# Transient growth in diabatic boundary layers with fluids at supercritical
pressure

Pietro Carlo Boldini<EMAIL_ADDRESS>Process and Energy
Department, Delft University of Technology, Leeghwaterstraat 39, 2628 CB
Delft, The Netherlands  Benjamin Bugeat  School of Engineering, University of
Leicester, University Road, Leicester, LE1 7RH, United Kingdom  Jurriaan W. R.
Peeters  Process and Energy Department, Delft University of Technology,
Leeghwaterstraat 39, 2628 CB Delft, The Netherlands  Markus Kloker  Institute
of Aerodynamics and Gas Dynamics, University of Stuttgart, Pfaffenwaldring 21,
70569 Stuttgart, Germany  Rene Pecnik<EMAIL_ADDRESS>Process and Energy
Department, Delft University of Technology, Leeghwaterstraat 39, 2628 CB
Delft, The Netherlands

###### Abstract

In the thermodynamic region close to the vapour–liquid critical point and in
the proximity of the Widom-line (pseudo-boiling line), strong property
variations extensively alter the growth of modal instabilities, as previously
revealed in Ren et al. (J. Fluid Mech., vol. 871, 2019, pp. 831–864). Here, we
study non-modal disturbances in the spatial framework using an eigenvector
decomposition of the linearized Navier-Stokes equations for locally parallel
flow. To account for non-ideality, a new energy norm is derived. At a constant
free-stream Mach number of $10^{-3}$ along an isobar, three different regimes
are considered by regulating the diabatic wall temperature: subcritical
(liquid-like), supercritical (gas-like), and transcritical (across the Widom
line). In the non-transcritical regimes, the resulting streamwise-independent
streaks originate from the lift-up effect. Wall cooling is found to enhance
the energy amplification for both subcritical and supercritical regimes. On
the other hand, when the temperature profile is increased beyond the Widom
line, a strong sub-optimal growth is observed over very short streamwise
distances due to the Orr mechanism. Moreover, the optimal energy growth is
found to arise from an interplay between lift-up and Orr mechanism due to the
additional presence of the transcritical Mode II. As a result, optimal
disturbance structures are streamwise-modulated streaks with strong thermal
components and with a propagation angle inversely proportional to the local
Reynolds number. When the same temperature difference is applied over the
Widom line, yet for cooling, the energy amplification is enhanced by
streamwise-independent optimal disturbance structures. Lastly, an $N$-factor
comparison with modal growth reveals that transient growth is scarcely
affected by a wall-temperature variation in the non-transcritical regimes. In
contrast, in the transcritical regime, transient growth is greatly enhanced,
especially when cooling over the Widom line. Here, non-modal $N$-factors are
found to resemble the imposition of an adverse pressure gradient in the ideal-
gas case.

††preprint: APS/123-QED

##  I Introduction

Supercritical fluids have recently gained interest in industrial applications
to enhance the thermodynamic efficiency of energy conversion systems. Their
application has been thoroughly investigated for various heat sources,
including nuclear energy, solar energy, coal power, fuel cell-power cycles,
and industrial waste heat [1]. At supercritical pressures, close to the
thermodynamic critical point, an ideal-gas law is inadequate to model the
thermodynamic state. Therefore, higher-order non-ideal equation of states are
necessary. Several studies have been previously conducted to explore how these
non-ideal gas effects impact turbulence [2, 3], heat transfer [4, 5], and more
recently, modal instability mechanisms [6, 7, 8, 9, 10, 11, 12]. Amongst these
studies, Ren et al. [8] conducted linear stability analyses of adiabatic flat-
plate boundary layers with a highly non-ideal fluid, namely carbon dioxide at
supercritical pressure. The flow undergoes stabilization for both the
subcritical (liquid-like) and supercritical (gas-like) temperature regimes,
specifically when the flow temperature approaches the Widom or pseudo-boiling
line. The latter is defined as $\max\\{c_{p}(T)\\}$, representing the maximum
specific heat at constant pressure [13]. Crossing the Widom line from a
liquid-like free stream, defined hereafter as transcritical heating
temperature regime, induces sharp gradients in thermo-physical-properties,
leading to the presence of two modal instabilities. The new unstable mode
(Mode II) is inviscid according to the generalized-inflection-point (GIP)
criterion and exhibits growth rates an order of magnitude larger than the
other conventional mode (Mode I), linked to Tollmien-Schlichting waves.
Recently, in Ref. [10], the presence of Mode II was associated with the
appearance of a minimum of the kinematic-viscosity base-flow profile at the
Widom line. Since all non-polar fluids at supercritical pressure and
transcritical temperature exhibit such a minimum, their corresponding boundary
layers are inviscidly unstable with a GIP in the proximity of the Widom line.
A physical mechanism of this instability was proposed by Ref. [11], based on
the interaction of shear and baroclinic waves.
In the case of compressible boundary-layer flows under the ideal-gas
assumption, it is well-known that various transition scenarios involve
instabilities preceding or bypassing the aforementioned exponential modal
growth [14]. Non-modal growth, also known as transient growth, has emerged as
a potential mechanism for explaining transition scenarios over a wide range of
parameter values, such as in the blunt-body paradox [15, 16]. Hanifi et al.
[17] conducted the first transient growth analysis in compressible boundary
layers in a temporal formulation. Similar to incompressible flows [18],
initial optimal perturbations took the form of local streamwise vorticity
disks, which evolve into linearly growing streamwise streaks driven by the
lift-up effect [19, 20]. Simultaneously, for compressible flows, density and
temperature fluctuations also increase linearly with time [21]. Analogous
conclusions were drawn for a spatial framework [22]. Optimal output
perturbations, in the form of streamwise velocity streaks, were found in
supersonic and hypersonic flows under the ideal-gas assumption [23, 24]. Here,
wall cooling below the adiabatic temperature reduces the level of non-modal
energy amplification. Conversely, at low Mach numbers (e.g., $M_{\infty}=0.5$)
and with highly cooled walls (e.g., $T_{w}/T_{\infty}=0.3$, where $T_{w}$ is
the wall temperature), no modal instabilities are found, and thus, transition
to turbulence is likely to be driven only by transient growth [23].
Another example, where large thermodynamic and transport property gradients
play a major role in flow instability, is in stratified flows [25]. The impact
of viscosity stratification on transient growth has been studied both for both
incompressible and compressible Couette [26, 27] and channel [28, 29, 30]
flows. In the latter, it was discovered that any viscosity stratification does
not significantly affect transient growth, whereas an increase in Prandtl
number can lead to a total-energy amplification rise by over an order of
magnitude. Recently, Jose et al. [31] studied the role of buoyancy on
stratified viscosity profiles in water and air. When the bottom layer is
denser than the top one, transient-growth stabilization occurs, and vice
versa. The optimal perturbation energy is found to be constantly localized in
low-viscosity regions, leading to the formation of strong localized streaks.
As the Richardson number is increased, these optimal perturbations are no
longer streamwise independent. A similar behavior was found in stably
stratified boundary layers [32] under the Boussinesq assumption, where the
optimal energy growth is a combination of the lift-up and the Orr mechanism.
Furthermore, as buoyancy effects become larger than shear production, the
optimal energy gain and optimal time decrease, while the optimal streamwise
wavenumber increases, confirming the existence of oblique optimal
perturbations.
The first study on non-modal growth with supercritical carbon dioxide at
$p=$80\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$$ was performed in a Poiseuille
flow [7]. Regardless of the temperature flow regime (subcritical,
supercritical, or transcritical), non-ideality always promoted transient
growth at a constant $T_{w}/T_{center}$, where $T_{center}$ is the centerline
temperature. The same behavior was discovered when the level of viscous
heating, controlled by the product of the Prandtl $Pr_{\infty}$ and Eckert
$Ec_{\infty}$ number, was increased in contrast to the ideal-gas assumption.
Optimal initial perturbations were found to be streamwise-independent
vortices, while optimal output perturbations were represented by velocity
streaks. In the transcritical temperature case, either when
$Pr_{\infty}Ec_{\infty}$ is sufficiently large or when the Widom line is
crossed, the most non-modally unstable condition appear, leading to the
additional formation of large thermal streaks in the dense near-wall region
close to the Widom line.
The main focus of this work is to investigate the non-modal growth of two-
dimensional zero-pressure-gradient (ZPG) flat-plate boundary layers with
fluids close to the critical point. Specifically, we aim to elucidate the role
of Mode II in transient growth, as first discussed by Ref. [33]. In order to
include non-ideal gas effects in the non-modal stability analysis, a new
energy norm has been derived differently from Ref. [7]. The work is organized
as follows: in Sec. II, the governing equations for base-flow calculation and
modal/non-modal stability analysis are unveiled. The derivation of the new
energy norm is then undertaken. Subsequently (Sec. III), base-flow properties
of eight flow cases at a constant supercritical pressure of
$80\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$ for carbon dioxide
($p^{*}/p^{*}_{c}=1.084$, with $p^{*}_{c}$ being the critical pressure) are
introduced. Given a diabatic wall, different temperature profiles are
considered, namely below (subcritical), above (supercritical), and crossing
(transcritical) the Widom line. Non-modal stability analysis results are then
reported in Sec. IV, with a special focus on the transcritical wall-heating
and -cooling cases. In conclusion, transient amplification and exponential
growth of unstable discrete eigenvalues are analyzed by means of a $N$-factor
comparison.

##  II Mathematical formulation

###  II.1 Flow-conservation equations

We consider a single-phase flow of a supercritical fluid governed by the fully
compressible Navier-Stokes equations (conservation of mass, momentum, and
energy) in differential and dimensionless form as

$\dfrac{D\rho}{Dt}+\rho\dfrac{\partial u_{j}}{\partial x_{j}}=0,$ (1a)
$\rho\dfrac{Du_{i}}{Dt}=-\dfrac{\partial p}{\partial
x_{i}}+\dfrac{1}{Re_{L}}\dfrac{\partial\tau_{ij}}{\partial x_{j}},$ (1b)
$\rho\dfrac{De}{Dt}=-Ec_{\infty}p\dfrac{\partial u_{j}}{\partial
x_{j}}+\dfrac{Ec_{\infty}}{Re_{L}}\dfrac{\partial\left(u_{i}\tau_{ij}\right)}{\partial
x_{j}}-\dfrac{1}{Re_{L}Pr_{\infty}}\dfrac{\partial q_{j}}{\partial x_{j}},$
(1c)

where $x_{i}=(x,y,z)$ are the Cartesian coordinates in the streamwise, wall-
normal, and spanwise directions, $t$ is the time, $\rho$ is the fluid density,
$u_{i}=(u,v,w)$ are the velocity components, $p$ is the pressure, and
$E=e+u_{i}u_{j}/2$ is the specific total energy with $e$ as the specific
internal energy. Under the assumption that the fluid is Newtonian, the viscous
stress tensor $\tau_{ij}$ is given as

$\displaystyle\tau_{ij}=\lambda\delta_{ij}\dfrac{\partial u_{k}}{\partial
x_{k}}+\mu\left(\dfrac{\partial u_{i}}{\partial x_{j}}+\dfrac{\partial
u_{j}}{\partial x_{i}}\right),\quad\lambda=\mu_{b}-\dfrac{2}{3}\mu,$ (2a,b)

where $\mu$ is the dynamic viscosity, $\lambda=\mu_{b}-2/3\mu$ is the second
viscosity coefficient, and $\delta_{ij}$ is the Kronecker delta. The bulk
viscosity $\mu_{b}$ has proven to have a very limited influence on the
stability of Poiseuille flows at supercritical pressure (see Ref. [7]), and
therefore it is set to $0$ in agreement with Ref. [8]. An additional
assumption is that buoyancy effects are not considered here. The convective
heat flux vector $q_{j}$ is modeled according to Fourier’s law as
$q_{j}=-\kappa\ \partial T/\partial x_{j}$, where $\kappa$ is the thermal
conductivity, and $T$ the fluid temperature. The above flow conservation
equations have been non-dimensionalized by the following reference values

$\displaystyle t=\frac{t^{*}U^{*}_{\infty}}{L^{*}},\quad
x_{i}=\frac{x^{*}_{i}}{L^{*}},\quad
u_{i}=\frac{u^{*}_{i}}{U^{*}_{\infty}},\quad\rho=\frac{\rho^{*}}{\rho^{*}_{\infty}},$
$\displaystyle p=\frac{p^{*}}{\rho^{*}_{\infty}{U^{*^{2}}_{\infty}}},\quad
T=\frac{T^{*}}{T^{*}_{\infty}},\quad
e=\frac{e^{*}}{c^{*}_{p,\infty}T^{*}_{\infty}},\quad\mu=\frac{\mu^{*}}{\mu^{*}_{\infty}},$
(3a-k)
$\displaystyle\mu_{b}=\frac{\mu^{*}_{b}}{\mu^{*}_{\infty}},\quad\kappa=\frac{\kappa^{*}}{\kappa^{*}_{\infty}},\quad\nu=\frac{\nu^{*}}{\nu^{*}_{\infty}},$

where $(\cdot)^{*}$ denotes dimensional quantities, and $(\cdot)_{\infty}$
corresponds to free-stream flow conditions. The corresponding non-dimensional
characteristic parameters are defined as

$\displaystyle
Re_{L}=\frac{\rho^{*}_{\infty}U^{*}_{\infty}L^{*}}{\mu^{*}_{\infty}},\quad
Ec_{\infty}=\frac{{U^{*2}_{\infty}}}{c^{*}_{p,\infty}T^{*}_{\infty}},\quad
Pr_{\infty}=\frac{c^{*}_{p,\infty}\mu^{*}_{\infty}}{\kappa^{*}_{\infty}},$
(4a–c)

where $c^{*}_{p,\infty}$ is specific heat at constant pressure, and $Re_{L}$
is the Reynolds number based on a chosen length scale $L^{*}$. We opt for the
local Blasius length scale $\delta^{*}$ such that

$\displaystyle
Re_{L}=Re_{\delta}=\frac{\rho^{*}_{\infty}U^{*}_{\infty}\delta^{*}}{\mu^{*}_{\infty}}=\sqrt{Re_{x}},\quad\delta^{*}=\sqrt{\dfrac{\mu^{*}_{\infty}x^{*}}{\rho^{*}_{\infty}U^{*}_{\infty}}}.$
(5a,b)

In Eq. (4a–c), $Ec_{\infty}$ is the Eckert number, and $Pr_{\infty}$ is the
Prandtl number (all based on free-stream conditions). The Mach number
$M_{\infty}=U^{*}_{\infty}/a^{*}_{\infty}$, with $a^{*}_{\infty}$ as the speed
of sound, can be obtained from $Ec_{\infty}$. In order to close the
conservation equations in Eqs. (1)(a–c), an equation of state (EoS) and
transport properties need to be defined. We choose
$80\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$ as the reference pressure of
supercritical $\text{CO}_{2}$ based on previous studies (see, for instance,
Ref. [8]). Its properties are summarized in Table 5. Both thermodynamic and
transport properties $\chi$ are obtained from the NIST REFPROP library (see
Ref. [34]), which is transformed into two-dimensional (2-D) look-up tables as
functions of $p$ and $T$. These properties are needed both for the laminar
base-flow calculation and the stability analysis.

###  II.2 Base-flow calculation

The calculation of the base-flow profiles is based on the compressible
boundary-layer equations for self-similar flow [8], combined with the
tabulated NIST REFPROP library and under the assumption of zero-pressure
gradient. Subsequently, a coordinate transformation based on the Lees-
Dorodnitsyn variables [35] is applied. Moreover, derivatives of the transport
and thermodynamic properties are numerically evaluated using a second-order
finite differences method as a function of $p$ and $T$ [7]. In this context,
isothermal boundary conditions (BC) are considered as

$\left.\begin{array}[]{cc}f(0)=0,\quad f_{\eta}(0)=0,\quad
f_{\eta}(\infty)=1,\\\ g(0)=g_{w},\quad g(\infty)=1,\end{array}\right\\}$ (6)

where $f_{\eta}$ and $g$ are the dimensionless streamwise velocity and static
specific enthalpy, respectively. The term $g_{w}(p,T_{w})$ denotes the
prescribed enthalpy at the wall. The system of boundary-layer equations is
solved numerically with a 4th order Runge-Kutta scheme, together with the
Newton-Raphson method to iteratively satisfy the BCs at the wall. Grid-
independent results are achieved with a wall-normal resolution of at least
10000 points and a domain size approximately equal to 10 times the boundary-
layer thickness.

