# Assessment of heat transfer and Mach number effects on high-speed turbulent
boundary layers

###### Abstract

High-speed vehicles experience a highly challenging environment in which the
free-stream Mach number and surface temperature greatly influence aerodynamic
drag and heat transfer. The interplay of these two parameters strongly affects
the near-wall dynamics of high-speed turbulent boundary layers in a non-
trivial way, breaking similarity arguments on velocity and temperature fields,
typically derived for adiabatic cases. In this work, we present direct
numerical simulations of flat-plate zero-pressure-gradient turbulent boundary
layers spanning three free-stream Mach numbers [2,4,6] and four wall
temperature conditions (from adiabatic to very cold walls), emphasising the
choice of the diabatic parameter $\mathit{\Theta}$ (Zhang et al., 2014) to
recover a similar flow organisation at different Mach numbers. We link
qualitative observations on flow patterns to first- and second-order
statistics to explain the strong decoupling of temperature-velocity
fluctuations that occurs at reduced wall temperatures and high Mach numbers.
For these cases, we find that the mean temperature gradient in the near-wall
region can reach such a strong intensity that it promotes the formation of a
secondary peak of thermal production in the viscous sublayer, which is in
direct contrast with the monotonic behaviour of adiabatic profiles. We propose
different physical mechanisms induced by wall-cooling and compressibility that
result in apparently similar flow features, such as a higher peak in the
streamwise velocity turbulence intensity, and distinct ones, such as the
separation of turbulent scales.

Michele Cogo1, Umberto Baù2, Mauro Chinappi3, Matteo Bernardini4, and
Francesco Picano1,5

1Centro di Ateneo di Studi e Attività Spaziali “Giuseppe Colombo”, via Venezia
15, 35131 Padova, Italy<EMAIL_ADDRESS>
2Institute of Fluid Mechanics and Heat Transfer, TU Wien, 1060 Vienna, Austria
3Department of Industrial Engineering, University of Rome Tor Vergata, via del
Politecnico 1, 00133 Rome, Italy
4Department of Mechanical and Aerospace Engineering, Sapienza University of
Rome, via Eudossiana 18, 00184 Rome, Italy
5Department of Industrial Engineering, Università degli Studi di Padova, via
Venezia 1, 35131 Padova, Italy

##  1 Introduction

The study of highly compressible turbulent boundary layers is of major
importance for high-speed turbulence research. Compressibility acts upon the
flow by influencing the mean and fluctuating fields of thermodynamic
quantities, which are in turn coupled to the momentum, promoting the energy
exchange between kinetic and thermal fields. This poses several difficulties
in the prediction of drag and wall heat transfer, which makes engineering
design more and more difficult as higher speeds are attained.

In recent decades, supersonic turbulent boundary layers have been extensively
studied and compared to their compressible counterparts at the same Reynolds
number, mainly focusing on the prediction of drag, e.g. Bernardini and
Pirozzoli (2011a); Wenzel et al. (2018); Duan et al. (2011). In fact, at
supersonic speeds the wall temperature can be considered for practical
purposes very close to the recovery temperature of the flow, implying a very
low heat exchange at the wall. The recovery temperature indicates the
temperature that is attained by the flow when it is brought to rest in a non-
isentropic manner, defined as

$T_{r}=T_{\infty}\left(1+r\frac{\gamma-1}{2}M_{\infty}^{2}\right),$ (1)

being $r=Pr^{1/3}$ the recovery factor (Zhang et al., 2018), where $Pr$ is the
Prandtl number. However, in hypersonic boundary layers, the recovery
temperatures are so high that the wall temperature is usually lower (Urzay and
Di Renzo, 2021), generating large heat fluxes to the wall. This affects the
flow dynamics in concurrency with the Mach number, enriching the physical
effects that have to be accounted for when developing theoretical relations
and reduced order models.

A renewed interest in hypersonic flight, along with the computational
advancements that render Direct Numerical Simulations more feasible, sparked
the attention on these problems, e.g. Zhang et al. (2018); Wenzel et al.
(2022), but there is still a lack of understanding of the individual effect of
different flow parameters.

The framework of theoretical relations applied to compressible flow for mean
velocity and fluctuating fields aims at mapping compressible profiles onto
incompressible reference by taking into account variations of mean properties
such as density and viscosity. When applied to the mean velocity field, these
relations are called compressibility transformations, first introduced by Van
Driest (1956) by accounting for mean density variations in the wall-normal
velocity profile. Among the plethora of relations proposed in recent years,
Volpiani et al. (2020) and Griffin et al. (2021) stand out as capable of
efficiently collapsing velocity profiles even at high Mach numbers. While
Volpiani et al. (2020) uses a mixed physical and data-driven approach to
determine the optimal parameters that define the weight of density and
viscosity, Griffin et al. (2021) base their arguments on the total-stress
equation, allowing for separate assumptions for the viscous sublayer and the
log layer.

Theoretical relations have also been derived to describe the interaction
between kinetic and thermal fields, classically referred to as Strong Reynolds
Analogy (SRA). First proposed by Morkovin (1962), SRA establishes a framework
based on the similarity between the momentum and total enthalpy equations,
from which a direct proportionality between velocity and total enthalpy can be
inferred. Under the more restrictive condition of wall adiabaticity, a set of
relations coupling velocity and temperature can be derived for both mean and
fluctuating fields, in which the temperature resembles a passive scalar field.

These relations have been extensively validated for adiabatic TBLs at
different Mach numbers, e.g. Bernardini and Pirozzoli (2011a); Wenzel et al.
(2018), although at hypersonic speeds ($M_{\infty}>5$) discrepancies start to
arise (Zhang et al., 2018). Subsequent extensions of the SRA accounting for
diabatic walls have been recently proposed (e.g. Zhang et al. (2014)), which
obtained promising results for different flow conditions, even when
thermochemical effects are present (Passiatore et al., 2021, 2022; Di Renzo
and Urzay, 2021).

A cold wall imposes a change in the sign of the mean temperature gradient near
the wall, affecting the production of temperature fluctuations, which may
result in a severe loss of similarity between velocity and temperature fields,
a building block of SRA, clearly visible in instantaneous snapshots of
turbulent structures (Cogo et al., 2022; Zhang et al., 2022). However, these
studies also noted that comparing cases with different Mach numbers at a fixed
wall-to-recovery temperature ratio $T_{w}/T_{r}$ ($<1$) resulted in vastly
different near-wall dynamics for temperature fluctuations, in a way that cold
cases at high $M_{\infty}$ seemed “more adiabatic” than their low $M_{\infty}$
counterparts. Recently, other definitions of the wall temperature condition
have been proposed, such as the Eckert number
$Ec=(\gamma-1)M_{\infty}^{2}T_{\infty}/(T_{r}-T_{w})$ (Wenzel et al., 2022) or
the diabatic parameter $\mathit{\Theta}=(T_{w}-T_{\infty})/(T_{r}-T_{\infty})$
(Zhang et al., 2014), which are capable to account for the indirect effect of
Mach number on the wall temperature condition. Although progress has been made
to incorporate the effects of compressibility and heat transfer on these
relations, their individual influence is still not well understood. While
compressibility effects induced by the increase in Mach number can be similar
to a change in wall temperature condition (and vice versa) for certain
mechanisms, such as redistribution of turbulent kinetic energy (Duan and
Martin, 2011), their relative role is still unclear in other aspects, such as
separations of turbulent scales (Huang et al., 2022). In this regard, wall-
cooling has been shown to reduce the separation between the large and small
turbulence scales in hypersonic flows (Huang et al., 2022; Fan et al., 2022),
but the specific role of the Mach number is still debated. Furthermore, while
wall-cooling has been shown to be dominant in regulating energy exchanges in
the near-wall region (Fan et al., 2022), the effect of the Mach number is
still not clear. These and other authors called upon the need for additional
computations to determine their individual effects.

The aim of this study is to unveil the physical mechanisms that yield
similarities and differences between the effect of compressibility and wall-
cooling. To pursue this objective, an extensive DNS database consisting of 12
simulations of zero-pressure-gradient TBLs has been computed fixing the
friction Reynolds number ($Re_{\tau}=443$), while spanning three Mach numbers
$M_{\infty}=[2,4,6]$ and four diabatic parameters
$\mathit{\Theta}=[0.25,0.5,0.75,1.0]$, going from extremely cold walls,
$\mathit{\Theta}=0.25$, to adiabatic case, $\mathit{\Theta}=1$. The computed
database is discussed in the present paper with the aim to be used for the
development of simplified models for high-speed wall-bounded flows with strong
heat flux.

The remainder of the paper is organised as follows. The numerical method and
details on the simulation setup are outlined in section §2. In section §3, a
general visualisation of instantaneous velocity and temperature fields is
given, describing the individual effect of Mach and wall temperature
conditions on the flow dynamics and turbulent structures. Then, first-order
statistics for mean velocity and temperature are presented in section §4,
which also compares different SRA formulations. Finally, second-order
statistics are presented in section §5, focusing on the effect of wall-cooling
on thermal production, and its implications on velocity-temperature
correlations.

