# The extended phase space thermodynamics of Planck-scale-corrected Reissner-
Nordström-anti-de Sitter black hole

Iarley P. Lobo<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Chemistry and Physics, Federal University of Paraíba, Rodovia BR 079 - Km 12,
58397-000 Areia-PB, Brazil  Physics Department, Federal University of Lavras,
Caixa Postal 3037, 37200-000 Lavras-MG, Brazil  Luis C. N. Santos
<EMAIL_ADDRESS>Physics Department, Federal University of Paraíba, Caixa
Postal 5008, 58059-900, João Pessoa, PB, Brazil  V. B. Bezerra
<EMAIL_ADDRESS>Physics Department, Federal University of Paraíba, Caixa
Postal 5008, 58059-900, João Pessoa, PB, Brazil  J. P. Morais Graça
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal do Rio de
Janeiro, 21.941-972 - Rio de Janeiro, RJ, Brazil  H. Moradpour
<EMAIL_ADDRESS>Research Institute for Astronomy and Astrophysics
of Maragha (RIAAM), University of Maragheh, P.O. Box 55136-553, Maragheh, Iran

###### Abstract

We analyze the effect of Planck-scale modified radiation equation of state on
the Reissner-Nodström-anti-de Sitter black hole inspired by Kiselev’s ansatz.
Deformed thermodynamic quantities are found, phase transitions and black holes
as heat engines are described for the Carnot and square cycles. Non-trivial
differences between linear and quadratic Planck-scale corrections are
discussed in detail.

##  I Introduction

A consistent quantization of gravitational degrees of freedom has been one of
the main quests of theoretical physics in the recent decades. Many proposals
based on very different approaches to this problem have emerged, where each of
them is responsible for filling an aspect of this general problem. Among some
of the most prominent programs we highlight, for instance, loop quantum
gravity (LQG) Rovelli:1997yv , causal dynamical triangulation Ambjorn:2010rx ,
causal sets approach Henson:2006kf , string theory Mukhi:2011zz . The maturity
of the quantum gravity problem and technological advances have led the
community to propose phenomenological approaches that can capture some of the
main properties of the proposed solutions of this problem in a bottom-up way,
where deformations of known concepts and equations of general relativity and
quantum mechanics are considered, governed by parameters that can become
important when quantum gravitational effects appear (we refer the reader to
the review AmelinoCamelia:2008qg ).

Amongst these phenomenological proposals, some of the main sources of
constraints are provided by deformations or violations of the Lorentz
transformations, induced by the quantum gravity energy scale (supposedly of
the order of the Planck energy) Ackermann:2009aa ; Acciari:2020kpi ;
Albert:2019nnn . In summary, the interaction of particles with quantum
gravitational degrees of freedom could be described kinematically by a
modified Hamiltonian or dispersion relation, such that trajectories would be
modified and symmetries could be deformed. Besides that, a fruitful way of
realizing the kinematics of the particles endowing corrections in intermediary
regime between the full quantum gravity theory and special/general relativity
consists in describing their motion as if they propagate in a spacetime that
depends on properties of individual particles themselves, like the ratio
between their energy/momentum and Planck energy Magueijo:2002xx ;
Girelli:2006fw ; Amelino-Camelia:2014rga ; Lobo:2016xzq ; Lobo:2016lxm ;
Lobo:2020qoa ; Barcaroli:2015xda ; Weinfurtner:2008if ; Assaniousssi:2014ota ;
Ling:2005bp ; Ling:2005bq ; Li:2008gs , or properties of fluids of particles,
like the ratio between its energy density and Planck energy Olmo:2011sw  or
the fluid’s temperature and Planck temperature Lobo:2019jdz .

These phenomenologically-inspired modifications have been demonstrated to be a
useful tool able to scrutinize the Planckian regime without the need of
passing through specific details of each quantum gravity proposal, i.e., this
approach allows one to derive known Planck-scale corrections to some
expressions, for instance, the logarithmic correction of Hawking-Bekenstein
black hole entropy formula AmelinoCamelia:2004xx ; Amelino:2006 , to
regularize the behavior of a black hole during its final stages of existence
Adler:2001vs , or to describe cosmological bounces derived from loop quantum
cosmology Gorji:2016laj .

When applying such procedure to the photon gas problem, it has been long known
that one detects non-trivial modifications of the equation of state parameter
Amelino:2006  proportional to the ratio between the temperature of the fluid
and Planck temperature. On the other hand, it has been found by Kiselev a
solution of the general relativity field equations Kiselev:2002dx  that is
able to modify the static black hole metric in a way that depends on an
averaged equation of state parameter of the fluid that surrounds the black
hole. In particular, the Reissner-Nordström solution has been shown to
correspond to the special case in which the average equation of state
parameter corresponds to that of a photon gas. Planck-scale corrections of the
Reissner-Nordström metric induced by the above mentioned modified
thermodynamics have been recently analyzed by some of the authors in
Lobo:2019jdz . Besides that, the impact of the trans-Planckian regime and the
novel notion of thermal dimension Amelino-Camelia:2016sru ; Husain:2013zda ;
Nozari:2015iba  on the evaporation of charged black holes have been studied in
Lobo:2020oqb .

On the other hand, the Reissner-Nordström black hole has proven to be an
important tool for research in thermodynamical aspects of gravity, when
considered along with a negative cosmological constant. In fact, the extended
phase space thermodynamics of the Reissner-Nordström-anti-de Sitter black hole
has been analyzed in recent years from the seminal papers Kastor:2009wy ;
Kubiznak:2012wp , and from then it has been explored in a large amount of
scenarios Brihaye:2018nta ; Ovgun:2017bgx ; Ma:2017pap ; Upadhyay:2017fiw ;
Fernando:2016sps ; Hendi:2015kza ; Dehghani:2014caa ; Mo:2014mba ;
Hendi:2016njy ; Liu:2016uyd ; Majhi:2016txt ; kiselevgb , in which the main
issue under analysis are phase transitions and heat engines Johnson:2014yja
induced by the presence of the electric charge or some other terms due to
different matter contents surrounding the black hole, like clouds of strings,
quintessence, etc, or alternative theories of gravity Balart:2019uok ;
Zhang:2018hms ; Fernando:2018fpq ; EslamPanah:2018ums ; Mo:2018hav ;
Rosso:2018acz ; Zhang:2018vqs ; Chakraborty:2017weq ; Wei:2017vqs ;
Hendi:2017bys ; Xu:2017ahm ; Liu:2017baz ; Hennigar:2017apu ;
Chakraborty:2016ssb ; Zhang:2016wek ; Bhamidipati:2016gel ; Johnson:2015fva ;
Johnson:2015ekr ; Setare:2015yra ; Lobo:2019put  (see Kubiznak:2016qmn  for a
comprehensive review on the subject).

Therefore, a natural question is imposed: what is the effect of quantum
corrections induced by departures of the classical equation of state parameter
of the radiation fluid $\omega=1/3$ in the scenario of black holes in the
extended phase space that includes a negative cosmological constant, i.e., how
are affected the thermodynamic quantities, phase transitions and efficiency of
this system operating as a heat engine? Answering these questions is the main
goal of this paper.

In this paper, we explore the non-trivial effect of pressure induced by a
negative cosmological constant on the thermodynamics of the Planck-scale
modified Reissner-Nordström-anti-de Sitter spacetime. In section II, we
revisit Kiselev’s solution of a black hole surrounded by matter content with
an average equation of state parameter. In section III, we calculate
modifications in the equation of state parameter induced by Planck-scale
modified dispersion relation. In section, IV, we derive the thermodynamic
quantities in the extended phase space assuming the results of the previous
sections and a negative cosmological constant. In section V, we describe the
impact of the quantum gravity parameter on phase transitions. In section VI,
we calculate the effect of the Planck scale parameter on the efficiency of the
black hole system when working as a heat engine. We draw our final remarks in
section VII.

We assume $c=\hbar=k_{B}=1$, for simplicity, and metric signature $(+---)$.

