A POD-based analysis of turbulence in the reduced nonlinear dynamics system

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1 Home Search Collections Journals About Contact us M IOPscience A POD-based analsis of turbulence in the reduced nonlinear dnamics sstem This content has been downloaded from IOPscience. Please scroll down to see the full tet. 6 J. Phs.: Conf. Ser. 78 ( View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: This content was downloaded on /5/6 at 7:56 Please note that terms and conditions appl.

2 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78// A POD-based analsis of turbulence in the reduced nonlinear dnamics sstem M-A Nikolaidis, B F Farrell, P J Ioannou, D F Game, A Loano-Durán 4, J Jiméne 4 Universit of Athens, Department of Phsics, Panepistimiopolis, Zografos, 5784 Athens, Greece Department of Earth and Planetar Sciences, Harvard Universit, Cambridge, MA 8, USA Department of Mechanical Engineering, Johns Hopkins Universit, Baltimore, MD 8, USA 4 School of Aeronautics, Universidad Politécnica de Madrid, 84 Madrid, Spain pjioannou@phs.uoa.gr Abstract. The structure of turbulence in a reduced model of turbulence (RNL) is analed b means of a Proper Orthogonal Decomposition (POD modes). POD analsis was carried out on two different components of the flow, the roll/streak and the perturbation structure. The POD structure in both RNL and direct numerical simulations (DNS) is similar and this correspondence suggests that the dnamics retained in RNL are the essential dnamical ingredients underling the self-sustaining mechanism of the turbulent state.. Introduction Proper Orthogonal Decomposition (POD) provides an orthogonal basis for the spatial structure, each component of which is ordered according to its contribution to the variance of the flow. This basis is optimall efficient for representing the variance in turbulent flows and allows a sstematic reduction in compleit of the representation of the flow. The POD analsis can be performed separatel on components of the flow in homogeneous directions including the streamwise-mean component of the flow (with streamwise wavenumber k = ) and the streamwise-varing components of the flow with k [ 4]. In this work we eploit this freedom to efficientl identif the structures that dominate the spatio-temporal variabilit of the streamwise-mean flow as well as the structures that dominate the streamwise-varing k components. POD modes obtained from DNS are compared with those obtained from a closel related reduced nonlinear (RNL) model [5]. The purpose of this comparison is in part to elucidate the similarities and differences of the DNS with the RNL. The dnamics of RNL model turbulence involve interaction of the streamwise mean flow with few, even as few as one, streamwise flow harmonic. Despite this simplification, salient aspects of the turbulent dnamics are captured [6 8]. Given the similarit between the structures that dominate the variance in DNS and RNL turbulence, it is argued that RNL and DNS turbulence are sustained b a dnamicall similar mechanism. RNL turbulence self-sustains through interaction between the time-dependent k = roll/streak structure with the streamwise varing flow components through a parametric time-dependent instabilit mechanism. It is argued that the same tpe of dnamics are responsible for sustaining DNS turbulence. Content from this work ma be used under the terms of the Creative Commons Attribution. licence. An further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence b Ltd

3 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78// Ck (T av) ŜCk (T av) 4 5 hk = hk = 4 hk = 6 hk = 8 /Tav T av 5 Figure. The -norm of the difference C k ŜC k between the covariance matri C k (6), and the covariance of the reflected flow about the plane (B.5), at the center of the flow ( = ) as a function of the averaging time, T av, for hk =, 4, 6, 8 (top to bottom). This plot verifies that reflection smmetr about the centerline is a statistical smmetr of the flow and that this smmetr is approached at the rate /T av consistent with the law of large numbers for quadratic statistics. Time is non-dimensionalied b h/u.. Procedure for obtaining the POD modes of the turbulent flow Second-order statistics of flow structure can be obtained from the two-point same-time spatial covariance of the flow-field variables. Adopting the notation for the time average and ( ) T for transposition we form the covariance of the k = components of the velocit, in which C = UU T, () U = [U s, V s, W s ] T, () is the column vector constructed of the deviations (U s (,, t), V s (,, t), W s (,, t)) of the three components of the k = Fourier mode of the streamwise (), cross-stream () and spanwise () velocities (U(,, t), V (,, t), W (,, t)) from their spanwise averages, ([U](, t), [V ](, t), [W ](, t)). In addition we form the covariance of the k components of the flow field, with c = U U T, () U = [u, v, w] T, (4) the column of the three components of the k Fourier component of the flow velocit epanded at all points of the domain. The POD basis is obtained b eigenanalsis of the two-point covariances, C and c, of velocit components. The resulting orthogonal set of eigenvectors is then ordered descending in the magnitude of their eigenvalue to form the POD basis. The eigenvalue of each POD reveals its time-averaged contribution to the energ of the flow. We have chosen for the stud a plane Poiseuille flow in a doubl periodic channel in the streamwise,, and spanwise,, direction of sie L /h = 4π, L /h = π, where h is the channel s half-width in the wall-normal direction,, in which Re τ = is retained in both the DNS and

