The mean velocity profile in the smooth wall turbulent

boundary layer : 1) viscous sublayer

the velocity varies linearly, as a Couette flow

(moving upper wall).

Thus, the shear stress is constant: 𝜏0

𝜏 = 𝜇𝑑𝑢

𝑑𝑦 𝑢 =

𝜏0 𝑦

𝜇

scaling near wall turbulence

We can define a velocity scale u* = 𝜏

𝜌 [m/s] characteristic of near wall turbulence

u* = shear velocity or friction velocity

we can rewrite the linear profile in the viscous sublayer as

where 𝜐

𝑢∗ is a length scale (very small, remember 𝜐 =O(10-5 10-6) m2/s,

while u* is a fraction (~10-20%) of the undisturbed velocity U0

𝑢

𝑢 ∗=

𝑦𝑢 ∗

𝜐

we already have 2 velocity scales:

1) u*

2) U0

How many length scale ?

1) 𝜐

𝑢∗

2) 𝛿

𝛿

boundary

layer

height

The smooth wall TBL

viscous sublayer continued

How thick is the viscous sublayer ?

it depends on the flow...

as u* and 𝜐 define the viscous length scale,

we can quantify the extension of the viscous sublayer

in terms of multiples of the viscous scale (viscous wall units)

𝛿𝜐 = 5 𝜐

𝑢∗

Note that as u* 𝛿𝜐 : the viscous sublayer becomes thinner

Note: roughness protrusion (fixed physical scale) may emerge from the viscous sublayer and change the

near wall structure of the flow

𝛿𝜐

The mean velocity profile in the smooth wall

turbulent boundary layer : 2) the logarithmic region

here is another velocity scale:

the standard deviation

or r.m.s. velocity

velocity scale

of the energy containing eddies

The mixing length theory:

fluid particles with a certain momentum are displaced throughout the

boundary layer by vertical velocity fluctuation.

This generate the so called Reynolds stresses 𝜏 = −𝜌𝑢′𝑣′

𝜏 = −𝜌𝑢′𝑣′

If we know the stress, we can obtain by integration the velocity profile

mixing length assumption (Prandtl: 𝑢′ = 𝑙 𝑑𝑢

𝑑𝑦 )

What does it mean?

A displaced fluid parcel (towards a faster moving fluid) will induce

a negative velocity u’ ~ v’ such that 𝜏 = −𝜌𝑢′𝑣′ = 𝜌𝑙2 𝑑𝑢/𝑑𝑦 2

l represent the scale of the eddy responsible for such fluctuation

very important:

we also assume that the size of the eddies l varies with the height

l=ky: very reasonable, farther from the wall eddies are larger

we thus have 𝜏 = −𝜌k2 y2 𝑑𝑢/𝑑𝑦 2

with u* = 𝜏

𝜌

integrating we obtain :

𝑢

𝑢∗=

1

𝑘ln

𝑦𝑢∗

𝜐 +C

Logarithmic law of the wall

where u* depends on the flow and the surface

k is the von Karman constant(?)=0.395-0.415 (k=0.41 is a good number)

C is the smooth wall constant(?) of integration (C=5.5 is a good number)

note that for a rough wall boundary layer 𝑢

𝑢∗=

1

𝑘ln

𝑦

𝑦0

where y0 is the aerodynamic roughness length:

it is a measure of aerodynamic roughness, not geometrical (surface) roughness

relating with y0 is complicate

The mean velocity profile: where is it valid ?

from about 60 viscous wall units to about 15% of he boundary layer height

it makes sense that the

extension of the log layer

has to be determined by both

inner scaling and outer scaling

What is turbulence ?

turbulence is a state of fluid motion where the velocity field is :

highly 3D, varying in space and time , hardly predictable, varying

over a wide range of scales non Gaussian, anisotropic but

somehow statistically organized

coherent structures +

Coherent structures in wall turbulence

Short term goal: understand and control near wall processes

(relevant for drag, lift, particle resuspension, near surface processes)

Long term goal: shift turbulent closure to larger scales, in order to solve

large domain accurately (atmosphere, rivers, oceans)

Smallest scale of the flow: kolmogorov scale

(in the near atmosphere about 1mm)

Largest scale of the flow: several times the boundary layer height

(in the atmosphere may go up to O(1-10 Km )

There are 6-7 orders of magnitude !

However IF, we understand how turbulent structures behave and IF

these structures truly play a major role (statistically) on momentum,

scalar and energy fluxes, mixing, etc. ...

Then we could propose low dimensional models, smart closures,

control systems

Flow visualization and sketches

Kline 1967 (near wall streaks) (log and outer layer)

F

L

O

W

(1981)

flow

flow

(towards the screen)

Acarlar and Smith, 1987, downstream of a fixed hemisphere

downstream of a low momentum

fluid ejection

Laminar flow upstream

flow

Robinson 1991

Kline 1967 Flow visualization (hydrogen bubbles, flow markers)

Hairpin vortex detection: track of a strongly 3D structure on the

2D streamwise - wall normal laser sheet: Adrian et al 2000

vortex Q2 event

Shear layer

The Biot-Savart law is used to calculate the velocity

induced by vortex lines.

