1. Introduction
Wall-bounded shear flows such as plane Poiseuille or channel flow (PPF) and plane Couette flow (PCF) subject to system rotation display many interesting physical phenomena, for example, turbulent–laminar patterns (Brethouwer, Duguet & Schlatter Reference Brethouwer, Duguet and Schlatter2012; Brethouwer Reference Brethouwer2017), recurring bursts of turbulence (Brethouwer et al. Reference Brethouwer, Schlatter, Duguet, Henningson and Johansson2014; Brethouwer Reference Brethouwer2016), large-scale structures (Gai et al. Reference Gai, Xia, Cai and Chen2016; Brethouwer Reference Brethouwer2017), multiple states (Yang & Xia Reference Yang and Xia2021) and strong increases in momentum and heat transfer (Brauckmann, Salewski & Eckhardt Reference Brauckmann, Salewski and Eckhardt2016; Brethouwer Reference Brethouwer2021, Reference Brethouwer2023). Studying the stability of such flows subject to system rotation in various directions may help to understand rotating shear flows in engineering applications.
The stability of PPF and PCF with and without spanwise system rotation has been studied extensively (see e.g. Hart Reference Hart1971; Hung, Joseph & Munson Reference Hung, Joseph and Munson1972; Lezius & Johnston Reference Lezius and Johnston1976; Schmid & Henningson Reference Schmid and Henningson2001; Wall & Nagata Reference Wall and Nagata2006; Daly et al. Reference Daly, Schneider, Schlatter and Peake2014; Nagata, Song & Wall Reference Nagata, Song and Wall2021). From now on, we abbreviate non-rotating PCF and PPF to NPCF and NPPF, respectively, and PCF and PPF subject to spanwise system rotation to ZPCF and ZPPF, respectively. The NPCF is linearly stable at any Reynolds number
$\textit{Re}$
, whereas in NPPF two-dimensional Tollmien–Schlichting (TS) modes with
$\beta =0$
are linearly unstable for
$\textit{Re} \geqslant 5772.3$
(Schmid & Henningson Reference Schmid and Henningson2001). Here and in the following
$\textit{Re} = U_{cl}\delta /\nu$
for PPF and
$\textit{Re} = U_w\delta /\nu$
for PCF, where
$U_{cl}$
is the centreline velocity,
$U_w$
is the velocity of the two walls moving in opposite directions,
$\delta$
is the half gap-width and
$\nu$
is the viscosity. Subscript
$c$
is used to denote values at critical condition for linear instabilities. Further,
$\alpha$
and
$\beta$
are the streamwise and spanwise wavenumbers, respectively, non-dimensionalised by
$\delta$
.
Spanwise rotation can drastically reduce the critical Reynolds number
$\textit{Re}_c$
of PPF. Lezius & Johnston (Reference Lezius and Johnston1976) and Alfredsson & Persson (Reference Alfredsson and Persson1989) did a linear stability analysis (LSA) of ZPPF assuming two-dimensional perturbations with
$\alpha =0$
and found that the minimum critical Reynolds number is
$\textit{Re}_c=66.40$
at
$\textit{Ro}_c=1/3$
. Here and in the following
$\textit{Ro}=2\varOmega \delta /U_{cl}$
for PPF and
$\textit{Ro}=2\varOmega \delta /U_w$
for PCF, where
$\varOmega$
is the imposed system rotation rate. Wall & Nagata (Reference Wall and Nagata2006) extended the LSA to three-dimensional perturbations and confirmed that at low
$\textit{Re}$
, ZPPF is most unstable to perturbations with
$\alpha =0$
. They recomputed the critical values and found the lowest
$\textit{Re}_c = 66.448$
at
$\textit{Ro}=0.3366$
with
$\beta _c=2.459$
.
Lezius & Johnston (Reference Lezius and Johnston1976) also pointed out the similarity between the linear perturbation equations of ZPCF and Rayleigh–Bénard convection between two flat plates. From that similarity follows
$16Re^2_c Ro(1-Ro)=Ra_c$
and
$\beta _c=3.117/2=1.558$
when
$\textit{Ro}\gt 0$
, where
$Ra_c=1707.762$
is the critical Rayleigh number (Chandrasekhar Reference Chandrasekhar1961) and
$\textit{Ro}\gt 0$
corresponds to anti-cyclonic rotation. The factors 16 and 2 in the relations for
$\textit{Re}_c$
and
$\beta _c$
arise when the half-gap width
$\delta$
is used for non-dimensionalisation instead of the gap width. The previous relation shows that the minimum
$\textit{Re}_c=20.6625$
of ZPCF occurs at
$\textit{Ro}=0.5$
. The non-normality of the linearised Navier–Stokes operator of PPF and PCF can explain the strong reduction of
$\textit{Re}_c$
by spanwise rotation (Jose & Govindarajan Reference Jose and Govindarajan2020).
Experiments (Alfredsson & Persson Reference Alfredsson and Persson1989; Tsukahara, Tillmark & Alfredsson Reference Tsukahara, Tillmark and Alfredsson2010) show that streamwise vortices develop in ZPPF and ZPCF slightly above
$\textit{Re}_c$
. The vortices are steady and turbulent motions are absent at these low
$\textit{Re}$
, but the vortices become three-dimensional and unstable when
$\textit{Re}$
increases (Yang & Kim Reference Yang and Kim1991; Finlay Reference Finlay1992; Nagata Reference Nagata1998; Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010; Daly et al. Reference Daly, Schneider, Schlatter and Peake2014; Nagata et al. Reference Nagata, Song and Wall2021), and turbulence sets in at sufficiently high
$\textit{Re}$
(Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010; Salewski & Eckhardt Reference Salewski and Eckhardt2015; Brethouwer Reference Brethouwer2017; Jose et al. Reference Jose, Kuzhimparampil, Pier and Govindarajan2017; Brethouwer Reference Brethouwer2021).
The effect of system rotation about axes other than the spanwise axis on PPF has also been investigated. Wu & Kasagi (Reference Wu and Kasagi2004) studied the effect of system rotation with various rotation axis directions on turbulent PPF using direct numerical simulation (DNS). The PPF subject to streamwise system rotation, abbreviated as XPPF, has also been investigated. Direct numerical simulation (Oberlack et al. Reference Oberlack, Cabot, Reif and Weller2006; Yang, Su & Wu Reference Yang, Su and Wu2010; Yang & Wang Reference Yang and Wang2018; Yang et al. Reference Yang, Deng, Wang and Shen2020; Yu et al. Reference Yu, Hu, Yan and Li2022; Hu et al. Reference Hu, Li and Yu2023, Reference Hu, Li and Yu2024) and experiments (Recktenwald et al. Reference Recktenwald, Weller, Schröder and Oberlack2007) of turbulent XPPF show a secondary mean flow and distinct Taylor–Görtler vortices, which are inclined to the streamwise direction.
Instabilities in XPPF have been studied via LSA and nonlinear analysis by Masuda, Fukuda & Nagata (Reference Masuda, Fukuda and Nagata2008). They used a non-dimensional rotation rate
$\varOmega ^* = 2\varOmega \delta ^2/\nu$
, which can be related to
$\textit{Ro}$
by noting that
$\varOmega ^*=Re\,Ro$
. Masuda et al. (Reference Masuda, Fukuda and Nagata2008) observed two asymptotic neutral stability regimes for three-dimensional perturbations in the LSA: one at low
$\textit{Ro}$
with
$\textit{Re}_c = 33.923/Ro$
, and one at high
$\textit{Ro}$
with
$\textit{Re}_c = 66.45$
and
$\beta _c\simeq 2.5$
and
$\alpha _c$
decreasing with
$\textit{Ro}$
. At very low
$\textit{Ro}$
, XPPF is most unstable to two-dimensional TS modes. The values of
$\textit{Re}_c$
and
$\beta _c$
in XPPF at high
$\textit{Ro}$
are remarkably similar to the minimum
$\textit{Re}_c = 66.448$
and
$\beta _c = 2.4592$
occurring at
$\textit{Ro}=0.3366$
in ZPPF (Wall & Nagata Reference Wall and Nagata2006). Masuda et al. (Reference Masuda, Fukuda and Nagata2008) did not comment on this similarity, but we show that it is not a coincidence.
