Laminar-turbulent coexistence in annular Couette flow
Kohei Kunii, Takahiro Ishida, Yohann Duguet, Takahiro Tsukahara
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Laminar-turbulent coexistencein annular Couette flow
Kohei Kunii , Takahiro Ishida † , Yohann Duguet ,and Takahiro Tsukahara ‡ Department of Mechanical Engineering, Tokyo University of Science, Chiba 278-8510, Japan LIMSI-CNRS, Universit´e Paris-Sud, Universit´e Paris-Saclay, F-91405 Orsay, France(Received xx; revised xx; accepted xx)
Annular Couette flow is the flow between two coaxial cylinders driven by the axialtranslation of the inner cylinder. It is investigated using direct numerical simulationin long domains, with an emphasis on the laminar-turbulent coexistence regime foundfor marginally low values of the Reynolds number. Three distinct flow regimes aredemonstrated as the radius ratio η is decreased from 0.8 to 0.5 and finally to 0.1. Thehigh- η regime features helically-shaped turbulent patches coexisting with laminar flow,as in planar shear flows. The moderate- η regime does not feature any marked laminar-turbulent coexistence. In an effort to discard confinement effects, proper patterning ishowever recovered by artificially extending the azimuthal span beyond 2 π . Eventually,the low- η regime features localised turbulent structures different from the puffs com-monly encountered in transitional pipe flow. In this new coexistence regime, turbulentfluctuations are surprisingly short-ranged. Implications are discussed in terms of phasetransition and critical scaling.
1. Introduction
Most wall-bounded shear flows possess a linearly stable base flow for parameters whereturbulence can also be observed. This is particularly true near the onset of turbulence,i.e. near the smallest Reynolds number Re g , ‘g’ for global, where turbulence is sustained.There is no contradiction between linear stability of the laminar base flow and theexistence of a turbulent regime, since the latter can be reached from well-chosen finite-amplitude perturbations to the base flow (Orszag 1971; Schmid & Henningson 2001).Such turbulent flows are called subcritical, and there is no general method to identifythe value of Re g . A large number of studies have demonstrated that turbulent regimesnear Re g are intermittent in space and time. At every instant in time they consist oftrains of localised coherent structures such as turbulent puffs in pipe flow (Wygnanski& Champagne 1973; Wygnanski et al. et al. et al. et al. a ; Fukudome& Iida 2012; Seki & Matsubara 2012).A turbulent puff is a turbulent region localised in the axial direction of the pipe.A mechanism for the self-sustenance of this coherent structure has been identified byShimizu & Kida (2009). However puffs do not persist indefinitely. As demonstrated byHof et al. (2006), puffs have finite lifetimes as long as they keep their localisation. Theonset Reynolds number Re g is determined for the case of pipe flow as the value of Re † Present address: Japan Aerospace Exploration Agency, Tokyo 182-8522, Japan ‡ Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] O c t K. Kunii, T. Ishida, Y. Duguet, and T. Tsukahara u w R i L (cid:1) L r = hR o L x x,u x r, u r (cid:1) , u (cid:1) Figure 1.
Configuration of annular Couette flow, with notations. at which their decay rate and proliferation rate exactly balance each other (Avila et al. ±
10 when Re is based on the flow rate and the pipe diameter.Turbulent stripes are large-scale coherent structures emerging spontaneously near Re g in a variety of planar wall-bounded flows such as plane Couette flow (pCf) with orwithout rotation (Barkley & Tuckerman 2007; Tsukahara et al. a ; Duguet et al. a ; Tsukahara et al. b ; Brethouwer et al. Re g ≈ ± Re ,from either experimental lifetime measurements (Bottin et al. et al. a ). Such structures were initially reported in counter-rotating regimes of the Taylor-Couette flow (TCf) (Coles 1965; Prigent et al. Re g for pipeflow can be described as one-dimensional, patterning in pCf is more genuinely two-dimensional. Establishing universal links between all the different sustained coherentstructures (puffs, stripes) in such flows is an ambitious task. By noting that these coherentstructures are (turbulent) steady states, one can borrow from bifurcation theories theconcept of continuation , i.e. a continuous deformation from one flow case to another one,hoping that there is indeed at least one path in parameter space which continuouslylinks one type of solution (here stripes) to the other one (puffs). If such a path existsthen puffs and stripes can be considered members of a unique family of subcritical flows.Continuation methods have shown to be useful for the tracking of nonlinear steady statesfrom one model problem to another. The first such instance was considered by Nagata(1990) who deformed tertiary solutions of TCf into non-trivial steady states of pCf. Inthe same spirit, Faisst & Eckhardt (2000) suggested a criterion to assess whether andwhen the curvature of TCf can be considered geometrically negligible.In this article, we focus on a new candidate flow for continuation between pipe flow puffsand plane Couette stripes : the annular Couette flow (aCf). It refers to the flow betweentwo infinitely long coaxial cylinders, driven by the axial translation of the inner rod withconstant velocity u w >
0. A sketch is displayed in figure 1. This flow is an idealisedacademic configuration inspired by thread-annular flows (Frei et al. η = R i /R o (where R i and R o denote respectively the innerradius and outer radius). The radius difference h = R o − R i , represents the gap betweenthe two cylinders. As η →
1, the wall curvature vanishes and the flow is asymptoticallyequivalent to pCf. At the other end, for η →
0, the geometry does not become a cylindricalpipe because of the presence of the inner rod (associated with no slip). There is hence aminar-turbulent coexistence in annular Couette flow h = 0.8 h = 0.5 h = 0.1 h = 10 –2 h = 10 –3 h = 10 –4 h = 10 –5 y U base ( y ) Figure 2.
Laminar velocity profiles of annular Couette flow at several radius ratios. The valuesof η investigated in this article ( η = 0 .
