the recent input–output approach developed by Liu & Gayme (Reference Liu and Gayme2020) to estimate the convection velocity based on the linearized Navier–Stokes equations, a range of large-scale structures was quantified with dimensions ${y^ + }\sim \lambda _z^ +{\sim} {(\lambda _x^ + )^{2/3}}$. They inferred that these structures are self-similar in the cross-plane and retain a wall-normal coherence with an inclination to the wall consistent with the attached eddy hypothesis of Townsend (Reference Townsend1976).

Remarkably, the average convection velocity of the pressure field ${C}_p^ + (y)$ is larger than the convection velocities of the three velocity components close to the wall (figure 3d). This larger value of the pressure convection velocity is consistent with the global features of the pressure fluctuations, as indicated by Poisson's equation for the pressure (Willmarth Reference Willmarth1975; Kim Reference Kim1989). Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2014) showed that pressure correlation retains a stronger correlation in the wall-normal direction than velocity correlations. From Poisson's pressure equation analysis by Green's function, Jiménez & Hoyas (Reference Jiménez and Hoyas2008) identified two mean structures concerned with the pressure fluctuations. The first one is an energetic localized structure in the near-wall and logarithmic regions, while the second one is a large-scale structure $O(h)$ that extends across the entire flow thickness. Sillero et al. (Reference Sillero, Jiménez and Moser2014) quantified the first and second structures in terms of strong and weak correlations, respectively. Therefore, the convection velocity of the pressure field is elevated in the near-wall region with multiple scale contributions. Pressure-relevant structures are discussed in § 5.

We investigated the average pressure convection velocity ${{C}_p}(y)$ across the entire flow. The frozen $\omega $-spectra $E_{pp}^F(k_x^F,y)$ in (3.8) are computed from ${{C}_p}(y)$ which is applied for ${U_T}$. The results of $\textbf{TH}\textrm{-}{{C}_p}$ are shown in figure 4. In comparison with figure 1, the collapse between both spectra in the near wall-region below ${y^ + } = 20$ is improved significantly and the difference between them decreased to a value of $O(10\,\%)$ for its maximum. This improvement is expected from figure 3(c) for the difference between ${{C}_p}(y)$ and $U(y)$. As indicated in figures 4(a-ii) and 4(b-ii), the difference exists around the ridges of the streamwise spectra and for larger scales. On the other hand, the small difference between the two velocities for the region above ${y^ + } = 20$ does not have an impact on the computed frozen $\omega $-spectra as their values are the same as in figure 1. Therefore, above ${y^ + } = 20$, $\textbf{TH}\textrm{-}{{C}_p}$ yields the same results as $\textbf{TH}\textrm{-}{\boldsymbol{U}}$, as shown in figure 4.

Figure 4.

Results of $\textbf{TH}\textrm{-}{{C}_p}$. Same as figure 1. However, the contour lines of the frozen spectra are indicated by red colour.

Based on the above observations, the average convection velocity ${{C}_p}(y)$ is better for representing the Taylor convection velocity for the pressure fluctuations compared with the local mean velocity. However, it is difficult to compute it from numerical simulations especially when Reynolds number increases. Besides, it cannot be obtained from experiments for static pressure. Therefore, we define a simply modelled convection velocity for the pressure field ${C}_p^M(y)$ derived from both the local mean velocity $U(y)$ and average convection velocity ${{C}_p}(y)$. It is discussed in the next subsection.

4.3. Taylor's frozen hypothesis with modelled convection velocity ${C}_p^M(y)$ (TH-${C}_{\boldsymbol{p}}^{\boldsymbol{M}}$)

The modelled convection velocity is defined as

(4.2)\begin{equation}{C}_p^M(y) = \left\{ {\begin{array}{*{20}{@{}ll}} {U({y^ + } = 20)}&{0 \le {y^ + } < 20,}\\ {U(y)}&{20 \le {y^ + } \le {h^ + }.} \end{array}} \right.\end{equation}

In this model, above ${y^ + } = 20$, the local mean velocity is adopted for the convection velocity. Below this position, the convection can be regarded as constant with a value that equals the local mean velocity at ${y^ + } = 20$. For this velocity, Taylor's hypothesis is defined as $\textbf{TH}\textrm{-}{{C}}_{\boldsymbol{p}}^{\boldsymbol{M}}$ and is considered in both wavenumber and frequency spaces.