Linear stability analysis does not always accurately predict a critical
$\textit{Re}$
for transition. Disturbances can exhibit transient energy growth in linearly stable flows due to the non-normality of the linearised Navier–Stokes operator (Grossmann Reference Grossmann2000), potentially triggering a subcritical transition (Orszag & Patera Reference Orszag and Patera1980; Daviaud, Hegseth & Bergé Reference Daviaud, Hegseth and Bergé1992). Consequently, the energy method has been employed to determine an energy-based Reynolds-number threshold
$\textit{Re}_E$
, below which all disturbances monotonically decay (Boeck, Brynjell-Rahkola & Duguet Reference Boeck, Brynjell-Rahkola and Duguet2024). This approach has been applied to NPPF and NPCF (Orr Reference Orr1907; Busse Reference Busse1969, Reference Busse1972; Joseph Reference Joseph1976; Falsaperla, Giacobbe & Mulone Reference Falsaperla, Giacobbe and Mulone2019), showing that
$\textit{Re}_E$
in NPPF is two orders of magnitude lower than
$\textit{Re}_c$
. However, even if transient growth occurs, a flow may relaminarise if disturbances do not grow sufficiently to trigger a subcritical transition. Hence,
$\textit{Re}_E$
can be significantly lower than the critical
$\textit{Re}$
below which a shear flow remains laminar (Fuentes, Goluskin & Chernyshenko Reference Fuentes, Goluskin and Chernyshenko2022).
Although
$\textit{Re}_E$
is a conservative measure, it is observed that turbulence can persist in NPPF at
$\textit{Re}$
much lower than
$\textit{Re}_c$
and in NPCF at finite
$\textit{Re}$
. However, below some
$\textit{Re}$
threshold, NPPF and NPCF are not uniformly turbulent but transitional. Intermittent turbulence, sometimes forming large-scale oblique bands with alternating laminar-like and turbulent-like flow, can develop in a range of
$\textit{Re}$
(Duguet, Schlatter & Henningson Reference Duguet, Schlatter and Henningson2010; Shimizu & Manneville Reference Shimizu and Manneville2019; Tuckerman, Chantry & Barkley Reference Tuckerman, Chantry and Barkley2020). The flows eventually become laminar at lower
$\textit{Re}$
, regardless of the initial conditions, and fully turbulent at higher
$\textit{Re}$
.
Subcritical transition has also been studied in rotating shear flows. Jose et al. (Reference Jose, Kuzhimparampil, Pier and Govindarajan2017) investigated transient growth in ZPPF and showed that the critical Reynolds number for such growth is almost independent of
$\textit{Ro}$
, decreasing from 51 at low rotation to 41 at high rotation. These values are far below
$\textit{Re}_c$
, both for
$\textit{Ro}\lesssim 10^{-2}$
and for large
$\textit{Ro}$
. Their DNS confirmed that subcritical transition can occur at low
$\textit{Ro}$
. Also, DNS and experiments of ZPCF with cyclonic rotation show subcritical transition and turbulent–laminar patterns in some
$(Re,Ro)$
range (Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010; Brethouwer et al. Reference Brethouwer, Duguet and Schlatter2012).
Subcritical transition to turbulence in XPPF has not, to our knowledge, been examined in detail. The values of
$\textit{Ro}$
considered in DNS of XPPF are
$\textit{Ro}\approx 0.1$
(Oberlack et al. Reference Oberlack, Cabot, Reif and Weller2006) and higher, where
$\textit{Re}_c$
is much lower than the
$\textit{Re}$
in the DNS, so the transition is supercritical. The energy and linear stability of spiral flow between concentric cylinders, rotating and sliding relative to one another, with or without a uniform axial pressure gradient, have been investigated by Joseph & Munson (Reference Joseph and Munson1970), Hung et al. (Reference Hung, Joseph and Munson1972) and Joseph (Reference Joseph1976). Both XPCF and XPPF represent two limiting narrow-gap cases of spiral flow with equal rotation rates of the inner and outer cylinders.
In summary, the stability of ZPPF and ZPCF has been extensively studied, whereas the stability of XPPF and XPCF has received much less attention. Only Masuda et al. (Reference Masuda, Fukuda and Nagata2008) and briefly Joseph & Munson (Reference Joseph and Munson1970), Hung et al. (Reference Hung, Joseph and Munson1972) and Joseph (Reference Joseph1976) have studied these two flow cases. Many aspects of the behaviour of the critical modes as well as the potential of subcritical transition remain unclear. In this study, we investigate the stability of XPPF and XPCF with the aim of obtaining a deeper physical understanding of these flows. We perform both LSA and DNS to examine the possibility of subcritical transition.
2. Methodology: linear stability analysis and direct numerical simulations
2.1. Configuration and governing equations
We investigate the stability of viscous incompressible PCF and pressure-driven PPF subject to constant system rotation about the streamwise axis, i.e. XPCF and XPPF, as illustrated in figure 1. The streamwise, wall-normal and spanwise coordinates non-dimensionalised by the half-gap width
$\delta$
are denoted by
$x$
,
$y$
and
$z$
, respectively, and the corresponding velocity components by
$u$
,
$v$
and
$w$
, respectively. The two infinite plane no-slip walls are at
$y=\pm 1$
. The velocity
$\boldsymbol{{u}}$
in the rotating frame of reference in both flow cases is governed by the non-dimensional Navier–Stokes equations:
\begin{equation} \frac {\partial {\boldsymbol{{u}}}}{\partial t} + {\boldsymbol{{u}}} \boldsymbol{\cdot }\mathbf{\boldsymbol{\nabla }} {\boldsymbol{{u}}} = - \mathbf{\boldsymbol{\nabla }} p + \frac {1}{Re} \mathbf{\boldsymbol{\nabla} }^2 {\boldsymbol{{u}}} - Ro (\hat {{\boldsymbol{{x}}}} \times {\boldsymbol{{u}}}),\quad \mathbf{\boldsymbol{\nabla }} \boldsymbol{\cdot }{\boldsymbol{{u}}} = 0, \end{equation}
where
$\hat {{\boldsymbol{{x}}}}$
is the unit vector in the
$x$
direction. The last term in the momentum equation is the Coriolis force and the centrifugal force is absorbed in a modified pressure
$p$
. The laminar streamwise velocity profile in XPPF, given by
$U=1-y^2$
, and in XPCF, given by
$U=y$
, is not affected by rotation.
Figure 1. Configurations for (a) XPPF and (b) XPCF.
2.2. Linear stability analysis
We use standard linear stability methodology and linearise the governing equation (2.1). Introducing wall-normal velocity
$v({\boldsymbol{{x}}},t)$
and wall-normal vorticity
$\eta ({\boldsymbol{{x}}},t)$
perturbations gives
\begin{align} \left [ \left ( \frac {\partial }{\partial t} + U \frac {\partial }{\partial x} \right ) \boldsymbol{\nabla} ^2 - U'' \frac {\partial }{\partial x} - \frac {1}{Re} \boldsymbol{\nabla} ^4 \right ] v + Ro \frac {\partial \eta }{\partial x} &= 0, \end{align}
\begin{align} \left [ \left ( \frac {\partial }{\partial t} + U \frac {\partial }{\partial x} \right ) - \frac {1}{Re} \boldsymbol{\nabla} ^2 \right ] \eta + \left [ U'\frac {\partial }{\partial z}- Ro \frac {\partial }{\partial x} \right ] v &= 0, \\[9pt] \nonumber \end{align}
where
$U' = \textrm {d} U/\textrm {d} y$
and
$U'' = \textrm {d}^2 U /\textrm {d} y^2$
and boundary conditions
$v = \partial v/\partial y = \eta = 0$
at the walls. Assuming wave-like perturbations
$v({\boldsymbol{{x}}},t) = \hat {v}(y) {\textrm e}^{{\textrm i}(\alpha x + \beta z - \omega t)}$
and
$\eta ({\boldsymbol{{x}}},t) = \hat {\eta }(y) {\textrm e}^{{\textrm i}(\alpha x + \beta z - \omega t)}$
with wavenumber vector
${\boldsymbol{{k}}}=(\alpha ,\beta )$
leads to the following eigenvalue problem in matrix form:
\begin{equation} -{\textrm i}\omega \begin{pmatrix} D^2-k^2 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \hat {v} \\ \hat {\eta } \end{pmatrix} + \begin{pmatrix} \mathcal{L}_\textit{OS} & \mathcal{L}_R \\ \mathcal{L}_C & \mathcal{L}_{SQ} \end{pmatrix} \begin{pmatrix} \hat {v} \\ \hat {\eta } \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}\!, \end{equation}
where the Orr–Sommerfeld and Squire operators
$\mathcal{L}_\textit{OS}$
and
$\mathcal{L}_{SQ}$
and operators
$\mathcal{L}_R$
and
$\mathcal{L}_C$
are given by
\begin{align} \mathcal{L}_\textit{OS} &= {\textrm i} \alpha U (D^2-k^2) - {\textrm i} \alpha U'' - \frac {1}{Re} (D^2-k^2)^2, \end{align}
\begin{align} \mathcal{L}_R &= {\textrm i}\alpha Ro, \end{align}
\begin{align} \mathcal{L}_C & = {\textrm i} (\beta U' - \alpha Ro), \end{align}
\begin{align} \mathcal{L}_{SQ} &= {\textrm i} \alpha U - \frac {1}{Re} (D^2-k^2), \end{align}
with
$k^2 = \alpha ^2 + \beta ^2$
,
$D(.) = \textrm {d}(.)/\textrm {d} y$
and boundary conditions
$\hat {v}(y) = D \hat {v}(y)= \hat {\eta }(y) =0$
at
$y=\pm 1$
. This eigenvalue problem (2.3) for
$\omega$
with eigenvalues
$\hat {v}(y)$
and
$\hat {\eta }(y)$
for XPPF and XPCF is discretised using Chebyshev polynomials and solved with Matlab routines. The imaginary part
$\omega _i$
of the complex eigenvalue
$\omega$
gives the non-dimensional growth rate of the perturbations. Convergence has been checked by changing the number of collocation points.
