2.1. Wall-units and mixed scalings

Because of this $\{\bar {\rho },\tilde {T}, \bar {\mu }\}$-stratification (figure 1), Huang, Coleman & Bradshaw (Reference Huang, Coleman and Bradshaw1995) realised that the standard wall-units inherited by incompressible flow analyses

(2.1a)\begin{equation} (\cdot)^+\ \text{units: } \{\bar{\tau}_w,\bar{\mu}_w,\bar{\rho}_w\}\implies y^+:=\dfrac{\bar{\rho}_w} {\bar{\mu}_w}\sqrt{\dfrac{\bar{\tau}_w}{\bar{\rho}_w}}(y-y_w);\quad Re_{\tau_w}:=\delta^+ \end{equation}

fail to correctly represent the major part of the flow, and suggested an alternative $y$-variable scaling which replaces $\{\bar {\mu }_w,\bar {\rho }_w\}$ by their local values

(2.1b)\begin{equation} (\cdot)^\star\ \text{units: } \{\bar{\tau}_w,\bar{\mu}(y),\bar{\rho}(y)\}\implies y^\star:=\dfrac{\bar{\rho}(y)}{\bar{\mu}(y)}\sqrt{\dfrac{\bar{\tau}_w}{\bar{\rho}(y)}}(y-y_w);\quad Re_{\tau^\star}:=\delta^\star. \end{equation}

This HCB friction Reynolds number $Re_{\tau ^\star }$ (2.1b) is the appropriate parameter determining the peak value of the shear Reynolds stress $-\overline {\rho u''v''}$ (Huang et al. Reference Huang, Coleman and Bradshaw1995) and is generally accepted as the representative Reynolds number for compressible TPC flow (Trettel & Larsson Reference Trettel and Larsson2016). However, both scalings (2.1a), (2.1b) become identical at the wall and, in general, neither is well adapted in the near-wall region $y^\star \lessapprox 10$. In that region, $(\cdot )^\star$-units vary rapidly in the wall-normal direction because of the strong temperature gradient (figure 1). The drawback of this rapid variation was particularly felt in the present work while studying the near-wall pressure field in relation with the near-wall low-speed streaks (§ 4.1). It is shown that, for a given flow defined by the couple $(Re_{\tau ^\star },\bar {M}_{{CL}_x})$, the average spanwise distance between streaks $\varLambda _{S}^{ ( z )}$ is practically $y$-independent in the near-wall region $y^\star \lessapprox 10$, so that studying the velocity and pressure fields in $(x^\star,z^\star )$ planes would misleadingly suggest strong $y$-normal variations in $(\cdot )^\star$-units (2.1b). On the other hand, it was also found that standard $(\cdot )^+$-units (2.1a) are completely inadequate in comparing results for different $\bar {M}_{{CL}_x}$ at constant $Re_{\tau ^\star }$. Noting that the strong variation of mean-flow properties essentially occurs in the region $y^\star \lessapprox 15$, we found that an alternative $y$-constant scaling which uses centreline thermodynamic properties $\{\bar {\mu }_{CL},\rho _{CL}\}$ everywhere performed better. In the TPC case, $\max _y \bar {T} = \bar {T}_{CL}$ occurs at the centreline, with monotonic decrease towards the wall (figure 1). On the other hand, in the case of adiabatic-wall ZPG TBLs, where $\max _y\bar {T} = \bar {T}_w$ occurs at the wall, with monotonic decrease towards the boundary-layer edge, standard wall-units (2.1a) perform quite well (Bernardini & Pirozzoli Reference Bernardini and Pirozzoli2011; Zhang et al. Reference Zhang, Duan and Choudhari2017). Both these choices can be accommodated, for the entire cold-wall range $\bar {T}_w/T_r\leqslant 1$, by

(2.1c)\begin{equation} (\cdot)^\ddagger\ \text{units: } \left\{\bar{\tau}_w,\max_y\bar{\mu},\min_y\bar{\rho}\right\} \implies y^\ddagger:=\dfrac{\min_y\bar{\rho}}{\max_y\bar{\mu}}\sqrt{\dfrac{\bar{\tau}_w}{\min_y\bar{\rho}}}(y-y_w);\quad Re_{\tau^\ddagger}:=\delta^\ddagger. \end{equation}

Obviously, for TPC (2.2a) or adiabatic-wall conditions (2.2b)

(2.2a)$$\begin{gather} \left(\max_y\bar{\mu},\min_y\bar{\rho}\right)=(\bar{\mu}_{CL},\bar{\rho}_{CL}) \stackrel{(2.1b),~(2.1c)}{\implies} \quad \delta^\ddagger = \delta^\star\iff Re_{\tau^\ddagger}= Re_{\tau^\star}, \end{gather}$$

(2.2b)$$\begin{gather}\left(\max_y\bar{\mu},\min_y\bar{\rho}\right)=(\bar{\mu}_w ,\bar{\rho}_w) \stackrel{(2.1a),~(2.1c)}{\implies} \quad \delta^\ddagger = \delta^+ \iff Re_{\tau^\ddagger}= Re_{\tau_w}. \end{gather}$$

Therefore, this $y$-independent system of mixed (friction/thermodynamics) scaling (2.1c) $(\cdot )^\ddagger$-units retains $(Re_{\tau ^\star },\bar {M}_{{CL}_x})$ as global flow parameters in canonical compressible TPC flow (2.2a). This point is discussed further in (§ 4).

Note that the unit for $p$ is $\bar {\tau }_w$ in all of these systems (2.1), i.e.

(2.3)\begin{equation} (\mbox{2.1})\implies p^\ddagger \equiv p^\star \equiv p^+:=\dfrac{p}{\bar{\tau}_w}{.} \end{equation}

2.2. Computational method and statistics

The computations were run using a very-high-order $O(\Delta \ell ^{17})$ upwind-biased scheme (Gerolymos, Sénéchal & Vallet Reference Gerolymos, Sénéchal and Vallet2009), implemented in a DNS solver (Gerolymos, Sénéchal & Vallet Reference Gerolymos, Sénéchal and Vallet2010) which has been validated extensively in previous compressible (Gerolymos & Vallet Reference Gerolymos and Vallet2014, Reference Gerolymos and Vallet2018) and low-$\bar {M}_{{CL}_x}$ (Gerolymos et al. Reference Gerolymos, Sénéchal and Vallet2013; Gerolymos & Vallet Reference Gerolymos and Vallet2016, Reference Gerolymos and Vallet2019) work, by comparison with standard DNS data (Coleman et al. Reference Coleman, Kim and Moser1995; Moser, Kim & Mansour Reference Moser, Kim and Mansour1999; Foysi et al. Reference Foysi, Sarkar and Friedrich2004; Vreman & Kuerten Reference Vreman and Kuerten2014), including spectra (Gerolymos et al. Reference Gerolymos, Sénéchal and Vallet2010) and higher-order-derivatives dissipation statistics (Gerolymos & Vallet Reference Gerolymos and Vallet2016).

The computational box was $L_x=8{\rm \pi} \delta$ long in the streamwise direction and $L_z=4{\rm \pi} \delta$ wide in the spanwise direction ($\delta$ is the channel half-height), which is in the upper range of DNS computational domains for channel flow both compressible (Modesti & Pirozzoli Reference Modesti and Pirozzoli2016; Trettel & Larsson Reference Trettel and Larsson2016; Tang et al. Reference Tang, Zhao, Wan and Liu2020; Yu et al. Reference Yu, Xu and Pirozzoli2019, Reference Yu, Xu and Pirozzoli2020; Yu & Xu Reference Yu and Xu2021) and incompressible (Kim, Moin & Moser Reference Kim, Moin and Moser1987; Moser et al. Reference Moser, Kim and Mansour1999; Hu & Sandham Reference Hu and Sandham2001; Hoyas & Jiménez Reference Hoyas and Jiménez2006; Lee & Moser Reference Lee and Moser2015; Hoyas et al. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022). Computational resolution in wall-units (table 1) is typical of compressible TPC flow DNS computations, in line with grid-resolution studies (Gerolymos et al. Reference Gerolymos, Sénéchal and Vallet2010; Modesti & Pirozzoli Reference Modesti and Pirozzoli2016; Trettel & Larsson Reference Trettel and Larsson2016).

