Received 16 August 2013, revised 30 April 2014Accepted for publication 6 September 2014Published 16 October 2014
Communicated by Y Hattori
AbstractThe quasi-cyclic evolution of turbulence driven by a steady force in a periodiccube is investigated by means of large-eddy simulations with vanishingkinematic viscosity. By constraining the domain size so that only a singleseries of energy cascade events can take place, quasi-cyclic motions of multi-scale coherent vortices with a period of about 20T are realized. (Here, Tdenotes the turnover time of the largest eddies.) The observed cycle is com-posed of four periods characterized by activities of the largest- and smallest-resolvable-scale eddies. Vigorous energy cascade events, which last for about2T, are observed between the two moments when large- and small-scale eddiesare active. Even though we have examined only a special case of steady forces,such cyclic behavior of turbulence is likely to capture the essential dynamicsof the regeneration cycle of multi-scale coherent structures, that is, the energycascade in homogeneous isotropic turbulence at high Reynolds numbers.
(Some figures may appear in colour only in the online journal)
1. Introduction
The Kolmogorov (1941) similarity hypothesis is a cornerstone theory of turbulence, whichstates that small-scale statistics of high-Reynolds-number turbulence are determined by themean energy dissipation rate and the kinematic viscosity of the fluid. This hypothesis issometimes explained in terms of the so-called energy cascade. That is, energy is passed downfrom larger to smaller vortices in a scale-by-scale manner, and subsequently it is converted tothermal energy when it reaches the smallest scales. This phenomenological explanationindeed provides a reason why the mean energy dissipation rate, which is equal to the meanenergy flux in the inertial range, is the key quantity in the statistics of turbulence. However,
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there is room for further investigation to connect the dynamics leading to the energy cascadeand the consequences of the Kolmogorov similarity hypothesis such as the−5 3 power law ofthe energy spectrum.
In order to deal with this fundamental problem of turbulence, it is essential to identify theelementary dynamics leading to the cascade of energy in the inertial range. Recent studies(Goto 2008, 2012) using direct numerical simulations (DNS) of homogeneous isotropicturbulence (HIT) have shown that high-Reynolds-number HIT consists of a hierarchy ofmulti-scale vortex tubes, and that an energy cascade can be caused by the successive creationof smaller-scale tubular vortices in larger-scale straining regions existing between pairs oflarger-scale tubular vortices. Although this picture is a candidate explanation of the ele-mentary process of an energy cascade, we cannot exclude the possibility of other processesrelevant to the energy cascade. This is because, due to the spatio-temporally chaotic behaviorof turbulence, it seems that various cascade events take place simultaneously.
A possible way of overcoming this difficulty was proposed by Hamilton et al (1995).They conducted DNS of plane Couette flow in a highly constrained computational domain(Jiménez and Moin 1991), and obtained a remarkably well-defined, quasi-periodic regen-eration cycle of near-wall coherent structures in the constrained Couette flow. This regen-eration cycle is known by the name of self-sustaining process (SSP), and has helped us tounderstand the elementary dynamical process sustaining coherent structures such as stream-wise vortices and low-speed streaks in near-wall turbulence.
The main purpose of the present paper is to numerically capture a regeneration cycle ofcoherent structures in high-Reynolds-number turbulence in a periodic cube, which is relevantto the cascade of energy in the inertial range. For this purpose, we have to pay specialattention to the properties of the external force. First, we choose a steady external body force.As will be shown in section 2.1, this choice is essential for obtaining the quasi-cyclic behaviorof turbulence. Secondly, we choose a force such that the integral scale of the driven turbu-lence is not much different from the system size. This idea resembles the spatial constraint ofminimal plane Couette flow (Jiménez and Moin 1991). Such a steady and highly-constrainedexternal force is contrary to the conventional forces used in previous DNS of HIT. Never-theless, at the sacrifice of homogeneity and isotropy of large-scale flow structures, it does leadto a regeneration cycle of coherent structures. Another key feature of the present study is thatwe conduct large-eddy simulations (LES) instead of DNS. Our preliminary DNS with thesteady external force (Goto et al 2012) show that the temporal evolution of turbulencesustained by the force is quasi-cyclic, and the period is about 20 turnover time of the largesteddies irrespective of the Reynolds number. As will be shown below, our LES with the forceshow the quasi-cyclic behavior similar to that of DNS. Since LES require much fewer degreesof freedom than DNS to simulate high-Reynolds-number turbulence, the quasi-cyclic motionof the turbulence in LES is useful for studies on the basis of a dynamical system approach,such as finding periodic solutions at sufficiently high Reynolds numbers to reproduce Kol-mogorovʼs inertial-range scaling, if any (see Kawahara et al 2012).
