Small scale exact coherent structures at large Reynolds numbers in plane Couette flow
SSmall scale exact coherent structures at largeReynolds numbers in plane Couette flow
Bruno Eckhardt
Fachbereich Physik, Philipps-Universt¨at Marburg, Renthof 6,35032 Marburg, GermanyE-mail: [email protected]
Stefan Zammert
Laboratory for Aero and Hydrodynamics, TU Delft, 2682 Delft, The NetherlandsJuly 2017
Abstract.
The transition to turbulence in plane Couette flow and several other shearflows is connected with saddle node bifurcations in which fully 3-d, nonlinear solutions,so-called exact coherent states (ECS), to the Navier-Stokes equation appear. As theReynolds number increases, the states undergo secondary bifurcations and their time-evolution becomes increasingly more complex. Their spatial complexity, in contrast,remains limited so that these states cannot contribute to the spatial complexity andcascade to smaller scales expected for higher Reynolds numbers. We here presentfamilies of scaling ECS that exist on ever smaller scales as the Reynolds number isincreased. We focus in particular on two such families for plane Couette flow, onecentered near the midplane and the other close to a wall. We discuss their scalingand localization properties and the bifurcation diagrams. All solutions are localizedin the wall-normal direction. In the spanwise and downstream direction, they areeither periodic or localized as well. The family of scaling ECS localized near a wall isreminiscent of attached eddies, and indicates how self-similar ECS can contribute tothe formation of boundary layer profiles.PACS numbers: 47.52.+j; 05.40.Jc a r X i v : . [ phy s i c s . f l u - dyn ] O c t mall scale exact coherent structures
1. Introduction
Pipe flow and various other parallel and non-parallel shear flows show a transitionto turbulence that is not connected to a linear instability of the laminar profile(Grossmann 2000). The transition can be triggered by finite amplitude bifurcationsand the new states that emerge are spatially and temporally fluctuating. The originof the transition and the subsequent dynamics cannot be understood within linearapproximations but require that the nonlinearity of the Navier-Stokes equation is takeninto account. Underlying the complex spatio-temporal patterns are exact coherent states(ECS) (Waleffe 1998, Waleffe 2001), i.e. velocity fields that are solutions to the Navier-Stokes equation with a relatively simple temporal dynamics: they can be fixed pointsof the equations of motion, travelling waves or more complex relative periodic orbits(Eckhardt 2007, Kerswell 2005, Eckhardt et al. 2007). ECS provide nuclei for theformation of turbulence. They typically appear in saddle-node bifurcations (Mellibovsky& Eckhardt 2011) and then undergo sequences of secondary bifurcations that givetemporally complex dynamics (Kreilos & Eckhardt 2012, Avila et al. 2013, Zammert& Eckhardt 2015). Crisis bifurcations can change the dynamics from persistent totransient, and collisions between different coherent structures can set up a networkthat sustains long-lived turbulent dynamics (Hof et al. 2006, Hof et al. 2008, Avilaet al. 2010, Schneider et al. 2010, Kreilos et al. 2014).The bifurcations just described follow the patterns familiar from the various routesto chaos and can explain the temporally complex dynamics (Eckmann 1981, Ott 2002).In order to realize the distribution of energy to ever smaller scales that are the hallmarkof fully developed turbulence (Frisch 1995) mechanisms that create structures on smallerscales are required. Steps towards developed turbulence are described in the studiesof (Kawahara & Kida 2001) where it is shown that ECS can capture some of theturbulent dynamics, and (van Veen et al. 2009), where coherent structures for models ofhomogeneous turbulence are described. In all cases the Reynolds numbers are moderateand the structures remain large-scale in the sense that they extend all the way acrossthe available volume. The examples presented below belong to families of states thatcan be scaled to ever finer spatial scales as the Reynolds number increases.All ECS are fully three-dimensional: all velocity components are active and theyvary in all three directions. Simpler structures, e.g. with translational invariance inthe downstream