Modeling the transition to turbulence in shear flows

Transcription

1 Modeling the transition to trblence in shear flows Dwight Barkley 1,2 1 Mathematics Institte, University of Warwick, Coventry CV4 7AL, United Kingdom 2 PMMH (UMR 7636 CNRS - ESPCI - Univ Paris 06 - Univ Paris 07), 10 re Vaelin, Paris, France D.Barkley@warwick.ac.k Abstract. One-dimensional models are presented for transitional shear flows. The models have two variables corresponding to trblence intensity and mean shear. These variables evolve according to simple eations based on known properties of transitional trblence. The first model considered is for pipe flow. A previos stdy modeled trblence sing a chaotic tent map. In the present work trblence is modeled instead as mltiplicative noise. This model captres the character of transitional pipe flow and contains metastable pffs, pff splitting, and slgs. These ideas are etended to a limited model of plane Coette flow. 1. Introdction The transition to trblence in shear flows has been the sbject of stdy since Reynolds pioneering stdies over a centry ago (Reynolds, 1883). The difficlty in nderstanding shearflow transition is largely attribtable to the sbcritical natre of the problem. In all of the classic cases pipe flow, channel flow, plane Coette flow, bondary layer flow and others trblence is fond at Reynolds nmbers for which laminar flow is linearly stable. In sch flows trblence appears abrptly following finite-sized distrbances of laminar flow and not throgh a seence of instabilities each increasing the dynamical compleity of the flow. This limits the applicability of linear and weakly nonlinear theories in addressing transition in these cases. One of the most intriging aspects of shear trblence is the intermittent form it takes in the transitional regime, near the minimm Reynolds nmber for which trblence can be triggered. In pipe flow, one observes localized trblent patches, known as pffs, srronded pstream and downstream by laminar flow (Wygnanski & Champagne, 1973; Nishi et al., 2008; Mllin, 2011). In planar cases, sch as plane Coette flow and bondary layer flow, one commonly observes trblent spots srronded by laminar flow (Wygnanski et al., 1976; Tillmark & Alfredsson, 1992). Even more intriging is the reglar alternation of trblent and laminar regions that is now known to arise spontaneosly in many shear flows with sfficiently large aspect ratio (Prigent et al., 2002; Barkley & Tckerman, 2005). Minimal models of spatiotemporal intermittency have been sefl in nderstanding generic featres of intermittent shear trblence (Chaté & Manneville, 1988; Bottin & Chaté, 1998). Here, I consider models that contain more of the physics specific to shear trblence and from this I obtain models that prodce ite realistic dynamics. For pipe flow it is possible to reprodce nearly all of the large-scale phenomena associated with transition sing only two scalar eations. Other shear flows are more difficlt, bt I point to some ideas for plane Coette flow.

