# Effect of outer cylinder rotation on the radially heated Taylor-Couette flow

Pratik Aghor<EMAIL_ADDRESS>Integrated Applied Mathematics
Program, University of New Hampshire (US)  Mohammad Atif<EMAIL_ADDRESS>Brookhaven National Laboratory, Upton, New York (US)

###### Abstract

A radially heated Taylor–Couette setup is considered where an incompressible,
Boussinesq fluid is sheared in the annular region between two concentric,
independently rotating cylinders maintained at different temperatures. The
radius ratios chosen for the current study are $0.6$ and $0.9$ corresponding
to wide–gap and thin–gap cases. A linear stability analysis assuming infinite
axial extent over a range of Grashof numbers with Taylor number as the control
parameter is performed. The fastest growing modes at the onset of instability
are found to be oscillatory for the parameters considered. Depending on the
ratio of angular velocities and Grashof numbers, both axisymmetric and non-
axisymmetric modes are found to be the fastest growing modes. Compared to the
case where the outer cylinder is held stationary, the rotation of the outer
cylinder is shown to have a general stabilizing effect on the linear stability
threshold, except for a few cases. Additionally, at a Grashof number of 1000,
unstable modes are located even in the Rayleigh-stable regime for both
wide–gap and thin–gap cases.

††preprint: APS/123-QED

##  I Introduction

Taylor-Couette flow (TCF) is the annular flow between two concentric,
independently rotating cylinders. Since the seminal work of Taylor [1], the
problem and its variants have remained of interest even after a century due to
experimental viability and a wide range of relevance from industrial
applications to geophysical and astrophysical fluid dynamics, see for example,
Gollub and Swinney [2], Aghor and Alam [3], Crowley  _et al._ [4], Ji and
Goodman [5] and references therein.

Linear stability analysis is often the first step to understanding the
parameter space and demarcating stable and unstable regions under various
probing conditions, see Chandrasekhar [6], Drazin and Reid [7]. Linear
stability analysis of an inviscid fluid by Rayleigh [8] postulated the so-
called Rayleigh criterion. According to the Rayleigh criterion, the inviscid
flow is stable (unstable) when the square of circulation increases (decreases)
monotonically with radial distance. In the $\Omega_{i}-\Omega_{o}$ parameter
space with $\Omega_{i},\Omega_{o}$ representing angular velocities of the
inner and outer cylinders respectively, the condition for inviscid instability
can be written as $\Omega_{i}/\Omega_{o}\leq\eta^{-2}$, where
$\eta=R_{i}/R_{o}$ is the radius ratio with $R_{i},R_{o}$ being the inner and
outer radii, respectively. In terms of the ratio of angular velocities
$\mu=\Omega_{o}/\Omega_{i}$, the Rayleigh line that marks the boundary for
inviscid instability can be expressed as $\mu=\eta^{2}$ (see Fig. 1(Right)).
When other effects such as magnetic fields (see Ji and Goodman [5], Nordsiek
_et al._ [9]) or density stratification (see for example, Shalybkov and
Rüdiger [10], Le Bars and Le Gal [11], Le Dizès and Riedinger [12], Robins
_et al._ [13] and references therein) are considered, the classical Rayleigh
criterion can be violated.

Another variant of the classical TCF is the radially heated TCF which is the
focus of this study. It is important in many problems ranging from rotating
machinery in industrial applications (Lee and Minkowycz [14]) to geophysical
and astrophysical flows, see for example Busse [15], Lopez  _et al._ [16],
Jiang  _et al._ [17].

Early efforts of studying radially heated TCF such as Roesner [18],
Soundalgekar  _et al._ [19], Takhar  _et al._ [20] considered only
axisymmetric disturbances. Ali and Weidman [21] considered radially heated
Taylor-Couette setup with stationary outer cylinder and rotating inner
cylinder along with non-axisymmetric (helical) disturbances as well. Other
studies such as Kedia  _et al._ [22], Yoshikawa  _et al._ [23], Guillerm  _et
al._ [24], Kang  _et al._ [25, 26, 27] also considered centrifugal buoyancy
which was ignored in earlier studies. Most of the investigations of radially
heated TCF so far have considered the case of rotating inner cylinder and
stationary outer cylinder with a few exceptions, see for example Meyer  _et
al._ [28] who investigated the effects of centrifugal buoyancy under the
assumption of micro-gravity conditions in the Rayleigh-stable regime.

Figure 1: (Left) Schematic of the problem setup. Inner and outer cylinders at
radii $R_{i},R_{o}$ rotating independently with angular speeds
$\Omega_{i},\Omega_{o}$ are maintained at constant temperatures $T_{i},T_{o}$,
respectively. Cylinders are assumed to be infinite in the axial ($z$)
direction.(Right) Rayleigh stability criterion.

In this article, we focus on the radially heated TCF for a Boussinesq fluid.
Specifically, this paper aims to examine the effects of outer cylinder
rotation ($\mu\neq 0$) on the linear stability threshold. We consider a fluid
of density $\rho$, dynamic viscosity $\nu$, thermal diffusivity $\kappa$, and
thermal expansion coefficient $\beta$ in a cylindrical annulus of infinite
length. The acceleration due to gravity ($g$) points in the negative axial
direction. The Prandtl number ($Pr$) for this study, defined as the ratio of
momentum diffusivity and thermal diffusivity, is fixed at unity unless stated
otherwise. A schematic of the setup is depicted in Fig. 1(left). The inner
cylinder with radius $R_{i}$ spinning with angular velocity $\Omega_{i}$ is
maintained at a temperature $T_{i}$, whereas the outer cylinder with radius
$R_{o}$ is spinning with angular velocity $\Omega_{o}$ and is maintained at
temperature $T_{o}$. To isolate the effect of outer cylinder rotation, we
neglect the centrifugal buoyancy term, corresponding to the limit of vanishing
Froude number defined by $Fr=\nu/\sqrt{gd^{3}}$, where $d=R_{o}-R_{i}$ is the
gap width, see Yoshikawa  _et al._ [23]. Thus, we arrive at a system similar
to Ali and Weidman [21], but with a general basic azimuthal velocity profile
that incorporates the effect of outer–cylinder rotation.

The rest of the paper is organized as follows. In Sec. II we formulate the
problem by non-dimensionalizing the governing equations and obtain the steady
base flow. We then linearize the system around the base flow and use modal
ansatz to reduce to an eigenvalue problem in the radial direction. We also
discuss the symmetries of the system and validate our methodology by comparing
results with published literature. In Sec. III we list the results of our
analysis for a range of parameters. We consider $\mu\in[-1,1]$ ranging from
counter-rotation ($\mu=-1$) to co-rotation ($\mu=1$) for radius ratios
$\eta=0.6$ (wide–gap case) and $\eta=0.9$ (thin–gap case). The Rayleigh line
for the wide–gap case is given by $\mu=\eta^{2}=0.36$, whereas it lies at
$\mu=\eta^{2}=0.81$ for the thin–gap case. For each of the radius ratios,
three typical values of Grashof numbers $G=\\{100,500,1000\\}$ and seven
intermediate values of $\mu\in[-1,1]$ are analyzed. In Sec. IV we discuss the
findings of our analysis followed by a conclusion in Sec. V.

##  II Problem Formulation and Numerical Method

The governing equations for the thermally active viscous Newtonian fluid that
control the dynamics of the system are the well-known Navier-Stokes equations
(continuity and momentum equations), and the energy equation. Following Ali
and Weidman [21] the governing equations are rendered dimensionless using gap
width $d=R_{o}-R_{i}$ as the characteristic length, velocity of the inner
cylinder $R_{i}\Omega_{i}$ as the characteristic azimuthal velocity,
$U_{0}=g\beta d^{2}\Delta T/\nu$ as the characteristic radial and vertical
velocity, $d^{2}/\nu$ as the time scale, $\rho U_{0}^{2}$ as the
characteristic pressure, and the temperature contrast $|\Delta
T|=|T_{i}-T_{o}|$ as the temperature scale.

