Transport between two fluids across their mutual flow interface: the streakline approach. Sanjeeva Balasuriya

Transcription

1 Transport between two fluids across their mutual flow interface: the streakline approach Sanjeeva Balasuriya

2 Steady (autonomous) flows Particle trajectories: dx dt = ẋ = u(x), x R2 Flow is steady (autonomous); velocities at each point remain the same in time

3 Steady (autonomous) flows Particle trajectories: dx dt = ẋ = u(x), x R2 Flow is steady (autonomous); velocities at each point remain the same in time Stagnation (fixed) points can be determined by u(a) = 0

4 Steady (autonomous) flows Particle trajectories: dx dt = ẋ = u(x), x R2 Flow is steady (autonomous); velocities at each point remain the same in time Stagnation (fixed) points can be determined by u(a) = 0 Their stable and unstable manifolds locally determined by eigensystem of Du(a)

5 Steady (autonomous) flows Particle trajectories: dx dt = ẋ = u(x), x R2 Flow is steady (autonomous); velocities at each point remain the same in time Stagnation (fixed) points can be determined by u(a) = 0 Their stable and unstable manifolds locally determined by eigensystem of Du(a) These are flow barriers, demarcating regions of distinct fluid motion

6 Steady (autonomous) flows Taylor-Green cellular flow An eddy Oceanic jet Hill s spherical vortex

7 Unsteady flow barriers Experimental vorticity field Nick Ouellette (Stanford) Gulf Stream rings Darryn Waugh (Johns Hopkins) Jupiter s Great Red Spot NASA/JPL/Caltech (NASA photo #PIA00343) Drifters and forbidden zone Josefina Olascoaga (Miami)

8 Unsteady flow barriers Flow is ẋ = u(x, t) nonautonomous

9 Unsteady flow barriers ( u(x, t)= sin 4t 2 + cos 4t 2 + cos 4t sin 4t ) x Haller (J Fluid Mech, 2005; Ann Rev Fluid Mech, 2015) Flow is ẋ = u(x, t) nonautonomous Fixed points Eigenvalues/vectors Eulerian structures

10 Unsteady flow barriers ( u(x, t)= sin 4t 2 + cos 4t 2 + cos 4t sin 4t ) x Haller (J Fluid Mech, 2005; Ann Rev Fluid Mech, 2015) Flow is ẋ = u(x, t) nonautonomous Fixed points Eigenvalues/vectors Eulerian structures Stable/unstable manifolds applicable, but may intersect in complicated ways or not at all

11 Unsteady flow barriers ( u(x, t)= sin 4t 2 + cos 4t 2 + cos 4t sin 4t ) x Haller (J Fluid Mech, 2005; Ann Rev Fluid Mech, 2015) Flow is ẋ = u(x, t) nonautonomous Fixed points Eigenvalues/vectors Eulerian structures Stable/unstable manifolds applicable, but may intersect in complicated ways or not at all There can be a velocity across these, and hence flux

12 Unsteady flow barriers ( u(x, t)= sin 4t 2 + cos 4t 2 + cos 4t sin 4t ) x Haller (J Fluid Mech, 2005; Ann Rev Fluid Mech, 2015) Flow is ẋ = u(x, t) nonautonomous Fixed points Eigenvalues/vectors Eulerian structures Stable/unstable manifolds applicable, but may intersect in complicated ways or not at all There can be a velocity across these, and hence flux Since these also move with time, how then, would one define flow barriers?

13 Unsteady flow barriers Diagnostic ways of defining flow barriers [Lagrangian Coherent Structures (LCSs)] Places where fluid piles up after advection Tél, Speetjens, Pierrehumbert Ridges of FTLEs/FSLEs and extensions Shadden, Tang, Peacock, Weinkauf, Haller, Karrasch, Peikert, Lapeyre, d Ovidio Eigen/Singular vectors associated with transfer operators Dellnitz, Junge, Froyland, Padberg-Gehle Clipped stable/unstable manifolds SB, Padberg-Gehle Averages along trajectories Mezić, Mancho, Rypina, Haller Curves/Surfaces of maximal stretching or shearing Haller, Beron-Vera of minimal flux Mackay, Polterovich, SB, Froyland, Santitissadeekorn of extremal attraction/repulsion Haller, Farazmand, Blazevski Ergodic/stochastic measures Mezić, Budisić, Tang, Rowley Topological entropy of trajectories Thiffeault, Allshouse Clustering Froyland, Padberg-Gehle, Haller, Karrasch, Dabiri Etc, etc Bollt, Rom-Kedar,...