##  II Kiselev’s solution

The solution of general relativity’s field equations describing a static,
spherically symmetric black hole surrounded by an average quintessence fluid
was found in the seminal paper Kiselev:2002dx  by Kiselev, in which the metric
and stress energy-tensor of the surrounding matter is given by

$\displaystyle ds^{2}=A(r)dt^{2}-\frac{dr^{2}}{A(r)}-r^{2}d\Omega^{2}\,,$ (1)
$\displaystyle T^{t}{}_{t}=T^{r}{}_{r}=-\rho(r)\,,$ (2) $\displaystyle
T^{\theta}{}_{\theta}=T^{\phi}{}_{\phi}=\frac{1}{2}(3\omega+1)\rho(r)\,,$ (3)

where $A(r)$ is a function of the radial coordinate $r$, $\rho(r)$ is the
energy density distribution and $\omega$ is an equation of state parameter
that relates the average pressure and energy density. This procedure allows us
to effectively capture the physical properties of the surrounding fluid and
will be of great usefulness for our purposes.

From Einstein’s field equations,

$G_{\mu\nu}=\kappa T_{\mu\nu}\,,$ (4)

we are able to find the shape of the energy-density profile $\rho(r)$ and the
metric function $A(r)$:

$\displaystyle\rho(r)=\frac{c}{\kappa}\frac{3w}{r^{3(w+1)}}\,,$ (5)
$\displaystyle A(r)=1-\frac{2GM}{r}+\frac{c}{r^{3w+1}}\,.$ (6)

From equations above, one can recognize a generalization of the Schwarzschild
metric, whose new contribution is null when the surrounding energy density is
absent ($c=0$) and depends on the average equation of state parameter
$\omega$, that labels different fluids that surround the black hole. This
solution obeys an additivity condition such that different fluids can, in
principle, contribute to the stress-energy tensor, which reflects in
contributions to the metric as $\sum_{n}(r_{n}/r)^{3\omega_{n}+1}$. From this
approach it is possible to reconstruct the Reissner-Nordström-anti-de Sitter
spacetime from contributions of a negative cosmological constant with equation
of state parameter $\omega_{1}=-1$, and a radiation fluid, which in principle
presents an equation of state parameter $\omega_{2}=1/3$. This gives

$A(r)=1-\frac{2GM}{r}-\frac{\Lambda}{3}r^{2}+\left(\frac{Q}{r}\right)^{2}\,.\\\
$ (7)

As we will see in the following sections, the presence of corrections in the
radiation equation of state parameter due to Planck scale effects,
$\omega\approx 1/3+{\cal O}(T_{P})$, (where $T_{P}$ is the Planck temperature)
will allow us to derive non-trivial corrections to the extended phase space
thermodynamics.

##  III Planck-scale-deformed equation of state parameter

In this section, we will revisit the problem of the photon gas by calculating
its internal energy and pressure from the occupation number of states with
given momenta and the partition function of this bosonic system. As we shall
see, Planck-scale corrections shall modify the relation between momenta and
energy, which will lead to a deformation of the equation of state parameter.

In fact, we can describe a special class of isotropic modified dispersion
relations of particles with mass $m$ by relying on deformation functions
$f(E/E_{QG})$ and $g(E/E_{QG})$ as follows

$m^{2}=E^{2}f^{2}(E/E_{QG})-p^{2}g^{2}(E/E_{QG})\,,$ (8)

where $p$ is the particle’s momentum, $E$ is its energy and $E_{QG}$ is the
quantum gravity energy scale, typically assumed to be of the order of the
Planck energy $E_{P}\approx 1.22\times 10^{19}\,\text{GeV}$.

These Planck-scale deformations are known to induce deformations in the
behavior of various gases with several applications from black hole physics to
cosmology Santos:2015sva ; Nozari:2006yia ; Alexander:2001ck ;
Alexander:2001dr ; Lobo:2020oqb ; Moradpour:2021ymp , where the nontrivial
rule for counting states within a given energy/momentum interval leads to
interesting phenomena, like inflation without an inflaton field and
dimensional reduction due to quantum gravity at the trans-Planckian regime.

The number of states with momentum between $p$ and $p+dp$ contained in a
volume $V$ is given by huang

$N(p)dp=\frac{V}{(2\pi)^{3}}4\pi p^{2}dp\,,$ (9)

and the relation between the momentum and the energy is given by the deformed
expression (8) when $m=0$. Assuming the degeneracy due to the two
polarizations of the photons, from Eq.(8) a straightforward calculation
implies that Lobo:2019jdz ; Santos:2015sva

$2N(p)dp\doteq
2\tilde{N}(E)dE=2\frac{V}{\pi^{2}}\left(\frac{f}{g}\right)^{3}\left(1+E\frac{f^{\prime}}{f}-E\frac{g^{\prime}}{g}\right)E^{2}dE\,,$
(10)

where prime (′) denotes differentiation with respect to the energy.

As usual, the equation of state parameter can be found from the average of the
energy density and the pressure, for which the Bose-Einstein statistical
distribution remains unmodified Alexander:2001ck :

$\omega(T)=\frac{P}{\rho}=-T\frac{\int\ln[1-e^{-E/T}]\tilde{N}(E)dE}{\int\frac{E}{\exp[E/T]-1}\tilde{N}(E)dE}\,,$
(11)

where $T$ is the fluid’s temperature. For an undeformed dispersion relation,
each of these integrals are proportional to the $4$-th power of the
temperature, which furnish an equation of state parameter given by the
constant value of $1/3$, which corresponds to the usual value associated to a
radiation fluid.

We assume a MDR of the form (8), for instance, one that has been considered in
several phenomenological scenarios of quantum gravity (as a way how to capture
a upper limit on the energy of fundamental particles when $\xi<0$ Ling:2005bp
; Ling:2005bq ; Li:2008gs ):

$f=1\ \ \ ,\ \ \ g=\sqrt{1+\xi\left(\frac{E}{E_{P}}\right)^{n}}\,,$ (12)

where $\xi$ is a dimensionless real parameter that labels different
phenomenological proposals, and $n$ is a natural number that describes the
leading order correction proportional to the ratio $(E/E_{P})^{n}$. In this
paper, we are considering the first and second order cases, i.e., $n=1$ and
$n=2$, since these are the orders of magnitude that have been mostly prominent
for observational constraints Acciari:2020kpi ; Amelino-Camelia:2016ohi , for
simplicity. As we shall see, this difference will be enough to produce
significantly distinctive scenarios.

By following the above procedure and considering only the first term in a
Taylor perturbation around $E/E_{P}$ in Eq.(11), we find a temperature-
dependent equation of state parameter:

$\omega_{n}(T)\approx\frac{1}{3}+\alpha_{n}\left(\frac{T}{T_{P}}\right)^{n}\,,$
(13)

where $T_{P}$ is the Planck temperature, $\alpha_{1}=\xi\,60\zeta(5)/\pi^{4}$
and $\alpha_{2}=\xi\,40\pi^{2}/63$ are dimensionless parameters that we shall
allow to vary in our analysis, and $\zeta(x)$ is Riemann zeta function, which
are typically expected in first order perturbations of the dispersion relation
Lobo:2019jdz . Now we proceed to applications in the Reissner-Nordström-anti-
de Sitter spacetime.

##  IV Extended phase space thermodynamics

The individual interaction of high energetic photons with a quantum spacetime
could be effectively described by the kinematics induced by a modified
dispersion relation, in such a way that when extended to a collection of
particles like a fluid, manifests as a temperature-dependent equation of state
parameter. Such temperature is the one assigned to this high energetic
radiation fluid. Now, if we suppose that such high energetic particles
surround a static, spherically symmetric configuration, in a similar way to
the one described in section II, we shall expect Planck scale deformations of
Kiselev’s approach to the Reissner-Nordström-anti-de Sitter spacetime.