2. Numerical method
2.1. External force
As mentioned in the introduction, the choice of external force is crucial for capturing quasi-cyclic motions of turbulence. In the previous numerical simulations of HIT, time-dependentforces were preferentially used in order to generate statistically stationary turbulence. In HITdriven by such a time-dependent force, however, large-scale structures tend to become
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spatio-temporally random as intended, and no clear cyclic behavior is observed. In the presentstudy, on the contrary, we employ a steady force. More concretely, we simulate turbulencedriven by spatially periodic and temporally steady force f given by
× = ( ) ( )f xx xsin sin ˆ , (1)1 2 3
where x3 is a unit vector in the x3-direction.When the Reynolds number Re is sufficiently low, this steady force sustains a steady
hyperbolic stagnation point flow as shown in figure 1. Our preliminary DNS (Goto et al 2012)also show that, as Re increases gradually, this steady flow bifurcates and becomes a stabletemporally periodic flow. The stable periodic flow displays the following cyclic behavior
→ → →((i) (ii) (iii) (i)): (i) quadruple large-scale vortices are intensified by the steadyforce. (ii) Around the quadruple large-scale vortices, smaller-scale vortices in the perpendi-cular direction to the large-scale vortices are stretched and created. (iii) Energy dissipates dueto the viscous effect around smaller vortices, and the whole flow becomes quiescent. Theexistence of this stable periodic solution at lower Re may be the reason why turbulencesustained at higher Re also shows quasi-cyclic behavior.
Melander and Hussain (1993), as well as Takahashi et al (2005), have investigated theinteractions between columnar vortices and fine-scale external turbulence by means of DNS,and observed that the columnar vortex wraps worm-like tubular vortices around itself. Theirobservations are likely to be related to phase (ii) of the cyclic motion.
2.2. Large-eddy simulation
For the LES, we employ the standard Smagorinsky model (Smagorinsky 1963), where thefiltered Navier–Stokes equation
ρρτ
ν ν∂∂
+∂∂
= − ∂∂
+ + ∂∂
+ +⎜ ⎟⎛⎝
⎞⎠ { }u
tu
u
x xp
xS f
1
32( ) , (2)i
ji
j i
kk
jT ij i
and the filtered continuity equation, =∂∂ 0,u
xi
iare numerically integrated. Here, ui, p, ρ, and ν
are the velocity, pressure, constant density, and kinematic viscosity, respectively; grid-scale
(GS) flow fields are denoted by ( · ). In (2), = +∂∂
∂∂
⎜ ⎟⎛⎝
⎞⎠Sij
u
x
u
x
1
2i
j
j
iis the GS strain-rate tensor,
τ = −u u u uij i j i j is the subgrid-scale (SGS) stress tensor, and ν Δ= C S S( ) 2T ij ijs2 is an eddy
viscosity. Here, Δ is the filter width. In our LES, we use the standard value of Smagorinskyconstant, Cs = 0.18 (Lilly 1967).
The periodic boundary condition with period π2 is imposed in all the x1-, x2- and x3-directions, and the LES equations are integrated numerically by using the Fourier spectralmethod. Aliasing errors of the nonlinear terms except for the eddy viscosity terms areremoved by the phase-shift method. Temporal integrations are made by the fourth orderRunge–Kutta scheme. We conduct the LES for five different resolutions (163, 323, 643, 1283
and 2563), and discuss the resolution effect on the results in section 3.1. In the following, wereport the results for the vanishing kinematic viscosity (ν = 0).
Figure 2 shows the temporal average of the energy spectrum E k t( , ) of the simulatedturbulence. The shown energy spectrum is normalized by ϵ⟨ ⟩ 2
3 , where
ϵ ν= S S2 (3)T ij ij
is the rate of the energy transfer from GS to SGS, and ⟨ ⟩· denotes the time average. As seenhere, ⟨ ⟩E k( ) obeys the −5 3 power law in a low-wavenumber range, while it deviates from
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the law for high wavenumbers. Such a deviation stems from the Smagorinsky model, and isconsistent with previous results (see Hughes et al 2001).