##  2 Simulation parameters and computational setup

The three-dimensional compressible Navier-Stokes equations are numerically
solved for a viscous, heat-conducting gas

$\begin{split}\frac{\partial\rho}{\partial t}+\frac{\partial(\rho
u_{j})}{\partial x_{j}}=0,\\\ \frac{\partial(\rho u_{i})}{\partial
t}+\frac{\partial(\rho u_{i}u_{j})}{\partial x_{j}}+\frac{\partial p}{\partial
x_{i}}-\frac{\partial\sigma_{ij}}{\partial x_{j}}=0,\\\ \frac{\partial(\rho
E)}{\partial t}+\frac{\partial(\rho Eu_{j}+pu_{j})}{\partial
x_{j}}-\frac{\partial(\sigma_{ij}u_{i}-q_{j})}{\partial x_{j}}=0,\end{split}$
(2)

where $\rho$ is the density, $u_{i}$ denotes the velocity component in the ith
Cartesian direction ($i=1,2,3$), $p$ is the thermodynamic pressure,
$E=c_{v}T+u_{i}u_{i}/2$ the total energy per unit mass and

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

represents the viscous stress tensor and the heat flux vector, respectively.
The molecular viscosity $\mu$ is assumed to follow Sutherland’s law

$\frac{\mu}{\mu_{\infty}}=\left(\frac{T}{T_{\infty}}\right)^{1/2}\frac{1+C/T_{\infty}}{1+C/T},$
(4)

where $C=110.4$ K and $T_{\infty}=220.0$ K, representing the typical
conditions that are met in the stratosphere. The thermal conductivity $k$ is
related to the viscosity by the expression $k=c_{p}\mu/Pr$, where $c_{p}$ is
the specific heat at constant pressure and the Prandtl number is $Pr=0.72$.
The thermodynamic variables are correlated to each other by means of the
equation of state for a calorically perfect gas. This choice was also assumed
for cases at $M_{\infty}=6$, after having verified that by introducing a
dependence of specific heat with temperature $c_{p}=f(T)$ differences in all
statistics were negligible. Moreover, gas dissociation effects are also not
expected in the present database, according to the observations of Passiatore
et al. (2022) which observed negligible effects with $T_{w}=1800K$ (our
highest imposed value is $T_{w}=1640K$ for $M_{\infty}=6$). The system of
equations is solved on a Cartesian grid using the in-house code STREAmS
(Bernardini et al., 2021, 2023), which has been extensively validated in
numerous canonical configurations (Bernardini et al., 2011; Bernardini and
Pirozzoli, 2011b; Cogo et al., 2022). Convective terms are discretised using
high-order, energy-preserving schemes applied in shock-free regions, while a
high-order shock capturing scheme (WENO) is applied when shock waves are
identified by the Ducros sensor (Ducros et al., 1999). Diffusive terms are
discretised using a locally conservative formulation (De Vanna et al., 2021),
expanded to Laplacian to ensure finite molecular dissipation at all resolved
wavelengths. The solver takes advantage of a multi-GPU architecture by means
of the CUDA Fortran paradigm. The domain is a rectangular box of length
$L_{x}=100\,\delta_{in}$, $L_{y}=15\,\delta_{in}$, $L_{z}=9\,\delta_{in}$,
where $\delta_{in}$ is the boundary layer thickness at the inflow station,
based on the $99\%$ of the freestream velocity $u_{\infty}$ (which is referred
for other stations as $\delta_{99}$). For each spatial direction, the number
of computational points employed for all cases is $N_{x}=5120$, $N_{y}=320$,
and $N_{z}=512$. Periodic boundary conditions are enforced in the spanwise
direction, purely non-reflecting boundary conditions are employed for the
outflow and the top boundary, and unsteady characteristic boundary conditions
are used at the bottom wall (Poinsot and Lele, 1992), where the isothermal
wall temperature condition is enforced. The recycling-rescaling procedure
(Pirozzoli et al., 2010) is applied at the inflow to reach a fully developed
state, the recycling length being placed at a distance of $80\,\delta_{in}$
from the inlet, ensuring a complete decorrelation of the fluctuations between
the recycling station and the inflow plane (Morgan et al., 2011).

Table 1 summarises the flow conditions and grid resolutions for each run,
where $M_{\infty}$ is the free-stream Mach number and $Re_{\tau}$ is the
friction Reynolds number, defined as the ratio between the boundary layer
thickness $\delta_{99}$ and the viscous length scale
$\delta_{\nu}=\nu_{w}/u_{\tau}$, where $u_{\tau}=\sqrt{\tau_{w}/\rho_{w}}$ is
the friction velocity, $\tau_{w}$ is the mean wall shear stress, and $\nu_{w}$
is the kinematic viscosity at the wall. $\Delta x^{+}=\Delta x/\delta_{\nu}$
and $\Delta z^{+}=\Delta z/\delta_{\nu}$ are the uniform grid spacings in the
streamwise and spanwise directions and $\Delta y^{+}=\Delta y/\delta_{\nu}$
represents the non-uniform wall-normal grid spacing (the wall and edge values
are reported). In the wall-normal direction, the stretching function of
Pirozzoli et al. (2010) is employed, which provides a more favourable scaling
of the number of grid points with the Reynolds number. Furthermore, this
function has the natural property of yielding precisely constant resolution in
terms of the local Kolmogorov length scale $\eta$ in the outer part of the
wall layer while maintaining a uniform near-wall spacing.
The present database is composed of a total of 12 simulations, spanning three
Mach numbers $M_{\infty}=[2,4,6]$ and four diabatic parameters
$\mathit{\Theta}=[0.25,0.5,0.75,1.0]$ (see Table 1). As discussed previously,
the way in which the wall temperature condition is defined is of great
importance to discern it from the Mach number effect. However, a formulation
in which $T_{w}$ is not a function of $T_{r}$ (which is, in turn, a quadratic
function of $M_{\infty}$) is not possible, because $T_{r}$ is the value
reached at the wall under adiabatic conditions ($T_{w}=T_{r}$). Therefore, the
goal of a suitable parameter is not to be independent of $M_{\infty}$, but to
incorporate it in order to have “the same integral behaviour between different
cases, regardless of whether its variation is caused by the change of the Mach
number or of the wall temperature” (Wenzel et al., 2022). This is the
rationale with which Wenzel et al. (2022) argued that the Eckert number
$Ec=(\gamma-1)M_{\infty}^{2}T_{\infty}/(T_{r}-T_{w})$ represents a more
suitable option than the conventional $T_{w}/T_{r}$ ratio. The Eckert number
happens to be directly related to the diabatic parameter
$\mathit{\Theta}=(T_{w}-T_{\infty})/(T_{r}-T_{\infty})$ proposed by Zhang et
al. (2014), since it can be shown that
$Ec=2/\left[r(1-\mathit{\Theta})\right]$. This parameter shows more clearly
the improvement over the conventional ratio $T_{w}/T_{r}$, showing that
$T_{\infty}$ needs to be subtracted from both $T_{w}$ and $T_{r}$ to compare
only the $\Delta T$ that is recovered when the flow is brought at rest, being
the only one responsible for kinetic-internal energy exchanges. In this study,
we choose to use the diabatic parameter $\mathit{\Theta}$ over $Ec$ given its
simplicity, but we also report the latter in Table 1. Table 2 summarises the
boundary layer parameters at selected locations where turbulence statistics
are gathered.

Throughout this study, we use the symbols $u$, $v$, and $w$ to denote the
streamwise, wall-normal and spanwise velocity components and the decomposition
of any variable is conducted using either the standard Reynolds decomposition
($f=\bar{f}+f^{\prime}$) or the density-weighted (Favre) representation
($f=\tilde{f}+f^{{}^{\prime\prime}}$), being $\tilde{f}=\overline{\rho
f}/\bar{\rho}$. Here, the averaging is conducted using multiple samples and
along the periodic direction $z$.

