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

ON THE NATURE OF THE TURBULENT/TURBULENT INTERFACE

Oliver R.H. Buxton
Department of Aeronautics
Imperial College London, UK
o.buxton@imperial.ac.uk

Krishna S. Kankanwadi
Department of Aeronautics
Imperial College London, UK
krishna.kankanwadi12@imperial.ac.uk

ABSTRACT

ture of turbulent flows is that they are vortical, i.e.
ω 2 = |∇ × u|2 ̸= 0, whereas non-turbulent flow is usually
irrotational, ω 2 = 0. Accordingly, the outermost boundary between the turbulent and non-turbulent fluid, theirrotational boundary, is an iso-surface of ω 2 = 0. The
only non-zero source term in (1) at the irrotationalboundary is therefore the viscous diffusion term hence
Corrsin & Kistler (1955) postulated the existence of athin viscous layer at the outer edge of the TNTI, whose
thickness ought to scale with the Kolmogorov lengthscale η = (ν 3 /ε )1/4 , with ε the dissipation rate of turbulent kinetic energy. This intuition, and scaling has
since been confirmed experimentally (Holzner & Lüthi,2011).
However, the majority of environmental and industrial flows exist within a turbulent background meaning that entrainment takes place between two adjacent streams of turbulent fluid, e.g. between the volcanic plume and the (turbulent) atmospheric boundary
layer. We refer to this as turbulent/turbulent entrainment (TTE) and in this case the intuition of Corrsin &Kistler (1955) breaks down since ω 2 ̸= 0 on both sides of
the interface between the two streams of fluid; the turbulent/turbulent interface (TTI). In fact the existenceof the TTI has not yet been definitively proven withthe review of da Silva et al. (2014) suggesting that the
TTI (if it exists) is likely to “break down” when the turbulence intensity becomes comparable on both sides of
the interface. In this paper we prove the existence of the
TTI and examine the physics governing TTE since viscous diffusion is no longer the only mechanism by which
entrainment can take place in the TTE paradigm.

We perform combined PIV and PLIF experiments
to investigate the nature of the interface between two
adjacent streams of fully-developed turbulence; the turbulent/turbulent interface (TTI). For the first time wedefinitively prove the existence of the TTI, somethingthat had been previously questioned for cases where theintensity of the turbulence in both streams is comparable. We then examine the dominant physics in the TTI.
Unlike for turbulent/non-turbulent interfaces viscosity
does not play an important role and the inertial vortex
stretching term is dominant in producing the discontinuity in enstrophy that is characteristic of the TTI. We
show that the particular organisation of the small-scale
turbulent strain rate and vorticity in the TTI is responsible for driving this vortex stretching and show that
freestream turbulence diminishes the rate of entrainment into the primary flow relative to a non-turbulent
freestream.

INTRODUCTION
The spatio-temporal processes by which turbulent
bodies of fluid expand into the background fluid with advected distance, such as the growth of a volcanic plumein the atmosphere, are collectively known as entrainment. The rate, and process, by which entrainment
occurs is governed by the turbulent dynamics within
an interfacial layer adjacent to the outermost boundary between the two regions of fluid. In the special
case where the background fluid is non-turbulent thislayer is known as the turbulent/non-turbulent interface(TNTI). The nature of the TNTI is best explained byexamining the behaviour of the terms of the enstrophy(ω 2 /2) transport equation
D ω2
Dt 2

=

ωi si j ω j| {z }
vorticity stretching+∂ (ω 2 /2)ν∂ x j∂ x j| {z }
viscous diffusion

(
+

ν
|

METHODOLOGY

)2

∂ ωi
∂xj
{z }

viscous dissipation(1)
where ωi are the components of the vorticity vector andsi j is the strain-rate tensor. A key distinguishing fea-

1

Examining the physics of TTE requires field velocity data and the ability to identify the TTI and trackit through time. Identification of the TTI is not as
straightforward as the TNTI due to the fact that ω 2 ̸= 0
on both sides of the interface. We thus use a passive
scalar, mixed into the primary turbulent flow (a wakeproduced by a circular cylinder), to identify the extent
of the primary flow (wake). In particular, a high Sc

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

3

PLIF camera
2.5

2

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PIV camera
PIV camera

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Figure 1: Experimental setup. Note that the originof the laboratory coordinate system has been moved
downstream for clarity.