2.3. Direct numerical simulations
We also carry out DNS to investigate the stability of XPPF and XPCF, using a pseudospectral code that solves (2.1) with Fourier expansions and periodic boundary conditions in
$x$
and
$z$
directions and Chebyshev polynomials in the
$y$
direction and no-slip conditions at the walls (Chevalier et al. Reference Chevalier, Schlatter, Lundbladh and Henningson2014). In the DNS of XPPF the flow rate is fixed. The code has been used in many previous studies (e.g. Brethouwer et al. Reference Brethouwer, Duguet and Schlatter2012; Brethouwer Reference Brethouwer2017, Reference Brethouwer2021).
2.4. Validation of the linear stability analysis
The LSA results for XPPF agree with those of Masuda et al. (Reference Masuda, Fukuda and Nagata2008), as we show later. To further validate our LSA we have carried out DNS of XPCF and XPPF with small initial perturbations at three
$\textit{Ro}$
and
$\textit{Re}$
slightly above
$\textit{Re}_c$
. We compared the growth rate of the velocity fluctuations with that of the most unstable mode predicted by LSA. The growth rates match, as shown in the Appendix.
3. Results: linear stability analysis
We first discuss the LSA results. Figure 2 shows the neutral stability curves of XPPF and XPCF. The most unstable mode in XPCF is three-dimensional due to system rotation and the same applies to XPPF, except at very low
$\textit{Ro}$
when a two-dimensional TS mode with
$\beta =0$
is most unstable. The neutral stability curve of this TS mode, which converges for
$\textit{Ro}\rightarrow 0$
to the critical Reynolds number
$\textit{Re}_c=5772.2$
of NPPF, is also shown. According to figure 2 we can distinguish two asymptotic neutral stability regimes for three-dimensional perturbations in XPCF and XPPF: a low-
$\textit{Ro}$
regime with
$\textit{Re}_c \propto 1/Ro$
at
$\textit{Ro}\rightarrow 0$
and a high-
$\textit{Ro}$
regime with
$\textit{Re}_c$
approaching a low constant value at
$\textit{Ro}\rightarrow \infty$
. The transition between these two regimes is at
$\textit{Ro} \sim O(1)$
. Since
$\textit{Ro}$
expresses the ratio of system rotation to mean shear rotation, we can call the regime with
$\textit{Ro}\gg 1$
a rotation-dominated regime and the regime with
$\textit{Ro}\ll 1$
a shear-dominated regime.
Figure 2. Neutral stability curves of three-dimensional (3-D) modes in XPCF and the two-dimensional (2-D)
$\beta =0$
and three-dimensional modes in XPPF. Horizontal dashed line,
$\textit{Re}=\sqrt {1707.762}/2$
; horizontal dash-dotted line,
$\textit{Re}=66.45$
. Sloped dashed line,
$\textit{Re}=17/Ro$
; sloped dash-dotted line,
$\textit{Re}=33.923/Ro$
.
Masuda et al. (Reference Masuda, Fukuda and Nagata2008) already identified these two asymptotic regimes for XPPF. They found
$\textit{Re}_c = 33.923/Ro$
in the low-
$\textit{Ro}$
regime and
$\textit{Re}_c = 66.45$
in the high-
$\textit{Ro}$
regime, shown by dash-dotted lines in figure 2, which match our LSA results. In XPCF,
$\textit{Re}_c \simeq 17/Ro$
when
$\textit{Ro}\rightarrow 0$
and
$\textit{Re}_c = 20.6625$
when
$\textit{Ro}\rightarrow \infty$
, shown by dashed lines in figure 2. The latter
$\textit{Re}_c$
is equal to the minimum critical Reynolds number
$\textit{Re}_c = \sqrt {1707.762}/2 = 20.6625$
in ZPCF occurring at
$\textit{Ro}=1/2$
, which is explained in the next section. In the high-
$\textit{Ro}$
regime of XPPF something similar happens since
$\textit{Re}_c$
approaches
$66.45$
, which is equal to the minimum
$\textit{Re}_c$
in ZPPF occurring at
$\textit{Ro}=0.3366$
(Wall & Nagata Reference Wall and Nagata2006).
Figure 3(a,b) shows the wavenumbers of the critical three-dimensional mode
$\alpha _c$
and
$\beta _c$
at neutral stability conditions, and figure 3(c) the angle
$\theta = \arctan (\alpha _c/\beta _c)$
of the wavenumber vector
${\boldsymbol{{k}}}_c=(\alpha _c,\beta _c)$
with the
$z$
axis as a function of
$\textit{Ro}$
in XPPF and XPCF. The critical spanwise wavenumber assumes a constant but different value in the low-
$\textit{Ro}$
and high-
$\textit{Ro}$
regimes, and changes at
$\textit{Ro}\sim O(1)$
. In XPPF,
$\beta _c = 2.459$
when
$\textit{Ro}\rightarrow \infty$
, which is the same
$\beta _c$
as in ZPPF at the minimum
$\textit{Re}_c$
at
$\textit{Ro}=0.3366$
(Wall & Nagata Reference Wall and Nagata2006). Similarly, in XPCF,
$\beta _c = 1.558$
when
$\textit{Ro}\rightarrow \infty$
, which is the same
$\beta _c$
as in ZPCF at the minimum
$\textit{Re}_c$
at
$\textit{Ro}=0.5$
(Lezius & Johnston Reference Lezius and Johnston1976), which in turn is the same critical wavenumber as in Rayleigh–Bénard convection (Chandrasekhar Reference Chandrasekhar1961). The angle
$\theta _c$
and
$\alpha _c$
assume a maximum value at
$\textit{Ro}\sim O(1)$
and decrease as
$\theta _c , \alpha _c \propto 1/Ro$
as
$\textit{Ro}\rightarrow \infty$
, and increase as
$\theta _c , \alpha _c \propto Ro$
at
$\textit{Ro}\rightarrow 0$
in XPPF and XPCF. The critical vortical structures have thus the largest inclination angle with respect to the streamwise direction when
$\textit{Ro} \sim O(1)$
and system rotation and mean shear rotation are of the same order, and become more aligned with the streamwise direction when
$\textit{Ro} \rightarrow 0$
and
$\textit{Ro}\rightarrow \infty$
. The observed alignment of the vortices with the
$x$
axis for
$\textit{Ro}\rightarrow \infty$
conforms to the Taylor–Proudman theorem.
Figure 3. The critical wavenumbers (a)
$\alpha _c$
and (b)
$\beta _c$
and (c) angle
$\theta$
of the wavenumber vector
${\boldsymbol{{k}}}_c=(\alpha _c,\beta _c)$
with the
$z$
axis as a function of
$\textit{Ro}$
in XPPF and XPCF. In (b), dashed lines,
$\beta =1.179$
and
$\beta =1.558$
; dash-dotted lines,
$\beta =1.917$
and
$\beta =2.459$
. In (c), dashed lines,
$\theta =0.5/Ro$
and
$\theta =0.8Ro$
; dash-dotted lines,
$\theta =0.3366/Ro$
and
$\theta =1.05Ro$
.
Figure 4 shows isocontours of the growth rate
$\omega _i$
in the
$(\alpha ,\beta$
) plane at neutral stability conditions in XPPF at high to low
$\textit{Ro}$
. Masuda et al. (Reference Masuda, Fukuda and Nagata2008) showed similar plots for XPPF, although only for cases with
$\textit{Ro}\sim O(1)$
. The isocontours are symmetric about the
$\beta =0$
axis since the modes with wavenumbers
$(\alpha ,\beta )$
and
$(\alpha ,-\beta )$
have the same growth rate
$\omega _i$
. This symmetry can be understood by considering the effective rotation rate
$\mathbf{\varOmega }^\textit{ef}=\varOmega \hat {{\boldsymbol{{x}}}}-(\partial U/2\partial y)\hat {{\boldsymbol{{z}}}}$
, where the last term is the rotation rate caused by mean shear and
$\hat {{\boldsymbol{{z}}}}$
is the unit vector in the
$z$
direction. In the bottom and top half of the channel,
$\mathbf{\varOmega }^\textit{ef}$
has a negative and positive inclination angle with the
$x$
axis, respectively, which leads to the same instability on both sides of the channel, but with opposite inclination angles to the
$x$
axis. To illustrate this, figure 6(a) visualises the vortical structure of the critical modes in XPPF at
$\textit{Re}=77.02$
and
$\textit{Ro}=1$
. In the bottom and top half of the channel the vortical structures have a negative and positive inclination angle to the
$x$
axis caused by modes with
$\beta _c \gt 0$
and
$\beta _c \lt 0$
, respectively. Modes with
$\beta _c \gt 0$
and
$\beta _c \lt 0$
also have a larger velocity disturbance and Reynolds shear stresses in the bottom and top half of the channel, respectively (see figure 6
c,e). The velocity disturbances and Reynolds shear stresses are obtained by averaging over
$xz$
planes. Due to streamwise rotation, all three Reynolds shear stress components become non-zero (Oberlack et al. Reference Oberlack, Cabot, Reif and Weller2006).
