TURBULENCE GENERATEDBY FRACTAL GRIDS

D. Hurst, R.E. Seoud & J.C. Vassilicos

Department of Aeronautics and

Institute for Mathematical Sciences

Imperial College London, U.K.

. – p.1/45

MOTIVATIONS

1. Many applications in environmental and geophysicalflows as well as in industry

of fractal-forced or fractal-generated turbulence,

e.g. polydisperse droplets/particles in turbulent carrier fluidthat are large enough to force the turbulence over a widerange of scales corresponding to a wide range of particlewake sizes (combustion applications, ocean wind-wavesprays); turbulent flows through trees, over plant canopies,over multi-sized breaking ocean waves, etc; various novelmixing devices for the process, oil and other industries aswel as novel ventilation systems (recent patents by ImperialCollege London) which can impact on the environment byrequiring less power to mix...

AT THE VERY LEAST, A REFERENCE LABORATORYEXPERIMENT IS REQUIRED . – p.2/45

MOTIVATIONS

2. How to create ideal turbulence experiments with(i) a very wide range of outer-to-inner scales(ii) fully controlled conditions in the laboratory

(iii) the possibility to accurately measure down to thesmallest scales

3. Better: how to tamper with the turbulence in thelaboratory?

Various theories exist where the exponents p, q inE(k) ∼ k−p, εL/u′3 ∼ Req are determined by

to one or many fractal dimensions of a fractal/multifractal,spiral/multispiral field:

is it possible to modify E(k) ∼ k−p and/or εL/u′3 ∼ Req awayfrom p = 5/3 and q = 0 by tampering with the fractal/spiral

field and changing these dimensions?. – p.3/45

MOTIVATIONS

4. Effects on drag properties?

5. How does a turbulence decay when it is generated bycreating many eddies of many different sizes at once?

6. How does a turbulent flow scale when it is generated bya fractal which has its own intrinsic scaling?

7. Multiscale flow control? in the present case, passive.

. – p.4/45

Wind tunnels

0.912m2 width; test section 4.8m; max speed 45m/s;background turbulence ≈ 0.25%.

0.462m2 width; test section ≈ 4.0m; max speed 33m/s;background turbulence ≈ 0.4%.

. – p.5/45

FRACTAL CROSS GRIDS

. – p.6/45

FRACTAL I GRIDS

. – p.7/45

FRACTAL SQUARE GRIDS

. – p.8/45

Three families of fractal grids

Three fractal-generating patterns

The fractal grids are totally characterised by(i) the number of fractal iterations N

(ii)the lengths Lj = RjLL0 and thicknesses tj = Rj

t t0,j = 0, ..., N − 1

(iii) the number Bj of patterns at iteration j: always here,B = 4 and RL ≤ 1/2, Rt ≤ 1

. – p.9/45

Important parameters

Fractal dimension of fractal perimeter: Df = logBlog(1/RL) .

1 ≤ Df ≤ 2.WE FIND THAT BEST MEAN FLOW HOMOGENEITY ISACHIEVED FOR MAXIMUM Df i.e. Df = 2:Thickness ratio tr ≡ t0/tN−1 ≡ tmax/tmin. (Note tr = R1−N

t .)WE FIND THAT THE TURBULENCE INTENSITYINCREASES WITH BOTH PRESSURE DROP (WHENINCREASING BLOCKADGE RATIO) AND THICKNESSRATIO tr (KEEPING BLOCKADGE RATIO CONSTANT).Effective mesh size Meff = 4T 2

P

√1 − σ where T = tunnel

width, P = fractal perimeter, σ = blockadge ratio.WE FIND THAT THE TURBULENCE SCALES WITH Meff

IN THE CASE OF CROSS AND I GRIDS. Statisticalhomogeneity can be as good as for classical grids, butfurther dowstream in multiples of Meff . . – p.10/45

Minimal complete description of grids

Cross grids require 4 parameters: e.g. T,N, tmax, Rt.(T = Lmax, RL = 1/2 hence Df = 2.)

I and Square grids require 5 parameters: e.g.

T,N,Lmax, tmax, tmin. (T ≈ Lmax1−RN

L

1−RL.)

