2. Characteristics of pressure fluctuations in turbulent channel and pipe flow

In an idealized incompressible flow, pressure fluctuations are theoretically transmitted instantaneously, as the assumption of constant density removes any constraints imposed by compressibility. However, in turbulent flows, even under the assumption of incompressibility, the propagation speed of pressure fluctuations deviates from this idealized infinite speed. This deviation arises from the intricate interactions between turbulent eddies and the surrounding flow, which introduce localized variations in pressure propagation.

Unlike its effect on fluid particle motion, viscosity does not directly influence the propagation of pressure fluctuations. This lack of direct viscosity impact is evident in the pressure fluctuation distribution, as shown in figure 2. For comparison, the figure also displays Reynolds stress distributions. In this study, we follow the notation system of Tennekes & Lumley (Reference Tennekes and Lumley1972), where uppercase letters represent mean quantities, and lowercase letters denote fluctuations. For example, $U$ is the mean velocity in the $x$ -direction, and $u$ is the corresponding velocity fluctuation. The Reynolds averaging operation is indicated by angle brackets. For instance, the Reynolds shear stress is expressed as $R_{uv} = -\langle u v \rangle$ , where $v$ denotes the velocity fluctuation in the $y$ -direction.

Figure 2.

Distribution of Reynolds stresses and r.m.s. of pressure fluctuation in a turbulent channel flow. The vertical dashed line denotes the location of the maximum Reynolds shear stress. The data are from the DNS of Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021, Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2022) at $Re_\tau =500$ .

As shown in figure 2, the Reynolds normal stresses $\langle uu \rangle$ , $\langle vv \rangle$ , $\langle ww \rangle$ , along with the Reynolds shear stress $-\langle u v \rangle$ , exhibit characteristic trends in wall-bounded turbulent flows. These stresses begin at zero at the wall, increase to their respective peaks in the near-wall region, then gradually decrease as the flow approaches the centre of the channel or pipe, or the edge of the boundary layer. The zero values of Reynolds stresses at the wall are a direct consequence of the no-slip and impermeable boundary conditions, coupled with the influence of viscosity, which smoothens velocity fluctuations in the immediate vicinity of the wall.

In contrast, the r.m.s. value of pressure fluctuations at the wall is non-zero, and remains constant in the immediate vicinity of the wall. This highlights a fundamental distinction: while viscosity directly smoothens velocity fluctuations near the wall, it does not similarly dampen pressure fluctuations.

This lack of direct influence of viscosity on pressure fluctuation propagation becomes evident when examining the Poisson equation for instantaneous pressure, where viscosity is not present explicitly (Townsend Reference Townsend1956; Monin & Yaglom Reference Monin and Yaglom1965):

(2.1) \begin{equation} \frac {1}{\rho }\, \nabla ^2 \tilde {p} = -\frac {\partial \tilde {u}_i}{\partial x_j}\,\frac {\partial \tilde {u}_j}{\partial x_i}. \end{equation}

Applying Reynolds decomposition, the Poisson equation for the pressure fluctuation $p$ is given as (see e.g. Pope Reference Pope2000)

(2.2) \begin{equation} \frac {1}{\rho }\, \nabla ^2 p = -2\,\frac {\partial U_i}{\partial x_j}\,\frac {\partial {u}_j}{\partial x_i} -\frac {\partial ^2}{\partial x_i\, \partial x_j}{(u_i u_j - \langle u_i u_j \rangle )}. \end{equation}

The first term, often referred to as the rapid term or mean-shear–turbulence interaction, involves the mean velocity gradient and the gradient of velocity fluctuations. In contrast, the second term originates from spatial variations in the correlation of fluctuating velocity components, referred to as the slow term or turbulence–turbulence term. Equation (2.2) clearly demonstrates that viscosity does not directly influence pressure fluctuations. However, this does not imply that pressure fluctuations are entirely unaffected by viscosity or the Reynolds number, as both the mean velocity gradient and the Reynolds stress $\langle u_i u_j \rangle$ are influenced by viscosity.

