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# Compressible Velocity Transformations for Various Noncanonical Wall-Bounded
Turbulent Flows

Tianyi Bai 111Ph.D. Candidate, Department of Mechanical and Aerospace
Engineering. The first two authors contributed equally. The Hong Kong
University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Kevin P. Griffin222Ph.D. Candidate, Center for Turbulence Research. The first
two authors contributed equally. Stanford University, Stanford, CA, USA  Lin
Fu333Assistant Professor, Department of Mechanical and Aerospace Engineering;
Department of Mathematics; Shenzhen Research Institute (The Hong Kong
University of Science and Technology, Shenzhen, China). Corresponding author.
Email<EMAIL_ADDRESS>The Hong Kong University of Science and Technology, Clear
Water Bay, Kowloon, Hong Kong

###### Abstract

While several velocity transformations for compressible zero-pressure-gradient
(ZPG) boundary layers have been proposed in the past decades, their
performance for non-canonical compressible wall-bounded turbulent flows has
not been systematically investigated. This work assesses several popular
transformations for the velocity profile through their application to several
types of non-canonical compressible wall-bounded turbulent flows.
Specifically, this work explores DNS databases of high-enthalpy boundary
layers with dissociation and vibrational excitation, supercritical channel and
boundary-layer flows, and adiabatic boundary layers with pressure gradients.
The transformations considered include the van Driest [Van Driest, J.
Aeronaut. Sci., 18(1951):145-216], Zhang et al. [Zhang et al., Phys. Rev.
Lett., 109(2012):054502], Trettel-Larsson [Trettel and Larsson, Phys. Fluids,
28(2016):026102], data-driven [Volpiani et al., Phys. Rev. Fluids,
5(2020):052602], and total-stress-based [Griffin et al., Proc. Natl. Acad.
Sci. U.S.A., 118(2021):e2111144118] transformations. The Trettel-Larsson
transformation collapses velocity profiles of high-enthalpy temporal boundary
layers but not the spatial boundary layers considered. For supercritical
channel flows, the Trettel-Larsson transformation also performs well over the
entire inner layer. None of the transformations above works for supercritical
boundary layers. For all the considered methods, the transformed velocity
profiles of boundary layers with weak pressure gradients coincide well with
the universal incompressible law of the wall. In summary, all these popular
methods fail to deliver uniform performance for non-canonical compressible
wall-bounded flows in the logarithmic region, and a more sophisticated
version, which accounts for these different physics, is needed. The data-
driven and total-stress-based transformations perform well in the viscous
sublayer for all the considered flows. Nevertheless, the present assessment
provides a useful guideline on the deployment of these transformations on
various non-canonical flows.

## Nomenclature

$\rho$ | = | density, $kg/m^{3}$
---|---|---
$p$ | = | pressure, $Pa$
$T$ | = | temperature, $K$
$u,v$ | = | streamwise and wall-normal velocity, $m/s$
$x$,$y$,$z$ | = | axes in the inertial coordinate system, $m$
$u_{\tau}$ | = | friction velocity, $m/s$
$\delta_{v}$ | = | viscous length scale, m
$\mu$ | = | dynamic viscosity, $Pa\cdot s$
$\tau$ | = | shear stress, $N/m^{2}$
${\rm Re}$ | = | Reynolds number
${\kappa}$ | = | Kármán constant
$\beta_{k}$ | = | Rotta-Clauser parameter

##  1 Introduction

The incompressible zero-pressure-gradient wall-bounded turbulent flow is a
canonical case that has been widely studied. In the inner layer, there exists
an inner scaling for the velocity profile, which is called the law of the wall
and was derived by Prandtl [1, 2] and von Kármán [3], also known as the
logarithmic law of the wall, which is valid for equilibrium, incompressible,
constant-property flows at high Reynolds numbers. The logarithmic law has been
a foundation in the wall-bounded turbulence field to validate simulations and
experiments and to construct reduced-order near-wall models. As for
compressible flows, the mean thermal properties become non-uniform and the
direct deployment of the incompressible law of the wall fails to describe mean
velocity profiles.

This motivates the idea of a compressible velocity transformation which maps a
compressible velocity profile to the incompressible law of the wall [4].
Similarly, a reliable compressible velocity transformation would be useful for
the validation of simulations and experiments and would enable the development
of new turbulence models in the near-wall region, which can provide accurate
mean flow prediction with reduced computational cost, e.g., for the aerospace
industry.

To extend the logarithmic law of incompressible flows to compressible flows,
Morkovin [5] first observes that at moderate Mach number, the effect of the
density fluctuations is negligible and concludes that “the essential dynamics
of these shear flows will follow the incompressible pattern”, which is known
as Morkovin’s hypothesis [5]. This hypothesis implies that the effects of
compressibility can be accounted for by the variation of mean flow properties.
In subsequent studies, many transformations have been developed following
Morkovin’s hypothesis [5].

One of the most widely used transformations is the van Driest [4]
transformation, which is derived based on the assumption of a constant stress
layer and Prandtl’s mixing-length model. This transformation implies that the
variation of mean density can account for the effect of compressibility, and
it performs well for flows over adiabatic walls.

Zhang et al. [6] argue against the assumption that Prandtl’s mixing length is
Mach-number invariant. A new Mach-number-invariant scaling is obtained by
considering the quasi-balance of the production and dissipation terms in the
turbulent kinetic energy (TKE) equation. Combining the assumptions of constant
total stress and a modified Prandtl’s mixing-length model, Zhang et al. [6]
derive a transformation based on turbulence equilibrium.

Huang et al. [7] proposed the semi-local scaling and Coleman et al. [8] showed
that when turbulence intensities are plotted versus the wall-normal coordinate
in semi-local units, a collapse is observed across various Mach numbers.
Trettel and Larsson [9] make an improvement of the van Driest [4]
transformation and derive the Trettel-Larsson [9] transformation based on the
semi-local scaling and the constant stress layer assumption. Patel et al. [10]
develop the same transformation based on the observation that the viscous
stress in the near-wall region collapses better in the semi-local scaling.
This transformation works in channel flows very well, indeed across the entire
inner layer.

A data-driven transformation has been developed by Volpiani et al. [11], by
postulating that the compressibility effect can be accounted for by the non-
dimensionalization with undetermined power laws of mean density and mean
viscosity. The power law coefficients are fitted using the DNS data of
boundary layers. This transformation was shown to succeed for both adiabatic
and diabatic ZPG boundary layers.

Griffin et al. [12] claim that there are two fundamental flow scalings in the
inner layer, i.e., one for the viscous sublayer and the other for the
logarithmic region. In the logarithmic region, Griffin et al. [12] generalize
the quasi-equilibirum argument of Zhang et al. [6] through introducing the
semi-local scaling. In the viscous sublayer, the transformation is the same as
the Trettel-Larsson transformation. Griffin et al. [6] further combine these
two scalings and propose the total-stress-based transformation, which works
for all the considered channel, pipe, and ZPG boundary-layer flows regardless
of the thermal boundary condition, Reynolds number and Mach number.

While these popular transformations have been validated comprehensively for
canonical ZPG boundary-layer flows, their performance for non-canonical flows
is rarely investigated. In this work, we assess them for a wide range of non-
canonical flows, including high-enthalpy flows with dissociation and
excitation, supercritical flows, and flows with pressure gradients. The paper
is organized as follows. In § 2, several transformations and their derivations
are introduced; the databases and their numerical configurations are outlined
in § 3; the numerical assessments are provided in § 4 with detailed analyses;
concluding remarks are given in § 5.

##  2 Various transformation methods

In this section, several established transformation methods will be introduced
comprehensively, as well as the embedded assumptions under which they are
derived. The connections and differences between these transformations will be
discussed in detail.

Hereafter, both the Reynolds ($\phi=\overline{\phi}+\phi^{\prime}$) and Favre
($\phi=\widetilde{\phi}+\phi^{\prime\prime}$) decompositions are used, where
$\overline{\phi}$ and $\widetilde{\phi}=\overline{\rho\phi}/\overline{\rho}$
denote the Reynolds- and Favre-averaged quantities, and $\phi^{\prime}$ and
$\phi^{\prime\prime}$ denote the fluctuations of the Reynolds and Favre
decompositions, respectively.

Throughout this paper, the superscript + indicates a non-dimensionalization by
the friction velocity $u_{\tau}=\sqrt{\tau_{w}/\overline{\rho}_{w}}$, the
viscous length scale
$\delta_{v}=\overline{\mu}_{w}/(u_{\tau}\overline{\rho}_{w})$, and $\rho_{w}$,
where $\tau_{w}=\overline{\mu}_{w}(\partial\widetilde{u}/\partial y)|_{w}$,
$\rho_{w}$, and $\mu_{w}$ are the shear stress, density, and dynamic viscosity
evaluated at the wall, respectively. For instance,
$\widetilde{u}^{+}=\widetilde{u}/u_{\tau}$, $y^{+}=y/\delta_{v}$,
$\overline{\mu}^{+}=\overline{\mu}/\overline{\mu}_{w}$,
$\overline{\rho}^{+}=\overline{\rho}/\overline{\rho}_{w}$, etc. The friction
Reynolds number is defined as $Re_{\tau}=\rho_{w}u_{\tau}\delta/\mu_{w}$,
where $\delta$ is the boundary layer thickness.

