2. The direct numerical simulation database

The analysis relies on DNS data of fully developed turbulent flow in a circular pipe. Friction Reynolds numbers up to ${\textit {Re}}_{\tau } \approx 12\,000$ are considered (Pirozzoli Reference Pirozzoli2024), which is the current upper limit for DNS. Here, ${\textit {Re}}_\tau = R u_{\tau } / \nu$ , with $R$ the pipe radius, $u_{\tau }$ the friction velocity and $\rho$ the fluid density. Inner normalisation by $u_{\tau }$ and $\delta _v = \nu / u_{\tau }$ is hereafter denoted with a $+$ superscript. Uppercase letters are used to denote mean values of the flow variables and lowercase letters to denote fluctuations thereof. The numerical method is described in detail by Verzicco & Orlandi (Reference Verzicco and Orlandi1996) and Orlandi (Reference Orlandi2000), and relies on an implementation of the fractional step method on a staggered cylindrical mesh, whereby uniform volumetric forcing is applied to the axial momentum equation to maintain constant mass flow rate in time. According to this method, pressure is obtained from discrete enforcement of the divergence-free constraint to the velocity field, which results in the solution of a discrete Poisson equation, as explained in detail by Kim & Moin (Reference Kim and Moin1985). Furthermore, since pressure is defined at the cell centres and the velocity components are defined at the cell faces, no numerical boundary condition for pressure is required, and pressure is automatically consistent with the discretised form of the momentum equation. This avoids uncertainties and ambiguities in the enforcement of proper boundary conditions for the pressure Poisson equation, which have been extensively discussed (e.g. Gresho & Sani Reference Gresho and Sani1987).

A list of the flow cases is reported in table 1, which includes basic information about the computational mesh and some key parameters. As can be seen in table 1, the largest DNS has run for less than 10 eddy turnover times, which is the commonly accepted limit to guarantee time convergence (Hoyas & Jiménez Reference Hoyas and Jiménez2006). Nevertheless, careful examination of the time convergence according to the method of Russo & Luchini (Reference Russo and Luchini2017) has shown that the estimated standard deviation in the prediction of the pressure and velocity variances in the range of wall distances under scrutiny here is at most $0.6\, \%$ . Power spectral densities of velocity and pressure are hereafter reported, which we obtained by taking Fourier transform in the azimuthal direction, and averaging in the axial direction and in time. The associated uncertainties are expected to be larger than for the basic flow statistics; however, the confidence bands shown in figure 5 convey that errors in the velocity spectra are mainly confined to the largest wavelengths, and they are barely visible in the pressure spectra. Details of the sensitivity of the flow statistics to the pipe length are provided in Appendix A.

Table 1. Flow parameters for DNS of pipe flow. Here $R$ is the pipe radius, $L_x$ is the pipe axial length, $N_{\theta }$ , $N_r$ and $N_x$ are the number of grid points in the azimuthal, radial and axial directions, respectively, ${\textit {Re}}_b = 2 u_b R / \nu$ is the bulk Reynolds number, $f = 8 \tau _w / (\rho u_b^2)$ is the friction factor, ${\textit {Re}}_{\tau } = u_{\tau } R / \nu$ is the friction Reynolds number, $T$ is the time interval used to collect the flow statistics and $\tau _t = R/u_{\tau }$ is the eddy turnover time. Cases C-L and C-LL were run to test the sensitivity of the results to the axial pipe length.

3. Pressure spectra and the attached-eddy model

Figure 1. Flow case G: pre-multiplied spanwise spectral densities of fluctuating streamwise velocity $k_{\theta }^+ E_u^+$ (a) and pressure $k_{\theta }^+ E_p^+$ (b), as a function of wavelength and wall distance. Wall distances and wavelengths are reported both in inner units (bottom, left axes) and in outer units (top, right axes). The diagonal line denotes the trend $y_s^+ = 0.11 \lambda _{\theta }^+$ , and the trapezoidal region bounded by the red dashed line marks the region of near-wall influence of attached eddies. Contour levels are shown from 0.36 to 3.6, in intervals of 0.36 in (a) and from 0.3 to 3, in intervals of 0.3 in (b).