Figure 4. Growth rate
$\omega _i$
as a function of
$(\alpha ,\beta )$
at neutral stability in XPPF: (a)
$\textit{Re}=66.47$
and
$\textit{Ro}=24$
; (b)
$\textit{Re}=77.03$
and
$\textit{Ro}=1$
; (c)
$\textit{Re}=682.8$
and
$\textit{Ro}=0.05$
; (d)
$\textit{Re}=5776$
and
$\textit{Ro}=0.000587$
. The neutrally stable modes are indicated by white stars.
Observations at other
$\textit{Ro}$
are qualitatively similar, although the inclination angle of the vortical structures with the
$x$
axis is smaller at lower and higher
$\textit{Ro}$
. When
$\textit{Ro}\rightarrow 0$
, two-dimensional modes with
$\beta =0$
become more prominent and are the most unstable modes if
$\textit{Ro}$
is sufficiently small (figure 4
d).
In contrast, the isocontours of the growth rate
$\omega _i$
in the
$(\alpha ,\beta$
) plane at neutral stability conditions in XPCF at four
$\textit{Ro}$
, shown in figure 5, are not symmetric about the
$\beta =0$
axis, with
$\omega _i$
generally being greater for
$\beta \gt 0$
. Vortical structures of the critical mode with
$\beta _c\gt 0$
in XPCF at
$\textit{Re}=28.14$
and
$\textit{Ro}=1$
, visualised in figure 6(b), are centred in the middle of the channel and have a negative inclination angle with the
$x$
axis, like the effective rotation rate
$\mathbf{\varOmega }^\textit{ef}$
. Figure 6(d, f) shows that this mode also has the largest streamwise and wall-normal velocity disturbance and Reynolds shear stress amplitudes in the centre of the channel. Observations at other
$\textit{Ro}$
are again quantitatively similar, with the differences that at lower and higher
$\textit{Ro}$
the inclination angle is smaller, and the wall-normal and spanwise velocity disturbances are negligible compared with the streamwise one if
$\textit{Ro}\ll 1$
(not shown here).
Figure 5. Growth rate
$\omega _i$
as a function of
$(\alpha ,\beta )$
at neutral stability in XPCF: (a)
$\textit{Re}=20.68$
and
$\textit{Ro}=24$
; (b)
$\textit{Re}=28.14$
and
$\textit{Ro}=1$
; (c)
$\textit{Re}=340.9$
and
$\textit{Ro}=0.05$
; (d)
$\textit{Re}=8496$
and
$\textit{Ro}=0.002$
. The neutrally stable mode is indicated by a white star.
Figure 6. (a) Visualisation using the Q criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988) and (c) root mean square of the velocity disturbances and (e) the Reynolds shear stresses of the two critical modes with
$\beta _c \gt 0$
and
$\beta _c \lt 0$
shown by the solid and dashed lines, respectively, in XPPF at
$\textit{Re}=77.02$
and
$\textit{Ro}=1$
. (b) Visualisation using the Q criterion (Hunt et al. Reference Hunt, Wray and Moin1988) and (d) root mean square of the velocity disturbances and (f) the Reynolds shear stresses of the critical mode in XPCF at
$\textit{Re}=28.14$
and
$\textit{Ro}=1$
. The vortices in (a,b) are coloured by the streamwise vorticity with blue and red denoting positive and negative values, respectively.
4. High-rotation-number asymptotics
We analyse the asymptotic behaviour seen in figures 2 and 3. For classical Rayleigh–Bénard convection between two horizontal flat plates and assuming wall-normal velocity perturbations
$v({\boldsymbol{{x}}},t) = \hat {v}(y) {\textrm e}^{{\textrm i}(\alpha x + \beta z - \omega t)}$
the linearised perturbation equation for a neutral stability mode with
$\omega =0$
can be written as
\begin{equation} \big(D^2-k^2 \big)^3 \hat {v} = - Ra\, k^2 \hat {v}, \end{equation}
where
$Ra$
is the Rayleigh number (Chandrasekhar Reference Chandrasekhar1961). The marginally stable mode is stationary with
$\omega =0$
. When the problem is non-dimensionalised with the gap width
$2\delta$
as length scale, it can be shown that the critical Rayleigh number
$Ra_c=1707.762$
and wavenumber
$k_c = 3.117$
(Chandrasekhar Reference Chandrasekhar1961).
We now consider ZPPF and ZPCF. In these cases, the eigenvalue problem of the LSA is the same as for XPPF and XPCF given by (2.3), except that the rotation term
$\alpha Ro$
is replaced by
$\beta Ro$
since the system rotation is about the
$z$
direction, that is,
$\mathcal{L}_R={\textrm i}\beta Ro$
and
$\mathcal{L}_C = {\textrm i}\beta (U'-Ro)$
. Since the most unstable mode is two-dimensional with
$\alpha =0$
in ZPPF and ZPCF (Lezius & Johnston Reference Lezius and Johnston1976; Wall & Nagata Reference Wall and Nagata2006), and the neutral stability mode is stationary with
$\omega =0$
, we can derive, from the eigenvalue problem (2.3) after substituting
$\mathcal{L}_R$
and
$\mathcal{L}_C$
, that
\begin{equation} \big( D^2-k^2 \big)^3 \hat {v} = - Re^2_d Ro(U'-Ro)\, \beta ^2 \hat {v}, \end{equation}
by eliminating
$\hat {\eta }$
from the eigenvalue problem. The boundary conditions for
$\hat {v}$
are the same as in Rayleigh–Bénard convection. The perturbation equation for PCF is non-dimensionalised using the gap width
$2\delta$
and the velocity difference between the walls
$2U_w$
, to retain the similarity with the Rayleigh–Bénard convection stability problem, so that
$\textit{Re}_d = 4U_w \delta /\nu = 4 Re$
. For PPF, we keep
$\textit{Re}_d=Re$
. Further,
$U'=1$
for PCF. The similarity between perturbation (4.1) for Rayleigh–Bénard convection and (4.2) for ZPCF then leads to
$16Re^2_c Ro(1-Ro)=Ra_c$
and
$\beta _c = 3.117/2 = 1.558$
if
$\delta$
is used as length scale (Lezius & Johnston Reference Lezius and Johnston1976). Consequently, the minimum
$\textit{Re}_c = \sqrt {Ra_c}/2=20.6625$
in ZPCF occurs at
$\textit{Ro}=1/2$
. In this case,
$\textit{Re}_c$
and
$\textit{Re}_E$
coincide, as shown by Joseph & Munson (Reference Joseph and Munson1970) and Busse (Reference Busse1970), similar to Rayleigh–Bénard flow.
We return to XPPF and XPCF and apply the same procedure. In XPPF the neutrally stable modes are not stationary; therefore, only
$\omega _i = 0$
and
$\omega = \omega _r$
with
$\omega _r$
the (real) wave frequency. Considering neutrally stable modes with
$\omega = \omega _r$
and eliminating
$\hat {\eta }$
from the eigenvalue problem (2.3), we find that
\begin{align} \big( D^2-k^2 \big)^3 \hat {v} &= - Re^2_d \alpha Ro \left (\beta U'-\alpha Ro \right ) \hat {v} + Re^2_d \alpha ^2 \mathcal{U}_X \mathcal{L}_X \hat {v} \nonumber\\ &\quad + {\textrm i} Re_d \alpha \big[ \mathcal{U}_X \big(D^2-k^2 \big)^2 + \big(D^2 - k^2 \big) \mathcal{L}_X \big] \hat {v}, \end{align}
where
$\mathcal{U}_X = U - \omega _r/\alpha$
and
$\mathcal{L}_X = \mathcal{U}_X (D^2-k^2 ) - U''$
. Note that the most unstable modes are three-dimensional. In XPCF,
$U''=0$
and
$\mathcal{U}_X = U$
since the neutrally stable modes are stationary (
$\omega _r=0$
) if the walls move with the same speed but in opposite directions.