Table 1. Parameters of the DNS computations: $L_x$, $L_y$, $L_z$ ($N_x$, $N_y$, $N_z$) are the dimensions (number of grid points) of the computational domain ($x=$ homogeneous streamwise, $y=$ wall-normal, $z=$ homogeneous spanwise direction); $u$, $v$, $w$ are the velocity components along $x$, $y$, $z$; $\delta$ is the channel half-height; $\Delta x^+$, $\Delta y_w^+$, $\Delta y_{CL}^+$, $\Delta z^+$ are the mesh sizes in wall-units (2.1a); $(\cdot )_w$ denotes wall and $(\cdot )_{CL}$ centreline values; $N_{y^+\leqslant 10}$ is the number of grid points between the wall and $y^+=10$; $Re_{\tau ^\star }$ is the friction Reynolds number in HCB scaling (2.1b); $\bar {M}_{{CL}_x}$ is the centreline Mach number (2.4); $Re_{\tau _w}$ is the friction Reynolds number (2.1a); $M_{{B}_w}$ is the bulk Mach number at wall sound speed (§ 2.3); $\varDelta t^+$ is the computational time step in wall-units; $t_{OBS}^+$ is the observation interval in wall-units over which single-point statistics were computed; $\Delta t_s^+$ is the sampling time step for the single-point statistics in wall-units.

Some early computations in the database (Gerolymos & Vallet Reference Gerolymos and Vallet2014, Reference Gerolymos and Vallet2018) were initialised following the procedure described in Gerolymos et al. (Reference Gerolymos, Sénéchal and Vallet2010, (44), p. 790). However, most of the computations were initialised by thermodynamic rescaling and linear interpolation of an instantaneous turbulent flow field at different $(Re_{\tau ^\star },\bar {M}_{{CL}_x})$-conditions.

Computational experience shows that $p'_{rms}$ reaches statistical convergence much slower than velocity covariances and other statistics. For this reason we were particularly careful to allow for a sufficiently long transient $t_{CNVRG}$. Upon reaching a statistically converged state, after the initial transient, single-point statistics were acquired at each time step and sampled over several thousands of time wall-units (table 1). For the $(Re_{\tau ^\star },\bar {M}_{{CL}_x}) = (965,1.50)$ flow, initialised from a statistically converged coarse-grid flow field, after a transient of $t^+_{CNVRG} = 9853$ ($t_{CNVRG}\tilde u_{CL}/\delta = 180$), statistics were acquired for an additional $t^+_{OBS} = 5346$ ($t_{OBS}\tilde u_{CL}/\delta = 98$).

Air thermodynamics were approximated by a perfect gas EoS with Sutherland laws for viscosity and heat conductivity (Gerolymos & Vallet Reference Gerolymos and Vallet2014, (2.1), p. 706). van Driest (Reference van Driest1951, figure 17, p. 157) highlights the overestimation of dynamic viscosity at high temperatures when using a power law in lieu of the more accurate Sutherland's law. Since temperature stratification is induced by viscous heating $\overline {S_{ij}\tau _{ij}}$ (Coleman et al. Reference Coleman, Kim and Moser1995) there are small differences at high $\bar {M}_{{CL}_x}$ in centreline-to-wall temperature ratio $\tilde {T}_{CL}/T_w$ and in the $M_{{B}_w}(Re_{\tau ^\star },\bar {M}_{{CL}_x})$ relation between DNS data using different laws for viscosity. This observation not withstanding, the present results using Sutherland's law are in very good agreement with the $p'_{rms}$ data of Modesti & Pirozzoli (Reference Modesti and Pirozzoli2016) who used a power law, for flows at similar $(Re_{\tau ^\star },\bar {M}_{{CL}_x})$ conditions.

2.3. Database

The target during the construction of the database was to obtain a matrix of $(Re_{\tau ^\star },\bar {M}_{{CL}_x})$ values, where $Re_{\tau ^\star }$ is the HCB friction Reynolds number (2.1b) and

(2.4)\begin{equation} \bar{M}_{{CL}_x}:=\overline{\left(\dfrac{u_{CL}}{a_{CL}}\right)} \end{equation}

is the average streamwise centreline Mach number. Note that the often used approximations $\bar {M}_{CL} \approxeq \bar {M}_{{CL}_x} \approxeq \tilde {u}_{CL}/a(\tilde {T}_{CL})$ are confirmed by the data with good accuracy because $v_{CL} \ll u_{CL} \gg w_{CL}$ and $(T'_{rms}/\bar {T})_{CL} \lll 1$. We use the exact average (2.4) in our work.

In general, $Re_{\tau ^\star }$ is adopted by workers in the field (§ 2.1), because it is the proper parameter for the scaling of the Reynolds shear stress $-\overline {\rho u''v''}$ and, hence, controls the mean velocity $\tilde u$ profile (Trettel & Larsson Reference Trettel and Larsson2016). We chose the centreline Mach number $\bar {M}_{{CL}_x}$ for the matrix construction because it allows comparison with boundary-layer data which are always referenced to the external flow Mach number, and because its physical significance is clear. Furthermore the coefficients of variation ${CV}_{({\cdot })'} := ({\cdot })'_{rms}/\overline {({\cdot })}$ of the thermodynamic fluctuations $\{p',\rho ',T',s'\}$ scale with $\bar {M}_{{CL}_x}^2$ (Gerolymos & Vallet Reference Gerolymos and Vallet2014, Reference Gerolymos and Vallet2018).

Neither $Re_{\tau ^\star }$ nor $\bar {M}_{{CL}_x}$ are fixed in the computations, the flow conditions being determined by the set $\{\rho _{B}, \dot {m}_{B}, \bar {T}_w\}$ (Gerolymos et al. Reference Gerolymos, Sénéchal and Vallet2010), where $\rho _{B} := \bar {\rho }^{xyzt}$ is the bulk density, $\dot {m}_{B} := L_yL_z\rho _{B} u_{B} := L_yL_z\overline {\rho u}^{xyzt}$ is the mass flow, and $T_w = {\rm const.}$ is the wall temperature (as a consequence the bulk-wall Mach number $M_{{B}_w} := u_{B}/\bar {a}_w$ is fixed).

However, we took particular care in the construction of the database to obtain the closest possible target $\bar {M}_{{CL}_x}\in \{0.33,0.80,1.50,2.00\}$ and $Re_{\tau ^\star }\in \{75,100,110,250,340,1000\}$ (table 1). The cost of the computations strongly increases with $\bar {M}_{{CL}_x}$ if nearly constant resolution in standard wall-units is sought (this is the most stringent resolution assessment), as a result of the rapid increase of the ratio $Re_{\tau _w}/Re_{\tau ^\star }$ with increasing $\bar {M}_{{CL}_x}$. This new database (table 1) extends available data at $\bar {M}_{{CL}_x} \in \{0.8,1.5\}$ to higher $Re_{\tau ^\star } \approxeq 1000$, which to the best of the authors' knowledge is the highest reported $Re_{\tau ^\star }$ for compressible TPC flow.

3.1. Wall-pressure-fluctuation amplitude

When plotting (figure 2) the compressible data vs $Re_{\tau _w}$ (2.1a) we observe similar curves for different $\bar {M}_{{CL}_x}$, shifted towards lower levels of $[p'_{rms}]_w^+$ with increasing $\bar {M}_{{CL}_x}$, highlighting a distinct $\bar {M}_{{CL}_x}$-dependence. Using instead the HCB friction Reynolds number $Re_{\tau ^\star }$ (2.1b) clusters the data much closer together (figure 2). Nonetheless, careful observation shows that there remains a weak $\bar {M}_{{CL}_x}$-dependence, which appears clearly when plotting the data for different nearly constant $Re_{\tau ^\star }$ versus $\bar {M}_{{CL}_x}$ (figure 2). At constant $Re_{\tau ^\star }$, $[p'_{rms}]_w^+$ slightly increases with increasing $\bar {M}_{{CL}_x}$, relative to the incompressible flow limit ($\bar {M}_{{CL}_x}\to 0$) level (figure 2).

Figure 2. Wall pressure fluctuation r.m.s. $[p'_{rms}]_w^+$, in wall-units (2.3), plotted against the friction Reynolds numbers $Re_{\tau _w}$ (2.1a) and $Re_{\tau ^\star }$ (2.1b), for varying Mach numbers $0.32 \leqslant \bar {M}_{{CL}_x} \leqslant 2.49$, and against the centreline Mach number $\bar {M}_{{CL}_x}$ (2.4), for varying Reynolds numbers $66 \leqslant Re_{\tau ^\star } \leqslant 983$, from the present database (table 1); also included are incompressible DNS data (Kim et al. Reference Kim, Moin and Moser1987; Moser et al. Reference Moser, Kim and Mansour1999; Hu & Sandham Reference Hu and Sandham2001; Hoyas & Jiménez Reference Hoyas and Jiménez2006; Lee & Moser Reference Lee and Moser2015; Hoyas et al. Reference Hoyas, Oberlack, Alcántara-Ávila, Kraheberger and Laux2022) and incompressible-flow correlation (Chen & Sreenivasan Reference Chen and Sreenivasan2022) using $Re_{\tau ^\star }$ (CS, 2022).

Carefully choosing the appropriate Reynolds and Mach number definitions, can only diminish the influence of one of the two parameters, but not altogether remove it: compressible TPC flow is biparametric.