To start with, we compare the spectra of streamwise velocity and pressure to highlight similarities and differences. Figure 1(a) shows the spectral density of the streamwise velocity ( $E_u$ ), as a function of the wall distance ( $y$ ) and of the spanwise wavelength ( $\lambda _{\theta }$ ), pre-multiplied by $k_{\theta }=2 \pi / \lambda _{\theta }$ , for flow case G. The figure highlights the presence of a prominent energetic buffer-layer peak scaling in wall units, and an outer energetic site showcasing $R$ -sized superstructures previously observed in experiments (Kim & Adrian Reference Kim and Adrian1999; Hellström & Smits Reference Hellström and Smits2014). Between these two primary locations, a band of energetic intermediate modes is observed, with lengths roughly proportional to their distance from the wall, as after the AEM (see Townsend Reference Townsend1976; Hwang Reference Hwang2015; Marusic & Monty Reference Marusic and Monty2019). In particular, we find that the ‘centre’ of attached eddies with size $\lambda _{\theta }$ resides at $y_{_S} \approx 0.11 \lambda _{\theta }$ . The separation between the two energetic sites is about two orders of magnitude in flow case G. The figure also clarifies that the influence of the attached eddies and the $O(R)$ eddies on streamwise velocity fluctuations extends down to the wall. Specifically, the spectral iso-lines tend to attain a triangular shape between the spectral ridge and the wall due to energy ‘leakage’ from the overlying eddies. This region influenced by wall-attached eddies is marked with dashed red lines in figure 1, indicating an upper wall distance beyond which the influence of outer eddies is not felt. Setting the boundary between attached eddies and large eddies at $\lambda _{\theta } = 0.5 R$ (past which isolated maxima of the spectral density emerge), it follows that the maximum wall distance where their influence is felt is $y_{\it max }^+ \approx 0.055 \, {\textit {Re}}_{\tau }$ , with $y_{\it max }^+ \approx 660$ at the highest Reynolds number under scrutiny here. This ‘near-wall’ region is the main subject of investigation in the present study.

The spectral map for pressure fluctuations is shown for contrast in figure 1(b). The overall structure seems to be similar to that of the streamwise velocity, with a clear inner energetic site centred in the buffer layer, and a diagonal ridge to be interpreted as the signature of wall-attached eddies. Some differences are, however, noteworthy. First, the velocity spectra reveal a distinct outer-layer spectral peak at $ \lambda _{\theta } \approx R$ , which is absent in the pressure spectra. This finding suggests that superstructures do not induce significant pressure variations. Second, along the primary spectral ridge (solid line in the figure), the behaviour differs markedly: the spectral density of pressure decreases monotonically, while the velocity spectral density exhibits a dip between the inner and outer energetic regions. Third, and most relevant to this study, the iso-lines of the pressure spectral density tend to extend down to the wall almost orthogonally. This feature is especially striking when comparing with figure 1(a), which instead indicates decreasing behaviour as the wall is approached. This essential difference in the near-wall behaviour of velocity and pressure has a large impact on the flow statistics, as discussed in the following.

Figure 2. Flow case G: pre-multiplied spectral density of axial velocity (a) and pressure (b) as a function of wall distance, corresponding to various wavelengths: $\lambda _{\theta }^+=455$ ( $y_s^+=50$ ) (purple), $\lambda _{\theta }^+=909$ ( $y_s^+=100$ ) (green), $\lambda _{\theta }^+=3636$ ( $y_s^+=400$ ) (orange), $\lambda _{\theta }^+=10970$ ( $y_s^+=1205$ ) (red). The filled circles denote the wall distance of the corresponding eddy centres ( $y_s$ ); see figure 1. The dashed lines in (b) denote predictions of Bradshaw (Reference Bradshaw1967). The directory including the data and the Jupyter notebook that generated this figure can be accessed at https://www.cambridge.org/S0022112025002642/JFM-Notebooks/files/figure_2.

In order to more clearly bring out this aspect, in figure 2 we show wall-normal profiles of $k_{\theta }^+ E_u^+$ and $k_{\theta }^+ E_p^+$ for various values of $\lambda _{\theta }$ in the range of wavelengths between the inner and the outer energetic sites. In the AEM interpretation, these characterise the variance of streamwise velocity and pressure fluctuations induced by attached eddies with size $\lambda _{\theta }$ and centred along the main spectral ridge of figure 1. As shown by Pirozzoli (Reference Pirozzoli2024), the wall-normal profiles of $E_u$ very well conform with the theoretically predicted distributions of the velocity variance resulting from ‘turbulent Stokes layers’, which are shown with dashed lines. Indeed, according to Bradshaw (Reference Bradshaw1967), the effect of ‘distant’ wall-attached eddies on the near-wall region is mainly a low-frequency modulation of the wall shear stress, to which turbulence has time to readjust, in such a way that the law-of-the-wall applies instantaneously. This simple scheme offers a compelling explanation for why the spectral density of streamwise velocity is influenced by the wall even well above the viscous sublayer. In contrast, the spectral densities of pressure extend down to the wall almost orthogonally, with little attenuation relative to the peak value observed at the centre of the corresponding attached eddies.

