2 nd UK-JAPAN Bilateral and 1 st ERCOFTAC Workshop, Imperial College London

PIV study of fractal grid turbulence

S. Discetti1, I. B. Ziskin2, R.J. Adrian2, K. Prestridge3

1DIAS, University of Naples Federico II, Naples (Italy)

2School for Engineering of Matter, Transport and Energy, Arizona State University, AZ (USA)

3Los Alamos National Laboratory, NM (USA)

2nd UK-JAPAN Bilateral and 1st ERCOFTAC Workshop, Imperial College London

S. Discetti, I.B. Ziskin, R.J. Adrian, K. Prestridge

Motivation

Multi-scale generated turbulence may lead to exciting new insights into turbulence theory as well as important new industrial applications:• Turbulence generated by injecting energy over a range of length

scales;• Uncommon fast turbulence kinetic energy decay, well modeled by a

self-preserving single lengthscale decay model (George and Wang, POF, 2009);

• About three times higher Reynolds number Reλ than turbulence generated by classical grids.

PIV can be a useful tool to obtain a better understanding of the fluid dynamics of fractal-generated turbulence.


Square space-filling fractal grids Despite of the geometrical complexity, the grid

features are defined by a few parameters:

• Lo an to, i.e. length and thickness of the largest square;

• RL and Rt, i.e. the scaling factors for the length and the thickness at each iteration (more often the thickness ratio tr between the largest and the smallest scale is considered as a main parameter).

tr RL Rt σ Meff [mm]

8.5 0.5 0.490 0.25 15.8

13 0.5 0.425 0.32 15.2

17 0.5 0.389 0.37 14.6


Experimental setup

The grids are tested in a low turbulence level open circuit wind tunnel, with a 1,524mm long and 152.4mm wide square test section. The fractal grid is placed at the inlet of the test section, immediately after the contraction.


Experimental setup

The grids are tested in a low turbulence level open circuit wind tunnel, with a 1,524mm long and 152.4mm wide square test section. The fractal grid is placed at the inlet of the test section, immediately after the contraction.

385mm

600mm

Measurement area

≈50mm


Particle Image Velocimetry

p

p

o y

x

tMv

u 1

Fig. from Raffel et al., Particle Image Velocimetry – A practical guide, Springer Ed. (2007)


• Optical calibration:• Images of a target with equally spaced dots with diameter of 250 m μ

and spacing of 1mm are recorded to determine the magnification map as a function of the physical space coordinates;• Images of the particle distributions are taken simultaneously by two

cameras in a pre-processing run to determine via disparity map computation the location of the laser sheet in the physical space to properly set the local magnification.

• Image acquisition (5000 samples to ensure satisfactorily statistical convergence)

• Image Processing• IW: 32 x 32 pixels (0.63 x 0.63 mm comparable to the laser sheet

thickness) with 75% overlap;• Multi-pass iterative window deformation with adoption of weighting

windows in cross-correlation to enhance the spatial resolution.

Test procedure


Mean flow features

U [m/s] U [m/s] U [m/s]

Mean streamwise velocity for tr=13 and ReM=3.5∙103


Mean flow features

Mean streamwise velocity for tr=13 and ReM=3.5∙103


Turbulent fluctuations

tr =13ReM =3.5∙103

u2 [m2/s2]

v2 [m2/s2]


Anisotropy of the Reynolds tensor

tr =13ReM =3.5∙103

u2/ v2


Two-point correlation

tr =13ReM =3.5∙103

The longitudinal correlation function changes only slightly moving downstream, suggesting that the integral lengthscale increases with the streamwise coordinate.

However, the narrow field of view does not enable to estimate it with good confidence.


Longitudinal and transverse correlation functions

xurxurR

1111

tr =13ReM =3.5∙103

x/x*=0.47

This is not equivalent to the longitudinal two-point correlation function (the velocity component is always the streamwise one).


Longitudinal and transverse correlation functions

xurxurR

2222

tr =13ReM =3.5∙103

x/x*=0.47

This is not equivalent to the transverse two-point correlation function (the velocity component is always the crosswise one).


2nd order structure functions

21111 xurxurD

tr =13ReM =3.5∙103

x/x*=0.47

This is not equivalent to the longitudinal 2nd order structure function (the velocity component is always the streamwise one).


2nd order structure functions

tr =13ReM =3.5∙103

x/x*=0.47

This is not equivalent to the transverse 2nd order structure function (the velocity component is always the crosswise one).

22222 xurxurD


Turbulent kinetic energy decay

Space filling square fractal grids are characterized by an unusually fast decay: is it governed by an exponential law or a power law?

