Material Barriers to Momentum and Vorticity Transport George Haller - PowerPoint PPT Presentation

Material Barriers to Momentum and Vorticity Transport George Haller ETH Zürich Collaborators : Stergios Katsanoulis & Markus Holzner (ETH), Davide Gatti & Bettina Frohnapfel (KIT)

Transport barriers: frequently discussed -- rarely defined (c (c) (d) (d) Available results: (1) Barriers to advective transport : (e) (e) (f) (f) Lagrangian coherent structures (LCS) (2) Barriers to passive scalar transport : material barriers to diffusion (g) (g) (h) (h H., Karrasch & Kogelbauer, PNAS [2018], SIADS [2020] Katsanoulis, Farazmand, Serra & H., JFM [2020] H., Ann. Rev. Fluid Mech. [2015] (3) Barriers to active vectorial transport ? surfaces impeding transport of Uniform Momentum Zones (UMZ) momentum, vorticity, … Requirement: experimentally verifiable à independent of observer à theory must be objective (frame-indifferent) de Silva, Hutchins & Marusic [2014] 2/16

Objectivity : indifference to the observer “One of the main axioms of continuum mechanics C is the requirement that material response must be independent of the observer .” B M. E. Gurtin, An Introduction to Continuum Mechanics. Academic Press (1981), p. 143 A H., Lagrangian Coherent Structures, Ann. Rev. Fluid Mech. [2015] 3/16

Classic views on transport barriers (as vortex boundaries) are not objective ( ) , ( ) T T W = 1 2 ∇ v − ∇ v ⎡ ⎤ S = 1 2 ∇ v + ∇ v ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ Spin Rate of strain Tensor tensor (non-objective) (objective) ( ) > 0 2 − S 2 NASA Q = 1 W • Q-criterion: 2 ∃ j : Im λ j ( W + S ) ≠ 0 Δ -criterion: • ( ) < 0 λ 2 W 2 + S 2 λ 2 -criterion: • | v | = const ., | v i | = const . • velocity level sets: SouVR Co. Passive tracers Example : Exact linear 2D Navier-Stokes solution H. [2005], Pedergnana, Oettinger, Langlois & H. [2020] ⎛ ⎞ sin 4 t 2 + cos4 t ⎟ ⎜ ⎟ x = v ( x , t ) = ⎟ x , ! ⎜ ⎟ ⎜ ⎟ ⎜ − 2 + cos4 t − sin 4 t ⎝ ⎠ Coherent vortex by all the above principles F. J. Beron-Vera 4/16

Available results for vorticity and momentum barriers u ( x , t ) = e � 4 π 2 νt ( a cos 2 πx 2 , 0) , ω ( x , t ) = 2 πae � 4 π 2 νt sin 2 πx 2 . Example: Decaying 2D channel flow x 2 1 4 u v ( x , t ) Normalized vorticity and its ! Normalized momentum and its ! observed transport barriers observed transport barriers ! 1 1 x 1 1/4 1/4 − 1 4 ρ u 1 ( x , t ) ω ( x , t ) 0 0 ω max ( t ) ρ u max ( t ) − 1 0 -1/4 -1/4 0 1 0 1 Prior prediction for ! Prior prediction for ! vorticity transport barriers momentum transport barriers 1/4 1/4 0 0 -1/4 -1/4 0 1 0 1 H., Karrasch & Kogelbauer, SIADS [2019] Meyers & Meneveau, JFM [2013] (objective) (nonobjective) 5/16

Assumptions on the active vector field f ( x , t ) Consider general velocity field v ( x ,t ) solving the momentum equation • compressible, possibly non-Newtonian • ρ ! v = −∇ p + ∇⋅ T vis + q T vis ( x , t ): viscous stress tensor • q ( x , t ) : external body forces • see, e.g., Gurtin, Fried & Anand [2013] ! f = h vis + h nonvis , ∂ T vis h nonvis = 0 Assume : - active vector field f ( x ,t ) satisfies: • x = Q ( t ) y + b ( t ) ⇒ ! h vis = Q T ( t ) h vis . - h vis is objective : • Examples: ! f : = ρ v → f = ∇ ⋅ T vis −∇ p + q − ! ρ v Lin. momentum: ⎡ ⎤ ! f : = ( x − ˆ f = ( x − ˆ vis + ( x − ˆ x ) × ρ v → x ) ×∇ ⋅ T x ) × ! ρ u −∇ p + q Ang. momentum: ⎢ ⎥ ⎣ ⎦ ( ) + ∇ u ( ) ( ) f − ∇ ⋅ u ( ) f + 1 ! f : = ω f = ν ∇× ρ ∇ ⋅ T ρ q → ρ 2 ∇ ρ ×∇ p + ∇× 1 1 Vorticity: vis 6/16

