# From Sheared Annular Centrifugal Rayleigh-Bénard Convection to Radially
Heated Taylor-Couette Flow: Exploring the Impact of Buoyancy and Shear on Heat
Transfer and Flow Structure

Jun Zhong1  Dongpu Wang1  Chao Sun1,2<EMAIL_ADDRESS>1 Center for
Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of
Ministry of Education, and Department of Energy and Power Engineering,
Tsinghua University, 100084 Beijing, China 2 Department of Engineering
Mechanics, School of Aerospace Engineering, Tsinghua University, 100084
Beijing, China

###### Abstract

We investigate the coupling effect of buoyancy and shear based on an annular
centrifugal Rayleigh-Bénard convection (ACRBC) system in which two cylinders
rotate with an angular velocity difference. Direct numerical simulations are
performed in a Rayleigh number range $10^{6}\leq Ra\leq 10^{8}$, at fixed
Prandtl number $Pr=4.3$, inversed Rossby number $Ro^{-1}=20$ and radius ratio
$\eta=0.5$. The shear, represented by the non-dimensional rotational speed
difference $\Omega$, varies from $0$ to $10$, corresponding to an ACRBC
without shear and a radially heated Taylor-Couette flow with only the inner
cylinder rotating, respectively. A stable regime is found in the middle part
of the interval of $\Omega$, and divides the whole parameter space into three
regimes: buoyancy-dominated regime, stable regime, and shear-dominated regime.
Clear boundaries between the regimes are given by linear stability analysis,
meaning the marginal state of the flow. In our parameter space, we find that
the marginal state is sensitive to $Ra$ at a low Rayleigh number while mainly
depending on $\Omega$ when $Ra\geq 10^{6}$. In the buoyancy-dominated regime,
the flow is a quasi-two-dimensional flow on the $r\varphi$ plane; as shear
increases, both the growth rate of instability and the heat transfer is
depressed. Firstly, the number of convection roll pairs decreases; then the
convection roll disappears with heat mainly transferred by strong plumes;
finally, plumes disappear as well. In the shear-dominated regime, the flow is
mainly on the $rz$ plane, first driven by buoyancy and then taken over by
shear quickly as $\Omega$ increases. The shear is so strong that the
temperature acts as a passive scalar, and the heat transfer is greatly
enhanced. The study shows shear can stabilize buoyancy-driven convection,
makes a detailed analysis of the flow characteristics in different regimes,
and reveals the complex coupling mechanism of shear and buoyancy, which may
have implications for fundamental studies and industrial designs.

###### keywords:

##  1 Introduction

Turbulent convection, one of the most complicated fluid motions, is ubiquitous
in nature and industrial processes. As two essential elements of fluid
dynamics, buoyancy, and shear play important roles in many kinds of turbulent
flows. Under gravity or other body force fields, buoyancy is generated by the
inhomogeneous density distribution of fluid. When the buoyancy force is large,
it can induce instability and drive the convection. Rayleigh-Bénard (RB)
convection is one typical paradigm of buoyancy-driven convection and has been
studied extensively in scientific research (Ahlers et al., 2009; Lohse & Xia,
2010; Chillà & Schumacher, 2012; Xia, 2013). In the RB cell, the fluid is
heated from below and cooled from above under gravity. Due to the thermal
expansion of the fluid, buoyancy appears and drives the convection. Some
manifold and involute flow structures are formed in the cell (Niemela et al.,
2001; Xi et al., 2004; Sun et al., 2005; Wang et al., 2021). Moreover, apart
from the classical RB model with rectangular cells, annular (Pitz et al.,
2017; Kang et al., 2019; Jiang et al., 2020; Rouhi et al., 2021; Wang et al.,
2023) and spherical (Gastine et al., 2015) RB cells also attract a lot of
interest. The uniform gravity is substituted by the centrifugal force or
gravity which varies in the direction of the temperature gradient. Different
from buoyancy, shear contributes to the flow mainly by the motion of the
system boundary. Taylor-Couette (TC) flow, the flow impelled by two concentric
cylinders rotating independently, is a widely used canonical model to study
the effect of shear. Secondary flows are caused by the centrifugal
instability, and then many complicated flow structures including turbulent
Taylor vortex flow and wavelets are generated at high Reynolds numbers (Bayly,
1988; Esser & Grossmann, 1996; Brauckmann & Eckhardt, 2013; Ostilla-Mónico et
al., 2014a; Grossmann et al., 2016).

Transport efficiency is an important quantity in both RB convection and TC
flow, as the physical quantities being transferred are temperature and angular
velocity, respectively (Eckhardt et al., 2007). In non-dimensional forms,
there exist scaling laws in RB convection, between the dimensionless heat flux
(measured by the Nusselt number $Nu_{h}$) and the dimensionless buoyancy-
driven strength (measured by the Rayleigh number $Ra$) (Castaing et al., 1989;
Shraiman & Siggia, 1990; Grossmann & Lohse, 2000); similar scaling laws occur
in TC flow as well, between the dimensionless angular velocity current
(measured by $Nu_{\omega}$) and the dimensionless shear (measured by the
Taylor number $Ta$) (van Gils et al., 2011; Merbold et al., 2013). Bradshaw
observed The high similarity between the RB convection and TC flow (Bradshaw,
1969); further, an exact analogy between the two flows is raised, extending
the Grossmann & Lohse’s scaling theory on RB convection well to TC flow
(Eckhardt et al., 2007; Busse, 2012). This analogy reveals that the transport
phenomena in the two systems have similar inner physics.

The coupling of buoyancy and shear is widespread in atmospheric motion and
oceanic flow (Deardorff, 1972; Khanna & Brasseur, 1998). For example, the
surface mesoscale eddies in the ocean are mainly generated by baroclinic
instabilities, which are relevant to the pole-equator temperature gradient and
a vertical shear (Hopfinger & van Heijst, 1993; Pierrehumbert & Swanson, 1995;
Vincze et al., 2014; Feng et al., 2022). By far, there are many attempts to
combine shear and buoyancy in one system and study their coupling effect. When
a radial temperature difference is applied to a TC system, it contributes to
the instability (Yoshikawa et al., 2013; Meyer et al., 2015; Kang et al.,
2015) and the momentum and heat transfer (Kang et al., 2017; Leng & Zhong,
2022). When the gravity or centrifugal force is considered as the body force
inducing buoyancy, the results are different. Leng et al. (Leng et al., 2021;
Leng & Zhong, 2022) apply radial or axial temperature difference on TC flow
with only the inner cylinder rotating, and find that with fixed temperature
difference and increasing rotational speed, the heat transfer is first
depressed by shear and then enhanced due to the development of turbulent TC
flow. Interestingly, similar phenomena occur when shear is applied on a
rectangular RB cell (Goluskin et al., 2014; Blass et al., 2020, 2021). With
increasing plane Couette shearing, the flow is first dominated by buoyancy and
then dominated by shear. The heat transport shows a similar trend in the two
systems, that heat transport is depressed under weak shearing until the shear
is strong enough to mix the system better than thermal plumes. The transition
of dominated regimes depends on the Richardson number $Ri$, defined as the
ratio between buoyancy and shear driving.

To explore the effect of shear on heat transfer in a wide parameter regime, we
adopt the annular centrifugal Rayleigh-Bénard convection (ACRBC). The
centrifugal force generated by rotating inner and outer cylinders offers
buoyancy and the Earth’s gravity is neglected, similar to the rotating
machines with high rotational speed. As the buoyancy is perpendicular to the
cylinder surface, the aspect ratio (here, the aspect ratio is defined as the
ratio of circumference to the gap length) in ACRBC is larger than that in RB
convection. Recent studies show that the ACRBC owns similar scaling laws with
classical RB convection, and is an efficient way to reach the ultimate regime
(Jiang et al., 2020, 2022). The angular velocity difference between the inner
and outer cylinders offers shear for the system, therefore the system can also
be regarded as a Taylor-Couette system with radial temperature difference.
With no shear, the inner and outer cylinders co-rotating at the same angular
speed; with strong shear, one of the cylinders can stop rotating or become
counter-rotating with another. Meyer et al. (2015) discusses the instability
of a similar system with a non-rotating outer cylinder but mainly focuses on
the instability at a low Taylor number. In our system, we work on a wide range
of the Rayleigh number and the Taylor number, concentrating on the instability
and scalar transport, aiming to bring a complete understanding of the coupling
effect of buoyancy and shear in an annular centrifugal system.

The rest of the paper is organized as follows: the establishment of the
numerical model and the numerical methods including linear stability analysis
(LSA) and direct numerical simulation (DNS) are introduced in section 2, and
the main results are discussed in section 3. Finally, conclusions are
presented in section 4.

##  2 Numerical model

A three-dimensional annular centrifugal RB cell bounded by two independent-
rotating concentric cylinders is considered, as shown in figure 1. The inner
cylinder with radius $R_{i}$ rotates about $z$ axis at angular velocity
$\Omega_{i}$ and the outer cylinder with radius $R_{o}$ rotates at the angular
velocity $\Omega_{o}$. $L=R_{o}-R_{i}$ is the gap width between the two
cylinders. The temperature difference between the hot outer cylinder and the
cold inner cylinder is $\Delta=\theta_{hot}-\theta_{cold}$. For the boundary,
no-slip and isothermal conditions are applied at two cylinder surfaces;
periodic boundary conditions are imposed on the velocity and temperature in
the axial direction, and we take a section of height $H$ as the computational
domain in DNS.

Figure 1:  Schematic diagram of the flow configuration in the stationary
reference frame. $R_{o}$,$R_{i}$, and $L$ are the inner radius of the outer
cylinder, the outer radius of the inner cylinder, and the gap width between
the two cylinders, respectively. $H$ is the height of the cylindrical annulus
in the computational domain of DNS. The outer cylinder rotates at the angular
velocity $\Omega_{o}$ and the inner cylinder rotates at the angular velocity
$\Omega_{i}$. $\theta_{hot}$ and $\theta_{cold}$ are the temperature of the
outer and inner walls.

