*Manuscript On the generation of rift flows in wall-boune flows transiting to turbulence Paul Manneville Hyroynamics Laboratory, CNRS-UMR 7646, École Polytechnique, F-91128 Palaiseau, France Abstract Despite recent progress, laminar-turbulent coexistence in transitional planar wall-boune shear flows is still not well unerstoo. Contrasting with the processes by which chaotic flow insie turbulent patches is sustaine at the local (minimal flow unit) scale, the mechanisms controlling the obliqueness of laminar-turbulent interfaces typically observe all along the coexistence range are still mysterious. An extension of Waleffe s approach [Phys. Fluis 9 (1997) 883 900] is use to show that, alreay at the local scale, rift flows breaking the problem s spanwise symmetry are generate just by slightly etuning the moes involve in the self-sustainment process. This opens perspectives for theorizing the formation of laminar-turbulent patterns. Keywors: wall-boune flows, plane Couette flow, Minimal Flow Unit, Self-Sustainment-Process, Galerkin moels 1. Context an purpose 5 Generically the transition to turbulence in flows along soli walls, so-calle wall-boune flows, can be triggere at Reynols numbers well below the value at which the laminar base flow profiles are linearly unstable [1, 2]. The existence of nontrivial solutions to the Navier Stokes equations (NSEs) competing with a the stable base flow, is believe to be well unerstoo in terms Email aress: paul.manneville@lahyx.polytechnique.fr (Paul Manneville) Preprint submitte to Journal of LATEX Templates September 25, 2017
10 15 20 25 30 35 of self-regenerating coherent structures [3]. The corresponing process, calle SSP [3, 4], involves streamwise vortices inucing streamwise streaks by lift-up, the so-prouce mean flow istortion being subsequently unstable in a way that closes the cycle by feeing the vortices present at start. At least at the moerate Reynols numbers where the transition to turbulence takes place, this plausible general sequence has a high egree of applicability. Such nontrivial flow regimes arise from sale-noe bifurcations in state space but their coexistence with laminar base flow has also to be appreciate in physical space. An important feature of transitional planar or nearly planar wall-boune flows is inee the separation of the full space into turbulent an laminar regions separate by sharp fluctuating interfaces, in the form of turbulent spots near the global stability threshol R g an more complicate laminarturbulent patterns at larger values of the Reynols number. Formally, this can be unerstoo as resulting from a moulation in space of the strength of the SSP mechanism, active in turbulent omains an switche off in the surrouning laminar flow. This moulation of the mechanism s intensity is in general etectable up to a limit where a regime of uniform turbulence calle featureless [5] is recovere. The transition from moulate to featureless turbulence is often marke by a well-efine threshol usually enote R t an laminar-turbulent coexistence is observe in an interval [R g, R t ] of finite with. For example, in plane Couette flow (PCF, the shear flow between counter-translating plates at istance 2h an relative spee 2U, for which R := Uh/ν), the transitional range extens from R g 325 to R t 415 an a perioic moulation of the turbulence intensity is observe in the form of bans alternatively laminar an turbulent, oblique with respect to the streamwise irection [6, 2]. Both aroun turbulent patches or between turbulent bans, near-laminar flow is the superposition of the base flow an large scale corrections believe to play an important role in the overall structure of turbulence [7]. These corrections have components that o not average to zero in the wall-normal normal irection, which makes them capable of transporting coherent structures as a whole, subsequently acting on oblique spot growth [8, 9]. For that reason, 2
