Passive scalar decay in bounded and unbounded fluid flows Jacques - PowerPoint PPT Presentation

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Passive scalar decay in bounded and unbounded fluid flows Jacques Vanneste School of Mathematics and Maxwell Institute University of Edinburgh, UK www.maths.ed.ac.uk/˜vanneste with P H Haynes, K Ngan, A Tzella

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Scalar decay Passive scalar (eg pollutant) released in fluid flow homogenise quickly through: ◮ differential advection, creating fine scales; ◮ molecular diffusion. Aims: ◮ characterise the decay of concentration fluctuations, ◮ relate it to flow and domain characteristics.

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Scalar decay Concentration C ( x , t ) obeys the advection–diffusion equation: ∂ t C + u · ∇ C = κ ∆ C , with a given flow u ( x , t ) that ◮ satisfies ∇ · u = 0, ◮ is smooth, � u ( x , t ) − u ( x ′ , t ) � ∝ | x − x ′ | as | x − x ′ | → 0. Also interpreted as the pdf of particles positions X ( t ) with √ ˙ 2 κ ˙ X = u ( X , t ) + W . Interested in the long-time limit: t → ∞ .

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Scalar decay Bounded domains ◮ C ( x , t ) → � C ( x , 0 ) d x = 0 as t → ∞ , ◮ for time-independent flows, ∂ t u = 0, C ( x , t ) ∼ e − λ ( κ ) t , with λ ( κ ) smallest e’value of u · ∇ − κ ∆ , ◮ for time-periodic flows, λ ( κ ) is a Floquet exponent, ◮ for stationary random flows, λ ( κ ) is a Lyapunov exponent. For ‘sufficiently mixing’ flows, dissipative anomaly: κ → 0 λ ( κ ) = λ ( 0 ) � = 0 , lim Relate λ ( 0 ) and corrections to flow characteristics. Part I

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Scalar decay Unbounded domains ◮ for t large enough, scalar scale ≫ tracer scale, ◮ homogenisation theory applies, ◮ ∂ t C ≈ κ eff ∆ C , ◮ Gaussian approximation C ( x , t ) ∝ e −| x | 2 / ( 4 κ eff t ) , ◮ breaks down in the tails, | x | ≫ O ( t 1 / 2 ) . Predict C ( x , t ) for | x | = O ( t ) using large deviations. Part II

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Part I

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Random flows Take u to be stationary random process: ◮ C ( x , t ) decays exponentially for t ≫ 1: C ( x , t ) ∼ D ( x , t ) e − λ ( κ ) t , ◮ λ is the deterministic Lyapunov exponent of the advection–diffusion equation, ◮ D ( x , t ) is a stationary random function, strange eigenmode, Pierrehumbert 1998 Lagrangian stretching theories Antonsen et 1996, Falkovitch et al 2000, Balkovsky & Fouxon 1999 Relate λ ( 0 ) to large-deviations of finite-time Lyapunov exponents of ˙ x = u ( x , t ) .

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Random flows Tangent dynamics: δ ˙ x = D u · δ x . Finite-time Lyapunov exponent h = t − 1 log � δ x � have pdf p ( h , t ) ≍ e − tg ( h ) , where g is the rate function . Prediction: scalar decay rate λ ( 0 ) = g ( 0 ) / 2 . ◮ numerical evidence inconclusive (finite- κ effects), ◮ prediction cannot always hold: for small-scale flows, homogenisation applies and λ ( κ ) ∝ L − 2 .

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Random flows Covariance equation Take u to be: ◮ homogeneous in x , in a periodic domain, ◮ white in time (Kraichnan flows), or iid in [ n τ, ( n + 1 ) τ ] , n = 1 , 2 · · · (renewing flows). Closed equation for the covariance Γ( x , t ) = E C ( x + y , t ) C ( y , t ) . For Kraichan flows, with E u i ( x + y , t ) u j ( y , t ′ ) = B ij ( x ) δ ( t − t ′ ) , ∂ t Γ = D ij ( x ) ∂ 2 ij Γ + κ ∆Γ , where D ij ( x ) = B ij ( x ) − B ij ( 0 ) . Note: D ij ( x ) ∼ S ijkl x k x l as | x | → 0.

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Random flows Covariance equation Eigenvalue of D ij ( x ) ∂ 2 ij + κ ∆ gives decay rate of Γ : t →∞ t − 1 log E � C � 2 = γ = smallest e’value . lim Numerical observation: γ ≈ 2 λ ( κ ) . Asymptotic analysis of the eigenvalue problem for κ → 0: Haynes & V 2005 ◮ singular perturbation problem, ◮ for κ = 0, continuous spectrum + discrete eigenvalues, ◮ relation to finite-time Lyapunov exponents: p ( δ x , t ) satisfies ∂ t p = S ijkl δ x k δ x l ∂ ij p ,

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Random flows Two regimes 1. locally controlled regime, associated with edge of continuous spectrum, γ local = g ( 0 ) + 2 π 2 1 � = 0 + o ( 1 / log 2 κ ) , log 2 κ x x g ′′ ( 0 ) 2. globally controlled regime, associated � = 0 / x x with isolated eigenvalues. 2 � � 1/log � 1/log � 0 γ global ∼ λ 0 + c κ σ , with σ related to g ( h ) .

