Turbulence, from onset to large scale structures

Björn Hof

Institute of Science and Technology Austria

Reynolds pipe experiments

O. Reynolds Phil. Trans. R. Soc. 1883

Increasing velocity

‚puffs‘ of turbulence

Turbulence is triggered by finite amplitude perturabtions Turbulence is observed at Re ~ 2000

However for an improved set up flow remained lamianr up to Re~13 000

Channel flow

Pipe flow (Re=2000)

Turbulent spots and stripes

Turbulence sets in despite the linear stability of the laminar flow.

Laminar and turbulent regions coexist

(Taylor) Couette flow

Dynamics of individual puffsDecay of turbulent puffs

Spreading of turbulent puffs

•Turbulent puffs decay suddenly after long tim•Memoryless process

P(t, Re)=exp(-t / τ (Re))

RE=2040: Turbulence becomes sustained at a non-equilibrium phase transition

Avila, Moxey, de Lozar, Avila, Barkley, Hof, Science 2011

Coupled map lattices:Kaneko Prog. Theoret. Phys.1985, Chate, Manneville PRL 1988Couette flow: S. Bottin, H. Chaté Eur. Phys J. 1998S. Bottin, F. Daviaud, P. Manneville, ODauchot,Europhys. Lett. 1998Manneville, PRE 2009Pipe flow:Moxey & Barkley, PNAS 2010

Spatio temporal intermittent regime

Re=2300

Direct numerical simulations by Marc Avila

Tim

e (D

/U)

Pipe axis / DPipe axis / D 0 150

Re=1900

Transition via spatio temporal intermittency:Coupled map lattice

time

spac

e

Kaneko Phys. Lett. A 1990

Directed Percolationtim

e

space

Single parameter: Probability P

Directed PercolationP < 0.64

Directed PercolationP < 0.64

Directed Percolation

Analogy to turbulencefirst suggested by Y. Pomeau 1986

P > 0.65

=0.276=-1.748 ∥=-1.841

Earlier Couette experimentTu

rbul

ent f

ract

ion

Length= 190 H

Wid

th=

35 H

Bottin, Daviaud, Manneville, Dauchot Europhys. Lett. 1998

The DP analogy• Replace P by Re• Laminar flow is the unique absorbing state• Time step ~ splitting/decay time• An active DP site is a puff (/stripe) including

laminar recovery region

Critical Exponents:1. Space correlation exponent:

Distribution of laminar gaps (spatial)

Best fit = -0.75DP: -0.748

Best fit = -0.84DP: -0.84

2. Time exponent:Distribution of laminar gaps in time

Cum

ulat

ive

Prob

abili

ty3. Critical exponent for theturbulent fraction

Re

Turb

ulen

t fra

ctio

n

Simulations by Liang Shi In collaboration with Marc Avila

Couette experimentsTaylor Couette apparatus Γθ = 5500 Γz = 16 and radius ratio= 0.998

Lem

oult,

Shi

, Avi

la, J

alik

ob, A

vila

& H

of, N

atur

e Ph

ysic

s 2

016

(arX

iv:1

504.

0330

4)

Re

Turb

ulen

t fra

ctio

n

The DP analogy• Replace P by Re• Laminar flow is the unique absorbing

state• Time step ~ splitting/decay time• An active DP site is a puff (/stripe)

Pipe flow?

Re=2020

Re=2060

Experiments: Mukund Vasudevan

Onset of sustained turbulence

Re=2020

Re=2060

Critical exponents?

Directed percolation Directed

percolationDirected percolation

Turb

ulen

t fra

ctio

n

# La

min

ar g

aps

(in s

pace

)

# La

min

ar g

aps

(in ti

me)

Re Gap size (space)

Gap size (time)

Other studiesChannel Flow: Sano Tamai Nature Physics 2016

Claim statistical steady state is assumed after 100 D/U (in pipes its order 108 D/U !!! )

Waleffe flow: Chantry, Tuckerman & Barkley JFM 2017

From localised to fully turbulent

From localised to fully turbulent

Nishi et al JFM 2008

(1973)

Fron

t sp

eed

Leading edge

Trailing edge

(Duguet, Willis & Kerswell JFM 2011)

From localised to fully turbulent

Re

Re

Barkley’s excitable media model(Barkley PRE 2011)

From localised to fully turbulent

Barkley, Song, Mukund, Lemoult, Avila & Hof Nature 2015

Emergence of Turbulence

Re=2300

Re

Laminar friction law

Re

What happens at larger Re?