Of the three terms on the right-hand side of (4.3) only the first contains
$\textit{Ro}$
and
$\textit{Ro}^2$
and therefore dominates if
$\textit{Ro} \gg 1$
. This has been verified by comparing the terms using the eigenvalue solver for the LSA. In that case, when
$\textit{Ro} \gg 1$
and only the first term is relevant, (4.2) and (4.3) are equivalent if
$\alpha Ro$
in (4.3) for XPPF/XPCF is equal to
$\beta Ro$
in (4.2) for ZPPF/ZPCF. That is, the perturbation equations are similar when the component of
$\boldsymbol{{k}}$
parallel to the rotation axis, multiplied by the rotation rate, is the same in the streamwise- and spanwise-rotating cases. This implies that the Coriolis force acting on a slightly oblique mode in a rapidly streamwise-rotating flow can have the same effect on the wall-normal velocity perturbation as the Coriolis force acting on a purely streamwise (longitudinal) mode in a spanwise-rotating flow. We know that the minimum critical
$\textit{Re}_c$
in ZPPF and ZPCF occurs at
$\textit{Ro}^c_\textit{ZPPF}=0.3366$
and
$\textit{Ro}^c_\textit{ZPCF}=0.5$
, respectively (Lezius & Johnston Reference Lezius and Johnston1976; Wall & Nagata Reference Wall and Nagata2006). The similarity of the perturbation equations when
$\alpha Ro$
in XPPF/XPCF is equal to
$\beta Ro$
in ZPPF/ZPCF means that
$\beta _c$
and
$\textit{Re}_c$
in the streamwise-rotating cases are the same as
$\beta _c$
and minimum
$\textit{Re}_c$
in the spanwise-rotating cases. Furthermore,
$\textit{Re}_c$
in XPPF and XPCF is found for that
$\theta$
when
$\textit{Ro} \tan \theta = Ro^c_\textit{ZPPF}=0.3366$
and
$\textit{Ro} \tan \theta = Ro^c_\textit{ZPCF}=0.5$
, respectively, where
$\theta = \arctan (\alpha / \beta )$
is again the angle of
$\boldsymbol{{k}}$
with the
$z$
axis in the streamwise-rotating case. We can approximate
$\tan \theta \simeq \theta$
when
$\textit{Ro}\gg 1$
, so that the critical mode in XPPF and XPCF obeys
$\theta = 0.3366/Ro$
and
$\theta = 0.5/Ro$
, respectively. With
$\textit{Ro} \tan \theta = Ro^c_\textit{ZPCF}=0.5$
, (4.3) for XPCF becomes
\begin{equation} \big( D^2-k^2 \big)^3 \hat {v} = - \frac {1}{4} Re^2_d \beta ^2 \hat {v}. \end{equation}
The similarity between (4.4) and (4.1) for Rayleigh–Bénard convection gives
$\beta _c = 1.558$
and
$\textit{Re}^2_d/4 = 4 Re^2 = Ra$
; therefore,
$\textit{Re}_c = \sqrt {Ra_c}/2 = 20.6625$
in XPCF when
$\textit{Ro} \rightarrow \infty$
.
The results of these considerations,
$\textit{Re}_c=20.6625$
,
$\beta _c=1.558$
and
$\theta _c=0.5/Ro$
in XPCF and
$\textit{Re}_c=66.45$
,
$\beta _c=2.459$
and
$\theta _c=0.3366/Ro$
in XPPF, are shown by dashed lines in figures 2 and 3(b,c), confirming that these values are approached for
$\textit{Ro}\gg 1$
.
In summary, the critical Reynolds number
$\textit{Re}_c$
and wavenumber
$\beta _c$
in XPPF and XPCF become independent of
$\textit{Ro}$
and approach the minimum
$\textit{Re}_c$
and corresponding
$\beta _c$
in ZPPF and ZPCF, respectively, for
$\textit{Ro}\rightarrow \infty$
. Moreover, the linear stability of ZPCF as well as XPCF at
$\textit{Ro}\rightarrow \infty$
share similarities with that of Rayleigh–Bénard convection. For
$\textit{Ro} \lesssim 5$
in the streamwise-rotating cases, the remaining terms on the right-hand side of (4.3) become significant, and the similarity with the spanwise rotating cases is lost.
In XPCF, the critical Reynolds number for energy instability
$\textit{Re}_E$
is identical to that in NPCF because energy stability is unaffected by rotation (Joseph & Munson Reference Joseph and Munson1970; Joseph Reference Joseph1976); the Coriolis term vanishes in the energy equation. In this case, the eigenvalue problem for energy instability is also equivalent to that of the LSA for Rayleigh–Bénard convection given by (4.1) (Joseph Reference Joseph1966), yielding
$\textit{Re}_E=\sqrt {Ra_c}/2=20.66$
for both NPCF and XPCF (Busse Reference Busse1970; Joseph & Munson Reference Joseph and Munson1970; Reddy & Henningson Reference Reddy and Henningson1993; Barletta & Mulone Reference Barletta and Mulone2024). The present analysis shows that
$\textit{Re}_c$
converges to this same value in the limit
$\textit{Ro}\rightarrow \infty$
, showing that linear and energy stability coincide, ruling out subcritical transition. Busse (Reference Busse1970) demonstrated the same result for ZPCF at
$\textit{Ro}=0.5$
and noted its extension to XPCF as
$\textit{Ro}\rightarrow \infty$
. Joseph & Munson (Reference Joseph and Munson1970) and Joseph (Reference Joseph1976), using a different approach within the framework of spiral flow between concentric cylinders, confirmed the coincidence of energy and linear stability in XPCF at
$\textit{Ro}\rightarrow \infty$
for
$\alpha \rightarrow 0$
,
$\theta Ro=0.5$
and
$\beta =1.558$
, consistent with the present results. This mode is the most susceptible to transient growth in NPCF due to the non-normality of the linearised Navier–Stokes operator (Reddy & Henningson Reference Reddy and Henningson1993).
Thus, streamwise rotation, like spanwise anti-cyclonic rotation, preferentially destabilises the mode showing maximal transient growth without rotation, explaining the strong destabilising effect of rotation and making the linearised Navier–Stokes operator effectively normal again. A similar argument applies to the XPPF case; further details can be found in the study by Jose & Govindarajan (Reference Jose and Govindarajan2020).
In the asymptotic limit
$\textit{Ro}\rightarrow 0$
of XPPF and XPCF the critical vortices also align with the
$x$
axis and thus
${\boldsymbol{{k}}}_c$
aligns with the
$z$
axis, giving
$\alpha = \beta \tan \theta \simeq \beta \theta$
. The first term on the right-hand side of (4.3) then approaches
$- Re^2_d \theta Ro U'\, \beta ^2 \hat {v}$
since
$\alpha Ro \ll \beta U'$
. When
$\theta \propto Ro$
,
$\textit{Re}_d \propto 1/Ro$
and
$\beta$
is constant, all three terms on the right-hand side of perturbation (4.3) remain constant and significant. This behaviour,
$\theta _c \propto Ro$
and
$\textit{Re}_c \propto 1/Ro$
(noting that
$\textit{Re}_c \propto Re_d$
) is observed in figures 2 and 3(c) in XPPF and XPCF in the limit
$\textit{Ro}\rightarrow 0$
.
5. Results: direct numerical simulations
Subcritical transition to turbulence can occur in NPPF and NPCF, resulting in stable coexisting laminar and turbulent states (Grossmann Reference Grossmann2000; Manneville Reference Manneville2015). However, when
$\textit{Re}$
is gradually reduced, uniformly turbulent NPPF and NPCF become transitional before relaminarising, and turbulent–laminar flow patterns develop if the flow domain is sufficiently large (Shimizu & Manneville Reference Shimizu and Manneville2019; Tuckerman et al. Reference Tuckerman, Chantry and Barkley2020). Subcritical transition and transitional regimes have also been observed in ZPCF, at higher
$\textit{Re}$
than in NPCF, when the rotation is cyclonic and
$\textit{Re}_c \rightarrow \infty$
(Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010; Brethouwer et al. Reference Brethouwer, Duguet and Schlatter2012), but not yet when the rotation is anti-cyclonic and destabilises the flow (Alfredsson & Persson Reference Alfredsson and Persson1989; Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010). Instead, regular and steady streamwise vortices are observed at
$\textit{Re}$
near
$\textit{Re}_c$
. In ZPPF subcritical transition has only been observed at very low
$\textit{Ro}$
(Jose et al. Reference Jose, Kuzhimparampil, Pier and Govindarajan2017), and turbulent–laminar patterns only appear in some
$\textit{Re}{-}Ro$
range on the channel side stabilised by rotation (Brethouwer Reference Brethouwer2017, Reference Brethouwer2019).
