Vertically localised equilibrium solutions in large-eddy simulations of homogeneous shear flow
UUnder consideration for publication in J. Fluid Mech. Vertically localised equilibrium solutions inlarge-eddy simulations of homogeneous shearflow
A T S U S H I S E K I M O T O † and J A V I E R J I M ´E N E Z
School of Aeronautics, Universidad Polit´ecnica de Madrid, 28040 Madrid, Spain(Received 21 October 2018)
Unstable equilibrium solutions in a homogeneous shear flow with sinuous (streamwise-shift-reflection and spanwise-shift-rotation) symmetry are numerically found in large-eddy simulations (LESes) with no kinetic viscosity. The small-scale properties are deter-mined by the mixing length scale l S used to define eddy viscosity, and the large-scalemotion is induced by the mean shear at the integral scale, which is limited by the span-wise box dimension L z . The fraction R S = L z /l S , which plays the role of a Reynoldsnumber, is used as a numerical continuation parameter. It is shown that equilibriumsolutions appear by a saddle-node bifurcation as R S increases, and that the flow struc-tures resemble those in plane Couette flow with the same sinuous symmetry. The vorticalstructures of both lower- and upper-branch solutions become spontaneously localised inthe vertical direction. The lower-branch solution is an edge state at low R S , and takesthe form of a thin critical layer as R S increases, as in the asymptotic theory of genericshear flow at high-Reynolds numbers. On the other hand, the upper-branch solutions arecharacterised by a tall velocity streak with multi-scale multiple vortical structures. Atthe higher end of R S , an incipient multiscale structure is found. The LES turbulence oc-casionally visits vertically localised states whose vortical structure resembles the presentvertically localised LES equilibria.
1. Introduction
Nonlinear invariant solutions of the incompressible Navier–Stokes (NS) equations, suchas equilibria (Nagata 1990) or periodic orbits (Kawahara & Kida 2001), are believed toplay an important role in transitional and self-sustaining turbulence. In particular, it hasbeen proposed that coherent structures in turbulent flows are incomplete representationsof such solutions, corresponding to times in which the flow approaches an invariant solu-tion in phase space (Jim´enez 1987). The solutions themselves could then be considered‘exact’ coherent structures (Waleffe 2001). Their properties and significance are reviewedin Kawahara et al. (2012).An additional advantage of invariant solutions in the description of turbulence is thatthey can be exactly reproduced numerically, potentially providing a well-defined dynam-ical ‘alphabet’ for the flow evolution, and chaotic fluid motions are created by homoclinicor heteroclinic entanglement of their stable/unstable manifolds. Their statistical agree-ment with turbulence at low Reynolds numbers has often been noted in the literature(Kawahara & Kida 2001; Jim´enez et al. † Email for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] J un A. Sekimoto and J. Jim´enez
Couette flow (Kawahara & Kida 2001; van Veen & Kawahara 2011; Kreilos & Eckhardt2012; Park & Graham 2015). From a practical point of view, they are hard to continueto higher Reynolds numbers partly because of their increasing complexity and instabil-ity, and by the limitations of the numerical resources. From the theoretical side, theirincreasing instability as the Reynolds number increases calls into question whether theflow would approach them often enough for them to be considered relevant. There has alsobeen a challenge to estimate turbulence statistics by using all of recurrent flows (Chan-dler & Kerswell 2013; Cvitanovi´c 2013), however, such an exhaustive study is still limitedat low Reynolds numbers. A a more promising method will be required to find invariantsolutions at high Reynolds numbers with a large number of degrees of freedom, such asthe multiple-shooting method (S´anchez & Net 2010; Van Veen et al. et al. et al. et al. et al. et al. a ; Deguchi 2015).The multiscale nature of fully-developed turbulence at high Reynolds number, and thepossible role of coherent structures in the energy cascade raise the question of whetherinvariant solutions may also be relevant in such processes. Van Veen et al. (2006) founda periodic orbit in highly symmetric turbulence that results in a k − / energy spectrum,and could be part of the generic energy cascade (Goto 2008). Similar attempts, however,have not been successful in wall-bounded turbulence at high Reynolds number, becauseit is difficult to accommodate the anisotropy and inhomogeneity of the length scalesintroduced by the wall.Here, we simplify the problem in two ways. In the first place we substitute shear-drivenwall-bounded flow by a homogeneous shear without walls, of which large-scale motion islimited by the spanwise box dimension in a long-term simulation (Sekimoto et al. et al. et al. et al. (2016) using directnumerical simulations (DNSes), from where the simulation code and parameters in thispaper are derived. They determined which computational boxes best mimic wall-boundedturbulence, and showed that the relevant limiting dimension is the spanwise box width. ertically localised equilibria in LES of homogeneous shear flow et al. et al. et al. et al. et al. §
3, focusing on lower-branch solutions. The dependenceof the lower- and upper-branch equilibria on the computational domain and on the LESparameters are investigated in §
4, while § § §
2. Numerical methods
The governing equations
We solve the incompressible LES momentum and continuity equations, ∂ t u i + u j ∂ j u i = − ∂ i p + ∂ j (2 ν t σ ij ) , (2.1) ∂ j u j = 0 , (2.2)where u i are the resolved (large scale) velocities, and p is a modified resolved kinematicpressure that includes the diagonal part of the SG stress tensor. The eddy viscosity iswritten as ν t = l S | σ | , in terms of a static length scale l S and of the local resolved strainrate (Smagorinsky 1963), where | σ | = 2 σ ij σ ij and σ ij = ( ∂ j u i + ∂ i u j ) / x, y, z ) to represent the streamwise, vertical(cross-shear) and spanwise coordinates, respectively, and ( u, v, w ) to denote the respectivevelocity components. Whenever convenient, as in (2.1), this notation is substituted bysubindices, in which case repeated indices imply summation. This is always the case for A. Sekimoto and J. Jim´enez
Run A xz A yz R S Re z Re λ L /L z η/L z u (cid:48) /SL z v (cid:48) /SL z w (cid:48) /SL z L32 (DNS) 3 2 104 2000 47 0.41 0.00965 0.208 0.159 0.164M32 (DNS) 3 2 367 12500 105 0.38 0.00273 0.182 0.131 0.136LES s m t1 t2 t3 Table 1.
