vorticity dynamics & the small scales of turbulence...
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Vorticity dynamics & the small scales of turbulence
Particle Tracking Velocimetry 3DPTV & DNS
Measurements along a single trajectory
velocity
vorticity
Luthi et al. JFM (528) pp 87, 2005
Guala et al. JFM (533) pp 339, 2005
Hoyer et al., Exp. in fluids (39) pp 923, 2005
Holzner et al. Phys of Fluids 22, 2010
Tsinober “informal introduction to turbulence” Elsevier
ABC of 3D-PTV technique
Turbulent box
From: Liberzon et al. Phys. of fluids 17, 2005
Some basic definitions...
Point wise check
• divergence free
• Lagrangian, convective, eulerian
acceleration
• enstrophy balance
Part I
On the validation of 3D-PTV results
= 0
point wise check: continuity
Luthi et al. JFM (528) pp 87, 2005
point wise check:
acceleration Luthi et al. JFM (528) pp 87, 2005
homogeneous turbulence properties Skewness of - sij sjk ski and i j sij
- sij sjk ski > 0 i j sij > 0
Strain production Enstrophy production
-
-
-
Luthi et al. JFM (528) pp 87, 2005
Experimental Numerical (Galanti e Tsinober)
D >0 one real, two complex conjugate eigenvalues swirling,
or vorticity doiminated regions
D < 0 three real conjugate strain dominated region
given the 3 x 3 tensor of velocity derivative Ai,j = 𝜕𝑢𝑖
𝜕𝑥𝑗 the
characteristic equations is given by
Strain production
Enstrophy production
Compression
Strain
Stretching
1>0
2
1
3
3< 0
2
Eigenvalues of sij
2
1
3
Alignment between and
the eigenframe j of sij
i j ijsi j ijs
• homogeneous turbulence properties
• alignments (vorticity – eigenframe of the strain tensor i)
• tear drop of RQ maps
• positiveness of < 2 >
• positiveness of < > and skewness of PDF( )
Some universal qualitative properties of turbulent flows
Let‘s consider now the versor of = / ||
Viscous tilting
Inviscid tilting
What is governing vorticity direction ?
Viscous and inviscid tilting of the vorticity vector
DIRECTION
MAGNITUDE
Holzner et al. Phys of Fluids 22, 2010
Holzner et al. Phys of Fluids 22, 2010
2 2 and = j ijs W ωi
Vortex
stretching
Viscous
tilting
Viscous
destruct. of
enstrophy
Vortex
reconnection Vortex
compression
Inviscid
tilting
Most of tilting
occurs when || 3
1: high 2 , low s2
2: high s2 , low 2
all values
Most of
stretching
occurs when
|| 1 or
|| 2
INVISCID VISCOUS
Holzner et al. Phys of Fluids 22, 2010
viscous contribution is on average weaker but it is responsible
for strong, though rare, tilting events
blue red
So....how is the small scale interaction working ?
When is on 1 vorticity is produced, the direction is not very stable, since
both the rotation of and the viscous term contribute to tilt towards 2
When is on 2 vorticity magnitude is not changing much due to the
balance between production by strain and the destruction by the viscous term.
Tilting contributions are weak and the diretion of vorticity is stable vortex
filaments
When is on 3 vorticity is destroyed by strain but its direction is strongly
unstable due to viscous contribution to tilting.
Case 1: Production in wall bounded flow 2D TBL:
the effect of the mean strain Gurka et al. 2003
for continuity we have one stretching
and one compression axis
wall distance from the wall (viscous units)
Remember the
hairpin vortex
model with 450
angle
inclination.