###  II.3 Stability calculation

To conduct the stability analysis, the flow field
$\mathbf{q}=[p,u,v,w,T]^{\mathrm{T}}$ is initially decomposed into a steady
laminar part $\bar{\mathbf{q}}$, obtained from the boundary-layer equations,
and a fluctuating component $\mathbf{q}^{\prime}$, infinitesimally small
compared to $\bar{\mathbf{q}}$. This decomposition results in

$\mathbf{q}(x,y,z,t)=\bar{\mathbf{q}}(y)+\epsilon\mathbf{q}^{\prime}(x,y,z,t),$
(7)

where $\epsilon\ll 1$. The base flow is assumed to be 2-D and locally parallel
in the streamwise direction. The fluctuating thermodynamic and transport
properties (e.g., $\rho^{\prime},\mu^{\prime},\kappa^{\prime}$) are determined
as functions of the two independent thermodynamic properties $p$ and $T$. For
instance, the viscosity perturbation $\mu^{\prime}$ is expressed as a first-
order Taylor series in terms of the base-flow properties:

$\displaystyle\mu^{\prime}=$
$\displaystyle\left.\dfrac{\partial\bar{\mu}}{\partial\bar{p}}\right|_{\bar{T}}p^{\prime}+\left.\dfrac{\partial\bar{\mu}}{\partial\bar{T}}\right|_{\bar{p}}T^{\prime}.$
(8)

Within the framework of linear stability theory (LST), the perturbation
$\mathbf{q}^{\prime}$ is assumed to depend solely on the wall-normal
direction, and be periodic in all other directions. Therefore, using the
classical Fourier ansatz, $\mathbf{q}^{\prime}$ is written as

$\mathbf{q}^{\prime}(x,y,z,t)=\hat{\mathbf{q}}(y)\exp[\mathrm{i}(\alpha
x+\beta z-\omega t)]+\text{c.c.},$ (9)

where $\hat{\mathbf{q}}(y)$ is the perturbation eigenfunction, $\alpha$ is the
non-dimensional streamwise wavenumber, $\beta\equiv\beta_{r}$ is the real
spanwise wavenumber, $\omega\equiv\omega_{r}$ is the real angular frequency,
and c.c. stands for the complex conjugate. In this work, the spatial problem
is considered by prescribing $\beta_{r}$ and $\omega_{r}$. The streamwise
wavenumber in Eq. (9) is set to be complex
($\alpha=\alpha_{r}+\mathrm{i}\alpha_{i}$), where $\alpha_{i}$ represents the
local spatial growth rate. Hence, modal amplification occurs for
$\alpha_{i}<0$.

####  II.3.1 Modal analysis

According to LST, Eq. (7) is substituted into Eq. (1), and non-linear terms of
order $O(\epsilon^{2})$ are neglected. Subsequently, after subtracting the
steady base flow, the linearized stability equations can be expressed in a
compact matrix form as follows:

$\displaystyle\mathbf{L_{t}}\frac{\partial\mathbf{q}^{\prime}}{\partial
t}+\mathbf{L_{x}}\frac{\partial\mathbf{q}^{\prime}}{\partial
x}+\mathbf{L_{y}}\frac{\partial\mathbf{q}^{\prime}}{\partial
y}+\mathbf{L_{z}}\frac{\partial\mathbf{q}^{\prime}}{\partial
z}+\mathbf{L_{q^{\prime}}}\mathbf{q}^{\prime}$
$\displaystyle+\mathbf{V_{xx}}\frac{\partial^{2}\mathbf{q}^{\prime}}{\partial
x^{2}}+\mathbf{V_{yy}}\frac{\partial^{2}\mathbf{q}^{\prime}}{\partial
y^{2}}+\mathbf{V_{zz}}\frac{\partial^{2}\mathbf{q}^{\prime}}{\partial
z^{2}}+\mathbf{V_{xy}}\frac{\partial^{2}\mathbf{q}^{\prime}}{\partial
x\partial y}+\mathbf{V_{xz}}\frac{\partial^{2}\mathbf{q}^{\prime}}{\partial
x\partial z}+\mathbf{V_{yz}}\frac{\partial^{2}\mathbf{q}^{\prime}}{\partial
y\partial z}=0.$ (10)

Here,
$\mathbf{q}^{\prime}=\left[p^{\prime},u^{\prime},v^{\prime},w^{\prime},T^{\prime}\right]^{\mathrm{T}}$
is the perturbation vector, and matrices $\mathbf{L_{t}}$, $\mathbf{L_{x}}$,
$\mathbf{L_{y}}$, $\mathbf{L_{z}}$, $\mathbf{L_{q^{\prime}}}$,
$\mathbf{V_{xx}}$, $\mathbf{V_{yy}}$, $\mathbf{V_{zz}}$, $\mathbf{V_{xy}}$,
$\mathbf{V_{xz}}$ and $\mathbf{V_{yz}}$ are functions of the base-flow
properties and dimensionless parameters in Eq. (4a–c). The expressions for
these matrices are documented in Ref. [7] for
$\bar{\chi}=f(\bar{\rho},\bar{T})$. However, in this study, $\bar{\chi}$ is a
function of $\bar{p}$ and $\bar{T}$. This modification results in new base-
flow matrices, detailed in Appendix B. The use of pressure, instead of
density, is motivated by the chosen Mach number in this study. Given
$M_{\infty}=10^{-3}$ (see Sec. III), density and temperature are directly
coupled, while pressure is independent. Consequently, two independent
variables must be chosen.
A pseudo-spectral collocation method based on $N$-Chebyshev collocation points
is employed with near-wall grid clustering in agreement with Ref. [17]. The
boundary conditions for $\hat{\mathbf{q}}$ are given as:

$\left.\begin{array}[]{cc}\hat{u}=\hat{v}=\hat{w}=\hat{T}=0,\quad\text{at}\ \
y=0\\\ \hat{u}=\hat{v}=\hat{w}=\hat{T}=0,\quad\text{at}\ \
y=y_{max}.\end{array}\right\\}$ (11)

Additionally, the pressure $\hat{p}$ is not prescribed at the wall. Finally,
the (non-linear) system in Eq. (10) is written as

$\mathbb{A}_{0}\hat{\mathbf{Q}}=\alpha\mathbb{A}_{\alpha}\hat{\mathbf{Q}}+\alpha^{2}\mathbb{A}_{\alpha^{2}}\hat{\mathbf{Q}},$
(12)

where $\mathbb{A}_{0}$, $\mathbb{A}_{\alpha}$, $\mathbb{A}_{\alpha^{2}}$ are
$5N\times 5N$ and
$\hat{\mathbf{Q}}=(\hat{\mathbf{q}}_{1},\ldots,\hat{\mathbf{q}}_{N})^{\mathrm{T}}$
is a $5N$-column vector containing all discretized perturbations. The
eigenvalue problem is then transformed into a (larger) linear system, which is
solved using the LAPACK implementation of the QZ algorithm.

####  II.3.2 Energy norm

Before addressing transient growth analysis, it is necessary to define an
inner product and an associated energy norm. In Ref. [7], the energy norm for
non-ideal gas flows was chosen to ensure convergence with respect to energy
amplification. Nevertheless, pressure-related energy terms persisted. Their
contribution to the total disturbance energy must vanish, as compression work
is conservative [17]. Thus, following Mack’s norm for ideal gas in Ref. [17],
a rigid definition of the norm for non-ideal gas flows must be formulated.
This norm will allow us to consider the characteristic non-orthogonality of
the eigenfunctions and to quantify the magnitude of transient-growth energy
amplification.
First, a scalar product based on the energy density, as proposed by Ref. [22]
is defined as

$(\mathbf{\hat{q}}_{k},\mathbf{\hat{q}}_{l})_{E}=\int_{0}^{\infty}\mathbf{\hat{q}}^{H}_{k}\mathbf{M}\mathbf{\hat{q}}_{l}\
dy,$ (13)

where the superscript $H$ denotes the complex conjugate transpose, and the
matrix $\mathbf{M}$ is the energy matrix. The associated norm is expressed as:

$2E=\|\mathbf{\hat{q}}\|^{2}_{E}=(\mathbf{\hat{q}},\mathbf{\hat{q}})_{E}.$
(14)

For compressible flows under the ideal-gas assumption [36, 37, 17], Eq. (14)
results in

$2E=\int_{\Omega}m_{u_{i}}u^{\prime}_{i}u^{\prime\dagger}_{i}+m_{\rho}\rho^{\prime}\rho^{\prime\dagger}+m_{T}T^{\prime}T^{\prime\dagger}d\Omega,$
(15)

where $(\cdot)^{\dagger}$ and $\Omega$ denote the complex conjugate and the
spatial domain, respectively. The coefficients $m_{u_{1,2,3}}$, $m_{\rho}$,
and $m_{T}$ of the symmetric positive-definite matrix $\mathbf{M}$ were
derived by Ref. [17] and take the form of

$\displaystyle m_{\rho}=\dfrac{R_{g}\bar{T}}{\bar{\rho}},\quad
m_{u}=\bar{\rho},\quad m_{v}=\bar{\rho},\quad m_{w}=\bar{\rho},\quad
m_{T}=\dfrac{\bar{\rho}c_{v}}{Ec_{\infty}\bar{T}},$ (16a–e)

with $R_{g}$ and $c_{v}$ being the gas constant and the specific heat at
constant volume, respectively. For non-ideal gas conditions in Ref. [7],
however, the coefficients $m_{\rho}$ and $m_{T}$ were assumed to be unity,
revealing a robust convergence behavior for energy amplification. Here,
instead, we follow the same procedure as outlined in Ref. [17].
As will be shown below, the coefficients in Eq. (15) are modified so that the
pressure-related term, appearing in the form
$\nabla\cdot\left(p^{\prime}\mathbf{u}^{\prime}\right)$, can be eliminated
after spatial integration. By multiplying each disturbance equation in Eq.
(10) with $m_{\rho}\rho^{\prime}$, $m_{u_{i}}u^{\prime}_{i}$,
$m_{T}T^{\prime}$, respectively, and combining the equations, the following
result is obtained

$\displaystyle\dfrac{\partial E}{\partial
t}=\dfrac{1}{2}\dfrac{\partial}{\partial t}\left(m_{\rho}\rho^{\prime
2}+m_{u_{i}}u^{\prime
2}_{i}+m_{T}\dfrac{\partial\bar{e}}{\partial\bar{T}}_{\bar{\rho}}T^{\prime
2}+2m_{T}\dfrac{\partial\bar{e}}{\partial\bar{\rho}}_{\bar{T}}\rho^{\prime}T^{\prime}\right)$
$\displaystyle=-m_{\rho}\bar{\rho}\rho^{\prime}\frac{\partial
u^{\prime}_{i}}{\partial
x_{i}}-\dfrac{m_{u_{i}}}{\bar{\rho}}u^{\prime}_{i}\frac{\partial
p^{\prime}}{\partial
x_{i}}-m_{T}\dfrac{Ec_{\infty}\bar{p}}{\bar{\rho}}T^{\prime}\frac{\partial
u^{\prime}_{i}}{\partial x_{i}}$ (17) $\displaystyle-
m_{T}\dfrac{\partial\bar{e}}{\partial\bar{\rho}}_{\bar{T}}\rho^{\prime}\frac{\partial
T^{\prime}}{\partial t}+\text{visc.},$

where visc. represents the viscous terms. When integrating over the spatial
domain, the pressure-work term in the RHS of Eq. (II.3.2) disappears, as both
of the following properties are satisfied [23, 38]: (1) the disturbances are
periodic in space, and (2) the wall-normal velocity is zero. Consequently,
using integration by parts yields:

$\displaystyle\int_{\Omega}\nabla\cdot\left(p^{\prime}\mathbf{u}^{\prime}\right)d\Omega=\int_{\Omega}m_{\rho}\bar{\rho}\rho^{\prime}\frac{\partial
u^{\prime}_{i}}{\partial
x_{i}}d\Omega-\int_{\Omega}m_{u_{i}}\dfrac{1}{\bar{\rho}}p^{\prime}\frac{\partial
u^{\prime}_{i}}{\partial
x_{i}}d\Omega+\int_{\Omega}m_{T}\dfrac{Ec_{\infty}\bar{p}}{\bar{\rho}}T^{\prime}\frac{\partial
u^{\prime}_{i}}{\partial x_{i}}d\Omega.$ (18)

The LHS of Eq. (II.3.2) represents the time rate of change of the disturbance
energy as:

$\dfrac{\partial E}{\partial t}=\dfrac{1}{2}\dfrac{\partial}{\partial
t}\int_{\Omega}\left(m_{\rho}\rho^{\prime 2}+m_{\mathbf{u}}u^{\prime
2}_{i}+m_{T}\dfrac{\partial\bar{e}}{\partial\bar{T}}_{\bar{\rho}}T^{\prime
2}+2m_{T}\dfrac{\partial\bar{e}}{\partial\bar{\rho}}_{\bar{T}}\rho^{\prime}T^{\prime}\right)d\Omega.$
(19)

By choosing $m_{\mathbf{u}}=\bar{\rho}$ similar to the ideal-gas norm [17],
and

$\displaystyle
m_{\rho}=\dfrac{1}{\bar{\rho}}\dfrac{\partial\bar{p}}{\partial\bar{\rho}}_{\bar{T}},\quad
m_{T}=\dfrac{\bar{\rho}}{Ec_{\infty}\bar{p}}\dfrac{\partial\bar{p}}{\partial\bar{T}}_{\bar{\rho}},$
(20a,b)

the pressure-work term is eliminated. Thus, the resulting energy norm for non-
ideal gas is written as

$\displaystyle 2E=$
$\displaystyle\int_{\Omega}\dfrac{1}{\bar{\rho}}\dfrac{\partial\bar{p}}{\partial\bar{\rho}}_{\bar{T}}\rho^{\prime
2}+\bar{\rho}u^{\prime
2}_{i}+\dfrac{\bar{\rho}}{Ec_{\infty}\bar{p}}\dfrac{\partial\bar{p}}{\partial\bar{T}}_{\bar{\rho}}\dfrac{\partial\bar{e}}{\partial\bar{T}}_{\bar{\rho}}T^{\prime
2}+2\dfrac{\bar{\rho}}{Ec_{\infty}\bar{p}}\dfrac{\partial\bar{p}}{\partial\bar{T}}_{\bar{\rho}}\dfrac{\partial\bar{e}}{\partial\bar{\rho}}_{\bar{T}}\rho^{\prime}T^{\prime}d\Omega.$
(21)

The kinetic energy of a perturbation is therefore

$\displaystyle 2E_{kin.}=$ $\displaystyle\int_{\Omega}\bar{\rho}u^{\prime
2}_{i}d\Omega,$ (22)

whereas its internal energy is given as

$\displaystyle
2E_{int.}=2\left(E_{\rho^{\prime}}+E_{T^{\prime}}+E_{\rho^{\prime}T^{\prime}}\right)=$
$\displaystyle\int_{\Omega}\dfrac{1}{\bar{\rho}}\dfrac{\partial\bar{p}}{\partial\bar{\rho}}_{\bar{T}}\rho^{\prime
2}+\dfrac{\bar{\rho}}{Ec_{\infty}\bar{p}}\dfrac{\partial\bar{p}}{\partial\bar{T}}_{\bar{\rho}}\dfrac{\partial\bar{e}}{\partial\bar{T}}_{\bar{\rho}}T^{\prime
2}$
$\displaystyle+2\dfrac{\bar{\rho}}{Ec_{\infty}\bar{p}}\dfrac{\partial\bar{p}}{\partial\bar{T}}_{\bar{\rho}}\dfrac{\partial\bar{e}}{\partial\bar{\rho}}_{\bar{T}}\rho^{\prime}T^{\prime}d\Omega.$
(23)

The inner product and the associated norm can then be expressed in symbolic
form as in Eqs. (13) and (14), respectively. The new energy matrix
$\mathbf{M}$ for non-ideal gas is

$\mathbf{M}=\begin{bmatrix}\dfrac{1}{\bar{\rho}}\dfrac{\partial\bar{p}}{\partial\bar{\rho}}_{\bar{T}}&0&0&0&\dfrac{\bar{\rho}}{Ec_{\infty}\bar{p}}\dfrac{\partial\bar{p}}{\partial\bar{T}}_{\bar{\rho}}\dfrac{\partial\bar{e}}{\partial\bar{\rho}}_{\bar{T}}\\\
0&\bar{\rho}&0&0&0\\\ 0&0&\bar{\rho}&0&0\\\ 0&0&0&\bar{\rho}&0\\\
\dfrac{\bar{\rho}}{Ec_{\infty}\bar{p}}\dfrac{\partial\bar{p}}{\partial\bar{T}}_{\bar{\rho}}\dfrac{\partial\bar{e}}{\partial\bar{\rho}}_{\bar{T}}&0&0&0&\dfrac{\bar{\rho}}{Ec_{\infty}\bar{p}}\dfrac{\partial\bar{p}}{\partial\bar{T}}_{\bar{\rho}}\dfrac{\partial\bar{e}}{\partial\bar{T}}_{\bar{\rho}}\\\
\end{bmatrix}.$ (24)

The matrix $\mathbf{M}$ is positive definite, meeting the requirement of the
energy-norm definition [36]. For an ideal gas, Eq. (24) simplifies to

$\mathbf{M}=\operatorname{diag}\left(\dfrac{R_{g}\bar{T}}{\bar{\rho}},\bar{\rho},\bar{\rho},\bar{\rho},\dfrac{\bar{\rho}\,c_{v}}{Ec_{\infty}\bar{T}}\right),$
(25)

recovering Chu’s [36] and Mack’s norm [17].

