Direct numerical simulations of spiral Taylor–Couette turbulence

We perform direct numerical simulations of spiral turbulent Taylor–Couette (TC) flow for $400\leqslant Re_{i}\leqslant 1200$ and $-2000\leqslant Re_{o}\leqslant -1000$, i.e. counter-rotation. The aspect ratio $\unicode[STIX]{x1D6E4}=\text{height}/\text{gap width}$ of the domain is $42\leqslant \unicode[STIX]{x1D6E4}\leqslant 125$, with periodic boundary conditions in the axial direction, and the radius ratio $\unicode[STIX]{x1D702}=r_{i}/r_{o}=0.91$. We show that, with decreasing $Re_{i}$ or with decreasing $Re_{o}$, the formation of a turbulent spiral from an initially ‘featureless turbulent’ flow can be described by the phenomenology of the Ginzburg–Landau equations, similar as seen in the experimental findings of Prigent et al. (Phys. Rev. Lett., vol. 89, 2002, 014501) for TC flow at $\unicode[STIX]{x1D702}=0.98$ an $\unicode[STIX]{x1D6E4}=430$ and in numerical simulations of oblique turbulent bands in plane Couette flow by Rolland & Manneville (Eur. Phys. J., vol. 80, 2011, pp. 529–544). We therefore conclude that the Ginzburg–Landau description also holds when curvature effects play a role, and that the finite-wavelength instability is not a consequence of the no-slip boundary conditions at the upper and lower plates in the experiments. The most unstable axial wavelength $\unicode[STIX]{x1D706}_{z,c}/d\approx 41$ in our simulations differs from findings in Prigent et al., where $\unicode[STIX]{x1D706}_{z,c}/d\approx 32$, and so we conclude that $\unicode[STIX]{x1D706}_{z,c}$ depends on the radius ratio $\unicode[STIX]{x1D702}$. Furthermore, we find that the turbulent spiral is stationary in the reference frame of the mean velocity in the gap, rather than the mean velocity of the two rotating cylinders.

of Esser & Grossmann (1996). The intermittent, spiral turbulence, and 'featureless turbulent' regimes are schematics, indicating the approximate locations of the phases at η = 0.91, similar to the phase diagram at lower radius ratio (η = 0.84) in Andereck, Liu & Swinney (1986). The horizontal and vertical dashed-dotted lines represent the simulations that are performed in this paper. The inset shows the stability boundaries for varying η.

Introduction
The coexistence of spatially and/or temporally intermittent turbulent and laminar flow regions is one of the most captivating phenomena in fluid mechanics (Barkley & Tuckerman 2005;Barkley 2016). In Taylor-Couette (TC) flow, the flow between two independently rotating concentric cylinders, not too far above the onset of instabilities these patterns manifest themselves as distinctive intertwined bands of laminar and turbulent spirals. Although already observed by Coles (1965) and Van Atta (1966), and famously commented on by Feynman (1964), the origin and dynamics of these patterns remain elusive. Figure 1 presents a simplified phase space of TC flow with inner cylinder Reynolds numbers Re i O(10 3 ), (where Re i = u i d/ν, with gap width d, inner cylinder velocity u i and kinematic viscosity ν) and a counter or co-rotating outer cylinder with −3000 Re o 1000 (where Re o = u o d/ν, with outer cylinder velocity u o ). In figure 1, at high inner cylinder Reynolds number Re i , the flow occupies the 'featureless turbulent' state. With decreasing Re i , a coherent, spatio-temporal intermittent, turbulent domain appears -spiral turbulence. On further decreasing Re i , the spiral structure loses coherence and breaks up into intermittent turbulent spots. Below the Taylor stability boundary (Taylor 1923) the flow becomes entirely laminar. The radius ratio dependence of the stability boundary Re i,cr at which the flow undergoes a transition from laminar to intermittently turbulent was derived by Esser & Grossmann (1996) as where the critical Reynolds number for the case of a resting outer cylinder is Re i,lc (η) = (1 + η) 2 /(2ηα 2 ((1 − η)(3 + η)) 1/2 ) with α = 0.1556, and r p = r i + (d/2)f (a(d n /d)), the radius of neutral stability r n = r o ((Re i − ηRe o )/(Re i − η −1 Re o )) 1/2 , d n = r n − r i , a(η) = (1 − η)( (1 + η) 3 /(2(1 + 3η)) − η) −1 and the function f (x) = x if x < 1 and f (x) = 1 if x 1. Equation (1.1) is shown as the blue line in figure 1. Similar diagrams at different η are found in figure 2(a) of Prigent et al. (2002) and figure 1 in Andereck et al. (1986).