Figure 3.

(a) Maximum Reynolds stresses and pressure fluctuation variance. These quantities are inner scaled, normalized by the wall shear stress, as commonly presented in studies of wall-bounded turbulent flows. (b) Ratio of the peak pressure fluctuation variance deficit $\langle pp \rangle _{{max}} - \langle pp \rangle _{{wall}}$ to the square of maximum Reynolds stresses. Red indicates DNS channel flow data (Lee & Moser Reference Lee and Moser2015), while blue represents DNS pipe flow data (Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021, Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2022; Pirozzoli Reference Pirozzoli2024).

Figure 3(a) displays the inner-normalized maximum Reynolds stresses alongside the maximum and wall values of the r.m.s. pressure fluctuation. It is known that the inner-normalized maximum Reynolds shear stress $ R_{uv}|_{{max}}/u_\tau ^2$ approaches 1 asymptotically, as indicated by the green dashed line in the figure, following the relationship $ R_{uv}^+|_{{max}} \approx 1 - O(1)/\sqrt {Re_\tau }$ (see e.g. Fife et al. Reference Fife, Klewicki and Wei2005b ).

Figure 3(a) shows a clear increase in $ \langle uu \rangle ^+_{{max}}$ , $ \langle ww \rangle ^+_{{max}}$ , $ \sqrt {\langle pp \rangle ^+_{{max}}}$ and $ \sqrt {\langle pp \rangle ^+_{{wall}}}$ with increasing Reynolds number. Furthermore, previous studies (e.g. Bradshaw Reference Bradshaw1967; Tsuji et al. Reference Tsuji, Marusic and Johansson2016; Panton et al. Reference Panton, Lee and Moser2017; Yu et al. Reference Yu, Ceci and Pirozzoli2022) have demonstrated that both $ \langle pp \rangle ^+_{{max}}$ and $ \langle pp \rangle ^+_{{wall}}$ exhibit a logarithmic increase with Reynolds number.

As noted in the Introduction, the pressure variance relative to its wall value serves as a more insightful metric for analysing its variations in the inner region. In figure 3(b), the ratio of peak pressure fluctuation variance difference from its wall value, $\langle pp \rangle _{{max}} - \langle pp \rangle _{{wall}}$ , to the square of the maximum Reynolds stress is shown. The difference $\langle pp \rangle _{{max}} - \langle pp \rangle _{{wall}}$ is smaller than the peak values of the squared Reynolds normal stresses in the streamwise and spanwise (circumferential) directions, but slightly larger than the peak values of the squared Reynolds normal stress in the wall-normal (radial) direction and the squared Reynolds shear stress.

Figure 4.

Ratio of Reynolds shear stress to r.m.s. of the pressure fluctuation: (a) channel flow, (b) pipe flow. The empirical curve-fitting functions are presented in (2.3). Red indicates DNS channel flow data (Lee & Moser Reference Lee and Moser2015), while blue represents DNS pipe flow data (Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021, Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2022; Pirozzoli Reference Pirozzoli2024).

Figure 4 illustrates the ratio between the Reynolds shear stress and the r.m.s. of pressure fluctuations at various Reynolds numbers. The results reveal a striking similarity in the profiles, which collapse onto a universal curve at sufficiently high Reynolds numbers ( $Re_\tau \gtrsim 500$ ), except in the near-wall region. This universal curve suggests a potential indirect method for estimating the r.m.s. of pressure fluctuations (excluding the near-wall region) using Reynolds shear stress data.