3.2. ${p'}^+_{rms}$ profiles

The profiles of $p'_{rms}$ at nearly constant $\bar {M}_{{CL}_x} \in \{0.33,0.80,1.50,2.00\}$ (figure 3) show an increase in the level of $p'$ with increasing $Re_{\tau ^\star }$, quite similar to the incompressible flow behaviour (Panton et al. Reference Panton, Lee and Moser2017; Chen & Sreenivasan Reference Chen and Sreenivasan2022). However, considering the effect of $\bar {M}_{{CL}_x}$ at nearly constant $Re_{\tau ^\star } \in \{100,110,250,340\}$ shows that in the near-wall region, which can be defined as $y^\star \lessapprox 15$, the level of $p'$ is strongly dependent on the Mach number.

Figure 3. Profiles of pressure-fluctuation r.m.s. $p'_{rms}$, in wall-units (2.3), plotted against the HCB-scaled (2.1b) non-dimensional distance from the wall $y^\star$ (logscale), for varying $73 \leqslant Re_{\tau ^\star } \leqslant 965$ at nearly constant $\bar {M}_{{CL}_x} \in \{0.33,0.80,1.50, 2.00\}$ and for varying $0.32 \leqslant \bar {M}_{{CL}_x} \leqslant 2.49$ at nearly constant $Re_{\tau ^\star } \in \{100,110,250,340\}$ (also scaled by its peak value, $[p'_{rms}]_{PEAK}$ (3.1), for the different nearly constant $Re_{\tau ^\star }$), from the present DNS database (table 1).

At the incompressible flow limit, the maximal value of $p'_{rms}$ is located in the range $29 \lessapprox y^\star \lessapprox 32$ (depending on the Reynolds number). We can define this peak value as a local maximum of the $p'_{rms}$ profile

(3.1)\begin{equation} [p'_{rms}]_{PEAK}^+:=\mathrm{local}\max_{y>0}{{p'}^+_{rms}}. \end{equation}

With increasing $\bar {M}_{{CL}_x}$ the wall level $[p'_{rms}]_w$ increases relative to the peak $[p'_{rms}]_{PEAK}$. The data for $Re_{\tau ^\star } \approxeq 100$ show (figure 3) that at $\bar {M}_{{CL}_x} \approxeq 1.9$ the decrease from $[p'_{rms}]_{PEAK}$ to the wall ceases to be monotonic, but instead the profile forms a local minimum before increasing again towards the wall. This increase becomes more pronounced with increasing $\bar {M}_{{CL}_x}$ and, at $\bar {M}_{{CL}_x} \gtrapprox 2.3$, $[p'_{rms}]_w$ rises slightly higher than $[p'_{rms}]_{PEAK}$ (figure 3). For the case $(Re_{\tau ^\star },\bar {M}_{{CL}_x}) = (113,2.49)$ (figure 3) the wall level is clearly higher than the local maximum $[p'_{rms}]_{PEAK}$ (3.1). This $\bar {M}_{{CL}_x}$ effect is confirmed by the higher $Re_{\tau ^\star } \in \{250,341\}$ data (figure 3) which follow the same trend. The $p'_{rms}$ profiles are better understood by non-dimensionalising by the peak value $[p'_{rms}]_{PEAK}$ (3.1), thereby correcting for the slight scatter imparted to the data by the small variations of $Re_{\tau ^\star }$ around the nominal target values of $\{100,110,250,341\}$ (figure 3).

Note that the centreline region (figure 3) shows a consistent $\bar {M}_{{CL}_x}$-effect, the centreline level $[p'_{rms}]_{CL}^+$ increasing with increasing $\bar {M}_{{CL}_x}$, at constant $Re_{\tau ^\star }$.

3.3. ${p'_{rms}}$ peak

The data for $[p'_{rms}]_{PEAK}^+$ (figure 4), with varying $Re_{\tau ^\star }$ at different $\bar {M}_{{CL}_x}$, are quite close to the incompressible DNS data (indeed closer compared with $[p'_{rms}]_w^+$; figure 2). There is, at constant $Re_{\tau ^\star }$, a small decrease with $\bar {M}_{{CL}_x}$ (figure 4), but notably smaller than the $\bar {M}_{{CL}_x}$-dependence of the wall value $[p'_{rms}]_w^+$ (figure 2).

Figure 4. Peak value of pressure-fluctuation r.m.s. $[p'_{rms}]_{PEAK}^+$ (3.1), in wall-units (2.3), plotted against $Re_{\tau ^\star }$ (2.1b), for varying $0.32 \leqslant \bar {M}_{{CL}_x} \leqslant 2.49$, and against $\bar {M}_{{CL}_x}$ (2.4), for varying $66 \leqslant Re_{\tau ^\star } \leqslant 983$, from the present database (table 1); also included are incompressible DNS data (Kim et al. Reference Kim, Moin and Moser1987; Moser et al. Reference Moser, Kim and Mansour1999; Hu & Sandham Reference Hu and Sandham2001; Hoyas & Jiménez Reference Hoyas and Jiménez2006; Lee & Moser Reference Lee and Moser2015) and incompressible-flow correlation (Chen & Sreenivasan Reference Chen and Sreenivasan2022) using $Re_{\tau ^\star }$ (CS, 2022).

The location of the peak

(3.2)\begin{equation} y^\star_{p'_{PEAK}}:=y^\star\big|_{[p'_{rms}]_{PEAK}} \end{equation}

was obtained by interpolating the discrete on-grid data by a degree-four polynomial in the neighbourhood of the discrete maximum, to avoid grid-dependent scatter ($[p'_{rms}]_{PEAK}^+$ is the value of the interpolating polynomial at $y^\star _{p'{PEAK}}$).

The location of the peak $y^\star _{p'{PEAK}}$ (3.2) varies both with $Re_{\tau ^\star }$ and $\bar {M}_{{CL}_x}$ (figure 5). The dependence on $Re_{\tau ^\star }$ is similar to that observed in the incompressible flow data (figure 5). In the transitional flow range $Re_{\tau ^\star } \lessapprox 100$ the peak moves away from the wall with increasing $Re_{\tau ^\star }$, reaching a local maximum around $100 \lessapprox Re_{\tau ^\star } \lessapprox 110$, i.e. at the $Re_{\tau ^\star }$-range which marks the end of transition. At higher $Re_{\tau ^\star }$, $y^\star _{p'{PEAK}}$ initially decreases towards a local minimum around $Re_{\tau ^\star } \approxeq 400$, to slowly increase again, possibly towards an asymptotic value (figure 5). The variation of $y^\star _{p'{PEAK}}$ vs $Re_{\tau ^\star }$ appears to be quite similar for different $\bar {M}_{{CL}_x}$ but shifted towards higher values with increasing $\bar {M}_{{CL}_x}$ (figure 5). The location of the peak $y^\star _{p'{PEAK}}$ (3.2) is much more sensitive to variations of $\bar {M}_{{CL}_x}$ than of $Re_{\tau ^\star }$ (figure 5). Note, however, that the $p'_{rms}$ profiles are quite flat in the neighbourhood of the peak (figure 3).

Figure 5. HCB-scaled (2.1b) non-dimensional distance from the wall of the $p'_{rms}$ peak (3.1) location, $y^\star _{p'{PEAK}}$ (3.2), plotted against $Re_{\tau ^\star }$ (2.1b), for varying Mach numbers $0.32 \leqslant \bar {M}_{{CL}_x} \leqslant 2.49$ and against $\bar {M}_{{CL}_x}$ (2.4), for varying $66 \leqslant Re_{\tau ^\star } \leqslant 983$, from the present database (table 1); also included are incompressible DNS data (Kim et al. Reference Kim, Moin and Moser1987; Moser et al. Reference Moser, Kim and Mansour1999; Hu & Sandham Reference Hu and Sandham2001; Hoyas & Jiménez Reference Hoyas and Jiménez2006; Lee & Moser Reference Lee and Moser2015).

3.4. Relative importance of $T_r/\bar {T}_w$ and $\bar {M}_{{CL}_x}$ on the near-wall field

The data for the $\bar {M}_{{CL}_x}$-dependance of $[p'_{rms}]_w^+$ (figure 2) reflect the combined influence of increasing Mach number $\bar {M}_{{CL}_x}$ and wall cooling $T_r/\bar {T}_w$, which increases very rapidly with $\bar {M}_{{CL}_x}$ in TPC where frictional heating can only be evacuated through the walls (figure 6). The data of Yu et al. (Reference Yu, Xu and Pirozzoli2020) with an artificial sink term in the energy equation extend the $\bar {M}_{{CL}_x}$ range with weakly cooled walls (figure 6).

