# Mach number and wall thermal boundary condition effects on near-wall
compressible turbulence

Akanksha Baranwal1<EMAIL_ADDRESS>Diego A. Donzis1  and Rodney D. W.
Bowersox1  1Department of Aerospace Engineering Texas A&M University, College
Station, Texas 77843, USA

###### Abstract

We investigate the effects of thermal boundary conditions and Mach number on
turbulence close to walls. In particular, we study the near-wall asymptotic
behavior for adiabatic and pseudo-adiabatic walls, and compare to the
asymptotic behavior recently found near isothermal cold walls (Baranwal et al.
(2022)). This is done by analyzing a new large database of highly-resolved
direct numerical simulations of turbulent channels with different wall thermal
conditions and centerline Mach numbers. We observe that the asymptotic power-
law behavior of Reynolds stresses as well as heat fluxes does change with both
centerline Mach number and thermal-condition at the wall. Power-law exponents
transition from their analytical expansion for solenoidal fields to those for
non-solenoidal field as the Mach number is increased, though this transition
is found to be dependent on the thermal boundary conditions. The correlation
coefficients between velocity and temperature are also found to be affected by
these factors. Consistent with recent proposals on universal behavior of
compressible turbulence, we find that dilatation at the wall is the key
scaling parameter for this power-law exponents providing a universal
functional law which can provide a basis for general models of near-wall
behavior.

##  1 Introduction

The detailed dynamics of turbulence near the wall has first-order effects on
phenomena such as heat transfer and viscous drag. When speeds are relatively
low, many aspects of these flows are relatively well understood such as
scaling laws for mean quantities and Reynolds stresses. The situation is more
challenging at higher speeds where compressibility effects become important
and the physics more involved due to the interaction of hydrodynamics with
thermodynamics. Understanding the detail dynamics in such regimes is critical
for accurate predictions and, ultimately, control of these flows. It is also
critical for model development in the context of Reydnolds Averaged Navier-
Stokes (RANS) approaches which are widely used in applications. Substantial
effort have been devoted to develop RANS models for compressible wall-bounded
flows, with adiabatic and weekly cooled walls (Menter, 1992; Spalart &
Allmaras, 1992; Catris & Aupoix, 2000). However, these models result in poor
prediction of statistics at high speeds (Roy & Blottner, 2006; Rumsey, 2010;
Aiken et al., 2020) due to the lack of an accurate representation of the
different physics and flow behavior in different conditions. One important
difference between different regimes is the wall thermal boundary condition
(WTBC) which, in general, is modeled as adiabatic at supersonic speeds but
cold-wall isothermal in hypersonic regimes. In certain situations, it is also
possible to have mixed boundary conditions which can again alter the flow
dynamics.

Direct numerical simulations (DNS) of a number of wall-bounded flows, such as
channels (Coleman et al., 1995; Huang et al., 1995; Foysi et al., 2004;
Morinishi et al., 2004; Gerolymos & Vallet, 2014; Sciacovelli et al., 2017; Yu
et al., 2019; Yao & Hussain, 2020) and flat plate boundary layers ((Smits &
Dussauge, 2006; Wenzel et al., 2018) and references therein) have been
conducted to try to understand compressibility effects on turbulent statistics
in high-speed regimes. Efforts have also been made to study the effects of
WTBC on the scaling of velocity and temperature statistics and the
relationship between them in high-speed regimes (Huang et al., 1995; Morinishi
et al., 2004; Tamano & Morinishi, 2006; Mader, 2000; Duan et al., 2010;
Shadloo et al., 2015; Hadjadj et al., 2015; Shahab et al., 2011; Zhang et al.,
2014, 2018, 2022a). Recent studies have investigated the effects of thermal
wall condition on pressure fluctuations (Zhang et al., 2017, 2022b), kinetic
energy transfer (Xu et al. (2021)), density and temperature resolvent mode
shapes (Bae et al. (2020)) highlighting WTBC effects on turbulent processes
and structures. Several studies focused on finding scaling laws and others on
using these scaling laws to collapse first and second order statistics in
high-speed regimes for different flow conditions and different WTBCs (Brun et
al., 2008; Zhang et al., 2012; Trettel & Larsson, 2016; Patel et al., 2015;
Volpiani et al., 2020; Griffin et al., 2021). These WTBCs can be broadly
characterized as isothermal (constant temperature) and isoflux (constant heat
flux) conditions. For the former, studies have been conducted to investigate
the effect of wall temperature, and for the latter, the effect of varying rate
of heat transfer. Some studies have also used the so-called pseudo-adiabatic
wall, a constant wall temperature (based on the recovery factor) whose value
is such that the mean heat transfer to the wall vanishes, mimicking an
adiabatic boundary. Some of the studies mentioned above (Shadloo et al.
(2015); Wenzel et al. (2018); Zhang et al. (2022a)) found that variation in
turbulent statistics (e.g., mean velocity, mean temperature, Reynolds
stresses) are not due to changes in the WTBC itself (i.e., change from
isothermal to isoflux), but instead due to change in the heat transfer at the
wall. However, direct effects of changing the boundary condition from
isothermal to isoflux were observed on temperature fluctuation statistics,
e.g. temperature fluxes in the near-wall region and these effects extended
beyond the viscous sublayer. Another important observation was the change in
asymptotic behavior of turbulent heat fluxes for different WTBCs.

From a fundamental and a modeling perspective, it is crucial to understand the
precise asymptotic behavior of turbulence close to the wall. Indeed, accurate
predictions necessitates models to satisfy the correct asymptotic scaling laws
(Lai & So, 1990; So et al., 1991a, b; Zhang et al., 1992; Durbin, 1993; Sommer
et al., 1993; So et al., 1998; Germano et al., 1991; Bowersox, 2009; Agrawal
et al., 2022), and thus many studies have reported the asymptotic behavior of
turbulent fluxes for both incompressible and compressible flows and under
different WTBCs (Morinishi et al. (2004); Shadloo et al. (2015); Hadjadj et
al. (2015); Zhang et al. (2022a); Li et al. (2009)). All these studies
compared their data to the theoritical asymptotes obtained from Taylor series
expansions in wall-normal direction and found good agreement. However, none of
these studies examined well-resolved wall asymptotes with a systematic
variation of Mach number for different WTBCs.

We have recently conducted DNS of turbulent channels with finer near-wall
resolution than the standard in the literature to capture true asymptotic
behavior (Baranwal et al. (2022)). In that study, which was done with cooled
isothermal walls, we systematically varied the centerline Mach number from
$M\gtrsim 0.2$ (virtually incompressible) to $M\lesssim 2.2$. We showed that
turbulent stresses and wall-normal heat flux comprising at least one wall-
normal velocity component do not collapse when the Mach number was changed as
suggested by widely used scaling laws which, thus, undermines Morkovin’s
hypothesis. In particular, due to the extremely high wall resolution, we were
able to unveil a new region very close to the wall where power-law scaling
exponents were found to differ from theoretical asymptotes and, furthermore,
depend on Mach number. Previous studies at the standard resolution are not
able to capture this region. We have also found that increasing the centerline
Mach number resulted in enhanced levels of dilatation motions at the wall
which is the key factor to understand changes in the power-law asymptotes
close to the wall. Dilatational levels at the wall were also found to be
affected by WTBCs in boundary layers (Xu et al., 2021; Zhang et al., 2022b).
Zhang et al. (2022b) further found that wall cooling effects on dilatation
depends also on the Mach number.

These complex dependencies on both WTBCs and Mach number is the motivation
behind the present work. In particular we investigate, for the first time, the
asymptotic behavior of various turbulent stresses and heat fluxes at different
Mach numbers and for different WTBCs. This systematic investigation is
possible due to extremely well resolved turbulent channels with centerline
Mach number ranging from 0.2 and 2.2 with isothermal, adiabatic and pseudo-
adiabatic walls. The new adiabatic and pseudo-adiabatic results complement the
isothermal data in Baranwal et al. (2022). This is also relevant in the
context of classical scaling laws based on Morkovin’s hypothesis which are
more effective at collapsing statistics when the walls are adiabatic or
weakly-cooled, than when they are isothermal in which case there is
significant wall cooling. Adiabatic walls, thus, possess the additional
advantage of isolating the effects of Mach number from wall cooling and
provide a more direct way to assess the effects of Mach number in isolation
and the validity of Morkovin’s hypothesis on the asymptotic scaling of
turbulent statistics.

The rest of the paper is organized as follows. We first present the numerical
method, configuration, and DNS database. Then, we present results on the
asymptotic behavior of Reynolds stresses and their dependency on centerline
Mach number and WTBCs. This analysis is then extended to temperature
fluctuations and heat fluxes. We conclude with a summary and some remarks on
the implications of the results presented here.