A conceptual sketch illustrating these observations is provided in figure 3. The main idea is that a wall-attached eddy produces both velocity and pressure fluctuations. However, due to differing boundary conditions, it leaves a direct pressure imprint at the wall, while its velocity influence primarily manifests as a change in local wall shear stress. We note that the wall signatures shown in the figure are indicative only, as they depend on the specifics of the particular AEM (see e.g. Ahn et al. Reference Ahn, Graham and Rizzi2010). The AEM (Townsend Reference Townsend1976; Marusic & Monty Reference Marusic and Monty2019) can be used to make predictions of the velocity and pressure statistics, under the critical assumptions that (a) the attached eddies are self-similar and (b) the eddies are not cross-correlated. Assuming for simplicity flow over a flat wall at $y=0$ , the velocity perturbation at point $\mathbf {x} = (x, y, z)$ from one representative eddy of height $h$ , rooted at $\mathbf {x}_e = (x_e , 0 , z_e)$ , and with a characteristic velocity scale $u_o$ is given by

(3.1) \begin{equation} u_i(\mathbf {x}) = u_o f_i \left ( \frac {\mathbf {x}-\mathbf {x}_e}{h} \right ), \end{equation}

where $u_1=u$ , $u_2=v$ and $u_3=w$ refer to the streamwise, wall-normal and spanwise components of fluctuating velocity. Adding up the contributions of wall-attached eddies of various heights, one then obtains the distribution of the Reynolds stresses:

(3.2) \begin{equation} \langle u_i u_j \rangle (y) = u_o^2 \int _{\delta _1}^{\delta } P(h) I_{ij} \left ( y/h \right ) \textrm {d} h, \end{equation}

Figure 3. Illustrative sketch of velocity and pressure perturbations associated with a wall-attached eddy ( $u$ , $p$ ), along with corresponding wall signatures ( $\tau _w$ , $p_w$ ). The cross identifies the tentative eddy centre.

where $P(h)$ is the probability distribution function of eddies of a given height;

(3.3) \begin{equation} I_{ij} ( \eta ) = \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } f_i (\xi , \eta , \zeta ) f_j (\xi , \eta , \zeta ) \textrm {d} \xi \,\textrm {d} \zeta , \end{equation}

with $\xi = x/h$ , $\eta =y/h$ , $\zeta =z/h$ , is the contribution from an individual eddy to the velocity correlations; and $\delta _1$ and $\delta$ are the smallest and largest eddy length scales, respectively. Townsend (Reference Townsend1976) reasoned that, on account of the no-penetration condition at the wall, $f_1$ and $f_3$ are asymptotically finite for $\eta \to 0$ , whereas $f_2 \sim \eta$ . This leads to the requirements that for small $\eta$ ,

(3.4) \begin{equation} I_{12} \simeq A_{12} \eta , \quad I_{11} \simeq A_{11}, \quad I_{22} \simeq A_{22} \eta ^2, \quad I_{33} \simeq A_{33}, \end{equation}

with $A_{12}$ , $A_{11}$ , $A_{22}$ , $A_{33}$ non-zero constants. Townsend then deduced the distribution of eddy sizes with wall distance necessary to produce invariance of the inner-scaled Reynolds shear stress ( $- \langle u v \rangle / u_{\tau }^2 \approx 1$ ) with distance from the wall, as nominally observed in the logarithmic layer. This analysis leads to setting $u_o = u_{\tau }$ and having $P(h) \sim 1/h$ . With this, it follows that the other components of the Reynolds stress tensor must have the form

(3.5) \begin{align} \langle u^2\rangle /u_{\tau }^2 = B_1 - A_1 \log (y/\delta ), \end{align}

(3.6) \begin{align} \langle v^2\rangle /u_{\tau }^2 = B_2, \qquad\quad \end{align}

(3.7) \begin{align} \langle w^2\rangle /u_{\tau }^2 = B_3 - A_2 \log (y/\delta ). \end{align}

We extend the AEM to pressure fluctuations by considering the Poisson equation that they satisfy, as given by (Kim Reference Kim1989)