40 45 50 55 60 65 they will be calle rift flows in the following. Oblique growth an the oblique laminar-turbulent organization clearly break the original spanwise symmetry of the problem statistically restore beyon R t. The aim of this note is to bring hints on the origin of these flows an their relation to symmetry breaking at a local level, i.e. the Minimal Flow Unit (MFU), the scale introuce by Jiménez an Moin [10] below which coherent structures with sizable lifetimes are no longer observe. The MFU is efine in the context of numerical simulations with wall-parallel perioic bounary conitions at istances l x an l z typically of the same orer of magnitue as the wall-normal characteristic size l y, viz. the gap 2h for PCF. It is the privilege scale at which transitional coherent structures are stuie [11] an arguments from ynamical systems an chaos theory are evelope [12]. Great progress in the unerstaning of the SSP has been obtaine thanks to Waleffe [4] who built an analytical, relatively simple, moel accounting for it within the Minimal Flow Unit (MFU) framework, calle Wa97 in the following. It was obtaine as a first-harmonic truncation of a Galerkin expansion of the NSEs with stress-free bounary conitions in a plane Couette-like geometry, sometimes calle Waleffe flow [13]. A straightforwar analysis using trigonometric basis functions yiels Wa97 as a system of 8 ifferential equations governing 8 moe amplitues that was next reuce to a 4-imensional system, the one stuie in greatest etail. The four amplitues retaine were explicitly associate to a mean flow istortion, the streaks an vortices amplitues an the amplitue of a combine moe effective in closing the system appropriately. By construction Wa97, whether reuce or not, preserves the spanwise symmetry of the flow configuration. My purpose will be to generalize Waleffe s approach to allow for rift flows observe in numerical simulations [8] or experiments [9]. In some sense, this can be viewe as an analytical counterpart to the numerical approach evelope by Kreilos et al. [14] who stuie rifting patterns at MFU size in relate transitional flows. The Galerkin approach to be use is a weighte-resiual metho analyzing the problem at han by expaning its fiels an governing equations onto func- 3
70 75 80 85 90 95 tional bases. When pushe at high orers, it can serve as a numerical simulation metho with goo convergence properties [15]. It is however usually not evelope as such in computational flui ynamics an alternate methos are use, e.g. [16], more straightforwar but rather working as black-boxes not amenable to analytical evelopments. Here, the aim is therefore not to apprehen the abstract structure of state space within the MFU framework in etail through the accurate etermination of exact solutions to the NSEs like in [11]. On the contrary, an much in the spirit of Waleffe s seminal work [4], I will attempt to uncover the concrete local source of mean-flow corrections involve in the symmetry breaking typically observe at transitional values of R. To this aim, I will consier the Galerkin metho rather as a systematic reuctive moeling strategy of the primitive equations, achieve by truncating the expansion at the lowest possible but still significant orer, so low that it can still be hanle analytically while clear physical significance can be given to the moe amplitues retaine. In accorance with the wie generality of the SSP for the base flows of interest, systems with similar structures can be erive with ifferences only appearing in the value of the coefficients. Since trigonometric relations between the basis functions use to eal with the stress-free bounary conitions for Waleffe flow [4] artificially kill some nonlinear interactions, in orer to work with a slightly less restrictive case, I will consier stanar PCF riven by no-slip conitions at the plates. On another han, I will follow Waleffe in his restriction to a first-harmonic approximation of the MFU ynamics to escribe the in-plane perioic epenence of the state variables. Section 2 gives a cursory presentation of the moel, the full expression of which is given in the Supplement [17]. Its main properties an virtues are then iscusse in 3 before presenting some perspectives on laminar-turbulent patterning in transitional wall-boune flows from a more general stanpoint in 4. 4