Parameters for the turbulence runs. L32, M32 are reference DNSes of SS-HST fromSekimoto et al. (2016). LES s , m , t1 , t2 , t3 are the present LESes in the symmetric subspace defined by(I)+(II) in (2.4)–(2.5). The effective Reynolds number, Re z , and the Kolmogorov viscous scale, η , are computed with the molecular viscosity in DNS, and with the averaged eddy viscosity inLES. The scale ratio R S is L z /l S in LES and L z /η in DNS. Re λ ≡ u λ/ (cid:104) ν t (cid:105) is the Reynoldsnumber based on the Taylor-microscale λ = √ u / (cid:104) ω i ω i (cid:105) / . the vorticity components ω i . There is no explicit filtering in our code, and the smallestflow scales are controlled either by the grid or by the eddy viscosity.It is usual in most LESes to include the molecular viscosity ν as part of the right-handside of (2.1), and to write the length l S in terms of some convenient grid spacing ∆ g andof a ‘universal’ Smagorinsky constant C S . None of those devices are needed here, and willnot be used. The main role of the molecular viscosity in LES is to enforce the boundaryconditions when ν t vanishes at the wall. Since there are no walls in our problem, we set ν = 0, and all our simulations run at infinite molecular Reynolds number.Similarly, the role of ∆ g in SG models is to accommodate variable grid spacings,assuming that most of the spectral content of the velocity gradients is concentrated atthe grid scale. Again, the statistical homogeneity of the flow implies that our grids areuniform, and makes this complication unnecessary.In fact, the essential role of the eddy viscosity is to introduce a minimum length scalethat takes the role of the Kolmogorov length, η , independently of the grid. If we estimatethe energy dissipation by the subgrid model as ε = (cid:104) ν t | σ | (cid:105) = l S (cid:104)| σ | (cid:105) , and define aneffective Kolmogorov length as η t = ( (cid:104) ν t (cid:105) /ε ) / , it follows from the definition of ν t that η t ≈ l S . Here (cid:104)·(cid:105) represents time and space average, and will occasionally be replaced by (cid:104)·(cid:105) xz to represent the y -dependent average over horizontal planes, or by (cid:104)·(cid:105) c to represent (cid:104)·(cid:105) xz particularised to the central plane y = 0. From now on, capital letters are reservedfor mean quantities and lower-case ones for fluctuations with respect to that mean, asin U = (cid:104) u (cid:105) xz and u = U + u . Primes denote root-mean-squared (rms) intensities of thefluctuations.Since most of the computational effort in DNSes is dedicated to resolving the smallestdissipative scales, the introduction of a cut-off length in LES drastically reduces the sizeof the linear systems that have to be solved to obtain invariant solutions.For the parallel flows which are the subject of this paper, (2.1) can be integrated to (cid:104)− uv + 2 ν t σ xy (cid:105) xz = u τ , (2.3)where u τ is independent of y and can be used to scale the velocities. Variables in thisscaling are denoted by a ‘+’ superscript.2.2. Large-eddy simulations with sinuous symmetry
We study a flow with a nominally uniform mean shear S = d U/ d y , which is a given con-stant, in a parallelepipedic computational domain that is periodic in the ( x, z ) directions, ertically localised equilibria in LES of homogeneous shear flow −6 −5 −4 −3 −2 −1 k x L z E uu ( k x ) / ( u ′ L z ) (a) −2 −1 −4 −3 −2 −1 k x l E ∗ uu (b) Figure 1.
Streamwise velocity spectra E uu ( k x ): (a) Large-scale normalisation, E uu / ( u (cid:48) L z )as a function of k x L z . (b) Small-scale normalisation, E ∗ uu ≡ ε − / l − / E uu ( k x l ), with l = l S for LES and l = η for DNS. – – (cid:79) – –(grey), DNS (L32); ——(grey), DNS (M32); – – • – –(blue), R S = 52 .
6; — ◦ —(red), R S = 101 .
6; — (cid:46) —(green), R S = 203. The slope of the chain-dotteddiagonal is − / and periodic between points of the lower and upper boundaries that are uniformly shiftedin time by the shear (Gerz et al. Sy plus the vertically periodic correction, which is an approximately linear profile as evalu-ated later in this section. The numerical code for DNS is described in detail in Sekimoto et al. (2016). The equations (2.1)–(2.2) for LES are formulated in terms of the verticalvorticity and of the Laplacian of the vertical velocity (Kim et al. et al. x, z ), and sixth-orderspectral-like compact finite differences in y , with the shear-periodic boundary conditionsembedded in the compact finite-difference matrices for each Fourier mode. As explainedin Sekimoto et al. (2016), this avoids recurrent remeshing and the resulting secular lossof enstrophy over long integration times.There are three dimensionless parameters: a Reynolds number, and two box aspectratios, A xz = L x /L z and A yz = L y /L z , where the L j are the dimensions of the compu-tational domain along the three coordinate directions. It was shown in Sekimoto et al. (2016) that the correct scales for the large-scale length and velocity of SS-HST arebased on the spanwise box dimension, L z and SL z , so that the Reynolds number is Re z ≡ SL z /ν in DNS. We will extend this definition to LES using the effective meaneddy viscosity, (cid:104) ν t (cid:105) , replacing the molecular viscosity ν . However, it is more convenientto use the length-scale ratio R S = L z /l S in LES. For example, Piomelli et al. (2015)recently developed a grid-independent LES using as parameter the ratio of the small(effective Kolmogorov) and integral scales, which is essentially the inverse of R S . It canbe extended to DNS by substituting η for l S ( L z /η ∼ Re / z ) and it turns out that, inDNS of SS-HST, Re z ≈ R / S (Sekimoto et al. L = u /ε , where u = (cid:104) u i u i (cid:105) / z = 0followed by a streamwise shift by L x / u, v, w ]( x, y, z ) = [ u, v, − w ]( x + L x / , y, − z ) , (2.4)and a rotation by π around x = y = 0 followed by a spanwise shift by L z / u, v, w ]( x, y, z ) = [ − u, − v, w ]( − x, − y, z + L z / . (2.5) A. Sekimoto and J. Jim´enez
Note that no translational symmetries are allowed in this subspace, so that travellingwaves are excluded. Moreover, (I)–(II) together with the boundary conditions enforcethat the instantaneous plane-averaged streamwise velocities at the top and bottom ofthe box are U ( ± L y /
2) = ± SL y / et al. (2016).Table 1 shows that the length and velocity scales found in DNS also work well in thesymmetric LESes. Although not included in the table, the effective Kolmogorov scale inthe LESes is found to be η t /l S ≈ .