The dimensionless governing equations thus obtained are

$\displaystyle\frac{1}{r}\frac{\partial{(ru)}}{\partial{r}}+\frac{1}{r}\frac{\partial{v}}{\partial{\phi}}+\frac{\partial{w}}{\partial{z}}$
$\displaystyle=0,$ (1)
$\displaystyle\frac{\partial{u}}{\partial{t}}+G\left[u\frac{\partial{u}}{\partial{r}}+S\frac{v}{r}\frac{\partial{u}}{\partial{\phi}}+w\frac{\partial{u}}{\partial{z}}-S^{2}\frac{v^{2}}{r}\right]$
$\displaystyle=-G\frac{\partial{p}}{\partial{r}}+\left[\left(\nabla^{2}-\frac{1}{r^{2}}\right)u-S\frac{2}{r^{2}}\frac{\partial{v}}{\partial{\phi}}\right],$
(2) $\displaystyle
S\left[\frac{\partial{v}}{\partial{t}}+G\left(u\frac{\partial{v}}{\partial{r}}+S\frac{v}{r}\frac{\partial{v}}{\partial{\phi}}+w\frac{\partial{v}}{\partial{z}}+\frac{uv}{r}\right)\right]$
$\displaystyle=-G\frac{1}{r}\frac{\partial{p}}{\partial{\phi}}+\left(\nabla^{2}-\frac{1}{r^{2}}\right)v+\frac{2}{r^{2}}\frac{\partial{u}}{\partial{\phi}},$
(3)
$\displaystyle\frac{\partial{w}}{\partial{t}}+G\left[u\frac{\partial{w}}{\partial{r}}+S\frac{v}{r}\frac{\partial{w}}{\partial{\phi}}+w\frac{\partial{w}}{\partial{z}}\right]$
$\displaystyle=-G\frac{\partial{p}}{\partial{z}}+\nabla^{2}w+T,$ (4)
$\displaystyle\frac{\partial{T}}{\partial{t}}+G\left[u\frac{\partial{T}}{\partial{r}}+S\frac{v}{r}\frac{\partial{T}}{\partial{\phi}}+w\frac{\partial{T}}{\partial{z}}\right]$
$\displaystyle=\frac{1}{Pr}\nabla^{2}T,$ (5)

where $u,v,w$ are the dimensionless velocities in the radial ($r$), azimuthal
($\phi$), and axial ($z$) directions respectively. Here, $p$ and $T$ are
dimensionless pressure and temperature respectively, and $\nabla^{2}$ is the
Laplacian operator defined as

$\displaystyle\nabla^{2}\equiv\frac{\partial^{2}{}}{\partial{r}^{2}}+\frac{1}{r}\frac{\partial{}}{\partial{r}}+\frac{1}{r^{2}}\frac{\partial^{2}{}}{\partial{\phi}^{2}}+\frac{\partial^{2}{}}{\partial{z}^{2}}.$
(6)

The nondimensionalization gives rise to the dimensionless parameters Prandtl
($Pr$), Taylor ($Ta$) and Grashof ($G$) numbers given by

$\displaystyle Pr=\frac{\nu}{\kappa},\quad
Ta=\frac{2\eta^{2}\Omega_{i}^{2}d^{4}}{\nu^{2}(1-\eta^{2})},\quad
G=\frac{g\beta\Delta Td^{3}}{\nu^{2}}.$ (7)

The Taylor number $Ta$ is proportional to the rotation rate of the inner
cylinder and can also be related to the Reynolds number based on inner
cylinder rotation as

$\displaystyle Ta=\frac{2(1-\eta)}{(1+\eta)}Re_{i}^{2},$ (8)

where $Re_{i}=\Omega_{i}R_{i}d/\nu$ is the inner Reynolds number. The swirl
parameter $S=\Omega_{i}R_{i}/U_{0}$ appears in the governing equations due to
the difference of scales between azimuthal ($v$) and meridional velocities
($u$ and $w$) and is not an independent parameter. It can be expressed in
terms of other dimensionless parameters as

$\displaystyle
S=\frac{\left(Ta(1+\eta)/2(1-\eta)\right)^{1/2}}{G}=\frac{Re_{i}}{G}.$ (9)

The classical isothermal TCF with all velocities scaled by $\Omega_{i}R_{i}$
is recovered by substituting $T=0$ and $S=1$, replacing $G$ with $Re_{i}$.
Infinite cylinders are assumed and Dirichlet boundary conditions in the radial
direction are imposed, i.e.,

$\displaystyle(u,v,w,T)=(0,1,0,1)\textrm{ at }r=r_{i}=\frac{\eta}{1-\eta},$
(10) $\displaystyle(u,v,w,T)=\left(0,\frac{\mu}{\eta},0,0\right)\textrm{ at
}r=r_{o}=\frac{1}{1-\eta},$ (11)

with $r_{i}=R_{i}/d$, $r_{o}=R_{o}/d$ being the dimensionless inner and outer
radii respectively. In the next section, we formulate the base state and
linearize the system in order to perform a linear stability analysis.

###  II.1 Base flow and linearization

The base state can be obtained analytically for this problem assuming radial
dependence of the steady basic fields. For the basic azimuthal flow ($v_{b}$),
the well-known general circular Couette flow profile in the azimuthal
direction valid for nonzero $\mu$ is obtained, see for example, Chandrasekhar
[6]. In the case of $\mu=0$ corresponding to the stationary outer cylinder,
the basic azimuthal velocity profile reduces to the one considered by Ali and
Weidman [21]. In the classical isothermal Taylor-Couette setup, there is no
axial flow in the base state. However, in the presence of a radial temperature
gradient an axial flow is induced. Since the axial flow $w_{b}$ is only
dependent on the basic temperature profile, outer cylinder rotation ($\mu\neq
0$) does not alter the basic axial flow and it is the same as reported in Ali
and Weidman [21]. Thus, the base flow is

$\displaystyle u_{b}=0,$ (12) $\displaystyle v_{b}=Ar+B/r,$ (13)
$\displaystyle T_{b}=\frac{\ln{\left[(1-\eta)r\right]}}{\ln{\eta}},$ (14)
$\displaystyle
w_{b}=\frac{C}{D}\left[(1-\eta)^{2}r^{2}-1+(1-\eta^{2})T_{b}\right]-\frac{1}{4}\left[(1-\eta)^{2}r^{2}-\eta^{2}\right]T_{b},$
(15)

with

$\displaystyle A=\frac{\mu-\eta^{2}}{\eta(1+\eta)},\quad
B=\frac{\eta(1-\mu)}{(1-\eta)(1-\eta^{2})},\quad
C=(1-\eta^{2})(1-3\eta^{2})-4\eta^{4}\ln{\eta},\quad
D=16\left[(1-\eta^{2})^{2}+(1-\eta^{4})\ln{\eta}\right].$

The above parameters represent the base state of the system used in the
subsequent analysis.

We perturb the primitive variables as

$\displaystyle\left[u,v,w,T,p\right]=[u_{b},v_{b},w_{b},T_{b},p_{b}](r)+\left[u^{\prime},v^{\prime},w^{\prime},T^{\prime},p^{\prime}\right],$
(16)

and substitute into the governing equations (1) – (5). Thereafter, imposing
the boundary conditions and neglecting higher-order terms in the perturbations
we obtain the linearized system of governing equations as (the primes are
dropped for brevity)

$\displaystyle\frac{1}{r}\frac{\partial{(ru)}}{\partial{r}}+\frac{S}{r}\frac{\partial{v}}{\partial{\phi}}+\frac{\partial{w}}{\partial{z}}$
$\displaystyle=0,$ (17)
$\displaystyle\frac{\partial{u}}{\partial{t}}+G\left[S\frac{v_{b}}{r}\frac{\partial{u}}{\partial{\phi}}+w_{b}\frac{\partial{u}}{\partial{z}}-S^{2}\frac{2v_{b}v}{r}\right]$
$\displaystyle=-G\frac{\partial{p}}{\partial{r}}+\nabla^{2}u-S\frac{2}{r^{2}}\frac{\partial{v}}{\partial{\phi}}-\frac{u}{r^{2}},$
(18) $\displaystyle
S\left[\frac{\partial{v}}{\partial{t}}+G\left(u\frac{dv_{b}}{dr}+S\frac{v_{b}}{r}\frac{\partial{v}}{\partial{\phi}}+w_{b}\frac{\partial{v}}{\partial{z}}+\frac{uv_{b}}{r}\right)\right]$
$\displaystyle=-G\frac{1}{r}\frac{\partial{p}}{\partial{\phi}}+S\nabla^{2}v-\frac{2}{r^{2}}\frac{\partial{u}}{\partial{\phi}}-S\frac{v}{r^{2}},$
(19)
$\displaystyle\frac{\partial{w}}{\partial{t}}+G\left[u\frac{dw_{b}}{dr}+S\frac{v_{b}}{r}\frac{\partial{w}}{\partial{\phi}}+w_{b}\frac{\partial{w}}{\partial{z}}\right]$
$\displaystyle=-G\frac{\partial{p}}{\partial{z}}+\nabla^{2}w+T,$ (20)
$\displaystyle\frac{\partial{T}}{\partial{t}}+G\left[u\frac{dT_{b}}{dr}+S\frac{v_{b}}{r}\frac{\partial{T}}{\partial{\phi}}+w_{b}\frac{\partial{T}}{\partial{z}}\right]$
$\displaystyle=\frac{1}{Pr}\nabla^{2}T,$ (21)

and the boundary conditions become

$\displaystyle(u,v,w,T)=(0,0,0,0)\textrm{ at }r=r_{i},r_{o}.$ (22)

