Variational identification of minimal seeds to trigger transition in plane Couette flow
aa r X i v : . [ phy s i c s . f l u - dyn ] N ov Under consideration for publication in J. Fluid Mech. Variational identification of minimal seeds totrigger transition in plane Couette flow
S. M. E. R A B I N † , C. P. C A U L F I E L D , A N D
R. R. K E R S W E L L Department of Applied Mathematics & Theoretical Physics, Centre for MathematicalSciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK BP Institute, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB30EZ, UK School of Mathematics, University of Bristol, BS8 1TW Bristol, UK(Received 1 November 2018)
A variational formulation incorporating the full Navier-Stokes equations is used toidentify initial perturbations with finite kinetic energy E which generate the largestgain in perturbation kinetic energy (across all possible time intervals) for plane Couetteflow. Two different representative flow geometries are chosen corresponding to thoseused previously by Butler & Farrell (1992) and Monokrousos et al. (2011). In the former(smaller geometry) case as E increases from 0, we find an optimal which is a smoothnonlinear continuation of the well-known linear result at E = 0. At E = E c , however,completely unrelated states are uncovered which trigger turbulence and our algorithmconsequently fails to converge. As E → E + c , we find good evidence that the turbulence-triggering initial conditions approach a ‘minimal seed’ which corresponds to the stateof lowest energy on the laminar-turbulent basin boundary or ‘edge’. This situation isrepeated in the Monokrousos et al. (2011) (larger) geometry albeit with one notablenew feature - the appearance of a nonlinear optimal (as found recently in pipe flow byPringle & Kerswell (2010) and boundary layer flow by Cherubini et al. (2010)) at finite E < E c which has a very different structure to the linear optimal. Again the minimalseed at E = E c does not resemble the linear or now the nonlinear optimal. Our resultssupport the first of two conjectures recently posed by Pringle et al. (2011) but contradictthe second. Importantly, their prediction that the form of the functional optimised is notimportant for identifying E c providing heightened values are produced by turbulent flowsis confirmed: we find the what looks to be the same E c and minimal seed using energygain as opposed to total dissipation in the Monokrousos et al. (2011) geometry.
1. Introduction
The investigation of hydrodynamic stability is one of the canonical problems of fluiddynamics. A particularly interesting archetypal flow is plane Couette flow (PCF), wherethe flow is between two parallel plates moving at a relative velocity 2 U , separatedby a distance 2 h , with thus a characteristic Reynolds number Re = U h/ν where ν is the kinematic viscosity. PCF is linearly stable for even large Re (Romanov (1973))yet turbulence has been observed experimentally as low as Re = 325 (Bottin & Chate(1998)). It has been hypothesized that transient perturbation growth, due to the non-normality of the underlying linear operator of the Navier-Stokes equations, may explainthis disconnect. Several authors (for example Gustavsson (1991); Butler & Farrell (1992); † Email address for correspondence: [email protected]
S. M. E. Rabin, C. P. Caulfield, and R. R. Kerswell
Reddy & Henningson (1993)) demonstrated that substantial transient kinetic energy gain G ( T ) = E ( T ) /E (0) (where E ( T ) is the infinitesimally small kinetic energy at the finaltime T ) could be achieved by a linear (infinitesimal) optimal perturbation (LOP) (seeSchmid (2007) for a review). Proponents of such an essentially linear mechanism for en-ergy growth point to the Reynolds-Orr equation (Schmid (2007)) as an indication thatenergy growth is a linear effect, as this equation shows that dE/dt is independent of thenonlinear advective terms in the Navier–Stokes equations. The argument is that if a realflow is seeded with a ‘small’, yet finite amplitude perturbation with the same structureas a LOP, the transient energy gain of the LOP could be sufficiently large to ‘push’ theperturbation into the ‘nonlinear regime’ and hence trigger transition.However, this line of thinking is based on a couple of implicit assumptions: that the‘entrance’ into the nonlinear regime of this perturbation will lead to turbulence; andthat the linear optimal perturbation (LOP) is still the ‘best’ choice for the growth ofnonlinear perturbations. The latter assumption can be explicitly probed by posing theoptimal growth problem for initial perturbations of finite amplitude which can affect thebase flow as they grow. This has been done recently for pipe flow (Pringle & Kerswell(2010)) and boundary layer flow (Cherubini et al. (2010)) with both studies discoveringthe existence of a nonlinear optimal perturbation (NLOP) which has a very differentstructure to the LOP and outgrows it beyond