Using DNS data from turbulent channel and pipe flows, the ratio of Reynolds shear stress to pressure fluctuations beyond its peak value can be fitted empirically with the following functions:

(2.3a) \begin{align} \frac {-\rho \langle u v \rangle }{\sqrt {\langle pp \rangle }} & \approx 0.38\left (1- \left (\frac {y}{\delta }-0.23\right )^{1.73} - 1.37\left (\frac {y}{\delta } - 0.23\right )^{5}\right ), \quad y/\delta \gtrsim 0.23 \quad \mathrm{(channel)}, \end{align}

(2.3b) \begin{align} \frac {\rho \langle u_z u_r \rangle }{\sqrt {\langle pp \rangle }} & \approx 0.366\left (1- \left (\frac {y}{R}-0.2\right )^{1.8} - \left (\frac {y}{R} - 0.2\right )^{5}\right ), \quad y/R \gtrsim 0.2 \quad \mathrm{(pipe)}. \end{align}

The curve-fitting functions in (2.3a ) and (2.3b ) are shown as solid curves in figure 4, demonstrating excellent agreement with the DNS data. The slight differences between the empirical functions for channel and pipe flows, as seen in figure 4(b), can be attributed to the geometric differences of these flow configurations.

Figure 1(a) demonstrates that the inner-normalized pressure fluctuation profiles $\langle pp \rangle ^+$ from different Reynolds numbers collapse onto a single curve in the outer region of turbulent channel and pipe flows. Additionally, it can be readily shown that the Reynolds shear stress in the outer region of turbulent channel or pipe flow decreases linearly as ${-\langle uv \rangle }/{u_\tau ^2} \approx 1 - {y}/{\delta }$ . As a result, both the Reynolds shear stress and the r.m.s. pressure fluctuations can be approximated as functions of $y/\delta$ in the outer region of turbulent channel or pipe flows. Incorporating the logarithmic decay of pressure fluctuations, the ratio of Reynolds shear stress to $p_{\mathrm{rms}}$ can be approximated as

(2.4) \begin{equation} \frac {-\rho \langle uv \rangle }{\sqrt {\langle pp \rangle }} \approx \frac {1 - y/\delta }{\left [A \ln (y/\delta ) + B\right ]^{1/2}}. \end{equation}

This approximation is shown as the dashed curves in figure 4. However, (2.4) (and related logarithmic expressions) loses validity near the core of the channel or pipe flow. In contrast, the empirical relations (2.3a ) and (2.3b ) provide superior fits to the DNS data in this region.

In physical experiments, directly measuring pressure fluctuations within turbulent flow is considerably more challenging than measuring Reynolds shear stress. Equation (2.3) can be used to verify measurement consistency when the r.m.s. of the pressure fluctuation is measured directly. If direct measurements of pressure fluctuations are not available, then (2.3) can be utilized to estimate r.m.s. pressure fluctuation values using Reynolds shear stress measurements.

Figure 4 suggests that the Reynolds shear stress and the r.m.s. of the pressure fluctuations are of the same order of magnitude, except in the near-wall region and near the core of the pipe or channel centre. Similarly, the ratios of Reynolds normal stresses to the r.m.s. of pressure fluctuation (see Appendix A) indicate that these quantities are also of the same order of magnitude, except in the near-wall region. This similar magnitude aligns with the balance of the Poisson equation, hinting that the characteristic scales for the pressure fluctuations can be determined from the characteristics of the Reynolds stresses distributions. Although these quantities are not known a priori, they can be determined more easily in physical experiments.

3. Scaling patch analysis of the pressure fluctuation equation

The scaling patch approach, originally developed for investigating the multi-layer and multi-scaling properties of wall-bounded turbulent flows (Fife et al. Reference Fife, Klewicki, McMurtry and Wei2005a ; Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005; Fife Reference Fife2006; Fife, Klewicki & Wei Reference Fife, Klewicki and Wei2009; Wei Reference Wei2020), offers a unique perspective compared to previous scaling methods. While some concepts and ideas in the scaling patch approach are similar to previous scaling approaches, the logical train of thought in the new approach is distinctly different.

One objective of a scaling patch analysis is to reveal naturally the relative magnitudes of different terms in an equation. Such an equation typically consists of more than two terms, with the terms contributing unequally to the balance of the equation. The relative magnitude of each term is often not obvious when the equation is presented in a dimensional form. Through a systematic transformation of the dimensional equation into a dimensionless form, the scaling patch approach is able to determine the scales for the various terms (Fife Reference Fife2006; Wei Reference Wei2020). A key component of the approach is to formulate an admissible scaling for the governing equation. The admissible scaling equation will be a dimensionless equation in which at least two pre-factors of the terms in the equation have nominal order 1, and the other pre-factors have nominal order either 1 or 0.