1.1. Spiral turbulence The first studies on spiral turbulence in TC flow go back to Coles (1965) and Van Atta (1966), who noticed a 'catastrophic' transition to turbulence if the outer cylinder rotates faster than the inner cylinder. In contrast, for pure inner cylinder rotation they observed a transition by 'spectral evolution', meaning that the complexity of the flow gradually increases with increasing inner cylinder Reynolds number. However, the 'catastrophic' transition does not lead to a 'featureless turbulent' flow directly, but rather forms a state of distinct turbulent and laminar domains, which at specific conditions form regular patterns -spirals. The angular velocity of the spiral was found to be very close to ω s = 2(ω i + ω o ), where ω i and ω o are the inner and outer cylinder angular velocities, respectively. Furthermore, Van Atta (1966) observed strong hysteresis of the spiral turbulence region when approaching the stability boundary from either the 'featureless turbulence' regime or the Couette flow regime. Later, Hegseth et al. (1989) observed that the pitch angle of the spiral is non-uniform, and showed that this fits well into a framework of phase dynamics, with the phase being represented by the mean azimuthal position of the spiral. The boundary conditions at the top and bottom play therein a crucial role.
More recently, the help of direct numerical simulations (DNSs) has led to a further understanding of the fluid flow inside the turbulent structure. Meseguer et al. (2009) discovered that the turbulent spiral originates at the inner cylinder, where vortical structures detach from the wall and spread out radially towards the outer cylinder. For smaller aspect ratios Γ 15 (with Γ = L/d the height of the cylinder divided by the gap width) where no turbulent spiral is formed, turbulent bursts were attributed to a secondary instability mechanism of the laminar flow (Coughlin & Marcus 1996). We note however that laminar spirals do form at low Γ , as they also play a central role in the bursting mechanism (Hamill 1995).
By means of conditional averaging over the spiral turbulence domain, Dong (2009) revealed that a strong angular gradient of the streamwise velocity prevails in the spiral structure. Subsequently, Dong & Zheng (2011) found that the spiral domain consists of elongated vortical structures and that the linearly unstable region of the laminar flow contains vortices with a streamwise vorticity. Furthermore, Burin & Czarnocki (2012) simulated spiral turbulence with a stationary inner cylinder, such that the entire flow is linearly stable. Finally, Barkley & Tuckerman (2007) and Tuckerman & Barkley (2011) studied, with the use of DNS, turbulent bands in transitional plane Couette (PC) flow.

A pattern forming turbulent spiral
A new and remarkable insight into the dynamics of the turbulent spiral came from Prigent et al. (2002). Their experimental observations of stripes in PC flow and spirals in TC flow at very high radius ratio η = r i /r o = 0.98 reveal a turbulence intensity modulation of these flow states that fits in every respect the phenomenology of the (complex) Ginzburg-Landau (GL) amplitude equations (van Saarloos 1994;Cross & Greenside 2009). Thereby, the spiral fits the dynamical behaviour of a finite-wavelength instability, originating from the 'featureless turbulent' state. The GL equation describes the time evolution of a complex amplitude A(x, t) of a physical field variable u(x, t) where τ 0 is the time scale, ξ 0 is the length scale and g 0 sets the magnitude scale of the structure; is the reduced bifurcation parameter = (Re i,c − Re i )/Re i,c , where Re i,c is the critical inner cylinder Reynolds number at which the bifurcation occurs upon reducing Re i and the pattern emerges. By rescaling (1.2), one finds that the intensity of the pattern |A| 2 = O( 1 ). Note that Prigent et al. (2002Prigent et al. ( , 2003 coupled two GL equations to account for the coexistence of spirals with opposing helicity and added a noise term, to account for the turbulent fluctuations in the background velocity field. In principle, for a bifurcation to travelling waves, the coefficients on the right-hand side of (1.2) are expected to acquire imaginary parts, in other words, to be complex. These imaginary parts model the shift of the frequency of the modes with , wavenumber and amplitude. We will not probe these effects here. In fact, as we shall see, the patterns are actually stationary in the frame moving with the mean flow, which indicates that it may be most appropriate to think of the patterns as stationary. This is natural to expect considering that the equations are invariant under continuous translation in the azimuthal and axial directions.
