# Intrinsic compressibility effects in near-wall turbulence

Asif Manzoor Hasan1<EMAIL_ADDRESS>Pedro Costa1  Johan Larsson2  Sergio
Pirozzoli3  Rene Pecnik1<EMAIL_ADDRESS>1 Process & Energy Department,
Delft University of Technology, Leeghwaterstraat 39, 2628 CB, Delft, The
Netherlands 2Department of Mechanical Engineering, University of Maryland,
College Park, MD 20742, USA 3 Dipartimento di Ingegneria Meccanica e
Aerospaziale, Sapienza Università di Roma, Via Eudossiana 18, 00184 Roma,
Italy

###### Abstract

The impact of intrinsic compressibility effects—changes in fluid volume due to
pressure variations—on high-speed wall-bounded turbulence has often been
overlooked or incorrectly attributed to mean property variations. To
unambiguously quantify these intrinsic compressibility effects, we perform
direct numerical simulations of compressible turbulent channel flows with
nearly uniform mean properties. Our simulations reveal that intrinsic
compressibility effects yield a significant upward shift in the logarithmic
mean velocity profile that can be attributed to the reduction in the turbulent
shear stress. This reduction stems from the weakening of the near-wall quasi-
streamwise vortices. We in turn attribute this weakening to the spontaneous
opposition of sweeps and ejections from the near-wall expansions and
contractions of the fluid, and provide a theoretical explanation for this
mechanism. Our results also demonstrate that intrinsic compressibility effects
are responsible for the increase in the inner-scaled streamwise turbulence
intensity in compressible flows compared to incompressible flows, previously
regarded to be an effect of mean property variations.

##  1 Introduction

Understanding the impact of compressibility effects on turbulent flow is
crucial for a wide range of engineering applications, as it influences the
performance and efficiency of aerospace vehicles, gas turbines, combustion
processes, and high-speed propulsion systems. Turbulence in compressible flow
involves effects related to heat transfer—also termed as variable-property
effects—and intrinsic compressibility (hereby IC) effects—also termed as
‘true’ compressibility effects (Smits & Dussauge, 2006), ‘genuine’
compressibility effects (Yu et al., 2019), or simply ‘compressibility’ effects
(Lele, 1994). Heat transfer is in turn responsible for two main effects.
First, heat transfer is associated with mean temperature variations and hence
variations in the mean density and viscosity. Second, it can cause
fluctuations in fluid volume (or density) as a result of a change in entropy
(Livescu, 2020). On the other hand, intrinsic compressibility effects are
associated with changes in fluid volume in response to changes in pressure
(Lele, 1994). While variable-property effects can be relevant at any (even
zero) Mach number, IC effects only become important at high Mach numbers.

In 1962, Morkovin postulated that the changes in fluid volume due to entropy
and pressure, mentioned above, are negligible such that only mean property
variations are important. This hypothesis is commonly referred to as
‘Morkovin’s hypothesis’ (Morkovin, 1962; Bradshaw, 1977; Coleman et al., 1995;
Smits & Dussauge, 2006). Some years later, Bradshaw (1977) performed a
detailed study on this hypothesis and provided an engineering estimate as to
when the hypothesis should hold. According to Bradshaw, Morkovin’s postulate
may be true in flows where the root-mean-square ($rms$) of the density
fluctuation is below 10% of the mean density. Subsequently, Coleman et al.
(1995) noted that most of these density fluctuations arise from passive mixing
across mean density gradients. Since Morkovin’s hypothesis implicitly assumes
that the spatial gradients of the mean density, and thus the fluctuations
resulting from them, are small, they argued that the density $rms$ is not a
rigorous evaluator of the hypothesis. Instead, they claimed that, consistent
with the original conjecture, the $rms$ of pressure and total temperature111We
note that pressure fluctuations scaled by mean pressure are a direct measure
of intrinsic compressibility effects. However, the justification for why total
temperature fluctuations should be small for the hypothesis to hold is unclear
(Lele, 1994). scaled by their respective means should be considered. To our
knowledge, there is no engineering estimate for these fluctuations such as the
one for density proposed by Bradshaw.

If Morkovin’s hypothesis holds, then turbulence statistics in compressible
flows can be collapsed onto their incompressible counterparts by simply
accounting for mean property variations. The first key contribution in
accounting for variable-property effects was proposed by Van Driest (1951),
who incorporated mean density variations in the mean shear formulation such
that

$\frac{d\bar{u}}{dy}=\frac{\sqrt{\tau_{w}/\bar{\rho}}}{\kappa y}\mathrm{,}$
(1)

where $u$ is the streamwise velocity, $\tau_{w}$ the wall shear stress, $\rho$
the fluid density, and $\kappa$ the von Kármán constant. The overbar denotes
Reynolds averaging and the subscript $w$ indicates wall values. Equation (1)
led to two major outcomes: (1) the Van Driest mean velocity transformation
(Van Driest, 1956a; Danberg, 1964) given as

$\bar{U}_{VD}^{+}=\int_{0}^{\bar{u}^{+}}{\sqrt{\frac{\bar{\rho}}{\rho_{w}}}}d\bar{u}^{+},$
(2)

where the supercript $+$ denotes wall scaling, and (2) the Van Driest skin-
friction theory (Van Driest, 1956b). These scaling breakthroughs are still
widely used, despite their known shortcomings (Bradshaw, 1977; Huang &
Coleman, 1994; Trettel & Larsson, 2016; Patel et al., 2016; Griffin et al.,
2021; Kumar & Larsson, 2022; Hasan et al., 2024).

Another key contribution is attributed to Morkovin (1962) who proposed scaling
the turbulent shear stress with $\bar{\rho}/\rho_{w}$ such that

$\widetilde{u^{\prime\prime}v^{\prime\prime}}^{*}=\frac{\bar{\rho}}{\rho_{w}}\frac{\widetilde{u^{\prime\prime}v^{\prime\prime}}}{u_{\tau}^{2}}$
(3)

collapses with the incompressible distributions. Here,
$u_{\tau}=\sqrt{\tau_{w}/\rho_{w}}$ is the friction velocity scale, the tilde
denotes density-weighted (Favre) averaging, and the double primes denote
fluctuations from Favre average. The contributions of Van Driest and Morkovin
can be consolidated by interpreting their corrections as if they were to
change the definition of the friction velocity scale from $u_{\tau}$ to
$u_{\tau}^{*}=\sqrt{\tau_{w}/\bar{\rho}}$ (termed ‘semi-local’ friction
velocity scale222The friction velocity scale is termed ‘semi-local’ instead of
‘local’ because the total shear stress in its definition is still taken at the
wall.), such that equations (1), (2), and (3) can be rewritten as

$\displaystyle\frac{d\bar{u}}{dy}=\frac{u_{\tau}^{*}}{\kappa y}\mathrm{,}$
$\displaystyle\bar{U}_{VD}^{+}=\int_{0}^{\bar{u}}\frac{1}{u_{\tau}^{*}}d\bar{u},$
$\displaystyle\widetilde{u^{\prime\prime}v^{\prime\prime}}^{*}=\frac{\widetilde{u^{\prime\prime}v^{\prime\prime}}}{u_{\tau}^{*2}}\mathrm{.}$
(4)

Similarly, efforts to account for mean density and viscosity variations in the
definition of the viscous length scale were made since the 1950s (Lobb et al.,
1955), giving rise to the well-known semi-local wall-normal coordinate
$y^{*}=y/\delta_{v}^{*}$ (where
$\delta_{v}^{*}=\bar{\mu}/(\bar{\rho}u_{\tau}^{*})$ is the semi-local viscous
length scale). Much later, the companion papers by Huang et al. (1995) and
Coleman et al. (1995) performed a comprehensive analysis where they showed
that turbulence quantities show a much better collapse when reported as a
function of $y^{*}$ rather than $y^{+}$. Another major consequence of using
the semi-local wall coordinate is reflected in velocity transformations. The
semi-local velocity transformation, derived independently by Trettel & Larsson
(2016) and Patel et al. (2016), is an extension to the Van Driest velocity
transformation accounting for variations in the semi-local viscous length
scale. This transformation (also known as the TL transformation) can be
written as

$\bar{U}_{TL}^{+}=\int_{0}^{\bar{u}^{+}}\left(1-\frac{y}{\delta_{v}^{*}}\frac{d\delta_{v}^{*}}{dy}\right)\underbrace{{\frac{u_{\tau}}{u_{\tau}^{*}}}}_{\sqrt{{\bar{\rho}}/{\rho_{w}}}}d\bar{u}^{+}.$
(5)

In short, the above-mentioned scaling theories in equations (4) and (5) show
that heat transfer effects associated with mean property variations can be
accounted for in terms of the semi-local friction velocity and viscous length
scales.

In addition to the studies mentioned above, many other studies have addressed
variable-property effects in low-Mach (Patel et al., 2015) and high-Mach
number flows (Maeder et al., 2001; Morinishi et al., 2004; Foysi et al., 2004;
Duan et al., 2010, 2011; Modesti & Pirozzoli, 2016; Zhang et al., 2018; Cogo
et al., 2022; Zhang et al., 2022; Wenzel et al., 2022; Cogo et al., 2023, to
name a few). However, less emphasis has been placed on studying intrinsic
compressibility effects, possibly due to the belief that Morkovin’s hypothesis
holds for wall-bounded flows even in the hypersonic regime (Duan et al., 2011;
Zhang et al., 2018).