4.3.1. Wavenumber space

The frozen $\omega $-spectra $E_{pp}^F(k_x^F,y)$ in (3.8) are computed from ${C}_p^M(y)$ which is applied for ${U_T}$. According to (4.2), we present the results for the region below ${y^ + } = 20$. Specifically, figures 5(a) and 5(b) show the comparison between the streamwise spectra ${k_x}{E_{pp}}/({\rho ^2}u_\tau ^4)$ and the frozen ones $k_x^FE_{pp}^F/({\rho ^2}u_\tau ^4)$ for R500 and R2000, respectively. The same results are obtained as in figure 4 where around the ridges of the streamwise spectra and for larger scales, the difference between both spectra still exists.

Figure 5.

Results of $\textbf{TH}\textrm{-}{{C}}_p^M$. Same as figure 1. However, the results are shown for the region $0 \le {y^ + } \le 20$.

This is further clarified in figure 6(a), where the wall pressure premultiplied streamwise and frozen spectra are displayed (streamwise spectrum: $k_x^ + {\; }E_{{p_w}{p_w}}^ += {k_x}{\; }{E_{pp}}({k_x},y = 0)/({\rho ^2}u_\tau ^4)$; frozen spectrum: $k_x^{F + }\; E_{{p_w}{p_w}}^{F + } = k_x^FE_{pp}^F(k_x^F,y = 0)/({\rho ^2}u_\tau ^4)$). The two dashed lines shown in the figure are associated with the two length scales $\lambda _{{x_{in}}}^ += 250$ and ${\lambda _{{x_{out}}}}/h = 1$ (for R2000). We define these length scales upon our analysis of the wall pressure structures classification in § 5. Several notes can be derived from the figure. First, both spectra, streamwise and frozen, peak at the same length scale of $\lambda _{{x_r}}^ + \textrm{(}y = 0\textrm{)} \approx 250$, which equals the length scale of the inner peak of the premultiplied spectra shown in figure 1(i) that occurs at $y_{pea{k_T}}^ +\approx 25$. This indicates that specific structures dominate the pressure field from the wall to ${y^ + } \approx 25$. Second, the frozen spectrum's peak exaggerates the streamwise one's peak, although both spectra peak at the same length scale. Third, an opposed behaviour for the large-scale structures is observed where an underestimation of the frozen spectrum with respect to the streamwise one can be discerned. These differences between the spectra arise from the various structures that contribute to the wall pressure fluctuations and propagate with different convection velocities as clarified in § 5. This yields a relative error ${D_{{p_w}{p_w}}}({k_x}) = {D_{pp}}({k_x},y = 0)$ between the spectra of $O(10\,{\%})$ for its maximum value as shown in figure 6(b).

Figure 6.

Results of $\textbf{TH}\textrm{-}{{C}}_p^M$. (a) The profiles of the premultiplied streamwise ${k_x}{E_{{p_w}{p_w}}}/({\rho ^2}u_\tau ^4) = {k_x}\; {E_{pp}}({k_x},y = 0)/({\rho ^2}u_\tau ^4)$ (solid curve) and frozen $k_x^F\; E_{{p_w}{p_w}}^F/({\rho ^2}u_\tau ^4) = k_x^F\; E_{pp}^F(k_x^F,y = 0)/({\rho ^2}u_\tau ^4)$ (dotted curve) spectra of the wall pressure fluctuations versus the streamwise wavelength in inner scaling $\lambda _x^ + $ for R500 (blue) and R2000 (brown). The top horizontal axis is the streamwise wavelength in outer scaling ${\lambda _x}/h$ for R2000. The two dashed vertical lines correspond to $\lambda _{{x_{in}}}^ += 250$ and ${\lambda _{{x_{out}}}}/h = 1$ ($\lambda _{{x_{out}}}^ += 2000$ for R2000). (b) The difference between the streamwise and frozen $\omega $-spectra of the wall pressure ${D_{{p_w}{p_w}}}({k_x}) = {D_{pp}}({k_x},y = 0)$ at R500 (blue) and R2000 (brown). The yellow circle in (a) indicates the ridge of the streamwise spectrum for R2000.