direction, decay (Moffatt 1990). Many of them share relatively stablerelations between their height, width, and downstream periodicity: if H denotes theheight, then the width of the structures is about πH and the downstream wavelengthis about 2 πH . For plane Couette flow, the exact optimal relations are documentedin (Clever & Busse 1997, Waleffe 2003), and the estimates for pipe flow are given in(Faisst & Eckhardt 2003, Eckhardt et al. 2008, Pringle et al. 2009). Similar results areavailable for plane Poiseuille flow, though the optimal wavelength described in (Zammert& Eckhardt 2016 a ) shows that there is some variability in the optimal ratios. All ECSjust described span across the entire height of the shear flow. mall scale exact coherent structures
2. Scaling in plane Couette flow
The flow we consider here is plane Couette flow, the flow between two parallel platesmoving relative to each other. With x the downstream direction, y the normal direction,and z the spanwise direction, the laminar profile is u ( x, y, z ) = Sy with the shear S = U /d for plates at y = ± d that move with velocity ± U . Deviations u from thelaminar profile then satisfy the equation ∂ t u + ( Sy e x · ∇ ) u + ( u · ∇ ) Sy e x + ( u · ∇ ) u + ∇ p = ν ∆ u . (1)with ν the kinematic viscosity. For stationary states, ∂ t u = 0 and only the spatialdegrees of freedom remain. For states moving with a constant phase velocity c in thedownstream direction, transition to a comoving frame ˜ x = x − ct gives ∂ t u = − c∂ x u and a time-independent equation in ˜ x .Let u ( x ) be a solution to the stationary equation for a viscosity ν . Then thescaled velocity field u λ ( x ) = u ( λ x ) /λ (2) mall scale exact coherent structures − c ∗ ∂ x u + ( Sy ∗ e x · ∇ ) u + ( u · e y ) S e x + ( u · ∇ ) u + ∇ p ∗ = ν ∗ ∆ u (3)in the scaled coordinates x ∗ = λ x with c ∗ = cλ and ν ∗ = ν/λ . That is to say, u λ is a solution at the modified viscosity ν ∗ . With this transformation, we also have toadjust the walls, and they move outwards at the same rate, d ∗ = dλ . However, if thestate is localized in the normal direction, the velocity fields will decay towards the wallsand the specific location of the walls will only have a small influence on the state. Bythe above heuristics, the state will be localized in the normal direction if the spanwiseand/or downstream periodicity are small compared to the initial distance between thewalls.To see the scaling in Reynolds number, we define Re = ( Sd ) d/ν , so that the rescaledstate u λ are equilibrium states forRe ∗ = λ Re . (4)or, alternatively, that a solution at Reynolds number Re ∗ can be obtained with therescaling λ = (cid:115) Re ∗ Re . (5)from a solution at Reynolds number Re. The scaling would be exact if the walls wereinfinitely far away. In the presence of the walls, the scaled states can be taken as initialconditions in a Newton refinement and ECS on the new scales can be obtained.For the numerical simulations we use Gibson’s Channelflow -code (Gibson 2012)and the optimized Newton methods for the determination of ECS. As a starting point,we use two equilibrium solution that are similar to Eq1 and Eq7 of (Gibson et al. 2009),which differ in their vortical content. They will collectively be referred to as EQ whenthey are turned into equilibrium solutions in the center and as TW when they are scaledas travelling waves near walls.
3. Families of scaling solutions in plane Couette flow
We begin with solutions that are localized in the center of the domain and stationary sothat c = 0. The initial computational domain has spanwise and streamwise periodicityof 0 . π and 1 . π , respectively, and a height of 2. Consistent with previous analysis, wecan determine the state accurately with a resolution N x × N y × N z = 32 × ×
48 near Re ∗ = 650. For higher Reynolds numbers, the resolution has to be increased, e.g. at Re = 10 we use a resolution N x × N y × N z = 24 × ×
56. At each Re we carefullychecked convergence and that the used resolution is sufficient.Although the rescaling works for any value of λ , we will here study powers of twoonly. Thus, using scaling factors λ of 2, 4 and 8, we can identify equilibrium solutionsthat are reduced in size by factors 2, 4, and 8 in all three directions. The corresponding mall scale exact coherent structures Figure 1.