###  2.1 The van Driest transformation [4]

The first successful transformation for compressible wall-bounded flows was
proposed by van Driest [4]. The assumptions that a constant stress layer
exists, and that the viscous stress is negligible, imply

$\widetilde{u^{\prime\prime}v^{\prime\prime}}\sim\left(\frac{\overline{\rho}_{w}}{\overline{\rho}}\right)u_{\tau}^{2}.$
(1)

Further assuming that Prandtl’s mixing-length model for the Reynolds shear
stress holds in a compressible flow, we have

$\overline{\rho}_{w}u_{\tau}^{2}=\overline{\rho}l_{m}^{2}\left(\frac{\partial\widetilde{u}}{\partial
y}\right)^{2},$ (2)

where $l_{m}$ denotes the characteristic length scale of eddies. Prandtl
assumes that the size of eddies is considered to be proportional to their
distance to the wall, leading to the relation $l_{m}=\kappa y$, where $\kappa$
is the von Kármán constant. Substituting the length scale of eddies above to
Eq. (2), the van Driest [4] transformation is obtained

$u_{VD}^{+}=\frac{1}{\kappa}\log y^{+}+C,$ (3)

where

$du_{VD}^{+}=\sqrt{\frac{\overline{\rho}}{\overline{\rho}_{w}}}d\widetilde{u}^{+},\quad
y^{+}=y\frac{u_{\tau}}{\overline{\nu}_{w}}.$ (4)

Here, $\overline{\nu}_{w}$ is the kinematic viscosity at the wall, and
$u_{VD}^{+}$ is the velocity transformed by the van Driest transformation [4].

The performance of the van Driest [4] transformation has been assessed by
experiments of turbulent boundary layers (Fernholz and Finley [13, 14],
Fernholz et al. [15]), DNS of adiabatic ZPG boundary layers (Pirozzoli and
Bernardini [16, 17], Pirozzoli et al. [18], Guarini et al. [19]), isothermal
ZPG boundary layers (Maeder et al. [20], Martín [21]), and isothermal (nearly
adiabatic) ZPG boundary layers (Duan et al. [22], Lagha et al. [23]). The
success of van Driest [4] transformation for adiabatic flows verifies the
validity of Morkovin’s hypothesis [5] and builds up the foundation for the
development of more advanced transformations. On the other hand, it has also
been widely recognized that the van Driest transformation does not function
well for boundary-layer flows with heat transfer [9, 12].

###  2.2 The Zhang et al. transformation [6]

Appealing to the established Mach-number-invariance of several quantities,
Zhang et al. [6] propose a velocity transformation for adiabatic boundary
layers. Zhang et al. [6] recognize that in the logarithmic region, the
dissipation term reduces to
$\overline{\mu}\overline{\omega^{\prime}_{i}\omega^{\prime}_{i}}$ at moderate
Mach numbers, and are in the quasi-balance with the turbulence production term
$-\overline{\rho}\widetilde{u^{\prime\prime}v^{\prime\prime}}(\partial\widetilde{u}/\partial
y)$ in the TKE equation, and the ratio of these two terms is Mach-number-
invariant. By further observing that the distributions of
$\overline{\omega^{\prime}_{i}\omega^{\prime}_{i}}^{+}$ and
$-\overline{\rho}^{+}\widetilde{u^{\prime\prime}v^{\prime\prime}}^{+}$ are
independent on the Mach number variations, one can conclude that
$(\partial\widetilde{u}/\partial y)^{+}/\overline{\mu}^{+}$ denoted by $S$ is
Mach-number-invariant and obtain the Mach-number-invariant scaling of the
mixing length

$l_{Z}^{+}=(\sqrt{\overline{\rho}^{+}}\overline{\mu}^{+})l_{m}^{+},$ (5)

where
$l_{m}^{+}=(-{\widetilde{u^{\prime\prime}v^{\prime\prime}}^{+})^{\frac{1}{2}}}/(\partial\widetilde{u}^{+}/\partial
y^{+})$. By combining the Mach-number-invariant mixing length and the
constant-stress-layer assumption, Zhang et al. [6] derive the following
velocity transformation

$\displaystyle
u_{Z}^{+}=\int_{0}^{\widetilde{u}^{+}}\frac{g}{\overline{\mu}^{+}}\,d\widetilde{u}^{+},$
(6) $\displaystyle
g=\frac{-\frac{S}{2}+\sqrt{(\frac{S}{2})^{2}+(1-\overline{\mu}^{+2}S)}}{(1-\overline{\mu}^{+2}S)},$

where $u_{Z}^{+}$ denotes the corresponding transformed velocity. Similar to
the van Driest transformation, the approach of Zhang et al. only works well
for wall-bounded turbulence with an adiabatic boundary condition [12].

###  2.3 The Trettel-Larsson transformation [9]

In the log region of incompressible wall-bounded flows, the relation
$du/dy=(\tau_{w}/\overline{\rho}_{w})^{1/2}/(\kappa y)$ holds, leading to the
well-known universal log law, i.e., $du^{+}/dy^{+}=1/(\kappa y^{+}$). For a
compressible flow, the key variables in the logarithmic region of compressible
flows are the wall shear $\tau_{w}$, the local flow density $\overline{\rho}$,
and the wall-normal distance $y$ [24]. By analogy to the incompressible flow,
the mean shear stress in a compressible flow is assumed by Trettel and Larsson
[9] to follow

$\frac{d\widetilde{u}}{dy}=\frac{1}{\kappa
y}(\frac{\tau_{w}}{\overline{\rho}})^{\frac{1}{2}},$ (7)

with the velocity scale $\sqrt{{\tau_{w}}/{\overline{\rho}}}$ according to
dimension analysis. In order for the velocity to follow the incompressible law
of the wall after the transformation is applied, Trettel and Larsson [9]
derive that the transformed velocity and length scale must follow the
condition

$\frac{du_{TL}}{d\widetilde{u}}=\frac{y}{y_{TL}}(\frac{\overline{\rho}}{\overline{\rho}_{w}})^{\frac{1}{2}}\frac{dy_{TL}}{dy},$
(8)

where $u_{TL}$ and $y_{TL}$ denote the “incompressible” velocity and wall-
normal distance mapped by the Trettel-Larsson [9] transformation. Then,
Trettel and Larsson [9] employ the constant-stress-layer assumption to find

$\overline{\mu}_{w}\frac{du_{TL}}{dy_{TL}}=\overline{\mu}\frac{d\widetilde{u}}{dy}.$
(9)

The velocity gradient derived from the two conditions above should be equal to
each other and yield the Trettel-Larsson transformation [9]

$\displaystyle
u_{TL}^{+}=\int_{0}^{\widetilde{u}^{+}}(\frac{\overline{\rho}}{\overline{\rho}_{w}})^{\frac{1}{2}}[1+\frac{1}{2}\frac{1}{\overline{\rho}}\frac{d\overline{\rho}}{dy}y-\frac{1}{\overline{\mu}}\frac{d\overline{\mu}}{dy}y]\,d\widetilde{u}^{+},$
(10) $\displaystyle
y^{*}=\frac{\overline{\rho}(\tau_{w}/\overline{\rho})^{1/2}y}{\overline{\mu}},$

where $y^{*}$ is often called the semi-local scaling of the wall-normal
coordinate.

Trettel and Larsson [9] validate this transformation with a compilation of
experimental data of turbulent boundary layers from Fernholz and Finley [13]
and Fernholz et al. [15], the DNS data of channel flows by Trettel and Larsson
[9], and one diabatic low-enthalpy temporal boundary layer from Duan and
Martín [25]. This transformation satisfactorily collapses the velocity profile
in the entire inner layer of channel flows. However, recent research reveals
that it typically fails in the log region when deployed to ZPG boundary layer
flows with heat transfer at the wall [12][26].

###  2.4 The data-driven transformation [11]

Following Morkovin [5], Volpiani et al. [11] postulate that the
compressibility effect can be accounted for by the variation of mean density
and mean viscosity, and they further assume the transformation is in the
integral form

$y_{V}=\int_{0}^{y}f_{I}\,dy,\quad
u_{V}=\int_{0}^{\widetilde{u}}g_{I}\,d\widetilde{u},$ (11)

where $f_{I}$ and $g_{I}$ are mapping functions, $y_{V}$ and $u_{V}$ denote
the resulting distributions from transformation $I$, and these profiles should
agree with incompressible distributions if the transformation is successful.
To be in line with the simplified constant-stress-layer assumption [9] in the
viscous sublayer, $f_{I}$ and $g_{I}$ must satisfy the relation
$(\overline{\mu}/{\overline{\mu}_{w}})f_{I}=g_{I}$, which implies

$f_{I}=(\frac{\overline{\rho}}{\overline{\rho}_{w}})^{b}(\frac{\overline{\mu}}{\overline{\mu}_{w}})^{-a},\quad
g_{I}=(\frac{\overline{\rho}}{\overline{\rho}_{w}})^{b}(\frac{\overline{\mu}}{\overline{\mu}_{w}})^{1-a}.$
(12)

Following the data-driven philosophy, the exponent parameters $a$ and $b$ are
calibrated by the DNS databases of adiabatic and diabatic boundary layers from
Volpiani et al. [27, 28], diabatic boundary layers from Volpiani et al. [11],
adiabatic and diabatic boundary layers from Zhang et al. [26], and a channel
flow from Modesti and Pirozzoli [29], and the optimal values turn out to be
$3/2$ and $1/2$, respectively. Therefore, the data-driven velocity
transformation is given as

$\displaystyle
u_{V}^{+}=\int_{0}^{\widetilde{u}^{+}}(\frac{\overline{\rho}}{\overline{\rho}_{w}})^{\frac{1}{2}}(\frac{\overline{\mu}}{\overline{\mu}_{w}})^{-\frac{1}{2}}\,d\widetilde{u}^{+},$
(13) $\displaystyle
y_{V}^{+}=\int_{0}^{y^{+}}(\frac{\overline{\rho}}{\overline{\rho}_{w}})^{\frac{1}{2}}(\frac{\overline{\mu}}{\overline{\mu}_{w}})^{-\frac{3}{2}}\,dy^{+},$

where $u_{V}^{+}$ and $y_{V}^{+}$ denote the transformed velocity and wall-
normal distance function. This transformation works well for boundary layers,
but its performance deteriorates for channel flows [12].

###  2.5 The total-stress-based transformation [12]

Inspired by the work mentioned above, Griffin et al. [12] argue that there
should be different scalings in the viscous sublayer and the logarithmic
region, respectively. In the logarithmic region, Griffin et al. [12] first
generalize the quasi-equilibirum argument by Zhang et al. [6] through
introducing the semi-local scaling. Griffin et al. [12] recognize the ratio of
turbulence production and viscous dissipation as a Mach-number-invariant
function of $y^{*}$. This weaker assumption leads to the following
dimensionless mean shear, i.e.,

$S_{eq}^{+}(y^{*})=\frac{1}{\overline{\mu}^{+}}\frac{\partial{\widetilde{u}^{+}}}{\partial{y^{*}}}=\frac{\partial{u_{eq}^{+}}}{\partial{y^{*}}},$
(14)

which is Mach-number-independent in the log region.