####  II.3.3 Non-modal analysis

Once the eigenmode spectrum is obtained, the spatial optimization method of
Ref. [22] is applied. The disturbance vector $\mathbf{q}^{\prime}$ is
projected onto the truncated eigenvector space spanned by the first $K$
spatial eigenvalues $\alpha_{k}$ and eigenfunctions $\hat{\mathbf{q}}_{k}$.
Consequently, the optimal transient growth, or optimal energy amplification,
at the specific downstream coordinate is then calculated as

$G=G(\omega,\beta,Re_{\delta},x)=\max_{\hat{\mathbf{q}}(0)}\dfrac{E(\hat{\mathbf{q}}(x))}{E(\hat{\mathbf{q}}(0))}=\max_{\hat{\mathbf{q}}(0)}\dfrac{\|\hat{\mathbf{q}}(x)\|^{2}_{E}}{\|\hat{\mathbf{q}}(0)\|^{2}_{E}},$
(26)

where $E(\hat{\mathbf{q}}(x))$ is the total perturbation energy defined in Eq.
(21), and $E(\hat{\mathbf{q}}(0))$ is its value at the initial location $x=0$.
The optimal transient growth can be expressed in terms of the
$L_{2}$-(Euclidean)-norm as

$G=\|\mathbf{F}\bm{\Lambda}\mathbf{F}^{-1}\|^{2}_{2},$ (27)

with

$\mathbf{\Lambda}=\operatorname{diag}(\exp(\mathrm{i}\alpha_{1}x),...,\exp(\mathrm{i}\alpha_{K}x)),\quad\mathbf{A}=\mathbf{F}^{H}\mathbf{F},\quad\mathbf{A}_{kl}=(\hat{\mathbf{q}}_{k},\hat{\mathbf{q}}_{l})=\int_{0}^{\infty}\hat{\mathbf{q}}^{H}_{k}\mathbf{M}\hat{\mathbf{q}}_{l}\
dy,$ (28)

and $\mathbf{M}$ as in Eq. (24). The $L_{2}$-norm of matrix
$\mathbf{F}\bm{\Lambda}\mathbf{F}^{-1}$ can be computed by means of singular
value decomposition (SVD). The value of $G$ is obtained by the square of the
largest singular value, whereas its optimal conditions, i.e., optimal
perturbations, can be computed via the right singular eigenvector [39].
Following the notation in Ref. [17], the maximum value of $G$ over all $x$ is
denoted as $G_{max}$, and its maximum over all $\beta$ and $\omega$, is
referred to as $G_{opt}$, with the optimal spanwise wavenumber $\beta_{opt}$,
optimal frequency $\omega_{opt}$, and optimal location $x_{opt}$,
respectively. For the purpose of code validation, transient-growth results in
Ref. [22] are reproduced and reported in Appendix C. Additionally, for the
non-ideal gas regime, a grid-independence study is performed for a highly non-
ideal reference case.

##  III Flow cases

Before conducting the non-modal stability analysis, base-flow profiles must be
known first. A constant pressure of
$p^{*}=$80\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$$, which is for
$\text{CO}_{2}$ $1.084$ times larger than the critical pressure $p^{*}_{c}$,
is selected. Note that the closer the ratio $p^{*}/p^{*}_{c}$ to unity, the
stronger the thermodynamic non-ideality. To better assess the thermodynamic
states considered in this study, a $T_{r}$–$p_{r}$ diagram is shown in Fig. 1
together with isolines of reduced specific volume
($\upsilon^{*}_{r}=1/\rho^{*}_{r}=\upsilon^{*}/\upsilon^{*}_{c}$). Along the
isobar (thick black line), two different configurations are considered, with
four having a subcritical temperature with respect to free-stream temperature
of $T^{*}_{\infty}/T^{*}_{pc}=0.90$ (marked by a black circle ($\CIRCLE$) in
Fig. 1), and four with a supercritical free-stream temperature of
$T^{*}_{\infty}/T^{*}_{pc}=1.10$ (marked by a black square ($\blacksquare$) in
Fig. 1), where $T^{*}_{pc}$ is the temperature at the Widom line
($T^{*}_{pc}/T^{*}_{c}=1.012$ at
$p^{*}=$80\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$$). Note that, while the
pressure is always supercritical, cases that are subcritical in temperature
are called subcritical, and the same applies for supercritical. By varying the
wall temperature per case, one can control the non-ideal gas effects, which
are maximal in the proximity of the Widom line. Therefore, for both free-
stream configurations, one case is considered in which the temperature within
the boundary layer crosses the Widom line (hereafter defined as
transcritical). All cases are listed in Table 1 and represented in Fig. 1.

Figure 1:  Reduced temperature-pressure $(T_{r}$–$p_{r})$ diagram with isolines of reduced specific volume ($\upsilon_{r}$): isobar at $p^{*}=$80\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$$ for $\text{CO}_{2}$ with the corresponding cases (lines) of Table 1. Note that the lines are offset for a better representation.  Table 1:  Flow parameters at $p^{*}=$80\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$$ for $\text{CO}_{2}$ and $M=10^{-3}$. $T^{*}_{w}$ is the wall temperature. Cases can be found in Fig. 1. Regime |  $T^{*}_{\infty}/T^{*}_{pc}$ |  $T^{*}_{w}/T^{*}_{pc}$ |  $T^{*}_{w}/T^{*}_{\infty}$ |  Wall |  Case |  Line style
---|---|---|---|---|---|---
[-1ex]Subcrit. temp. |  0.90 |  0.85 |  0.944 |  cooling |  T09w085 |
0.90 |  1.0 |  isotherm |  T09w090 |
0.95 |  1.056 |  heating |  T09w095 |
[1ex]Transcrit. temp. |  0.90 |  1.05 |  1.167 |  heating |  T09w105 |
|  1.10 |  0.95 |  0.864 |  cooling |  T11w095 |
[0.1ex]Supercrit. temp. |  1.10 |  1.05 |  0.955 |  cooling |  T11w105 |
1.10 |  1.0 |  isotherm |  T11w110 |
1.20 |  1.091 |  heating |  T11w120 |

In all configurations, a Mach number of $10^{-3}$ is chosen to reduce the
complexity of the physical problem, ruling out compressibility effects on the
flow stability and allowing only non-ideality. On the upper third of Table 1,
cases with $T^{*}_{\infty}/T^{*}_{pc}=0.90$ (liquid-like free stream) are
presented. The temperature profiles for these cases remain subcritical, except
for case T09w105, which becomes transcritical, i.e.,
$T^{*}_{\infty}/T^{*}_{pc}<1$ and $T^{*}_{w}/T^{*}_{pc}>1$. Furthermore, it
holds: $Ec_{\infty}=5.27\times 10^{-7}$, $Pr_{\infty}=2.11$, and a unit
Reynolds number of
$Re_{u}=$5.317\text{\,}\times\mathrm{1}\mathrm{0}^{6}\,\mathrm{m}^{-1}$$.
Cases with $T^{*}_{\infty}/T^{*}_{pc}=1.10$ (gas-like free stream) are shown
in the lower third of Table 1. By mirroring case T09w105 on the Widom line,
while maintaining the same $\Delta T$, case T11w095 results as the
transcritical case for a gas-like free stream. Furthermore, it is:
$Ec_{\infty}=9.92\times 10^{-8}$, $Pr_{\infty}=1.22$, and a unit Reynolds
number of
$Re_{u}=$2.209\text{\,}\times\mathrm{1}\mathrm{0}^{6}\,\mathrm{m}^{-1}$$.
The base-flow profiles for all cases of Table 1 are shown in Fig. 2.
Dimensional quantities are made dimensionless by the corresponding Widom-line
quantities, except for the streamwise velocity $u$ and kinematic viscosity
$\nu$ (see Eq. (3)). Temperature, streamwise velocity, density and kinematic
viscosity are plotted over the wall-normal coordinate $y$, which is made
dimensionless by the local Blasius length scale $\delta^{*}$ (see Eq. (5a,b)).
As the temperature profile crosses the Widom line (see Fig. 2(a)), regardless
of the free-stream temperature, large variations in thermodynamic and
transport properties are noticeable (e.g., density in Fig. 2(d)). As a result,
both transcritical cases T09w105 and T11w095 exhibit inflectional base-flow
profiles according to the definition of the generalized inflection point,
i.e., $d(\bar{\rho}\ d\bar{u}/dy)/dy=0$. This GIP is located in the proximity
of the Widom line, as indicated in Fig. 2(b–d) with a black circle ($\circ$)
symbol. This leads to a minimum near the Widom line of kinematic viscosity
$\nu$ (see Figs. 2(e,f)), as already observed in Ref. [10]. Moreover, for case
T11w095, an inflectional $\bar{u}(y)$-profile, i.e., $d^{2}\bar{u}/dy^{2}=0$,
is found above the Widom line (marked by a black star ($\smallstar$) symbol),
due to a strong wall cooling similar to Ref. [12].

Figure 2:  Base-flow profiles for the considered cases: (a) temperature, (b,c)
streamwise velocity, (d) density and (e,f) kinematic viscosity over the
dimensionless wall-normal coordinate $y$. See Tab. 1 for line legend. The red
solid line ( ) in (a,d) indicates the Widom line. The location of the
generalized inflection point is marked by a black circle ($\circ$) symbol in
(b–d). In (c), the location of the inflection point is marked by a black star
($\smallstar$) symbol. The location of the Widom line is indicated with
$y_{WL}$ in (b,c) and (e,f).

##  IV Results

In this section, we present the non-modal and modal stability analyses of the
boundary-layer flows. After calculating the maximum and optimal energy
amplifications in Sec. IV.1, Section IV.2 provides profiles of the optimal
disturbances for selected cases, with special regard to the transcritical
regime. Subsequently, the influence of Reynolds number and wall temperature on
transient growth is investigated. Finally, in Sec. IV.4, a comparison between
non-modal and modal growth is provided.

###  IV.1 Optimal amplifications

Optimal growth has been studied for the subcritical (T09w090, T09w095,
T09w085), supercritical (T11w110, T11w105, T11w120), and transcritical
(T09w105, T11w095) cases. An initial local Reynolds number of
$Re_{\delta}=300$ is chosen, and the optimization process is conducted up to
$x=3000$ until the maximum value of $G(x)$, i.e., $G_{max}$, is found. Each
contour plot in Fig. 3 has been computed on a grid of $\Delta\omega=0.001$
$\times$ $\Delta\beta=0.01$. The colored contour represents
$G_{max}(\omega,\beta)$, and the largest $G_{max}$, or rather $G_{opt}$, is
marked with a black star ($\filledstar$). Secondary contour regions display an
unstable mode of spatial growth rate $-\alpha_{i}$ with contour lines equally
spaced between 0 and $\max\\{-\alpha_{i}\\}$. Here, the maximum energy
amplification is infinite, and the optimization process is stopped.
Furthermore, the integral amplification, e.g., $N$-factor, contours are
indicated. For their calculation, the local growth rate has been integrated
from $Re_{\delta}=300$ to the optimal transient location at $Re_{\delta,opt}$,
given by $x_{opt}$. A more comprehensive comparison between non-modal and
modal $N$-factors is later performed in Sec. IV.4.

Figure 3: Contour plot of $G_{max}$ and growth rate $-\alpha_{i}$ (modally
unstable regions) against the angular frequency $\omega$ and spanwise
wavenumber $\beta$ at $Re_{\delta}=300$ for the subcritical and supercritical
cases: (a) T09w085, (b) T09w090, (c) T09w095, (d) T11w105, (e) T11w110, (f)
T11w120, (g) T09w105, (h) T11w095. $G_{opt}$, i.e., $\max\\{G_{max}\\}$, is
denoted with a black star ($\filledstar$) symbol. $N$-factor iso-contours are
indicated in gray.

In the non-transcritical cases, as the wall temperature increases towards to
the Widom line, either through an increase in $T_{w}$ (subcritical regime) or
a reduction in $T_{w}$ (supercritical regime), modal amplification is
completely damped. The unstable regions in Fig. 3(a) and Fig. 3(f) correspond
to the only modal instability present in the subcritical and supercritical
regimes, according to Ref. [8]. They are of Tollmien-Schlichting-wave type.
Overall, wall cooling in the liquid-like region (see T09w085 in Fig. 3(a))
shows similar transient growth as wall cooling in the gas-like region (see
T11w105 in Tab. 3(d)). Cases T09w090 and T11w110 retain an identical energy
amplification, reproducing the isothermal incompressible limit, which is
independent of the considered gas law.
With respect to the transcritical cases in Figs. 3(g,h), $G_{opt}$
significantly increases compared to the non-transcritical cases, especially in
the case of cooling over the Widom line (T11w095) with $G_{opt}=1750$. Note
that, while for case T09w105 modal instabilities are not present at
$Re_{\delta}=300<Re_{c}$ (with $Re_{c}$ being the critical Reynolds number),
case T11w095 is highly modally unstable (only one single mode) already at
$Re_{\delta}=300$. In contrast to the non-transcritical cases and case
T11w095, wall heating over the Widom line (case T09w105) causes a shift of
$\omega_{opt}$ from 0 to finite values. For instance, in Fig. 3(g), it is
$\omega_{opt}=0.013$. A similar shift is observed when a temporal analysis is
performed in Appendix D. The physical mechanism responsible for the shift is
explained in Sec. IV.2. Differently than in the non-transcritical cases, case
T09w105 unveils a weakly decaying behavior of $G_{max}$ for large angular
frequency $\omega$ around $\beta_{opt}$. It is interesting to note that at
$\beta=0$, there is a distinct increase in $G_{max}$ along the $\omega$-axis
with a sub-optimal amplification peak of $G_{max}\approx 282$ at
$\omega\approx 0.23$. A further analysis of the sub-optimal energy
amplification is performed in Sec. IV.2.5.
Contours of the maximum location $x_{max}$ are displayed in Fig. 4 for the
same $\Delta\omega$ $\times$ $\Delta\beta$ of Fig. 3. The optimum transient
growth is always located at large streamwise locations (long-distance
amplification) in agreement with the observations of Refs. [40] and [23] for
incompressible and compressible boundary layers. Similarly, for small $x$
(short-distance amplification), $G_{opt}$ is always detected at large
$\omega$-values before shifting to $\omega=0$ as $x$ increases. This
phenomenon is observed for all non-transcritical cases and case T11w095,
except for the non-trivial case T09w105 in Fig. 4(g) where $\omega_{opt}$ is
finite.

Figure 4: Contour plot of $x_{max}$ and growth rate $-\alpha_{i}$ (modally
unstable regions) against the angular frequency $\omega$ and spanwise
wavenumber $\beta$ at $Re_{\delta}=300$ for the subcritical and supercritical
cases: (a) T09w085, (b) T09w090, (c) T09w095, (d) T11w105, (e) T11w110, (f)
T11w120, (g) T09w105, (h) T11w095. $G_{opt}$, i.e., $\max\\{G_{max}\\}$, is
denoted with a black star ($\filledstar$) symbol. $N$-factor iso-contours are
indicated in gray.

A summary of transient growth for the subcritical, transcritical, and
supercritical regimes is reported in Table 2. Note that the value of
$G_{kin.opt}$ is calculated using the kinetic part and a frozen internal
energy of $\mathbf{M}$ in Eq. (24), with
$\mathbf{M}=\operatorname{diag}\left(1,\bar{\rho},\bar{\rho},\bar{\rho},1\right)$
(see Appendix E). Furthermore, the effect of the energy norm on the optimal
growth is analyzed.

Table 2: Summary of spatial transient growth characteristics at $Re_{\delta}=300$: subcritical, transcritical and supercritical cases. Case | $G_{opt}$ | $G_{kin.opt}$ | $x_{opt}$ | $Re_{\delta,opt}$ | $\omega_{opt}$ | $\beta_{opt}$ | $\max\\{-\alpha_{i}\\}$
---|---|---|---|---|---|---|---
T09w085 | 540 | 476 | 464 | 478.7 | 0 | 0.377 | $0.569\times 10^{-2}$
T09w090 | 437 | 437 | 462 | 478.1 | 0.381 | $0.238\times 10^{-3}$
T09w095 | 460 | 397 | 434 | 469.2 | 0.408 | -
T09w105 | 758 | 490 | 422 | 465.4 | 0.013 | 0.45 | -
T11w095 | 1750 | 1.6704 $\times 10^{4}$ | 346 | 440.2 | 0 | 0.46 | 0.089
T11w105 | 466 | 457 | 447 | 473.4 | 0 | 0.391 | -
T11w110 | 437 | 437 | 462 | 478.1 | 0.381 | $0.238\times 10^{-3}$
T11w120 | 434 | 414 | 490 | 486.8 | 0.364 | $0.566\times 10^{-2}$

From Table 2, it is evident that, given the same temperature ratio
$T^{*}_{w}/T^{*}_{\infty}$, the optimal energy amplification increases more
when the wall is cooled compared to the incompressible reference case, as
density increases for both regimes. In fact, the largest $G_{opt}$ is found
for case T09w085, with the largest density ratio, i.e.,
$\rho^{*}_{w}/\rho^{*}_{\infty}=1.0834$, among all non-transcritical cases of
Tab. 1. This confirms the trend encountered in the ideal-gas results of Ref.
[22] at $M_{\infty}=0.5$. Regarding the optimal energy amplification with
frozen internal energy $G_{kin.opt}$, its greater impact in noticed in the
subcritical regime, where $G_{opt}$ drops by more than $10\%$. Additionally,
the optimal spanwise wavenumber $\beta_{opt}$ shows a dependency on the wall
temperature. This is a consequence of the modification of the boundary-layer
thickness (see $u^{*}/u^{*}_{\infty}=0.99$ in Figs. 2(b,c)): wall cooling in
the subcritical regime, and vice versa heating in the supercritical regime,
increases $\delta_{99}$, and thus $\beta_{opt}$ reduces.
For case T09w105, $G_{opt}$ significantly increases due to non-ideal gas
effects, whereas $G_{kin.opt}$ consistently drops (density ratio at the wall:
$\rho^{*}_{w}/\rho^{*}_{\infty}\approx 0.23$). Furthermore, as the velocity
profile is pushed closer to the wall (see Fig. 2(b)), $\beta_{opt}$ further
rises, and $x_{opt}$ decreases. When the gas-like free stream is cooled down
at the wall to a liquid-like regime (case T11w095 with
$\rho^{*}_{w}/\rho^{*}_{\infty}\approx 4.61$), the optimal energy
amplification is strongly dampened by the non-ideal gas effects, i.e.,
$G_{opt}\ll G_{kin.opt}$, while $\omega_{opt}$ is found at $\omega=0$. A
similar behavior regarding $\beta_{opt}$ and $x_{opt}$ can be observed as in
case T09w105. Concerning the ratio between kinetic $\hat{E}_{kin.}$ and total
energy $\hat{E}$, obtained by evaluating Eqs. (22) and (21), respectively, for
$\hat{q}=\hat{q}_{opt}$, a value of $32\%$ is found for case T09w105, whereas
of $47\%$ for case T11w095. This demonstrates how optimal thermal fluctuations
become relevant whenever the Widom line is crossed, i.e., for large property
variations.