4 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78// its reduced nonlinear model simulations (cf. Table ). In order to obtain a converged POD basis a ver long time series of a turbulent flow field is required. This convergence is facilitated b taking into account the statistical smmetries of the flow: homogeneit in the and direction, mirror smmetr in about the plane at the center of the channel, and mirror smmetr in about the plane at the center of the spanwise direction. The statistics of the turbulent flow have been verified to asmptote to the above statistical smmetries as the averaging time increases; convergence to the statistical smmetr is demonstrated for the case of the mirror-smmetr in Fig.. These statistical smmetries are not necessar consequences of the translation and mirror smmetr of the NS equations in the periodic channel; the solutions ma undergo smmetr breaking and be inhomogeneous either in or or mirror asmmetric. An eample of smmetr breaking of the statistical homogeneit in the meridional direction is apparent in the nearl time invariant onal flows that are maintained in the turbulent atmosphere of Jupiter. This smmetr breaking in the pipe flow would amount to a turbulent state with temporall constant and spatiall fied streak structures that are aligned parallel to the streamwise ais (cf. Ref. [9]). Statistical homogeneit in implies that the eigenvectors of C are single harmonics of the form: Φ k = A k () B k () Γ k () e ik, (5) where A k (), B k () and Γ k () are N dimensional column vectors, with N, the number of discretiation points in []. At each time instant the N column vector of a k Fourier component, U k (t), of the instantaneous flow field U is obtained and used to form N k average covariances: C k = U k (t)u k (t), (6) where N k is the number of k Fourier components retained in the simulation and is the Hermitian transpose. Eigenanalsis of these covariances determines N N k eigenvectors and the POD orthogonal basis of the k = flow field is this basis ordered decreasing in eigenvalue. The POD modes for the k component of the flow are obtained similarl. The eigenvectors of c are again single harmonics because of the statistical homogeneit in both and directions, of the form: φ kk = α k k () β kk () e i(k+k), (7) γ kk () where α kk (), β kk () and γ kk () are N dimensional column vectors. To obtain the POD basis we determine at each instant the N column vector corresponding to the k, k Fourier component of U k k (t), of the instantaneous flow field U and then form N k N k average covariances: c kk = U k k (t)u k k (t), (8) where N k is the number of the k Fourier components retained in the simulation. Eigenanalsis of these covariances determines N N k N k eigenvectors and the POD orthogonal basis of the k flow field is this basis ordered decreasing in eigenvalue. A simpler calculation was carried out in this stud, restricting the k components POD analsis to a single plane, n =.66. Note that because the phsical flow field is real the Fourier components satisf U k = Uk where denotes the comple conjugate. Consequentl C k = C k and covariances C k and C k have the same eigenvalues, as the covariances are positive definite Hermitian matrices with real eigenvalues. Their corresponding eigenfunctions