This approach allows to propose an effective way to describe a spacetime
endowed with these properties that resembles a known approach called “rainbow
gravity” Magueijo:2002xx . In that case, the propagation of individual
particles on a curved spacetime could be described by a metric that depends on
the energy/momentum of the particles itself. More recent versions of this idea
are realized by Finsler Girelli:2006fw ; Amelino-Camelia:2014rga ;
Lobo:2016xzq ; Lobo:2016lxm  and Hamilton geometries Barcaroli:2015xda . Here,
we propose that the spacetime probed by a fluid of massless particles could be
described in a temperature-dependent way $g_{\mu\nu}(T/T_{P})$, whose
realization can be done by the Kiselev approach.

In fact, we realize this idea by having a black hole spacetime of mass $M$
surrounded by a negative cosmological constant $\Lambda$ and a radiation fluid
defining a deformed Reissner-Nordström-anti-de Sitter black hole of charge
$Q$:

$A(r)=1-\frac{2GM}{r}-\frac{\Lambda}{3}r^{2}+\left(\frac{Q}{r}\right)^{3\omega(T)+1}\,,$
(14)

that depends on the ratio $T/T_{P}$ induced by the equation of state parameter
(11). The main difference of this approach in comparison to the previous ones
consists in the fact that we are inducing Planck scale correction in geometry
through the matter sector of the gravitational field equations, instead of
accommodating the kinematics of particles in a certain non-Riemannian
spacetime.

In order to connect with black hole thermodynamics, we realize that the
photons emitted via Hawking radiation present energies of the order of the
black hole’s temperature $E\sim T$ Ling:2005bp ; Adler:2001vs . For this
reason, we analyze a stable scenario in which the temperature of the radiation
that surrounds the black hole is of the order of the Hawking temperature as
well. This is a common ansatz assumed in energy-dependent metric approaches
Ling:2005bp  and also for quantum-gravity-induced generalized uncertainty
principles Adler:2001vs . This allows us to find the Hawking temperature as a
function of the outer horizon radius $r_{+}$ (solution of $A(r_{+})=0$):

$\displaystyle T_{H}(r_{+})=\frac{1}{4\pi}\frac{\partial}{\partial
r}A(r_{+})=\frac{1}{4\pi}\left[\frac{1}{r_{+}}+8\pi
Pr_{+}-\frac{3\omega(T)}{r_{+}}\left(\frac{Q}{r_{+}}\right)^{3\omega(T)+1}\right]\,,$
(15)

where we have, in fact, the temperature implicitly defined, since we are
making $T=T_{H}$ and the pressure is identified from the cosmological
constant, as usual in the extended phase space thermodynamics Kubiznak:2016qmn
, as

$P=-\frac{\Lambda}{8\pi}\,.$ (16)

Concretely, from the dispersion relations used in the previous section, we use
Eq.(13) in Eq.(15), replacing $\omega\rightarrow\omega_{n}$, and expanding in
powers of the inverse of the Planck temperature, to find

$T=T^{\circ}-\alpha_{n}\left(\frac{T^{\circ}}{T_{P}}\right)^{n}\frac{Q^{2}}{V^{\circ}}\left[1+\ln\left(\frac{Q}{r_{+}}\right)\right]\,,$
(17)

where $V^{\circ}=4\pi r_{+}^{3}/3$ is the Euclidean, undeformed volume and

$T^{\circ}=\frac{1}{4\pi}\left[\frac{1}{r_{+}}+8\pi
Pr_{+}-\frac{Q^{2}}{r_{+}^{3}}\right]$ (18)

is the undeformed temperature of the Reissner-Nordström-anti-de Sitter black
hole.

This ansatz for the temperature modifies the thermodynamics of the black hole
as we shall see. In fact, the mass $M$, that plays the role of enthalpy of
this thermodynamic system, is defined by:

$GM(r_{+},P,Q)=\frac{r_{+}}{2}+\frac{Q^{2}}{2r_{+}}+\frac{4\pi}{3}Pr_{+}^{3}+\frac{3\alpha_{n}Q^{2}}{2r_{+}}\left(\frac{T^{\circ}(r_{+},Q,P)}{T_{P}}\right)^{n}\ln\left(\frac{Q}{r_{+}}\right)\,,$
(19)

where $T^{\circ}=T^{\circ}(r_{+},Q,P)$ is assumed as a function of the horizon
radius, charge and pressure according to (18).

We describe the variation of the black hole’s mass due to variations of the
horizon radius $r_{+}$, pressure $P$ and electric charge $Q$ as usual

$dM=\frac{\partial M}{\partial r_{+}}\Bigg{|}_{P,Q}\delta r_{+}+\frac{\partial
M}{\partial Q}\Bigg{|}_{r_{+},P}\delta Q+\frac{\partial M}{\partial
P}\Bigg{|}_{r_{+},Q}\delta P.$ (20)

We also define the entropy, as usual, for static black holes from variation of
the enthalpy with respect to the horizon radius as follows:

$dS\doteq\frac{1}{T}\frac{\partial M}{\partial r_{+}}\Bigg{|}_{P,Q}dr_{+}\,.$
(21)

This leads to the following expression for the linear correction ($n=1$) after
integration:

$\displaystyle S_{1}=\frac{\pi
r_{+}^{2}}{G}+\frac{3\alpha_{1}}{2\sqrt{2}G\,T_{P}}\frac{Q}{r_{+}}\left\\{3\sqrt{2}Q\left[\ln\left(\frac{Q}{r_{+}}\right)-1\right]-2r_{+}\sqrt{a+1}\ln\left(\frac{Q}{r_{+}}\right)\tanh^{-1}\left(\frac{4r_{+}\sqrt{\pi
P}}{\sqrt{a-1}}\right)\right.$ (22)
$\displaystyle+2r_{+}\sqrt{a-1}\ln\left(\frac{Q}{r_{+}}\right)\tan^{-1}\left(\frac{4r_{+}\sqrt{\pi
P}}{\sqrt{a+1}}\right)-2r_{+}\sqrt{a+1}\,\chi_{2}\left(\frac{4r_{+}\sqrt{\pi
P}}{\sqrt{a-1}}\right)\left.+2r_{+}\sqrt{a-1}\text{Ti}_{2}\left(\frac{4r_{+}\sqrt{\pi
P}}{\sqrt{a+1}}\right)\right\\}\,,$

where $\alpha_{1}=\xi 60\zeta(5)/\pi^{4}$, $a=\sqrt{1+32\pi PQ^{2}}>1$, and
the terms $\chi_{\nu}(z)=\frac{1}{2}[\text{Li}_{\nu}(z)-\text{Li}_{\nu}(-z)]$
and $\text{Ti}_{2}(z)=\frac{1}{2i}[\text{Li}_{2}(iz)-\text{Li}_{2}(-iz)]$ are
called Legendre’s chi function and inverse tangent integral function,
respectively, and are defined in terms of the polylogarithm function
$\text{Li}_{\nu}(z)=\sum_{k=1}^{\infty}z^{k}/k^{\nu}$. This expression reduces
to the one found in Lobo:2019jdz  when $P\rightarrow 0$.

This entropy $S_{1}$ presents the peculiar property of assuming complex values
for a non-null pressure $P>0$. The concept of complex entropy has been
recently studied in the literature c:entropy1 ; c:entropy2 ; c:entropy3  and
it is suggested that its imaginary part describes the entropy transferred to
the background, which would mean that such black hole is an unstable system
c:entropy1 . This dissipation is driven by the pressure term given by the
negative cosmological constant, i.e., when $P\rightarrow 0$, we recover a real
valued entropy and stable system as considered in Lobo:2019jdz . The
conservation of energy described by the first law of thermodynamics needs to
involve also the energy lost due to the pressure term. This information is
already included in the nature of the entropy. As we shall see, this is a
property that shall not be present when we assume quadratic terms as the
leading order corrections.

In fact, for $n=2$, we have the following entropy after integrating Eq.(21):

$\displaystyle S_{2}=\frac{\pi r_{+}^{2}}{G}-\frac{3\alpha_{2}}{64\pi
G\,T_{P}^{2}}\frac{Q^{2}}{r_{+}^{4}}\left\\{4r_{+}^{2}-3Q^{2}+4\left[3Q^{2}-2r_{+}^{2}+16\pi
P\,r_{+}^{4}\ln\left(\frac{Q}{r_{+}}\right)\right]\ln\left(\frac{Q}{r_{+}}\right)\right\\}\,,$
(23)

where $\alpha_{2}=\xi 40\pi^{2}/63$. This is a real-valued quantity that
reduces to the case studied in Lobo:2019jdz  when $P\rightarrow 0$. In these
two cases, we see logarithmic corrections to the entropy, whose Planck-scale
contributions vanish when $Q\rightarrow 0$.