Substituting the modal perturbation ansatz

$\displaystyle[u,v,w,T,p]=[\hat{u},\hat{v},\hat{w},\hat{T},\hat{p}](r)\exp{\left[i(kz+m\phi)+\sigma
t\right]}+c.c.,$ (23)

in Eqs. (17)-(22), a generalized eigenvalue problem of the form
$\bm{A}\bm{x}=\sigma\bm{B}\bm{x}$ in $r$ is obtained, with $\sigma$ as the
eigenvalue and $\bm{x}=[\hat{u},\hat{v},\hat{w},\hat{T},\hat{p}](r)$. Here
‘c.c.’ stands for ‘complex conjugate’. Since the domain in the azimuthal
direction has a natural periodicity of $2\pi$, the azimuthal wavenumber $m$
only takes integer values. On the other hand, since the cylinders are assumed
to be infinite in the axial direction, the axial perturbation wavenumber $k$
is a continuous parameter and can take real values. The eigenvalue $\sigma$ is
in general a complex number of the form $\sigma=\sigma_{r}+i\sigma_{i}$, with
$\sigma_{r},\sigma_{i}\in\mathcal{R}$. The real part of the eigenvalue
$\sigma_{r}$ represents the growth rate and the imaginary part $\sigma_{i}$
represents the frequency of the evolution of perturbations according to Eq.
(23). The linearly stable perturbations are characterized by $\sigma_{r}<0$,
whereas linearly unstable perturbations have $\sigma_{r}>0$. The locus of
points satisfying $\sigma_{r}=0$ marks the neutral stability boundary which
denotes the transition between stable and unstable regimes, see for example
Drazin and Reid [7]. As mentioned in Ali and Weidman [21], critical
perturbation modes with $\sigma_{r}=0$ can be completely described by the
triplet $(m,k,\sigma_{ic})$. The non-dimensional axial propagation speed of
the phase lines $C$, wavelength $\lambda$ normal to lines of constant phase,
and inclination of the phase lines $\psi$ with respect to the horizontal are
given by

$\displaystyle
C=-\frac{\sigma_{i}}{k},\quad\lambda=\frac{2\pi}{\left(m^{2}/r^{2}+k^{2}\right)^{1/2}},\quad\psi=-\tan^{-1}\left({\frac{m}{rk}}\right).$
(24)

As the linearized equations (18) – (21) each have a second derivative term
with respect to $r$, the resulting eigenvalue problem is $8$-dimensional and
is closed by $8$ boundary conditions given in Eqs. (25).

In this study, we solve the resulting eigenvalue problem using Dedalus’s (see
Burns  _et al._ [29]) excellent package ‘eigentools’. Eigentools is equipped
with automatic rejection of spurious modes by calculating the drift ratio as
given in Boyd [30]. This is performed by comparing the eigenvalues at a
specified resolution and a higher resolution of 1.5 times the original
resolution, see Oishi  _et al._ [31] for details. We use Chebyshev polynomials
to discretize in the radial direction. Substituting the modal ansatz Eq. (23)
into Eq. (22) appropriate homogeneous Dirichlet boundary conditions are
obtained as

$\displaystyle(\hat{u},\hat{v},\hat{w},\hat{T})=(0,0,0,0)\textrm{ at
}r=r_{i},r_{o}.$ (25)

###  II.2 Symmetries

We briefly discuss the symmetries of the linearized problem, their
consequences and justify our choice of numerical parameters in this section.
Two symmetries of the linearized perturbation equations for the stationary
outer cylinder ($\mu=0$) were identified previously in by Ali and Weidman
[21]. We generalize these symmetries to include any nonzero $\mu$ as

$\displaystyle
S_{1}=S^{c/r}(\Omega):[\hat{u},\hat{v},\hat{w},\hat{T},\hat{p};Ta,G,k,\sigma,\mu,m]\rightarrow[\hat{u},-\hat{v},\hat{w},\hat{T},\hat{p};Ta,G,k,\sigma,\mu,-m],$
(26) $\displaystyle S_{2}=S^{c/r}(\Delta
T):[\hat{u},\hat{v},\hat{w},\hat{T},\hat{p};Ta,G,k,\sigma,\mu,m]\rightarrow[-\hat{u}^{*},\hat{v}^{*},\hat{w}^{*},\hat{T}^{*},\hat{p}^{*};Ta,-G,k,\sigma^{*},\mu,-m].$
(27)

The symmetry $S^{c/r}(\Omega)$ corresponds to a situation where the inner
cylinder is rotated in the opposite direction, i.e.,
$\Omega_{i}\rightarrow-\Omega_{i}$. For the general case considered in our
study with $\mu$ allowed to be nonzero, if we consider
$\Omega_{o}\rightarrow-\Omega_{o}$, keeping $\mu$ invariant, the modified
version of $S^{c/r}(\Omega)$ given in Eq. (26) can be shown to hold. It is
seen that only $v_{b}\rightarrow-v_{b}$ is sufficient to quantify the effect
of this transformation on the base state whereas boundary conditions for the
perturbation fields remain the same as Eq. (22).

The symmetry $S^{c/r}(\Delta T)$ represents the scenario with temperature
gradient reversed, i.e., $G\rightarrow-G$. We first note that $G\rightarrow-G$
does not alter the basic temperature profile $T_{b}$ as it is
nondimensionalized using the temperature contrast $|\Delta T|=|T_{i}-T_{o}|$.
As a consequence, $w_{b}$ (which is completely determined by $T_{b}$) is also
invariant under $G\rightarrow-G$. Substituting $S^{c/r}(\Delta
T)[\hat{u},\hat{v},\hat{w},\hat{T},\hat{p};Ta,G,k,\sigma,\mu,m]$ in Eq. (23)
and Eqs. (17)-(21), we obtain the complex conjugated version of the original
system. This is independent of whether $\mu$ is zero or otherwise. Hence the
$S^{c/r}(\Delta T)$ symmetry can be extended for nonzero $\mu$ as well,
implying that for each $G>0$ there exists an equivalent case of $G<0$ with the
inner wall being cooler than the outer wall.

Using a combination of these two symmetries, a third symmetry could also be
identified, where $S_{3}=S_{1}\cdot S_{2}$, given by

$\displaystyle
S_{3}:[\hat{u},\hat{v},\hat{w},\hat{T},\hat{p};Ta,G,k,\sigma,\mu,m]\rightarrow[-\hat{u}^{*},-\hat{v}^{*},\hat{w}^{*},\hat{T}^{*},\hat{p}^{*};Ta,-G,k,\sigma^{*},\mu,m].$
(28)

The effect of these symmetries can be explained as follows. For each solution
of the linearized equations with $(\Omega_{i},\Delta T,\mu)$,, there are 3
equivalent solutions for the cases $(-\Omega_{i},\Delta T,\mu)$,
$(\Omega_{i},-\Delta T,\mu)$ and $(-\Omega_{i},-\Delta T,\mu)$. If the first
case corresponds to spirals having phase speed and inclination with respect to
the horizontal given by $(C,\psi)$, then the $S_{1},S_{2},S_{3}$ symmetries
imply that there also exist solutions with $(C,-\psi)$, $(-C,-\psi)$ and
$(-C,\psi)$. We have therefore generalized the symmetries identified by Ali
and Weidman [21] to include nonzero outer-cylinder rotation.

As noted by Yoshikawa  _et al._ [23], Kang  _et al._ [25, 26], the inclusion
of the centrifugal buoyancy breaks these symmetries even in the case where the
outer cylinder is held stationary and the effects of inward and outward
heating are different. However, since we neglect centrifugal buoyancy, we only
consider $G>0$ in our analysis. Other equivalent cases can be constructed for
$G<0$ using symmetry arguments.

###  II.3 Validation

As mentioned earlier, we use eigentools package from Dedalus to solve the
resulting eigenvalue problem. To verify our linear stability routine, We
compare neutral stability curves obtained from our code to those reported in
Ali and Weidman [21]. A sample calculation for $m=-2$ mode is shown in Fig. 2
at $Pr=15,G=300$ and $\eta=0.6$. It can be seen that the two curves match
well.

