Scaling Laws of Turbulent Diffusion – An Experimental Validationltces.dem.ist.utl.pt/lxlaser/lxlaser2008/papers/11.1_1.pdf · 2008-06-09 · 3: Department of Fluid Dynamics (FDY),

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, [email protected]

2: Institute of Environmental Engineering (IfU), Swiss Federal Institute of Technology, Zurich, Switzerland,

[email protected]

3: Department of Fluid Dynamics (FDY), Technische Universitaet of Darmstadt, Darmstadt, Germany,

[email protected]

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


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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


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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


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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


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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.


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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


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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).


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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.


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6. References

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Documents

Scaling Laws of Turbulent Diffusion – An Experimental Validationltces.dem.ist.utl.pt/lxlaser/lxlaser2008/papers/11.1_1.pdf · 2008-06-09 · 3: Department of Fluid Dynamics (FDY),