a small but finite energy threshold.With more of an eye on reaching turbulence, Monokrousos et al. (2011) posed a dif-ferent problem for PCF by maximising the total energy dissipation over a long but fixedtime period. They purposely looked for a turbulent end state at the end of their opti-misation window and then worked downwards in initial energy to identify the thresholdfor transition. Earlier, Pringle & Kerswell (2010) had failed to identify this thresholdby working upwards in initial energy because of convergence issues. However, a follow-up study (Pringle et al. (2011), henceforth referred to as PWK11) with a more efficientcode run at higher resolution succeeded in identifying a (converged) nonlinear optimalfor larger initial energies. At E = E fail , they again found a failure to converge butnoticed that this corresponded to their optimisation algorithm encountering turbulent(end) flows. The conclusion was that this failure energy E fail is sufficiently large to en-able an initial perturbation to undergo the transition to turbulence: i.e. E fail > E c , theenergy threshold of transition (PWK11 actually conjectured that E fail = E c : see Con-jecture 1 below). In dynamical systems parlance, this initial perturbation is then in thebasin of attraction of the turbulence or more generally (if the turbulence is actually notan attractor but a chaotic saddle), has crossed the ‘edge’, a hypersurface which separatesinitial conditions which become turbulent from those with relaminarise (Itano & Toh(2001); Skufca et al. (2006); Schneider et al. (2007); Duguet et al. (2008)). The picturethen put forward by PWK11 is that as E increases from 0, the E = E hypersurfacein phase space intersects the edge for the first time at E = E c and that the initialperturbation which corresponds to their (generically unique) intersection at E = E c isthe ‘minimal seed’ for triggering turbulence. This seed is ‘minimal’ in the sense that itis the lowest energy state on the edge and therefore represents the most energy efficientway of triggering turbulence by adding an infinitesimal perturbation to it. PWK11 alsofind evidence to suggest that the NLOP tends to the minimal seed associated with thisloss of convergence (and transition to turbulence) as E → E − c . They summarise theirthinking as two conjectures.“ Conjecture 1 : For T sufficiently large, the initial energy value E fail at which theenergy growth problem first fails (as E is increased) to have a smooth optimal solutionwill correspond exactly to E c . Conjecture 2 : For T sufficiently large, the optimal initial condition for maximal en- inimal seeds in plane Couette flow E = E c − ǫ converges to the minimal seed at E c as ǫ →
0. ”In this paper, we wish to investigate the validity of these conjectures in the context ofPCF. Two sets of geometry and Reynolds numbers are considered for PCF, one discussedin each of Butler & Farrell (1992) (henceforth referred to as BF92) - a relatively narrowspanwise domain with Re = 1000 - and Monokrousos et al. (2011) (henceforth referredto as M11) who used a domain with double the spanwise extent at Re = 1500. Byconsidering these different situations, we are able to investigate whether there is anythinggeneric that can be said about the progression of optimal perturbations starting with the(infinitesimal) LOPs at E = 0, through NLOPs as E increases to the minimal seedat E = E c . Of principal interest will be whether this approach can identify E c andthe form of the minimal seed either directly (by smooth evolution of the optimal as E → E − c ) or indirectly (by failing to converge). By considering the M11 geometry butchoosing to maximise the energy gain rather than total energy dissipation, we also assessthe sensitivity of the procedure to the exact choice of optimising functional.From the technical perspective, we also take this opportunity to further develop thevariational formulation to include optimisation over T , the duration of the observationwindow or ‘target time’. This means we are then able to identify the initial perturba-tion which achieves the highest gain possible over all T with the corresponding optimalfinal time now an interesting output. All previous studies (Pringle & Kerswell (2010);Pringle et al. (2011); Cherubini et al. (2010); Monokrousos et al. (2011)) chose to workwith a pre-defined T over which to perform their optimization. Conceptually, letting T bean output of the optimisation seems a significant advance yet operationally, it is requiresonly a small adjustment in the algorithm.The plan of the paper is as follows. In section 2 we briefly present the variationalframework and discuss how the new target time optimisation is carried out. The followingtwo sections, 3 and 4, discuss the results obtained for the BF92 and M11 situationsrespectively. A final section 5 then discusses the results in light of the above-quotedconjectures of PWK11 and draws a number of conclusions.