It is not yet known whether a subcritical transition can occur, and whether the transitional regime exists in XPPF and XPCF. We carry out DNS of XPPF and XPCF to address these questions. The DNS cover the range
$\textit{Re} \leqslant 2000$
in XPPF and
$\textit{Re} \leqslant 1000$
in XPCF, and
$0 \leqslant Ro \leqslant 0.8$
, and also include non-rotating and rapidly rotating cases at
$\textit{Ro}=24$
. A computational domain
$L_x/\delta \times L_z/\delta$
of
$110 \times 50$
and
$250 \times 125$
is used in the DNS of XPPF and XPCF, respectively, and a spatial resolution of
$512 \times 33 \times 512$
in the streamwise, wall-normal and spanwise directions, unless otherwise specified. These computational domain sizes are similar to those used in DNS by Brethouwer et al. (Reference Brethouwer, Duguet and Schlatter2012) and large enough to accommodate large-scale turbulent–laminar patterns. The resolution in the XPCF cases is finer than that determined by Manneville & Rolland (Reference Manneville and Rolland2011) for relatively well-resolved uniformly turbulent and transitional regimes in NPCF. The formation of turbulent–laminar patterns in NPCF is not very sensitive to resolution, with only a gradual downward shift of the
$\textit{Re}$
threshold for patterns when resolution becomes coarse (Manneville & Rolland Reference Manneville and Rolland2011).
To determine the lower
$\textit{Re}$
threshold of sustained vortices or turbulence at a given
$\textit{Ro}$
, we initialise the DNS with turbulent flow at sufficiently high
$\textit{Re}$
and reduce
$\textit{Re}$
in small steps until the flow becomes laminar. The lower threshold for sustained turbulence, called
$\textit{Re}_t$
, is defined as the lowest
$\textit{Re}$
at which turbulence or turbulent patterns persist for a time period of at least
$2\times 10^5(\delta / U_{cl,w})$
in our DNS. This does not preclude that turbulence or turbulent patterns eventually disappear on longer time scales.
Figure 7. Visualisation of the streamwise velocity field in XPPF in an
$xz$
plane at
$y=-0.9$
.
Firstly, we discuss DNS results of PPF using visualisations of the instantaneous streamwise velocity field in an
$xz$
plane near the wall at
$y=-0.9$
shown in figure 7. Additional visualisations of the velocity field are presented in the supplementary material available at https://doi.org/10.1017/jfm.2025.10723. Hereafter,
$\textit{Re}= U_{cl}\delta /\nu$
for PPF, where
$U_{cl}$
is the centreline velocity of the laminar base flow. The bulk Reynolds number
$\textit{Re}_b=U_b \delta /\nu = 2 Re/3$
, where
$U_b$
is the bulk mean velocity. We observe in NPPF (
$\textit{Ro}=0$
) uniform turbulence at
$\textit{Re}=2000$
, and transitional turbulence with oblique turbulent–laminar patterns at
$\textit{Re}=1400$
(figure 7
a). When
$\textit{Re}$
decreases, the turbulent patterns become less structured and at
$\textit{Re}_t=1000$
one oblique turbulent band persists in a laminar-like flow environment (figure 7
b), while below
$\textit{Re}_t$
the flow relaminarises.
These results for NPPF are broadly consistent with those of Shimizu & Manneville (Reference Shimizu and Manneville2019). They observed local relaminarisation at
$\textit{Re} \approx 1800$
and turbulent patterns at lower
$\textit{Re}$
until about
$800$
in NPPF. This
$\textit{Re}$
threshold for turbulent patterns is lower than in our DNS, which may be a result of the larger computational domain in their study, different simulation time period and other flow forcing (constant pressure gradient in their study versus constant mass flow in our study). However, using a larger computational domain in our DNS is prohibitively expensive when covering a wide range of
$\textit{Ro}$
, which requires many simulations.
In XPPF we also observe at low
$\textit{Ro} \lesssim 0.05$
a transitional regime with sustained turbulent–laminar patterns (figure 7
c,d), sometimes forming oblique bands, at low
$\textit{Re}$
until
$\textit{Re}_t=1000$
at
$\textit{Ro} \leqslant 0.04$
and
$\textit{Re}_t=950$
at
$\textit{Ro}=0.05$
(figure 7
e, f). The observed patterns span the whole channel gap width, as in NPPF, but in the present configuration we observe differences in the DNS at low
$\textit{Ro}$
. At
$\textit{Ro}=0$
,
$0.025$
and
$0.0356$
the flow relaminarises if
$\textit{Re}\lt Re_t$
, while at
$\textit{Ro}=0.04$
and
$0.05$
the turbulent patterns disappear if
$\textit{Re} \lt Re_t$
, but the flow does not relaminarise since
$\textit{Re}_t \gt Re_c$
. Instead, we observe regular vortices nearly aligned with the streamwise direction without signs of turbulence (figure 7
g,h). When
$\textit{Re}$
is further reduced the flow only relaminarises once
$\textit{Re} \leqslant Re_c$
. At
$\textit{Ro}=0.07$
we observe spotty turbulent structures at low
$\textit{Re}$
until
$\textit{Re} \approx 900$
(figure 7
i), and more regular vortices at lower
$\textit{Re}$
until
$\textit{Re}_c$
when the flow relaminarises. When
$\textit{Ro}$
increases, the spotty structures gradually disappear and turbulence becomes more uniform (figure 7
j,k). The flow becomes less turbulent when
$\textit{Re}$
approaches
$\textit{Re}_c$
(figure 7
l) and fully relaminarises when
$\textit{Re}\lt Re_c$
.
We now study XPCF using visualisations of the instantaneous streamwise velocity field in an
$xz$
plane at the centre at
$y=0$
shown in figure 8. Additional visualisations of the velocity field are again presented in the supplementary material. The behaviour of XPCF is qualitatively similar to that of XPPF. In DNS of NPCF (
$\textit{Ro}=0$
) we observe uniform turbulence at
$\textit{Re} \gt 400$
, local relaminarisation at
$\textit{Re} \simeq 400$
and turbulent–laminar patterns at lower
$\textit{Re}$
, which are sustained until
$\textit{Re}_t=340$
(figure 8
a). The patterns form clearer structured oblique bands than in NPPF. These observations are consistent with previous studies (Prigent et al. Reference Prigent, Grégoire, Chaté and Dauchot2003; Duguet et al. Reference Duguet, Schlatter and Henningson2010), although in DNS by Duguet et al. (Reference Duguet, Schlatter and Henningson2010) turbulent–laminar patterns could also be sustained at somewhat lower
$\textit{Re} \simeq 324$
. This may be caused by a difference in the computational domain size and simulation time period, which was
$2\times 10^4 (\delta /U_w)$
in the DNS by Duguet et al. (Reference Duguet, Schlatter and Henningson2010). In our DNS, turbulent patterns persist for such a time period at
$\textit{Re}=330$
, but after a time period of nearly
$10^5(\delta /U_w)$
the flow relaminarises.
Figure 8. Visualisation of the streamwise velocity field in XPCF in an
$xz$
plane at
$y=0$
.
Observations in DNS of XPCF at
$\textit{Ro}=0.025$
,
$0.05$
,
$0.07$
and
$0.1$
are similar. We observe uniform turbulence at
$\textit{Re} \gt 400$
, local flow relaminarisation at
$\textit{Re} \simeq 400$
and turbulent patterns and oblique bands develop when
$\textit{Re}$
is gradually reduced (figure 8
b,c,e). Full relaminarisation of the flow happens when
$\textit{Re}\lt 340$
at
$\textit{Ro}=0.025$
and
$\textit{Re}\lt 330$
at
$\textit{Ro}=0.05$
. The oblique bands span the whole channel gap width, as in NPCF and ZPCF at low cyclonic rotation rates (Brethouwer et al. Reference Brethouwer, Duguet and Schlatter2012). At
$\textit{Ro}=0.07$
and
$0.1$
the turbulent pattern disappears when
$\textit{Re} \lt 310$
and
$\textit{Re} \lt 300$
, respectively, but the flow does not relaminarise when
$\textit{Re}$
is reduced as long as
$\textit{Re} \gt Re_c$
, since regular vortices persist with localised disturbances but without larger turbulent patterns (figure 8
d, f). The flow relaminarises once
$\textit{Re} \lt Re_c$
.
Oblique band-like structures appear in XPCF at
$\textit{Ro}=0.14$
if
$\textit{Re}\lesssim 450$
. These bands become more distinct when
$\textit{Re}$
is further lowered (figure 8
g), but between the turbulent bands we see streamwise vortices and not the clear laminar-like flow regions, as at lower
$\textit{Ro}$
. The turbulent bands disappear when
$\textit{Re} \lt 280$
. Localised disturbances and vortical motions persist at
$\textit{Re}$
near
$\textit{Re}_t$
(figure 8
h), while only streamwise vortices persist at lower
$\textit{Re}$
(figure 8
i) until
$\textit{Re} \lt Re_c$
and the flow relaminarises. At
$\textit{Ro}=0.2$
we observe oblique patterns if
$250 \lesssim Re \lesssim 600$
with different turbulence activity but without laminar-like flow regions (figure 8
j). With increasing
$\textit{Ro}$
the oblique patterns gradually disappear (figure 8
k) and we only see uniform turbulence or regular vortices when
$\textit{Re} \gt Re_c$
(figure 8
l).