The electric potential for $n=1$ and $n=2$ reads:

$\displaystyle\Phi_{n}\doteq\frac{\partial M}{\partial
Q}\Bigg{|}_{r_{+},P}=\frac{Q}{Gr_{+}}+\frac{3\alpha_{n}Q}{2r_{+}}\frac{\left(T^{\circ}\right)^{n-1}}{G\,T_{P}^{n}}\left[T^{\circ}+\ln\left(\frac{Q}{r_{+}}\right)\left(2T^{\circ}-\frac{nQ^{2}}{2\pi
r_{+}^{3}}\right)\right]\,,$ (24)

thus also presenting logarithmic corrections and, as usual, $T^{\circ}$ is a
function of $r_{+}$, $Q$ and $P$, and $\alpha_{n}$ are given by the quantities
below Eq.(13).

Besides that, a notion that is common in this so called black hole chemistry
approach is the existence of a thermodynamic volume, which is read from the
first law of thermodynamics and not necessarily is given by the Euclidean
volume for cases beyond general relativity (for further discussions on this
topic, we refer the reader to the review Kubiznak:2016qmn ). In our case, the
thermodynamic volume also gains logarithmic correction as

$V\doteq\frac{\partial M}{\partial P}\Bigg{|}_{r_{+},Q}=\frac{4\pi
r_{+}^{3}}{3G}+\frac{3n\alpha_{n}Q^{2}}{G\,T_{P}^{n}}\left(T^{\circ}\right)^{n-1}\ln\left(\frac{Q}{r_{+}}\right)\,.$
(25)

##  V Critical behavior

Endowed with these quantities, we can study how quantum corrections affect
phase transitions due to the electric charge. As a first step, we need to
establish an equation of state for the pressure, which can be
straightforwardly calculated from Eqs.(17) and (18):

$P=\frac{T}{2r_{+}}-\frac{1}{8\pi r_{+}^{2}}+\frac{Q^{2}}{8\pi
r_{+}^{4}}+\frac{3\alpha_{n}Q^{2}}{8\pi
r_{+}^{4}}\left(\frac{T}{T_{P}}\right)^{n}\left[1+\ln\left(\frac{Q}{r_{+}}\right)\right]\,.$
(26)

We find the critical quantities $(r_{c},T_{c},P_{c})$ by solving the following
system of equations

$\frac{\partial P}{\partial
r_{+}}\Bigg{|}_{T,Q}=0=\frac{\partial^{2}P}{\partial
r_{+}^{2}}\Bigg{|}_{T,Q}\,,$ (27)

where $P_{c}\doteq P(r_{+}=r_{c},T=T_{c})$. This gives the following solution:

$\displaystyle T_{c}=\frac{\sqrt{6}}{18\pi
Q}-\frac{\alpha_{n}}{A_{n}\,T_{P}^{n}}\left[5-\ln(36)\right],$ (28)
$\displaystyle
r_{c}=\sqrt{6}Q+\frac{\alpha_{n}}{B_{n}\,T_{P}^{n}}\left[19-\ln(46656)\right],$
(29) $\displaystyle P_{c}=\frac{1}{96\pi
Q^{2}}-\frac{\alpha_{n}}{C_{n}\,T_{P}^{n}}\left[8-\ln(216)\right]\,,$ (30)

where $n=(1,2)$, and we have
$A_{n}=(144\pi^{2}Q^{2},432\sqrt{6}\pi^{3}Q^{3})$,
$B_{n}=(24\pi,72\sqrt{6}\pi^{2}Q)$,
$C_{n}=(576\sqrt{6}\pi^{2}Q^{3},10368\pi^{3}Q^{4})$. The quantities between
square brackets in the above equations are positive.

From these expressions, we immediately verify the role of the Planck scale in
increasing or reducing the critical temperature depending on the sign of the
dimensionless parameter $\alpha_{n}$, which on the other hand, is proportional
to the parameter $\xi$ that is present in the modified dispersion relation
(12). Regardless of the order of the first Planck-scale correction $n$, we see
that negative values of $\xi$ are responsible for the phenomenology of
investigating an upper threshold in the energy of fundamental particles, which
from this approach is reflecting in a raising of the critical temperature.
Besides that, $\xi<0$ describes the subluminal propagation of massless
particles, i.e., UV photons are propagating with a speed $v<1$. So, we can
conclude that the critical temperature, from which phase transition can occur,
is raised if we have Planck-scale subluminal propagation. The opposite
behavior occurs for superluminal propagation $\xi>0$.

This behavior can be seen in Figs. (1) and (2), where we depicted the pressure
as a function of the outer horizon for some values of the quantum gravity
parameter $\alpha_{n}$, assuming the critical temperature
$T_{c}(\alpha_{n}=0)$. The blue (solid) curve corresponds to the critical
isotherm when $\alpha_{n}=0$, i.e., it corresponds to a portrayal of the
equation of state (26) for $T=T_{c}(\alpha_{n}=0)$ (where we assumed constant
charge). The green (dotted) curve is a depiction of the deformed equation of
state parameter (26) for $\alpha_{n}<0$ (subluminal propagation), but assuming
the undeformed critical temperature $T_{c}(\alpha_{n}=0)$. Since
$T_{c}(\alpha_{n}=0)<T_{c}(\alpha_{n}<0)$, this means that such deformed
isotherm lies within the region in the $P-r_{+}$ diagram that presents phase
transitions, which corresponds to temperatures below the critical one for each
case. In contrast, the orange (dashed) isotherm corresponds to the case
$\alpha_{n}>0$ (superluminal propagation). Now, it can be seen that
$T_{c}(\alpha_{n}=0)>T_{c}(\alpha_{n}>0)$, which means that this deformed
depicted isotherm lies above the critical curve for the superluminal case,
thus not manifesting phase transitions.

From the previous considerations, one realizes that the observation or absence
of phase transitions in regions of the phase space in which they should be
presented can be a manifestation of Planck scale effects, which implies that
quantum gravity parameters can be inductors or complicators of phase
transitions in Reissner-Nordström-anti-de Sitter black holes.

Figure 1: Pressure versus horizon radius for $n=1$ and $Q=0.03$ at constant
temperature, $T_{c}(\alpha_{1}=0)=\sqrt{6}/18\pi Q$. We assumed natural units
in which $T_{P}=1$. Here it can be seen the induction of criticality by the
parameter $\alpha_{1}$. Figure 2: Pressure versus horizon radius for $n=2$ and
$Q=0.03$ at constant temperature, $T_{c}(\alpha_{1}=0)=\sqrt{6}/18\pi Q$. We
assumed natural units in which $T_{P}=1$. Here it can be seen the induction of
criticality by the parameter $\alpha_{2}$.

##  VI Black hole as heat engine

Thermodynamics was born as a theoretical tool to study the efficiency of heat
engines. A necessary component that a thermodynamic system needs to have in
order to be analyzed under this perspective is a work term in its conservation
of energy equation (first law of thermodynamics). Since the black hole system
developed in this paper presents such term due to the identification of the
cosmological constant with pressure, it is natural to wonder how this system
would behave under a thermodynamic cycle, and what would be its efficiency.
This approach, was initially analyzed as a holographic heat engine
Johnson:2014yja  due to the presence of an anti-de Sitter term, that is also a
common ingredient of holographic scenarios due to the AdS/CFT conjecture.

