vorticity dynamics & the small scales of turbulence...

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