Bimolecular chemical reaction in two-dimensional Navier-Stokes flow

Bimolecular chemical reaction in two-dimensionalNavier-Stokes flow

Farid Ait-Chaalal

Under the supervision ofProf. Peter Bartello and Prof. Michel Bourqui

McGill UniversityDepartment of Atmospheric and Oceanic Sciences

PhD defenseApril 18, 2012

Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 1 / 27

Motivation: mixing of chemicals in the stratosphere.Stratospheric dynamics.

Average atmosphere

temperature profile. From

Zonal mean dynamics of the stratosphere from

Haynes (2000). The isolines from 300 to 850

indicate the potential temperature of the

isentropes in Kelvin.Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 2 / 27

Motivation: mixing of chemicals in the stratosphere.

Tracer distribution in thestratosphere advected by wind from

reanalysis (January 1992) on anisentrope (450K). The tracer are

initiated as potential vorticity (PV)contours and the integration is runfor 12 days. From Waugh (1994).

1 Exponential lengthening oftracer filaments (e-foldingtime of about 5 days, whichcorresponds to a strain of 0.2day−1).

2 Small scales processes can beparametrized by an effectivediffusivity of about 103 m2/sto 104 m2/s: typical width offilaments O(10km)

Motivation: Ozone depletion in the winter-timestratospheric surf zone.

1 Depletion controlled by the deactivation of activated chlorineoriginating from the polar vortex with NOx originating from lowlatitudes (Tan et al., 1998)

2 Stirring of chemicals into filaments and subsequent mixing essential:strong dependence of the chemical concentration on resolution inclimate-chemistry models (Tan et al., 1998)

3 Only a few studies tackle this problem from a theoretical point ofview (Thuburn and Tan, 1997; Wonhas an Vassilicos, 2003)

Objectives and methodology.

1 Our flow: doubly periodic barotropic two-dimensional flow as asimplified model for isentropic stirring in the stratospheric surf zone.

2 Our chemical reaction A + B −→ C is infinitely fast (controlled bydiffusion).

3 Numerical simulations: ensemble of simulations for various diffusioncoefficients 1 ≤ Pr = diffusion κ

viscosity ν≤ 128 in a doubly periodic box

[−π, π]2. The viscosity ν is kept constant.

4 Theoretical approach: local Lagrangian straining theory (LLST,Antonsen, 1996). How does the chemical production depend on thetracer diffusion? Can we relate the concentration of the chemicals tothe Lagrangian straining properties of the flow?

An infinitely fast chemical reaction.

We study the bimolecular reaction: A + B −→ CEulerian equations for the concentrations Ci (x, t), i = A,B,C ,in the flow u:

∂t+ u · ∇CA = κ∇2CA − kcCACB

∂t+ u · ∇CB = κ∇2CB − kcCACB

∂t+ u · ∇CC = κ∇2CC + kcCACB ,

φ = CA − CB is a passive tracer:

∂t+ u · ∇φ = κ∇2φ

A and B react instantaneously: they cannot coexist. The passive tracerφ = CA − CB gives CA and CB through:{

CA(x, t) = φ(x, t) and CB(x, t) = 0 if φ(x, t) > 0CB(x, t) = −φ(x, t) and CA(x, t) = 0 if φ(x, t) < 0

The space average of the concentrations are:

CA = CB =|φ|2

CC =|φ(t = 0)| − |φ|

A and B react instantaneously: they cannot coexist. The passive tracerφ = CA − CB gives CA and CB through:{

CA(x, t) = φ(x, t) and CB(x, t) = 0 if φ(x, t) > 0CB(x, t) = −φ(x, t) and CA(x, t) = 0 if φ(x, t) < 0

The space average of the concentrations are:

CA = CB =|φ|2

CC =|φ(t = 0)| − |φ|

Results: general approach.

We want to understand the behavior of d |φ|dt , how it depends on diffusion,

what determines its time evolution. We call it the chemical speed.

We can write:

2Ad |φ|

dt= −κ

∫L(t)∇φ · n dl = −κ

∫L(t)|∇φ| dl

Where the contact line L is the set {x|φ(x) = 0} (A is the total area ofthe domain).Three regimes:

Initial regime: L is a clearly defined material line. It does not dependon diffusion. Increase of the chemical speed.

Intermediate regime: merging of tracer filaments. The chemical speedreaches a maximum.

Long time decay of the tracer fluctuations and of the chemical speed.

what determines its time evolution. We call it the chemical speed.We can write:

2Ad |φ|

dt= −κ

∫L(t)∇φ · n dl = −κ

∫L(t)|∇φ| dl

Where the contact line L is the set {x|φ(x) = 0} (A is the total area ofthe domain).

Three regimes:

what determines its time evolution. We call it the chemical speed.We can write:

2Ad |φ|

dt= −κ

∫L(t)∇φ · n dl = −κ

∫L(t)|∇φ| dl

Where the contact line L is the set {x|φ(x) = 0} (A is the total area ofthe domain).Three regimes:

Three regimes:

Results: general approach.Initial regime.