Within the viscous sublayer, the viscous stress is assumed to be Mach-number-
independent, leading to the non-dimensionalized mean shear

$S_{TL}^{+}(y^{*})=\overline{\mu}^{+}\frac{\partial{\widetilde{u}^{+}}}{\partial{y^{+}}}=\frac{\partial{u_{TL}^{+}}}{\partial{y^{*}}},$
(15)

which is also known as the Trettel-Larsson transformation [9].

To combine these two scalings to derive a composite transformation, Griffin et
al. [12] rewrite the total stress $\tau^{+}=\tau_{v}^{+}+\tau_{R}^{+}$ as

$\tau^{+}=S_{t}^{+}(\frac{\tau_{v}^{+}}{S_{TL}^{+}}+\frac{\tau_{R}^{+}}{S_{eq}^{+}}),$
(16)

where $\tau^{+}$ denotes the non-dimensional total stress, $\tau_{v}^{+}$ the
viscous stress, $\tau_{R}^{+}$ the Reynolds shear stress, and $S_{t}^{+}$ is a
generalized non-dimensional mean shear, which is defined such that the
equation holds. This construction allows $S_{t}^{+}$ to recover $S_{TL}^{+}$
as $\tau^{+}\rightarrow\tau_{v}^{+}$ in the viscous sublayer and to recover
$S_{eq}^{+}$ as $\tau^{+}\rightarrow\tau_{R}^{+}$ in the logarithmic region.
Noticing that $\tau_{v}^{+}=S_{TL}^{+}$, and plugging
$\tau_{R}^{+}=\tau^{+}-\tau_{v}^{+}$ into Eq. (16), $S_{t}^{+}$ can be
analytically written as

$S_{t}^{+}=\frac{\tau^{+}S_{eq}^{+}}{\tau^{+}+S_{eq}^{+}-S_{TL}^{+}},$ (17)

and the transformed velocity $u_{t}^{+}$ is defined correspondingly as

$u_{t}^{+}=\int_{0}^{y^{*}}S_{t}^{+}\,dy^{*}.$ (18)

Based on the constant-stress-layer assumption, a simplified version can be
derived with a similar performance, see Eq. (5) of [12].

This total-stress-based transformation [12] has been assessed by DNS databases
of channel flows (Trettel and Larsson [9], Modesti and Pirozzoli [29], Yao and
Hussain [30]), pipe flows (Modesti and Pirozzoli [31]), both adiabatic
boundary layers (Pirozzoli and Bernardini [17], Zhang et al. [26], Volpiani et
al. [27]) and diabatic boundary layers (Zhang et al. [26], Volpiani et al.
[27, 28]), and the high-Mach turbulent boundary layers downstream of the
shock/boundary-layer interaction (Fu et al. [32]). These extensive numerical
validations reveal that this new transformation performs very well for all the
considered canonical wall-bounded flows, irrespective of the Mach number,
Reynolds number, and the thermal boundary condition.

##  3 Database descriptions

The DNS databases utilized to assess different velocity transformations
include high-enthalpy and low-enthalpy turbulent temporal boundary layers with
dissociation and vibrational excitation by Duan and Martín [25], the high-
enthalpy spatial boundary layer of di Renzo and Urzay [33], diabatic channel
flows at supercritical pressure by Wan et al. [34] and Toki et al. [35],
heated and unheated boundary layers at supercritical pressure by Kawai [36],
and adiabatic boundary layers with pressure gradients by Wenzel et al. [37]
and Gibis et al. [38].

###  3.1 High-enthalpy turbulent boundary layers

For flying vehicles at a high Mach number, the shock heating can lead to
extremely hot boundary layers downstream. This high-enthalpy condition can
cause internal excitation and dissociation of air molecules. The specific heat
capacity varies with temperature as a consequence, and both the thermal and
chemical equilibrium may be disrupted.

Duan and Martín [25] conduct the DNS of high- and low-enthalpy wall-bounded
flows. Zero-pressure-gradient temporal boundary-layer simulations are
conducted for various enthalpy conditions and Mach numbers. The two low
enthalpy cases assume a calorically perfect gas. The four high-enthalpy cases
match the conditions downstream of the oblique shock for wedges of
$35^{\circ}$ and $8^{\circ}$ (denoted by Wedge35 and Wedge8). To investigate
the effect of species boundary conditions, ‘supercatalytic’ and ‘non-
catalytic’ surface models are considered for each wedge angle. For the non-
catalytic model, there is no atom recombination and there exists minimal
enthalpy recovery at the wall. It can be written as $(\partial Y/\partial
n)_{w}=0$, where $Y$ is the species concentration and $n$ is the wall-normal
coordinate. On the contrary, the supercatalytic model means infinitely fast
atom recombination and maximum enthalpy recovery implying $Y_{w}=Y_{\infty}$.
For the low-enthalpy flows, their edge Mach number and the ratio of wall to
recovery temperature are carefully chosen to approximately match those for
high-enthalpy cases. The corresponding low-enthalpy cases are denoted as
LowH_M3 and LowH_M10 for Wedge35 and Wedge8, respectively.

The wall boundary conditions are no-slip and isothermal (cold wall). The flow
variables on the top boundary are fixed and obtained from a RANS calculation
of a larger domain that includes the wedge and associated oblique shock.
Periodic conditions are enforced for the streamwise and spanwise boundaries.

The DNS of di Renzo and Urzay [33] is a spatially evolving boundary layer
which includes the transition from a laminar to a fully turbulent boundary
layer. Only profiles from the fully turbulent section of the domain
($Re_{\tau}>600$) are considered in this work. A non-catalytic, flat, cold
wall is used. The freestream Mach number is 10, and the stagnation enthalpy is
21.6 MJ/kg, which corresponds to the post-shock conditions of a $9^{\circ}$
wedge flying at Mach 23 in Earth’s atmosphere at 25 km. Vibrational excitation
and dissociation are considered.

###  3.2 Flows at supercritical pressure

####  3.2.1 Channel flows at supercritical pressure

Toki et al. [35] conduct DNSs of planar turbulent channel flows driven by a
body force with a wall-normal temperature gradient. The working fluid is
nitrogen at a supercritical pressure, which is approximately 1.3 times its
critical pressure. Since the ideal-gas assumption becomes invalid, the Soave-
Redlich-Kwong (SRK) [39] equation of state is employed. The bottom wall (116
K) is colder than its pseudocritical temperature (131.8 K at 4.4 MPa) while
the top wall is hotter. Toki et al. [35] evaluate the thermal conductivity
$\lambda$ and the viscosity coefficient $\mu$ using the formula by Chung [40].
Two cases named R4 and R8 are considered, whose density ratio of the two walls
is 4 and 8, respectively, and their corresponding temperatures at the top wall
are 149 K and 219 K. The friction Reynolds number $Re_{\tau}$ is approximately
390 at the cold wall, 268 and 122 at the top wall for R4 and R8 respectively.
The channel is $3\pi H$ in length and $\pi H$ in the spanwise direction, where
$H$ is the half-height of the channel. The no-slip condition is applied on the
top and bottom walls. Periodic boundary conditions are applied in both the
spanwise and streamwise directions.

Wan et al. [34] perform a similar investigation. The working fluid is carbon
dioxide at 8 MPa and the corresponding pseudocritical temperature
$T_{pc}=307.8$ K. The thermodynamic properties are taken from the property
table of the REFPROF 9.1 database [41]. The channel is $4H$ and $16H$ in the
spanwise and streamwise directions, respectively, where $H$ is the half-height
of the channel. The no-slip boundary condition is applied at both walls with
constant temperature ($T_{pc}-T_{w,cold}=T_{w,hot}-T_{pc}$). Periodic boundary
conditions are applied in both the spanwise and streamwise directions. Wan et
al. [34] conduct 5 simulations whose temperature difference between the two
walls increases and one extra case with constant properties as a reference
(only three of them will be considered in this work). The maximum temperature
difference is selected to cover the significant variation range of all
properties. All cases share the same bulk Reynolds numbers $Re_{b}=5705$.

####  3.2.2 Boundary layers at supercritical pressure

Kawai [36] conducts boundary layer simulations at two different supercritical
pressures, i.e., 2 MPa and 4 MPa, for the substance parahydrogen, and their
corresponding pseudocritical temperatures are approximately 36.4 K and 43.3 K,
respectively. Thermodynamic and fluid transport properties are evaluated using
REFPROP (NIST Standard Reference Database 23, Version 9.0). The flat plate
consists of two parts, i.e., the unheated part ($0<x<20\delta_{0}$) and the
heated part ($20\delta_{0}<x<45\delta_{0}$), where $\delta_{0}$ is
approximately 99$\%$ of the boundary layer thickness at the inlet of the
heated domain. Six boundary layer profiles are considered, i.e., two of them
from unheated section of the wall ($T_{w}=T_{\infty}=25$ K) and the rest four
from the heated section ($T_{w}=100$ K and 200 K). The plate is isothermal and
no-slip, and the periodic condition is adopted for the spanwise boundaries.
The friction Reynolds numbers are $Re_{\tau}\approx 390$ for the unheated
cases, $Re_{\tau}\approx$ 210 and 90 for cases with $T_{w}=100$ K and 200 K at
the pressure of 2 Mpa, and $Re_{\tau}\approx$ 360 and 150 for cases with
$T_{w}=100$ K and 200 K at the pressure of 4 Mpa, respectively.

