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

BALANCED NON-EQUILIBRIUM TURBULENCE

Konstantinos Steiros
Department of Aeronautics
Imperial College London
London SW7 2AZ, UK
k.steiros13@imperial.ac.uk

ABSTRACTtion which greatly resembles the dissipation equation of thek-epsilon model for decaying homogenous turbulence. The
assumptions and conclusions of this work are validated using
data from laboratory and numerical experiments. A more thorough presentation of this work can be found in Steiros (2022).

The K41 framework remains central to the understanding of turbulent flows. However, in unsteady turbulence,
K41’s critical equilibrium assumption is expected to hold in
an asymptotic manner, as the Reynolds number and wavenumbers tend to infinity, rendering K41 not strictly valid at finitewavenumbers. This work proposes a generalization of K41
for non-equilibrium effects, in decaying turbulence cascades
far from initial conditions. The main result is a correction to
the -5/3 law for out-of-equilibrium eddies, unrelated to intermittency effects. Experimental and numerical evidence is provided in support of the theoretical results.

Balanced cascadeA starting point for studies on homogenous decaying turbulence is often the scale-by-scale energy budget∂ K > (k,t)= Π(k,t) − ε > (k,t) ,∂t

Introduction
A landmark of out-of-equilibrium physics remains Kolmogorov’s 1941 (K41) theory of turbulence (Kolmogorov,1941a,c,b), regulating many quantities of engineering interest.
A cornerstone assumption of K41 is that of cascade equilibrium (Pope, 2001). Note that the latter is used here in the context of stationarity and is not related to the concept of detailedbalance encountered in statistical physics. While cascade equilibrium is exactly fulfilled in statistically static cascades, it
is generally expected to be valid only asymptotically in unsteady ones, as turbulent eddies become vanishingly small,and Reynolds number infinite (Pope, 2001; Vassilicos, 2015).
Therefore, for such unsteady cascades K41 cannot exactly describe finite, out-of-equilibrium wavenumbers, the physics ofwhich remain obscure.
The recent investigation of Goto & Vassilicos (2016) has
indicated that out-of-equilibrium eddies exhibit collective dynamics, at least in the case of freely decaying homogeneous
turbulence. In particular, the turbulence cascade was shown
to reach a “balanced” non-equilibrium state far from initialconditions, in the sense that the terms of the energy budget
equation, expressed in spectral space, scale with each other at
non-equilibrium wavenumbers. It is noted that the possibilityof an asymptotic approach to a balanced state, far from initialconditions, was initially postulated by George (1989, 1992) for
the whole spectral range, while Goto & Vassilicos (2016) observed it, in their direct numerical simulations (DNS), only foreddies comparable to the integral length scale of the cascade.
The above observation of balance thus allows the modelling of the behaviour of the large, out-of-equilibrium scalesof the turbulence cascade, at least in the special case of decaying turbulence far from initial and boundary conditions. Usingthat information, this work proposes an extension of the K41
theory, for out-of-equilibrium effects. The main results area correction to the -5/3 law and the formulation of an equa-(1)where K > (k,t) = k∞ E(k,t)dk and ε > (k,t) = k∞ k2 E(k,t)dk
are the high pass filtered turbulence kinetic energy and dissipation rate, respectively, with E(k,t) being the energy spectrum. Π(k,t) is the interscale flux from wavenumbers smaller,
to wavenumbers larger than k. In the following, when the superscript > is dropped, the integration occurs from 0 to infinity.
Kolmogorov’s equilibrium assumption states that for sufficiently large Reynolds numbers and for eddies which are ofnegligible
size compared to the integral length scale L(t) =3π R ∞ −1
4 0 k E(k,t)dk/K(t), the cascade is approximately staK>
tionary, i.e. ∂∂t≈ 0. If in addition to the above assumption
one considers wavenumbers k which are not too large and sothat the local dissipation is negligible, then ε > (k,t) ≈ ε(t).
Combining the above two arguments one then obtains
RRΠ(k,t) ≈ ε(t) ,(2)which is the well-known repercussion of the equilibrium assumption. This result is foundational, as it directly leads to the-5/3 law for the energy spectrum. Indeed, dimensional analysis
arguments lead to E(k,t) ∝ Π2/3 k−5/3 . Utilization of expression 2 then readily leads to E(k,t) ∝ ε 2/3 k−5/3 , i.e. the -5/3
law (Pope, 2001). As the equilibrium assumption (i.e. expression 2) was used in its derivation, we might expect that there
will be deviations from the -5/3 law in cases of non-stationarity
in the cascade.
To account for out-of-equilibrium phenomena one has tofind an alternative expression for the energy flux Π, without
K>. To do that, wehaving to discard the time-dependent term ∂∂t1

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

DΠ/Dt = ∂ Π/∂t +V ∂ Π/∂ κ = 0 .