5 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78// Table. Simulation parameters. [L, L ]/h is the domain sie in the streamwise, spanwise direction. N, N are the number of Fourier components after dealiasing and N is the number of Chebshev components. Re τ is the Renolds number of the simulation based on the friction velocit. Abbreviation [L, L ]/h N N N Re τ Re NS [4π, π] RNL [4π, π] are conjugate. This implies that the eigenvalues of the POD modes come in pairs corresponding to the sin(k ) and cos(k ) POD structure. When presenting the POD modes for each k onl one of this pair will be shown. Similarl for the POD modes of the k flow field: there is a four-fold degenerac corresponding to the common eigenvalues of c kk, c kk, c k k, c k k and the POD modes can be represented with (, ) structure either in the checkerboard form sin(k ) sin(k ), cos(k ) sin(k ), sin(k ) cos(k ),cos(k ) cos(k ), or in the oblique wave form: sin(k + k ), cos(k + k ), sin(k k ),cos(k k ), which will be chosen in our figures. The complete set of POD modes form a basis orthogonal in the energ norm that spans the three components of the flow field. Although the POD basis is optimall efficient for representing perturbation energ, this basis is not optimal for representing the dnamics of the flow and care must be eercised in relating them to the dnamics of the turbulent state (for a discussion of the relation of the POD basis to the dnamics refer to Ref. []; construction of a basis for optimal representation of flow dnamics refer to Ref. []). In addition to non-optimalit of the POD basis for representing dnamics is the potential for Galerkin projection of the dnamics using this basis to produce dnamical interaction inconsistent with NS which leads inter alia to support of unphsical instabilities [4]. What the POD basis does is to provide important information about flow structure that can be used to reconsrtuct the velocit fields with significantl lower degrees of freedom and therefore constrain the dnamics.. DNS and its RNL approimation In this section we briefl describe the construction of the reduced nonlinear model (RNL) of Navier-Stokes plane channel turbulence. The incompressible Navier-Stokes equations in a constant mass-flu channel flow, with laminar flow in the streamwise direction, is decomposed into equations for the streamwise average flow, U, and the perturbations, u, as follows: t U + U U G(t)ˆ + P ν U = u u, t u + U u + u U + p ν u = (u u u u ). (9a) (9b) U =, u =. (9c) The pressure gradient G(t) that drives the flow maintains constant mass flu. Quantities with an overline u u impl averaging over. Capital letters indicate streamwise averaged quantities. No-slip, impermeable boundaries are placed at = and = h. The corresponding RNL equations are obtained b suppressing nonlinear interactions among streamwise non-constant flow components in the perturbation equations resulting in the right 4

6 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78//.8 DNS RNL.8 DNS RNL /h [U] 5 5 d[u]/d Figure. Left panel: The mean velocit profile of the DNS and RNL simulations. Right panel: The corresponding mean shear in the two simulations. hand side of (9b) being set to. The RNL equations are: t U + U U G(t)ˆ + P ν U = u u, t u + U u + u U + p ν u =. (a) (b) U =, u =. (c) Under this restriction, the perturbation field interacts nonlinearl onl with the mean, U, flow and not with itself. This quasi-linear restriction of the dnamics results in a turbulent state that is supported b a greatl restricted subset of streamwise Fourier components. The components that are retained b the dnamics identif the dnamicall active components sustaining the turbulence. These components reveal the subset of streamwise harmonics that are are energeticall active in the parametric instabilit that sustains the turbulent state [5 8]. Turbulence with realistic mean statistics is obtained with RNL dnamics even with the perturbation field further truncated to comprise a single wavenumber component. Simulations reported here retained onl perturbations with streamwise wavelength λ /h = 4π, which correspond to the gravest streamwise mode in our computational channel of streamwise length L /h = 4π and spanwise width L /h = π. These RNL simulations are compared with DNS simulations in this channel at Re τ = u τ h/ν = with u τ = ν d[u]/d w (d[u]/d w is the shear at the wall) is the friction velocit. In order for the single component RNL simulation to obtain comparable Re τ with DNS the pressure gradient is increased appropriatel. If the perturbation field in the RNL were not constrained to a single streamwise component but were allowed to evolve until its natural support in streamwise wavenumbers had been obtained, it would have retained three streamwise components, with wavelengths λ /h = 4π, π, 4π/, and would sustain turbulence with the Re τ of the DNS without requiring adjustment of the pressure gradient. Numerical details can be found in Appendi A and a summar of the parameters of the simulations in Table. The time averaged streamwise mean flow [U] and its associated shear in the DNS is shown in Fig.. 4. DNS and RNL POD modes The POD basis for the k = component of the flow in DNS and the RNL simulation will now be described. As eplained earlier because of the statistical homogeneit of the flow in the direction, these POD modes come in sin(k ) and cos(k ) pairs. The first 8 POD modes of the 5