VARIOUS WIND TUNNEL TESTS WERE CARRIED OUTWITH A NUMBER OF GRIDS FROM EACH FAMILY.GROUPS OF GRIDS FROM GIVEN FAMILIES WERECHOSEN SO AS TO HAVE THE SAME VALUES OFPARAMETERS BUT ONE, IN ORDER TO DETERMINETHIS ONE PARAMETER’S EFFECT WHEN EVERYTHINGELSE IS KEPT CONSTANT:E.G. KEEPING BLOCKADGE RATIO, AND/OR NUMBEROF ITERATIONS AND/OR Meff AND/OR tmin CONSTANT,ETC, ETC, ETC...

. – p.11/45

I grids: N = 6 and Df = 1.98, 1.87, 1.79, 1.68

Equal σ = 25 %, tmin = 1mm, T = 0.91m tunnel.

. – p.12/45

I grid: N = 5 and Df = 2.0

σ = 31%, tmin = 4mm, T = 0.91m tunnel.

. – p.13/45

Df = 2 fractal I grids; T = 0.46m tunnel

Equal N = 4, σ = 25%, Meff between 36mm and 37mm.tr = 2.5, 5.0, 8.5, 13.0, 17.0

. – p.14/45

Results: turbulence decay

Many possible ways to collapse the I grid data have beentried. It is found that

(u′/U)2 = trC∆P (T/Lmax)2fct(x/Meff )

collapses the turbulence decay data generated by all fractalI grids in both wind tunnels.

0 50 100 150 2000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

x/Meff

(u’/U

) no

rmal

ised

Df = 1.98

Df = 1.87

Df = 1.79

Df = 1.68

Df = 2.00

tr = 2.5

tr = 5.0

tr = 8.5

tr = 13.0

tr = 17.0

. – p.15/45

Df = 2, σ = 25% fractal square grids

and equal Meff ≈ 2.6cm, Lmax ≈ 24cm, Lmin ≈ 3cm, N = 4,T = 0.46m.

BUT tr = 2.5, 5.0, 8.5, 13.0, 17.0

. – p.16/45

Profiles at x = 3.25m in T = 0.46m tunnel

−0.5 0 0.50.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

y/T

U/U

∞

tr = 2.5

tr = 5.0

tr = 8.5

tr = 13.0

tr = 17.0

−0.5 0 0.50

1

2

3

4

5

6

y/T

u’/U

(%

)

−0.5 0 0.50

1

2

3

4

5

6

y/T

v’/U

(%

)

. – p.17/45

Results: turbulence intensity

0 1 2 3 40

2

4

6

8

10

x (m)

u’/U

(%

)

tr = 2.5

tr = 5.0

tr = 8.5

tr = 13.0

tr = 17.0

0 1 2 3 40

2

4

6

8

10

x (m)

v’/U

(%

)

0 50 100 150 2000

2

4

6

8

10

(x Lmin

) / (tmin

T)

u’/U

(%

)

tr = 2.5

tr = 5.0

tr = 8.5

tr = 13.0

tr = 17.0

0 50 100 150 2000

2

4

6

8

10

(x Lmin

) / (tmin

T)

v’/U

(%

)

tr = 2.5

tr = 5.0

tr = 8.5

tr = 13.0

tr = 17.0

xpeak = 75 tminTLmin

(Hurst & V PoF 2007) but xpeak = 1.2L2

max

tmax

(Mazellier, Bruera & V (to appear)) . – p.18/45

Isotropy collapse using xpeak

xpeak helps collapse u′/v′ as fct of x

0 1 2 3 40.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

x (m)

u’/v

’

tr = 2.5

tr = 5.0

tr = 8.5

tr = 13.0

tr = 17.0

0 0.5 1 1.5 2 2.5 30.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

x/xpeak

u’/v

’

tr = 2.5

tr = 5.0

tr = 8.5

tr = 13.0

tr = 17.0

T = 0.46m tunnel with U∞ = 10m/s.

. – p.19/45

Results: power-law turbulence decay?

How does the Taylor microscale evolve?

0 1 2 3 40

1

2

3

4

5

6

7

8

x (m)

λ (m

m)

tr = 2.5

tr = 5.0

tr = 8.5

tr = 13.0

tr = 17.0

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

8

x/xpeak

λ (m

m)

tr = 2.5

tr = 5.0

tr = 8.5

tr = 13.0

tr = 17.0

T = 0.46m tunnel and U∞ = 10m/s

. – p.20/45

Results: integral length-scales

0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

0.06

x/xpeak

L u (m

)

tr = 2.5

tr = 5.0

tr = 8.5

tr = 13.0

tr = 17.0

0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

0.06

x/xpeak

L v (m

)

tr = 2.5

tr = 5.0

tr = 8.5

tr = 13.0

tr = 17.0

At U∞ = 10m/s, for all x � xpeak and for all grids, λ ≈ 6mm,Lu ≈ 48mm, Lv ≈ 22mm (about Lu/2 as required byisotropy) all � T = 0.46m.