A heat engine works between two reservoirs of temperature, a hot and a cold
one, and heat flows between these reservoirs. The overall result of this
process is work done by the engine at cost at some waste of energy in the form
of heat flow to the cold reservoir. This allows one to define the efficiency
of the machine as

$\eta=\frac{W}{Q_{H}}\,,$ (31)

where $W$ is the work done by the system and $Q_{H}$ is the amount of heat
coming from the hot source. The most efficient heat engine is performed
following the so called Carnot cycle, constructed by two isothermals and two
adiabatics. The most important fact about the Carnot cycle is that the
efficiency of the heat engine depends only on the temperature of the
reservoirs, and it is given by

$\eta_{C}=1-\frac{T_{C}}{T_{H}}\,,$ (32)

where $T_{C}$ and $T_{H}$ are the temperatures of the cold and hot reservoirs.
As can be seen, the efficiency cannot be equal to unity, since a reservoir
cannot have zero temperature.

Since we have a well-defined equation of state involving our thermodynamic
variables, given by equation (17) and (18), the efficiency of the most
efficient black hole can be easily obtained. If one wants to calculate the
efficiency of an arbitrary cycle, then one needs to use, in general, Eq. (31).

The efficiency of the Carnot cycle for both orders of perturbation
($n=1\,,\,2$) is depicted in Figs. (3) and (4) for some values of the quantum
gravity deformation parameter. We are allowed to perform this analysis as long
as we do not split too much the three efficiencies in a way that we would need
to move to higher order corrections in each case. One sees a universal
behavior independently on $n$, in which if UV photons present subluminal
propagation ($\alpha_{n}<0$) the efficiency is raised (green, dotted line),
and the opposite behavior occurs for superluminal propagation ($\alpha_{n}>0$,
orange, dashed line). Therefore, Planck scale corrections can improve or
worsen the efficiency of the Carnot cycle of the Reissner-Nordström-anti-de
Sitter black hole.

Figure 3: Carnot efficiency versus electric charge for $n=1$. Values of the
geometrical quantities are $r_{+}^{C}=1.13$, $P_{C}=0.01$ and
$r_{+}^{H}=0.78$, $P_{H}=0.05$. We assumed natural units in which $T_{P}=1$.
Figure 4: Carnot efficiency versus electric charge for $n=2$. Values of the
geometrical quantities are $r_{+}^{C}=1.13$, $P_{C}=0.01$ and
$r_{+}^{H}=0.78$, $P_{H}=0.05$. We assumed natural units in which $T_{P}=1$.

As proposed in the pioneering work Johnson:2014yja , we also consider the
square circle, depicted in Fig. (5). Since the pressure is constant, the work
done by the engine is the variation of the enthalpy, which is simply the mass
of the black hole. This gives us a straightforward formula for the efficiency
as follows:

$\eta_{S}=1-\frac{M_{3}-M_{4}}{M_{2}-M_{1}}\,.$ (33)

Figure 5: A square cycle in the $P-r_{+}$ diagram. Points $2$ and $4$ are
those of highest and lowest temperatures, respectively, between which the
black hole as a heat engine works.

For $n=1$, this efficiency is depicted in Fig. (6), and one verifies a
different qualitative behavior in comparison to the Carnot cycle. In this
case, the subluminal case ($\alpha_{1}<0$, green, dotted line) is still more
efficient, and moves away from the undeformed efficiency as the charge grows.
However, the superluminal case ($\alpha_{1}>0$, orange, dashed line), instead
of staying less efficient than the undeformed one (but growing with the
electric charge) at a certain point manifests an inflection towards decreasing
the efficiency with the growth of the charge. This behavior was not observed
in the Carnot cycle, and the point in which not only the superluminal case
begins to decrease, but also the subluminal one starts to increase at a more
intense way corresponds to $Q\sim 1$ for our choice of points in phase space.

The order of the Planck scale perturbation plays a significant role in the
square cycle, as can be verified in Fig. (7). In this case, for low charges,
we have the same ordering of the previous cases, i.e., with the less efficient
engine being the superluminal case ($\alpha_{2}>0$, orange, dashed line),
passing through the undeformed case ($\alpha_{2}=0$, blue, solid line), until
the subluminal case ($\alpha_{2}<0$, green, dotted line) provides the most
efficient heat engine. However, we show here that this is not an universal
behavior, since, for the geometric values assigned in this cycle (which were
the same considered in Fig. (6)) when the black hole’s electric charge reaches
the value $Q\sim 2.5$, we verify an inversion of the behavior, in which the
subluminal case starts to behave in a way similar to the superluminal one. The
opposite behavior occurs for the superluminal case, in which $\alpha_{2}>0$
describes the most efficient way of performing work from the black hole
system, and $\alpha_{2}<0$ becomes the less efficient one.

Figure 6: Square cycle efficiency versus electric charge for $n=1$. Values of
the geometrical quantities are $r_{+}^{2}=r_{+}^{3}=1.13$,
$r_{+}^{1}=r_{+}^{4}=0.78$ and $P_{1}=P_{2}=2$, $P_{3}=P_{4}=1$. We assumed
natural units in which $T_{P}=1$. Figure 7: Square cycle efficiency versus
electric charge for $n=2$. Values of the geometrical quantities are
$r_{+}^{2}=r_{+}^{3}=1.13$, $r_{+}^{1}=r_{+}^{4}=0.78$ and $P_{1}=P_{2}=2$,
$P_{3}=P_{4}=1$. We assumed natural units in which $T_{P}=1$.

##  VII Final remarks

Endowed with the Kiselev proposal for describing a black hole surrounded by
matter fluids, which would depend on their equation of state parameters, we
propose an effective generalization of the Reissner-Nordström-anti-de Sitter
black hole spacetime depending on Planck-scale corrections. This approach
allows us to analyze the black hole extended phase space thermodynamics and
its quantum gravity effects.

The introduction of a negative cosmological constant as a pressure term
allowed us to derive some non-trivial results, like the presence of
dissipation and instabilities in the linear regime (a property that is absent
in the quadratic one), and logarithmic corrections for some the thermodynamic
quantities, like the black hole’s electric potential, entropy, thermodynamic
volume, temperature and mass.

The extended phase space thermodynamics of the Reissner-Nordström-anti-de
Sitter spacetime is sometimes referred as black hole chemistry, due to the
resemblance of its equations of state with a Van der Waals gas, which are able
to manifest phase transitions. In fact, we analyzed the effect of the quantum
gravity parameters in the critical behavior of this black hole system assuming
linear and quadratic Planck scale contributions. Among our findings, we
verified that if UV photons have subluminal (superluminal) propagation, the
critical temperature below which phase transitions can occur is raised
(lowered). Besides that, critical radius and pressure were calculated.

We also analyzed the black hole as a heat engine and found that the efficiency
of the Carnot cycle presents properties that are common to the linear and
quadratic corrections by raising (lowering) the efficiency of this engine if
UV photons present subluminal (superluminal) propagation. However, this is not
a universal behavior for all cycles, as we demonstrated using the square cycle
in a $P-r_{+}$ diagram, in which the linear case presents a more drastic
splitting in the behavior of the efficiency for a given critical value of the
electric charge (however still obeying the same hierarchy of efficiencies of
the Carnot cycle), and the quadratic case presents an inversion in the
behavior of the efficiency. Therefore, at a given value of the electric
charge, the hierarchy of efficiencies reverses and the superluminal case
becomes dominant over the undeformed and subluminal ones, respectively.

The analyses done in the paper show that Planck scale corrections on the
thermodynamics of the radiation that works as a source of the Reissner-
Nordström-anti-de Sitter black hole can present unexpected properties: like
pressure-driven dissipation for linear corrections, logarithmic contributions
in other thermodynamic quantities, not only the entropy, deformations of the
phase transition regions, and alternating efficiency hierarchies involving
subluminal and superluminal propagation of UV photons depending on the nature
of the Planck-scale corrections (if linear or quadratic) and on the considered
cycles.

## Acknowledgements

I. P. L. would like to acknowledge the contribution of the COST Action
CA18108. I. P. L. was partially supported by the National Council for
Scientific and Technological Development - CNPq grant 306414/2020-1. J. P. M.
G. was partially supported by the National Council for Scientific and
Technological Development - CNPq grant 151701/2020-2. V. B. B. was partially
supported by the National Council for Scientific and Technological Development
- CNPq grant 307211/2020-7.

