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14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008
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Scaling Laws of Turbulent Diffusion – An Experimental Validation
Matthias Kinzel1, Markus Holzner
2, Beat Lüthi
2, Cameron Tropea
1, Wolfgang
Kinzelbach2, Martin Oberlack
3
1: Department of Fluid Mechanics and Aerodynamics (SLA), Technische Universitaet of Darmstadt, Darmstadt,
Germany, m.kinzel@sla.tu-darmstadt.de
2: Institute of Environmental Engineering (IfU), Swiss Federal Institute of Technology, Zurich, Switzerland,
markus.holzner@ifu.baug.ethz.ch
3: Department of Fluid Dynamics (FDY), Technische Universitaet of Darmstadt, Darmstadt, Germany,
oberlack@fdy.tu-darmstadt.de
Abstract From Lie-group (symmetry) analysis of the multi-point correlation equation Oberlack and Guenther [Fluid Dyn. Res. 33, 453-476 (2003)] found three different solutions for the diffusion of shear-free turbulence: (i) a heat equation like solution when the turbulence diffuses freely into the adjacent calm fluid, (ii) a deceleration wave like solution when there is an upper bound for the integral length scale and (iii) a finite domain solution for the case when rotation is applied to the system. This paper deals with the experimental validation of the theory. We use an oscillating grid to generate turbulence in a water tank and PIV (Particle Image Velocimetry) to determine the two-dimensional velocity and out-of-plane vorticity components. The whole setup is placed on a rotating table. After the forcing is initiated, a turbulent layer develops that is separated from the initially irrotational fluid by a sharp interface, the so-called turbulent/non-turbulent interface (TNTI). The turbulent region grows in time through entrainment of surrounding fluid. We measure the propagation of the TNTI and find good agreement with the theoretical prediction for all three cases.
1. Introduction
Many flows observed in nature are partly turbulent, e.g. Scorer (1978), where the turbulent regions
are separated from surrounding irrotational (non-turbulent) regions by a sharp interface, the so-
called turbulent/non-turbulent interface (TNTI). Common examples are smoke plumes from
chimneys, effluents from pollution outlets, clouds, volcanic eruptions, seafloor hydrothermal vents
and many others. Typically in these flows the turbulence diffuses and the TNTI advances into the
ambient, while calm fluid is entrained into the turbulent flow regions. This mixing process is of
utmost importance for the dispersion of contaminants. The problem is also important for industry,
where some examples are combustion chambers, chemical technology, jets and wakes of aircrafts,
missiles, ships and submarines.
Traditionally, most of the attention was dedicated to flows with significant mean shear, like
canonical free shear flows (e.g., jets, wakes or mixing layers), see e.g. Townsend (1976), Pope
(2000) and Tsinober (2001). Ideally, these flows develop into an undisturbed ‘infinite’ environment.
However, most real flows do not develop freely, for example they can be bounded (by walls,
stratification, etc.) so that there is an upper limit for the integral length scale (e.g., stratified layers in
the ocean) or they can be subject to rotation (e.g., geophysical flows). The understanding of these
effects is up to now incomplete, see for example the recent review by Hunt et al. (2006).
We consider the problem of shear-free turbulence that diffuses from a planar source of energy.
Despite its fundamental importance, much less attention was dedicated to this problem compared to
canonical shear flows. Hopfinger and Toly (1976) reported experiments, where shear-free
turbulence was generated by a planar grid that oscillates normally to its plane in a water tank. The
14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008
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flow produced by an oscillating grid is the result of interactions between the individual jets and
wakes created by the motion of the grid bars. At sufficient distance from the grid, these jets and
wakes interact and break into turbulence that propagates away from the grid. Ideally, no mean flow
exists and the turbulence is considered nearly isotropic and homogeneous in planes parallel to the
driving grid. For the steady problem, Hopfinger and Toly (1976) (see also De Silva and Fernando,
1994 and references therein) measured that the r.m.s. velocity, urms, and the integral length scale, L,
scales with the distance to the source, y, as urms ~ y -1
and L ~ y. Based on dimensional analysis,
Long (1978) predicted that the mean depth of the TNTI, H(t) grows in time, t, according to a power
law H ~ t ½
. This was confirmed experimentally by Dickinson and Long (1978). In a second
study, Dickinson and Long (1982) found a t1 dependency when rotation is applied to the system. In
both studies, the TNTI was detected visually from the recorded images. Recently, Holzner et al.
(2006) used detailed flow measurements and a detection algorithm for the TNTI and they confirmed
the same propagation law H ~ t ½
.