8, 0.5, and 0.1) are the three top-most curves (blackonline), while smaller values of η down to 10 − are displayed using a clearer font (blue online).See § y . no possible continuation from aCf to pipe flow. Besides the difference in topology, thevelocity distributions of pipe flow and aCf in the vanishing η limit strongly differ. Thisis readily seen by analysing the analytical laminar solutions shown in figure 2. For η → η → Re = u w h/ ν , where ν is the kinematic viscosity of the fluid. That definitionis consistent with the pCf limit as η approaches 1.A number of stability investigations have focused on the linear stability of the baseflow as a function of η and Re . According to Gittler (1993), aCf is linearly stable againstaxisymmetric perturbations for η > . Re > ) for η < . η and for values of Re much below the linear stability threshold. Byconsidering non-axisymmetric perturbations and tracking numerically exact nonlineartravelling waves, Deguchi & Nagata (2011) determined the values of Re g of 255.4, 256.6,and 288.6 for η = 1 − , 0.5, and 0.1, respectively. To our knowledge, no study has trackedturbulent coherent structures for fixed η all the way towards their extinction at Re g .It is the goal of the present paper to describe the distinct representative regimes oflocalised turbulence in aCf near Re g . The mechanisms that can take aCf away from pCffor decreasing η are twofold : either azimuthal confinement prevents large-scale coherentstructures from forming, or the wall curvature is such that the coherent structuresthemselves change form. These two mechanisms, though both linked to finite η , aredifferent. We take inspiration from a cousin of aCf, namely aPf (annular Poiseuille flow),the flow occurring in the same geometry, yet with a non-moving inner rod and a pressuregradient applied against the axial direction. The different transitional regimes of aPfwere described by Ishida et al. (2016). Three types of coherent structures were identifieddepending on the value of η , all sketched in figure 3 : helical stripes for η close to butbelow unity, straight puffs for low enough η , and an intermediate state occurring near η ≈ . et al. (2017) performed a statistical analysis of the transition from puffs to stripesas η is varied, by investigating statistically the large-scale flows present at the laminar- K. Kunii, T. Ishida, Y. Duguet, and T. Tsukahara (a) ≈ 25 h — hh Flow (b)
Flow (c)
Flow ≈ 25 h Figure 3.
Typical transitional structures in annular flow geometries depending on η , accordingto Ishida et al. (2016). Blue regions represent localised turbulence. The aspect ratio (gap vs streamwise length) is changed for each figure. (a) helical turbulence for high η , (b) helical pufffor intermediate η , and (c) straight puff for low η . turbulent interfaces. They revealed a statistical cross-over from straight puffs (figure 3c)to oblique laminar-turbulent interfaces (figure 3b), for radius ratios η ≈ . h , and by expressing alllengths in units of h , “large-scale” flows can only be accommodated inside the gap for η close enough to unity. The common geometry to aCf and aPf leads to the expectationthat transitional structures occurring in aCf are similar to those found in aPf. To thisend we have chosen to focus on the transitional regimes occurring for the same values of η = 0.8, 0.5 and 0.1. For each case, we report whether or not turbulent stripes form as inpCf. In the absence of stripes, the mechanical reason for it, either azimuthal confinementor wall curvature, will be analysed. As will be described throughout the paper, the planarpicture is valid for large η ≈
1, where the flow stays similar to pCf, but it breaks downfor lower η (cid:54) . §
3, the intermittent regimes are describedfor three different values of η = 0 .
8, 0.5, and 0.1 in decreasing order, with a focus onwhether the flow patterns are influenced by the azimuthal confinement or not. Section 4contains temporal statistics of the flow field as well as measures of intermittency. Finallythe universal features of transitional aCf are discussed in the concluding § aminar-turbulent coexistence in annular Couette flow
2. Governing equations and numerical methods
Governing equations
We consider an incompressible Newtonian fluid flow between two coaxial cylinders.The outer cylinder of radius R o is fixed whereas the inner cylinder of radius R i moves inthe axial direction x with velocity u w >
0. The gap has size h = R o − R i . The quantities h and u w are used to non-dimensionalise the governing equations. The flow configurationis sketched in figure 1. The non-dimensional coordinates r and θ are, where necessary,converted respectively into y = r − r i and z = rθ where r i (cid:54) r (cid:54) r o , with r i = R i /h and r o = R o /h . The non-dimensional velocity field u ≡ ( u x , u r , u θ ) in cylindrical coordinatesand the pressure field are governed by the incompressible Navier–Stokes equations : ∇ · u = 0 , (2.1) ∂ u ∂t + ( u · ∇ ) u = −∇ p + 1 Re ∇ u , (2.2)where p stands for the reduced pressure field. The fluid density is unity.The Reynolds number is defined as Re ≡ u w h/ ν , with ν the kinematic viscosity of thefluid. The total wall shear stress is linked to the friction Reynolds number Re τ , definedas Re τ = u τ h ν , (2.3)It is based on the friction velocity u τ evaluated at each wall using the formula u τ = ηu τ, inner + u τ, outer η + 1 , (2.4)where each u τ is defined by the corresponding wall shear stress τ w and by the relation τ w = ρu τ , from which inner units can be defined (see table 1).The boundary conditions are periodicity in the streamwise ( x ) and the azimuthalcoordinates ( θ ), and no-slip at the walls : u = (1 , ,
0) at r = r i (or y = 0) , (2.5) u = (0 , ,
0) at r = r o (or y = 1) , (2.6) u (0 , r, θ ) = u ( L x , r, θ ) , (2.7) u ( x, r,
0) = u ( x, r, nπ ) . (2.8)In the last boundary condition (2.8) expressing the azimuthal periodicity, the valueof n is usually n = 1. Choosing n as a fraction of the form 1 /m , with m a positiveinteger, would correspond to imposing an additional discrete azimuthal symmetry. In §
3, we will take the original choice of considering n as a positive integer such as n = 1 , , , ... . Such a boundary condition leads to an azimuthal coordinate ranging from0 to 2 πn , which for n > U base ( y ) = ln [ y (1 − η ) + η ]ln η , (2.9)interpreted as the laminar base flow. For η → U base converges in a non-singular way K. Kunii, T. Ishida, Y. Duguet, and T. Tsukahara η Re →
390 750 → → Re τ → → → R i , R o h , 1.11 h h , 2 h h , 5 hL x × L r × L θ . h × h × π . h × h × π . h × h × πL zi , L zo h , 6.98 h h , 12.6 h h , 31.4 hN x × N r × N θ × ×
256 1024 × ×
256 1024 × × ∆x . h . h . h∆r . h –0 . h . h –0 . h . h –0 . h∆z i , ∆z o . h , 0 . h . h , 0 . h . h , 0 . h∆x + ∆r + ∆z + Table 1.