Figure 2: A sample comparison of the present code with published literature at
$Pr=15,\eta=0.6,G=300,m=-2$ in our notation. Dashed green lines with vertical
markers represent neutral stability data obtained from the current methodology
and black circles represent data extracted from Ali and Weidman [21].

##  III Linear Stability Analysis

In this section, we present the results of linear stability analysis. As
mentioned earlier, we limit our analysis to $Pr=1$ and radius ratios
$\eta=0.6,0.9$ which correspond to wide–gap and thin–gap cases respectively.
We sweep over the angular velocity ratio ($\mu$) and azimuthal wave number
($m$) for each of three chosen Grashof numbers ($G\in\\{100,500,1000\\}$).
While all $\mu<0$ lie in the counter-rotation regime and $\mu>0$ in the co-
rotation regime, we refer to $\mu=-1$ as counter-rotation and $\mu=1$ as co-
rotation in the discussion below.

###  III.1 Wide–gap case, $\eta=0.6$

$\mu$ | $m$ | $G=100$ | $G=500$ | $G=1000$
---|---|---|---|---
|  | $Ta_{c}$ | $k_{c}$ | $\sigma_{ic}$ | $Ta_{c}$ | $k_{c}$ | $\sigma_{ic}$ | $Ta_{c}$ | $k_{c}$ | $\sigma_{ic}$
-1 | -3 | 25118.05 | 6.39508 | 1.30793e+02 | 24148.42 | 6.1814 | 1.10312e+02 | 23060.40 | 5.93754 | 8.7977e+01
-1 | -2 | 15273.00 | 4.39 | 7.08178e+01 | 16353.32 | 4.09188 | 6.49581e+01 | 18572.05 | 5.85807 | 4.61341e+01
-1 | -1 | 16939.48 | 6.00516 | 3.95910e+01 | 17312.73 | 6.02229 | 2.2851e+01 | 17780.14 | 6.15256 | 7.72213e-01
-1 | 0 | 17165.92 | 6.24542 | -4.6002e+00 | 17148.87 | 6.26010 | -2.305226e+01 | 17092.65 | 6.23491 | -4.59031e+01
-1 | 1 | 16758.57 | 5.98348 | -4.78934e+01 | 16403.20 | 5.94661 | -6.43627e+01 | 15823.30 | 5.83604 | -8.42856e+01
-1 | 2 | 14929.30 | 4.52047 | -7.49310e+01 | 14634.19 | 4.70764 | -8.37993e+01 | 14933.89 | 4.86187 | -1.94001e+01
-1 | 3 | 25599.09 | 6.46334 | -1.41442e+02 | 26479.39 | 6.65983 | -1.64198e+02 | 27408.38 | 6.89704 | -1.94146e+02
-0.5 | -3 | 16290.56 | 5.578030 | 1.18575e+02 | 13273.0 | 5.277 | 8.891255e+01 | 9814.35 | 4.89194 | 5.7900e+01
-0.5 | -2 | 6685.0 | 4.07 | 4.23920e+01 | 6139.0 | 3.94 | 3.30286e+01 | 5875.06 | 3.69844 | 2.3731e+01
-0.5 | -1 | 5416.0 | 3.77 | 2.01051e+01 | 5584.0 | 3.825 | 1.211934e+01 | 5701.57 | 4.05867 | -6.7819e-01
-0.5 | 0 | 5562.95 | 4.05 | -2.53420e+00 | 5429.77 | 4.0156 | -1.23594e+01 | 5135.82 | 3.88670 | -2.2838e+01
-0.5 | 1 | 5374.90 | 3.76223 | -2.38180e+01 | 5405.41 | 3.75904 | -3.09215e+01 | 5713.14 | 3.70639 | -3.95978e+01
-0.5 | 2 | 7086.65 | 4.09744 | -4.76606e+01 | 8219.31 | 4.22546 | -6.02115e+01 | 10030.80 | 4.32672 | -7.83492e+01
-0.5 | 3 | 17637.11 | 5.73391 | -1.33707e+02 | 19932.88 | 6.02522 | -1.63989e+02 | 22210.20 | 6.36873 | -2.01605e+02
-0.2 | -3 | 14429.64 | 5.060069 | 1.2858295e+02 | 7744.0 | 4.50 | 7.55399e+01 | 3338.60 | 3.64635 | 3.8570e+01
-0.2 | -2 | 4332.97 | 3.71887 | 3.7878027e+01 | 3123.89 | 3.438 | 2.67254e+01 | 2292.02 | 2.87532 | 1.8710e+01
-0.2 | -1 | 2883.28 | 3.31861 | 1.42900e+01 | 2550.0 | 3.10 | 9.73440e+00 | 2497.02 | 2.80735 | 5.9476e+00
-0.2 | 0 | 2651.51 | 3.21339 | -9.380406e-01 | 2833.81 | 3.159 | -4.8007e+00 | 3287.03 | 3.20385 | -1.0691e+01
-0.2 | 1 | 3187.72 | 3.37146 | -1.73120e+01 | 3836.48 | 3.46079 | -2.41583e+01 | 4815.42 | 3.44300 | -3.33710e+01
-0.2 | 2 | 5089.19 | 3.78475 | -4.45581e+01 | 6916.91 | 3.78029 | -5.92347e+01 | - | - | -
-0.2 | 3 | 17022.15 | 5.34569 | -1.52330e+02 | - | - | - | - | - | -
0 | -3 | 18451.30 | 5.0238 | 1.707085e+02 | 4508.63 | 3.8036 | 6.828717e+01 | 1863.57 | 2.75454 | 4.0911e+01
0 | -2 | 3828.30 | 3.47265 | 4.369439e+01 | 2529.79 | 3.0979 | 3.262131e+01 | 1750.89 | 2.40775 | 2.5263e+01
0 | -1 | 2724.30 | 3.23457 | 1.791782e+01 | 2351.88 | 2.9376 | 1.421454e+01 | 2279.48 | 2.52391 | 1.1727e+01
0 | 0 | 2581.52 | 3.14863 | -6.67643e-01 | 2801.97 | 3.1362 | -3.47543e+00 | 3413.10 | 3.05819 | -7.5626e+00
0 | 1 | 3050.19 | 3.28778 | -2.05192e+01 | 3966.48 | 3.38219 | -2.68902e+01 | 5447.53 | 3.36079 | -3.63104e+01
0 | 2 | 4764.73 | 3.61355 | -5.09802e+01 | 7535.30 | 3.57944 | -6.9569e+01 | - | - | -
0 | 3 | 23020.28 | 5.45039 | -2.07582e+02 | 29117.84 | 5.91596 | -2.62032e+02 | - | - | -
0.2 | -4 | 214085.0 | 7.01 | 6.089696e+02 | 191047.20 | 6.80716 | 6.010557e+02 | 142487.28 | 6.43302 | 5.5745e+02
0.2 | -3 | 69724.5 | 5.56392 | 3.029782e+02 | 5703.13 | 3.49568 | 1.000056e+02 | 1541.34 | 2.18362 | 5.1182e+01
0.2 | -2 | 6217.0 | 3.46 | 7.103152e+01 | 3371.83 | 2.83704 | 5.056856e+01 | 1781.54 | 1.99982 | 3.5861e+01
0.2 | -1 | 4365.0 | 3.16 | 2.944755e+01 | 3446.134 | 2.76240 | 2.435666e+01 | 2971.26 | 2.16111 | 2.1219e+01
0.2 | 0 | 4190.03 | 3.12791 | -5.469713e-01 | 4569.57 | 3.10585 | -2.82116e+00 | 5655.64 | 3.00812 | -6.0502e+00
0.2 | 1 | 5074.48 | 3.31003 | -3.29040e+01 | 7034.51 | 3.43711 | -4.14196e+01 | - | - | -
0.2 | 2 | 8501.25 | 3.62811 | -8.45812e+01 | 16297.41 | 3.50979 | -1.206837e+02 | - | - | -
0.5 | -3 | stable | stable | stable | stable | stable | stable | 1706.85 | 1.50802 | 7.7196e+01
0.5 | -2 | stable | stable | stable | stable | stable | stable | stable | stable | stable
0.5 | -1 | stable | stable | stable | stable | stable | stable | stable | stable | stable
0.5 | 0 | stable | stable | stable | stable | stable | stable | stable | stable | stable
0.5 | 1 | stable | stable | stable | stable | stable | stable | stable | stable | stable
1 | -4 | stable | stable | stable | stable | stable | stable | 6209.99 | 0.56951 | 2.9684e+02
1 | -3 | stable | stable | stable | stable | stable | stable | stable | stable | stable
1 | -2 | stable | stable | stable | stable | stable | stable | stable | stable | stable
1 | -1 | stable | stable | stable | stable | stable | stable | stable | stable | stable
1 | 0 | stable | stable | stable | stable | stable | stable | stable | stable | stable
1 | 1 | stable | stable | stable | stable | stable | stable | stable | stable | stable
Table 1: Critical values for $\eta=0.6$. The ‘$-$’ entries represent cases for
which the fastest growing mode has already been located and ‘stable’ entries
represent cases where the reported modes are found to be stable up to a Taylor
number of $10^{5}$. The highlighted values correspond to the fastest growing
modes.