CBxu 2 C

Bx

eu

2

tr =13 ReM =11.5∙103

B=0.526 C=0.262R2=0.91

B=-0.481 C=-3.90R2=0.91


Turbulence decay

C

Bx

eu

2

Exponential decay

tr B C R2

8.5 -0.075 0.281 0.89

13 -0.074 0.244 0.90

17 -0.127 0.261 0.85

tr B C R2

8.5 0.596 0.265 0.90

13 0.526 0.262 0.91

17 0.523 0.269 0.90

ReM =3.5∙103

ReM =11.5∙103


Turbulence decay

ReM =3.5∙103

ReM =11.5∙103 CBxu 2

Power-law decay

tr B C R2

8.5 -0.956 -5.50 0.91

13 -0.962 -6.24 0.91

17 -0.991 -6.15 0.91

tr B C R2

8.5 -0.410 -3.55 0.92

13 -0.481 -3.90 0.91

17 -0.484 -3.88 0.87


Turbulence decay

C

Bx

eu

2ReM =11.5∙103


Turbulence decay

CBxu 2ReM =11.5∙103


Turbulence decay - remarks

It is not possible to conclude if the decay law is exponential or a power law by visual inspection or using the correlation factor of the fitting law;

More confidence in specifying the decay law can be obtained by increasing the measurement area in the streamwise direction;

Even if the decay was not exponential, it is still governed by a power law with an unusually high exponent!


Single lengthscale turbulence decay

W.K. George (Physics of Fluids, 1992) showed that single length scale power law solutions of the spectral equations for the decay of isotropic turbulence is possible;

W.K. George and H. Wang (Physics of Fluids, 2009) proposed a viscous solution (exponential decay) and an inviscid one (power law decay);

An important feature is that the turbulent statistics collapse if normalized with respect u2 and λT.


Single lengthscale turbulence decay

tr = 13 X = 385 mm


Taylor lengthscale measurement

DIRECT METHOD:2

2

xu

uT

INDIRECT METHOD:

211

2

2

xR

uT



DIRECT METHOD:

Very simple and straightforward application;

The noise effects are amplified by the derivative operator;The data need to be filtered: the choice of the filter intensity is critical.

2

2

xu

uT



INDIRECT METHOD:

211

2

2

xR

uT

The peak of the two-point correlation is fitted with a parabolic function:

2

22

11 21

T

rurR



INDIRECT METHOD:

211

2

2

xR

uT

The peak of the two-point correlation is fitted with a parabolic function:

2

22

11 21

T

rurR



INDIRECT METHOD:

211

2

2

xR

uT

According to Adrian and Westerweel (2011):

21111

~ rRrRE Idr ,



INDIRECT METHOD:

211

2

2

xR

uT

According to Adrian and Westerweel (2011):

21111

~ rRrRE Idr ,

At least the first 3 measurement points have to be excluded in the fitting, since we are using interrogation windows with 75% overlap.



A least square fitting may lead to a more accurate estimate of both the Taylor lengthscale and the turbulent fluctuations;

A Signal to Noise criterion can be introduced by considering the ratio of the estimate peak of the two-point correlation, and the one obtained by best fitting of the peak.

INDIRECT METHOD:

211

2

2

xR

uT

2

2

211

2

0~

u

uR

uSNR



2

2

211

2

0~

u

uR

uSNR



Direct method Indirect method

tr =13 ReM =3.5∙103



tr =13 ReM =3.5∙103 tr =13 ReM =11.5∙103


Dissipation rate

DIRECT METHOD:

ENERGY BALANCE:

212

222

211 123 SSS

x

qU

2

2

1

One can use the relations of the power law and exponential decay.

INDIRECT METHOD: 2

2

15T

u

One can use the Taylor lengthscale estimated with the indirect method.


Dissipation rate

tr =13

ReM =3.5∙103


Conclusion

PIV performances are assessed for measurements in nearly isotropic and homogeneous low-intensity turbulence; the results are in close agreement with the literature;

PIV complements pointwise measurement techniques by its capability of detecting inhomogeneity and anisotropy;

The results confirm the presence of a single lengthscale decay; it is not possible to conclude on the nature of the decay;

Fitting of the peak of the two-point correlation enables a more accurate estimate of the Taylor length scale and of the dissipation;

Three dimensional measurements (Tomo-PIV) are planned to get a better understanding of the underlying dynamics (QR pdf, dissipation, etc.).

Thank you for your attention

ACKNOWLEDGMENTS

This research was supported in part by Contract 79419-001-09, Los Alamos National Laboratory.

Documents

2 nd UK-JAPAN Bilateral and 1 st ERCOFTAC Workshop, Imperial College London