Laminar-turbulent patterning in wall-bounded shear flows: a Galerkin model
LLaminar-turbulent patterning in wall-boundedshear flows: a Galerkin model
K. Seshasayanan and P. Manneville Laboratoire de Physique Statistique, CNRS UMR 8550,´Ecole Normale Sup´erieure, F-75005 Paris, France Laboratoire d’Hydrodynamique, CNRS UMR7646,´Ecole Polytechnique, F-91128, Palaiseau, Franceto appear in Fluid Dyn. Res.
Abstract
On its way to turbulence, plane Couette flow – the flow between counter-translatingparallel plates – displays a puzzling steady oblique laminar-turbulent pattern. We ap-proach this problem via
Galerkin modelling of the Navier–Stokes equations. The wall-normal dependence of the hydrodynamic field is treated by means of expansions on func-tional bases fitting the boundary conditions exactly. This yields a set of partial differentialequations for the spatiotemporal dynamics in the plane of the flow. Truncating this setbeyond lowest nontrivial order is numerically shown to produce the expected pattern,therefore improving over what was obtained at cruder effective wall-normal resolution.Perspectives opened by the approach are discussed.
Keywords : wall-bounded flow, laminar-turbulent transition, Galerkin modelling
Turbulent flows display transport properties strongly enhanced with respect to those of laminarflows, a feature that has particularly important consequences in configurations of engineeringinterest. Understanding how a given laminar flow becomes turbulent or a turbulent flow decaysto laminar is therefore of great interest, both conceptual and practical. In this respect, thecase of wall-bounded flows is of utmost concern since the transition can be direct, withoutthe intermediate steps observed, e.g. in free shear flows (Huerre & Rossi 1998, Manneville2015). This direct transition is a result of the local stability of laminar flow in competitionwith nontrivial solutions to the Navier–Stokes equation (NSE for short) in the range of controlparameter where the transition effectively takes place. As analysed by Waleffe (1997), themechanism by which such nontrivial solutions exist, the self-sustainment process (SSP), is nowthought to be well understood, but the laminar-turbulent coexistence still raises important1 a r X i v : . [ phy s i c s . f l u - dyn ] M a r uestions. In Hagen–Poiseuille flow (HPF), the flow under pressure gradient through a circularpipe, the transition takes place when turbulent puffs , the nontrivial states alluded to above,split and propagate turbulence before they have time to decay, a scenario well-reproducedby a reaction-diffusion model introduced by Barkley (2011a). In its own transitional range,plane Couette flow (PCF), the simple shear flow developing between counter-translating plates,experiences laminar-turbulent coexistence in the form of steady oblique bands (Prigent et al. et al. R = U h/ν where U is the speed of the plates, h the halfgap width between them, and ν the kinematic viscosity of the fluid. The bands are observedfor R g < R < R t . Below the global stability threshold R g turbulence is only transient, in theform of finite-lifetime spots , and the laminar base flow is always recovered after the spots havedecayed. Beyond the upper threshold R t turbulence is essentially featureless , i.e. uniform. Amodel, also of reaction-diffusion type, was proposed by one of us (Manneville 2012) to accountfor this pattern formation, in which a Turing mechanism was proposed to be responsible forthe bands when R is decreased below R t . Such explicative models are analogical in essence.Trying to support them directly from a reliable simplification of the primitive problem in orderfind out the physical mechanisms behind laminar-turbulent coexistence is the actual purposeof the work presented here.In the transitional range, the nontrivial solutions appear to be strongly coherent at the scaleof the distance to the wall, pipe diameter (Hof et al. etal. The derivation follows previous work in (Lagha & Manneville 2007a) with the difference that, inorder to avoid difficulties in the treatment of the pressure field, the NSE governing the departurefrom laminar flow is now written in a velocity-vorticity formulation as described in (Schmid &Henningson 2001), p.155ff, i.e. the (nonlinear) Orr–Sommerfeld equation for the wall-normalvelocity component v : ( ∂ t + u b ∂ x ) ∇ v − u (cid:48)(cid:48) b ∂ x v + N v = ν ∇ v , (1)and the Squire equation for the wall-normal vorticity component ζ = ∂ z u − ∂ x w , where u and w are the streamwise ( x ) and spanwise ( z ) velocity components, respectively:( ∂ t + u b ∂ x ) ζ + u (cid:48) b ∂ z v + N ζ = ν ∇ ζ . (2)In these general equations the base flow is v b = u b ( y ) e x . When dealing with PCF, using thehalf-gap width h as length unit and h/U as time unit, with U the speed of the plates drivingthe flow at y = ± u b ( y ) ≡ y . In that system of units the Reynolds number R isjust numerically equal to 1 /ν , i.e. the inverse of the kinematic viscosity of the fluid. Primesdenote the differentiation with respect to the wall-normal coordinate ( y ), hence u (cid:48) b ≡ u (cid:48)(cid:48) b ≡