###  IV.2 Optimal perturbations

####  IV.2.1 Subcritical and supercritical regimes

For all subcritical and supercritical cases in Fig. 3 and Tab. 2, $G_{opt}$ is
found at $\omega=0$. This condition corresponds to a streamwise vortex,
typical of the ideal-gas regime [17, 22, 23], as displayed for case T09w095 in
Fig. 5(a). Very similar profiles are also obtained for the other non-
transcritical cases of Tab. 2 (not shown here for the sake of brevity), as the
initial perturbation energy, i.e., Eq. (21), is entirely of kinetic nature.

Figure 5: Absolute value of the optimal disturbances at $Re_{\delta}=300$ over
wall-normal distance $y$ for T09w095 of Tab. 2: (a) initial, (b) output, (c)
output energy amplitude obtained by Eqs. (22) and (23) with respect to the
total energy. The boundary-layer thickness is indicated by $\delta_{99}$.

Figure 5(b) represents the flow response to the streamwise vortex at
$x=x_{opt}$ for case T09w095. The shape of the output optimal disturbance
consists mainly of streamwise velocity, taking the form of streamwise (high-
low velocity) streaks. Temperature and density disturbances are almost
negligible in this regime due to small corresponding mean-flow gradients.
Overall, streamwise velocity streaks are recovered for all subcritical and
supercritical cases of Tab. 2 (here only T09w095 is displayed for brevity).
The physical mechanism behind the formation of velocity streaks has been
explained by the lift-up effect [19]. Note that the total amount of kinetic
energy to the total energy $\hat{E}_{kin.}/\hat{E}$ after amplification has
reduced from $99.9\%$ to $86\%$ for case T09w095. On the other hand, for the
supercritical cases, the drop of kinetic energy reaches only up to $5\%$. By
plotting in Fig. 3(c) the energy amplitude of the different disturbance
components, i.e., $\hat{E}_{kin.}/\hat{E}$, $\hat{E}_{\hat{\rho}}/\hat{E}$,
$\hat{E}_{\hat{T}}/\hat{E}$ and $\hat{E}_{\hat{\rho}\hat{T}}/\hat{E}$, over
the wall-normal direction, it is evident that the total energy is dominated by
its kinetic part. Conversely, the contribution of the non-diagonal terms
$\hat{E}_{\rho^{\prime}T^{\prime}}$ in matrix $\mathbf{M}$ of Eq. (24) to the
total energy, which are non-zero for a non-ideal gas, are not more than $20\%$
due to the small base-flow thermodynamic derivatives (i.e.,
$\partial\bar{p}/\partial\bar{\rho}|_{\bar{T}}$ ,
$\partial\bar{p}/\partial\bar{T}|_{\bar{\rho}}$,
$\partial\bar{e}/\partial\bar{\rho}|_{\bar{T}}$). This confirms that the level
of non-ideality is weak in the subcritical and supercritical regimes. The
optimal perturbations, therefore, resemble the ones found in the
incompressible ideal-gas regime [18, 22].

####  IV.2.2 Transcritical regime

In Figs. 6(a,b), the optimal initial and output perturbations for the
transcritical case T09w105 are plotted, respectively. As the initial optimal
disturbance at $x=0$ takes the form of counter-rotating vortices in agreement
with the other regimes, the resulting velocity streaks at maximum
amplification are also characterized by strong thermal $\hat{\rho}$ and
$\hat{T}$ components. More importantly, a strong density disturbance, which is
absent in the non-transcritical cases (see e.g., Fig. 5(b)), peaks at the
Widom line due to the abrupt mean density gradient observed in Fig. 2(d). By
plotting in Fig. 5(c) the energy amplitude of the disturbance components
normalized by the total energy, one can recognize that at the Widom line
$\hat{E}_{\hat{\rho}\hat{T}}$ reaches almost unity, whereas the kinetic energy
contribution is negligible. When crossing the Widom line, non-ideal gas
effects are strongly induced. On the contrary, the kinetic energy peak is
found to occur in the liquid-like regime (high-viscosity region) from
$y\approx 0.35$, along with the maximum of the temperature disturbance and the
second peak of $\hat{E}_{\hat{\rho}\hat{T}}$. Just as in the non-transcritical
cases of Fig. 5, the resulting optimal streaks are regions of excess and
defect of streamwise momentum in the streamwise direction.

Figure 6: Absolute value of the optimal disturbances at $Re_{\delta}=300$ over
wall-normal distance $y$ for T09w105 of Tab. 2: (a) initial, (b) output, (c)
output energy amplitude obtained by Eqs. (22) and (23) with respect to the
total energy amplitude. The boundary-layer thickness and the location of the
Widom line are indicated by $\delta_{99}$ and $y_{WL}$, respectively.

The optimal perturbations for the other transcritical case T11w095, in which
the temperature profile is cooled over the Widom line, are now examined. As
revealed in Fig. 3(h), streamwise-independent streaks ($\omega_{opt}=0$,
$\beta_{opt}=0.46$) are the optimal disturbances, in contrast to case T09w105.
This is confirmed in Fig. 7(a) and Fig. 7(b), where wall-normal profiles of
optimal perturbations are plotted. The lift-up effect is the active mechanism.
Unlike all the other cases, thermal streaks, i.e., $\hat{\rho}$ and $\hat{T}$,
become even more relevant here. In particular, the density disturbance reaches
a factor 18 at the Widom line, due to the even larger mean density gradient
compared to the one of case T09w105. In Fig. 7(c), the disturbance energy
amplitude is displayed. The maximum total energy amount is located at the
Widom line, due to the abrupt gradients of the thermo-physical properties (see
Fig. 2). Here, the non-ideality is largest, whereas the kinetic energy amount
is almost negligible throughout the boundary layer.

Figure 7: Absolute value of the optimal disturbances at $Re_{\delta}=300$ over
wall-normal distance $y$ for T11w095 of Tab. 2: (a) initial, (b) output, (c)
output energy amplitude obtained by Eqs. (22) and (23) with respect to the
total energy amplitude. The boundary-layer thickness and the location of the
Widom line are indicated by $\delta_{99}$ and $y_{WL}$, respectively.

####  IV.2.3 Transcritical regime: vortex tilting

The mechanism of streak generation can be best explained by the lift-up effect
or vortex tilting [41, 42]. The latter is here further considered for a more
accurate physical interpretation of the effect of the Widom line on the
optimal perturbation profiles. Firstly, the inviscid compressible vorticity,
denoted as $\omega_{i}$, equation is considered as

$\dfrac{\partial\omega_{i}}{\partial
t}+u_{j}\dfrac{\partial\omega_{i}}{\partial x_{j}}=-\omega_{i}\dfrac{\partial
u_{j}}{\partial
x_{j}}+\dfrac{\epsilon_{ijk}}{\rho^{2}}\dfrac{\partial\rho}{\partial
x_{j}}\dfrac{\partial p}{\partial x_{k}},$ (29)

where $\epsilon_{ijk}$ is the Levi-Civita symbol, and the second term on the
RHS is the baroclinic term. After decomposition and linearization of Eq. (29)
(see Appendix F), the wall-normal vorticity perturbation $\omega^{\prime}_{y}$
component can be written as

$\dfrac{\partial\omega^{\prime}_{y}}{\partial
t}+\bar{u}\dfrac{\partial\omega^{\prime}_{y}}{\partial x}=-\dfrac{\partial
v^{\prime}}{\partial z}\dfrac{\partial\bar{u}}{\partial y},$ (30)

where the baroclinic term is zero, $\partial\bar{u}/\partial y$ is the
spanwise mean-flow vorticity, and $\partial v^{\prime}/\partial z$ is the
perturbation strain rate of the initial vortex [43]. The latter tilts the
mean-flow vorticity into the wall-normal $y$ direction, causing an increase in
$\omega^{\prime}_{y}$. Thus, the vortex-tilting mechanism is a function of the
initial disturbance strain rate $\partial v^{\prime}_{in}/\partial z$, which
can be assessed by examining the initial streamwise vorticity
$\omega^{\prime}_{x,in}=\partial w^{\prime}_{in}/\partial y-\partial
v^{\prime}_{in}/\partial z$.
In Figs. 8(a,b), the initial streamwise vorticity with the perturbation strain
rate and the base-flow streamwise momentum are plotted for the subcritical
case T09w095 and the transcritical case T09w105, respectively. First of all,
the streamwise vorticity $|\hat{\omega}_{x,in}|$ exhibits two peaks. The lower
one is determined by the wall conditions, as $\hat{v}$ and $\hat{w}$ are zero
at the wall, and does not correspond to a streamwise vortex. On the other
hand, the upper peak, located far from the wall, corresponds to the vortex
structure responsible for the largest amount of wall-normal redistribution, or
alternatively where $\partial\hat{v}_{in}/\partial z$ is largest.
Consequently, it is here where $\hat{\omega}_{y,out}$ (black continuous line)
has its maximum, not just for case T09w095 but also for all the other non-
transcritical cases of Tab. 1 (not shown here for the sake of brevity).
In the transcritical case T09w105, as shown in Fig. 8(b), the impact of the
Widom line on $\hat{\omega}_{x,in}$ is evident. Here, the streamwise vorticity
reaches its wall-normal maximum due to the large $y$-gradient of spanwise
perturbation velocity $\hat{w}_{in}$ (not displayed here). However, in this
region, the wall-normal displacement (see $\hat{v}_{in}$ in Fig. 6(a)) of a
larger streamwise momentum (see blue dotted line at $y=y_{WL}$ in comparison
to case T09w095), is again low, and thus this peak does not contribute to the
lift-up effect. Additionally, the perturbation strain rate remains unaffected
by the non-ideality, resembling the subcritical case shown in Fig. 8(a).
However, what is strongly affected by the Widom line, is the base-flow shear
near the wall (see red dash-dotted line in Fig. 8(b)). This enhancement has,
in turn, a considerable effect on the vortex-tilting term
($|\partial\bar{u}/\partial y\cdot\partial\hat{v}_{in}/\partial z|$). As a
result, $\hat{\omega}_{y,out}$ is substantially altered at the Widom line,
exhibiting a second peak, as shown in the inset of Fig. 8(b). This peak
corresponds to the one observed in $\hat{u}_{out}$ in Fig. 6(b). Overall, in
case T09w105, the streamwise-velocity streaks show two distinct peaks, with
the largest one located near the maximum perturbation strain rate and the less
pronounced one at the Widom line. The latter modifies the classical streaks’
shape compared to the non-transcritical cases. On the other hand, it is
important to note that the $\hat{u}_{out}$-peak at the Widom line in Fig. 6(b)
has a negligible influence on the energy amplification, since the internal
energy is the dominant factor in the $G$-increase of this transcritical case,
and not the kinetic energy (see difference between $G_{opt}$ and $G_{kin.opt}$
in Tab. 2).
Analogous to the transcritical wall-heating case shown in Fig. 8(b), the
vortex tilting of the transcritical wall-cooling case is investigated in Fig.
8(c). Similar to case T09w105, the perturbation strain rate
$\partial\hat{v}_{in}/\partial z$ in Fig. 8(c) is not altered by the presence
of the Widom line. Instead, its effect is evident in $\hat{\omega}_{x,in}$,
leading to its substantial reduction of the lower peak near the wall.
Simultaneously, due to the liquid-like regime below the Widom line (see the
large $\bar{\rho}_{w}$ in Fig. 2(d)), we observe, in the vicinity of $y_{WL}$,
a considerable increase in the horizontal momentum (see blue dotted line in
Fig. 8(c)), and thus expecting a stronger lift up in this region. In fact, we
notice that the resulting peak in $\hat{\omega}_{y,out}$ occurs just above the
Widom line, where the vortex-tilting term $|\partial\bar{u}/\partial
y\cdot\partial\hat{v}_{in}/\partial z|$ is largest. Conversely, near the wall,
the mean-flow vorticity $\partial\bar{u}/\partial y$ is small; therefore,
unlike in Fig. 8(b), there is no second peak at the Widom line in this case.
Based on these observations, we can argue that the resulting streaks have a
higher amplitude than those in case T09w105, attributed to a stronger vortex-
tilting term $|\partial\bar{u}/\partial y\cdot\partial\hat{v}_{in}/\partial
z|$. This behavior is supported by a greater optimal energy amplification of
case T11w095 compared to case T09w105: from Tab. 2, it holds
$G_{opt,\mathrm{T11w095}}>G_{opt,\mathrm{T09w105}}$.

Figure 8: Vortex tilting: (a) case T09w095, (b) case T09w105, (c) case
T11w095. All disturbance terms $(\hat{\cdot})$ are normalized by
$\max\\{|\hat{\omega}_{x}|\\}$. The boundary-layer thickness and the location
of the Widom line are indicated by $\delta_{99}$ and $y_{WL}$, respectively.

####  IV.2.4 Transcritical regime: lift-up effect and Orr mechanism

For both transcritical cases, to better understand the structure of the
optimal amplification, its spatial evolution is analyzed. Starting with case
T09w105, the transient growth $G(x)$ at $\omega_{opt}$ and $\beta_{opt}$ is
plotted in Fig. 9(a) according to Eq. (27). This represents the optimal gain
envelope over all possible initial conditions. However, in order to calculate
disturbance profiles over the streamwise distance $x$, we assume the same
initial condition at $x=0$ for the optimal perturbation and perform space
marching up to $x=x_{opt}$, i.e., we follow $g(x)$. At this location (dotted
red line), the energy amplification $g(x)$ (dashed red line) is equal to the
maximum value of the $G$-envelope (solid blue line), while remaining lower for
$x\neq x_{opt}$. For the calculation of $g(x)$, Eq. (27) is used with
$\mathbf{F}$ associated with the right singular eigenvector at $x=x_{opt}$.
A three-dimensional (3-D) view of the streamwise velocity perturbations is
reported in Fig. 9(b). This representation is independent of time, i.e.,
steady oblique elongated structures. For their computation, the optimal
perturbation is chosen based on $g(x)$ of Fig. 9(a).

Figure 9: Case T09w105 ($Re_{\delta}=300$, $\omega_{opt}=0.013$,
$\beta_{opt}=0.45$): (a) envelope of the optimal transient growth $G(x)$ ( )
and transient growth $g(x)$ of the optimal perturbation ( ) over the
streamwise direction $x$, the red dotted line indicates the location of the
optimal perturbation ($g(x_{opt})=G(x_{opt})$); (b) iso-contours of the
streamwise velocity perturbations ($\Re(u^{\prime}_{out})=\pm 4.0$) for
$\beta=\beta_{opt}$ corresponding to their transient growth $g$; (c) iso-
contours of the streamwise velocity perturbations ($\Re(u^{\prime}_{out})=\pm
4.0$) for $\beta=-\beta_{opt}$; (d) superposition of iso-contours
($\Re(u^{\prime}_{out})=\pm 8.0$) with $\pm\beta_{opt}$. The red dashed line
indicates the optimal growth location ($x=x_{opt}$). Planes: A, B, C and D of
Fig. 10.

The physical mechanism of the lift-up effect can be clearly seen in the cross-
stream slices ($y$-$z$ plane) of Figs. 10(a–c), where contours of the output
streamwise velocity perturbation $u^{\prime}_{out}$ are superimposed over the
velocity vectors of the cross-stream perturbation velocity
$|\vec{V^{\prime}_{out}}|=\sqrt{v^{\prime 2}_{out}+w^{\prime 2}_{out}}$ at
three different streamwise locations. The resulting high-velocity streaks
($\Re(u^{\prime}_{out})>0$) are located where counter-rotating vortices pull
high momentum fluid downward, and vice versa for the low-velocity streaks
($\Re(u^{\prime}_{out})<0$). The same behavior is observed for the density and
temperature streaks: streamwise vortices act with the mean-density and against
the mean-temperature profile, respectively. At $x=x_{opt}=425$, the lift-up
mechanism causes the largest amplification of the streaks. Due to the crossing
of the Widom line, secondary $u^{\prime}_{out}$-peaks are visible in the near-
wall region, especially at early $x$, as already explained by the vortex
tilting in Fig. 8(b).