For the wide–gap case, $\eta=0.6$, the critical Taylor numbers $Ta_{c}$ and
corresponding critical axial wave numbers ($k_{c}$) are reported in Table 1.
In all the cases reported the modes at the onset of instability were found to
be oscillatory as is evident by nonzero critical frequency $\sigma_{ic}$.

We start by discussing the fastest growing mode for each of the three values
of $G\in\\{100,500,1000\\}$. To this effect, we plot eigenfunctions only for
$\mu=-1$ (counter-rotating cylinders) and the largest positive $\mu$ at which
we could find instability in the co-rotating regime for each $G$. Figure 3
plots the radial variation of critical eigenmodes (top row) and visualizes the
normalized critical eigenfunctions (bottom row) for the counter-rotation case
with $\mu=-1$ at $G=100$ for which the fastest growing mode is $m=2$. The $1d$
radial eigenmodes are defined up to a phase. Here, we choose the phase of
eigenmode of the azimuthal velocity perturbation $v$ to be zero. This choice
is reflected in the fact that radial eigenmodes for $v$ are purely real (see
Oishi  _et al._ [31] for more details). Slices of the full $3D$ normalized
eigenfunctions are constructed using critical radial eigenmodes and
corresponding azimuthal and axial wavenumbers. The height of the periodic
domain $L_{z}$ can be obtained by

$L_{z}=\frac{2\pi}{k_{c}},$ (29)

with $k_{c}$ being the corresponding critical axial wavenumber reported in
Table 1. To conserve space, the $r-z$ slices are scaled to a square domain.
The negative inclination of the spirals with respect to the horizontal is
apparent from the contour plots of the azimuthal velocity perturbation in the
$r-z$ plane, which is consistent with the expression in Eq. (24). Furthermore,
$m=2$ mode is visible from the $r-\phi$ slices where contours of $v$ and $T$
can be seen to have two maxima and two minima in the azimuthal
$\phi$-direction. Figure 4 plots the radial variation of critical eigenmodes
(top row) and visualizes the normalized critical eigenfunctions (bottom row)
for the co-rotation case with $\mu=0.2$ at $G=100$ for which the fastest
growing mode is found to be axisymmetric ($m=0$).

For the counter-rotation case of $\mu=-1$, the temperature and azimuthal
velocity eigenmodes can be seen to be confined near the inner cylinder for all
values of $G$ considered here. At $G=100$, $\mu\in[-0.2,0.2]$ axisymmetric
toroidal modes are found to be the fastest growing, while for other values of
$\mu$, non-axisymmetric helical modes were found to be the most dangerous. The
critical mode for $\mu=0.2$ in the co-rotation regime was found to be
axisymmetric at $\eta=0.6$ and $G=100$, which can be seen from the $r-\phi$
slices in Fig. 4. We did not find any unstable mode below the Rayleigh-line
for $\mu>\eta^{2}=0.36$ at $G=100$.

Figure 3: (Top row) Unnormalized 1d critical eigenfunctions for
$\eta=0.6,G=100$ for counter-rotation case with $\mu=-1,m=2$. (Bottom row)
Normalized critical eigenfunctions – slices of contour plots for the critical
azimuthal velocity eigenmode in the $r-z$ and $r-\phi$ planes, followed by
contour plots of temperature eigenmode in the $r-z$ and $r-\phi$ planes. The
quiver plots in the $r-z$ slices are made up of corresponding eigenmodes for
meridional velocity perturbations ($u$ and $w$).

Figure 4: (Top row) Unnormalized 1d critical eigenfunctions for
$\eta=0.6,G=100$ for co-rotation case with $\mu=0.2,m=0$. (Bottom row)
Normalized critical eigenfunctions – colors and slices to be interpreted as in
Fig. 3.

Figures 5 and 6 depict the fastest growing modes for $(\eta,G)=(0.6,500)$ for
counter-rotation $\mu=-1$ and $\mu=0.2$ in the co-rotation regime,
respectively. The fastest growing mode for $\mu=-1$ is found to be $m=2$.
Similar to $G=100$, for the counter-rotation case for $G=500$ the temperature
and azimuthal velocity eigenmodes can be seen to be confined near the inner
cylinder. The critical mode for the $\mu=0.2$ case was found to be a non-
axisymmetric spiral mode with $m=-2$ at $\eta=0.6$ and $G=500$. However, from
Fig. 6 (top row) and $2D$ slices in Fig. 6 (bottom row) it can be seen that
unlike the critical eigenmode for the counter-rotation case, this mode is not
confined near the inner cylinder. No unstable modes were found below the
Rayleigh-line for $\mu>\eta^{2}=0.36$ at $G=500$.

Figure 5: (Top row) Unnormalized 1d critical eigenfunctions for
$\eta=0.6,G=500$ for counter-rotation case with $\mu=-1,m=2$. (Bottom row)
Normalized critical eigenfunctions – colors and slices to be interpreted as in
Fig. 3.

Figure 6: (Top row) Unnormalized 1d critical eigenfunctions for
$\eta=0.6,G=500$ for co-rotation case with $\mu=0.2,m=-2$. (Bottom row)
Normalized critical eigenfunctions – colors and slices to be interpreted as in
Fig. 3.

Figures 7 and 8 depict the fastest growing modes at $(\eta,G)=(0.6,1000)$ for
the two extremes among the cases considered in this article, i.e., counter-
rotation ($\mu=-1$) and co-rotation ($\mu=1$). They also show corresponding
slices of the entire $3D$ eigenfunctions constructed using radial eigenmodes
and corresponding critical azimuthal and axial wavenumbers. For the counter-
rotation case, like other values of $G$, the temperature and azimuthal
velocity eigenmodes can be seen to be confined near the inner cylinder. For
the co-rotation case, although the temperature and velocity modes reach all
the way to the outer cylinder, an interesting bimodal structure of the
azimuthal velocity eigenmode is observed. The effect of bimodal structure is
reflected in the bottom panel of Fig. 8. Moreover, at $G=1000$ we find
unstable modes even in the Rayleigh-stable regime $\mu>\eta^{2}=0.36$. Here,
except for the case of $\mu=-0.5$ in the counter-rotation regime, the fastest
growing modes are non-axisymmetric. We also plot neutral stability curves for
$G=1000$ for four selected values of $\mu$ because we find unstable modes in
the Rayleigh-stable regime for $G=1000$. For $\mu=0.5$, closed regions of
instability are found, see Fig. 9 (bottom left). However, for the co-rotation
case ($\mu=1$) we do not observe the closed region and find $m=-4$ to be the
fastest growing mode.

Figure 7: (Top row) Unnormalized 1d critical eigenfunctions for
$\eta=0.6,G=1000$ for counter-rotation case with $\mu=-1,m=2$. (Bottom row)
Normalized critical eigenfunctions – colors and slices to be interpreted as in
Fig. 3.

Figure 8: (Top row) Unnormalized 1d critical eigenfunctions for
$\eta=0.6,G=1000$ for co-rotation case with $\mu=1,m=-4$. (Bottom row)
Normalized critical eigenfunctions – colors and slices to be interpreted as in
Fig. 3.

Figure 9: Neutral stability curves for $\eta=0.6,G=1000$ for different
azimuthal perturbation numbers with $\mu=$ (top left) $-1$, (top right)
$-0.5$, (bottom left) $0.5$ and (bottom right) $1$. The critical wavenumber of
the perturbations $k_{c}$ decreases with increasing $\mu$, corresponding to
increasing wavelength of instabilities at the onset, see Table 1 for details.