###  2.1 Governing equations of the flow

The system is at a rotating frame with angular velocity $\Omega_{c}$, and the
buoyancy is induced by the centrifugal force
$(\Omega_{c}r+u_{\varphi})^{2}/r$. The motion of the flow is governed by the
non-dimensional Oberbeck–Boussinesq equations under cylindrical coordinate
system $(r,\varphi,z)$ (Jiang et al., 2020; Wang et al., 2022a):

$\displaystyle\boldsymbol{\nabla}\cdot\boldsymbol{u}$ $\displaystyle=0,$ (1)
$\displaystyle\frac{\partial\boldsymbol{u}}{\partial
t}+\boldsymbol{u}\cdot\boldsymbol{\nabla
u}=-\boldsymbol{\nabla}p-Ro^{-1}\boldsymbol{e_{z}}\times\boldsymbol{u}$
$\displaystyle+\sqrt{\frac{Pr}{Ra}}\nabla^{2}\boldsymbol{u}-\theta\frac{2(1-\eta)}{1+\eta}(1+\frac{2u_{\varphi}}{Ro^{-1}r})^{2}\boldsymbol{r},$
$\displaystyle\frac{\partial\theta}{\partial
t}+\boldsymbol{u}\cdot\boldsymbol{\nabla}\theta$
$\displaystyle=\sqrt{\frac{1}{Ra\cdot Pr}}\nabla^{2}\theta,$

where $\boldsymbol{u}=(u_{r},u_{\varphi},u_{z})$ is the velocity vector,
$\theta\in[0,1]$ is the temperature, $p$ is the pressure, $\boldsymbol{e_{z}}$
is the unit vector along the axial direction and $\eta=R_{i}/R_{o}$ is the
radius ratio. Scaled quantities, including $L=R_{o}-R_{i}$ for length,
$\Delta$ for temperature,
$U=\sqrt{\alpha\Delta\Omega_{c}^{2}\frac{(R_{i}+R_{o})}{2}L}$ for velocity,
and $L/U$ for time are used to non-dimensionalize the governing equation,
where $\alpha$ is the coefficient of thermal expansion of the fluid.
$\Omega_{c}$ is the rotational speed of the rotating reference frame, and we
suppose it to represent the strength of centrifugal buoyancy, as reflected in
the expression of free-fall velocity $U$. In our system, considering both
physical meaning and simplicity, we set
$\Omega_{c}=(\Omega_{i}+\Omega_{o})/2$. It is a good estimate for free-fall
velocity when shearing is relatively small, which means
$\Omega_{i}-\Omega_{o}\ll\Omega_{c}$; when shearing is strong enough and
dominates the flow, buoyancy contributes little and can be neglected, which
will be discussed later in section 3. Therefore, the selected $\Omega_{c}$ is
reasonable in most of the parameter space. Under that rotating frame, the
inner cylinder rotates at non-dimensional angular velocity
$\Omega=(\Omega_{i}-\Omega_{c})L/U$ and the outer cylinder rotates at
$-\Omega$.

Three non-dimensional parameters are generated through non-dimensionalization,
they are the Rayleigh number (measuring the buoyancy-driving strength) $Ra$,
the inverse Rossby number (measuring Coriolis effects) $Ro^{-1}$ and the
Prandtl number (fluid property) $Pr$, expressed as:

$\displaystyle Ra=\frac{(UL)^{2}}{\nu\kappa}$
$\displaystyle=\frac{\alpha\Delta\Omega_{c}^{2}\frac{(R_{i}+R_{o})}{2}L^{3}}{\nu\kappa},$
(2) $\displaystyle Ro^{-1}=\frac{2\Omega_{c}L}{U}$
$\displaystyle=2(\frac{\alpha\Delta(R_{i}+R_{o})}{2L})^{-1/2},$ $\displaystyle
Pr$ $\displaystyle=\frac{\nu}{\kappa},$

where $\nu$ is the kinematic viscosity and $\kappa$ is the thermal diffusivity
of the fluid. Therefore, there are five independent control parameters in the
system: the Rayleigh number $Ra$, the inverse Rossby number $Ro^{-1}$, the
Prandtl number $Pr$, the rotational speed difference of two cylinders
$\Omega=(\Omega_{i}-\Omega_{c})L/U$ and the radius ratio $\eta$. Moreover,
since the system can also be regarded as a TC system with radial buoyancy, the
Taylor number $Ta$ can be defined as:

$Ta=\frac{(1+\eta)^{4}}{64\eta^{2}}\frac{(R_{o}-R_{i})^{2}(R_{i}+R_{o})^{2}(\Omega_{i}-\Omega_{o})^{2}}{\nu^{2}}=\frac{(1+\eta)^{6}}{16\eta^{2}(1-\eta)^{2}}\frac{\Omega^{2}Ra}{Pr}.$
(3)

Therefore, $Ta$ is not an independent control parameter in our system,
proportional to the $Ra$ and $\Omega^{2}$. The Richardson number $Ri$,
measuring the ratio between buoyancy and shear driving, can also be calculated
by the ratio of characteristic buoyancy-velocity and the characteristic
shearing-velocity:

$Ri=\frac{U^{2}}{(\Omega_{i}-\Omega_{c})^{2}r_{i}^{2}}=(\frac{1-\eta}{\eta})^{2}\Omega^{-2}.$
(4)

As we can see, the non-dimensional rotational speed difference $\Omega$ can
represent the strength of shear relative to buoyancy.

In addition, two key response parameters are the Nusselt numbers measuring the
efficiency of heat transport and momentum transport, given by the ratio of
radius-independent currents to the currents in laminar and nonvortical cases,
respectively (Eckhardt et al., 2007; Wang et al., 2022a), as
$Nu_{h}=J^{\theta}/J^{\theta}_{lam}$,
$Nu_{\omega}=J^{\omega}/J^{\omega}_{lam}$. More about the Nusselt numbers and
how they respond to $Ra$ and $\Omega$ will be discussed in section 3.

###  2.2 Linear stability analysis

LSA is a good approach to studying the instability of the flow. At the stable
state, the flow is laminar and nonvortical TC flow and the heat is transferred
by pure conduction. The azimuthal velocity $V$ and temperature $\Theta$ depend
only on $r$, and the radial and axial velocities are zero. The functions of
$V$ and $\Theta$ are given by (Ali & Weidman, 1990)

$\displaystyle V(r)=Ar+\frac{B}{r},A$
$\displaystyle=-\frac{1+\eta^{2}}{1-\eta^{2}}\Omega,B=\frac{2r_{i}^{2}}{1-\eta^{2}}\Omega,$
(5) $\displaystyle\Theta(r)$
$\displaystyle=\frac{ln(r/r_{i})}{ln(r_{o}/r_{i})},$

where $r_{i}=\eta/(1-\eta)$ and $r_{o}=1/(1-\eta)$ are the non-dimensional
radii of the inner and outer cylinders, respectively.

To perform LSA on this problem, we superimpose infinitesimal perturbations
$(u^{\prime}_{r},u^{\prime}_{\varphi},u^{\prime}_{z},p^{\prime},\theta^{\prime})$
on the base flow state. After substituting the perturbation fields into
equations (1), doing linearization, and expanding the perturbations into
normal modes (Meyer et al., 2015; Kang et al., 2017),

$(u^{\prime}_{r},u^{\prime}_{\varphi},u^{\prime}_{z},p^{\prime},\theta^{\prime})=(\hat{u_{r}}(r),\hat{u_{\varphi}}(r),\hat{u_{z}}(r),\hat{p}(r),\hat{\theta}(r))exp(st+i(n\varphi+kz)),$
(6)

we obtain the resulting ordinary differential equations for $r-$ dependent
normal mode quantities:

$\displaystyle(D+r^{-1})\hat{u_{r}}+$ $\displaystyle
inr^{-1}\hat{u_{\varphi}}+ik\hat{u_{z}}=0,$ (7)
$\displaystyle(s+\frac{inV}{r})\hat{u_{r}}-\frac{2V}{r}\hat{u_{\varphi}}=$
$\displaystyle-D\hat{p}+Ro^{-1}\hat{u_{\varphi}}+\sqrt{\frac{Pr}{Ra}}(\nabla^{2}\hat{u_{r}}-\frac{\hat{u_{r}}}{r^{2}}-\frac{2in\hat{u_{\varphi}}}{r^{2}})$
$\displaystyle-\frac{2(1-\eta)}{(1+\eta)}r[(1+\frac{2V}{Ro^{-1}r})^{2}\hat{\theta}+\frac{4\Theta}{Ro^{-1}r}(1+\frac{2V}{Ro^{-1}r})\hat{u_{\varphi}}],$
$\displaystyle(s+\frac{inV}{r})\hat{u_{\varphi}}+(DV+\frac{V}{r})\hat{u_{r}}=$
$\displaystyle-\frac{in}{r}\hat{p}-Ro^{-1}\hat{u_{r}}+\sqrt{\frac{Pr}{Ra}}(\nabla^{2}\hat{u_{\varphi}}-\frac{\hat{u_{\varphi}}}{r^{2}}+\frac{2in\hat{u_{r}}}{r^{2}}),$
$\displaystyle(s+\frac{inV}{r})\hat{u_{z}}=$ $\displaystyle-
ik\hat{p}+\sqrt{\frac{Pr}{Ra}}\nabla^{2}\hat{u_{z}},$
$\displaystyle(s+\frac{inV}{r})\hat{\theta}+(D\Theta)\hat{u_{r}}=$
$\displaystyle\frac{1}{\sqrt{RaPr}}\nabla^{2}\hat{\theta},$

where operators $D=d/dr$ and $\nabla^{2}=D^{2}+D/r-n^{2}/r^{2}-k^{2}$ are
introduced for simplification. $s$ is the temporal growth rate of
perturbation, $n$ is the azimuthal mode number and $k$ is the axial
wavenumber. Due to the infinite axial length and $2\pi$ period in the
azimuthal direction, $k$ must be real and $n$ must be an integer. The boundary
conditions of the perturbations are homogeneous:

$\displaystyle r=r_{i}:\hat{u_{r}}$
$\displaystyle=\hat{u_{\varphi}}=\hat{u_{z}}=\hat{p}=\hat{\theta}=0,$ (8)
$\displaystyle r=r_{o}:\hat{u_{r}}$
$\displaystyle=\hat{u_{\varphi}}=\hat{u_{z}}=\hat{p}=\hat{\theta}=0.$

Then, the instability problem has been transformed into an eigenvalue problem,
described by equations (7)-(8). This eigenvalue problem is solved by Chebyshev
spectral collocation method. The equations (7) are discretized on Chebyshev-
Gauss-Lobatto collocation points with Chebyshev differentiation matrices. In
our work, the number of collocation points range from $200$ to $400$ for good
convergence. Then the growth rate of perturbations $s$ becomes the eigenvalue
of the generalized eigenvalue problems in the matrix form, and the
corresponding perturbation normal modes are the eigenvectors. For certain
parameters $(Ra,Pr,Ro^{-1},\Omega,\eta)$, the flow is stable if for all
$(n,k)$, the growth rate $\sigma=real(s)$ is always negative, which means all
perturbation modes decay with time.

###  2.3 Direct numerical simulation

Direct numerical simulations (DNS) are performed using an energy-conserving
second-order finite-difference code based on a Chebyshev-clustered staggered
grid. The time-stepping of the explicit terms is based on a fractional-step
third-order Runge–Kutta scheme, and the implicit terms are based on a
Crank–Nicolson scheme with a pressure correction step set following. For more
details on the numerical schemes of the governing equations, we refer the
reader to this literature (Verzicco & Orlandi, 1996; van der Poel et al.,
2015; Zhu et al., 2018).

Adequate resolutions are ensured for all simulations and we have performed
posterior checks of spatial and temporal resolutions to guarantee the
resolution of all relevant scales. The ratios of maximum grid spacing
$\Delta_{g}$ to the Kolmogorov scale estimated by the global criterion
$\eta_{K}=(\nu^{3}/\varepsilon)^{1/4}$, where $\varepsilon$ is the mean
viscous dissipation rate via exact relation, and the Batchelor scale
$\eta_{B}=\eta_{K}Pr^{-1/2}$ (Silano et al., 2010) are checked, as shown in
Appendix A. Furthermore, the clipped Chebychev-type clustering grids adopted
in the radial direction ensure the spatial resolution within boundary layers
(BLs), as at least $10$ grid points inside the thermal boundary layers. As for
temporal resolution, we use the Courant-Friedrichs-Lewy (CFL) conditions and
set $CFL\leq 0.7$ to guarantee the computational stability (Ostilla et al.,
2013; van der Poel et al., 2015; Zhang et al., 2017). The simulations are run
over enough time after the system has reached the statistically stationary
state to obtain good statistical convergence. The relative difference of
$Nu_{h}$ based on the first and second halves of the simulations is generally
less than $1\%$. And for $Nu_{\omega}$, as its absolute value is small and
close to $1$ in many weak shear cases, the relative difference is controlled
generally less than $4\%$ under weak shear, and less than $2\%$ under strong
shear. All those details are provided in Appendix A.

###  2.4 Other numerical details

In the present study, we aim at the coupling effect of shear and buoyancy on
the flow structure and heat and momentum transfer, that is
$(Nu_{h},Nu_{\omega})=f(Ra,\Omega)$. Therefore, referring to the results of
our previous experiments and simulations of ACRBC (Jiang et al., 2020, 2022;
Wang et al., 2022b), $Pr=4.3$ is taken for water at $40^{\circ}C$, and $\eta$
is set to be $0.5$. Considering the Boussinesq approximation, $\alpha\Delta\ll
1$ is required. According to equations (2), we choose $\alpha\Delta=6.67\times
10^{-3}$ and then $Ro^{-1}=20$, which is also in the parameter range of our
previous experiments, corresponding to a temperature difference $\Delta\approx
17.3K$ for water (Jiang et al., 2020, 2022). The Non-Oberbeck-Boussinesq
effect is not obvious at this temperature difference (Ahlers et al., 2006) and
the Oberbeck-Boussinesq conditions are well satisfied. Meanwhile, considering
the definition of $Ro^{-1}$, in the non-dimensional system, the rotational
speed of the rotating reference frame is $(\Omega_{c}L/U)=Ro^{-1}/2=10$.

As for instability, we focus on $Ra\in[10^{3},10^{9}]$ and
$\Omega\in[10^{-2},10]$, where positive $\Omega$ means the inner cylinder
rotates at a faster angular velocity than the outer cylinder in the stationary
reference frame. For example, $\Omega=10$ corresponds to the state that the
inner cylinder is rotating with the non-dimensional angular velocity
$\omega=20$ and the outer cylinder is static in the stationary reference
frame. The DNS for heat tranfer analysis covers a $Ra$ range $[10^{6},10^{8}]$
and a rotational speed difference $\Omega$ range $[10^{-2},10]$.

##  3 Results and discussion

Previous studies have discussed or implied how the flow and heat transfer are
at both ends of our parameter range. When $\Omega=0$, Jiang et al. (Jiang et
al., 2020) shows that the flow will be quasi-two-dimensional at high $Ro^{-1}$
number ($Ro^{-1}\geq 10$), due to the constraint of Taylor-Proudman theorem.
The scaling law of heat transfer in this region follows the Grossman-Lhose
(GL) theory, with a power-law relationship of $Nu\sim Ra^{0.27}$ in the
classical regime. At the other end, $\Omega=10$, only the inner cylinder
rotating, previous work (Leng et al., 2021; Leng & Zhong, 2022) implies that
the shear is so strong that the flow is dominated by the TC vortex and the
temperature behaves like a passive scalar. In this section, we will reveal how
the flow undergoes such a large transformation, from the RB flow in the
$r\varphi$ plane to the TC flow developed in the $rz$ plane.

###  3.1 Flow regimes

When increasing shear converts the flow from the RB flow to the TC flow, a
stable regime is found on the way. To determine the boundaries of the stable
regime, the approach of linear stability analysis is applied and the results
are checked by the direct numerical simulation. As shown in figure 2(a), two
unstable regimes are distributed on the two sides of the parameter domain and
a stable regime locates in the middle. The results of DNS agree well with the
boundaries given by LSA, guaranteeing the validity of the stable regime. We
denote the three regimes as RB-dominated regime (Regime I), stable regime
(Regime II), and TC-dominated regime (Regime III), with the shear strength
increasing.

Figure 2: (a) Instability regimes divided by LSA (the bottle green lines) and
checked by DNS (the red circles and the blue triangles) in the $(Ra,\Omega)$
parameter domain. (b) Instantaneous temperature fields from DNS at
$Ra=10^{6},\Omega=10^{-2}$ (Regime I). (c) Instantaneous temperature fields
from DNS at $Ra=10^{6},\Omega=1$ (Regime II). (d) Instantaneous temperature
fields from DNS at $Ra=10^{6},\Omega=7$ (Regime III).

In the stable regime (Regime II), the typical instantaneous temperature field
is illustrated in figure 2(c). The system is governed by steady laminar and
nonvortical TC flow, as described by equations (5). In the other two regimes,
the typical instantaneous temperature fields exhibit significant differences,
depicted in figures 2(b,d). In Regime I, the flow is quasi-two-dimensional on
the $r\varphi$ plane. Plumes are detached from the boundary layers, move
across the bulk region, and transfer heat to the other side. Several
convection roll pairs are generated, the number depending on the radius ratio
$\eta$ when there is no shear (Pitz et al., 2017; Wang et al., 2022a). In
Regime III, the flow is three-dimensional, and the roll pairs, or called
Taylor vortexes, are mainly on the $rz$ plane, which can be seen clearly on
the vertical $\varphi$ slice in figure 2(c). There are large differences among
the flow structures in these three regimes. For a fixed $Ra$, as the shear
increases from zero, the flow first stabilizes from a quasi-two-dimensional RB
flow into a laminar flow and then destabilizes into a three-dimensional flow
similar to the TC flow. It’s worth noting that as $\eta=0.5$ in our system,
the Richardson number $Ri=\Omega^{-2}$, based on the equation (4). Therefore,
at high $Ra$, the stable regime locates at $Ri\sim O(1)$, which means the
buoyancy is in the same order with shear; this is consistent with the
transition region from buoyancy-dominated to shear-dominated obtained from
previous work (Blass et al., 2020; Leng et al., 2021).

The curves meaning the marginal state between regimes contain important
information. The boundary between Regimes I and II has a horizontal asymptote
on the left side for $\Omega\rightarrow 0$, that is the onset $Ra$ of the
ACRBC at $\eta=0.5$. As $\Omega$ increases, the instability of the flow is
suppressed and a stronger buoyancy force is required to drive the convection.
The critical Rayleigh number $Ra_{c}$ of the marginal state grows faster and
faster with increasing $\Omega$, and finally, when $\Omega$ exceeds a certain
value (about $0.6$ with our parameter settings), the flow is always stable no
matter how much the Rayleigh number increases in our parameter range.
According to the marginal-state curve in figure 2(a), there may also exist a
vertical asymptote for $Ra\rightarrow\infty$. With the participation of the
shear, the flow instability problem is different from the onset of the RB
convection. For the onset $Ra$ of the ACRBC without shear, there exists a
perturbation energy balance between the work performed by the buoyancy and the
energy dissipation by the fluid viscosity; when the shear is imposed, the
movement of the boundaries changes the balance. In addition, we notice that
for the sheared classical rectangular RB system, the flow does not become
stable with increasing shear (Blass et al., 2020). This indicates that the
effect of curvature and the rotation in sheared ACRBC may also contribute to
the stabilization of the flow. We will discuss the physics of the instability
between Regime I and II in depth in section 3.2.