0. The nonlinear terms N v and N ζ are complicated, formally quadratic, expressions of thevelocity components and their derivatives that can be found in (Schmid & Henningson 2001).It will turn out convenient to use a poloidal-toroidal decomposition of the hydrodynamic fieldsby introducing a velocity potential φ and a stream function ψ such that: v = − ∆ φ , (3) u = − ∂ z ψ + ∂ xy φ, w = ∂ x ψ + ∂ zy φ , ζ = − ∆ ψ , (4)∆ denoting the in-plane Laplacian ∂ xx + ∂ zz .The Galerkin approach used in (Lagha & Manneville 2007a) separates the in-plane spacedependence of the hydrodynamic field from its wall-normal dependence by expanding it onto apolynomial basis in y , yielding amplitudes functions of x, z and time t . The no-slip boundaryconditions to be fulfilled read u = v = w = ζ = ψ = φ = 0 at y = ± ∂ x u + ∂ y v + ∂ z w = 0, ∂ y v = ∂ y φ = 0 at y = ±
1. The basis functions are chosen so asto fulfil these boundary conditions exactly. The functions chosen for u, w, ζ , and ψ are in theform f i ( y ) = (1 − y ) R i ( y ), i = 0 , . . . , i max ; the R i are polynomials of degree i , and i max is the3 (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) Basis functions for (cid:99) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239)
Basis functions for v
Figure 1: Basis functions for the wall-normal vorticity ζ (left) and the wall-normal velocity v .Functions used in our model are displayed with thick lines; f : black, dotted; ( f , g ): green,continuous; ( f , g ): red, dashed; ( f , g ): blue, dash-dotted. Higher-order functions are shownwith thin lines. The work in (Lagha & Manneville 2007a,b) made use of { f , f , g } only.truncation order. For v and φ the functions are taken as g i ( i ) = (1 − y ) S i ( y ), i = 1 , . . . , i max ;the S i are polynomials of degree i − i max . The bases { f i } and { g i } are separately made orthonormal via a standard Gram–Schmidtprocedure using the canonical scalar product (cid:104) r | s (cid:105) = (cid:82) +1 − r ( y ) s ( y )d y . Basis functions areshown in figure 1 from which it is clearly understood how ( i ) the resolution close to the platesis improved by increasing the truncation order, and ( ii ) the profiles chosen for v incorporatethe boundary condition ∂ y v ( x, ± , z, t ) = 0. The analytic expressions of basis functions up to i max = 5 are given in A.1. As pointed out by Rolland (2012) in Appendix B of his PhD thesis,the chosen basis { f i , g i } is related to Jacobi polynomials of alternate possible use in standardspectral methods for the NSE (Canuto et al . 2007).According to the standard Galerkin procedure (Finlayson 1972), the expansions { v, φ } = (cid:80) i { V i , Φ i } ( x, z, t ) g i ( y ) and { u, w, ζ, ψ } = (cid:80) i { U i , W i , Z i , Ψ i } ( x, z, t ) f i ( y ) are inserted in theequations which are then projected onto the relevant bases, Eq. 1 for v onto { g i } , and Eq. 2 for ζ onto { f i } . The concrete derivation is straightforward and can be automated once the orderof truncation i max has been fixed. The formal expression of the model reads: (cid:8) ( I ∆ + A ) ∂ t + (cid:0) ¯ I ∆ + ¯ A (cid:1) ∂ x − ν (cid:0) I ∆ + 2 A ∆ + P (cid:1)(cid:9) ∆Φ = N ( V ) , (5) (cid:8) ( I ∂ t + B ∂ x ) − ν (cid:0) I ∆ + ¯ P (cid:1)(cid:9) ∆Ψ + ¯ B ∂ z ∆Φ = N ( Z ) . (6)In these expressions Φ and Ψ respectively stand for arrays { Φ , . . . , Φ i max ) } t and { Ψ , . . . , Ψ i max ) } t , superscript ‘t’ denoting transposition. I is the identity matrix of order i max in Eq. 5 and i max + 1 in Eq. 6. ¯ I is a square but non-diagonal matrix of order i max + 1, playinga role similar to I . All other matrices, A , ¯ A , B , ¯ B , P , and ¯ P are either square or rectangu-lar, with coefficients straightforwardly obtained by integration over [ − ,
