12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

LOW-ORDER MODELLING OF TURBULENT QUASI-CYCLIC BEHAVIOUR
AND SUDDEN RELAMINARISATION

Ryo Araki
Wouter J. T. Bos
LMFA
LMFA
École Centrale de Lyon
École Centrale de Lyon, CNRS, Univ Lyon
69134 Écully cedex, France
69134 Écully cedex, France
Graduate School of Engineering Science
wouter.bos@ec-lyon.fr
Osaka University
1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japanaraki.ryo@ec-lyon.fr

Susumu Goto
Graduate School of Engineering ScienceOsaka University
1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japans.goto.es@osaka-u.ac.jp

ABSTRACTGreen forcing. The features of the periodic flow are similar todeveloped turbulence at higher Reynolds numbers. Our goal
is to illustrate the similarity between the temporal dynamics ofthe periodic and turbulent flow and model them with a minimal
degree of freedom to understand the origin of the QCB.
Thereto, we conduct DNS of the Navier-Stokes equations,

We propose a three-equation model which reproduces thequasi-cyclic behaviour of turbulent flow [Araki et al. (2022)].
We examine a 3D flow driven by steady forcing at various
Reynolds numbers (Re) to find a perfectly time-periodic flow
at a specific low Re and quasi-cyclic flow at higher Re. The
two states are continuously connected when changing Re. A
mode-by-mode analysis of the periodic flow allows us to formulate the minimal model, which describes the evolution of
three families of scales with their nonlinear interactions, dissipation, and driving terms. By calibrating the model parameters
to the DNS results, when possible, we find a qualitatively similar periodic solution to the DNS. By increasing the “Reynoldsnumber” of the model, we find permanent chaos, which wecompare to turbulence. The same model also reproduces probabilistic transitions between chaotic and steady states for thedifferent parameter sets. The scaling of this sudden relaminarisation agrees with the ones observed in turbulence to someextent. We consider that our minimal model reproduces several key properties of turbulence.

(∂t u + (u · ∇)u = −∇p + ν∇2 u + f,
f = (− f0 sin x cos y, f0 cos x sin y, 0),

(1)
(2)

along with the incompressibility condition ∇ · u = 0. Here,u, p, and f denote velocity, pressure, and steady Taylor–Green
body force fields, respectively. The kinematic viscosity ν isthe control parameter of the flow and we define the Reynolds√
3/2
ν with the forced wavenumber
number Re ≡ f0 / k fk f = (1, 1, 0). We set the forcing coefficient f0 to unity.
We summarise both 3D Periodic Flow (3DPF) at Re =5.83 and turbulence at Re = 29.7 in Fig. 1. Visualisations ofvorticity magnitude (Fig. 1 (a,c)) depict similar spatial structures for both Re, namely, the forcing-induced large-scalecolumnar vortices and smaller vortices in antiparallel orientation.
Here, low-pass filtered quantity is defined by u< (x) ≡R
dr G(r/σ )u(x√ + r), where G(r/σ ) is Gaussian function andwe set σ = 2 k f = 2. Parametric time series of energy input rate P ≡ hf · ui and energy dissipation rate ε = νh|ω|2 i,
where ω = ∇ × u and h·i is the spatial average, show counterclockwise cycle in the 3DPF (Fig. 1 (b)). Here, the timeseries are normalised by the characteristic time scale T ≡√−1k f f0= 0.840, where k f = 2 and f0 = 1. The turbulent time series in Fig. 1 (d) also displays intrinsic QCB
after a phase-averaging process. See Araki et al. (2022) for
the detail of this procedure. We note that for intermediate
5.83 < Re < 29.7, we observe continuously changing phaseaveraged hPiphase –hεiphase plots from Fig. 1 (b) to Fig. 1 (d).

BACKGROUND
Quasi-Cyclic Behaviour (QCB) is observed in many turbulent flows and is sometimes related to the energy cascademechanism. For example, QCB is observed in periodic box
turbulence driven by a steady forcing and found to be associated with vortex stretching of hierarchical coherent vortical structures responsible for the energy cascade from large
to small scales (Goto et al., 2017). Furthermore, QCB is associated with the concept of unstable periodic orbits and has
been extensively studied (Lucas & Kerswell, 2017; van Veenet al., 2019). Another important example is the self-sustainingprocess in near-wall turbulence (Waleffe, 1997).
This study finds a non-trivial periodic flow in Direct Numerical Simulation (DNS), using a time-independent Taylor–1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

(a)

(b)

(c)

zy(d)

zx

yx

Figure 1. (a,c) Visualisations of vortical structures and (b,d) parametric time series of energy dissipation rate ε(t) against energy
input rate P(t) in (a,b) low Reynolds number periodic flow at Re = 5.83 and (c,d) high Reynolds number turbulent flow at Re = 29.7.
Visualisations show isosurfaces of (a) |ω| in cyan and (c) |ω| in cyan and low-pass filtered |ω < | in red. Parametric time series show(b) original time series for 40T and (d) phase averaged time series for 20T . The original time series in panel (b) are normalised by themaximum Pmax of P(t). The phase-averaged time series are also normalised during the process. Thus the maximum of hPiphase is alsounity. Dark to light colours represent the time evolution, and the gap between two consecutive dots denotes 5T .