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Random flows ◮ local control consistent with prediction λ ( κ ) ≈ γ/ 2 = g ( 0 ) / 2, ◮ global control when L � L flow flow scale; recovers homogenisation for L ≫ L flow , ◮ confirmed numerically in 2D, Haynes & V 2005, Tsan et al 2005 ◮ in 3D, γ is the same for a flow and its time reverse (exact result). Numerical simulations Ngan & V 2011 3D sine map in periodic domain [ 0 , 2 π P ] 3 for P = 1 , 2 , · · · . x n + 1 = x n + π sin ( y n + φ n ) , y n + 1 = y n + π sin ( z n + ψ n ) , z n + 1 = z n + π sin ( x n + 1 + ϕ n )

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Random flows Numerical results Forward map, P = 1, κ = 10 − 4 , n = 2 , 4 , 6 Inverse map, P = 1, κ = 10 − 4 , n = 2 , 4 , 6

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Random flows Numerical results Results for P = 1: forward and inverse maps. 2.0 2.0 1.8 1.5 1.6 1.0 γ γ 1.4 0.5 1.2 0.0 1.0 0 2 4 6 8 10 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 P κ Decay rate γ vs κ Decay rate γ vs P Estimate transition P : γ local ∼ γ global ∼ γ hom ⇒ P = 2 . 2

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Bounded domain: random flows Numerical results Forward map, n = 30, P = 2 , 3 , 4, κ = 10 − 4

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Part II

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Large deviations Localised release of passive scalar in an unbounded domain: examine spread of C ( x , t ) . For t ≫ 1 , scalar scale ≫ flow scale: homogenisation applies: can compute an effective diffusivity κ eff : ◮ E X ⊗ X ∼ 2 κ eff t , ◮ C ≍ exp ( − x · κ − 1 eff · x / ( 4 t )) : Gaussian distribution, ◮ effective equation ∂ t C = ∇ · ( κ eff · ∇ C ) . In simple flows: κ eff can be computed explicitly. ◮ shear flows (Taylor dispersion), ◮ periodic flows. e.g. Majda & Kramer 1999

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Large deviations Shear dispersion: dye in pipe flows spreads along the pipe. �� y � 2 � + κ ∝ κ − 1 . κ eff = κ − 1 � U ( y ′ ) d y ′ Taylor 1953 − 1 6 Cellular flow: ψ = − sin x sin y 4 2 0 y κ eff ∼ 2 νκ 1 / 2 for κ ≪ 1 , � 2 � 4 � 6 � 6 � 4 � 2 0 2 4 6 x with ν ∼ 0 . 5327407 · · · Shraiman, Rosenbluth et al, Childress, Soward. ..

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Large deviations Diffusive approximation assumes x / t 1 / 2 = O ( 1 ) as t → ∞ . It cannot describes the tails of C ( x , t ) which are non-Gaussian. Large deviations: ◮ obtain C ( x , t ) for x / t = O ( 1 ) , ◮ recover homogenisation as a limiting case. Interest: ◮ Low concentrations can be important: ◮ anecdotally: highly toxic chemicals, ◮ exactly: FKPP fronts. Tzella’s talk ◮ Unifies ‘improvements’ to homogenisation. ◮ Example of extreme-event statistics.

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Large deviations For t ≫ 1, the concentration takes the large-deviation form C ( x , t ) ≍ exp ( − tg ( ξ )) ξ = x / t = O ( 1 ) , for with g the rate function, convex with g ( 0 ) = g ′ ( 0 ) = 0. Computing g : define f ( q ) by e tf ( q ) ≍ E e q · X . f and g are a Legendre transform pair. Eigenvalue problem: f ( q ) is e’value, φ e’function: Pe u · q + | q | 2 � � ∆ φ − ( Pe u + 2 q ) · ∇ φ + φ = f ( q ) φ. Solve for q ∈ R d . Deduce g ( q ) by Legendre transform. Haynes & V 2014a

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Large deviations: cellular flow Cellular flow u = ( − ψ y , ψ x ) , with ψ = − sin x sin y . For Pe ≫ 1, par- 20 20 ticles are trapped 10 10 y y inside cells, with 0 0 rare exits across −10 −10 separatrices. −20 −20 −20 −10 0 10 20 −20 −10 0 10 20 x x log C at t = 2 , 4 for Pe = 250.

S CALAR DECAY B OUNDED DOMAINS U NBOUNDED DOMAINS C ONCLUSIONS Large deviations: cellular flow 4 Solve the e’value problem 3 ◮ numerically, 2 ◮ asymptotically for 1 Pe = UL /κ ≪ 1. 0 − 1 − 0.5 0 0.5 1 x/t Three regimes: 1. | x | / t = O ( Pe − 3 / 4 ) : averaging in interior + boundary layers, 2. | x | / t = O ( log Pe ) : empty interior, boundary layers + corners, 3. | x | / t = O ( Pe ) : action-minimising trajectories (Freidlin–Wentzel). Haynes & V 2014b More in Tzella’s talk.