##  2 Numerical Method

We perform direct numerical simulations of the equations governing mass,
momentum, and energy conservation for a compressible channel flow. The
equations are discretized on a uniform mesh in the streamwise ($x$) and
spanwise ($z$) directions. In the wall-normal ($y$) direction, the grid is
clustered close to the wall using a hyperbolic tangent function. We use sixth-
order compact schemes to compute spatial derivatives in the $x$ and $z$
directions. For the $y$ direction, we utlize the sixth-order compact scheme in
interior points and the order is reduced to fourth and third at the last two
grid points in the domain. The variables are marched in time using a third-
order low-storage Runge-Kutta scheme. More details on simulations can be found
in Baranwal et al. (2022) where we also present detailed grid convergence
studies and validations against other DNS databases in the literature (e.g.
Coleman et al., 1995). The simulations presented here satisfy those resolution
criteria which are summarized in table 1.

Periodic boundary conditions are used in the streamwise and spanwise
directions. At the walls, we apply no-slip boundary conditions for all
velocity components. The boundary condition for pressure is obtained by
evaluating the momentum equation in the normal direction at the wall which was
found to have a greater numerical stability than the commonly used zero-
pressure gradient (Baranwal et al., 2022). In all the simulations presented
here, the bottom wall ($y=0$) is isothermal with $T=300$. For the top wall,
three different thermal boundary conditions are investigated, namely,
isothermal, adiabatic and pseudo-adiabatic cases denoted by I, A and PA
respectively. For isothermal cases, the top wall is kept at the same
temperature as the bottom wall ($T=300$). These simulations, which were
studied in our previous study (Baranwal et al. (2022)), act as base case to
compare with other thermal wall conditions. For adiabatic cases, we specify
zero temperature gradient at the top wall. This approach with mixed boundary
conditions in a channel have been used before (Morinishi et al., 2004; Tamano
& Morinishi, 2006; Zhang et al., 2022a; Lusher & Coleman, 2022; Baranwal et
al., 2023). Finally, the pseudo-adiabatic case consists of imposing an
isothermal boundary condition at the average temperature obtained from the
adiabatic simulation with all other flow parameters kept the same as in the
adiabatic case.

Following standard notation, the bulk, wall and centerline values of a
variable $f$ are denoted by $f_{b}$, $f_{w}$ and $f_{c}$, respectively.
Reynolds and Favre decompositions are denoted by $\overline{q}+q^{\prime}$ and
$\widetilde{q}+q^{\prime\prime}$, respectively. The averages in these
decompositions are taken along the homogeneous directions (i.e. $x$-$z$
planes) and time. As done in Baranwal et al. (2022), snapshots of all fields
are saved at time intervals of $5h/\overline{u_{b}}$ for all simulations ,
where $h$ is the channel half width and $u$ is the streamwise velocity
component. This time scale ($h/\overline{u_{b}}$) is commensurate with the
eddy-turnover time of the turbulence in the center of the channel and thus
representative of the largest turbulent structures. Our temporal averages
involved 25 snapshots for velocity, density and temperature fields.

(a)

$Re_{c}$

$M_{c}$

$Re_{\tau}$

$\theta_{w,rms}^{+}$

$tu_{b}/h$

(b)

$u_{rms}/u_{\tau}$

$tu_{b}/h$

(c)

$v_{rms}/u_{\tau}$

$tu_{b}/h$

(d)

$T_{rms}/\overline{T}$

$tu_{b}/h$

Figure 1: Time evolution of (a) $Re_{c}$, $M_{c}$, $Re_{\tau}$,
$\theta_{w,rms}^{+}$, (b) $u_{rms}/u_{\tau}$, (c) $v_{rms}/u_{\tau}$, (d)
$T_{rms}/\overline{T}$ for $M_{b}\approx 1.2$, (**** - $y^{+}=0$), (….-
$y^{+}=0.5$), (xxxx - $y^{+}=2.0$), (xxxx - $y^{+}=5.6$), (xxxx - $y^{+}=11$),
(xxxx - $y^{+}=54$), (xxxx - $y^{+}=100$), (xxxx - $y^{+}=173$).

Consider a forced, periodic channel with Dirichlet boundary conditions for
temperature at the walls, that is isothermal walls. If initialized with zero
velocity and constant temperature, the flow will accelerate and develop
velocity gradients that lead to viscous dissipation. This leads to an increase
in temperature inside the channel which is higher than that imposed at the
walls. Because of the thermal gradient that forms at the wall, there is a flux
of energy from the fluid to the wall and the flow eventually reaches a
statistically steady state where the rate of production of internal energy due
to viscous dissipation is compensated by the energy transfer through the wall.
If, on the other hand, we apply a Neumann boundary condition for temperature
at the wall, in particular zero temperature gradient, then the heat transfer
to the walls is identically zero. In this case, the increase in temperature
due to dissipation maintained by the forcing in the momentum equation is not
balanced by heat flux through the walls. Therefore, the internal energy in the
channel increases continuously leading to a time-dependent mean thermodynamic
state. Alternatively, one can apply a (cold) isothermal condition to one wall
and an adiabatic condition the other wall. This allows for heat transfer
through one wall and results in a decreased rate of change of mean
thermodynamic parameters. In this case, the flow also achieves a pseudo-steady
state where statistics (at least to second order) are in a statistically
steady state when normalized by their corresponding (slowly varying) means.
This can be seen in figure 1(b)(c)(d) where we show the temporal evolution of
the root-mean-square (r.m.s.) of several variables normalized by their
respective time-varying means for $M_{b}\approx 1.2$ and very long simulation
time ($\approx$ $200h/u_{b}$). While global quantities (Reynolds and Mach
numbers in panel (a)) are seen to decrease slowly, normalized fluctuations
statistics are virtually in a statistical steady state. This is, in fact,
consistent with observation in forced isotropic flows (Kida & Orszag, 1990).
We do note that there seems to be a (very weak) increase in $T_{rms}$ at the
centerline ($y^{+}=173$, red symbols). Because our interest lies close to the
wall, we have verified this trend very far from the wall is not a concern in
this study. The normalized r.m.s. dilatation at the wall, as shown in figure 1
(a), $\theta_{w,rms}^{+}=\overline{(\partial v^{\prime}/\partial
y)_{w}^{2}}\nu_{w}^{2}/u_{\tau}^{4}$ is another quantity of interest which
also exhibits a steady-state behavior. We take advantage of this pseudo-steady
state to find averages over the simulation time. The statistics below are
based on this averaging.

The friction Reynolds numbers based on wall quantities and the friction
Reynolds numbers based on centerline viscosity and density, are defined as
$Re_{\tau}=\overline{\rho_{w}}u_{\tau}h/\overline{\mu_{w}}$ and
$Re_{\tau}^{*}\equiv\overline{\rho_{c}}{(\tau_{w}/\overline{\rho_{c}}})^{1/2}h/\overline{\mu_{c}}$
respectively, with $u_{\tau}\equiv\sqrt{\tau_{w}/\overline{\rho_{w}}}$ being
the friction velocity. The centerline Reynolds number and centerline Mach
numbers are $Re_{c}\equiv\overline{\rho_{c}}$
$\overline{u_{c}}h/\overline{\mu_{c}}$ and
$M_{c}\equiv\overline{u_{c}}/\sqrt{\gamma R\overline{T_{c}}}$, respectively.
Our domain has dimensions $4\pi h\times 2h\times 4\pi/3h$ for all our
simulations. This is larger than widely used in literateure ((e.g. Trettel &
Larsson, 2016; Yu et al., 2019)). Finally, as a direct assessment of boundary
conditions effects on the quantities studied here, we have run additional
simulations with a domain which is 20% shorter and confirmed that the near-
wall scaling laws are unaffected. Table 1 summarizes the important parameters
for the DNS database used here.

Wall |  $M_{c}$ |  $Re_{c}$ |  $Re_{\tau}$ |  $Re_{\tau}^{*}$ |  $\triangle y_{min}^{+}$ |  $\triangle y_{max}^{+}$ |  $\triangle x^{+}$ |  $\triangle z^{+}$ | Line style
---|---|---|---|---|---|---|---|---|---
Isothermal |  0.23 |  5692 |  295 |  293 |  0.08 |  2.9 |  14.5 |  4.8 |
Adiabatic |  0.23 |  5684 |  296 |  292 |  0.08 |  2.9 |  14.5 |  4.8 |
Isothermal |  0.35 |  5638 |  294 |  289 |  0.08 |  2.9 |  14.4 |  4.8 |
Isothermal |  0.46 |  5582 |  294 |  286 |  0.08 |  2.9 |  14.4 |  4.8 |
Isothermal |  0.57 |  5476 |  293 |  281 |  0.05 |  3.2 |  14.4 |  4.8 |
Adiabatic |  0.57 |  5476 |  293 |  281 |  0.05 |  3.2 |  14.4 |  4.8 |
Isothermal |  0.68 |  5498 |  301 |  283 |  0.05 |  3.3 |  14.8 |  4.9 |
Isothermal |  0.89 |  5371 |  307 |  276 |  0.05 |  3.4 |  15.1 |  5.0 |
Adiabatic |  0.84 |  5099 |  225 |  260 |  0.05 |  2.2 |  11.0 |  3.6 |
Isothermal |  1.26 |  5022 |  325 |  259 |  0.05 |  3.6 |  15.9 |  5.3 |
Adiabatic |  1.12 |  4513 |  177 |  226 |  0.05 |  1.7 |  8.7 |  2.9 |
Pseudo-adiabatic |  1.12 |  4306 |  179 |  220 |  0.05 |  1.7 |  8.8 |  2.9 |
Isothermal |  1.50 |  5489 |  393 |  277 |  0.10 |  4.0 |  19.3 |  6.4 |
Isothermal |  1.98 |  5631 |  572 |  279 |  0.10 |  6.2 |  14.0 |  4.7 |
Adiabatic |  1.9 |  5092 |  138 |  236 |  0.05 |  1.0 |  3.4 |  1.0 |
Isothermal |  2.22 |  5666 |  745 |  273 |  0.09 |  8.8 |  14.8 |  6.1 |
Table 1: Details of flow conditions and grid resolutions