2. The moel The moeling approach starts from the velocity-vorticity formulation of the NSEs written for the perturbation to the base flow as etaile in [18], p. 155ff. Though the wall-normal an wall-parallel irections can be treate simultaneously in the Galerkin approach as originally one by Waleffe [4], here I will first eal with the wall-normal irection making use of results in [19, 20], an next with the wall-parallel irection an the perioic conitions corresponing to the MFU efinition. Simulations of PCF have shown that a representation of the flow at lowest significant orer contains about 90% of the perturbation energy for transitional Reynols numbers [21, App.B], accoringly I will just consier the corresponing minimal functional set, like for Wa97: {u, w} = {U 0, W 0 } f 0 (y) + {U 1, W 1 } f 1 (y), an v = V 1 g 1 (y), (1) 100 105 110 {u, w} an v being the in-plane an wall-normal perturbation velocity components, respectively. Polynomial bases introuce in [19, 20] are particularly well aapte to the no-slip bounary conitions at y = ±1, base flow u b = y, an low-orer truncation [19], namely 1 f 0 (1 y 2 ), f 1 y (1 y 2 ), an g 1 (1 y 2 ) 2. Amplitues U 0 an W 0 are attache to parabolic flow components that o not average to zero over [ 1, 1] an clearly contribute to the rift flows mentione in the introuction. At this stage, amplitues {U 0,1, W 0,1 } an V 1, an the wall-normal vorticity components Z 0,1 = z U 0,1 x W 0,1, are still functions of space (x, z) an time t. The partial ifferential equations expressing the NSEs, or rather their Orr Sommerfel part for V 1 an Squire part for Z 0,1 are given in the Supplement. The next step is the MFU reuction, treate by a Fourier series expansion expressing the wall-parallel perioic bounary conitions at l x = 2π/α an l z = 2π/γ, where α an γ are the funamental wavevectors of the wall-parallel MFU space epenence. Here the expansion is truncate beyon the first harmonic as 1 In the stress-free case, one has f 0 = 1/ 2, f 1 = sin πy/2, g 1 = cos πy/2, an u b f 1. 5
in [4] since, as inicate earlier, we are not primarily intereste in an accurate representation of the solution. The generalization of Walleffe s ansatz reas: Ψ 0 = U 0 z + W 0 x + X 1 sin αx + X 2 sin γz + X1 u cos αx + X1 w cos γz + X 3 cos αx cos γz + X2 u sin αx cos γz + X2 w cos αx sin γz + X1 o sin αx sin γz, (2) Ψ 1 = U 1 z + W 1 x + X 4 cos αx + X3 u sin αx + X4 u sin γz + X2 o cos γz + X 5 sin αx cos γz + X5 u cos αx cos γz + X3 w sin αx sin γz + X3 o cos αx sin γz, (3) Φ 1 = X 6 cos γz + X4 w sin αx + X5 w sin γz + X4 o cos αx + X 7 cos αx sin γz + X6 u sin αx sin γz + X6 w cos αx cos γz + X5 o sin αx cos γz. (4) The velocity components are retrieve from the expression of the streamfunctions Ψ 0,1 an velocity potential Φ 1 through: U 0 = z Ψ 0, W 0 = x Ψ 0, U 1 = x Φ 1 z Ψ 1, W 1 = z Φ 1 + x Ψ 1, (5) so that Ψ 0,1 = Z 0,1 an Φ 1 = βv 1, 115 120 125 where = xx + zz is the Laplacian in the plane of the flow an β plays the role of a wall-normal wavevector (no-slip: β = 3, stress-free: β = π/2). The terms U 0,1 z + W 0,1 x in (2,3) correspon to the non-oscillatory mean-flow components, governe by appropriately average equations as iscusse in [18]. In Wa97, only U 1 an the set {X 1,..., X 7 } are present uner ifferent names (equations (8,9) in [4]), specifically: M = 1+U 1 (mean flow), U = γx 2 (streak amplitue), V = γx 6 (streamwise vortex amplitue), A = αx 1, B = X 3, C = αx 4, D = X 5, an E = X 7. The justification for superscripts u, w, an o, ecorating the other sets of amplitues amplitues will appear in the next section. Amplitues U 0 an W 0 are the key ingreients in the extension of Wa97. A set of 28 equations for the 28 unknowns is obtaine by mere separation of harmonics. It isplays all the properties, lift-up, viscous issipation, quaratic avection nonlinearities, linear stability of the base flow, expecte from NSEs 6