Computational conditions for aCf with nominal azimuthal extent L θ = 2 π : radiusratio η , Reynolds number Re , friction Reynolds number Re τ , inner radius R i , outer radius R o ,computational domain dimensions L x × L r × L θ , inner perimeter L zi , outer perimeter L zo ,number of grid points N x × N r × N θ , and range of values for the increments ∆j ( j = x, r, θ (or z )). The superscript ( + ) denotes the normalisation by wall units based on u τ and ν . towards the linear profile U base ( y ) | η =1 ∼ (1 − y ) which is consistent with the pCf limit.Figure 2 displays profiles of U base ( y ) parameterised by η ranging down to 10 − , for whichthe profile unambiguously differs from that in cylindrical pipe flow.2.2. Numerical tools
Equation (2.2) is discretised in space using finite differences. The time discretisationis carried out using a second-order Crank–Nicolson scheme, and an Adams–Bashforthscheme for the wall-normal viscous term and the other terms, respectively. Further detailsabout the numerical methods used here can be found in Abe et al. (2001).The tested radius ratios are 0.8, 0.5, and 0.1, and the other numerical parametersare shown in table 1. The number of grid points in the azimuthal direction is 256 for η = 0 . η = 0 . h . The strategy to isolate thetransitional regimes of aCf is similar to that used in aPf. First, using an arbitrary finite-amplitude initial condition, a turbulent flow is computed at large enough Re for whichno laminar-turbulent coexistence is expected. In a second phase, Re is decreased in smallsteps until the flow can be considered statistically steady based on energy time series.Visual inspection of the flow fields at mid-gap is used to decide whether or not the flowdisplays laminar-turbulent coexistence.
3. Morphology of coherent structures
High η = 0 . η = 0 .
8, for which the curvature is weak and aCf is expected to behave like planeCouette flow (pCf). Figure 4 shows the wall-normal velocity of a typical flow field for η = 0 . Re = 350. The unfolded plane of two-dimensional contours in figure 4b liesat mid-gap ( y = 0 .
5) where the corresponding spanwise width is 28 . h . Measurements ofthe friction Reynolds number yield Re τ = 25 .
2. Expressed in wall units, the domain size aminar-turbulent coexistence in annular Couette flow (a) h Flow (b) (2 (cid:1) ) (3 (cid:1) /2) ( (cid:1) ) ( (cid:1) /2) z/h ( (cid:1) )
0 10 20 30 40 50 60 70 80 90 100 x/h u r (cid:1) −0.03+0.03 ◦ ❑ Figure 4.
Instantaneous flow field of aCf for η = 0 . Re = 350, accompanied byhelical turbulence. (a) Three-dimensional visualisation by iso-surfaces of wall-normal velocityfluctuations ( u (cid:48) r = − .
03, blue; u (cid:48) r = 0 .
03, red); (b) two-dimensional contour of u (cid:48) r in the x - z (or x - θ ) plane at mid-gap. The pattern propagates steadily towards x > visualised in figure 4b is equivalent to L + x × L + z = 5160 × ◦ with respect to the x axis. The laminarstreamwise intervals have lengths of about 50 h , consistently with those in pCf (Prigent et al. et al. a ; Brethouwer et al. et al. Re , the helical turbulence sustains until Re = 337 . Re τ = 24 .
0) and eventually decaysat 325. This scenario and the estimated value for Re g are fully consistent with those forpCf, despite the curvature of the walls.3.2. Moderate η = 0 . Nominal parameters
Let us now focus on the moderate η regime by analysing the case η = 0 .
5. Note thatin the context of aPf, the transitional regime for η = 0 . η = 0 .
8. The analysis in Ishida et al. (2017) shows that for η > η c ≈ .
3, the perimeteris sufficiently large (in units of h ) for large-scale flows to form, and as a consequencelocalised turbulence can only take the shape of helical puffs as in figure 3b or helicalstripes for even higher η . At these marginal Reynolds numbers, the smallest vorticalstructures occupy the whole gap and h is hence the convenient reference lengthscale. The K. Kunii, T. Ishida, Y. Duguet, and T. Tsukahara (a) h Flow (b) z/h ( (cid:1) )
0 10 20 30 40 50 60 70 80 90 1009.4 (2 (cid:1) ) ( (cid:1) ) x/h u r (cid:1) −0.03+0.03 Figure 5.
Same as figure 4 for η = 0 . Re = 387 .
5, featuring laminar patches. fact that h defines the order of magnitude for the size of the smallest lengthscales, hasbeen suggested as the condition defining marginality (Alfredsson & Matsubara 2000).Whereas helical stripes prevail for η = 0 .
8, they are absent from the simulationsat η = 0 .
5. A typical flow regime obtained for Re = 387 . . h at mid-gap, which is shorter than the(intrinsic) laminar interval of turbulent bands. As a result, despite a furtive occurrence,proper helical turbulent bands cannot sustain as organised structures. Some disorganisedlaminar patches can be observed in practice near Re = 380 ( Re τ = 27 . Re further, the turbulent regions become intermittent in the streamwise direction but noband-like or puff-like structure forms. The flow eventually relaminarises at Re = 377 . η = 0 . Azimuthally extended system
As mentioned in the introduction, two intermingled factors can influence the formationof large-scale structures in comparison with the planar case : the azimuthal extent andthe wall curvature. We use a numerical trick to test whether the obtained flows persistin the absence of azimuthal confinement without modifying the wall curvature, namelyby simulating the flow in a domain where the azimuthal variable describes the range(0 , nπ ) with n >
1. The hypothesis L θ > π should be seen as a deliberate validationtechnique rather than as the introduction of an esoteric parameter.The strategy to seek transitional regimes featuring laminar subdomains is similar tothat for L θ = 2 π . For the case η = 0 .
5, we have tried several integer values of n untilvisual inspection of the flow fields reveals some clear changes with respect to n = 1. Thecase n = 4 corresponds to a periphery of L θ = 8 π , i.e. 4 nominal peripheries, and theperimeter values at the inner wall, at mid-gap and at the outer wall are ( L zi , L zc , L zo ) =(25 . h, . h, . h ), respectively. The number of azimuthal grid points is increased to N θ = 512, hence doubled in resolution with respect to n = 1. In other words there wasa trade-off in the numerical resolution, resulting probably in a quantitative shift of theReynolds numbers. The extended azimuthal grid spacings on the outer and inner walls aminar-turbulent coexistence in annular Couette flow (a) (8 (cid:1) ) (6 (cid:1) ) (4 (cid:1) ) (2 (cid:1) ) z/h ( (cid:1) )
0 10 20 30 40 50 60 70 80 90 100 x/h u r (cid:1) −0.03+0.03 (b) (8 (cid:1) ) (6 (cid:1) ) (4 (cid:1) ) (2 (cid:1) ) z/h ( (cid:1) )
0 10 20 30 40 50 60 70 80 90 100 x/h u r (cid:1) −0.03+0.03 ◦ ✠ Figure 6.