In subsequent sections, we investigate various statistics near isothermal
(solid lines) and adiabatic walls (dash-dotted lines) for three different
centerline Mach numbers, $M_{c}\approx 0.23$ (red), $M_{c}\approx 1.2$ (black)
and $M_{c}\approx 1.9$ (magenta) and near pseudo-adiabatic walls (dashed
lines) for $M_{c}\approx 1.2$ using our DNS database. The adiabatic and
pseudo-adiabatic results are taken from the upper halves of channel from the A
and PA simulations respectively where bottom walls are isothermal. The
isothermal case throughout the work refers to simulations where both walls are
isothermal, unless specifically noted otherwise. We note that wall quantities
for a particular case refer to the statistics at the wall with that particular
thermal boundary condition (e.g. for pseudo-adiabatic case, wall quantities
refer to statistics at pseudo-adiabatic wall).

(a)

$\overline{u}/u_{{\tau}}$

$M_{c}\approx 0.23$(I)\- $\cdot$ -$M_{c}\approx 0.23$(A)$M_{c}\approx
1.2$(I)\- $\cdot$ \- $M_{c}\approx 1.2$ (A)\- - - $M_{c}\approx 1.2$
(PA)$M_{c}\approx 1.9$(I)\- $\cdot$ -$M_{c}\approx 1.9$(A)$y^{+}$

(b)

$\overline{T}/\overline{T}_{w}$

$y^{+}$

Figure 2: (a) Streamwise mean velocity normalized by the friction velocity (b)
Mean temperature normalized by the mean wall temperature plotted against wall-
normal coordinate in viscous units for isothermal (—), adiabatic (- $\cdot$ -)
and pseudo-adiabatic (- - -) cases. Red, black and magenta correspond to
$M_{c}\approx 0.23$, $M_{c}\approx 1.2$ and $M_{c}\approx 1.9$ respectively.
Dotted red lines represent viscous and log layer scalings.

##  3 First-order statistics

The mean streamwise velocity normalized by the friction velocity and the mean
temperature normalized by the wall temperature are shown in figure 2 (a) and
(b) respectively. Consistent with the literature, wall-normalization
($u_{{\tau}}$) performs well in collapsing velocity for all presented cases in
the viscous sublayer and the majority of the buffer layer ($y^{+}\lesssim 20$)
as shown in figure 2 (a). In the log-law region, such collapse is not observed
and Mach number and WTBC effects exist on the mean velocity. In particular,
Mach number effects are more pronounced in isothermal than in adiabatic cases
due to the increased heat transfer to the wall as the Mach number increases
for the former. In figure 2 (b) we find that, for isothermal cases, wall
cooling leads to a temperature inside the channel which is higher than the
wall temperature with a maximum at the centerline. This maximum temperature
along with the wall cooling rate increase with the Mach number. For adiabatic
cases, because of the zero heat flux at the top wall, there is a rise of
temperature across the channel, with the maximum temperature at the upper
wall. The temperature gradient for adiabatic and pseudo-adiabatic cases is not
zero at the centerline, which may result in non-zero temperature fluxes at the
channel half-width, an effect that will be discussed later.

(a)

$\overline{\mu}/\overline{\mu}_{w}$

$\overline{u}/u_{{\tau}}$

$y^{+}$(b)

$\overline{p}/\overline{p}_{w}$

$y^{+}$$M_{c}\approx 0.23$(I)\- $\cdot$ -$M_{c}\approx 0.23$(A)$M_{c}\approx
1.2$(I)\- $\cdot$ \- $M_{c}\approx 1.2$ (A)\- - - $M_{c}\approx 1.2$
(PA)$M_{c}\approx 1.9$(I)\- $\cdot$ -$M_{c}\approx 1.9$(A)(c)

$\overline{\rho}/\overline{\rho}_{w}$

$y^{+}$(d)

$\overline{\mu}/\overline{\mu}_{w}$

$y^{*}$(e)

$\overline{p}/\overline{p}_{w}$

$y^{*}$(f)

$\overline{\rho}/\overline{\rho}_{w}$

$y^{*}$

Figure 3: Mean viscosity (a,d), pressure (b,e), and density (c,f) normalized
by their corresponding wall values and plotted versus wall-normal coordinate
in viscous units (a-c) and semi-local units (d-f) for isothermal, (—)
adiabatic (- $\cdot$ -) and pseudo-adiabatic (- - -) cases. Red, black and
magenta correspond to $M_{c}\approx 0.23$, $M_{c}\approx 1.2$ and
$M_{c}\approx 1.9$ respectively.

The mean viscosity, mean pressure and mean density are shown in figure 3 (a-f)
against wall-normalized (a-c) and semi-local (d-f) wall-normal coordinates. On
comparing figure 3 (a) with (d), (b) with (e), and (c) with (f), we observe
that some features become independent of Mach number or WTBC when the
statistics are plotted against the semi-local wall-normal coordinate
$y^{*}\equiv\overline{\rho}{(\tau_{w}/\overline{\rho}})^{1/2}y/\overline{\mu}$
as opposed to $y^{+}$. For example, in figure 3 (e) we see that pressure start
decreasing significantly only at $y^{*}\approx 5$, reaching a minimum at
$y^{*}\approx 65$ for all $M_{c}$, and then increasing towards the channel
centerline. Similar observations can be made for viscosity and density for
isothermal cases (figure 3 (d, f)). As expected, the mean viscosity follows a
similar trend as the mean temperature (figure 2 (b)). Pressure is relatively
constant across the channel with a small dip outside of the viscous sublayer
which increases with $M_{c}$ to about 1.5% at the highest Mach number shown
($M_{c}\approx 1.9$). In figure 3 (c) and (f) we show the mean density
normalized by the mean density at the wall. Because the mean pressure is
roughly constant across the channel, the mean density is inversely
proportional to the mean temperature which is what we observe in these plots.

We can also see opposite trends depending on WTBC. For isothermal cases, the
density decreases as one moves away from the wall or when the Mach number
increases. For adiabatic and pseudo-adiabatic cases, on the other hand, the
density increases as one moves away from the wall or when the Mach number
decreases. A result of these trends is that, close to the wall, the density
gradients are higher for isothermal cases at higher Mach numbers indicating
that the statistics will change more rapidly from their wall values in the
near-wall region when the level of compressibility and heat transfer to the
wall are increased.

$y^{*}$

$y^{+}$(Adiabatic)$M_{c}$(Isothermal)$M_{c}$$M_{c}$$M_{c}$

Figure 4:  Wall-normal coordinate in semi-local units versus wall-normal
coordinate in viscous units for isothermal (—), and adiabatic (- $\cdot$ -)
cases. Colors as in table 1. Black and blue arrows indicate increase in
$M_{c}$ for isothermal and adiabatic cases respectively.

Because of the different scaling observed with wall and semilocal units, a
natural question is, thus, on the relation between these two normalized
distances to the wall. In figure 4, we show the wall-normal coordinate in
semi-local units ($y^{*}$) against the wall-normal coordinate in viscous units
($y^{+}$). The two normalizations are virtually the same in the viscous
sublayer for the given range of Mach numbers. Further away from the wall,
isothermal and adiabatic walls lead to opposite trends when the Mach number is
increased. These can be explained as follows. From figure 3 (a) and (c), we
found that away from the wall, viscosity increases and density decreases as
the Mach number is increased for isothermal cases while opposite trends are
observed for adiabatic cases. Thus, the ratio
$\sqrt{\overline{\rho}}/\overline{\mu}$ decreases with increasing Mach number
for isothermal cases, while this ratio increases for adiabatic cases when
$M_{c}$ is increased. Thus, following this trend, $y^{*}$ decreases with
$M_{c}$ for fixed $y^{+}$ for isothermal cases while it increases with $M_{c}$
for adiabatic cases as shown in figure 4(a). We can also see that $y^{*}$ at
the centerline ($y^{*}_{c}$) reaches a range of values of $y^{*}_{c}\approx
220-290$ for all Mach numbers and WTBCs. This range is much wider for wall
units, $y^{+}_{c}\approx 140-750$, which seems to support the idea that semi-
local units provide a better self-similar normalization than wall units.
However, we note that this may be, in part, due to the fact that simulations
were conducted with an approximately constant $Re_{\tau}^{*}$ (Trettel &
Larsson, 2016). Further simulations at a wide range of $Re_{\tau}^{*}$ are
needed to provide a more definite assessment of this claim.