Further work on the GL description of laminar Taylor spirals has been carried out by Goharzadeh & Mutabazi (2010), who measured the GL coefficients. Rolland & Manneville (2011) carried out underresolved simulations of oblique bands in Couette flow. Also, the amplitude in their simulations, defined as the modulus of the first Fourier mode of the streamwise velocity, does obey the phenomenology of the GL model.
In this paper, we set out to investigate spiral turbulent TC flow by means of DNS. In contrast to the experiments of Prigent et al. (2002), which were carried out in the limit of very small curvature (η = 0.98), in our DNSs, curvature effects do play a role (η = 0.91). To quantify this, we refer to the curvature Obukhov length L c = u τ /κω i , with u τ the friction velocity and κ = 0.39 the von Kármán constant, as defined in Bradshaw (1969). This length differentiates the flow regions in a turbulent flow where the production of turbulent kinetic energy is governed by shear and where it is governed by curvature of the streamlines. For values above approximately (r − r i )/L c 0.1, the effects of curvature are pronounced. We find that for η = 0.91, 0.5d/L c ≈ 0.2. For η = 0.98, 0.5d/L c ≈ O(10 −3 ).
Further, we employ periodic boundary conditions in the axial direction, and thereby exclude effects originating from the end plates, which are responsible for the no-slip axial boundary conditions in experiments. We study the fluid mechanics of the turbulent spirals and investigate whether the GL description of the spirals holds in our simulations.
The paper is organized as follows: in § 2 we present the description of TC flow and the details of the numerical solver. In § 3.1 we study the appearance and disappearance of the spiral structure, followed by § 3.2, in which we look at the impact of the turbulent spiral on the global transport of momentum. In § 3.3 we study the velocity of the spiral pattern. Section 3.4 then presents the GL description of the spiral. The paper ends with conclusions ( § 4).

Taylor-Couette flow and numerical procedure
The TC set-up consists of two concentric, independently rotating cylinders with radii r i and r o . The gap width d is defined as d = r o − r i , the radius ratio η = r i /r o and the aspect ratio as Γ = L/d, where L is the height of the cylinders. In this paper we keep the radius ratio fixed at η = 0.91. The aspect ratio Γ varies in the range 42 Γ 125. For every DNS we simulate the full azimuthal circumference of the TC set-up.
The Navier-Stokes equations that govern the shear-driven fluid flow in between the two concentric rotating cylinders are formulated in cylindrical coordinates and dimensionless form, namely The differential operators are defined as: In the axial direction we employ periodic boundary conditions. Here, Re i = r i ω i d/ν and Re o = r o ω o d/ν are the inner and outer cylinder Reynolds numbers, respectively, where by definition ω i 0. Note that the equations can also be written in terms of the Taylor number and a geometric Prandtl number, highlighting the analogy with Rayleigh-Bénard convection (Grossmann, Lohse & Sun 2016). The velocity vector u comprises (u θ , u z , u r ), respectively the streamwise, spanwise and wall-normal velocity, which in this paper are normalized by the inner cylinder azimuthal velocity u θ ,i . Hatted symbols represent dimensionless variables, where velocity, length and time are made dimensionless as, respectively, u = r i |ω i − ω o |û, r = dr and t = (d/(r i |ω i − ω o |))t and pressure is made dimensionless accordingly, P = ρr 2 i |ω i − ω o | 2P . The equations are spatially discretized to second order and are solved on a finite difference grid (Verzicco & Orlandi 1996). Time integration is performed with a fractional-step third-order Runge-Kutta scheme. For more details of the numerical code we refer the reader to van der Poel et al. (2015) and Ostilla-Mónico et al. (2013). Grid independence checks are carried out to ensure sufficient numerical resolution. Time convergence is controlled by monitoring the torque on both cylinders, which is expressed in dimensionless form as the Nusselt number Nu ω , is the angular velocity flux, . A(r),t represents averaging over a cylinder surface A(r) and time t, is the laminar angular velocity flux and T is the torque.
Alternatively to Nu ω , we define the friction factor C f , ( 2.6) where G = T /(ρν 2 L) is the dimensionless torque; C f is thus the ratio of the wall stress over the kinetic energy of the flow.