2 2. Pipe flow In this section I consider pipe flow. As concerns the large-scale featres, pipe flow is effectively a one-dimensional system and this makes it a particlarly good problem to tackle first. I will smmarize basic featres of pipe flow and recall the modeling proposed in Barkley (2011). Then I will consider an alternative approach to that in Barkley (2011) and here model trblence by mltiplicative noise Phenomenology Figre 1 smmarizes the three important dynamical regimes of transitional pipe flow from direct nmerical simlations (DNS) (Blackbrn & Sherwin, 2004; Moey & Barkley, 2010). Qantities are nondimensionalized by the pipe diameter D and the mean (blk) velocity Ū. The Reynolds nmbers is Re = DŪ/ν, where ν is kinematic viscosity. Flows are well represented by two antities, the trblence intensity and the aial (streamwise) velocity, sampled on the pipe ais. Specifically, is the magnitde of transverse flid velocity (scaled p by a factor of 6). The centerline velocity is relative to the mean velocity and is a proy for the state of the mean shear that conveniently lies between 0 and 1. At low Re, as in Fig. 1(a), trblence occrs in localized patches propagating downstream with nearly constant shape and speed. These are called eilibrim pffs (Wygnanski et al., 1975; Darbyshire & Mllin, 1995; Nishi et al., 2008), a misnomer since at low Re pffs are only metastable and eventally revert to laminar flow, i.e. decay (Faisst & Eckhardt, 2004; Peiinho & Mllin, 2006; Hof et al., 2006; Willis & Kerswell, 2007; Schneider & Eckhardt, 2008; Hof et al., 2008; Avila et al., 2010; Kik et al., 2010). Asymptotically the flow will be laminar parabolic flow, ( = 0, = 1), throghot the pipe. For intermediate Re, as in Fig. 1(b), pff splitting freently occrs (Wygnanski et al., 1975; Nishi et al., 2008; Moey & Barkley, 2010; Avila et al., 2011). New pffs are spontaneosly generated downstream from eisting ones and the reslting pairs move downstream with approimately fied separation. Frther splittings will occr and interactions will lead asymptotically to a highly intermittent mitre of trblent and laminar flow (Rotta, 1956; Moey & Barkley, 2010). At yet higher Re, trblence is no longer confined to localized patches, bt spreads aggressively in so-called slg flow (Wygnanski & Champagne, 1973; Nishi et al., 2008; Mllin, 2011), as illstrated in Fig. 1(c). The asymptotic state is niform, featreless trblence throghot the pipe (Moey & Barkley, 2010) PDE model The modeling in Barkley (2011) is based on the following physical featres of transitional trblence in pipes. At the pstream (left in Fig. 1) edge of trblent patches, laminar flow abrptly becomes trblent. Energy from the laminar shear is rapidly converted into trblent motion and this reslts in a rapid change to the mean shear profile (Wygnanski & Champagne, 1973; Hof et al., 2010). In the case of pffs, the trblent profile is not able to sstain trblence and ths there is a reverse transition (Wygnanski & Champagne, 1973; Narasimha & Sreenivasan, 1979) from trblent to laminar flow on the downstream side of a pff. In the case of slgs, the trblent shear profile can sstain trblence indefinitely; there is no reverse transition and slgs grow to arbitrary streamwise length (Wygnanski & Champagne, 1973; Nishi et al., 2008). On the downstream side of trblent patches the mean shear profile recovers slowly (Narasimha & Sreenivasan, 1979), seen in the behavior of in Fig. 1. The degree of recovery dictates how ssceptible the flow is to re-ecitation into trblence (Hof et al., 2010). The following partial-differential eation (PDE) model captres the essence of these physical featres: t + U = ( + r 1 (r + δ)( 1) 2) +, (1) t + U = ɛ 1 (1 ) ɛ 2. (2)

3 (a) eilibrim pff downstream (d) nll (a) nll (b) nll nll (b) pff splitting split nll (e) nll (c) (d) (c) slg epanding (f) pipe ais Figre 1. Regimes of transitional pipe flow from simlations in a periodic pipe 200D long. Shown are instantaneos vales of trblence intensity and aial velocity along the pipe ais. (a) Eilibrim pff at Re = (b) Pff splitting at Re = (c) Slg flow at Re = Figre 2. The distinction between pffs and slgs seen as the difference between ecitability and bistablilty in the PDE model, Es. (1)-(2). Phase planes show nllclines at (a) r = 0.7 and (b) r = 1. i (space) The fied point (1, 0) corresponds to stable laminar flow. In (a) this is the only fied point. In (b) the additional stable fied point corresponds to stable trblence. Soltion snapshots show (c) a pff at r = 0.7 and (d) a slg at r = 1. These soltions are plotted in the phase planes with arrows indicating increasing. In the model, the parameter r plays the role of Reynolds nmber Re. U acconts for downstream advection by the mean velocity, and is otherwise dynamically irrelevant since it can be removed by a change of reference frame. The model incldes minimm derivatives, and, needed for trblent regions to ecite adjacent laminar ones and for left-right symmetry breaking. The core of the model is seen in the - phase plane in Fig. 2. The trajectories are organized by the nllclines: crve where = 0 and = 0 for the local dynamics ( = = = 0). For all r the nllclines intersect in a stable, bt ecitable, fied point corresponding to laminar parabolic flow. The dynamics with ɛ 2 > ɛ 1 captres in the simplest way the behavior of the mean shear. In the absence of trblence ( = 0), relaes to = 1 at rate ɛ 1, while in response to trblence ( > 0), decreases at a faster rate dominated by ɛ 2. Vales ɛ 1 = 0.04 and ɛ 2 = 0.2 give reasonable agreement with pipe flow. The -nllcline consists of = 0 (trblence is not spontaneosly generated from laminar flow) together with a parabolic crve whose nose varies with r, while maintaining a fied intersection with = 0 at = 1+δ, (δ = 0.1 is sed here). The pper branch is attractive, while the lower branch is repelling and sets the nonlinear stability threshold for laminar flow. If laminar flow is pertrbed beyond the threshold (which decreases