1] of the appropriateproducts of f i , g j and their derivatives. About one half of the possible combinations canceldue to parity considerations. Remarkably enough, matrices A , P and ¯ P are diagonally dom-inant [( i, i ) (cid:29) ( i, i + k )] and the absolute value of the diagonal terms increases rapidly withthe position of the coefficient [( i, i ) (cid:28) ( i + 1 , i + 1)], which suggests possible simplificationsin the equations governing the dynamics of the field amplitudes. Finally, N ( V ) and N ( Z ) are4omplicated, formally quadratic expressions of the ( U i , V i , W i )’s that have to be derived fromthe (Φ i , Ψ i )’s introduced upon elimination of the pressure. Their explicit expressions are givenin A.2.Equations (5–6) only involve ∆Φ and ∆Ψ, which implies that some care is needed whendealing with spatially averaged terms corresponding to Fourier modes at ( k x , k z ) = (0 , u and w explicitly by assuming: u = ¯ u − ∂ z ˜ ψ + ∂ xy φ, w = ¯ w + ∂ x ˜ ψ + ∂ zy φ , with ¯ u and ¯ w still function of y and t but independent of x and z , while ˜ ψ refers to the ( x, z )-varying part of ψ . Notations being unambiguous, the tilde will be dropped in the following.On general grounds, the mean flow components ¯ u and ¯ w are governed by ∂ t ¯ u − ν (¯ u ) (cid:48)(cid:48) = − ( uv ) (cid:48) , ∂ t ¯ w − ν ( ¯ w ) (cid:48)(cid:48) = − ( wv ) (cid:48) , (7)where the overline means averaging over the in-plane coordinates. In the model, this is treatedby expanding ¯ u and ¯ w onto basis { f i } . From the continuity equation we get: U = U − ∂ z Ψ + C ∂ x Φ , W = W + ∂ x Ψ + C ∂ z Φ (8)where U and W stand for arrays { U , . . . U i max } t and { W , . . . W i max } t while matrix C arisesfrom the projection of ∂ y v onto the basis used to expand u and w in the continuity equation.Upon projection, equations 7 read: I ∂ t U − ν ¯ P U = N ( x )0 , I ∂ t W − ν ¯ P W = N ( z )0 (9)where N ( x )0 and N ( z )0 are the projections of the ( x, z ) spatially averaged nonlinear terms inequations 7. The model is now complete and ready for use. By construction, the model possesses all the properties requested to account for the transitionalregime of PCF: it can be checked that laminar flow is linearly stable for all Reynolds numbers,despite transient energy growth linked to lift-up, and that its nonlinearities redistribute butconserve the kinetic energy contained in finite amplitude perturbations. A numerical solverwas developed in order to examine whether bands can be recovered beyond lowest nontrivialtruncation order i max = 1. Wanting to add higher modes of both parities, we chose i max = 3,i.e. 7 fields: Ψ i , i = (0 : 3), and Φ i , i = (1 : 3). In-plane space dependence was handled usinga Fourier pseudo-spectral scheme that gets rid of aliasing via the usual 3 / etal. { x, z } , the numbers of evolving modes are N { x,z } and nonlinearterms are evaluated via back-and-fro FFTs with solutions reconstructed on N { x,z } points.Time marching was treated by formally rewriting the initial problem ∂ t X = L X + N ( X ) as ∂ t [exp( − t L ) X ] = exp( − t L ) N ( X ) and solving the new problem using a Runge–Kutta schemeof order 4.In parallel to the study in (Manneville & Rolland 2011) dealing with DNS at reduced wall-normal resolution, a first numerical experiment was devoted to the recovery of the featureless5
10 20 30 40 50 60 70 (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) n z s pe c t r a l po w e r ( l og sc a l e ) streaks andstreamwise vortices n x = 0 (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) n x n z = 0 Figure 2: Fourier spectral power of the streamwise perturbation velocity component u in themid-plane y = 0 as functions of wave-numbers n z for n x = 0 (left) and n x for n z = 0 (right). Thecorresponding Fourier wave-vectors read k { x,z } = 2 πn { x,z } /L { x,z } , where L { x,z } are the stream-wise ( x ) and spanwise ( z ) dimensions of the computational domain. The curves correspond tothe different resolutions studied, in the form N x ∗ N z , where N { x,z } are the maximum runningwave-numbers.turbulent state belonging to the nontrivial branch at high R in a domain of size L x × L z = 32 × u asa function of wave-numbers n z for n x = 0 (left) and of n x for n z = 0 (right). Normalisation bythe total number of modes N x N z makes the curves corresponding to the different resolutions lieon top of each other. In the left panel, the peak generated by the spanwise statistical periodicityof streaks and streamwise vortices is clearly identified for all the resolutions considered but morepronounced for N x × N z = 128 ×