Figure 1. Steady flow driven by the steady force (1) at low Reynolds number. Velocityvectors on the plane π=x3 and isosurfaces of enstrophy are shown.
Figure 2. Temporal average of the energy spectrum in the case of ν = 0. Thin dashed,thin solid, thick gray, thick dashed and thick solid lines represent the average spectrumfor 163, 323, 643, 1283 and 2563 grid points, respectively. Thin straight dashed lineindicates −5 3 slope. ϵ is the rate of energy transfer from GS to SGS, and L is theintegral scale.
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3. Results
3.1. Quasi-cyclic evolution of energy and its transfer rate
One of the most remarkable features of turbulence driven by the steady force (1) is its quasi-cyclic behavior. This is demonstrated in figure 3, where long-term evolutions of kineticenergy K(t) and energy transfer rate ϵ t( ) are plotted in the case of 1283 grid points. This figureshows that both K(t) and ϵ t( ) fluctuate in time with large amplitudes, and the typical period ofits quasi-cyclic behavior is about 20T. Here
=TL
K(4)
23
is the turnover time of the largest eddies, with L being the integral scale. We emphasize thatthe quasi-periodic behavior is always observed irrespective of the resolutions of LES. In orderto verify this resolution independence quantitatively, we conduct the LES with the fivedifferent resolutions for sufficiently long time (about 2000T for the resolutions 163, 323, 643
and 1283, and about 200T for 2563), and estimate the average period by the followingprocedure. First, we divide (with overlaps) the whole time series of K(t) into shorter intervalsof 80T; the number of the intervals is 75 and 8 for the resolutions 163, 323, 643, 1283 and for2563, respectively. Secondly, we estimate the local period of the cyclic motion in the eachinterval from the first peak of the auto-correlation function of K(t). Finally, we take theensemble average of the local periods. The resultant ensemble average of the local periods is
±T T19 6 , ±T T18 7 , ±T T20 7 , ±T T26 11 and ±T T19 2 for 163, 323, 643, 1283 and 2563,respectively. The error-bar indicates the standard deviation among the 75 or the 8 realizations.There is no significant dependence on the resolution, and therefore we will show, in thefollowing, results with the resolution of 1283 only. Results with other resolutions are similarnot only qualitatively but also quantitatively. It is worth mentioning, in passing, that the localperiod has a relatively large fluctuation as large as T10 around its average about 20T.
We can observe in figure 3 that K(t) attains a local maximum (or a local minimum) at atime which is earlier than ϵ t( ) by about T2 . This time lag is also estimated on the basis of thecorrelation function of K(t) and ϵ t( ).
The cyclic behavior is also confirmed by plotting the orbit of the numerical solution onthe K-ϵ plane (figure 4). The orbit always evolves in the anti-clockwise direction in figure 4.
Figure 3. Long-term evolution of the simulated turbulence in the case of 1283 gridpoints. Solid line and dashed line denote kinetic energy K and its transfer rate ϵ to SGS,respectively.
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This is consistent with the observation in figure 3, that the temporal evolution of ϵ t( ) followsthat of K(t).
Since we have set ν = 0 in our LES, the above results imply that the quasi-cyclicbehavior observed in DNS (Goto et al 2012) at moderate Reynolds numbers is also relevant
Figure 4. Two-dimensional projection of a quasi-cyclic orbit in the case of 1283 gridpoints. Horizontal and vertical axes respectively represent kinetic energy K and itstransfer rate ϵ to SGS. The thin line corresponds to time evolution in figure 3. The thickline corresponds to a single cycle shown in figure 5. The open circle, open triangle,open square, and open inverted triangle correspond to times =t t1, t2, t3, and t4 infigure 5, respectively. Temporally averaged values for the long-term evolution and forthe single cycle are plotted by the filled square and the filled triangle, respectively.
