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

SCALINGS OF THE TEMPORALLY DEVELOPING TURBULENT PLANARJET AND ITS TURBULENT/NON-TURBULENT INTERFACE

Sarp ER
Univ. Lille, CNRS, ONERA, Arts et MetiersInstitute of Technology, Centrale Lille Institut,
UMR 9014 - LMFL, F-59000 Lille, France
sarp.er.etu@univ-lille.fr

Jean-Philppe LAVAL
Univ. Lille, CNRS, ONERA, Arts et MetiersInstitute of Technology, Centrale Lille Institut,
UMR 9014 - LMFL, F-59000 Lille, France
jean-philippe.laval@univ-lille.fr

J. Christos VASSILICOS
Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology,
Centrale Lille Institut, UMR 9014 - LMFL, F-59000 Lille, Francejohn-christos.vassilicos@centralelille.fr

ABSTRACTThe aim of the present work is to establish the scalingsof the average TNTI propagation speed in temporally developing planar turbulent jets and to see how they depend on turbulence dissipation. We start by obtaining a relation for the TNTI
propagation velocity by considering the fractal nature of theinterface and the generalized form of the dissipation scaling.
We then present our DNS results on the mean flow and TNTI,
and verify the relation obtained from our theoretical analysis
against our DNS.

The temporally developing turbulent planar jet is examined starting from the fundamental conservation equations. We
demonstrate that this flow is exceptional in that its mean flow
scalings are the same for both equilibrium and non-equilibrium
dissipation scalings. We also show that the average propagation velocity of the turbulent/non-turbulent interface (TNTI)
is proportional to the global Reynolds number to a power
which is inversely related to the fractal dimension of the isoenstrophy surfaces which are all packed within the thin spacecomposing the TNTI. Using a Direct Numerical Simulation(DNS) we confirm our theoretical predictions and show thatthe fractal dimension of these iso-enstrophy surfaces decreases
from about 2.18 at the higher enstrophy values to between 2.1
and 2.05 at the lower ones. The average propagation velocity
therefore increases from the higher to the lower enstrophy values, i.e. from the TNTI’s inner, turbulent, edge to its purely
viscous outer edge, in agreement with the constant mass flux
through these inner layers.

THEORETICAL ANALYSISIn order to gain insight into the average propagation velocity vn of the interface, we use the equality between the expansion rate of the turbulent jet volume and the flux throughthe TNTI,
dVJ= Svn ,
dt

(1)

where S is the TNTI surface area and VJ is the volume of the
turbulent jet. Following Zhou & Vassilicos (2017), we express
dVJ /dt in terms of the jet half-width δ (t) and therefore write

INTRODUCTION
Turbulent shear flows are present in many natural and engineering processes. The TNTI delineating their turbulent extent is a central aspect of the entrainment mechanism whereby
ambient potential fluid acquires vorticity when entrained into
the turbulent flow. Investigation of the structure of this verythin interface and the local mechanisms driving the entrainment has been the subject of various studies (Corrsin & Kistler,
1955; Turner, 1986). The advent of DNS and Particle Image
Velocimetry has brought a revival of studies of the TNTI overthe past 20 years (Reeuwijk & Holzner, 2014; da Silva et al.,2014; Mistry et al., 2019).
Previous studies (Zhou & Vassilicos, 2017; Cafiero &Vassilicos, 2019) have shown that the scalings of the average TNTI propagation speed in spatially developing turbulent
wakes and jets are different for different turbulence dissipation
scalings. Two different types of turbulence dissipation scalings
have been identified (Vassilicos, 2015), the non-equilibrium
dissipation scalings and the classical Taylor-Kolmogorov dissipation scalings.

dδ
2Lx Lz ∼ Svn .
dt(2)Here, Lx and Lz are the domain extents in x and z directions. Taking into account the fractal character of the interface(Sreenivasan et al., 1989; Miller & Dimotakis, 1991; Flohr &Olivari, 1994; Mistry et al., 2016), its surface area at spatial
resolution r can be estimated as;S(r) ∼ Lx Lz

r
δ (t)