Simulations with artificial azimuthal extension L θ = 8 π . Contours show instantaneouswall-normal velocity fluctuations u (cid:48) r in the x - z (or x - θ ) plane at mid-gap for η = 0.5. These flowsare in a statistically steady state. (a) Re = 387 .
5, irregular laminar patches; in (b), Re = 350,helical turbulence. are, respectively, ∆z = 0 . h and 0 . h , which remain adequate resolutions whenexpressed in inner units since they verify ∆z + (cid:54) x - z (or x - θ ) plane at mid-gap. At Re = 387 .
5, laminar patches can be found, and there areno strong differences between the cases L θ = 2 π (figure 5b) and L θ = 8 π (figure 6a). Byreducing Re further down to 350, helical turbulence occurred for L θ = 8 π (figure 6b),whereas the flow was fully laminar at the same value of Re for L θ = 2 π . The pitchangle α θ is 36 ◦ with respect to the streamwise direction, which appears comparable tothe value of 29 ◦ reported for η = 0 .
8. Such a slight difference is trivial in the presentstudy, because α θ is dictated by the domain aspect ratio via the geometric relation α θ = tan − ( pL z /mL x ), with p and m integers. This helical turbulence regime is foundto relaminarise at Re = 325, which is similar to the case η = 0 . η = 0 .
5. Nevertheless, artificial azimuthal extension of thenumerical domain, without change of curvature, reveals that helical stripes can in factbe sustained if the perimeter reaches 8 π . This suggests that the lack of occurrence ofhelical large-scale structures in aCf with η = 0 . K. Kunii, T. Ishida, Y. Duguet, and T. Tsukahara (a) h Flow (b) z/h ( (cid:1) )
0 10 20 30 40 50 60 70 80 90 1003.8 (2 (cid:1) ) x/h u r (cid:1) −0.03+0.03 Figure 7.
Same as figure 4, but for η = 0 . Re = 400, featuring puff-like structures. Onlyhalf the axial length of the computational domain is displayed. Low η = 0 . Nominal parameters
For η = 0 .
1, no oblique laminar-turbulent interface was observed in our simulationseither, neither as a permanent regime nor even transiently. By direct analogy with aPf,for which no helical structure was ever detected below η (cid:54) .
3, the immediate conclusionis that η = 0 . Re strictly below400. These structures are referred to as “puff-like” because there are both common pointsand dissimilarities with the puffs found in pipe flow intermittency. Several such puff-likestructures are shown in figure 7 to coexist for Re = 400. Their long-time dynamicssuggests that they are in equilibrium; however, if these coherent structures decay after asufficiently longer time, the flow goes back to the globally laminar state and puffs are onlymetastable objects. At slightly higher Re , splitting of some of the puffs is also observed.All these features have been reported as robust statistical properties of cylindrical pipeflow (Shimizu et al. η = 0 .
1. Two space-time diagrams for Re = 400 (a) and 395 (b) show the azimuthally-averaged streamwisevelocity (cid:104) u x (cid:105) θ = (cid:82) u x d θ/ π , evaluated at mid-gap as a function of x and t . In both cases,perturbations appear close to the laminar-turbulent interface. They can propagate inboth upstream and downstream directions from the core of the puff towards the laminarflow. This makes the puff appear more symmetric than its pipe flow counterpart, wheremost perturbations are advected towards decreasing pressure (Shimizu et al. Re = 400 (figure 8a), the flow apparently settles to a statistically steady statefor t (cid:62) h/u w . One can identify locally transient quasi-laminar regions, but no clearwavelength or patterning emerges. It should be pointed out that no quasi-1D shear flow,has ever shown convincing patterning properties with well-defined wavelengths.For Re = 395, all turbulent puff-like structures are found to decay. According to themeasurements above, the critical point for η = 0 . Re = 400 and 395. Ifnormalised by the hydraulic diameter (2 h ) and the “centreline velocity” u w as in pipe aminar-turbulent coexistence in annular Couette flow x / h − . t/ ( h / u w )
0 650 1300 1950 2600 3250 3900 4550204.85200 5850 6500 7150 7800 8450 9100 9750 10400 t/ ( h/u w ) x / h − . t/ ( h / u w ) (cid:1) u x (cid:2) (cid:1) (cid:1) u x (cid:2) (cid:1) (a) Re = 400 (cid:1) u x (cid:2) (cid:1)
0 400 800 1200 1600 2000 2400 2800 t/ ( h/u w ) x / h − . t/ ( h / u w ) (b) Re = 395 Figure 8.
Spatiotemporal diagrams of the azimuthally-averaged streamwise velocity (cid:104) u x (cid:105) θ atmid-gap for η = 0 . L θ = 2 π . The vertical axis shows the streamwise distance in a frame ofreference moving at streamwise speed 0 . u w . flow, this critical point corresponds to 4 Re = 1590 ±
10 lower than the value of 2040 ± et al. Azimuthally extended system
As before, we wish to go beyond the mere observation that no oblique pattern forms for η = 0 . η = 0 . L θ = 16 π ( n = 8 times longer than the nominal circumference). In this case, ( L zi , L zc , L zo ) =(5 . h, . h, . h ) and the resolution is kept to N θ = 512 with ∆z + (cid:54)
3. We note thatthe concept of ‘extended system’ depends here highly on the radial position. This is anunavoidable geometric consequence of the relative smallness of the inner rod. One effectof the azimuthal extension is again a marked shift of Re g : the flow does not return tolaminar even at Re = 275, a value much lower than Re g for L θ = 2 π . Since the numericalresolution is still satisfying according to table 1, the shift in Re g is not attributed to thecoarser resolution but to the change in boundary conditions. The nominal aCf with η = 0 . Re ≈ L θ = 16 π (figure 9a) since relaminarisation in finite time has been observed for all Re (cid:54) .
5. For the values of Re where turbulence was detected, transiently or sustained,the flow displays laminar-turbulent coexistence, but never displays any helically-shapedturbulence, as can be seen in figure 9b (see also the supplementary movie availableat https://doi.org/10.1017/jfm.2019.666, which highlights the dynamical behaviour of2 K. Kunii, T. Ishida, Y. Duguet, and T. Tsukahara (a) (16 (cid:1) ) (12 (cid:1) ) (8 (cid:1) ) (4 (cid:1) ) z/h ( (cid:1) )
0 10 20 30 40 50 60 70 80 90 100 x/h u r (cid:1) −0.03+0.03 (b) (16 (cid:1) ) (12 (cid:1) ) (8 (cid:1) ) (4 (cid:1) ) z/h ( (cid:1) )
0 10 20 30 40 50 60 70 80 90 100 x/h u r (cid:1) −0.03+0.03 Figure 9.