For a vortex line of infinite length, the induced velocity

at a point is given by:

V = 2 πΓ /d where

Γ is the strength of the vortex

d is the shortest distance from a point P to the

vortex line

For a arch-like vortex line, there is a combined

induction towards its center

(ejection of low momentum fluid u’v’ Q2 event

• Single hairpin vortices can explain the observed

features of low and high speed streaks, bursting

phenomena and lift up of structures (viscous &

buffer layer)

• What is still missing so far is the outer layer,

• Structures were observed to form bulges with

ramp-like features.

A brief summary . . .

Numerical Simulation (Zhou, Adrian et al. 1996, 1999)

isovorticity surface

Self sustaining mechanism

(see also Waleffe 1990) and

vortex alignment

Limitation : low Re

with initial perturbation

Experimental evidence of hairpin packets

in smooth wall turbulence (Adrian, Meinhart, Tomkins JFM, 2000)

Instantaneous flow fields: U-Uc (convection velocity)

Vortex marker: swirling strength

Ramp packet

Q2

Q4 Q4

Detection of zones of uniform

momentum associated to the

streamwise alignment of

hairpin vortex: mutual

induction of Q2 event

Vortex identification Okubo-Weiss parameter Swirling Strength analysis

zu

xw

s

zw

xu

n

sn

S

S

where

SSS

:

222

22

zSQ

z

w

x

wz

u

x

u

u

From the local velocity

gradient tensor

Imaginary eigenvalues

cicrc i

We select the region

where

0ciSee also Chong & Perry, 1990

Jeong and Hussain 1995

Numerical simulation Adrian, PoF 2008 multimedia appendix

Flow visualization,

flow

Statistical Signature

1)Relevance

2)Physical mechanisms

3)Connection with quadrant analysis (Lu &

Willmart, 1973, Wallace 1972, Nezu &

Nakagava 1977)

4)Vortex identification in 2D and 3D

5)Zones of uniform momentum

6)Consistency with observed resuspension

events (strong correlation between c’w’ and u’w’ events)

Besides instantaneous realizations…

Is it possible to obtain some quantitative

information about turbulent structures ?

2 point correlation

vu, ji,for

y'σyσ

',,ρ

d)(normalize

t coefficienn correlatio

', ,',,

n tensorcorrelatiopoint 2

ji

ij

*

yyrR

yrxuyxuyyrR

xij

xjixij

Linear stochastic estimate

Estimate of the flow field

Statistically conditioned

To the realization of a

known event :

1) II quadrant (u < 0, v > 0)

2) IV quadrant (u > 0, v < 0)

3) Vortex

identified by the swirling strength :

complex part of the eigenvalue of

the local velocity gradient tensor.

See also Proper

Orthogonal

Decomposition

(Holmes & Lumley )

A B

Comparison A B center

(reduction of the streamwise lengthscale:

lost of coherence within the structures

of the packets)

(see also Krogstad e Antonia 1994 rough wall)

Two point correlation

streamwise velocity fluctuation

Comparison A B center

B

A

Linear Stochastic Estimate:

Question:

What is the average flow field statistically conditioned to the realization of a

vortex with a spanwise axe (identified as the signature of the hairpin vortex

On the laser sheet)?

The best (linear) estimate is given by

Adrian, Moin & Moser, 1987

Adrian 1988, Christensen 2000

Note:

Information about conditioned

probabilistic variables are obtained from

unconditioned statistical moments

y,rxu y'x,λ'xu xλ

),(xcon

xλxλ xλ

'xu xλ xλL xλ'xu

xj

j

jj

yx

Linear stochastic

Estimate :

known event

assumed at a fixed y’

Flow field obtained from a statistical

analysis (conditioned to the realization of

a E event) E

E

See Christensen 2000

Kim and Adrian 1999

Spanwise alignment of hairpin structures leading to

long coherent regions of uniform momentum Kim & Adrian 1999

VLSM Contribution :

turbulent kinetic energy and Reynolds stresses

Guala et al. 06

Pipe : Guala et al, 06

channel:

• Pipe flow

• Turbulent Boundary layers

• channel flow

Turb. B. layer: Balakumar, (2007)

Net force exerted by Reynold stress in the mean momentum equation

• Large scale motion participate significantly to the Reynolds stress, thus contribute not only to TKE but also to TKE production.

• In terms of momentum balance, close to the wall, VLSM push the flow forward, while smaller scales slow down the flow.

• Such features are observed for turbulent pipe, channel and boundary layers flows

A brief summary . . .

