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
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