Results and discussion
3.1. Turbulent spirals: their formation and disappearance When the destabilizing effects of inner cylinder rotation become large compared to the stabilizing effects of outer cylinder rotation, TC flow can enter the 'featureless turbulent' regime, see figure 1. Although the flow is turbulent throughout the domain, the term 'featureless turbulence', as mentioned in Andereck et al. (1986), is deceiving, since the mean flow does contain structures with streamwise vorticity, i.e. the turbulent Taylor vortex (Huisman et al. 2014).
We run a DNS of the 'featureless turbulent' regime, with Re i = 800 and Re o = −1200, of which three snapshots of the streamwise/azimuthal velocity at varying radii are presented in figure 2 (a-c). As initial conditions for all our simulations we use the laminar solution of TC flow (Grossmann et al. 2016) plus small spatial perturbations to the radial velocity component u r = 0.1r i ω i sin(2πz/L), with z the axial coordinate and the axial velocity component u with ν the mean kinetic dissipation rate. The spacing of the finite size grid at the wall in the radial direction is 0.44y 0 , where y 0 is the viscous length scale y 0 = ν/ √ τ w /ρ, with ν the kinematic velocity, ρ the fluid density and τ w the wall shear stress. With varying Γ , we change N z = 1280(Γ /64) while N θ × N r = 768 × 80 for all simulations.
In figure 2 we observe long, thin, meandering patterns in the azimuthal velocity u θ . The structures have a length scale ≈ (0.5 − 1.0)d in the axial direction and are more distinct close to the inner cylinder at r = r i + 0.25d, whereas they become more diffuse closer to the outer cylinder at r = r i + 0.75d. The nodal plane of neutral stability (defined by the laminar azimuthal velocity being zero u θ,lam = 0), is at r n = r o ((Re i − ηRe o )/(Re i − η −1 Re o )) ≈ 1.04r i for Re i = 800 and Re o = −1200, such that the inner region of the domain r < 1.04r i is linearly unstable and the outer region of the domain r > 1.04r i is linearly stable (Lord Rayleigh 1916). This is confirmed in figure 3(a), where we present contours of the instantaneous azimuthal velocity u θ = u θ (θ , z, r, t) in a small portion of the axial-radial plane at Re i = 800 and Re o = −1200. We find vortical structures, with a streamwise vorticity and length scale ≈ (0.5 − 1.0)d close to the inner cylinder. These structures resemble Taylor vortices, although they do not close the entire circumferential direction of the domain, see figure 2. The region r > r n does exhibit chaotic fluid flow motion, triggered by the instabilities from the inner region. This is reminiscent of the so-called inner-outer region interaction as described by Coughlin & Marcus (1996). As such, we find that, for Re i = 800, the entire domain is filled with chaotic (turbulent) fluid motion, and hence falls into the 'featureless turbulent' part of the above phase space, see figure 1.
Starting from this state (figure 2a-c), we decrease Re i to Re i = 750 (figure 2d-f ), while keeping Re o = −1200. We run the simulations until the flow is statistically stationary, i.e. when the dimensionless torque Nu ω t , calculated at both cylinders, is constant to within 1 %, where . t indicates time averaging over 50t. Resulting from the decrease in shear, the flow starts to laminarize at the outer cylinder, see figure 3(b). However, the flow remains turbulent throughout the majority of the domain, as seen in figure 2(d-f ). Interestingly, we find in these turbulent parts of the flow, in the contours  FIGURE 2. Snapshots of the azimuthal velocity u θ , close to the inner cylinder (r = r i + d/4), at the centre (r = r i + d/2) and close to the outer cylinder (r = r i + 3d/4) for Re o = −1200 and for different Re i = 800 (a-c), 750 (d-f ), 700 (g-i) and 524 ( j-l). The horizontal axis gives the angle θ . On the vertical axis the axial coordinate z is normalized with the gap width. From a chaotic turbulent base flow (a-c), the finite-wavelength instability forms (panels d-f and in particular g-i). Further away from the transition an isolated stripe (spiral) breaks down in connected and isolated turbulent spots ( j-l).
of u θ (θ , z, r, t), the appearance of diagonal coherent patterns. At this Re i , the patterns exist at opposing angles and 'nucleate' at varying places.