for wall-boune shear flows within the MFU framework. It formally reas: tz + L Z = N (Z, Z), (6) where the variable set Z can further be ecompose into: Z = {Y, Y u, Y w, Y o } {{ U 1, X }, { U 0, X u}, { W 0, X w}, { W 1, X o}}, (7) 130 where X = {X 1,... X 7 }, X u = {X u 1,... X u 6 }, X w = {X w 1,... X w 6 }, an X o = {X o 1,... X o 5}. Y = { U 1, X } is precisely the set corresponing to Wa97. The full expression of System (6) is given in the Supplement where equations labelle (n) are here name (Sn). An immeiate inspection of this system shows that the subspace spanne by { U 1, X } is close, which means that, Z Wa97 { Y, 0, 0, 0 } is a consistent assumption solving the problem with equations for Y u, Y w, an Y o ientically cancelling. Here are four sample equations: The first one (S27) governs the streamwise mean flow correction: t U 1 + ν p 1 U 1 = 1 4 γ s 1[ 2γ 2 (X 6 X 2 X w 1 X w 4 ) + κ 2 (X w 6 X w 2 + X o 5X o 1 X u 2 X u 6 X 3 X 7 ) ] (8) where κ 2 = α 2 + γ 2. Once reuce reuce to Wa97, it closely correspons to Waleffe s equation (10-1) for M = 1 + U 1 [4], in the present notations: t U 1 + ν p 1 U 1 = 1 4 γ s 1[ 2γ 2 X 6 X 2 κ 2 X 3 X 7 ]. (9) 135 My secon sample is (S26) governing W 0, a spanwise mean flow correction absent from Wa97: t W 0 + ν p 0 W 0 = 1 4 α s 0[2α 2 (X 4 X w 4 X o 4X u 3 ) + κ 2 (X o 5X u 5 + X o 3X u 6 X 5 X w 6 X 7 X w 3 )]. (10) The two last equations in this group, (S25) for U 0 an (S28) for W 1, follow the same simple pattern. The thir sample is (S2), the equation governing the streak amplitue X 2 : t X 2 + νκ γ 0 X 2 = b X 6 + s 0 γ ( X1 w W 0 + 1 2 α(x 1X 3 X1 u X2 u ) ) + s 1 [ γ ( X o 2 W 1 + βx 6 U 1 ) + 1 2 αγ( (X u 3 X u 5 X 4 X 5 ) + β 2 (X o 4X o 5 X w 5 X w 6 ) ) 1 2 α2 β(x 4 X 7 + X o 4X o 3 + X u 3 X u 6 + X w 5 X w 3 ) )], (11) 7
with κ γ 0 = γ2 + p 0. The last sample is (S17) for the streamwise vortex amplitue 140 X 6 ( V in Wa97) that generates the streaks by lift-up through the term b X 6 in (11): µ 2 γ t X 6 + νκ 4 γ X 6 = α 2 r(x u 1 X u 5 + X 4 X 3 + X 1 X 5 + X u 3 X u 2 ) + γ (X w 5 X w 2 X o 4X o 1) + e γ (X u 1 X u 6 X 1 X 7 ) c γ X w 6 W 0, (12) 145 with µ 2 γ = γ 2 +β 2 an κ 4 γ = γ 4 +2β 2 γ 2 +p 1. The values of constants appearing in the equations, p 1, p 0,1, etc. erive from the wall-normal part of the moeling of the consiere flow configuration, thus here epening on whether no-slip or stress-free bounary conitions are use. Once reuce to Wa97, (11) an (12) rea: t X 2 + νκ γ 0 X 2 = b X 6 + 1 2 γαs 0X 1 X 3 + s 1 ( βγx6 U 1 1 2 αγx 4X 5 1 2 α2 βx 4 X 7 ) (13) an µ 2 γ t X 6 + νκ 4 γ X 6 = α 2 r(x 4 X 3 + X 1 X 5 ) e γ X 1 X 7 (14) strictly corresponing to Waleffe s (10-2) an (10-3). When comparing his system to the corresponing one extracte from (S1) (S28), a single ifference appears in equation (S9) for X 4 that reas t X 4 + νκ 2 αx 4 = αbx 1 + s 1 (αγx 2 X 5 2αX 1 U 1 βγ 2 X 2 X 7 ) 1 2 s 4γ 2 X 6 X 3. (15) 150 155 In Wa97, X 4 is variable C an the corresponing equation is (10-6) with the same terms as in (15) but lacks the last one, X 3 X 6, i.e. BV, that isappears owing to an acciental cancellation from trigonometric relations as notice earlier. The etaile consequences of this observation have however not been scrutinize in etail. Before consiering the virtues an limitations of moel (6) in the following sections, let me stress that, within the 1st-harmonic MFU assumption, its expression an etaile structure are quite general so that its applicability is 8
160 not restricte to PCF or Waleffe flow. On the contrary, it shoul rather be unerstoo as implementing the SSP on an extene footing that inclues rift flows. 