Moreover, the straight boundary between Regimes II and III seems to behave
differently. On this boundary, $Ta$ is much larger than $Ra$: according to the
equation (3), when $Ra=10^{3}$, $Ta\approx 1\times 10^{5}$. Therefore, the
marginal-state curve between Regime II and III is similar to the line given by
Rayleigh’s inviscid criterion (Ali & Weidman, 1990; Drazin & Reid, 2004;
Yoshikawa et al., 2013), with the thermal effect enhancing the instability
slightly. As the instability of TC flow with radial temperature difference has
been widely discussed (Ali & Weidman, 1990; Yoshikawa et al., 2013; Kang et
al., 2015; Meyer et al., 2015; Yoshikawa et al., 2015), we mainly focus on the
instability in Regime I and II in the following discussion.

###  3.2 Instability

#### Mode analysis

To investigate the effect of shear on the flow and how the stable regime
arises, we begin with linear instability. In Regime I, the flow is unstable,
and each pair of azimuthal and axial wavenumber $(n,k)$ is associated with a
perturbation mode of a growth rate $\sigma$. We notice that for every $n$,
$\sigma$ reaches a maximum all at $k=0$, corresponding to the quasi-2D flow
characteristic in Regime I, that the flow is limited on the $r\varphi$ plane.
Therefore, only $k=0$ is considered in subsequent analysis on Regime I.

Figure 3: (a) Growth rate $\sigma$ as a function of the azimuthal wavenumber
$n$ for $\Omega=0,0.1,0.3,0.5$ and $0.57$., at $Ra=10^{7}$. The black dashed
line denotes $\sigma=0$. (b) The azimuthal wavenumber of the main mode
$n_{main}$ as a function of $\Omega$, at $Ra=10^{7}$. (c) The critical
azimuthal wavenumber $n_{c}$ as a function of the critical Rayleigh number
$Ra_{c}$ at the marginal state.

Wondering how shear influences the flow instability, we analyze the growth
rate $\sigma$ as a function of the azimuthal wavenumber $n$ for increasing
$\Omega$ at $Ra=10^{7}$, and the results are plotted in figure 3(a). In the no
shear case $\Omega=0$, $\sigma$ grows with $n$ quickly at first and then
reaches a plateau. Modes of high azimuthal frequency $n\gtrsim 5$ are
preferred, having a higher growth rate than the low azimuthal wavenumber
modes. With the shear imposed, the growth rates of all modes decrease
synchronously, while the high-frequency modes are suppressed more. This means
that shear suppresses the growth of linear instability and further influences
the generation of the convection flow. Meanwhile, perturbations with higher
azimuthal frequency modes are suppressed more, and the dominated mode becomes
the low-frequency mode gradually, as an increasing shear is imposed. Figure
3(b) demonstrates this trend. The azimuthal wavenumber of the main mode (the
mode owning maximum growth rate $\sigma$) $n_{main}$ decreases from $15$ to
$1$ with the increasing shear. When the flow approaches the marginal state,
$n_{main}$ drops to $1$ and holds on. Figure 3(c) shows at the marginal state,
how the critical azimuthal number $n_{c}$ changes with the critical Rayleigh
number $Ra_{c}$. At no shear state, which is the left asymptote in figure
2(a), $n_{c}=5$, giving the mode of five roll pairs. As the Rayleigh number
and the shear increase, $n_{c}$ drops from $5$ to $1$, and holds on at high
Rayleigh number $(Ra_{c}\geq 10^{6})$. It’s worth noting that in figure 2(a),
the critical rotating difference $\Omega_{cr}$ reaches a maximum near
$Ra=10^{6}$ as well, and then the $\Omega_{cr}$ required to stabilize the flow
decreases slightly with increasing $Ra$. The slight decrease of $\Omega_{cr}$
may be related to the fact that the critical azimuthal wavenumber reaches the
minimum.

The temperature and the velocity eigenfunctions of certain modes normalized by
the maximum temperature perturbations are illustrated in figure 4. For $n=1$,
there is only one hot-cold perturbation pair, while for $n=5$ there are five
pairs. In all perturbation modes, the colder flow goes outward from the inner
side with a clockwise azimuthal velocity, in the opposite direction of the
basic flow. When the system is under low shear, as shown in figure 4(a), the
azimuthal motion of the flow is strong at the junction of the cold and hot
flow, while relatively weak at the center of cold and hot perturbations. Near
the inner wall, two flows with opposing azimuthal velocity converge at the
center region of cold temperature perturbations. As a result of the mass
conservation, the flow undergoes an outward deflection, thereby transporting
the strong cold temperature perturbations toward the outer wall. Similar
structures exist extensively in the modes under various conditions, as
illustrated in figures 4(b-d). Comparing figure 4(b) to (a), as the shear
becomes stronger, we notice that the cold and hot perturbations are more
stretched azimuthally and are narrower radially, which is similar to the
plumes in sheared RB convection (Goluskin et al., 2014; Blass et al., 2020).
In figure 4(b), a strong azimuthal velocity exists everywhere in the cold and
hot flow, even in the center region of the cold flow, different from the low-
shear case. The main motion of the flow is in the azimuthal direction, which
has a negative effect on heat transfer and the work done by buoyancy. Under
stronger shear, it seems more difficult to transport heat from one side to the
other side. Both larger dissipation and less work done by buoyancy depress the
growth of the instability.

Figure 4: Eigenfunctions $(\boldsymbol{u^{\prime}},\theta^{\prime})$ of the
unstable modes at corresponding conditions: (a) $Ra=10^{6},\Omega=0.1,n=1$,
(b) $Ra=10^{6},\Omega=0.5,n=1$, (c) $Ra=10^{6},\Omega=0.5,n=5$, (d)
$Ra=10^{8},\Omega=0.5,n=1$. The maximum temperature perturbation is set to be
the same in all conditions, $|\theta^{\prime}|_{max}=10^{-2}$. The velocity
perturbation vectors are scaled by the maximum velocity magnitude of each, as
$|\boldsymbol{u^{\prime}}|_{max}$ equals to (a) $7.3\times 10^{-3}$, (b)
$4.0\times 10^{-3}$, (c) $4.2\times 10^{-3}$, (d) $4.0\times 10^{-3}$.

Moreover, for the modes of high azimuthal frequency under strong shear, more
pairs occupy the azimuthal length of $2\pi$, an example of which is shown in
figure 4(c). The temperature perturbation pairs and flow roll pairs are
denser, presenting a pattern that extends counterclockwise from the inside
out. As can be expected, this mode has higher dissipation and more significant
buoyancy work than the low azimuthal frequency mode $n=1$ in figure 4(b). In
addition, we are interested in the instability case when moving to high
Rayleigh numbers. A mode near the marginal state at $Ra=10^{8}$ is presented
in figure 4(d). Interestingly, the perturbations exhibit a pronounced
concentration within an exceedingly narrow annulus, approximately located at
the intermediate radius of the system. Inside and outside of this annulus,
both temperature and velocity perturbations are weak. As $Ra$ escalates, the
annulus progressively diminishes in width, tending to converge to a certain
radius, which may be associated with the limit of $\Omega_{cr}$ for an
infinitely-great Rayleigh number.

#### Energy analysis

In order to get a comprehensive explanation of the instability in Regime I, we
perform the energy analysis on perturbations. Multiply the velocity
perturbations $u^{\prime}_{r},u^{\prime}_{\varphi},u^{\prime}_{z}$ by the
linearized momentum equations in their respective directions and take their
summation, then the kinetic energy equation of perturbations is obtained
(Yoshikawa et al., 2013, 2015; Meyer et al., 2015):

$\frac{dK}{dt}=W_{Ta}+W_{cB}-D_{\nu},$ (9)

where the kinetic energy $K$, the power of inertial forces $W_{Ta}$, the power
of centrifugal buoyancy $W_{cB}$ and the dissipation
$D_{\nu}=\langle\Phi\rangle$ are expressed as:

$\displaystyle K=\frac{1}{2}\langle|\boldsymbol{u^{\prime}}|^{2}\rangle,\
W_{Ta}$
$\displaystyle=\langle-u^{\prime}_{r}u^{\prime}_{\varphi}(\frac{dV}{dr}-\frac{V}{r})\rangle,$
(10) $\displaystyle
W_{cB}=\langle-\frac{2(1-\eta)ru^{\prime}_{r}}{1+\eta}[\theta^{\prime}(1$
$\displaystyle+\frac{2V}{Ro^{-1}r})^{2}+\frac{4\Theta
u^{\prime}_{\varphi}}{Ro^{-1}r}(1+\frac{2V}{Ro^{-1}r})]\rangle,$
$\displaystyle D_{\nu}=\sqrt{\frac{Pr}{Ra}}\langle 2[(\frac{\partial
u^{\prime}_{r}}{\partial r})^{2}$ $\displaystyle+(\frac{1}{r}\frac{\partial
u^{\prime}_{\varphi}}{\partial{r}}+\frac{u^{\prime}_{r}}{r})^{2}+(\frac{\partial
u^{\prime}_{z}}{\partial z})^{2}]+[r\frac{\partial}{\partial
r}(\frac{u^{\prime}_{\varphi}}{r})+\frac{1}{r}\frac{\partial
u^{\prime}_{r}}{\partial\varphi}]^{2}$
$\displaystyle+[\frac{1}{r}\frac{\partial
u^{\prime}_{z}}{\partial\varphi}+\frac{\partial u^{\prime}_{\varphi}}{\partial
z}]^{2}+[\frac{\partial u^{\prime}_{r}}{\partial z}+\frac{\partial
u^{\prime}_{z}}{\partial r}]^{2}\rangle.$

The angle brackets $\langle\cdot\rangle$ denote the average over the whole
space $r,\varphi$, and $z$. A mode is stable if the right-hand side of the
equation (9) is negative. The kinetic energy generation terms
$W_{Ta},W_{cB},D_{\nu}$ normalized by the kinetic energy $K$ are illustrated
in figure 5 as functions of $\Omega$ at $Ra=10^{6}$ and $10^{8}$. As all terms
are normalized by the averaged kinetic energy $K$, the left side of the
equation (9) $dK/dt$ is twice the growth rate, derived from equation (6). As
shown in figure 5(a), $Ra=10^{6}$ and $n=1$, with $\Omega$ increasing, the
power of centrifugal buoyancy $W_{cB}$ decreases and the dissipation term
$D_{\nu}$ increases slightly, in line with our analysis of the modes in figure
4. The largest change is in the inertial force action term, $W_{Ta}$, which is
zero under no shear but its magnitude increases rapidly as $\Omega$ increases.
According to the formula of $W_{Ta}$ in equations (10), this term is closely
related to the shear action. When the system reaches the marginal state, three
energy generation terms exhibit equilibrium. The presence of $W_{Ta}$
constitutes the primary differentiating factor between the flow instability
with shear and the flow instability without shear.