. – p.21/45

Exponential turbulence decay at x � xpeak

u′2 = u′2peakexp[−(x − xpeak)/lturb]

wherexpeak = 75 tminT

Lminand lturb = 0.1λ0

Uλ0

ν

0 1 2 3 44

5

6

7

8

9

10

11

x (m)

ln(U

/u’)2

tr = 2.5

tr = 5.0

tr = 8.5

tr = 13.0

tr = 17.0

0 1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

x (m)

u’2 (

m/s

)2

tr 17

exponential theory

. – p.22/45

Comparison with classical grid turbulence

(measurements taken by N. Mazellier)

0 50 1000

100

200

300

400

(x − x0) / M

eff

Rλ

3 3.5 4 4.5 5−12

−11.5

−11

−10.5

−10

−9.5

−9

ln( (x − x0) / M

eff)

ln(

(λ /

T)2 )

3 3.5 4 4.5 5−9

−8.5

−8

−7.5

−7

−6.5

−6

ln( (x − x0) / M

eff)

ln (

<u2 >

/ U

2 )

0 50 100−6

−5.5

−5

(x − x0) / M

eff

ln (

<u2 >

/ U

2 )

. – p.23/45

Dissipation during exponential u′2 decay

u′2 = u′2peakexp[−(x − xpeak)/lturb]

and

Lu, Lv independent of x

ARE INCOMPATIBLE WITH

−32U d

dxu′2 = ε = Cεu′3/Lu

with Cε a universal constant.All the results presented in what follows have been obtainedin the decay region (x > xpeak) of the turbulence generatedby fractal square grids in the T = 0.46cm tunnel.

. – p.24/45

In fact, Lu, Lv, λ, Lu/λ and Lv/λ

are independent of x, tr and U∞ during decay.

100 200 300 400 500 600 700 800 900 10005

10

15

20

25

Reλ

L 11 /

λ

L11

/ λ v Reλ

7 ≤ Uinf

/ m / s ≤ 19

Grids = tr17 , tr 13 and tr 8.5

tr 17 7.0 m/s tr 17 10.5 m/s tr 17 16.2 m/s tr 17 19 m/s tr 13 7 m/s tr 13 16.3 m/s tr 13 19 m/s tr 8.5 7.3 m/s tr 8.5 16 m/s

Hence, ε = 15ν u′2

λ2 = 15Re−1λ

u′3

λ ∼ Re−1λ

u′3

Lu

VERY DIFFERENT FROM TAYLOR-KOLMOGOROVSCALING WHERE ε ∼ u′3

LuAND Lu/λ ∼ Reλ

. – p.25/45

εLu/u′3 ∼ Re−1

λ

100 200 300 400 500 600 700 800 900 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Reλ

Cε

Cε ( ε.L

11/u

rms3 ) v Re

λ

7 ≤ U

inf / m/s ≤ 19

Grids = tr17 , tr 13 and tr 8.5

tr 17 7.0 m/s tr 17 10.5 m/s tr 17 16.2 m/s tr 17 19 m/s tr 13 7 m/s tr 13 16.3 m/s tr 13 19 m/s tr 8.5 7.3 m/s tr 8.5 16 m/s

Cε = 140 /Reλ

Measurements taken on the centreline at x/xpeak betweenabout 1 and 3, i.e. x/Meff between about 50 and 110 (end

of test section). NOTE HIGH Reλ VALUES IN SMALLTUNNEL. . – p.26/45

Statistical homogeneity at x > xpeak

0 2 4 6 8 10 12 14 16 18

0.4

0.6

0.8

1

1.2

1.4

1.6

Ulocal

/ Uinf

v y / cm U

inf = 10.5 m/s, 180 ≤ x/cm ≤ 370

Grid =tr17

y / cm

Ulo

cal /

Uin

f

370320300280210180

. – p.27/45

Statistical homogeneity at x > xpeak

0 2 4 6 8 10 12 14 16 18

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

urms

/ Ulocal

v y / cm U

inf = 16.2 m/s, 180 ≤ x / cm ≤ 370

Grid = tr 17

y / cm

urm

s / U

loca

l

180210280300320370

0 2 4 6 8 10 12 14 16 18

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

vrms

/ Ulocal

v y / cm U

inf = 16.2 m/s , 180 ≤ x / cm ≤ 370

Grid = tr 17

y / cm

vrm

s / U

loca

l

180210280300320370

. – p.28/45

Statistical isotropy at x > xpeak

10−4

10−3

10−2

10−1

100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cxy

( |E12

(k)| 2 / [E11

(k).E22

(k)] ) v kη U

inf = 16 . 2 m/s , CL data, Grid = tr 17

kη

Cxy

, Cxy

(45

)