Instantaneous visualisation of wall-normal velocity fluctuations u (cid:48) r at mid-gap for η = 0.1, L θ = 16 π , and at (a) Re = 400 and (b) Re = 275. The supplementary movie shows thetime evolution during an arbitrary 800 h/u w of the flow visualised in (b), demonstrating a newregime of dynamic laminar-turbulent coexistence. the new flow regime). Note that the average streak width remains unaffected by theazimuthal extension beyond 2 π . Given the very large perimeter of the outer wall inunits of h , large enough in principle to accommodate one wavelength of a turbulentband, the absence of helical band, or equivalently of azimuthal large-scale flow, cannotbe assigned to the azimuthal confinement, even though the inner perimeter is itself toosmall to accommodate this wavelength. We thus deduce that, unlike for η = 0 .
5, the wallcurvature is responsible for the inability of the geometry to sustain helical stripe patternsfor η = 0 . η = 0 . L z /h ) differ between the three panels, theobserved irregular patterns of localised turbulence are very similar to each other andmatch well when plotted versus θ . As also confirmed in the supplementary movie, small-scale structures in turbulent patches seem to penetrate the entire gap from the innerto the outer wall. The intensity of velocity fluctuations in the region near the outerwall is weaker than near the inner wall, as also shown in figure 10 (using differentcolourbars). The figure also makes the absence of large-scale helical pattern clear atany radial position. There is always a possibility that for sufficiently large azimuthalextension, both inner and outer perimeters become large enough to accommodate long-wavelength turbulent patterns again, yet with a very small pitch angle that can not becaptured for smaller L θ . However, given the present low value of η , this would imply evenlonger computational domains and hence very expensive simulation. This hypothesis ishence not considered in what follows.Among the many implications of wall curvature, an interesting one which can bechecked for in simulations, in analogy with the corresponding laminar profiles, is thestatistical asymmetry of the turbulent flow with respect to the mid-gap (this symmetry aminar-turbulent coexistence in annular Couette flow (a) (16 (cid:1) ) (8 (cid:1) )
00 20 40 60 80 100 x/hz/h ( (cid:1) ) u r (cid:1) −0.03+0.03 (b)
0 20 40 60 80 100 x/hz/h ( (cid:1) ) 30.4 (16 (cid:1) ) (8 (cid:1) ) u r (cid:1) −0.03+0.03 (c) (16 (cid:1) ) (12 (cid:1) ) (8 (cid:1) ) (4 (cid:1) ) z/h ( (cid:1) ) 0 20 40 60 80 100 x/h u r (cid:1) −0.01+0.01 Figure 10.
Instantaneous visualisation of wall-normal velocity fluctuations u (cid:48) r in x - θ planes atthree different radial positions for η = 0 . L θ = 16 π , and Re = 300. From top to bottom : (a) y ≈ .
15 (near the inner wall) , (b) y ≈ . y ≈ . is exact only in the limit η → x - r plane. The moving inner rod is located at the topof the figure while the outer rod lies at its bottom. Strong mean gradients of azimuthalvorticity (the “boundary layer”) are found near the inner wall only. This boundary layertriggers ejections of vorticity towards the outer wall, at various places corresponding tothe location of the turbulent patch. This calls for a comparison with the regenerationmechanisms discussed in the context of pipe flow puffs (Shimizu & Kida 2009; Duguet et al. b ; Hof et al. η = 0 .
1, the perturbationsemanating from the shear layer at the inner rod are advected upstream in the framemoving with the inner wall. This suggests a turbulence regeneration process of a differentkind. More work is needed to relate rigorously this observation to the short coherencelength of these patches, as will be discussed in the next section. Section 4 is devoted toa deeper quantitative analysis of the statistical properties of all the laminar-turbulentcoexistence regimes identified in this section.4
K. Kunii, T. Ishida, Y. Duguet, and T. Tsukahara (cid:1) (cid:1) (cid:2)
0 20 40 60 80 100 120 140 160 180 2000 ( r i ) y ( r )1 ( r o ) x/h u x (cid:2)
30 35 40 45 50 55 60 65 700 ( r i ) y ( r )1 ( r o )0 ( r i ) y ( r )1 ( r o ) x/h Figure 11.
Typical snapshots of azimuthal vorticity ω (cid:48) θ (two top rows) and streamwise velocityfluctuation u (cid:48) x (bottom row) distributions in an x - r plane of aCf for η = 0 . Re = 395 withnominal L θ = 2 π . The inner rod (top boundary) moves towards the right of the figure.
4. Statistics
The statistics presented in this section have been gathered from the numerical runsdescribed in the previous section. The cases simulated with L θ = 2 π are summed upin table 1, while the two azimuthally extended simulations, η = 0 . L θ = 8 π and η = 0 . L θ = 16 π are described in § Mean velocity profile
The mean streamwise velocity profiles are displayed in figure 12. Here, the verticalaxis y (= r − r i ) is the radial distance to the wall measured from the inner cylinder.The overbar denotes time-averaging in space over the two variables x and z , and in timeover a time horizon T . Let Re g be the lowest value of Re where turbulence is foundfor each value η , and let us assume by convention that it indicates relaminarisation for Re < Re g , while in the other cases turbulence could sustain at least until time T , with T limited here to O (10 ). Our results indicate that Re g has a strong dependence on η .Mean velocity profiles, evaluated here only for Re close to Re g ( η ), are also affected by η .The dashed-dotted lines indicate the laminar profiles, in agreement with the theoreticalsolutions given in equation (2.9) as well as in figure 2. We note that for η = 0 . Re = 390 deviates from the cluster formed by the other profilesreported for 400 (cid:54) Re (cid:54) de facto ofthe laminar parts in the spatial averaging : any variation in the mean turbulent fraction(which will be documented in § y = 0 .
5. Thesymmetric property, which the limiting case of pCf also possesses, apparently persists inall aCf configurations despite the broken symmetry of the system with respect to the mid-gap. This does not necessarily imply shear instabilities yet. Recalling the novel kind oflaminar-turbulent coexistence identified for low η = 0 .