(3.8) \begin{equation} \frac {1}{\rho } \frac {\partial ^2 p}{\partial x_j \partial x_j} = -2 \frac {\textrm {d} U}{\textrm {d} y} \frac {\partial v}{\partial x} - \frac {\partial u_i}{\partial x_j} \frac {\partial u_j}{\partial x_i}, \end{equation}

where $ x_1 = x$ , $ x_2 = y$ and $ x_3 = z$ . The appropriate wall boundary condition for pressure is derived from enforcing momentum balance in the wall-normal direction, resulting in

(3.9) \begin{equation} \frac 1{\rho } \left . \frac {\partial p}{\partial y} \right |_{y=0} = \nu \left . \frac {\partial ^2 v}{\partial y^2}\right |_{y=0}. \end{equation}

Additional background and details of the pressure Poisson equation for turbulent wall-bounded flows can be found in Appendix B. Assuming a similarity solution for pressure fluctuations of the form

(3.10) \begin{equation} p(x, y, z) = \rho u_{\tau }^2 f_p(\xi , \eta , \zeta ), \end{equation}

and noting that in the logarithmic layer $ \textrm {d} U / \textrm {d} y = u_{\tau } / \kappa y$ , (3.8) can be approximated with

(3.11) \begin{equation} \frac {\partial ^2 f_p}{\partial \xi _j \partial \xi _j} = -\frac {2}{\kappa \eta } \frac {\partial f_2}{\partial \xi } - \frac {\partial f_i}{\partial \xi _j} \frac {\partial f_j}{\partial \xi _i}. \end{equation}

As for the boundary condition (3.9), in terms of similarity variables it becomes

(3.12) \begin{equation} \left . \frac {\partial f_p}{\partial \eta } \right |_{\eta =0} = \frac 1{h^+} \left . \frac {\partial ^2 f_2}{\partial \eta ^2}\right |_{\eta =0} , \end{equation}

with $h^+ = h u_{\tau } / \nu$ the typical shear Reynolds number for a given attached eddy. For eddies whose centre is sufficiently distant from the wall, to which the AEM is supposed to apply, $h^+ \gg 1$ , and hence the previous boundary condition becomes

(3.13) \begin{equation} \left . \frac {\partial f_p}{\partial \eta } \right |_{\eta =0} = 0 . \end{equation}

Equations (3.11) and (3.13) show that wall-attached similarity solutions for the pressure field are possible if the velocity field is self-similar. This is a key result which allows one to extend Townsend’s formalism to pressure in straightforward manner, yielding

(3.14) \begin{equation} \langle p^2 \rangle (y) = \tau _w^2 \int _{\delta _1}^{\delta } P(h) I_{p} (\eta ) \textrm {d} h, \end{equation}

with

(3.15) \begin{equation} I_{p} \left ( \eta \right ) = \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } f_p^2 (\xi , \eta , \zeta ) \textrm {d} \xi \,\textrm {d} \zeta . \end{equation}

The approximate boundary condition (3.13) implies that $f_p$ at the wall is finite and independent of $h$ , whence it follows that, for small $\eta$ ,

(3.16) \begin{equation} I_p \simeq A_p , \end{equation}

with $A_p$ a non-zero constant, which figure 2(b) well supports. Substitution into (3.14) and carrying out the integral as suggested by Perry & Chong (Reference Perry and Chong1982, Appendix B) yields

(3.17) \begin{equation} \langle p^2 \rangle / \tau _w^2 = B_p - C_p \log (y/\delta ). \end{equation}

The prediction of (3.17) is very well verified in the pipe flow DNS data, with constants $B_p = 1.13$ , $C_p=2.28$ (Yu et al. Reference Yu, Ceci and Pirozzoli2022). Indeed, figure 4 shows the presence of an extremely long logarithmic layer ranging from the buffer layer all the way up to $y/R \approx 0.5$ . This evidence is far more convincing than that supporting the existence of a logarithmic layer for $\langle u^2\rangle$ , which is relatively thin and still debated (e.g. Monkewitz Reference Monkewitz2022). We interpret this as a result of near-wall viscous effects on velocity, which undermine the inviscid asymptotic predictions of (3.4). Therefore, we propose that (3.17) is the most robust prediction derived from the AEM.

Figure 4. Distribution of pressure variance as a function of outer-scaled wall distance. The colour codes correspond to different Reynolds numbers, as defined in table 1. The dashed grey line indicates the prediction of (3.17), with $B_p=1.13$ , $C_p=2.28$ . The directory including the data and the Jupyter notebook that generated this figure can be accessed at https://www.cambridge.org/S0022112025002642/JFM-Notebooks/files/figure_4.