Marusic & Hutchins 2008

Atmospheric Surface Layer

Reτ=O(106)

Hutchins & Marusic 2007

Large scale influence on the near

Wall turbulence intensity:

Amplitude modulation

Note that in different research field

some type of very large scale

motions are addressed with

different names

e.g. streamwise rolls (atmospheric

science) or secondary current

(river hydraulics)

Low Re

High Re

High Re

Low Re

VLSM : A visual inspection

Lekakis 88, Guala et al. 06 Metzger et al. 07; Guala, Metzger, McKeon 08

PIPE FLOW ATMOSPHERIC

SURFACE LAYER (ASL)

Summary: dominant structural populations in

turbulent boundary layers

(above the near wall streaks / roughness sublayer, i.e.

where vortex structure organization really matters)

Ramp like structures (widely accepted) hairpin vortex (exact shape? transitional?)

Schlatter & Orlu 2010

Adrian et al .

2000, 2007

Hommema 2001

Ret = 5884

Ret = 14380

Ret = 106

Ret = 520

Ret = 106

Ret = 106

Very large

scale motion

Super-

structures

Kim & Adrian 99

Balakumar 07

Guala et al. 06, 10, 11

Hutchins et al 2007, 2013

Monty et al 2009

Mathis et al. 2009

Marusic, et al. 2011

micrometeorological process

e.g. surface hoar formation and related avalanche risk (Stoessel et al. WRR, 2010)

and wind turbine siting optimization and turbine lifetime

e.g. Howard et al. WE, 2015-16

How far can we represent the atmospheric

surface layer in wind tunnel study ?

Among the many relevant issues: 1) Inner – Outer Scale separation

In general, scaling of structural types

2) Thermal stability effects

The scientific and engineering relevance of large scale structures

is due to their dominant contribution to the Reynolds stresses and

to TKE production:

1) About scaling and structural

types

1) how far from the wall do we expect

hairpin and hairpin packets to extend ?

2) what is the correct scaling ?

do we expect ramp like coherent

structures to extend

up to =50m in the ASL ?

At high Reynolds we do not know for sure, mixed

evidence Morris 2007, Hommema 2001, Marusic 2007

Ramp structures YES – single hairpins ???

At moderate-low Reynolds number they scrape the

boundary layer height (Adrian 2000, Christensen

2001, Adrian 2007, Wu 2009)

Adrian et al, 2000

Comparison between

the Atmospheric Surface Layer (ASL)

the flat plate turbulent boundary layer (TBL)

Re = 5 * 105

Re = 4 * 102

Z [

m]

y/ = 0.06

T UMAX / = 10

y/ = 1.2

T UMAX / = 30

Lehew et al .2011, 2013,

Guala et al .2010, 2011

Experimental methods, Statistics, Spectra

• ASL: 29 simultaneous hotwire probes (1 velocity component , coarse vertical resolution, time resolved)

Symbols

Metzger and Klewicki, 2001-2002

Guala Metzger McKeon 2009

TBL: Production - dissipation ASL: Production - dissipation

Two point correlation of the streamwise velocity

fluctuation:

the signature of ramp like structures.

ASL

TBL

y/ = 0.0003 y/ = 0.01

y/ = 0.07 y/ = 0.24

However, recent evidence may provide a different view (Hutchins et al. 2013)

Zref Condition at 2.14m , corresponding to z+~ 104

From conditional average: evidence of dominant roll mode

It is legitimate to ask if these structures are

attached or non- attached (in the sense of Towsend)

CS are responsible for most of the Reynolds stresses, thus contributing to near

surface processes, such as momentum heat and vapor fluxes.

A non-ordinary example on the relevance of coherent structures in the Atmospheric

Surface Layer.

Stoessel et al. 2010

Images of deployed

sonic anemometer

Coherent structures vs vortices Some Questions

1) What are the relevant scales for CS (inner, outer)?

2) How VLSM relate to hairpin packets (is it Reynolds number dependent)?

3) why near wall peak can be affected by outer layer structures?

4) How roughness in general can perturb CS self organization, how about complex

terrain ?

5) How CS grow in size ?

6) Is there a hope to reproduce CS in a non-Navier-Stokes environment?

7) Can we really define a coherent structure ?

8) Can we describe coherent structures evolution in

unambiguous quantitative (not handwavy) terms ?

8) can we go beyond geometrical characteristics (exp) and vorticity contour (num) ?

More questions:

1) How spanwise mean vorticity relates to streamwise fluctuating vorticity ?

2) Do CS both scale with Kolmogorov (core) and the integral lengthscale ?

3) Are CS more or less stable as compared to worms (vortex filaments) in isotropic

3D turbulence ? What is the effect of a non zero mean strain

( and perhaps also mean vorticity) ?

Chacin & Cantwell 2000 (Turb. Boundary Layer) Soria 94

Chong & Perry 90

Luthi 2005

PTV isotropic turbulence

A different

view:

the small

scales of

turbulence