###  3.3 Boundary layers with pressure gradients

The DNS of boundary layers with adverse (APG) and favourable (FPG) pressure
gradients have been systematically explored by Wenzel et al. [37] and Gibis et
al. [38]. The kinematic Rotta-Clauser parameter
$\beta_{k}=(\delta^{*}/\tau_{w})(d\overline{p}_{e}/dx)$, where
$\delta_{K}^{*}=\int_{0}^{\delta_{99}}(1-\overline{u}/\overline{u}_{e})\,dy$
denotes the kinematic displacement thickness, and $d\overline{p}_{e}/dx$ the
pressure gradient at the edge of the boundary layer, is adopted as the
appropriate parameter representing the streamwise self-similarity for boundary
layers with pressure gradients. Both subsonic ($M_{\infty,0}=0.5$) and
supersonic ($M_{\infty,0}=2$) flows are considered, where $M_{\infty,0}$ is
the free-stream Mach number at the inlet. For subsonic APG boundary layers,
Rotta-Clauser parameters are set to be $\beta_{k}=$0, 0.19, 0.58, and 1.05,
respectively. For hypersonic cases, Rotta-Clauser parameters are
$\beta_{k}=$0, 0.15, 0.42, 0.55 and 0.62 for APG boundary layers and -0.18 for
FPG boundary layer. For each simulation, the Rotta-Clauser parameter remains
constant in the streamwise direction. The adiabatic no-slip condition is
applied to the wall, and the periodic condition is adopted for spanwise
boundaries.

##  4 Assessment and analysis

In this section, various transformations will be assessed by the databases
mentioned above. Also, distributions of the various non-dimensionalizations of
the mean shear will be analyzed as diagnostic functions indicating the
accuracy of the transformations as a function of wall-normal distance. The
results are presented case by case and then summarized in Tables 1 and 2
afterwards.

###  4.1 High-enthalpy turbulent boundary layers

Figure 1:  Transformed velocity distributions versus the non-dimensional wall-
normal coordinate. The database of high-enthalpy and low-enthalpy turbulent
temporal boundary layers by Duan and Martín [25] is used.

Figure 2:  Transformed velocity distributions versus the non-dimensional wall-
normal coordinate. The database of high-enthalpy turbulent spatial boundary
layers by di Renzo and Urzay [33] is used. Figure 3:  Three types of non-
dimensional mean shear multiplied by the semi-local wall-normal coordinate are
plotted with respect to the semi-local wall-normal coordinate. The database of
high-enthalpy and low-enthalpy turbulent temporal boundary layers by Duan and
Martín [25] is used.

Figure 4: The quantity $\epsilon^{\ddagger}=\epsilon^{+}/\mu^{+}$ (a) and the
ratio of the modified production
$P^{\ddagger}=-\overline{\rho}^{+}\overline{u^{\prime}v^{\prime}}^{+}(\partial\widetilde{u}^{+}/\partial
y^{*})$ and dissipation [12] (b). The incompressible channel flow by Lee and
Moser [42] at $Re_{\tau}\approx 5200$ is employed as reference in the dashed
black line. The database of high-enthalpy and low-enthalpy turbulent temporal
boundary layers by Duan and Martín [25] is used. Figure 5: The approximate
Reynolds shear stress profiles normalized by the wall stress for the high-
enthalpy and low-enthalpy turbulent boundary layers by Duan and Martín [25].
The incompressible boundary layer by Sillero et al. [43] at $Re_{\tau}\approx
2000$ is employed as reference in the dashed black line.

In Fig. 1, transformed velocity distributions are plotted versus the non-
dimensional wall-normal coordinate for the high- and low-enthalpy temporal
boundary layers. The transformations employed include the van Driest [4] (a),
Zhang et al. [6] (b), Trettel-Larsson [9] (c), data-driven [11] (d), total-
stress-based [12] (e), and constant-stress-layer version of the total-stress-
based [12] (f). The color of lines indicates the friction Reynolds number
$Re_{\tau}$. Two incompressible DNS mean velocity profiles, i.e., from the
channel flow (black dashed line) by Lee and Moser [42] at $Re_{\tau}\approx
5200$ and the ZPG boundary layer (black dotted line) by Sillero et al. [43] at
$Re_{\tau}\approx 2000$, are also shown for comparisons. The van Driest [4]
transformation fails, which may be due to the heat flux at the wall. There is
an obvious increase of the log-law intercept for the van Driest [4] and Zhang
et al. [6] transformed velocity profiles, and similar behavior has also been
noticed by Duan and Martín [25], and Maeder [44] for the cold-wall boundary
layers. The data-driven approach [11] and the total-stress-based
transformation perform well in the viscous sublayer, but their performance
deteriorates significantly in the logarithmic region. It is worth mentioning
that there are two cases, i.e., LowH_M3 and Wedge35noncata (the readers are
referred to [25] for details), whose transformed profiles from the total-
stress-based approach coincide well with the incompressible reference by
Sillero et al. [43]. Overall speaking, the Trettel-Larsson [9] transformation
works the best among all the others.

In Fig. 2, the velocity transformations are shown for the high-enthalpy
spatial boundary layer of di Renzo and Urzay [33]. Contrary to the temporal
boundary layer, the data-driven and total-stress-based transformations are in
better agreement with the incompressible reference than the Trettel-Larsson
transformation is. Like the temporal boundary layer, the constant-stress-layer
assumption does not visibly affect the results, so $u_{t}^{+}$ is not shown
for brevity. Also, the van Driest transformation is plotted by di Renzo and
Urzay [33]. It is worth noting that Passiatore [45] performed a direct
numerical simulation of a hypersonic, spatially developing, fully turbulent
boundary layer with thermochemical effects. They also observed that the total-
stress-based transformation out-performs the Trettel-Larsson transformation in
the logarithmic region.

Returning to the temporal boundary layer, in Fig. 3, three types of non-
dimensional mean shear multiplied by the semi-local wall-normal coordinate are
plotted with respect to the semi-local wall-normal coordinate, where the one
derived from Trettel-Larsson [9] is in green, the one based on turbulence
quasi-equilibrium in cyan (see Eq. (14)), the one from the total-stress-based
[12] transformation in black (see Eq. (17)), and the last one $y^{+}(\partial
u^{+}/\partial y^{+})$ for the incompressible channel flow by Lee and Moser
[42] at $Re_{\tau}\approx 5200$ is in magenta. The non-dimensional mean shear
derived by Trettel-Larsson [9] has a slight downward translation in the
viscous sublayer and an upward translation in the buffer layer and the
logarithmic region compared with the incompressible reference. And these
errors cancel out during the integration (see Eq. (10)), leading to the
success of the Trettel-Larsson [9] transformation in the log region. The
total-stress-based transformation is inaccurate in the log region because the
modified equilibrium transformation is failing. This transformation makes
three assumptions which are assessed individually. (1) In Fig. 4(a), the
quantity $\epsilon^{\ddagger}=\epsilon^{+}/\mu^{+}$ does not collapse to the
incompressible reference data in the log region, which disrupts the first
assumption. (2) In Fig. 4(b), the ratio of the modified production and the
dissipation does not collapse to the incompressible reference data, which
invalidates the assumption of the Mach invariance of this ratio. (3) In Fig. 5
(the Reynolds-averaged Reynolds stress is approximated as
$\overline{\rho}^{+}\overline{u^{\prime}v^{\prime}}^{+}$ since only this
statistic is available for this database), the Reynolds shear stress is also
not Mach invariant in the log region; this is another reason why the modified
equilibrium transformation (and thus also the one based on the total shear
stress) is failing. The disagreement of these three non-dimensional turbulence
statistics with incompressible data suggests that the non-dimensionalizations
do not accurately account for high-enthalpy effects. Specifically, the near-
wall non-dimensional dissipation is much lower than the incompressible data.

###  4.2 Flows at supercritical pressure

At supercritical pressure, the distinction between the gas and liquid phase
blurs, and the flow changes continuously from the liquid phase to the gas
phase. The thermodynamic and molecular transport properties have very large
gradients near the pseudocritical temperature, known as pseudo-boiling
phenomena. These real-fluid effects have inevitable impacts on the near-wall
turbulence dynamics and the boundary-layer velocity profile. Additionally, the
diabatic wall condition also influences the flow field. As a consequence, it
is much more challenging to propose an appropriate near-wall scaling for
supercritical flows than for ideal-gas boundary layers.

####  4.2.1 Channel flows at supercritical pressure

Figure 6:  Transformed velocity distributions versus the non-dimensional wall-
normal coordinate. The database is for channel flows at supercritical pressure
by Toki et al. [35].

In Fig. 6, transformed velocity distributions versus the non-dimensional wall-
normal coordinate are shown for the van Driest [4] (a), Zhang et al. [6] (b),
Trettel-Larsson [9] (c), data-driven [11] (d), and total-stress-based [12] (e)
transformations. The color of lines indicates the friction Reynolds number
$Re_{\tau}$. Two incompressible DNS mean velocity profiles, i.e., from the
channel flow (black dashed line) by Lee and Moser [42] at $Re_{\tau}\approx
5200$ and the ZPG boundary layer (black dotted line) by Sillero et al. [43] at
$Re_{\tau}\approx 2000$, are also shown for comparisons. The performance of
various transformations is compared for the database from Toki et al. [35].
Surprisingly, the Trettel-Larsson [9] and van Driest [4] transformed mean
velocity profiles show good agreement with the incompressible law of the wall
even for supercritical channel flows, especially when the Trettel-Larsson
transformation is deployed. The rest three transformations perform well in the
viscous sublayer but fail in the buffer layer and the logarithmic region. The
Zhang et al. [6] and total-stress-based [12] transformations fail in the
logarithmic region in these cases, which can be further confirmed by checking
the diagnostic function in Fig. 7(a). The non-dimensional mean shear derived
from Trettel-Larsson [9] is in green, the one based on turbulence quasi-
equilibrium in cyan (see Eq. (14)), the one from the total-stress-based [12]
transformation in black (see Eq. (17)), and the last one $y^{+}(\partial
u^{+}/\partial y^{+})$ for the incompressible channel flow by Lee and Moser
[42] at $Re_{\tau}\approx 5200$ in magenta. In Fig. 8, the Reynolds shear
stress profiles are shown to depart from the incompressible reference, which
violates an assumption made by Zhang et al. and Griffin et al., and could
explain the failure of these transformations. Since data for the dissipation
is not available, the other assumptions made by these transformations can not
be interrogated for this data set.