9, in approximate agreement with our analysis in § η t /l S ≈ . . L /L z ≈ . . | U − Sy | ≈ . SL z ) and approximately constant fluctuation profiles.We will see below that the same is not generally true for the profiles of the equilibriumsolutions.Figure 1 displays the longitudinal streamwise-velocity spectrum of the symmetric LESturbulence, compared to that of the reference DNSes. Given the different Reynolds num-bers and techniques, the agreement is good, showing that symmetry does not influenceturbulence greatly. Note that the three LESes in figure 1(b) collapse well in terms of l S , and that the resolution of the simulations is fine enough to resolve the smallest LESscales.The size of the collocation grids for the turbulence LESes and for typical equilibriumsolutions are given in table 2 of §
3. It follows from the definition of the different quantitiesthat the collocation resolution is ∆ x/l S = R S A xz /N x , ∆ y/l S = R S A yz /N y and ∆ z/l S = R S /N z , where the N j are the collocation points along the three coordinate directions.For example, in the lowest-Reynolds number LES s in tables 1 and 2, this results into(∆ x, ∆ y, ∆ z ) = (3 . , . , . l S . If our SG model is interpreted in terms of the usualSmagorinsky formula ν t = ( C S ∆) | σ | , with ∆ a representative grid spacing, it followsthat C S = l S / ∆ = 0 .
42, which is a relatively high value for usual LES practice. In thissense, all our LESes are overdamped. Their dynamics are not controlled by the grid, butby the SG model, and can be expected to be approximately independent of the resolution.They also only resolved the integral scales of the flow, and very little of its inertial range.The effective Smagorinsky constant for the different simulations is included in table 2.At box aspect ratios ( A xz , A yz ) = (3 , . R S (cid:38)
65. There is a transitional range, 45 (cid:46) R S (cid:46)
65, in which the kineticenergy of the flow increases and decreases several times before decaying to laminar after St = O (1000). Both the transitional range and the decay time depend somewhat on A yz .We will see in § A yz ≈ (cid:46) R S (cid:46)
60, but collapses intermittently to vertically localised turbulence.2.3.
Searching for invariant solutions in LES
Our main interest is to characterise invariant solutions in LES of SS-HST. Strictly equi-librium solutions are technically impossible in this system. The shear-periodic boundarycondition slides the upper computational boundary with a velocity SL y with respect tothe lower one, and the numerical configuration only repeats itself after a ‘box’ period T s ≡ L x / ( SL y ). It was shown in Sekimoto et al. (2016) that this periodic forcing doesnot interfere strongly with the turbulent solutions as long as the aspect ratios are keptin the range 2 < A xz (cid:46) A xz (cid:46) A yz . We will approximately respect these con- ertically localised equilibria in LES of homogeneous shear flow A yz → ∞ , since they areindependent of the shear-periodic boundary condition, but it should be remembered thatall of them are conceptually periodic orbits in a finite computational domain. All of thestatistics discussed below are averages over a box period.Solutions are computed using the Newton–Krylov–hookstep method (Viswanath 2007)on f T ( x ) − x = 0, where x is the vector of independent variables, and f T : x (0) → x ( T )is the integration of (2.1) over time T , using the evolution code described in the previoussection. Because of the periodicity mentioned above, the search time is always an integermultiple of the box period, T m = mT s . The convergence criterion for the relative errorof the Newton method is generally taken (cid:107) f T m ( x ) − x (cid:107) / (cid:107) x (cid:107) < − , where (cid:107) · (cid:107) is theL norm.The initial condition for the search was taken from snapshots of the LES symmetricturbulence in the above-mentioned marginal range in which turbulence eventually decaysto laminar, R S < A xz = 3 and A yz = 1 .
33. In order to find a solution, we fix theaspect ratios and the time period T m ≈
10 ( m = 5), then change R S . For R S ≈
35, theNewton search always converges to a trivial laminar state. The first non-trivial solutionis obtained spontaneously at R S = 38 .
6, and a trial with m = 7 converges to the samesolution. It is, therefore, a periodic orbit with the period of T m = T s ( m = 1) close tothe bifurcation point that marks the lowest R S of our family of solutions. The solutionsare always unstable and some example of the linear stability analysis are shown in thelater section and in the Appendix A.From this initial solution, lower and upper branches are continued along the parameters A xz , A yz and R S , overcoming the turning points with a pseudo-arclength method. Wehave seen that all these solutions are periodic orbits in which the period is T s . Theperiodicity is especially noticeable for the upper branch solutions in flat boxes, A yz =1 . .
64 and R S ≈
45, where the temporal oscillation of the integrated kinetic energyis of order 1%. They asymptotically approach a fixed point in the limit of A yz → ∞ ,and the oscillations become negligible, O (10 − ) for A yz >
2, for the vertically localisedsolutions described later in taller boxes.
3. Lower-branch solutions in LES
Here, we preview the obtained equilibria, and characterise lower-branch solutions in acomputational box, ( A xz , A yz ) = (3 , . A xz (cid:46) A yz , in which unphysical strong linear bursts appear as described inSekimoto et al. (2016), in spite of which the turbulence statistics are reasonably closeto the logarithmic layer of channel turbulence. We investigate in this section the R S dependence of the equilibria. The effect of the aspect ratios is discussed in § R S turbulence LES s in table 2, ( N x , N y , N z ) = (64 , , R S as longas convergence is achieved. This happens in R S (cid:46)
100 for the lower-branch solutions, andin R S (cid:46)
60 for the upper branch. We emphasise that R S depends on the eddy-viscosityparameter l S , rather than on the numerical grid. As long as the numerics resolves featuresof the order of η t ≈ l S , all grids should converge to the same solution. We saw in theprevious section that this is true for most of our simulations. As a further check, finergrids are used in § − , a few were A. Sekimoto and J. Jim´enez
40 50 60 70 80 9000.050.10.150.20.25 u ′ c / S L z R S (a)
40 50 60 70 80 9000.010.020.03 v ′ c / S L z R S (b)
40 50 60 70 80 9000.010.020.03 w ′ c / S L z R S (c) Figure 2. (a-c) Velocity fluctuation intensities at the centre plane y = 0, as functions of R S = L z /l S , for ( A xz , A yz ) = (3 , . u (cid:48) c . (b) Vertical, v (cid:48) c . (c)Spanwise w (cid:48) c . ( ◦ , black) the upper branch; ( (cid:79) , blue) the lower branch. Solid symbols are casesplotted in figure 3. confirmed to a tighter tolerance. For example, the lower- and upper-branch solutions at R S = 50 . − .Figures 2(a–c) are continuation diagrams of the lower- and upper-branch solutionsfor the box mentioned above, showing the fluctuation intensity of the three velocitycomponents at y = 0. The lower-branch solutions are characterised by weaker streamwisevelocity fluctuations and stronger transverse velocities than those in the upper branch,but note that the even those more intense transverse velocity fluctuations, v (cid:48) and w (cid:48) , arean order of magnitude smaller than in the LES and DNS turbulence at a comparableReynolds number in table 1 ( v (cid:48) ≈ w (cid:48) ≈ . SL z ).The identification of which branch is the lower one is not straightforward, but we willdenote as such the lower branch in figure 2(a). This is actually consistent with the factthat the lower-branch solution typically has low drag, while the upper one has high drag(figure 6 c) as we shall discuss later in § R S reveals that, as this branchextends towards higher R S its solutions concentrate in a relatively thin critical layer for R S >
70. This is a common feature of lower-branch solutions in wall-bounded flows athigh Reynolds numbers (Wang et al. et al. et al. | ω x | = 0 . | ω x | max , andof u = 0, representing the geometry of the vortical structures and of the velocity streakin the upper- and lower-branch solutions, showing that they are localised around y = 0.The streamwise-velocity streak of the lower-branch solutions meanders more deeply thanthe one in the upper-branch, leading to stronger cross-flow velocity fluctuations. As R S increases, the vortical structure of lower-branch solutions becomes flatter. This occursdrastically, but smoothly, at around R S ≈
60, after which it becomes a critical-layer-typesolution similar to those described by the vortex-wave interaction (VWI) theory for the ertically localised equilibria in LES of homogeneous shear flow − − − − − − − − − − − − − − − − − − − − Figure 3.