Oberlack and Guenther studied three different cases of shear free turbulent diffusion applying Lie-
group (symmetry) analysis to the multiple point correlation function, Oberlack and Guenther
(2003). In their analysis, the turbulence is generated by a planar source of energy and diffuses
normally to this plane, similar to the oscillating grid experiments described before. They derived (i)
a heat equation like solution for the case in which the turbulence can diffuse freely into the adjacent
calm fluid, which means that H(t) evolves according to a power law,
0( ) ( )nH t A t t B= − + (1)
(ii) a deceleration wave like solution when there is an upper bound for the integral length scale,
0( ) ln( )H t A t t B= − + (2)
and (iii) a finite domain solution for the case when rotation is applied to the system,
0( ) exp( / )H t A t t B= − + (3)
The aim of this study is the experimental validation of these predictions. We use the experimental
setup and techniques described in Holzner et al. (2006). In section II we describe the Method and
present the results in Section III followed by the conclusions.
2. Method
A sketch of the oscillating grid setup is shown in Fig. 1. For the measurements the experimental
setup described in Holzner et al. (2006) was used. A screen of squared bars (of d = 1 mm, mesh-size
d0 = 4 mm) is installed near the upper edge of a water filled glass tank with dimensions 200 x 200 x
300 mm³. The grid is connected to a linear motor, which drives the vertical oscillation on a
supporting frame connected to the grid through four rods of 4 mm in diameter. The motor, operated
in a closed loop with feedback from a linear encoder, runs at a frequency of 9 Hz and an amplitude
ε = ± 4 mm for all the experiments. The whole setup is placed on a rotating table, see Fig. 1.
The PIV experiments were conducted by using a high-speed camera (Photron Ultima APX, 1,024 x
1,024 pixels) at a frame rate of 50 Hz. The maximum recording time at this frame rate is 80 s. The
camera is triggered by the onset of grid motion. The beam of a continuous 25 Watt Ar-Ion laser is
expanded through a cylindrical lens and forms a planar laser sheet about 1 mm thick, which passes
14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008
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vertically through the mid-plane of the tank, as shown schematically in Fig. 1. The camera recorded
the light scattered by neutrally buoyant Polystyrene tracer particles with a diameter of 40 µm. The
PIV images were processed with an interrogation window size of 32 x 32 pixels, 50% overlap,
yielding approximately 4000 two-component velocity vectors per realization.
Fig. 1 Schematic of the experimental setup
We detect the TNTI by using the method based on the out-of-plane vorticity component described
in Holzner et al. (2006). For each time instant t and for each x, the position of the TNTI is the
lowest point, y*(x,t), in which the magnitude of the vorticity signal exceeds a fixed (for all times
and x locations) threshold. Similar methods were used by Westerweel et al. (2002) (see also
Westerweel et al. 2005 and references therein). The mean position of the TNTI for a given time
instance is the average over x of the detected points, i.e. H(t)=< y*(x,t) >x. Fig. 2 shows an example
of an instantaneous PIV realization. The contours in Fig. 2a show the magnitude of ωz,, the vectors
in Fig. 2b show the direction and the magnitude of the velocity field and the superimposed black
line marks the detected TNTI.
Fig. 2 Vorticity magnitude map, velocity vector field and TNTI for t=10s
First we conducted experiments for the case (i). For case (ii) (confined integral length scale) we
placed a thin-walled transparent tube in the centre of the tank, so that the length scale of the flow
inside the tube was confined by its diameter. Three different diameters were tested. Finally, for case
14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008
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(iii) the table was rotating at three different constant angular velocities. The experimental
parameters are summarized in Table 1.
Type of experiment Tube diameter / angular
velocity Number of runs Symbol
case (i) - 10 +
case (ii) 20mm 10 30mm 10 ∆
40mm 10 case (iii) 0.79 rad/s 5
0.39 rad/s 5 ∆
0.29 rad/s 5 Table 1 Experimental parameters for the three types of experiments (i) free diffusion, (ii) diffusion with
constant integral length scale and (iii) diffusion under the influence of rotation
3. Results
The present study focuses on the propagation of the TNTI in time for the three cases, (i) free
diffusion, (ii) fixed integral length scale and (iii) rotation.