Recently, by isolating intrinsic compressibility effects, Hasan et al. (2023)
found that Morkovin’s hypothesis is inaccurate at high Mach numbers. These
compressibility effects modify the mean velocity scaling, leading to an upward
shift in the logarithmic profile. The authors attributed this shift to the
modified near-wall damping of turbulence and proposed a mean velocity
transformation based on a modification of the Van Driest damping function as

$\bar{U}_{HLPP}^{+}=\int_{0}^{\bar{u}^{+}}\\!\\!\left({\frac{1+\kappa
y^{*}{D(y^{*},M_{\tau})}}{1+\kappa{y^{*}}{D(y^{*},0)}}}\right){\left({1-\frac{y}{\delta_{v}^{*}}\frac{d\delta_{v}^{*}}{dy}}\right)}\sqrt{\frac{\bar{\rho}}{\rho_{w}}}\,{d\bar{u}^{+}}.$
(6)

This transformation was found to be accurate for a wide variety of flows
including (but not limited to) adiabatic and cooled boundary layers, adiabatic
and cooled channels, supercritical flows, and flows with non-air-like
viscosity laws. The modified damping function in (6) reads

$D(y^{*},M_{\tau})=\left[1-\mathrm{exp}\left({\frac{-y^{*}}{A^{+}+f(M_{\tau})}}\right)\right]^{2},$
(7)

with $f(M_{\tau})=19.3M_{\tau}$. Despite the evidence that intrinsic
compressibility effects modify the damping, the underlying physical mechanism
is still unknown.

More evidence on the importance of intrinsic (or ‘genuine’) compressibility
effects has been provided in a series of recent publications by Yu and co-
workers (Yu et al., 2019, 2020; Yu & Xu, 2021), who analysed these effects in
channel flows through direct numerical simulations (DNS). They performed a
Helmholtz decomposition of the velocity field and mainly focused on
dilatational motions and their direct contribution to several turbulence
statistics. Their main observations were: (1) intrinsic compressibility
effects, if present, are likely concentrated in the near-wall region, where
the wall-normal dilatational velocity field exceeds the solenoidal
counterpart; (2) the correlation between the solenoidal streamwise and the
dilatational wall normal velocity is negative and can constitute up to 10% of
the total shear stress; (3) this negative correlation was attributed to the
opposition of sweeps near the wall by dilatational motions; and (4) the
dilatation field (and thus the dilatational velocity) exhibits a travelling
wave-packet-like structure, whose origin is yet unknown (see also Tang et al.,
2020; Gerolymos & Vallet, 2023; Yu et al., 2024).

In this paper, we will focus mainly on the indirect effects of intrinsic
compressibility, namely, those that do not result directly from contributions
by dilatational motions but result as a consequence of changes in the
solenoidal dynamics of turbulence. To achieve this, we first perform direct
numerical simulations employing the methodology described in Coleman et al.
(1995), whereby variable-property effects are essentially removed by
cancelling the aerodynamic heating term in the energy equation. These
simulations will allow us to study intrinsic compressibility effects by
isolating them. With this approach, our main goal is to answer why the near-
wall damping of turbulence changes with increasing Mach number, as observed in
Hasan et al. (2023). Since this is also observed for conventional flows, we
believe that the knowledge obtained from our simplified cases is directly
applicable to those flows. With the simulated cases, we look into various
fundamental statistics of turbulence such as turbulent stresses, pressure-
strain correlation, and into coherent structures, eventually tracing back the
change in near-wall damping of the turbulent shear stress to the weakening of
quasi-streamwise vortices. Subsequently, with the help of what is known from
the incompressible turbulence literature, we provide a theoretical explanation
as to why the vortices weaken.

The paper is structured as follows. §2 describes the cases and methodology
used in this paper. §3 explains the change in damping of near-wall turbulence
as a result of the change in turbulent stress anisotropy, caused by a
reduction in the pressure-strain correlation. §4 connects this reduced
correlation with the weakening of quasi-streamwise vortices, which is then
explained using conditional averaging. Finally, the summary and conclusions
are presented in §5.

##  2 Computational approach and case description

In order to investigate turbulence in high-speed wall-bounded channel flows
with uniform mean temperature (internal energy) in the domain, we perform
direct numerical simulations by solving the compressible Navier-Stokes
equations in conservative form, given as

$\displaystyle\frac{\partial\rho}{\partial t}+\frac{\partial\rho
u_{i}}{\partial x_{i}}$ $\displaystyle=0,$ (8)
$\displaystyle\frac{\partial\rho u_{i}}{\partial t}+\frac{\partial\rho
u_{i}u_{j}}{\partial x_{j}}$ $\displaystyle=-\frac{\partial p}{\partial
x_{i}}+\frac{\partial\tau_{ij}}{\partial x_{j}}+f\delta_{i1},$
$\displaystyle\frac{\partial\rho E}{\partial t}+\frac{\partial\rho
u_{j}E}{\partial x_{j}}$ $\displaystyle=-\frac{\partial pu_{j}}{\partial
x_{j}}-\frac{\partial q_{j}}{\partial
x_{j}}+\frac{\partial\tau_{ij}u_{i}}{\partial x_{j}}+fu_{1}+\Phi.$

The viscous stress tensor and the heat flux vector are given as

$\tau_{ij}=\mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial
u_{j}}{\partial x_{i}}-\frac{2}{3}\frac{\partial u_{k}}{\partial
x_{k}}\delta_{ij}\right),~{}q_{j}=-\lambda\frac{\partial T}{\partial x_{j}},$
(9)

where $u_{i}$ is the velocity component in the $i^{th}$ direction, and where
$i=1,2,3$ corresponds to the streamwise ($x$), wall-normal ($y$) and spanwise
($z$) directions, respectively. $\rho$ is the density, $p$ the pressure,
$E=c_{v}T+u_{i}u_{i}/2$ the total energy per unit mass, $\mu$ the viscosity,
$\lambda$ the thermal conductivity and $Pr=\mu c_{p}/\lambda$ the Prandtl
number. $c_{p}$ and $c_{v}$ indicate specific heats at constant pressure and
constant volume, respectively. $f$ is a uniform body force that is adjusted in
time to maintain a constant total mass flux in periodic flows (e.g., a fully
developed turbulent channel or pipe).

As outlined in the introduction, herein we attempt to remove mean property
gradients to isolate intrinsic compressibility effects. For that purpose, we
follow the approach presented by Coleman et al. (1995), whereby the energy
equation is augmented with a source term

$\Phi=-\tau_{ij}\frac{\partial u_{i}}{\partial x_{j}}$ (10)

that counteracts the effects of viscous dissipation. Consequently, the mean
internal energy remains approximately uniform across the entire domain. For an
ideal gas, this implies that the mean temperature is also approximately
constant, which, when combined with a uniform mean pressure, leads to a nearly
uniform mean density. Furthermore, the mean dynamic viscosity and mean thermal
conductivity are also uniform. However, it is important to note that the
simulations still permit fluctuations of these properties—primarily along
isentropes, as we will see below.

Using this approach, four cases with increasing Mach numbers are simulated, as
presented in table 1. These simulations are performed with STREAmS (Bernardini
et al., 2021) using the assumption of a calorically perfect ideal gas
(constant specific heat capacities), a constant Prandtl number of $0.7$, and a
power law for the viscosity with an exponent of $0.75$. The domain is periodic
in the streamwise and spanwise directions, while at the walls an isothermal
boundary condition is used for temperature, and a zero normal gradient is
specified for pressure. Since the four cases have similar $Re_{\tau}$ values,
we use the same grid for all simulations. The computational grid consists of
$n_{x}=1280$, $n_{y}=480$ and $n_{z}=384$ points for a domain of size
$L_{x}=10h$, $L_{y}=2h$ and $L_{z}=3h$, where $h$ is the channel half-height.
This gives a near-wall resolution of $\Delta x^{+}=4.3$ and $\Delta
z^{+}=4.3$. The grid in the wall-normal direction is stretched in such a way
that $y^{+}\leq 1$ is achieved for the first grid point.

Case name |  $M_{b}$ |  $M_{cl}$ |  $M_{\tau}$ |  $Re_{\tau}$ |  $Re_{\tau_{c}}$ |  Line colour
---|---|---|---|---|---|---
Mach 0.3 |  0.3 |  0.34 |  0.0162 |  556 |  556 |
Mach 2.28 |  2.28 |  2.59 |  0.1185 |  546 |  539 |
Mach 3 |  3.0 |  3.37 |  0.1526 |  547 |  527 |
Mach 4 |  4.0 |  4.47 |  0.1968 |  544 |  513 |
Table 1: Description of the cases. $M_{b}=U_{b}/\sqrt{\gamma RT_{w}}$ is the
bulk Mach number, $M_{cl}=U_{c}/\sqrt{\gamma RT_{c}}$ is the channel
centreline Mach number and $M_{\tau}=u_{\tau}/\sqrt{\gamma RT_{w}}$ is the
wall friction Mach number. $Re_{\tau}=\rho_{w}u_{\tau}h/\mu_{w}$ is the
friction Reynolds number based on the channel half-height $h$ and
$Re_{\tau_{c}}$ corresponds to the value of the semi-local friction Reynolds
number ($Re_{\tau}^{*}=\bar{\rho}u_{\tau}^{*}h/\bar{\mu}$) at the channel
centre.

Figure 1 shows the mean density, viscosity, and semi-local Reynolds number
profiles for the four cases introduced in table 1. The figure also shows the
profiles of a conventional boundary layer at a free-stream Mach number of 14,
taken from Zhang et al. (2018). Compared to the conventional $M_{\infty}=14$
boundary layer case, our cases show little to no variation in mean properties.
This implies that mean heat transfer effects are indeed negligible in the
present cases.

To determine whether other heat transfer effects associated with changes in
fluid volume as a result of changes in entropy are important, we compute
density fluctuations using the isentropic relation

$\frac{\rho^{is}_{rms}}{\bar{\rho}}\approx\frac{1}{\gamma}\frac{p_{rms}}{\bar{p}},$
(11)

and compare it with the density fluctuations obtained from DNS in figure 2(a).
With the exception of the viscous sublayer, the two distributions appear to
collapse, which implies that entropic heat transfer effects are negligible in
the present cases. Hence, any deviations from incompressible flows observed in
these cases should be attributed to intrinsic compressibility effects.