##  4 Effects of thermal boundary conditions on turbulent stresses

The wall-normal coordinate in semi-local units, $y^{*}$ along with local
density-weighted averaging have been widely used to try to collapse turbulent
stresses in compressible wall-bounded flow with varying WTBC, with their
incompressible counterparts (Huang et al., 1995; Foysi et al., 2004; Morinishi
et al., 2004; Trettel & Larsson, 2016; Modesti & Pirozzoli, 2016; Zhang et
al., 2018). We have recently shown Baranwal et al. (2022), however, that semi-
local scaling is not able to collapse turbulent stresses
$R_{\alpha\beta}^{*}\equiv\overline{\rho}\widetilde{\alpha^{\prime\prime}\beta^{\prime\prime}}/\tau_{w}$
($\alpha$ and $\beta$ are velocity components; e.g.,
$R_{uv}^{*}\equiv\overline{\rho}\widetilde{u^{\prime\prime}v^{\prime\prime}}/\tau_{w}$)
or the wall-normal turbulent heat flux,
$R_{vT}^{*}=\overline{\rho}\widetilde{v^{\prime\prime}T^{\prime\prime}}/(\rho_{w}u_{\tau}T_{\tau})$
close to an isothermal wall in turbulent channels for centerline Mach numbers
ranging from the incompressible limit to supersonic regimes. This can also be
observed here in, e.g., figure 5 (a)(b).

(a)

$\overline{\rho}\widetilde{v^{\prime\prime}v^{\prime\prime}}/\tau_{w}$

$y^{*}$$M_{c}$R1R2(b)

$-\overline{\rho}\widetilde{u^{\prime\prime}v^{\prime\prime}}/\tau_{w}$

$y^{*}$$M_{c}$R1R2

Figure 5: (a)-(b) Density-scaled Reynolds stresses distributions versus semi-
local wall-normal coordinate for isothermal (—), adiabatic (- $\cdot$ -) and
pseudo-adiabatic (- - -) cases. Red, black and magenta correspond to
$M_{c}\approx 0.23$, $M_{c}\approx 1.2$ and $M_{c}\approx 1.9$ respectively.
Insets show same profiles up to $y^{*}\approx 300$.

In figure 5 (a) and (b), we show $R_{vv}^{*}$ and $R_{uv}^{*}$ respectively
for three Mach numbers, $M_{c}\approx 0.23,1.2$ and $1.9$, for both isothermal
(solid lines) and adiabatic (dash-dotted lines) walls. The figure also include
one pseudo-adiabatic case (dashed line) at $M_{c}\approx 1.2$. At the lowest
Mach number ($M_{c}\approx 0.23$), turbulent stresses ($R_{vv}^{*}$,
$R_{uv}^{*}$) collapse well for isothermal and adiabatic walls suggesting no
appreciable WTBC effect as one approaches the incompressible limit. As the
Mach number is increased, however, we can clearly observe differences between
isothermal, adiabatic and pseudo-adiabatic cases for $R_{vv}^{*}$ and
$R_{uv}^{*}$ which are apparent for $M_{c}\approx 1.2$ and beyond. This effect
is especially strong in the viscous sub-layer where we can clearly see higher
normal Reynolds stresses close to isothermal (solid line) than to adiabatic
(dashed-dotted line) walls. However, one can also observe that some Mach
number effects are similar in isothermal cases and adiabatic cases.
Investigating these differences and similarities are the main focus of the
current work.

Three observations can be made. First, in the region adjacent to the wall,
indicated by R1 in figure 5, we can see power-law behavior for both
$R_{uv}^{*}$ and $R_{vv}^{*}$ with exponents that decrease with $M_{c}$ for
adiabatic cases and, as observed before, isothermal cases (Baranwal et al.
(2022)). The slope of $R_{uv}^{*}$ in R1, however, does change with WTBC when
$M_{c}$ is kept constant. This WTBC effect is much weaker for $R_{vv}^{*}$.
Second, $R_{uv}^{*}$ and $R_{vv}^{*}$ transition to another scaling regime,
indicated as R2 in figure 5, with much weaker WTBC and $M_{c}$ effects.
Finally, the transition location changes with both $M_{c}$ and WTBC. Taken
together, these general observations suggest that significant WTBC and Mach
number effects are observed close the wall as Mach number increases.

| $\overline{v^{\prime}v^{\prime}}$ | $\overline{u^{\prime}v^{\prime}}$ | $\overline{v^{\prime}T^{\prime}}$
---|---|---|---
|  |  | isothermal adiabatic pseudo-adiabatic
solenoidal | 4 | 3 | 3 2 3
non-solenoidal | 2 | 2 | 2 1 2
Table 2:  Exponents $\gamma_{\alpha\beta}$ for near wall asymptotic behavior for $R_{\alpha\beta}$ ($\alpha$ and $\beta$ are $u$, $v$ or $T$).  | Isothermal | Adiabatic | Pseudo-adiabatic
---|---|---|---
| R1 R2 | R1 R2 | R1 R2
wall |  |  |
semi-local |  |  |
Table 3:  Marker styles used for exponents ${\gamma_{vv}}$ and ${\gamma_{uv}}$
for different WTBCs and scaling regimes.

The near-wall asymptotic behavior of turbulent stresses can be theoretically
estimated by expanding the constituent velocity components as Taylor series
expansions in $y$:

$u^{\prime}=a_{u}+b_{u}y+c_{u}y^{2}+\dots,\quad
v^{\prime}=a_{v}+b_{v}y+c_{v}y^{2}+\dots$ (1)

The coefficients $a_{\alpha}$ for $\alpha=u$ and $v$ are identically zero due
to the no-slip boundary condition at the wall. The other coefficients are
given by $b_{v}=\partial v^{\prime}/\partial y$, and
$c_{v}=(1/2)\partial^{2}v^{\prime}/\partial y^{2}$, and similarly for $u$. If
the flow is incompressible (solenoidal), mass conservation combined with the
no-slip condition at the wall leads to an additional constraint in the wall-
normal velocity component, namely, $\partial v^{\prime}/\partial y=b_{v}=0$.
On the other hand, if the flow is non-solenoidal, $b_{v}\neq 0$. By taking the
product between the expansions of different components and averaging, one can
formulate Reynolds averaged turbulent stresses
($R_{\alpha\beta}\equiv\overline{\alpha^{\prime}\beta^{\prime}}/u_{\tau}^{2}$),
resulting in near-wall scaling laws of the form
$R_{\alpha\beta}\approx\sigma_{\alpha\beta}y^{\gamma_{\alpha\beta}}$ with
exponents summarized in table 2. These theoretical exponents are the same for
$R_{\alpha\beta}$ and $R_{\alpha\beta}^{*}$ given that density has a finite
value at the wall. From table 2, we see that the solenoidal and non-solenoidal
exponents are different for turbulent stresses containing a wall-normal
velocity component. As in Baranwal et al. (2022), we investigate exponents
($\gamma_{\alpha\beta}$) and pre-factors ($\sigma_{\alpha\beta}$) but
extending the analysis to include WTBC effects.

(a)

$\gamma_{vv}$

$M_{c}$(b)

$\gamma_{uv}$

$M_{c}$

Figure 6:  Power law exponents for (a) wall-normal Reynolds stress (b)
Reynolds shear stress plotted against centerline Mach number. Horizontal gray
lines for solenoidal (- - -) and non-solenoidal (- $\cdot$ -) asymptotic
exponents (table 2). Markers in all panels (table 3): $\square$ indicates wall
normalizations
($R_{\alpha\beta}=\sigma_{\alpha\beta}^{+}(y^{+})^{\gamma_{\alpha\beta}^{+}}$),
$\triangle$ indicates semi-local normalizations
($R_{\alpha\beta}^{*}=\sigma_{\alpha\beta}^{*}(y^{*})^{\gamma_{\alpha\beta}^{*}}$)
for isothermal (empty markers), pseudo-adiabatic (light-filled markers) and
adiabatic (dark-filled markers) cases. Blue and red markers correspond to R1
and R2 regions, respectively. The solid line in all panels connects isothermal
data for comparison.