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Results sec:results

[width=]figures/3Dflow_inst_shock.png

Figure : Instantaneous visualisation of the turbulent and shock structures.
Isosurface of the swirling strength coloured by the streamwise velocity
component ($\lambda_{ci}L_{sep}^{u}/U_{\infty}=60$), isosurface of the shock
sensor in pink ($\theta=0.9$), numerical schlieren on the xy and yz slices in
the background.

##  Qualitative flow organisation

To first understand the qualitative flow organisation, figure  reports an
instantaneous visualisation of the turbulent and shock structures in the flow
field using an isosurface of the swirling strength coloured by the streamwise
velocity and an isosurface of the shock sensor. A video of the flow, generated
using in situ visualisation at runtime bna2023insitu, is linked in the
supplementary material. In addition to the shocks associated with the
uncontrolled sbli, another shock system is visible in correspondence with the
microramp, which has been extensively characterised in previous works
dellaposta2023direct_reynolds. In particular, an almost-planar shock wave is
generated at the leading edge of the ramp, while a conical shock wave
surrounds the wake from the trailing edge. These weak shocks interact with the
main impinging shock far from the wall. The turbulent structures show,
instead, how the incoming flow is first captured at the sides of the
microramp, thus generating the primary vortex pair and then the arch-like
vortices around the ramp wake. However, the most notable differences with the
traditional uncontrolled sbli take place in correspondence with the foot of
the reflected shock wave, whose shape is significantly altered by the incoming
flow. The lower part of the shock is completely disrupted at the centre of the
domain, where the ramp wake hits the interaction, and bulges are periodically
generated when the kh vortices around the wake pass through the shock. The
spanwise alteration of the separation region, which is, however, still present
as it can be observed from the presence of reversed flow between the shocks,
also affects the flow downstream of the interaction region, where the
compression wave after the separation is no longer homogeneous in the spanwise
direction. The differences along the span of the microramp-controlled sbli are
also visible in the instantaneous temperature contours at the symmetry and
lateral xy planes (figure ). While the lateral slice reminds of a traditional
2D sbli, only with the addition of two extra shocks upstream, the billows of
the ramp wake completely alter the incoming flow (as observed in
bo2012experimental), the interaction region, and the downstream flow at the
symmetry plane, with a significantly thicker boundary layer. As anticipated,
the presence of the mvg radically affects the incoming boundary layer, which
shapes the interaction region in turn. Typical near-wall streaks, observable
in the velocity contours on xz planes in figure , are now overlapped to the
large-scale trace of the ramp wake. On the one hand, the low-momentum wake
decelerates the flow at the centre of the domain, corresponding to an increase
in the reversed flow extent. On the other hand, the vortical motion of the
primary vortices and the transversal mixing associated with the arch-like
vortices promote local accelerations of the flow, corresponding to a decrease
in the reversed flow. Before and even after the interaction, the meandering
motion of the ramp wake and the spanwise alteration of the separation generate
the formation of alternated regions of accelerated and decelerated flows,
similar to large-scale streak structures.

[width=0.36]figures/T_ist_xy_colors.png
[] [width=]figures/T_ist_xy_z0.3.png
[] [width=]figures/T_ist_xy_z0.png

Figure : Instantaneous temperature on xy planes at (a) $z^{*}=-0.3$, (b)
$z^{*}=0.0$. The white lines indicate $u/U_{\infty}=0$, while the yellow lines
indicate points with $M=1$. Zoom into the separation region in the boxes.

[width=0.34]figures/streaks_colors.png
[] [width=]figures/streaks_yplus_1.png
[] [width=]figures/streaks_yh_0.5.png
[] [width=]figures/streaks_yh_1.png

Figure : Instantaneous streamwise velocity on xz planes at (a) $y^{+}=1$, (b)
$y/h=0.5$, (c) $y/h=1$. The white lines indicate $u/U_{\infty}=0$.

##  A 2D view: a classic 2D SBLI analysis

Although the qualitative flow description confirmed the literature findings
about the increased geometrical complexity of the 3D flow field for the
controlled SBLI, figure  shows that the streamwise distribution of the wall-
pressure rise is homogeneous in the span, as also observed in the experimental
measurements of [babinsky2009microramp]. The figure reports the wall pressure
at three notable sections in the span, which are: the symmetry plane
$z^{*}=0$, the lateral plane $z^{*}\approx-0.30$, and the spanwise section
corresponding to the minimum streamwise extent of the separation,
$z^{*}=-0.0513$ (see section ). The curves show that, compared with the USBLI
case, the presence of the microramp induces a forward shift in the separation
onset (indicated with circles) and a slight backward shift in the reattachment
(indicated with squares), which thus entails an overall shorter separation
along the span. The wall-pressure standard deviation $\sigma_{p_{w}}$ at the
same sections shows instead that wall-pressure fluctuations are stronger for
CSBLI. As in the case of non-adiabatic wall conditions bernardini2016heat or
transitional sbli quadros2018numerical, a reduction (increase) of the
separation length corresponds to an increase (decrease) in the intensity of
the wall-pressure fluctuations. The CSBLI curves also present another peak in
the rear part of the separation and, most importantly, vary considerably with
the span (see section ). In particular, the standard deviation distribution is
minimum at the symmetry plane and maximum at the lateral plane. After the
separation, the decay rate of $\sigma_{p_{w}}/p_{\infty}$ is larger in CSBLI
than in USBLI, and rather homogeneous in the span.

[] [width=6cm]figures/pwall_x.eps [] [width=6cm]figures/sigmapwall_x.eps

Figure : Streamwise distribution of (a) mean wall pressure and (b) wall-
pressure standard deviation. Spanwise average of the uncontrolled case (solid
black line), controlled case at
$z^{*}=-0.2986$ (solid orange line), $z^{*}=-0.0513$ (solid light blue line),
$z^{*}=0.0$ (solid green line). Symbols indicate the location of the
separation point (circles), the reattachment point (squares), and the point
with minimum streamwise pressure gradient (diamonds) for the curve of the
corresponding colour.

Figure  reports instead the distribution of the time-averaged streamwise skin
friction component $\overline{C_{f,x}}$ at different sections in the span. As
shown by the spanwise-averaged curve of the CSBLI case, the extent of the
region with negative $\overline{C_{f,x}}$ is significantly shortened compared
to the USBLI case. However, we also notice in this case that conditions vary
considerably with the span when the microramp is present, suggesting that the
streamwise skin friction component alone may provide misrepresentative
information regarding the extent and the nature of the separation, because of
the non-negligible contribution of the spanwise skin friction component.

[width=6cm]figures/Cfx_x.eps

Figure : Streamwise distribution of the streamwise skin friction coefficient.
Spanwise average of the uncontrolled case (solid black line), spanwise average
of the controlled case (dashed-dotted blue line), controlled case at
$z^{*}=-0.2986$ (solid orange line), $z^{*}=-0.0513$ (solid light blue line),
$z^{*}=0.0$ (solid green line).

The significant spanwise modulation of the flow is also observed in the xy-
slices of figure , which compares the Favre-averaged streamwise velocity
component for the USBLI case (spanwise-averaged) and for the three above-
defined spanwise sections of the CSBLI case. To consider compressibility,
Favre averages are used, which are defined as
$\widetilde{\phi}=\overline{\rho\,\phi}/\overline{\rho}$ for a generic
variable $\phi$. The reversed flow region is highlighted and hints that for
the two inner sections, the separation is reduced relevantly. However, it is
important to notice that especially for strongly three-dimensional flow, as in
the case under consideration, the reversed flow is not indicative of the
actual separation taking place. In conclusion, the wall-pressure standard
deviation, the streamwise skin friction component, and the xy slices of the
mean velocity proved that for many features – but not all – the flow
organisation is fully three-dimensional, especially close to the wall and in
the interaction region. The analysis of single streamwise slices of the
streamwise components of vectorial quantities may be thus insufficient and
even misleading to understand the overall interaction topology and dynamics.