4.3.2. Frequency domain

To test the application of $\textbf{TH}\textrm{-}{{C}}_p^M$ further, a comparison with the experimental wall pressure results of Tsuji et al. (Reference Tsuji, Fransson, Alfredsson and Johansson2007) is performed. For this, we move to the frequency domain. Figure 7 presents the frequency spectrum of the wall pressure ${E_{{p_w}{p_w}}}(\omega ) = {E_{pp}}(\omega ,y = 0)$ (equation (3.7)) for R500 and R2000. In addition, figure 7 presents the results of the frozen ${k_x}$-spectrum (equation (3.9) with ${C}_p^M(y)$ applied for ${U_T}$) for the wall pressure $E_{{p_w}{p_w}}^F({\omega ^F}) = E_{pp}^F({\omega ^F},y = 0)$ for cases R500, R2000, R4000 and R8000. The frequency spectrum of the wall pressure for a turbulent boundary layer at $R{e_\tau } = 1760$, 2100 and 3883 from Tsuji et al. (Reference Tsuji, Fransson, Alfredsson and Johansson2007) is also shown in the figure. All spectra are normalized with the inner variables ${u_\tau }$ and $\nu $, and are presented as functions of $\omega \nu /u_\tau ^2$. Thus, the collapse of the spectra is observed in the high-frequency range for all Reynolds numbers. This indicates that near-wall-region structures dominate the wall pressure fluctuations within this frequency range (Farabee & Casarella Reference Farabee and Casarella1991; Tsuji et al. Reference Tsuji, Fransson, Alfredsson and Johansson2007).

Figure 7.

Results of $\textbf{TH}\textrm{-}{{C}}_p^M$. Wall pressure frequency spectra ${E_{{p_w}{p_w}}}(\omega )/({\rho ^2}u_\tau ^2\nu ) = {E_{pp}}(\omega ,y = 0)/({\rho ^2}u_\tau ^2\nu )$ (dashed) and frozen ${k_x}$-spectra $E_{{p_w}{p_w}}^F\; ({\omega ^F})/({\rho ^2}u_\tau ^2\nu ) = E_{pp}^F\; ({\omega ^F},y = 0)/({\rho ^2}u_\tau ^2\nu )$ (solid) versus $\omega \nu /u_\tau ^2$ for R500 (blue), R2000 (brown), R4000 (green) and R8000 (magenta). Wall pressure spectra at $R{e_\tau } = 1760$, 2100 and 3883 from Tsuji et al. (Reference Tsuji, Fransson, Alfredsson and Johansson2007) are shown in grey. The two solid, vertical grey lines correspond to $\omega \nu /u_\tau ^2 = 0.013$ and 0.189, while the two dashed ones correspond to $\omega \nu /u_\tau ^2 = {\omega _{in}}\nu /u_\tau ^2$ and ${\omega _{out}}\nu /u_\tau ^2$ with ${\omega _{in}} = (2{\rm \pi}/{\lambda _{{x_{in}}}}){C}_{{p_w}}^M$ and ${\omega _{out}} = (2{\rm \pi}/{\lambda _{{x_{out}}}}){C}_{{p_w}}^M$, where ${C}_{{p_w}}^M = {C}_p^M(y = 0)$, $\lambda _{{x_{in}}}^ += 250$ and ${\lambda _{{x_{out}}}}/h = 1$ ($\lambda _{{x_{out}}}^ += 2000$ for R2000).