Figure 7:  Three types of non-dimensional mean shear multiplied by the semi-
local wall-normal coordinate are plotted with respect to the semi-local wall-
normal coordinate. (a) utilizes the DNS data by Toki et al. [35] and (b) Wan
et al. [34]. Figure 8: The Reynolds shear stress normalized by the wall stress
for the database of channel flows at supercritical pressure by Toki et al.
[35]. The incompressible channel flow by Lee and Moser [42] at
$Re_{\tau}\approx 5200$ is employed as reference in the dashed black line.

In Fig. 9, transformed velocity distributions are plotted versus the non-
dimensional wall-normal coordinate for the channel flows at supercritical
pressure by Wan et al. [34]. The velocity transformations plotted include that
of van Driest [4] (a), Zhang et al. [6] (b), Trettel-Larsson [9] (c), data-
driven [11] (d), and total-stress-based [12] (e) transformations. The color of
lines indicates the friction Reynolds number $Re_{\tau}$. Two incompressible
DNS mean velocity profiles, i.e., from the channel flow (black dashed line) by
Lee and Moser [42] at $Re_{\tau}\approx 5200$ and the ZPG boundary layer
(black dotted line) by Sillero et al. [43] at $Re_{\tau}\approx 2000$, are
also shown for comparisons. It is evident that results from cool walls (low
$Re_{\tau}$) and hot walls (high $Re_{\tau}$) follow different patterns, which
can be seen in Fig. 6 as well. The van Driest [4] and Trettel-Larsson [9]
transformations perform better, and an increase of the log-law intercept can
be observed for hot walls in these two transformations. The data-driven
approach works well for the cases with hot walls, but fails for those with
cold walls. Velocity profiles obtained through the Zhang’s [6] and total-
stress-based [12] transformations collapse to the incompressible law of the
wall only in the viscous sublayer, and the non-dimensional mean shears plotted
in Fig. 7(b) imply their failure in the logarithmic region. As shown in Fig.
10, a conclusion similar to that for the database from Toki et al. [35] can be
obtained.

Figure 9:  Transformed velocity distributions versus the non-dimensional wall-
normal coordinate. The database is for channel flows at supercritical pressure
by Wan et al. [34]. Figure 10: The Reynolds shear stress normalized by the
wall stress for the database of channel flows at supercritical pressure by Wan
et al. [34]. The incompressible channel flow by Lee and Moser [42] at
$Re_{\tau}\approx 5200$ is employed as reference in the dashed black line.

####  4.2.2 Boundary layers at supercritical pressure

In Fig. 11, the transformed velocity distributions are plotted versus the non-
dimensional wall-normal coordinate for boundary layer flows at supercritical
pressure. The velocity transformations include the van Driest [4] (a), Zhang
et al. [6] (b), Trettel-Larsson [9] (c), data-driven [11] (d), and total-
stress-based [12] (e) transformations. The color of lines indicates the
friction Reynolds number $Re_{\tau}$. Two incompressible DNS mean velocity
profiles, i.e., from the channel flow (black dashed line) by Lee and Moser
[42] at $Re_{\tau}\approx 5200$ and the ZPG boundary layer (black dotted line)
by Sillero et al. [43] at $Re_{\tau}\approx 2000$, are also shown for
comparisons. There are two cases which can be mapped to the incompressible law
of the wall by all the transformations tested. The wall conditions in these
two cases are unheated such that the flow temperature remains below the
pseudo-critical temperature, and the pseudo-boiling phenomenon does not
appear. For the heated cases, the Zhang et al. [6], data-driven [11], and
total-stress-based transformations only work in the viscous sublayer, and the
performance of all the transformations tested deteriorates in the logarithmic
region. In Fig. 12, three types of non-dimensional mean shear multiplied by
the semi-local wall-normal coordinate are plotted with respect to the semi-
local wall-normal coordinate. The non-dimensionalization implied by the
Trettel-Larsson [9] transformation is in green, the one based on turbulence
quasi-equilibrium in cyan (see Eq. (14)), the one from the total-stress-based
[12] transformation in black (see Eq. (17)), and the last one $y^{+}(\partial
u^{+}/\partial y^{+})$ for the incompressible channel flow by Lee and Moser
[42] at $Re_{\tau}\approx 5200$ in magenta. The DNS data by Kawai [36] is
used. Fig. 12 also shows that none of the proposed non-dimensional mean shears
collapses to the incompressible reference in the entire inner region. The
failure of the van Driest [4], Trettel-Larsson [9], and Zhang et al. [6]
transformation may be due to the effect of heat transfer at the wall since
these transformations also do not perform well in perfect gas boundary layers
with heat transfer [12]. Also, the pseudo-boiling phenomenon could be a source
of error, and it might compound with the aforementioned error. The data-driven
approach [11] and the total-stress-based transformation [12] fail because of
the influence of the pseudo-boiling phenomenon, since they show excellent
performance in canonical diabatic boundary layers without supercritical
effects. In Fig. 13, the Reynolds shear stress profiles are shown to depart
from the incompressible reference, which violates an assumption made by Zhang
et al. and Griffin et al., and could explain the failure of these
transformations. Unlike canonical boundary layers, the Favre-averaged
Reynolds-shear stress data exceeds unity for some cases due to the excessive
positive contribution from the mean-density-related term. Large changes in
density arise in cases where wall heating leads to transcritical thermal
conditions (see page 585 in [36] for details). As a result, the Reynolds-shear
stress exceeds unity in only the cases with wall heating.

Figure 11:  Transformed velocity distributions versus the non-dimensional
wall-normal coordinate. The database is for boundary layers at supercritical
pressure by Kawai [36]. Figure 12:  Three types of non-dimensional mean shear
multiplied by the semi-local wall-normal coordinate are plotted with respect
to the semi-local wall-normal coordinate. Figure 13: The Reynolds shear stress
normalized by the wall stress for the database of turbulent boundary layers at
supercritical pressure by Kawai [36]. The incompressible boundary layer by
Sillero et al. [43] at $Re_{\tau}\approx 2000$ is employed as reference in the
dashed black line.

In summary, although the Trettel-Larsson [9] transformation is widely
recognized as one of the most appropriate transformations for channel flows,
it is unexpected that it is also successful in supercritical channel flows.
For boundary layers at supercritical pressure, none of the transformations is
able to collapse these mean velocity profiles and the appropriate
transformation still remains unclear.

###  4.3 Boundary layers with pressure gradients

Figure 14:  Transformed velocity distributions versus the non-dimensional
wall-normal coordinate. The database is for boundary layers with the pressure
gradient by Wenzel et al. [37]. Figure 15:  Three types of non-dimensional
mean shear multiplied by the semi-local wall-normal coordinate are plotted
with respect to the semi-local wall-normal coordinate. The DNS data of
turbulent boundary layers by Wenzel et al. [37] is used.

In Fig. 14, transformed velocity distributions are plotted versus the non-
dimensional wall-normal coordinate for the database is for boundary layers
with the pressure gradient by Wenzel et al. [37]. The transformaitons include
the van Driest [4] (a), Zhang et al. [6] (b), Trettel-Larsson [9] (c), data-
driven [11] (d), and total-stress-based [12] (e) transformations. The color of
lines indicates the friction Reynolds number $Re_{\tau}$. Two incompressible
DNS mean velocity profiles, i.e., from the channel flow (black dashed line) by
Lee and Moser [42] at $Re_{\tau}\approx 5200$ and the ZPG boundary layer
(black dotted line) by Sillero et al. [43] at $Re_{\tau}\approx 2000$, are
also shown for comparisons. The performance of all the considered
transformations is similar for boundary layers with pressure gradients, and
good agreement is observed between the transformed velocity distributions and
the incompressible law of the wall, although the log-law intercept varies in a
small range due to the pressure gradient effects characterized by the Rotta-
Clauser parameter $\beta_{k}$. Such a good collapse can be further confirmed
by Fig. 15, which shows that the three non-dimensional mean shears agree well
with the incompressible reference. The three non-dimensional mean shears
include the one derived from the Trettel-Larsson [9] transformation in green,
the one based on turbulence quasi-equilibrium in cyan (see Eq. (14)), the one
from the total-stress-based [12] transformation in black (see Eq. (17)), and
the last one $y^{+}(\partial u^{+}/\partial y^{+})$ for the incompressible
channel flow by Lee and Moser [42] at $Re_{\tau}\approx 5200$ in magenta. It
is worth mentioning that the pressure gradients are relatively weak for these
cases and, moreover, the wall boundary condition is adiabatic, which further
simplifies the near-wall dynamics [37]. Such an observation indicates that,
for wall-bounded turbulence with a weak pressure gradient, the transformations
developed for ZPG boundary layers can be sufficiently accurate.

Table 1:  Summary of the performance in the viscous sublayer of various
velocity transformations.

Transformations | BLHE [25, 33] | CSPT [35] | CSPW [34] | BLSP [36] | BLPG [37, 38]
---|---|---|---|---|---
van Driest [4] | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$
Zhang et al. [6] | $\times$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
Trettel and Larsson [9] |  $\checkmark$ , $\checkmark$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$
Volpiani et al. [11] |  $\checkmark$ , $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
Griffin et al. [12] |  $\checkmark$ , $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$

$\checkmark$ indicates adequate performance and $\times$ denotes
unsatisfactory performance. BLHE denotes high-enthalpy and low-enthalpy
turbulent temporal and spatial boundary layers, CSPT denotes channel flows at
supercritical pressure, CSPW denotes channel flows at supercritical pressure,
BLSP denotes boundary layers at supercritical pressure, and BLPG denotes
boundary layers with pressure gradients

Table 2:  Summary of the performance in the logarithmic region of various
velocity transformations.

