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 . . . . . . 

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

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

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 . . . . . . 

Not only drag reduction . . . . . . 

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. . . . . . . 

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 .... . . . . . . 

• 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 . . . . . . 

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. . . . . . . 

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 . . . . . . 

Experimental method . . . . . . 

Experimental principles . . . . . . 

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 . . . . . . 

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); . . . . . . 

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 . . . . . . 

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 . . . . . . 

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 . . . . . . 

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 ⊥ . . . . . . 

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 . . . . . . 

Alignment of al and ac − 1 − 0.5 0 0.5 1 0 1 2 3 4 5 6 7 PDF water polymer . . . . . . 

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) . . . . . . 

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 . . . . . . 

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 . . . . . . 

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. . . . . . . 

Stretching rates . . . . . . 

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

−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)) . . . . . . 

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

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 . . . . . . 

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. . . . . . . 

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