We notice that for all $G$ considered here, the fastest growing modes for
$\mu=-1$ and $-0.5$ are confined near the inner cylinder in the radial
direction due to strong counter-rotation. For $\mu=-0.2$, however, counter-
rotation is not strong enough to confine the unstable modes near the inner
cylinder and we observe that the temperature contours reach all the way to the
outer wall. It should also be noted that the wavelength at the onset of
instability, given by $2\pi/k_{c}$, is longer for the Rayleigh-stable case of
co-rotation in comparison to the cases in the Rayleigh-unstable regime,
evident from values of $k_{c}$ listed in Table 1.

###  III.2 Thin–gap case, $\eta=0.9$

We now discuss the thin–gap case of $\eta=0.9$. Table
LABEL:tbl:eta_0.9_neutral summarizes the results of the linear stability
analysis. As seen from Table LABEL:tbl:eta_0.9_neutral, one often needs to go
beyond $m=-4$ to find the fastest growing mode for many cases. Here, the
Rayleigh-line is given by $\mu=\eta^{2}=0.81$ and the Rayleigh stable regime
is given by $\mu>\eta^{2}=0.81$. Similar to the wide–gap case, all the modes
at the onset of instability are found to be oscillatory, evident from nonzero
critical frequencies $\sigma_{ic}$ in Table LABEL:tbl:eta_0.9_neutral.

Table 2: Critical values for $\eta=0.9$. The ‘$-$’ entries represent cases for which the fastest growing mode has already been located and ‘stable’ entries represent cases where the reported modes are found to be stable up to a Taylor number of $10^{5}$. The highlighted values correspond to the fastest growing modes. $\mu$ | $m$ | $G=100$ | $G=500$ | $G=1000$
---|---|---|---|---
|  | $Ta_{c}$ | $k_{c}$ | $\sigma_{ic}$ | $Ta_{c}$ | $k_{c}$ | $\sigma_{ic}$ | $Ta_{c}$ | $k_{c}$ | $\sigma_{ic}$
-1 | -5 | - | - | - | 5831.82 | 3.99733 | 3.38104e+01 | 5822.94 | 3.79609 | 2.66661e+01
-1 | -4 | 5482.20 | 3.83496 | 3.206719e+01 | 5532.12 | 3.72849 | 2.65773e+01 | 5904.26 | 3.58278 | 2.06109e+01
-1 | -3 | 5392.41 | 3.71162 | 2.55659e+01 | 5738.36 | 3.71826 | 2.01363e+01 | 6232.08 | 4.16401 | 9.10370e+00
-1 | -2 | 5730.44 | 3.94898 | 1.85877e+01 | 6064.42 | 4.22732 | 9.64255e+01 | 6213.97 | 4.47134 | -4.03754e+00
-1 | -1 | 6018.53 | 4.32370 | 0.84736e+01 | 6106.44 | 4.42460 | -1.97869e+01 | 5975.70 | 4.44946 | -1.52341e+01
-1 | 0 | 6066.49 | 4.40708 | -0.26149e+01 | 5940.15 | 4.35794 | -1.28211e+01 | 5558.93 | 4.19639 | -2.00041e+00
-1 | 1 | 5907.36 | 4.24389 | -1.33376e+01 | 5570.33 | 4.05455 | -2.17057e+01 | 5165.36 | 3.85050 | -2.94438e+01
-1 | 2 | 5531.36 | 3.85554 | -2.204124e+01 | 5220.56 | 3.76885 | -2.78578e+01 | 5138.40 | 3.68113 | -3.42262e+01
-1 | 3 | 5296.46 | 3.73086 | -2.82281e+01 | 5285.11 | 3.74952 | -3.37210e+01 | 5622.58 | 3.70697 | -4.08248e+01
-1 | 4 | 5555.81 | 3.87060 | -3.51942e+01 | 5915.95 | 3.90781 | -4.20588e+01 | 6773.24 | 3.92187 | -5.20952e+01
-0.5 | -5 | - | - | - | - | - | - | 2545.10 | 3.34599 | 2.10345e+01
-0.5 | -4 | 3051.24 | 3.56707 | 2.5800136e+01 | 2657.00 | 3.43838 | 1.98820e+01 | 2460.55 | 3.18859 | 1.51461e+01
-0.5 | -3 | 2736.75 | 3.42973 | 1.77802766e+01 | 2496.04 | 3.29327 | 1.33584e+01 | 2476.22 | 3.11815 | 9.64734e+00
-0.5 | -2 | 2545.95 | 3.32201 | 1.1002425e+01 | 2440.74 | 3.22178 | 7.50762e+00 | 2568.18 | 3.14012 | 3.88648e+00
-0.5 | -1 | 2442.72 | 3.24678 | 5.02669e+00 | 2472.68 | 3.21181 | 1.79701e+01 | 2714.67 | 3.21703 | -2.42134e+00
-0.5 | 0 | 2433.55 | 3.23508 | -0.804563e+00 | 2577.56 | 3.25750 | -4.21114e+01 | 2906.45 | 3.28911 | -9.21665e+00
-0.5 | 1 | 2512.54 | 3.28153 | -6.70696e+00 | 2750.64 | 3.32981 | -1.07711e+01 | 3161.99 | 3.33462 | -1.64688e+01
-0.5 | 2 | 2666.79 | 3.36284 | -1.31431e+01 | 2998.31 | 3.40612 | -1.80470e+01 | 3520.28 | 3.36384 | -2.44289e+01
-0.2 | -6 | - | - | - | - | - | - | 1854.91 | 3.10887 | 3.26209e+01
-0.2 | -5 | - | - | - | - | - | - | 1832.02 | 2.99341 | 2.64812e+01
-0.2 | -4 | 2243.46 | 3.35995 | 2.69389e+01 | 1920.12 | 3.20221 | 2.28615e+01 | 1864.91 | 2.92531 | 2.06957e+01
-0.2 | -3 | 2049.10 | 3.25993 | 1.90404e+01 | 1869.51 | 3.12952 | 1.63868e+01 | 1946.64 | 2.90953 | 1.49354e+01
-0.2 | -2 | 1934.94 | 3.19006 | 1.21307e+01 | 1876.37 | 3.08874 | 1.02807e+01 | 2075.38 | 2.93694 | 8.88949e+01
-0.2 | -1 | 1883.14 | 3.15123 | 5.76155e+00 | 1933.74 | 3.08625 | 4.17958e+01 | 2250.29 | 2.99935 | 2.27335e+00
-0.2 | 0 | 1884.55 | 3.14454 | -0.41986e+00 | 2041.83 | 3.11848 | -0.90786e+01 | 2476.38 | 3.07785 | -5.09879e+00
-0.2 | 1 | 1938.04 | 3.16825 | -6.73047e+00 | 2207.88 | 3.17427 | -9.20190e+00 | 2758.99 | 3.15405 | -1.33962e+01
0 | -7 | - | - | - | - | - | - | 1728.44 | 2.99275 | 4.67167e+01
0 | -6 | - | - | - | - | - | - | 1719.71 | 2.89803 | 3.96288e+01
0 | -5 | - | - | - | - | - | - | 1751.73 | 2.84231 | 3.29572e+01
0 | -4 | 2062.37 | 3.27138 | 3.07824e+01 | 1808.34 | 3.11261 | 2.75088e+01 | 1818.71 | 2.82611 | 2.63673e+01
0 | -3 | 1933.43 | 3.20292 | 2.22342e+01 | 1798.71 | 3.06760 | 2.02250e+01 | 1920.80 | 2.84045 | 1.96343e+01
0 | -2 | 1858.27 | 3.15565 | 1.44181e+01 | 1826.42 | 3.05273 | 1.31041e+01 | 2061.02 | 2.88257 | 1.25388e+01
0 | -1 | 1825.13 | 3.13005 | 6.99383e+00 | 1890.85 | 3.05874 | 5.91841e+00 | 2238.61 | 2.94673 | 4.84410e+00
0 | 0 | 1830.91 | 3.12687 | -0.30047e+00 | 1994.41 | 3.09325 | -0.84847e+00 | 2461.40 | 3.01898 | -3.62633e+00
0 | 1 | 1875.32 | 3.14563 | -7.69738e+00 | 2141.51 | 3.13838 | -9.63904e+00 | 2738.91 | 3.08702 | -1.30737e+01
0 | 2 | 1960.60 | 3.18430 | -1.54656e+01 | 2344.49 | 3.19623 | -1.85269e+01 | 3083.91 | 3.14313 | -2.37368e+01
0 | 3 | 2095.43 | 3.24171 | -2.38975e+01 | 2616.65 | 3.25743 | -2.85936e+01 | 3531.08 | 3.19391 | -3.60056e+01
0.2 | -9 | - | - | - | - | - | - | 1791.74 | 2.95175 | 7.48679e+01
0.2 | -8 | - | - | - | - | - | - | 1765.43 | 2.85565 | 6.58774e+01
0.2 | -7 | - | - | - | 2091.46 | 3.23062 | 6.35495e+01 | 1773.26 | 2.78668 | 5.75972e+01
0.2 | -6 | - | - | - | 1991.73 | 3.14385 | 5.29988e+01 | 1810.61 | 2.74096 | 4.96448e+01
0.2 | -5 | - | - | - | 1938.26 | 3.08387 | 4.33720e+01 | 1879.73 | 2.73007 | 4.18579e+01
0.2 | -4 | 2145.97 | 3.21388 | 3.72520e+01 | 1921.66 | 3.04438 | 3.43461e+01 | 1978.18 | 2.74627 | 3.39657e+01
0.2 | -3 | 2052.70 | 3.16668 | 2.72453e+01 | 1936.51 | 3.02583 | 2.55912e+01 | 2106.27 | 2.78474 | 2.57563e+01
0.2 | -2 | 1997.26 | 3.13337 | 1.78375e+01 | 1982.01 | 3.02826 | 1.68982e+01 | 2268.16 | 2.84706 | 1.70398e+01
0.2 | -1 | 1975.93 | 3.12012 | 8.75900e+00 | 2059.61 | 3.04636 | 8.05065e+00 | 2463.15 | 2.91645 | 7.62532e+00
0.2 | 0 | 1985.34 | 3.11973 | -0.22458e+00 | 2168.80 | 3.08035 | -0.084338e+00 | 2697.45 | 2.99092 | -2.66574e+00
0.2 | 1 | 2025.32 | 3.13379 | -9.32600e+00 | 2313.39 | 3.12309 | -1.09903e+01 | 2983.03 | 3.05846 | -1.4019e+01
0.2 | 2 | 2100.08 | 3.16375 | -1.87725e+01 | 2499.05 | 3.17433 | -2.16170e+01 | 3324.88 | 3.11715 | -2.66710e+01
0.5 | -12 | - | - | - | - | - | - | 2264.83 | 2.67458 | 1.43227e+02
0.5 | -11 | - | - | - | - | - | - | 2232.38 | 2.57911 | 1.30267e+02
0.5 | -10 | - | - | - | - | - | - | 2235.43 | 2.52124 | 1.18428e+02
0.5 | -9 | - | - | - | - | - | - | 2273.36 | 2.48403 | 1.07387e+02
0.5 | -8 | - | - | - | - | - | - | 2342.76 | 2.46297 | 9.68018e+01
0.5 | -7 | - | - | - | 2852.53 | 2.98707 | 9.38135e+01 | 2447.51 | 2.45833 | 8.64365e+01
0.5 | -6 | - | - | - | 2820.96 | 2.96112 | 7.98614e+01 | 2586.92 | 2.47868 | 7.60139e+01
0.5 | -5 | - | - | - | 2829.01 | 2.94556 | 6.65324e+01 | 2760.59 | 2.52177 | 6.52176e+01
0.5 | -4 | 3170.00 | 3.15653 | 5.67924e+01 | 2867.41 | 2.94416 | 5.34498e+01 | 2970.68 | 2.59625 | 5.38514e+01
0.5 | -3 | 3091.99 | 3.12965 | 4.20318e+01 | 2939.39 | 2.96026 | 4.04158e+01 | 3215.85 | 2.68364 | 4.16453e+01
0.5 | -2 | 3049.01 | 3.11603 | 2.77781e+01 | 3041.51 | 2.98711 | 2.71574e+01 | 3497.51 | 2.78294 | 2.84293e+01
0.5 | -1 | 3038.39 | 3.11115 | 1.37886e+01 | 3175.77 | 3.02882 | 1.34916e+01 | 3818.61 | 2.88102 | 1.40207e+01
0.5 | 0 | 3058.69 | 3.11628 | -1.5190e-01 | 3344.77 | 3.07307 | -0.78757e+00 | 4182.63 | 2.97346 | -1.73913e+00
0.5 | 1 | 3110.92 | 3.12984 | -1.42605e+01 | 3547.32 | 3.12814 | -1.58875e+01 | 4593.67 | 3.05210 | -1.90168e+01
0.5 | 2 | 3197.90 | 3.15422 | -2.87577e+01 | 3803.97 | 3.17478 | -3.20692e+01 | 5083.80 | 3.11946 | -3.80876e+01
0.5 | 3 | 3320.96 | 3.18589 | -4.38833e+01 | 4107.42 | 3.22521 | -4.95635e+01 | 5643.29 | 3.16986 | -5.91703e+01
0.5 | 4 | 3488.70 | 3.22545 | -5.99210e+01 | 4484.63 | 3.27365 | -6.87547e+01 | 6340.51 | 3.20545 | -8.29205e+01
1 | -22 | stable | stable | stable | stable | stable | stable | 48851.97 | 0.583363 | 1.66520e+03
1 | -21 | stable | stable | stable | stable | stable | stable | 45648.53 | 0.58320 | 1.53650e+03
1 | -20 | stable | stable | stable | stable | stable | stable | 43411.00 | 0.58277 | 1.42701e+03
1 | -19 | stable | stable | stable | stable | stable | stable | 44753.92 | 0.58333 | 1.37646e+03
1 | -18 | stable | stable | stable | stable | stable | stable | 48272.25 | 0.53983 | 1.35431e+03