Lift-up
mechanism
peak at 0.71
corresponding to
a 450 angle
Liberzon, , Luethi, Guala, Tsinober, Phys Fluids 2006
Case 2: TKE production in a convection cell
(Ra 109)
Decomposition along the main strain axis
Pxk =−<ukuj>Skj
major
differences
along vertical
x2= y direction
along which
buoyancy act
Decomposition along the Cartesian axis
forced convection
shear flow
+ TKEP > 0
- TKEP > 0
+ TKEP > 0
- TKEP > 0
B
T U
f´u´ FU
s 2
sss s2
T
f´u´ FU
s 2
sss s2
WEAK
T U TKEP < 0
s 2
Sss s2 WEAK
TKEP > 0
CASE A:
Buoyancy
T s 2
sss s2 STRONG
STRONG TKEP < 0
TKEP > 0
B
U U
TKEP < 0
TKEP > 0
II I
III IV
CASE B:
Shear
B = buoyancy
U = mean flow
T = turbulence
TKEP= -<uiuj>Sij
B T U maintain a
delicate equilibrium: turbulence starts the roller by converting and
transferring energy from buoyancy to mean flow.
Once the roller is established T and U exchange energy periodically: when T
has a surplus it feeds U , when T cannot sustain itself it is fed by U. Different
regions (due to the vertical profile of T) occur to have different B and so
TKEP ><0.
Vortex lines vs. Material lines
many l‘s
Part II
On the evolution of vorticity and material lines
Compression
Strain
Stretching
1>0
2
1
3
3< 0
2 Many randomly oriented l‘s
Vs. One vorticity vector
Stretching of vorticity:
The productive interaction between vorticity and strain
continuously feeds the vorticity field in order to balance
the viscous destruction term
Stretching of material lines:
two material points (fluid particles) A & B
initially close, are driven apart from each other.
• active quantities:
vorticity and strain
strongly coupled and thus
interacting
• passive quantities:
material elements
stretched / compressed by
strain and tilted by vorticity
without giving any feed
back to the flow, e.g. A
material line li connecting
two
fluid particles
l(t0+ t)
l(t = t0)
A
A
B
B
To investigate turbulence and turbulent mixing we can
distinguish between active and passive quantities.
The line l connecting
them is on average stretched
The faster l grows,
i.e. l is stretched, the more
efficient is the mixing.
21
2i j ij
Dll l s
Dt
Girimaji & Pope, J. Fluid Mech. 220, 1990.
l+Δl
Lagrangian evolution of
the alignment between l and
the strain eigenframe 1,2,3
l 3
l || 1
2
mixing
Guala et al. JFM (533) pp 339, 2005
Qualitative universal aspects of turbulent flows:
are on average positive
Viscous destruction and production
of enstrophy are non local quantities
but their balance is maintained at any time Tsinober (2001) and reference herein
What is telling the strain field that vorticity
is too much stretched or compressed ???
stays mostly aligned with 2
Stretching >0 Compression <0
both and changes
l parallel to
How can a small scale quiet portion of the vorticity field - fluid blob - which is
strongly compressed , tilted by the local strain eigenvector escape, such not too
feel its persistent action?
We found that the different orientatoin of l‘s is not enough to
justify their stronger stretching as compared to vorticity
Switch and persistent alignment
|| 3
|| 1
|| 2
Let us consider the change of orientation or “tilting” of the
vorticity vector
2 ( / ) / ( / ) /i iD Dt D Dt
Vorticity tilting
TOTAL INVISCID
what is the viscous contribution to tilting ?
How can we compute the third derivatives of the velocity field ?
Navier Stokes Where a is the
Lagrangian acceleration
Main question: what is responsible for the change of direction of ?
l parallel to
Let‘s consider now the versor of = / ||
Viscous tilting
Inviscid tilting
Viscous and inviscid tilting of the vorticity vector
DIRECTION
MAGNITUDE
Holzner et al. Phys of Fluids 22, 2010
Holzner et al. Phys of Fluids 22, 2010
2 2 and = j ijs W ωi
Holzner et al. Phys of Fluids 22, 2010
ω // λ3
2 2 and = j ijs W Holzner et al. Phys of Fluids 22, 2010
ω // λ3
Vortex
stretching Viscous
tilting
Viscous
destruct. of
enstrophy
Vortex
reconnection Vortex
compression
Inviscid
tilting
Most of tilting
occurs when || 3
1: high 2 , low s2
2: high s2 , low 2
all values
Most of
stretching
occurs when
|| 1 or
|| 2