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

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

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

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

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.

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Scalar decay

Concentration C(x, t) obeys the advection–diffusion equation: ∂tC + 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 ˙ X = u(X, t) + √ 2κ ˙ W . Interested in the long-time limit: t → ∞ .

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Scalar decay

Bounded domains

◮ C(x, t) →

• C(x, 0) dx = 0 as t → ∞,

◮ for time-independent flows, ∂tu = 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: lim

κ→0 λ(κ) = λ(0) = 0 ,

Relate λ(0) and corrections to flow characteristics. Part I

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Scalar decay

Unbounded domains

◮ for t large enough, scalar scale ≫ tracer scale, ◮ homogenisation theory applies, ◮ ∂tC ≈ κeff∆C, ◮ Gaussian approximation

C(x, t) ∝ e−|x|2/(4κefft) ,

◮ breaks down in the tails, |x| ≫ O(t1/2).

Predict C(x, t) for |x| = O(t) using large deviations. Part II

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Part I

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

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).

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Random flows

Tangent dynamics: δ˙ x = Du · δx. Finite-time Lyapunov exponent h = t−1 log δx 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.

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Random flows

Covariance equation

Take u to be:

◮ homogeneous in x, in a periodic domain, ◮ white in time (Kraichnan flows), or iid in [nτ, (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 ui(x + y, t)uj(y, t′) = Bij(x)δ(t − t′), ∂tΓ = Dij(x)∂2

ijΓ + κ∆Γ ,

where Dij(x) = Bij(x) − Bij(0). Note: Dij(x) ∼ Sijklxkxl as |x| → 0.

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Random flows

Covariance equation

Eigenvalue of Dij(x)∂2

ij + κ∆ gives decay rate of Γ:

lim

t→∞ t−1 log E C2 = γ = smallest e’value.

Numerical observation: γ ≈ 2λ(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(δx, t)

satisfies ∂tp = Sijklδxkδxl∂ijp,

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Random flows

Two regimes

• 1. locally controlled regime, associated with

edge of continuous spectrum, γlocal = g(0) + 2π2 g′′(0) 1 log2 κ + o(1/ log2 κ),

• 2. globally controlled regime, associated

with isolated eigenvalues. γglobal ∼ λ0 + cκσ, with σ related to g(h).

x x

• = 0

= 0 /

2

1/log 1/log

x

• x

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Random flows

◮ local control consistent with prediction

λ(κ) ≈ γ/2 = g(0)/2,

◮ global control when L Lflow flow scale; recovers

homogenisation for L ≫ Lflow,

◮ 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, 2πP]3 for P = 1, 2, · · · . xn+1 = xn+π sin(yn + φn), yn+1 = yn + π sin(zn + ψn), zn+1 = zn + π sin(xn+1 + ϕn)

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

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

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Random flows

Numerical results

Results for P = 1: forward and inverse maps.

10−6 10−5 10−4 10−3 10−2 10−1 κ 1.0 1.2 1.4 1.6 1.8 2.0 γ

Decay rate γ vs κ

2 4 6 8 10 P 0.0 0.5 1.0 1.5 2.0 γ

Decay rate γ vs P Estimate transition P: γlocal ∼ γglobal ∼ γhom ⇒ P = 2.2

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Bounded domain: random flows

Numerical results

Forward map, n = 30, P = 2, 3, 4, κ = 10−4

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Part II

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

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 κeff :

◮ E X ⊗ X ∼ 2κefft, ◮ C ≍ exp(−x · κ−1 eff · x/(4t)): Gaussian distribution, ◮ effective equation

∂tC = ∇ · (κeff · ∇C). In simple flows: κeff can be computed explicitly.

◮ shear flows (Taylor dispersion), ◮ periodic flows. e.g. Majda & Kramer 1999

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Large deviations

Shear dispersion: dye in pipe flows spreads along the pipe. κeff = κ−1 y

−1

U(y′) dy′ 2 + κ ∝ κ−1 .

Taylor 1953

Cellular flow: ψ = − sin x sin y κeff ∼ 2νκ1/2 for κ ≪ 1,

x y 6 4 2 2 4 6 6 4 2 2 4 6

with ν ∼ 0.5327407 · · ·

Shraiman, Rosenbluth et al, Childress, Soward. ..

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Large deviations

Diffusive approximation assumes x/t1/2 = 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.

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Large deviations

For t ≫ 1, the concentration takes the large-deviation form C(x, t) ≍ exp(−tg(ξ)) for ξ = x/t = O(1), with g the rate function, convex with g(0) = g′(0) = 0. Computing g: define f(q) by etf(q) ≍ E eq·X . f and g are a Legendre transform pair. Eigenvalue problem: f(q) is e’value, φ e’function: ∆φ − (Pe u + 2q) · ∇φ +

• Pe u · q + |q|2

φ = f(q)φ. Solve for q ∈ Rd. Deduce g(q) by Legendre transform.

Haynes & V 2014a

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Large deviations: cellular flow

Cellular flow u = (−ψy, ψx), with ψ = − sin x sin y. For Pe ≫ 1, par- ticles are trapped inside cells, with rare exits across separatrices.

x y

−20 −10 10 20 −20 −10 10 20

x y

−20 −10 10 20 −20 −10 10 20

log C at t = 2, 4 for Pe = 250.

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Large deviations: cellular flow

Solve the e’value problem

◮ numerically, ◮ asymptotically for

Pe = UL/κ ≪ 1.

−1 −0.5 0.5 1 1 2 3 4

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.

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Large deviations: rectangular network

O L βL βL U V

Non-Gaussian behaviour induced by geometry.

Tzella & V 2016

Rate function g:

◮ for U = V = 0: from g ∼ |ξ|2/2 to g ∼ (|ξ1| + |ξ2|)2/4, ◮ for U, V ≫ 1: g independent of κ, topological dispersion.

U = V = 0

−5 5 −5 5 5 5 5 10 15 20 25

U = V = 5

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Large deviations: rectangular network

◮ Monte Carlo

simulations vs large deviations

◮ t = 1, 5, ◮ large deviations

work well at moderately large time.

SCALAR DECAY BOUNDED DOMAINS UNBOUNDED DOMAINS CONCLUSIONS

Conclusions

◮ Scalar decay in bounded random flows

◮ exponential decay: of C ∼ exp(−λ(κ)t), ◮ dissipative anomaly: limκ→0 λ(κ) = 0

(under which conditions?)

◮ E C2 ∼ exp(−γ(κ)t): local vs global control of γ(κ)

(but is λ(κ) = γ/2?)

◮ Scalar decay in unbounded flows

◮ homogenisation, C ≍ exp(−x|2/(4κt) for |x| = O(t1/2), ◮ large deviations, C ≍ exp(−tg(x/t)) for |x| = O(t), ◮ rate function g(·) found by solving a family of e’value

problems,

◮ large-Pe asymptotics for cellular flows, ◮ mixing in ‘interesting’ topologies.