For a fully developed turbulent channel flow, where the mean flow is confined to the streamwise direction, the Poisson equation (2.2) can be simplified as

(3.1) \begin{align} 0 & = \frac {1}{\rho } \frac {\partial ^2 p}{\partial x_i \partial x_i} + 2 \frac {\partial U}{\partial y} \frac {\partial v}{\partial x} \nonumber \\ & \quad + \bigg \{ \bigg ( \frac {\partial ^2 (u^2 - \langle u^2 \rangle )}{\partial x^2} + \frac {\partial ^2 (v^2 - \langle v^2 \rangle )}{\partial y^2} + \frac {\partial ^2 (w^2 - \langle w^2 \rangle )}{\partial z^2} \bigg ) \nonumber \\ & \quad + 2\bigg ( \frac {\partial ^2 ( uv -\langle uv \rangle )}{\partial x \partial y} + \frac {\partial ^2 ( uw )}{\partial x \partial z} + \frac {\partial ^2 ( vw )}{\partial y \partial z}\bigg ) \bigg \}. \end{align}

In this simplified form, the rapid term corresponds to the second term on the right-hand side of the equation. The remaining terms, spanning the second and third lines, represent the slow term. Note that in turbulent channel or pipe flows, the Reynolds shear stresses $ \langle uw \rangle$ and $ \langle vw \rangle$ are zero.

In conventional scaling analysis of equations, a reference scale is typically assigned to each flow variable and spatial coordinate. For example, a reference velocity scale $ u_{{ref}}$ is used to characterize and scale the mean flow, while $ x_{\textit{ref}}$ and $ y_{\textit{ref}}$ serve as characteristic length scales in the streamwise and wall-normal directions, respectively. However, in (3.1), the rapid and slow terms involve combinations of derivatives of multiple flow variables. As a result, it is more practical to group the characteristic scales rather than assigning separate scales to each variable. For instance, the scaled rapid term can be expressed as

(3.2) \begin{equation}\displaystyle 2 \left ( \frac {\partial U}{\partial y} \frac {\partial v}{\partial x}\right )^* = 2\left [\frac {1}{{u^2_{\textit{ref}, r}}/{l^2_{{\textit{ref}, r}}}}\right ]{\frac {\partial U}{\partial y} \frac {\partial v}{\partial x}}, \end{equation}

where $ u^2_{{{\textit {ref}}, r}}$ represents the product of characteristic velocity scales associated with the rapid term, while $ l^2_{{\textit{ref},r}}$ denotes the combined characteristic length scale. Specifically, $ l^2_{{\textit{ref}, r}}$ serves as a lumped scale encapsulating the scaling of spatial derivatives $ \partial x$ and $ \partial y$ . Alternatively, the length scale can be expressed as $ x_{{\textit{ref}, r}}\, y_{{\textit{ref}, r}}$ , where $ x_{{\textit{ref}, r}}$ and $ y_{{\textit{ref}, r}}$ correspond to the characteristic length scales for the streamwise and wall-normal derivatives, respectively. However, since the analysis does not require separate treatment of the streamwise and wall-normal directions, it is unnecessary to distinguish between $ x_{{\textit{ref}, r}}$ and $ y_{{\textit{ref}, r}}$ . Instead, a single lumped scale $ l^2_{{\textit{ref}, r}}$ is sufficient for the purposes of this study.