Figure 1: Wall-normal distributions of (a) density $\overline{\rho}$, (b)
viscosity $\overline{\mu}$, and (c) the semi-local friction Reynolds number
$Re_{\tau}^{*}=\bar{\rho}u_{\tau}^{*}h/\bar{\mu}$ for the cases described in
table 1. The red lines represent the $M_{\infty}=14$ case of Zhang et al.
(2018). These quantities are plotted as a function of the wall-normal
coordinate scaled by the channel half-height for the channel flow cases, and
by boundary layer thickness ($\delta_{99}$) for the $M_{\infty}=14$ boundary
layer case.

Figure 2(a) also shows the total and isentropic density fluctuations for the
$M_{\infty}=14$ flow case computed by Zhang et al. (2018). As can be seen, the
total density fluctuations are much higher than the isentropic ones in the
buffer layer and beyond, corroborating that both heat transfer and intrinsic
compressibility effects are important. Interestingly, our highest Mach number
case (Mach 4) and Zhang’s $M_{\infty}=14$ boundary layer have similar
isentropic density $rms$ (or similar pressure $rms$). Given that the pressure
$rms$ scaled by mean pressure is an effective measure of intrinsic
compressibility effects (Coleman et al., 1995), we can expect that these
effects are of comparable magnitude for our Mach 4 case and the conventional
$M_{\infty}=14$ boundary layer.

Figure 2: Wall-normal distributions of (a) the root-mean-square ($rms$) of the
total (solid) and isentropic (dashed) density fluctuations [equation (11)];
(b) the turbulence Mach number $M_{t}=\sqrt{2k}/\sqrt{\gamma R\bar{T}}$; and
(c) the semi-local friction Mach number
$M_{\tau}^{*}=u_{\tau}^{*}/\sqrt{\gamma R\bar{T}}$ for the cases described in
table 1. The red lines represent the $M_{\infty}=14$ case of Zhang et al.
(2018).

In addition to the pressure $rms$, intrinsic compressibility effects can also
be quantified in terms of Mach numbers. Figure 2(b) shows the turbulence Mach
number, defined as $M_{t}=\sqrt{2k}/\sqrt{\gamma R\bar{T}}$, where
$k=\overline{\rho u_{i}^{\prime\prime}u_{i}^{\prime\prime}}/2$ is the
turbulence kinetic energy (TKE) and the denominator is the local speed of
sound for ideal gases. Three out of four cases are above the threshold of
$M_{t}=0.3$, above which intrinsic compressibility effects are considered
important (Smits & Dussauge, 2006). Due to the inhomogeneous nature of wall-
bounded flows, $M_{t}$ is not constant throughout the domain, becoming zero at
the wall where the pressure and density $rms$ are the strongest as shown in
figure 2(a).

Other parameters have been proposed in the literature as a better measure of
intrinsic compressibility effects in wall-bounded flows, most prominently the
friction Mach number $M_{\tau}=u_{\tau}/\sqrt{\gamma RT_{w}}$ (Bradshaw, 1977;
Smits & Dussauge, 2006; Yu et al., 2022; Hasan et al., 2023). When defined in
terms of local properties, one obtains the semi-local friction Mach number
$M_{\tau}^{*}=u_{\tau}^{*}/\sqrt{\gamma R\bar{T}}$. Figure 2(c) shows that, in
contrast to $M_{t}$, the distribution of $M_{\tau}^{*}$ is nearly constant,
even for flows with mean property variations. The reason why $M_{\tau}^{*}$ is
constant for flows with ideal gases is because
$\bar{T}/T_{w}\approx\rho_{w}/\bar{\rho}$ such that

$M_{\tau}^{*}=\frac{u_{\tau}^{*}}{\sqrt{\gamma
R\bar{T}}}=\frac{u_{\tau}\sqrt{\rho_{w}/\bar{\rho}}}{\sqrt{\gamma
R\bar{T}}}\approx\frac{u_{\tau}\sqrt{\bar{T}/T_{w}}}{\sqrt{\gamma
R\bar{T}}}=\frac{u_{\tau}}{\sqrt{\gamma RT_{w}}}=M_{\tau}.$ (12)

As seen in figure 2(b) and (c), the profiles of $M_{t}$ and $M_{\tau}^{*}$ are
equivalent for the Mach 4 constant-property and the $M_{\infty}=14$
conventional cases, further supporting the statement made above that the IC
effects in these cases are comparable.

##  3 Intrinsic compressibility effects on turbulence statistics

Having introduced the flow cases, we first discuss the modified near-wall
damping of the turbulent shear stress and its consequence on the mean velocity
scaling. Unless otherwise stated, all quantities will be presented in their
semi-locally scaled form. Nevertheless, since the cases have approximately
constant mean properties, there is no major difference between the classical
wall scaling (denoted by the superscript ‘$+$’) and the semi-local scaling
(denoted by the superscript ‘$*$’).

###  3.1 Outward shift in viscous and turbulent shear stresses

In the inner layer of parallel (or quasi-parallel) shear flows, integration of
the mean streamwise momentum equation implies that the sum of viscous and
turbulent shear stresses is equal to the total shear stress, given as

$\overline{\mu\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial
x}\right)}-\overline{\rho u^{\prime\prime}v^{\prime\prime}}={\tau_{tot}},$
(13)

where $\tau_{tot}\approx\tau_{w}$ in zero-pressure-gradient boundary layers,
whereas it decreases linearly with the wall distance in channel flows.
Neglecting terms due to viscosity fluctuations and normalizing equation (13)
by $\tau_{w}$, we get for the latter case

$\frac{\bar{\mu}}{\mu_{w}}\frac{d\bar{u}^{+}}{dy^{+}}-\widetilde{u^{\prime\prime}v^{\prime\prime}}^{*}\approx
1-\frac{y}{h},$ (14)

where $h$ is the channel half-height.

Integrating the viscous shear stress yields the TL-transformed mean velocity
profile (Trettel & Larsson, 2016; Patel et al., 2016) as

$\bar{U}^{+}_{TL}=\int_{0}^{y^{*}}\frac{\bar{\mu}}{\mu_{w}}\frac{d\bar{u}^{+}}{dy^{+}}dy^{*}\mathrm{.}$
(15)

Figure 3(a) shows the transformed velocity profiles for the cases listed in
table 1 (or simply $\bar{u}^{+}$, since the mean flow properties are nearly
constant). A clear shift in the logarithmic profile is seen that increases
with the Mach number. Based on equation (15), an upward shift in the mean
velocity profile corresponds to an equivalent upward shift (or increase) in
the viscous shear stress. This is evident from figure 3(b). Since the total
shear stress is universal for the four flow cases under inspection, an
increase in the viscous shear stress directly implies a decrease in the
turbulent shear stress. Indeed, figure 3(b) shows that the turbulent shear
stress reduces with increasing Mach number.

Figure 3: (a) TL-transformed mean velocity profiles [equations (5), (15)], and
(b) viscous and turbulent shear stresses for the cases described in table 1.

In other words, the log-law shift observed in figure 3(a) is a consequence of
the modified damping of the turbulent shear stress, as also noted by Hasan et
al. (2023).

###  3.2 Outward shift in wall-normal turbulent stress: change in turbulence
anisotropy

The outward shift in the turbulent shear stress corresponds to an outward
shift in the wall-normal turbulent stress, because wall-normal motions
directly contribute to turbulent shear stress by transporting momentum across
the mean shear (Townsend, 1961; Deshpande et al., 2021). This is also
reflected in the turbulent shear stress budget, whose production is controlled
by the wall-normal turbulent stress (Pope, 2001).

Figure 4: Wall-normal distributions of (a) streamwise, (b) wall-normal and (c)
spanwise turbulent stresses, and (d) the turbulence kinetic energy for the
cases described in table 1.

Figure 4(b) shows profiles of the wall-normal turbulent stress. A clear
outward shift is evident, which is consistent with the observed outward shift
in the turbulent shear stress. Now, the decrease in the wall-normal stress can
either be due to less energy being received from the streamwise component
(inter-component energy transfer), or due to an overall reduction of the
turbulence kinetic energy. In order to clarify this, we report the streamwise
and the spanwise turbulent stresses, along with the turbulence kinetic energy
in panels (a), (c) and (d) of figure 4, respectively.

Figure 4(a) shows that the streamwise turbulent stress becomes stronger with
increasing Mach number. The increase in the peak streamwise turbulence
intensity in compressible flows, compared to incompressible flows at similar
Reynolds numbers, has also been observed in several other studies (Gatski &
Erlebacher, 2002; Pirozzoli et al., 2004; Foysi et al., 2004; Duan et al.,
2010; Modesti & Pirozzoli, 2016; Zhang et al., 2018; Trettel, 2019; Cogo et
al., 2022, 2023). However, none of these studies assessed whether intrinsic
compressibility effects play a role in peak strengthening. In fact, the higher
peak observed in the $M_{\infty}=14$ boundary layer was attributed to
variable-property effects by Zhang et al. (2018). Our results instead
demonstrate unambiguously that intrinsic compressibility effects play a
central role in the strengthening of streamwise turbulence intensity, since
our flow cases are essentially free of variable-property effects.

Similar to the wall-normal stress, the spanwise turbulent stress also
decreases with increasing Mach number, shown in figure 4(c). The increase in
the streamwise stress and the decrease in the wall-normal and spanwise
stresses imply suppression of inter-component energy transfer with increasing
Mach number. However, before discussing this in more detail in the next
subsection, we first note that the increase in the streamwise turbulent stress
is much more pronounced than the decrease in the other two components, which
essentially results in an increase in the turbulence kinetic energy with Mach
number as shown in figure 4(d). This suggests that, in addition to the change
in intercomponent energy transfer, there is also a change in the production of
$\widetilde{u^{\prime\prime}u^{\prime\prime}}^{*}$. This change in production
can be attributed to the changes in viscous and turbulent shear stresses
observed in figure 3, since it is their product that governs the production
term. This is further discussed in detail in Appendix A, where we present the
budget of the streamwise turbulence stress, and provide a phenomenological
explanation for the increase in
$\widetilde{u^{\prime\prime}u^{\prime\prime}}^{*}$.