Transformations | BLHE [25, 33] | CSPT [35] | CSPW [34] | BLSP [36] | BLPG [37, 38]
---|---|---|---|---|---
van Driest [4] | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$
Zhang et al. [6] | $\times$ | $\times$ | $\times$ | $\times$ | $\checkmark$
Trettel and Larsson [9] |  $\checkmark$ , $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$
Volpiani et al. [11] |  $\times$ , $\checkmark$ | $\times$ | $\times$ | $\times$ | $\checkmark$
Griffin et al. [12] |  $\times$ , $\checkmark$ | $\times$ | $\times$ | $\times$ | $\checkmark$

$\checkmark$ indicates adequate performance and $\times$ denotes
unsatisfactory performance. BLHE denotes high-enthalpy and low-enthalpy
turbulent temporal and spatial boundary layers, CSPT denotes channel flows at
supercritical pressure, CSPW denotes channel flows at supercritical pressure,
BLSP denotes boundary layers at supercritical pressure, and BLPG denotes
boundary layers with pressure gradients.

##  5 Conclusions

The objective of this paper is to assess several popular compressible velocity
transformations in non-canonical flows including boundary layers with high-
enthalpy thermochemical effects wall-bounded turbulent flows at supercritical
pressures, and boundary layers with streamwise pressure gradients.
Specifically, the following transformations are analyzed, i.e., the van Driest
transformation [4], the transformation of Zhang et al. [6], the Trettel-
Larsson transformation [9], the data-driven transformation [11], and the
total-stress-based transformation [12]. The performance of these methods has
been summarized in Tables 1 and 2.

The Trettel-Larsson [9] transformation works well for the high- and low-
enthalpy temporal boundary-layer flows by Duan and Martín [25] because the
discrepancy between its non-dimensional mean shear and the incompressible
reference cancels out during integration. On the other hand, the data-driven
approach [11] and the total-stress-based transformation [12] are approximately
valid only in the viscous sublayer. For the high-enthalpy spatial boundary
layer of di Renzo and Urzay [33], the conclusion is different. The total-
stress-based and data-driven transformations collapse the data well even in
the log region and the Trettel-Larsson transformation is only accurate in the
viscous sublayer. However, more data is required to further substantiate these
observations; as such, they should be regarded as preliminary.

For supercritical flows, the significant gradient of fluid properties near the
pseudocritical temperature results in real-fluid effects. And the interaction
of real-fluid effects with the wall-bounded turbulence dynamics remains
unclear, which increases the difficulty of proposing an appropriate
transformation.

The Trettel-Larsson [9] transformation works the best in supercritical channel
flows and followed by the van Driest [4] transformation. The performance
deterioration of the total-stress-based [12] and Zhang et al. [6]
transformation in the log layer may be due to the failure of the assumption of
quasi-equilibrium, the Mach-number-invariance of the dissipation profile, or
the Mach-number-invariance of the Reynolds shear stress.

For supercritical boundary layers, none of the transformations mentioned is
valid. Both the heat transfer at the wall and the real-fluid effect could
result in the performance deterioration.

All transformations satisfactorily collapse the velocity profiles of the
boundary layers with pressure gradients to the incompressible reference. This
might be due to the relatively weak pressure gradient imposed in the DNS by
Wenzel et al. [37] and the adiabatic wall boundary condition. Further
validation of flows with stronger pressure gradients and diabatic wall
boundary condition is needed.

In summary, all these considered velocity transformations fail to deliver a
uniform performance for non-canonical compressible wall-bounded flows in the
log region, and a more sophisticated version, which accounts for these
different physics, is needed. The data-driven transformation [11] and the
total-stress-based transformation [12] function well in the viscous sublayer
for all the considered flows. Nevertheless, the present assessment provides a
useful guideline on the deployment of these transformations on various non-
canonical flows.

These observations might inspire future developments of mean velocity
transformations. Since none of the transformations are valid in the buffer and
logarithmic layers of supercritical boundary layers with heat transfer, this
regime is an opportunity for further research. Each of the existing methods
makes assumptions which are apparently not appropriate for this flow. Such
assumptions include the constant stress layer, the validity of Prandtl’s
incompressible mixing length, the balance of production and dissipation, etc.
It may be possible to modify or generalize these physical assumptions by
appealing to DNS data. Then, the mean velocity transformations could be
reformulated accordingly.

An alternative avenue for future study is to develop data-driven
transformations for non-canonical flows. Volpiani et al. [11] proposed a
successful mean velocity transformation for subcritical boundary layers with
heat transfer and constant specific heats. In principle, data-driven methods
could also be applied to the non-canonical cases considered in this work.
Since there are only a limited number of high-fidelity databases on non-
canonical flows, it is expected that a data-driven method would have only
limited generality. That is, it might perform poorly in cases with parameters
that are different from existing data. This may motivate additional high-
fidelity simulations of non-canonical flows.

## Acknowledgments

K.P.G. acknowledges support from the National Defense Science and Engineering
Graduate Fellowship and the Stanford Graduate Fellowship. L.F. acknowledges
the fund from Shenzhen Municipal Central Government Guides Local Science and
Technology Development Special Funds Funded Projects (NO. 2021Szvup138).

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††thanks: Present address: Department of Chemistry, Princeton University,
Princeton, New Jersey 08544, USA††thanks<EMAIL_ADDRESS>
# Signature of topological non-trivial band structure in Ta3SiTe6

Shubhankar Roy  Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar,
Kolkata 700 064, India  Vidyasagar Metropolitan College, 39, Sankar Ghosh
Lane, Kolkata 700 006, India  Ratnadwip Singha  Saha Institute of Nuclear
Physics, HBNI, 1/AF Bidhannagar, Kolkata 700 064, India  Arup Ghosh  Saha
Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Kolkata 700 064, India
Prabhat Mandal  Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar,
Kolkata 700 064, India

###### Abstract

The study of topology protected electronic properties is a fascinating topic
in present day condensed matter physics research. New topological materials
are frequently being proposed and explored through various experimental
techniques. Ta3SiTe6 is a newly predicted topological semimetal with fourfold
degenerate nodal-line crossing in absence of spin-orbit coupling (SOC) and an
hourglass Dirac loop, when SOC is included. Recent angle-resolved
photoemission spectroscopy study in this material, has also confirmed Dirac
like dispersions and two nodal-lines near the Fermi energy, protected by
nonsymmorphic glide mirror symmetry. In this work, we present the detailed
magnetotransport properties of single crystalline Ta3SiTe6. A nonsaturating
magnetoresistance has been observed. Hall measurements reveal hole type charge
carriers with high carrier density and a moderate value of carrier mobility.
Furthermore, we report a robust planar Hall effect, which persists up to high
temperatures. These results validate the nontrivial nature of the electronic
band structure.

##  I Introduction

In last few years, the nontrivial topological phases of matter have introduced
a new paradigm of understanding the solids in condensed matter physics
research. The discoveries of topological insulators (TI) Xia ; Zhang ; Chen ,
Dirac and Weyl semimetals Liu ; Liu2 ; Lv ; Xu , nodal-line materials Bian ;
Singha , and topological superconductors Xu2 ; Matano , have paved the way to
realize and understand the dynamics of the relativistic quasiparticles such as
Dirac, Weyl, and Majorana fermions in low-energy electronic systems.
Topological materials, which host nodal-line fermions, are recently receiving
considerable attention due to their various interesting properties like
anisotropic electron transport Mullen , possible surface
magnetism/superconductivity Bian ; Heikkila , anomalous Landau level spectrum,
etc Rhim ; Lim . Nodal-line materials with nonsymmorphic symmetry are of
greater interest as they are robust against spin-orbit coupling (SOC), which
often gaps out band crossing points. Moreover, nonsymmorphic symmetry
protected band structure are expected to show more exotic quantum phases such
as M$\ddot{o}$bius-twist surface states, hourglass fermions, and nodal chains,
etc. Bzdusek ; Wang4 ; Wang5 ; Ma ; Shiozaki .

Recently, from the first-principles calculations, $X_{3}$SiTe6 ($X$=Ta, Nb)
are predicted to show rich band crossing features with nodal fermions
protected by nonsymmorphic space-group symmetry. In the absence of SOC, these
materials host an essential fourfold nodal-line and an accidental nodal-loop
Li6 . Furthermore, when the SOC is included, an hourglass Dirac-loop appears
in the close vicinity of EF. Even when the materials are thinned down to
monolayers, the nontrivial band structure is sustained; predicted to have
either a two-dimensional (2D) nodal-lines or 2D Dirac points in the absence or
presence of SOC, respectively. An angle-resolved photoemission spectroscopy
(ARPES) study on the layered ternary telluride Ta3SiTe6 has reported Dirac-
like dispersions and the existence of two nodal-lines near the Fermi energy
protected by the nonsymmorphic glide mirror symmetry of the crystal Sato . On
the other hand, from recent quantum oscillation measurements, it was not
possible to confirm the non-trivial Berry phase due to the comparatively high
oscillatory frequency and the presence of multiple hole pockets Naveed . A
candidate material with nodal-line band crossings near Fermi energy enabled by
nonsymmorphic symmetry is still rare. Therefore, materials like $X_{3}$SiTe6
with such intriguing predictions are highly desirable as they provide a
fantastic platform to explore the exotic topological properties via various
theoretical and experimental tools.