Run | $M_{\infty}$ | $Re_{\tau}$ | $\mathit{\Theta}$ | $T_{w}/T_{r}$ | Ec | $\Delta x^{+}$ | $\Delta y^{+}_{w,edge}$ |  $\Delta z^{+}$
---|---|---|---|---|---|---|---|---
M2T025 | 2.00 | 436 – 579 | 0.25 | 0.69 | 2.975 | 4.51 | 0.71-4.64 | 4.07
M2T050 | 2.00 | 427 – 564 | 0.5 | 0.79 | 4.463 | 4.52 | 0.71-4.64 | 4.07
M2T075 | 2.00 | 424 – 561 | 0.75 | 0.9 | 8.926 | 4.53 | 0.71-4.65 | 4.08
M2T100 | 2.00 | 415 – 584 | 1.0 | 1.0 | $\infty$ | 4.52 | 0.71-4.64 | 4.07
M4T025 | 4.00 | 404 – 535 | 0.25 | 0.44 | 2.975 | 4.36 | 0.68-4.52 | 3.93
M4T050 | 4.00 | 391 – 521 | 0.5 | 0.63 | 4.463 | 4.38 | 0.68-4.53 | 3.94
M4T075 | 4.00 | 379 – 507 | 0.75 | 0.81 | 8.926 | 4.37 | 0.68-4.53 | 3.94
M4T100 | 4.00 | 371 – 494 | 1.0 | 1.0. | $\infty$ | 4.38 | 0.68-4.53 | 3.95
M6T025 | 6.00 | 376 – 500 | 0.25 | 0.35 | 2.975 | 4.24 | 0.66-4.42 | 3.82
M6T050 | 6.00 | 351 – 470 | 0.5 | 0.57 | 4.463 | 4.21 | 0.66-4.40 | 3.80
M6T075 | 6.00 | 343 – 462 | 0.75 | 0.78 | 8.926 | 4.24 | 0.66-4.43 | 3.83
M6T100 | 6.00 | 337 – 451 | 1.0 | 1.0 | $\infty$ | 4.26 | 0.67-4.44 | 3.84
Table 1: Summary of parameters for DNS study. Grid spacings are given in wall-units according to the stations selected in table 2. The values of $\Delta y^{+}_{w}$ and $\Delta y^{+}_{edge}$ refer to the wall-normal spacing at the wall and at the boundary layer edge, respectively. The range of $Re_{\tau}$ is representative of the statistical growth of the boundary layer’s thickness along $x$.  Station | $Re_{\tau}$ | $Re_{\theta}$ | $Re_{\delta 2}$ | $Re_{\tau}^{*}$ | $\delta^{*}/\delta$ | $\theta/\delta$ | $H$ | $-B_{q}(\cdot 10^{-2})$ |  $C_{f}(\cdot 10^{-3})$
---|---|---|---|---|---|---|---|---|---
M2T025 | 443 | 1179 | 1043 | 541 | 0.210 | 0.092 | 2.291 | 2.29 | 3.40
M2T050 | 443 | 1404 | 1155 | 645 | 0.224 | 0.088 | 2.528 | 1.34 | 3.16
M2T075 | 443 | 1613 | 1169 | 750 | 0.234 | 0.085 | 2.756 | 0.57 | 2.99
M2T100 | 443 | 1848 | 1239 | 856 | 0.246 | 0.083 | 2.979 | 0.08 | 2.79
M4T025 | 443 | 1607 | 1101 | 825 | 0.314 | 0.067 | 4.669 | 6.10 | 2.19
M4T050 | 443 | 2331 | 1267 | 1228 | 0.331 | 0.061 | 5.460 | 3.07 | 1.85
M4T075 | 443 | 3090 | 1429 | 1638 | 0.355 | 0.056 | 6.297 | 1.17 | 1.61
M4T100 | 443 | 3718 | 1530 | 2024 | 0.367 | 0.052 | 7.030 | 0.17 | 1.43
M6T025 | 443 | 2169 | 1156 | 1273 | 0.399 | 0.048 | 8.279 | 8.89 | 1.39
M6T050 | 443 | 3586 | 1449 | 2103 | 0.426 | 0.043 | 9.813 | 4.05 | 1.09
M6T075 | 443 | 4835 | 1629 | 2951 | 0.443 | 0.039 | 11.485 | 1.49 | 0.93
M6T100 | 443 | 5893 | 1764 | 3701 | 0.454 | 0.036 | 12.800 | 0.16 | 0.81
Table 2: Boundary layer properties averaged at the selected station.
$Re_{\tau}=\rho_{w}u_{\tau}\delta/\mu_{w}$;
$Re_{\theta}=\rho_{\infty}u_{\infty}\theta/\mu_{\infty}$;
$Re_{\delta_{2}}=\rho_{\infty}u_{\infty}\theta/\mu_{w}$;
$Re_{\tau}^{*}=\sqrt{\rho_{\infty}\tau_{w}}\delta/\mu_{\infty}$;
$H=\delta^{*}/\theta$ ($\delta^{*}$ and $\theta$ are the boundary layer
displacement and momentum thickness, respectively).
$B_{q}=q_{w}/(\rho_{w}C_{p}u_{\tau}T_{w})$ and
$C_{f}=\tau_{w}/(1/2\rho_{\infty}u_{\infty}^{2})$ are the nondimensional wall
heat transfer $q_{w}=-\bar{k}\ \partial\tilde{T}/\partial y$ and skin friction
coefficient $\tau_{w}=\bar{\mu}\ \partial\tilde{u}/\partial y$, respectively.

##  3 Instantaneous visualisation

To highlight the emerging features of the flow in a qualitative way, we
selected the two extreme cases with regard to the wall-cooling condition,
$\mathit{\Theta}=0.25$ and $\mathit{\Theta}=1.0$, for each Mach number in our
database. Figure 1 shows a portion of wall-normal $x$-$y$ planes coloured with
the instantaneous density, whose variability is a clear sign of the degree of
compressibility. The effect of Mach number is clearly apparent for all cases
moving from top to bottom with a decrease of the minimum value of density and
an increase of the general level of acoustic disturbances, generated in the
boundary layer and emanated towards the far field. However, a stronger wall-
cooling (lower $\mathit{\Theta}$, left column of figure 1) attenuates this
effect, since lower wall temperatures generate higher density fields in the
near wall region.

(a) $M_{\infty}=2,\mathit{\Theta}=0.25$ (b) $M_{\infty}=2,\mathit{\Theta}=1.0$
(c) $M_{\infty}=4,\mathit{\Theta}=0.25$ (d) $M_{\infty}=4,\mathit{\Theta}=1.0$
(e) $M_{\infty}=6,\mathit{\Theta}=0.25$ (f) $M_{\infty}=6,\mathit{\Theta}=1.0$

Figure 1: Instantaneous density in wall-normal slices ($x$-$y$ plane), with a
window size of $\Delta x=20\delta-30\delta$ and $\Delta y=0\delta-2\delta$.
Here, all Mach numbers are shown while the two extremes are chosen with regard
to wall-cooling ($\mathit{\Theta}=0.25$ and $\mathit{\Theta}=1.0$).

The intensity of wall-cooling strongly affects the coupling between velocity
and temperature fluctuations, especially in the near-wall region. This is
apparent in figure 2, which compares these quantities in wall-parallel slices
located at approximately $y^{*}\approx 10$, representing the onset turbulence
activity after the viscous sub-layer. Here, $y^{*}=y/\delta_{\nu,SL}$ is the
semilocal scaled wall-normal coordinate, with
$\delta_{\nu,SL}=\bar{\nu}/\sqrt{\tau_{w}/\bar{\rho}}$. The chosen $x$-$z$
planes are centred at the selected stations of table 1 spanning a window of
$\Delta x^{*}=4000$ and $\Delta z^{*}=600$. Velocity fluctuations
$\sqrt{\bar{\rho}}u^{{}^{\prime}}/\sqrt{\tau_{w}}$ are scaled according to the
Morkovin’s transformation (Morkovin, 1962) (also used in section 5.1), which
enables comparison across different Mach numbers and wall temperature values
by accounting for the variation of the mean properties of the flow. In other
words, velocity fluctuations are scaled by the semilocal friction velocity
$u_{\tau,SL}=\sqrt{\tau_{w}/\bar{\rho}}$, which differs from the conventional
one by employing the mean density $\bar{\rho}$ instead of the wall density
$\rho_{w}$. Temperature fluctuations $\bar{\rho}T^{{}^{\prime}}/(R\tau_{w})$
are scaled in a similar fashion, assuming $\tau_{w}$ a proper parameter to
scale pressure fluctuations, then $\tau_{w}/(R\bar{\rho})$ can be used to
scale temperature (for further details refer to section 5.2). A general look
at the velocity fluctuations shows the presence of near-wall streaks for all
cases, representative of the near-wall self-sustaining cycle of turbulence.
Similar values of intensities appear across all cases. This result is not
shared with temperature fluctuations, where cold cases ($\mathit{\Theta}<1$)
show reduced intensity and a clear breakdown of elongated streaks, appearing
more isotropic when compared to their adiabatic counterpart. Adiabatic cases
maintain a streaky pattern, which shows a clear coupling with the velocity
field. Although this behaviour will be further discussed in the following
sections by analysing temperature fluctuations and thermal production
profiles, these qualitative results are consistent with the discussion of
Wenzel et al. (2022), which states that the same general behaviour due to the
effect of wall-cooling is expected when comparing flows with the same Eckert
number (or diabatic parameter $\mathit{\Theta}$).

(a) $M_{\infty}=2,\mathit{\Theta}=0.25$ (b) $M_{\infty}=2,\mathit{\Theta}=1.0$
(c) $M_{\infty}=4,\mathit{\Theta}=0.25$ (d) $M_{\infty}=4,\mathit{\Theta}=1.0$
(e) $M_{\infty}=6,\mathit{\Theta}=0.25$ (f) $M_{\infty}=6,\mathit{\Theta}=1.0$

Figure 2: Temperature fluctuations $\bar{\rho}T^{\prime}/\tau_{w}$ (top) and
streamwise velocity fluctuations $\sqrt{\bar{\rho}}u^{\prime}/\sqrt{\tau_{w}}$
(bottom) in wall-parallel slices ($x$-$z$ plane) selected at $y^{*}\approx
10$. Here, all Mach numbers are shown while the two extremes are chosen with
regard to wall-cooling ($\mathit{\Theta}=0.25$ and $\mathit{\Theta}=1.0$).
Here, $x_{ref}$ is the streamwise location of the selected station.