###  3.3 Reduced inter-component energy transfer

The strengthening of the streamwise turbulent stress and the weakening of the
other two components, as observed in figures 4(a) - (c), imply an increase in
turbulence anisotropy, which was also previously observed in several studies
on compressible wall-bounded flows (Foysi et al., 2004; Duan et al., 2010;
Zhang et al., 2018; Cogo et al., 2022, 2023), mainly regarded as a variable-
property effect.

From turbulence theory, one can argue that the change in turbulence anisotropy
is due to reduced inter-component energy transfer. Since the negative of the
streamwise pressure-strain correlation
($-\Pi_{11}=-2\,\overline{p^{\prime}\partial u^{\prime\prime}/\partial x}$) is
a measure of the energy transferred from the streamwise turbulent stress to
the cross-stream components, we expect it to decrease with increasing Mach
number for our cases. To verify this, figure 5 shows $-\Pi_{11}$ scaled by the
TKE production (Duan et al., 2010; Patel et al., 2015; Cogo et al., 2023), for
(a) Mach 2.28, (b) Mach 3 and (c) Mach 4 cases, compared to the Mach 0.3 case.

Figure 5: Wall-normal distributions of the streamwise pressure-strain
correlation ($-\Pi_{11}$) scaled by the production term ($P_{11}$) for (a)
Mach 2.28, (b) Mach 3 and (c) Mach 4 cases described in table 1, compared to
the Mach 0.3 case.

The figure clearly corroborates our claims. We further note that $\Pi_{11}$
scaled by semi-local units ($\bar{\rho}u_{\tau}^{*3}/\delta_{v}^{*}$) also
reduces for the three high-Mach-number cases compared to the Mach 0.3 case
(not shown).

###  3.4 Identifying direct and indirect effects of intrinsic compressibility

So far we have observed strong intrinsic compressibility effects on various
turbulence statistics. Are these strong effects due to a direct contribution
from the dilatational motions or due to IC effects on the solenoidal motions?
To answer this, we apply Helmholtz decomposition to the velocity field
obtained from DNS to isolate the solenoidal (divergence-free) and dilatational
(curl-free) parts, namely

$u^{\prime\prime}_{i}={u_{i}^{s}}^{\prime\prime}+{u_{i}^{d}}^{\prime\prime}.$
(16)

Appendix B reports details on how the decomposition is actually performed.
Following Yu et al. (2019), the turbulent stresses are then split as

$\widetilde{u_{i}^{\prime\prime}u_{j}^{\prime\prime}}^{*}=\widetilde{{u_{i}^{s}}^{\prime\prime}{u_{j}^{s}}^{\prime\prime}}^{*}+\widetilde{{u_{i}^{d}}^{\prime\prime}{u_{j}^{s}}^{\prime\prime}}^{*}+\widetilde{{u_{j}^{s}}^{\prime\prime}{u_{j}^{d}}^{\prime\prime}}^{*}+\widetilde{{u_{i}^{d}}^{\prime\prime}{u_{j}^{d}}^{\prime\prime}}^{*}.$
(17)

The terms involving dilatational motions are absent in incompressible flows,
and thus any contribution from them is regarded as a _direct_ effect. However,
the first term on the right-hand side is also present in incompressible flows.
Thus, any effect of compressibility on this term will be regarded as an
_indirect_ effect.

Figure 6: Wall-normal distributions of the total and solenoidal (a)
streamwise, (b) wall-normal and (c) spanwise turbulent stresses as per
equation (17), for the cases described in table 1. Inset: profiles of the
terms $\widetilde{{v^{d}}^{\prime\prime}{v^{d}}^{\prime\prime}}^{*}$ (dotted)
and $\widetilde{{v^{s}}^{\prime\prime}{v^{d}}^{\prime\prime}}^{*}$ (dash-
dotted).

Figure 6 shows the first term on the right-hand side of equation (17),
associated with solenoidal velocity fluctuations, for the normal turbulent
stresses. They are seen to almost overlap with the total turbulent stresses,
which is shown in grey. This implies that any change in the total stresses as
a function of the Mach number is reflected in their respective solenoidal
components, and thus intrinsic compressibility effects on turbulence
statistics are mainly indirect. The collapse of the total and solenoidal
stresses also implies that the correlations involving
${u_{i}^{d}}^{\prime\prime}$ are small. However, there are some exceptions,
particularly the terms
$\widetilde{{v^{d}}^{\prime\prime}{v^{d}}^{\prime\prime}}^{*}$ and
$\widetilde{{v^{s}}^{\prime\prime}{v^{d}}^{\prime\prime}}^{*}$, that can have
large contributions in the near-wall region as shown in the inset of figure
6(b). Negative values of
$\widetilde{{v^{s}}^{\prime\prime}{v^{d}}^{\prime\prime}}^{*}$ physically
represent opposition of solenoidal motions (sweeps/ejections) from
dilatational wall-normal velocity. This opposition was first observed by Yu et
al. (2019), and plays a key role in the forthcoming discussion.

##  4 Weakening of the quasi-streamwise vortices

Quasi-streamwise vortices play an important role in transferring energy from
the streamwise to the wall-normal and spanwise components (Jeong et al.,
1997). Thus, any reduction in this inter-component energy transfer (see figure
5), and hence any weakening of the wall-normal and spanwise velocity
fluctuations (see figure 4) is directly related to the weakening of those
vortices. To verify this claim, the root-mean-square of the streamwise
vorticity is shown in figure 7(a). This quantity indeed decreases with
increasing Mach number, implying weakening of the quasi-streamwise vortices.
In contrast, the root-mean-square of the wall-normal and spanwise vorticity
shows a weak Mach number dependence, as seen in figure 7(b) and (c).

Figure 7: Wall-normal distributions of the root-mean-square of (a) streamwise,
(b) wall-normal, and (c) spanwise vorticity fluctuations, scaled by
$u_{\tau}^{*}/\delta_{v}^{*}$, for the cases described in table 1.

Choi et al. (1994) showed that active opposition of sweeps and ejections is
effective in weakening the quasi-streamwise vortices. As noted in §3.4, a
similar opposition also occurs spontaneously in compressible flows, in which
solenoidal motions like sweeps and ejections are opposed by wall-normal
dilatational motions.

To explain the physical origin of near-wall opposition of sweeps and
ejections, and hence the weakening of the quasi-streamwise vortices, we
perform a conditional averaging procedure that identifies shear layers. Shear
layers are in fact inherently associated with quasi-streamwise vortices, being
formed as a consequence of sweeps and ejections initiated by those vortical
structures (Jeong et al., 1997). To educe shear layers, we rely on the
variable interval space averaging (VISA) technique introduced by Kim (1985),
which is the spatial counterpart of the variable interval time averaging
(VITA) technique developed by Blackwelder & Kaplan (1976). Since only the
solenoidal motions carry the imprint of incompressible turbulent structures,
like shear layers, the VISA detection criterion is directly applied to the
solenoidal velocity field. More details on the implementation of the VISA
technique are provided in Appendix C.

###  4.1 Results from the variable interval space averaging technique

Figure 8 shows the conditionally averaged $\xi^{*}-y^{*}$ planes, at
$\zeta^{*}=0$, of various quantities for the Mach 2.28, 3 and 4 cases, only
considering acceleration events. A similar plot with deceleration events is
not shown since they are much less frequent (Johansson et al., 1987). $\xi$
and $\zeta$ indicate streamwise and spanwise coordinates, respectively,
centred at the locations of the detected events.

The first row in figure 8 shows the contours of the conditionally averaged
solenoidal streamwise velocity fluctuations
$\left<{u^{s}}^{\prime\prime}\right>^{*}$, which clearly represent a shear
layer. The second row of the plot shows the contours of the conditionally
averaged solenoidal wall-normal velocity fluctuations
$\left<{v^{s}}^{\prime\prime}\right>^{*}$. Positive streamwise velocity
fluctuations are associated with negative wall-normal fluctuations, resulting
in a sweep event. Similarly, negative streamwise fluctuations are associated
with positive wall-normal velocity, resulting in an ejection event. For
greater clarity, we also show streamlines constructed using
$\left<{u^{s}}^{\prime\prime}\right>^{*}$ and
$\left<{v^{s}}^{\prime\prime}\right>^{*}$, with their thickness being
proportional to the local magnitude of
$\left<{v^{s}}^{\prime\prime}\right>^{*}$.

Figure 8: Conditionally averaged quantities, based on VISA applied to
streamwise velocity fluctuations at $y^{*}\approx 15$ (see Appendix C), for
the Mach 2.28 (left column), Mach 3 (centre column), and Mach 4 (right column)
cases in table 1. The $\xi^{*}-y^{*}$ planes are taken at the centre of the
shear layer ($\zeta^{*}=0$). The velocity contours (first, second and fifth
rows) are scaled by the semi-local friction velocity $u_{\tau}^{*}$, the
pressure contours (third row) are scaled by $\tau_{w}$, and the dilatation
contours (fourth row) are scaled by $u_{\tau}^{*}/\delta_{v}^{*}$. The
overlaying streamlines are constructed using
$\left<{u^{s}}^{\prime\prime}\right>^{*}$ and
$\left<{v^{s}}^{\prime\prime}\right>^{*}$, and their thickness is scaled by
the magnitude of $\left<{v^{s}}^{\prime\prime}\right>^{*}$. The solid black
line indicates $y^{*}\approx 15$ and the dashed black line indicates
$\xi^{*}=0$.