128 than for 32 ×
32. This corresponds to N x,z /L x,z = 4 and1, with effective space steps δ x,z = 0 .
25 or 1, respectively, to be compared to the period ofthe streaks λ z ∼ L z /n str with n str ≈
7, hence about λ z ∼ . L x L z N x /L x N z /L z
108 48 2 4128 84 2 4680 340 1 1Our main result is that, in all cases, steady oblique patterns of alternately laminar andturbulent domains were observed in a limited range of Reynolds numbers, between R g ≈ R t ≈ R = 151 illustrates the two different possible orientations ofa single band pattern in the domain 108 ×
48. Orientation fluctuations are known to exist inDNSs at such an intermediate size. They are also present in the model as seen in the bottompanel showing the alternative dominance of modes (1 , +1) and (1 , − t s pe c t r a l po w e r (1,+1)(1, (cid:239) (cid:239) (cid:239) (cid:239) Figure 3: Patterning in a domain of size 108 ×
48 at R = 151. Top: the two different orien-tations (1 , +1) (left) and (1 , −
1) (right) with perturbation energy field averaged over the gapin grey levels, black = laminar, white = largest local energy. Bottom: Orientation fluctuationsevidenced by the spectral power in modes with wave-numbers ( n x , n z ), n x = 1 , n z = ± , ± ×
84 domain.The average of the turbulent energy over the volume V of the domain, E t = V (cid:82) V ( u + v + w ) d x d y d z , has been measured through the whole transitional range. The bifurcation diagramdisplayed in figure 4 (right) is again as expected, however the occurrence of large-scale laminar-turbulent coexistence in the form of oblique bands, easily detected visually and permitting theidentification of R g ≈
150 and R t ≈ E t with R than in DNSs for which a marked break at R t and a linear decrease below were observed.Here the smoother variation of E t ( R ) and the absence of clear-cut change at R t between theband regime (open circles) and uniform turbulence (filled circles) are presumably again a directconsequence of the higher level of fluctuations. Below R = 150, turbulence is only transient buta mean energy, roughly constant before the decay stage, can still be measured (open square).In principle R g should be located using a statistical study in line with the approach in terms ofchaotic transients, like in (Lagha & Manneville, 2007a). Its detection via a single experimentwhere R was progressively decreased by small steps has been judged sufficient for the presentpurpose.Finally, in a very wide domain 680 ×
340 of size comparable to that of the largest experimental7
48 150 152 154 156 158 160 162 16400.0050.0100.0150.020
Reynolds number E t Figure 4: Patterning in a domain of size 128 ×
84. Left: Patterns at R = 154. Right: Bifurcationdiagram (distance to laminar flow as a function of Reynolds number, see text).Figure 5: Snapshot of the solution in a domain of size 680 ×
340 at R = 151. To reach sucha size, the in-plane resolution has been lowered to N x = L x , N z = L z , without destroying thepattern.setups (Prigent et al. R , or equivalently a similarturbulence level at lower R , than in better resolved simulations and laboratory experimentswhere energy is transferred to and efficiently dissipated in much smaller scales. (There islittle or no trade-off for the in-plane dissipation that is treated like in the full-3D DNSs.)For a concrete comparison, experiments (Prigent et al. et al. R t ≈
410 and thelower threshold (global stability) at R g ≈ N y being the number of Chebyshev polynomials used in the representationof the wall-normal dependence, these values are shifted down to 350 and 270 for N y = 15 andto 275 and 215 for N y = 11. Here we have R t ≈
159 and R g ≈
150 but the pattern is stillwell rendered. This larger shift can therefore be understood because the effective wall-normalresolution is much lower. The fact that a physically relevant solution is obtained here while theChebyshev implementation breaks down with similarly few modes is due the optimal renderingof boundary conditions on v achieved by our basis choice (see below).Second, though the angle between the bands and the streamwise direction is correct, thewavelengths of the pattern, both streamwise and spanwise, are too short by a factor of 1.5 to 2,and the pattern’s orientation in domains of intermediate size fluctuates more than in the DNSs.The amount of enhancement is however difficult to appreciate quantitatively. These phenomenaremain unexplained for the moment but might relate to the effect of the wall-normal resolutionon the streamwise coherence that was shown to play an important role on the existence of thepattern (Philip & Manneville 2011). A hand-waving confirmation of this