3. Translational invariance an the generation of rift flows 165 170 175 180 185 The 28 variables in Z is the most general ensemble compatible with the first-harmonic approximation. As etaile in the Supplement, equations in (6) are iniviually rather complicate but with clear physical meanings. Terms with ν in factor of specific expressions of the Laplacian obviously account for viscous issipation an lift-up, alreay ientifie as bx 6 on the right han sie of (11) acts similarly on other moe pairs explicitly perioic in z, e.g. X 7 as a source term for X 3. Conservation of the kinetic energy by quaratic terms is expecte from the way the moel is erive. It is inee fulfille but the check remains technically cumbersome. More importantly, the choice Z Wa97 = {Y, 0, 0, 0 } is associate with specific spatial resonances between the ifferent flow components. This resonance conition can be retrieve in each an every solution to the full system, whatever its time epenence, by performing an arbitrary time-inepenent translation x x + x 0, z z + z 0. Similar observations have been mae in the literature, see [14] an references quote. Here I will take a own-to-earth but instructive viewpoint an first note that this implies relations between the equations of the full system. For example, a translation by l x /4 amounts to performing the changes cos αx sin αx an sin αx cos αx, which straightforwarly explains the similarity of equations for X 1 an X1 u, (S1) an (S3), X 3 an X2 u, (S5) an (S6), etc. with ientical coefficients an signs moifications linke to the minus sign in the secon change. An immeiate consequence is that the ynamics restricte to Z Wal97,u = {Y, Y u, 0, 0} is also close. The case of z-translation can be treate in the same way, showing that the subspace Z Wal97,w = {Y, 0, Y w, 0} is similarly invariant. Inspecting the full expression of (S1 S28) finally shows that the subspace Z Wa97,o = {Y, 0, 0, Y o } is also in- 9
190 195 variant but with no obvious relation to translational properties. On general grouns, the knowlege of the structure of phase space takes avantage of the stability properties of solutions known. As a consequence of the ientification of invariant subspaces above an the quaratic nature of the nonlinearities, it follows from stanar linear analysis that the stability operator aroun a solution in Z Wa97 has a block iagonal structure. The first 8 8 block accounts for the stability of the solution as if the system was restricte to { U 1, X }, as ealt with in [4]. It correspons to amplitue perturbations. The two next 7 7 blocks are for infinitesimal perturbations living in { U 0, X u } an { W 0, X w }. These subspaces being associate with translations as iscusse above, the relate linear moes correspon to phase perturbations. For example, let us consier the effect of an infinitesimal translation z z + δ z on an arbritrary solution Z = {Y, 0, 0, 0}. At leaing orer the solution reas Z + δz with δz = {0, 0, ɛ z Y w, 0} with ɛ z = γδ z. The components of Y w are W 0 = 0, X w 1 = X 2, X w 2 = X 3, X w 3 = X 5, X w 4 = 0, X w 5 = X 6, an X w 6 = X 7. It is inee reaily checke that the right han sie of equation equation (S18) governing X w 4 cancels ientically for such a perturbation, which implies X w 4 0. Next for X w 4 = 0, it is verifie that the right han sie of (10) also vanishes, so that W 0 0. In aition, the equations governing all the non-zero components of Y w are ientical to the equations for the corresponing component of Y apart from appropriate sign changes an, finally, insertion of perturbation ɛ z Y w in equation (8) for U 1 yiels t U 1 + ν p 1 U 1 = 1 4 γ s 1[ 2γ 2 X 6 X 2 κ 2 X 3 X 7 (1 + ɛ 2 z) ], (16) which shows that the ynamics of U 1 is preserve at leaing orer. Accoringly, perturbations corresponing to an infinitesimal z-translation are neutral an o not generate rift flow W 0, as expecte. The same argument can be evelope for infinitesimal streamwise translations, with ientification of the corresponing perturbation moe Y u an proof of the absence of relate U 0. However, perturbations along the so-obtaine eigenvectors Y w an Y u, while neutral, are extremely special an it is immeiately seen that arbitrary perturbations 10