Vortical structures and the velocity streaks in a box ( A xz , A yz ) = (3 , . R S = 50 .
5, upper branch. (b) R S = 38 . R S = 50 .
5, lower branch.(d) R S = 69 .
8, lower branch. The shaded background in the left figures are isocontours thestreamwise-averaged ω x ( y, z ) /S = [ − . .
03 : 0 . u = 0. The right figures shows the isosurfaces of: Red (dark-grey in black/white), ω x = 0 . | ω x | max ; blue (black), ω x = − . | ω x | max ; yellow (light-grey), u = 0 gradually colouredby y . lower-branch solutions in plane Couette flow (Blackburn et al. R S , but tracking them is difficult in our flow. For example, a newcomplex pair appears around R S = 55, but the associated branch cannot be followed bythe present method because its period is not a simple multiple of the box period T s .0 A. Sekimoto and J. Jim´enez −0.5 0 0.5−0.6−0.4−0.200.20.40.6 y / L z U/SL z (a) h| σ |i xz /S (b) −0.05 0 0.05−0.2−0.100.10.2 µ i / S µ r /S (c) u / ( S L z ) St (d) Figure 4.
LES equilibrium solutions, ( A xz , A yz ) = (3 , .
33) and R S = 50 .
5. In (a,b): ——, Lowerbranch, as in figure 3(a); – – – –, upper branch, as in figure 3(c). (a) Mean streamwise velocity.The thin diagonal is U = Sy . (b) Resolved strain rate. (c) Stability eigenvalues ( µ r + i µ i ) /S ofthe equilibria in (a,b): (cid:79) (blue), lower branch; ◦ (red), upper branch. (d) Temporal evolutionof the fluctuation velocity magnitude of symmetric LES initialised from the equilibria in (a,b).The darker lines are initialised without any disturbance except numerical inaccuracies. ——(darkblue), Initialised from the lower branch; – – – –(dark red), from the upper branch. The lighter greylines are initialised from the lower-branch with a small disturbance along the unstable directioncorresponding to the real unstable mode. ——, attracted to the turbulence state; – – – –, attractedto the laminar state. All attempts to perturb the upper branch along its unstable modes ledto laminarisation. The short thick dotted line represents the fluctuation intensity of LES s intable 1, using the same box and Reynolds number. In the high-Reynolds number limit, Blackburn et al. (2013) have shown that the VWIstates begin to localise at y = 0 as spanwise wavenumbers increasing, i.e. as L z narrows.Waleffe (1997) showed that equilibrium solutions similar to those of Couette flow aregeneric to many shear flows, and Deguchi & Hall (2014 a ); Deguchi (2015) have describedmore recently how a VWI state can be embedded in any shear flow at high-Reynoldsnumber. There is an inviscid mechanism as in VWI-type solutions of the Navier-Stokesequation, whose streamwise velocity structure is shown to be thinner as increasing Re .“The singularity occurs where the wave propagates downstream with the local fluidvelocity and defines the location of a critical layer in which viscosity smooths out thesingularity” (Deguchi & Hall 2016). The critical layers in LES equilibria must be similarto those in DNS and the singularity occurs, but the eddy viscosity smooths it out.Upper-branch solutions are characterised by their taller streamwise-velocity streaks.Their height increases with increasing R S , while their quasi-streamwise vortices become ertically localised equilibria in LES of homogeneous shear flow R S are shown infigures 4(a,b). The lower branch solutions is the one shown in figure 3(c), and the upperbranch one is in figure 3(a). Both solutions are concentrated around y = 0, but theconcentration is more pronounced in the lower branch. This is seen in the more horizontalquasi-streamwise rollers in figure 3(c), and in the shallower mean velocity profile near y = 0 in figure 4(a).Since it follows from the momentum conservation equation (2.3) that the total shearstress has to be independent of y , the shallower profile of the lower branch suggests ahigher eddy viscosity, and consequently a higher total strain. The opposite turns out tobe the case. Figure 4(b) displays the mean profile of the mean total strain rate for the twosolutions, which can also be interpreted as a profile of eddy viscosity. The flatter profileof the lower branch is due to higher resolved Reynolds stresses. Beyond | y | /L z ≈ .