Figure 6. Variation of wall-cooling ratio $1< T_r/\bar {T}_w \lessapprox 7.35$ ($0.136\lessapprox \bar {T}_w/T_r<1$) and centreline Mach number $0.32\lessapprox \bar {M}_{{CL}_x}\lessapprox 6.97$ for the present data (table 1) and for the Yu et al. (Reference Yu, Xu and Pirozzoli2019, Reference Yu, Xu and Pirozzoli2020) DNS data (YXP, 2020), for varying $66 \leqslant Re_{\tau ^\ddagger } \leqslant 983$.

A working correlation should be a function of the three parameters $\{Re_{\tau ^\ddagger },\bar {M}_{{CL}_x},T_r/\bar {T}_w\}$. The correlation (figure 7)

(3.3)\begin{equation} [p'_{rms}]_w^+ = \underbrace{\left(4.4-10.5Re_{\tau^\ddagger}^{-{1}/{4}}\right)}_{[p'_{rms}]_{w, {INC}}^+(Re_{\tau^\ddagger})} \underbrace{\left(\dfrac{1+0.020\bar{M}_{{CL}_x}^2}{1+0.011\bar{M}_{{CL}_x}^2}\right)\left(1+ 0.020\left(\dfrac{T_r}{\bar{T}_w}-1\right)\right)}_{g_{p'_w}(\bar{M}_{{CL}_x},T_r/\bar{T}_w)} \end{equation}

scales the incompressible-flow correlation of Chen & Sreenivasan (Reference Chen and Sreenivasan2022) $[p'_{rms}]_{w,{INC}}^+(Re_{\tau ^\ddagger })$ with a separable function $g_{p'_w}(\bar {M}_{{CL}_x},T_r/\bar {T}_w)=g_{M,p'_w}(\bar {M}_{{CL}_x})g_{T,p'_w}(T_r/\bar {T}_w)$. The choice of $Re_{\tau ^\ddagger }$ (2.1c) is made to bridge the present very-cold-wall TPC data with the mildly-cold-wall data of Yu et al. (Reference Yu, Xu and Pirozzoli2020), $\max _y \bar {T}$ progressively moving from the centreline towards the wall as adiabatic conditions are approached. The pure Mach-number effect $g_{M,p'_w}(\bar {M}_{{CL}_x})$ cannot be isolated from the available TPC data, and was instead evaluated from adiabatic TBL data (Bernardini & Pirozzoli Reference Bernardini and Pirozzoli2011), the rational function ensuring $\bar {M}_{{CL}_x}^2$-dependence at low-Mach conditions while avoiding unbounded increase at high-Mach conditions. Dividing the TPC data by $g_{M,p'_w}(\bar {M}_{{CL}_x})$ indicated a nearly linear trend with increasing wall cooling (with $\pm 5\,\%$ scatter), suggesting the form of $g_{T,p'_w}(T_r/\bar {T}_w)$. The final correlation (3.3) fits the available data reasonably well (figure 7). Although available data are limited regarding the independent effects of Mach number and wall cooling, the satisfactory collapse of the compressible data on the incompressible correlation supports the view that both effects are important, and both tend to increase $[p'_{rms}]_w^+$ relative to its incompressible-isothermal level at the same $Re_{\tau ^\ddagger }$.

Figure 7. Wall-pressure-fluctuation r.m.s. $[p'_{rms}]_w^+$, in wall-units (2.3) and scaled by $g_{p'_w}(\bar {M}_{{CL}_x},T_r/\bar {T}_w)$ (3.3), plotted against the modified friction Reynolds number $66 \leqslant Re_{\tau ^\star } \leqslant 983$ (2.1c), for varying Mach numbers $0.32 \leqslant \bar {M}_{{CL}_x} \leqslant 6.97$ and wall-cooling ratio $1< T_r/\bar {T}_w \lessapprox 7.35$ ($0.136\lessapprox \bar {T}_w/T_r<1$), from the present database (table 1) and for the Yu et al. (Reference Yu, Xu and Pirozzoli2019, Reference Yu, Xu and Pirozzoli2020) DNS data (YXP, 2020); also included is incompressible-flow correlation (Chen & Sreenivasan Reference Chen and Sreenivasan2022) using $Re_{\tau ^\ddagger }$ (CS, 2022).

Regarding the data of Yu et al. (Reference Yu, Xu and Pirozzoli2020) the actual $T_r:=\bar {T}_{CL}(1+r_f(({\gamma -1})/{2})\bar {M}_{{CL}_x}^2)$ of the statistically converged flow was used (the tabulated data refer to $T_r$ of a virtual flow used for the initialisation and boundary conditions).

3.5. Pressure–velocity correlations

Modifications of near-wall turbulence structure (Bradshaw Reference Bradshaw1977) associated with these $\bar {M}_{{CL}_x}$-effects on $p'_{rms}$ (§ 3) can be assessed by considering correlation coefficients between fluctuating pressure and velocity components

(3.4a,b)\begin{equation} c_{p'u'}:=\dfrac{\overline{p'u'}}{\sqrt{\overline{p'^2}}\sqrt{\overline{u'^2}}} \equiv\dfrac{\overline{p'u'}}{p'_{rms}u'_{rms}};\quad c_{p'v'}:=\dfrac{\overline{p'v'}}{\sqrt{\overline{p'^2}}\sqrt{\overline{v'^2}}}\equiv\dfrac{\overline{p'v'}}{p'_{rms}v'_{rms}}. \end{equation}

In general, for all of the $(Re_{\tau ^\star },\bar {M}_{{CL}_x})$ flow conditions in the database (table 1), $c_{p'u'} > 0$ for all $y^\star \in ]0,\delta ^\star ]$ (figure 8), whereas $c_{p'v'} < 0$ for all $y^\star \in ]0,\delta ^\star [$ (figure 9), with $[c_{p'v'}]_{CL} = 0$ because of symmetry.

Figure 8. Profiles of correlation coefficient $c_{p'u'}$ of streamwise pressure transport (3.4a,b), plotted against the HCB-scaled non-dimensional distance from the wall $y^\star$ (logscale and linear wall-zoom), for varying HCB Reynolds numbers $73 \leqslant Re_{\tau ^\star } \leqslant 983$ at constant centreline Mach numbers $\bar {M}_{{CL}_x} \in \{0.33,0.80,1.50, 2.00\}$, and for varying centreline Mach numbers $0.32 \leqslant \bar {M}_{{CL}_x} \leqslant 2.49$ at constant HCB Reynolds numbers $Re_{\tau ^\star } \in \{100,110,250,340\}$, from the present DNS database (table 1).

Figure 9. Profiles of correlation coefficient $c_{p'v'}$ of wall-normal pressure transport (3.4a,b), plotted against the HCB-scaled non-dimensional distance from the wall $y^\star$ (logscale and linear wall-zoom), for varying HCB Reynolds numbers $73 \leqslant Re_{\tau ^\star } \leqslant 983$ at constant centreline Mach-numbers $\bar {M}_{{CL}_x} \in \{0.33,0.80,1.50, 2.00\}$, and for varying centreline Mach numbers $0.32 \leqslant \bar {M}_{{CL}_x} \leqslant 2.49$ at constant HCB Reynolds numbers $Re_{\tau ^\star } \in \{100,110,250,340\}$, from the present DNS database (table 1).

In the near-wall region ($y^\star \lessapprox 15$) $c_{p'u'}$, at constant $\bar {M}_{{CL}_x}$, shows weak dependence on $Re_{\tau ^\star }$ (figure 8). On the other hand, as $\bar {M}_{{CL}_x}$ increases above $0.8$, at constant $Re_{\tau ^\star }$, the near-wall level of $c_{p'u'}$ also increases indicating that compressibility effects enhance the positive correlation of $p'$ with the streamwise velocity fluctuation $u'$ near the wall (figure 8). This region ($y^\star \lessapprox 15$) corresponds precisely to the region where an increase of ${p'}^+_{rms}$ was observed with increasing $\bar {M}_{{CL}_x}$ (figure 3).