(2−D f )(3)

where D f is the fractal dimension of the interface (2 ≤ D f <3). To estimate the actual surface area of the interface we need1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
As for the Taylor length scale λ , the relation ε0 ∼ νK0 /λ 2 isemployed with the Townsend’s relation K0 ∼ R0 (Townsend,1976);

to estimate the smallest relevant length-scale rmin . Consideringthat a surface with a finite thickness cannot fold with a radius
smaller than its thickness, the interface thickness can be usedas the minimal length scale. We assume that this length scale is
proportional to the Corrsin length-scale ηI ∼ ν/vn (Corrsin &Kistler, 1955), where ν is the kinematic viscosity of the fluid.
We therefore arrive at
dδ
∼
dt



ν
vn δ

−1/2

λ ∼ (UJ HJ )1/2 (t − t0 )1/2 ReG

The velocities related to these length scales can be written as−1/4uη ∼ (UJ HJ )1/2 (t − t0 )−1/2 ReG
and uλ ∼ (UJ HJ )1/2 (t −

(2−D f )
(4)

vn

−1/2

t0 )−1/2 ReG . Comparing these inner velocity scalings witheq. 7, it can be seen that vn has similar ReG -dependence withuη and uλ for different values of D f , i.e. vn ∼ uη when D f =7/3 and vn ∼ uλ when D f = 3.

from eq. 2. We now need to obtain the time-evolution of thejet half-width δ (t), in order to proceed with eq. 4 and obtainan expression for vn .
We conduct the analysis of the temporally developing turbulent jet flow starting from the conservation of mass and momentum by assuming self-similarity of turbulent profiles in
the form φ = φ0 (t) f (y/`(t)) for a variable φ , where φ0 (t) isthe amplitude and `(t) is the time varying length scale. The
volume flux is conserved
for the temporally developing pla∂ R∞
nar jet case, i.e. ∂thuidy= 0, unlike spatially developing−∞
jets. It was shown that mean flow scalings differ for different dissipation scaling regimes by Cafiero & Vassilicos (2019)
for spatially developing planar jets. In order to investigate if
there is a similar behavior in temporally developing planar jets,
we pursue our analysis by using the turbulent kinetic energym 3/2K0
Gequation and the general dissipation scaling ε0 ∼ Re,
Re0
δ
where ReG is a global Reynolds number, determined by the initial jet velocity and jet width (ReG = UJ HJ /ν) and Re0 (t) =K0 (t)δ (t)/ν is a local Reynolds number which varies in time.
The classical equilibrium dissipation scaling is retained for
m = 0, and m 6= 0 leads to non-equilibrium dissipation scalings. The following mean flow scalings are found as a result;u0 ∼ (UJ HJ )1/2 (t − t0 )−1/2 ,
1/2δ ∼ (UJ HJ )1/2(t − t0 )SIMULATION DETAILSA set of simulations of temporally developing planar jets
have been carried out with a pseudo-spectral solver with periodic boundary conditions in all three directions. The simulations were run in a domain of size (8HJ , 12HJ , 8HJ ) with resolution 1024x1536x1024 in directions x, y and z respectively,
such that the spatial resolution is the same in all the three directions and is smaller than the Kolmogorov length scale calculated on the center-plane throughout the time evolution of thejet as shown in the figure 1, where Tre f = HJ /(2UJ ) is used
for the non-dimesionalization of time. The time integration
was carried out with a second order Runge-Kutta time stepping scheme. Apart from the 2/3 truncation for de-aliasing, afiltering function which is only effective at the highest resolved
wavenumbers was also applied to obtain smooth enstrophy isosurfaces at the outer edge of the TNTI. This additional filtering was found to be very effective for eliminating numericaloscillations at near-zero enstrophy levels. The initial velocity
profile of the jet is (Reeuwijk & Holzner, 2014; da Silva &Pereira, 2008);

(5)

,




UJ UJ
HJ
2|y|
U(y) =
−
tanh
1−
,
2
2
4θ0
HJ

(6)

where u0 is the center-plane mean flow velocity and t0 is avirtual origin. A peculiar feature of this problem is that themean flow scalings are independent of the exponent m in thedissipation relation, thus insensitive to the different dissipation
regimes. The fact that the mean flow scalings are the same forboth equilibrium and non-equilibrium dissipation scalings is
exceptional as this is not to the case in turbulent axisymmetric wakes and spatially developing turbulent planar jets, where
different scalings for u0 and δ are shown to hold for different
dissipation regimes in Dairay et al. (2015); Cafiero & Vassilicos (2019).
From eq. 4 and eq. 6 one can obtain the following relationfor vn :(2−D f )/(D f −1)

vn ∼ (UJ HJ )1/2 (t − t0 )−1/2 ReG

.