4. Analysis of the spectra

Figure 5. Pre-multiplied spanwise spectral densities of streamwise velocity (a,c,e,g,i,k) (from Pirozzoli Reference Pirozzoli2024) and pressure (b,d,f,h,j,l) in inner scaling, at various inner-scaled wall distances: $y^+=1$ (a,b), $y^+=15$ (c,d), $y^+=50$ (e,f), $y^+=100$ (g,h), $y^+=200$ (i,j), $y^+=400$ (k,l). The colour codes indicate different Reynolds numbers as given in table 1. The shaded grey regions denote the expected range of uncertainty for flow case G. The dashed black lines denote the ${\lambda _{\theta }^+}^{-0.18}$ trend. The directory including the data and the Jupyter notebook that generated this figure can be accessed at https://www.cambridge.org/S0022112025002642/JFM-Notebooks/files/figure_5.

The arguments presented in § 1 for the pressure fluctuation spectra by Bradshaw (Reference Bradshaw1967), which lead to the $E_p \sim \tau _w^2 k^{-1}$ spectrum, can similarly be applied to velocity fluctuations, resulting in the prediction $E_u \sim u_{\tau }^2 k^{-1}$ . This is the key contribution of the work of Perry & Abell (Reference Perry and Abell1977). In figure 5, we present the spanwise spectral densities of streamwise velocity and pressure fluctuations at various locations with fixed $ y^+ \lt y_{\it max }^+$ . Uncertainty bars for flow case G are shown in grey, illustrating that the effects of limited time convergence on the velocity spectra are mainly concentrated at scales $ \lambda _{\theta } \gtrsim R$ . Time convergence of the pressure spectra appears to be very nearly perfect. The figure provides clear evidence for the tendency towards universality at the small-scale end of the spectra when inner normalisation is applied. However, this tendency manifests more rapidly for the velocity field ( $ {\textit {Re}}_{\tau } \gtrsim 2000$ ) than for the pressure field, which appears to attain universal behaviour only at the highest Reynolds number examined. In the velocity field, the universal inner layer extends towards larger scales. In contrast, the large-scale end of the pressure spectra still seems far from achieving a universal behaviour. The general trend for the pressure field is an increase in spectral density, potentially forming a plateau at the highest Reynolds numbers. This observation aligns with Panton et al. (Reference Panton, Lee and Moser2017), who inferred a similar behaviour from channel flow data, suggesting a $ k^{-1}$ spectral range.

Spectral overlap arguments as those used by Bradshaw (Reference Bradshaw1967) and Perry & Abell (Reference Perry and Abell1977) were used by Pirozzoli (Reference Pirozzoli2024) to infer the plausible behaviour of the velocity spectra, which can also be applied to the pressure spectra. By assuming that: (i) the typical velocity of all eddies is the friction velocity and (ii) the typical length of the smaller eddies is $\delta _v$ , whereas the typical length of the larger eddies is $R$ , it follows that the spectral signature of the smaller eddies can be expressed as

(4.1) \begin{align} k_{\theta }^+ E^+ = f(\lambda _{\theta }^+), \end{align}

while the larger eddies are expected to follow a spectrum of the form

(4.2) \begin{align} k_{\theta }^+ E^+ = g(\lambda _{\theta }/R). \end{align}

Assuming an overlap spectral layer exists between these inner- and outer-scaled regions, the functions $f$ and $g$ should take the form of either power laws or logarithmic functions. Based on the DNS data, the power-law description appears to be more appropriate. Hence, the spectral densities in the overlap layer are expected to behave as

(4.3) \begin{equation} k_{\theta }^+ E^+ = C \left ( \lambda _{\theta }^+ \right )^{-\alpha } = C {\textit {Re}}_{\tau }^{-\alpha } \left ( \frac {\lambda _{\theta }}R \right )^{-\alpha }, \end{equation}

holding in inner and outer scaling, respectively, where $\alpha$ and $C$ could in general depend on $y^+$ . Equation (4.3) includes the $k_{\theta }^{-1}$ spectral scaling as a special case, occurring for $\alpha =0$ . In that case, both the small-scale and the large-scale ends of the spectrum would be universal.