Regarding $c_{p'v'}$ (figure 9) we observe near the wall ($y^\star \lessapprox 15$) a very marked influence of $\bar {M}_{{CL}_x}$, at constant $Re_{\tau ^\star }$, both in level and slope. For all of the $(Re_{\tau ^\star },\bar {M}_{{CL}_x})$ flow conditions in the database (table 1) we have roughly $c_{p'v'}(y^\star \approxeq y^\star _{p'{PEAK}}) \approxeq -0.05$, which progressively develops to a plateau with increasing $Re_{\tau ^\star }$. At low $\bar {M}_{{CL}_x} \approxeq 0.33$, $c_{p'v'}$ decreases monotonically for $y^\star \lessapprox y^\star _{p'{PEAK}}$ towards the wall where it reaches a high negative value $-[c_{p'v'}]_w \approxeq 0.4$ (depending on $Re_{\tau ^\star }$), in line with incompressible flow data (Hoyas & Jiménez Reference Hoyas and Jiménez2006; Lee & Moser Reference Lee and Moser2015). At the incompressible flow limit, which is quite well represented by the $\bar {M}_{{CL}_x}\approxeq 0.33$ data (Gerolymos et al. Reference Gerolymos, Sénéchal and Vallet2013; Gerolymos & Vallet Reference Gerolymos and Vallet2016), the region $y^\star \lessapprox y^\star _{p'{PEAK}}$ is dominated by $(p',v')$ events of opposite sign (figure 9), the more so as the wall is approached. The data at $\bar {M}_{{CL}_x} \approxeq 0.8$ indicate similar behaviour, up to the deep viscous sublayer, where we observe a very slight decrease of the negative correlation $-c_{p'v'} = \left \lvert c _{p'v'}\right \rvert$, for $y^\star \lessapprox \tfrac {1}{2}$, at $Re_{\tau ^\star } \lessapprox 340$ (figure 9). This decrease of $-c_{p'v'}$ with increasing $\bar {M}_{{CL}_x}$ is an important trend, as confirmed by the data for $\bar {M}_{{CL}_x} \approxeq 1.5$ (figure 9), where it starts at $y^\star \lessapprox 1.5$ reaching wall-values of $-[c_{p'v'}]_w \lessapprox 0.1$, nearly four times less than at $\bar {M}_{{CL}_x} \approxeq 0.33$. Note also that the $\bar {M}_{{CL}_x} \approxeq 1.5$ data at constant $Re_{\tau ^\star }$ deviate from the lower $\bar {M}_{{CL}_x} \approxeq 0.8$ data already at $y^\star \lessapprox 15$. For $\bar {M}_{{CL}_x} \approxeq 2$ this deviation is larger and the negative peak (NP) of $c_{p'v'}$ occurs at $y^\star \approxeq 2.5$, with a very strong positive slope nearer to the wall, where near-zero values are observed (figure 9). This compressibility effect becomes more pronounced at higher $\bar {M}_{{CL}_x}$ as shown by the $Re_{\tau ^\star } \approxeq 100$ data at progressively increasing $\bar {M}_{{CL}_x} \in \{1.5,1.8,1.9,2.0,2.1,2.2\}$ (figure 9). For the case $(Re_{\tau ^\star },\bar {M}_{{CL}_x}) = (113,2.49)$ there is a monotonic increase of $c_{p'v'}$ towards $[c_{p'v'}]_w \lessapprox 0$ in the entire region $y^\star \lessapprox 15$ (figure 9).

There is, nonetheless, a $Re_{\tau ^\star }$-effect as well, the decrease of $-c_{p'v'}$ becoming weaker and occurring closer to the wall with increasing $Re_{\tau ^\star }$ (figure 9).

This substantial modification in the pressure–velocity correlations suggests a noticeable modification of the near-wall pressure field by compressibility. With increasing $\bar {M}_{{CL}_x}$, both the extent of the near-wall region impacted by the compressibility effects and the difference from the incompressible behaviour increase significantly (figures 8, 9). Note that the region $y^\star \lessapprox 10$ is also the region where both wall-echo effects and the Stokes term in the $\nabla ^2p'$ splitting are important (Gerolymos et al. Reference Gerolymos, Sénéchal and Vallet2013).

Regarding the outer part of the flow, note that $c_{p'v'}$ profiles (figure 9) have little $\bar {M}_{{CL}_x}$ influence, whereas there is a weak $\bar {M}_{{CL}_x}$ effect on $c_{p'u'}$ (figure 8).

4.1. Scaling of the spanwise distance between streaks

A good surrogate for this distance in an average sense can be defined via the spanwise two-point correlation coefficient of the streamwise velocity (Chernyshenko & Baig Reference Chernyshenko and Baig2005b)

(4.2a)\begin{equation} C^{(z)}_{u'u'}(y,\xi_z):=\dfrac{\overline{u'(x,y,z,t)\,u'(x,y,z+\xi_z,t)}}{\overline{u'^2}(y)}, \end{equation}

where the footprint of the streaks is the appearance of a strong ($\sim-0.2$) negative peak (NP) before the correlation increases again towards $0$ (figure 10). This NP is attributed to the negative correlation between low velocities ($u'<0$) inside the streaks with the high velocities ($u'>0$) in the adjoining region. It ‘may be used to assign a length scale to the turbulence’ (Tritton Reference Tritton1988, p. 307), namely, the distance of the first NP from the origin of $\xi _z$ (4.2a)

(4.2b)\begin{equation} \varLambda_{u'}^{(z;{NP})}:=\min\left\lvert\xi_z\right\rvert\left|\begin{matrix}\partial_{\xi_z} C^{(z)}_{u'u'}(y,\xi_z)=0,\\[3pt] C^{(z)}_{u'u'}(y,\xi_z)<0. \end{matrix}\right. \end{equation}

Chernyshenko & Baig (Reference Chernyshenko and Baig2005b) suggest that twice this length scale is a good estimate for the average distance between streaks

(4.2c)\begin{equation} \varLambda_{S}^{(z)}:=2\varLambda_{u'}^{(z;{NP})}. \end{equation}

Of course there are much more advanced techniques for identifying the streaks as coherent structures, either on each $(xz)$-plane (Wang et al. Reference Wang, Pan and Wang2019) or as three-dimensional structures (Bernard Reference Bernard2019; Bae & Lee Reference Bae and Lee2021), allowing a more accurate statistical determination of an average distance between streaks, but the surrogate (4.2c) is sufficient for the purposes of the present work.

Figure 10. Two-point spanwise correlation coefficient of streamwise velocity fluctuations $C_{u'u'}^{(z)}(y;\xi _z)$ (4.2a), at different HCB-scaled non-dimensional distances from the wall $y^\star \in \{1,5,10,15,20\}$, plotted against different non-dimensionalisations (2.1) of the distance between the two points $\{\xi _z^+,\xi _z^\star,\xi _z^\ddagger \}$, for varying centreline Mach numbers $0.32 \leqslant \bar {M}_{{CL}_x} \leqslant 2.49$ at nearly constant $Re_{\tau ^\star } \approxeq 110$ (table 2); the average distance between streaks surrogate $\varLambda _{S}^{ ( z )\ddagger }$ (4.2b), (4.2c) is indicated at $y^\star \approxeq 20$.

Plotting the correlation coefficient $C^{(z)}_{u'u'}(y,\xi _z)$ (4.2a) against $\xi _z^+$ in standard wall-units (2.1a) at different wall-normal stations $y^\star \in \{1,5,10,15,20\}$ (figure 10) clearly shows little variation with wall distance for given flow conditions $(Re_{\tau ^\star },\bar {M}_{{CL}_x})$. As in standard wall-units, the length-unit does not depend on wall distance, this suggests that the spanwise spacing of the streaks, for given flow conditions $(Re_{\tau ^\star },\bar {M}_{{CL}_x})$, varies little with $y$ in the near-wall region ($y^\star \lessapprox 15$). On the other hand, expectedly, standard wall-units (2.1a) scaling of $\xi _z^+$ shows very important differences with increasing $\bar {M}_{{CL}_x}$ at nearly constant $Re_{\tau ^\star }$, exactly analogous to the strong $\bar {M}_{{CL}_x}$-dependence of the ratio $Re_{\tau _w}/Re_{\tau ^\star }$ (table 1).

Plotting the correlation coefficient $C^{(z)}_{u'u'}(y,\xi _z)$ (4.2a) against $\xi _z^\star$ in HCB-scaled units (2.1b) illustrates both the successes and shortcomings of this scaling. Sufficiently away from the wall ($y^\star \approxeq 20$) the two-point correlation coefficients $C^{(z)}_{u'u'}$ for different $\bar {M}_{{CL}_x}$ at nearly constant $Re_{\tau ^\star }$ practically collapse on a single curve when plotted against $\xi _x^\star$ (figure 10). However, closer to the wall, as progressively $y \to 0\implies \xi _z^\star \to \xi _z^+$ (4.1), the substantial differences between different $\bar {M}_{{CL}_x}$ are recovered, e.g. at $C^{(z)}_{u'u'}(y^\star \approxeq 1,\xi _z^+) \approxeq C^{(z)}_{u'u'}(y^\star \approxeq 1,\xi _z^\star )$ (figure 10).