Figure 10: Case T09w105. Contours of the optimal streamwise velocity
perturbation $u^{\prime}_{out}$ and velocity vectors of the resulting cross-
stream perturbation velocity $|\vec{V^{\prime}_{out}}|=\sqrt{v^{\prime
2}_{out}+w^{\prime 2}_{out}}$ on an $z$-$y$ plane: (a) $x=0$, (b) $x=50$, (c)
$x=x_{opt}=425$. Contours of the optimal perturbation on an $x$-$y$ plane at
$z=0$: (d) streamwise velocity $u^{\prime}_{out}$, (e) Reynolds stress
$u^{\prime}_{out}v^{\prime}_{out}$. Note that $\delta_{99}$ and $y_{WL}$ are
constant due to the parallel mean-flow assumption used in the transient-growth
analysis. Capital A, B, C, D correspond to planes in Fig. 9.

However, despite the 2-D parallel flow assumption, vortices are not
symmetrical with respect to the wall-normal axis as evident in Figs. 10(a–c).
They are inclined instead in the spanwise direction. Within a very short
distance ($x\approx 50$), they are first erected before being tilted in the
other direction farther downstream. It is clear that another mechanism is
present, which yields an additional amplification of the energy in the cross-
plane spanwise component [44]. Due to the shift of the optimal amplification
$G_{opt}$ to finite frequencies, visible in Fig. 3(d), this behavior is now
the optimal one.
In fact, when examining a wall-normal slice ($x$-$y$ plane) in Figs. 10(d,e),
the presence of the Orr mechanism is observed [45]. The output streamwise
velocity perturbations in Fig. 10(d) are initially misaligned with the mean
shear, but, as they move downstream, they incline in the direction of shear.
In this process, the mean-shear energy is firstly extracted by the disturbance
through the action of the Reynolds stress $u^{\prime}_{out}v^{\prime}_{out}$
(see Fig. 10(e)), which is increased due to the conservation of circulation in
an $x$-$y$ plane [32]. Subsequently, the energy is returned to the mean flow,
and viscosity effects become relevant at large $x$. The largest absolute value
of $u^{\prime}_{out}v^{\prime}_{out}$ is found at $x\approx 240$, farther
upstream than $x_{opt}$ at $x=425$. The enhancement of the cross-stream
velocity by the Orr mechanism produces an additional rise of the streamwise
velocity perturbations [43], which peak at $x=x_{opt}$. This positive
interplay of Orr mechanism, originally located at $\beta=0$ and $\omega>0$,
and lift-up effect, originally located at $\beta=0$ and $\omega>0$, leads to
the optimal transient growth at finite frequencies or wavenumbers, e.g.
$\omega_{opt}>0,\beta_{opt}>0$. This physical situation was already discovered
in constant shear flows [41, 43], and more recently in strongly stratified
flows [27, 32] under the Boussinesq assumption.
When observing the oblique perturbations in Fig. 9(b), one may wonder why the
optimal streaks have such an orientation with a propagation angle $\Psi$, with
$\tan(\Psi)=\beta/\alpha$. Given the two-dimensional spanwise-periodic
underlying base flow, the same optimal growth can be achieved with a spanwise
wavenumber $\beta$ as well as with $-\beta$ (see also [41]). This means that
the same 3-D structures are obtained with the opposite inclination to the
wall-normal axis (see Fig. 9(c)). In addition, the Reynolds stress mechanism
in Figs. 10(d–e) is exactly recovered. Since we are considering eigenvalues
and eigenvectors of a linear amplification problem [20], a superposition of
the two oblique waves can be calculated. As shown in Fig. 9(d), a checkered
wave pattern is produced, which results from multiple standing waves in the
spanwise direction with a resulting zero propagation angle. These streamwise
and spanwise alternating structures resemble the initial stage of an oblique
transition (see, for instance, Ref. [46]). In fact, the (non-linear)
generation of a streamwise vortex by two least-damped Orr-Sommerfeld oblique
waves with $(\omega,\pm\beta)$ is followed by the transient growth of vortex-
generated streaks. An analysis of the superposed cross-stream slices in Figs.
10(a–c) reveals symmetrical structures in the spanwise direction, with zero
cross-stream velocity.
Oblique elongated structures are not only found for the dynamic streaks but
also for the thermal streaks, which were previously observed in Fig. 6. Thus,
slices of 3-D density perturbations are shown in Fig. 11(a). Note that there
is a phase difference of $\pi$ between high-low-density and -temperature
streaks, as already witnessed in three-dimensional compressible ideal-gas
boundary layers [47]. Similar perturbation shapes are observed for the
$x$-momentum in Fig. 11(b). Here, regions of high and low density are found in
combination with regions of low and high streamwise velocity, respectively.

Figure 11: Case T09w105 ($Re_{\delta}=300$, $\omega_{opt}=0.013$,
$\beta_{opt}=0.45$). Slices of 3-D contours of optimal disturbances: (a)
density, (b) $x$-momentum.

With respect to the other transcritical case T11w095, a 3-D visualization of
the optimal disturbances (streamwise velocity streaks and density streaks) is
unveiled in Fig. 12. For their calculation, the same procedure as in Fig. 9(a)
is performed. As expected from its energy-amplification contours of Fig. 3(h),
case T11w095 exhibit streamwise-independent structures in Fig. 12, analogously
to the subcritical and supercritical regimes.

Figure 12: Case T11w095 ($Re_{\delta}=300$, $\omega_{opt}=0$,
$\beta_{opt}=0.46$). Slices of 3-D contours of optimal disturbances: (a)
streamwise velocity, (b) density.

####  IV.2.5 Sub-optimal growth

Despite optimal transient growth due to a combined Orr and lift-up mechanism
at $x_{opt}\gg 1$, contours of $G_{max}$ for case T09w105 in Fig. 3(g)
indicate strong sub-optimal growth occuring at small distances from the
initial location. This non-trivial phenomenon is further investigated
hereafter.
For this purpose, in Fig. 13, the maximum energy amplification $G_{max}$ is
presented again in the $\omega$–$\beta$ space for the wall-heating cases
T09w090 (isothermal incompressible limit), T09w095, and T09w105. Additionally,
black dash-dotted isolines corresponding to $G_{max}/G_{opt}=2/3$ and
$G_{max}/G_{opt}=1/3$ are included. It is evident that the region enclosed by
these two lines is significantly larger when the base-flow temperature crosses
the Widom line, i.e., in case T09w105. Energy amplifications larger than
$G_{opt}/3$ are even located along the $\omega$-axis for spanwise-uniform
disturbances, demonstrating the substantial contribution of the Orr mechanism
to the energy amplification for the transcritical wall-heating case. In fact,
the maximum value of $G_{max}$ along the $\omega$-axis, i.e.,
$\max\\{G_{max}(\omega,\beta=0)\\}$ (marked by a black square ($\blacksquare$)
symbol), is a factor of $7.8$ and $12.3$ larger than for cases T09w095 and
T09w090, respectively. Likewise, in Fig. 13(c) for case T09w105, the non-
negligible sub-optimal growth peaks much earlier in space than the global
optimal (marked by a black star ($\filledstar$) symbol) at large $\omega$ and
low $\beta$. Here, grey dashed isolines indicate a ratio of $x_{max}$ to
$x_{opt}$ of about $1/8$, with $x_{opt}=422$ according to Tab. 2. This
confirms the observations of Refs. [40] and [41] in the temporal setting,
where sub-optimal disturbances, associated with short time scales, are no
longer streamwise uniform but reach their maximum amplitude more rapidly.
Moreover, if we consider the streamwise evolution of $G_{max}$ by tracking the
grey stars in Fig. 13(c), we observe that, at small $x$, disturbances are
highly oblique, given the large $\omega$, and the initial disturbance is not
any more a streamwise vortex. Only at large $x$ similar disturbance shapes as
in Fig. 6 are recovered. This behavior was also previously witnessed for ideal
gas at $M_{\infty}=5.0$ [23].

Figure 13: Contour plot of $G_{max}$ in the $\omega$–$\beta$ space at
$Re_{\delta}=300$ for cases: (a) T09w090, (b) T09w095, and (c) T09w105.
$G_{opt}$, i.e., $\max\\{G_{max}\\}$, is denoted with a black star
($\filledstar$) symbol. The black dash-dotted lines represent
$G_{max}/G_{opt}=2/3$ and $G_{max}/G_{opt}=1/3$, whereas the grey dashed lines
represent $x_{max}/x_{opt}=1/3$, $x_{max}/x_{opt}=1/5$, and
$x_{max}/x_{opt}=1/8$. The black square ($\blacksquare$) symbols denote
$\max\\{G_{max}(\omega,\beta=0)\\}$, wheres the grey star ($\filledstar$)
symbols denote $\max\\{G_{max}(x_{max}/x_{opt}=1/3)\\}$,
$\max\\{G_{max}(x_{max}/x_{opt}=1/5)\\}$, and
$\max\\{G_{max}(x_{max}/x_{opt}=1/8)\\}$.

A comparison of the optimal and sub-optimal energy amplification over the
streamwise distance is presented in Fig. 14(a). The (global) optimal transient
growth, marked by a black star symbol in Fig. 13(c), is considered alongside
the largest sub-optimal transient growths, peaking at
$x=[x_{opt}/3,x_{opt}/5,x_{opt}/8]$ (marked by grey star symbols in Fig.
13(c)). At $x\approx 52$ from the initial location, the local maximum energy
amplification $G_{max}(x_{max}/x_{opt}=1/8)$ is $2.4$ larger than the optimal
energy amplification at $x_{opt}$. In other words, a near-optimal
amplification of $G_{max}/G_{opt}\approx 0.43$ can be achieved at an eight
times shorter distance than $x_{opt}$, making sub-optimal energy growth non-
negligible.
Examining the initial condition that yields maximum sub-optimal gain at
$x_{max}/x_{opt}=1/8$, denoted as $g(x_{max}/x_{opt}=1/8)$ in Fig. 14(a),
enables us to employ the same procedure introduced in Sec. IV.2.4 for the
spatial evolution of the sub-optimal streamwise velocity perturbations. Here
the spanwise wavenumber and frequency are $0.38$ and $0.204$, respectively. As
shown in Fig. 14(b), an even stronger tilting of the flow structures,
characteristic of the Orr mechanism, is evident with larger resulting spanwise
velocity components than those observed in Fig. 10. This leads to oblique
structures having a propagation angle $\Psi$ of approximately $42^{\circ}$, in
contrast to the (global) optimal structures in Fig. 9(b,c) with only
$\Psi\approx 2^{\circ}$. In other words, we can confirm the observation of
Ref. [40], where larger local optima can be achieved at $x\ll x_{opt}$ by
efficiently extracting energy from the mean flow via the Orr mechanism. As $x$
increases, the lift-up mechanism becomes dominant, and (nearly) streamwise-
independent structures are recovered.

Figure 14: Case T09w105. (a) Over the streamwise direction $x$: envelope of
the largest sub-optimal transient growth at $x_{max}/x_{opt}=1/8$ ( ),
$x_{max}/x_{opt}=1/5$ ( ), $x_{max}/x_{opt}=1/3$ ( ), sub-optimal transient
growth $g(x_{max}/x_{opt}=1/8)$ of the optimal perturbation ( ), and envelope
of the (global) optimal transient growth at $\omega_{opt}=0.013$ and
$\beta=0.45$ ( , black star indicates the global maximum as in Fig. 13(c));
the grey dotted line indicates the location of the maximum energy growth (grey
star as in Fig. 13(c)). (b) Contours of the output streamwise velocity
perturbations corresponding to the black dotted line $g(x_{max}/x_{opt}=1/8)$
($x$-$y$ plane at $z=0$).

###  IV.3 Optimal disturbance: effects of initial Reynolds number and wall
temperature

####  IV.3.1 Initial Reynolds number

In Sec. IV.2, all investigations were performed at a constant initial Reynolds
number of $Re_{\delta}=300$. Yet, for incompressible and compressible
boundary-layer flows under the ideal-gas assumption, the following scaling
holds for streamwise-independent modes [48, 21, 22]: for the spatial theory,
it is $G$ that varies quadratically with the local Reynolds number, and it is
$x$ that scales linearly with $Re_{\delta}$. Thus, a similar investigation is
performed for the current non-ideal gas cases of Tab. 1. The optimal energy
amplification is calculated at three different initial Reynolds numbers
$Re_{\delta}=300,1000$ and $3000$ with the same $\beta_{opt}$ equal to
$\beta_{opt}(Re_{\delta}=300)$ and $\omega=0$. Figs. 15(a) and (b) show the
scaling relations for cases with wall heating from a liquid-like free stream
and wall cooling from a gas-like free stream, respectively. Clearly, curves at
different $Re_{\delta}$ collapse for the same temperature difference. This
behavior proves that the scaling law is valid not only at subcritical and
supercritical temperatures, but also with the presence of a strong non-
ideality across the Widom line.

Figure 15: Streamwise-independent disturbances. Energy amplification
$G(x)/Re^{2}_{\delta}$ over the streamwise distance $x/Re_{\delta}$ at
$\beta=\beta_{opt}(Re_{\delta}=300)$ and $\omega=0$. Lines in (a): T09w095,
T09w105, T09w090, T11w120. Lines in (b): T11w105, T11w095, T11w110, T09w085.
Symbols: circle ($\circ$) ($Re_{\delta}=300$), asterisk ($\ast$)
($Re_{\delta}=1000$), triangle ($\triangledown$) ($Re_{\delta}=3000$). Cases
T09w090, T11w120, T11w110, T09w085 are displayed without symbols for better
representation, but they also obey the scaling. In the insets,
$G_{max}/Re^{2}_{\delta}$ is marked by a colored star ($\filledstar$) symbol.

In the following, the transcritical case T09w105, for which streamwise-
modulated streaks were previously found at $Re_{\delta}=300$, is here
considered further. In Figs. 16(a) and (b), the optimal frequency
$\omega_{opt}$ and optimal location $t_{opt}$ are plotted as a function of the
initial Reynolds number for the optimal amplification $G_{opt}$ in Fig. 16(c).
As $Re_{\delta}$ is increased, the optimal energy amplification shifts to
lower frequencies, while the spanwise wavenumber remains nearly unaltered (not
displayed here for the sake of brevity). Therefore, one can interpret this
behavior as a concurrent reduction of the Orr mechanism being overcome by the
mere lift-up effect, i.e., $\omega\rightarrow 0$. The decrease in
$\omega_{opt}$ can be well approximated by $\omega_{opt}\propto
Re^{n}_{\delta}$, with $n=-1$. This value resembles the relation obtained by
Ref. [44], i.e., $\omega_{opt}\propto Re^{-0.8}_{\delta}$, when including non-
parallel effects. This trend additionally suggests that in the inviscid limit
of $Re\rightarrow\infty$ the optimal perturbations will have zero frequency
[44]. Moreover, analogous power laws for $G_{opt}$ and $x_{opt}$, similar to
the ones of streamwise-independent perturbations (see Fig. 15), can be
achieved even when the Orr mechanism is at play. Hence, the relations
$x_{opt}\propto Re_{\delta}$ and $G_{opt}\propto Re^{2}_{\delta}$ remain
valid, regardless of the underlying physical mechanism.

Figure 16: Case T09w105: (a) optimal frequency, (b) optimal location, (c)
optimal energy amplification. The dashed black lines indicate the power-law
approximations with corresponding exponent $n$.

At $Re_{\delta}=50$ and $1000$, the optimal perturbations are presented in
Figs. 17(a,b) and (c,d), respectively. As expected, the spanwise and wall-
normal velocity components are most pronounced for the initial condition.
However, a non-negligible input streamwise velocity amplitude at
$Re_{\delta}=50$, especially in the proximity of the Widom line, is visible.
This behavior underscores the influence of the Orr mechanism at low Reynolds
numbers. In contrast, at $Re_{\delta}=1000$, the lift-up effect dominates,
with $\hat{u}_{in}\approx 0$. Additionally, when examining the output spanwise
velocity perturbation $\hat{w}_{out}$, it becomes evident that at
$Re_{\delta}=50$ with $\omega_{opt}\gg 0$, the amplification of the spanwise
velocity induced by the Orr mechanism is significant, consistent with the
findings in Ref. [49].

Figure 17: Case T09w105. Wall-normal profiles of optimal input (a,c) and
output (b,d) disturbances (real part) at two different input Reynolds numbers:
(a,b) $Re_{\delta}=50$, (c,d) $Re_{\delta}=1000$. The boundary-layer thickness
and the location of the Widom line are indicated by $\delta_{99}$ and
$y_{WL}$, respectively.