Similar to the velocity field, we also split pressure into solenoidal and
dilatational parts, namely

${p}^{\prime}={p^{s}}^{\prime}+{p^{d}}^{\prime}.$ (18)

Unlike the Helmholtz decomposition for velocities, this splitting is not
unique. In this work, we adhere to the definition of solenoidal pressure given
for homogeneous flows by Ristorcelli (1997); Jagannathan & Donzis (2016); Wang
et al. (2017), which we extend to inhomogeneous flows as follows:

$\frac{\partial^{2}{p^{s}}^{\prime}}{\partial x_{i}\partial
x_{i}}=-\frac{\partial(\overline{\rho}{u_{i}^{s}}^{\prime\prime}{u_{j}^{s}}^{\prime\prime}-\overline{\rho}\overline{{u_{i}^{s}}^{\prime\prime}{u_{j}^{s}}^{\prime\prime}})}{\partial
x_{i}\partial
x_{j}}-2\overline{\rho}\frac{d\widetilde{u}}{dy}\frac{\partial{v^{s}}^{\prime\prime}}{\partial
x}.$ (19)

This part of the pressure field is also referred to as pseudo-pressure
(Ristorcelli, 1997), as it propagates with the flow speed. Looking at the
source terms on the right-hand side of equation (19), the solenoidal pressure
can be interpreted as being generated from vortices and shear layers, similar
to incompressible flows (Bradshaw & Koh, 1981).

The third row of figure 8 shows the conditionally averaged solenoidal pressure
as per equation (19). Clearly, the pressure maxima occur approximately in
between the high-velocity regions, which suggests a phase shift between
velocity and pressure. To shed further light on this point, in figure 9 we
plot the wall-normal velocity at $y^{*}\approx 15$, and the solenoidal
pressure at the wall as a function of the streamwise coordinate ($\xi^{*}$).
Since the wall pressure is mainly contributed by the buffer-layer eddies (Kim,
1989; Johansson et al., 1987; Kim & Hussain, 1993; Luhar et al., 2014), its
convection velocity is comparable with the speed of the buffer-layer coherent
structures (Kim & Hussain, 1993). Using this information and Taylor’s
hypothesis, one can transform the spatial axis in figure 9 to a temporal axis
($\tau$) by taking the mean velocity at $y^{*}\approx 15$ as the propagation
velocity. Reading figure 9 using the temporal axis (axis on the top), we note
that the high negative sweep velocity corresponds to a high negative rate of
change of the wall pressure, and likewise for the ejection velocity, i.e.,

$\frac{\partial\left<{p^{s}_{w}}^{\prime}\right>^{*}}{\partial\tau^{*}}\sim\left<{v^{s}}^{\prime\prime}\right>^{*}.$
(20)

Similar observations were made by Johansson et al. (1987), using the VITA
technique, and by Luhar et al. (2014), using the resolvent analysis. Other
interesting observations can be made from figure 9. First, the magnitude of
the conditionally averaged streamwise fluctuations increases, whereas the
magnitude of the conditionally averaged wall-normal fluctuations decreases
with increasing Mach number, as also seen in the first two rows of figure 8.
This is consistent with the strengthening of the streamwise and weakening of
the wall-normal turbulent stresses observed in figure 6. Second, the wall
pressure maximum shifts upstream with increasing Mach number, as also seen in
the third row of figure 8. While we know that such shift is attributed to the
Mach number dependence of the solenoidal motions that contribute to the source
terms in equation (19), at the moment we cannot provide a detailed explanation
for this and leave it for future studies.

Figure 9: Conditionally averaged profiles of solenoidal streamwise and wall-
normal velocities at $y^{*}\approx 15$, and wall pressure as a function of
space ($\xi^{*}$, at $\zeta^{*}$=0; bottom-axis) and time
($\tau^{*}=\tau/(u_{\tau}^{*}/\delta_{v}^{*})$; top-axis), for (a) Mach 2.28,
(b) Mach 3 and (c) Mach 4 cases in table 1.

After establishing the relation between the solenoidal wall-normal velocity
and the rate of change of the solenoidal pressure in equation (20), we
continue in our attempt to relate the solenoidal and the dilatational velocity
fields. For that purpose, we first isolate the dilatation generated from the
solenoidal pressure—also referred to as ‘pseudo-sound’ dilatation (superscript
$ps$) in the literature (Ristorcelli, 1997; Wang et al., 2017)—, as follows

$d^{ps}\approx\frac{-1}{\gamma\bar{P}}\left(\frac{\partial{p^{s}}^{\prime}}{\partial
t}+u_{j}\frac{\partial{p^{s}}^{\prime}}{\partial x_{j}}\right).$ (21)

Pseudo-sound dilatation represents the volume changes of fluid elements caused
by pressure changes associated with solenoidal turbulent structures such as
vortices and shear layers. Normalization by the wall shear stress yields

$d^{ps}\approx\frac{-\tau_{w}}{\gamma\bar{P}}\left(\frac{\partial{p^{s}}^{\prime*}}{\partial
t}+u_{j}\frac{\partial{p^{s}}^{\prime*}}{\partial x_{j}}\right),$ (22)

where the factor ${\tau_{w}}/({\gamma\bar{P}})$ is equal to the square of the
semi-local friction Mach number for ideal gas flows. Using
$M_{\tau}^{*}\approx M_{\tau}$ (see equation (12) and figure 2), we then
rewrite equation (22) as

$d^{ps}\approx-M_{\tau}^{2}\left(\frac{\partial{p^{s}}^{\prime*}}{\partial
t}+u_{j}\frac{\partial{p^{s}}^{\prime*}}{\partial x_{j}}\right).$ (23)

According to the pseudo-sound theory (Ristorcelli, 1997), the inner-scaled
scaled solenoidal pressure is assumed to be unaffected by compressibility
effects. Thus, from equation (23), one would expect $d^{ps}$ to increase with
the square of the friction Mach number. However, as noted in the discussion
following figure 9, the solenoidal motions change as a function of the Mach
number, thereby affecting the solenoidal pressure as per equation (19). This
suggests that $d^{ps}$ could increase with an exponent that is close to two
but not necessarily equal to two. To assess the correct scaling, in table 2 we
report the root-mean-square of $d^{ps}$ at the wall. Data fitting yields
$d^{ps}\sim M_{\tau}^{2.42}$, hence close to what was suggested by equation
(23).

Case name |  $M_{b_{w}}$ |  $M_{\tau}$ |  $\left(d_{w}^{ps}\right)^{*}_{rms}$ |  $\left(v_{p}^{d}\right)^{*}_{rms}$ |  $\left(v_{p}^{d,ps}\right)^{*}_{rms}$ |  $\left(v_{p}^{d,nps}\right)^{*}_{rms}$
---|---|---|---|---|---|---
Mach 2.28 |  2.28 |  0.1185 |  0.0096 |  0.066 |  0.047 |  0.059
Mach 3 |  3 |  0.1526 |  0.0160 |  0.153 |  0.078 |  0.140
Mach 4 |  4 |  0.1968 |  0.0311 |  0.332 |  0.150 |  0.323
|  |  $b$ |  2.42 |  3.1 |  2.37 |  3.3
Table 2: Root-mean-square ($rms$) of the pseudo-sound dilatation at the wall
and the peak $rms$ value of the total, pseudo-sound and non-pseudo-sound wall-
normal dilatational velocities. ‘$b$’ is the exponent obtained from power-law
fitting ($aM_{\tau}^{b}$) of the data.

Continuing on our path to relate solenoidal and dilatational motions, close to
the wall we can write

${d_{w}^{ps}}^{*}\approx-
M_{\tau}^{2}\frac{\partial{p_{w}^{s}}^{\prime*}}{\partial t^{*}},$ (24)

where ${d_{w}^{ps}}^{*}={d_{w}^{ps}}/(u_{\tau}^{*}/\delta_{v}^{*})$. This
equation, when conditionally averaged and combined with equation (20), leads
to

$\left<d_{w}^{ps}\right>^{*}\sim-
M_{\tau}^{2}\left<{v^{s}}^{\prime\prime}\right>^{*}.$ (25)

Using this result, we expect positive dilatation events (expansions) to be
mainly associated with sweeps and negative dilatation events (compressions) to
be associated with ejections. The fourth row in figure 8 shows the contours of
conditionally averaged pseudo-sound dilatation defined in equation (23).
Consistent with our expectation, positive dilatation is indeed found to be
associated with sweeps and negative dilatation with ejections, and its
magnitude increases with the Mach number.

Having related the pseudo-sound dilatation and the solenoidal velocity in
equation (25), the next step is to introduce the pseudo-sound dilatational
velocity as

$\displaystyle\frac{\partial^{2}\phi^{ps}}{\partial x_{j}\partial x_{j}}$
$\displaystyle=d^{ps},$ (26) $\displaystyle v^{d,ps}$
$\displaystyle=\dfrac{\partial\phi^{ps}}{\partial y},$

where $\phi^{ps}$ is the scalar potential. Note that this equation is similar
to equations (39) and (40) used to solve for the total dilatational velocity,
as reported in Appendix B. Based on equation (26), one would expect $v^{d,ps}$
to increase with the Mach number at a similar rate as $d^{ps}$. Power-law
fitting of the data reported in table 2 indeed yields $v^{d,ps}\sim
M_{\tau}^{2.37}$, hence close to what was found for $d^{ps}$.

Equation (26) stipulates that the conditionally averaged pseudo-sound
dilatational velocity in the buffer layer should be proportional to and in
phase with the dilatation at the wall. Thus, we can write

$\left<{v^{d,ps}}\right>^{*}\sim\left<{d_{w}^{ps}}\right>^{*}.$ (27)

Using equation (27) and (25) we can finally develop a relation between the
solenoidal and the pseudo-sound dilatational velocity, namely

$\left<{v^{d,ps}}\right>^{*}\sim-
M_{\tau}^{2}\left<{v^{s}}^{\prime\prime}\right>^{*}.$ (28)

In our opinion, this relation is quite meaningful as it theoretically supports
near-wall opposition of sweeps and ejections by dilatational motions.
Moreover, it suggests that the opposition effect should approximately increase
with the square of $M_{\tau}$.