Visualization of the exact solutions localized in the center of the channelfor Reynolds number 100 , Figure 2.
Visualization of the lower branch states of exact solutions near the lowerwall, at Reynolds number Re = 100 , Reynolds numbers increase by factors of 4, 16, and 64. The stationary state is initiallyidentified at Reynolds number near 3000, and then scaled up in Reynolds number tohigher values up to 192 , , mall scale exact coherent structures Re . . . . . D ( ~ u ) Eq c Eq c λ = 2 Eq c λ = 4 Eq c λ = 8 TW w TW w λ = 2 TW w λ = 4 TW w λ = 8 a) Re/λ . . . . . . D λ − /λ − − D λ b)Figure 3. Bifurcation diagrams for the scaled ECS in the center (derived from EQ ,full lines) and near the wall (derived from T W , dashed lines). (a) shows the unscaledbifurcation diagrams and (b) the rescaled ones where a collapse of the data is observed.The inset in (b) shows the rescaled dissipation of the wall states at Re = 10 versusthe scaling parameter λ . For the structures localized in the center, the midplane where c = 0 is a good point ofreference, and scaling by λ moves the boundary planes further away. For states close toa wall, the point of reference has to be the wall. Eventually, the state will move closerto the wall and the phase speed will approach ± U , the speed of the wall. Accordingly,we shift the domain upwards by d and change to a co-moving frame of reference where U ( y = 0) = 0 and U ( y = 2 d ) = 2 U . The equation for the stationary state remainssimilar to (3).The spanwise and streamwise wavelengths of the initial domain are 0 . π and0 . π , respectively. The scaled states are shown in Figure 2. This family of ECS isreminiscent of the structures used in attached eddy models for the logarithmic layerin wall turbulence, where the flow field is modeled by a hierarchy of eddies whichare attached to the wall and whose dimensions increase with the distance to the wall(Woodcock & Marusic 2015). A bifurcation diagram using the volume averaged dissipation, D ( u ) = 12 L x L z (cid:90) L z (cid:90) − (cid:90) L x (cid:107)∇ × u (cid:107) dxdydz, (6)along the ordinate is given in figure 3a. If one uses the rescaled dissipation on the abscissaand the rescaled Reynolds number Re ∗ = λ Re on the ordinate the bifurcation curvesshould collapse. Indeed, in the rescaled bifurcation diagram shown in figure 3b) thecollapse of the bifurcation curves for different λ is very good. mall scale exact coherent structures − . − . . . . y − . − . − . − . . u a) y τ − − − u τ b) . . . . . y e C F · c) y τ . . . e C F , τ d)Figure 4. Profiles for
T W w in the original coordinates (left column) and in wall units(right column). The linear laminar profile has been subtracted. (a) Profile of thestreamwise velocity. (b) Rescaled velocity profile. c) Profiles in the cross flow energydensity e CF as measure for the location of vortices. d) Rescaled profile of e CF in wallunits. At a fixed Reynolds number the volume averaged dissipation (eqn. 6) decreaseswith λ . But the states also become smaller with increasing λ , filling only a fraction1 /λ of the domain in the wall-normal direction, so that the rescaled dissipation λD isa better measure for the dissipation and its scaling. At fixed value of Re the rescaleddissipation increases with λ , as shown for T W and Re = 10 and 1 . · in the insetin Figure 3b). Scaling of the solutions requires that in the normal direction they are not or only weaklyinfluenced by the walls. In wall bounded flows, distances and velocities are usuallymeasured in wall units, based on the viscosity and the friction velocity u τ . The wallfriction is given by τ = ν (cid:42) ∂u x ∂ y (cid:43) w (7)where the index w indicates an average at the wall. Then the units for velocity are