M ( t ) What is the flux of f ( x , t ) through a material surface ? ( ) = ∫ Vorticity flux: Flux ω M ( t ) ω ⋅ n dA • M ( t ) - not the physical flux of vorticity (units!) - not objective à vortex tubes are observer-dependent ( ) = ( ) dA v ∫ Flux ρ v M ( t ) ρ v v ⋅ n Momentum flux: • M ( t ) t M ( ) M ( t ) = F - not the physical flux of momentum (units!) t 0 0 - no advection through a material surface - not objective ! f ⎡ ⎤ ( ) = ! ∫ ∫ Φ f M ( t ) f ⋅ n dA = h vis ⋅ n dA ⎢ ⎥ • Diffusive flux of f : ⎢ ⎥ vis ⎣ ⎦ M ( t ) M ( t ) - units OK ✓ à Time-normalized - objective ✓ t 1 ( ) = t 1 M ∫ ∫ ψ t 0 h vis ⋅ n dAdt 1 diffusive transport of f : 0 t 1 − t 0 t 0 M ( t ) 7/16

Active barriers : material surfaces minimizing diffusive transport of f ( ) = t 1 M ∫ t 1 ψ t 0 b t 0 ( x 0 ) ⋅ n 0 ( x 0 ) dA Theorem 1: 0 0 Notation : M 0 t 1 1 ( ) : = ( ) ∫ dt with the objective Lagrangian vector field t 1 − t 0 t 0 ( ) * − 1 ( ) ⎡ t x 0 ⎤ t x 0 ( ) ( ) , t F t h vis = ∇ F h vis F ( ) * t F ⎢ ⎥ ⎣ ⎦ t 0 t 0 t 0 t 1 ( x 0 ) : = det ∇ F t b t 0 h vis t 0 t 0 M n 0 ( x 0 ) 0 t 1 ( x 0 ) à Perfect active barriers:= b t 0 x 0 robust material surfaces with pointwise zero active transport Theorem 2: Active barriers are structurally stable 2D invariant manifolds of • Objective, steady , ′ t 1 ( x 0 ) x 0 = b t 0 Material (Lagrangian) barrier equation volume-preserving ′ x = h vis ( x ; t , v , f ) Instantaneous (Eulerian) barrier equation • Active LCS methods: passive LCS methods applied to barrier equations 2D stable and unstable manifolds ! 2D stable and unstable manifolds ! 2D invariant tori of fixed points of periodic orbits GH, Katsanoulis, Holzner, Frohnapfel & Gatti, Objective material barriers to the transport of momentum and vorticity, JFM, in revision 8/16

Example 1: Active barriers in directionally steady 3D Beltrami flows ω = k ( t ) v , v ( x , t ) = α ( t ) v 0 ( x ). 3D, unsteady, viscous e.g., unsteady ABC flow: v = e − ν t v 0 ( x ), v 0 = ( A sin x 3 + C cos x 2 , B sin x 1 + A cos x 3 , C sin x 2 + B cos x 1 ) Theorem: In all directionally steady, 3D Beltrami flows: active barriers = classic LCS t 1 ∫ k 2 νρ α ( t ) dt t 0 ′ x 0 = − v 0 ( x 0 ) Lagrangian barrier eq. vorticity norm t 1 − t 0 sectional streamlines ′ x = − νρ k 2 α ( t ) v 0 ( x ) Eulerian barrier eq. values of the Q parameter active Poincaré maps GH, Katsanoulis, Holzner, Frohnapfel & Gatti, Objective material barriers to the transport of momentum and vorticity, JFM, in revision 9/16