2. Lagrangian framework
We seek the initial disturbance of kinetic energy E to the laminar flow which attainsthe largest energy growth G ( T ) := E ( T ) /E a time T later while evolving under theNavier-Stokes equations, remaining incompressible and respecting the applied boundaryconditions. Here E ( T ) := h u ( T ) , u ( T ) i , with the angle brackets denoting, h v , u i := 1 V Z D v † u dV , (2.1)where † denotes the Hermitian conjugate, and V is the volume of the domain D . Theflow configuration considered is PCF with coordinate system such that the streamwisedirection is x , the wall normal direction is y and the spanwise direction z . The x and z directions are assumed to be periodic and the separation between the walls (2 h ) is usedto scale length so that their positions are given by y = ±
1. The speed difference betweenthe walls (2 U ) scales the velocity so that the non-dimensionalised background Couetteflow is U ( y ) = y e x and the Reynolds number Re := U h/ν .The functional to be extremised is the energy gain which, when constrained by theNavier Stokes equations, the initial energy value E (0) = E and incompressibility, pro- S. M. E. Rabin, C. P. Caulfield, and R. R. Kerswell duces the Lagrangian L := E ( T ) E − [ ∂ t u + N ( u ) + ∇ p, v ] − [ ∇ . u , q ] − (cid:18) h u , u i − E (cid:19) c + h u − u (0) , v i . (2 . a ) N is the nonlinear operator N ( u i ) := U j ∂ j u i + u i ∂ i U j + u j ∂ j u i − Re ∂ j ∂ j u i , (2 . b )and square brackets denote a time average of the inner product,[ v , u ] := 1 T Z T h v , u i dt . (2.3)In the Lagrangian, v , q , v and c are Lagrange multipliers, u is the initial value of theperturbation velocity u and U the background Couette flow. While not strictly necessaryto divide our cost functional by E we found it easier to tune our algorithm by doing so.Taking first variations of the Lagrangian with respect to v , q , v and c and settingthem to zero recovers (respectively) the constraints of the Navier Stokes equations, in-compressibility, the initial kinetic energy of E and initial state u = u (0), δ L δ v = ∂ t u + N ( u ) + ∇ p := 0 , (2.4) δ L δq = ∇ . u := 0 , (2.5) δ L δ v = u − u (0) := 0 . (2.6) δ L δc = 12 h u , u i − E := 0 . (2.7)First variations with respect to the physical variables yields a complementary set ofadjoint equations, δ L δ u = ∂ t v + N † ( v , u ) + ∇ q + (cid:18) u E − v (cid:19) | t = T + ( v − v ) | t =0 := 0 , (2.8) δ L δp = ∇ . v := 0 , (2.9) δ L δ u = v − c u := 0 . (2.10)Here, N † ( v i , u ) := ∂ j ( u j v i ) − v j ∂ i u j + ∂ j ( U j v i ) − v j ∂ i U j + 1 Re ∂ j ∂ j v i (2.11)can be identified as the adjoint of N and v , v and q are the adjoint variables of u , u and p . Equation (2.8) is in reality three equations. The first part, ∂ t v + N † ( v , u ) + ∇ q = ,must be satisfied at all times and is the adjoint Navier Stokes equation. Since the fullNavier-Stokes equations have been imposed, the adjoint operator depends on the velocityfield u . The sign of the diffusion term is also reversed and therefore the adjoint equationcan only be solved backwards in time. The second part of (2.8), ( u /E − v ) | t = T = 0,is a terminal condition, linking our physical and adjoint variables and needs only to besatisfied at time T . The third part, ( v − v ) | t =0 = , is a condition linking v and v (0),which must be satisfied at t = 0.Further to previous recent formulations (Pringle & Kerswell (2010); Pringle et al. (2011); inimal seeds in plane Couette flow et al. (2010); Monokrousos et al. (2011)), we also optimize over the target time T . The first variation with respect to T yields the simple relation ∂ L ∂T := 1 E ddT E ( T ) = 0 (2.12)provided u is incompressible and satisfies the Navier-Stokes equations at t = T . Ouralgorithm then proceeds as follows. We first start with a suitable guess for the optimalinitial condition, u and a target time T . We then time march our initial condition totime T using the Navier-Stokes equations and use ( u /E − v ) | t = T = 0 to ‘initialise’the adjoint equations which are then solved backwards in time to calculate v . Thisprocedure ensures that all the variational equations are satisfied apart from (2.10) and(2.12). If the current value for u is optimal then (2.10) will be satisfied: on the otherhand if (2.10) is not satisfied, it provides an estimate for the gradient δ L /δ u . Using thisgradient we then use a method of steepest ascent to update our guess for u , while c is simultaneously calculated by ensuring that our new initial condition has an energy of E . Once a new value of u is obtained, T is updated by integrating the Navier-Stokesequations forward in time using the updated u as an initial condition until a maximumvalue of E ( t ) is reached. The time of this maximum is taken as the new value of T and(2.12) is then satisfied.