u τ = √ τ and (cid:96) τ = ν/u τ . With the scaling of the ECS given by (2) and the scaling ofthe viscosity, one finds that the scales at two Reynolds numbers Re ∗ and Re (as in (5))are related by τ ∗ = 1 λ τ u ∗ τ = 1 λ u (0) τ (cid:96) ∗ τ = 1 λ (cid:96) (0) τ (8) mall scale exact coherent structures λ is equivalent to a rescaling to wallunits if the solutions scale exactly. Since we have to adjust the solutions a little bit inorder to obtained converged states at the respective Reynolds numbers, there are smalldeviations in the friction factors, and hence in the wall units. Specifically, for the casesshown here, l τ varies between 3 . · − for the largest state with λ = 1 and 3 . · − for the smallest state with λ = 8, all evaluated at Reynolds number Re = 100 , λ the states become ever more localized near the walland the maximal amplitude becomes smaller. However, from the maximum to the upperwall, the decay is very slow and essentially linear, as shown in the rescaled solution inthe right column.Other measures provide a much clearer signal for the localization. For instance, thecross flow energy density, e CF ( y ) = 1 L x L z (cid:90) L x (cid:90) L x (cid:16) v + w (cid:17) dxdz, (9)which is shown in Figure 4c, decays much faster outside the region where the vorticesare located. The rescaled curves for the cross flow energy density (Figure 4d) collapseperfectly and reveal the similarity of the solutions. In the normal direction, the ECSmodify the mean velocity within the structure, but do not provide any forces furtheraway. In the absence of forces, the laminar shear profile is linear, which shows that thelinear profile in the outer region is a consequence of the viscous mediation between thedownstream velocity at the outer edge of the ECS and the velocity at the upper wall.In the other directions, one can adapt the model for streamwise localization inplane Couette flow (Brand & Gibson 2014 a , Barnett et al. 2016) to show that ECS areexponentially localized in the streamwise direction. In the spanwise direction, the decayseems to be somewhat stronger, as also noticed for large scale ECS (Schneider, Marinc& Eckhardt 2010).The upper branch states for both solutions have a much larger wall-normalextension than the lower branches states (see Figure 5). Thus, they are more stronglyinfluenced by the wall which causes an imperfect scaling, especially for low values of λ .For larger λ the range of the upper branch states in the wall normal direction decreases,resulting in better scaling.For both the states in the center and near the wall, the lower branch shows lessvariation in streamwise direction with increasing distance to the bifurcation point, whichis a common feature of lower branch states (Wang et al. 2007, Gibson & Brand 2014). In order to analyze the stability properties of the scales states the eigenvalues of theECS are calculated in computational domains with spatial periodicities equal to thoseof the states. The leading eigenvalues are shown in Figure 6. All states are unstable,but the number of unstable eigenvalues is rather low. The result show that the leading mall scale exact coherent structures Figure 5.
Comparison of the lower (a,c) and upper (b,d) branch states for the solutionslocalized near the lower wall. For the left pair of figures (a,b) the Reynolds number is18 ,
200 and for the right pair (c,d) it is 52 , .
002 for (a,b) and Q = 0 .
005 for (c,d).In the back-plane the streamwise velocity component is color-coded. The minimal(blue) and maximal (red) values of u are − . .
05 in (a,b) and − .
16 and 0 . .
00 0 .
02 0 . Re ( σ )-3-2-10123 I m ( σ ) · Re=5850Re=23400Re=93600 a) .
00 0 .
02 0 . Re ( σ )-2-1012 I m ( σ ) · Re=4000Re=16000Re=64000 b)Figure 6.