−3/4

.

(10)

where y = 0 is the center-plane of the planar jet and θ0 is theinitial momentum thickness. HJ /θ0 = 35 is chosen similarly toother studies, as this value leads to a faster transition comparedto lower HJ /θ0 values when perturbed by high amplitude noise(da Silva and Pereira, 2008). A white-noise is added on top of
the mean velocity profile and the noise is filtered by the velocity profile, thus the region outside the turbulent jet is free fromrandom oscillations. The amount of enstrophy added with thewhite noise corresponds to 4% of the enstrophy peaks of themean profile.

RESULTS
Scaling of the Mean Flow(7)

The analysis of the DNS data starts with the investigationof the mean flow characteristics of the planar jet, in order todetermine when the jet becomes fully turbulent and the selfsimilar regime is established.
The self-similar regime is the range, where the dynamics of the problem evolve with a local amplitude scaling anda local length scale, i.e. φ0 (t) and `(t), so that the time dependent mean profile of a variable φ can be written in the form
φ = φ0 (t) f (y/`(t)) (Townsend, 1976). For the investigation of
the self-similarity of the mean flow profiles, we start by normalizing the profiles by using the local length scale, jet halfwidth δ (t), in figure 2a.

Eq. 7 shows that the average TNTI propagation velocity
is proportional to the global Reynolds number to the power
2−D f1
D f −1 = D f −1 − 1 which is inversely related to D f .
The scalings of the inner turbulent length scales can also
1/4
be obtained. By using the relation; η = ν 3 /ε0
and thedissipation scaling relation, the Kolmogorov length scale η
can be written as;η ∼ (UJ HJ )1/2 (t − t0 )1/2 ReG

(9)

(8)

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
3b. The outer length scale of the problem, δ , has also the sametime dependency as λ and η.

(a)

Figure 1: Spatial resolution dy/η of the simulations plotted versus normalized time.

(b)
The expansion of the jet can be seen in figure 2b, whereδ 2 /HJ2 ∼ t behavior is compatible with eq.6. Here the jet halfwidth δ (t) is defined by the condition that the mean steamwise velocity at a given y location is half of the mean streamwise velocity at the center-plane, U(y = δ ,t) = Uc (t)/2, whereUc (t) = U(y = 0,t). An important feature of the temporally
Figure 3: (a) Time evolution of Reλ calculated at thecentre-plane (y = 0). (b) Ratios of length scales λ /ηand δ /λ throughout time.

(a)

Looking at the mean flow scalings, it is also observed that
the U uH0 δ/2 ∼ 1 for all the investigated time range, attesting to
J J
the constant volume flux throughout the time evolution of theflow. Townsend’s relation, K0 ∼ R0 holds until T /Tre f = 80,
and after that the ratio K0 /R0 starts to decrease. This is a probable demarcation of boundary effects on the jet as it expands
significantly compared to the domain extent in y-direction.

Identification of the TNTIThe TNTI is a very sharp interface between the turbulent
and non-turbulent regions of the flow. Thus, a quantity related
to the turbulent flow should be used to distinguish the two flow
regimes. In this study we use the enstrophy as the TNTI de2 , similar to other studies intection criterion, ω 2 (x, y, z) = ωththe literature (Reeuwijk & Holzner, 2014; Zhou & Vassilicos,2017). Unlike some other studies, various threshold values areused to locate the different iso-surfaces constituting the TNTI.
In order to see the enstrophy values marking the TNTI, the2 is plotted for a wide range ofvolume where ω 2 (x, y, z) > ωththreshold values in figure 4. The plateau over a wide rangeof ω 2 (x, y, z)th /ω 2 (x, y, z)re f marks the TNTI region. This
plateau forms due to the fact that the enstrophy iso-surfaces
are positioned very close to each other spatially. By picking athreshold value, which falls inside the plateau, one can detect
the iso-surface which envelopes the turbulent jet but also theirrotational fluid pockets which are completely engulfed into
the turbulent region. In this study, we detect these pockets,and the iso-surfaces related to these pockets are not taken into
account. Thus, we are only left with the continuous iso-surface
that envelopes the jet from outside.