effect on the robustnessof the bands comes from the continuous trend observed as the resolution is decreased, here asthe truncation level i max is lowered, in rough correspondence with what was observed whenreducing the wall-normal resolution in DNSs. First, a conspicuous steady pattern is observedwith i max = 3. Next, for i max = 2 (not reported here but studied in parallel) coexistence offluctuating, wide, laminar and turbulent domains are observed in an even narrower Reynoldsrange; these domains remain disorganised and do not form bands. Finally, for i max = 1 (Lagha& Manneville 2007a), streaks stay short, the transitional range seems to be reduced to a pointat a somewhat larger value ( R g ≈ N y = 9 and 7 and blow-up occurred for N y <
7, which give amarked advantage to our separately optimal wall-normal representations of v and ζ . Theseobservations should contain some physics that warrant to be elucidated, as suggested in thenext section. Understanding the transition to turbulence in wall-bounded flows, and especially PCF that islinearly stable for all R and displays alternating laminar and turbulent oblique stripes on itsway to fully developed turbulence, is a hard problem when starting from the NSE. Some sim-plification can be expected by taking a key ingredient into account: the transition takes placeat moderate values of the Reynolds number for which the flow is controlled by the presenceof coherent structures (Hof al al. et al. i max , which raises the interesting question of the rate of conver-9ence of the approximation. Such a study would possibly be rewarding because, to be precise,DNSs treating the three space directions on a similar footing are computationally extremelydemanding (Duguet et al. i max = 1, 3, and 5, respectively. This does not prove that the global dynamics of the systemwould be equally well captured quantitatively by increasing the number 2 i max + 1 of fieldsin the model but hints at such a convergence, as generally expected for spectral approaches,here relying on specific complete series of Jacobi polynomials (Rolland 2012). At the price ofa pre-treatment of the problem that amounts to the once-for-all automated derivation of aneffective set of equations of sufficiently high order, it might be found interesting to replace thefull 3 D numerical simulations of the NSE by a finite set of two-dimensional partial differentialequations already taking the continuity condition fully into account and managing with wall-normal coherence in the transitional range. A quantitative estimate of the expected gain interms of memory requirements and time steps definitely warrants further study.In a complementary perspective, one can rather think of analysing the properties of themodel. First, in-plane coherence may be added to the wall-normal coherence inherent in thederivation. This can be done by inserting specific assumptions about the ( x, z )-dependence offields φ and ψ , in particular strict periodicity in space at the scale of the MFU (Jim´enez &Moin 1991). With i max = 1 and further limiting the in-plane expansion to the first harmonic, itis then straightforward to recover Waleffe’s models (Waleffe 1997) by making the correspondingeducated guess. A system of eight equations for eight amplitudes is obtained, identical to hissystem (10) but with a different set of coefficients acknowledging the difference in boundaryconditions (which, in passing, shows the structural genericity of that model). For example, theequation for the streamwise mean-flow component called M by Waleffe and governed by hisequation (10a) here reads: dd t U − ν ¯ p U = γ ¯ s (cid:2) ( α + γ ) BE − U V (cid:3) where U ≡ M −
1, while other symbols have the same definition as in (Waleffe 1997), especiallythe streamwise and spanwise wave-vectors α = 2 π/(cid:96) x and γ = 2 π/(cid:96) z , (cid:96) x and (cid:96) z being thedimensions of the MFU. The numerical values of coefficients in the equation above can beobtained from the formal expressions in A. Following the very same line, a study to be presentedelsewhere (Manneville, in preparation) shows that uniform large scale flows are generated justby shifting the phase of specific ingredients of Waleffe’s eight-equation model. Combining thisto the introduction of appropriately weighted in-plane second harmonics should help us toaccount for oblique coherent structures like those recently found by Daly and Schneider (2014),though the actual derivation of a model possessing them as fixed points would be cumbersome.Beyond the simple hypotheses corresponding to strictly periodic coherent structures, thenext step is to describe spatially slow turbulence