In order to verify this, the final row in figure 8 reports the conditionally
averaged contours of the pseudo-sound wall-normal dilatational velocity given
in equation (26). As suggested from equation (27), the contours of $v^{d,ps}$
appear to be in phase with those of $d^{ps}$. Thus, consistent with the
observations made for the pseudo-sound dilatation, the wall-normal
dilatational velocity is positive during sweeps and negative during ejections,
and its magnitude increases with the Mach number. This opposition is also
clearly seen in figure 10, which shows the conditionally averaged profiles of
${v^{s}}^{\prime\prime}$ and $v^{d,ps}$ at $y^{*}\approx 15$. Additionally, in
figures 8 and 10 we note that the pseudo-sound dilatational velocity contour
(or profile) shifts upstream (leftward) with increasing Mach number. This is
due to the upstream shift in the pressure contour mentioned above.

Figure 10: Conditionally averaged profiles of solenoidal and pseudo-sound
dilatational wall-normal velocities at $y^{*}\approx 15$ as a function of
$\xi^{*}$ (at $\zeta^{*}$=0) for (a) Mach 2.28, (b) Mach 3 and (c) Mach 4
cases in table 1. Figure 11: (a) Conditionally averaged and integrated
[equation (29)] correlations between solenoidal and dilatational velocities.
(b) Conditionally averaged pseudo-sound correlation coefficient ($C^{ps}$) as
defined in equation (30).

To further quantify the opposition effect, we analyse the conditionally
averaged correlation between solenoidal and pseudo-sound dilatational wall-
normal velocity, i.e. $\left<v^{s}v^{d,ps}\right>$. The correlation is
integrated over a window of 300 viscous units in the streamwise direction and
40 viscous units in the spanwise direction (Johansson et al., 1991), at each
wall-normal location as

$\left<{v^{s}}^{\prime\prime}v^{d,ps}\right>_{\xi\zeta}(y^{*})=\int_{\zeta^{*}=-20}^{20}\int_{\xi^{*}=-150}^{150}\left<{v^{s}}^{\prime\prime}v^{d,ps}\right>(\xi^{*},y^{*},\zeta^{*})d\xi^{*}d\zeta^{*}.$
(29)

The integrated correlation, scaled by the squared semi-local friction
velocity, is reported in figure 11 with dashed lines. Figure 11 also shows the
pseudo-sound correlation coefficient defined as

$C^{ps}=\frac{\left<{v^{s}}^{\prime\prime}v^{d,ps}\right>_{\xi\zeta}}{\sqrt{\left<{v^{s}}^{\prime\prime}{v^{s}}^{\prime\prime}\right>_{\xi\zeta}\left<v^{d,ps}v^{d,ps}\right>_{\xi\zeta}}}.$
(30)

The correlation and its coefficient are negative as expected. The magnitude of
the correlation increases approximately with the square of Mach number, as
expected. However, the correlation coefficient almost collapses across all
Mach numbers.

The association of the opposition effect with the quasi-streamwise vortices is
visualised in figure 12 for the Mach 2.28 case, all other cases being
qualitatively similar. Indeed, the figure insets illustrate that sweeps and
ejections initiated by quasi-streamwise vortices are opposed by the near-wall
pseudo-sound dilatational velocity, thereby resulting in their weakening.

Figure 12: Opposition of sweeps and ejections by wall-normal pseudo-sound
dilatational velocity in the context of quasi-streamwise vortices. The shaded
three-dimensional isosurfaces represent quasi-streamwise vortices identified
by applying the Q-criterion to the conditionally averaged velocity field.
Their shadows are also plotted on the wall below, showing that the vortices
are inclined and tilted. Underneath the vortices, the contours of solenoidal
wall pressure are shown. The transparent planes mark regions of high rate of
change of wall pressure and hence high wall-normal pseudo-sound dilatational
velocity $\left<v^{d,ps}\right>^{*}$ (see discussion related to equations (20)
- (28)). The arrows between the vortices indicate $\left<v^{d,ps}\right>^{*}$
as a function of $\xi^{*}$ at $\zeta^{*}=0$ and $y^{*}\approx 20$. Note that
the line along which the arrows are plotted is slightly shifted away from the
wall for better visibility. Insets: contours of pseudo-sound dilatation
$\left<d^{ps}\right>^{*}$ along the transparent planes, overlaid with the
streamlines generated by quasi-streamwise vortices. These streamlines are
constructed using the wall-normal and spanwise solenoidal velocities, i.e.
$\left<v^{s}\right>^{*}$ and $\left<w^{s}\right>^{*}$, with their thickness
being proportional to the magnitude of the local planar velocity.
$\left<v^{d,ps}\right>^{*}$ at $y^{*}\approx 15$ and $y^{*}\approx 25$ is also
shown using arrows in the left and right planes, respectively. These wall-
normal locations correspond to the maximum value of
$\left<v^{d,ps}\right>^{*}$ in those planes. The red and blue colours in the
contour plots indicate positive and negative values, respectively. An
interactive version of this figure can be accessed here.

###  4.2 Role of non-pseudo-sound dilatational velocity in near-wall
opposition

So far we have looked into the pseudo-sound dilatational velocity and provided
an explanation for why they are out-of-phase with respect to the solenoidal
motions. However, from table 2, we see that the peak root-mean-square value of
$v^{d,ps}$ is much smaller than that of the total dilatational velocity.
Hence, a large portion of the dilatational velocity and its correlation (if
any) with the solenoidal velocity is still unexplained. To address this point,
figure 11(a) shows the integrated correlation between solenoidal and total
dilatational velocities, i.e.
$\left<{v^{s}}^{\prime\prime}v^{d}\right>^{*}_{\xi\zeta}$, denoted by solid
grey lines. Except very close to the wall, the total and pseudo-sound
correlations almost overlap. This implies that the contribution from the
remaining portion of the dilatational velocity, referred to as the ‘non-
pseudo-sound’ component and given by

$v^{d,nps}=v^{d}-v^{d,ps},$ (31)

is small. In other words, despite being stronger in magnitude than the pseudo-
sound component, the non-pseudo-sound dilatational velocity does not play an
important role in opposing sweeps and ejections.

Before concluding, we would like to comment on the travelling wave-packet-like
structures, first identified by Yu et al. (2019) and later studied in Yu et
al. (2020); Yu & Xu (2021); Tang et al. (2020); Gerolymos & Vallet (2023); Yu
et al. (2024). Figure 13 shows the $x^{*}-z^{*}$ plane with the instantaneous
contours of the pseudo-sound and non-pseudo-sound dilatational velocity at
$y^{*}\approx 11$, for the Mach 3 case in table 1. The wave-packet structures
are predominantly present in the non-pseudo-sound component, whereas the
pseudo-sound component shows a spotty structure similar to that observed for
the streamwise gradient of wall pressure in incompressible flows (Kim, 1989).
Combining the observation above that the non-pseudo-sound component hardly
contributes to the opposition effect, and that the wave-packet-like structures
are present mainly in this component, one can argue that these structures do
not play an important role in opposing sweeps and ejections.

Figure 13: Instantaneous $x^{*}-z^{*}$ planes at $y^{*}\approx 11$ of (top)
the non-pseudo-sound and (bottom) the pseudo-sound wall-normal dilatational
velocities (see the text for definitions) scaled by their respective root-
mean-squares for the Mach 3 case in table 1. Note that, for clarity, the
colour bar is adjusted such that structures stronger than 1.33 times the root-
mean-square value are highlighted.

##  5 Conclusions

In this paper, we have attempted to provide an explanation for the underlying
mechanism through which intrinsic compressibility effects modulate the near-
wall dynamics of turbulence. To rigorously assess these effects, we have
devised four DNS cases of fully developed high-Mach-number channel flows with
approximately constant mean properties, whereby intrinsic compressibility
effects are isolated. Our findings, sketched as a flow chart in figure 14, are
summarised as follows.

Figure 14: A graphical summary of the present findings. Note that the arrows
are meant to indicate the chain of arguments made in this paper, not relations
of causality.

First, we have decomposed the velocity field into solenoidal and dilatational
parts and educed shear layers by applying conditional averaging to the
solenoidal component. We have noticed that there exists a streamwise phase
shift between the buffer-layer sweeps and ejections that form shear layers,
and the associated ‘solenoidal’ wall pressure. Equivalent observations were
made for incompressible flows by Johansson et al. (1987) and Luhar et al.
(2014). By using Taylor’s hypothesis, this streamwise shift in phase can be
interpreted as a phase shift in time, such that regions of high positive rate
of change of wall pressure correspond to regions of high positive wall-normal
velocity. Similarly, regions of high negative rate of change of wall pressure
correspond to the regions of high negative wall-normal velocity. Close to the
wall, the high rate of change of the solenoidal pressure results in large
dilatation values with an opposite sign (also referred to as pseudo-sound
dilatation), which upon integration results in a wall-normal dilatational
velocity that inherently opposes sweeps and ejections. Since sweeps and
ejections are initiated by quasi-streamwise vortices, their opposition
directly affects the evolution of those vortices, causing their weakening.
This is schematically depicted in figure 12.

Interestingly, we also found that the remaining portion of the dilatational
velocity (also referred to as the non-pseudo-sound component) does not play an
important role in the opposition mechanism. Moreover, we have observed that
the majority of the travelling wave-packet-like structures, recently
discovered in the literature, are present in this non-pseudo-sound component.

The weakening of quasi-streamwise vortices directly hinders the energy
transfer from the streamwise velocity component to the other two components,
resulting in an outward shift (reduction) in the wall-normal turbulent stress
with increasing Mach number. Since the wall-normal motions actively contribute
to the transport of momentum across mean shear, thereby generating turbulent
shear stress, the outward shift in the wall-normal turbulent stress results in
a corresponding outward shift in the turbulent shear stress. This reduction in
the turbulent shear stress is in turn responsible for an upward shift in the
logarithmic mean velocity profile (Hasan et al., 2023).

A longstanding question in the compressible flow community is why the inner-
scaled streamwise turbulent stress is higher in compressible flows than in
incompressible flows, with similar Reynolds numbers. In this respect, our
results suggest that intrinsic compressibility effects play a dominant role.
Specifically, the increase in the peak value is a consequence of the outward
shift in the turbulent and viscous shear stresses, since their product yields
the production of the streamwise turbulence stress. This implies that the
near-wall opposition mechanism outlined above is also responsible for the
strengthening of the streamwise turbulence intensity.