370320300280210180

Cxy

(45)

Coherence spectrum at various x positions on thecentreline y = 0. Coherence spectra are very much thesame off centreline at y = 3cm and y = 6cm.

. – p.29/45

Statistical local isotropy at x > xpeak

150 200 250 300 350 400 4500.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

K1 v Reλ

Uinf

= 10.5 m/s and 16.2 m/s

Grid = tr17, 280 ≤ x / cm ≤ 370 K

1

Reλ

0 cm 10.5 m/s3 cm 10.5 m/s6 cm 10.5 m/s 0 cm 16.2 m/s3 cm 16.2 m/s6 cm 16.2 m/s

Derivative ratio K1 ≡ 2 < (∂u∂x)2 > / < (∂v

∂x)2 > as function ofReλ at locations (x, y) downstream from the tr = 17 fractalgrid where x is larger than 2xpeak and y = 0, 3, 6cm.

Local isotropy implies K1 = 1. . – p.30/45

εLu/u′3 ∼ Re−1

λ at x > xpeak and |y| < Lmax/2

100 200 300 400 500 600 700 800 900 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Reλ

Cε

Cε ( ε.L11

/urms3 ) v Reλ

7 ≤ Uinf

/ m/s ≤ 16, CL + 3 cm / + 6 cm

Grids = tr17 , tr 13 and tr 8.5

tr 17 7.0 m/s CL + 3cm tr 17 10.5 m/s CL + 3 cm tr 17 16.2 m/s CL + 3 cm tr 17 7.0 m/s CL + 6 cm tr 17 10.5 m/s CL + 6 cm tr 17 16.2 m/s CL + 6 cm tr 13 16.3 m/s CL + 3 cm tr 13 16.3 m/s CL + 6cm tr 8.5 16 m/s CL + 3 cm tr 8.5 16 m/s CL + 6 cm

Cε = 143 / Reλ

Measurements taken at x/xpeak between about 1 and 3, i.e.x/Meff between about 50 and 110 (end of test section), at y

positions between -12cm and +12cm (Lmax = 24cm,Lu ≈ 5cm) . – p.31/45

Does Kolmogorov scaling hold here?

We still get a power-law range where E11(k1) ∼ k−5/31 at

high enough Reλ even though ε is Reλ-dependent!But we can collapse spectra E11(k1) at different x with onlyone length-scale: e.g. E11(k1) = u′2Luf(k1Lu).

10−1

100

101

102

103

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

E11

(k) / urms2 L

11 v kL

11

Uinf

= 16.2 m/s, CL data

Grid = tr 17

kL11

E11

(k)

/ urm

s2

L11

370320300280210180

(kL11

)−1.67

. – p.32/45

or E11(k1) = u′2λf (k1λ)

10−2

10−1

100

101

102

103

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

E11

(k) / urms2 λ v kλ

< Uinf

> = 16.2 m/s, CL data

Grid = tr 17

kλ

E11

(k)

/ urm

s2

λ

370320300280210180

(kλ)−1.67

. – p.33/45

Non-Kolmogorov -5/3

The energy spectrum of decaying turbulence generated byspace-filling fractal square grids scales with only onelength-scale l(x), i.e. E11(k1) = u′2lf(k11l).

This implies that Lu ∼ l and λ ∼ l, hence L ∼ λ asobserved.

This also implies that ε ∼ Re−1λ u′3/Lu as also observed.

And it also implies that in the power-law range, if a -5/3

spectrum exists, then E11(k1) ∼ (u′3

Lu)2/3k

−5/31 instead of

E11(k1) ∼ ε2/3k−5/31 .

There exist fractal, i.e. multiscale, generators of turbulencewhich lock the turbulence into a single length-scale! Yet, the-5/3 is present even though the dissipation anomaly is not.

. – p.34/45

Vortex Stretching?

The nonlinear rate of change of the enstrophy results fromvortex stretching and equals < ω · sω >.