Figure 9 shows a survey of the observed flow regimes as a function of
$\textit{Re}$
and
$\textit{Ro}$
in the DNS of XPPF and XPCF. We distinguish between four flow regimes: a fully laminar regime; a transitional regime with local relaminarisation or large-scale turbulent–laminar patterns; a regime with a less clear distinction between turbulent and laminar flow regions but with large-scale patterns; and a regime with uniform turbulence or vortical motions. In XPPF and XPCF, organised and steady vortices appear at higher
$\textit{Ro}$
near
$\textit{Re}_c$
. As
$\textit{Re}$
increases further, these vortices gradually become more unsteady and chaotic, ultimately transitioning into a uniformly turbulent flow. As a result, it was not possible to define a sharp transition between the regimes characterised by uniform vortices and uniform turbulence; therefore, these regimes are not treated separately.
Figure 9. Flow regimes as a function of
$\textit{Ro}$
and
$\textit{Re}$
in (a) XPPF and (b) XPCF. Four flow regimes are distinguished (each marked by a different colour): a regime with (i) uniform/featureless turbulence or vortices, (ii) laminar flow, (iii) turbulent–laminar (TL) patterns and (iv) spotty structures or spots (XPPF)/band-like structures (XPCF) but no clear turbulent and laminar flow regions. Also shown are the neutral stability curve (solid line), subcritical threshold
$\textit{Re}_t$
(dashed line), conditions at which two stable non-laminar flow states coexist (yellow diamonds), conditions at which DNS were performed (white circles) and conditions corresponding to the visualisations shown in figures 7 and 8 (blue stars).
Furthermore, we have not observed the variety of vortical structures reported previously for ZPCF (Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010; Suryadi, Segalini & Alfredsson Reference Suryadi, Segalini and Alfredsson2014) and Taylor–Couette flow (Andereck, Liu & Swinney Reference Andereck, Liu and Swinney1986). Also, no clear qualitative differences were evident in the uniform turbulence regime of XPCF for
$\textit{Ro}\lesssim 0.14$
and only at higher rotation rates (e.g.
$\textit{Ro}=0.8$
) do the effects of rotation on the vortical structures become noticeable (not shown here). Developing a more detailed and refined regime map would require extensive additional simulations and analysis and is therefore beyond the scope of the present study.
At some
$\textit{Ro}$
we observe two coexisting stable regimes in XPPF and XPCF at a fixed
$\textit{Re}$
somewhat higher than
$\textit{Re}_c$
: a regime with regular vortices nearly aligned with the streamwise direction but without turbulent motions, and a regime with transitional or turbulent flow. This regime with regular vortices appears when the DNS is initialised with a laminar flow with small noise. These two coexisting non-laminar flow regimes are only observed in a narrow
$\textit{Ro}$
range (see figure 9). This differs from NPPF and NPCF and XPPF and XPCF at low
$\textit{Ro}$
when under subcritical conditions only transitional and laminar flow regimes are stable.
Figure 9 shows that subcritical transition can be triggered at low
$\textit{Ro}$
in XPPF and XPCF since
$\textit{Re}_t \lt Re_c$
, while the flow relaminarises if
$\textit{Re}\lt Re_t$
. At higher
$\textit{Ro}$
, when
$\textit{Re}_t \gt Re_c$
or when the transitional regime is absent, we cannot find evidence of subcritical transition since in all our DNS, XPPF and XPCF then relaminarise if
$\textit{Re} \lt Re_c$
. This absence of subcritical transition in XPPF and XPCF at higher
$\textit{Ro}$
was checked by initialising the DNS in two different ways: (i) with a uniformly or transitional turbulent flow at higher
$\textit{Re}$
and subsequently reducing
$\textit{Re}$
in steps until
$\textit{Re}$
was slightly below
$\textit{Re}_c$
and (ii) with a flow with strong disturbances at
$\textit{Re}$
slightly below
$\textit{Re}_c$
. In both cases, the flow relaminarised in the DNS. The crossover from the low-
$\textit{Ro}$
range with subcritical transition to high-
$\textit{Ro}$
range without subcritical transition is at
$\textit{Ro} \simeq 0.034$
in XPPF and
$\textit{Ro}\simeq 0.05$
in XPCF. Observations do not change fundamentally for
$\textit{Ro}\gt 1$
, that is, turbulent motions or vortices only develop if
$\textit{Re} \gt Re_c$
. In ZPPF and ZPCF, there is likewise no evidence of subcritical transition once rotation has substantially reduced
$\textit{Re}_c$
(Alfredsson & Persson Reference Alfredsson and Persson1989; Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010). Moreover, in XPCF, subcritical transition must vanish entirely in the limit
$\textit{Ro}\rightarrow \infty$
, since in this limit
$\textit{Re}_c$
and
$\textit{Re}_E$
coincide, implying that transient growth cannot occur for
$\textit{Re} \lt Re_c$
. Figure 9 further shows that a transitional regime with turbulent–laminar patterns is observed in XPPF and XPCF, as in NPPF and NPCF, at low
$\textit{Ro}$
but not at higher
$\textit{Ro}$
. These patterns develop even though streamwise rotation acts as destabilising and lowers
$\textit{Re}_c$
, while in ZPCF turbulent–laminar patterns are so far only observed when rotation is cyclonic and stabilises the flow (Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010; Brethouwer et al. Reference Brethouwer, Duguet and Schlatter2012). The
$\textit{Re}$
range with turbulent–laminar patterns is fairly constant with
$\textit{Ro}$
. In XPCF we observe a transitional regime at
$340 \lesssim Re \lesssim 400$
, and in XPPF at
$\textit{Re}\gtrsim 1000$
with the upper bound not determined here. Subcritical transition in XPPF and XPCF is thus only observed when
$\textit{Re}_c$
is higher than the lower bound for turbulent–laminar patterns, that is, when
$\textit{Re}_c \gtrsim 1000$
in XPPF and
$\textit{Re}_c \gtrsim 340$
in XPCF.
Interestingly, we observe a transitional regime with turbulent–laminar patterns in XPPF and XPCF in a small range of
$\textit{Ro}$
when
$\textit{Re}\gt Re_c$
, unlike in NPPF and NPCF where this regime only appears if the flow is subcritical. This suggests that in this small
$\textit{Ro}$
range, patterns can emerge by lowering and raising
$\textit{Re}$
. Indeed, in XPPF at
$\textit{Ro}=0.04$
and
$0.05$
turbulent laminar patterns emerge in our DNS starting not only from a turbulent flow at higher
$\textit{Re}$
and subsequently lowering
$\textit{Re}$
, but also from a flow with vortices but without turbulence at lower
$\textit{Re}$
and subsequently increasing
$\textit{Re}$
.
In fact, we can observe the formation of turbulent–laminar patterns at
$\textit{Re} \gt Re_c$
in XPPF and XPCF with laminar flow and some noise as initial condition. To show this, we carry out DNS of XPPF at
$\textit{Re}=1200$
and
$1500$
and
$\textit{Ro}=0.04$
(
$\textit{Re}_c =852$
) and
$\textit{Ro}=0.05$
(
$\textit{Re}_c =683$
) with a computational domain size of
$242 \times 2 \times 110$
and resolution of
$1536 \times 65 \times 1536$
in the streamwise, wall-normal and spanwise directions, respectively, as well as DNS of XPCF at
$\textit{Re}=350$
and
$\textit{Ro}=0.07$
(
$\textit{Re}_c =244$
) and
$\textit{Ro}=0.1$
(
$\textit{Re}_c =172$
) with a computational domain size of
$750 \times 2 \times 375$
and resolution of
$1920 \times 49 \times 1920$
in the streamwise, wall-normal and spanwise directions, respectively. These domains and resolutions are larger than in our other DNS of XPPF and XPCF to show the robustness of the observations. The initial condition is a laminar base flow with small noise.