##  4 Mean flow statistics

In this section, we present the wall-normal profiles of averaged quantities
such as velocity and temperature, selected at stations listed in table 2. We
consider mean velocity profiles in the framework of compressibility
transformations, which aim at incorporating compressibility effects in wall-
bounded flow statistics in order to recover the incompressible behaviour.
Since the pioneering work of Van Driest (1951), several relations have been
proposed to account for the variations of mean fluid properties, such as
density and viscosity. These relations can be cast in terms of mapping
functions $f_{I}$ and $g_{I}$ for wall distance $y_{I}$ and mean velocity
$u_{I}$, which denote the equivalent incompressible distributions obtained
from the transformation $I$:

$y_{I}=\int_{0}^{y}f_{I}dy,\quad\quad
u_{I}=\int_{0}^{\tilde{u}}g_{I}d\tilde{u}.$ (5)

Table 3 shows the relative values of $f_{I}$ and $g_{I}$ for Van Driest (1951)
and the recent transformation of Volpiani et al. (2020), which employs a
partially data-driven approach to derive the mapping exponents.

Transformation | Wall distance ($f_{I}$) | Mean velocity ($g_{I}$)
---|---|---
Van Driest (1951) | $f_{VD}=1$ |  $g_{VD}=R^{1/2}$
Volpiani et al. (2020) | $f_{VI}=\frac{R^{1/2}}{M^{3/2}}$ |  $g_{VI}=\frac{R^{1/2}}{M^{1/2}}$
Table 3: Compressibility transformations for the wall distance and the mean
velocity according to Eq. (5), where $R=\bar{\rho}/\bar{\rho}_{w}$ and
$M=\bar{\mu}/\bar{\mu}_{w}$.

Griffin et al. (2021) transformation, instead, is based on the total stress
equation, which reads:

$\tau^{+}=S_{t}^{+}\left(\frac{\tau_{v}^{+}}{S_{TL}^{+}}+\frac{\tau_{R}^{+}}{S_{eq}^{+}}\right)$
(6)

where $\tau_{v}^{+}$ and $\tau_{R}^{+}$ are the scaled viscous and Reynolds
shear stresses (whose sum is equal to $\tau^{+}$), while $S_{TL}^{+}=\partial
U_{TL}^{+}/\partial y^{*}$ and $S_{eq}^{+}=\partial U_{eq}^{+}/\partial y^{*}$
are the generalised nondimensional mean shear stresses derived for the viscous
region (the subscript $TL$ indicated the accordance with the Trettel and
Larsson (2016) velocity transformation) and for the log layer (the subscript
$eq$ indicates the assumption of turbulence quasi-equilibrium). The
generalised nondimensional mean shear $S_{t}^{+}=\partial U_{t}^{+}/\partial
y^{*}$ is the unknown and once computed it can be integrated with respect to
the semilocal wall-normal coordinate $y^{*}$, leading to the transformed
velocity $u^{+}_{GR}=\int S_{t}^{+}dy^{*}$.

We report in figure 3 the scaled profiles according to the classical law of
Van Driest (1951) (which has been the standard for several decades and widely
employed in wall modelling) and the latest transformations of Volpiani et al.
(2020) and Griffin et al. (2021). Panel 3(a) reveals the main weaknesses of
the Van Driest (1951) scaling, whose accuracy is affected both by the increase
of the Mach number and wall-cooling. In particular, non-adiabatic cases at
$M_{\infty}=4,6$ show a clear departure from the linear law of the wall, while
even adiabatic cases show a positive shift from the log-law as compressibility
increases. Panels 3(b) and 3(c) show a great improvement in collapsing all
profiles to the laws of the wall, the only minor discrepancy being present in
the log layer for extremely cold cases at high Mach numbers. Overall, our
database supports Volpiani et al. (2020) and Griffin et al. (2021)
transformations, proving their wide range of applicability.

(a) Van Driest et al.  (b) Volpiani et al.  (c) Griffin et al.

Figure 3: Mean velocity profiles at stations listed in table 2 scaled
according to (a) Van Driest (1951), (b) Volpiani et al. (2020) and (c) Griffin
et al. (2021) compressibility transformations. Profiles have been translated
along the $x$ axis according to the law $10^{M_{\infty}/2-1}$ to enable better
comparison.

Figures 4(a) to 4(e) show the mean temperature profiles throughout the height
of the boundary layer and in the near-wall region, respectively. In
particular, the latter profiles are scaled using the wall temperature $T_{w}$.
As expected, the adiabatic wall temperature greatly increases with the Mach
number, while enhanced wall-cooling (lower $\mathit{\Theta}$) forces the mean
temperature profiles to slant towards lower wall temperatures ($T_{w}<T_{r}$).
The combination of these two conditions imposes a change in the sign of
temperature gradient near the wall, which is necessary to adjust to a wall
temperature lower than the recovery value. Thus, a local peak arises, whose
prominence and location are directly connected to the phenomenon of
aerodynamic heating, generating a net heat flux from the flow to the solid
boundary. Local temperature peaks are marked in figures 4(b) to 4(e) with
dots. An increase in the Mach number generates more intense gradients and
higher peak temperatures for non-adiabatic cases, enhancing aerodynamic
heating. However, the wall-normal position of the peaks seems to be mainly
affected by the change $\mathit{\Theta}$, and weakly dependent on the Mach
number. This is apparent in Figure 4(f), which shows a progressive departure
from the wall of the peak location as the wall-cooling increases, with a mild
downward shift at high Mach numbers. As anticipated in section §3, the
position of the local maximum of the temperature profile has major
implications in the generation of temperature fluctuations, which affect both
their overall intensity and their spatial organisation (breakdown of near-wall
streaks). Actually, the departure from a monotonic adiabatic profile to
increasingly prominent local peaks of mean temperature profiles prevents the
formation of organised temperature streaks that are generated by thermal
production (see section §5.2).

(a)

(b) $\mathit{\Theta}=0.25$ (c) $\mathit{\Theta}=0.5$ (d)
$\mathit{\Theta}=0.75$ (e) $\mathit{\Theta}=1.$ (f)

Figure 4: Panel (a): Mean temperature profiles for all cases of table 1 as a
function of the wall-normal coordinate $y/\delta_{99}$. Panels (b-c-d-e): Mean
temperature profiles and relative peaks as a function of the wall-normal
coordinate $y^{*}$ scaled with $T_{w}$. Panel (f): Wall-normal position of
mean temperature peaks as a function of the wall-cooling $\mathit{\Theta}$
parameter.

###  4.1 Reynolds analogy

In this section, the coupling between velocity and temperature is discussed
for both mean and fluctuating fields. The DNS results are compared with the
classical relations of Walz (1969)

$\frac{\bar{T}}{T_{\infty}}=\frac{T_{w}}{T_{\infty}}+\frac{T_{r}-T_{w}}{T_{\infty}}\frac{\bar{u}}{U_{\infty}}+\frac{T_{\infty}-T_{r}}{T_{\infty}}\left(\frac{\bar{u}}{U_{\infty}}\right)^{2}$
(7)

and the modified relation of Zhang et al. (2014)

$\frac{\bar{T}}{T_{\infty}}=\frac{T_{w}}{T_{\infty}}+\frac{T_{rg}-T_{w}}{T_{\infty}}\frac{\bar{u}}{U_{\infty}}+\frac{T_{\infty}-T_{rg}}{T_{\infty}}\left(\frac{\bar{u}}{U_{\infty}}\right)^{2}$
(8)

where $T_{rg}=T_{\infty}+r_{g}U_{\infty}^{2}/(2c_{p})$ and
$r_{g}=2c_{p}(T_{w}-T_{\infty})/U_{\infty}^{2}-2\,Pr\,q_{w}/(U_{\infty}\tau_{w})$.

Figure 5 compares the relations (7) and (8) with the present database. As
expected, Walz (1969) relation greatly degrades its accuracy when the wall-
cooling is increased, while Zhang et al. (2014) is able to better perform
under these conditions, the only minor deviations being present for the case
M6T025. However, we note that Walz (1969) law still excellently holds for
adiabatic cases at high Mach numbers, while being incapable of correctly
capturing wall-cooling effects.

(a) $M_{\infty}=2$ (b) $M_{\infty}=4$ (c) $M_{\infty}=6$

Figure 5: Mean temperature profiles against mean velocity compared with the
classical law of Walz (1969) (Eq. (7)) and the modified relation of Zhang et
al. (2014) (Eq. (8)).