5, themean strain tends to (cid:104)| σ |(cid:105) xz ≈ d U/ d y , and most of the momentum flux is carried by theSG term.The simplest interpretation is that the Reynolds stresses created by the transversevelocities of the equilibrium state flatten the profile into a local region of lower shear.This results in a locally lower eddy viscosity and a locally higher Reynolds number, thathelps to sustain the solution. However, the requirement from the boundary conditionthat the total velocity difference across the domain is constant prevents the low-shearlayer from spreading over the whole box, and results in a local high-Reynolds number‘turbulent’ layer within a box in which all other fluctuations are damped by the model.The linear stability eigenvalues of the two solutions in figures 4(a,b) are shown in fig-ure 4(c). The upper-branch has two unstable complex-conjugate pairs, while the lower-branch solution has a pair of unstable complex-conjugate modes and a real unstablemode. Since we have already noted that all solutions are periodic orbits, all these eigen-values are actually Floquet exponents that have an underlying periodic component. Theperiod of the real unstable mode of the lower branch in figure 4(c) is the box period,representing an exponentially growing oscillation synchronous with the numerics. Thecomplex conjugates pairs have periods that are not simple multiples of the box period,and represent bifurcations into a torus. Further details of the distribution of the unstablemodes and their dependency on A yz are in the Appendix A.Figure 4(d) shows the results of initialising symmetric LESes from the equilibria justdiscussed. At first, the LES is initiated from the equilibrium without adding any distur-bances beyond numerical errors. The result is that the lower branch transitions ratherquickly to a turbulence-looking bursting state, while the upper branch does not separatefrom equilibrium during the time plotted in the figure. This is consistent with the sta-bility analysis in figure 4(c), which shows that the instability eigenvalues of the upperbranch are weaker than for the lower one.Next, the LESes are initiated by perturbing the equilibria along the eigenfunction ofindividual unstable modes. The grey lines in figure 4(d) show the result of perturbationsof the lower branch along the eigenfunction of its real unstable eigenvalue. One directionleads to exponential growth of the kinetic energy into a burst and chaotic turbulence,while the opposite direction laminarises. None of the LESes initialised from the unstablecomplex modes of the lower branch leads to bursts or to self-sustaining turbulence, andneither do the perturbations of the upper branch. The lower-branch solution thus behavesas a torus ‘edge’, which not only has a single unstable real mode, like simple ‘edgestates’, but also two complex unstable modes. However, the most interesting part of this2 A. Sekimoto and J. Jim´enez
Run A xz A yz R S N x , N y , N z C S ∆ g /L z EQ(figure 4) 3 1 .
33 50 . , ,
32 0.42 0.0471EQs(figure 5a) 3 1 . . .
60 64 , . .
29 3 38 .
95 64 , ,
32 0.508–0.649 0.0396–0.0505tall EQ(L) 3 3 37 . . , ,
32 0.274–0.539 0.0489tall EQ(L) 3 3 77 . . , ,
64 0.334–0.430 0.0301tall EQ(L) 3 3 99 . . , ,
64 0.360–0.368 0.0273tall EQ(U) 3 3 37 . . , ,
32 0.431–0.538 0.0490tall EQ(U) 3 3 47 . . , ,
64 0.646–0.702 0.0301tall EQ(U) 3 3 52 . . , ,
64 0.695–0.698 0.0273tall EQ(U) 3 3 55 . . , ,
64 0.649–0.754 0.0239LES s .
33 50 . , ,
32 0.42 0.0471LES m .
33 91 . , ,
32 0.256 0.0428LES t1 . , ,
64 0.838 0.0259LES t2 , ,
64 0.519 0.0190LES t3 , ,
64 0.260 0.0190
Table 2.
Grid information for the sinuous-symmetric equilibrium and turbulence LES. EQ(L)and (U) represent lower- and upper-branch equilibrium solutions, respectively, and the turbu-lence LESes are defined in table 1. C S is the effective Smagorinsky constant discussed in § g ≡ √ ∆ x ∆ y ∆ z . −3 − h u v i c / ( S L z ) A yz (a) −3 A xz (b) Figure 5.
Reynolds stress averaged over the y = 0 plane, R S = 38 . N x = 64 and N z = 32. (a)As a function of A yz for A xz = 3. ——, N y = 48; (cid:3) – – – – (cid:3) , N y = 48. (b) As in (a), a functionof A xz for A yz = 3, N y = 64. observation is not the detail of this ‘edge’, but the burst originating from the unstablemanifold of the real saddle. A similar behaviour was found by van Veen & Kawahara(2011) in Couette flow.
4. Vertical localised upper-branch equilibria, and localised turbulence
The effect of box aspect ratios and characterisation of equilibria
Figure 5(a) shows the dependence on A yz of the Reynolds stress at the central plane for R S = 38 . A xz = 3, close to the initial bifurcation. Solutions only exist for A yz > . A yz increases, and may approach a non-zero constantin the limit A yz → ∞ . The same is true for the velocity fluctuations (not shown). Notethat, since the grid becomes relatively coarser as A yz increases, a finer y grid is used for ertically localised equilibria in LES of homogeneous shear flow
40 60 80 10000.10.20.30.4 u ′ c / S L z R S (a)
40 60 80 10000.050.10.150.2 v ′ c / S L z R S (b)
40 60 80 10010 −4 −3 −2 −1 − h u v i c / ( S L z ) R S (c)
40 60 80 10010 h | σ | i c / S R S (d) Figure 6.
Equilibrium solutions for A xz = 3 and different A yz : (solid with (cid:79) , blue ) A yz = 1 . (cid:47) , blue) 1 .
5; (dashed with (cid:52) , green) 1 .
64; (solid with (cid:46) , green) 1 .
8; (dashed with (cid:3) , black) 2 .
0; (solid with ◦ , black) 3 .
0. (a) u (cid:48) c /SL z . (b) v (cid:48) c /SL z . (c) Resolved-scale Reynoldsstress −(cid:104) uv (cid:105) c . (d) (cid:104)| σ |(cid:105) c /S ≡ R S / (cid:104) Re z (cid:105) c . The filled and open symbols represent lower andupper branches, respectively. The red diamonds are box-averaged statistics of symmetric LESturbulence in a box ( A xz , A yz ) = (3 , taller boxes (dashed lines), and that the good agreement of the results whenever the twogrids overlap confirms numerical convergence.The dependence on the streamwise aspect ratio A xz is shown in figure 5(b). Solutionsexist for 1 . (cid:46) A xz (cid:46) .
29, which covers the range of box aspect ratios ( A xz ≈ et al. (2016) to be good models for wall-bounded shear turbulence.It is interesting that the minimum aspect ratio for steady Nagata equilibrium solutionsin plane Couette flow is A xz ≈ .
62 at low Reynolds numbers (Jim´enez et al. A xz (cid:38) .