Figure 5. Temporal evolution of kinetic energy K (solid curve), energy transfer rate ϵ toSGS (thin dashed curve), and energy input rate I (thick dashed curve) of quasi-cyclicflow in the case of 1283 grid points. An example of a single cycle is shown. Times
=t t1, t2, t3 and t4 correspond to (a), (b), (c) and (d) in figure 6, respectively.
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even in the high-Reynolds-number limit. It is also interesting to observe that the averagedperiod (≈ T20 ) of quasi-cyclic motions in DNS (Goto and Kawahara 2013) is almost the sameas that in the present LES.
3.2. Regeneration cycle of multi-scale vortices
In this subsection, we investigate flow structures. As previously shown (figures 3 and 4), theflow shows quasi-cyclic behavior. Therefore, we focus on a single cycle of this quasi-cyclicmotion. An example of such a cycle is shown in figure 5, where the temporal evolutions ofK(t), ϵ t( ) and the energy input rate I(t) are plotted. Note that averaged quantities ⟨ ⟩K s and ϵ⟨ ⟩ s
over the single cycle are close to ⟨ ⟩K and ϵ⟨ ⟩ for the long-term evolution, as shown in figure 4.Hereafter, ⟨ ⟩· s denotes the temporal average over the single cycle shown in figure 5.
An interesting observation is that coherent vortical structures also change drastically inthis cycle. This is demonstrated in figure 6. Here, large-scale vortices (light-colored objects)are identified by isosurfaces of coarse-grained enstrophy, which are obtained by the low-passfiltering of Fourier components of vorticity with a sharp cut-off wavenumber at kc, where
⟨ ⟩ =k L 4.69c . On the other hand, small-scale vortices (dark-colored objects) are identified byisosurfaces of GS enstrophy. Snapshots (a)–(d) in figure 6 correspond to =t t1 (when K
Figure 6. Snapshots of high-Reynolds-number quasi-cyclic turbulence (ν = 0) drivenby the external steady force (1). (a), (b), (c) and (d) correspond to times =t t1, t2, t3 andt4 in figure 5, respectively. The smallest-resolvable-scale and largest-scale vorticalstructures are represented by dark-colored and light-colored isosurfaces, respectively.
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attains its maximum), t2 (when ϵ attains its maximum), t3 (when K attains its minimum) and t4(when ϵ attains its minimum) in figure 5, respectively.
At =t t1, quadruple large-scale vortices sustained by the external force are energetic.From =t t1 to t2, a number of smaller-scale vortices are created due to vortex stretching, sothat kinetic energy is transferred from larger to smaller scales. At the same time, large-scalevortices become weaker while passing some of their energy on to smaller-scale vortices. From
=t t2 to t3, energy is transferred from GS to SGS, and less energy is injected into GS by theforce, so that the entire flow becomes quiescent. At =t t4, quadruple large-scale vortices areintensified by the external force, so that they go back to an active state. This quasi-cyclicbehavior of multi-scale coherent vortices is schematically illustrated in figure 7. It is worthnoting here that such cyclic behavior is observed even in an LES field without kinematicviscosity.
Our simulations demonstrate that by restricting the spatial domain and using the steadyforce, we can capture quasi-cyclic behavior of coherent structures. In this cycle, we alsoidentify vigorous energy cascade events which take place for the duration between =t t1 andt2. Figure 6 shows that the observed cascade event is consistent with the notion that smaller-scale vortices are stretched and created around larger vortex tubes (Goto 2008, 2012).Although we have demonstrated only one example for the specific external force (1), we mayapply this technique to a variety of forcing schemes. An important part of our future work willbe to verify whether or not there is a universal elementary process of energy cascade in HIT,which may be reminiscent of the SSP (Hamilton et al 1995) of near-wall structures.
3.3. Temporal evolution of energy spectrum
In the preceding subsection, we have observed the quasi-cyclic behavior of coherent struc-tures (see figure 6) in the simulated LES turbulence. Now let us turn to the question of howthe observed dynamics is related to universal statistics of turbulence. As a first step in thisdirection, let us consider the temporal evolution of the energy spectrum E k t( , ).