14 Unsteady flow barriers Variational LCSs George Haller (ETH), Mohammad Farazmand (MIT) Transfer operator / ocean separation Gary Froyland (UNSW), Erik van Sebille (Imperial) FTLEs for double gyre SB (Adelaide), Nick Ouellette (Stanford) Clustering in ABC flow George Haller (ETH)

15 Unsteady flow barriers Nearby steady flow I Interface splits into stable and unstable manifolds I Time-varying mixing area I Causes transport between gyres FTLEs for double gyre

16 My related work unsteady LCSs 1. Locating stable/unstable manifolds SB (SIAM J Appl Dyn Sys, 2011; Phys Fluids, 2012) 2. Defining and quantifying flux in 2D SB (Nonlinearity, 2006; SIAM J Appl Dyn Sys, 2005; in: Erg Theor Open Dyn & Coherent Struct, Springer, 2014) 3. Optimising flux across LCS interfaces, and design of micromixers SB (Phys D, 2005; Phys Fluids, 2005; Phys Rev Lett, 2010, Phys Fluids, 2015), SB & Finn (Phys Rev Lett, 2012), SB (J Micromech Microeng, 2015) 4. Controlling stable and unstable manifolds SB & Padberg-Gehle (Phys D, 2014; SIAM J Appl Math, 2013; Phys Rev E, 2014), SB (J Nonlin Sci, 2016) 5. Controlling structures in turbulent experimental data Ouellette, SB & Kalampattel (sbmt, 2016), Ouellette & SB (ongoing) 6. Manifolds under impulses (e.g., tapping) SB (sbmt, 2016) 7. LCSs and ocean transport SB (Nonlin Proc Geophys, 2001; Geophys Astrophys Fluid Dyn, 2004; in progress), SB & Jones (Nonlin Proc Geophys, 2001) 8. Manifold locations used to determine front speeds and profiles SB et al (SIAM J Appl Math, 2007), SB & Volpert (Combust Theor Model, 2008), SB & Gottwald (J Math Biol, 2010), SB (J Theor Biol, 2010), SB & Binder (Phys D, 2014), Binder, Blyth & SB (J Fluid Mech, 2015) 9. Book: SB, Melnikov methods for barriers and transport in unsteady flows, SIAM Press, in press (2016)

17 My related work unsteady LCSs All these flow barrier definitions: Analyse the unsteady velocity field by itself Using LCS (stable/unstable manifold) ideas Just think of one fluid

18 Mixing of two fluids I Mixing of a sample and a reagent is a typical problem in microfluidics (DNA synthesis, biochemical assay,...)

19 Mixing of two fluids I I Mixing of a sample and a reagent is a typical problem in microfluidics (DNA synthesis, biochemical assay,...) Low Re flow, so no turbulent mixing

20 Mixing of two fluids I I I Mixing of a sample and a reagent is a typical problem in microfluidics (DNA synthesis, biochemical assay,...) Low Re flow, so no turbulent mixing Agitations may lead to chaotic mixing Aref (J Fluid Mech, 1984)

21 Mixing of two fluids I I I Mixing of a sample and a reagent is a typical problem in microfluidics (DNA synthesis, biochemical assay,...) Low Re flow, so no turbulent mixing Agitations may lead to chaotic mixing Aref (J Fluid Mech, 1984) Cordero et al (New J Phys, 2009) I I Experiment of mixing of two fluids within a microdroplet Mixing across flow interface achieved using thermocapillary forces (lasers)

22 Flow interface Velocity? 1 a b 2 a b fluids in channel Anomalous fluid in vortex

23 Flow interface Velocity? 1 a b 2 a b fluids in channel Anomalous fluid in vortex Flow interface is a physical entity

24 Flow interface Velocity? 1 a b 2 a b fluids in channel Anomalous fluid in vortex Flow interface is a physical entity Methods of LCSs, FTLEs, etc, obtained from Lagrangian advection under the velocity field, cannot obtain these interfaces

25 Flow interface Velocity? 1 a b 2 a b fluids in channel Anomalous fluid in vortex Under a weak unsteady velocity agitation, can we Determine unsteady flow interface? Quantify transport across it?

26 Why streaklines? n a x p 1 x p 2 x p 3 x p n U b 2 d 1 2 d 2 2 d 3 2 d n

27 Mathematical setup a x p u n p ẋ = u(x) x p b x p d Γ = { x(p), p R} { } Γ = x(p), p [p u, p d ]

28 Mathematical setup a x p u n p ẋ = u(x) x p b x p d Γ = { x(p), p R} { } Γ = x(p), p [p u, p d ] u := ( ) u, ˆn(p) := u ( x(p)) u ( x(p))

29 Mathematical setup a x p u n p ẋ = u(x) x p b x p d Γ = { x(p), p R} { } Γ = x(p), p [p u, p d ] upstream streakline Γ u 0(t) = p R { } x0 u (p, t) solving with x0 u (p, t p+p u ) = a downstream streakline Γ d 0 (t) = p R { } x0 d (p, t) solving with x0 d (p, t p+p d ) = b