We fit six of nine parameters of the model by DNS of3DPF. We investigate the energy equation of the model andcompare each term with the DNS result to estimate the parameters. The scale-local nonlinear coefficients A1 and A2 arecompared to the average energy transfer from the forced modeto the primary energetic modes while ignoring the scale nonlocal terms, A3 = A4 = 0. The viscous coefficients Kα2 areevaluated by the time average of εα /2νEα , where Eα and εαare energy and energy dissipation rate of scale α = {X,Y,√ Z}
of the 3DPF. The forcing term F is defined from P/ 2EX .
The obtained parameters are, A1 = 0.4, A2 = 4, F = 0.7, KX2 =2, KY2 = 5, and KZ2 = 15. This analysis leaves two unset parameters A3 and A4 . The model has one control parameterRe ≡ 1/ν.

Thus, we conclude that the current flow exhibits QCB regardless of the Reynolds number (See also Fig. 12 of Gotoet al. (2017) for results with three different forcing schemes athigher Re).

THREE-EQUATION MODEL
We consider a minimal model to reproduce such robust
QCB irrespective of Re. By investigating the Fourier mode dynamics of the 3DPF, we find a few modes energetically dominate the 3DPF. Thus, we can decompose the flow into three“families” of Fourier modes. Figure 2 shows the schematic ofsuch a grouping. The first family corresponds to the forced
mode k f = (1, 1, 0). Next, we define the primary energetic
scale with six Fourier modes. The forced plus the primaryscales (seven Fourier modes in total) reproduce more than
98 % of the time average of energy E = h|u|2 i/2 in the 3DPF.
Last, we introduce the secondary scale, which consists of therest of the Fourier modes. Although the energy of these modes(large-scale representative) is small, their enstrophy (smallscale representative) is non-negligible.
After assessing the possible triad interactions among
these three scales (triangles in Fig. 2), we propose the following three-equation model,

RESULTS
By setting A3 = −0.5 and A4 = −0.95, we obtain a bifurcation diagram shown in Fig. 3 (a). The first bifurcation issupercritical from steady to periodic solutions at Re ≈ 12.0,followed by a subcritical one to a more complex periodic solution at Re ≈ 14.0. Then, we identify a critical Re of thepermanently chaotic solution at 14.060 < Recr < 14.061. We
realize that the periodic and chaotic solutions take similar orbits in the phase space as shown in Fig. 3 (b). Thus, the chaotic
orbit exhibits QCB.
We compare the 3DPF and the periodic solution of themodel in Fig. 4 by computing two quantities. One is the energy of the forced mode from which we subtract the laminarbase flow, defined by EX−X0 ≡ |u(k f ) − f/2νk2f |2 /2 for theDNS and EX−X0 ≡ (X − F Re /KX2 )2 /2 for the model. The
other is the energy contained by the remaining modes EY +Z .
In the DNS, EY +Z means the energy of the flow without theforced mode k f . In the model, EY +Z ≡ Y 2 /2 + Z 2 /2. Figure 4 shows the qualitatively same periodic fluctuations interms of the relation of amplitude and phase of peaks between
two quantities. Notably, both EX−X0 and EY +Z show exponential growth and decay, reminiscent of the predator-prey dynamics. Although our three-equation model (3) is different
from the standard two-species Lotka-Volterra equations, thissimilarity suggests the link between the QCB in turbulent flow
and a kind of predator-prey relation between large and small
scale vortex structures. We stress here that the model without
the non-local triad, A3 = 0, A4 = 0, or A3 + A4 = 0, does not
excite permanent fluctuations, indicating the necessity of thed X = −A1Y 2+ A3Y Z− νKX2 X + F,
 t
dt Y = +A1 XY − A2 Z 2 + A4 XZ− νKY2Y,(3)2dt Z =+ A2Y Z − (A3 + A4 )XY − νKZ Z,
where X,Y , and Z denote the characteristic velocity amplitude of the forced, primary, and secondary scales, respectively. The right-hand side of Eq. (3) consists of nonlinear
terms with coefficients Ai with i = 1–4, viscous damping terms
multiplied by νKα2 where Kα is a characteristic scale factor
with α = {X,Y, Z}, and a forcing term F acting solely on theforced scale. We define the Reynolds number of the model byRe ≡ 1/ν to analyse the model. Note that sign of Ai is set
to satisfy the detailed energy conservation in the triad interactions. The coefficients A1 and A2 represent the scale-local
forward energy transfer from X to Y and Y to Z, respectively.
The remaining A3 and A4 terms represent non-local interactions that give the model additional freedom in the parameter
space.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Forced scale

Primary scale
Secondary scale

Figure 2. Schematic of three different scales. We visualize |ω| distributions of typical Fourier modes in each scale. The forced wave
vector is k f = (1, 1, 0). In the primary scale, we visualize k = (0, 0, 2) and (0, 2, 2) modes. For the secondary scale, we visualizek = (3, 1, 0) and (2, 2, 2) modes for example. Triangles denote triad interactions between different scales satisfying triad interaction
conditions, k + p + q = 0.