We follow individual contact line elements, calculate the evolution oftheir length in a Lagrangian framework, and of the correspondingadvected gradient ∇φ.

This is justified by a separation of scale: the contact line is clearlydefined and the contact zone is small compared to the flow scale aslong as t . Tmix ≈ T ln Pe = T ln RePr , where Tmix is the mixingtime scale from the large scale to the diffusive scale.

Lagrangian straining properties (LSP) of the flow:finite time Lyapunov exponent (FTLE).

Definition:

λ(x, t) =1

tmaxα

lim|δl0|→0

|δl(t)||δl0|

}The maximum is calculated over all the possible orientations α of |δl0|.

We define a singular vector ψ+(x, t) corresponding to the direction wherethis maximum is reached.

Lagrangian straining properties (LSP) of the flow:finite time Lyapunov exponent (FTLE).

Definition:

λ(x, t) =1

tmaxα

lim|δl0|→0

|δl(t)||δl0|

}The maximum is calculated over all the possible orientations α of |δl0|.

We define a singular vector ψ+(x, t) corresponding to the direction wherethis maximum is reached.

Lagrangian straining properties (LSP) of the flow:FTLE probability density function (pdf).

FTLE PDFs shown at different times

In ergodic chaotic flow, the pdf of the FTLE is given, forsufficiently large times, by the large deviation theory result (e.g.Balkovsky(1999), Ott(2002)):

ePλ(t, λ) =

stG ′′(λ0)

2πexp(−tG(λ)), (3)

where G(λ) is the Cramer function. It is concave, minimum atλ0 with G(λ0) = G ′(λ0) = 0.λ0 is the infinite time Lyapunov exponent (slow algebraicconvergence).

ePλ(t, λ) =

stG ′′(λ0)

2πexp(−tG(λ)), (3)

where G(λ) is the Cramer function. It is concave, minimum atλ0 with G(λ0) = G ′(λ0) = 0.

λ0 is the infinite time Lyapunov exponent (slow algebraicconvergence).

ePλ(t, λ) =

stG ′′(λ0)

2πexp(−tG(λ)), (3)

ePλ(t, λ) =

stG ′′(λ0)

2πexp(−tG(λ)), (3)

Example of strain map (FTLE as time goes to 0).

Time evolution of the FTLE maps (length of the animation: 20T ), plotted at theinitial positions of the trajectories. This is the map of future stretching rates ofLagrangian parcels.

The singular vectors converge to the (forward) Lyapunov vector exponentially fastin time: it will be assumed constant (when needed).

Time evolution of the FTLE maps (length of the simulation: 20T ), plotted at theinitial positions of the trajectories. This is the map of future stretching rates ofLagrangian parcels.

The singular vectors converge to the (forward) Lyapunov vector exponentially fastin time: it will be assumed constant (when needed).

Lagrangian straining properties of the flow:equivalent times.

We define two equivalent times:

∫ t0 e2uλ(u)du

e2tλ(t)and τ =

0e−2uλ(u)du.

The time τ measures the inverse of the Lagrangian stretching over ashort time before t on a chaotic orbit (Antonsen (1996), Haynes andVanneste (2004)), for a long enough integration time t.

The time τ measures the inverse of the Lagrangian stretching over ashort after t = 0, for a long enough integration time t.

∫ t0 e2uλ(u)du

e2tλ(t)and τ =

0e−2uλ(u)du.

∫ t0 e2uλ(u)du

e2tλ(t)and τ =

0e−2uλ(u)du.

Probability density function of 1/τ :

Joint probability density function of λ and 1/τ :

Probability density function of 1/τ :

Joint probability density function of λ and 1/τ :

Contact line lengthening.

Ensemble average 〈L〉 ofthe length of the contactline.

With γ the angle betweenthe contact line and thesingular vector at theinitial time, we have:

〈L〉 = L0

∫ ∞λ=0

∫ 2π

γ=0Pλ(t, λ)

[e2λt cos2 γ + e−2λt sin2 γ

dλdγ

∼t�T

LE =2L0

∫ ∞0

Pλ(t, λ)eλtdλ

Asymptotically LE ≈ eλ1t with λ1 = maxλ[λ− G (λ)].Farid Ait-Chaalal (McGill) Chemistry in 2D flow PhD defense April 18, 2012 18 / 27

Advection of gradients along the contact line.

Ensemble average of thegradients advected withthe contact line,multiplied by

√κπ

In the limit of infinite initial gradients, wecan relate them to the LSP:

〈|∇φL|〉 =A0√πκ

〈L〉

ZZZZe2λt cos2 γ + e−2λt sin2 γpτe2λt cos2 γ + eτ sin2 γ

eP(t, λ, τ, eτ) dλ dτ d eτ dγ

2πCalculation from the LSP

∼t�T

2A0√π3κ

ZZeλt

Pλ,τ (t, λ, τ) dλ dτ Calculation from the LSP

contact line equilibrated with the flow

With the statistical independence between λ and τ , we could approximate, at large times (t � T ), 〈|∇φL|〉 by A0/√πκτ .