3. BF92 geometry
The underlying objective of this paper is to investigate how optimal initial conditionsfor energy growth some T later change as a function of E . One specific issue is whetherthere is always an energy range below E c where a NLOP is the optimal (a NLOP beingan initial condition qualitatively different in structure and gain from the LOP). A secondis whether E fail = E c and a third is examining the form of the optimal as E → E c fromabove or below.Results are first presented from a geometry studied previously in the linear regime( E →
0) in BF92: a periodic box with dimensions L x = 2 π / .
49 = 13 . L y = 2and L z = 2 π / . .
31 (or 4 . π × × . π ) with Re = 1000. A modified version ofthe Diablo CFD solver, (Taylor & Sarkar (2008)) which is spectral in x and z and finitedifference in y , was used to solve the forward and adjoint equations using a resolution of128 × ×
32 in x , y and z respectively. For sufficiently low energies we found that theoptimal perturbation was extremely similar to the LOP in both gain and structure and asresult was named a ‘quasi linear optimal perturbation’ (QLOP). The QLOP achieved amaximum gain of approximately 1100 at T = 125 (in units of h/U ). With increasing butstill small E , the gain and optimal time of the QLOP remains fairly constant as shownin figure 1(a). However, beyond a certain energy threshold, (approximately 2 . × − )there is a sudden and large jump in the gain achievable, as shown in figure 1(b) (notethe change of ordinate scale). In addition to the much higher gain at E = 2 . × − ,it is noticeable that the optimal time tends to very large values as E approaches thistransition energy from above. The initial conditions found by our algorithm there clearlyevolve into a turbulent state given the much higher target-time kinetic energy valuesand the highly disordered endstate. This implies that E c . . × − is the thresholdenergy for transition. To examine convergence, h δ L /δ u , δ L /δ u i /G is plotted in figure2 against iteration for E = 5 . × − < E c and E = 5 . × − > E c .For the smaller initial energy case (figure 2(a)) we see that after 10 iterations the gainhas plateaued and that the value of the normalised gradient has dropped by 10 orders ofmagnitude, which suggests that the QLOP is converging well. S. M. E. Rabin, C. P. Caulfield, and R. R. Kerswell −8 −6 G a i n E −8 −6 O p t i m a l T −8 −6 −6 −5 −4 G a i n E −6 −5 −4 O p t i m a l T Figure 1. ( a ) Gain (blue circles) and associated optimal time (red crosses) against E . E c ≈ . × − is marked by vertical green dashed line. LOP gain is the horizontal blue solidline, LOP optimal time is the horizontal red dashed line. ( b ) Gain (blue circles) and associatedoptimal time (red crosses) against E for E > . × − . G a i n Iterations −5 G r a d i e n t / G G a i n Iterations G r a d i e n t / G Figure 2.
G (blue solid line) and h δ L /δ u , δ L /δ u i /G , (red dashed line) plotted againstiteration for ( a ) E = 5 . × − ( b ) E = 5 . × − . The QLOP presented in ( a ) appears tobe converging well, whereas there is no convergence in ( b ). Conversely, for the higher energy case (figure 2(b)) while the gain appears to plateau,the gradient is failing to decrease in size indicating the algorithm is not converging.In reality, because of the turbulent nature of the flow at time T we would not expectconvergence to be possible, as a very small change in the initial condition is likely toproduce a significant change in the final state. Despite this lack of convergence, thealgorithm is successful in finding initial states which trigger turbulence when E > E c .In the picture of PWK11, one unique initial condition - the minimal seed - shouldemerge as the limiting state for turbulence-triggering initial conditions as E → E + c . Toexamine the dynamical route states close to the minimal seed take to turbulent disorder,we have considered in detail a turbulent seed found at E = 2 . × − as it evolvesin time. Figure 3(a) plots the kinetic energy and dissipation rate against time and in3(b) the same quantities are plotted for the ‘rescaled’ turbulent seed whose initial energy inimal seeds in plane Couette flow −2 G a i n Time0 100 200 300 40000.00250.0050.00750.01 D i ss i p a t i on −2 G a i n Time0 100 200 300 40000.00250.0050.00750.01 D i ss i p a t i on Figure 3.