As in the wide–gap case, we plot eigenfunctions only at $\mu=-1$ (counter-
rotating cylinders) and the largest positive $\mu$ at which we could find
instability in the co-rotating regime for each $G$. Here too, we start by
discussing the fastest growing modes for each of the three values of
$G\in\\{100,500,1000\\}$. Figure 10 plots the radial variation of critical
eigenmodes (top row) and the corresponding normalized critical eigenfunctions
(bottom row) for the counter-rotation case with $\mu=-1$ at $G=100$ for which
the fastest growing mode is $m=3$. Figure 11 the fastest growing marginal mode
$m=-1$ for $\mu=0.5$ in the co-rotation regime at $G=100$. Figures 12 and 13
plot the fastest growing marginal modes at $G=500$ for $\mu=-1,0.5$ with
$m=2,-6$, respectively. We do not find any unstable mode in the co-rotation
case for $G=100,500$.

Figure 10: (Top row) Unnormalized 1d critical eigenfunctions for
$\eta=0.9,G=100$ for counter-rotation case with $\mu=-1,m=3$. (Bottom row)
Normalized critical eigenfunctions – colors and slices to be interpreted as in
Fig. 3.

Figure 11: (Top row) Unnormalized 1d critical eigenfunctions for
$\eta=0.9,G=100$ for co-rotation case with $\mu=0.5,m=-1$. (Bottom row)
Normalized critical eigenfunctions – colors and slices to be interpreted as in
Fig. 3.

Figure 12: (Top row) Unnormalized 1d critical eigenfunctions for
$\eta=0.9,G=500$ for counter-rotation case with $\mu=-1,m=2$. (Bottom row)
Normalized critical eigenfunctions – colors and slices to be interpreted as in
Fig. 3.

Figure 13: (Top row) Unnormalized 1d critical eigenfunctions for
$\eta=0.9,G=500$ for co-rotation case with $\mu=0.5,m=-6$. (Bottom row)
Normalized critical eigenfunctions – colors and slices to be interpreted as in
Fig. 3.

Figures 14 and 15 plot the fastest growing marginal modes at $G=1000$ with
$m=2,-20$ for $\mu=-1,1$, respectively. Similar to the wide–gap case, here too
for all $G$ considered, the fastest growing modes for $\mu=-1$ are confined
near the inner cylinder in the radial direction due to strong counter-
rotation. For $(\mu,G)=(1,1000)$ in the Rayleigh-stable co-rotation regime,
the fastest growing mode can be observed to be $m=-20$, which again has a
similar radial structure as the corresponding wide–gap case, i.e., a bimodal
radial variation of the azimuthal velocity perturbation $v$ is observed (see
Figs. 15 and 8).

Figure 14: (Top row) Unnormalized 1d critical eigenfunctions for
$\eta=0.9,G=1000$ for counter-rotation case with $\mu=-1,m=2$. (Bottom row)
Normalized critical eigenfunctions – colors and slices to be interpreted as in
Fig. 3.