Figure 5: Variation of energy generation terms $-W_{Ta}$,$D_{\nu}$,$W_{cB}$
and the growth rate of kinetic energy $\frac{dK}{dt}$ with $\Omega$, at
conditions (a) $Ra=10^{6},n=1$, (b) $Ra=10^{6},n=5$, (c) $Ra=10^{8},n=1$. All
terms are normalized by $K$. The modes in the light orange region (left side)
are unstable, while in the nattier blue region (right side) are stable.

The energy analysis method can also explain the reasons behind the greater
suppression of high azimuthal frequency modes in comparison to low-frequency
modes to a certain degree. Figure 5(b) shows the variation of energy
generation terms under $Ra=10^{6},n=5$. At $\Omega=0$, the centrifugal
buoyancy term $W_{cB}$ is much larger than dissipation, injecting a lot of
energy into the development of the instability. However, with shear imposed,
both $-W_{Ta}$ and $D_{\nu}$ rise quickly, while the buoyancy term $W_{cB}$
drops faster than in the low-frequency mode $n=1$. This reflects the strong
suppression of the high-frequency modes by shear. Unlike the low-frequency
mode in figure 5(a), the dissipation term $D_{\nu}$ plays a more important
role, increasing significantly with shear, which is closely related to the
tight distribution of velocity perturbations rolls. Therefore, although the
high-frequency mode offers a high growth rate at no shear, it reaches the
marginal state earlier as the shear increases.

At high $Ra$, the dissipation term decreases at first with increasing shear,
as illustrated in figure 5(c). The buoyancy terms drop more than the mode at
$Ra=10^{6}$, as the perturbations are concentrated in a narrow annulus and the
azimuthal motion occupies more kinetic energy. Meanwhile, we notice that at
the marginal state, it is still a balance of three items
$W_{Ta},W_{cB},{D_{\nu}}$, which can be further observed at higher
$Ra=10^{13}$. Therefore, the limit of instability as the Rayleigh number tends
to infinity can not be explained by an inviscid solution, since the
dissipation term $D_{\nu}$ is not negligible.

Moreover, at $Ra=10^{6}$, the distribution of energy generation terms,
$w_{Ta},w_{cB},\Phi$ are compared under different shear strengths with the
azimuthal wavenumber $n=1$, as shown in figure 6. The densities of energy
generation terms are also normalized by $K$. By comparing figure 6(b) and
figure 6(f), one can find that the power of inertial term $w_{Ta}$ increases
substantially with the enhancement of shear. (Note: the scales of the color
bars in these two figures are different.) Negative $w_{Ta}$ is derived from
the rightward deflection of outward or inward flow
($u^{\prime}_{r}u^{\prime}_{\varphi}<0$); while the positive part mainly comes
from the limit of mass conservation near the boundary and occurs at the
junction of cold and hot currents. As for the buoyancy power $w_{cB}$, the
increase of shear amplifies its maximum intensity while making it more
concentrated in distribution, as the cold and hot perturbations are more
stretched azimuthally. Dissipation term $\Phi$ mainly occurs near the boundary
and the regions with opposing flows of cold and hot currents, where a strong
velocity gradient exists. As shear increases, $\Phi$ increases significantly
at the region where two flows meet, contributing to a final increase of
$D_{\nu}$. These qualitative interpretations are in relatively good agreement
with the curves in figure 5(a).

Figure 6: Distribution of perturbations (a,e)
$(\boldsymbol{u^{\prime}},\theta^{\prime})$ and the densities of energy
generation terms (b,f) $w_{Ta}$, (c,g) $w_{cB}$ and (d,h) $\Phi$ under
different shear (a-d) $\Omega=0.1$, (e-f) $\Omega$=0.5 when $Ra=10^{6},n=1$.
All the densities of energy generation terms are normalized by the averaged
kinetic energy $K$. Note that the scales of $w_{Ta}$ in (b) and (f) are
different.

###  3.3 Heat and momentum transfer

In this subsection, we focus on both heat and momentum transport efficiency
and discuss their behaviors in different regimes. Under the dimensionless
formulation, the heat transfer efficiency and the momentum transfer efficiency
are measured by two Nusselt numbers: $Nu_{h}$ and $Nu_{\omega}$, defined as
the ratios of the corresponding currents of the system to the currents in the
laminar and nonvortical flow case (Eckhardt et al., 2007; Wang et al., 2022a):

$\displaystyle Nu_{h}$ $\displaystyle=\frac{\sqrt{RaPr}\langle
u_{r}\theta\rangle_{t,\varphi,z}-\partial_{r}\langle\theta\rangle_{t,\varphi,z}}{(rln(\eta))^{-1}},$
(11) $\displaystyle Nu_{\omega}$ $\displaystyle=\frac{r^{3}[Ra/Pr\langle
u_{r}\omega\rangle_{t,\varphi,z}-\sqrt{Ra/Pr}\partial_{r}\langle\omega\rangle_{t,\varphi,z}]}{2B},$

where $\omega=u_{\varphi}/r$ is the angular velocity of the fluid, and $B$ is
the parameter of the base flow defined in equations (5). The variation of the
two Nusselt numbers with the shear strength $\Omega$ under different Rayleigh
numbers are illustrated in figure 7. The different regimes in the figure are
distinguished using different background colors. As the Taylor number is much
larger than $Ra$ in Regime III ($Ta=2.65\times 10^{9}$ at
$Ra=10^{7},\Omega=10$, calculated from the equation (3)), the flow is too
drastic and computationally expensive, so we only calculate the cases of
$Ra=10^{6}$ in the TC-dominated regime.

Figure 7: Variation of (a) $Nu_{h}$ and (b) $Nu_{\omega}$ with $\Omega$, at
$Ra=10^{6}$, $10^{7}$ and $10^{8}$.

In Regime I, as the imposed shear increases, $Nu_{h}$ decreases slowly at
first and then rapidly when $\Omega>0.3$. The curves of different $Ra$ behave
in a similar trend. The heat transfer efficiency $Nu_{h}$ increases with $Ra$
in Regime I, but as the shear increases and flow turn stable, $Nu_{h}$ under
different $Ra$ all drop to $1$. These results are similar to the results in
the sheared rectangular RB cell (Blass et al., 2020), where the heat transfer
is also first depressed by shear. However, in sheared rectangular RB cell,
shear can not turn the flow laminar. While for momentum transfer, we notice
that the system gives a negative $Nu_{\omega}$ under a weak shear, which means
the large-scale circulation (LSC) in the RB convection pushes the two
cylinders to rotate relative to each other. According to the definition of
$Nu_{\omega}$, when there is no relative rotation between two cylinders, LSC
also imposes shear stress to the wall, and $Nu_{\omega}$ is negative infinity.
$Nu_{\omega}$ increases quickly as $\Omega$ increases, and the growth trend
slows down when its value is close to $1$. Similar to the heat transfer, a
larger $Ra$ gives a stronger torque.

In Regime II, the flow field is laminar, described by equations (5). Both
$Nu_{h}$ and $Nu_{\omega}$ equal to $1$. As $\Omega$ increases, the system
reaches the marginal state between Regime II and III. If the buoyancy is not
considered, at high $Ta$, the marginal-state curve of Taylor instability is
close to the solution given by Rayleigh inviscid theory (Drazin & Reid, 2004),
which gives $\Omega=6$. Considering the buoyancy, the instability is
strengthened and the flow at $\Omega=6$ gives a heat transfer efficiency close
to $Nu_{h}$ under weak shear. Meanwhile, $Nu_{\omega}$ remains almost
unchanged, which means the flow driven by buoyancy is hard to mix the angular
velocity profile. As the shear strength continues to increase, the heat
transfer is enhanced more, and the momentum transfer increases gradually as
well. It is suggested that this regime appears in a similar system with
gravity-driven buoyancy, where the system is mainly controlled by the Taylor
vortex and temperature acts as a passive scalar (Leng et al., 2021). To check
if this idea is also valid in our system, we perform another set of
simulations under the same conditions, where the velocity is decoupled from
temperature (buoyancy not considered) and the temperature really acts as a
passive scalar. The comparison of the two is displayed in figure 8(a). As
there is no buoyancy, the system is still stable at $\Omega=6$, but then it
behaves almost the same as the velocity-temperature-coupled system, at
$\Omega=6.5$ in the figure. Therefore, in most of Regime III, the temperature
can be regarded as a passive scalar in our system; it is the shear that
dominates the flow.