(b)

Figure 2: (a)Self-similar, normalized profiles of mean

flow U, stream-wise turbulence intensity urms , Reynolds
shear stress Ruv and turbulent kinetic energy K. (b) Jet
half-width squared, δ 2 /HJ2 , versus time.

Fractal Dimension of the TNTI

developing planar jet is that Reλ remains constant due to theinverse relationship between the scalings of u0 and λ , which
also suggests that all length scales evolve identically as time
progresses. Constancy of Reλ can be seen in figure 3a and thetime evolution of the ratios λ /η and δ /λ is given in the figure

An important term in the relation 7 for the mean TNTI
propagation velocity vn involves the fractal dimension of thesurface D f . D f is present in the term responsible for the Re
dependence of vn . After the detection of the 3D TNTI surface,3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
data points included. But it is found that this way of calculation of D f , shown as D f 1 in the figure legend, leads to inaccurate values due to the variation of the slope with r. In order to find the linear region of the log2 N - log2 1/r plot, local
slopes of the data points are calculated by fitting nine points
at a time. Linear fits are made for each set, and their slope isplotted as the local slope at that location in figure 5b. Considering a wide range of thresholds and the time range after thetransition to turbulence of the jet, it is observed that a plateauof the local slope is present between the scales δ and λ (ascan be observed for example from figure 5b). For the scalesr < λ the local slope reduces towards the value of 2. Thus theaverage of local slope values in this region (marked in red infigure 5b) is used for the calculation of D f , denoted as D f 2 ,with the exception of the first point from the side of δ as it introduces oscillations because N is not well converged for large
box sizes. Figure 6a shows D f 2 values calculated for various

Figure 4: The volume which satisfies the condition of
2 is plotted for a wide range of thresholdsω 2 (x, y, z) > ωth
2
ωth .

(a)
a box-counting algorithm is utilized for the calculation of its
fractal dimension. Briefly stated, the surface S is covered withcubes with varying edge sizes r (Mandelbrot, 1982). By looking at the log − log variation of the number N of boxes needed
to cover the surface, with varying box-size r, the fractal di2 at amension D f is calculated for a chosen threshold value ωthgiven instant. The independent parameters which D f may de-

(a)

(b)

(b)

Figure 6: (a) Variation of the fractal dimension through-the TNTI surface, versus inverse box-size 1/r plotted ina log − log scale and (b) the local slope of the log2 N log2 (1/r).

2 /ω 2 values and theirout the TNTI with different ωthre fconstancy in time. (b) a section of the TNTI, where
2 are shown on top ofdifferent iso-surfaces of ωth2
2
log10 (ωth /ωre f ) contours. Iso-contours from cyan to
2 /ω 2 = 10−6 , 10−5 , 10−4 and 10−3 .
pink are ωthre fpend on are the measurement length scale r, time T /Tre f and
2 /ω 2 . The dependence on r is thethe enstrophy threshold ωth
re ffirst to investigate. Even though the TNTI surface shows fractal characteristics, it is also known that for very small scales,diffusive processes smooth out the contortions. Thus the slopeof the log2 N - log2 (1/r) plot is expected to reduce to D f = 2.
Looking at figure 5a, the orange line is the slope with all the

enstrophy thresholds for a range of times. It is observed thatthe D f values remain approximately constant in time with avery slight variation. D f is seen to vary significantly with enstrophy thresholds. It tends to have a higher value on the turbu2 /ω 2 = 10−3 andlent side of the TNTI, i.e. D f ≈ 2.18 for ωth
re f
22
reduces to D f ≈ 2.085 for ωth /ωre f = 10−6 . This trend canbe seen qualitatively from figure 6b, where the iso-surfaces of

Figure 5: (a) The number N of boxes needed to cover4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
turbulent region, other mechanisms begin to be as active as theviscous diffusion, thus causing the deviation observed in the
figure 7b.

higher enstrophy values towards the inner side of the TNTI are
more contorted compared to the iso-surfaces at the outer edge.

