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
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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