modulations corresponding to the patternsobserved experimentally through the formal introduction of a slow dependence of the amplitudeof the local bifurcated state, in the spirit of the derivation of standard multiple-scale envelopeequations. The approach cannot be made as rigorous as, e.g., for convection the since thebifurcated state stays at finite distance from the laminar-flow base, which leaves room for10urther modelling. Of the two scales introduced, the fast one accounts for mechanisms at theMFU scale and the slow one corresponds to the modulations. The slow variables are drivenby source terms arising from a filtering of the Reynolds stresses, like in (Lagha & Manneville2007b) and it can be seen that the modulation of the uniform large scale flows alluded toabove generates nonlocal contributions of the class identified by Hayot and Pomeau (1994) asplaying an essential role in the balance between laminar and turbulent regions responsible forpatterning. But, in contrast with their phenomenological introduction of such contributions,here they directly arise from the equations and are therefore sensitive to the local orientationof the flow with respect to the streamwise direction, hopefully giving a microscopic support tothe empirical observations of Duguet and Schlatter (2013).Finally, large scale flows are present already with i max = 1 (Lagha & Manneville 2007b;Manneville, in preparation), though steady patterns are not observed in that case (Manneville2009). Taking smaller wall-normal scales into account ( i max >
1) is therefore necessary for atheoretical interpretation of the stabilisation of long-wave modulations observed with i max = 3,as reported in §
3. Simplification of models with higher truncation levels would then takeadvantage of the diagonal dominance of matrices A , P , and ¯ P noticed earlier to perform theadiabatic elimination of terms of least relevance yielding an effective model for the slowlyevolving terms. Such a heavy work could however possibly not be necessary and consideringseven fields might be sufficient up to an optimisation of the model’s coefficients. As a matter offact, the three first amplitudes (Ψ , Ψ , Φ ) are the most appropriate to deal with the nontrivialproperties of the in-plane flow dependence. So, if one is willing to include more of the wall-normal dependence, it should suffice to consider that the pairs (Ψ , Φ ) and (Ψ , Φ ) collectall the higher order contributions of each parity and, owing to its generic structure, to restrictoneself to the consideration of the seven-field model as an effective system replacing the NSE. Inthis perspective, except as a starting guess, sticking to the values of the coefficients obtained inthe strict Galerkin expansion is not advisable and introducing some multiplicative randomnessat appropriate strategic places like in (Barkley 2011b) seems profitable. Applying the programsketched in the previous paragraph to this new primitive problem is currently developed, whichis expected to improve over the one-dimensional phenomenological approaches of Manneville(2012) and Hayot & Pomeau (1994). The subcritical coexistence of different regimes forming laminar and turbulent patterns in PCFand other wall-bounded flow configurations is a difficult problem in which the interplay ofmean flow corrections and finite amplitude perturbations plays a crucial role. Our approach via
Galerkin decomposition yields explicit models replacing the NSE by coupled systems gov-erning amplitudes that encode the gross features of the flow. The derivation is systematicand the structure of the obtained models is generic, reflecting that of the primitive equations.Simulations of those models reproduce the patterning provided that the truncation level isnot too low. They are next amenable to further analysis, especially through in-plane spacedependence assumptions and explicit scale separation. Here, this program has been developedfor PCF but its adaptation to other flows such as plane Poiseuille or Couette–Poiseuille flow,cylindrical Couette–Taylor flow, etc. is straightforward. Obtaining a Barkley-like model forHagen–Poiseuille flow from first principles can also be considered along similar lines, using ba-sis functions adapted to the tube geometry and the no-slip condition. The extension to the11ess trivial case of boundary layer flows of various kinds, with their free-stream boundariesat infinity, remains a stimulating challenge. Once obtained such models offer tools to scruti-nise laminar-turbulent coexistence and provide us with detailed physical explanations of thisphenomenon of great conceptual and practical importance.
