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

TURBULENT/NON-TURBULENT INTERFACES IN EQUILIBRIUM AND
NON-EQUILIBRIUM TURBULENCE

Marco Zecchetto
Instituto Superior Técnico, Universidade de Lisboa
Av. Rovisco Pais, 1, Lisboa 1049-001, Portugal
marco.zecchetto@tecnico.ulisboa.pt

Kouji Nagata
Department of Aerospace Engineering
Nagoya University,
464-8603 Nagoya, Japannagata@c.nagoya-u.jp

Tomoaki Watanabe
Department of Aerospace Engineering
Nagoya University,
464-8603 Nagoya, Japanwatanabe.tomoaki@c.nagoya-u.jp

Carlos B. da Silva
Instituto Superior Técnico, Universidade de Lisboa
Av. Rovisco Pais, 1, Lisboa 1049-001, Portugal
carlos.silva@ist.utl.pt

ABSTRACTCε = const in the near and far field regions of grid-generated
turbulence.
The goal of the present work is to assess the effects of
non-equilibrium turbulence in the characteristics of the socalled turbulent/non-turbulence interface layer that separates
many flow fields e.g. jets, into an irrotational (or non turbulent) an a turbulent region. For this purpose several direct
numerical simulations (DNS) of turbulent fronts are carriedout, where the turbulent core region is in an unbalanced (nonequilibrium) state.

Direct numerical simulations (DNS) of turbulent fronts
created by non-equilibrium (unbalanced) turbulence are used
to assess the characteristics of the turbulent/non-turbulent interface (TNTI) that separates regions of turbulent from nonturbulent (or irrotational) flow. The effects of the unbalancecan be observed in the detailed shape of the conditional enstophy magnitude, and in the magnitude of the maximum enstrophy but do not affect its main features. In particular the mean
thickness of the TNTI is equal in classical or non-equilibrium
turbulence i.e. it scales with the Kolmogorov micro-scale aspreviously observed at high Reynolds numbers.

NUMERICAL METHOD
Direct numerical simulations (DNS) of forced homogeneous isotropic turbulence (HIT), decaying isotropic turbulence (DEC), and shear free turbulence (SFT) are carriedout employing classical pseudo-spectral methods (collocationmethod) for spatial discretization, and a three-step third order explicit Runge-Kutta scheme for temporal advancement;
the 2/3 rule is used for full de-aliasing. Periodic boundary
conditions are imposed in the x, y, and z directions. A uniform (∆x = ∆y = ∆z) grid of 10243 collocation points is created for all the cases in a cubic box of 2π sides. A DNS offorced homogeneous isotropic turbulence, was carried out using the forcing scheme by Alvelius (1999) with a peak forcing concentrated in the three wavenumbers centred at k p = 5,and the flow has a Reynolds number of Reλ = 230 and resolution of kmax η = 1.5 once the statistically stationary is attained. From this point onward the forcing scheme is turned
off creating an unbalance in turbulence dynamics. Two different condition are simulated: in one case turbulence is let toevolve freely (decaying turbulence); in the other simulation thevelocity fields are convoluted by an hyperbolic tangent profilethat essentially conserves the velocity of the HIT simulation,and thus its vorticity, only in a central zone of π/3 width, suppressing it in the rest of the domain (shear free turbulence). In
shear free turbulence (figure 1) two sharp and highly convoluted turbulent/non-turbulent interfaces (TNTI) are generated
by the remaining core turbulence. As time progresses, the initial isotropic turbulence region (where |ω| ̸= 0) spreads into