For engineering design purposes, the value of $r_{g}$ can be difficult to
evaluate given its dependence on the wall temperature $T_{w}$ and the ratio of
the wall heat flux $q_{w}$ and the wall shear stress $\tau_{w}$. Following the
discussion of Zhang et al. (2014), the Reynolds analogy factor $s$ comes into
play to greatly simplify the calculation, since $r_{g}$ can be rewritten in
terms of $s$

$r_{g}=r[sPr+(1-sPr)\mathit{\Theta}]$ (9)

being $s$ defined as

$s=\frac{2C_{h}}{C_{f}}=\frac{q_{w}u_{\infty}}{\tau_{w}c_{p}(T_{w}-T_{r})}$
(10)

where $C_{f}=\tau_{w}/(1/2\rho_{\infty}u_{\infty}^{2})$ is the skin friction
coefficient and $C_{h}=q_{w}/(\rho_{\infty}u_{\infty}c_{p}(T_{w}-T_{r}))$ the
Stanton number. The simplification consists in the fact that several authors
(Duan et al., 2010; Zhang et al., 2014) identified the term $sPr$ to be an
empirical constant around the value of $0.8\pm 0.03$ (data fitting of Zhang et
al. (2014)) over several different flow cases, meaning that only $T_{w}$ would
be needed to be evaluated to compute $r_{g}$.

Figure 6 reports the computed values of $sPr$ in our database showing a good
agreement to Zhang et al. (2014) fit. A slight decreasing trend with
$\mathit{\Theta}$ can be observed, and it is interesting to note that at a
given $\mathit{\Theta}$ the values appear to be independent of $M_{\infty}$.

The data reported in figure 6 have a mean value and standard deviation of
$0.78\pm 0.03$, which is close to the value reported by (Zhang et al., 2014).
By approximating $r_{g}$ in Eq. (9) with the mean value of $sPr$ and comparing
it with DNS data, we obtain a maximum error of $5\%$ (for the case M6T025),
which can be considered acceptable for engineering purposes.

Figure 6: Reynolds analogy factor $s=2C_{h}/C_{f}Pr$ as a function of the
diabatic parameter $\mathit{\Theta}$ for different Mach numbers. The grey band
refers to the data fitting of $0.8\pm 0.03$ of Zhang et al. (2014).

Another important set of theoretical relations that couple the thermodynamic
and kinetic fluctuating fields is given by the Strong Reynolds Analogy (SRA)
(Morkovin, 1962). Originally derived for an adiabatic case, the three main
relations can be expressed as

$\begin{gathered}\frac{\left(\widetilde{T^{{}^{\prime\prime}2}}\right)^{1/2}/\tilde{T}}{(\gamma-1)\tilde{M}^{2}\left(\widetilde{u^{{}^{\prime\prime}2}}\right)^{1/2}/\tilde{u}}\approx
1,\\\
R_{u^{{}^{\prime\prime}}T^{{}^{\prime\prime}}}=\frac{\widetilde{u^{{}^{\prime\prime}}T^{{}^{\prime\prime}}}}{\sqrt{\widetilde{u^{{}^{\prime\prime}2}}}\sqrt{\widetilde{T^{{}^{\prime\prime}2}}}}\approx
1,\\\ Pr_{t}=\frac{\overline{\rho
u^{\prime}v^{\prime}}(\partial\tilde{T}/\partial y)}{\overline{\rho
T^{\prime}v^{\prime}}(\partial\tilde{u}/\partial y)}\approx 1.\end{gathered}$
(11)

where we remind the Favre average definition $\tilde{f}=\overline{\rho
f}/\bar{\rho}$ and that $f^{{}^{\prime\prime}}=f-\tilde{f}$.

Figure 7: Velocity and temperature correlation
$R_{u^{{}^{\prime\prime}}T^{{}^{\prime\prime}}}$ as function of
$y/\delta_{99}$. Full lines ( ) indicate $M_{\infty}=2$, dashed lines ( )
$M_{\infty}=4$ and dotted lines ( ) $M_{\infty}=6$.

Figure 7 shows the profiles of
$R_{u^{{}^{\prime\prime}}T^{{}^{\prime\prime}}}$ that clearly deviate from
unity, which is expected since it was derived assuming zero total temperature
fluctuation (Morkovin, 1962). All profiles collapse around the value
$-R_{u^{{}^{\prime\prime}}T^{{}^{\prime\prime}}}=0.6$, except in the near-wall
region, which is marked with an inset (Duan et al., 2010). The inset of Figure
7 shows that the crossover location, where
$R_{u^{{}^{\prime\prime}}T^{{}^{\prime\prime}}}=0$, corresponds approximately
to the location of the maximum mean temperature. Here, we observe that as the
wall gets progressively cold, the crossover location moves at higher $y^{*}$
values, indicating a temperature-velocity decorrelation that is progressively
moved farther from the wall. Our database also shows that this location is
almost independent of the Mach number when $\mathit{\Theta}$ is fixed, whereas
distinct Mach and wall-cooling effects are visible on the near-wall peak
intensity and position of $R_{u^{{}^{\prime\prime}}T^{{}^{\prime\prime}}}$.

(a) Original SRA (b) PARAMETRI-3 (c) Modified HSRA (d) PARAMETRI-3

Figure 8: Comparison of the Strong Reynolds Analogies in the (a,b) original
form (Eq. (11)), (c,d) Modified HSRA (Zhang et al., 2014)) (Eqs. (12), (13)).
Full lines ( ) indicates $M_{\infty}=2$, dashed lines ( ) $M_{\infty}=4$ and
dotted lines ( ) $M_{\infty}=6$.

The remaining two relations of equation (11) have been modified over the years
to account for finite heat flux at the wall and remove wall temperature
dependence (Huang et al., 1995) (HSRA). The most recent improvement has been
made by Zhang et al. (2014), who proposed another definition of the turbulent
Prandtl number $\overline{Pr_{t}}$ which should perform better at high Mach
numbers, yielding the following expression of the modified strong Reynolds
analogy (modified HRSA):

$\frac{\left(\widetilde{T^{{}^{\prime\prime}2}}\right)^{1/2}/\tilde{T}}{(\gamma-1)\tilde{M}^{2}\left(\widetilde{u^{{}^{\prime\prime}2}}\right)^{1/2}/\tilde{u}}\overline{Pr_{t}}\left(1-(\partial\tilde{T}_{t}/\partial\tilde{T})\right)\approx
1$ (12)

where the proposed definition of $\overline{Pr_{t}}$ is

$\overline{Pr_{t}}=\frac{\overline{(\rho
v)^{\prime}u^{{}^{\prime}}}\partial\tilde{T}/\partial y}{\overline{(\rho
v^{{}^{\prime}})T^{{}^{\prime}}}\partial\tilde{u}/\partial
y}=Pr_{t}\frac{1+\bar{v}\overline{\rho^{\prime}u^{\prime}}/\overline{\rho
v^{\prime}u^{\prime}}}{1+\bar{v}\overline{\rho^{\prime}T^{\prime}}/\overline{\rho
v^{\prime}T^{\prime}}}$ (13)

in which the difference from the classical definition is notable when both
$\bar{v}$ and $\rho^{\prime}$ are nonzero.

Figure 8 compares the wall-normal profiles obtained with the original SRA and
the modified version of Eq. (12), as well as the profiles of $Pr_{t}$ and the
modified $\overline{Pr_{t}}$ of Eq. (13). Figure 8(c) shows that the modified
version of Zhang et al. (2014) clearly improves the insensitivity to the
freestream Mach number and wall temperature condition, with only slight
deviations at the edge of the boundary layer. It is also interesting to note
the excellent collapse that the original SRA of panel 8(a) exhibits for
profiles at fixed $\mathit{\Theta}$, independently of the Mach number,
highlighting the relevance of the diabatic parameter $\mathit{\Theta}$ in
accounting for the effects of different wall temperatures independently of the
Mach number.

##  5 Fluctuation statistics

###  5.1 Velocity fluctuations and length scales

The distribution of velocity fluctuation intensities and Reynolds shear stress
is reported in the left panels of figures 9 and 10, using the classical
transformation of Morkovin (1962):

$\left(u_{i}^{*}\right)^{2}=\frac{\widetilde{u_{i}^{\prime\prime
2}}}{u_{\tau}^{2}}\frac{\bar{\rho}}{\bar{\rho}_{w}},\quad(uv)^{*}=\frac{\widetilde{u^{\prime\prime}v^{\prime\prime}}}{u_{\tau}^{2}}\frac{\bar{\rho}}{\bar{\rho}_{w}}.$
(14)

The profiles are shown as a function of the wall-normal distance in semilocal
scaling $y^{*}$ (Huang et al., 1995), considering its ability to collapse
compressible profiles of different Mach numbers and wall temperature
conditions, in particular with respect to the peak positions (Zhang et al.,
2018, 2022). Right panels of figures 9 and 10 show the corresponding turbulent
kinetic energy budget terms, (being
$k=\widetilde{u_{i}^{\prime\prime}u_{i}^{{}^{\prime\prime}}}/2$ the turbulent
kinetic energy, TKE) according to the derivation of Zhang et al. (2018), where
$P$ is the production term, $TT$ represents the turbulent transport, $\Pi$
includes the pressure diffusion and dilatation, $-\phi$ is the viscous
dissipation and $D$ is the viscous diffusion. For these results, semilocal
scaling is also employed in the normalisation of budget terms (refer to Zhang
et al. (2018)) and for the wall-normal distance $y^{*}$, enabling a good
collapse between different profiles (Zhang et al., 2018; Cogo et al., 2022).