Some questions related to the findings made in this paper remain unanswered as
of yet. First, why do the solenoidal pressure maxima shift upstream with
increasing Mach number (see figures 8 and 9)? Second, what is the Mach number
scaling of the turbulence statistics presented in the paper? This could help
explain the quasi-linear increase in the log-law constant observed by Hasan et
al. (2023). Moreover, knowing the Mach number scaling of the peak streamwise
turbulence intensity would help in developing empirical scaling laws. Third,
why is the dissipation of turbulence kinetic energy, and thus the small scales
of turbulence, not affected by intrinsic compressibility effects (see Appendix
A)? A spectral analysis of the velocity field could shed more light on this
important issue.

##  Appendix A Increase in the streamwise turbulence intensity

In order to explain the increase in the streamwise turbulent stress and hence
in the turbulence kinetic energy, we consider the streamwise turbulent stress
budget for a fully-developed compressible channel flow:

$P_{11}+\epsilon_{11}+T_{11}^{\nu}+T_{11}^{u}+\Pi_{11}+C_{11}=0\mathrm{,}$
(32)

where

$\displaystyle
P_{11}=-2\overline{\rho{u}^{\prime\prime}{v}^{\prime\prime}}\frac{\partial\widetilde{u}}{\partial
y},~{}\epsilon_{11}=-2\overline{\tau^{\prime}_{1j}\frac{\partial{u}^{\prime\prime}}{\partial
x_{j}}},$ $\displaystyle T^{\nu}_{11}=2\frac{\partial}{\partial
y}\left(\overline{{\tau^{\prime}_{12}}{u}^{\prime\prime}}\right),~{}T_{11}^{u}=-\frac{\partial}{\partial
y}\left(\overline{\rho{u}^{\prime\prime}{u}^{\prime\prime}{v}^{\prime\prime}}\right),~{}$
$\displaystyle\Pi_{11}=2\overline{p^{\prime}\frac{\partial
u^{\prime}}{\partial
x}},~{}C_{11}=2\overline{u^{\prime\prime}}\frac{\partial\bar{\tau}_{12}}{\partial
y}.$ (33)

The distributions of the production, viscous and turbulent diffusion terms,
and the sum of dissipation and pressure-strain correlation, are shown in
figure A.1, scaled by $\bar{\rho}{u_{\tau}^{*}}^{3}/\bar{\mu}$. The
compressibility term $C_{11}$ is omitted because of its negligible magnitude.

Figure A.1: Wall-normal distributions of (a) the streamwise turbulent stress
budget [see equation (32)] scaled in semi-local units, and (b) the sum of
viscous and turbulent fluxes obtained upon integrating the semi-locally scaled
viscous and turbulent diffusion terms [see equation (36)], for the cases
described in table 1.

Three observations can be made. First, there is an outward shift in
$P_{11}^{*}$ with increasing Mach number. Since the production term in scaled
form is simply the product of the turbulent and viscous shear stresses, its
outward shift is explained by the corresponding shift in the shear stresses in
figure 3(b) as follows. Assuming that the total stress is approximately equal
to $\tau_{w}$, such that the sum of the scaled stresses is unity, one can
substitute the viscous shear by the turbulent shear stress in $P_{11}$ to
obtain (Pope, 2001)

$P_{11}^{*}\approx-2\widetilde{u^{\prime\prime}v^{\prime\prime}}^{*}\left(1+\widetilde{u^{\prime\prime}v^{\prime\prime}}^{*}\right)=2\left(-\widetilde{u^{\prime\prime}v^{\prime\prime}}^{*}\right)-2\left(-\widetilde{u^{\prime\prime}v^{\prime\prime}}^{*}\right)^{2}.$
(34)

Taking the derivative of $P^{*}_{11}$ with respect to the turbulent shear
stress yields

$\frac{dP_{11}^{*}}{d\left(-\widetilde{u^{\prime\prime}v^{\prime\prime}}^{*}\right)}\approx
2-\,4\left(-\widetilde{u^{\prime\prime}v^{\prime\prime}}^{*}\right).$ (35)

Between the wall and the location where
$-\widetilde{u^{\prime\prime}v^{\prime\prime}}^{*}$ is equal to 0.5 the
derivative is positive, while it is negative above this location. On the other
hand, from figure 3(b), we observe that the rate of change of the turbulent
shear stress with the Mach number, i.e.
$\partial(-\widetilde{u^{\prime\prime}v^{\prime\prime}}^{*})/\partial M_{b}$,
at a fixed $y^{*}$ is negative. Combining these two observations, we can
conclude that the rate of change of production of the streamwise turbulent
stress with the Mach number, i.e. $\partial P_{11}^{*}/\partial M_{b}$, is
negative close to the wall and becomes positive away from it, resulting in an
effective outward shift.

Second, except very close to the wall, the sum of the two sink terms in the
budget of the streamwise turbulent stress (32), namely $\epsilon_{11}^{*}$ and
$\Pi_{11}^{*}$, show a weak Mach number dependence. Interestingly, the TKE
dissipation
($2\epsilon_{k}^{*}=\epsilon^{*}_{11}+\epsilon^{*}_{22}+\epsilon^{*}_{33}$),
reported with grey dashed lines in figure A.1, also shows marginal dependence
on the Mach number. This is consistent with the observation made by Hasan et
al. (2023) regarding the universality of the local Kolmogorov length scale.
The universality of $\epsilon_{11}^{*}+\Pi_{11}^{*}$ and $\epsilon_{k}^{*}$
are related as follows. Any Mach-number-dependent reduction in $\Pi_{11}^{*}$
would imply that less energy is being received by the lateral turbulent
stresses, and hence, less TKE is being dissipated through the terms
$\epsilon^{*}_{22}+\epsilon^{*}_{33}$. This suggests that the Mach-number-
dependence of $\Pi_{11}^{*}$ and $\epsilon^{*}_{22}+\epsilon^{*}_{33}$ is
linked, such that the universality of $\epsilon^{*}_{11}+\Pi^{*}_{11}$ is
connected with the universality of the TKE dissipation.

Third, above $y^{*}\approx 12$, the production term is higher at higher Mach
numbers, which combined with the observation that the total sink
$\epsilon^{*}_{11}+\Pi^{*}_{11}$ is universal, implies more negative values of
the diffusion term. This means that the surplus production is transported away
from the buffer layer towards the wall. For further insight, figure A.1(b)
shows the sum of the viscous and turbulent fluxes obtained by integrating the
transport terms as

$\Phi^{*}_{11}=\int_{0}^{y^{*}}\left(T_{11}^{\nu*}+T_{11}^{u*}\right)dy^{*},$
(36)

such that positive values signify that energy is transported towards the wall,
and negative values signify the opposite. As one can observe, the flux is
positive close to the wall and increases with the Mach number. This implies
that more energy is being carried towards the wall at higher Mach numbers.
Between the wall and the peak location of the streamwise turbulence intensity,
the total flux is mainly controlled by the viscous flux, which can be
approximated as $d\widetilde{u^{\prime\prime}u^{\prime\prime}}^{*}/dy^{*}$.
Thus, a higher positive flux at increasing Mach numbers implies a higher
gradient of the streamwise turbulent stress, which results in a higher peak
value upon integration.

The strengthening of the streamwise velocity fluctuations can also be
explained based on a phenomenological mixing-length model. The semi-locally
scaled streamwise stress can be written as

$\left(\overline{u^{\prime}u^{\prime}}^{*}\right)^{1/2}\sim\ell^{*}\frac{d\bar{U}^{*}}{dy^{*}},$
(37)

where $\ell^{*}$ is the mixing length scaled by $\delta_{v}^{*}$, and
${d\bar{U}^{*}}/{dy^{*}}$ is the semi-locally scaled mean velocity gradient,
which is equivalent to $d\bar{U}^{+}_{TL}/{dy^{*}}$. Note that the streamwise
stress is written in the Reynolds averaged form, since we observe that the
peak of both Reynolds and Favre averaged stresses increases alike (not shown),
and therefore the error incurred by excluding density fluctuations from
equation (37) is small. The mixing length is determined as
$\ell^{*}\sim\sqrt{\overline{v^{\prime}v^{\prime}}^{*}}\,\mathcal{T}$ (Durbin,
1991), where $\mathcal{T}\sim k^{*}/\epsilon^{*}$. For the present cases this
definition of mixing length yields universal distributions across the Mach
number range (not shown). This is because the velocity with which a fluid
parcel travels reduces with increasing Mach number. However, the time scale
over which it retains its streamwise momentum increases with the Mach number
(due to higher TKE and almost universal dissipation), thus effectively
travelling the same distance. Due to the universality of the mixing length,
equation (37) implies that the increase in mean shear observed in figure 3 is
directly responsible for an increase in the peak streamwise turbulence
intensity. Interestingly, an increase in the mean shear was also found to be
responsible for higher production in the buffer layer (see figure A.1) that
formed the basis of our explanation above, making the phenomenological model
consistent.

##  Appendix B Helmholtz decomposition of the velocity field

The Helmholtz decomposition in compressible flows is the representation of the
velocity field as the sum of divergence-free ‘solenoidal’ and curl-free
‘dilatational’ components. This is mathematically written as

${u}_{i}={u_{i}^{s}}+{u_{i}^{d}},$ (38)

where superscripts ‘$s$’ and ‘$d$’ stand for solenoidal and dilatational
components. This equation is similar to equation (16) in the main text, the
only difference being that there the decomposition was written explicitly for
the fluctuating velocity field.