are expecte to generate some non trivial rift flow ( U 0, W 0 ). It suffices to look at (10), for convenience rewritten by ropping all irrelevant higher orer terms as: t W 0 + ν p 0 W 0 = 1 4 α s 0[2α 2 X 4 X w 4 κ 2 (X 5 X w 6 + X 7 X w 3 )] (17) 200 205 210 215 220 to see how perturbations within subspace { W 0, X w} but with X w ɛ z X w, i.e. X4 w = ɛ 1 0, X6 w = γδ z X 7 + ɛ 2, X5 w = γδ z X 6 + ɛ 3, introuce sources terms for W 0, generating a response of the same orer of magnitue that comes an fees back into the whole system. Going back to the efinitions of the ifferent variables, assuming ɛ 1,2,3 0 means that arbitrary infinitesimal perturbations X4 w, X6 w, an X3 w can resonate with alreay present flow components, X 4, X 5, an X 7 to prouce some rift flow W 0 as soon as they o not strictly erive from an infinitesimal spanwise translation. Let us just consier the contribution of X 4 X4 w to the r.h.s. of (17) since X4 w 0 for an infinitesimal translation: Returning to (3), we see that X 4 is the amplitue of a spanwise velocity component αx 4 sin αx of the nontrivial state of interest, hence of orer one, that interacts resonantly by lift-up with an infinitesimal wall-normal velocity component ( α 2 /β)x4 w as obtaine from (4) to prouce a istortion W 0. The others contributions X6 w an X3 w eparting from strict infinitesimal translation woul be analyze in the same way with prouction of some W 0 as a net result. In turn the so-prouce W 0 of orer ɛ fees back into the ynamics of the set X w also of orer ɛ, while corrections to ynamics of set X are of orer ɛ 2, as a consequence of the block structure of the linear stability operator. Unfortunately, without specifying the nontrivial state of interest which is clearly beyon the scope of this stuy it is not possible to go further an ecie whether system Y w = { W 0, X w} is stable or unstable, i.e. whether it has exponentially growing solutions in aition to the neutral phase moe that exists in all circumstances. The same reasoning woul also separately apply to streamwise perturbations. What precees leas us to suspect that the stuy of solutions obtaine in a MFU context, either in a moel like (S1 S28) or in the full NSEs, lacks an important ingreient if symmetries are impose that forbi the existence of rift flows. 11
4. Discussion an perspectives Perioic bounary conitions inherent in the MFU assumption maintain the fiction of a solution that woul be uniformly evelope in space. In actual systems with wall-parallel imensions much larger than the wall-normal scale, itself typical of the size of the MFU, the intensity of the SSP can be moulate, especially in the turbulent spot regime aroun R g where the turbulence level varies from 0 to 1 in space, an in oblique bane laminar-turbulent patterns up to R t. In the stability analysis sketche above, spanwise an streamwise translations coul be treate separately. This is no longer the case more generally since the corresponing rift flows are couple by the continuity equation, an important conition at the heart of the Duguet Schlatter argument about the obliqueness of laminar-turbulent interfaces [8], here expresse as [19]: x U 0 + z W 0 = 0, 225 230 an automatically fulfille thanks to (5). Moulations to the SSP intensity have to be unerstoo as perturbations brought to a Wa97 solution that, as is reaily verifie, must inclue all the components of its extene representation in the present moel. For convenience (6) can be rewritten by separating the mean flows U = { } U 0, W 0, U 1, W 1 from the rest of the amplitues X = { X, X u, X w, X o} : t X + L X + M( U ) X = N ( X, X ), (18) t U + L U = N ( X, X ), (19) 235 highlighting the origin an role of U. Assuming that this system only escribes the small-scale (MFU) flow in a simplifie way an amitting further that this local solution can experience moulations that perturb the fine tuning of SSP moes, we see that rift flows inevitably appear as nontrivial responses to Reynols stresses on the r.h.s. of (19) inuce by resonance mismatches pointe out in 3 above. These rift flows then fee back into the rest of the solution via the term M ( U ) X in (18); 12