5, even at high Reynoldsnumbers. On the other hand Kawahara & Kida (2001) found unstable periodic orbits(UPOs) for somewhat shorter boxes, A xz = 1 . R S for A xz = 3 and several A yz . The box-averaged statistics of LES (symmetric) turbulence are also shown for comparison. For R S (cid:46)
60, those turbulence simulations occasionally become vertically localised around y = 0, but they spread again to fill the whole domain (see § v (cid:48) and w (cid:48) fluctuations than would otherwise be obtained at the central plane. When R S (cid:46)
50, LES turbulence often decays to laminar after these localisation events.The velocity fluctuations for the upper-branch solutions are quite large in flat boxesand low Reynolds number, A yz = 1 . R S ≈
45, but that behaviour disappears fortaller boxes and saturates beyond A yz ≈
3. These large fluctuations are thus probablyan effect of the shear-periodic boundary condition. This is also the range in which theshear-periodicity results in the strongest temporal oscillations of the kinetic energy, of4
A. Sekimoto and J. Jim´enez
40 60 80 10012345 u ′ + c R S (a)
35 40 451.522.53 u ′ + c R S (b)
40 60 80 10000.511.5 v ′ + c R S (c)
40 60 80 10000.511.5 w ′ + c R S (d) Figure 7. (a,c,d) As in figure 6, but scaled by u τ at y = 0. (a,b) u (cid:48) + c , (c) v (cid:48) + c , (d) w (cid:48) + c . (b) Detailof (a): (solid with (cid:52) , red) A xz = 2, (solid with (cid:3) , blue) A xz = 2 .
5, (black with ◦ ) A xz = 3. the order of 1%, but we tested that the fluctuations in figure 6(a, c) are not due tothe temporal variability. They are also present in the spacial average of instantaneoussnapshots. The temporal oscillation of solutions with A yz > − .The relatively poor scaling of the velocity fluctuations of the LES equilibria with SL z can be traced to the poor scaling of | σ | /S . Figure 6(d) shows that the dimensionlessstrain rate of the upper-branch equilibria is roughly unity for R S = 50–100, while that ofturbulence is in the range of 2–5. It was shown by Sekimoto et al. (2016) that, although SL z is the natural velocity scale for ‘good’ DNS boxes, the fluctuations in non-optimalboxes scale better with the friction velocity obtained from their total measured stress.In essence, the Reynolds stress and the velocity fluctuations scale with each other.The same is true for LES and for equilibrium solutions. Figures 7(a,c,d) show that thefluctuations collapse to a common curve in these ‘wall’ units. The vorticity componentsalso scale well with u τ /l S (not shown). In this normalisation, the velocity fluctuationsof symmetric LES turbulence at R S = 100 are u (cid:48) + ≈ v (cid:48) + ≈ . w (cid:48) + ≈ .
4, whichagrees well with those at the top of the logarithmic layer of turbulent channels. When R S <
60, the velocity fluctuations of the LES equilibria are not very different from thoseof turbulence, but we have seen that the lower-branch solutions tend to get concentratedaround the critical layer as R S increases. Their statistics then become very different fromturbulence.Figure 7(b) is an enlargement of figure 7(a), showing the dependence on A xz of theminimum bifurcation Reynolds number of equilibria with A yz = 3. It turns out that thebifurcation is more dependent on A xz than on A yz . At a fixed A xz = 3, the minimum R S is (38 . , . , . , . , . , .
90) for A yz = (1 . , . , . , . , . , . A yz is fixed at 3, R S changes by8%, from 35.05 to 37.90, as A xz changes from 2 to 3. ertically localised equilibria in LES of homogeneous shear flow −1.5 −1 −0.5 0 0.5 1 1.510 −6 −5 −4 −3 −2 −1 u ′ / ( S L z ) y/L z (a) −1.5 −1 −0.5 0 0.5 1 1.510 −6 −5 −4 −3 −2 −1 y/L z (b) Figure 8.
Velocity fluctuations for A xz = A yz = 3. (a) R S = 50, (b) R S = 62 . u (cid:48) ; ——(blue), v (cid:48) ; — · —(red), w (cid:48) . −1.5 −1 −0.5 0 0.5 1 1.500.20.40.60.81 − h u v i + x z , h ν t S i + x z y/L z (a) −1.5 −1 −0.5 0 0.5 1 1.500.20.40.60.81 y/L z (b) Figure 9. (a,b) As in figure 8, for the momentum balance. ——(blue), (cid:104)− uv (cid:105) xz /u τ ;– – – –(red), (cid:104) ν t σ xy (cid:105) xz /u τ ; ——(black), total stress.
40 50 60 7000.511.5 d s / L z R S (a)
40 50 60 7000.20.40.60.81 R S (b) Figure 10.
Height of the velocity streak, defined by the distance d s between the peaks of u (cid:48) in the core part of the solutions in figure 8. Only peaks in the inner core are considered,and the secondary peaks in the outer one-component layer are ignored. (a) A xz = 3 and A yz = 1 . , . , . , . , ,
3. Lines and symbols as in figure 6. (b) A xz = 2 , . , A yz = 3. Lines and symbols as in figure 7(b). The structure of the upper-branch equilibria
We focus next on the flow structure of the vertically localised upper-branch equilibriain a box with A xz = 3, A yz = 3. The right-hand side of the two panels in figure 8shows that the velocity fluctuations of these solutions decay exponentially away from y = 0, which is a common feature of localised solutions in other shear flows (Schneider et al. u (cid:48) is much stronger than those of v (cid:48) and w (cid:48) . This can be shown to be due to a streamwise-constant ‘streak’ of the streamwise6 A. Sekimoto and J. Jim´enez
Figure 11.
Vertically localised upper-branch solutions: As in figure 3, for A xz = 3 , A yz = 3and: (a) R S = 42 . R S = 62 . Figure 12.
Isosurfaces of the second invariant of the velocity gradient ( Q = 0 . Q max ) colouredby ω x , with the same colour code as in the streamwise-averaged ω x maps in the left parts offigures 11(a,b): (dark-grey) ω x <
0, (light-grey) ω x >
0. From top to bottom, each figure shows:a three-dimensional view, a side view of the region 0 . ≤ z/L z ≤
1, and a cross section ofcross-flow velocity vectors at x/L z = 2 .
25, marked as a chain-dotted vertical line in the sideviews. The primary streamwise rollers visible in these cross-sections are not always compactenough to appear in the Q isosurfces and are marked as (A) in the lateral views. Flow as infigures 11(a,b). Only the active part, | y | /L z < .
5, is shown in all cases. −1 −5 −4 −3 −2 −1 E ∗ uu k x l S Figure 13.
Streamwise velocity spectrum E ∗ uu = E uu ( k x ) ε − / l − / S as function of k x l S : (cid:79) (blue), upper-branch solution at R S = 42 . ◦ (black), upper-branch solution at R S = 62 . (cid:52) (red), turbulence LES at R S = 62 .