Similarly, the slow term can be normalized as

(3.3) \begin{align} (\mathrm{slow\, term})^* & = \left [ \frac {1}{u^2_{{\textit{ref},s}}/l^2_{{\textit{ref},s}}} \right ]\left \{ \left ( \frac {\partial ^2 (u^2 - \langle u^2 \rangle )}{\partial x^2} + \frac {\partial ^2 (v^2 - \langle v^2 \rangle )}{\partial y^2} + \frac {\partial ^2 (w^2 - \langle w^2 \rangle )}{\partial z^2} \right ) \right.\nonumber \\ & \quad{}\left.+ 2\left ( \frac {\partial ^2 ( uv -\langle uv \rangle )}{\partial x\, \partial y} + \frac {\partial ^2 ( uw )}{\partial x\, \partial z} + \frac {\partial ^2 ( vw )}{\partial y\, \partial z}\right ) \right \}, \end{align}

where $ u^2_{{\textit{ref}, s}}$ represents the product of characteristic velocity scales associated with the slow term, while $ l^2_{{\textit{ref},s}}$ denotes the product of characteristic length scales for the slow term.

To establish an admissible scaling within the framework of the scaling patch approach (Fife et al. Reference Fife, Klewicki, McMurtry and Wei2005a ; Fife Reference Fife2006; Wei Reference Wei2020), one can select $ u_{{\textit{ref},s}}$ , $ l_{{\textit{ref},s}}$ , $ u_{{\textit{ref},r}}$ and $ l_{{\textit{ref},r}}$ such that the dimensionless equation contains at least two terms with pre-factors of $ O(1)$ . The remaining terms should either have pre-factors of $ O(1)$ or represent higher-order contributions.

3.1. New outer scaling

In the outer region, the mean streamwise velocity deficit serves as a more appropriate measure of relative mean flow variation. Similarly, the deficit of the pressure fluctuation from its centreline value provides a more suitable representation of its relative variation in this region. Consequently, these deficits are used in scaling the mean Poisson equations:

(3.4) \begin{align} 0 ={}& \left [\frac {p_{{\textit{ref}-\textit{out}}}}{{\rho}l^2_{{\textit{ref}-\textit{out}}}} \right ] \left (\frac {\partial ^2 (p - \sqrt {\langle pp \rangle _{\mathrm{ctr}}})}{\partial x_i\, \partial x_i}\right )^* - 2 \left [ \frac {u^2_{{\textit{ref},\textit{r}-\textit{out}}}}{l^2_{{\textit{ref},\textit{r}-\textit{out}}}} \right ] \left ( \frac {\partial (U_c - U)}{\partial y} \frac {\partial v}{\partial x} \right )^* \nonumber \\ & {}+ \left [ \frac {u^2_{{\textit{ref},\textit{s}-\textit{out}}}}{l^2_{{\textit{ref},\textit{s}-\textit{out}}}} \right ] \left ( \left \{ \frac {\partial ^2 (u^2 - \langle u^2 \rangle )}{\partial x^2} + \underline { \frac {\partial ^2 (v^2 - \langle v^2 \rangle )}{\partial y^2} } + \frac {\partial ^2 (w^2 - \langle w^2 \rangle )}{\partial z^2} \right \} \right.\nonumber \\ & {}\left.+ 2\left \{ \underline {\frac {\partial ^2 ( uv -\langle uv \rangle )}{\partial x\, \partial y}} + \frac {\partial ^2 ( uw )}{\partial x\, \partial z} + \frac {\partial ^2 ( vw )}{\partial y\, \partial z}\right \} \right )^*. \end{align}

Here, $_{{-out}}$ in a subscript denotes characteristic scales associated with the outer region.