Equation 4 poses a constraint which can be used to calculate the function g but to do that, we first need an expression for the interscale energy speed V = dκ/dt. Following the dimensional arguments of Pao (1965), we expect thatdk/dt = Cv Π1/3 k5/3 , where Cv is a coefficient of proportionality. Using equation 3 and the Kolmogorov dissipation scaling
ε = Cε ( 32 K)3/2 /L we end up with

Figure 1. g ≡ Π/ε (solid lines) and Cε (dashed line) as afunction of time, in decaying periodic box turbulence. t = 0
marks the onset of decay. Note that κ = kL.

dk/dt =
may use the original idea of George (1989, 1992), who postulated that, far from initial conditions, transient turbulent systems (including the turbulence cascade) will approach a stateof “balance” where the dynamic terms of their characteristic
budget differential equation should scale with each other.
Under the assumption of balance, the budget equation 1∂ K > (kL,t)then leads to∝ Π(kL,t) ∝ ε > (kL,t). For wavenum∂tbers which are not too small, one has ε > (k,t) ≈ ε(t), and thusbalance implies

Π(k,t) = g(kL)ε(t) .

(4)

3 Cv
ε(t)
g(κ)1/3 κ 5/3 .
2/3
2C
L(t)K(t)εThe speed of the energy flux across normalized wavenumbers
dkdL
is dκ
dt = L dt + k dt . It is expected that the second term on theright hand will be negligible compared to the first, i.e. dκdt ≈L dk,
asthetimescaleofthenonlinear
energytransportof
andt
individual eddy must be much smaller than that of the dilationof the entire cascade. This postulation was verified using theDNS data, and the term k dL
dt was found to be at least an orderof magnitude smaller than L dk
dt , after κ = 2.
Under the above assumption, the spectral velocity becomes(3)
V≈

Figure 1 validates equation 3 using decaying periodic box turbulence DNS (see section “Methodology” for details on thedataset). At time t = 0 turbulence is left to decay, and after asufficient time has passed (i.e. so that turbulence is far fromits initial conditions) g ≡ Π/ε becomes independent of time
for a variety of κ = kL > 1. Figure 1 also shows that spectral balance coincides with the establishment of Kolmogorov’s
dissipation scaling ε = Cε ( 32 K)3/2 /L, with Cε a constant. Thatis, because combination of equation 3 with the well-known expression for the large scale energy flux Π ∝ ( 32 K)3/2 /L directly
yields the dissipation scaling.
It can be readily seen that the above self-preserving expression for Π is an (out-of-equilibrium) generalization of Kolmogorov’s equilibrium assumption, i.e. equation 2. The arbitrary function g(kL) will be smaller than unity for decaying
turbulence, and is expected to approach unity (i.e. equilibrium)
as kL → ∞ and Re → ∞, in agreement with Kolmogorov’s
ideas (Pope, 2001). The main objective of this article is to determine g(kL) for sufficiently large, but smaller-than-infinite
wavenumbers, i.e. as equilibrium is being approached but has
not yet been reached. Combination of that with the dimensional analysis expression E(k,t) ∝ Π2/3 k−5/3 will then yield
the out-of-equilibrium correction to the -5/3 law.

3 Cv ε(t)g(κ)1/3 κ 5/3 .
2 C2/3 K(t)εInjecting the above result to equation 4 and using equation 3one obtains
K dε
3 Cv g0 5/3
κ= −C0 ,=−2
2 C2/3 g2/3ε dt(5)εwith g0 = dg/dκ. The left hand side of the above equation isa function of time only, while the right hand side a functionof the normalized wavenumber κ only. Thus, both sides must
equal a constant which is dependent only on initial conditions.
In figure 2 we validate the above result with the DNS data.
Approximately 3.5 time units after the onset of decay (t = 0),
both sides of the equation indeed become relatively constant,independent of time or wavenumber.

Dissipation equation of the k − ε model
The left hand side of equation 5 isε2dε= −C0 ,dtK

Simple transportation of energy
To calculate g(kL) we follow the original idea of Lumley (1992) who postulated that, at sufficiently high Reynolds
numbers, a range of scales will form in the cascade, where effects linked to production or dissipation of turbulence kinetic
energy will be negligible. Consequently, the energy there isconserved, simply transported downwards in the cascade. We
interpret the above statement as the substantial derivative of
the energy flux being zero in the self-preserving coordinate
system (κ,t), i.e.