Subsequently, we further lower Re i to Re i = 700 (figure 2, g-l) for which one welldefined turbulent spiral pattern emerges. At r = r i + 0.75d, the spiral contains turbulent motion, whereas the region in between the turbulent spiral bands is laminarized and contains no vorticity, see figure 3(c). Note that the regions of maximal intensity of the banded structure is shifted to the right somewhat as one moves outwards (this is most clearly visible for Re i = 700). This is in agreement with the results of Meseguer et al. (2009) for the structure of turbulent bands. Close to the inner cylinder, the spiral also contains turbulent motions, however, the region in between the turbulent spiral bands does contain Taylor-like vortices. Figure 3(c) presents a section of the turbulent spiral corresponding to 0 z/d 3 in figure 2 (third row). For an extensive range of 530 Re i 700, the spiral remains present in the flow. Lowering Re i even further, e.g. to Re i = 524 (see figure 2j-  focus of this study. In the remainder, we will study the dynamics in that part of the phase space where the spiral is still coherent.

Dynamics of the turbulent spiral
The influence of the turbulent spiral structure on the momentum transport is not addressed in the literature. In this section we will study the global response for both varying spiral wavelength λ z and varying Re i , while we keep Re o = −1200. To vary λ z , we simulate 20 cases of varying aspect ratio 42 Γ = L/d 125 of the TC set-up. In the axial direction we employ periodic boundary conditions, such that λ z = Γ /n, with n a positive integer that represents the number of windings of the spiral around the inner cylinder. We find a range of 26 λ z /d 45, whereby up to three spirals fit into the domain for a given Γ . When we examine the mean velocity and the mean turbulent intensity, we do not find any influence of Γ on the flow for fixed λ z /d. In other words, the flow at Γ = 84 with two spirals of λ z /d = 42 is statistically identical to the flow at Γ = 42 with one spiral of the same λ z /d = 42. There is, however, a strong dependence of the global angular momentum transport -expressed in dimensionless form in (2.5) -on the axial wavelength of the spiral. Figure 4(a) presents Nu ω versus λ z /d for varying Re i . With increasing λ z /d, the turbulence fraction in the domain decreases, resulting in a predominantly diffusive, less efficient, transport of momentum. Figure 4(b) presents the corresponding C f versus λ z /d, see (2.6).
The graphical representation of C f (Re i ) is commonly referred to as the 'Moody' diagram (Moody 1944). Whereas the laminar part of the diagram can be derived analytically, the turbulent part is empirically fitted with the celebrated Prandtl's friction law. For inner cylinder rotation, linearly unstable, TC flow, C f (Re i ) is monotonically decreasing in the transition region between the laminar and turbulent flow (Lathrop, Fineberg & Swinney 1992). For counter-rotating TC flow, where the transition to turbulence is subcritical and 'catastrophic', we find that C f (Re i ) is increasing in the transitional region, see figure 4(c). This is reminiscent of the transition scenario in pipe flow, where the sudden appearance of spatio-temporally intermittent turbulence leads to an increase in C f (Re) (Pope 2000).
3.3. Angular velocity of the spiral Surprisingly, the angular velocity of the spiral ω s has hitherto received only little attention in literature. Van Atta (1966) was the first to investigate ω s and found that it scales with the mean angular velocity of the two cylinders ω s ≈ 1 2 (ω i + ω o ) for The time dependent signal. To obtain the converged amplitude, we employ time averaging and we average over spatial coordinates (r i + d/4) < r < (r i + 3d/4) and for all z. This particular signal is acquired for Re i = 660 and Re o = −1200.
position of the maximum turbulence intensity in the spiral and from the translation of that position in time extract ω s , see figure 5. We find that ω s is not equal to the mean rotation rate, 1 2 (ω i + ω o ), but instead equals the mean angular velocity in the domain ω = ω θ ,z,r,t . Note that the difference between ω and ω s is only minor and could easily be missed in experiments. In fact, we think that the consistent mismatch between ω m and ω s , as found in figure 5 in Prigent & Dauchot (2000), is explained by the mismatch between ω m and ω .