The DNS data (Kim Reference Kim1989; Hoyas & Jiménez Reference Hoyas and Jiménez2006; Gerolymos et al. Reference Gerolymos, Sénéchal and Vallet2013; Panton et al. Reference Panton, Lee and Moser2017) confirm that the rapid and slow terms are of comparable magnitudes, indicating that both contribute significantly to achieving an admissible scaling. In von Kármán’s defect law (von Kármán Reference von Kármán1934), the mean velocity deficit in the outer region is scaled by the friction velocity $u_\tau$ . Zagarola & Smits (Reference Zagarola and Smits1998) demonstrated that scaling the mean velocity deficit with $U_{\mathrm{_{ZS}}}= U_{\mathrm{ctr}} - U_{\mathrm{b}}$ provides a better collapse of data. Here, $U_{\mathrm{ctr}}$ is the mean velocity at the centreline, and $U_{\mathrm{b}}$ is the bulk mean velocity. Using a logarithmic approximation for the mean velocity deficit, it can be shown that $U_{\mathrm{ctr}} - U_{\mathrm{b}} \approx 3.75 u_\tau$ (see p. 609 in Schlichting Reference Schlichting1979). Experimental data from the Princeton-ONR Superpipe experiment reveal that $U_{\mathrm{ctr}} - U_{\mathrm{b}}$ decreases from approximately $4.55 u_\tau$ at $Re_\tau = 1000$ to approximately $4.25 u_\tau$ at $Re_\tau = 10\,000$ , and subsequently remains nearly constant at $4.25 u_\tau$ as the Reynolds number continues to increase (Zagarola & Smits Reference Zagarola and Smits1998; McKeon & Smits Reference McKeon and Smits2002). In the outer region, the velocity fluctuation $v$ in the rapid term is effectively scaled by $u_\tau$ , making $u^2_{{\textit{ref,r-out}}}$ appropriately scaled by $u^2_\tau$ .

For the slow term, given the homogeneity in the streamwise and spanwise directions, it is reasonable to hypothesize that the leading-order contributions are the underlined components. Furthermore, considering the gradual variation of $\langle v^2 \rangle$ in the outer region (see figure 2), the dominant component in the slow term would be $\partial ^2(uv - \langle uv \rangle )/(\partial x\, \partial y)$ . Therefore, a suitable scale for $u^2_{{\textit{ref,s}-\mathrm{out}}}$ is $R_{uv}|_{{max}}$ . Additionally, $l^2_{{\textit{ref,r-out}}}$ and $l^2_{{\textit{ref,s-out}}}$ are expected to be similar. In other words, in the outer region of turbulent channel or pipe flows, the rapid and slow terms share the same scaling. Following the scaling patch analysis, the pressure fluctuations are also expected to adhere to this scaling.

Recently, Wei & Knopp (Reference Wei and Knopp2023) proposed an outer scaling framework for the mean momentum equation in turbulent boundary layer flows subjected to both zero and non-zero pressure gradients. In this framework, the lower bound of the outer layer is defined by the location of the maximum Reynolds shear stress, denoted as $y_{\mathrm{m}}$ , which corresponds to the centre of the meso layer in the four-layer structure of wall-bounded turbulence proposed by Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005). The characteristic scale for the Reynolds shear stress in the outer region is given by its maximum value $R_{uv}|_{{max}}$ . In turbulent boundary layers under pressure gradients, the r.m.s. of pressure fluctuations and the maximum Reynolds shear stress can differ significantly from the wall shear stress, as reported by Simpson et al. (Reference Simpson, Ghodbane and McGrath1987). However, for turbulent internal flows, $R_{uv}|_{{max}}$ closely approximates the friction velocity squared, $u^2_\tau$ , at high Reynolds numbers (see e.g. Pope Reference Pope2000; Fife et al. Reference Fife, Klewicki, McMurtry and Wei2005b ).

Based on these findings, we propose an outer scaling for the pressure fluctuation in the outer region of turbulent channel and pipe flows. The reference scales are defined as $y_{\textit{ref}} = \delta -y_{\mathrm{m}}$ , $p_{\textit{ref}-\textit{out}} = \rho (-\langle uv \rangle )_{{max}}$ . The corresponding outer normalized variables are given by

(3.5) \begin{align} y^- =& \, \frac {y-y_{m{m}}}{\delta - y_{m{m}}}, \end{align}

(3.6) \begin{align} \langle pp \rangle ^- = & \, \frac {\langle pp \rangle - \langle pp \rangle _{{ctr}}}{(\rho (-\langle uv \rangle )_{{max}})^2}. \end{align}

Figure 5.