1, it is interesting to compare howthe mean profile evolves with decreasing η . Other features of interest can be extractedfrom figure 12, such as the mean shear profile, given by S ( y ) := − ∂u x /∂y . As η decreases,this velocity gradient becomes steeper at the inner cylinder than at the outer one. Theprofile also becomes more and more asymmetric with respect to the mid-gap y = 0 .
5. A aminar-turbulent coexistence in annular Couette flow h = 0.8 Re = 750 Re = 337.5 Re = 325 u x ( y ) y h = 0.5 Re = 750 Re = 380 Re = 377.5 h = 0.1 Re = 750 Re = 400 Re = 390Turbulent pCf Re = 750 0 2 4 6 8 10 S ( y ) h = 0.8 Re = 750 Re = 337.5 h = 0.5 Re = 750 Re = 380 h = 0.1 Re = 750 Re = 400Turbulent pCf Re = 7500 0.2 0.4 0.600.51 Figure 12.
Mean profiles of the streamwise velocity (left) and its velocity gradient S (right)for various η and Re . Black, red, and blue lines show η = 0.8, 0.5, and 0.1, respectively. Thedashed-dotted lines are of laminar and in agreement with the theoretical solution. For reference,a velocity profile of turbulent pCf (Tsukahara et al. quantitative comparison is given for Re = 750, a case for which statistics are not affectedby laminar-turbulent coexistence. While S ( y ) is approximately 0.2 at mid-gap for allvalues of η , its value ranges from 1 to 3 at the outer wall (decreasing with decreasing η ) and from 4 to 11 at the inner wall (now increasing with decreasing η ). The trend ishence such that, as η decreases towards 0.1, the mean profile gets more asymmetric, andresembles more and more a simple linear boundary layer profile located near the innerrod, while the gradients at the outer rod are increasingly weak.4.2. Friction factor
We present here calculations of the friction coefficient C f versus the Re , displayed infigure 13. To the authors’ knowledge, no experimental study has examined C f of aCfexcept for Shands et al. (1980), who reported measurements in a wide range of Reynoldsnumber including laminar and fully-developed turbulent regimes for low η < . C f forthe present study is defined as : C f = 2 τ w ( ξu w ) , (4.1)where the coefficient ξ stands for the ratio between the dimensional bulk velocity and thedimensional wall velocity u w . Note that τ w and ξ are both defined as temporal averages.The introduction of the ratio ξ is intended to capture the dependence of the bulk-meanvelocity on η . For instance, ξ should be 0.5 for the case η = 1. In this series of numericalexperiments, we obtained ξ = 0.46 ( η = 0 . ξ = 0.39 ( η = 0 . ξ = 0.20 ( η = 0 . C f, avg is a weighted average between the inner andouter values C f , i.e., C f, avg = ηC f, inner + C f, outer η + 1 , (4.2)where each C f is computed from the wall shear stress τ w measured locally. The solid lineshows the laminar law given by C f, laminar = 4 ξ − ηRe (1 + η ) | ln η | , (4.3)6 K. Kunii, T. Ishida, Y. Duguet, and T. Tsukahara -2 -1 Re C f , a vg ( R e ) L a m i n a r ( h = . ) L q = 2 ph = 0.8 h = 0.5 h = 0.1 L q = 8 ph = 0.5Robertson (1959)(Turbulent pCf) G =
G =
G = L q = 16 ph = 0.1 L a m i n a r ( h = . ) L a m i n a r( h = . ) Figure 13.
Friction coefficient as a function of Re including the cases of artificially extended L θ . Solid and dashed-dotted lines show equations (4.3) and (4.4), respectively. and the dashed-dotted line is the empirical law suggested by Robertson (1959) forturbulent pCf : (cid:114) C f G log Re . (4.4)Circles and crosses label the results with nominal and extended L θ , respectively. Theconstant G is determined by a logarithmic fit of (cid:112) C f directly from the simulations.We start by describing the results obtained for standard numerical domains with L θ = 2 π . We first note that Robertson’s formula, borrowed from the plane Couettecase, captures surprisingly well the turbulent regime for all values of η as long as Re is large enough (above 500). A ‘transitional’ region, i.e. a range comprising both non-laminar and non-fully-turbulent regions, emerges for 325 < Re <
400 (for η = 0 .
8) and395 < Re <
412 (for η = 0 . η = 0 .
5, the turbulent flow becomes laminar at Re = 377 . C f value between laminar and turbulentvalues was identified. From figure 13, Re g can be approximated by 325, 380, and 400 for η = 0 .
8, 0.5, and 0.1, respectively.In the cases of artificially extended L θ (crosses in figure 13), the variations of C f exhibitdifferent but still interesting trends. For η = 0 . Re g of L θ = 8 π (red crosses) is lowerthan that of L θ = 2 π and the new value of Re g = 325 is similar to the threshold Reynoldsnumber reported in pCf by Duguet et al. (2010 a ). For η = 0 .
1, the lower value of Re g has already been reported in the previous section. It is surprising that C f is very closeto the fully turbulent value (see the blue crosses), even at Re = 300, although the flowfield is much more intermittent than at higher Re (as will be quantified by the turbulentfraction in § Two-point correlations
After describing statistics involving streamwise and azimuthal averages, we now de-scribe the spatial correlations in the case of laminar-turbulent coexistence (i.e. close to Re g ), restrained to the wall-parallel variables x and θ . The correlations of the velocityperturbations, evaluated at a constant value of y = y ref are classically defined, after aminar-turbulent coexistence in annular Couette flow D x / h R xx ( D x ) h = 0.1 L q = 2 p (Re = 400) L q = 16 p (Re = 280) h = 0.5 L q = 8 p (Re = 350) (a) D x / h R rr ( D x ) (b) D x / h R qq ( D x ) (c) D q /p R xx ( D q ) h = 0.1 L q = 2 p (Re = 400) L q = 16 p (Re = 280) h = 0.5 L q = 8 p (Re = 350) (d) D q /p R rr ( D q ) (e) D q /p R qq ( D q ) (f) Figure 14.