Recalling the streaks’ shape in Fig. 11 and the evolution of the optimal
frequency in Fig. 16, we might anticipate that the propagation angle of the
oblique streaks is dependent on the initial Reynolds number. To gain a better
understanding, snapshots of the resulting optimal streak tangential velocity
$|\vec{V}^{\prime}_{res}|=\sqrt{u^{\prime 2}+w^{\prime 2}}$ on an $x$-$z$
plane are illustrated in Figs. 18(a,e,h), Figs. (b,f,i), and (c,g,j) at
$Re_{\delta}=50$, $300$, and $1000$, respectively. Optimal perturbations are
obtained with $\beta=\beta_{opt}$ in Figs. (a–c), with $\beta=-\beta_{opt}$ in
Figs. (e–g), and with $\beta=\pm\beta_{opt}$ in Figs. (h–j). To calculate the
streaks’ propagation angle $\tan(\Psi)=\beta_{opt}/\alpha$ (indicated in Fig.
18), the optimal spanwise wavenumber is provided from
$G_{opt}(\beta_{opt},\omega_{opt})$ (see, for instance, Fig. 3(d)). However,
the streamwise wavenumber $\alpha$ cannot be computed, because of the spatial
framework that is used in this study. Its extraction is performed as follows:
the spatial evolution of the wall-normal maximum of the streamwise
perturbation velocity is first evaluated at $z=0$ (see plane D in Fig. 9(b)),
the streamwise wavelength $\lambda_{x}$ is then calculated according to the
wave period of the oscillating streamwise velocity. Thus, the streamwise
wavenumber $\alpha$ is obtained as $\alpha=2\pi/\lambda_{x}$. A remarkable
good agreement between the calculated value of $\Psi$ and the actual streaks’
orientation is achieved. This procedure is repeated at different $Re_{\delta}$
and, as a result, the dependence of $\Psi$ on the Reynolds number is displayed
on a logarithmic axis in Fig. 18(d). As the optimization procedure is carried
out further downstream, streaks progressively evolve into 2-D structures due
to the frequency shift of their optimal transient growth, as illustrated in
Fig. 16. Since $\beta_{opt}$ remains nearly unchanged over the optimization
location, while $\omega_{opt}$ is proportional to $Re^{-n}_{\delta}$, the
propagation angle follows the same power law with an exponent of $n=-1$.
Hence, it is predicted that, as $Re_{\delta}\rightarrow\infty$,
$\Psi\rightarrow 0$ . Note that the current analysis is based on a locally-
parallel spatial optimization procedure with different initial Reynolds
numbers. Therefore, generalizing the actual streak structure to the entire
flat-plate streamwise distance is not feasible. For this purpose, more
advanced non-local methods, such as a PSE-based formulation [47, 24], need to
be employed. As highlighted in Fig. 9(c), oblique streaks can have both
positive and negative propagation angle with the exact same patterns. The
superposition in Figs. (i-k) depends on the initial Reynolds number. When
moving downstream, the checkerboard wave pattern stretches in the streamwise
direction due to a larger streamwise wavenumber.

Figure 18: Case T09w105. In (a–c) and (e–j), contour plots in the $x$-$z$
plane ($y=0.77$) of the resulting streaks’ tangential velocity
$|\vec{V}^{\prime}_{res}|$: (a,e,h) $Re_{\delta}=50$, (b,f,i)
$Re_{\delta}=300$, and (c,g,j) $Re_{\delta}=1000$; (a–c) $\beta=\beta_{opt}$,
(e–g) $\beta=-\beta_{opt}$, and (h–j) $\beta=\pm\beta_{opt}$. The arrow
represents the velocity vector with propagation angle $\Psi$. In (d),
development of the streaks’ propagation angle $\Psi$ as a function of the
initial Reynolds number. The dashed black line indicates the power-law
approximation with exponent $n=-1$.

In order to further understand the reason of the frequency shift in the
optimal energy amplification of the transcritical wall-heating case T09w105,
we analyze the underlying eigenspectrum, which constitutes the eigenvector
basis for matrix $\mathbf{F}$ (see Sec. II.3.3), used for the calculation of
the energy growth $G$ in Eq. (27). In Figs. 19(a–c), the eigenspectra at
$Re_{\delta}=50$, $300$, and $10^{4}$ with the corresponding
$\omega=\omega_{opt}$ are illustrated. This spectrum is obtained by solving
the eigenvalue problem in Eq. (12). As previously discovered by Ref. [8], two
discrete modes (Mode I and II), here damped, are present when the base-flow
temperature crosses the Widom line from a liquid-like free-stream to a gas-
like wall. Note that Mode I is the Tollmien-Schlichting-like mode and will
always exist independently of the thermodynamic regime.
In the eigenvalue spectra of Figs. 19(a–c), the only continuous branches
present at $M_{\infty}=10^{-3}$ are the ones belonging to the vorticity and
entropy modes (indicated by the vertical dash-dotted line with
$\alpha=\omega$). As we are dealing with 3-D disturbances, i.e.,
$\beta=\beta_{opt}>0$, two vorticity modes and one entropy mode are found in
agreement with Ref. [50]. When moving downstream, the continuous branches
become more distinct, whereas the two discrete modes move closer to the
vorticity/entropy modes. At $Re_{\delta}=10^{4}$, in fact, both Mode I and II
are almost synchronized with the continuous spectrum at a phase velocity of
$c_{r}\approx 1$.
As the process of synchronization progresses from $Re_{\delta}=50$ to
$10^{4}$, the two eigenmodes contributing to the largest optimal amplification
are tracked (see purple circles in Figs. 19(a–c)). They are computed via the
right singular eigenvector associated to the largest singular value of
$\mathbf{F}\bm{\Lambda}\mathbf{F}^{-1}$ in Eq. (27). Interestingly, the two
with the highest transient-growth contribution correspond always to Mode I and
II. This implies that, regardless of the initial Reynolds number, the main
input to the energy amplification is attributed to these two slowly-decaying
modes, while the other modes of the continuous spectrum rapidly decay and play
a minor role. For ideal gas at $M_{\infty}=5.0$, a similar process for
discrete modes was witnessed in Ref. [23], where optimal disturbances were
oblique for $x<x_{opt}$, differently than the continuous-spectrum interaction
of [14]. Note that, in case of Ref. [23], the least-damped discrete mode
corresponds to Mack’s second mode, which is different from the current Mode II
in supercritical fluids [8].
Since transient growth is mathematically associated with the non-orthogonality
of the eigenvectors that span the initial value problem, resolving the
underlying eigenspectrum is essential in our study [40]. Moreover, it has been
observed in Ref. [48] that the individual and least-damped Orr-Sommerfeld mode
can interfere with the continuous branches and lead to the largest energy
amplification. Based on these observations, we can investigate how the two
discrete least-damped modes contribute to transient growth and influence the
energy amplification in a spatial framework.
First of all, $G_{opt}$ at finite $\beta_{opt}$ and $\omega_{opt}$, which is
based on the full spectrum of Figs. 19(a–c), is marked by a blue continuous
line in Fig. 19(d). This distribution coincides with the one in Fig. 16(c).
Secondly, Mode II is intentionally excluded from the optimization process. By
doing so, an eigenspectrum complementary to the one of the non-transcritical
cases is obtained. While maintaining the same optimal frequency and spanwise
wavenumber as the previous $G_{opt}$, the modified optimal amplification
$G_{\mathrm{w.o.II}}$ is then calculated and marked by a red continuous line
in Fig. 19(d). When excluding Mode II, one can observe the large drop of $G$,
with $G_{opt}/G_{\mathrm{w.o.II}}\approx 5$. Subsequently, both Mode I and II
are discarded. As a result, the energy amplification $G_{\mathrm{w.o.I+II}}$
(marked in Fig. 19(d) by a yellow continuous line) only experiences a slight
decrease compared to the $G$-value without Mode II. Hence, the substantial
drop in $G$ can be primarily attributed to the abrupt increase in non-ideality
when crossing the Widom line, and consequently, the presence of Mode II. For a
clearer understanding, the maximum energy amplification is displayed in Fig.
19(e) over the frequency at a constant spanwise wavenumber of
$\beta=\beta_{opt}$. When Mode II is omitted from the initial value problem,
the peak optimal growth (marked by a blue star ($\filledstar$) symbol) shifts
from finite frequency values (continuous blue line) to $\omega=0$ (continuous
red line). This behavior resembles the non-transcritical cases in Fig. 3,
where streamwise-independent disturbances were detected. Furthermore, a
comparison between $G_{opt,\mathrm{w.o.II}}$ at $\omega=0$ with $G_{opt}$ of
case T09w095 (marked by a black ($\blacktriangledown$) symbol) reveals similar
values, given the spectra similarity.

Figure 19: Case T09w105. Eigenspectrum at (a) $Re_{\delta}=50$, (b)
$Re_{\delta}=300$, and (c) $Re_{\delta}=10^{4}$ (higher mesh resolution
needed), $\omega=\omega_{opt}$. Blue and red dots represent Mode I and II,
respectively. The 2 modes with the largest contribution to the optimal growth
are circled in purple. The black dash-dotted line refers to the continuous
spectra according to Ref. [50]. In (d), the optimal energy amplification is
plotted over the local Reynolds number for the full spectrum (blue line); note
that $G$ for the full spectrum without Mode II (red line) and full spectrum
without Mode I and II (yellow line) is not anymore $G_{opt}$ for these cases.
In (e), the maximum energy amplification is plotted over the frequency for the
full spectrum (blue line with $G_{opt}$ marked by a blue ($\filledstar$)
symbol), full spectrum w.o. Mode II (red line), and full spectrum w.o. Mode I
and II (yellow line). $G_{opt}$ of case T09w095 (marked by a black
($\blacktriangledown$) symbol.

Physically, if we consider the results presented in Sec. IV.1 and IV.2 in
relation to the role of Mode II, one might infer that the Orr mechanism
disappears once Mode II is removed from the eigenspectrum. Nevertheless, even
in the non-transcritical cases where $G_{opt}$ is located at $\omega=0$, the
Orr mechanism is still present at $\omega>0$ [43], but it only results in sub-
optimal energy growth (see Fig. 13). This situation is actually typical of
wall-bounded shear flows, where the dominant energy growth is associated with
2-D streaks [41]. Yet, for unbounded infinite-shear flows, the combination of
Orr and lift-up mechanism has revealed to be optimal. In fact, transient
growth is inherently three-dimensional and relies on the interplay of Orr
mechanism and lift-up effect [51].
From a modal perspective, with the appearance of Mode II and its
aforementioned highest contribution to transient growth, an additional degree
of non-orthogonality is introduced into the initial value problem [14],
enhancing the interaction between the lift-up effect and Orr mechanism. The
latter significantly amplifies the spanwise velocity [49], as observed in
Figs. 10(a–c) with inclined structures in the spanwise direction and in Fig.
17(b,d) with larger $\hat{w}_{out}$ at a lower Reynolds number. This
particular behavior is supported by the fact that, in all the other cases of
Tab. 2, where only one discrete least-damped mode is present, the interplay of
lift-up and Orr mechanism only produces sub-optimal energy growth (see Fig.
13).

####  IV.3.2 Wall temperature

In the following, the influence of the wall temperature on transient growth is
examined. For the non-transcritical cases in Tab. 2, a preliminary trend was
evident in Fig. 3. Above and below the Widom line, wall cooling has shown
higher amplification levels than wall heating. A more detailed analysis is
conducted in Fig. 20(a) and Fig. 20(c), where the maximum energy amplification
$G_{max}/Re^{2}_{\delta}$ is plotted over the spanwise wavenumber at a fixed
frequency. Streamwise-independent modes, i.e., $\omega=0$, are considered, as
they have proved to be the most non-modally amplified perturbations in Fig. 3.
The wall temperature is modified for subcritical free-stream conditions with a
fixed $T^{*}_{\infty}/T^{*}_{pc}$ at $0.90$, as shown in Fig. 3(a), and for
supercritical free-stream conditions with a fixed $T^{*}_{\infty}/T^{*}_{pc}$
at $1.10$, as shown in Fig. 3(b).

Figure 20: Maximum energy amplification $G_{max}/Re^{2}_{\delta}$ at different
$T^{*}_{w}/T^{*}_{pc}$: (a,b) $T^{*}_{\infty}/T^{*}_{pc}=0.90$, (c,d)
$T^{*}_{\infty}/T^{*}_{pc}=1.10$. In (b), $G/Re^{2}_{\delta}$ is plotted over
$\omega$ at constant $\beta_{opt}$. The optimal amplification $G_{opt}$ is
indicated with a colored star ($\smallstar$) symbol.

The impact of a diabatic wall is more pronounced in the subcritical regime of
Fig. 20(a), where both wall cooling and wall heating cause $G$ to increase
more than in the supercritical regime of Fig. 20(c). Interestingly, it is
found that the lowest value of $G_{opt}$ is not obtained for an isothermal
boundary layer but with a slightly negative wall-temperature gradient in the
supercritical regime. With respect to the optimal growth, it is evident how
$\beta_{opt}$ shifts to higher or lower values in case of wall heating in the
subcritical or supercritical regime, respectively. Instead, $\beta_{opt}$
remains unaffected by a reduction of the wall temperature, especially in the
subcritical regime. Values of $\beta_{opt}$ and $x_{opt}$ reveal an opposite
behavior for the subcritical and supercritical regimes, as they are directly
dependent on the boundary-layer thickness.
A similar investigation is performed for the transcritical wall-heating case
T09w105, which was previously presented in Fig. 3(d). Here, the free-stream
temperature is kept constant at $T^{*}_{\infty}/T^{*}_{pc}=0.90$, whereas the
wall temperature is increased in the gas-like regime. Consequently, the height
at which the Widom line is crossed in relation to the boundary-layer
thickness, i.e., $y_{WL}/\delta_{99}$, rises. Results are reported in Fig.
20(b) at four different wall temperatures. Note that the optimal spanwise
wavenumber is not varying for the oblique streaks, maintaining the value of
case T09w105. With respect to the optimal location, a shift to higher values
is noticed with enhanced wall heating.
From Fig. 20(b), we recognize that a wall-temperature increase above
$T^{*}{pc}$, is only associated to a moderate $G_{opt}$-gain despite the
stronger mean-flow gradients. Remarkably, just after the Widom line is
crossed, optimal growth shifts to finite frequencies. Moreover, we observe
streamwise-modulated streaks up to $T^{*}_{w}/T^{*}_{pc}=1.10$, with a
marginal increase in optimal frequency. From there on, the energy
amplification at very low frequency begins to dominate before $G_{opt}$ is
found at $\omega=0$. Hence, optimal streamwise-independent streaks with
$\Psi=$$$ are obtained for a wall-heating factor $T^{*}_{w}/T^{*}_{\infty}$
greater than $1.22$.
The wall-normal profiles of optimal input and output disturbances at
$T^{*}_{w}/T^{*}_{pc}=1.01$ and $T^{*}_{w}/T^{*}_{pc}=1.15$ are displayed in
Figs. 21(a,b) and (c,d), respectively. The chosen Reynolds number is
$Re_{\delta}=300$. For $T^{*}_{w}/T^{*}_{pc}=1.01$, since $G_{opt}$ is found
at $\omega=\omega_{opt}\approx 0.01$, the Orr mechanism is still at play,
resulting in a small, yet non-zero, streamwise velocity and a larger spanwise
velocity for the input perturbations. The latter, in turn, leads to cross-
stream inclination of the perturbation structures, as observed in Figs.
10(a–c). Conversely, in case $T^{*}_{w}/T^{*}_{pc}=1.15$, optimal
amplifications are found at zero frequency with $\hat{u}_{in}=0$ and a smaller
$\hat{w}_{in}$ than in Fig. 21(a). Despite the Widom line moving away from the
wall, the shape of the input disturbances remains nearly identical, as
previously discovered in Ref. [52] for compressible ideal-gas boundary layers.
On the other hand, the output profiles are greatly affected by the different
transcritical wall heating. The amplitude of the density perturbation (purple
dash-dotted line) at the Widom line increases with its local base-flow
gradient, i.e., $\partial\bar{\rho}/\partial y|_{WL}$ at
$T^{*}_{w}/T^{*}_{pc}=1.15$ is $1.5$ times larger than at
$T^{*}_{w}/T^{*}_{pc}=1.01$. As a consequence, the stronger the wall heating
beyond the Widom line, the larger the strength of the density streaks.
Regarding the output streamwise velocity perturbation (yellow dotted line), we
recognize that its largest peak has moved from the middle of the boundary
layer in Fig. 21(b) to the Widom line in Fig. 21(d). This is due to a stronger
vortex-tilting term ($|\partial\bar{u}/\partial
y\cdot\partial\hat{v}_{in}/\partial z|$) around the Widom line, caused by much
larger perturbation strain rate away from the wall. Thus, the highest increase
in $\hat{\omega}_{y}$ takes place around the Widom line. The second peak of
$\hat{u}_{out}$ in the middle of the boundary layer is now less amplified
compared to case $T^{*}_{w}/T^{*}_{pc}=1.01$.

Figure 21: Transcritical wall heating at $Re_{\delta}=300$. Wall-normal
profiles of optimal input (a,c) and output (b,d) disturbance at two different
wall temperatures: (a,b) $T^{*}_{w}/T^{*}_{pc}=1.01$, (c,d)
$T^{*}_{w}/T^{*}_{pc}=1.15$. The boundary-layer thickness and the location of
the Widom line are indicated by $\delta_{99}$ and $y_{WL}$, respectively.