0
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5

10

15

Figure 2: {L12 , T I} (length-scale - turbulence inten-sity) parameter space of the freestream turbulence
explored. The red vertical line corresponds to theturbulence intensity within the wake for the no-gridcase (Run 1a).

scalar is used to ensure the effects of molecular diffusion
are negligible. We thus deploy a combined high-speed
particle image velocimetry (PIV) and planar laser induced fluorescence (PLIF) approach. Two experimental campaigns were conducted, one in which planar PIV
was used and one in which cinematographic stereoscopic
PIV was used (in conjunction with Taylor’s hypothesis)
to measure all of the three-dimensional terms of (1).
A circular cylinder of diameter d is placed in a water
flume, such that Red = 4 000, a variable distance downstream of a turbulence generating grid to ensure thatthe background is fully turbulent. The type of grid andgrid - cylinder spacing are varied such that the integral
length scale L12 and turbulence intensity T I = u′ /U∞ ,where u′ is the r.m.s. velocity fluctuation, of the background turbulence can be independently varied. The
experimental setup for the stereoscopic PIV campaign
is illustrated in figure 1. Note that the origin of thecoordinate system is the centre of the cylinder, however
this has been shifted downstream for reasons of clarity
in the figure. The PIV cameras interrogate a necessarily small field of view approximately 40d downstream of
the cylinder (i.e. the wake was fully developed) in orderto ensure that the spatial resolution of approximately
3.6η was sufficient to resolve the various terms of (1).
The turbulent Reynolds number at this field of view was
Reλ = 94. Figure 2 shows the explored {L12 , T I} parameter space of the freestream turbulence. Of particularnote are the runs categorised as “group 3” where the T Iof the background turbulence is greater than that within
the wake (denoted with the solid red line). Full details of
the experimental methods, including the identificationof the interface location using a threshold on gradient of
the PLIF light intensity |∇ϕ |, are given in Kankanwadi& Buxton (2020) and Kankanwadi & Buxton (2022).

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Figure 3: Mean enstrophy conditioned on interface-normal location. γ = 0 corresponds to the wakeboundary with γ > 0 being the freestream and γ < 0being within the wake.
tance in a result that is directly analogous to what is observed in a TNTI (e.g. da Silva et al., 2014). This holds
true in even the most extreme cases of background turbulence where the T I of the background is greater thanthat in the wake (group 3). This represents a repeatably
observed fluidic discontinuity (independently of the artificially introduced scalar) and thus constitutes proofof the existence of the TTI, even in cases where the T Iof the background is comparable (or even greater) thanthat in the primary turbulent flow.
We verify the existence of the TTI through additional consideration of the two-point statistics of the interfacial region crossing the wake boundary γ = 0. Figure 4 depicts the correlation of the fluctuating spanwise
component of vorticity in the interface-normal direction,
Rω3′ γ ,RESULTS AND DISCUSSIONFirst of all we address the zeroth-order questionof whether the turbulent/turbulent interface even exists, which has been previously questioned (e.g. da Silva
et al., 2014)? Figure 3 shows the mean enstrophy conditioned on normal distance γ from the wake boundary (γ = 0, analogous to the irrotational boundary for a
TNTI) for the various experimental runs considered. In
all cases it can be seen that the enstrophy adjusts itself
from a fixed level in the freestream (γ > 0) to some otherfixed level within the wake (γ < 0) over some short dis-

⟨ω3′ (γ )ω3′ (γ + r)⟩√
Rω3′ γ = √.
⟨ω3′ 2 (γ )⟩ ⟨ω3′ 2 (γ + r)⟩2(2)

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

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Figure 4: Correlation function for the spanwise vor-

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(a)