INTRODUCTION
According to the classical Richardson-Kolmogorov energy cascade picture the rate of viscous dissipation of kinetic
energy at high Reynolds numbers is imposed by the inviscid
dynamics of the large scale, energy carrying eddy motions(Richardson, 1922; Kolmogorov, 1941a,b). This concept however, relies on the assumption that the rate of energy transferfrom the large-scales is balanced by the rate of viscous dissipation caused by the smallest scale eddies. In this case thenon-dimensional dissipation rate Cε , defined by
Cε = ⟨ε⟩L /U 3 ,(1)is constant (Davidson, 2004), where ⟨ε⟩ is the mean dissipation rate. However, several experimental and numerical works(Valente et al., 2014; Watanabe et al., 2019; Takamure et al.,2019) have shown that there are important flow cases/flow regions where Cε is not constant even if the flow is at very
high Reynolds number and displaying the classic −5/3 energy
spectrum over a wide wave-number range region. Indeed apower law of Cε ∼ Renl , with n ≈ −1 and Rel a local Reynolds
number, is often observed (Vassilicos, 2015; Goto & Vassilicos, 2016) attesting a lack of equilibrium between the large andsmall scales of motion e.g. Valente & Vassilicos (2012); Isazaet al. (2014) reported a sudden transition from Cε ∼ Re−1l to1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
the irrotational region (where |ω| = 0) without mean shear.
For comparison, another shear free turbulent simulation is performed employing the same procedure but starting from (classical) forced homogeneous isotropic turbulence in equilibrium.
In order to isolate data collected at particular locations within
the TNTI layer, conditional statistics in relation to the distance from a specific position within the TNTI layer are used.
Due to the highly convoluted shape of the TNTI and its intermittent position, plane-by-plane statistics will invariably contain samples corresponding both to irrotational and to turbulent flow events masking the detailed local dynamics of turbulence. Conditional statistical data is obtained with a 3 steps
procedure: i) the outer surface of the TNTI layer - the irrotational boundary (IB) - which consists in an iso-surface of√vorticity magnitude |ω| = ωi ωi is detected by analysing thehistogram of the turbulent flow fraction of the flow; ii) a local 3D normal to the IB is defined at each one of the Nx × Nzpoints in the upper and lower shear layers that delimit a interface envelope;iii) any flow quantity of interest is collected
at fixed distances yI , by trilinear interpolation, into a grid defined on this local 3D normal, that points into the interior of
the turbulent core region, so that yI < 0 and yI > 0 correspondto the irrotational and turbulent regions, respectively (the IB islocated at yI = 0). Once the procedure is completed statisticalanalysis is performed on the collected data set. Furthermore,
the temporal evolution of Cε is calculated in a stack of planeson the turbulent core around the central plane using equation
1. Due to the flow geometry, instead of the integral scale one
uses the longitudinal integral length scale L11 computed fromthe one dimensional energy spectrum E11 (κ1 ) and the mean
streamwise fluctuating velocity u′1 norm is replaced by U . The
mean dissipation is computed inside the turbulent core region
using a stack of 2d planes.

enstrophy with no normalisation (figure 3 a,c) show the typical sharp rise bridging the regions of irrotational (yI < 0) andturbulent (yI > 0) flow, respectively, and the expected decay ofenstrophy in the turbulent core region as the flow evolves withtime.
The normalisation of the conditional enstrophy profiles
using the procedure outlined in Zecchetto & da Silva (2021) isshown in figure 3 (b,d). Recall that in that paper it has beenshown that the shape of the conditional enstrophy, when normalised with the local Kolmogorov velocity and time scales, isuniversal i.e. equal in all flows, provided the Reynolds number is sufficiently high. The evolution of the profiles with time
is similar whether the starting instant is in equilibrium or not.
In the first 3-4 instants the flow is still evolving from the SFT
initial state. For the later times the profiles begin to collapse
and the shape recovered in the 2 cases is essentially the sameobtained in Zecchetto & da Silva (2021). Indeed, a sharp peakof 1.5 is attained at a distance of about 10η from the turbulent
non-turbulent interface and the theoretical value of 1 is attained
after approximately 40η. This ‘self-similar’ state is reachedfaster in equilibrium turbulence, which can be ascribed to different small scales dynamics. However, the mean thickness ofthe TNTI is equal in classical or non-equilibrium turbulence
i.e. it scales with the Kolmogorov micro-scale as previouslyobserved at high Reynolds numbers.
More details will be provided during the presentation.