Gain (blue solid line) and dissipation (red dashed line) against time for ( a ) theturbulent seed at E = 2 . × − and ( b ) rescaled turbulent seed at E = 2 . × − . Bothinitially behave similarly achieving a gain of approximately 1000. They maintain this energylevel for an extended period of time, until t ∼ is E = 2 . × − . The behaviours of the turbulent seed and rescaled turbulent seedare very similar up to t ∼ h/U . Beyond this time, however, the kinetic energies ofthe two initial perturbations begin to differ significantly, with the rescaled turbulentseed eventually decaying so the flow relaminarises whereas the turbulent seed triggerstransition at T ∼ h/U . (Reassuringly, if the rescaled turbulent seed is used to initiatethe optimizing procedure at E = 2 . × − , the algorithm converges to the expectedQLOP result.)We observe in figure 3 that both flows spend an extended period of time at an interme-diate (perturbation) energy level before going their separate ways. This is because bothinitial states are close to the edge (but on ‘opposite sides’) and spend some time trackingit while being gently repelled (in opposite directions). To confirm this, the turbulentseed and its rescaling can be used to refine the initial condition so that it stays nearerto the edge for longer (Itano & Toh (2001); Skufca et al. (2006); Schneider et al. (2007);Duguet et al. (2008)). In figure 4, this refinement is carried out to track the edge for t = 400 h/U showing that the edge state (attracting state for edge-confined dynamics)has constant energy (consistent with the work of Schneider et al. (2008) who treat a PCFsystem 4 π × × π albeit at Re = 400 and find a steady edge state).Figure 5 shows a side by side comparison of the contours of the perturbation stream-wise velocity at x = 0 at four different times for the QLOP at E = 2 . × − , theminimal seed at E = E c (at least to the accuracy of figure 4) and the turbulent seed(which clearly is close to the minimal seed) at E c . E = 2 . × − . These plots demon-strate that while the minimal seed evolves towards the edge state, a small increase inits initial energy will lead to transition. As E → E c from above (below), the time totransition (relaminarisation) tends to ∞ due to the extra time needed to evolve upwards(downwards) in energy away from the edge. It is also clear that the QLOP is completelydifferent from the minimal seed.Finally it is worth examining the 3D structure of the time-evolving QLOP, the minimalseed and the turbulent seed just above the edge. In figure 6 we plot iso-contours of thestreamwise velocity for times 0, 150, 250 and 350 (in unit of h/U ). From the plots it is S. M. E. Rabin, C. P. Caulfield, and R. R. Kerswell −4 −3 −2 −1 Time E n e r g y Figure 4.
Perturbation energy against time for various initial states close to the edge. Upperbound of edge state blue, lower bound red. Every 100 time units, the edge state is rescaled toproduce new upper and lower bounds. The minimal seed at E c (blue line) stays on the edge(over this time period) and gets attracted to the edge state which emerges as having constantenergy. The turbulent seed at E = 2 . × − is shown as the first blue/orange dashed line andthe rescaled turbulent seed is the first red/green dashed line. clear that the minimal seed is initially quite localized but quickly ‘unpacks’ itself intoa series of a streamwise streaks. This unpacking process appears to be achieved by thewell-known Orr mechanism followed by the lift-up mechanism. The minimal seed flowthen remains in this configuration, whereas the streaks destabilise and there is transitionto turbulence for the higher energy turbulent seed ‘above’ the edge.To summarise, in this geometry and at this Re , it appears that a well-converged NLOPdoes not exist prior to our algorithm uncovering turbulence-triggering initial conditionsat E = E fail . Our algorithm only fails to converge if there are turbulent seeds presentso E fail > E c and we find no evidence for inequality consistent with PWK11’s firstconjecture. The minimal seed (as is apparent in figures 5 and 6) is qualitatively differentfrom the QLOP, and so the optimals do not converge to the minimal seed as E → E − c ,a clear counterexample to PWK11’s second conjecture.