Figure 8 shows instantaneous energy spectra at times =t t1, t2, t3 and t4 indicated infigure 5. At =t t1, when K attains its maximum, for all the wavenumbers, E k t( , )1 is largerthan its temporal average ⟨ ⟩E k( )s over the single cycle shown in figure 5. Note that
= − ⟨ ⟩ =E k t E k( 1, ) ( 1)1 s has a large value, where k = 1 corresponds to k⟨L⟩ = 1.17. It isconsistent with the observation that largest-scale vortices are vigorous at =t t1 (see figure 6(a)for a visualization). At =t t2, when ϵ attains its maximum, − ⟨ ⟩E k t E k( , ) ( )2 s becomesmuch larger than − ⟨ ⟩E k t E k( , ) ( )1 s in a higher wavenumber range, while
= − ⟨ ⟩ =E k t E k( 1, ) ( 1)2 s becomes negative. This is because a number of smaller vorticesare created through the vigorous energy cascade events (figure 6(b)). At =t t3, when K attainsits minimum, that is, when largest-scale vortices become quiescent after a period during
Figure 7. Schematic diagram of the quasi-cyclic behavior of multi-scale vorticalstructures in HIT driven by the steady force (1).
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which the energy transfer rate ϵ t( ) is larger than the energy input rate I(t) (figure 6(c)),E k t( , )3 is smaller than ⟨ ⟩E k( )s for all the wavenumbers. At =t t4, when ϵ attains itsminimum, E k t( , )4 is much smaller than E k t( , )3 for high wavenumbers. On the other hand,
=E k t( 1, )4 becomes larger than =E k t( 1, )3 . This is also consistent with the observationthat largest-scale vortices start becoming active at =t t4 (figure 6(d)).
In order to investigate cascade dynamics in terms of E k t( , ), the deviation of the energyspectrum E k t( , ) from its long-term average ⟨ ⟩E k( ) was considered by Kida and Ohkitani(1992) and van Veen et al (2006) for HIT with a high symmetry, and by Horiuti andOzawa (2011) for homogeneous shear flow. We use a similar approach to investigate therelationship between the dynamics and statistics of the quasi-cyclic motion observed in thepresent LES turbulence. Figure 9 shows the contour of the excess energy spectrum
Figure 8. Unsteadiness of the energy spectrum. Thick dashed lines denote the temporalaverage ⟨ ⟩E k( )s of the energy spectrum over the single cycle shown in figure 5. Thesolid line and thin dashed line respectively denote (a) E k t( , )1 and E k t( , )2 , (b) E k t( , )3
and E k t( , )4 , (c) − ⟨ ⟩E k t E k( , ) ( )1 s and − ⟨ ⟩E k t E k( , ) ( )2 s , (d) − ⟨ ⟩E k t E k( , ) ( )3 s
and − ⟨ ⟩E k t E k( , ) ( )4 s .
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− ⟨ ⟩E k t E k( , ) ( )s . Horizontal and vertical axes of this figure respectively denote time t andwavenumber k, and we can see how the excess energy spectrum evolves in time.
A filled triangle and an open triangle denote the time when the excess energy spectrum,− ⟨ ⟩E k t E k( , ) ( )s , takes the maximum and the minimum over the single cycle, respectively,
for a given wavenumber k. Since filled triangles tell us how energy is transferred from large tosmaller scales, let us first focus on them. The excess energy spectrum for the lowest wave-number attains its maximum approximately when K attains its maximum =t t1. After thistime, the wavenumber at which the excess energy spectrum attains the maximum in the cyclegradually increases because energy is transferred from larger to smaller scales. Once theenergy reaches the cut-off wavenumber, it is transferred to SGS. Approximately when theexcess energy spectrum at the cut-off wavenumber attains the maximum in this cycle, ϵattains its maximum ( =t t2). Figure 9 shows that the time lag to attain the maximum isshorter for higher wavenumbers. This is consistent with the known statistics, i.e., that theturnover time of eddies at scale −k 1 is proportional to −k 2 3 in the inertial range.
It is also interesting to observe that the graph of open triangles, which denote time for theminima of the excess energy spectrum, is quite similar to that of filled triangles for themaxima. This is consistent with the idea that the energy cascade takes place in a scale-by-scale manner. More concretely, the graph of maxima shows that ‘the situation that energysupply from larger-scale excesses its transfer to smaller-scale’ is transferred from larger tosmaller scales. Similarly, the graph of minima shows that ‘the situation that energy supplyfrom larger-scale is insufficient compared with its transfer to smaller-scale’ is transferred fromlarger to smaller scales. It may be natural to expect that the time-scale of both of these twoinformation transfers is identically determined by the eddy turnover times in the inertial range.