(6)which is the dissipation equation of the (otherwise empiricallyderived) k − ε turbulence model, for homogenous decaying
turbulence (Pope, 2001). It is noteworthy to mention that integration of the above equation yields a power law decay forthe kinetic energy without the need of assuming turbulence invariants (Pope, 2001). If an invariant were to be assumed, itwould yield a value for C0 . The DNS dataset indicates a valueof C0 ≈ 1.7 which is the value obtained from Loitsyanskii’sinvariant.

2

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

Figure 3. Validation of expression 7 using DNS. The compensated spectra become flat using the out-of-equilibrium correction, and not the classical K41 formulation.

Figure 2. Evolution of the two sides of equation 5 over time,using the DNS dataset (forcing stops at t = 0). g2/3 κ −5/3 /g0(solid lines) is plotted for κ = 3, 4, 5, 6, 7, 8, 10.

-5/3 law correction
Integration of the right hand side of equation 5 yields
32/3
Cε−2/3where the constant of integration wasg = 1 − C03C
κv
calculated using the fact that we expect equilibrium, i.e. g → 1,
as κ → ∞ and Re → ∞. Using the self preserving expression 3
along with the dimensional analysis result E(k,t) ∝ Π2/3 k−5/3
one obtains

hi2E(κ,t)≈ Ck 1 − cκ −2/3 κ −5/3 ,
2/3
5/3
ε(t) L(t)Figure 4. Validation of expression 8 using hot wire anemometry. The compensated spectra become flat using the out-ofequilibrium correction, and not the classical K41 formulation.

(7)
point where Kolmogorov’s equilibrium can be thought approximately valid. Indeed, at this normalized wavenumber expression 7 produces almost identical results to the -5/3 slope, indicating that out-of-equilibrium effects have started to become
negligible.
Figure 4 validates expression 8 using hot wire measurements of grid turbulence at various stages downstream of thegrid (see section “Methodology” for details). Much similar
to the three-dimensional spectra of the DNS, the experimental
one-dimensional spectra require the novel out-of-equilibrium
correction to the -5/3 law to flatten. As wavenumbers grow,the new correction tends to become indistinguishable to the-5/3 law, a sign that out-of-equilibrium effects have becomenegligible.

2/3C0Cε3Cvwith c =. Determination of the constant c requires
knowledge of Cε , Cv and C0 . However, only the latter canbe calculated analytically, by assuming a turbulence invariant. Thus, Cv and Cε need to be determined from observations. Our DNS results (i.e. figures 1 and 2) suggest Cε ≈ 0.9,
Cv ≈ 1 and C0 ≈ 1.7 for the DNS, which lead to c = 0.53.
To calculate the out-of-equilibrium correction of the -5/3 law
for the one-dimensional spectrum E11 (useful for experimental
hot-wire measurements), we may utilize the following expression which links the two types of spectra (see Comte-Bellot &Corrsin (1971))


1
∂ 1 ∂
E11 (κ,t) .
E(κ,t) = κ 32 ∂κ κ ∂κMethodology
For validation purposes, a data-set of periodic-box decaying turbulence is used, the details of which are presented in Goto & Vassilicos (2016).
A forcing f =(− sin(k f x) cos(k f y), cos(k f x) sin(k f y), 0) with k f = 4 is imposed on the Navier-Stokes equations, and is turned off at t = 0
when dissipation is maximum, allowing the turbulence to decay. The simulation size was N 3 = 20483 . The spatial resolution kmax η was slightly larger than one at t0 , while kmax η
increased during decay. The Taylor scale Reynolds number
was naturally decaying, but was roughly equal to 120.
The experiments were conducted in the wind tunnel ofthe Lille Fluid Mechanics Laboratory (LMFL) which has atest section of 1 × 2 m2 and a length of 21 m. Hot wire measurements were conducted using a single Pt-W 5 micron wire,3 mm long, with a 1 mm sensing element, driven by a TSI
IFA300 anemometer at 50 kHz acquisition frequency with alow-pass filter at 20 kHz. Two planar grids were tested withdifferent bar thicknesses, but identical blockage (30 %). Three
inlet velocities were tested 4, 5 and 7 ms−1 , and all mea-A double integration then yields