30 Mathematical setup a x Ε u p,t x p Ε u t b ẋ = u(x) + v(x, t) v(x, t) < ε u(x) Ε d t v = 0 on Γ \ Γ

31 Mathematical setup a x Ε u p,t x p Ε u t b ẋ = u(x) + v(x, t) v(x, t) < ε u(x) Ε d t v = 0 on Γ \ Γ upstream streakline Γ u ε(t) = p [ P,P] downstream streakline Γ d ε (t) = p [ P,P] { } xε u (p, t) solving with xε u (p, t p+p u ) = a { } xε d (p, t) solving with xε d (p, t p+p d ) = b

32 Streakline expressions a x Ε u p,t x p Ε u t b upstream streakline Ε d t [x u ε (p, t) x(p)] ˆn(p) = Mu (p, t) u ( x(p)) + O(ε2 ) M u (p, t) = {p > p u} min{p,p d } p e τ [ u]( x(ξ))dξ u ( x(τ)) v ( x(τ), τ +t p) dτ p u SB (Phys Rev Fluids, submitted)

33 Streakline expressions a x Ε u p,t x p Ε u t b downstream streakline Ε d t [ ] xε d (p, t) x(p) ˆn(p) = Md (p, t) u ( x(p)) + O(ε2 ) M d (p, t) = {p < p d} p d p e τ [ u]( x(ξ))dξ u ( x(τ)) v ( x(τ), τ +t p) dτ max{p,p u } SB (Phys Rev Fluids, submitted)

34 y y y y Validation: cross-channel micromixer Stills from the previous streakline movie (ε = 0.1) t= t= t= t= x x x x

35 y y y y y y y y Validation: cross-channel micromixer Stills from the previous streakline movie (ε = 0.1) t= t= t= t= x x x x Computations using the theory (ε = 0.1) 0.04 t t t t

36 Validation: cross-channel micromixer y x 0.02 ε = 0.1, t = 7.854

37 Validation: cross-channel micromixer y x ε = 0.1, t = Backwards FTLE

38 Validation: cross-channel micromixer y x ε = 0.1, t = ln E 0 Backwards FTLE ln E ln Ε L 2 -error ln Ε

39 Transport across flow interface a x p u x p n p b x p d Γ separates the two fluids (upstream and downstream)

40 Transport across flow interface Ε u t a n p x u Ε p,t x p x Ε d p,t p,t Ε d t b Γ u ε separates from upstream Γ d ε separates downstream

41 Transport across flow interface Ε u t a n p x u Ε p,t x p x Ε d p,t p,t Ε d t b 1 2 a n p Φ p,t b 1' 2' Uses idea of a gate (cf. Haller & Poje (Phys D, 1998))

42 Transport across flow interface Ε u t a n p x u Ε p,t x p x Ε d p,t p,t Ε d t b 1 2 a n p Φ p,t b 1' 2' Transport of fluid from side 2 to 1 at time t will be defined as the instantaneous transport across the pseudo-streakline.

43 Transport across flow interface Ε u t a n p x u Ε p,t x p x Ε d p,t p,t Ε d t b 1 2 a n p Φ p,t b 1' 2' Transport of fluid from side 2 to 1 at time t will be defined as the instantaneous transport across the pseudo-streakline. For p [p u, p d ] φ(p, t) = M(p, t) + O(ε 2 ) M(p, t) = p d p u p e τ [ u]( x(ξ))dξ u ( x(τ)) v ( x(τ), τ + t p) dτ SB (Phys Rev Fluids, submitted)

44 y y y y Transport in cross-channel mixer Pseudo-streaklines constructed by numerical streaklines t= t= x x t= t= x x

45 Transport in cross-channel mixer Comparison between numerical flux (obtained by integrating across gate) [red] and theory [blue] 0.03 ǫ= φ t ε = 0.1

46 Transport in cross-channel mixer Comparison between numerical flux (obtained by integrating across gate) [red] and theory [blue] 0.03 ǫ= ǫ= φ 0 φ t t ε = 0.1 ε = 1

47 Closed streakline situation Γ is closed (a = b) Θ n Θ x Θ a

48 Closed streakline situation Γ is closed (a = b) Require v(a, t) 0 Θ n Θ x Θ a

49 Closed streakline situation Γ is closed (a = b) Require v(a, t) 0 Θ n Θ x Θ a Perturbed streaklines will continue to wrap around