Amplitude modulation 3.4.1. The amplitude
To describe the turbulent spiral as a pattern forming instability above a critical bifurcation point = 0, we introduce a perturbation A(z, t)e (ikz−iωt) to the base flow u b . Here, we treat the instability in the axial coordinate direction only. Note that the instability contains both parity symmetry (flipping of the streamwise or axial coordinate directions) and translational symmetry. We define the perturbed physical field u as the root mean square (r.m.s.) of the radial velocity component (in contrast to Prigent et al. (2002), who define u as the r.m.s. of the axial velocity). Figure 6(a) exhibits the instantaneous u r,rms over a line encircling the inner cylinder at r = r i + 0.5d and z = L/d. We define the amplitude from the radial component of velocity u r as A = ( u 2 r ) max − ( u 2 r ) min which is a function of (r, z; t). We calculate A at runtime and average over half the gap width (r i + d/4) < r < (r i + 3d/4) and the full height z. Figure 6(b) presents the time signal of the spatially averaged squared amplitude A 2 z,r . Significant fluctuations of A 2 z,r force us to also take long averages ofτ av ≈ 50. The width of the turbulent spiral s is approximately (r i + 0.5d)π.
We measure the amplitude versus the bifurcation parameter at a fixed aspect ratio Γ = 64, and hence a fixed λ z /d = 32. For a stationary, finite-wavelength instability, for which the modulation of the amplitude is described by (1.2), we expect (for different η = 0.98) extracted a critical Re i,c = 857 ± 5 at which the instability occurs. Figure 7(a) shows that we also observe A 2 ∝ 1 over a range of , close to the bifurcation point. With Re o = −1200 we obtain Re i,c = 863, in very close agreement with Prigent et al. (2002), in spite of different η. For Re i > 780, nucleation of domains of opposing helicity, and the appearance of turbulence in the laminar spiral regions, obscures the precise measurements of the amplitude of the turbulence intensity signal. Prigent et al. (2002) were still able to extract the amplitude for these Re due to the very long runtime in the experiments, which at these Γ are not accessible for DNSs. As for Re i = 770, the amplitude is still 0.04, we conclude that the noise term is certainly high.
In a similar manner, we also approach the boundary of the spiral turbulent regime in figure 1 from the horizontal direction, i.e. by varying Re o and maintaining Re i = 680. For decreasing Re o , the stabilizing stratification of centrifugal pressure increases, the flow laminarizes and the spiral emerges. As such, we define = (Re o,c − Re o )/Re o . Figure 7(b) convincingly indicates that, also for varying Re o , A 2 ∝ 1 , with Re o,c = −713. This indicates that the turbulent spiral behaves as a finite-wavelength instability, in spite of the curvature effects.

The instability diagram
By carefully varying the aspect ratio Γ we are able to study a variation in the axial wavelength λ z of the spiral in the domain 26 λ z /d 45. Considering that we do probe the amplitude of a stationary turbulent spiral versus the wavelength, and the influence of the wavelength of the momentum transport, we do not imply that for a certain aspect ratio only one such wavelength exists. In fact, it is most likely that the wavelength of the spiral is sensitive to initial conditions, as also the wavelength of the Taylor-Vortex is. With axially periodic boundary conditions, we do not find any Reynolds number dependence of λ z , neither on Re i nor on Re o , since λ z = Γ /n. In contrast, such a dependence was experimentally observed by Prigent  Note that for large aspect ratios multiple spirals exist. Therefore, we cannot simulate smaller wavenumbers than indicated in the graph. We observe consistent behaviour of the amplitude with increasing , following the phenomenology of a finite-wavelength instability. We fit a second-order polynomial through the data. From this fit we obtain the most unstable wavelength for each thus five values in total. We obtain 38.91d < λ c < 42.52d with mean(λ c ) = 41.38d and var(λ c ) = 1.74d. As we do not observe a systematic trend in the difference between the data and the fit, but rather find that the error is of similar order for all wavenumbersk, we conclude that a parabolic fit is justified.
(2000) for no-slip boundary conditions at the plates and a much larger system Γ = 430. Figure 8 presents the amplitude versus the wavenumber k = 2π/λ z for increasing . Note that Re i and Re o are identical to those in figure 4, except for Re i = 720, which is excluded here due to the very high noise originating from the 'featureless' turbulent flow.
The parabolic shape, with a maximum around λ z,c /d = 41 ± 2, represents the characteristic band of unstable wavenumbers for a finite-wavelength instability above onset. Thereby, we do observe the dependence of A on the bifurcation parameter , such that an increasing band of wavenumbers becomes unstable with increasing . We conclude that the most unstable wavelength λ z,c /d ≈ 41 for our simulations at η = 0.91 differs from the most unstable wavelength λ z,c /d ≈ 32 found in Prigent et al. (2002) at η = 0.98.