Sophisticated techniques like ARPES are frequently used as a direct
experimental probe to reveal the topologically non-trivial band structure in
materials. Whereas, albeit indirectly, by tailoring the temperature at a very
low value and modulating the magnetic field, some transport measurements can
also reveal non-trivial electronic structure close to the Fermi energy. In
recent times, unusual experimental observations, e.g., large and non-
saturating positive magnetoresistance (MR), ultrahigh carrier mobility, and
low carrier effective mass, have been treated as the general characteristics
of topological materials Singha ; Ali ; Shekhar ; Roy . The transport
phenomena not only provide the signature of relativistic excitations but also
evaluate the viability of the material for technological applications.
Negative longitudinal MR (LMR) driven by Adler-Bell-Jackiw (ABJ) chiral
anomaly under parallel electric and magnetic fields, is commonly regarded as
definitive evidence of Dirac/Weyl fermions Huang ; Li ; Li2 ; Pariari ; Wang .
However, because of the large positive MR at transverse electric and magnetic
field configuration, a small misalignment between them can easily mask the
weak negative LMR component Huang . Furthermore, it has been reported that
current jetting Hu  and weak localization effect Ulmet  can also produce
negative LMR. Recently a new phenomenon, called the planar Hall effect (PHE)
has been proposed to be directly related to the ABJ chiral anomaly and
nontrivial Berry curvature Burkov ; Nandy . In PHE, the Hall voltage
terminals, electric and magnetic fields are coplanar, whereas they are
mutually perpendicular to each other in a conventional Hall measurement. As
PHE is entirely distinct from conventional Hall effect, both in experimental
configuration and angle dependence, it is easier to identify and confirm the
topological non-trivial band structure in a material. PHE has already been
observed in topological insulator ZrTe5 Li3 ; Dirac semimetals Cd3As2 Li4  and
VAl3 Singha3 ; Weyl semimetals WTe2 Wang2  and GdPtBi Kumar . Although
conventional metals and semiconductors are not expected to show PHE, a weak
PHE can be observed in ferromagnetic compounds. However, for ferromagnets, it
appears due to anisotropic MR, which originates from different in-plane and
out-of-plane spin scattering Nazmul .

In this paper, we have grown single crystal of proposed topological semimetal
Ta3SiTe6 and measured the magnetotransport properties. Nonsaturating MR has
been observed. The positive and linear field dependence of Hall resistivity
confirms the presence hole type carriers. A prominent PHE has been observed,
which is detectable up to high temperature and reveals the nontrivial nature
of the band structure in this material.

##  II Experimental details

Single crystals of Ta3SiTe6 were grown by standard chemical vapor transport
technique. Iodine was used as a transport medium. The stoichiometric mixture
of high-purity Ta powder (Alfa aesar 99.9%), Si pieces (Alfa aesar 99.9999%)
and Te pieces (Alfa aesar 99.999%) along with a small amount of iodine (Alfa
aesar 99.5%) was sealed in a quartz tube under high vacuum. The quartz tube
was then placed in a two-zone horizontal furnace. The source end of the quartz
tube containing the mixture and iodine was heated to 950∘C, while the other
end was kept at 850∘C. This temperature difference was maintained for 7 days.
The furnace was then cooled very slowly to room temperature. Several shiny,
plate-like crystals formed at the cold end of the tube were mechanically
extracted for transport measurements. Phase purity and the structural analysis
on crashed single crystals of Ta3SiTe6 were done using the powder x-ray
diffraction (XRD) technique with Cu-Kα radiation in a high resolution Rigaku
x-ray diffractometer (TTRAX III). Within the resolution of XRD, we did not see
any peak due to the impurity phase (Fig. 1). From Rietveld profile refinement,
we have extracted the lattice parameters $a$=6.328 Å, $b$=11.412 Å, $c$=14.017
Å with space group symmetry $Pnma$. The elemental composition was checked by
energy dispersive x-ray (EDX) spectroscopy in a field emission scanning
electron microscope [FESEM SUPRA 35 VP, Carl Zeiss (Germany)]. Mapping of the
elemental concentrations at different randomly selected regions of the grown
crystals provides detailed information about the overall chemical composition
and confirms almost perfect stoichiometry [Ta:Si:Te=3:1.1:6.05] of the
Ta3SiTe6 crystals. The relative error in the calculated atomic ratio is about
5 %. The transport measurements were performed using standard four-probe
technique in a 9 T physical property measurement system (Quantum Design) using
the ac-transport option and sample rotator. Freshly cleaved crystals were used
for transport measurements. Electrical contacts were made using gold wire and
conducting silver paste.

Figure 1:  (a) Rietveld profile refinement of the powder XRD pattern for
Ta3SiTe6. Black circles are experimental data (Yobs), red line is the
calculated pattern (Ycal), blue line is the difference between experimental
and calculated intensities (Yobs-Ycal), and magenta vertical lines are the
Bragg positions. Inset shows a typical single crystal with scale bar for
reference. Figure 2:  (a) Temperature dependence of resistivity ($\rho_{xx}$)
both in presence and absence of transverse magnetic field. Inset shows the
low-temperature region, fitted with $\rho_{xx}(T)=a+bT^{2}$. (b) Magnetic
field dependence of transverse magnetoresistance of Ta3SiTe6 at different
temperatures.

##  III Results and Discussions

In Fig. 2(a), the longitudinal resistivity ($\rho_{xx}$) of a representative
Ta3SiTe6 single crystal is plotted as a function of temperature, which shows
metallic behavior down to lowest measured temperature. At room temperature,
$\rho_{xx}$ is 139 $\mu$$\Omega$ cm. With decrease in temperature, $\rho_{xx}$
deceases monotonically and becomes 6.8 $\mu$$\Omega$ cm at 2 K. The residual
resistivity ratio (RRR), $\rho_{xx}$(300 K)/$\rho_{xx}$(2 K), is $\sim$20,
which indicates the good metallicity of the single crystal. As temperature
decreases below 50 K, $\rho_{xx}$($T$) curve exhibits a weak upward curvature.
This indicates a crossover in charge scattering mechanism with decrease in
temperature. As shown in the inset, the zero-field resistivity at low
temperature below 50 K can be fitted well with the equation,
$\rho_{xx}$($T$)=$\rho_{0}$+$aT^{2}$. The quadratic temperature dependence
implies pure electronic correlation dominated scattering in the low-
temperature region Ziman . The coefficient $a$ determines the strength of
electron-electron scattering in the system. The low-temperature fit reveals
$a$$\sim$ 6$\times$10-3 $\mu$$\Omega$ K-2. We note that usually, the zero-
field resistivity in topological semimetals shows quite different temperature
dependence. $\rho_{xx}$ is either very weakly dependent on $T$ or follows the
relation $\rho_{xx}(T)\propto T^{n}$ with $n\geq$3 Singha ; Shekhar ; Tafti ;
Singha4 . Nonetheless, $T^{2}$ dependence of $\rho_{xx}$ has been reported in
few topological semimetals Roy ; Wang3 . Often the low-temperature resistivity
of topological semimetals is strongly influenced by magnetic field. When the
magnetic field is applied perpendicular to the current direction, the
resistivity is observed to increase at low temperature [Fig. 2(a)]. However,
there is no field-induced metal to semiconductor-like crossover as observed in
most topological semimetals Singha ; Ali ; Shekhar ; Roy ; Pariari ; Singha4 .

Figure 3: (upper panel) Angular dependence of transverse magnetoresistance of
Ta3SiTe6 single crystal at different representative temperatures for applied
field 9 T. (lower panel) Schematic shows the experimental set-up. $\theta$ is
the angle between $z$-axis and magnetic field.

Next, we have measured MR under transverse electric and magnetic field
configuration at several fixed temperatures for understanding the exact nature
of field dependence of MR. As shown in Fig. 2(b), the transverse MR is
positive and shows monotonic increment with field without any sign of
saturation. At 2 K and 9 T, a MR of $\sim$32% is obtained. With the increase
in temperature, however, the MR decreases rapidly. As mentioned before, the
ARPES results for Ta3SiTe6 Sato  confirm the presence of nonsymmorphic
symmetry-protected nodal-line crossing near EF. In addition, it is shown that
there is a large trivial Fermi pocket in this material. The contribution from
this trivial pocket may suppress the MR in Ta3SiTe6 as compared to several
other topological semimetals. However, the non-saturating nature of the MR in
this compound still cannot be explained using semiclassical theory for trivial
band structureSingha5 ; McKenzie . Moreover, the observed planar Hall effect
(to be discussed later on), which can only originate from relativistic chiral
anomaly in a non-magnetic system, provides an indirect signature of the non-
trivial band structure in Ta3SiTe6.

In Fig. 3, the angular dependence of transverse MR is shown for different
temperatures at 9 T. In this experimental set up, the current ($I$) direction
has been kept fixed along the $x$-axis within the plane of the rectangular
plate-like crystal, and the magnetic field ($B$) is rotated from out-of-plane
(along $z$-axis) to in-plane direction ($y$-axis), i.e., $B$ is rotated about
$x$-axis [illustrated in the schematic of Fig. 3]. We observe that the MR is
maximum, when the magnetic field direction is along the out-of-plane
direction, i.e., ($\theta\simeq$0) and becomes minimum, when the field is
along in-plane direction ($\theta\simeq$90∘). The polar plot shows a two-fold
symmetric pattern. At 2 K and 9 T, the anisotropy ratio is about 1.5, which is
quite small given the predicted quasi two-dimensional electronic structure of
Ta3SiTe6.

Figure 4:  Hall resistivity for Ta3SiTe6 as a function of magnetic field at
different temperatures.

To determine the type and density of charge carriers and their mobility, we
have performed the Hall resistivity ($\rho_{yx}$) measurement as functions of
temperature and magnetic field. In Fig. 4, $\rho_{yx}$ has been plotted as a
function of magnetic field at some representative temperatures. At all
temperatures, $\rho_{yx}$ is almost linear in field with a positive slope,
which indicates hole dominated conduction. From the semiclassical one-band
model, the calculated hole density ($n_{h}$) at 2 K and 9 T is
$\sim$1.7$\times$1021 cm-3 and mobility ($\mu_{h}$) is $\sim$0.6$\times$103
cm2 V-1 s-1. In a nodal-line semimetal, the band crossing occurs along a line
in momentum space in contrast to discrete points in a Dirac/Weyl semimetal. As
a result, if such a crossing resides at the Fermi energy, the corresponding
Fermi pocket would be quite larger compared to a tiny pocket originated from
the Dirac/Weyl point crossing. Hence, the carrier density of a nodal-line
semimetal ($\sim$1021-1022 cm-3) is usually quite higher than that of a
Dirac/Weyl semimetal ($\sim$1017-1019 cm-3).

The obtained value of carrier density in Ta3SiTe6 is comparable to that
observed in some topological nodal-line semimetals Laha ; Sankar . The high
carrier density might provide an additional support for the possible nodal-
line state in Ta3SiTe6.