Figure 15: (Top row) Unnormalized 1d critical eigenfunctions for
$\eta=0.9,G=1000$ for co-rotation case with $\mu=1,m=-20$. (Bottom row)
Normalized critical eigenfunctions – colors and slices to be interpreted as in
Fig. 3.

Figure 16 shows calculated neutral stability curves for
$\mu=\\{-1,-0.5,0.5,1\\}$ at $G=1000$ for various azimuthal perturbation
wavenumbers $m$. The critical axial perturbation wavenumber $k_{c}$ is seen to
decrease with increasing $\mu$ in Table LABEL:tbl:eta_0.9_neutral,
corresponding to longer wavelength instabilities at the onset. We plot neutral
stability curves for $G=1000$ for four selected values of $\mu$ in Fig. 16. We
select this case because we find unstable modes in the case of co-rotating
cylinders ($\mu=1$). As shown in Fig. 16, we find unstable modes in the
Rayleigh-stable regime at $G=1000,\mu=1$ at higher azimuthal wavenumbers,
beyond $m=-15$. Azimuthal mode with $m=-20$ was found to be the fastest
growing, with $m=-18,-19,-21$ being close to the onset of $m=-20$, indicating
a possibility of mode competition in the weakly nonlinear regime.

Figure 16: Neutral stability curves for $\eta=0.9,G=1000$ for different
azimuthal perturbation numbers with $\mu=$ (top left) $-1$, (top right)
$-0.5$, (bottom left) $0.5$, (bottom right) $1.0$. From the neutral stability
curves at $G=1000$ for $\mu=\\{-1,-0.5,0.5,1\\}$.

##  IV Discussion

In this section, we discuss observations of our analysis focusing on the
fastest growing modes, the effect of outer cylinder rotation, and the impact
of imposed temperature gradient and increased radius ratio on the linear
stability thresholds.

We begin with a qualitative description of the fastest growing modes at the
onset of instability. In all cases reported in this article, neutral modes at
the onset of instability are found to be oscillatory. Depending on the
combination of parameters $\\{\eta,G,\mu\\}$, both axisymmetric toroidal modes
($m=0$) and non-axisymmetric spiral modes can grow at the fastest rate. For
all the critical modes reported in Tables 1 and LABEL:tbl:eta_0.9_neutral, we
observe that critical modes with $m<0$ have a positive oscillation frequency
($\sigma_{ic}$) and $m\geq 0$ have a negative $\sigma_{ic}$. Therefore, all
the critical non-axisymmetric modes with $m\geq 0$ travel with an upward axial
speed, while $m<0$ travel with a downward axial speed. Furthermore, we find
both positive and negative $m$ to be the fastest growing modes depending on
different parameters, indicating negative and positive inclination $\psi$,
respectively, of the lines of constant phase with respect to the horizontal
direction according to Eq. (24). For both radius ratios considered here, the
fastest growing modes at the onset of instability are seen to be confined near
the inner cylinder for $\mu=-0.5,-0.1$. This effect is more pronounced in the
wide–gap case ($\eta=0.6$) evident from the $2D$ slices of the reconstructed
$3D$ critical eigenmodes. At $G=1000$ in the wide–gap case ($\eta=0.6$), we
observe closed loops of unstable regions for $\mu=0.5$ and an unstable mode
$m=-4$ for the co-rotation case $\mu=1$. At $G=1000$ in the thin–gap case
($\eta=0.9$), we find $m=-20$ to be the fastest growing mode with $m=-19,-21$
having close critical Taylor numbers $Ta_{c}$ for the onset of instability.

We now shift the focus of our discussion from qualitative to quantitative and
describe the effects of varying one of $\\{\eta,G,\mu\\}$ while the other two
control parameters are held constant. First, we fix $\\{\eta,G\\}$ and observe
the effect of changing $\mu$. This amounts to choosing either of the Tables 1
or LABEL:tbl:eta_0.9_neutral, fixing a value for $G$ and then comparing the
highlighted values, moving in the vertical direction to see the effect of
changing $\mu$. We find that the outer cylinder rotation has a general
stabilizing effect in contrast to scenarios where the outer cylinder is held
stationary, except for the case $\\{\eta,G\\}=\\{0.6,1000\\}$. This can be
seen by comparing $Ta_{c}$ of the fastest growing mode for different $\mu$ at
a fixed set of $\\{\eta,G\\}$. In general, $Ta_{c}$ for $\mu\neq 0$ is larger
in comparison to $Ta_{c}$ for $\mu=0$, indicating a general stabilizing
effect. In the case $\\{\eta,G\\}=\\{0.6,1000\\}$, counter-rotation stabilizes
the flow (as $\mu$ is lowered below $0$, $Ta_{c}$ increases in comparison to
when $\mu=0$), but co-rotation has a destabilizing effect, shown by the
decrease in $Ta_{c}$ when $\mu$ is increased from $0$ to $0.2$.

Next, to assess the effect of the increased temperature gradient we fix
$\\{\eta,\mu\\}$ and compare critical $Ta_{c}$ as $G$ is varied. This amounts
to choosing one of the tables 1 or LABEL:tbl:eta_0.9_neutral, fixing a value
for $\mu$ and then comparing the highlighted values, moving in the horizontal
direction to see the effect of changing $G$. Here, in the wide–gap case
($\eta=0.6$) increasing $G$ decreases the critical Taylor number $Ta_{c}$
required to trigger instability for all $\mu\in[-0.2,0.2]$. For other cases,
the effect of increasing $G$ is not monotonous. Therefore, increasing $G$ is
in general destabilizing for $\mu\in[-0.2,0.2]$ but has a non–monotonous
response for other values of $\mu$ considered here. Similarly, for the thin
gap case ($\eta=0.9$), increasing $G$ generally destabilizes the flow except
for $\mu\in\\{-0.5\\}$, where its effect is observed to be stabilizing.

Finally, we compare critical Taylor numbers at $\eta=0.6$ and $0.9$ for a
constant $\\{G,\mu\\}$ to assess the impact of increasing radius ratio. This
amounts to comparing highlighted values across Tables 1 and
LABEL:tbl:eta_0.9_neutral for a pair of values of $\\{G,\mu\\}$. It is seen
that the critical Taylor numbers for the thin–gap case are lower than those
for the wide–gap case for $\mu\leq 0$, thus suggesting that increasing $\eta$
has a destabilizing effect in the Rayleigh-unstable counter-rotation regime.
For $\mu=0.2$, we see a destabilizing effect of increasing $\eta$ (evidenced
by decreased $Ta_{c}$) for $G=\\{100,500\\}$, but a stabilizing effect of
increasing $\eta$ (evidenced by increased $Ta_{c}$) for $G=1000$. The case of
$\mu=0.5$ is qualitatively different for the two radius ratios, i.e., it is in
the Rayleigh-stable regime for $\eta=0.6$, but in the Rayleigh-unstable regime
for $\eta=0.9$. Lastly, the co-rotation case ($\mu=1$) at $G=1000$, although
lying in the Rayleigh-stable regime, is found to be unstable at $Ta\sim
6\times 10^{3}$ at $\eta=0.6$, while at $\eta=0.9$ is found to be unstable at
$Ta\sim 5\times 10^{4}$, thus implying a stabilizing effect of increasing
$\eta$ for the co-rotation case in the Rayleigh-stable regime.

##  V Conclusion

We conducted a systematic linear stability analysis of the radially–heated
Taylor–Couette flow and observed the effects of outer–cylinder rotation. The
selected range of parameters includes three Grashof numbers
$G\in\\{100,500,1000\\}$ and two radius ratios $\eta\in\\{0.6,0.9\\}$,
representing wide–gap and thin–gap cases. The Prandtl number was fixed to a
constant value $Pr=1$. For each combination of $(\eta,G)$, seven angular
velocity ratios were considered $\mu\in[-1,1]$, ranging from counter-rotation
to co-rotation. We focused on the small Froude number limit $Fr\rightarrow 0$
where centrifugal buoyancy can be neglected. We extended symmetries of the
linearized system identified by Ali and Weidman [21] to include nonzero $\mu$
as defined in Eqs. (26), (27), and (28). We also discussed the observations of
our analysis focusing on the the description of the fastest –growing modes,
the effects of outer cylinder rotation, of imposed temperature gradient and of
increasing radius ratio on the linear stability thresholds. In the future, we
plan to extend the analysis to examine the effects of Prandtl numbers and
centrifugal buoyancy (non-vanishing Froude numbers). The linear stability
analysis also provides insight into the interesting range of parameters for
performing direct numerical simulations of the governing equations which could
shed light on the rich nonlinear dynamics of the system.

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