Vorticity dynamics & the small scales of turbulence Particle Tracking Velocimetry 3DPTV & DNS

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1 Vorticity dynamics & the small scales of turbulence Particle Tracking Velocimetry 3DPTV & DNS 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 Measurements along a single trajectory velocity vorticity

2 ABC of 3D-PTV technique

3 From: Liberzon et al. Phys. of fluids 17, 2005 Turbulent box

4 Some basic definitions...

5

6 Part I On the validation of 3D-PTV results Point wise check divergence free = 0 Lagrangian, convective, eulerian acceleration enstrophy balance

7 point wise check: continuity Luthi et al. JFM (528) pp 87, 2005

8 point wise check: acceleration Luthi et al. JFM (528) pp 87, 2005

9 homogeneous turbulence properties Skewness of - s ij s jk s ki and i j s ij - s ij s jk s ki > 0 i j s ij > 0 Strain production Enstrophy production - - -

10 Luthi et al. JFM (528) pp 87, 2005

11 Experimental Numerical (Galanti e Tsinober)

12 given the 3 x 3 tensor of velocity derivative A i,j = u i x j the characteristic equations is given by D >0 one real, two complex conjugate eigenvalues swirling, or vorticity doiminated regions D < 0 three real conjugate strain dominated region

13 Enstrophy production Strain production

14 3 < 0 Strain 1 >0 1 2 Compression 2 Stretching 3 Alignment between and the eigenframe j of s ij Eigenvalues of s ij 2 3 1

15 Some universal qualitative properties of turbulent flows homogeneous turbulence properties alignments (vorticity eigenframe of the strain tensor i ) tear drop of RQ maps positiveness of < 2 > i jsij i jsij positiveness of < > and skewness of PDF( )

16 What is governing vorticity direction? Viscous and inviscid tilting of the vorticity vector Let s consider now the versor of = / DIRECTION Inviscid tilting Viscous tilting MAGNITUDE

17 Holzner et al. Phys of Fluids 22, 2010

18 Holzner et al. Phys of Fluids 22, ω i jsij W and =

19 Vortex compression INVISCID Inviscid tilting Vortex stretching Viscous destruct. of enstrophy VISCOUS Viscous tilting Vortex reconnection 1: high 2, low s 2 2: high s 2, low 2 all values Most of stretching occurs when 1 or 2 Most of tilting occurs when 3

20 Holzner et al. Phys of Fluids 22, 2010 red blue viscous contribution is on average weaker but it is responsible for strong, though rare, tilting events

21 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.

22 Case 1: Production in wall bounded flow 2D TBL: the effect of the mean strain Gurka et al wall distance from the wall (viscous units) for continuity we have one stretching and one compression axis

23 peak at 0.71 corresponding to a 45 0 angle Remember the hairpin vortex model with 45 0 angle inclination. Lift-up mechanism

24 Case 2: TKE production in a convection cell (Ra 10 9 ) Liberzon,, Luethi, Guala, Tsinober, Phys Fluids 2006

25 Decomposition along the main strain axis

26 Decomposition along the Cartesian axis P xk = <u k u j >S kj major differences along vertical x 2 = y direction along which buoyancy act

27 forced convection + TKEP > 0 - TKEP > 0 shear flow + TKEP > 0 - TKEP > 0

28 FU B f u TKEP < 0 U CASE A: Buoyancy FU TKEP < 0 B T f u s 2 STRONG sss s 2 II III I IV U TKEP < 0 T s 2 WEAK Sss s 2 CASE B: Shear U TKEP > 0 T s 2 WEAK sss s 2 U TKEP > 0 T s 2 STRONG sss s 2 TKEP > 0 B = buoyancy U = mean flow T = turbulence TKEP= -<u i u j >S ij 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.

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30 Part II On the evolution of vorticity and material lines Vortex lines vs. Material lines many l s 1 >0 3 < 0 Strain 1 Compression 2 2 Stretching 3 Many randomly oriented l s Vs. One vorticity vector

31 To investigate turbulence and turbulent mixing we can distinguish between active and passive quantities. active quantities: vorticity and strain strongly coupled and thus interacting Stretching of vorticity: The productive interaction between vorticity and strain continuously feeds the vorticity field in order to balance the viscous destruction term 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 l i connecting two fluid particles Stretching of material lines: two material points (fluid particles) A & B initially close, are driven apart from each other. 1 2 Dl Dt 2 l l s i j ij l+δl A A l(t = t 0 ) B B The line l connecting them is on average stretched The faster l grows, i.e. l is stretched, the more efficient is the mixing. l(t 0 + t) Girimaji & Pope, J. Fluid Mech. 220, 1990.

32 l 1 2 l 3 Lagrangian evolution of the alignment between l and the strain eigenframe 1,2,3

33 Guala et al. JFM (533) pp 339, 2005 mixing

34 Qualitative universal aspects of turbulent flows: are on average positive stays mostly aligned with 2 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??? Stretching >0 Compression <0

35 l parallel to both and changes

36 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?

37 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

38 Let us consider the change of orientation or tilting of the vorticity vector Vorticity tilting INVISCID 2 D( / ) / Dt D( / ) / Dt i TOTAL i what is the viscous contribution to tilting?

39 Main question: what is responsible for the change of direction of? l parallel to How can we compute the third derivatives of the velocity field? Navier Stokes Where a is the Lagrangian acceleration

40 Viscous and inviscid tilting of the vorticity vector Let s consider now the versor of = / DIRECTION Inviscid tilting Viscous tilting MAGNITUDE

41 Holzner et al. Phys of Fluids 22, 2010

42 Holzner et al. Phys of Fluids 22, ω i jsij W and =

43 ω // λ 3 Holzner et al. Phys of Fluids 22, 2010

44 Holzner et al. Phys of Fluids 22, jsij W and = ω // λ 3

45 Vortex compression Inviscid tilting Vortex stretching Viscous destruct. of enstrophy Viscous tilting Vortex reconnection 1: high 2, low s 2 2: high s 2, low 2 all values Most of stretching occurs when 1 or 2 Most of tilting occurs when 3