Example 1: Active LCS methods for the ABC flow 10 ( x 0 ; ω ) 15 ( x 0 ; ω ) 5 ( x 0 ) FTLE 0 aFTLE 0,5 aFTLE 0,5 x 15 ( x 0 ; ω ) 50 ( x 0 ; ω ) 5 ( x 0 ) PRA 0 aPRA 0,5 aPRA 0,5 GH, Katsanoulis, Holzner, Frohnapfel & Gatti, Objective material barriers to the transport of momentum and vorticity, JFM, in revision 10/16

Example 2: Active transport barriers in 2D incompressible Navier-Stokes flows data set: Mohammad Farazmand (NCS) Eulerian momentum barriers at time t =0 0.05 ( x ; ρ u ) 0.15 ( x ; ρ u ) 0 ( x ) FTLE 0 FTLE 0,0 FTLE 0,0 y y y y y y In 2D: Eulerian momentum barrier eq. is an x x x autonomous Hamiltonian system! 0.1 ( x ; ρ u ) 0.15 ( x ; ρ u ) 0 ( x ) ′ PRA 0 PRA 0,0 x = J ∇ H ( x ), PRA 0,0 y y y y y y H ( x ) = νρ ! ω z ( x ; t ). x x x GH, Katsanoulis, Holzner, Frohnapfel & Gatti, Objective material barriers to the transport of momentum and vorticity, JFM, in revision 11/16

Example 2: Lagrangian momentum and vorticity barriers over [t 0 ,t 1 ] = [0,25] 0.35 ( x 0 ; ρ u ) 0.05 ( x 0 ; ω ) 25 ( x 0 ) FTLE 0 FTLE 0,25 FTLE 0,25 y y y y y y y y y y y y In 2D: Lagrangian momentum barrier eq. is an autonomous Hamiltonian system! ′ x = J ∇ 0 H ( x 0 ), t ( x 0 ), t ). H ( x 0 ) = νρ ω z ( F x x x x x x t 0 0.35 ( x 0 ; ρ u ) 0.05 ( x 0 ; ω ) 25 ( x 0 ) PRA 0,25 PRA 0,25 PRA 0 y y y y y y In 2D: Lagrangian vorticity barrier eq. is an autonomous Hamiltonian system! ′ x = J ∇ 0 H ( x 0 ), t 1 ( x 0 ), t 1 ) − ω z ( x 0 , t 0 )]. H ( x 0 ) = νρ [ ω z ( F t 0 x x x GH, Katsanoulis, Holzner, Frohnapfel & Gatti, Objective material barriers to the transport of momentum and vorticity, JFM, in revision 12/16

Example 2: Coherence of material barriers to momentum transport Momentum-barrier evolution and momentum norm | ρ u ( x , t ) | t = 25 t = 0 y y y y y y y y in Eulerian ! coordinates x x | ρ u ( F t = 0 t = 25 t ( x 0 ), t ) | y y y y y y y y 0 in Lagrangian ! coordinates x x GH, Katsanoulis, Holzner, Frohnapfel & Gatti, Objective material barriers to the transport of momentum and vorticity, JFM, in revision 13/16

Example 3: Active transport barriers in 3D channel flow (Re=3,000) Eulerian active barriers at time t =0 from FTLE (a) 2 400 FTLE 0 0 ( x ) 6 1 . 5 300 4 2 y/h y + 1 200 2 1 0 . 5 100 0.3 0 30 0 0 0 (b) aFTLE 31 2 400 0 , 0 ( x ; ρ u ) 6 1 . 5 300 4 4 y/h y + 1 200 2 2 0 . 5 100 0.3 30 0 0 0 0 (c) 2 400 aFTLE 0 . 62 0 , 0 ( x ; ω ) 1 . 5 300 6 4 y/h y + 1 200 4 2 0 . 5 100 0.3 2 30 0 0 0 0 1 2 3 4 5 6 0 z/h GH, Katsanoulis, Holzner, Frohnapfel & Gatti, Objective material barriers to the transport of momentum and vorticity, JFM, in revision 14/16