(a)(b)

Figure 3. (a) Bifurcation diagram of the model (3) by the local extrema of Y . See main text for the parameter set-up. Blue, red, andyellow points denote steady, periodic, and chaotic solutions, respectively. Inset: close-up in the range shown by the red rectangle in themain plot. (b) Periodic (red, Re = 14.05 shown by the red vertical dashed line in panel (a)) and chaotic (black, Re = 14.1) orbits of themodel (3). The chaotic orbit is tracked over 100 periods. The yellow arrow indicates the direction of the orbit.

scale non-local triad interactions to maintain the QCB. Notethat we observe fast oscillations in the model (Fig. 4 (b)) andnot in the DNS result (Fig. 4 (a)). By the mode-by-mode analysis, we find that there are also rapid fluctuations in specificFourier modes of the 3DPF. However, such behaviour is compensated between the symmetric modes and does not appear inFig. 4 (b).

the exponential scaling with the characteristic time scale τ,

PRe (t) = c exp[−t/τ(Re)],(4)

in which τ also displays exponential scaling against Re,

τ(Re) = c0 exp[a Re],

Another intriguing property of the model is that it exhibitsa sudden transition from a chaotic to steady states as shownin Fig. 5 (a). Here, we employ a different parameter set-up(A3 = 0.4 and A4 = −0.5 with other parameters being as sameas in Fig. 3 and Fig. 4 (b)). This probabilistic process reminds
us of the sudden relaminarisation observed in linearly forced
box turbulence (Linkmann & Morozov, 2015). We computethe survival probability PRe (t) of how likely the solution in achaotic regime at a given time t to plot in Fig. 5 (b). We find(5)as shown in Fig. 5 (b) inset. Although Eq. (4) is consistent
with the literature (Linkmann & Morozov (2015)), the exponential scaling of τ(Re) by Eq. (5) crucially differs from asuper-exponential behaviour observed in Linkmann & Morozov (2015). We speculate that this reflects the minimum degreeof freedom in the model.

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

(a)(b)

Figure 4. Time series of fluctuating energy EX−X0 (t) of the forced scale and residual energy EY +Z (t) of (a) 3DPF of the Navier-Stokes
equations (1) and (b) the model (3) at Re = 14.05. See main text for the parameter set-up and the definition of the quantities. Note thattime in panel (a) is normalised by the characteristic time scale T .

(a)

(b)

Figure 5. (a) Time series of (X,Y, Z) of the model (3) at Re = 32 with a random initial condition. See main text for the parameter
set-up. (b) Survival probability PRe (t) of the transient chaos of the model (3) evaluated from 10,000 samples for each Re. The parameter
set is the same as panel (a). Different symbols correspond to different values of Re. Dashed line denotes exponential fitting (4) using
0.01 ≤ PRe (t) ≤ 0.9 data. Inset: Escape rate 1/τ as a function of Re. Dashed line denotes the exponential fitting (5). Symbols are thesame as the main plot.

CONCLUSIONS

ture of the model at the conference.

We investigate the Quasi-Cyclic Behaviour (QCB) observed in a periodic box flow driven by the steady TaylorGreen forcing (2). We reveal that there is intrinsic periodicity
in turbulent flow by phase-averaging the complex time series(Fig. 1 (d)), which is similar to the 3D periodic flow (3DPF) atlow Re (Fig. 1 (b)). By conducting the mode-by-mode analysisof the 3DPF, we propose the three-equation model (3) describing the evolution of the velocity amplitude of the three distinct
scales, namely forced, primary (energy-containing), and secondary (the rest) Fourier modes (Fig. 2). We fit the model
parameters from the 3DPF to find the qualitatively similar periodic solutions (Fig. 4). The model exhibits permanent chaos
at higher Re, which corresponds to statistically steady turbulence (Fig. 3). In a different parameter set-up, we find sudden
relaminarisation of transient chaos (Fig. 5 (a)) with the exponential scaling of the survival probability (Fig. 5 (b)). We conclude that the proposed minimal model with only three variables contains fundamental properties of turbulence. We willfurther investigate the model analytically and numerically andits similarity with the existing model to present a complete pic-REFERENCES
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2 (6), 64603.
Linkmann, Moritz F. & Morozov, Alexander 2015 Sudden relaminarization and lifetimes in forced isotropic turbulence.
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Lucas, Dan & Kerswell, Rich R 2017 Sustaining processes
from recurrent flows in body-forced turbulence. Journal ofFluid Mechanics 817 (4), 178–205.
van Veen, Lennaert, Vela-Martı́n, Alberto & Kawahara, Genta
2019 Time-Periodic Inertial Range Dynamics. Physical Review Letters 123 (13), 134502.
Waleffe, Fabian 1997 On a self-sustaining process in shearflows. Physics of Fluids 9 (4), 883–900.

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