REFERENCES
Alvelius, K. 1999 Random forcing of three-dimensional homogeneous turbulence. Physics of Fluids 11 (7), 1880–1889.
Davidson, P. A. 2004 Turbulence, an introduction for scientists
and engineers. Oxford University Press.
Goto, Susumu & Vassilicos, J. C. 2016 Unsteady turbulence
cascades. Phys. Rev. E 94, 053108.
Isaza, J.C., Salazar, R. & Warhaft, Z. 2014 On grid-generated
turbulence in the near and far field regions. Journal of Fluid
Mechanics 753, 402–426.
Kolmogorov, A. N. 1941a Local structure of turbulence in
an incompressible fluid for very large Reynolds numbers.
Dokl. Aka. Nauk SSSR 30, 301–305.
Kolmogorov, A. N. 1941b On degeneration of isotropic turbulence in an incompressible viscous liquid. Dokl. Akad.
Nauk. SSSR 31, 538–541.
Richardson, L. F. 1922 Weather prediction by numerical process. Cambridge university press.
Takamure, K., Sakai, Y., Ito, Y., Iwano, K. & Hayase, T. 2019
Dissipation scaling in the transition region of turbulent mixing layer. International Journal of Heat and Fluid Flow 75,77–85.
Valente, P. C., Onishi, R. & da Silva, C. B. 2014 Origin ofthe imbalance between energy cascade and dissipation inturbulence. Phys. Rev. E 90, 023003.
Valente, P. C. & Vassilicos, J. C. 2012 Universal dissipation
scaling for nonequilibrium turbulence. Phys. Rev. Lett. 108,214503.
Vassilicos, J. C. 2015 Dissipation in turbulent flows. Annual
Review of Fluid Mechanics 47 (1), 95–114.
Watanabe, T., da Silva, C. B. & Nagata, K. 2019 Nondimensional energy dissipation rate near the turbulent/nonturbulent interfacial layer in free shear flows and shear freeturbulence. Journal of Fluid Mechanics 875, 321–344.
Zecchetto, M. & da Silva, C. B. 2021 Universality of small-RESULTS
Figure 1 shows contours of vorticity magnitude near theturbulent/non-turbulent interface layer at the start of the simulation, when the turbulent core region has characteristics ofnon-equilibrium turbulence (Fig. 1 a), and later during thesimulation, when the flow recovers classical turbulent statistics(Fig. 1 b). As can be seen, a larger number of convolutions isimprinted in the TNTI layer at a later stage, as expected, because the integral scale of the flow typically increases withtime, typical of classical turbulence. Moreover, the shape of
the TNTI seems to be sharper, even if regions of more intense
vorticity seem to be placed further away from the TNTI in theunbalanced (non-classical) interface layer.
The non-dimensional dissipation Cε evolves with thetime, from the typical value of Cε ≈ 0.5 from forced isotropicturbulence until Cε ≈ 1.0, typical of free decaying turbulence(Fig. 2 a). During this time the non-dimensional dissipation
exhibits the power law Cε ∼ Re−1.2
typical of non-equilibrium
λturbulence. Figure 2 b) shows the temporal evolution of theturbulent kinetic energy and viscous dissipation in a simulation of isotropic turbulence right after the power input is turned
off. The time lag observed initially in the evolution of the twoquantities is typically observed in non-equilibrium turbulence
as described in Valente et al. (2014).
Figures 3 show conditional mean profiles of enstrophy
ωi ωi , in relation to the distance from the TNTI, at differentinstants. In figures 3 (a,b) the first instants the flow field in
the turbulent core region is characteristic of equilibrium turbulence, whereas in figures 3 (c,d) the evolution starts from
non-equilibrium turbulence. The conditional mean profiles of2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
scale motions within the turbulent/non-turbulent interface
layer. Journal of Fluid Mechanics 916.

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

(a)

(b)

Figure 1. Contours of vorticity magnitude for the shear free turbulence simulations for two instants corresponding to non-equilibrium
turbulence (a) t = 3.08 and equilibrium turbulence (b) t = 3.80.

1.1

1.4
DECSF Tk/k0
ε/ε01

1.2

0.9
0.8

1Cε

0.7
0.8

0.6
0.5

0.6
0.4
0.4
2.9

3

3.1

3.2

3.3

3.4

3.5

0.3

3.6

3

3.1

3.2

3.3

3.4

3.5

t[s]

t[s]

(a)

(b)

Figure 2. (a) Evolution of the non-dimensional dissipation rate Cε with time in decaying isotropic turbulence and in shear freeturbulence (computed near the centerline of the flow region); (b) Time evolution of turbulent kinetic energy and dissipation in decaying
isotropic turbulence, once the forcing is turned off. The time lag observed initially in the evolution of the two quantities is typicallyobserved in non-equilibrium turbulence.

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

(a)

(b)

(c)

(d)

Figure 3. Conditional mean profiles of enstrophy ωi ωi , in relation to the distance from the TNTI, at different instants (a,c) withoutnormalization and (b,d) normalised by the local Kolmogorov scale. The irrotational boundary is placed at yI = 0 while the non-turbulent(irrotational) and turbulent flow regions are at yI < 0 and yI > 0, respectively. The top figures refer to SFT starting from turbulence inequilibrium state whereas the bottom to SFT starting from non-equilibrium turbulence.

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