Figure 5 indeed supports the validity of (4.3) for the streamwise velocity, suggesting the occurrence of an overlap layer with power-law scaling with exponent $\alpha \approx 0.18$ , as we have estimated by fitting the DNS data for flow case G in the range of wavelengths $1000 \leqslant \lambda _{\theta }^+ \leqslant 10\,000$ (Pirozzoli Reference Pirozzoli2024). Notably, we confirm that the spectral power-law exponent remains universal with respect to changes in wall distance and Reynolds number, and, within numerical uncertainty, is consistent even in channel flow (Pirozzoli Reference Pirozzoli2024). The behaviour of pressure spectra is not so clear-cut, although the decay exponent is certainly less than for the streamwise velocity, and energy pile-up is still visible as the Reynolds number increases, reflecting imperfect universality of the small scales of motion.

In figure 6, the spectral densities of velocity and pressure are shown as a function of the outer-scaled wavelength. The universality of the velocity spectra is clearly not as pronounced as seen for the small wavelengths in figure 5. The energy associated with the large scales of motion becomes slightly but consistently less at higher Reynolds numbers for a given $\lambda _{\theta }/R$ . Inspection of (4.3) shows that this behaviour is consistent with the occurrence of a negative power law in the intermediate scales. On the other hand, the large-scale end of the pressure spectra appears to be very nearly independent of both the Reynolds number and of the wall distance, given that the corresponding iso-lines of the spectral map in figure 1(b) are vertically straight. Based on the theoretical prediction (4.3), the universality at the large scales and the tendency towards (though imperfect) universality at the small scales, the only asymptotically possible behaviour for pressure is a $k^{-1}$ spectrum, as predicted by Bradshaw (Reference Bradshaw1967).

Figure 6. Pre-multiplied spanwise spectral densities of streamwise velocity (a,c,e,g,i,k) (from Pirozzoli Reference Pirozzoli2024) and pressure (b,d,f,h,j,l) in outer scaling, at various inner-scaled wall distances: $y^+=1$ (a,b), $y^+=15$ (c,d), $y^+=50$ (e,f), $y^+=100$ (g,h), $y^+=200$ (i,j), $y^+=400$ (k,l). The colour codes indicate different Reynolds numbers as given in table 1. The directory including the data and the Jupyter notebook that generated this figure can be accessed at https://www.cambridge.org/S0022112025002642/JFM-Notebooks/files/figure_6.

Figure 7. Pre-multiplied spanwise spectral densities of streamwise velocity (a,c,e,g) (from Pirozzoli Reference Pirozzoli2024) and pressure (b,d,f,h) in outer scaling, at various outer-scaled wall distances: $y/R=0.1$ (a,b), $y/R=0.2$ (c,d), $y/R=0.3$ (e,f), $y/R=0.5$ (g,h). The colour codes indicate different Reynolds numbers as given in table 1. The directory including the data and the Jupyter notebook that generated this figure can be accessed at https://www.cambridge.org/S0022112025002642/JFM-Notebooks/files/figure_7.

For completeness, in figure 7 we also report the spectral densities of velocity and pressure as a function of the outer-scaled wavelength at fixed values of $y/R$ , to highlight scaling in the outer part of the wall layer. At these wall distances the contamination from viscous effects is minimal, and universality across Reynolds-number variation is effectively achieved. However, whereas some scatter at the largest scales remains in the velocity spectra, the pressure spectra are very nearly universal. This is in our opinion a very important result as it speaks about tendency for the near-wall flow to achieve universality in viscous units, but also for the flow in the pipe core to become universal when expressed in $R$ units, in support of the scenario envisaged by Dennis & Sogaro (Reference Dennis and Sogaro2014). Universality of the large scales in the near-wall region is retained in the pressure spectra, but it is lost in the velocity spectra on account of viscous effects, recalling the discussion of figure 2.

5. The pressure variance

The information derived from the analysis of the velocity and pressure spectra can be distilled to infer the behaviour of the velocity and pressure variances, taking inspiration from Hwang (Reference Hwang2024). Let $\lambda _s$ and $\lambda _{\ell }$ be, respectively, indicative lower and upper limits for the observed overlap spectral range. The velocity variance can be expressed as

(5.1) \begin{equation} \langle u^2 \rangle ^+ = \underbrace {\int _{-\infty }^{\log \lambda _s^+} k_{\theta }^+ E_{u}^+ \textrm {d} \log \lambda _{\theta }^+}_{\langle u^2\rangle ^+_s} + \underbrace {\int _{\log \lambda _s^+}^{\log \lambda _{\ell }^+} k_{\theta }^+ E_{u}^+ \textrm {d} \log \lambda _{\theta }^+}_{\langle u^2\rangle ^+_o} + \underbrace {\int _{\log \lambda _{\ell }^+}^{\infty } k_{\theta }^+ E_{u}^+ \textrm {d} \log \lambda _{\theta }^+}_{\langle u^2\rangle ^+_{\ell }}, \end{equation}

where the subscripts $s$ , $\ell$ and $o$ denote, respectively, the contributions of the smallest scales, the largest scales and the intermediate, overlap-layer scales. Likewise, the pressure variance can be expressed as