Stability properties of the rescaled ECS. (a) Real and imaginary parts ofthe leading eigenvalues for EQ at Re = 5850 and the scaled solutions for λ = 2 and 4at Re = 23400 and Re = 93600. b) Leading eigenvalues for T W at Re = 4000 and thescaled solutions for λ = 2 and 4 at Re = 16 ,
000 and Re = 64 , eigenvalues of states whose Reynolds numbers differ by the scaling factor λ , the leadingeigenvalues are almost identical. This is a consequence of the scaling in (3), which leavethe time-dependence invariant. Thus, the dynamics close to the ECS is similar in theadjusted domains. mall scale exact coherent structures
4. Spanwise and streamwise localization
In addition to the ECS in domains that are periodic in the downstream and spanwisedirection, we also tracked states that are localized in these directions (Schneider, Marinc& Eckhardt 2010, Schneider, Gibson & Burke 2010). As in other cases, good initialguesses can be be obtained by applying suitable window functions to extract nuclei forlocalized states from spatially extended states (Gibson & Brand 2014). We demonstratethis for one streamwise and two spanwise localized equilibrium solutions which arerelated to EQ and T W .Figure 7 shows visualizations of a streamwise localized equilibrium state relatedto EQ , and of the corresponding scaled solutions. As for the streamwise extendedsolutions, the visualizations for the different values of λ look very similar because thescaling works quite well. The bifurcation diagram of the streamwise localized states ismore complicated than for the spatially extended states. In particular, it has not beenpossible to trace the family of scaled states to a common fixed value of Re. They aretherefore visualized at different Reynolds numbers that differ approximately by a factor λ = 4.The figure shows that the vortex tubes are oriented in a V-shape which is also afeature of the recently identified doubly localized equilibrium states of PCF (Brand &Gibson 2014 b ). The states show an exponential decay of the velocity components intheir tails. Models for the streamwise decay length (cid:96) in an exponential representation u ∝ exp( −| x | /(cid:96) ) of the downstream variation show that (cid:96) increases with Reynoldsnumber but decreases with spanwise wavelength (Brand & Gibson 2014 b , Zammert &Eckhardt 2016 b ). For the rescaled ECS studied here this means that the stretching of (cid:96) due to the increase in Reynolds number is compensated by the reduction of the lateralscales so that the overall all directions can be rescaled by λ .
5. Conclusions
The tracking of ECS from large scales to ever smaller scales at increasing Reynoldsnumbers show that a multitude of small-scale ECS populate the state space of flowsat high Reynolds numbers. Their localization in the wall-normal direction show thatsimilar states can also appear in shear flows with curvature in the mean profile, suchas Poiseuille flow, since eventually the states will only probe the local shear gradient(Deguchi 2015).The two cases studied here are located at the midplane of the domain, where themean velocity is 0, and near the walls, where the mean advection speed approaches thespeed of the wall. For states at some distance to the wall, one can keep that distancefixed and scale the states so that they become localized at that height. Initially, forlow Reynolds numbers, there will be some influence of the walls, but then for higherReynolds numbers and more localized states, the influence from the walls will becomesmaller, and one can anticipate that the states become similar to the ones in the center. mall scale exact coherent structures a) − −
20 0 20 40 x . . . . . . . . M ( v ) b) − −
20 0 20 40 λx . . . . . . . . λ M ( v ) c)Figure 7. Streamwise localized equilibrium solutions. (a) Visualizations of the flowfields at Reynolds numbers 900, 3750, and 15600. The streamwise and spanwise lenghtsof the largest domain are 7 . π and 0 . π , respectively. (b) Maximum of the wall-normalvelocity component M ( v ) versus the streamwise position x for the three localizedequilibrium solutions. (c) M ( v ) rescaled by λ . Note that the green line is covered bythe blue one, showing the almost perfect scaling for high Reynolds number. The localization in the normal direction, and also in the other directions, impliesthat sets of states can be combined to form ECS of more complex spatial structures:for superpositions of localized states the nonlinear terms are weak if they are very farapart, and even if the interactions between the two states are stronger, the superpositionprovides a good starting point for a Newton method. In the few cases where weattempted such superpositions, the Newton method was able to adjust the flow fieldsso that converged ECS that are have two (or more) centers of localization could beobtained.For the staggered attached eddies used to represent boundary layers (Perryet al. 1991, Perry et al. 1994) a simple superposition will not work because the statesoverlap not only in their fringes but in their core. The interactions will then be morecomplicated than the simple perturbative adjustment that worked for spatially separatedstructures, and remains a challenge for computations. Without that interaction, the wall mall scale exact coherent structures
Acknowledgements
We thank the participants of the 2017 KITP Workshop ”Recurrent Flows: TheClockwork Behind Turbulence” for discussions, and the National Science Foundationfor partial support of KITP under Grant No. NSF PHY11-25915. This work wasalso supported in part by the Deutsche Forschungsgemeinschaft within FOR 1182 andby Stichting voor Fundamenteel Onderzoek der Materie (FOM) within the program”Towards ultimate turbulence”.
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