Following Baranwal et al. (2022), we fit power laws in regions R1 and R2, as
shown in figure 5 for both wall ($R_{\alpha\beta}$ versus $y^{+}$: $\square$)
and semi-local ($R_{\alpha\beta}^{*}$ versus $y^{*}$: $\triangle$)
normalizations, to obtain ($\gamma^{+}_{\alpha\beta}$,
$\sigma^{+}_{\alpha\beta}$) and ($\gamma^{*}_{\alpha\beta}$,
$\sigma^{*}_{\alpha\beta}$) respectively for all cases in our database. In
figure 6 (a), we show the exponent $\gamma_{vv}$ for isothermal (empty
markers), adiabatic (dark-filled markers) and pseudo-adiabatic (light-filled
markers) wall conditions as a function of $M_{c}$. The theoretical asymptotic
values in table 2 are expected to be attained for exponents in R1 (blue
symbols) which are the closest to the wall. On changing thermal wall
conditions, the difference between $\gamma_{vv}$ in R1 is small for the same
centerline Mach number except for $M_{c}=0.5$ where the adiabatic case has a
slightly larger exponent. The exponent $\gamma_{vv}$ approaches its solenoidal
and non-solenoidal limiting behavior (see table 2) for $M_{c}\lesssim 0.2$ and
$M_{c}\gtrsim 0.8$, respectively. Between these two limits there is a smooth
transition with $M_{c}$ for both isothermal and adiabatic cases.

In figure 6(b) we show the exponents for the shear Reynolds stress,
$\gamma_{uv}$, versus $M_{c}$ and observe a much stronger influence of thermal
boundary conditions with larger values of $\gamma_{uv}$ in R1 for adiabatic
cases at all Mach numbers. The pseudo-adiabatic case appears to match the
isothermal case, which may not be completely unexpected given that in this
case we also impose a constant temperature at the wall. This may indicate that
$\gamma_{uv}$ is independent of $T_{w}$ since exponents for isothermal and
pseudo-adiabatic are very close to each other even though wall temperature is
markedly different. Furthermore, this may also suggest that differences in
exponents for isothermal and adiabatic cases are not due to differences in
wall temperature. The values obtained for the R1 exponents, however, are
independent of whether one uses wall or semilocal units for all WTBCs. This is
in line with the theoretical behavior discussed earlier.

In R2, semi-local normalization provides a better collapse of exponents with
different WTBCs. This can be seen in figure 6(a) where we see that, for a
fixed $M_{c}$, there are negligible differences between ${\gamma^{*}_{vv}}$
for isothermal (red empty triangles), adiabatic (red filled triangles), and
pseudo-adiabatic (light red filled triangles) cases. Note also that
${\gamma^{+}_{vv}}$ and ${\gamma^{*}_{vv}}$ in R2 are the same for all Mach
numbers for adiabatic cases but not for isothermal cases. For isothermal
cases, when $M_{c}$ is roughly above unity, ${\gamma^{+}_{vv}}$ and
${\gamma^{*}_{vv}}$ differ. This can be understood by noting that the
temperature and density gradients are higher near the isothermal wall than the
adiabatic wall. Therefore, in adiabatic cases, local density and viscosity are
closer to wall values as compared to those in isothermal cases (also seen in
figure 3(a)(c)). Similar behavior is observed for ${\gamma_{uv}}$. In figure
5, we found that turbulent stresses in R2 are less affected by variations in
Mach number when semi-local normalizations are used for different WTBCs. This
is consistent with the results in figure 6(a)(b), where we see a very weak
$M_{c}$ effect on ${\gamma^{*}_{uv}}$ (and to a lesser degree on
${\gamma^{*}_{vv}}$) for all WTBCs. In general, though, we observe a weaker
$M_{c}$ dependence for adiabatic than isothermal walls for exponents in wall
units.

In addition to obtaining exponents for isothermal cases from simulations where
both walls are isothermal and at the same temperature, we also obtain the
exponents close to the isothermal wall from simulations with different thermal
boundary conditions (pseudo-adiabatic or adiabatic) on the other wall. The
exponents $\gamma_{vv}$ and $\gamma_{uv}$ in R1 near the isothermal wall was
found to, in fact, be independent of the boundary condition of the other wall,
indicating that the near-wall asymptotic behavior is not significantly
affected by the WTBC on the non-identical wall.

(a)

$\sigma_{vv}$

$M_{c}$(b)

$\theta_{rms}^{+}$

$M_{c}$$y^{+}$

Figure 7: (a) Prefactor $\sigma_{vv}$ (squares) in R1 and normalized coefficients in Taylor expansion for $R_{vv}$, non-solenoidal, $\overline{b_{v}^{2}}\nu_{w}^{2}/u_{\tau}^{4}$ (circles) and solenoidal, $\overline{c_{v}^{2}}\nu_{w}^{4}/4u_{\tau}^{6}$ (triangles) against $M_{c}$ for different WTBCs. Markers are given in table 4. (b) Distribution of root-mean-squared dilatation with wall-normal coordinate for isothermal (—), adiabatic (- $\cdot$ -) and pseudo-adiabatic (- - -) cases. Red, black and magenta correspond to $M_{c}\approx 0.23$, $M_{c}\approx 1.2$ and $M_{c}\approx 1.9$ respectively.  | Isothermal | Adiabatic | Pseudo-adiabatic
---|---|---|---
$\sigma_{vv}$ |  |  |
$\overline{b_{v}^{2}}\nu_{w}^{2}/u_{\tau}^{4}$ |  |  |
$\overline{c_{v}^{2}}\nu_{w}^{4}/4u_{\tau}^{6}$ |  |  |
Table 4:  Marker styles used for prefactors and coefficients in Taylor series
expansion for $R_{vv}$ for different WTBCs.

Written out explicitly, the first three terms in the expansion of the wall-
normal Reynolds stress is
$\overline{v^{\prime}v^{\prime}}=\overline{b_{v}^{2}}y^{2}+(\overline{b_{v}c_{v}}/2)y^{3}+(\overline{c_{v}^{2}}/4+\overline{b_{v}d_{v}}/6)y^{4}+{\cal
O}(y^{5})$. As discussed above, when the flow is incompressible $b_{v}=0$ and
the $y^{4}$ term dominates; when the flow is compressible, one expects the
$y^{2}$ term to dominate. However, this would also depend on the prefactors
involved, $\overline{b_{v}^{2}}$ and $\overline{c_{v}^{2}}$, which in a
particular region may make one term dominates over the other. In figure 7(a)
we show these prefactors normalized with wall units, that is,
$\overline{b_{v}^{2}}\nu_{w}^{2}/u_{\tau}^{4}=\overline{(\partial
v^{\prime}/\partial y)_{w}^{2}}\nu_{w}^{2}/u_{\tau}^{4}$ (circles) and
$\overline{c_{v}^{2}}\nu_{w}^{4}/4u_{\tau}^{6}=\overline{(\partial^{2}v^{\prime}/\partial
y^{2})_{w}^{2}}\nu_{w}^{4}/4u_{\tau}^{6}$ (triangles) computed using the
derivatives from DNS data at the wall. In the same plot, we also include the
prefactor $\sigma^{+}_{vv}$ obtained from the fits
$R_{vv}\approx\sigma^{+}_{vv}(y^{+})^{\gamma^{+}_{vv}}$ as described above. We
can clearly see that $\sigma^{+}_{vv}$ (squares) tends to the solenoidal
(triangles) and non-solenoidal (circles) analytical values for $M_{c}\lesssim
0.2$ and $M_{c}\gtrsim 0.8$, respectively for isothermal (empty markers) and
adiabatic (dark-filled markers) wall conditions. Pseudo-adiabatic (light-
filled marker) case with $M_{c}\approx 1.2$ also follows analytical non-
solenoidal value. These observations are consistent with the behavior of
exponents obtained from the fit. The value of $\sigma^{+}_{vv}$ (squares) is
also found to be lower for adiabatic (dark-filled) than isothermal (empty)
cases at $M_{c}\gtrsim 0.8$. We finally note that, at high Mach numbers, the
dominant prefactor is the one involving $b_{v}$ which for no-slip walls, is
equal to the level of dilatation motions at the wall (Baranwal et al., 2022).
Thus, from a purely kinematic standpoint, the particular scaling laws observed
will depend only on dilatation (i.e. $b_{v}$) regardless of how those
dilatations are generated.

It is known that different levels of dilatation at the wall can be generated
either by changing the centerline Mach number (Baranwal et al., 2022) or
thermal boundary condition at the wall (Xu et al. (2021)). This is also clear
in figure 7(b), where we observe that the level of dilatational motions at the
wall is different for different Mach numbers and WTBCs. Dilatation levels are
weaker for adiabatic than isothermal walls with the same $M_{c}$. Pseudo-
adiabatic walls have intermediate dilatation levels close to the wall. As
previously stated, dilatation is a key factor governing the scaling laws, and
one may, thus, expect better collapse of different statistics when using the
dilatational content as a normalizing parameter. This general concept of
universality based on the level of dilatational motions independent of the
specific mechanism that generated them was indeed recently proposed (Donzis &
Panickacheril, 2020) though only for homogeneous flows.

(a)

$\gamma_{vv}$

$\theta_{w,rms}^{+}$(b)

$\gamma_{uv}$

$\theta_{w,rms}^{+}$

Figure 8: Power-law exponents in R1 for (a) wall-normal turbulent stress
(b)turbulent shear stress plotted against r.m.s of dilatation at the wall.
Markers as in table 3. Horizontal gray lines for solenoidal (- - -) and non-
solenoidal (- · -) asymptotic exponents (table 2). Solid lines are scalings,
(a) $2+2\exp(-10^{10}{\theta_{w,rms}^{+}}^{1.69})$ (b)
$2+\exp(-126{\theta_{w,rms}^{+}}^{0.45})$.

