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Lagrangian and Eulerian aspects of turbulent flows with polymers - representative results

Lagrangian and Eulerian aspects of turbulent flows with polymers - representative results

This presentation summarizes our work on turbulent flow with dilute polymers in both Lagrangian and Eulerian settings, emphasizing the similarity and differences. The results were obtained experimentally, using particle image velocimetry and particle tracking velocimetry. We have measured Lagrangian evolution of material lines and surfaces as well as rates of decay of turbulent kinetic energy and vorticity-velocity mixed quantities. The results point to a clear route of effect of minute amounts of dilute polymers on turbulence, outside of the drag reduction effect.

Alex Liberzon

April 20, 2013
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  1. Lagrangian and Eulerian aspects of turbulent flows
    with dilute polymers - some representative results
    Alex Liberzon and Arkady Tsinober
    Turbulence Structure Laboratory, Tel Aviv University
    WPI Workshop, May 7-10, 2012
    . . . . . .

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  2. Outline
    • Background
    • Motivation
    • Experimental study - 3D-PTV
    • Lagrangian/Eulerian results
    • Discussion
    . . . . . .

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  3. Turbulence Structure Laboratory
    2XU´SKLORVRSK\µ² learn from the
    change
    7
    Turbulence
    Polymers
    Particles
    Forcing
    Lagrangian
    Eulerian
    . . . . . .

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  4. Motivation is both basic and practical
    • Drag reduction has been studied since
    1948 Toms effect
    • Body of literature is huge, important
    contributions of the present in this room
    The turbulence which occurs in the presence of drag-reducing
    additives is different from the turbulence which occurs in the solvent
    alone. Indeed, in some cases of very dilute polymer solutions, the
    anomalous (i.e. less dissipative) turbulence is probably the only
    detectable non-Newtonian effect. McComb 1990
    . . . . . .

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  5. Not only drag reduction
    . . . . . .

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  6. Phenomenology of polymer effects
    • Fluctuating and complex strain field
    is necessary to “turn the effect on”
    • Reaction back changes the field of
    strain, e.g. resistance to large strain,
    suppression of strong events, bursts
    • The flow could be considered
    intermittently rheological and not
    evenly distributed (networks)
    • The polymer drag reduction is not
    necessarily associated with
    suppression of turbulence, but with
    qualitative changes of some of its
    structure and production. In other
    words, there exist turbulent flows
    with strongly reduced drag and
    consequently dissipation and strain.
    . . . . . .

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  7. Motivation
    • Turbulent flows with polymer solutions - important
    example where the Lagrangian approach is unavoidable:
    .
    .
    1 The material elements (Lagrangian objects) are not
    passive;
    .
    .
    2 There are no equations reliably describing flows of
    polymer solutions (such as NSE for Newtonian fluids).
    There is a need for Lagrangian experimentation with such
    turbulent flows (and any other active additives), but ....
    . . . . . .

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  8. • Lagrangian methods alone are limited - there is a necessity of
    Eulerian approaches in parallel:
    .
    .
    1 The fluid particle acceleration a ≡ Du/Dt (Lagrangian) and the
    Eulerian components.
    .
    .
    2 Evolution of small scales via Lagrangian approaches using strain
    and vorticity in Eulerian form.
    .
    .
    3 Dealing with the material elements one needs again quantities such
    as strain and vorticity in Eulerian form.
    .
    .
    4 Eulerian approaches are needed for large scale issues as Reynolds
    stresses and TKE production.
    .
    .
    5 Direct interaction of small and large scales may be exhibited by
    mixed quantities: aL = ω × u
    . . . . . .

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  9. Representative results
    The results presented will cover the following topics:
    .
    .
    1 Accelerations
    .
    .
    2 Velocity derivatives
    .
    .
    3 Material elements
    .
    .
    4 Large scale stuff (RS and TKE)
    .
    .
    5 Direct interaction of SS and LS as may be exhibited by aL = ω × u
    and perhaps something else available (ω · u) and (doubtfully) in
    the spirit of Brasseur.
    . . . . . .

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  10. Experimental setup
    120 mm
    120 mm
    140 mm
    dia. 40 mm
    Light
    from
    20W
    CW
    Ar-Ion laser
    Stereoscopic view
    from
    four CCD cameras
    Observation volume
    of 10 x 10 x 10 mm
    Schematic drawing
    of a disk with 6 baffles
    Front view
    Top view
    . . . . . .

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  11. Experimental method
    . . . . . .

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  12. Experimental principles
    . . . . . .

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  13. PTV algorithm
    9

    2-5 ± PT V processing scheme
    (Willneff, 2003).