In knowing that the amplitude in figure 8 is described by A 2 = ( − ξ 2 0 (k − k c ) 2 )/g 0 , we can also extract the interaction strength coefficient g 0 = /max k (A 2 ) and the coherence length coefficient ξ 0 from the fits for varying . We find that g 0 = 2.80 ± 0.11 and does not depend on , i.e. it does not depend on the magnitude of the noise, as also found in Prigent et al. (2003). The curvature we find at k c (i.e. ξ 2 0 /g 0 ≈ 1.5) is comparable to the values observed by Prigent et al. (2002) (i.e. ξ 2 0 /g 0 ≈ 0.5), but unfortunately our data are not precise enough to extract the coherence length ξ 0 of the pattern with sufficient accuracy to study the trends with . It is interesting to note that, in doing so, we are in principle able to extract properties of the full amplitude and pattern (like ξ 0 ) within the GL framework from an analysis of a single mode with fixed wavenumber (the Landau-Hopf or 887 A18-14

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Stuart-Landau framework) induced by varying the aspect ratio, and that the resulting coherence length is much larger than our system size.

Summary and conclusions
In conclusion, we performed direct numerical simulations of counter-rotating Taylor-Couette flow at 400 Re i 1200 and −2000 Re o −1000. For the aspect ratio Γ of the domain 42 Γ = L/d 125 with periodic boundary conditions in the axial direction, and the radius ratio η = r i /r o = 0.91. In this regime we found the coexistence of spatio-temporal intermittent laminar and turbulent domains, commonly referred to as spiral turbulence (Coles 1965;Van Atta 1966).
The formation of the turbulent structure (i.e. spiral in TC) out of a turbulent base flow is similar to the phenomenology of the (complex) Ginzburg-Landau equations with a noise term (Prigent et al. 2002), in the limit of very low curvature (η = 0.98). With these fully resolved simulations, we showed that the GL phenomenology also holds for the spirals when curvature effects do play a role. We found that the GL description for pattern formation holds close to the bifurcation point for both varying Re i and varying Re o . This, once more, suggests the existence of a finite-wavelength instability in a fully turbulent flow.
Also, we found that the pattern is stationary in the reference frame of the mean angular velocity in the domain ω s = ω . This is in contrast to findings by Prigent & Dauchot (2000) and Andereck et al. (1986), who experimentally found that the spiral moves with the mean velocity of the two rotating cylinders, i.e. ω s = 1 2 (ω i + ω o ). However, note that the small difference between ω and 1 2 (ω i + ω o ) is hard to observe in experiments at these low Reynolds numbers.
In contrast with Prigent et al. (2002) we have periodic boundary conditions in the axial direction. As such, we only allow a wavelength which is an integer division of the aspect ratio Γ . It is therefore very likely that we do not observe a Reynolds number dependence of λ z . In their experiment, Prigent et al. (2002), however, employ no-slip boundary conditions on the vertical axis, and as such, allow for any wavelength to exist. Despite the different boundary conditions we find strong agreement in the critical Reynolds number at which the instability occurs (i.e. Re i,c = 863 for our DNS and Re i,c = 857 for the experiments in Prigent et al. (2002)) as we also find similar values of the interaction strength ξ 2 0 ≈ 1 for both. Whereas we obtain k c by altering the aspect ratio, and Prigent et al. (2002) from analysis of monodomain regions in the flow, we both find a parabolic trend of the amplitude with k, where k c does not appear to depend on Re.
The most unstable wavelength of the instability is found to be λ z,c ≈ 41d, thereby differing from findings in plane Couette flow and in TC flow, where it is λ z,c ≈ 32d at very high η = 0.98. Apparently, λ z,c is a function of the radius ratio η. This finding may be an important clue for further theoretical investigations, and it may point the way towards understanding the modulated turbulent states in terms of a stationary bifurcation.
Whereas a formal derivation of the instability from the turbulent base flow seems hopeless, since it requires a closed description of the turbulence, fully resolved simulations help to uncover the characteristics and ingredients of the instability when it appears or disappears in the flow. Future work could continue on this road, by e.g. studying the response of the structure to abrupt changes in the boundary conditions, or by studying the effects of initial conditions on the turbulent spiral.