In isotropic homogeneous turbulence,

< (∂u∂x)3 >= − 2

35 < ω · sω >

and< (∂u

∂x)2 >= 115 < ω2 >.

Hence, the derivative skewnessS ≡< (∂u

∂x)3 > / < (∂u∂x)2 >3/2

is a normalised dimensionless measure of the averagevortex stretching rate and can be obtained from a single hotwire if use is made of Taylor’s frozen flow hypothesis.

. – p.35/45

Vortex stretching in single-scale turbulence

The scale-by-scale energy balance∂∂tE(k, t) = T (k, t) − 2νk2E(k, t)

implies that∫∞

0 k2T (k)dk is the rate of change of theaverage enstrophy

∫∞

0 k2E(k)dk as a result of nonlinearinteractions.

Hence, in isotropic homogeneous turbulence,

S = − 235(15

2 )3/2R

∞

0k2T (k)dk

(R

∞

0k2E(k)dk)3/2

= −(135/98)1/2R

∞

0k2T (k)dk

(R

∞

0k2E(k)dk)3/2

which can be evaluated from the scale-by-scale energybudget and the single-scale spectrum propertyE(k, t) = u′2λf(kλ) to give:

S = ARe−1λ + B

u′

ddtλ

in terms of two dimensionless constants A and B. . – p.36/45

Mean vortex stretching drops as Reλ grows.

. – p.37/45

but as S ∼ Re−0.15λ rather than S ∼ Re−1

λ

This apparent -0.15 scaling is caused by the smalltime-dependence of λ. Indeed

S = ARe−1λ + B

u′

ddtλ

which can be recast asSReλ = A + B Ulocal

νddxλ2

if use is made of Taylor’s frozen flow hypothesisUlocaldt = dx.

This slow increase of λ with x, if fitted by λ ∼ (x − x0)s with

0 < s < 1/2, implies a stretched exponential decay of u′2

instead of the exponential form mentioned earlier. We leavethis correction for future detailed measurements andstudies.

. – p.38/45

SReλ = A + BUlocalν

ddxλ

2 for grid tr = 17.0

. – p.39/45

SReλ = A + BUlocalν

ddxλ

2 for grid tr = 13.0

. – p.40/45

SReλ = A + BUlocalν

ddxλ

2 for grid tr = 8.5

. – p.41/45

Can intermittency grow with L/λ constant?

F ≡< (∂u∂x)4 > / < (∂u

∂x)2 >2

. – p.42/45

CONCLUSIONS

Df , tr and Meff are important fractal grid parameters. Besthomogeneity is obtained for Df = 2. For space-filling fractalI and square grids, homogeneity can be further improved byincreasing tr. In all cases of fractal grids, turbulenceintensity and Reynolds number can also be increased byincreasing tr.Turbulence decay, fractal I grids:

(u′/U)2 = trC∆P (T/Lmax)2fct(x/Meff )

Turbulence decay, fractal square grids, at x � xpeak:

u′2 = u′2peakexp[−(x − xpeak)/lturb]

wherexpeak = 75 tminT

Lminand lturb = 0.1λ0

Uλ0

ν

u′2peak increases linearly with tr. The Taylor microscale

λ = λ0 and the integral length scales are independent of trand U∞ and remain approx constant during decay. . – p.43/45

CONCLUSIONS

In the decay region of space-filling fractal square grids theturbulence is approximately homogeneous and locallyisotropic and such that (see W.K. George, PoF 1992):

E11(k1) = u′2Luf(k11Lu) = u′2λf(k11λ)

L/λ = Const independent of x, tr and U∞

ε ∼ Re−1λ u′3/Lu

A -5/3 power-law range exists where E11(k1) ∼ (u′3

Lu)2/3k

−5/31

instead of E11(k1) ∼ ε2/3k−5/31 .

. – p.44/45

CONCLUSIONS

Furthermore, in this decay region of space-filling fractalsquare grids where turbulence is approximatelyhomogeneous and locally isotropic, the turbulence is alsosuch that(i) vortex stretching decreases in the mean as the Reynoldsnumber is increased(ii) and “intermittency” does not grow but remains constantwith increasing Reynolds number.

It is possible to tamper with the deepest properties ofhomogeneous isotropic turbulence: the dissipationanomaly, vortex stretching and intermittency. This points atnew possibilities for turbulence control. Also, if you cantamper with something, you can start understanding it.

. – p.45/45