To test these concepts, in figure 8 (a)(b) we show the exponents as a function
of the r.m.s. of dilatation at the wall normalized with wall units,
$\theta_{w,rms}^{+}$. We clearly see a better collapse of exponents than in
the corresponding panels (a) and (b) of figure 6, supporting the idea that
dilatational levels, regardless of how they are generated, provide the
appropriate scaling parameter for near-wall behavior at high speeds. This is
consistent with Donzis & Panickacheril (2020) where the use of dilatational
content as a governing parameter yielded a universal behavior for a number of
statistics including pressure variance, dissipation, and skewness of the
velocity gradients. From a modeling perspective, it may be useful to
parametrize these seemingly universal curves. We have found that these curves
can be represented reasonably well with simple exponentials in
$\theta_{w,rms}^{+}$, which are included in figure 8(a)(b) and noted in its
caption.

(a)

$-C_{uv}$

$y^{*}$$M_{c}$(b)

$-C_{uv}$

$y^{*}$

Figure 9:  Correlation coefficient for $R_{uv}$ (a) Isothermal wall (b)
Isothermal (—), adiabatic (- $\cdot$ -) and pseudo-adiabatic (- - -) walls.
Inset contains the same data in linear scales. Colors as in table 1.

On comparing figure 6(a) with (b), we find that the transition from the low to
the high Mach number limit in R1 for $\gamma_{uv}$ is smoother than that of
$\gamma_{vv}$ for isothermal as well as adiabatic cases (adiabtic cases
exhibit an even slower transition than isothermal cases). A similar
observation can also be made from figure 8 where the transition (with levels
of dilatation at the wall in this case) is smoother for ${\gamma_{uv}}$ as
compared to ${\gamma_{vv}}$. This suggests a slow decorrelation between
$u^{\prime}$ and $v^{\prime}$ as compressibility levels increase close to the
wall. To study this, we show in figure 9(a) the correlation coefficient
$C_{uv}\equiv\overline{u^{\prime}v^{\prime}}/u_{rms}v_{rms}$ for all
isothermal cases in the database. We similarly define the correlation
coefficient $C_{\alpha\beta}$ for arbitrary variables $\alpha$ and $\beta$ as

$C_{\alpha\beta}\equiv\frac{\overline{\alpha^{\prime}\beta^{\prime}}}{\alpha_{rms}\beta_{rms}}$
(2)

We see that for the lowest Mach numbers, $C_{uv}$ is relatively constant close
to the wall ($y^{*}\lesssim 1$). As $M_{c}$ increases, the overall magnitude
of the correlation is reduced in this region, though all the lines seem to
approach, a region of relatively constant correlation of about 0.45, a value
consistent with those observed in supersonic boundary layers (Shadloo et al.
(2015)). The distance from the wall at which this region starts, however,
increases with $M_{c}$, indicating that compressibility effects are felt at
increasing distance from the wall as the Mach number increases. The increasing
decorrelation close to the wall with $M_{c}$ has also been observed in
Sciacovelli et al. (2017), an effect that was also found to be independent of
Reynolds number. This near-wall decorrelation that becomes stronger as $M_{c}$
increases suggests that while a simple product of Taylor expansions can
describe diagonal stresses (e.g. $R_{uu}$ or $R_{vv}$), this is not the case
for off-diagonal stresses ($R_{uv}$) which comprise the correlation between
two different variables. In particular, we see that for low and high $M_{c}$,
the correlation $C_{uv}$ is relatively constant close to the wall, though at
different levels. It is at intermediate Mach numbers that $C_{uv}$ presents a
positive slope in this region. Thus, because
$\overline{u^{\prime}v^{\prime}}=C_{uv}u_{rms}v_{rms}$ we can see how the R1
exponent for $R_{uv}$, would be close to the sum of the exponents for
$u_{rms}$ and $v_{rms}$ for low and high $M_{c}$ while it would be larger at
intermediate $M_{c}$. This explains, then, why the transition from the
solenoidal to the non-solenoidal asymptotes is smoother for $R_{uv}$ than for
the case of diagonal stresses. At the centerline of the channel, $C_{uv}$
vanishes due to reflective symmetry across the centerline plane, which is seen
as a rapid decrease in the correlation in the figure at high values of
$y^{*}$.

To assess the effect of WTBC, in figure 9(b) we show the correlation
coefficient for different boundary conditions and three Mach numbers,
$M_{c}\approx 0.23$, $1.2$, and $1.9$. As before, we see that $C_{uv}$ is
relatively flat at the lowest $M_{c}\approx 0.23$ and for distances below
$y^{*}\sim O(1)$, with very little WTBC effect. The same weak dependence on
WTBC is observed at $y^{*}$ beyond, say, 4, where $C_{uv}$ approaches the
constant value discussed above. As the Mach number is increased, however,
there are observable differences between isothermal, adiabatic and pseudo-
adiabatic walls. In particular, we see that isothermal walls (black solid
line) create a stronger decorrelation between $u$ and $v$ than adiabatic walls
(black dashed-dotted line) for $M_{c}\gtrsim 1.2$ and pseudo-adiabatic (black
dashed line) for $M_{c}\approx 1.2$. In addition, there are differences in the
slope for $C_{uv}$ close to the wall between adiabatic and isothermal cases,
especially for $M_{c}\approx 1.2$ which also seem to contribute to the
difference in power-law behavior for these two WTBCs. This is clearly evident
in figure 6(b), where ${\gamma_{vv}}$ for $M_{c}\approx 1.2$ seems to have the
largest difference between isothermal and adiabatic cases. Moreover, the
distance from the wall at which constant region of $C_{uv}$ starts, is larger
for isothermal than adiabatic cases.

(a)

$y_{tr}$

$M_{c}$(b)

$y_{tr}$

$\theta_{w,rms}^{+}$

Figure 10: Transition location of scaling exponents plotted versus (a)
centerline Mach number (b) r.m.s dilatation at the wall. Markers in all
panels: ($\square$, $y^{+}$; $\triangle$, $y^{*}$) Black and Blue colored
markers correspond to wall-normal Reynolds stress and shear Reynolds stress
respectively for isothermal (empty markers), adiabatic (dark-filled markers)
and pseudo-adiabatic (light-filled markers) cases.

Finally, in figure 10 we show the wall-normal location where $R_{vv}$ and
$R_{uv}$ transition from region R1 to region R2, which is denoted as $y_{tr}$.
Consistent with the results in Baranwal et al. (2022) we see in panel (a) that
$y_{tr}$ moves away from the wall as $M_{c}$ is increased. However, we also
observe clear WTBC effects. In particular, we see that for adiabatic walls
(dark-filled symbols) the transition moves closer to the wall compared to
isothermal (empty symbols) and pseudo-adiabatic (light-filled symbols) walls.
For example, for high $M_{c}$, we see close to order-of-magnitude differences
in $y_{tr}$ between isothermal and adiabatic cases for $R_{vv}$. As before
(figure 8(a)(b)), we can explore the suggestion in Donzis & Panickacheril
(2020) that a higher degree of universal behavior will be observed when
dilatational motions are used to scale statistics of interest. This is indeed
supported by the data in figure 10(b) where we show $y_{tr}$ as a function of
$\theta^{+}_{w,rms}$. Data for both $R_{vv}$ and $R_{uv}$ appear to be closer
to exhibiting universal scaling (though not perfect) under this normalization.

We can then conclude that by increasing the centerline Mach number or changing
any other flow condition which results in enhancing dilatation levels at the
wall, an enlarged region close to the wall will develop where compressibility
effects are significant. This is also the region where Morkovin’s hypothesis
is found to be inadequate to collapse Reynolds stresses as shown before.

##  5 Effects of thermal boundary conditions on temperature fluxes

(a)

${T_{rms}}/T_{w}$

$y^{*}$$M_{c}$$\propto y^{*}$

(b)

${T_{rms}}/T_{\tau}$

$y^{*}$$\propto y^{*}$

(c)

${T_{rms}}/T_{w,rms}$

$y^{*}$

Figure 11: Root-mean-squared temperature fluctuations (a) normalized with wall
temperature for isothermal (—), adiabatic (- $\cdot$ -) and pseudo-adiabatic
(- - -) cases. (b) normalized with friction temperature for isothermal cases.
(b) normalized with r.m.s temperature at the adiabatic wall for adiabatic
cases. Red, black and magenta correspond to $M_{c}\approx 0.23$, $M_{c}\approx
1.2$ and $M_{c}\approx 1.9$ respectively.

