Turbulence, Heat and Mass Transfer 6
K. Hanjalic´, Y. Nagano and S. Jakirlic´ (Editors)
The local entrainment velocity is a viscous quantity
M. Holzner1, B. Lu¨thi1, A. Liberzon2, A. Tsinober2,3, W. Kinzelbach1
1Institute of Environmental Engineering, ETH Zurich, CH-8093 Zurich, Switzerland,
holzner@ifu.baug.ethz.ch
2 School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Ramat Aviv
69978, Israel.
3 Institute for Mathematical Sciences and Department of Aeronautics, Imperial College, SW7
2AZ London, United Kingdom.
Abstract —
In this contribution we analyze properties of the turbulent/nonturbulent interface (TNTI) in a flow without
mean shear. The approach is numerical and based on a Direct Numerical Simulation (DNS). For the final
paper the results will be complemented with Particle Tracking Velocimetry measurements. The focus
is on the ‘local’ entrainment velocity, vn, defined as the velocity of the TNTI relative to the fluid and
directed towards the irrotational ambient. We provide an expression for vn that consists of two parts: a
viscous and an inviscid one. The results show that the viscous contribution is stronger than the inviscid
part and governs the local entrainment velocity towards the ambient. We also propose a measure to
characterize the characteristic length scale or ‘thickness’ of the TNTI.
1. Introduction
An ubiquitous phenomenon in fluid flows observed in nature and industrial applications is the
occurrence of sharp interfaces or fronts that separate vortical from (irrotational) ambient flow.
Clouds, smoke plumes from chimneys, effluents from pollution outlets, volcanic eruptions,
seafloor hydrothermal vents or combustion chambers are just a few common examples. An
interesting process associated with these so-called ‘turbulent/nonturbulent interfaces’ (TNTI) is
the entrainment and mixing of ambient fluid into the turbulent regions. This process can also
be understood from the viewpoint of an observer moving with the interface as motion of the
TNTI relative to the fluid and directed towards the ambient. In their pioneering work, [2] and
[3] postulated that this relative velocity, vn, depends on two parameters: the kinematic fluid
viscosity, ν, and the intensity of straining in the adjacent turbulent region, ǫ, where ǫ = 2νsijsij
is the dissipation and sij are the components of the rate of strain tensor. The characteristic ve-
locity and length scales associated with these two parameters are the Kolmogorov velocity and
length scales, uη and η, respectively. On the other hand, it is well know that - globally - the
entrainment velocity, ue, scales with the large scale velocity field, either a conveniently chosen
mean velocity or r.m.s. of a fluctuating velocity, u′, e.g., [10]. The two different views can
be reconciled by arguing that the global entrainment flux, Q, occurs through a large scale (or
projected) interface area A0, i.e. Q = ueA0, and the strongly convoluted total interface area
Aη adjusts itself to account for the same flux with a much smaller characteristic velocity so that
Q = ueA0 = vnAη(∝ uηAη), see [9] and references therein. However, it is not known how
precisely the total area will adjust itself and so far there are no direct measurements of the local
interface velocity, only indirect evidence for vn ∝ uη was provided in [7]. Moreover, recent
work seems to indicate that the thickness of the TNTI is of order λ rather than η, e.g., [1, 4, 11],
but the reason for it is presently unclear.