Two-point correlation of velocity fluctuations as a function of either an axial lag ∆x (top row) determined by equation (4.5) [in (a–c)] or an azimuthal lag ∆θ (bottom row)determined by equation (4.6) [in (d–f)] : R xx in (a, d), R rr in (b, e), and R θθ in (c, f) representautocorrelations of u (cid:48) x , u (cid:48) r , and u (cid:48) θ , respectively. The legend in (a) is also valid for (b–f). normalisation, by : R ii ( ∆x ) = u (cid:48) i ( x, y ref , θ ) · u (cid:48) i ( x + ∆x, y ref , θ ) u (cid:48) i ( x, y ref , θ ) , (4.5) R ii ( ∆θ ) = u (cid:48) i ( x, y ref , θ ) · u (cid:48) i ( x, y ref , θ + ∆θ ) u (cid:48) i ( x, y ref , θ ) , (4.6)where the overbar ( · ) again denotes averaging over x , θ , and time. The value of y ref isapproximately 0.5, i.e. close to mid-gap. The present focus is on a neater characterisationof the new regime identified for η = 0 . L θ = 16 π . To thisend, we compare in figure 14 the autocorrelations of each velocity component as functionsof the streamwise and azimuthal spatial lags ∆x and ∆θ , computed for η = 0 .
1, to theircounterparts for L θ = 2 π , as well as to the other extended case with η = 0 . L θ = 8 π .The results obtained for L θ = 2 π are not significantly different between η = 0 . h . This appears significantly shorter than the minimum puff spacing away fromcriticality, i.e. 20–30 diameters or larger values for lower Reynolds numbers (Samanta et al. η = 0 . η = 0 . L θ yield similar lag angles of approximately π .8 K. Kunii, T. Ishida, Y. Duguet, and T. Tsukahara
200 300 400 500 600 700 80000.20.40.60.81 L q = 2 p h = 0.8 h = 0.5 h = 0.1 L q = 8 p h = 0.5 L q = 16 p h = 0.1 Re F t ( R e ) Figure 15.
Time-averaged turbulent fraction F t as a function of Re , nominal and artificiallyextended numerical domains with L θ = 2 π, π and 16 π . The second trend (panels c, f) is for the correlations of the azimuthal velocity field :the correlation lengths in x and θ for η = 0 . h and 2 π ,because of the large-scale structure of the turbulent bands. For η = 0 . ≈ h and less than π/
2. For comparison with the case η = 0 . Re = 2000) display wider azimuthalcorrelations and longer streamwise correlations (Willis & Kerswell 2008).These observations shed a new light on the regimes identified for η = 0 .
1, regardlessof the choice of L θ : as far as the azimuthal component is concerned, the transitionalregime for η = 0 . short-range correlations not found for the higher valuesof η investigated in this paper. The fact that the correlation decay is not modifiedbetween L θ = 2 π and 16 π suggests that this regime is robust and not an artefact ofeither confinement by boundary conditions or of the artificial numerical extension in θ ,though confirmation for even larger η would be welcome. A rapid comparison with thecorrelations found in aPf (Ishida et al. η = 0 . § Turbulent fraction
The turbulent fraction F t ( t ) measures the instantaneous amount of turbulence in theflow independently of its local intensity, it is the natural way to quantify robust laminar-turbulent coexistence. If F t = 1, the flow is turbulent everywhere, while F t = 0 meansthat the flow is fully laminar. Figure 15 shows temporal averages of F t , evaluated from x - z plane data evaluated at mid-gap. The local criterion for extracting laminar regionsis based on thresholding the radial velocity : | u (cid:48) r | < .
015 indicates locally laminar flow,while | u (cid:48) r | > .
015 indicates locally turbulent flow. As shown in figure 15, for Re = 750, F t > . η , when the flow can be unambiguously described as fullyturbulent. F t monotonically decreases in average as Re decreases, with a more significantdecrease around Re = 400–500. For η = 0 . F t approaches an aminar-turbulent coexistence in annular Couette flow η = 0 .
5, however, F t directly jumps from 0.6 to 0 due to the lack oflocalised turbulent structure.We focus then on the azimuthally extended systems. For η = 0 . L θ = 8 π ), helicalturbulence occurs, but the slope of the curve F t ( Re ) and the pointwise values are directlycomparable to the case L θ = 2 π above the critical point. Only the few additional non-zerovalues of F t near the onset Reynolds number suggest a change in the curve of F t ( Re ),which can be linked to the release in azimuthal confinement. That change is compatiblewith continuous F t even at the origin, although the trend cannot be confirmed at thatstage. The situation becomes clearer for η = 0 .
1. For η = 0 . L θ = 16 π ), F t showsvalues similar to the case η = 0 . L θ = 2 π ) when Re is in the range 400–750. However,below Re = 400, F t displays a much smoother decrease from values of 0.3–0.4 down tozero for Re ≈
275 (the last non-zero point has in fact F t ≈ . η = 0 . η = 0 . L θ = 8 π ), whereas in our simulationsfor η = 0 .
1, intermittent patches of turbulence have not managed to self-organise intolarge-scale coherent structures.
5. Discussion
The numerical results from § η but also on the value of the numerical parameter L θ :i) large-scale patterned turbulence at η = 0 .
8, similar to all the planar regimesdescribed in the literatureii) frustrated patterned turbulence at η = 0 .
5, similar to the previous regime butemergent in simulations only when the azimuthal confinement is releasediii) a short-range laminar-turbulent coexistence regime at η = 0 .
1, with no obliquepatterning even for artificially large values of L θ , at least up to 16 π .The first regime has a direct equivalent in aPf above η (cid:62) . F t with Re is continuous or discontinuous, the answer for the patterned regime clearly followsthe answer given for all Couette-like flows : in ‘affordable’ numerical simulations, thetransition at Re g appears discontinuous (Bottin et al. et al. a ; Chantry et al. F t ∼ ε β , where ε is the normalised distance to the onset,i.e. ε = ( Re − Re g ) /Re g , and b) the scaling in time of the unsteady turbulent fractionduring relaminarisation F t ( t ) ∼ t − α . The consensus at the moment is that this transitionin shear flows is continuous. Besides it falls apparently into the universality class ofdirected percolation (DP), for which β and α only depend on the effective dimensionof the problem (Lemoult et al. et al. β ≈ .
583 and α ≈ . β ≈ . α ≈ . Re g . This unfortunately implies that the determination of Re g itself, as wellas that of exponents like β and α , also require divergent domain sizes and observationtimes, at the risk for the simulation or the experiment to become rapidly unfeasible.In practice however, the exponents are determined from finite- ε simulations in finitedomains over finite observation times, provided the domain length is several times the0 K. Kunii, T. Ishida, Y. Duguet, and T. Tsukahara -2 -1 -1 L q = 16 p h = 0.1 ( Re g = 264) e = ( Re – Re g )/ Re g F t ( R e ) S l o p e = . S l o p e = . L q = 2 p h = 0.8 ( Re g = 325) h = 0.1 ( Re g = 390) 250 300 350 40000.20.40.6 Re Re g Figure 16.