This behaviour is better understood by considering the profiles (against $y^\star$) of the distance between streaks (4.2c) in the two scalings, $\varLambda _{S}^{ ( z )+}$ and $\varLambda _{S}^{ ( z )\star }$, in the entire buffer region (figure 11). Data at $Re_{\tau ^\star } \approxeq 110$ for different $\bar {M}_{{CL}_x} \in \{0.32,0.79,1.51,2.02,2.49\}$ show that the HCB-scaled $\varLambda _{S}^{ ( z )\star }$ is not particularly sensitive to $\bar {M}_{{CL}_x}$ for $y^\star \gtrapprox 15$, but also reveal increasingly large differences (with increasing $\bar {M}_{{CL}_x}$) when approaching the wall (figure 11), in line with (4.1). Note, in particular, how, for $(Re_{\tau ^\star },\bar {M}_{{CL}_x}) = (113,249)$, $\varLambda _{S}^{ ( z )\star }(y^\star \to 0)$ increases sharply towards values approaching $\varLambda _{S}^{ ( z )+}$ at the wall which are outside of the plot range (figure 11).

Figure 11. Different non-dimensionalisations (2.1) $\{\varLambda _{{S}}^{ ( z )+},\varLambda _{{S}}^{ ( z )\star },\varLambda _{{S}}^{ ( z )\ddagger }\}$ of the surrogate for the average spanwise distance (4.2b), (4.2c), plotted against the HCB-scaled non-dimensional distance from the wall $y^\star$ (2.1b), at nearly constant $Re_{\tau ^\star }\in\{110\pm 5,341\pm 1\}$ for different $\bar {M}_{{CL}_x}\in \{0.32,0.79,1.51, 2.00\pm 0.02,2.49\}$ (table 2); enlarged view of the region $y^{\star } \leq 50$.

As we are interested in examining flow fields in the near-wall region, we need a length unit that does not vary in the wall-normal direction, to conform with the fact that in physical space the spanwise distance between streaks (which is a physically relevant length scale) varies little wall-normal-wise (figures 10, 11). On the other hand, this length-unit should approach $\ell _{UNIT}^\star$ for $y^\star \gtrapprox 15$ where HCB scaling is successful in accounting for $\bar {M}_{{CL}_x}$ effects. One possible, but rather difficult to use in practice, choice would be to use $\ell _{UNIT}^\star (y)$ for all $y^\star \geqslant y^\star _{p'{PEAK}}$, and use a constant $\ell _{UNIT}^\star (y_{p'{PEAK}})$ for all $y^\star \leqslant y^\star _{p'{PEAK}}$ from the $p'_{rms}$ peak (3.2) down to the wall. However, this complexity is not necessary. The $y$-dependence of $\ell _{UNIT}^\star (y)$ (2.1b) stems essentially from the mean-temperature stratification of the flow (figure 1) which induces the variations of $\bar {\mu }(y)$ and $\bar {\rho }(y)$ (§ 2.1). Closer observation of the $\tilde {T}(y)$ profiles (figure 1) reveals that strong temperature variations actually occur only in the near-wall region ($y^\star \lessapprox 20$), for a given $(Re_{\tau ^\star },\bar {M}_{{CL}_x})$ flow, whereas in the outer region ($y^\star \gtrapprox 20$) temperature variations are much smaller ($\tilde {T}_{CL}-\tilde {T}(y) \ll \tilde {T}_{CL}-\tilde {T}_w\;\forall y^\star \gtrapprox 20$). Therefore, a system of units based on $\bar {\tau }_w$, which was found to be an appropriate scale for $p'$ (§ 3), and centreline thermodynamic quantities $\{\bar {\rho }_{CL},\bar {\mu }_{CL}\}$ should be very close to $(\cdot )^\star$ units for $y^\star \gtrapprox 15$, while being $y$-independent. For TPC flow (2.2b), these are precisely the $(\cdot )^\ddagger$ units (2.1c) introduced in § 2.1, which retain with a very good accuracy the advantages of the HCB scaling in the outer part of the flow, while respecting the very small $y$ variation of the $(xz)$ organisation of the flow very near the wall. Furthermore, in TPC flow, this system of units also maintains $Re_{\tau ^\star } = Re_{\tau ^\ddagger }$ (2.2a) as the relevant Reynolds number.

Plotting the correlation coefficient $C^{(z)}_{u'u'}(y,\xi _z)$ (4.2a) against $\xi _z^\ddagger$ practically collapses the curves for the different $\bar {M}_{{CL}_x} \in \{1.51, 2.02,2.49\}$ at $Re_{\tau ^\star } \approxeq 110$, for all $y^\star \in \{1,5,10,15,20\}$ (figure 10). The applicability of the $(\cdot )^\ddagger$ units (2.1c) is further confirmed by the profiles of $\varLambda _{S}^{ ( z )\ddagger }(y)$ (figure 11) which are clustered closely together for the different $\bar {M}_{{CL}_x} \in \{0.32,0.79,1.51, 2.00\pm 0.02 ,2.49\}$ at $Re_{\tau ^\star } \approxeq 110$. These profiles of $\varLambda _{S}^{ ( z )\ddagger }(y)$ are very similar with those obtained in incompressible flow (Chernyshenko & Baig Reference Chernyshenko and Baig2005b, figure 1, p. 1100).

The scaling of the streamwise two-point correlation

(4.2d)\begin{equation} C^{(x)}_{u'u'}(y,\xi_x):=\dfrac{\overline{u'(x,y,z,t)\,u'(x+\xi_x,y,z,t)}}{\overline{u'^2}(y)} \end{equation}

with $\xi _x^\ddagger$ (2.1c) is also quite satisfactory, compared with $\xi _x^+$ (2.1a) or $\xi _x^\star$ (2.1b), in that it very efficiently clusters together the curves for different $\bar {M}_{{CL}_x}$ at nearly constant $Re_{\tau ^\star }$ (figure 10). Nonetheless, there remains, expectedly, a small $\bar {M}_{{CL}_x}$-dependence which slightly increases as $y$ decreases towards the wall (figure 10). This observation highlights the fact that both $(\cdot )^\star$ units (2.1b) and $(\cdot )^\ddagger$ units (2.1c) may show $\bar {M}_{{CL}_x}$-dependence very near the wall, but the latter are less $\bar {M}_{{CL}_x}$-sensitive and have the advantage of being $y$-independent and of not tending to $(\cdot )^+$ units (2.1a) at the wall.

4.2. Appearance of organised pressure waves with increasing $\bar {M}_{{CL}_x}$

The low-speed streaks in the near-wall region can be identified (figure 13) using the conditional average (truncated distribution)

(4.3)\begin{equation} \overline{M_x|_{M_x'<0}}:=\dfrac{\displaystyle\int\nolimits_{M_{x_{min}}}^{\bar{M}_x} M_x\,f_{M_x}(M_x)\,dM_x} {\displaystyle\int\nolimits_{M_{x_{min}}}^{\bar{M}_x} \ f_{M_x}(M_x)\,dM_x} , \end{equation}

where $M_{x_{min}}$ is the minimum observed $M_x$ and $f_{M_x}$ the corresponding p.d.f. This simple parameter-free choice (only requiring sampling the p.d.f. of $M_x$) performs quite well (figure 13), thus avoiding user-defined thresholds (Wang et al. Reference Wang, Pan and Wang2019) or percolation techniques (Bae & Lee Reference Bae and Lee2021). Note that the spanwise distance between the instantaneous streaks in the $(x^\ddagger,z^\ddagger )$-plane (figure 13) varies little with $\bar {M}_{{CL}_x}$, in line with the previous analysis (§ 4.1) which resulted in the introduction of $(\cdot )^\ddagger$ units (2.1c).

Figure 12. Two-point streamwise correlation coefficient of streamwise velocity fluctuations $C_{u'u'}^{(x)}(y;\xi _x)$ (4.2d), at different HCB-scaled non-dimensional distances from the wall $y^\star \in \{1,5,10,15,20\}$, plotted against different non-dimensionalisations (2.1) of the distance between the two points $\{\xi _x^+,\xi _x^\star,\xi _x^\ddagger \}$, for varying centreline Mach numbers $0.32 \leqslant \bar {M}_{{CL}_x} \leqslant 2.49$ at nearly constant $Re_{\tau ^\star } \approxeq 110$ (table 2).

Figure 13. Instantaneous fluctuating streamwise Mach number $M_x'$ (colourmap $-2M'_{x_{rms}} \leqslant M_x' \leqslant +2M'_{x_{rms}}$), on $(x^\ddagger,z^\ddagger )$ planes very near the wall ($y^\star \approxeq 1$), for different $\bar {M}_{{CL}_x} \in \{1.51, 2.02,2.49\}$ at nearly constant $Re_{\tau ^\star } \approxeq 110$ and approximate streak boundaries (black contours $M_x = \overline {M_x|_{M_{x}^{'}<0}}$); also included is the surrogate of the average spanwise distance between streaks $\varLambda _{S}^{ ( z )\ddagger }$ (4.2b), (4.2c).