[width=0.32]figures/FaVelX_colors.png
[] [width=0.8]figures/FaVelX_USBLI_kav.png
[] [width=0.8]figures/FaVelX_k1.png
[] [width=0.8]figures/FaVelX_k160.png
[] [width=0.8]figures/FaVelX_k193.png

Figure : Favre-averaged streamwise velocity on xy planes: (a) USBLI, spanwise-
averaged, and CSBLI at (b) $z^{*}\approx-0.3$, (c) $z^{*}=-0.05$, (c)
$z^{*}=0$. Yellow lines indicate points with $\widetilde{M_{\infty}}=1$, white
lines indicate points with $\tilde{u}/U_{\infty}=0$.

##  More than 2D: the controlled SBLI separation

Given the three-dimensionality of the separation, we must resort to more
sophisticated tools for its analysis. Following the work of Legendre and Werlé
delery2001robert, delery2013three, the topology of the separation can be
examined by studying the organisation of the skin friction lines at the wall.
Skin friction lines can be viewed as the limit towards the wall of the
streamlines and, if a mean flow is well-defined, they can provide insightful
information about the actual location of the separation and reattachment.
Figure  reports the skin friction lines for the CSBLI case on half of the
domain, taking advantage of the flow case symmetry, and highlights the
formation of curved separation and reattachment lines along the span, where
skin friction lines converge to and depart from respectively. Critical points
($C_{f,x}=C_{f,z}=0$) are also highlighted according to their nature: saddle
points indicated as light green circles, nodes as light blue squares, and foci
as orange diamonds. For the sake of completeness, we remind the reader that
only two skin friction lines run through a saddle point, while all the others
avoid it adopting the shape of a hyperbolic curve, that all the skin friction
lines have a common tangent at a node, except for one of them, and that skin
friction lines end at a focus after spiralling around it. Considering also the
symmetric half not shown in the figure, the periodic boundary conditions, and
the two nodes at infinite upstream and downstream, our configuration for the
interaction region (figure ) has a total of five nodes ($n=5$), four foci
($f=4$), and seven saddle points ($s=7$), which satisfies the necessary
topological condition $n+f-s=2$ delery2013three. Our arrangement has one node,
two foci, and three saddle points more than the one in grebert2023microramp,
as the separation line is split into more than one single segment in our case.

[width=0.34]figures/sigmapw_skinfriction_lines_colors.png
[] [width=]figures/sigmapw_skinfriction_lines_overall.png
[] [width=0.9]figures/sigmapw_skinfriction_lines_ramp_dense_width.png
[] [width=0.9]figures/sigmapw_skinfriction_lines_separation_dense_peak.png

Figure : Skin friction lines overlapped to a contour of the wall-pressure
standard deviation on the xz plane for the CSBLI case: (a) overall domain, (b)
zoom into ramp region, (c) zoom into separation region. Half of the domain is
shown for symmetry. Saddle points are indicated with circles in magenta, nodes
with squares in light blue, and foci with diamonds in orange. The main
separation and reattachment lines are highlighted in white, other relevant
critical lines in blue, while the foremost peak of $\sigma_{p_{w}}$ is
indicated with the dashed black line.

The focus, already observed also in oil-flow visualisations
babinsky2009microramp, indicates the presence of a tornado-like structure
which captures the flow close to the wall and pushes it upwards. A
visualisation of the streamlines associated with the tornado in figure b shows
that this structure tilts outwards to join the wall region closer to the
symmetry plane and the external sides of the separation bubble. Indeed, its
axis is first normal to the wall in correspondence with the focus and ends up
being aligned with the spanwise direction towards the side. Compared to the
classical 2D recirculation bubble, the region of separated flow is now more
complicated and not identified by the simple condition of negative
$\overline{C_{f,x}}$. Moreover, reversed flow is now even less indicative of
the separation, which makes it difficult to identify the edge of the
recirculating bubble. Figure a uses streamlines to show the qualitative shape
of the bubble, which presents a characteristic saddle shape with a higher
separated region at the symmetry and the lateral planes and a lower separation
region in correspondence with the minimal streamwise separation extent (see
figure ).

[width=]figures/streamlines_3D.png

Figure : Streamlines in the separation region of the CSBLI case: qualitative
edge of the recirculation bubble (streamlines coloured by mean vertical
velocity) (a), the tornado-like structure (b), and the internal structure of
the bubble (c). Vertical slices report the Favre-averaged Mach number
$\widetilde{M}$, yellow lines indicate $\widetilde{M}=1$, and blue lines
indicate $\tilde{u}/U_{\infty}=0$ on the vertical slices. Half of the domain
is shown for symmetry.

Significant 3D effects are also visible in the distribution of the wall-
pressure standard deviation $\sigma_{p,w}/p_{\infty}$ in figure . Close to the
microramp, we can notice regions with relevant pressure fluctuations
associated with i) the impingement of the flow of the primary vortices at the
sides of the ramp towards the wall, ii) the reattachment of the flow after the
trailing edge and the following formation of the parallel, primary vortex
pair. In the interaction region instead, we observe a first, stronger peak
after the separation onset, whose streamwise position follows the separation
front and whose magnitude is greater close to the lateral boundaries. The
distribution is compatible with the observation that the shock is disrupted at
the centre of the domain and, hence, weaker on average. As a result of the
reduced shock intensity, the amplification of the pressure fluctuations
induced by the shock wave is weaker in the central region of the domain. A
second peak, with a spanwise-modulated amplitude as well, takes place at
approximately half of the separation, in correspondence with the peak height
of the bubble. After the second peak, the standard deviation decays,
maintaining a non-uniform spanwise distribution even far from the interaction.
Given the separation and reattachment line in figure , it is possible to
estimate the extent of the time-averaged streamwise separation
$\overline{L_{s}}$ along the span reported in figure . The spanwise average
$\langle\overline{L_{s}}\rangle/\delta_{shk}$ shows that the mean separation
in the presence of the microramp shrinks by 26.47% the value observed in the
USBLI case. Moreover, the distribution shows that the separation is largely
modulated in the span,
$(\overline{L_{s}}_{max}-\overline{L_{s}}_{min})/\langle\overline{L_{s}}\rangle=26.31\%$
along the span, with a minimal separation at $z^{*}\approx-0.05$
($\overline{L_{s}}_{min}/\delta_{shk}=3.12$),

[width=6cm]figures/Lsep_z.eps

Figure : Spanwise distribution of the time-averaged separation length with
respect to the boundary layer thickness at the shock impingement location.
Spanwise average of the uncontrolled case (solid black line), local controlled
case (solid red line), and spanwise average of the controlled case (dashed red
line).

To understand the relationship between the spanwise modulation of the
separation and the addition of momentum by the primary vortex pair, figure
reports the distribution of the compressible added momentum in the xz plane.
First defined in its incompressible version by giepman2014flow, the
compressible added momentum is defined as

$E_{add}=\int_{0}^{\overline{y^{*}}}\frac{\overline{\rho
u^{2}}_{CSBLI}-\overline{\rho
u^{2}}_{USBLI}}{\rho_{\infty}U_{\infty}^{2}}\,\mathrm{d}y$ (1)

and tracks the addition of streamwise momentum towards the wall. Following
giepman2014flow, the upper bound of integration $\overline{y^{*}}$ is taken
equal to $0.43\,\delta$. The distribution of $E_{add}/h$ along the entire
wall-parallel plane allows us to appreciate that the primary vortex pair
brings fresh momentum towards the wall in a large spanwise range, even quite
far from the symmetry plane. The peak added momentum just before the
interaction is at $z^{*}\approx-0.1$, just at the side of the minimum
separation, which thus suggests that the onset of the separation – and its
local extent – follows strictly the addition of momentum associated with the
primary vortex pair helical motion.

[width=0.34]figures/E_add_h_colors.png
[] [width=]figures/E_add_h.png
[] [width=]figures/E_add_h_zoom.png

Figure : Normalised compressible added momentum $E_{add}/h$ along the xz plane
(a) and zoom into the separation region (b). Skin friction lines in the
separation region are indicated in grey. Half of the domain is shown for
symmetry.