ticity component in the interface-normal direction.
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The fixed probe is located 0.05d into the freestreamside of the wake boundary, i.e. γ = 0.05d, and r denotes the distance from the fixed probe in the interfacenormal direction. Solid lines in the figure represent
correlations in which r points in the direction of thewake, whereas dashed lines indicate r pointing into thefreestream. In the interests of clarity only one example
from each group has been presented in figure 4. As expected, an asymmetric correlation function is observedwith more rapid de-correlation witnessed as the moving
probe crosses the wake boundary (solid lines), in comparison to a slower de-correlation as the moving probe
traverses the freestream (dashed lines). The level ofasymmetry between the correlation functions pointing
in different directions relative to the interface is seen
to drop for cases with increased background turbulence
intensity, however this may be expected since more intense turbulence is expected to de-correlate over a physically shorter distance. We thus confirm the existenceof a turbulent/turbulent interface where the enstrophy
adjusts itself between the levels found on either side of
the interface, similarly to a TNTI. We finish by drawing an analogy between the TTI and internal shear layers (across which there is a discontinuity in, and decorrelation of, turbulent statistics) which are thought tobe a feature of high-Reynolds number turbulence (e.g.
Hunt et al., 2010; Ishihara et al., 2013).
Whilst it is not the intention of this paper to establish a scaling for the identified TTI thickness, it canbe useful to represent the thickness of the interface as afunction of known turbulent length scales. The vorticity
adjustment region of run 1a (our closest approximation
to a TNTI) may be estimated to have a thickness of 0.2d.
Using Taylor and Kolmogorov scales calculated for theno-grid case, 0.2d is equivalent to approximately 0.7λ
and 12.5η . These are in line with the TNTI thicknesses
observed in literature (e.g Silva et al., 2018). We also
note that the solid and dashed correlations of figure 4,
i.e. those pointing towards the wake and the freestream,
begin to diverge at r ≈ 0.12d, i.e. ≈ 0.07d into the interface. This corresponds to ≈ 4.5η , which is very close to
the widely accepted thickness for the viscous superlayerof a TNTI, although we make no claims as to having
identified the viscous superlayer from these correlation
functions.

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(b)
Figure 5: The interface conditioned plot of the vis-

cous diffusion term of (1) for cases in (a) groups 1& 2 and (b) group 3. The inset displays 95% confidence intervals for the no-grid (a) and run 3a (b)cases.

We now consider the relevant flow physics in the
TTI. Since ω 2 ̸= 0 on both sides of the TTI, the inertial
vortex stretching term ωi si j ω j of (1) is no longer constrained to be zero at the wake boundary and so can
act as a source term to generate this enstrophy jump.
Accordingly, there is no requirement for the viscous diffusion term to be the only source term of (1) at the wakeboundary as there is for a TNTI. We therefore seek todetermine the role of viscous diffusion in a TTI throughconsideration of figure 5 which shows the mean viscous
diffusion term conditioned on distance from the wakeboundary γ for (a) groups 1 and 2 and (b) group 3 (areminder that these groups are classified based on the
turbulence intensity of the background turbulence asshown in figure 2). In both sub-figures the behaviour of
the no-grid case, our closest approximation to a TNTI,
is presented for comparison as the solid green line. The
behaviour of the viscous term from the no-grid case isremarkably similar to results for TNTIs obtained via
direct numerical simulation (e.g. Taveira & da Silva,
2014; Watanabe et al., 2014; Silva et al., 2018), acting
as a source at the wake (irrotational) boundary and at-3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
taining a maximum a short distance inside the interface
before then acting as a sink and eventually returning
to zero, giving us further confidence in our experimental methodology. The viscous diffusion term acts in asimilar manner to the no-grid case for runs in groups 1
and 2 with minimal activity in the freestream followedby source-like behaviour in the outer portion of the interface. All cases with background turbulence exhibit
increased statistical noise which may be expected due
to diminishing sample sizes related to the interface position fluctuating more when exposed to freestream turbulence, often outside of the small-scale PIV field ofview. However, even with the increased noise the underlying behaviour can be established for groups 1 and2. This is not the case for runs that lie in group 3 where
all of the similarity to the no-grid case is lost. The levelof noise is also greater as the characteristic production
peak is no longer distinctly visible.

case and is thus our closest approximation to a TNTI.
Figure 6(b) shows both the conditional mean vorticity stretching term and the conditional mean viscous
diffusion term for the no-grid case, zoomed in aroundthe region γ ≈ 0. The behaviour of ωi si j ω j also closely
follows that reported in literature for TNTIs, with themagnitude of ωi si j ω j being smaller than that of the viscous diffusion in the outer-most portion of the interface before exceeding it some short distance away fromthe wake (irrotational) boundary and then completely
dominating in the inner part of the interface (and bulk
portion of the flow) (e.g. da Silva et al., 2014; Watanabe et al., 2014). This behaviour is also in accordancewith the intuition of Corrsin & Kistler (1955). For allcases with freestream turbulence present in figure 6(a)the vorticity stretching term is non-zero at the wakeboundary, since ⟨ωi si j ω j ⟩ > 0 is a feature of homogeneous turbulence (Taylor, 1938). However, it is clearthat the vorticity stretching term starts to increase at
the wake boundary and increases throughout the thickness of the TTI for all cases with freestream turbulence
present. In fact, the magnitude of this term dwarfs thatof viscous diffusion (c.f. figure 5) and so we conclude
that viscosity is an unimportant source of enstrophy
amplification in a TTI - a result of some consequence
for modelling strategies that frequently model entrainment through diffusion, inspired by the phenomenologyof TNTIs.