It is not easy to identify structures by simply visualising instantaneous $p'$-fields (figure 14). For the higher $\bar {M}_{{CL}_x} = 2.49$ case, the ($+ / -$)-$p'$ regions, which were identified and analysed for the same flow conditions by Tang et al. (Reference Tang, Zhao, Wan and Liu2020), are visible inside the streaks (figure 14). Note also that we can sometimes discern bow-like ($+ / -$)-$p'$ regions connecting corresponding regions between neighbouring streaks. At $\bar {M}_{{CL}_x} = 2.02$ the same pattern is observed (figure 14), but with lower spatial density (probability of occurrence), and at lower $\bar {M}_{{CL}_x} = 1.51$, ($+ / -$)-$p'$ regions are visible essentially at the boundaries of the streaks, with much less and weaker streamwise events or connecting bow structures (figure 14).

Figure 14. Instantaneous fluctuating pressure field $p'$ (X-ray colourmap $-3p'_{rms} \leqslant p'\leqslant +3p'_{rms}$), on $(x^\ddagger,z^\ddagger )$ planes very near the wall ($y^\star \approxeq 1$), for different $\bar {M}_{{CL}_x} \in \{1.51, 2.02,2.49\}$ at nearly constant $Re_{\tau ^\star } \approxeq 110$ and approximate streak boundaries (cyan contours $M_x = \overline {M_x|_{M_{x}^{'}<0}}$); also included is the surrogate of the average spanwise distance between streaks $\varLambda _{S}^{ ( z )\ddagger }$ (4.2b), (4.2c).

To better identify the $p'$-structures it is preferable to use gradients or Laplacians similarly to standard optical visualisation techniques (Liepmann & Roshko Reference Liepmann and Roshko1957, pp. 157–163). After testing different options it appeared that the in-plane Laplacian

(4.4)\begin{equation} \nabla^2_{xz}p':=\dfrac{\partial^2 p'}{\partial x^2}+\dfrac{\partial^2 p'}{\partial z^2} \end{equation}

is a very good choice. Note that for the present $xz$-invariant-in-the-mean flow, $\nabla ^2_{xz}p'=\nabla ^2_{xz}p$, so that (4.4) can be evaluated without the knowledge of $\bar p(y)$. Furthermore, $\nabla ^2_{xz}p'$ only requires in-plane data which allows $y$-subsampling in data extraction. Following the previous analysis (§ 4.1) of the two-point correlation coefficients $C_{u'u'}^{(z)}(z;\xi _z)$ (figure 10) and $C_{u'u'}^{(x)}(y;\xi _x)$ (figure 12), and of the surrogate distance between streaks $\varLambda _{S}^{ ( z )\ddagger }(y)$ (figure 11), we used $(\cdot )^\ddagger$ units (2.1c) to non-dimensionalise $(\nabla ^2_{xz}p')^\ddagger$. Recall that the unit for $p$ is $\bar {\tau }_w$ in all systems (2.3).

By analogy with harmonic waves, we may identify very approximately the high negative values of $(\nabla ^2_{xz}p')^\ddagger$ with $p'$-maxima and the high positive values with $p'$-minima. The wavefronts can actually be identified to the contours corresponding to $(\nabla ^2_{xz}p')^\ddagger = 0$ which separate the ($+ / -$)-$p'$ regions. The waves, which are very clearly visible for the higher $\bar {M}_{{CL}_x}\in \{2.02,2.49\}$ (figure 15), are essentially in the streamwise direction ($x$-wise), but the fronts (essentially $z$-wise for $x$-wise waves) seem to be faster in the high-$M_x$ regions between steaks and slower within the low-$M_x$ streaks, taking a bow shape in the high-$M_x$ regions. More importantly, these bow fronts cross the streaks and seem to have spanwise extent over several streaks (figure 15). A more detailed time-dependent analysis would be necessary to identify precisely the nature and statistics of these coherent structures, but this is beyond the scope of the paper and should be the subject of future research. It does seem plausible however that these cross-streak $p'$ bow waves are related with the previously observed effects of $\bar {M}_{{CL}_x}$ on $p'$-statistics and correlations with the velocity field near the wall.

Figure 15. Instantaneous in-plane Laplacian of fluctuating pressure in mixed inner/outer scaling (2.1c) $(\boldsymbol {\nabla }_{xz}p')^\ddagger$ (X-ray colourmap $-\tfrac {1}{100} \leqslant (\boldsymbol {\nabla }_{xz}p')^\ddagger \leqslant +\tfrac {1}{100}$), on $(x^\ddagger,z^\ddagger )$ planes very near the wall ($y^\star \approxeq 1$), for different $\bar {M}_{{CL}_x} \in \{1.51, 2.02,2.49\}$ at nearly constant $Re_{\tau ^\star } \approxeq 110$ and approximate streak boundaries (cyan contours $M_x = \overline {M_x|_{M_{x}^{'}<0}}$); also included is the surrogate of the average spanwise distance between streaks $\varLambda _{S}^{ ( z )\ddagger }$ (4.2b), (4.2c).

Note that these bow waves are precisely connected to the ($+ / -$)-$p'$ structures inside the streaks (figure 14) which were identified and analysed by Tang et al. (Reference Tang, Zhao, Wan and Liu2020). However, using $(\nabla ^2_{xz}p')^\ddagger$ has definite advantages in identifying the spanwise extent and spanwise coherence of these $p'$-waves, which is not as obvious by simply considering $p'$ X-rays (figure 14) or $p'$-level plots (Tang et al. Reference Tang, Zhao, Wan and Liu2020). The ($+ / -$)-$p'$ regions inside the streaks are possibly the effect of the interaction of these pressure waves with the three-dimensional vortical structures responsible for the appearance of the streaks. In three dimensions the streaks are identified by pairs of counter-rotating structures of wall-normal vorticity $\omega _y$ which induce highly negative $u' < 0$ in the centre region of the streaks (Adrian Reference Adrian2007; Bernard Reference Bernard2019). Therefore the spanwise cross-streak coherence of the bow waves observed by studying $(\nabla ^2_{xz}p')^\ddagger$ (figure 15) is a feature of the near-wall $p'$ field which requires further study. Note that with the hindsight of the detection of the wavefronts using $(\nabla ^2_{xz}p')^\ddagger$ (figure 15), the spanwise persistence of the waves between streaks is observable in the $p'$-field as well, especially with increasing $\bar {M}_{{CL}_x}$ (figure 14).

These waves are essentially present in the near-wall region ($y^\star \lessapprox 15$), and have substantial amplitude with increasing $\bar {M}_{{CL}_x}$, so at the higher $\bar {M}_{{CL}_x} \in \{2.02,2.49\}$ they are the dominant feature in the $(\nabla ^2_{xz}p')^\ddagger$ X-rays (figure 15). At $\bar {M}_{{CL}_x} = 1.51$ they no longer dominate the very-near-wall $p'$-field (figure 15), although a few weak streamwise events (spanwise fronts) are discernible. However, the fronts between ($+ / -$)-$p'$ regions have become mostly parallel to the streaks, implying that with decreasing $\bar {M}_{{CL}_x}$ the amplitude of streamwise waves decreases, so that they no longer dominate the background $p'$-field at the wall, which becomes essentially associated with (induced by) the vortical structures responsible for the streaks. This is also the case at lower $\bar {M}_{{CL}_x} \in \{0.32,0.79\}$. Although not shown, for conciseness, $(\nabla ^2_{xz}p')^\ddagger$ X-rays at different distances from the wall $y^\star \in \{3,5,10,15,20,y^\star _{p'_{PEAK}}\}$ indicate that the density and intensity of the cross-streak streamwise bow waves, at constant high $\bar {M}_{{CL}_x} \in \{2.02,2.49\}$ diminishes with increasing distance from the wall, and at $y^\star \approxeq 20$ they are hardly discernible even for the highest $\bar {M}_{{CL}_x} = 2.49$. It is therefore plausible to associate this major modification of the $p'$-field, which is strengthened with increasing $M_{{CL}_x}$ and with decreasing distance from the wall, to near-wall compressibility effects, precisely in the region where differences were observed in $p'_{rms}$ (figure 3) and in the correlation coefficients $\{c_{p'u'},c_{p'v'}\}$ (figures 8 and 9).