4 with r like r 1 ), is nonlinearly amplified and decreases in response. The (ecitable) pff regime occrs for r < r c ɛ 2 /(ɛ 1 +ɛ 2 ), Figs. 2(a) and (c). The pstream side of a pff is a trigger front (Tyson & Keener, 1988) where abrpt laminar to trblent transition takes place. However, trblence cannot be maintained locally following the drop in the mean shear. The system relaminarizes (reverse transition) on the downstream side in a phase front (Tyson & Keener, 1988) whose speed is set by the pstream front. Following relaminarization, relaes and laminar flow regains ssceptibility to trblent pertrbations. The slg regime occrs for r > r c, Figs. 2(b) and (d). The nllclines intersect in additional fied points. The system is bistable and trblence can be maintained indefinitely in the presence of modified shear. Both the pstream and downstream sides are trigger fronts, moving at different speeds, giving rise to an epansion of trblence SPDE model While the PDE model captres the essence of the pff-slg transition, the model of trblence is too simple to captre featres sch as pff decay and pff splitting. In Barkley (2011), a more realistic model was obtained by employing a tent map to mimic shear trblence. The map was designed to give a local phase-space strctre similar to the nllcline pictre for the PDE seen in Fig. 2, with the eception that the pper trblent branch is instead a region of transient chaos. This approach was motivated by the view that shear trblence is locally a chaotic saddle (Eckhardt et al., 2007) and it natrally etends previos ideas of modeling chaotic transients with maps (Chaté & Manneville, 1988; Bottin & Chaté, 1998; Vollmer et al., 2009). The reslting model has the advantage of being deterministic, as is flid flow, at least at the level of the Navier-Stokes eations. Here I consider an alternative approach and model trblence as noise. This is at the other etreme from the low-dimensional map. Here the dynamics is infinite dimensional and not deterministic. The simplest approach is to apply noise to the eation and assme it is proportional to itself. This leads to the following stochastic PDE (SPDE) model: t + U = ( + r 1 (r + δ)( 1) 2) + + ση, (3) t + U = ɛ 1 (1 ) ɛ 2. (4) where η = η(, t) is Gassian noise. The parameter σ controls the noise strength. In reality, shear trblence has significant correlations on the scale of a pff, bt these correlations are not considered here and η(, t) taken here to be space-time white. A large advantage of modeling the effect of trblence throgh a noise term is that one has a direct connection to the simple PDE model. Moreover, analysis of the SPDE is likely to be easier than analysis of the deterministic map model. The price is the loss of deterministic dynamics. Figres 3 and 4 show the regimes of transitional pipe flow from simlations of Es. (3)-(4). The deterministic parameters are as before: ɛ 1 = 0.04, ɛ 2 = 0.2, and δ = 0.1. The noise strength is σ = 1.4. Figre 3 shows soltion snapshots in terms of the model variable and. Pffs, pff splitting, and slgs are fond very similar to those observed in fll DNS (see Fig. 1) and in the deterministic map model (see Barkley, 2011). The dynamics of the different regimes is seen in the space-time plots of Fig. 4. At low r, pffs are metastable. They persist for long times before abrptly decaying. For intermediate r, pff splitting occrs. New pffs are spontaneosly ncleated downstream of eisting pffs and the system evolves to an intermittent mitre of trblent and laminar phases. At larger r, slgs are observed which differ from the deterministic PDE mainly in that they first occr at larger r and the pper branch is noisy rather than constant. An investigation of the lifetime statistics of pff decay and pff splitting in the SPDE is crrently nderway. While all three regions shown in Figs. 3 and 4 strongly resemble their conterparts in fll DNS and eperiment, the splitting regime is particlarly significant and worthy of frther comment.

5 (a) (a) eilibrim pff downstream (d) (b) (b) pff splitting split (e) t (c) slg 1000 epanding (f) Figre 3. Three regimes of transitional pipe flow from simlations of the SPDE (3)-(4). Shown are instantaneos vales of and. (a) Pff at r = 0.7. (b) Pff splitting at r = (c) Slg at r = 1.2. (c) i (space) Figre 4. Space-time diagrams illstrate (a) decaying pff at r = 0.7, (b) pff splitting at r = 0.94, and (c) slg formation at r = 1.2. Trblence intensity is plotted on a logarithmic scale in a frame co-moving with strctres. Unlike for pffs and slgs, which are essentially contained in the model by constrction, splitting is seen to arise natrally from the elementary pff-slg transition in the presence of comple trblent dynamics (either noise as here or chaotic dynamics as in Barkley (2011)). The spacetime plot in Fig. 4(b) cold easily be mistaken for the corresponding plot from fll DNS (e.g. see Avila et al., 2011). Sfficient trblence occasionally escapes from the irreglar downstream side of a pff to ncleate a new pff downstream. Visally, this is jst as in real pipe flow and is a strong alitative validation of this modeling approach.