Average Propagation Velocity of the TNTIThe average propagation velocity of the TNTI into the
non-turbulent region is obtained by using eq.7. The time
derivative of the jet half-width dδ /dt is obtained from themean flow profiles and the surface area S is measured by thebox-counting algorithm. The vn profiles calculated over enstrophy thresholds are shown in figure 7a. The propagation
velocity is observed to be higher in the non-turbulent side
2 /ω 2 takes smaller values and it is reduced towards
where ωthre f
the higher thresholds. This is compatible with the constancyof the flux throughout the TNTI, Q = Svn , and thus the relation between the surface area and propagation velocity becomes S ∼ 1/vn . As the inner iso-surfaces are more contorted, their surface area S is higher than the outer iso-surfaces.
Figure 7b shows the TNTI propagation velocity vn normal-

CONCLUSIONS AND DISCUSSION
The temporally developing planar jet has been analyzed
starting from the basic conservation equations. The evolutionof the time-developing turbulent planar jet is fundamentallydifferent from the evolution of its space-developing counterpart. It is the volume flux which is being conserved throughout
the time evolution of the temporal jet. More importantly, Reλremains constant after the jet becomes fully turbulent. Thisimplies that the inner turbulent length scales evolve together intime. In addition, the mean flow length scale, i.e. the jet halfwidth, varies in a similar manner, thus all the length scales ofthe problem have identical t 1/2 power law evolutions in time.
The corresponding velocity scales also decay identically witheach other as t −1/2 . As shown in (Cafiero & Vassilicos, 2019),
the non-equilibrium dissipation scaling leads to different scalings of the mean flow than the equilibrium dissipation scaling
for spatially developing planar jets. Similar analysis has been
conducted to discover the implications of the non-equilibrium
dissipation scaling on the evolution of the temporally developing jet, but it has been found that the time evolution of themean flow parameters are insensitive to the turbulence dissipation scalings. By considering the fractal nature of the TNTI
and the self-similarity of the turbulent jet, a relation is obtained
for the average propagation velocity of the interface vn . This
relation shows that vn is proportional to the global Reynolds
number raised to a power which is inversely related to the fractal dimension of the surface. Our DNS supports this relationand the assumptions and predictions of our theory and reveals
an interesting inner structure of the TNTI with a dependence ofthe fractal dimension on iso-enstrophy levels. It is shown thatthe fractal dimension D f of the enstrophy iso-surfaces varythroughout the thickness of the TNTI. D f values are smaller atthe outer edge of the TNTI, and increase towards the turbulent
region. Thus the outer edge of the TNTI is less contorted compared to the turbulent side where the surface is more irregularand the surface is higher. The TNTI propagation velocity isfound to be higher at the outer edge and lower on the turbulent
side, which agrees with the constancy of the entrainment flux
throughout the thickness of the TNTI, Q = Svn . The vn normalized by the term responsible for the D f dependence remains
approximately constant for the range of thresholds falling into
the TNTI, supporting the relation found for vn .

(a)

(b)

Figure 7: (a) Average interface propagation velocity,vn , normalized by uη (solid lines, left axis) and by uλ(dashed lines, right axis) for different threshold values.
(b) vn pre-multiplied by the Re f (D f ) with same normalizations.

ACKNOWLEDGEMENTS
This work is supported by JCV’s Chair of Excellence CoPreFlo funded by I-SITE-ULNE (grant number R-TALENT19-001-VASSILICOS); MEL (grant number CONVENTION219-ESR-06) and Region Hauts de France (grant number
20003862). This work was granted access to the HPC resources of IDRIS under the allocation 021741 made by GENCI(Grand Equipment National de Calcul Intensif).

ized by the term responsible for the dependence on D f , i.e.
(2−D )/(D −1)(2−D )/(D −1)

f
f
ReG f
in the eq. 7. vn /ReG fremains
approximately constant for the thresholds which fall into the
enstrophy range of TNTI, which confirms the D f dependence
found with the theoretical analysis. It is observed that the constancy of the normalized vn starts to deviate at the higher enstrophy values. To better understand this, it must be remembered that the Corrsin length ηI ∼ ν/vn is used as the smallest
length scale on the TNTI for obtaining eq. 7. This lengthscale is obtained by assuming that it is the viscous diffusion
mechanism that drives the process of irrotational fluid packets
obtaining vorticity at the TNTI. As we go further inside the

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