4. M11 geometry
In this section we examine a second, larger geometry of dimensions 4 π × × π (es-sentially twice as wide as that in BF92) at a higher Reynolds number Re = 1500. Wedemonstrate in this geometry that now a NLOP exists at energies below E c and inves-tigate whether in the limit E → E − c it converges to the minimal seed. Choosing thegeometry and Reynolds number used by M11 has the added benefit that we can com-pare our results to those obtained using an entirely different functional. M11 optimizedthe total dissipation over a long time interval rather than the energy gain achieved at aspecific target time. We find a critical energy value E c = 3 . × − , plotted in figures 7(a) and (b), which agrees well with M11, who find 3 × − < E c < × − (see theirfigure 1). Note ǫ in M11 is E here as || || E in their equation (1) is strictly a kineticnorm with a included (Monokrousos, personal communication). Our calculated timefor transition at E = 4 . × − is approximately 200 not too dissimilar from the timeof 150 in M11. This suggests that the particular choice of optimizing functional is notimportant for the calculation of a minimal seed (or more accurately to lose convergence), inimal seeds in plane Couette flow z y t=0 −5 z y t=0 −3 z y t=0 −3 z y t=150 z y t=150 z y t=150 z y t=250 z y t=250 z y t=250 z y t=350 z y t=350 z y t=350 Figure 5.
Contours of streamwise velocity u at times 0, 150, 250, 350 forQLOP (left), minimal seed (centre) with E = E c and turbulent seed (right)for E = 2 . × − & E c . Contour levels are: going down the left column(min,spacing,max)=( − , , × − , ( − . , . , . − . , . , .
1) and ( − . , . , . − , . , . × − and ( − . , . , .
1) subse-quently; going down the right column (min,spacing,max)=( − , . , . × − , ( − . , . , . − . , . , .
5) and ( − . , . , . provided the functional attains heightened values for turbulent flows (as discussed inPWK11).Figure 7( a ) indicates that three energy regimes exist in this geometry rather than thetwo in BF92. As before, below a certain initial energy value, a QLOP is selected andabove a critical energy E c initial conditions significantly different from the QLOP triggerturbulence. Between these two energy regions, however, there now exists a range of initial0 S. M. E. Rabin, C. P. Caulfield, and R. R. Kerswell
Figure 6.
Iso surfaces of streamwise velocity u , at 60% of maximum and minimum values, forthe QLOP at E = 2 . × − (left), the minimal seed (centre) and a turbulent seed above theedge at E = 2 . × − (right), at times 0, 150 ,250, 350. energies where our algorithm generates an initial condition different from the QLOP (inthe sense that it appears to have a qualitatively different spatial structure) - see figure 8.Using the nomenclature described in the introduction, we call this qualitatively differentoptimal perturbation a NLOP after Pringle & Kerswell (2010) and PWK11. Figure 9contrasts the convergence for the NLOP with the non-convergence in the turbulent seedregion E > E c . Only five points are plotted in figure 9 as for additional iterations(however small we made our step size in the direction of the gradient) it was not possibleto find a new u with a gain that improved on the previous iteration.As a consequence of the kinetic energy gains of the NLOP and QLOP becoming verysimilar around E = 1 . × − , the cross-over between the NLOP and QLOP is hard topinpoint. If the NLOP is used to initialise the algorithm for energies slightly above E c ,the algorithm is found to converge to an initial condition very similar to the NLOP. Infact, the algorithm started with random noise will still sometimes converge to the NLOP inimal seeds in plane Couette flow −8 −7 G a i n E −8 −7 O p t i m a l T −8 −7 −8 −7 −6 G a i n E −6 O p t i m a l T Figure 7. ( a ) Gain, E ( T ) /E (0), of QLOP (blue circles) and NLOP (orange circles) and as-sociated optimal time of QLOP (red crosses) and NLOP (green crosses), T , against E . E c ismarked by vertical green dashed line. LOP gain horizontal blue solid line, LOP optimal timehorizontal red dashed line. ( b ) Gain, E ( T ) /E (0), (blue circles) and associated optimal time (redcrosses), T , against E . E c is marked by vertical green dashed line. Figure 8.