5. The dash-doted line represents the inertial theory E ∗ uu ( k x ) = 0 . l S k x ) − / . The spectra and dissipation rates of the upper-branch solutions areaveraged over | y | /L z < . ertically localised equilibria in LES of homogeneous shear flow u s ∼ sin(2 πz/L z ). This weak perturbation is the far tail of the strongersinuous streak of u concentrated near y = 0 and, because it is almost independent of x ,is essentially independent of v , w and of the shear. The left-hand part of both panels infigure 8 represent lower-branch solutions, which share many of the characteristics of theupper branch. As the Reynolds number increases, all structures become more complex,as seen in figure 8(b) and later in figures 11 and 12, and the core of the structuresdevelop substructures that could perhaps be interpreted as a first indication of a turbulentcascade. The momentum balance of these flows is shown in figure 9. The Reynolds stressis dominant in the core, | y | /L z < .
5, but only the mean shear d U/ d y contributes to theeddy viscosity in the outer part.Figure 10 presents the height of the velocity streak for the different equilibria, definedas the twice the distance to y = 0 of the first maximum of the u (cid:48) profile of the solutionsin figure 8. The height of lower-branch streak stays roughly constant below R S ≈ R S ≈
45. This is also the range in whichthe velocity fluctuations become very strong in figure 6, and only appears in flat boxeswith A yz = 1 . − .
64. We argued in the discussion of figure 6 that upper-branchsolutions tend to be limited by the height of these flat boxes, and figure 10(a) confirmsthis interpretation by showing that the maximum height of the streak reaches the boxheight. On the other hand, this interaction does not take place when A yz (cid:38)
2, confirmingthe independence of those solutions from the box dimensions.The flow structures of the upper-branch solutions at R S = 42 . . u = 0 isosurface that spans | y | /L z < . ω x in figure 11. The vortical structures are shown by themselves in figure 12,and are surprisingly complex for an equilibrium solution. This is especially true for thehigher Reynolds number case in figures 11(b) and 12(b), which appear to include adouble structure that is nevertheless in equilibrium. In fact, there is a first indicationof two separate scales in these flow fields. The smaller tube-like vortical structures areisosurfaces of the second invariant of the velocity gradient tensor (Q criterion), and arecoloured by the streamwise vorticity. They do not always coincide with the larger-scalestructures of the cross-stream velocity, which are visible in the cross sections in thebottom part of figure 12. These ‘rollers’ are too diffuse to appear in the Q-map, whichmay actually appear empty in the region in which the roller is dominant (see the regionlabelled as (A) in the side view in figure 12b, and note that, because of the symmetry ofthe flow, a mirror-symmetric arrangement is present in the upstream half of the box). Onthe other hand, it is clear from the cross sections that the roller dominates the velocityfield.The low-Reynolds number solution in figure 12(a) shows that the Q-vortices are partsof the larger streamwise rollers that have been sheared and stretched by the mean flow.The cross sections in the bottom part of figure 12 shows that these vortices are longenough to spill into neighbouring periodic boxes, so that they appear as double in thecross section. The inclination angle of these streamwise rollers and vortices is roughly15 ◦ at all Reynolds numbers. In the higher-Reynolds number case in figure 12(b) thestreamwise vortices are strong enough to create new vortices, roughly perpendicular tothem, which are labelled as (B) in the side view in figure 12(b). They rotate in theopposite sense to the primary streamwise rollers and are aligned in the direction of the8 A. Sekimoto and J. Jim´enez (a)
500 1000 1500 2000−1.5−1−0.500.51 00.020.040.06 St y / L z (b) (c) Figure 14. (a) Vertical velocity fluctuation intensity profiles, (cid:104) v (cid:105) / xz / ( SL z ), as a function of y/L z and St . The contours are [0.02:0.02:0.08], and the mean value of v (cid:48) is approximately 0.07 SL z (see figure 6). A xz = 3, A yz = 3, R S = 50. (b) Homogeneous turbulence at St = 1318, (c)Localised state at St = 1525. The left panels in (b,c) show the streamwise-averaged ω x , and theright ones show isosurfaces of ω x and u , as in figure 3. −4 −3 −2 −1 St u ⊥ / ( S L z ) (a) t u /SL z ( − γ ) (b) Figure 15. (a) The local cross-flow velocity fluctuation intensity u ⊥ defined in (4.1) at: (black) y = L y / u t ), (grey) y/L z = 0 ( u c ). The black dashed line is the mean value, (cid:104) u ⊥ (cid:105) , and the reddashed lines are (cid:104) u ⊥ (cid:105) ± σ , where σ is the standard deviation. They are defined as averages overthe centre and top of the box, as explained in the text. (b) The intermittency factor (1 − γ ),where γ is defined in (4.2): — ◦ —, R S = 50 .
0; — • —, R S = 52 .
6; — (cid:79) —(grey), R S = 55 . (cid:77) – –, R S = 62 .
5; — (cid:3) —(grey), R S = 101 . strain produced by them. A similar generation mechanism of secondary smaller vorticeshas been investigated in the homogeneous isotropic turbulence (Goto 2012).The velocity spectra in figure 13 show that the upper-branch solutions acquires moresmall-scale structures as R S increases, approaching the spectrum of turbulence LES atsimilar Reynolds numbers. Even though the turbulence state has more small scale, thelarge-scale end of its spectrum is quantitatively similar to the upper-branch solutions.4.3. Intermittent visiting of turbulence to vertically-localised states
We mentioned in § R S is relatively low, LES turbulence collapses inter-mittently to a localised state around y = 0, and that these states persist for a long time ertically localised equilibria in LES of homogeneous shear flow R S (cid:38)
50. These are the statistics plotted as diamonds in figure 6. Figure 14(a)shows the temporal evolution of the profile of local v (cid:48) . Light colours represent activeturbulent regions and dark ones are overdamped ‘laminar’ areas. Localised turbulenceoccurs when laminarisation does not extend over the whole height of the box, such as in St = 1500 − y = 0, but that the symmetry allows localisationboth at the centre plane, y = 0, or at the top or bottom of the box, y = ± L y /
2. Thestrongest event in figure 14(a) is localised at y = 0 ( St = 1500 − St ≈ § u ⊥ ( y ) ≡ (cid:0) (cid:104) v (cid:105) xz + (cid:104) w (cid:105) xz (cid:1) / . (4.1)The darker line is the intensity at y = 0, and the lighter one is the intensity at y = ± L y / (cid:104) u ⊥ (cid:105) , and deviation, σ , whichare the averages of these two positions. By defining u c = u ⊥ (0), and u t = u ⊥ ( L y / u c ≈ u t ≈ (cid:104) u ⊥ (cid:105) , while localised states arecharacterised by having one of these intensities much weaker than the fully turbulentone. The quantity γ ≡ (cid:90) ∞ t u (cid:90) ∞ t u P ( u c , u t ) d u c d u t , (4.2)where P ( u c , u t ) is the joint probability density function of both intensities, and t u isa threshold, measures the probability that the points y = 0 or L y / u c ≥ t u or u t ≥ t u . Since the localised state has the overdamped laminarwhich intermittently appears as shown in figure 15(a), (1 − γ ) indicates the intermittencyfactor of the localised turbulence. Figure 15(b) shows that the frequency of localisationdecreases as R S increases.