New outer scaling for the variance of pressure fluctuation: (a) channel flow, (b) pipe flow.

The new outer-scaled variance of pressure fluctuations in turbulent channel and pipe flows is presented in figure 5, showing good data collapse at high Reynolds numbers in the outer region. At high Reynolds numbers, it is well established that $y_m \sim \sqrt {\delta \nu /u_\tau } = \delta /\sqrt {Re_\tau }$ , as demonstrated in Afzal (Reference Afzal1982), Long & Chen (Reference Long and Chen1981), Sreenivasan & Sahay (Reference Sreenivasan and Sahay1997) and Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005). Consequently, at sufficiently high Reynolds numbers, $y_{\mathrm{m}} \ll \delta$ and $(-\langle uv \rangle )_{{max}} \approx u^2_\tau$ , and the new outer scaling approaches the traditional outer scaling shown in figure 1(a), represented by the dashed green curve in figure 5.

3.2. Inner scaling of the pressure fluctuation

As highlighted in the Introduction, an appropriate scaling for the mean Poisson equation in the near-wall region involves using the difference in pressure fluctuations relative to their wall value. Accordingly, the inner-scaled form of the mean Poisson equation is expressed as

(3.7) \begin{align} 0 =& \bigg [\frac {p_{\textit{ref}-\textit{in}}}{\rho l^2_{\textit{ref}-\textit{in}}} \bigg ] \bigg (\frac {\partial ^2 (p - \sqrt {\langle pp \rangle _{{wall}}})}{\partial x_i \partial x_i}\bigg )^* + 2 \bigg [ \frac {u^2_{\textit{ref},r-\mathrm{in}}}{l^2_{{\textit{ref}},r-\mathrm{in}}} \bigg ] \bigg ( \frac {\partial U}{\partial y} \frac {\partial v}{\partial x} \bigg )^* \nonumber \\ & + \bigg [ \frac {u^2_{{\textit{ref}},s-\mathrm{in}}}{l^2_{{\textit{ref}},s-\mathrm{in}}} \bigg ] \bigg ( \bigg \{ \frac {\partial ^2 (u^2 - \langle u^2 \rangle )}{\partial x^2} + \underline { \frac {\partial ^2 (v^2 - \langle v^2 \rangle )}{\partial y^2} } + \frac {\partial ^2 (w^2 - \langle w^2 \rangle )}{\partial z^2} \bigg \} \nonumber \\ & + 2\bigg \{ \underline {\frac {\partial ^2 ( uv -\langle uv \rangle )}{\partial x \partial y}} + \frac {\partial ^2 ( uw )}{\partial x \partial z} + \frac {\partial ^2 ( vw )}{\partial y \partial z}\bigg \} \bigg )^* \end{align}

For the rapid term, a suitable scale for $u^2_{{\textit{ref}},r-\mathrm{in}}$ would also be $u^2_\tau$ , as both $U$ and $v$ are properly scaled by $u_\tau$ in the near-wall region. Figure 2 shows that the wall-normal variations of $\langle v^2 \rangle$ and $\langle uv \rangle$ are similar in this region. Given that streamwise variations are generally smaller than wall-normal variations, it is reasonable to hypothesize that the leading-order component in the slow term for the near-wall region is $\partial ^2(v^2-\langle v^2 \rangle )/\partial y^2$ . While the Reynolds shear stress is bounded by $u^2_\tau$ , the peak values of $\langle v^2 \rangle$ increase with the Reynolds number, though very slowly. Thus a suitable scale for $u^2_{{\textit{ref}},s-\mathrm{in}}$ is $ \langle vv \rangle _{{max}}$ .

In contrast to the outer region, where the rapid and slow terms share the same scaling, the rapid and slow terms in the near-wall region exhibit distinct scalings. As a result, the pressure fluctuations in the near-wall region are governed by a mixed scaling. This observation aligns with the trends shown in figure 1(b), where the pressure fluctuation difference $\langle pp \rangle - \langle pp \rangle _{{wall}}$ normalized by $(\rho u^2_\tau )^2$ exhibits a gradual increase with Reynolds number.