A visualization of the 3-D optimal density perturbation structures is
displayed in Fig. 22 for $T^{*}_{w}/T^{*}_{pc}=1.01,1.10,1.15$. For the first
two cases, the superposition with $\beta=\pm\beta_{opt}$ is considered,
whereas for the strongest wall-heating case optimal disturbances are found to
be streamwise-independent. Due to a higher value of $\omega_{opt}$ when the
wall temperature is increased (until $T^{*}_{w}/T^{*}_{pc}=1.10$), the oblique
structures have a larger propagation angle with a shorter streamwise
wavelength, resulting in a strong alternate checkered wave pattern.
Additionally, while the largest density structures are confined to the Widom-
line region at $T^{*}_{w}/T^{*}_{pc}=1.01$, they also extend to the mid-
section of the boundary layer at $T^{*}_{w}/T^{*}_{pc}=1.10$ with significant
amplitudes. In the case of $T^{*}_{w}/T^{*}_{pc}=1.15$ in Fig. 22(c), even
stronger density perturbations are observed, yet limited to the Widom-line
region.

Figure 22: Transcritical wall heating at $Re_{\delta}=300$. Iso-contours of
the optimal density perturbations ($\Re(\rho^{\prime}_{out})=\pm 8.0$): (a)
$T^{*}_{w}/T^{*}_{pc}=1.01$ (superposition), (b) $T^{*}_{w}/T^{*}_{pc}=1.10$
(superposition), (c) $T^{*}_{w}/T^{*}_{pc}=1.15$. The red dashed line
indicates the optimal growth location ($x=x_{opt}$).

The same analysis is repeated for the transcritical case T11w095 (see Tab. 2).
Now, the free-stream temperature is kept constant at
$T^{*}_{\infty}/T^{*}_{pc}=1.10$, whereas the wall temperature is varied in
the liquid-like regime from $T^{*}_{w}/T^{*}_{pc}=0.99$ to
$T^{*}_{w}/T^{*}_{pc}=0.85$. First of all, the height of the Widom-line
crossing is subsequently increased by wall cooling. Note also that with
stronger transcritical wall cooling, the boundary-layer thickness grows. In
relation to an equivalent increase in the wall-heating case of Fig. 20(b), we
can recognize that a similar growth of the Widom-line crossing, with respect
to the boundary-layer thickness, is achieved with the same $\Delta
T=|T^{*}_{\infty}-T^{*}_{w}|$ for both transcritical wall heating and cooling.
Nevertheless, the wall-normal gradients of the base-flow properties at the
Widom line, e.g., $|\partial\bar{\rho}/\partial y|_{WL}$, are much larger for
the transcritical wall-cooling case, given the same aforementioned $\Delta T$.
Therefore, higher energy amplification rates are expected to be reached as the
wall temperature is further reduced in the liquid-like regime.
The transient-growth analysis is performed at a Reynolds number of
$Re_{\delta}=300$. Values of $G_{max}/Re^{2}_{\delta}$ are reported in Fig.
20(d) as a function of the spanwise wavenumber for streamwise-invariant
perturbations. The latter are always the optimal ones regardless of the wall
temperature, i.e., $G_{opt}$ is always located at $\omega=\omega_{opt}=0$ in
the transcritical wall-cooling case. Similar to Fig. 20(b), an increase in the
mean-flow gradients, i.e., wall-cooling enhancement, is associated with a rise
of $G$. The rate of increase in $G_{opt}$ (marked by a colored star
($\smallstar$) symbol) over $T^{*}_{w}/T^{*}_{pc}$ is found to be directly
dependent on the $y_{WL}/\delta_{99}$-growth. Thus, a $5\%$ decrease in the
wall temperature results in a $G_{opt}$-gain between $30\%$ and $70\%$
depending on the considered $T^{*}_{w}/T^{*}_{pc}$. Additionally, there is no
significant shift in the optimal spanwise wavenumber, while $x_{opt}$ slightly
decreases. Wall-normal profiles of optimal perturbations exhibit similar
properties as the transcritical wall-heating cases in Fig. 21 (not shown here
for the sake of brevity). Yet, due to a larger $\partial\bar{\rho}/\partial
y|_{WL}$, the output density perturbations reveal much larger amplitudes than
the output velocity perturbations. With respect to the vortex-tilting term
$|\partial\bar{u}/\partial y\cdot\partial\hat{v}_{in}/\partial z|$, both mean-
flow vorticity and perturbation strain rate are greatly enhanced by wall
cooling. Nevertheless, the vortex-tilting term always peaks above $y_{WL}$,
which, in turn, increases with decreasing $T^{*}_{w}/T^{*}_{pc}$. As a result,
the wall-normal vorticity perturbation $\hat{\omega}_{y,out}$ shows a similar
profile as in Fig. 8(c), but more pronounced around the Widom line.
In Fig. 23, iso-contours of optimal density and streamwise velocity
perturbations at $T^{*}_{w}/T^{*}_{pc}=0.85$ are shown. As mentioned earlier,
large-amplitude density streaks can be detected around the Widom line. They
reveal a very similar shape as the superposed structures at
$T^{*}_{w}/T^{*}_{pc}=1.15$ for the transcritical heating case in Fig. 22(c),
yet with a shorter optimal distance.

Figure 23: Transcritical wall cooling at $Re_{\delta}=300$. Iso-contours of
the optimal perturbations at $T^{*}_{w}/T^{*}_{pc}=0.85$: (a) density
($\Re(\rho^{\prime}_{out})=\pm 20.0$), (b) streamwise velocity
($\Re(u^{\prime}_{out})=\pm 5.0$). The red dashed line indicates the optimal
growth location ($x=x_{opt}$).

###  IV.4 Non-modal vs. modal growth

So far in this study, we have maximized the energy amplification over all
possible streamwise distances, e.g., $G_{opt}$ at $x_{opt}$. Nevertheless,
since the streamwise distance is non-dimensionalized by the Blasius length
scale $\delta$, the optimal locations found in the previous sections might be
too large to justify the locally parallel assumption in this study. It holds
for every case in Tab. 1 that the optimal location is proportional to
$Re_{\delta,0}$, the initial local Reynolds number (see Sec. IV.3.1). This
implies that the optimal disturbance needs $O(Re_{\delta,0}\cdot\delta)$ to
grow from the initial location $x_{0}$. Simultaneously, as
$\delta\propto\sqrt{x}\propto Re_{\delta}$, the growth of the boundary-layer
thickness, which is a multiple of the Blasius length scale [35], corresponds
to a factor $\sqrt{x_{opt}}/\sqrt{x_{0}}$ or $Re_{\delta,opt}/Re_{\delta,0}$
from $x_{0}$ to $x_{opt}$. Consequently, to achieve the same increase in
$\delta$, a shorter streamwise distance is required near the leading edge, and
thus, it is here where the locally parallel as well as the boundary-layer
assumption are more likely to fail. Yet, previous studies on the non-parallel
assumption [18, 52] have also revealed similar optimal distances to those of
this study, to Ref. [41] for incompressible flows, and to Ref. [23] for
hypersonic flows. For this reason, further investigations are not undertaken
in the context of this study.
Another important aspect is that, with large values of $x_{opt}$, transient
growth might be irrelevant for transition to turbulence due to a much earlier
exponential growth of modal instabilities. Therefore, it is necessary to
assess which mechanism is more likely to lead to non-linear breakdown for the
cases of Tab. 1. In order to properly compare non-modal with modal
calculations, the integral amplification $N$ needs to be evaluated, since $G$
in Eq. (27) is optimized over a prescribed space interval [14]. Hence, the
integral spatial amplification rate, namely the $N$-factor, is used. The
$N$-factor for modal instabilities is calculated as

$\displaystyle N=\int_{x_{0}}^{x}-\alpha_{i}(\omega/Re_{\delta},\beta)dx,\quad
N_{modal}(x)=\max_{\omega/Re_{\delta},\beta}\\{N(x,\omega/Re_{\delta},\beta)\\},$
(31a,b)

where $x_{0}$ is the streamwise location for neutral instability (branch I),
and $N_{modal}(x)$ represents the envelope of all possible modal $N$-factors
over all frequencies and spanwise wavenumbers. An analysis of the modal growth
of 3-D disturbances for cases in Tab. 1 is reported in Sec. G. To summarize,
for all cases, the maximum growth rate at a constant frequency, i.e.,
$\max_{\omega=\text{const.}}\\{-\alpha_{i}\\}$, is dependent on $\beta$, and
thus 2-D modes are not always the most unstable ones. Yet, the maximum
integral amplification $N$ is always found for 2-D modes. Hence, the modal
$N$-factor, $N_{modal}$, is only dependent on $x$, i.e.,
$N_{modal}(x)=\max_{\forall\omega}\\{N(\beta=0)\\}$.
On the contrary, the $N$-factor for non-modal (optimal) growth aligns with the
calculations of [23, 47, 53] and is expressed as:

$\displaystyle N_{opt}(x,Re_{\delta,0})=0.5\ln\left(G\right).$ (32)

Here, $N_{opt}$ is calculated along the streamwise direction and depends on
the initial Reynolds number $Re_{\delta,0}$. For all cases, $G$ is chosen as
the optimal amplification $G_{opt}$. Note that, very short-distance energy
amplification, for instance in Fig. 14(a), is larger than $G_{opt}(x)$ at
$x\ll x_{opt}$, yet, as $Re_{\delta,0}$ is continuously varied over the flat
plate, $G_{opt}$ will be always the global maximum.
For both non-transcritical and transcritical wall-cooling cases,
$\omega_{opt}$ is always zero. However, in the transcritical wall-heating
case, due to a streaks’ propagation angle that varies with $Re_{\delta,0}$,
$\omega_{opt}$ is not constant (see Fig. 16(a)), and thus, $G$ needs to be
adapted in the streamwise direction. With regards to the finite optimal
spanwise wavenumber $\beta_{opt}$, as was previously found in Sec. IV.3.1, it
remains almost constant with respect to the chosen initial Reynolds number.
Thus, $\beta_{opt}$ is initially selected as the optimal one at
$Re_{\delta,0}=300$ (see Tab. 2) before being optimized for the initial
perturbation at $Re_{\delta,0,N}$ (initial optimal location) resulting in
$N_{modal}=N_{opt}$. Furthermore, note that, for the sake of comparison, the
calculation of $N_{opt}$ does not include modal growth. Thus, the infinite
energy amplification, caused by an unstable mode in the eigenspectrum (see,
for instance, the modally unstable region in Fig. 3(a)), is discarded here.
This approach ensures that only finite $G_{opt}$, caused only by stable
eigenmodes, are considered, enabling a comparison of the spatial energy
evolution due to transient growth with the integral amplification in Eq.
(31a,b) arising from modal instabilities.
The envelope of $N_{opt}$, denoted as $N_{opt,env.}$, represents the maximum
level of transient growth achievable at every streamwise location. Despite the
actual growth of the boundary-layer, non-modal calculations are performed
under the aforementioned locally parallel assumption. However, non-parallel
results of Ref. [52] indicated a difference in $N_{opt}$ of only $0.4$–$0.5$
[23], which is deemed acceptable for the purpose of comparison to the modal
growth.
In Figs. 24(a,b), the comparison between non-modal and modal growth is
displayed for the subcritical case T09w095 and supercritical case T11w120,
respectively. Transient-growth calculations are performed for $Re_{\delta,0}$
varying from 100 to 2000 with a step size of $\Delta Re_{\delta,0}=100$. The
corresponding $N$-factor, denoted as $N_{opt}$ (continuous light blue line),
is calculated as in Eq. (32). The envelope of $N_{opt}$ is represented by a
continuous dark blue line, while the modal $N$-factor curve, obtained from Eq.
(31a,b), is illustrated by a continuous red line.

Figure 24: $N$-factor calculation: $N_{modal}$ ( ) is the $N$-factor obtained
from modal stability analysis, $N_{opt}$ ( ) is the $N$-factor corresponding
to transient growth, and $N_{opt,env.}$ ( ) is the envelope of $N_{opt}$. (a)
T09w095, (b) T11w120. The optimal growth originating from the optimal initial
position $x_{0,N}$ is indicated with a dashed blue line. The location $x_{N}$
at which $N_{modal}=N_{opt}$ is denoted with a black star ($\filledstar$)
symbol. In the inset, the optimal growth is varied over the spanwise
wavenumber (black arrow); black dashed line illustrates the maximum optimal
growth at $\beta_{N}$.

In the case T09w095, modal growth overtakes non-modal growth at
$Re_{\delta}\approx 2010$, whereas for case T11w120, due to wall heating in
the gas-like regime, modal growth is enhanced and prevails over transient
growth already at $Re_{\delta}\approx 595$. Given the final optimization
position $x_{N}$ by the location at which $N_{modal}=N_{opt}$ (see black star
($\filledstar$) symbol in Fig. 24), we can investigate the optimal initial
position $x_{0,N}$ for transient growth. It is observed that close to the
leading edge, optimal disturbances grow at a faster rate than further
downstream. Consequently, for case T09w095, an optimal growth of
$N=N_{opt}=N_{modal}=4.6$ is reached within $\Delta Re_{\delta}\approx 550$ or
$\Delta x\approx 0.35$, while for case T11w120, it occurs within $\Delta
Re_{\delta}\approx 170$ or $\Delta x\approx 0.08$ with a lower optimal
$N$-value. The insets in Fig. 24 illustrate the effect of an increasing
spanwise wavenumber on the optimal $N$-factor (depicted by the dashed blue
line), considering that all the non-modal calculations in Fig. 24 were
performed at a fixed $\beta=\beta_{opt}(Re_{\delta}=300)$. Despite the shift
of $\beta_{opt}$, the increase in $N$ is marginal for all the non-
transcritical cases.
Tab. 3 provides a summary of the comparison between non-modal and modal growth
for the non-transcritical cases.

Table 3: Summary of the $N$-factor comparison between modal and non-modal growth for subcritical and supercritical cases. $Re_{\delta,0,N}$, $Re_{\delta,N}$, and $\beta_{N}$ indicate the initial optimal Reynolds number, the local Reynolds number at which $N_{modal}$ exceeds $N_{opt}$, and the optimal spanwise wavenumber at $Re_{\delta,N}$, respectively. $N(Re_{\delta,N})$ is the $N$-factor at this position, with its corresponding energy amplification $G_{opt}(Re_{\delta,N})=\exp\\{N(Re_{\delta,N})\\}^{2}$. Cases T09w090 and T11w110 are identical. Case | $Re_{\delta,0,N}$ | $Re_{\delta,N}$ | $\beta_{N}$ | $N(Re_{\delta,N})$ | $G_{opt}(Re_{\delta,N})$
---|---|---|---|---|---
T09w085 | 440 | 605 | 0.45 | 3.4 | 947
T09w090 | 760 | 1040 | 0.46 | 3.9 | 2314
T09w095 | 1470 | 2010 | 0.49 | 4.6 | 9725
T11w105 | 1240 | 1705 | 0.47 | 4.4 | 6966
T11w120 | 425 | 595 | 0.44 | 3.3 | 739

Transient growth, which is also present in the subcritical flow region
($Re<Re_{c}$), yields a maximum $N$-factor not larger than 4.6 up to
$Re_{\delta}=2000$, corresponding to energy amplifications of about $10^{4}$,
with $G=\exp(N)^{2}$. This happens only when the wall is heated in the liquid-
like regime or cooled in the gas-like regime towards the Widom line. On the
other hand, it is apparent that modal growth dominates the transition to
turbulence once the wall is cooled in the subcritical regime or heated in the
supercritical regime.
In order to draw any transition prediction, the initial disturbance energy and
the underlying receptivity process need to be known. Values of $N$ for non-
linear breakdown or transition to turbulence are dependent on various factors,
such as the disturbance sources [23]. For instance, transient growth can play
a key role in a noisy environment with high free-stream turbulence levels
$\mathrm{Tu}$ [53] or with discrete wall roughness elements [54]. Despite the
similarities in the here-obtained disturbance structures with other ideal-gas
transient-growth studies, the establishment of accurate analogies is extremely
limited due to a lack of transition experiments and the unknown receptivity
mechanisms in boundary layers at supercritical pressure. Hence, hypotheses on
whether transient growth is critical for the cases selected in this study
remain questionable.

Figure 25: $N$-factor calculation: $N_{modal}$ ( ) and ( ) are the $N$-factors
obtained from Mode I and II, respectively, according to the modal stability
analysis; $N_{opt}$ ( ) is the $N$-factor corresponding to transient growth,
and $N_{opt,env.}$ ( ) is the envelope of $N_{opt}$. (a) T09w105, (b) T11w095.
The optimal growth originating from the optimal initial position $x_{0,N}$ is
indicated with a dashed blue line. The location $x_{N}$ at which
$N_{modal}=N_{opt}$ is denoted with a black star ($\filledstar$). In the
inset, the optimal growth is varied over the spanwise wavenumber (black
arrow); black dashed line illustrates the maximum optimal growth at
$\beta_{N}$.