A Explicit expressions
A.1 Basis functions • In-plane velocity components and wall-normal vorticity component: f (0) = √ (cid:0) − y (cid:1) ,f (1) = √ (cid:0) − y (cid:1) y ,f (2) = √ (cid:0) − y (cid:1) (cid:0) y − (cid:1) ,f (3) = √ (cid:0) − y (cid:1) (cid:0) y − (cid:1) y ,f (4) = √ (cid:0) − y (cid:1) (cid:0) y − y + (cid:1) ,f (5) = √ (cid:0) − y (cid:1) (cid:0) y − y + (cid:1) y , . . . • Wall-normal velocity component: g (1) = √ (cid:0) − y (cid:1) ,g (2) = √ (cid:0) − y (cid:1) y ,g (3) = √ (cid:0) − y (cid:1) (cid:0) y − (cid:1) ,g (4) = √ (cid:0) y − (cid:1) (cid:0) y − (cid:1) y ,g (5) = √ (cid:0) y − (cid:1) (cid:0) y − y + (cid:1) , . . . A.2 Coefficients in the evolution equations
Taking care of the order of the subscripts introduced, symmetries within the sets of coefficientsare easily detected, directly or via integration by parts. Energy conservation relies on thesymmetries of coefficients introduced in the expressions of nonlinear terms. The elements ofmatrix C appearing in (8) are straightforwardly obtained as c ji = (cid:82) +1 − d y f j g (cid:48) i . A.2.1 Equation (5) for Φ j • linear terms:matrix ¯ I : ¯ δ ji = (cid:82) +1 − d y g j ( yg i ),matrices A and ¯ A : a ji = (cid:82) +1 − d y g j g (cid:48)(cid:48) i , ¯ a ji = (cid:82) +1 − d y g j ( yg (cid:48)(cid:48) i ),matrix P : p ji = (cid:82) +1 − d y g j g (cid:48)(cid:48)(cid:48)(cid:48) i ; • nonlinear terms, for j ∈ (1 : i max ): N ( V ) j = i max (cid:88) i =0 i max (cid:88) k =1 q jik ∆ [ ∂ x ( U i V k ) + ∂ z ( W i V k )] + i max (cid:88) i =1 i max (cid:88) k =1 ¯ q jik ∆( V i V k )12 i max (cid:88) i =0 i max (cid:88) k =0 r jik [ ∂ xx ( U i U k ) + 2 ∂ xz ( U i W k ) + ∂ zz ( W i W k )] − i max (cid:88) i =0 i max (cid:88) k =1 ¯ r jik [ ∂ x ( U i V k ) + ∂ z ( W i V k )] , (10)with: q jik = (cid:82) +1 − d y g j f i g k , ¯ q jik = (cid:82) +1 − d y g j ( g i g k ) (cid:48) , r jik = (cid:82) +1 − d y g j ( f i f k ) (cid:48) , ¯ r jik = (cid:82) +1 − d y g j ( f i g k ) (cid:48)(cid:48) . A.2.2 Equation (6) for Ψ j • linear terms: matrices B , ¯ B , and ¯ P : b ji = (cid:82) +1 − d y f j ( yf i ), ¯ b ji = (cid:82) +1 − d y f j g i , ¯ p ji = (cid:82) +1 − d y f j f (cid:48)(cid:48) i , • nonlinear terms, for j ∈ (0 : i max ): N ( Z ) j = i max (cid:88) i =0 i max (cid:88) k =0 s jik [ ∂ xz ( U i U k − W i W k ) + ( ∂ zz − ∂ xx )( U i W k )]+ i max (cid:88) i =0 i max (cid:88) k =1 ¯ s jik [ ∂ z ( U i V k ) − ∂ x ( W i V k ))] , (11)with s jik = (cid:82) +1 − d y f j f i f k , ¯ s jik = (cid:82) +1 − d y f j ( f i g k ) (cid:48) . Acknowledgements.
Results described here have been obtained by K.S. within the frame-work of program “Fluid Mechanics, Fundamental & Applications” of ´Ecole Polytechnique’sMaster of Mechanics under the supervision of P.M. Thanks are due to Y. Duguet (LIMSI, Or-say, France) for interesting discussions about the topics treated here and his critical reading ofthe manuscript. Constructive remarks of the Referees are also deeply acknowledged.
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