50 Closed streakline situation Γ is closed (a = b) Require v(a, t) 0 Perturbed streaklines will continue to wrap around Θ n Θ x Θ a Upstream/downstream expressions change slightly: M u (p, t) = M d (p, t) = p p u e p d p p τ [ u]( x(ξ))dξ u ( x(τ)) v ( x(τ), τ +t p) dτ e p τ [ u]( x(ξ))dξ u ( x(τ)) v ( x(τ), τ +t p) dτ

51 Kirchhoff s elliptic vortex x Θ ẋ 1 = 2x 1 m 2, ẋ 2 = 2x 1 l 2 Θ n Θ a

52 Kirchhoff s elliptic vortex x Θ ẋ 1 = 2x 1 m 2, ẋ 2 = 2x 1 l 2 Θ n Θ a { x 2 Suppose different fluid inside Γ := 1 l 2 + x 2 2 } m 2 = 1

53 Kirchhoff s elliptic vortex x Θ ẋ 1 = 2x 1 m 2, ẋ 2 = 2x 1 l 2 Θ n Θ a { x 2 Suppose different fluid inside Γ := 1 l 2 + x 2 2 } m 2 = 1 Examine effect of weak external strain on the vortex Turner (Phys Fluids, 2014)

54 Kirchhoff s elliptic vortex x Θ ẋ 1 = 2x 1 m 2, ẋ 2 = 2x 1 l 2 Θ n Θ a { x 2 Suppose different fluid inside Γ := 1 l 2 + x 2 2 } m 2 = 1 Examine effect of weak external strain on the vortex Turner (Phys Fluids, 2014) ( 0 Choose v(x 1, x 2, t) = ε sin (x 1 l) 1 ) tanh (t 5)

55 Kirchhoff s elliptic vortex x Θ ẋ 1 = 2x 1 m 2, ẋ 2 = 2x 1 l 2 Θ n Θ a { x 2 Suppose different fluid inside Γ := 1 l 2 + x 2 2 } m 2 = 1 Examine effect of weak external strain on the vortex Turner (Phys Fluids, 2014) ( 0 Choose v(x 1, x 2, t) = ε sin (x 1 l) 1 ) tanh (t 5) Theoretical expressions can be explicitly determined

56 Kirchhoff s elliptic vortex x Θ Θ n Θ a

57 y y y y Kirchoff s elliptic vortex Stills from the previous streakline movie (ε = 0.2) 1.5 t=3 1.5 t=5 1.5 t=7 1.5 t= x x x x

58 y y y y Kirchoff s elliptic vortex Stills from the previous streakline movie (ε = 0.2) 1.5 t=3 1.5 t=5 1.5 t=7 1.5 t= x x x x Computations using the theory (ε = 0.2) 1.5 t t t t

59 y y y Kirchoff s elliptic vortex 1.5 t=3 1.5 t= x x 1.5 t= x Pseudo-streaklines numerics [solid] theory [dashed]

60 y y y Kirchoff s elliptic vortex 1.5 t=3 1.5 t= x 1.5 t= x Mv Π,t,Φv Π,t t x Ε 0.2 Pseudo-streaklines numerics [solid] theory [dashed] Flux entering eddy numerics [red] theory [blue]

61 In summary... Physical flow barrier (interface between two fluids) discussed

62 In summary... Physical flow barrier (interface between two fluids) discussed These are not associated with standard LCS/FTLE/etc methods obtained purely from the velocity field

63 In summary... Physical flow barrier (interface between two fluids) discussed These are not associated with standard LCS/FTLE/etc methods obtained purely from the velocity field Streaklines are used to define these

64 In summary... Physical flow barrier (interface between two fluids) discussed These are not associated with standard LCS/FTLE/etc methods obtained purely from the velocity field Streaklines are used to define these Obtained numerically in cross-channel mixer and elliptic vortex examples

65 In summary... Physical flow barrier (interface between two fluids) discussed These are not associated with standard LCS/FTLE/etc methods obtained purely from the velocity field Streaklines are used to define these Obtained numerically in cross-channel mixer and elliptic vortex examples Theory developed for these entities, and transport across them

66 In summary... Physical flow barrier (interface between two fluids) discussed These are not associated with standard LCS/FTLE/etc methods obtained purely from the velocity field Streaklines are used to define these Obtained numerically in cross-channel mixer and elliptic vortex examples Theory developed for these entities, and transport across them In progress: enhancing transport

67 In summary... Physical flow barrier (interface between two fluids) discussed These are not associated with standard LCS/FTLE/etc methods obtained purely from the velocity field Streaklines are used to define these Obtained numerically in cross-channel mixer and elliptic vortex examples Theory developed for these entities, and transport across them In progress: enhancing transport Potential for CFD validation?

68 Thank You! Floating liquid marble (radius 0.5mm) for robust cell culture Nam-Trung Nguyen (QMNC, Griffith Univ)