6 3. Model for plane Coette flow One of the main difficlties in etending these ideas to other shear flows, sch as channel flow or plane Coette flow, is the compleity of the mean flow in these cases (Barkley & Tckerman, 2007). It is not clear at the present time whether one can adeately model the mean shear in these flows sing simple scalar fields. Nevertheless, I discss here some preliminary ideas on how plane Coette flow might be approached within this modeling framework. Figre 5 shows a sketch for plane Coette flow. A trblent patch (red) is shown srronded to the left and right by laminar flow. One can view this as a ct throgh a single trblent band in the striped regime or throgh a localized patch of trblence (Prigent et al., 2002; Barkley & Tckerman, 2005). However, the model is rather crde at present and the important three-dimensional aspects of the problem are not taken into accont. (a) y Figre 5. Sketch of plane Coette flow. The shaded region (red) represents a trblent patch srronded by laminar flow. Arrows indicate the motion of the bonding plates and the direction of the mean flow in the pper and lower halves of the domain. Three shear profiles are sketched. The -coordinate is centered on the time-averaged flow which has centro-symmetry. (b) Figre 6. Soltions of the plane Coette model, Es. (5)-(8). (a) Localized state at r = 0.7 and (b) periodic state at r = 0.9. Solid (dotted) crves represent variables in the pper (lower) half of the domain. The mean trblence, ( )/2, is plotted in bold (red). Recall that the geometry of plane Coette flow has translation symmetry in the streamwise direction and also centro-symmetry (rotation by π) abot any point on the midplane y = 0. Taking the point to be at = 0, the centro-symmetry is the transformation (, y) (, y). Time averaged trblent-laminar patterns break translational symmetry, bt do not break centro-symmetry (Barkley & Tckerman, 2007). The idea is to model intermittent trblence in plane Coette flow as two layers, each described by one-dimensional eations similar to those for pipe flow. The trblence in the two layers is assmed to be copled. Taking into accont that the mean advection in the bottom

7 layer is opposite to that in the top layer, I propose the following PDE model t 1 + U 1 = 1 ( 1 + r 1 (r + δ)( 1 1) 2) κ( 2 1 ), (5) t 1 + U 1 = ɛ 1 (1 1 ) ɛ , (6) t 2 U 2 = 2 ( 2 + r 1 (r + δ)( 2 1) 2) κ( 1 2 ), (7) t 2 U 2 = ɛ 2 (1 2 ) ɛ (8) where 1 and 1 are the trblence and mean shear in the pper layer and 2 and 2 are the trblence and mean shear in the lower layer. These eations are symmetric nder translation in and the reflection defined by (, 1, 1, 2, 2 ) (, 2, 2, 1, 1 ), which is the model eivalent of centro-symmetry. Figre 6 shows soltions to Es. (5)-(8) with U = 1 and the copling constant κ = 0.1. Other parameter vales are the same as for the model pipe simlations. For small r, stable localized states appear from localized pertrbations of laminar flow. At r 0.75 the localized states became nstable and spread to form a periodic alternation of trblent and laminar phases. The localized and periodic strctres both have centro-symmetry. The choice of U and the copling parameter κ are probably rather important in obtaining steady patterns. The effect of noise is also not yet flly nderstood as this investigation is still in a preliminary stage. 4. Conclsion I have presented models of parallel shear flows in two scalar variables trblence and mean shear. The model for pipe flow is based closely on physical featres of transitional trblence and it reprodces nearly all large-scale featres of transitional pipe flow. The model for plane Coette flow necessarily misses many featres of the real flow since the model is only one dimensional, whereas plane Coette flow has two etended dimensions. Nevertheless, the plane Coette flow model is an important starting point for frther investigations of this and other parallel shear flows sch as plane channel and bondary layer flow. References Avila, K., Moey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B Onset of sstained trblence in pipe flow. Science 333, Avila, M., Willis, A. P. & Hof, B On the transient natre of localized pipe flow trblence. J. Flid Mech. 646, Barkley, D Simplifying the compleity of pipe flow. Phys. Rev. E 84, Barkley, D. & Tckerman, L Comptational stdy of trblent laminar patterns in Coette flow. Phys. Rev. Lett. 94, Barkley, D. & Tckerman, L. S Mean flow of trblent laminar patterns in plane Coette flow. J. Flid Mech. 576, Blackbrn, H.M. & Sherwin, S.J Formlation of a Galerkin spectral element-forier method for three-dimensional incompressible flows in cylindrical geometries. J. Compt. Phys. 197, Bottin, S. & Chaté, H Statistical analysis of the transition to trblence in plane Coette flow. Er. Phys. J. B 6, Chaté, H. & Manneville, P Spatio-temporal intermittency in copled map lattices. Physica D 32,

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