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Figure 7: Small-scale anisotropy ratio ⟨Σ⟩ condi-

tioned on interface-normal location. Values greaterthan unity correspond to small-scale anisotropy
with enhanced velocity gradients normal to the interface.

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Our results show that the role of viscosity, in termsof the relative strength of the viscous diffusion term
to the inertial vorticity stretching term, is reduced toinsignificance in a turbulent/turbulent interface, andthat it is this vorticity stretching term that is responsible for the conditional mean enstrophy jump, characteristic of the turbulent/turbulent interface. We now
seek to address the origins of this enhanced vorticity
stretching term on the wake-side of the TTI only andto determine why a discontinuity in enstrophy exists,
with a jump in enstrophy existing even in cases where
the T I of the background turbulence exceeds that in-(b)
Figure 6: (a) Mean ωi si j ω j conditioned on interface-normal location. (b) Zoomed in behaviour of theinertial and viscous source terms of (1) close to thewake boundary for the no-grid case, run 1a.

Figure 6(a) shows the conditional mean (inertial)
vorticity stretching term of (1) ωi si j ω j as a function of
γ . The thick green line is produced from the no-grid412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
for which γ̂ · êi is maximum, i.e. the term in which thestrain rate is best aligned with the interface-normal direction, and plot its contribution to ωi si j ω j overall as afunction of γ in figure 8. This should reflect the relative
contribution of the RHS term of (3) that is best able
to exploit the enhanced strain rate normal to the interface, i.e. the small-scale anisotropy. It can be seen thatthe contribution of this term is ωi si j ω j /3 far away fromthe wake boundary (all three terms of (3) contributeequally to ωi si j ω j ) but spikes to larger values at thesame locations where the enstrophy jump and enhanced
small-scale anisotropy are located. We thus conclude
from figure 8 that the vorticity in the wake-side of the
TTI is “organised” to take advantage of the enhanced
strain rate in the interface-normal direction and produce
the enstrophy jump that is characteristic of the TTI.

side the wake. Previous work on TNTIs has identified
strong small-scale anisotropy in which the presence ofa turbulent interface enhances the interface-normal velocity gradients (e.g. Cimarelli et al., 2015). We now
address whether a similar small-scale anisotropy exists
in TTIs. To do so we resolve the velocity vector intoa (pseudo two-dimensional) interface coordinate systemv = (vγ , vξ , vx ), where vγ and vξ are the components of velocity normal and parallel to the interface locally (fromour planar data), i.e. in the ξ and γ directions, andvx is the out-of-plane component (which is the streamwise component from our cinematographic stereoscopic
PIV experiment). We now define an anisotropy ratio⟨Σ⟩ = ⟨(∂ vγ /∂ γ )2 /(∂ vξ /∂ ξ )2 ⟩ as a function of γ . Notethat the interface-conditioned anisotropy has been calculated two-dimensionally in the transverse y − z plane
using only the v and w velocity components. Figure 7 reveals that far away from the interface ⟨Σ⟩ is unity, symptomatic of turbulence exhibiting small-scale isotropy,
but in the TTI region on the wake side only, smallscale anisotropy is introduced with enhanced turbulent
strain rates normal to the interface. This is a result thatis also mirrored in the turbulent sublayer of TNTIs (e.g.
Buxton et al., 2019).

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Figure 9: Entrainment mass flux as a function of T Iof the background turbulence.

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Finally, we explore the effects of background turbulence on the entrainment rate itself into the far wakeregion, x ≈ 40d. The entrainment rate is computed byintegrating the entrainment velocity vE along the captured interface within our field of view. Due to thelimitations of our planar data we define a normalised
entrainment flux as

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Figure 8: Proportion of overall ωi si j ω j contributedby the principal strain rate most closely aligned with
the interface-normal direction.