2 Turbulence, Heat and Mass Transfer 6
x2
x1
x3
L1
L2
L3
/
/
/
1 2 3 4 50
4
2
1
2
3
Figure 1: Snapshot of an enstrophy
iso-surface obtained from DNS. The
value of the iso-surface normalized
over its mean in the turbulent region
is ω2/〈ω2〉 ∼10−3.
2. Method
Direct numerical simulation (DNS) was performed in a box (side-length 5L1, 3L2, 5L3) of
fluid initially at rest [7]. Random (in space and time) velocity perturbations are applied at
the boundary x2=0. The method of boundary velocity assignment determines the velocity scale,
V = max(Vi) and the length scale ∆l. Together with the viscosity of a fluid, ν, these parameters
define the Reynolds number Re = V ∆l/ν = 1000 of the simulation. The Navier-Stokes
equations were solved with shear-free conditions ∂u1/∂x2 = ∂u3/∂x2 = u2 = 0 imposed
at the boundary x2 = 3L2. The numerical scheme is finite differences with time advancement
computed by a semi-implicit Runge-Kutta method [7, 8]. The resolution is 256×256×256 grid
points in x1, x2 and x3 direction and the local Taylor-scale Reynolds number is Reλ=50. The
analysis was carried out for times when the turbulent/non-turbulent interface is about half a box
size away from the source, see Fig. 2.. The local Kolmogorov and Taylor length scales are about
η ∼ 2∆xk and λ ∼ 28∆xk respectively, where ∆xk is the grid spacing. This relatively high
resolution was necessary to achieve sufficient accuracy at the level of higher order derivatives
of the velocity.
3. Results and Discussion
Similar to, e.g., [1, 11, 5], the TNTI is detected by using a threshold on enstrophy, ω2 =ω·ω,
where ω is the vorticity vector. The value of the threshold normalized over the mean enstrophy
in the turbulent region is ω2/〈ω2〉 ∼10−3. Figure 2.shows a snapshot of the iso-surface associ-
ated with the selected threshold. In the frame of reference moving with an iso-surface element
the material change of ω2 will obviously be zero. By writing the velocity of the iso-surface
element, ui, as a sum of fluid velocity, u, and velocity of the area relative to the fluid, V, i.e.,
ui = u + V, one can see that the iso-surface will evolve according to the following equation
∂ω2
∂t + uj
∂ω2
∂xj
= −Vj
∂ω2
∂xj
= −vn|∇ω2|, (1)
where vn is the component of V normal to the surface, vn = V · nˆ and nˆ is the surface normal,
defined as nˆ = ∇ω2/|∇ω2|. With the use of the enstrophy transport equation,
∂ω2/2
∂t + uj
∂ω2/2
∂xj
= ωiωjsij + νωi∇2ωi (2)

M. Holzner et al. 3
−1 0 1
0
1
2
3
4
y/uη
P
D
F


−2 −1 0 1
10−2
10−1
100
101
y/uη
P
D
F


y = vinvn
y = vvisny = vn
y = vinvn
y = vvisny = vn
a) b)
Figure 2: PDFs of vn and its inviscid vinvn and viscous vvisn components normalized by the
Kolmogorov velocity, uη, on linear (a) and semi-logarithmic axes (b).
we obtain an equation for vn, written as a sum of an inviscid and a viscous contribution:
vn = −
2ωiωjsij
|∇ω2| −
2νωi∇2ωi
|∇ω2| = v
inv
n + vvisn . (3)
Figure 2 shows PDFs of of vn and its inviscid vinvn and viscous vvisn components, normalized
by the Kolmogorov velocity. The data used for the statistics is from the iso-surface locations
and comprises 3·105 points and the values of vn are negative when the velocity points towards
the irrotational ambient. The figure shows that all three curves are negatively skewed, but,
compared to vinvn , the PDF of vvisn is more concentrated at negative values, consistent with the
positiveness and strong skewness of the term νωi∇2ωi in the proximity of the TNTI shown in
[6, 7]. Therefore the viscous contribution is mostly responsible for the local (relative) advance-
ment of the TNTI towards the irrotational ambient. The mean values are 〈vinvn 〉/uη ≃-0.08,
〈vvisn 〉/uη ≃-0.36 and 〈vn〉/uη ≃-0.47. It is a bit puzzling that 〈vn〉 is about a factor two smaller
than uη and it will be interesting to compare this value to measurements of velocity gradients
via Particle Tracking Velocimetry as done in, e.g., [7]. However, the figure provides quite clear
evidence that the Kolmogorov velocity is the characteristic velocity scale for the quantity vn,
consistent with the indirect estimate in [7]. One can use the above terms to obtain a charac-
teristic length scale. We define a viscous length scale as lvis = ν/vvisn and an inviscid length
scale as linv = ν/vinvn and obtain lvis ≃2.78η and lvis ≃12.52η ∼ λ. In [6], enstrophy was
shown to drop with distance from the TNTI by an order of magnitude within a few η, which
is in agreement with the estimate lvis. However, if one considers the full span from the point
where ω2 starts to drop to the point where a numerical ‘noise’ level is reached, this distance is in
fact comparable to one λ. Hence we infer that the two length scales provide a lower and upper
measure for the thickness of the TNTI.

4 Turbulence, Heat and Mass Transfer 6
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