Initially, a linear instability occurs in all six DNS since
$\textit{Re} \gt Re_c$
, leading to an exponential growth of
$u'$
and
$v'$
. See figures 10(a) and 10(b) for XPPF at
$\textit{Ro}=0.04$
and XPCF at
$\textit{Ro}=0.07$
, respectively. Here,
$u'$
and
$v'$
are the streamwise and wall-normal velocity fluctuations integrated over the whole volume. After this transient period
$u'$
and
$v'$
saturate, and turbulent–laminar patterns appear, which develop into sustained oblique turbulent–laminar bands in our DNS of XPPF and XPCF (see figure 10
c–f), with similar results for the other two XPPF cases. These results show that in a limited
$\textit{Ro}$
range turbulent–laminar patterns develop under subcritical and supercritical conditions in XPPF and XPCF. By contrast, in DNS at lower
$\textit{Ro}$
with
$\textit{Re}_t \lt Re\lt Re_c$
, that is, DNS of XPPF at
$\textit{Re}=1500$
,
$\textit{Ro}=0$
(
$\textit{Re}_c=5772$
) and
$\textit{Re}=1200$
,
$\textit{Ro}=0.025$
(
$\textit{Re}_c=1359$
), and DNS of XPCF at
$\textit{Re}=350$
,
$\textit{Ro}=0$
(
$\textit{Re}_c \rightarrow \infty$
) and
$\textit{Re}=350$
,
$\textit{Ro}=0.025$
(
$\textit{Re}_c = 680$
) (not shown here), transition to turbulence and formation of turbulent–laminar patterns only occur when the initial noise levels are finite.
Figure 10. Time series of the streamwise (red line) and wall-normal (pink line) velocity fluctuations in (a) XPPF at
$\textit{Ro}=0.05$
and
$\textit{Re}=1200$
and (b) XPCF at
$\textit{Ro}=0.1$
and
$\textit{Re}=350$
. Visualisation of the streamwise velocity field in an
$xz$
plane (c,d) at
$y=-0.9$
in XPPF and (e, f) at
$y=0$
in XPCF.
Turbulent–laminar patterns have not yet been observed in ZPCF under supercritical conditions (Tsukahara et al. Reference Tsukahara, Tillmark and Alfredsson2010), whereas such patterns can develop in ZPPF, though only on the channel side stabilised by rotation (Brethouwer Reference Brethouwer2017). It is possible that the dominant streamwise roll cells, triggered by the strongly destabilising effect of anti-cyclonic spanwise rotation, inhibit the formation of turbulent–laminar patterns. In Taylor–Couette flow, turbulent–laminar bands appear as spiral patterns under subcritical (Meseguer et al. Reference Meseguer, Mellibovsky, Avila and Marques2009a ; Burin & Czarnocki Reference Burin and Czarnocki2012) and supercritical (Meseguer et al. Reference Meseguer, Mellibovsky, Avila and Marques2009b ; Wang et al. Reference Wang, Ayats, Deguchi, Mellibovsky and Meseguer2022) conditions. We note similarities with the XPPF and XPCF cases considered here, in which turbulent–laminar patterns also appear under subcritical and supercritical conditions.
Berghout et al. (Reference Berghout, Dingemans, Zhu, Verzicco, Stevens, van Saarloos and Lohse2020) and Wang et al. (Reference Wang, Mellibovsky, Ayats, Deguchi and Meseguer2023) performed DNS of Taylor–Couette flow with counter-rotating cylinders in the supercritical regime, analysing the formation and statistical characteristics of these spiral patterns. Their observed spiral patterns closely resemble those found in subcritical NPCF (Wang et al. Reference Wang, Mellibovsky, Ayats, Deguchi and Meseguer2023). However, in Taylor–Couette flow, weak vortices persist near the inner cylinder within the laminar-like regions due to the centrifugal instability of the base flow. A similar phenomenon occurs in XPCF under supercritical conditions exhibiting turbulent–laminar patterns; despite significantly weaker fluctuations, streamwise-oriented vortices remain visible within the laminar-like flow regions (see e.g. figure 8 e,g).
6. Conclusions
We carried out LSA of PPF and PCF subject to streamwise system rotation. Linear stability analysis of streamwise-rotating PPF has already been performed by Masuda et al. (Reference Masuda, Fukuda and Nagata2008), but we have extended it and compared it with the PCF case. Three-dimensional perturbations are considered since the most unstable modes are inclined to the streamwise direction, in contrast to spanwise-rotating PPF and PCF in which two-dimensional perturbations with streamwise wavenumber
$\alpha =0$
are most unstable (Lezius & Johnston Reference Lezius and Johnston1976; Wall & Nagata Reference Wall and Nagata2006).
Linear stability analysis of streamwise-rotating PCF shows an asymptotic regime at
$\textit{Ro} \ll 1$
with
$\textit{Re}_c \propto Ro$
, and another asymptotic regime at
$\textit{Ro} \gg 1$
with
$\textit{Re}_c$
approaching a constant value, as in streamwise-rotating PPF (Masuda et al. Reference Masuda, Fukuda and Nagata2008). In both asymptotic regimes, the critical spanwise wavenumber
$\beta _c$
approaches a constant value, and the critical vortices become increasingly streamwise-aligned.
The minimum critical Reynolds number
$\textit{Re}_c = 20.66$
of streamwise-rotating PCF at
$\textit{Ro} \rightarrow \infty$
is equal to the minimum
$\textit{Re}_c$
of spanwise-rotating PCF at
$\textit{Ro}=0.5$
(Lezius & Johnston Reference Lezius and Johnston1976). Likewise,
$\textit{Re}_c=66.45$
of streamwise-rotating PPF at
$\textit{Ro} \rightarrow \infty$
is equal to the minimum
$\textit{Re}_c$
of spanwise-rotating PPF occurring at
$\textit{Ro}=0.3366$
(Wall & Nagata Reference Wall and Nagata2006). These results follow from the equation for the wall-normal velocity perturbation. We also show that the linear stability of streamwise-rotating PCF is related to Rayleigh–Bénard convection, like that of spanwise-rotating PCF. In all cases,
$\beta _c = 1.558$
and the minimum
$\textit{Re}_c$
in streamwise- and spanwise-rotating PCF at
$\textit{Ro}\rightarrow \infty$
and
$\textit{Ro}=0.5$
, respectively, is related to the critical Raleigh number
$Ra_c$
as
$\textit{Re}_c = \sqrt {Ra_c}/2$
.
We carried out DNS of streamwise-rotating PPF and PCF in a range of
$\textit{Re}$
and
$\textit{Ro}$
to investigate flow characteristics at low
$\textit{Re}$
and whether a subcritical transition can occur. Our DNS show that a subcritical transition can occur in both flow cases at low
$\textit{Ro}$
but not at higher
$\textit{Ro}$
, since in all simulations the flow then fully relaminarises once
$\textit{Re} \lt Re_c$
. We find that at low
$\textit{Ro}$
the flow can become transitional and sustained large-scale turbulent–laminar patterns can develop at sufficiently low
$\textit{Re}$
. These turbulent–laminar patterns can, especially in streamwise-rotating PCFs, form clear band-like structures. In a small
$\textit{Ro}$
range, turbulent–laminar patterns emerge under supercritical conditions when
$\textit{Re} \gt Re_c$
. We have carried out DNS of streamwise-rotating PPF and PCF to show that under such conditions turbulent–laminar patterns can develop from a growing linear instability when the DNS are initialised by a laminar flow with small noise.
Supplementary material
Supplementary material is available at https://doi.org/10.1017/jfm.2025.10723
Acknowledgements
NAISS is acknowledged for providing computational resources in Sweden.
Funding
This research received financial support from the Swedish Research Council through grant number 2021-03967.
Declaration of interests
The author reports no conflict of interest.
Appendix Comparison of LSA and DNS
To validate the LSA we have performed DNS of XPCF and XPPF with small initial perturbations at
$\textit{Ro}=0.05$
,
$0.8$
and
$24$
and
$\textit{Re}$
slightly above
$\textit{Re}_c$
. Figure 11 shows the growth of the root mean square of the velocity fluctuations in the DNS of XPPF and XPCF as well as the growth rate of the most unstable mode predicted by LSA at the same
$\textit{Ro}$
and
$\textit{Re}$
as in the DNS. The computational domain size in all DNS is taken as
$L_x = 4\pi /\alpha$
and
$L_z = 8\pi /\beta$
, where
$\alpha$
and
$\beta$
are the streamwise and spanwise wavenumber of the most unstable mode, as predicted by LSA. The resolution is
$128 \times 97 \times 96$
and
$128 \times 65 \times 96$
in the DNS of XPPF and XPCF, respectively. Figure 11 shows that the DNS and LSA results coincide.
Figure 11. Comparison between the growth of the streamwise velocity fluctuation in DNS of XPPF at (a)
$\textit{Ro}=0.05$
and
$\textit{Re}=725$
(pink line),
$\textit{Ro}=0.8$
and
$\textit{Re}=82.2$
(light blue line),
$\textit{Ro}=24$
and
$\textit{Re}=66.57$
(amber line), and DNS of XPCF at (b)
$\textit{Ro}=0.05$
and
$\textit{Re}=370$
(pink line),
$\textit{Ro}=0.8$
and
$\textit{Re}=31.35$
(light blue line),
$\textit{Ro}=24$
and
$\textit{Re}=20.7$
(amber line), and the growth rate predicted by LSA at the same
$\textit{Ro}$
and
$\textit{Re}$
(dashed lines).
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