    2-6 ± Stereo-matching
    -matching is based on epipolar geometry (see
    2.6
    2.6 below
    2.6).
    The important thing is that we measure directly the
    full gradient tensor along the particle trajectories:
    ∂ui/∂xj
    and its evolution in time.
    5.4. Object space based tracking techniques
    Est
    con
    nat
    tim
    Fig. 15: Main processing steps
    . . . . . .

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  14. Quality checks: Lagrange vs Euler
    Lagrangian acceleration, the material derivative of velocity vector, a,
    a ≡
    Du
    Dt
    =
    ∂u
    ∂t
    + (u · ∇)u = −
    1
    ρ
    ∇p + ν∇2u
    is studied in conjunction with its physically important Eulerian
    decompositions:
    a = al + ac = a∥
    + a⊥ = aL + aB
    where al = ∂u/∂t is the local acceleration, ac = (u · ∇)u is the
    convective acceleration, a∥
    = (a · u)u is the acceleration component
    parallel to the velocity vector, a⊥ = a − a∥
    is the acceleration
    component normal to the velocity vector, aL = ω × u is the Lamb
    vector and aB = ∇(u2/2);
    . . . . . .

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  15. Joint PDF of a and al + ac
    a
    x
    a
    lx
    +a
    cx
    −0.01 −0.005 0 0.005 0.01 0.015
    −0.01
    −0.005
    0
    0.005
    0.01
    0.015
    200
    400
    600
    800
    1000
    1200
    1400
    Solid line - water, dashed line - polymers
    . . . . . .

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  16. PDFs of acceleration components
    0 0.02 0.04 0.06 0.08 0.1
    0
    10
    20
    30
    40
    50
    60
    a, a
    l
    , a
    c
    p(a), p(a
    l
    ), p(a
    c
    )
    a
    a
    l
    a
    c
    101
    102
    103
    104
    0
    1
    2
    3
    4
    5
    x 10−3
    a
    a
    l
    a
    c
    PDFs of the magnitudes of the acceleration vector (|a|) and of its components
    for water ( solid lines) and polymer (dashed lines). (left) dimensional form
    (right) dimensionless form, normalized with ε3/2ν−1/2
    . . . . . .

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  17. PDFs of acceleration components
    (cont.)
    0 0.02 0.04 0.06 0.08 0.1
    0
    10
    20
    30
    40
    50
    60
    a, a
    Lamb
    , a
    B
    p(a), p(a
    Lamb
    ), p(a
    B
    )
    a
    a
    Lamb
    a
    B
    101
    102
    103
    104
    0
    1
    2
    3
    4
    5
    x 10−3
    a
    a
    Lamb
    a
    B
    . . . . . .

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  18. PDFs of acceleration components
    (cont.)
    0 0.02 0.04 0.06 0.08 0.1
    0
    20
    40
    60
    80
    100
    a, a
    ||
    , a

    p(a), p(a
    ||
    ), p(a

    )
    a
    a
    ||
    a

    10−1
    100
    101
    102
    103
    104
    0
    0.002
    0.004
    0.006
    0.008
    0.01
    a
    a
    ||
    a

    . . . . . .

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  19. Ratios of PDFs of polymer to water
    10−2
    10−1
    1
    4
    a [ms−2]
    Ratios of polymer/water pdfs
    a
    a
    l
    a
    c
    a
    ||
    a

    a
    L
    a
    B
    102
    0.9
    1
    1.1
    1.2
    1.3
    1.4
    a/ε3/2 ν−1/2
    Ratios of polymer/water pdf
    a
    a
    l
    a
    c
    a
    ||
    a

    a
    L
    a
    B
    . . . . . .

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  20. Alignment of al
    and ac

    1 −
    0.5 0 0.5 1
    0
    1
    2
    3
    4
    5
    6
    7
    PDF
    water
    polymer
    . . . . . .

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  21. Lagrangian information on the
    evolution of material elements
    Infinitesimal material lines, li
    evolve according to a purely kinematic
    equation :
    Dli
    Dt
    = Wl
    i
    Wl
    i
    = lj
    sij + (1/2)εijk j
    lk ≡ (s · l)i + (1/2)(ω × l)i
    Term 1) Change of magnitude of l, and Term 2) the tilting of l . More
    details in Liberzon et al. PoF (2005)
    . . . . . .