Figure 8: (a) Variation of $Nu_{h}$ with $\Omega$ in Regime III, for the cases
that the temperature and velocity are coupled (the purple line) and uncoupled
(the red line). $Ra=10^{6}$. (b) Variation of $|\gamma|$ with $\Omega$ at
$Ra=10^{6},10^{7},10^{8}$. $\gamma<0$ in Regime I and $\gamma>0$ in Regime
III.

The exact relation gives us an interesting perspective to understand the
relation between $Nu_{h}$ and $Nu_{\omega}$. Derivated by the kinetic energy
balance equation, the exact relation represents the energy balance between
buoyant energy injection, shear energy injection, and mechanical dissipation.
In the dimensional form, referring to derivation from Eckhardt, Grossmann, and
Lohse (Eckhardt et al., 2007) and Wang et al. (Wang et al., 2022b), if we
assume
$\langle\alpha\theta(\Omega_{c}+u_{\varphi}/r)^{2}ru_{r}\rangle_{V,t}=C\alpha\Omega_{c}^{2}\langle
ru_{r}\theta\rangle_{V,t}$, the exact relation of our system is:

$\varepsilon-\varepsilon_{lam}=\nu^{3}L^{-4}[\sigma_{r}^{-2}Ta(Nu_{\omega}-1)+Cf(\eta)Pr^{-2}Ra(Nu_{h}-1)],$
(12)

where
$\varepsilon=\nu\langle(\partial_{i}u_{j}+\partial_{j}u_{i})^{2}\rangle_{V,t}$
is the mean energy dissipation rate, $\varepsilon_{lam}$ is the mean energy
dissipation rate of the laminar and nonvortical flow,
$\sigma_{r}=(1+\eta)^{4}/16\eta^{2}$ is the quasi-Prandtl number, and
$f(\eta)=\frac{2(\eta-1)}{(1+\eta)ln(\eta)}$ is a correction factor. In Regime
I, $u_{\varphi}/r\ll\Omega_{c}$, then $C\approx 1$. In Regime III, the flow is
dominated by shear; therefore, referring to the angular velocity profile of
the TC flow (Grossmann et al., 2016), $C$ is still at the order $O(1)$.

The terms on the right of the equation (12) represent the energy injection by
the wall shear and the buoyancy, respectively. The TC flow considers the first
term while the RB convection concentrates on the second. In Regime I, as the
$Nu_{h}>1$ and $Nu_{\omega}<1$, only buoyancy provides energy for turbulent
dissipation, while the wall motion consumes energy instead. In Regime III,
buoyancy and shear provide energy to the fluid together. Moreover, considering
the ratio of the two terms, we can verify the dominant regimes of the flow.
The ratio $\gamma$ can be simplified by take $\eta=0.5$, $Pr=4.3$ and
$C\approx 1$:

$\gamma=\frac{\sigma_{r}^{-2}Ta(Nu_{\omega}-1)}{Cf(\eta)Pr^{-2}Ra(Nu_{h}-1)}\approx
31.79\Omega^{2}\frac{Nu_{\omega}-1}{Nu_{h}-1},$ (13)

where $Ta/Ra$ is simplified using the equation (3). How $\gamma$ varies with
$\Omega$ is shown in figure 8(b). In Regime I, $\gamma<0$; in Regime II,
$\gamma$ is not calculated as both terms are zero; in Regime III, $\gamma>0$.
When the system is under weak shear, such as $\Omega=10^{-2}$, a very small
fraction of the energy is consumed by shearing. As $\Omega$ increases, more
energy is transferred out of the system by the wall motion, as $|\gamma|$
grows approximately linearly with $\Omega$ in logarithmic coordinates. At the
point near Regime II, $\Omega=0.5$, about half of the energy injected by
buoyancy is consumed by the wall shear. Such a large percentage of energy
consumption may be responsible for bringing the system to a stable state.
Crossing Regime II into Regime III, $\gamma\gg 1$, the energy input by the
wall shear is much more than the energy injection of the buoyancy. As $\Omega$
increases, the dominance of shear is reinforced. Additionally, the data of the
buoyancy-dominated part in Regime III ($\Omega=6$) is not given in figure
8(b), because the accuracy of $\gamma$ is difficult to guarantee as
$Nu_{\omega}$ is very close to $1$. The factor in front of the ratio of the
Nusselt numbers, $31.79\Omega^{2}$, is large, therefore the buoyancy-driven
flow ($|\gamma|<1$) is hard to influence $Nu_{\omega}$. From an alternative
perspective, this can be attributed to the fact that the energy difference
induced by angular velocity difference is much larger than the energy
difference induced by density difference since the density variation is
limited as $\alpha\Delta\ll 1$.

###  3.4 Flow structures

To further investigate the dynamics behind the behaviors of heat and momentum
transfer, we focus on the evolution of the flow structures. As the flow is
stable in Regime II, the typical two-dimensional slice of temperature and
angular velocity fields for Regimes I and III are visualized in figures 9 and
11, respectively. Next, we will explore and analyze the flow structures of the
systems in Regimes I and III sequentially, to figure out how heat transfer is
inhibited and facilitated, and how the system is stabilized.

#### Regime I

In Regime I, the flow is quasi-two-dimensional under a large inversed Rossby
number, $Ro^{-1}=20$. Therefore, two-dimensional slices on the $r\varphi$
plane are taken for the temperature and angular velocity fields under
different $\Omega$ at $Ra=10^{6}$, and the visualizations are displayed in
figure 9. Without shear, four pairs of convection rolls are formed, which can
be clearly observed in the temperature and angular velocity snapshots. Due to
the effect of Coriolis force, hot plumes detached from the outer cylinder turn
right when rising, therefore one roll pair consists of a bigger roll on the
right of the hot plume and a smaller roll on the left. This effect also
results in the zonal flow in annular centrifugal RB convection, which is
explained well by Jiang et al. (Jiang et al., 2020) and Wang et al. (Wang et
al., 2022a). The flow structure of the system is stable and the heat is
continuously transferred through the plume with the LSC. Meanwhile, due to the
asymmetry of the two rolls, the bigger roll takes more area and has a larger
impact on the boundaries than the other roll, which pushes the inner cylinder
to rotate anticlockwise and the outer cylinder to rotate clockwise. Hence,
once the boundary is rotating slowly, a negative $Nu_{\omega}$ is generated.

It is conceivable that the flow structure shifts continuously with shear
enhancement. As $\Omega$ increases to $0.1$, there is a notable alteration in
the flow structure, whereby the quantity of convection roll pairs exhibits a
decline. Only two pairs can be observed, and each pair consists of a large
roll covering nearly half of the annulus and a quite small roll. These can be
seen especially clearly in the angular velocity field. It can be speculated
that the shear of two boundaries stretches the plumes and promotes the
azimuthal motion of the plume in the shear direction. The stretched plumes
under shear are also observed in sheared RB convection (Goluskin et al., 2014;
Blass et al., 2020). The flow motion induced by shear is in the same direction
as the motion induced by the Coriolis force, including that hot plumes turn
anticlockwise near the inner cylinder and the cold plumes turn clockwise near
the outer cylinder; hence, the imposed shear increases the size of the large
roll further. Meanwhile, the other roll becomes smaller. The number of roll
pairs decreases as the system is not large enough to accommodate four roll
pairs. Moreover, at this stage, the angular velocity of the fluid is still
larger than the rotational speed of the boundary, and stronger than the
angular velocity of the fluid without shear as well. This means, on the one
hand, the azimuthal motion of the fluid is strengthened by shear; on the other
hand, the shear is not strong enough and the azimuthal movement of the fluid
driven by heat is still maintained. In the small rolls, the flow against the
direction of boundary motion remains strong. Due to that the basic convection
structures, the LSC and plumes, are maintained, and the heat transfer is
reduced by only a small fraction, about $6\%$ compared to the heat transfer at
$\Omega=0$.

As $\Omega$ continues to increase to $0.3$, one big roll covering near
$3\pi/2$ is generated. In this case, the number of convection roll pairs is
hard to define, as the roll against the direction of boundary motion nearly
disappears. Actually, though we can see two strong cold plumes in the
temperature field, there is no fluid rotating anticlockwise near the outer
cylinder; while near the inner cylinder, only two small pieces of fluid rotate
clockwise, against the direction of boundary motion. For all the fluid in the
system, $|\omega|\leq\Omega$, fluid can not rotate faster than the boundary
wall and the $Nu_{\omega}$ turns positive but is still smaller than $1$. The
rotating velocity of the boundaries exceeds the azimuthal velocity of the
fluid induced by LSC. At this stage, the convection roll pairs are not
maintained, while plumes remain working, transporting heat from one side to
the other. Therefore, the heat transfer drops not too much but noticeably,
reducing about $28\%$ compared to the heat transfer at $\Omega=0$.

As $\Omega$ continues to increase to $0.5$, near the marginal state, the flow
is close to the laminar and nonvortical flow, and the plumes disappear in the
end. Some temperature fluctuations are observed in the temperature field,
while the angular velocity fluctuations are hard to recognize. At this stage,
the convection and heat transfer are highly suppressed. Plumes, as important
carriers of heat transfer, are hard to exist. The Nusselt number $Nu_{h}$
drops significantly, about $83\%$ compared to the $Nu_{h}$ at $\Omega=0$.

Figure 9: Typical snapshots of the instantaneous temperature $\theta$ (the
first row) and angular velocity $\omega=u_{\varphi}/r$ (the second row), at
$\Omega=0,0.1,0.3,0.5$ in Regime I. $Ra=10^{6}$.