4.3. Pressure–velocity correlations near the wall

With the hindsight of the results above (§ 4.2) regarding the effect of $\bar {M}_{{CL}_x}$ on the near-wall $p'$-field we reexamine the pressure–velocity correlations $c_{p'u'}$ and $c_{p'v'}$ (figures 8 and 9), by studying (figure 16) the integrands in the evaluation of $\{c_{p'u'},c_{p'v'}\}$ (3.4a,b) from the joint p.d.f.s $\{f_{p'u'}f_{p'v'}\}$ (table 2)

(4.5a)$$\begin{gather} c_{p'u'}\stackrel{(3.4a,b)}= \int_{p'_{min}}^{p'_{max}} \int_{u'_{min}}^{u'_{max}} p'u'\,f_{p'u'}(p',u')\,d\left(\dfrac{p'}{p'_{rms}}\right) d\left(\dfrac{u'}{u'_{rms}}\right),\end{gather}$$

(4.5b)$$\begin{gather}c_{p'v'}\stackrel{(3.4a,b)}= \int_{p'_{min}}^{p'_{max}} \int_{v'_{min}}^{v'_{max}} p'v'\,f_{p'v'}(p',v')\,d\left(\dfrac{p'}{p'_{rms}}\right)d\left(\dfrac{v'}{v'_{rms}}\right) ,\end{gather}$$

integrated with respect to the standardised variables

(4.5c)\begin{equation} p'_{STD}:=p'/p'_{rms};\quad u'_{STD}:=u'/u'_{rms};\quad v'_{STD}:=v'/v'_{rms}. \end{equation}

Figure 16. Integrands in the calculation of the correlation coefficients $c_{p'u'}$ (4.5a) and $c_{p'v'}$ (4.5b) from the corresponding joint p.d.f.s, plotted against the standardised variables $\{p'/p'_{rms},u'/u'_{rms},v'/v'_{rms}\}$, very near the wall ($y^\star \approxeq 1$), at nearly constant $Re_{\tau ^\star } \approxeq 110$ for different $\bar {M}_{{CL}_x} \in \{0.32,0.79,1.51, 2.02,2.49\}$.

Table 2. Sampling parameters for p.d.f.s, joint p.d.f.s and two-point correlations: $Re_{\tau ^\star }$ is the friction Reynolds number in HCB-scaling (2.1b); $\bar {M}_{{CL}_z}$ is the centreline Mach number (2.4); $\Delta t$ is the computational time step (table 1); $\Delta t_{s_q}$, $\varDelta t_{s_{2q}}$, $\varDelta t_{s_{R2}}$ are the sampling time steps for p.d.f.s, joint p.d.f.s and two-point correlations; $t_{{OBS}_q}$, $t_{{OBS}_{2q}}$, $t_{{OBS}_{R2}}$ are the sampling observation intervals for p.d.f.s, joint p.d.f.s and two-point correlations; $N_{{bins}_q}$, $N_{{bins}_{2q}}$ are the sampling bins for the calculation of p.d.f.s and joint p.d.f.s; $N_{s_q}$, $N_{s_{2q}}$ is the number of samples involved in the calculation of p.d.f.s and joint p.d.f.s; $(\cdot )^+$ denotes wall-units (2.1a).

Near the wall, the streamwise correlation $c_{p'u'}$ (figure 16) is dominated by $u' < 0$ events, essentially inside the streaks. The positive/negative correlation peaks occur at $u' \approxeq -u'_{rms}$ (figure 16), which is located approximately on or inside the streaks boundaries determined by $M_x = \overline {M_x|_{M_{x}^{'}<0}}$. At the lower $\bar {M}_{{CL}_x} \in \{0.32,0.79,1.51\}$ the integral of the positive $p'u' > 0$ quadrant $Q_3$ ($u_{STD}' < 0 > p_{STD}'$) is slightly higher than the integral of the negative $p'u' < 0$ quadrant $Q_4$ ($u_{STD}' < 0 < p_{STD}'$). With increasing $\bar {M}_{{CL}_x}$, the $Q_3$ contribution increases with respect to $Q_4$, enhanced by $p' < 0$ events well inside the streaks ($u_{STD}' < -1 > p_{STD}'$). These statistical results suggest that the increase of $c_{p'u'}$ with increasing $\bar {M}_{{CL}_x}$ in the near-wall region (figure 8) is probably the result of the interaction of the cross-streak pressure waves with the vortical structures inside the streaks, with little contribution from the high-speed regions between streaks.

Regarding the wall-normal correlation $c_{p'v'}$, the effect of $\bar {M}_{{CL}_x}$ is more pronounced (figure 16), confirming that the decrease towards $0$ of $-c_{p'v'} > 0$ with increasing $\bar {M}_{{CL}_x}$ in the near-wall region (figure 9) is the result of structural changes in the flow. The positive $p'v' > 0$ quadrants $Q_1$ ($v_{STD}' > 0 < p_{STD}'$) and $Q_3$ ($v_{STD}' < 0 > p_{STD}'$) are nearly symmetric around the origin. With increasing $\bar {M}_{{CL}_x}$ their weight increases relative to the negative quadrants $Q_2$ ($v_{STD}' > 0 > p_{STD}'$) and $Q_4$ ($v_{STD}' < 0 < p_{STD}'$), so that at the higher $\bar {M}_{{CL}_x} \in \{2.02,2.49\}$ the integral approaches $0$. This is in complete contrast with the lower $\bar {M}_{{CL}_x} \in \{0.32,0.79,1.51\}$ where the negative quadrants are clearly dominant (figure 16), the more so with decreasing $\bar {M}_{{CL}_x}$, with the maximum of $Q_4$ ($v_{STD}' < 0 < p_{STD}'$) being higher than that of $Q_2$ ($v_{STD}' > 0 > p_{STD}'$). Note also that there are increasing contributions from the positive quadrants ($Q_1$ and $Q_3$) with increasing $\bar {M}_{{CL}_x}$. At the higher $\bar {M}_{{CL}_x} = 2.49$ the integrands of $c_{p'v'}$ have become practically antisymmetric with respect to the zero axes (figure 16), explaining by cancellation the very low value of $-c_{p'v'}$ (figure 9).

The instantaneous contours $\nabla ^2_{xz}p' = 0$ are also the boundaries (fronts) between ($+ / -$)-$p'$ events, whereas the instantaneous contours $M_x = \overline {M_x|_{M_{x}^{'}<0}}$ highlight the low-$u$ streaks (generally the statistics of $M_x$ and $u$ are very similar). The correlation of this $\{p',u'\}$-fields organisation with the wall-normal velocity $v'$ can be visualised by superposing the instantaneous $p'$-fronts and the low-speed streaks boundaries with the instantaneous $M_y'$-field (figure 17). At the higher $M_{{CL}_x} = 2.49$, high-intensity ($+ / -$)-$M_y'$ events occur invariably on the $p'$-fronts ($\nabla ^2_{xz}p' = 0$ contours), and have precisely the same spanwise coherence and organization as the cross-streak $p'$-bow-waves (figure 17). At $\bar {M}_{{CL}_x} = 2.02$, in the regions where cross-streak $p'$-bow-fronts are present, the ($+ / -$)-$M_y'$ events coincide with the fronts (figure 17). However, in the regions where the $p'$-fronts have started to align themselves with the streaks the ($+ / -$)-$M_y'$ events appear elongated in the streamwise direction, and no longer coincide with the location of the fronts (figure 17). At $M_{{CL}_x} = 1.51$, the majority of $p'$-fronts are aligned with the streamwise direction, and ($+ / -$)-$M_y'$ events are elongated in the streamwise direction and no longer coincide with the $p'$-fronts (figure 17).

Figure 17. Instantaneous wall-normal Mach number fluctuation $M_y'$ (colourmap $-M'_{y_{rms}} \leqslant M_y' \leqslant +M'_{y_{rms}}$), on $(x^\ddagger,z^\ddagger )$ planes very near the wall ($y^\star \approxeq 1$), for different $\bar {M}_{{CL}_x} \in \{1.51, 2.02,2.49\}$ at nearly constant $Re_{\tau ^\star }\approxeq 110$; also included are approximate streak boundaries (white contours $M_x = \overline {M_x|_{M_{x}^{'}<0}}$) and $p'$-fronts (black contours $\nabla _{xz}p'=0$).

Explaining the $\bar {M}_{{CL}_x}$-induced structural modifications of the wall-normal correlation $c_{p'v'}$ is obviously more complicated than for the streamwise correlation $c_{p'u'}$. The strong negative $c_{p'v'}$ correlation coefficients observed at low $\bar {M}_{{CL}_x}$ can be associated to $(+ / -)$-$v'$ events that form relatively elongated streamwise structures which do not coincide with the $p'$-fronts, as, e.g., in the $\bar {M}_{{CL}_x} = 1.51$ case (figure 17). At higher $\bar {M}_{{CL}_x} = 2.49$ the flow structure is completely masked by high-intensity $(+ / -)$-$v'$ events which coincide with the fronts of the cross-streak bow-waves (figure 17), and which symmetrise the $c_{p'v'}$-integrand (figure 16). It is not clear whether it is possible to analyse this modification as an additive compressible effect, via linear approaches (Tang et al. Reference Tang, Zhao, Wan and Liu2020), or if there are important nonlinear effects (Chu & Kovásznay Reference Chu and Kovásznay1958) that become dominant with increasing $\bar {M}_{{CL}_x}$.