Case (i): free diffusion
Fig. 3 Vorticity magnitude maps for three snap shots a) t=2s b) t=8s and c) t=16s
In the experiments, the turbulence produced by the oscillating grid diffuses freely until the whole
tank is in turbulent motion. Fig. 3 shows magnitude maps of the vorticity for three snap shots, a) t =
2 s, b) t = 8 s and c) t = 16 s. The locations where the vorticity magnitude exceeds the threshold
value are marked in white color and represent the turbulent regions, whereas the gray areas
represent irrotational regions. We see that in the initial stage of the experiment the turbulence is
mainly confined within small regions in proximity of the grid (Fig. 3a), whereas a few seconds later
turbulent motion has noticeably spread out (Fig. 3b) and at t=16s the turbulent flow reached the
lower end of the field of view (Fig. 3c)
The propagation of the TNTI is depicted in Fig. 4, which shows its mean position, H(t), as a
function of time. All experimental runs are included in the figure, but in this and all following
figures of this type not all data points are plotted to allow for a clearer representation. We recall that
the prediction of Oberlack and Guenther (2003) is H(t)=A(t-t0) n+B in this case. We take B and t0
both equal to zero so that H(t=0)=0. We see from the figure that H evolves indeed according to a
power law and we estimate the exponent n by using a regression analysis (see Holzner et al. 2006)
with H(t)=A t
n and obtain n=0.5+/-0.1, A=26.9 +/-4.6 mm²/s and a mean R
=0.94. The regression
analysis was conducted with the data of the ten runs. The obtained values are consistent with the
14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008
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results in Holzner et al. (2006).
Fig. 4 Mean position of the TNTI, H(t),versus time in linear (a) and logarithmic (b) axes, respectively.
Symbols (+) are experimental data, lines are the best fit to the data
Case (ii): turbulent diffusion with a constant integral length scale
Next, we placed a transparent cylinder vertically in the center of the tank and observed the
propagation of the TNTI inside the cylinder. The purpose is to limit the growth of the integral
length scale of the turbulence, L, through the confinement of the domain. In the initial period of the
experiment the turbulent length scales are comparable to the mesh size of the oscillating grid and as
the experiment evolves the length scales grow. Initially, L is free to grow and the turbulence is
expected to diffuse like in case (i), but at some point L will become comparable to the diameter of
the cylinder and from there one it will remain constant. For a constant integral length scale,
Oberlack and Guenther (2003) predicted H(t) ~ ln(t).
Fig. 5 Vorticity magnitude maps for three snap shots a) t=2s b) t=16s and c) t=32s. The diameter of the
cylinder placed in the center of the field of view is D=30mm
We tested three different cylinder diameters, D = 20, 30 and 40 mm. Fig. 5 shows magnitude maps
of the vorticity for three snap shots of an experiment with D = 30mm: a) t = 2 s, b) t = 16 s and c) t
= 32 s. Already in the initial stage (Fig. 5a) we notice differences between the flow in the cylinder
and the outer field. While in the outer field turbulent flow regions are visible close to the grid, the
flow in the cylinder is mostly irrotational. The difference becomes clearer for later times. In Fig. 5b
we observe that in the cylinder the TNTI reached approximately one third of the total length, while
outside we clearly distinguish turbulent regions beyond that distance. In Fig. 5c the TNTI reached
about one half of the total length, while outside the flow is in turbulent motion everywhere.
14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008
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Fig. 6 Mean position of the TNTI, H(t),versus time for the three diameters, D=20mm ( ), D=30mm (∆) and
D=40mm ( ) in linear (a) and logarithmic (b) axes, respectively. The lines represent the best fits to the data
The propagation of the TNTI in the cylinders is presented in Fig. 6. All experimental runs for the
three tube sizes are included in the figure. Note that the considered time span of the experiment is
about one decade longer than the previous case, because the diffusion of the turbulence inside the
tube is much slower than free diffusion. We notice that the propagation law is indeed of logarithmic
type and we use H(t) = A ln (t - t0)+B for the regression analysis. The parameters t0 and B account
for the initial period, where the behavior is still of power law type. The transition from power law to
logarithmic behavior appears to occur very quickly in a period within, say, 0-5 s. In this preliminary
analysis we assume for simplicity that the logarithmic law holds from the onset of the experiment
and therefore take t0 = 1 s and B=0 mm, in this way H(t=0)=0 mm.
From regression analysis we obtain A=8.6, 21.4 and 27.6 mm for the three cases D=20, 30 and 40
mm, respectively with a mean R=0.87, 0.95 and 0.95 thus confirming that the experiments agree
very well with the prediction of Oberlack and Guenther (2003). We note that A increases
approximately linearly with D. This is consistent with the following observation that the integral
length scale increases linearly with distance from the source.