(5.2) \begin{equation} \langle p^2 \rangle ^+ = \underbrace {\int _{-\infty }^{\log \lambda _s^+} k_{\theta }^+ E_{p}^+ \textrm {d} \log \lambda _{\theta }^+}_{\langle p^2\rangle ^+_s} + \underbrace {\int _{\log \lambda _s^+}^{\log \lambda _{\ell }^+} k_{\theta }^+ E_{p}^+ \textrm {d} \log \lambda _{\theta }^+}_{\langle p^2\rangle ^+_o} + \underbrace {\int _{\log \lambda _{\ell }^+}^{\infty } k_{\theta }^+ E_{p}^+ \textrm {d} \log \lambda _{\theta }^+}_{\langle p^2\rangle ^+_{\ell }}. \end{equation}

Although the precise values of the limits in (5.1) are not important, it is crucial that the lower limit scales in wall units, hence $\lambda _{s}^+ = \textrm {const.}$ , and that the upper limit scales in outer units, hence $\lambda _{\ell }/R = \textrm {const}$ . Based on the previous analysis of the velocity spectra, Pirozzoli (Reference Pirozzoli2024) showed that the behaviour of the velocity variance should be as follows:

(5.3) \begin{equation} \langle u^2 \rangle ^+ = A_u(y^+) - B_u(y^+) {\textit {Re}}_{\tau }^{-\alpha } , \end{equation}

with $A_u$ and $B_u$ functions of the wall distance to be determined by fitting the DNS data.

Based on evidence previously given that the pressure spectra at the small and large scales tend to be universal across the ${\textit {Re}}_{\tau }$ range, the associated contribution to the pressure variance is expected to be asymptotically constant, and hence

(5.4) \begin{equation} {\langle p^2 \rangle ^+_{s}} = A_{s} (y^+) , \quad {\langle p^2 \rangle ^+_{\ell }} = A_{\ell } (y/R) \approx \textrm {const.} \end{equation}

Last, the contribution of the overlap layer can be evaluated by integrating the power-law spectrum given in (4.3) corresponding to $\alpha =0$ , thus obtaining

(5.5) \begin{equation} {\langle p^2 \rangle ^+_o} = A_o(y^+) + B_o(y^+) \, \log {\textit {Re}}_{\tau }. \end{equation}

Combining the previous equations, it follows that overall pressure variance should vary as

(5.6) \begin{equation} \langle p^2 \rangle ^+ = A_p(y^+) + B_p(y^+) \, \log {\textit {Re}}_{\tau } , \end{equation}

where the functions $A_p$ and $B_p$ do not depend explicitly on the Reynolds number, given the assumptions made to define $\lambda _s$ and $\lambda _{\ell }$ . This formula predicts that, at any fixed $y^+$ , the pressure variance should increase logarithmically with ${\textit {Re}}_{\tau }$ , in agreement with the arguments of Bradshaw (Reference Bradshaw1967) and Perry & Abell (Reference Perry and Abell1977), and with the predictions of the AEM presented in § 3. Equation (5.6) is tested in figure 8, where we show the contributions to the pressure variance from small, large and intermediate scales, having set $\lambda _s^+ = 400$ , $\lambda _\ell = R/2$ . In agreement with the pressure spectra, the contribution from the large scales (figure 8 b) is found to be very nearly universal when reported as a function of the outer-scaled coordinate $y/R$ . The contributions from the small scales (figure 8 a) show deviations from universality at relatively low $\textit {Re}$ ; however, saturation is close to complete at ${\textit {Re}}_{\tau } = 12\,000$ (black line). The contribution from the intermediate scales is instead found to grow continuously, as after (5.5); see figure 8(c). Comparison of the various contributions for flow case G (figure 8 d) shows that all of them tend to assume flat behaviour near the wall, again consistent with the notion that pressure fluctuations are not directly affected by viscosity. The contribution from the small scales (dashed lines) has a peak in the buffer layer, which is reflected in the peak of the overall variance. The contribution from the superstructures (dotted lines) is nearly constant across the wall layer with a peak at $y/R \approx 0.1$ , whereas the contribution from the intermediate scales (dot-dashed lines) has a peak at an intermediate distance from the wall, here $y^+ \approx 200$ . Values for the asymptotic constants $A_p$ , $B_p$ in (5.6) determined from fitting the DNS data at representative off-wall locations are reported in table 2, along with the corresponding standard deviations.