Gradients along the contact line:probability density function.

Using the expression for the gradient advected along each contact lineelement, we can show that the probability density function of

G =√πκ

A0|∇φL| is:

PG ,L(t, g) =

∫∫ dγπ dlPG ,λ(t, g , l)

√e2lt cos2 γ + e−2lt sin2 γ∫∫ dγ

π dlPλ(t, l)√

e2lt cos2 γ + e−2lt sin2 γ

For a contact line equilibrated with the flow:

PG ,L(t, g) ∼ PG ,L,∞(t, g) =

∫dlP 1√

τ,λ(t, g , l)e lt∫

dlPλ(t, l)e lt

If τ and λ were independent the pdf of G along the contact line would beequal to the pdf of 1√

Gradients along the contact line:probability density function.

Using the expression for the gradient advected along each contact lineelement, we can show that the probability density function of

G =√πκ

A0|∇φL| is:

PG ,L(t, g) =

∫∫ dγπ dlPG ,λ(t, g , l)

√e2lt cos2 γ + e−2lt sin2 γ∫∫ dγ

π dlPλ(t, l)√

e2lt cos2 γ + e−2lt sin2 γ

For a contact line equilibrated with the flow:

PG ,L(t, g) ∼ PG ,L,∞(t, g) =

∫dlP 1√

τ,λ(t, g , l)e lt∫

dlPλ(t, l)e lt

If τ and λ were independent the pdf of G along the contact line would beequal to the pdf of 1√

Gradients along the contact lineprobability density function.

Comparison between the pdf PG ,Pr

of G along the contact lineapproximated from the directnumerical simulations and thecalculation from the LSP PG ,L,PG ,L,∞ and the pdf of 1√

τ. We have

only plotted the curves PG ,Pr

corresponding to direct numericalsimulations consistent with theinfinite initial gradient hypothesis.

Chemical speed.

Ensemble average of thechemical speed dividedby the diffusion κ.

−〈d|φ|dt〉 =

L0A0√πA√κ

ZZZZe2λt cos2 γ + e−2λt sin2 γpτe2λt cos2 γ + eτ sin2 γ

eP(t, λ, τ, eτ) dλ dτ d eτ dγ

2πCalculation from the LSP

∼t�T

2L0A0√π3A

ZZeλt

Pλ,τ (t, λ, τ) dλ dτ Calculation from the LSP

contact line equilibrated with the flow

Probability distribution function of |φ|.

As long as there is separation of scale between the contact zone and the flow, we canfind a small ε such that the pdf PΦ of φ is given by:

PΦ(φ) =4

√κ〈L〉〈 1

G〉Erf−1′ ` φ

´for φ ∈ [0,A0(1− ε)]. (4)

(It is normalized when considering values of Φ larger than A0(1− ε)).

Time evolution of the pdf of A0 − φ,multiplied by

√Pr 1〈1/G〉〈L〉 , calculated from

trajectories. The red curve (theoretical

prediction) corresponds to 4√ν

AA0Erf−1′

where Erf is the Gauss error function.Log-log scale.

Long-term decay.

Decay of 〈φ2〉 for various Prandtl numbers:

Decay of 〈φ2〉 at Pr = 128 for different members:

Two successive exponential decays:

In the first regime, the decay seemsglobally controlled, and not predictedby local Lagrangian straining theories(LLST). However LLST are successfulin predicting the shape of the pdf ofφ, away from the tails, and somefeatures of the variance spectrum.

In the second regime, the systemkeeps a memory of the initialconditions and the decay is moresensitive to them than in regime firstregime. However the decay of 〈φ2〉might not be controlled by anymechanism described in the literature(global or local). Tracer captured invortices and ejected with tracerfilaments seem to be an importantprocess for the control of the decay.

Conclusions

LLST adapted to the study of an initial regime (5 to 10 T). 〈d |φ|dt 〉scales like

√κ. Rare events in the FTLE pdf determine the global

chemical production.

LLST also give how gradients pdf along the contact line andchemicals pdf scale with diffusion.

In the long-term decay, previous theories or frameworks (LLST,strange eigenmode) do not seem directly applicable to flows solutionof the Navier-Stokes equation, in particular because they do notcapture the role of coherent transient structures.

Conclusions

What next?

What about the assumption of a stationary singular vector?

Intermediate regime. Use of LLST and of the fractal structure of thecontact line?

Slower chemistry, other kinds of reactions.

More realistic flows: critical layers, vertical structure.

LSP: time evolution of the FTLE pdf, existence of a Cramer function,statistical dependence between FTLE and equivalent times, equivalenttimes pdf.

Mechanisms for the decay of tracer fluctuations in two-dimensionalNavier-Stokes flows.

What next?

Thanks!

Bimolecular chemical reaction in two-dimensional Navier-Stokes flow

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