5. Conclusions
We have performed large-eddy simulations of statistically stationary homogeneousshear turbulence (SS-HST) in a subspace with sinuous symmetry, and found equilibriumsolutions with the same symmetry. We use a Smagorinsky-type model with no molecularviscosity, which is not required because of the absence of walls, so that the eddy-viscosity ν t ≡ l S | S | acts as the only energy sink. It is parametrised by a mixing length l S thatplays the role of the Kolmogorov scale in LES, independent of the numerical grid. Forthe grids used in this study, the flow is independent of the numerical resolution. Theintegral scale in the LES of the SS-HST is comparable to the spanwise box dimension, L ≈ . L z , as in the DNS of the same flow in Sekimoto et al. (2016) and as in wall-0 A. Sekimoto and J. Jim´enez bounded turbulence in spanwise-limited boxes (Flores & Jim´enez 2010). The effectiveKolmogorov scale is η t ≈ . l S , and the velocity fluctuations scale well with SL z , as inDNS. Even if we introduce the molecular viscosity in our LES, considering the effectiveKolmogorov length, (cid:101) η t ≈ (cid:101) l S ≡ l S (1 + ν/ν t ) / (see Pope (2000)), the results would notchange as long as ν is enough small with respect to the mean ν t .The length-scale ratio R S = L z /l S is used as a continuation parameter for LES equi-libria, playing the role of a Reynolds number, and it is found that vertically localisedequilibrium solutions appear by a saddle-node bifurcation at R S = 37 . A xz = 3. Thedependence of the equilibrium solutions on the box aspect ratios has been investigated,and it is revealed that both lower- and upper-branch solutions tend to localise verticallyaround the central plane of box. The initial bifurcation point is roughly independent of A yz , and much more dependent on A xz . These solutions exist in 1 . (cid:46) A xz (cid:46) .
29 and A yz > . R S = 39 .
0. This range of aspect ratios spans those found by Sekimoto et al. (2016) to be good models for unconstrained shear flows in general. The minimumlimit of A xz (cid:38) . R S increases, the lower-branchsolutions take the form of a critical layer, such as those found in previous works on wall-bounded flows (Wang et al. b ), and describedby vortex-wave interaction (VWI) theory (Hall & Smith 1991; Hall & Sherwin 2010).The length scales of the LES equilibria are l S for the small scale and L z for the largeone, as in LES turbulent. The comparison of the contribution of the eddy viscosity and ofthe Reynolds stress to the momentum balance reveals that the former is weak within thelocalised equilibrium solutions, and predominate outside it. The velocity fluctuations ofthe present equilibria are substantially smaller than those of self-sustaining turbulence,especially in the VWI limit, and do not scale well with SL z . However, they scale well withtheir own u τ , with similar values to those of wall-bounded flows expressed in wall-units.At low Reynolds numbers, lower-branch solutions act as edge states. Although theireigenvalue structure is more complicated than a simple saddle, one of the unstable di-rections of the saddle leads to an exponential burst and to chaotic turbulence, whilethe other laminarises. In turbulent LESes, the flow occasionally collapses to a localisedstate which resembles the equilibrium solutions. Depending on the Reynolds number, theoutcome of these events is more often reinjection to turbulence, or laminarisation.Upper-branch solutions have tall velocity streaks associated with small-scale vortices,whose complication increases with increasing R S . It is interesting that, even at the rela-tively low R S ≈
62, small secondary vortices begin to appear in these solutions, alignedperpendicularly to the primary streamwise rollers in a manner strongly reminiscent ofthe multiscale process frequently invoked as models for the turbulent cascade. Furthercontinuations to higher R S are hardly successful probably because of their increasingcomplexity and instability, which is the similar limitation to the one we encounter whensearching dynamically important invariant solutions in the Navier-Stokes computationsat high Reynolds numbers.In all, the localised LES equilibria discovered here represent a promising model forgeneric isolated turbulent structures in shear flows. Most intriguingly, the higher Reynoldsnumbers contain what appear to be the first stages of a multiscale cascade. ertically localised equilibria in LES of homogeneous shear flow
38 40 4200.020.040.060.080.10.12 µ r / S R S (a)
38 40 4200.020.040.060.080.10.12 R S (b)
38 40 4200.020.040.060.080.10.12 R S (c) Figure 16.
The real-part of nondimensional eigenvalues obtained by the linear stability analysisof equilibria for A xz = 3 and (a) A yz = 1 .
5, (b) A yz = 2, (c) A yz = 3. (cid:79) , lower branch; ◦ , upperbranch. The filled (open) symbols represent real (complex conjugate) modes. Positive valuesrepresent unstable modes, and the square is the bifurcation point. Acknowledgements
This research has been funded by the European Research Council grants ERC-2010.AdG-20100224 and ERC-2014.AdG-669505. We are grateful to G. Kawahara for early discus-sions.
Appendix A. Linear stability analysis of equilibria in LES
We discuss in this section the linear stability of the vertically localised symmetricequilibria described in the body of the paper. Even if we saw in figure 7(a) that thecontinuation diagram of these solutions is roughly independent of the vertical box aspectratio, it turns out that this aspect ratio affects the stability of the equilibria, especiallythat of the upper-branch solutions. Since we showed in figure 10 that these solutions aretall enough to span the full height of the box, especially in the case of the flatter boxeswith A yz ≤ .
5, the artificial interactions with the shear-periodic copies in y is inevitable.Figure 16 shows the distribution of a few of the least stable eigenvalues obtained from thelinear stability analysis of equilibria with A xz = 3 and A yz = 1 . , R S increases.The upper branch always has at least one pair of unstable complex conjugate modes, andbecomes more stable as A yz decreases.In taller boxes, with A yz = 2 −
3, the distribution of unstable eigenvalues of the upper-branch solutions is not strongly affected by the box aspect ratio, in rough agreementwith the criterion, A xz (cid:46) A yz , found in previous DNS studies of SS-HST to be requiredfor box independence (Sekimoto et al. REFERENCESAvila, M., Mellibovsky, F., Roland, N. & Hof, B.
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