Panton et al. (Reference Panton, Lee and Moser2017) analysed the rapid and slow terms using DNS data from Lee & Moser (Reference Lee and Moser2015), revealing that the increase of pressure fluctuation in the near-wall region is primarily attributed to the slow term. Based on these findings, we propose scaling the pressure fluctuations in the near-wall region using the slow-term scale, specifically the peak value $\rho \langle vv \rangle _{{max}}$ . The inner-scaled pressure fluctuation difference from its wall value can then be expressed as

(3.8) \begin{equation} \frac {\langle pp \rangle - \langle pp \rangle _{{wall}}}{(\rho \langle vv \rangle _{{max}})^2}. \end{equation}

Empirical observations indicate that the maximum pressure fluctuation variance consistently occurs at $y^+ \approx 30$ , with little to no dependence on Reynolds number. Similarly, the peak of $\langle uu \rangle$ is observed at $y^+ \approx 15$ , where any Reynolds number dependence appears to be minimal. In contrast, the peak locations of $\langle vv \rangle$ , $\langle ww \rangle$ and $-\langle u v \rangle$ , when scaled by $\nu /u_\tau$ , clearly increase with Reynolds number. These observations suggest that the peak locations of $\langle pp \rangle$ or $\langle uu \rangle$  (as in figure 6(a)), or simply the traditional inner-scaled wall-normal distance $y^+$ , can be used to present pressure fluctuations in the near-wall region (as in figure 6(b)).

Figure 6.

New inner scaling for the variance of pressure fluctuation: (a) channel flow, (b) pipe flow.

Figure 6 presents the new inner scaling of the pressure fluctuation variance from DNS data of turbulent pipe and channel flows. To emphasize the near-wall region, the plot employs semi-logarithmic axes and focuses exclusively on this area. The figure demonstrates that the new scaling effectively collapses the pressure fluctuation variance profiles across a range of Reynolds numbers, with the exception of cases at lower Reynolds numbers.

4. Summary

This study introduces novel scaling methods for pressure fluctuations in turbulent flows, improving our understanding of these fundamental and complex phenomena. Using the scaling patch analysis of the Poisson equation, we investigate the scaling of the rapid and slow terms in both the outer and inner regions of the wall layer. We find that in the outer region, the rapid and slow terms exhibit similar scaling behaviours, while in the inner region, they follow distinct scaling patterns. Consequently, pressure fluctuations in the inner region are governed by a mixed scaling. Building on these findings, we propose new scalings for pressure fluctuations in both the outer and inner regions. The new outer scaling is based on the location and magnitude of the maximum Reynolds shear stress, while the inner scaling incorporates the maximum Reynolds normal stress in the wall-normal direction. These new scaling methods address key limitations of traditional approaches, which depend primarily on viscosity and wall shear stress, and are grounded in the characteristics of the mean velocity profiles and momentum equations.

Figure 7.

Ratios between the Reynolds normal stresses and r.m.s. of pressure fluctuations. (a,c,e) Turbulent channel flow, showing the streamwise, wall-normal and spanwise directions, respectively. (b,d,f) Turbulent pipe flow, showing the axial, radial and azimuthal directions, respectively.

The new scalings provide a more accurate representation of pressure fluctuations across a range of turbulent flow regimes, offering greater precision in predictive models, and deeper insights into the mechanisms driving pressure fluctuation dynamics. Additionally, a strong correlation between the r.m.s. of pressure fluctuations and Reynolds shear stress has been identified, and an empirical equation is proposed to capture this similarity. This empirical relationship is particularly valuable for estimating the r.m.s. values of pressure fluctuations in the flow, which are difficult to measure in physical experiments. These advancements offer powerful new tools for analysing and predicting pressure fluctuations in wall-bounded turbulent flows, enhancing both theoretical understanding and practical applications in fluid mechanics.