7 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78// POD() Var=6.468, ma(u/v) = 9.89 POD() Var=5.94, ma(u/v) = POD() Var=5., ma(u/v) = POD(4) Var=.4666, ma(u/v) = POD(5) Var=8.79, ma(u/v) = 7.4 POD(6) Var=7.9, ma(u/v) = POD(7) Var=.98, ma(u/v) =.87 POD(8) Var=.686, ma(u/v) = Figure. The first 8 POD modes of the turbulent channel flow. This basis has been obtained from data of a 7k advective (ht/u) time units simulation b eploiting the full smmetries of the flow as described in the tet. There is a two-fold degenerac such that each POD corresponds to two identical structures in with sin(k ) and cos(k ) structure in. The contours show levels of the streamwise U velocit and the arrows indicate the the vector velocit (V, W ). These structures and this ordering was verified to be converged b doubling and quadrupling the averaging time. DNS simulation are shown in Fig. and the corresponding POD modes for the RNL simulation in Fig. 4. Shown are both the streak velocit and the corresponding (V, W ) velocit field for each POD. Note that both the DNS and the RNL simulation produce rolls and streaks similarl collocated and with similar structure. The percentage variance eplained b the first POD modes shown in Fig. 5 is similar in the DNS and RNL simulation. Note that ever POD component comes with its streamwise, wall-normal and spanwise velocit component. There is no a priori reason that these fields should be sstematicall correlated. However, a remarkable sstematic correlation is seen between the wall-normal velocit V of the roll and the streak velocit: positive V is correlated with low speed streaks (defects in the streamwise flow) and vice versa. This correlation etends through to higher order POD modes. Each POD shows this remarkable correlation which indicates that the rolls and the streaks form a self-organied structure in which the lift-up mechanism is acting to maintain the streak. If instead the streaks were maintained directl b Renolds stress forcing from the perturbation field no such sstematic correlation between roll velocities and streak velocities would be epected. Analsis of the self-sustaining process (SSP) in RNL reveals that the streak is maintained b 6

8 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78// POD() Var=6.594, ma(u/v) =.7968 POD() Var=.895, ma(u/v) = POD() Var=.796, ma(u/v) =.74 POD(4) Var=.6, ma(u/v) = POD(5) Var=5., ma(u/v) = POD(6) Var=.946, ma(u/v) = POD(7) Var=.74, ma(u/v) =.558 POD(8) Var=.7, ma(u/v) = Figure 4. The first 8 POD modes of the turbulent channel flow under the corresponding RNL simulation as in Fig.. PODs of k = flow DNS RNL 9 PODs of k = flow 5 8 Variance (%) 5 Cumulative variance (%) Inde Inde Figure 5. Left panel: Percentage energ (variance) of the k = flow eplained b each POD doublet in the DNS and RNL simulation. Right panel: The cumulative variance accounted for b the POD modes in the DNS and RNL simulation as a function of the number of POD modes included in the sum. The POD modes have been placed in descending order of the percentage variance that the account. 7