###  5.1 Temperature fluxes

The asymptotic behavior of temperature can be analyzed in a similar way to the
velocity field by considering separately isothermal and adiabatic conditions.
The Taylor series expansion of temperature fluctuations is given by

$T^{\prime}=a_{T}+b_{T}y+c_{T}y^{2}+\dots$ (3)

For the isothermal case, temperature is fixed at the wall and one has
$a_{T}=0$ (but $b_{T}\neq 0$). For the adiabatic case, there are fluctuations
at the wall but its normal gradient vanishes, in which case $b_{T}=0$ (but
$a_{T}\neq 0$). One would thus expect different near-wall asymptotic behavior
based on thermal boundary conditions. In figure 11 (a), we plot the r.m.s. of
temperature, $T_{rms}$, normalized by the mean wall temperature against
$y^{*}$ for all cases. We find that the asymptotic behavior of $T_{rms}$ is
qualitatively different for isothermal and adiabatic walls. Near the
isothermal wall, $T_{rms}$ follows a power-law increase while the adiabatic
cases are flat. Similar asymptotic behavior was observed in incompressible and
low-Mach number flows for isothermal and isoflux conditions (Tiselj et al.,
2001; Li et al., 2009). The asymptotic power-law scaling for isothermal cases
is equal to its theoretical asymptote ($\gamma_{T}=1$) for all $M_{c}$. For
adiabatic cases, the profile is constant for most of the viscous sublayer
(until $y^{*}\approx 2$) and that constant increases with increase in the
$M_{c}$. Interestingly, the pseudo-adiabatic case (black dashed line), which
has been extensively used in the literature to model adiabatic walls, exhibits
an isothermal-like power-law behavior close to the wall.

$\overline{\rho}\widetilde{u^{\prime\prime}T^{\prime\prime}}/(\rho_{w}u_{\tau}T_{\tau})$

$\propto y^{2}$(a)$y^{*}$(b)

$\overline{\rho}\widetilde{u^{\prime\prime}T^{\prime\prime}}/(\rho_{w}u_{\tau}T_{w})$

$y^{*}$$\propto y^{2}$$\propto y$

Figure 12: Streamwise turbulent heat flux close to (a) isothermal walls
normalized by friction temperature. (b) isothermal (—), adiabatic (- $\cdot$
-) and pseudo-adiabatic (- - -) cases normalized by their respective wall
temperature. Inset contains the same data in linear scales upto $y^{*}\approx
300$. Red, black and magenta correspond to $M_{c}\approx 0.23$, $M_{c}\approx
1.2$ and $M_{c}\approx 1.9$ respectively.

An alternative normalization for temperature, in analogy with the Reynolds
stresses, is through the so-called friction temperature,
$T_{\tau}\equiv-\kappa(\partial{\overline{T}}/\partial{y})_{w}/\overline{\rho_{w}}c_{p}u_{\tau}$,
where $\kappa$ is the thermal conductivity. It is clear, however, that this
normalization can only be applied to isothermal walls since adiabatic (and
pseudo-adiabatic) walls present zero conductive heat transfer to the wall
($\partial\overline{T}/\partial y|_{w}=0$). In figure 11 (b), we show all
isothermal cases which do, in fact, collapse in the near-wall region following
its asymptotic scaling of $\sim y^{*}$. A collapse of adiabatic cases is also
obtained when $T_{rms}$ is normalized with their respective wall values
($T_{w,rms}$) as seen in figure 11 (c). Since $T_{w,rms}=0$ for isothermal
cases, it is clear that neither normalization provides universal scaling
across different WTBCs.

####  5.1.1 Streamwise turbulent heat flux

The streamwise component of the turbulent heat-flux ($R_{uT}$) is an important
quantity in wall-bounded flows which needs to be correctly modeled in order to
make accurate predictions. In fact, this heat-flux component has been found to
be even larger than the wall-normal turbulent heat flux (Huang et al. (2020)).
Current Boussinesq or constant $Pr_{T}$ based RANS models, however, cannot
capture its behavior accurately (Bowersox, 2009; Huang et al., 2019;
Broslawski et al., 2022).

In figure 12 (a) we show the density-scaled streamwise turbulent heat flux,
$\overline{\rho}\widetilde{u^{\prime\prime}T^{\prime\prime}}/(\rho_{w}u_{\tau}T_{\tau})$
in the near-wall region of an isothermal wall for different Mach numbers
against $y^{*}$. We observe very good collapse for all Mach numbers along the
theoretical asymptotic power law given by $\gamma_{uT}=2$ (table 2). The
adiabatic and pseudo-adiabatic cases are included in figure 12 (b) (normalized
with wall temperature) along with the isothermal cases for comparison. The
temperature fluctuations at the wall for adiabatic cases result in
$\gamma_{uT}=1$ and again conforms to the theoretical behavior. Following
$T_{rms}$, we observe that power-law behavior for pseudo-adiabatic streamwise
heat flux follows isothermal-like behavior and thus, also matches with the
isothermal theoretical exponent. The streamwise heat flux becomes negative for
$y^{*}\gtrsim 1$ in adiabatic and pseudo-adiabatic cases and thus can not be
shown in logarithmic scales. Again, this indicates that fine resolution close
to the wall is required to capture correct near-wall asymptotic behavior. In
inset of figure 12 (b), we also include streamwise heat-flux along the channel
in linear scales. Similar to $T_{rms}$, we find that any scaling, either by
normalization using $T_{w}$ or $\overline{T}$ (not shown here), $R_{uT}$ do
not collapse for different $M_{c}$ and WTBCs in high-speed regime.

-$C_{uT}$

$y^{*}$

Figure 13: Correlation coefficient for $R_{uT}$ for isothermal (—), adiabatic
(- $\cdot$ -) and pseudo-adiabatic (- - -) cases. Colors as in table 1. Inset
contains the same data in linear scales.

Similar to the case of $R_{uv}$ discussed above, the near-wall asymptotic
behavior of $R_{uT}$ will depend not only on the scaling of the r.m.s. of the
two variables involved in the flux, but also on their cross-correlation. The
excellent agreement seen for $\gamma_{uT}$ for all cases with their respective
theoretical scaling, then, implies that the correlation coefficient, $C_{uT}$
does not vary in $y$ in this region and is evident in figure 13. For
$y^{*}\lesssim 1$, $C_{uT}$ is constant with $y^{*}$ for all $M_{c}$ and
WTBCs. However, we see interesting differences between different WTBCs. First,
the absolute value of $C_{uT}$ is minimum near the wall for adiabatic cases
while for isothermal cases, the absolute value of $C_{uT}$ is maximum close to
the wall. Some $M_{c}$ effects can be observed for adiabatic cases close to
the wall while -$C_{uT}$ for different $M_{c}$ collapses to a constant value
of $-1$ near isothermal walls. For adiabatic cases, the decorrelation
decreases on moving away from the wall until $y^{*}\approx 15$, while $C_{uT}$
for isothermal cases remains constant in this region, with -$C_{uT}\approx-1$.
Interestingly, the pseudo-adiabatic case resembles isothermal-like behavior
near the wall and adiabatic-like behavior beyond $y^{*}\approx 10$. At further
distance, $y^{*}\gtrsim 15$, decorrelation increases for all WTBCs with
isothermal case maintaining a positive correlation, while adiabatic and
pseudo-adiabatic maintaining negative correlations ($C_{uT}$). The correlation
approaches zero as one moves towards the centerline. For $y^{*}\gtrsim 100$,
Mach number effects can be seen for isothermal and adiabatic cases. Another
interesting observation from figure 13 is that $C_{uT}$ for adiabatic and
pseudo-adiabatic cases, as shown in the inset of figure 13 resemble $C_{uT}$
profile for a flat-plate boundary layer (Duan et al. (2010)) with isothermal
or pseudo-adiabatic walls.

####  5.1.2 Wall-normal turbulent heat flux

(a)

$-\overline{\rho}\widetilde{v^{\prime\prime}T^{\prime\prime}}/(\rho_{w}u_{\tau}T_{w})$

$y^{*}$$M_{c}$

(b)

$-\overline{\rho}\widetilde{v^{\prime\prime}T^{\prime\prime}}/(\rho_{w}u_{\tau}T_{w})$

$y^{*}$$M_{c}$

Figure 14: Wall-normal turbulent heat flux close to (a) isothermal, (—),
adiabatic (- $\cdot$ -) and pseudo-adiabatic (- - -) walls in logarithmic
scale. Inset contains the same data in linear scales upto $y^{*}\approx 300$.
Red, black and magenta correspond to $M_{c}\approx 0.23$, $M_{c}\approx 1.2$
and $M_{c}\approx 1.9$ respectively. (b) adiabatic walls in linear scales.
Colors as in table 1.

(a)

-$C_{vT}$

$y^{*}$$M_{c}$(b)

-$C_{vT}$

$y^{*}$$M_{c}$

Figure 15:  Correlation coefficient for $R_{vT}$ near (a) isothermal walls (b)
adiabatic (- $\cdot$ -) and pseudo-adiabatic (- - -) walls. Inset has the same
data in linear scales. Colors as in table 1.