240 245 250 255 260 265 typical examples are the terms with a factor W 0 in the equations governing the amplitues of the two most crucial ingreients to the SSP, (11) for the streaks X 2 an (12), for the streamwise vortices X 6. When ealing with moulations, we have to face the ifficulty that there is no systematic multiscale approach available owing to the sub-critical character of transition. Nontrivial solutions emerge abruptly an steep interfaces in physical space separate ifferent flow regimes, in sharp contrast with the case of supercritical bifurcations as pointe out by Pomeau [22, 4]. The first reason is that there is no linear marginal stability conition to work with: the coherence length that controls the iffusion of moulations near threshol is inee irectly obtaine from its curvature at the critical point. The secon reason is that supercriticality implies a controllable saturation of the solution s amplitue. Both circumstances permit a rigorous an systematic perturbation approach [23, Chaps. 8 10], an none hols in the present case. The spatial coexistence of laminar an turbulent flows is particularly ifficult to apprehen from the primitive equations. An approach via analogical moeling in terms of reaction-iffusion (RD) systems [24], evelope by Barkley [25, 26], has been particularly fruitful to account for the transitional range of pipe flow. In that moel, the prouction of turbulence was consiere as the result of a reaction an iffusion was introuce phenomenologically to treat the spatial coexistence of the two states, laminar or turbulent. Soon after the earliest evelopments of this work, I use the same RD framework but in the context of a Turing instability, i.e. a pattern-forming mechanism controlle by iffusion rates with sufficiently ifferent orers of magnitue [24]. In my moel [27], the local reaction terms were expresse using the reuce (4- imensional) Wa97 moel [4], while its variables were allowe to iffuse with wiely ifferent turbulent viscosities in one irection of space. As a result, a Turing bifurcation was obtaine at ecreasing R, efining a threshol R t below which a pattern was present own to some R g corresponing to a general breakown towar laminar flow. Whereas it seems reasonable to use the variables in X to treat turbulence prouction at a local scale, the structure of (18,19) 13
270 clearly shows that the simple heuristic assumption of a iffusion via turbulent viscosities is unable to properly rener the possible role of rift flows on pat- tern formation. On the other han, a moel equivalent to the spatiotemporal Galerkin system escribe in 1 of the Supplement was numerically stuie by Lagha an myself in [28]. Filtering out the small scales, we coul next etermine the ynamics of large-scale flows, in particular their rift-flow component ( U 0, W 0 ) aroun turbulent spots. They were obtaine analytically as a response 275 to Reynols stresses given from the outsie, not as stemming from some local 280 285 290 295 ynamics possibly obtaine within a MFU framework as examine here. We can therefore infer that a combination of the two approaches, small scale ynamics incluing the feeback of large scales flows, shoul provie a satisfactory, now self-containe, escription of laminar-turbulent coexistence in the transitional regime. Numerical simulations of Galerkin moels truncate at ifferent levels however suggest that the lowest nontrivial, three-fiel, level is insufficient to recover an organize laminar-turbulent ban pattern, for PCF [20] as well as for Waleffe flow [13], an that we are requeste to consier at least seven fiels in orer to obtain oblique bans in a [R g, R t ] range of finite with [20]. Working with a higher level moel at the MFU scale, further incorporating the effect of space moulations, an of course simplifying the cumbersome soobtaine moel appropriately, e.g. through aiabatic elimination of fast variables, is likely the only way to really explain the occurrence of laminar-turbulent pattern analytically. A RD picture [25, 26, 27] woul emerge, mostly irecte at the unerstaning of the transition from featureless turbulence to bane turbulence at R t upon ecreasing R. It woul be erive from the NSEs an woul replace the naive introuction of turbulent viscosities by a clean account of rift flows, hopefully containing the mechanism for a Turing instability. The approach is not limite to PCF or Waleffe flow an shoul provie a generic interpretation to laminar-turbulent coexistence in the transitional range for a wie range of wall-boune flows of practical interest, as can be anticipate from the universal structure of Galerkin approximations to the NSEs, the relevance of the SSP in proucing nontrivial states alreay at the MFU scale an 14