Friction in pipes: early experiments

Blasius (1913): power law friction factor for

Darcy-Weisbach friction factor

Fit on experimental data

Friction in pipes: higher Reynolds numbers

Prandtl-von Karman

New data at high Re (Nikuradse,

1930)

Velocity log law proposed (von

Karman, 1930)

Friction factor formula from the

log law (Prandtl, 1932)Blasius

Friction in pipes: higher Reynolds numbers

Prandtl-von Karman

Blasius

New data at high Re (Nikuradse,

1930)

Velocity log law proposed (von

Karman, 1930)

Friction factor formula from the

log law (Prandtl, 1932)

von Karman’s view of power laws

“The resistance law is no power law,”Stockholm congress 1930 (von Karman):

Von Karman 1930:

(see Bodenschatz Eckert 2011)

Stockholm congress 1930 (von Karman):

It is known that over broad ranges the friction law can be approximated by functions of the type:

The exponent reduces with Re.

I believe this riddle has a simple explanation.

Power laws are approximations to a logarithmic function

Zagarola and Smits 1998 (◊), Swanson et al. 2002 (○) and Furuichi et al. 2015 (□)

Friction factor Deviation wrt Blasius (%)

Friction in the Blasius regime: state of the art

Zagarola and Smits 1998 (◊), Swanson et al. 2002 (○) and Furuichi et al. 2015 (□)

Available data is scattered and insufficient to draw conclusions

Friction factor Deviation wrt Blasius (%)

Friction in the Blasius regime: state of the art

Pressure drop measurements

Temperature controller unit

Servomotor driven syringe pump

Precision bore glass pipe

Differential pressure sensor

Run length friction factor accuracy `

Friction factor Deviation wrt Blasius (%)

Friction in the Blasius regime: results

Direct Numerical Simulations

streamwise azimuthal radial (wall) radial (max)

Pseudo-spectral code

8th

order finite differences in radial direction (4th

order at the wall)

Domain length

Resolution (in wall units):

A power law emerges for

Friction factor Deviation wrt Blasius (%)

Friction in the Blasius regime: resultsDNS ( ) + experiments ( )

Kolmogorov theory applied to pipe flow

● Local isotropy

● First similarity hypothesis

Kolmogorov theory applied to pipe flow

● Local isotropy

● First similarity hypothesis

Can we apply the theory at moderate ?

Kolmogorov theory applied to pipe flow

● Local isotropy

● First similarity hypothesis

● Schumacher et al. (2014):

● Small-scale universality in different flows at low

● Developed inertial range not required

Can we apply the theory at moderate ?

Deriving a friction law from Kolmogorov theory

● Start by multiplying the scaling by (see also: Gioia and Chakraborty, 2006)

Deriving a friction law from Kolmogorov theory

● Start by multiplying the scaling by

● Rewrite and compare with the wall shear stress

(see also: Gioia and Chakraborty, 2006)

Deriving a friction law from Kolmogorov theory

● Start by multiplying the scaling by

● Rewrite and compare with the wall shear stress

● Let

(see also: Gioia and Chakraborty, 2006)

Deriving a friction law from Kolmogorov theory

• -1/4 power law simplest scaling from Kolmogorov theory

What happens at larger Re ?

Velocity RMS in wall units

Inner peak location constant

Progressive development of a

second peak

Second peak associated with

large scale motions

How do the increasing LSMs affect the friction factor?

Contribution of LSM to the friction factor

● Fukagata et al. 2002

Contribution of LSM to the friction factor

● Fukagata et al. 2002

● Compute contributions of LSM and SSM (Chin et al., 2014)

Contribution of LSM to the friction factor

● Fukagata et al. 2002

● Compute contributions of LSM and SSM (Chin et al., 2014)

● Filter cutoff wavelength

Contribution of LSM to the friction factor

● Fukagata et al. 2002

● Compute contributions of LSM and SSM (Chin et al., 2014)

● Filter cutoff wavelength

Contribution of LSM to the friction factor

Contribution of LSMs to the friction factor

: LSMs dominate and friction deviates from a power law

Jose M. Lopez, Davide Scarselli, Balachandra Suri & BH in preparation

Conclusions

Friction factor scaling starts as a power law

Blasius scaling can be rationalized from Kolmogorov theory

Deviation of friction related to increasing dominance of LSMs in

turbulent momentum transport