In figure 14 (a), we plot the density averaged wall-normal turbulent heat flux
($R_{vT}$),
$\overline{\rho}\widetilde{v^{\prime\prime}T^{\prime\prime}}/(\rho_{w}u_{\tau}T_{w})$
close to the wall for isothermal cases (solid lines) with
$M_{c}\approx{0.23,1.2,1.9}$, low Mach adiabatic case with $M_{c}\approx 0.23$
and pseudo-adiabatic case with $M_{c}\approx 1.2$. It can be seen that close
to the isothermal wall, a Mach number dependent power-law exists for the wall-
normal turbulent heat-flux. A detailed study of asymptotic power-law for wall-
normal turbulent heat flux close to isothermal walls was performed in Baranwal
et al. (2022), where power-law exponents were observed to transition from its
theoretical low Mach to high Mach asymptotes. This transition was found to be
similar to that of $\gamma_{uv}$. The asymptotic behavior of heat-flux close
to the pseudo-adiabatic wall exhibits a power-law behavior with
$\gamma_{vT}\approx 2.1$ and matches closely to the theoretical limit of
isothermal asymptotic power law. This is in line with the behavior of all
other statistics close to the pseudo-adiabatic wall which behave like those in
isothermal cases. For Mach number in the near-incompressible range
$M_{c}=0.23$, a power-law behavior with exponent equals its theoretical value
is observed close to the adiabatic wall. Similar to $R_{uT}$, $R_{vT}$ for
this adiabatic case ($M_{c}=0.23$), and pseudo-adiabatic case changes sign
moving away from the wall and therefore can not be shown in logarithmic
scales. For adiabatic cases with $M_{c}>0.23$, we find that a well-defined
power-law behavior is not observed close to the wall and hence the data is
plotted in linear scales in figure 14 (b). Similar to $T_{rms}$ and $R_{uT}$,
we find that any scaling, either by normalization using $T_{w}$ or
$\overline{T}$ (not shown here), $R_{vT}$ do not collapse for different
$M_{c}$ and WTBC in high-speed regimes.

Finally, we plot $-C_{vT}$ as a function of $y^{*}$ in figure 15 (a) and (b).
For isothermal cases as shown in figure 15(a), $C_{vT}$ closely resembles
$C_{uv}$ as shown in figure 9 indicating that increasing $M_{c}$ has similar
effects on $C_{vT}$ as were observed for $C_{uv}$. In the inset of figure
15(a), we plot $C_{vT}$ in linear scales where moving towards the centerline
($y^{*}\gtrsim 100$), some Mach number effects can be observed. For adiabatic
walls, as is shown in figure 15 (b), a trend with the Mach number close to the
wall is observed for $C_{vT}$. On moving away from the wall, the decorrelation
between $v^{\prime}$ and $T^{\prime}$ decreases. Furthermore, the effect of
mixed boundary condition can be observed close to the channel centerline where
$C_{vT}$ does not vanish. This is because of the finite mean temperature
gradients at the channel half-width resulting in the non-zero wall-normal heat
flux at $h$. On comparing figure 15 (a) and (b), we observe that $C_{vT}$
assumes opposite signs for isothermal and adiabatic cases in regions away from
the wall. Like the previous observation for other statistics close to the
wall, pseudo adiabatic exhibits isothermal-like near-wall behavior but follows
adiabatic in regions away from the wall.

##  6 Conclusions

The asymptotic behavior of turbulent stresses and turbulent heat fluxes close
to the wall were investigated using a large DNS database of turbulent channel
flows with centerline Mach numbers spanning from 0.23 to 2.22. The dataset
comprises of simulations with three different wall thermal boundary conditions
(WTBC), namely isothermal, adiabatic and pseudo-adiabatic. A distinguishing
feature of the present DNS is the near-wall resolution which is much finer
than those typically found in the literature. We show this is essential to
capture near-wall behavior for different flow and wall boundary conditions.

Turbulent stresses containing wall-normal velocity component do not exhibit a
universal behavior close to the wall when normalized using either wall or
semi-local units. Interestingly, some statistics behave differently for
different WTBCs while others behave similarly. Similarities include Mach
number effects on statistics close to the wall for isothermal and adiabatic
cases. In both cases, turbulent stresses exhibited asymptotic power-law
behavior in the near-wall region (which we call R1) for all Mach numbers and
WTBCs. With increase in Mach number, smooth transition of asymptotic power-law
exponents from the solenoidal limit to the high-speed limit was observed.
Consistent with previous findings, a second scaling regime (R2) with a steeper
exponent and a weaker Mach number dependence beyond R1 was observed. The
transition location between R1 and R2 was dependent on Mach number.

A notable difference between cases with different WTBCs is the change in
power-law exponents for turbulent stresses with changing WTBC at high Mach
numbers. This effect is stronger for $R_{uv}$ than for $R_{vv}$. In general,
$R_{uv}$ was found to be more sensitive to changes in $M_{c}$ or WTBC. This
was linked to a decorrelation between $u^{\prime}$ and $v^{\prime}$ when
$M_{c}$ is increased or when the WTBC changes from isothermal to adiabatic.

Inspired by a recent proposal based on homogeneous flows, we found that
universality can be indeed recovered if dilatational motions are incorporated
as a governing parameter regardless of the mechanism that generated them. In
particular, asymptotic power-law exponents and the transition location between
the two scaling regimes R1 and R2 do collapse on a universal curve which
depends uniquely on $\theta_{w,rms}$, the r.m.s. of dilatation at the wall. If
one uses the (perhaps more intuitive) centerline Mach number one can clearly
see differences in exponents and transition location for different WTBCs. This
clearly supports the idea that dilatational levels, regardless of how they are
generated, provide the appropriate scaling parameter for near-wall behavior at
high speeds furthering the idea of some universality of statistics in
compressible wall-bounded flows. This also support the previously found
conclusion that Morkovin’s hypothesis does not take into consideration all the
effects associated with compressibility at higher Mach numbers.

We also investigated statistics of temperature fluctuations, wall-normal and
streamwise turbulent heat fluxes for varying $M_{c}$ and WTBC. For isothermal
cases we found that $T_{rms}$ follows a power-law behavior predicted by the
analytical form of its Taylor expansion. For adiabatic cases, on the other
hand, $T_{rms}$ remains constant in the viscous sublayer followed by an almost
universal increase with $y^{*}$.

The streamwise heat flux, $R_{uT}$, exhibits a power-law behavior close to the
wall with exponents given by theoretical predictions for both isothermal and
adiabatic cases. In general, it was found that temperature statistics
($T_{rms}$, $R_{uT}$) can be collapsed separately for isothermal and adiabatic
cases by normalizing temperature with $T_{\tau}$ and $T_{w,rms}$,
respectively. However, no general scaling laws were found that could collapse
statistics containing temperature fluctuations for both WTBCs.

As with Reynolds stresses, the wall-normal turbulent heat flux ($R_{vT}$) for
isothermal cases exhibits power-law behavior with exponents that depend on
$M_{c}$. A well-defined power-law behavior cannot be unambiguously identified
for adiabatic cases with $M_{c}>0.23$. Pseudo-adiabatic walls, which are often
used to mimic an adiabatic wall by imposing an isothermal condition at the
adiabatic temperature, displayed isothermal-like behavior close to the wall as
$M_{c}$ increases. A rich interplay between Mach number and WTBC effects was
observed for correlation coefficients between $v^{\prime}$ and $T^{\prime}$,
and between $u^{\prime}$ and $T^{\prime}$ indicating a complex dynamics
between velocity and temperature fluctuations.

Mach number effects were observed in the viscous sublayer for the correlation
between $v^{\prime}$ and $T^{\prime}$ for all WTBCs, but only in the adiabatic
case for $u^{\prime}$ and $T^{\prime}$. The strong WTBC effect is evident by
the fact that these correlations possess different signs in most of the region
across the channel. In these regions, $v^{\prime}$ and $T^{\prime}$ are
negatively correlated for isothermal walls while they are positively
correlated for adiabatic cases. In contrast, $u^{\prime}$ and $T^{\prime}$ are
positively correlated for isothermal walls while negatively correlated for
adiabatic cases. Moreover, in the region close to the wall, the magnitude of
these $u^{\prime}$ and $T^{\prime}$ correlations are very different for
isothermal and adiabatic cases, with the former being much stronger than the
latter. Similar to all other statistics, pseudo-adiabatic case exhibits
isothermal-like near-wall behavior but resembles the adiabatic profile away
from the wall.

We close by pointing out that, overall, Morkovin’s hypothesis and semi-local
normalizations do not collapse data for all the flow and boundary conditions.
Universal scaling laws for wall-bounded compressible flows, thus, requires
more general scaling laws.

Acknowledgments. The authors acknowledge support from (1) the National Science
Foundation (Grant No. 1605914), (2) DoD Vannevar Bush Faculty Fellows (ONR
Grant No. N00014-18-1-3020), and (3) the Extreme Science and Engineering
Discovery Environment (XSEDE) for computational resources. The opinions,
findings, views, conclusions, or recommendations contained herein are those of
the authors and should not be interpreted as necessarily representing the
official policies or endorsements, either expressed or implied, of the U.S.
Government.

Competing interests: The authors declare none.

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