Iso surfaces of streamwise velocity u , at 60% of maximum and minimum values, for( a ) QLOP at E = 5 . × − and ( b ) NLOP at E = 3 . × − . It is clear that the NLOP isdistinct from the QLOP. for values of E twice as large as E c . As a consequence, approaching E c from aboveproved a better strategy. Random noise was used at E ∼ . × − ≈ . E c to find a turbulent seed and then this was used sequentially to initiate the algorithm as E wasgradually decreased. This experience clearly emphasizes the main hazard of nonlinearoptimisation: it is easy to get stuck near local maxima. Although not a cure, an obviousstrategy to reduce this possibility is to look for robustness of result over a suite of initialconditions.Figure 7 also indicates that the turbulent seeds show the same trend, as in the BF92geometry, with regards to the optimal target time, namely that it increases drastically as E → E + c . We conclude that the turbulent seeds remain near the edge for an even greaterperiod of time than the turbulent seeds in the BF92 geometry. This may be because theyare closer to the edge and/or that the edge is less repelling. As before, we have tracedthe edge up to t = 400 h/U using a slightly rescaled turbulent seed to find a similar2 S. M. E. Rabin, C. P. Caulfield, and R. R. Kerswell G a i n Iterations G r a d i e n t / G G a i n Iterations G r a d i e n t / G Figure 9.
G (blue solid line) and h δ L /δ u , δ L /δ u i /G (red dashed line) plotted against iter-ation for (left) E = 1 . × − and (right) E = 3 . × − . The NLOP (left) appears to beconverging well, whereas the calculation which throws up turbulent seeds does not. plot to figure 4 (not shown). A comparison of cross sectional streamwise velocities forthe NLOP at E = 3 . × − , the minimal seed and the turbulent seed at 3 . × − in figure 10 again emphasizes their different temporal evolutions despite being so closeenergetically. Also interestingly, the NLOP is localised in the cross-stream direction andis not dissimilar from the minimal seed although they are clearly not the same. This ismade obvious by comparing their streamwise structure: see the top row of figure 11.Figure 11 also shows how the NLOP unpacks into a series of streamwise streaks.An examination of the early time suggests that it is a combination of the Orr andlift-up mechanisms (as discussed in PWK11) that is responsible for the localized flowunpacking into streamwise streaks. The minimal and turbulent seeds also unpack inthe streamwise and cross-stream direction producing streamwise streaks which are stillspanwise localised. If there is sufficient energy in these streaks they are unstable (theturbulent seed) otherwise not (the minimal seed). By comparing the isosurface of theQLOP at time zero (figure 8) and the isosurfaces depicting the time evolution of theNLOP it is clear that the NLOP evolves into a structure which is very similar to theQLOP. This suggests that while at early time there exists a distinct NLOP structure,which is able to extract enhanced gain from the base flow by ‘unpacking’, it later exploitsthe same ‘lift-up’ mechanism as the QLOP at intermediate times.It is significant that the minimal and turbulent seeds are spanwise-localised (at leastuntil the turbulence is reached) in this 2 π wide geometry and not in the 1 . π widegeometry of BF92. Of course the higher Re must be a contributory factor but thespanwise dimension does seem important. Pringle & Kerswell (2010) originally foundan azimuthally-localised (and radially-localised) NLOP in a short pipe where one couldtalk about a ‘spanwise’ (azimuthal) lengthscale of 2 π (radii or half-channel heights). Theemergence of a NLOP in the wider geometry is also noteworthy. The LOP, and by def-inition QLOP, are global periodic states which are largely insensitive to the geometry,whereas the gathering evidence is that the NLOP is an attempt by the fluid to localisein order to maximise the energy gain for given global kinetic energy. As a result, thegeneral trend should be for the energy cross-over from QLOP to NLOP to de crease withincreasing domain size. Clearly this cross-over is above E c for the BF92 geometry at Re = 1000 and below E c for the M11 geometry at Re = 1500. inimal seeds in plane Couette flow z y t=0 −4 z y t=0 −4 z y t=0 −4 z y t=75 z y t=75 z y t=75 z y t=150 z y t=150 z y t=150 z y t=250 z y t=250 z y t=250 z y t=400 z y t=400 z y t=400 Figure 10.