The enhanced strain rates normal to the interface
offer a potential explanation for the enhanced enstrophy
production yielding the enstrophy jump in TTIs. However, enstrophy production through vorticity stretchingdepends not only on the magnitudes of the strain-rate
tensor and vorticity vector, but also their alignment.
This is illustrated in the following decomposition of thevorticity stretching term (Betchov, 1956)

Normalised mass flux =

∫

vE dξ

(4)

where vE = vI − v0 , i.e. the entrainment velocity is thedifference between the local interface velocity vI andthe local fluid velocity v0 . Figure 9 shows that the entrainment mass flux decreases as a function of T I of
the background turbulence. Consideration of the probability density functions (PDFs) of the normalised massflux (with one flux computed from each PIV snapshot)shows that this is largely driven by an increased prevalence of rare, but high-magnitude (i.e. intermittent) detrainment events as the background turbulence intensity
is increased (Kankanwadi & Buxton, 2020). It is wellknown that ωi si j ω j is an intermittent quantity in turbulent flows at moderate/high Reynolds number (as is
symptomatic of turbulent velocity gradient quantities).
We previously showed that the turbulent physics of the3

ωi si j ω j = ∑ ω 2 si (ω̂ · êi )2

1
dU∞

(3)

i=1

where si are the eigenvalues of si j with corresponding
unit eigenvectors êi . (3) illustrates that it is both themagnitude and alignment of the vorticity vector (withrespect to the strain rates) that determines ωi si j ω j . We
choose the particular term on the right hand side of (3)512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
TTI are dominated by ωi si j ω j which offers an explanation for why intermittency is a powerful driver of the
reduction of entrainment rate as freestream turbulence
intensity is increased in the far-wake region.

far wake (i.e. a fully-developed turbulent shear flow)
it is assumed that the generality of the study is increased, with the analysis being more applicable to otherflows such as jets and even boundary layers. Whilst
it is not possible to claim any universality, since thatwould require experiments conducted over a large range
of Reynolds numbers using several flow types, the results are robust within the extensive parametric envelope that was investigated. The considered freestream
turbulence conditions had length scales both larger andsmaller than the wake-generating object as well as intensities smaller and larger than within the wake.

CONCLUSIONS
We verify the existence of a distinct interface separating the far field of a turbulent wake and a turbulent
background; the turbulent/turbulent interface (TTI).
The TTI was examined in detail through interfaceconditioned statistics of the enstrophy transport equation (1). Unlike for a TNTI, with rotational fluid
available in the free-stream side of a TTI the vorticity stretching term is free to contribute to enstrophy
production throughout the entire thickness of the interface, including at the wake boundary. Results showedthat the magnitude of enstrophy production through thevorticity stretching term dwarfed that of the viscous diffusion term hence it is possible to conclude that in a
TTI the role of viscosity is reduced to insignificance.
Similarly to a wall in wall-bounded turbulence, the TTI
acts to enhance strain-rates in the interface-normal direction close to the boundary. Decomposing the vorticity stretching term into three terms corresponding to
the three principal strain directions highlighted that thecomponent most aligned to the interface-normal direction contributes the largest share of enstrophy production on the wake side of the interface. This is indicativeof better “organised” vorticity on the wake side thattakes advantage of the strong interface-normal strain
rates to produce the enstrophy jump observed in a TTI.
These results have important implications with regards
to modelling entrainment behaviour in a turbulent environment as the role of vorticity stretching is instrumental in the entrainment process, something typically neglected when only considering entrainment from a nonturbulent background. Finally, we show that the presence of freestream turbulence has the effect of reducing
the entrainment mass flux relative to a non-turbulent
background. This is largely driven by intermittency,with rare but powerful detrainment events becoming
more common when the background is turbulent.
Our results lead to open questions regarding thephenomenology of a TTI. The structure of a TNTI iswell known, as there are distinct layers in which, respectively, viscous and inertial processes dominate theflow physics. For example, it is possible to claim thatthe viscous super-layer is bounded between the irrotational boundary and the location where the inertial vorticity stretching term produces a larger share of enstrophy than the viscous term (Taveira & da Silva, 2014).
However, these bounds lack any physical relevance whenapplied to the TTI. It is interesting to note that evenwith turbulence available in the background, the viscous
diffusion term for groups 1 and 2 produces a production
peak similar to the no-grid case. Hence, it is possible toraise the question: is this is the remnant of a defunct viscous super-layer? Additionally, with a diminished roleof viscosity there is also no longer a relevant physicalargument supporting the scaling of the thickness of the
TTI with the Kolmogorov length scale.
Lastly, it should be noted that by considering the

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