Figure 5:  (a) Angle dependence of the planar Hall resistivity
($\rho_{yx}^{PHE}$) for Ta3SiTe6 for different magnetic fields at 2 K. Inset
shows the schematic for experimental set-up. $\theta$ is the angle between
current and magnetic field.(b) Angle dependence of $\rho_{yx}^{PHE}$ at
different temperatures at 9 T.

In Fig. 5(a), we have shown the angle dependence of the planar Hall
resistivity ($\rho_{yx}^{PHE}$) for Ta3SiTe6 at 2 K for different magnetic
field strengths. In the inset, the schematic illustrates the experimental
setup for PHE measurement, where the magnetic field lies on the same plane of
the current and voltage ($V_{yx}$). The magnetic field is rotated within the
plane. Due to the nature of Lorenz force, the conventional Hall resistivity is
zero when the magnetic field and electric field (i.e. current) are coplanar
and is maximum, when magnetic field, current, and Hall voltage are mutually
perpendicular. The experimental configuration of PHE suggests that there
should not be any Lorenz force. However, to exclude any contribution from
conventional Hall voltage due to small misalignment, $\rho_{yx}^{PHE}$
measurements have been performed with both positive and negative field
directions and $\rho_{yx}^{PHE}$ has been calculated from the average. As
demonstrated in Fig. 5(a), $\rho_{yx}^{PHE}$ shows a periodic nature with a
period of $\pi$ and the amplitude increases with the increasing field. It
shows maxima at 135∘ and 315∘, whereas minima at 45∘ and 225∘. The positions
of extrema are consistent with the theoretically predicted PHE Nandy . As
shown in Fig. 5(b), though $\rho_{yx}^{PHE}$ decreases with increasing
temperature, the amplitude shows weak temperature dependence.
$\rho_{yx}^{PHE}$ remains detectable at 100 K and above. This is one of the
remarkable differences between PHE and chiral anomaly induced negative LMR as
it vanishes quickly with increasing temperature Huang ; Li ; Li2 ; Pariari ;
Wang .

Figure 6:  (a) Angle dependence of the planar resistivity
($\rho_{xx}^{planar}$) for Ta3SiTe6 for different magnetic fields at 2 K. (b)
Angle dependence of $\rho_{xx}^{planar}$ at different temperatures at 9 T. The
measurement set-up is shown schematically in the inset.

The planar resistivity ($\rho_{xx}^{planar}$) for different temperatures and
magnetic fields have been shown in Fig. 6(a) and Fig. 6(b). The inset
illustrates the experimental setup. Likewise $\rho_{yx}^{PHE}$,
$\rho_{xx}^{planar}$ also shows a periodicity of $\pi$. However, the positions
of extrema are different. $\rho_{xx}^{planar}$ is maximum at 90∘ and 270∘,
i.e., when magnetic field and current are in transverse configuration. PHE in
topological systems can be mathematically formulated from the semiclassical
Boltzmann theory Burkov ; Nandy  as

$\rho_{yx}^{PHE}=-\Delta\rho_{chiral}\sin\theta\cos\theta,$ (1)
$\rho_{xx}^{planar}=\rho_{\bot}-\Delta\rho_{chiral}\cos^{2}\theta,$ (2)

where $\Delta\rho_{chiral}=\rho_{\bot}-\rho_{\|}$ is the chiral anomaly
induced resistivity component. $\rho_{\bot}$ and $\rho_{\|}$ are the
resistivity for transverse and longitudinal configuration, respectively.
Although, this theoretical model of PHE is formulated for Dirac/Weyl
semimetals, the fundamental arguments remain also valid in nodal-line
semimetal state; hence we have used this model to analyze the PHE in Ta3SiTe6.

Figure 7:  Fittings of (a) $\rho_{yx}^{PHE}$ and (b) $\rho_{xx}^{planar}$ for
Ta3SiTe6 using Eqs. (1) and (2), respectively. (c) The temperature dependence
of the extracted chiral anomaly induced resistivity component
($\Delta\rho_{chiral}$) at 9 T. Inset shows magnetic field dependence of
$\Delta\rho_{chiral}$ at 2 K. (d) The temperature dependence of the extracted
transverse resistivity ($\rho_{\bot}$) at 9 T. Inset shows magnetic field
dependence of $\rho_{\bot}$ at 2 K.

Using the theoretical Eqs. (1) and (2), we have fitted $\rho_{yx}^{PHE}$ and
$\rho_{xx}^{planar}$ in Figs. 7(a) and 7(b), respectively. From the fitting
parameters, the chiral resistivity and transverse resistivity components have
been extracted. In Figs. 7(c) and 7(d), the temperature dependence of
$\Delta\rho_{chiral}$ and $\rho_{\bot}$ have been plotted. At 9 T,
$\Delta\rho_{chiral}$ decreases with increasing temperature, showing a
$\Delta\rho_{chiral}=A_{1}-A_{2}T^{n},(n\sim 1/2)$ type power-law behavior,
where $A_{1}$ and $A_{2}$ are constants, while, $\rho_{\bot}$ increases almost
quadratically with temperature. Such monotonic decrease of
$\Delta\rho_{chiral}$ with the increase in temperature has been observed in
several Dirac/Weyl semimetals Kumar ; Wang2 ; Singha3 ; Li3 ; Li4 . Both
$\Delta\rho_{chiral}$ and $\rho_{\bot}$ have been fitted to a power-law
function of magnetic field ($B$), i.e., $\Delta\rho_{chiral}\propto B^{2}$ and
$\rho_{\bot}\propto B^{1.6}$ [Insets of Figs. 7(b) and 7(d)]. The deduced
values of exponents are very close to those observed in other systems Singha3
; Li4 ; Li5 ; Liang ; Wang2 . However, theoretically it was predicted that
$\Delta\rho_{chiral}$ does not follow a simple linear or quadratic field
dependence rather its dependency can be divided into two different magnetic
field regions Burkov .

$\Delta\rho_{chiral}\propto(\frac{L_{c}}{L_{a}})^{2}\propto B^{2},$ (3)

for $L_{a}\gg L_{c}$; weak magnetic-field region.

$\Delta\rho_{chiral}\propto\frac{1}{\sigma}(1-\frac{2L_{a}}{L_{x}}),$ (4)

for $L_{a}\ll L_{c}$, $L_{a}<L_{x}<\frac{L_{c}^{2}}{L_{a}}$; strong magnetic
field region and

$\Delta\rho_{chiral}\propto\frac{1}{\sigma}(1-\frac{L_{a}^{2}}{L_{c}^{2}}),$
(5)

for $L_{a}\ll L_{c}$, $L_{x}>\frac{L_{c}^{2}}{L_{a}}$; strong magnetic field
region.

where, $\sigma$, $L_{a}$, $L_{c}$, $L_{x}$ are conductivity, magnetic length
($L_{a}\propto B^{-1}$), chiral charge diffusion length, and sample length,
respectively Burkov . $\Delta\rho_{chiral}$ follows a $B^{2}$ dependence in
the low field region while at strong field the amplitude of PHE will follow
either $B^{-1}$ [Eq. (4)] or $B^{-2}$ [Eq. (5)], depending on the sample
length $L_{x}$. As we can see from Fig. 7(b)(inset) that $\Delta\rho_{chiral}$
follows almost $B^{2}$ dependence up to 9 T, which is satisfied in low field
region limit. It is important to note that the quantum oscillation frequency
for Ta3SiTe6 has been reported to be very high, $\sim$1383 T Naveed . For such
a large Fermi pocket, it is expected that one would require really high
magnetic field to reach the quantum limit. Hence, the low field approximation
should be valid in the present case. Our results call for further
investigation in high magnetic field. A few previous reports Kumar ; Liang ;
Li5  on PHE suggest a possible dominance of strong anisotropic orbital MR
(OMR) over the chiral anomaly. It is well known that the transport properties
of materials are closely related to the Fermi surface topology. If the
morphology of Fermi pockets is complicated enough, transport parameters such
as effective mass, mean scattering time, mobility, etc. would also vary along
different directions. The same feature is also noticed in magnetoresistance in
different directions. This effect may be commonly observed in high MR along
with highly anisotropic systems. However, previous reports on Ta3SiTe6 Naveed
as well as our present study show that the transverse MR is not very large and
highly anisotropic (e.g. TMR is $\sim$32% and anisotropy is about 1.5 for
transverse MR, whereas for planar MR it is about 1.06) compared to the
systems, where such OMR dominant PHE was reported Kumar ; Liang ; Li5 .

##  IV Conclusion

In conclusion, we have performed magnetotransport measurements on single
crystals of Ta3SiTe6. A nonsaturating magnetoresistance has been observed. The
linear field dependence of Hall resistivity shows the presence of hole type
charge carrier in the system. In spite of having comparatively high carrier
density, the mobility of charge carriers is moderately high. In our
measurements, prominent planar Hall effect has been observed, which is robust
and persists up to high temperature. Planar Hall originates from the
relativistic Adler-Bell-Jackiw chiral anomaly and nontrivial Berry curvature.
Therefore, these results support the nontrivial nature of the band structure
in the material and validate the presence of Dirac-like dispersions. As the
bulk materials of layered ternary telluride $X_{3}$SiTe6 ($X$=Ta, Nb) have
been predicted to host accidental Dirac-loops and essential fourfold nodal-
lines in the absence of SOC and fourfold degenerate hourglass Dirac-loop in
presence of SOC, a family of isostructural materials with identical electronic
band structures provides a great advantage for tunability of their electronic
states. For example, by substitution of atoms of different atomic numbers
($Z$) from Nb ($Z$=41) to Ta ($Z$=73), one can tune SOC in these materials,
hence their electronic properties. Likewise, by doping a small amount holes or
electrons, one can move the nodal-line or the necks of the hourglass
dispersions nearer to Fermi energy, resulting more exotic transport
properties. So, our investigations not only ratify the presence of topological
nontrivial band structure in Ta3SiTe6, but also offer an excellent platform
for further investigations by chemical substitution/doping.

##  V ACKNOWLEDGMENTS

We thank Dr. Dipten Bhattacharya for his help in EDAX analysis. We also thank
Arun Kumar Paul for his help during sample preparation and measurements.

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