In figure 7 we have referred to the energy cascade events from t3 to t4, which arecharacterized by the insufficient energy supply from the larger scales, as inactive energycascade events. As observed in section 3.2, the creation of the large-scale vortices is followedby the vigorous energy cascade events, while the inactive energy cascade events follow the
Figure 9. Temporal evolution of energy spectrum. Excess from the temporal average⟨ ⟩E k( )s over the single cycle shown in figure 5 is plotted. The value is normalized byits standard deviation. Filled triangles and open triangles indicate the time when theexcess energy spectrum takes maximum and minimum over a single cycle,respectively, for a given wavenumber.
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decay of the large-scale vortices, i.e., the insufficient energy supply from the largest scale.Both the vigorous and inactive energy cascade events are key elements of the cycle with alarge-amplitude energy oscillation.
It may be worth mentioning that E k t( , ) is never steady in time, even in turbulencesustained by the steady force, and it varies drastically during the cycle. This is clearlyobserved in figure 8. Note that the unsteadiness of E k t( , ) implies that the energy flux at eachwavenumber is also unsteady. It was the well-known footnote remark by Landau (section 33of Landau and Lifshitz 1959; see also section 6.5 of Frisch 1995) that first pointed out theunsteadiness of energy flux (and dissipation rate). He cast doubt on the universality of ⟨ ⟩E k( )normalized by ϵ⟨ ⟩, because the unsteadiness of ϵ is likely to depend on large-scale motions.The unsteadiness of ϵ is indeed observed in our turbulence (figure 3). Recall that the period isabout T20 . Although Landauʼs objection has been extensively discussed in terms of thespatial intermittency, there seems room for further investigation on this feature of unstea-diness in statistically stationary turbulence.
4. Concluding remarks
We have conducted the LES of turbulence in a periodic cube driven by the spatially periodicand temporally steady force (1) in the case that kinematic viscosity vanishes. We havesubsequently shown that the simulated turbulence behaves in a temporally quasi-cyclicmanner (figure 3) with a period of about T20 irrespective of the LES resolutions. Theobserved quasi-cyclic motion was summarized in figure 7.
Thanks to the quasi-cyclic behavior, we were able to concentrate on a single cycle toinvestigate the temporal evolution of coherent structures (section 3.2) and the energy spec-trum (section 3.3) of the simulated turbulence. Our research indicates that vigorous energycascade events and inactive energy cascade events take place for a relatively short duration(about T2 ) in a cycle. By visualizing multi-scale coherent vortices (figure 6), we have proventhat the cascade events can be explained in terms of vortex stretching, and verified that theevolution of the excess energy spectrum shown in figure 9 is consistent with the scale-by-scale energy cascades.
This quasi-cyclic motion shows the unsteadiness of the energy spectrum. The unstea-diness is not our finding, but was noticed by many previous authors (for example, Yoshi-zawa 1994, Woodruff and Rubinstein 2006, Horiuti and Ozawa 2011). Even the well-knownfootnote by Landau and Lifshitz (1959) stated that ϵ t( ) varies temporally with a period ofO T( ). The cyclic behavior is caused by the vigorous energy cascade events triggered by theinstability of the largest-scale vortices. Therefore, we must be cautious about the universalityof energy spectrum, for example, in terms of ϵ⟨ ⟩. It is worth noting here that our observationsare quite similar to those in homogeneous shear turbulence demonstrated by Horiuti andOzawa (2011).
Some problems are left for future studies. First, we have conducted simulations only on asingle case of steady forces. It would be, therefore, interesting to conduct simulations withother types of steady forces to identify the universal cascade events, if any. This issue is alsorelevant to the universality problem stated in the preceding paragraph. Secondly, the origin ofthe quasi-cyclic behavior of stationary turbulence is likely to be explained by a dynamicalsystem approach. Indeed, we have found unstable periodic solutions to the LES equationsgiven by (2). The results will be reported elsewhere in the near future.
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Acknowledgements
This work was partly supported by Grants-in-Aid for Scientific Research from the JapanSociety for the Promotion of Science (22360079, 25249014).
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