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  22. Stretching related quantities -
    Cauchy-Green tensor eigenvalues
    and the stretching vector Wi
    l,s=lj
    sij
    are weaker at later
    in the polymer solution, compared to the flow of pure
    ͓Figs. 3͑c͒ and 3͑d͔͒. This effect becomes stronger
    conditioned on large strain. Vorticity makes a consid-
    contribution to the mutual geometry of material lines
    he eigenframe of the rate of strain due to tilting of the
    ial lines,15 though this contribution is not changed sub-
    terial lines from some initial moment till some chosen
    and its properties are expected to be changed in a tu
    flow of dilute polymer solution as compared to that
    water.
    The simplest information is contained in the C
    Green tensor Wij
    =Bik
    Bkj
    . The eigenvalues wi
    of the
    Wij
    reflect the deformation of elementary material
    ͑a͒ Time evolution of the mean values of the eigenvalues of the Cauchy–Green tensor, ln͑wi
    ͒, and ͑b͒ PDF of the second invaria
    –Green tensor Q͑W͒, for different time moments, for water ͑solid lines͒ and polymer solution ͑dashed lines͒.
    nloaded 18 Apr 2008 to 132.66.7.211. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyri
    ℓi(t) = Bij(t)ℓ(0), dBij/dt = (∂ui/∂xk ) Bkj, Bij(0) = δij
    , Wij = Bik
    Bkj
    . . . . . .

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  23. Stretching dynamics of infinitesimal
    material lines through a single tensor
    ℓi(t) = Bij(t) ℓj(0), dBij/dt = (∂ui/∂xk ) Bkj
    Bij(0) = δij
    ℓiℓj
    sij = Bik
    Bjm
    sijℓk (0)ℓm(0) ≡ Tkm(t)ℓk (0)ℓm(0)
    Tkm(t)ℓ(0)ℓm(0) = ℓ2(0)
    [
    Ti
    cos2(ℓ(0), τi)
    ]
    ⟨ℓiℓj
    sij⟩ = ⟨Ti⟩ × ⟨cos2(ℓ0, τi)⟩ =
    1
    3
    ⟨ℓ2(0)⟩⟨T1 + T2 + T3⟩
    .
    .
    1 trace tr(T ) is positive on average
    .
    .
    2 empirically found that one eigenvalue is three orders of magnitude
    larger than others
    .
    .
    3 it was shown to be strongly reduced in dilute polymers flow
    . . . . . .

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  24. Strong reduction of the “stretching
    eigenvalue” in polymers
    Reynol
    ing fro
    derivati
    lief is t
    be true
    results
    and we
    further
    turbulen
    come p
    and dir
    conform
    Thi
    der Gra
    1B. A. T
    FIG. 5. PDF of the first eigenvalue ⌼1
    of the T matrix for water ͑solid lines͒
    and polymer solution ͑dashed lines͒ for different time moments.
    031707-4 Liberzon et al.
    . . . . . .

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  25. Stretching rates
    . . . . . .

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  26. Stretching rates - time evolution
    Notice the “delay” of polymer stretching rate - could explain the
    resistance to strong strain via mis-alignment or tilting.
    . . . . . .

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  27. −1 −0.5 0
    0.4
    0.5
    0.6
    0.7
    0.8
    0.9
    cos(ω(t),λ
    1
    (t=0))
    −1 −0.5 0
    0.3
    0.4
    0.5
    0.6
    0.7
    0.8
    cos(ω(t),λ
    2
    (t=0))
    −1 −0.5 0
    0
    0.2
    0.4
    0.6
    0.8
    1
    cos(ω(t),λ
    3
    (t=0))
    −1 −0.5 0
    0.4
    0.5
    0.6
    0.7
    0.8
    cos(ω(t=0),λ
    1
    (t))
    −1 −0.5 0
    0.4
    0.6
    0.8
    1
    cos(ω(t=0),λ
    2
    (t))
    −1 −0.5 0
    0.2
    0.4
    0.6
    0.8
    cos(ω(t=0),λ
    3
    (t))
    . . . . . .

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  28. Large scale effects, TKE production
    . . . . . .

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  29. PDF of alignment
    −0 .5 0 0. 5
    0
    0. 5
    1
    1. 5
    2
    2. 5
    3
    3. 5
    4
    4. 5
    5
    cos( ui
    uj
    , Sij
    )
    .
    water, smooth
    polymer, smooth
    water, baffles
    polymer, baffles
    . . . . . .

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  30. Discussion
    .
    .
    1 Lagrangian information is crucial in the case of dilute polymers
    (and probably particles, bubbles, fibers, colloids, etc.)
    .
    .
    2 Eulerian information is crucial, maybe because our Lagrangian
    formulation is very limited and we need dynamics explained by
    strain, vorticity, etc.
    .
    .
    3 Mixing Lagrangian and Eulerian information could help to get
    some new ideas.
    . . . . . .

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  31. Acknowledgments
    .
    .
    1 Institute of Environmental Engineering (IfU), ETH Zurich
    .
    .
    2 Turbulence Structure Laboratory team
    .
    .
    3 Funding agencies: SNF, ISF, GIF, BSF
    . . . . . .

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