The dilatational component is computed as the gradient of a scalar potential
$\phi$, namely

${u_{i}^{d}}=\frac{\partial\phi}{\partial x_{i}},$ (39)

where $\phi$ is obtained by solving a Poisson equation as

$\displaystyle\frac{\partial^{2}\phi}{\partial x_{j}\partial
x_{j}}=\frac{\partial u_{i}}{\partial x_{i}}.$ (40)

Equation (40) is solved using a second-order accurate FFT-based Poisson solver
(see Costa (2018) for example) with periodic boundary conditions in the
streamwise and spanwise directions, and no-penetration boundary condition
$\partial\phi/\partial y=0$ (or $v^{d}=0$) at the wall. Note that with these
boundary conditions, no-slip is not satisfied at the wall, that is $u^{d}$ and
$w^{d}$ are not equal to zero. While seemingly counter-intuitive at first
glance, this is not unphysical, as pointed out in Sharma & Girimaji (2023).

Likewise, the solenoidal component can be obtained using the vorticity field
as described in Yu et al. (2019) and Sharma & Girimaji (2023). However, here
we will make use of the fact that the total velocity field is available from
the direct numerical simulation. Thus, the solenoidal field is simply computed
using equation (38) as

${u_{i}^{s}}={u_{i}}-{u_{i}^{d}}.$ (41)

##  Appendix C Steps to perform variable interval space averaging

In this conditional average technique, strong sweep and ejection events
resulting in a shear layer are said to occur when the short-space variance,
given by

$\mathrm{var}(x,z,t)=\frac{1}{L}\int_{-\frac{L}{2}}^{\frac{L}{2}}[{u^{s}}^{\prime\prime}(x+s,y_{ref},z,t)]^{2}\,\mathrm{d}s-\left(\frac{1}{L}\int_{-\frac{L}{2}}^{\frac{L}{2}}{u^{s}}^{\prime\prime}(x+s,y_{ref},z,t)\,\mathrm{d}s\right)^{2},$
(42)

exceeds $K[u^{s}_{rms}(y_{ref})]^{2}$, where $K$ is the threshold level. Here,
$y_{ref}$ is the location of the reference $x-z$ plane where the detection
criteria is applied, and $L$ is the size of the averaging window,
representative of the length scale of the shear layer identified by this
technique (Johansson et al., 1987). Following Johansson et al. (1991), we take
$K=1$, $y^{*}_{ref}\approx 15$, and $L^{*}\approx 200$.

Having computed the short-space variance at the reference plane, a condition
variable C is set to non-zero values in regions where the variance exceeds the
threshold, and zero otherwise. The assigned non-zero value is 1 for
acceleration events and -1 for deceleration events. Mathematically, this is
written as

$\mathrm{C}(x,z,t)=\begin{cases}1,&\text{ for
}\operatorname{var}>K[u^{s}_{rms}(y_{ref})]^{2}\text{ and
}\dfrac{\partial{u^{s}}^{\prime\prime}}{\partial x}<0\\\ \\\ -1,&\text{ for
}\operatorname{var}>K[u^{s}_{rms}(y_{ref})]^{2}\text{ and
}\dfrac{\partial{u^{s}}^{\prime\prime}}{\partial x}>0\\\ \\\ 0,&\text{
otherwise, }\end{cases}$ (43)

where $\partial{u^{s}}^{\prime\prime}/{\partial x}<0$ implies
$\partial{u^{s}}^{\prime\prime}/{\partial t}>0$ and vice-versa, as per
Taylor’s hypothesis. This will result in patches on the reference $x-z$ plane
with values of 1 and -1 as shown in figure C.1. Within these patches, the
location where the short-space variance is locally maximum is also shown. Let
the coordinates of these locations be denoted by $(x_{o},z_{o})$. These
coordinates, detected at $y^{*}\approx 15$, will form the basis around which
conditional averaging is performed at all wall-normal locations.

Figure C.1: (Top) $x^{*}-z^{*}$ contour plot of the instantaneous solenoidal
streamwise velocity fluctuations at $y^{*}_{ref}\approx 15$ for the Mach 2.28
case. Boundaries of patches where the short-space variance exceeds the
Reynolds averaged value [see equation (43)] are overlaid on the contour plot.
Additionally, the location inside each patch where the short-space variance is
locally maximum is also displayed by a black circle or a grey square for
acceleration and deceleration events, respectively. (Bottom) Instantaneous
solenoidal streamwise velocity fluctuation along the horizontal line indicated
in the top plot. The black circle is the same point as in the top plot. The
short-space variance [equation (42)] is also shown using a grey dashed line.

With the detected VISA locations, the conditional average of any variable
$\Psi$ is then given as:

$\left<\Psi\right>(\xi,y,\zeta)=\frac{1}{N}\sum_{f=1}^{N_{f}}\sum_{n=1}^{N_{e}}\Psi(x^{n}_{o}+\xi,y,z_{o}^{n}+\zeta,t^{f}),$
(44)

where $\xi$ and $\zeta$ are the streamwise and spanwise displacements with
respect to the reference or detected locations $(x_{o},z_{o})$, and they vary
from $-L_{x}/2$ to $L_{x}/2$ and $-L_{z}/2$ to $L_{z}/2$, respectively. The
inner sum is over the number of detected events ($N_{e}$) in a particular
snapshot $f$ (at time instant $t^{f}$), whereas the outer sum is over the
number of snapshots ($N_{f}$), such that the global sum of the detected events
over all the snapshots is $N$.

Note that equation (44) leads to a conditional average from which phase jitter
is yet to be removed (Johansson et al., 1987). The concept of phase-jitter is
explained with an example as follows. It is known that an acceleration VISA
event detected at the location ($x_{o},y^{+}\approx 15,z_{o}$) corresponds to
a wall pressure peak directly underneath, i.e. at ($x_{o},y^{+}\approx
0,z_{o}$). However, there can be a small and random phase lag or lead. This
means that in reality, the pressure peak may occur at a location that is
randomly shifted in the streamwise-spanwise direction with respect to the
detected location, i.e. it may occur at ($x_{o}+\Delta_{x},y^{+}\approx
0,z_{o}+\Delta_{z}$). This misalignment leads to a reduction in the magnitude
of the pressure peak obtained after conditional averaging.

To fix this issue, we employ a cross-correlation technique (Johansson et al.,
1987) that is described using the above example as follows. We first compute
the conditional average of wall pressure as usual without fixing the phase-
jitter issue. We then cross-correlate this conditionally averaged wall
pressure plane with the instantaneous wall pressure plane using the Fourier
transform. Having done this, we should obtain a $x-z$ plane of correlation
coefficients on the wall that displays a local maximum close to but not
necessarily at the point of detection, i.e. $(x_{o},z_{o})$. This maximum
implies that the conditionally averaged wall pressure profile has its imprint
in the instantaneous plane around the detection location. The shift between
the detection location $(x_{o},z_{o})$ and the local maximum around
$(x_{o},z_{o})$ gives the amount of phase lag or lead in the streamwise and
spanwise directions, i.e. $\Delta_{x}$ and $\Delta_{y}$ discussed above. In
order to remove the phase lag or lead, we compute a new conditional average by
shifting the instantaneous planes by this $\Delta_{x}$ and $\Delta_{y}$ around
the detection points, thereby aligning them. Mathematically, equation (44) is
modified for wall pressure as

$\left<p^{\prime}\right>(\xi,0,\zeta)=\frac{1}{N}\sum_{f=1}^{N_{f}}\sum_{n=1}^{N_{e}}p^{\prime}(x^{n}_{o}+\Delta_{x}^{n}+\xi,\,0,\,z_{o}^{n}+\Delta_{z}^{n}+\zeta,t^{f}).$
(45)

Now, the same procedure described for pressure at the wall can be repeated for
pressure at any wall-normal location. Doing this results in $\Delta_{x}$ and
$\Delta_{y}$ that depend on $y$ for each detected event. With this, equation
(45) can be rewritten for the entire pressure field as

$\left<p^{\prime}\right>(\xi,y,\zeta)=\frac{1}{N}\sum_{f=1}^{N_{f}}\sum_{n=1}^{N_{e}}p^{\prime}(x^{n}_{o}+\Delta_{x}^{n}(y)+\xi,\,y,\,z_{o}^{n}+\Delta_{z}^{n}(y)+\zeta,t^{f}).$
(46)

Although this gives more control on the alignment of three-dimensional
conditionally averaged structures, it may result in a conditionally averaged
profile that may not be very smooth in the wall-normal direction, such that
layering is observed (some layering can be seen in figure 8).

In the phase-jitter removal procedure, events for which the required shift is
greater than approximately 40 viscous lengths in the streamwise or spanwise
directions are excluded from the averaging procedure, and the total number of
detected events ($N$) is reduced accordingly. Since the applied shifts are
wall-normal dependent, the excluded number of events would also be wall-normal
dependent.

Figure C.2 shows the $\xi^{*}-y^{*}$ pressure contours taken at the centre of
the shear layer, i.e. at $\zeta^{*}=0$, after no alignment (equation (44)) and
after one iteration of alignment (equation (46)). As seen, the pressure
contours remain qualitatively similar in both the cases, however, the
magnitude after one iteration of alignment has increased substantially.

Figure C.2: Contours of the solenoidal pressure along the $\xi^{*}-y^{*}$
plane at $\zeta^{*}=0$ after (top row) equation (44) (no alignment), and after
(bottom row) equation (46) (first alignment iteration). The left, middle and
right columns correspond to the Mach 2.28, 3 and 4 cases in table 1,
respectively.

The conditionally averaged profile obtained from equation (46) can be cross-
correlated again with the instantaneous field, and the procedure above can be
repeated to further improve the alignment. However, as noted in Johansson et
al. (1987), and also verified for our cases, the maximum jitter is eliminated
in the first iteration. Thus, the results presented in the main text are
obtained after one iteration.

## Acknowledgments

This work was supported by the European Research Council grant no.
ERC-2019-CoG-864660, Critical; and the Air Force Office of Scientific Research
under grants FA9550-23-1-0228 and FA8655-23-1-7016. The authors acknowledge
the use of computational resources of the Dutch National Supercomputer
Snellius (grant no. 2022/ENW/01251049), and of the DelftBlue supercomputer,
provided by the Delft High Performance Computing Centre.

[Declaration of Interests]The authors report no conflict of interest.

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