The averaged temperature and angular velocity profiles under different
$\Omega$ are shown in figure 10. A uniform bulk temperature $\theta_{c}$ is
observed for $\Omega\leq 0.3$, which deviates from the arithmetic mean
temperature of two boundaries $\theta_{m}=0.5$. The asymmetry temperature
profiles without shear can be explained by the radially dependent gravity
effects and the bulk temperature $\theta_{c}$ depends on the radius ratio
$\eta$, which are discussed by Wang et al. (Wang et al., 2022a). The bulk
temperature remains unchanged until $\Omega$ exceeds $0.3$. As the shear
increases, the uniform bulk temperature vanishes and the temperature profile
gradually evolves toward a laminar and nonvortical flow profile (the dashed
line). For the angular velocity profile, a stage similar to the bulk
temperature is not found. At $\Omega=0$, the averaged angular velocity is
small as the asymmetry of the flow structure is not strong; as the shear is
imposed, the averaged angular velocity varies from $\Omega$ at the inner
cylinder to $-\Omega$ at the outer cylinder. It should be noted that when the
shear is not strong (such as $\Omega=0.1$), the two extremums of the profile
occur inside the cell rather than on the boundaries. While for $\Omega\geq
0.3$, the extremums of the profile move to boundaries. As $\Omega$ increases,
the angular velocity profile gradually evolves towards a laminar flow profile
as well. At $\Omega=0.5$, it coincides almost exactly with the laminar and
nonvortical flow profile, while the temperature profile still differs. Thus,
we can see that as shear is enhanced, the changes in the angular velocity
profile are not synchronized with the changes in the temperature profile; the
former tends to occur earlier, either for the transition at $\Omega=0.3$ or at
$\Omega=0.5$.

Figure 10: Radial distribution of azimuthally, axially and time-averaged (a)
temperature $\langle\theta\rangle_{t,\varphi,z}$ and (b) angular velocity
$\langle\omega\rangle_{t,\varphi,z}$ profiles for different $\Omega$ in Regime
I. $Ra=10^{6}$. The dashed lines mean the profiles of the laminar and
nonvortical flow.

#### Regime III

In Regime III, though the flow is three-dimensional, the main structure is on
the $rz$ plane (Grossmann et al., 2016). Therefore, two-dimensional slices are
taken for the temperature and angular velocity fields, illustrated in figure
11. At $\Omega=6$, of which the inviscid TC system without buoyancy is at the
marginal state, the flow shows a classical RB temperature field with a typical
LSC. However, the angular velocity distribution is still almost laminar and
nonvortical though convection occurs. As explained in subsection 3.3, the
convection driven by heat is hard to transfer angular momentum at large
$\Omega$. With $\Omega$ increasing, the flow becomes more turbulent, with the
angular velocity field better mixed and the plumes get thinner, meaning the
transport efficiency is enhanced. At this stage, because the temperature acts
as a passive scalar, the temperature fields and the angular velocity fields
are highly relevant, which can be roughly seen by the highly similar plume
morphology.

Figure 11: Typical snapshots of the instantaneous temperature $\theta$ (the
first row) and angular velocity $\omega=u_{\varphi}/r$ (the second row), at
$\Omega=6,7,8.5,10$ in Regime III. $Ra=10^{6}$. The corresponding Taylor
numbers are $Ta=9.54\times 10^{7},1.30\times 10^{8},1.91\times
10^{8},2.65\times 10^{8}$, respectively.

The averaged temperature and angular velocity profiles of Regime III are shown
in figure 12. A uniform bulk temperature is observed for all the cases, both
the buoyancy-dominant case ($\Omega=6$) and the shear-dominated cases.
Interestingly, this temperature does not change with $\Omega$, but is
different from the arithmetic mean temperature $\theta_{m}=0.5$ and the bulk
temperature in Regime I. As $\Omega$ increases, the profile becomes sharper at
two boundaries, which also implies an improved efficiency of heat transfer.
The angular velocity profile is in line with the laminar and nonvortical
profile at $\Omega=6$, and then transits to a turbulent profile as the Taylor
number is increasing. At $\Omega=10$, the outer cylinder is at rest, and the
Taylor number $Ta=2.65\times 10^{8}$, therefore the system is still in the
classical regime of the TC flow (Ostilla-Mónico et al., 2014b; Grossmann et
al., 2016); the shear-dominated flow in the Regime III behaves similar to the
TC flow.

Figure 12: Radial distribution of azimuthally, axially and time-averaged (a)
temperature $\langle\theta\rangle_{t,\varphi,z}$ and (b) normalized angular
velocity $\langle\omega\rangle_{t,\varphi,z}/\Omega$ profiles for different
$\Omega$ in Regime III. $Ra=10^{6}$. The dashed lines mean the profiles of the
laminar and nonvortical flow.

##  4 Conclusion

In the present study, we investigate the evolution of the flow regimes in an
annular centrifugal Rayleigh-Bénard system with increasing imposed shear,
which is represented by the angular velocity difference between the inner and
outer cylinders, $\Omega$. At first, $Ri\ll 1$ under weak shear, the flow is
buoyancy-dominated and manifests itself as quasi-two-dimensional convection on
the $r\varphi$ plane. As $\Omega$ increases, the convection and heat transfer
are depressed, and the flow finally turns to stable laminar and nonvortical
flow. Through linear instability analysis, we notice that the marginal state
depends on both $Ra$ and $\Omega$, formed by the energy balance among viscous
dissipation, the work of buoyancy, and the work of inertial terms. At low
$Ra\sim 10^{3}$, the marginal state mainly depends on $Ra$ and only a weak
shear is required to stabilize the flow. At high $Ra\geq 10^{6}$, the marginal
state seems insensitive to $Ra$, locating at $Ri=\Omega^{-2}\sim O(1)$. In the
parameter region $Ra\in[10^{6},10^{8}]$, as shear increases, the growth rate
of the instability is suppressed and the low azimuthal frequency modes become
the main mode. Similarly, the heat transfer efficiency $Nu_{h}$ is also
depressed gradually, and the number of roll pairs decreases as the shear
enhances the asymmetry between the two opposing rolls and enlarges the bigger
one. The plumes are stretched by the shear, and the temperature perturbations
are also stretched in LSA as well. The LSA and DNS give many comparable
findings.

With the continued enhancement of shear, the flow becomes unstable again, but
this time the flow is completely different from before. Initially, when the
system has not crossed the marginal state derivated by Rayleigh’s inviscid
theory for pure TC flow, the instability of the flow is driven by buoyancy,
and the thermal convection occurs on the $rz$ plane, similar to the classical
RB convection. As $\Omega$ increases and the centrifugal instability takes
over, the flow quickly becomes shear-dominated, with the temperature acting as
a passive scalar. At this moment, the flow is similar to the TC flow, and
still in the classical regime for our parameter sets.

On the whole, as the flow transits from an annular centrifugal RB convection
to a radially heated TC flow of only the inner cylinder rotating, it passes
through three regimes. Regime I is mainly buoyancy dominated, Regime II is the
stable regime while Regime III is mainly shear dominated. There is an obvious
dividing line between the three regimes, that is the marginal state of the
instability. Furthermore, the energy analysis through the exact relation
implies more information. In Regime I, the energy injected by buoyancy is
consumed by both turbulent dissipation and wall shear. As the shear increases,
more percentages of injected energy are taken by the wall motion, and near the
marginal state, it takes about $50\%$. Therefore, shear plays an important
role in Regime I as well, diverting buoyant injections of energy and
inhibiting the convection. Meanwhile, on the other end of the stable regime,
buoyancy drives the flow at first. Convection is observed in the temperature
profile while the angular velocity profile is still close to the profile in
the laminar and nonvortical case. When near the marginal state, both shear and
buoyancy are important, and the exact relation (the equation (12)) is a good
approach to analyzing the flow characteristics.

The traditional Rayleigh-Bénard system sheared by wall motion in a plane
Couette flow is remarkably close to our system. Blass and Yerragolam have
conducted a series of meaningful studies on this topic recently (Blass et al.,
2020, 2021; Yerragolam et al., 2022a, b). As shear increases from $0$, the
flow regime moves from the buoyancy-dominated regime, through the transitional
regime, and finally into the shear-dominated regime. A non-monotonic
progression of $Nu$ is observed as well, with a minimum in the transitional
regime, where $Ri\sim 1$. These observations are consistent with ours,
suggesting that the dominance of buoyancy or shear is a key factor in
determining the flow state. Moreover, the coupling effects of buoyancy and
shear on the heat transfer are also close between the two systems. However,
due to the effect of curvature and rotation, the evolution of flow structures
in the sheared ACRBC exhibits a lot of differences. The appearance of the
stable regime is a very special finding. Meanwhile, because of the differences
between the Taylor-Couette flow and the plane Couette flow, the flow in the
shear-dominated regime is quite dissimilar. The similarities and differences
between the sheared ACRBC and sheared RB are interesting points that deserve
our future research.

The current system properly couples the TC flow with the RB convection through
a couple of differential rotation centrifugal cylinders and is a feasible
approach to analyze the coupling effect of shear and buoyancy. Limited by the
parameters space, there are plenty of questions to be further explored. In the
region of high Rayleigh number, how does shear affect the $Nu\sim Ra^{\gamma}$
scaling? Meanwhile, the effect of other parameters on the stable region,
including $Ro^{-1}$ and $Pr$, is also an interesting project. Moreover, some
studies show that as the radius ratio $\eta$ increases and approaches $1$, the
convection in the annulus system behaves more and more similar to the
convection between two planes (Pitz et al., 2017; Wang et al., 2022a). As we
know, there is no stable regime for a sheared RB convection between two
horizontal planes (Blass et al., 2020). The disappearance of the stable regime
is an attractive phenomenon that is deserved to be explored in the future.

[Acknowledgements] We thank Detlef Lohse and Hechuan Jiang for the helpful
discussions.

[Funding]This work was supported by the National Natural Science Foundation of
China under grant no. 11988102, and the New Cornerstone Science Foundation
through the XPLORER PRIZE.