After a transitory region following the interaction, we can observe that the
trace of the primary streamwise vortices becomes once again more visible,
indicating that these are able to withstand the shock waves and that their
signature lasts for a long streamwise distance (figure ). Even far downstream
of the separation, we thus have non-uniform conditions as highlighted by
figure  reporting the spanwise distribution at $x^{*}=2$ of the incompressible
shape factors and the streamwise skin friction coefficient for the USBLI and
CSBLI cases. The shape factor denotes a fuller and healthier boundary layer
even after the interaction at the sides of the symmetry plane, which however
corresponds to increased skin friction. We point out that the associated
increased drag and the potential consequences of a long-lasting, non-uniform
flow for the engine components following the inlet may limit the benefits of a
reduced separation and thus that these aspects should be addressed carefully
when considering mvg for real applications.

[] [width=0.45]figures/H_inc_xstar2.eps [] [width=0.445]figures/Cfx_xstar2.eps

Figure : Spanwise distribution at $x^{*}=2$ of (a) incompressible shape factor
and (b) streamwise skin friction coefficient. USBLI (dashed black line) and
CSBLI (solid red line).

##  Vortical structures: a mean view

An interesting issue regarding the effects of microramps on sbli is the mutual
interaction between the arch-like vortical structures induced by mvg and the
interaction region. According to the literature and the results in the
previous sections, the main effect on the sbli of the arch-like vortices is
that they periodically disrupt the impinging and reflected shock waves,
leading to regions of large curvature in the shock surfaces (the “bump” in
figure ). The effect of the shock waves on the arch-like vortices is instead
less clear. Some works yan2013study suggest that these vortices travel rather
undisturbed across the shock, however, quantitative information supporting
this statement, regarding for example the effect of sbli on the trajectory and
intensity of the arch-like vortices, is lacking. To assess the streamwise
development of the vortical structures related to the microramp and the sbli,
we consider the behaviour of the Favre-averaged vorticity, which we define in
this work as the curl of the Favre-averaged velocity
($\widetilde{\boldsymbol{\zeta}}=\nabla\times\widetilde{\boldsymbol{u}}$). In
particular, we can assume that, at the symmetry plane, the negative peaks of
the spanwise component $\widetilde{\zeta_{z}}$ correspond to the trace of the
kh vortices associated with the arch-like vortices and with the shear layer
delimiting the separation. Locating these minima and recording the magnitude
of the vorticity along these loci allows us to track the trajectory of the
main vortical structures and investigate the streamwise evolution of their
intensity. Figure  reports the wall-normal coordinate of the points with
minimal Favre-averaged spanwise vorticity for a given streamwise section,
whereas figure  reports the corresponding value of the mean spanwise
vorticity.
Observing the last figure, the peaks in correspondence with the shock position
suggest an increase in the intensity of the external shear layers that
directly intersect the impinging shock in both the USBLI and CSBLI cases (red
and blue curves). Whether this increase is associated with the generation of
vorticity induced by baroclinic, diffusive, turbulent, or other effects is not
clear at this stage. However, if we observe the equation for the evolution of
the Favre-averaged vorticity (see Appendix LABEL:sec:appendix), we obtain that

$\frac{\mathrm{D}}{\mathrm{D}t}\,\left(\frac{\widetilde{\boldsymbol{\zeta}}}{\overline{\rho}}\right)=\left(\frac{\widetilde{\boldsymbol{\zeta}}}{\overline{\rho}}\right)\cdot\nabla\widetilde{\boldsymbol{u}}+\frac{\nabla\,\overline{\rho}\times\nabla(\overline{p}+2/3\,\overline{\rho}\,\widetilde{k})}{\overline{\rho}^{3}}+\frac{1}{\overline{\rho}}\nabla\times\left(\frac{\nabla\cdot\tau^{t,d}}{\overline{\rho}}\right)$
(2)

where the first term at the right-hand side is the compressible vortex
stretching and tilting, the second term is the baroclinic term, and the third
term is the diffusion of vorticity by the action of viscous and turbulent
stresses. We indicated with $\mathrm{D}/\mathrm{D}t$ the material derivative,
and with $\tau_{ij}^{t}=\overline{\tau_{ij}}+\widetilde{\tau_{ij}}^{R}$ the
total stress tensor including both the Reynolds-averaged molecular stress
tensor $\overline{\tau_{ij}}$ and the Favre-averaged Reynolds stress tensor
$\widetilde{\tau_{ij}}^{R}=-\overline{\rho
u_{i}^{\prime\prime}u_{j}^{\prime\prime}}$, while the Reynolds-averaged
pressure is indicated as $\overline{p}$, the Favre-averaged turbulent kinetic
energy as $\widetilde{k}$ and the deviatoric part of the total stress tensor
as $\tau_{ij}^{t,d}$. Compressibility effects in the conservation of mass for
a generic fluid element are thus correctly accounted for in the Favre-averaged
vorticity equation only if we introduce density-weighting. Indeed, by dividing
the Favre-averaged vorticity by the Reynolds-averaged density, the classical
equations describing vorticity dynamics are recovered. Therefore, if we
observe the contours of the specific vorticity
$\widetilde{\zeta_{z}}/\overline{\rho}$ (figure ), we can see that the sudden
jump observed in correspondence with the incident shock is mainly an effect of
the conservation of mass associated with the sudden density rise, which is
also observable in the USBLI case. Figure  further confirms this conclusion,
suggesting that only a limited generation of mean vorticity in the top shear
layer of the CSBLI case can be related to baroclinic or turbulent effects
across the shock, as the vortex stretching and tilting term is null for the
spanwise vorticity component because of the flow symmetry at $z^{*}=0$.
Looking at the specific vorticity, the decay of the shear layer intensity is
now almost undisturbed by the presence of the shock, which confirms the
qualitative belief that the arch-like vortices are robust enough not to be
affected by sbli. Figure , however, shows that there is no relevant difference
between the trajectories of the wall-normal coordinates of the
$\widetilde{\zeta_{z}}$ and $\widetilde{\zeta_{z}}/\overline{\rho}$ peaks.
Considering the CSBLI case, the bottom shear layer surrounding the separation
region follows the behaviour of the one in the USBLI case up to reattachment
($x^{*}\approx 0$). Conversely, the upper shear layer defined by the arch-like
vortices and delimiting the edge of the boundary layer, first rises slowly
until the interaction region, because of the known lift-up at the symmetry
plane in the microramp wake, and then follows the triangular shape of the
separation region with a slow recover after reattachment. Indeed, the wall-
normal position is affected by the separation up to $x^{*}\approx 1$. After
the reattachment point, the bottom shear layer converges to the upper one,
which thus delimits the new boundary layer downstream of the interaction,
significantly thicker than in the USBLI case.

[] [width=]figures/zetaz_USBLI.png
[] [width=]figures/zetaz_CSBLI.png
[] [width=]figures/zetaz_rho_USBLI.png
[] [width=]figures/zetaz_rho_CSBLI.png

Figure : Contours of the spanwise component of the Favre-averaged vorticity
(a-b) and the density-weighted Favre-averaged vorticity (c-d) for the USBLI
(a-c) and the CSBLI (b-d) cases. Yellow lines indicate points at
$\widetilde{M}=1$, blue and black lines indicate respectively the position of
the top and bottom shear layers of the CSBLI case, while red lines indicate
the shear layer of the USBLI case.

[] [width=0.45]figures/yzeta_all_shears.png []
[width=0.45]figures/yzetarho_all_shears.png

Figure : Wall-normal position of shear layers at the symmetry plane based on
the spanwise Favre-averaged vorticity without (a) and with (b) density
weighting: USBLI shear layer (red), CSBLI bottom shear layer (black), CSBLI
top shear layer (blue).

[] [width=0.45]figures/zeta_all_shear.png []
[width=0.45]figures/zetarho_all_shear.png

Figure : Spanwise component of the Favre-averaged vorticity without (a) and
with (b) density weighting along the respective shear layers at the symmetry
plane: USBLI shear layer (red), CSBLI bottom shear layer (black), CSBLI top
shear layer (blue).