(a) $M_{\infty}=2$ (b) $M_{\infty}=2$

(c) $M_{\infty}=4$ (d) $M_{\infty}=4$

(e) $M_{\infty}=6$ (f) $M_{\infty}=6$

Figure 9: Semilocal-scaled turbulent velocity fluctuations (a,c,e) and
turbulent kinetic budget (b,d,f) as function of the wall-normal distance
$y^{*}$. Here, different diabatic parameters $\mathit{\Theta}$ are compared at
a given Mach number $M_{\infty}$.

The effect of wall-cooling on velocity fluctuations, shown in the left panels
of figure 9, is apparent as an increase in the peak of the streamwise
component located at $y^{*}\approx 15$ that is more prominent at high Mach
numbers. In contrast, the spanwise component of highly cooled cases shows the
opposite behaviour, being reduced in intensity compared to the adiabatic
reference. This implies an increase in the anisotropy of normal components of
Reynolds stresses in the near-wall region, which is discussed in more detail
at the end of this section. The semilocal scaling provides an excellent
collapse of the peak positions for all cases, preventing the outward shift
that is present for cold cases when plotted in wall units (not shown). This is
also true for the position of the turbulent production peak (right panels of
figure 9), which would move farther from the wall if displayed in wall units.
In general, the effect of wall-cooling on the turbulent kinetic energy budget
is marked in the very near-wall region, especially at high Mach numbers, while
all profiles progressively collapse in the outer layer.

(a) $\mathit{\Theta}=0.25$ (b) $\mathit{\Theta}=0.25$

(c) $\mathit{\Theta}=0.5$ (d) $\mathit{\Theta}=0.5$

(e) $\mathit{\Theta}=0.75$ (f) $\mathit{\Theta}=0.75$

(g) $\mathit{\Theta}=1.0$ (h) $\mathit{\Theta}=1.0$

Figure 10: Semilocal-scaled turbulent velocity fluctuations (a,c,e) and
turbulent kinetic budget (b,d,f) as function of of the wall-normal distance
$y^{*}$. Here, different Mach number $M_{\infty}$ are compared at a given
diabatic parameter $\mathit{\Theta}$.

The effect of the Mach number on velocity fluctuations is reported in the left
panels of figure 10, where an increase of the streamwise component peak with
the Mach number is apparent, while the other normal components intensities are
observed to weakly decrease until $y^{*}<40$. Unlike the wall-cooling effect,
all normal components increase in the log layer as $M_{\infty}$ increases. We
note that this effect could be reduced at the BL edge by matching the
semilocal friction Reynolds number $Re_{\tau}^{*}$ in place of the
conventional definition (see table 1), which would allow all profiles to
collapse when $y^{*}\approx Re_{\tau}^{*}$. In fact, $Re_{\tau}^{*}$ has been
shown by several authors to better incorporate compressibility and wall-
cooling effects on the separation of scales in highly-compressible flows (e.g.
Griffin et al. (2021); Hirai et al. (2021)). This suggests that
compressibility acts in the direction of increasing the scale separation in
the outer layer, while wall-cooling has the opposite effect (Fan et al.,
2022). For all values of $\mathit{\Theta}$, the turbulent kinetic energy
budget (right panels of figure 10), shows an increase of the production term
$P$ in the buffer and log layers as the Mach number increases and a
corresponding decrease of diffusion $D$ and turbulent transport $TT$ in the
same regions, consistently with Cogo et al. (2022), who noted the presence of
this effect also in the outer region at higher $Re_{\tau}$.

While the effect of wall-cooling on the TKE budget seems confined in the near-
wall region, the influence of Mach number is more prominent after the peak of
production and throughout the log layer.

Further insights on the mechanism of redistribution of turbulent kinetic
energy in the near-wall region can be gained by looking at the ratio between
the streamwise component of the pressure-strain term and the streamwise
component of turbulent production (Duan et al., 2010):

$\mathcal{R}=\left(\overline{p^{{}^{\prime}}\frac{\partial
u^{{}^{\prime\prime}}}{\partial x}}\right)/\left(\overline{\rho
u^{{}^{\prime\prime}}v^{{}^{\prime\prime}}}\frac{\partial\tilde{u}}{\partial
y}\right)$ (15)

which is a measure of the energy transfer from the streamwise velocity
fluctuations to the others.

(a) $\mathit{\Theta}=1.0$ (b) $M_{\infty}=6$

Figure 11: Ratio of streamwise components of pressure-strain and turbulent
production terms for cases at (a) $\mathit{\Theta}=1.0$ and (b)
$M_{\infty}=6$, as function of the wall-normal distance in semilocal units.

The role of the pressure-strain term in increasing turbulence anisotropy was
also noted for other flows (e.g. Foysi et al. (2004)). To gauge the respective
effects of the Mach number and the wall temperature condition, figure 11
compares $-\mathcal{R}$ for cases at $\mathit{\Theta}=1.0$ (figure 11(a)) and
at $M_{\infty}=6$ (figure 11(b)). In panel 11(a), profiles of $-\mathcal{R}$
are reduced in magnitude as compressibility increases, with greater intensity
farther from the wall.

This is consistent with the less efficient redistribution of turbulent kinetic
energy discussed before, and is attributed to the absence of a solenoidal
condition for the velocity field for highly-compressible cases preventing an
efficient energy transfer between velocity components. Looking at panel
11(b)), we observe that wall-cooling acts similarly to an increase of
compressibility, strongly decreasing the profiles of $-\mathcal{R}$, but its
effect is localised in the near-wall region and strongly reduced after the
buffer layer.

We attribute this effect to a localised stratification of flow properties in
the near-wall region. As wall-cooling is increased and the location of the
mean temperature peak approaches the buffer layer, the flow above and below
the peak location is relatively colder and denser. This is true for all Mach
numbers in our database (although with different intensities), since the
temperature peak location remains unaffected (see figure 4(f)). This localised
stratification forces turbulent fluctuations to be active almost only in the
streamwise direction, while the other components tend to be suppressed.

(a) $\mathit{\Theta}=1.0$ (b) $M_{\infty}=6$

Figure 12: Diagram of the anisotropy invariant map of Lumley and Newman (1977)
for cases at (a) $\mathit{\Theta}=1.0$ and (b) $M_{\infty}=6$.

This effect is quantified in figure 12 by showing the anisotropic invariant
map of Lumley and Newman (1977), also known as the Lumley triangle, which uses
the second ($II$) and third ($III$) principal components of turbulence
anisotropy. The invariant map is composed of three limiting states: one-
component ($x_{1c}$), axisymmetric two-component ($x_{2c}$) and isotropic
($x_{3c}$); which are representative of the relative strengths of the
fluctuating velocity components. Looking at figure 12(a), we note that the
cusp point, which coincides approximately with the peak of velocity
fluctuations in the buffer layer, shifts towards a one-component behaviour
($x_{1c}$) as $M_{\infty}$ increases. This effect is strongly enhanced by wall
cooling, figure 12(b), which further promotes the one-dimensional state of the
flow.

We note that although this effect resembles a promotion of compressibility, as
noted by several authors (Duan and Martin, 2011; Chu et al., 2013), the
underlying mechanism is strongly different and relevant only when $M_{\infty}$
is high. In fact, different wall-cooling and compressibility signatures are
clearly noted for other effects, such as their effect on scale separation and
their region of influence through the BL.

To provide further insights on these differences, we analyse the
characteristic turbulent lengths. We consider the length scale characterising
large eddies as $L=\bar{\rho}k^{3/2}/\phi$ (Pope, 2000), and the Kolmogorov
length scale
$\eta=\left[(\bar{\mu}/\bar{\rho})^{3}/(\phi/\bar{\rho})\right]^{0.25}$ for
the smallest ones, with $\phi$ being the local dissipation rate of TKE. The
ratio of these two scales, reported in figure 13, measures the separation
between large and small scales, which in our discussion can be ascribed to the
effect of $M_{\infty}$ and $\mathit{\Theta}$ numbers (since $Re_{\tau}$ is
fixed). In agreement with previous observations, figure 13(a) shows that the
separation of scales in the outer layer increases with the Mach number, while
the opposite behaviour is found reducing the diabatic parameter
$\mathit{\Theta}$, see figure 13(b). The insets in figures 13(a) and 13(b)
show the individual change of $L^{+}=L/\delta_{\nu}$ and
$\eta^{+}=\eta/\delta_{\nu}$, revealing that $M_{\infty}$ and
$\mathit{\Theta}$ strongly affect the Kolmogorov length $\eta^{+}$, with a
minor impact on large scales $L^{+}$, influencing the separation of large to
small scales $L/\eta$ in the outer layer.