Instead of observing the power-law spectrum ${\omega ^{ - 1.0}}$ predicted by Bradshaw (Reference Bradshaw1967), Tsuji et al. (Reference Tsuji, Fransson, Alfredsson and Johansson2007) inferred the existence of a power-law spectrum corresponding to ${\omega ^{ - 0.7}}$ within the frequency range of $0.013 \le \omega \nu /u_\tau ^2 \le 0.189$. This frequency range is presented in figure 7. Tsuji et al. (Reference Tsuji, Fransson, Alfredsson and Johansson2007) concluded that the wall pressure spectra with an exponent of $( - 1.0)$ occur only at high Reynolds numbers $(R{e_\tau } \approx O({10^6}))$, which were examined later by Klewicki, Priyadarshana & Metzger (Reference Klewicki, Priyadarshana and Metzger2008). Our DNS results in figure 7 indicate that both frequency and frozen ${k_x}$-spectra exhibit ${\omega ^{ - 0.7}}$ spectra over the frequency range ${\omega _{out}}\nu /u_\tau ^2 \le \omega \nu /u_\tau ^2 \le {\omega _{in}}\nu /u_\tau ^2$ for R2000. The bounds ${\omega _{in}}$ and ${\omega _{out}}$ correspond to the two length scales ${\lambda _{{x_{in}}}}$ and ${\lambda _{{x_{out}}}}$ in figure 6 such that ${\omega _{in}} = (2{\rm \pi}/{\lambda _{{x_{in}}}}){C}_{{p_w}}^M$ and ${\omega _{out}} = (2{\rm \pi}/{\lambda _{{x_{out}}}}){C}_{{p_w}}^M$, where ${C}_{{p_w}}^M = {C}_p^M(y = 0)$. It is noted that R500 exhibits ${\omega ^{ - 0.7}}$ spectra over a very narrow range. The reasoning behind this narrow range is clarified later in the discussion about classifying the wall pressure structures in § 5.

Using the modelled convection velocity for the wall pressure allows ${\omega ^{ - 0.7}}$ spectra to be identified within our DNS for R4000, consistent with the experimental results at the nearly same Reynolds number. As expected, the spectral region corresponding to ${\omega ^{ - 0.7}}$ spectra bounded by ${\omega _{in}}$ and ${\omega _{out}}$ increases with Reynolds number. On the other hand, for the higher Reynolds number, R8000, the slope increases slightly towards $( - 1.0)$, although we do not reach the theoretical slope discussed by Bradshaw (Reference Bradshaw1967). Panton, Lee & Moser (Reference Panton, Lee and Moser2017) also observed an increase in the slope of the wall pressure spectrum for their high Reynolds number of $R{e_\tau } = 5200$.

In the low-frequency range, there is a difference between the experimental boundary layer results and the channel results from DNS at the same Reynolds number. In figure 7, the results of channel flow of R2000 (dashed brown line from present DNS) and the results of boundary layer at $R{e_\tau } = 2100$ from Tsuji et al. (Reference Tsuji, Fransson, Alfredsson and Johansson2007) are almost at the same Reynolds number. Despite that, there is no collapse between both spectra in the low-frequency range. The low-frequency range of the spectra is associated with structures from the outer layer (Farabee & Casarella Reference Farabee and Casarella1991). As a result, this difference is expected since the outer layer large-scale structures in boundary layers (with outside potential field) are different from those in turbulent channels (Sillero et al. Reference Sillero, Jiménez and Moser2014), and they affect the pressure statistics (Panton et al. Reference Panton, Lee and Moser2017). Besides, for the channel DNS results, there is an obvious difference in the low-frequency range between frequency and frozen ${k_x}$-spectra for either R500 (solid and dashed blue lines) or R2000 (solid and dashed brown lines). Since both spectra (frequency and frozen ${k_x}$-spectra) are at the same Reynolds number (either $R{e_\tau } = 500$ or 2000) for the same flow field (channel) from the same dataset (DNS), the only reasoning we have here is applying Taylor's hypothesis with the convection velocity ${C}_{{p_w}}^M$.

In summary, $\textbf{TH}\textrm{-}{{C}}_p^M$ is better than $\textbf{TH}\textrm{-}{\boldsymbol{U}}$ in the near-wall region below ${y^ + } = 20$. However, $\textbf{TH}\textrm{-}{{C}}_p^M$ produces an error of $O(10\,{\%})$ for its maximum value, and it increases with Reynolds number. Also, $\textbf{TH}\textrm{-}{{C}}_p^M$ produces an error of $O (5\,{\%})$ for its maximum in the region between ${y^ + } = 20$ and $y/h = 0.2$ for structures with wavelengths associated with the ridges of the premultiplied streamwise spectra and larger ones (same as $\textbf{TH}\textrm{-}{\boldsymbol{U}}$ according to (4.2)). Therefore, in the next section, we examine the scale-dependent convection velocity for the conversion between time and space domains.