hi18
E11−5/3−2/3
2 −4/3=Cκ
1−1.209cκ
+0.407cκ.
k55
ε 2/3 L5/3(8)In both equations 7 and 8 a power-law out-of-equilibrium
correction (brackets) has been superimposed on the usual -5/3
law.
Figure 3 validates the above conclusions using appropriately compensated spectra obtained from decaying turbulence
DNS. Since turbulence is decaying, its large scales will beout-of-equilibrium, and the -5/3 will not be exact. Indeed,
figure 3 shows that the compensated spectrum becomes flat
only when utilizing the out-of-equilibrium expression 7. Atvery high normalized wavenumbers, close to 50, there is a tendency for the spectrum to approach the -5/3 law. This is the312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
that quasi-isotropic wavenumbers are characterized by a simple transportation of energy. Both of these assumptions aresupportd by periodic box DNS. The main result of the above
framework is a correction to the −5/3 law for non-equilibriumand quasi-isotropic wavenumbers, in cascades far from initialconditions.
We end this work with a quick comparison of the above
result with the K62 refined theory of Kolmogorov (Kolmogorov, 1962) and the spectrum of Pao (1965), both of which
provide corrections to the K41 framework. K62 accounts forthe spatial fluctuations of the small scales in ε. Its physical argument is therefore different from the current theory which assumes a “balanced” non-equilibrium. Indeed, K62 produces adrastically different correction to the -5/3 law, than the current
theory. Pao’s correction (Pao, 1965) on the other hand is basedon the speed V that energy is transferred across wavenumbers,
but yields corrections only for the dissipative scales of turbulence, leaving larger eddies, and thus the -5/3 law, unaffected.
These corrections thus differ from the one proposed in thiswork and in fact we cannot think of any reason why all threecorrections cannot apply simultaneously.

1.2
1
0.8
0.6
0.4
0.2
040

50

60

70

80

Figure 5. Measured dissipation coefficient for the two grids
tested (circles and squares) for inlet velocities of 7, 5 and 4
ms−1 (blue, red and black colours).

REFERENCES
Comte-Bellot, G & Corrsin, S 1971 Simple eulerian time correlation of full-and narrow-band velocity signals in gridgenerated, ‘isotropic’ turbulence. Journal of Fluid Mechanics 48 (2), 273–337.
George, W K 1989 The self-preservation of turbulent flows
and its relation to initial conditions and coherent structures.
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George, William K 1992 The decay of homogeneous isotropicturbulence. Physics of Fluids A: Fluid Dynamics 4 (7),1492–1509.
Goto, S & Vassilicos, J C 2016 Unsteady turbulence cascades.
Physical Review E 94 (5), 053108.
Kolmogorov, A N 1941a Dissipation of energy in the locallyisotropic turbulence. In Dokl. Akad. Nauk SSSR A, , vol. 32,pp. 16–18.
Kolmogorov, A N 1941b The local structure of turbulence inincompressible viscous fluid for very large Reynolds numbers. In Dokl. Akad. Nauk SSSR A, , vol. 30, pp. 301–5.
Kolmogorov, A N 1941c On degeneration (decay) of isotropic
turbulence in an incompressible viscous fluid. In Dokl.
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Pao, Y H 1965 Structure of turbulent velocity and scalar fields
at large wavenumbers. The Physics of Fluids 8 (6), 1063–1075.
Pope, S B 2001 Turbulent flows. IOP Publishing.
Sinhuber, M, Bodenschatz, E & Bewley, G P 2015 Decay ofturbulence at high Reynolds numbers. Physical Review Letters 114 (3), 034501.
Steiros, K 2022 Balanced nonstationary turbulence. PhysicalReview E 105 (3), 035109.
Vassilicos, J C 2015 Dissipation in turbulent flows. Annual Review of Fluid Mechanics 47, 95–114.

Figure 6. Measured turbulence intensities for the two grids
tested (circles and squares) for inlet velocities of 7, 5 and 4
ms−1 (blue, red and black colours).

surements were conducted at the region of the flow whereCε = const, i.e. sufficiently far from the grid (in our casethis occurred approximately ten mesh-sizes downstream of thegrid). The Taylor scale Reynolds number ranged from 75 to150, depending on the case. All tested cases demonstrated aqualitatively similar behaviour to the one shown in figure 4.
Figures 5 and 6 show the measured dissipation coefficient andturbulence intensities for the various cases tested. It is notedthat for the calculation of c in expression 8 the current value ofCε ≈ 1 was used, while Cv was assumed unity, as in the DNS.
C0 was taken to be 1.83, which corresponds to Saffman’s invariant, using the fact that this invariant has been often linked
to grid turbulence (Sinhuber et al., 2015).

Concluding discussion
This work generalizes the K41 framework for unsteadycascades far from initial conditions, on the basis of two assumptions. First, that far from initial conditions the cascade
exhibits a self-preserving approach to equilibrium, and second4