[Declaration of interests] The authors report no conflict of interest.

##  Appendix A Simulation parameters

No. | $Ra$ | $\Omega$ | $Ta$ | $N_{\varphi}\times N_{z}\times N_{r}$ | $\Delta_{g}/\eta_{K}$ | $Nu_{h}$ | $\epsilon_{Nu_{h}}$ | $Nu_{\omega}$ | $\epsilon_{Nu_{\omega}}$ | $\Gamma$ | $\phi_{0}$
---|---|---|---|---|---|---|---|---|---|---|---
$1$ | $1.0\times 10^{6}$ | $0$ | $0$ | $1024\times 32\times 128$ | $0.29$ | $7.30$ | $0.30\%$ | $-$ | $-$ | $0.25$ | $1$
$2$ | $1.0\times 10^{6}$ | $0.01$ | $2.65\times 10^{2}$ | $1024\times 32\times 128$ | $0.29$ | $6.96$ | $0.86\%$ | $-7.22$ | $2.83\%$ | $0.25$ | $1$
$3$ | $1.0\times 10^{6}$ | $0.1$ | $2.65\times 10^{4}$ | $1024\times 32\times 128$ | $0.28$ | $6.85$ | $0.99\%$ | $-0.39$ | $2.29\%$ | $0.25$ | $1$
$4$ | $1.0\times 10^{6}$ | $0.2$ | $1.06\times 10^{5}$ | $1024\times 32\times 128$ | $0.27$ | $6.35$ | $0.49\%$ | $0.18$ | $3.84\%$ | $0.25$ | $1$
$5$ | $1.0\times 10^{6}$ | $0.3$ | $2.38\times 10^{5}$ | $1024\times 32\times 128$ | $0.24$ | $5.27$ | $0.22\%$ | $0.54$ | $2.73\%$ | $0.25$ | $1$
$6$ | $1.0\times 10^{6}$ | $0.35$ | $3.25\times 10^{5}$ | $1024\times 32\times 128$ | $0.20$ | $3.08$ | $2.79\%$ | $0.78$ | $3.44\%$ | $0.25$ | $1$
$7$ | $1.0\times 10^{6}$ | $0.4$ | $4.24\times 10^{5}$ | $1024\times 32\times 128$ | $0.17$ | $2.09$ | $3.24\%$ | $0.91$ | $3.77\%$ | $0.25$ | $1$
$8$ | $1.0\times 10^{6}$ | $0.5$ | $6.62\times 10^{5}$ | $1024\times 32\times 128$ | $0.12$ | $1.28$ | $2.02\%$ | $0.98$ | $8.62\%$ | $0.25$ | $1$
$9$ | $1.0\times 10^{7}$ | $0$ | $0$ | $1024\times 32\times 128$ | $0.62$ | $13.29$ | $0.36\%$ | $-$ | $-$ | $0.25$ | $1$
$10$ | $1.0\times 10^{7}$ | $0$ | $0$ | $1536\times 32\times 192$ | $0.41$ | $13.28$ | $0.54\%$ | $-$ | $-$ | $0.25$ | $1$
$11$ | $1.0\times 10^{7}$ | $0.01$ | $2.65\times 10^{3}$ | $1024\times 32\times 128$ | $0.62$ | $12.81$ | $0.10\%$ | $-18.49$ | $1.87\%$ | $0.25$ | $1$
$12$ | $1.0\times 10^{7}$ | $0.01$ | $2.65\times 10^{3}$ | $1536\times 32\times 192$ | $0.41$ | $12.73$ | $0.30\%$ | $-18.42$ | $0.38\%$ | $0.25$ | $1$
$13$ | $1.0\times 10^{7}$ | $0.1$ | $2.65\times 10^{5}$ | $1024\times 32\times 128$ | $0.59$ | $12.42$ | $0.57\%$ | $-1.55$ | $0.03\%$ | $0.25$ | $1$
$14$ | $1.0\times 10^{7}$ | $0.2$ | $1.06\times 10^{6}$ | $1024\times 32\times 128$ | $0.58$ | $12.20$ | $1.55\%$ | $-0.47$ | $2.42\%$ | $0.25$ | $1$
$15$ | $1.0\times 10^{7}$ | $0.3$ | $2.38\times 10^{6}$ | $1024\times 32\times 128$ | $0.54$ | $10.33$ | $1.01\%$ | $0.35$ | $1.37\%$ | $0.25$ | $1$
$16$ | $1.0\times 10^{7}$ | $0.4$ | $4.24\times 10^{6}$ | $1024\times 32\times 128$ | $0.47$ | $7.51$ | $1.09\%$ | $0.50$ | $3.15\%$ | $0.25$ | $1$
$17$ | $1.0\times 10^{7}$ | $0.45$ | $5.36\times 10^{6}$ | $1024\times 32\times 128$ | $0.34$ | $3.12$ | $0.74\%$ | $0.84$ | $0.86\%$ | $0.25$ | $1$
$18$ | $1.0\times 10^{8}$ | $0$ | $0$ | $2048\times 32\times 256$ | $0.66$ | $26.06$ | $0.04\%$ | $-$ | $-$ | $0.125$ | $1$
$19$ | $1.0\times 10^{8}$ | $0$ | $0$ | $2560\times 32\times 270$ | $0.58$ | $26.12$ | $1.02\%$ | $-$ | $-$ | $0.125$ | $1$
$20$ | $1.0\times 10^{8}$ | $0.01$ | $2.65\times 10^{4}$ | $2048\times 32\times 256$ | $0.65$ | $24.70$ | $0.73\%$ | $-25.07$ | $4.40\%$ | $0.125$ | $1$
$21$ | $1.0\times 10^{8}$ | $0.1$ | $2.65\times 10^{6}$ | $2048\times 32\times 256$ | $0.64$ | $25.11$ | $0.04\%$ | $-1.33$ | $5.18\%$ | $0.125$ | $1$
$22$ | $1.0\times 10^{8}$ | $0.2$ | $1.06\times 10^{7}$ | $2048\times 32\times 256$ | $0.60$ | $21.24$ | $1.69\%$ | $-0.63$ | $4.03\%$ | $0.125$ | $1$
$23$ | $1.0\times 10^{8}$ | $0.3$ | $2.38\times 10^{7}$ | $2048\times 32\times 256$ | $0.57$ | $17.75$ | $0.63\%$ | $0.14$ | $5.09\%$ | $0.125$ | $1$
$24$ | $1.0\times 10^{8}$ | $0.4$ | $4.24\times 10^{7}$ | $2048\times 32\times 256$ | $0.57$ | $17.35$ | $1.69\%$ | $0.56$ | $6.19\%$ | $0.125$ | $1$
$25$ | $1.0\times 10^{8}$ | $0.5$ | $6.62\times 10^{7}$ | $2048\times 32\times 256$ | $0.42$ | $10.11$ | $0.71\%$ | $0.40$ | $4.47\%$ | $0.125$ | $1$
$26$ | $1.0\times 10^{6}$ | $6$ | $9.54\times 10^{7}$ | $384\times 256\times 256$ | $0.19$ | $6.50$ | $2.10\%$ | $1.00$ | $0$ | $2/3\pi$ | $1/4$
$27$ | $1.0\times 10^{6}$ | $6.5$ | $1.12\times 10^{8}$ | $384\times 256\times 256$ | $0.82$ | $19.97$ | $0.67\%$ | $2.42$ | $1.41\%$ | $2/3\pi$ | $1/4$
$28$ | $1.0\times 10^{6}$ | $7$ | $1.30\times 10^{8}$ | $384\times 256\times 256$ | $1.01$ | $23.58$ | $0.81\%$ | $3.90$ | $0.79\%$ | $2/3\pi$ | $1/4$
$29$ | $1.0\times 10^{6}$ | $7$ | $1.30\times 10^{8}$ | $768\times 256\times 256$ | $1.01$ | $23.38$ | $1.00\%$ | $3.88$ | $1.35\%$ | $2/3\pi$ | $1/2$
$30$ | $1.0\times 10^{6}$ | $8.5$ | $1.91\times 10^{8}$ | $432\times 324\times 324$ | $1.35$ | $28.61$ | $0.74\%$ | $7.27$ | $0.77\%$ | $2/3\pi$ | $1/4$
$31$ | $1.0\times 10^{6}$ | $10$ | $2.65\times 10^{8}$ | $432\times 324\times 324$ | $1.61$ | $31.63$ | $1.75\%$ | $9.98$ | $1.25\%$ | $2/3\pi$ | $1/4$
$32$ | $1.0\times 10^{6}$ | $10$ | $2.65\times 10^{8}$ | $432\times 324\times 432$ | $1.26$ | $30.83$ | $0.71\%$ | $9.83$ | $0.36\%$ | $2/3\pi$ | $1/6$
Table 1:  Simulation parameters. The columns dispaly the Rayleigh number $Ra$,
the non-dimensional rotational speed difference $\Omega$, the Taylor number
$Ta$, the resolution employed, the maximum grid spacing $\Delta_{g}$ compared
with the Kolmogorov scale estimated by the global criterion
$\eta_{K}=(\nu^{3}/\varepsilon)^{1/4}$, the calculated Nusselt numbers
$Nu_{h}$, $Nu_{\omega}$ and their relative difference of two halves
$\epsilon_{Nu}=|(\langle Nu\rangle_{0-T/2}-\langle
Nu\rangle_{T/2-T})/(Nu-1)|$, the aspect ratio $\Gamma=H/L$, and the reduced
azimuthal domain $\phi_{0}$. $\varepsilon$ is the mean energy dissipation rate
calculated by the equation (12) of $C\approx 1$. Note that the azimuthal
resolution corresponds to the resolution of the segment. For example,
$N_{\varphi}=768$ for $\phi_{0}=1/2$ means $768$ points for one half of the
annulus.

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