Figure 8. Contributions to pressure variance in pipe flow from: small scales (a), large scales (b) and intermediate scales (c). (d) The pressure variance for flow case G ( ${\textit {Re}}_{\tau } \approx 12\,000$ ), and the contributions from small scales (dashed lines), from large scales (dotted lines) and from intermediate scales (dash-dotted lines). The colour codes correspond to different Reynolds numbers, as defined in table 1. The directory including the data and the Jupyter notebook that generated this figure can be accessed at https://www.cambridge.org/S0022112025002642/JFM-Notebooks/files/figure_8.

Table 2. Fitting parameters to use in (5.6), based on DNS data fitting, at several off-wall positions, with accompanying asymptotic standard errors.

Figure 9. Variances of (a,c) streamwise velocity and (b,d) pressure, as a function of ${\textit {Re}}_{\tau }^{-0.18}$ (a,b) and as a function of $\log {\textit {Re}}_{\tau }$ (c,d), at various off-wall positions: $y^+=1$ (purple), $y^+=15$ (green), $y^+=50$ (cyan), $y^+=100$ (orange), $y^+=200$ (red), $y^+=400$ (blue). (e) The ratio of pressure to streamwise velocity variances as a function of ${\textit {Re}}_{\tau }$ . The directory including the data and the Jupyter notebook that generated this figure can be accessed at https://www.cambridge.org/S0022112025002642/JFM-Notebooks/files/figure_9.

The resulting distributions of the pressure variances as a function of ${\textit {Re}}_{\tau }$ are shown in figure 9, at various $y^+$ locations. In the figure we contrast the logarithmic fits predicted from (5.6) with the defect power law given in (5.3). For comparison, fits of the streamwise velocity variance are also displayed. As shown by Pirozzoli (Reference Pirozzoli2024), velocity variances are clearly better fitted with the defect power law than from the logarithmic law. Whereas the defect power law could well be mistaken for logarithmic growth at $y^+=15$ , the trends at positions farther from the wall are distinctly different from logarithmic. As for pressure, the logarithmic law (5.6) yields perfect fit of the DNS data, across the entire Reynolds-number range, and at all off-wall stations. Larger scatter is observed when data are fitted with a defect power-law distribution. To provide a quantitative impression, the standard deviation for the logarithmic fit of the pressure variance at $ y^+ = 50$ is $ 0.045$ , whereas the defect power-law fit yields a standard deviation of $ 0.24$ . The different behaviour of velocity and pressure is made manifest in figure 9(e), where we show the ratio of pressure to streamwise velocity variances. Note that the data are not shown at $y^+=1$ as the velocity variance there is very small, so this indicator is off scale. Whereas at low-to-moderate Reynolds number there is no obvious trend, with logarithmic increase at near-wall stations ( $y^+=15, 50$ ) and flat behaviour farther from the wall, there is very little ambiguity for ${\textit {Re}}_{\tau } \gtrsim 3000$ , at which an increasing logarithmic trend is observed at all $y^+$ , with the same growth rate. In our opinion, this is very convincing evidence that streamwise velocity variance and pressure variance behave in a different way, the streamwise velocity variance tending to saturate with the Reynolds number following a defect power law, and pressure variance following an ever-increasing logarithmic trend.

Figure 10. Predicted distributions of pressure variances at various ${\textit {Re}}_{\tau }$ , according to (5.6). The symbols denote the DNS data used to determine the fitting coefficients $A_p(y^+)$ , $B_p(y^+)$ (see table 1 for the colour codes). The directory including the data and the Jupyter notebook that generated this figure can be accessed at https://www.cambridge.org/S0022112025002642/JFM-Notebooks/files/figure_10.

Figure 10 shows extrapolated distributions of the pressure variance as a function of wall distance for various ${\textit {Re}}_{\tau }$ . In addition to confirming that the predictive formula (5.6) accurately represents the range of Reynolds numbers used to inform the model, the figure also illustrates extrapolations beyond this range. Unlike the streamwise velocity variance (Pirozzoli Reference Pirozzoli2024), the growth does not saturate, and the shape of the pressure variance distribution remains consistent. Instead, there is continuous growth, primarily driven by the expanding range of scales associated with wall-attached eddies, as predicted by (5.6).