Critical behaviour of the turbulent fraction as a function of the normalised distance ε to the critical threshold Re g (whose value is specific to each case), in log-log scale. The twostraight lines represent power law fits of the form F t ∼ ε β with β ≈ .
583 as in (2+1)-DP(solid, blue online) and β ≈ .
276 as in (1+1)-DP (dotted, red online). The inset shows thedata for η = 0 . L θ = 16 π in linear scale as in figure 15. correlation length (and similarly with time). In the patterning regime one can assumethat the wavelength, for Re away from Re g , yields a decent estimate of the correlationlength; then the estimation of Re g requires a domain size at least one order of magnitudetimes this wavelength. Given that the patterning wavelength for η = 0 . h (cf figures 4b and 6b), this suggests that no exponent can be trusted for sizes below L x ≈ O (10 ), as confirmed in recent investigations of other shear flows (Lemoult et al. et al. η = 0 . η = 0 .
1, with short-range correlations in alldirections, changes the picture. The short correlation lengths reported in § L x ≈ h is more ‘extended’, and in some sense closer to the thermodynamic limitof the related problem. We thus expect, for similar computational efforts as for othervalues of η , stronger evidence for continuous transition and easier measurements of theassociated critical exponents. We thus replot the data for η = 0 . F t = O ( ε β ) holds and, in case it does,which value β takes. This is displayed in figure 16 with L θ = 16 π , and ε defined based onthe choice Re g = 264. The estimation of Re g is performed classically by trial and error,checking that the curve F t ( Re ) displays algebraic decay/growth for the fitted value of Re = Re g . A linear fit emerges over little less than a decade in F t , which validates thenotion of critical scaling. The fit for β is consistent with the theoretical value β ≈ . β ). Also plotted are thenominal cases of L θ = 2 π for η = 0 . β ≈ .
276 rather than 0.583(although a steep slope that deviates from β at ε < .
06 might be due to shortagein the streamwise domain length). This alone is not sufficient to validate the (2+1)-DPpicture of the new regime as two other independent exponents need to be validated aswell. Among these exponents, we can have a rough approximation of the critical exponent α by monitoring in log-log plot the decay of F t ( t ) to laminar, starting from a noisy initial aminar-turbulent coexistence in annular Couette flow -2 -1 Re = 300 Re = 280 Re = 260 (Run 1) Time, t F t ( t ) t –0.4505 t –0.159 -2 -1 Re = 260 Run 1 Run 2 Run 3 Avg. Time, t F t ( t ) t –0.4505 t –0.159 (a) Re = 260–300 (b) Three runs for Re = 260 Figure 17.
Instantaneous turbulent fraction F t ( t ) as a function of time for η = 0 . L θ = 16 π near the critical Reynolds number. Two dashed lines represent power laws of F t ∝ t − α with thetheoretical exponent either of α = 0 . α = 0 .
159 for (1+1)-DP. condition with finite F t at t = 0. As can be seen in figure 17a, the decay of an individualrun for Re ≈ Re g ≈
260 alone cannot confirm the algebraic decay of F t . There is alwaysa possibility to get an improved critical scaling range by choosing more specific initialconditions with higher turbulent fraction. Ensemble averaging over several such runs ishowever helpful since an unambiguously algebraic decay for F t ( t ) emerges over almosta decade in figure 17b. The measured exponent is then consistent with the theoreticalvalue of α ≈ .
45. The current data appears hence consistent with (2+1)-DP, thoughmore data and more exponents would be needed to properly confirm this trend. Note infigure 17 that the scaling regime is reached after a relatively short time of O (10 h/u w ),especially compared with other similar studies (e.g., Lemoult et al. et al. η = 0 . η =0.1 a new dynamical regime with short-range correlations, for whichverification of the DP property seems computationally feasible using realistic domainsizes. This is in marked contrast to the majority of planar subcritical shear flows inwhich a similar task would still today appear as computationally hopeless.
6. Conclusion
Direct numerical simulation of annular Couette flow (aCf) is reported using finitedifferences in long computational domains. The so-called transitional regimes of aCf,featuring coexistence of laminar and turbulent flow, have been investigated depending onthe radius ratio η . The influence of an additional numerical parameter, the azimuthal L θ (usually fixed to 2 π ), has also been considered as a way to question the direct influence ofazimuthal confinement on the possible formation of large-scale flows and thus of organisedlaminar-turbulent coexistence. Three different regimes have been identified. For η close tounity (e.g. here 0.8) large-scale helical bands form as in the planar limit of plane Couette2 K. Kunii, T. Ishida, Y. Duguet, and T. Tsukahara flow (pCf). For moderate η (e.g. here 0.5), these helical bands do not have enough spaceto form because of the azimuthal confinement, this is confirmed by their occurrencefor L θ sufficiently large with respect to 2 π . Eventually, for low enough η , turbulencenear its onset takes the form of disorganised patches; their localisation and interactionare reminiscent of turbulent puffs in pipe flow but their structure displays a strongasymmetry. Importantly, the correlation length between these puffs is shorter by one orderof magnitude than the coherence length of organised patterns. This new regime appearsas a potential candidate for directed percolation, the short-range property even suggeststhat critical exponents could be measured with significantly less computational effortthan for the other long-range regimes usually encountered in subcritical shear flows. Itremains an open question whether it is possible to identify other shear flow geometries inwhich turbulent patches would also display short-range correlations, preferably a realisticflow that can be released experimentally.In this study, aCf has been introduced as a continuation prototype linking pCf toa one-dimensional flow geometry, with cylindrical pipe flow as the canonical examplefor a one-dimensional geometry. The present results indicate that high- η aCf connectssmoothly with pCf, however it fails at connecting with the pipe flow for low η and is thusno relevant candidate for this continuation. The presence of the inner rod, together withthe no-slip condition on it, induces a mean flow differing strongly from the one expectedfor a pipe flow. In particular, the statistics for η = 0 . η is reduced. This corresponds to theopposite situation to the pipe flow, where the shear is expected to vanish near the axis.The strong shear near the inner rod is responsible for the occurrence of a new dynamicalregime characterised by short-range correlations and no large-scale organisation. Acknowledgements
This work was supported by Grant-in-Aid for JSPS (Japan Society for the Promotionof Science) Fellowship 16H06066, 16H00813, and 19H02071. Numerical simulations wereperformed on SX-ACE supercomputers at the Cybermedia Centre of Osaka Universityand the Cyberscience Centre of Tohoku University.
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