Contours of streamwise velocity u for NLOP E = 3 . × − (left), approxi-mated minimal seed and turbulent seed at E = 3 . × − (right), at times 0, 75, 150, 250and 400. At intermediate times, the minimal seed flow on the edge and that initiated fromthe turbulent seed remain relatively similar. Contour levels are: going down the left column(min,spacing,max)=( − , , × − , ( − . , . , . − . , . , . − . , . , . − . , . , . − , , × − ,( − . , . , . − . , . , . − . , . , .
05) and ( − . , . , . − , , × − , ( − . , . , . − . , . , . − . , . , .
1) and ( − . , . , . The evolution shown in the right column of figure 11 looks very similar to that shownin figure 4 of M11 (albeit at a slightly different initial energy) indicating that we havegenerated an approximation to the same (presumably unique) minimal seed. This isfurther supported by the aforementioned correspondence in their and our estimates for E c . Beyond validating each others results (which is important for nonlinear optimisationproblems), this points to an insensitivity in the choice of the functional to be maximisedfor finding the minimal seed. There is one proviso, of course, that the functional mustbe selected so that it detects turbulent flows by assuming large values as discussed inPWK11.4 S. M. E. Rabin, C. P. Caulfield, and R. R. Kerswell
Figure 11.
Iso surfaces of streamwise velocity u , at 60% of maximum and minimum value,for NLOP E = 3 . × − (left), approximated minimal seed (centre) and turbulent seed at E = 3 . × − (right), at times 0, 75, 150, 250 and 400. The minimal and turbulent seeds areinitially localized but quickly unpack into streamwise streaks, which are stable for the minimalseed but unstable for the turbulent seed ultimately leading to breakdown. inimal seeds in plane Couette flow
5. Discussion
In this paper, we have sought the disturbance to plane Couette flow of a given finite ki-netic energy E which will experience the largest subsequent energy gain G = E ( T ) /E where the time of maximum gain T is an output of our variational formulation. Twogeometry- Re situations have been considered: ( L x × L y × L z , Re ) = (4 . π × × . π, π × × π, et al. (2011). Our results can be summarised as follows.( a ) A nonlinear optimal (NLOP) distinct from the ‘nonlinearised’ linear optimal (QLOP)exists only in the wider geometry at the Re considered.( b ) In both situations, there exists an energy E fail beyond which the variational algo-rithm no longer converges due to the existence of turbulence-triggering initial conditions.This means E fail > E c , the energy above which turbulence can be triggered. PWK11’sfirst conjecture is that E fail = E c if the energy hypersurface is sufficiently sampled andwe find nothing to contradict this.( c ) The QLOP or NLOP are not found to converge to the minimal seed (the distur-bance of lowest energy which can trigger turbulence) as E → E − c contradicting PWK11’ssecond conjecture.( d ) The failure of our variational algorithm to optimise energy gain appears to givethe same estimate for E c and the minimal seed as optimising the total dissipation over along time period Monokrousos et al. (2011). This confirms the ‘robustness of failure’ ofthe variational approach discussed by PWK11 providing the functional to be optimisedassumes large values for turbulent flows.The underlying motivation for all these types of optimising calculations is the hopethat the ‘optimal’ perturbation at E < E c for an appropriately selected functionalbears some relation to disturbances of lowest energy which actually trigger turbulence.By optimising the energy gain, PWK11 found this was at least approximately the case fortheir pipe flow set-up. Here though, this is clearly not true in either PCF situation studied.Ironically, however, the variational procedure does identify these turbulence-triggeringdisturbances and the critical energy for transition E c but only indirectly by failing toconverge. Importantly, we have also found evidence that this revealing failure to convergeis insensitive to the exact functional selected providing the functional takes on heightenedvalues for turbulent flows as argued in PWK11. Taken together, the variational approachadopted here seems to offer a fairly robust new theoretical tool to examine the nonlinear stability of fluid flows.Many applications suggest themselves, but here we just note one - assessing the stabi-lizing or destabilizing influence of applied flow perturbations or controls. Normally, thiswould be attempted either by investigating the linearised operator around the base (lami-nar) flow or by carrying our exhaustive numerical simulations. The current work suggestsa third way where the movement (in phase space) of the laminar-turbulent boundary to-wards (destabilisation) or away (stabilisation) from the base flow is investigated. We hopesoon to report on some calculations along these lines.6 S. M. E. Rabin, C. P. Caulfield, and R. R. KerswellAcknowledgements
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