The same analysis of Fig. 24 is undertaken for the transcritical cases in Fig.
25. Note that for case T09w105, the calculation of $N_{opt}$ involves
modifying the optimal-disturbance frequency as a function of $Re_{\delta,0}$,
as illustrated in Fig. 16(a). Overall, the transcritical cases exhibit a much
stronger modal growth compared to the non-transcritical cases. With
transcritical wall heating (case T09w105), the highly unstable Mode II,
resulting from an inviscid instability (see GIP in Fig. 2(b), despite the
fuller velocity profile), is present (marked by a yellow line). It is worth
noting that Mode I (marked by a red line) is also more unstable than its
counterpart in the liquid-like regime, given the same wall-to-free-stream
temperature ratio $T^{*}_{w}/T^{*}_{\infty}$ of $1.167$. For case T09w105, the
larger growth rates of Mode II compared to Mode I and the low level of non-
modal energy amplification let the modal instability dominate further
downstream around $Re_{\delta}\approx 590$. Consequently, this suggests that
at these transcritical conditions, subcritical growth is present, yet with
modest amplification rates. As pointed out before for the non-transcritical
cases, transient growth might be critical here only if a trivial analogy is
drawn with the other ideal-gas transient-growth studies.
When the wall temperature is cooled over the Widom line (case T11w095), an
even stronger modal instability, also of inviscid nature, is found. Note that
in this transcritical wall-cooling case, only one mode is unstable in the
eigenspectrum regardless of the considered frequency and Reynolds number. In
this scenario, the GIP moves away from the wall, given the immense viscosity
gradient at the wall [12], while the $\bar{u}(y)$-profile becomes inflectional
above the Widom line, leading to a strong inviscid instability. As evident in
Fig. 25(b), the exponential growth (marked by a yellow line) overtakes the
algebraic growth already close to the flat-plate leading edge due to the
aforementioned inviscid instability. Thus, subcritical disturbance growth is
irrelevant here. Note that the boundary-layer assumption might be not adequate
here due to leading-edge effects. Regarding the optimal spanwise wavenumber in
the insets of Fig. 25, one can recognize that the optimal $N$-factor is not
affected by a $\beta$-change. A recap of the $N$-factor analysis is showcased
in Tab. 4.

Table 4: Summary of the $N$-factor comparison between modal and non-modal growth for transcritical cases. $Re_{\delta,0,N}$, $Re_{\delta,N}$, and $\beta_{N}$ indicate the initial optimal Reynolds number, the local Reynolds number at which $N_{modal}$ exceeds $N_{opt}$, and the optimal spanwise wavenumber at $Re_{\delta,N}$, respectively. $N(Re_{\delta,N})$ is the $N$-factor at this position, with its corresponding energy amplification $G_{opt}(Re_{\delta,N})=\exp\\{N(Re_{\delta,N})\\}^{2}$. Case | $Re_{\delta,0,N}$ | $Re_{\delta,N}$ | $\beta_{N}$ | $N(Re_{\delta,N})$ | $G_{opt}(Re_{\delta,N})$
---|---|---|---|---|---
T09w105 | 420 | 590 | 0.53 | 3.5 | 1097
T11w095 | 30 | 52 | 0.40 | 1.7 | 30

In Fig. 26, a final comparison among all cases of this study at supercritical
pressure is unveiled, i.e., Fig. 26(a) for a liquid-like free stream (cases
T09w090, T09w085, T09w095, T09w105) and Fig. 26(b) for a gas-like free stream
(cases T11w110, T11w105, T11w120, T11w095). As a function of $Re_{x}$ in
logarithmic scaling, non-modal growth is represented as straight line. Non-
transcritical cases exhibit $N_{opt,env.}$-factors that are essentially
independent of any small wall-to-free-stream temperature variation in the
proximity of the Widom line. Regarding the transcritical cases, cooling over
the Widom line in Fig. 26(b) significantly increases $N_{opt,env.}$. For
example, $N_{opt,env.,\mathrm{T11w095}}/N_{opt,env.,\mathrm{T11w110}}\approx
1.34$. Nevertheless, this rise is much more pronounced for modal growth (see
dashed lines). When examining the $N$-factors in Fig. 26 for a Falkner-Skan
boundary layer with adverse (APG, Hartree parameter $\beta_{H}=-0.1$) and
zero-pressure (ZPG) gradient according to Ref. [53], three main conclusions
can be drawn. First, it is observed that an increase in the APG (black circle
($\circ$) symbol) has a similar effect on transient growth as heating or
cooling over the Widom line in a ZPG boundary layer at supercritical pressure,
with transcritical wall cooling revealing a larger amplification. Yet, in case
T11w095, exponential growth with a strong inviscid instability dominates from
the leading edge. Secondly, it was predicted in Ref. [53] that, as the APG
increases, the required level of free-stream turbulence for bypass transition
decreases. In fact, when $\mathrm{Tu}>3$, $Re_{x,tr}$ was seen to shift to
values below $10^{5}$. If this scenario was also confirmed in a non-ideal
boundary-layer flow at supercritical pressure, then we would assume that
transient growth could be critical for transition in case T09w095, as non-
modal growth in Fig. 26(a) still dominates over modal growth below
$Re_{x}\approx 3\times 10^{5}$. Lastly, if the ZPG-case of Ref. [53] (black
square ($\square$) symbol) is considered, a difference of $N\approx 0.4$–0.5
to case T09w090 is detected. This comparison verifies the robustness of the
locally parallel assumption used in this study for non-modal optimization with
respect to an adjoint-based optimization algorithm with parabolized stability
equations in Ref. [53].

Figure 26: Envelope curves of optimal non-modal and modal growth: (a) liquid-
like free-stream cases with $T^{*}_{\infty}/T^{*}_{pc}=0.90$, (b) gas-like
free-stream cases with $T^{*}_{\infty}/T^{*}_{pc}=1.10$. $N_{opt,env.}$:
continuous line, $N_{modal}$: dashed line. For case T09w105, one common modal
$N$-factor is calculated for Mode I and II. The black circle ($\circ$) and
square ($\square$) symbols refer to the adverse-pressure and zero-pressure
gradient case of Ref. [53], respectively.

Lastly, the influence of the relative pressure, i.e., $p_{r}=p^{*}/p^{*}_{c}$,
on the transient growth is investigated. It is known that, as one approaches
the critical point, the mean-flow gradients of thermo-physical properties are
larger in the proximity of the Widom line (see, for instance, $c^{*}_{p}$ in
Fig. 29), and thus, the base flow becomes even more inflectional. This leads
to a stronger modal growth [8], making subcritical transition less likely to
occur. Instead, the relative pressure is increased hereafter to assess whether
transient growth could be the critical transition mechanism. Therefore, the
supercritical pressure, constant at
$p^{*}=$80\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$$ (i.e., $p_{r}=1.083$) so
far, is modified to $81.15\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$
($p_{r}=1.10$) and $84.84\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$
($p_{r}=1.15$) for both cases T09w105 and T11w095. With respect to the other
base-flow parameters in Tab. 1, they are marginally influenced by the pressure
variation and remain nearly constant, given the Mach number of
$M_{\infty}=10^{-3}$. In Fig. 27, the corresponding base-flow profiles of
$d(\bar{\rho}\,d\bar{u}/dy)dy$ decrease for the two transcritical cases with
increasing relative pressure. In this regard, the location of the generalized
IP is barely modified. At the same time, for case T11w095 in Fig. 27(b) the
location of the IP (colored star ($\smallstar$) symbol) moves slightly away
both from the wall and the Widom line, at which $y_{WL}\approx y_{GIP}$.

Figure 27: Generalized inflection point of the boundary-layer profile at
different relative pressures ($p_{r}=1.083,1.10,1.15$): (a) case T09w105, (b)
case T11w095. Note that $\delta_{99}$ is not altered by a pressure change.

For each relative pressure, the same analysis conducted in Fig. 26 is
performed and displayed in Fig. 28. Firstly, regardless of the selected
supercritical pressure, the same optimal growth mechanisms for the
transcritical regime previously analyzed in Sec. IV.2 are observed: lift-up
and Orr mechanism for T09w105, and only lift-up effect for T11w095.

Figure 28: Effect of $p_{r}$-increase on the envelope curves of optimal non-
modal and modal growth: case T09w105 ( ), case T11w095 ( ). In (a),
$p_{r}=1.083$: continuous lines, $p_{r}=1.10$: dashed lines, and $p_{r}=1.15$:
dash-dotted lines. In (a), black arrows indicate an increase in relative
pressure. The black circle ($\circ$) symbol refers to the adverse-pressure of
Ref. [53]. In (b), $N$-factors are indicated at the Reynolds number $Re_{x,N}$
where $N_{modal}=N_{opt}$.

Fig. 28(a) presents the envelope curves for modal and non-modal growth. The
robustness of transient growth with respect to a thermodynamic variation is
again verified here. $N_{opt,env}$ is only slightly affected by the pressure
change, being negligible for the transcritical wall-heating case T09w105.
Regarding modal growth, the following observations can be noted. In case
T09w105, as shown in Fig. 27(a), the base-flow profile becomes less
inflectional with a relative-pressure increase, leading to a strong
stabilization of Mode II (note that $Re_{x}$ is plotted on a logarithmic
scale). Simultaneously, Mode I remains almost invariant to the pressure
change. Therefore, at $p_{r}=1.15$, Mode I dominates the modal instability
until $Re_{x}\approx 2.2\times 10^{6}$, where a kink in the $N$-factor curve
is visible (marked by a colored ($\square$) symbol). Subsequently, the
Reynolds number $Re_{x,N}$ at which $N_{modal}=N_{opt}$ is displayed in in
Fig. 28(b). $Re_{x,N}$ grows linearly by approximately a factor of $3$ from
$p_{r}=1.084$ to $1.15$ (continuous red line). Yet, the level of non-modal
amplification remains moderate, with a chance of subcritical transition by
strong free-stream disturbances or surface roughness. In case T11w095, a
similar behavior of the GIP to case T09w105 is witnessed in Fig. 27(b) as a
function of $p_{r}$. Nevertheless, the increase of the inflectional location,
at which $d^{2}\bar{u}/dy^{2}=0$, leads to a weakly stabilized modal
instability. In contrast, non-modal amplification is again almost unaffected
by the pressure change. Hence, we can notice in Fig. 28(b) that $Re_{x,N}$ has
more than doubled from $p_{r}=1.083$ to $1.15$ (continuous blue line), but
near the flat-plate leading edge, $N_{opt}$ remains still too low for
subcritical transition. In conclusion, when considering a transcritical wall-
cooled boundary layer, modal growth dominates in the neighborhood of the
critical point. For transcritical wall-heating, under the assumption of large
free-stream disturbances as in [53], bypassing the exponential growth of Mode
I and II might be much more likely when the gradients of thermo-physical
properties are no longer abrupt.

##  V Conclusion

Transient growth of diabatic flat-plate boundary layers at supercritical
pressure is studied. Supercritical carbon dioxide ($\text{CO}_{2}$) at a
constant pressure of $80\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$ (critical
pressure $73.9\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$) is selected as a
representative non-ideal fluid in the proximity of the Widom line. Optimal
energy amplifications and disturbances are calculated in the spatial framework
by means of a singular value decomposition of the locally-parallel, linearized
compressible Navier-Stokes equations. To account for non-ideality and strong
thermo-physical property variations around the Widom line, a new non-ideal
energy norm is developed such that pressure work is eliminated. The new norm
is equal to Chu’s [36] and Mack’s [17] norms under the ideal-gas assumption.
This study explores the subcritical, supercritical, and transcritical regimes
(with respect to the $\text{CO}_{2}$ pseudo-boiling temperature of
$T^{*}_{pc}=$307.7\text{\,}\mathrm{K}$$), determined by prescribing free-
stream and wall temperatures at a fixed Mach number of $10^{-3}$. In the
subcritical (liquid-like: $T^{*}_{\infty}/T^{*}_{pc}=0.90$ with
$T^{*}_{w}<T^{*}_{pc}$) and supercritical (gas-like:
$T^{*}_{\infty}/T^{*}_{pc}=1.10$ with $T^{*}_{w}>T^{*}_{pc}$) regimes, wall
cooling enhances transient growth. For the subcritical cases, the optimal
streamwise distance $x_{opt}$ decreases, while the optimal spanwise wavenumber
$\beta_{opt}$ increases as the wall temperature approaches the Widom line.
Conversely, in the supercritical regime, the opposite trend is observed. The
optimal energy amplification is detected at finite spanwise wavenumbers, yet
the optimal frequency is zero, indicating that the optimal disturbances
correspond to counter-rotating vortices, which evolve into streamwise-
independent streaks. This corresponds to the well-known lift-up mechanism, in
agreement with ideal-gas results. Moreover, scaling laws, e.g., $G\propto
Re_{\delta}^{2}$ and $x_{max}\propto Re_{\delta}$, consistently hold.
When the wall temperature crosses the Widom line (transcritical wall-heating:
$T^{*}_{\infty}/T^{*}_{pc}=0.90$ with $T^{*}_{w}>T^{*}_{pc}$; transcritical
wall-cooling: $T^{*}_{\infty}/T^{*}_{pc}=1.10$ with $T^{*}_{w}<T^{*}_{pc}$),
the optimal structures are no longer always streamwise independent. In the
transcritical wall-heating regime, the optimal frequency is finite, suggesting
the involvement of another mechanism: the Orr mechanism. In fact, through the
analysis of the Reynolds stress, we observe that optimal disturbance
structures are initially tilted against the mean shear before being re-
oriented to it. A strong contribution of the Orr mechanism is observed for
sub-optimal transient growth, where larger local energy-amplification peaks
than the global optimal one can be achieved for $x\ll x_{opt}$. This leads to
highly-oblique sub-optimal disturbance structures, which do not originate from
a streamwise vortex. The Orr mechanism exists also for perturbations with a
negative spanwise wavenumber. Thus, the resulting superposition forms a
checkerboard pattern, analogous to the oblique-transition mechanism in Ref.
[46]. Moreover, streamwise-modulated streaks in the transcritical wall-heating
regime exhibit significant streamwise velocity disturbances and strong thermal
components ($\hat{\rho}$ and $\hat{T}$) around the Widom line. Similar large
density streaks are also observed in the transcritical wall-cooling regime,
but they are not oblique. Given the same $|T^{*}_{w}-T^{*}_{\infty}|$, larger
energy amplification rates are achieved in the transcritical wall-cooling
regime. This increase in $G$ can be explained by analyzing the vortex-tilting
mechanism, which highlights the alignment above the Widom line of the output
wall-normal vorticity with the highest wall-normal displacement. Conversely, a
misalignment of the two is present in the transcritical wall-heating regime,
causing a smaller $G$ but with a second peak of the optimal streamwise
velocity at the Widom line.
The effect of initial Reynolds number and wall temperature for the
transcritical cases is also investigated in this study. Scaling laws hold for
both optimal streamwise-independent and -modulated disturbances. Particularly,
when considering the case of optimal oblique streaks in transcritical wall-
heating, we observe a dependency of the optimal frequency $\omega_{opt}$ on
$Re^{-1}_{\delta}$, while the optimal energy $G_{opt}$ scales with
$Re^{2}_{\delta}$. This implies that, as $Re_{\delta}\rightarrow\infty$, the
Orr mechanism is not active, and streamwise-independent structures are
recovered. Consequently, the streaks’ propagation angle $\Psi$ becomes
proportional to $Re^{-1}_{\delta}$. When investigating the underlying
eigenspectrum in transcritical wall-heating, it is noticeable how the non-
orthogonal eigenfunctions can influence the optimal growth. In fact, the here
stable transcritical Mode II is found to actively participate in the interplay
of lift-up and Orr mechanism. If this mode is intentionally excluded from the
optimization procedure, both mechanisms yield a significantly reduced sub-
optimal growth only. As a result, the modified optimal transient growth is
solely associated with streamwise-independent disturbances, which grow via the
lift-up mechanism.
When the wall temperature is increased in the transcritical wall-heating
regime, there is first a shift to higher optimal frequencies. However, once
the Widom line moves away from the near-wall layer, the optimal energy growth
is found at $\omega=0$. Stronger wall heating induces, in turn, a rise in the
amplitude of density perturbations around the Widom line due to greater mean-
flow density gradients. On the contrary, in transcritical wall-cooling,
$G_{opt}$ is always located at $\omega_{opt}=0$, while $\beta_{opt}$ slightly
increases. Notably, by changing the wall temperature by only 5%, one can
achieve a gain in $G_{opt}$ of 30% to 70%. Furthermore, similar optimal
disturbance structures are detected when, for both transcritical regimes, the
optimal perturbations are streamwise independent.
Ultimately, a comparison between modal and non-modal growth is presented. For
this purpose, the modal instability of oblique perturbations is first studied
for both non-transcritical and transcritical cases. At the low Mach number of
$10^{-3}$ in this study, 2-D modes exhibit the highest amplifications, both
locally (across most of the frequency spectrum) and integrally. This suggests
that, following the oblique modal stability analysis at finite Mach numbers in
Ref. [8], decreasing the Mach number (compressibility effects) causes a shift
of the largest possible local and integral amplification from a 3-D to a 2-D
mode, irrespective of the considered thermodynamic regime at supercritical
pressure.
Subsequently, modal and non-modal $N$-factors are compared for all regimes. We
note that no critical $N$-factors of transition have been experimentally
measured for flows at supercritical pressure so far, and hence transition-