On this aspect, we remark that while an increase in compressibility, i.e.
$M_{\infty}$, reduces the Kolmogorov length, the opposite holds decreasing the
wall temperature, i.e. $\mathit{\Theta}$.

The variation of $L/\eta$ in the outer layer is effectively captured by the
change of
$Re_{\tau}^{*}=\mu_{w}/\mu_{\infty}\sqrt{\rho_{\infty}/\rho_{w}}Re_{\tau}$
(see table 2), which better account for density and viscosity variations in
the outer layer. It should be noted, however, that the definition of a single
similarity parameter among different flow cases concerning the scale
separation is prevented by the strong change of flow properties across the BL.
In particular, while $Re_{\tau}$ essentially regulates the outer-inner scale
separation, i.e. $L^{+}$, $Re_{\tau}^{*}$ controls the large-small scale
separation in the outer layer, i.e. $L/\eta$. These two variables are strongly
related in incompressible flows and both growing functions of $y^{+}$ in the
log-layer (Pope, 2000), while they appear to be decoupled for highly
compressible flows due to the influence of $M_{\infty}$ and $\mathit{\Theta}$
numbers. For this reason, specific flow features associated with the outer-
inner scales separation, such as the enhancement of outer layer motions at
high $Re_{\tau}$ (Cogo et al., 2022), are not visible here, even though
$L/\eta$ actually increases in the outer layer.

(a) $\mathit{\Theta}=1.0$ (b) $M_{\infty}=6$

Figure 13: Ratio of integral length scale $L$ and Kolmorov scale $\eta$ for
cases at (a) $\mathit{\Theta}=1.0$ and (b) $M_{\infty}=6$, as function of the
wall-normal distance $y^{+}$. The inset shows separately $L^{+}$ and
$\eta^{+}$, normalized with the viscous length $\delta_{\nu}$.

###  5.2 Thermodynamic quantities

Important insights into the respective roles of Mach number and wall-cooling
can also be attained by looking at root-mean-square profiles of temperature
and pressure fluctuations shown in figures 14 and 15. The semilocal scaling is
used to better account for fluid property variations across the boundary layer
and rms quantities are scaled accordingly. In particular, rms profiles of
pressure are scaled with the wall shear stress $\tau_{w}$, while the resulting
scaling for temperature is obtained using the ideal gas law $P=R\rho T$:

$\frac{\tau_{w}}{R\bar{\rho}}=\frac{\bar{\rho}u_{\tau,SL}^{2}}{R\bar{\rho}}=\frac{u_{\tau,SL}^{2}}{R}=\gamma\bar{T}\frac{u_{\tau,SL}^{2}}{R\gamma\bar{T}}=\gamma\bar{T}M_{\tau,SL}^{2}$
(16)

being $u_{\tau,SL}=\sqrt{\tau_{w}/\bar{\rho}}$ the semilocal friction velocity
and $M_{\tau,SL}=u_{\tau,SL}/\sqrt{\gamma R\bar{T}}$ the semilocal friction
Mach number. First, the effect of $\mathit{\Theta}$ at a given Mach number is
presented in figure 14. Considering the region starting from $y^{*}>10$, both
temperature and pressure fluctuations show a reduction in intensity as
$\mathit{\Theta}$ decreases, although more intense for the temperature. In
particular, the suppression of temperature fluctuations by wall-cooling forms
a plateau for the coldest case that is due to the great attenuation of the
turbulent heat flux in the log-layer, consistently with Fan et al. (2022).
Around $y^{*}\approx 10$, the aforementioned attenuation of temperature
fluctuations reaches its maximum for highly cooled cases
($\mathit{\Theta}=0.25$), which is the point where mean temperature gradients
are close to zero. In the near-wall region ($y^{*}<10$), strongly cooled cases
exhibit a peculiar behaviour, which goes in direct contrast to the monotonic
attenuation of adiabatic profiles. In fact, in this region, there is an
increase in the intensity of the temperature fluctuations that forms a local
peak. We attribute this phenomenon to the large increase of conductive heat
flux close to the wall, which is able to overcome the expected attenuation of
turbulent heat flux that concurs with the generation of thermal production
(see figure 16(d)). This is due to the large increase of near-wall temperature
gradients that generate steeper mean profiles and for a wider region of
$y^{*}$ values (before reaching the temperature peak), as visible in figure
4(a) of section §4. The increase in pressure fluctuations in this region is
shared only by the high-Mach number case, showing that additional physical
interpretations are needed on the distinct role of vorticity and acoustic
modes, for which we remind to the recent study of Zhang et al. (2022). Figure
15 shows the effect of Mach number at a given $\mathit{\Theta}$. Here,
temperature fluctuation profiles are very similar up to roughly $y^{*}<15$,
while the main differences are present in the outer layer, where at higher
Mach numbers a peak is formed. This result indicates that $\mathit{\Theta}$ is
an adequate parameter to recover the same general behaviour with respect to
wall-cooling at different Mach numbers, as noted by Wenzel et al. (2022) (we
remark the similarity between $\mathit{\Theta}$ and Eckert number). As
discussed for velocity fluctuations in section §5.1, we note the tendency of
compressibility to increase the separation of scales (figure 14), while the
opposite is true for enhanced wall-cooling (figure 15). This effect would be
greatly reduced if profiles were compared at the same $Re_{\tau}^{*}$, which
incorporates these effects (not shown). Pressure fluctuations exhibit a good
collapse at the peak location around $y^{*}\approx 30$, in accordance with
Zhang et al. (2022), but do not share the collapse between profiles in the
near-wall region noted for temperatures.

(a) $M_{\infty}=2$ (b) $M_{\infty}=2$

(c) $M_{\infty}=4$ (d) $M_{\infty}=4$

(e) $M_{\infty}=6$ (f) $M_{\infty}=6$

Figure 14: Profiles of RMS temperature (left) and pressure (right) in
semilocal scaling. Here, different diabatic parameters $\mathit{\Theta}$ are
compared at a given Mach number $M_{\infty}$.

(a) $\mathit{\Theta}=0.25$ (b) $\mathit{\Theta}=0.25$

(c) $\mathit{\Theta}=0.5$ (d) $\mathit{\Theta}=0.5$

(e) $\mathit{\Theta}=0.75$ (f) $\mathit{\Theta}=0.75$

(g) $\mathit{\Theta}=1.0$ (h) $\mathit{\Theta}=1.0$

Figure 15: Profiles of RMS temperature (left) and pressure (right) in
semilocal scaling. Here, different Mach number $M_{\infty}$ are compared at a
given diabatic parameter $\mathit{\Theta}$.

Further insights on the sources of production of temperature fluctuations,
which are highly influenced by wall-cooling, can be gained by considering the
temperature variance budget $K_{T}=\widetilde{T^{{}^{\prime\prime}2}}$, which
can be written as (ref. Gatski and Bonnet (2013)):

$\displaystyle\bar{\rho}\frac{DK_{T}}{Dt}=$
$\displaystyle-\bar{\rho}\widetilde{u_{k}^{\prime\prime}T^{\prime\prime}}\frac{\partial\widetilde{T}}{\partial
x_{k}}-\frac{\partial}{\partial
x_{k}}\left(\frac{\bar{\rho}\widetilde{u_{k}^{\prime\prime}T^{\prime\prime
2}}}{2}\right)+\gamma\overline{T^{\prime\prime}}\frac{\partial}{\partial
x_{k}}\left(\frac{\bar{k}_{T}}{c_{p}}\frac{\partial\bar{T}}{\partial
x_{k}}\right)$ (17)
$\displaystyle+\bar{\rho}D_{T}-\bar{\rho}\varepsilon_{T}+\bar{\rho}C_{T},$

where the terms on the right-hand side are in order of appearance: thermal
production, turbulent velocity transport, mean thermal conduction, thermal
diffusion, thermal dissipation rate, and contributions due to
pressure–dilatation and viscous dissipation, respectively. Details on the
composition of each term can be found on Gatski and Bonnet (2013). Here, we
analyse the thermal production term, which acts in a similar way to turbulent
production, transferring internal energy from the mean field to the
fluctuating one Fan et al. (2022). For turbulent boundary layers, its wall-
normal component is the main contributor, especially as we approach the wall,
which we refer to as $\mathcal{P}_{T}$:

$\mathcal{P}_{T}=-\bar{\rho}\widetilde{v^{{}^{\prime\prime}}T^{{}^{\prime\prime}}}\frac{\partial\widetilde{T}}{\partial
y}$ (18)

Here, two terms concur to the heat exchange between different flow regions by
two distinct processes: the first part
$\bar{\rho}\widetilde{v^{{}^{\prime\prime}}T^{{}^{\prime\prime}}}$ is
dominated by turbulence with the velocity-temperature fluctuations
correlation, while $\partial\widetilde{T}/\partial y$ represent the conductive
part and is related to the mean temperature gradient. Profiles of
$\mathcal{P}_{T}$ are reported in figures 16, showing the effect of wall-
cooling at different Mach numbers. Similarly to temperature fluctuations, cold
profiles behave differently before and after $y^{*}\approx 10$, where mean
temperature gradients change after the peaks. While adiabatic profiles
monotonically rise from zero to a clear peak at around $y^{*}\approx 15$,
proving their coupling with velocity fluctuations, cold cases progressively
exhibit a reduction and outward shift of the main peak, with the creation of
another peak in the viscous sub-layer. An insight to understand this process,
which is more apparent at $M_{\infty}=6$, can be gained by analysing the
individual behaviour of turbulent and convective heat exchange terms in
thermal production (Fan et al., 2022), which are shown in figure 16(d). Here,
it can be seen that while the turbulent term is significantly far from the
wall with reduced intensity for cold cases, convective heat exchange dominates
the near-wall region as wall-cooling increases. In this region, even though
$\bar{\rho}\widetilde{v^{{}^{\prime\prime}}T^{{}^{\prime\prime}}}$ is close to
zero for all cases, the temperature gradient raises considerably for
$\mathit{\Theta}=0.25$ which result in a non-zero product that is visible in
plots of thermal production. Thus, the formation of a peak of thermal
fluctuation production in the viscous sublayer is promoted. The vanishing mean
temperature gradient in the buffer layer reduces the production of temperature
fluctuations and promotes a decorrelation with velocity fluctuations, as
discussed in the previous sections.