9 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78// POD() Var=5.69, ma(u/v) =.867,(n, n)=(,).8 POD() Var=.699, ma(u/v) = 8.76,(n, n)=(,) POD() Var=7.656, ma(u/v) = 9.544,(n, n)=(,4). POD(4) Var=6.578, ma(u/v) = 9.9,(n, n)=(,) Figure 6. The first 4 POD modes of the DNS simulation at constant =.66. Contours are levels of the cross-flow velocit v on the plane, while the arrows indicate the vector velocit (u, w ) on the plane. Note that u > is associated with v < and vice versa an indication that the underling dnamics are the lift-up mechanism as it operates in k perturbations. the lift-up mechanism and that the eddies oppose the streak rather than maintain it. Note that the first 6 DNS POD modes of the roll/streak structure are reproduced in the RNL model, albeit with some reordering. The similarit of the RNL and DNS POD modes suggests that the same dnamics is operating and that the rolls and streaks self-organie so that the rolls form where the streaks are, in such a manner so as to reinforce the preeisting streak structure. This mechanism is evidentl scale independent given that all the POD modes have this roll/streak velocit correlation. This scale independence suggests that the mechanism of maintaining the roll/streak structure is also scale independent. The wall-normal V velocities of the roll component of the POD modes are about / of the streak velocit which, assuming an average non-dimensional mean flow shear of magnitude (cf. Fig. ) is consistent with the emergence of the streak through the lift-up mechanism over an average 5 time units. Note that in the RNL the streak/roll velocit ratio increases, which is to be epected because RNL turbulence is less disrupted due to having far fewer perturbation structures and streaks can reach larger amplitude. We now consider the k POD modes of the flow. These assume the form of harmonics in and given the statistical homogeneit in and and there is fourfold degenerac in the POD spectrum as each of the four combinations of waves (±k, ±k ) accounts for the same amount of energ. These POD modes were calculated at =.66. A plane wave representative of the top 4 POD quartets for the DNS and the RNL are shown in Fig. 6 and Fig. 7 and the variance accounted b the POD modes is shown in Fig. 8. As the RNL simulation was restricted to a single mode, hk =.5, the RNL and DNS POD sets are not as similar as the would be if the full support of RNL turbulence had been retained. Nevertheless, the first 4 POD quartets of both the RNL and the DNS are hk =.5 structures. For the RNL we can see in Table that 8

10 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78// POD() Var=.79, ma(u/v) = ,(n, n)=(,). POD() Var=9.968, ma(u/v) = ,(n, n)=(,) POD() Var=9., ma(u/v) = 9.99,(n, n)=(,).5 POD(4) Var=9.8, ma(u/v) = 9.67,(n, n)=(,4) Figure 7. The first 4 plane POD modes of the RNL simulation as in Fig. 6. PODs of k = flow,=.66 5 DNS RNL PODs of k = flow,=.66 9 Variance (%) 5 5 Cumulative Variance (%) Inde Inde Figure 8. Left panel: Percentage energ (variance) of the k flow eplained b each POD quartet in the DNS and RNL simulation. Right panel: The cumulative variance accounted for b the POD modes in the DNS and RNL simulation as a function of the number of POD modes included in the sum. over 9% of the variance is eplained b the first 4 modes, in fact the structure of the hk =.5 mode formed b using the first ten POD modes eplains 99% of the variance. As epected, the DNS is more varied,with the hk = mode making a comparable contribution (cf. Table ). As in the the case of the POD s at k = the velocities u and v are correlated indicating that the lift-up mechanism is again at work, here however the lift-up mechanism is supercharged b the Orr-mechanism, which creates enhanced wall-normal velocit as the perturbations are sheared [ 5]. This supercharging mechanism associated with the eruptions in the flow does not operate at k = and is distinct from the lift-up mechanism at k =. This distinction between the lift-up mechanisms operating at k = and k is reflected in the ratio of the u /v of the POD modes being approimatel double the U/V ratio in the k = POD modes. 9

11 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78// Table. POD modes of =.66 plane for DNS Inde (±)n (±)n Variance(%) Cumulative Variance(%) DNS RNL.5 E k time Figure 9. A segment of the energ time-series of hk = on the plane =.66. Also note that because the supercharge lift-up mechanism operating in the k perturbations scales inversel with the square of the total wavenumber of the perturbations the induced u is larger for the lower wavenumber POD modes (cf. Refs. [, 5]). Due to the statistical homogeneit, the k POD modes of individual planes give information onl about the relations between the velocit fields in the assumed single wave harmonic form. In order to obtain an understanding of the perturbation structure it is necessar to project the instantaneous flow on the POD basis. The time evolution of the perturbation energ at the =.66 plane for a segment of the DNS and RNL simulation is shown in Fig. 9. Figure shows instantaneous snapshots of the perturbation structure projected on the top POD modes of the hk =.5 basis, at the start of an energ rising event in the DNS and RNL simulation. Regions of u v < with alternating v and u signs are collocated with the most prominent streak defects which in both cases are negative. On this plane the interaction of the streak with the perturbations is connected with the product w U s, generating perturbation velocit in the process. The perturbation structure consists of oblique waves at the wings of the spanwise shear of the streak.