300 moerate Reynols numbers, an the ubiquitous presence of rift flows. Acknowlegments: I woul like to thank Profs. G. Kawahara an M. Shimizu (Osaka University, Japan) an Dr. Y. Duguet (LIMSI, France) for iscussions relate to this work within the framework of the TransTurb JSPS CNRS exchange program. 305 References [1] S. Grossmann, The onset of shear flow turbulence, Rev. Mo. Phys. 72 (2000) 603 618. [2] P. Manneville, Transition to turbulence in wall-boune flows: Where o we stan?, Mech. Eng. Rev. Bull. JSME 3. oi:10.1299/mer.15-00684. 310 [3] J. Hamilton, J. Kim, F. Waleffe, Regeneration mechanisms of near-wall turbulence structures, J. Flui Mech. 287 (1995) 317 348. [4] F. Waleffe, On a self-sustaining process in shear flows, Phys. Fluis 9 (1997) 883 900. 315 [5] C. Anereck, S. Liu, H. Swinney, Flow regines in a circular Couette system with inepenently rotating cyliners, J. Flui Mech. 164 (1986) 155 183. [6] A. Prigent, G. Grégoire, H. Chaté, O. Dauchot, Long-wavelength moulation of turbulent shear flows, Physica D 174 (2003) 100 113. [7] D. Barkley, L. Tuckerman, Mean flow of turbulent-laminar patterns in plane Couette flow, J. Flui Mech. 576 (2007) 109 137. 320 [8] Y. Duguet, P. Schlatter, Oblique laminar-turbulent interfaces in plane shear flows, Phys. Rev. Lett. 110 (2013) 034502. [9] M. Couliou, R. Monchaux, Growth ynamics of turbulent spots in plane Couette flow, J. Flui Mech. 819 (2017) 1 20. 15
325 [10] J. Jimenez, P. Moin, The minimal flow unit in near wall turbulence, J. Flui Mech. 225 (1991) 213 240. [11] J. Gibson, J. Halcrow, P. Cvitanović, Equilibrium an travelling-wave solutions of plane Couette flow, J. Flui Mech. 638 (2009) 243 266. [12] G. Kawahara, M. Uhlmann, L. van Veen, The significance of simple invariant solutions in turbulent flows, Ann. Rev. Flui Mech. 44 (2012) 203 225. 330 [13] M. Chantry, L. Tuckerman, D. Barkley, Turbulent-laminar patterns in shear flows without walls, J. Flui Mech. 791 (2016) R8. [14] T. Kreilos, S. Zammert, B. Eckhart, Comoving frames an symmetryrelate motions in parallel shear flows, J. Flui Mech. 751 (2014) 685 697. 335 [15] B. Finlayson, The metho of weighte resiuals an variational principles, with applications in flui mechanics, heat an mass transfer, Acaemic Press, 1972. [16] J. Gibson, Channelflow: a spectral Navier Stokes simulator in C++, Tech. rep., Georgia Institute of Technology (2008). URL http://www.channelflow.org 340 [17] P. Manneville, On the generation of rift flows in wall-boune flows transiting to turbulence: Supplement, Tech. rep., this Journal s repository (2017). [18] P. Schmi, D. Henningson, Stability an transition in shear slows, Springer, 2001. 345 [19] M. Lagha, P. Manneville, Moeling transitional plane Couette flow, Eur. Phys. J. B 58 (2007) 433 447. [20] K. Seshasayanan, P. Manneville, Laminar-turbulent patterning in wallboune shear flows: a Galerkin moel, Flui Dyn. Res. 45 (2015) 035512. 16
350 [21] P. Manneville, On the transition to turbulence of wall-boune flows in general, an plane Couette flow in particular, Eur. J. Mech. B/Fluis 49 (2015) 345 362. [22] Y. Pomeau, The transition to turbulence in parallel flow: a personal view, C.R. Mécanique 343 (2015) 210 218. 355 [23] P.Manneville, Dissipative structures an weak turbulence, Acaemic Press, 1990. [24] B. Murray, Mathematical biology, Springer-Verlag, 1993. [25] D. Barkley, Simplifying the complexity of pipe flow, Phys. Rev. E 84 (2011) 016309. 360 [26] D. Barkley, Theoretical perspective on the route to turbulence in a pipe, J. Flui Mech. 803 (2016) P1. [27] P. Manneville, Turbulent patterns in wall-boune flows: a Turing instability?, Eur. Phys. Lett. 98 (2012) 64001. [28] M. Lagha, P. Manneville, Moeling of plane Couette flow: I. large scale flow aroun turbulent spots, Phys. Fluis 19 (2007) 095105. 17
Supplemental Files (for review) Click here to ownloa Supplemental Files (for review): Supplement.pf