At this point, it is possible to reconsider the qualitative results presented
in figure 2 in a more quantitative way. Wall parallel slices of velocity and
temperature fluctuations were taken approximately at $y^{*}=10$, where
$\partial\tilde{T}/\partial y\approx 0$ for extremely cold cases. It is now
apparent that the decorrelation between
$\bar{\rho}T^{{}^{\prime\prime}}/\tau_{w}$ and
$\sqrt{\bar{\rho}}u^{{}^{\prime\prime}}/\sqrt{\tau_{w}}$ can be explained with
the interplay of the mean temperature gradient and
$\bar{\rho}\widetilde{v^{{}^{\prime\prime}}T^{{}^{\prime\prime}}}$, which
entirely damps the production of temperature fluctuations. This is also
visible in figure 17, which shows the joint probability density function
between velocity and temperature fluctuations only for extreme cases at
$M_{\infty}=6$ (other cases are similar). Here, a direct contrast is present
between figures 17(a) and 17(b)). While the latter (M6T100) shows a good
correlation between the two fields, supporting their similarity, the former
(M6T025) shows a strong decorrelation, especially when velocity fluctuations
are negative, which is due to the influence of wall-cooling.

(a) $M_{\infty}=2$ (b) $M_{\infty}=4$

(c) $M_{\infty}=6$ (d) $M_{\infty}=6$

Figure 16: Panels (a$\,\to\,$c): Production of temperature variance
$\mathcal{P}_{T}$ as function of $y^{*}$ and scaled by
$\bar{\rho}u_{\tau,SL}\widetilde{T}^{2}/\delta_{\nu,SL}$. Here, different
wall-cooling conditions are compared for each Mach number. Panel (d):
Turbulent $\bar{\rho}\widetilde{v^{{}^{\prime\prime}}T^{{}^{\prime\prime}}}$
and conductive $\partial\widetilde{T}/\partial y$ heat transfer terms in the
thermal production. Here, different wall-cooling conditions are compared for
case $M_{\infty}=6$.

(a) $M_{\infty}=6,\mathit{\Theta}=0.25$ (b) $M_{\infty}=6,\mathit{\Theta}=1.0$

Figure 17: Scatter plot of $\bar{\rho}T^{{}^{\prime\prime}}/\tau_{w}$ vs
$\sqrt{\bar{\rho}}u^{{}^{\prime\prime}}/\sqrt{\tau_{w}}$ . Here, only the
cases M6T025 and M6T100 are shown. Data were collected in the same plane shown
in figure 2 ($y^{*}\approx 10$).

##  6 Conclusions

In this paper, we have presented a systematic study on the effect of the Mach
number and wall-cooling on zero-pressure-gradient TBLs using direct numerical
simulations. A total of 12 computations have been carried out spanning three
Mach numbers and four values of the diabatic parameter $\mathit{\Theta}$,
while the friction Reynolds number has been kept constant. In this parameter
space, we put emphasis on the choice of the wall-cooling parameter
$\mathit{\Theta}$, first proposed by Zhang et al. (2014), which can better
incorporate the indirect effects of the Mach number on wall-cooling, yielding
the same integral behaviour between different cases. It is worth noting that
$\mathit{\Theta}$ can be directly related to the Eckert number $Ec$, whose
relevance has recently been discussed by Wenzel et al. (2022). These
parameters show an improved ability to account for the wall-cooling effect at
different Mach numbers with respect to the more classically used wall-to-
recovery temperature ratio $T_{w}/T_{r}$, which has been shown to produce
vastly different effects of wall temperature on the flow dynamics in the near-
wall region across different Mach numbers (e.g. Cogo et al. (2022); Zhang et
al. (2022)).

A summary of the most important remarks is presented below.

  1. 1.

The instantaneous flow organisation of temperature fluctuations near the wall,
which for adiabatic cases is clearly discernible with the presence of near-
wall elongated streaks highly correlated to streamwise velocity, breaks down
as the wall temperature is progressively lowered, showing an isotropic
behaviour for extremely cold cases without organised patterns. Nevertheless, a
similar flow organisation is attained at different Mach numbers when
$\mathit{\Theta}$ is fixed, a first sign of the aptness of this parameter to
yield the same wall-cooling effects across different $M_{\infty}$.

  2. 2.

The recent compressibility transformations of Volpiani et al. (2020) and
Griffin et al. (2021) correctly collapse all mean velocity profiles of our
database to the incompressible laws of the wall. Similarly, Zhang et al.
(2014) mean velocity-temperature relations are able to capture non-adiabatic
and compressibility effects in an excellent manner. When this relation is
approximated with the computed mean value of the Reynolds analogy factor
$s=0.78\pm 0.03$ (which is close to the fit of Zhang et al. (2014)), an
excellent estimate is recovered, with maximum errors of $5\%$ from the DNS
data.

  3. 3.

As the Mach number increases, we observe an increased separation between large
and small scales in the outer layer measured by the ratio $L/\eta$, which is
mainly regulated by the strong reduction of the Kolmogorov length $\eta$, and
only weakly by a growth of the largest scale $L$. This effect can be
effectively described by the growth of the semilocal friction Reynolds number
$Re_{\tau}^{*}$, even though the resulting flow dynamics is different from a
pure increase of the friction Reynolds number $Re_{\tau}$, the latter also
leading to an increase of the inner-outer scale separation $L^{+}$, feeding
outer layer motions. In the near-wall region, compressibility enables a less
efficient redistribution of turbulent kinetic energy, which results in a
promotion of the peak of the streamwise velocity component and a decrease of
the others.

  4. 4.

The enhancement of wall-cooling appears as a reduction of the large-small
scale separation in the outer layer (as opposed to the effect of Mach number),
which is mainly due to an increase of the Kolmogorov length scale $\eta$ that
occurs throughout the whole BL thickness. Lower wall temperatures force the
rise of the mean temperature peak, inducing a stratification of flow
properties localised around the buffer layer. This effect is visible as an
apparent promotion of compressibility, since velocity fluctuations are
enhanced in the streamwise direction, while the other components are damped.

  5. 5.

In the near-wall region, a dominant effect of wall-cooling is present in the
RMS temperature profiles and TKE budget, while the Mach number exerts its
influence mainly through the buffer and log layers. When the diabatic
parameter $\mathit{\Theta}$ is kept constant, the RMS temperature profiles at
different $M_{\infty}$ collapse into each other near the wall, displaying a
similar wall-cooling effect.

  6. 6.

For extremely cold cases (in our database $\mathit{\Theta}=0.25$), the effect
of wall-cooling is so marked that temperature fluctuations are massively
damped at the point where the mean temperature gradient is zero (thus thermal
production is also zero), and a second (minor) peak arises in the viscous
sublayer. This phenomenon completely decorrelates velocity and temperature
fields in the near-wall region, and is more pronounced at high Mach numbers.
The different behaviour of thermal production for cold cases can be explained
by looking at the mean temperature gradient, which persists with a positive
value for a wider wall-normal region (before reaching the mean temperature
peak at $y^{*}_{peak}$), and with a much stronger intensity than for adiabatic
cases.

It should be noted that points (1), (5), and (6) are closely connected since
they are representative of the same physical phenomenon (aerodynamic heating),
which is a key point in this study. Moreover, points (3) and (4) consist in
novel observations on the modulation of scales separation in compressible
flows that require future studies. In the authors’ opinion, the observed
decorrelation between velocity and temperature fields due to wall-cooling
poses a major challenge for the development of simplified wall models, since
any similarity between the two fields is lost. In this context, we believe
that the present database and the relative discussion can help to gain
physical insights on the important mechanisms that can be captured by a
physics-informed model for Reynolds-averaged Navier-Stokes (RANS) or large-
eddy simulation (LES), which is certainly of great interest to the scientific
community given the recent advancements on the topic (e.g. Kawai and Larsson
(2010); Yang et al. (2018)).

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