12 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78// v projected, DNS v projected, RNL Figure. Instantaneous velocit fields of DNS and RNL projected on the first POD quartets with hk =,showing the onset of an energ growth event. The contours show levels of the wall-normal velocit, the arrows the (u, w) vectors of the perturbation field on the plane =.66. The black dashed line at = π is the of the streak, U s, at this plane and the black solid line is times the value of the streak velocit U s. DNS is taken from time= 56 and RNL from time= 74. The corresponding perturbation energ is shown in Fig. 9. Table. POD modes of =.66 plane for RNL Inde (±) n (±) n Variance (%) Cumulative Variance (%) Further truncation of the RNL dnamics The possibilit of reducing the RNL dnamics support even further b using projection on the POD s will now be eamined. Projection on a basis comprised of all three components (U s, V, W ) is constraining the dnamics in an undesirable wa, since the streak U s U [U] is the result of the (V, W ) operating on the mean profile [U] and the U s itself. Therefore the streak dominated POD modes would not be able to correctl describe the roll dnamics since an roll is instantaneousl associated with a much bigger streak [4]. For this reason we calculate the POD modes of onl the streak portion of the flow and inquire whether we can obtain self-sustaining turbulence with a low order POD representation of the streak structure. It could be argued that a single pair of the POD basis cannot describe correctl the dnamics of the flow, since it would enforce smmetr between high and low speed streaks, and the mean flow would be consequentl characteried b a nearl laminar unblunted profile.

13 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78// Re τ 9 8 POD6 POD POD POD6 POD POD time Figure. Friction Renolds number Re τ as a function of advective time for simulations truncating streak velocit U s to 6,,, 6, and 4 POD pairs. Figure shows the time series of Re τ in simulations with a selection of POD pairs used for the truncation. The RNL sstem is otherwise unmodified in comparison to the one used for the calculation of the POD basis. When the lowest number of POD pairs is retained, the simulation laminaries without ehibiting an dnamical behavior. An increase in the number of POD modes allows the sstem to retain its turbulent behavior. The lifetime of each simulation appears to be uncorrelated with the increase of the POD pair number. This suggests that the truncation likel does not cause relaminariation due to failure to represent the energ of the streak, which would be epected to produce laminariation far more quickl. 6. Discussion Numerical data were obtained from a DNS of a turbulent channel and the corresponding RNL simulation. POD analsis was carried out on two different components of the flow, the roll/streak and the perturbation structure. Striking resemblance of the relations between the velocit fields was seen in the two simulations. This correspondence suggests a similar dnamics is at work in maintaining turbulence in these sstems. Remarkabl, the mechanism maintaining turbulence in the RNL sstem has been comprehensivel characteried [5 8]. The close correspondence in structure and dnamics between RNL and NS dnamics invites the eploitation of the comprehensivel characteried RNL turbulence to advance understanding of NS turbulence in channel flow. In particular, the lift-up mechanism operating at k = and the associated supercharge mechanism operating at k have been clearl seen operating in both RNL and DNS including using the POD projections. As a further probe of the mechanism of RNL turbulence the POD modes were used to truncate the k = velocit fields in a different RNL simulation, showing that the sstem sustains at a fairl low number of POD modes. This result provides an even more reduced turbulent dnamics in terms of the dimension of its support than the alread highl reduced RNL turbulence which operates realisticall supported b a single streamwise wavenumber. The purpose of this comparison is in part to elucidate the similarities and differences of the DNS with the RNL. The dnamics of RNL model turbulence involve interaction of the streamwise mean flow with few, even as few as one, streamwise flow harmonic. Despite this simplification salient aspects of the turbulent dnamics are captured [5 8].