Fig. 7 Spatial (horizontal) autocorrelation of the vertical velocity component for increasing distance from the
grid (a) and resulting integral length scale with dashed trend line (b)
The integral length scale of the turbulence for the case of free diffusion was estimated through
spatial autocorrelation of the vertical velocity component. The result is shown in Fig. 7a for varying
14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008
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vertical coordinate. Fig. 7b displays the resulting integral length scale as a function of the distance
to the grid and we note that L increases linearly with distance. This is consistent with the similarity
hypothesis (Long 1978) and related experiments in literature (e.g. Hopfinger and Toly 1976)
Case (iii): turbulent diffusion under the influence of rotation
Finally, we consider the case of rotation applied to the system. Similar to the observations in
Dickinson and Long (1982), the turbulence spreads in the initial moments like in case (i), but very
quickly tube-like structures start to form at the edge of the turbulent layer. These structures, also
known as ‘Taylor columns’ (e.g. Hopfinger et al. 1982), propagate towards the bottom of the tank at
constant speed, while the turbulence remains confined within a small region close to the grid. The
flow reaches an equilibrium state with the turbulent region, where the motion is fully three-
dimensional, clearly separated by a region governed by waves, where the flow is essentially two-
dimensional. Typically in the experiments the turbulent region was observed to grow at first (at t ~
1 – 10 s), then retract slightly (at t ~ 10-30 s) and finally reach an equilibrium depth. This finite
domain, where the motion is fully turbulent is exactly as Oberlack and Guenther (2003) obtained
and they predict H(t)=A exp(-t/t0)+B.
Fig. 8 Vorticity magnitude maps for three snap shots for the experiment with angular velocity 0.39 rad/s, a)
t=2s b) t=16s and c) t=32s
Three different constant angular velocities were used in the experiment, 0.39, 0.29 and 0.21 rad/s.
Fig. 8 shows magnitude maps of the vorticity for three snap shots of an experiment with angular
velocity of 0.39 rad/s for three time instances, t=2, 16 and 32 s. In the initial period (Fig.8a) we
observe that the diffusion of the turbulence is similar to case (i). In a second stage inertial waves
start to form. Their axis of rotation is parallel to the axis of rotation of the table, i.e. parallel to the
vertical coordinate. Therefore, the out-of-plane vorticity component remains almost unaffected by
the waves and can be used to detect the outer edge of the turbulent region, where the motion is fully
three-dimensional. In Fig. 8b we note that the turbulent region reaches about one third of the
vertical extent of the field of view. As mentioned before, in a second stage the turbulent region
retracts slightly and reaches an equilibrium position (Fig. 8c).
14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008
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Fig. 9 Mean position of the TNTI, H(t),versus time with the table rotating at three constant angular
velocities, 0.29rad/s ( ), 0.39rad/s (∆) and 0.79rad/s ( ) in linear (a) and logarithmic (b) axes, respectively.
The lines represent the best fits to the data
Fig. 9 depicts the propagation of the outer edge of the turbulent region in time for the three angular
velocities. We note that after about 10-30 s, depending on the angular velocity, the turbulent region
reaches an equilibrium depth. The faster the rotation, the smaller is the final equilibrium depth. The
equilibrium depth is well described by an exponential function. We fit H(t)=A exp(-t/t0)+B to the
data and obtain A=-32, -22 and -4 mm, t0=2.5, 1.43 and 1 s-1
, B=32, 22 and 4 mm for 0.29, 0.39 and
0.79 rad/s. Hence we confirm the prediction of Oberlack and Guenther for the equilibrium stage of
the experiment. The high scatter of the data of the 0.79 rad/s experiments is due to the proximity
between the TNTI and the grid. For the transient period we need a more detailed analysis for a
better understanding of the processes involved.
4. Conclusions
In summary, the diffusion of shear-free turbulence away from a planar source of energy was
investigated experimentally for three different cases: (i) the turbulence diffuses freely into the
adjacent calm fluid, (ii) there is an upper bound for the integral length scale and (iii) rotation is
applied to the system. An oscillating grid drives the turbulence and the flow is analyzed by using
time resolved PIV measurements. We measure the propagation of the TNTI for the three cases and
confirm all three predictions obtained via symmetry analysis by Oberlack and Guenther (2003):
In particular, for case (i) we observe that the TNTI propagates according to a power law, H ~ t n,
where n is estimated to be n=1/2, in agreement with the results in Holzner (2006). For case (ii) the
behaviour changed to a ln-law with H ~ A ln(t) and we note that there is a linear correlation between
the parameter A and the upper limit for the integral scale, D. Finally, for case (iii) we measure that
the turbulence remains confined within a finite domain and the behaviour is of exponential type. We
note that the equilibrium depth decreases with increasing angular velocity, consistent with the
results of Dickinson and Long (1982).
5. Acknowledgements
The financial support by the German Research Foundation (DFG) is gratefully acknowledged.
Furthermore, the authors would like to thank K. W. Hoyer for his contributions to this work.
14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008
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6. References
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