14 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78// Acknowledgments This work was funded in part b the Second Multiflow Program of the European Research Council. Brian Farrell was also supported b NSF AGS We would like to thank Daniel Chung for his reviewing comments. Appendi A. The data were obtained from a DNS of () and from the RNL that is directl associated with the DNS. Both the DNS and its directl associated RNL are integrated with no-slip boundar conditions in the wall-normal direction and periodic boundar conditions in the streamwise and spanwise directions. The dnamics were epressed in the form of evolution equations for the wall-normal vorticit and the Laplacian of the wall-normal velocit, with spatial discretiation and Fourier dealiasing in the two wall-parallel directions and Chebchev polnomials in the wall-normal direction [6]. Time stepping was implemented using the third-order semi-implicit Runge-Kutta method. Appendi B. Calculating covariances for smmetrical states Smmetries in reflections in about the centerline plane at = and reflections in about the plane at = π/ add in the time series of the simulation independent combinations of A k (), B k () and Γ k () : Ŝ Φ k = A k ( ) B k ( ) Γ k ( ) e ik, ŜΦ k = A k () B k () Γ k () e ik, Ŝ Ŝ Φ k = A k ( ) B k ( ) Γ k ( ) e ik. (B.) To recalculate the additional covariances for the above arrangements of the velocit fields we can use the following separation of the covariance C k = Ck uu Ck vu Ck wu Ck uv Ck vv Ck wv Ck uw Ck vw Ck ww, (B.) with C u iu j k = (C u ju i k ). In the following the k subscript will be omitted and instead of u i u j the superscript ij will be used. So the covariance is written as: C = C C C C C C. (B.) C C C Reflections of the velocities in result in the conjugation of each individual covariance, and placement of a minus sign in the w components of the covariance, so that: Ŝ C = (C ) (C ) (C ) (C ) (C ) (C ) (C ) (C ) (C ). (B.4)

15 nd Multiflow Summer School on Turbulence Journal of Phsics: Conference Series 78 (6) doi:.88/ /78// Reflections in require to reverse the order of the row and column indees in each individual covariance and if this operation is noted as ŜC ij = C ij R we have: Ŝ C = C R CR CR CR CR CR C R C R C R. (B.5) Finall combined and reflections are obtained b the successive action of the individual operators on the initial covariance, and thus: Ŝ Ŝ C = (C R ) (CR ) (CR ) (CR ) (CR ) (CR ) (C R ) (C R ) (C R ). (B.6) References [] Moin P and Moser R D 989 Characteristic-edd decomposition of turbulence in a channel J. Fluid Mech [] Berkoo G, Holmes P and Lumle J L 99 The proper orthogonal decomposition in the analsis of turbulent flows Ann. Rev. Fluid Mech [] Sirovich L, Ball K S and Keefe L R 99 Plane waves and structures in turbulent channel flow Phs. Fluids A 7 6 [4] Moehlis J, Smith T R, Holmes P and Faisst H Models for turbulent plane Couette flow using the proper orthogonal decomposition Phs. Fluids [5] Farrell B F and Ioannou P J Dnamics of streamwise rolls and streaks in turbulent wall-bounded shear flow J. Fluid Mech [6] Thomas V, Lieu B K, Jovanović M R, Farrell B F, Ioannou P J and Game D F 4 Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow Phs. Fluids 6 5 [7] Constantinou N C, Loano-Durán A, Nikolaidis M A, Farrell B F, Ioannou P J and Jiméne J 4 Turbulence in the highl restricted dnamics of a closure at second order: comparison with DNS J. Phs. Conf. Ser [8] Thomas V, Farrell B F, Ioannou P J and Game D F 5 A minimal model of self-sustaining turbulence Phs. Fluids 7 54 [9] Avsarkisov V, Hoas S, Oberlack M and Garcia-Galache J P 4 Turbulent plane Couette flow at moderatel high Renolds number J. Fluid Mech. 75 R [] Farrell B F and Ioannou P J 99 Stochastic forcing of the linearied Navier Stokes equations Phs. Fluids A [] Farrell B F and Ioannou P J Accurate low-dimensional approimation of the linear dnamics of fluid flow J. Atmos. Sci [] Farrell B F and Ioannou P J 99 Optimal ecitation of three-dimensional perturbations in viscous constant shear flow Phs. Fluids A [] Farrell B F and Ioannou P J 99 Perturbation growth in shear flow ehibits universalit Phs. Fluids A 5 98 [4] Jiméne J, Del Álamo J and Flores O 4 The large-scale dnamics of near-wall turbulence J. Fluid Mech [5] Jiméne J How linear is wall-bounded turbulence? Phs. Fluids 5 84 [6] Kim J, Moin P and Moser R 987 Turbulence statistics in full developed channel flow at low Renolds number J. Fluid Mech