Improving global stability analysis of Kolmogorov fl ows using - - PowerPoint PPT Presentation

Improving global stability analysis of Kolmogorov flows using enstrophy Yue-Kin Tsang School of Mathematics and Statistics University of St Andrews William R. Young (SIO, UCSD) Richard R. Kerswell (Mathematics, Bristol)

Kolmogorov Flows

ζt + uζx + vζy + βψx = −µζ + cos x + ν∇2ζ velocity: (u, v) = (−ψy, ψx) (2-D periodic domain) vorticity: ζ(x, y) = vx − uy = ∇2ψ single-scaled sinusoidal body force (at kf = 1)

initially used by Kolmogorov (with β = µ = 0) to study bifurcations as Reynolds number increases subsequent work by others: viscous linear stability (Meshalkin & Sinai), weakly nonlinear theory (Sivashinsky), energy stability (Fukuta & Murakami) quasi-2D laboratory experiments with approximate sinusoidal forcing (e.g. Rivera & Wu, Burgess et al.)

Kolmogorov Flows

ζt + uζx + vζy + βψx = −µζ + cos x + ν∇2ζ velocity: (u, v) = (−ψy, ψx) (2-D periodic domain) vorticity: ζ(x, y) = vx − uy = ∇2ψ Two-dimensional flows:

dual conservation of energy E and enstrophy Z, E = 1 2

• |∇ψ|2

, Z = 1 2

• (∇2ψ)2

nonlinear interactions transfer E simultaneously up (k < 1) and down (k > 1) scale large-scale dissipation (e.g. sidewall drag, Ekman friction) needed to achieve statistically steady state

Kolmogorov Flows

ζt + uζx + vζy + βψx = −µζ + cos x + ν∇2ζ velocity: (u, v) = (−ψy, ψx) (2-D periodic domain) vorticity: ζ(x, y) = vx − uy = ∇2ψ Geophysical flows:

variation of the Coriolis parameter with latitude modeled using the β-plane approximation Kolmogorov flow on a β-plane as a model of zonal jet formation

Kolmogorov Flows

ζt + uζx + vζy + βψx = −µζ + cos x + ν∇2ζ velocity: (u, v) = (−ψy, ψx) (2-D periodic domain) vorticity: ζ(x, y) = vx − uy = ∇2ψ Geophysical flows:

variation of the Coriolis parameter with latitude modeled using the β-plane approximation Kolmogorov flow on a β-plane as a model of zonal jet formation

We shall first consider the inviscid case: ν = 0

Stability of the Laminar Solution

ζL(x) = a cos(x − xβ)

a = 1

• β2 + µ2

, xβ = tan−1 β µ

−1 −0.5 0.5 1

x y µ = 0.5 β = 1.0

−10 −5 5 10

x y µ = 0.1 β = 0.0

Stability of the Laminar Solution

ζL(x) = a cos(x − xβ)

a = 1

• β2 + µ2

, xβ = tan−1 β µ

−1 −0.5 0.5 1

x y µ = 0.5 β = 1.0

−10 −5 5 10

x y µ = 0.1 β = 0.0

Stability of the Laminar Solution

ζL(x) = a cos(x − xβ)

a = 1

• β2 + µ2

, xβ = tan−1 β µ

−1 −0.5 0.5 1

x y stable µ = 0.5 β = 1.0

−10 −5 5 10

x y unstable µ = 0.1 β = 0.0

Goal: Neutral Curve

ζt + uζx + vζy + βψx = −µζ + cos x

µ

• STABLE

UNSTABLE

Stability Analysis

ψ(x, y, t) = ψL(x) + ϕ(x, y, t) Linear Instability assume infinitesimal disturbance ϕ ∼ e−iωt {ω} > 0 ⇒ ψL is unstable gives sufficient condition for instability Global Stability (Asymptotic Stability) ϕ is not assumed to be small disturbance energy Eϕ(t) = 1 2

• |∇ϕ|2

→ 0 as t → ∞ gives sufficient condition for stability

Energy Method

∇2ψt + J(ψ, ∇2ψ) + βψx = −µ∇2ψ + cos x ψL(x) = −a cos(x − xβ) Time evolution equation for ϕ (ψ = ψL + ϕ) : ∇2ϕt + J(ψL, ∇2ϕ) + J(ϕ, ∇2ψL) + J(ϕ, ∇2ϕ) + βϕx = −µ∇2ϕ ∴ dEϕ dt =

• ϕJ(ψL, ∇2ϕ)
• − µ
• |∇ϕ|2

= a

• ϕxϕy cos x
• − 2µEϕ

Energy Method

Φ ϕ(t) dEϕ dt = 2

• aR[ϕ] − µ
• Eϕ

where R[ϕ] ≡

• ϕx ϕy cos x
• |∇ϕ|2

Now define R∗ ≡ max

ϕ∈Φ R[ϕ]

Φ : set of all functions satisfying periodic boundary conditions Then, dEϕ dt < 2 (aR∗ − µ) Eϕ

Energy Method

By Gronwall’s inequality, dEϕ dt < 2 (aR∗ − µ) Eϕ ⇒ Eϕ(t) < Eϕ(0) e2(aR∗−µ)t ∴ Eϕ(t→∞) → 0 if aR∗ − µ < 0 Neutral condition a = 1 R∗ µ ⇒ β =

• R2

∗

µ2 − µ2

Variational Results

x y

Maximize: R[ϕ] ≡

• ϕx ϕy cos x
• |∇ϕ|2
• ver the set Φ.

Optimal solution R∗ = R[ϕ∗] = 1 2 ϕ∗(x, y) ≈ lim

l→∞ cos l(y + sin x) exp

• l

2 cos 2x

Energy Stability Curve

β =

• 1

4µ2 − µ2 (a = 2µ)

0.0 0.2 0.4 0.6 0.8 1 2 3

Energy stability

B

STABLE

C D

?

β µ

(a > 2µ) (a < 2µ)

Energy Stability and Linear Stability Curve

β =

• 1

4µ2 − µ2 (a = 2µ)

0.0 0.2 0.4 0.6 0.8 1 2 3

Energy stability Linear stability

B

STABLE

?

UNSTABLE

β µ

Limitations of the Energy Method

requires Eϕ(t) to decrease monotonically for all ϕ, thus excludes transient growth of Eϕ(t)

t E

the most efficient energy-releasing disturbance ϕ∗(x, y) is unphysical: l → ∞ a gap between the energy stability curve and the neutral curve from linear stability analysis

Energy-Enstrophy Balance

Disturbance enstrophy: Zϕ = 1

2

• (∇2ϕ)2

dZϕ dt = a

• ϕx ϕy cos x
• − 2µZϕ

Recall, dEϕ dt = a

• ϕx ϕy cos x
• − 2µEϕ

Then, d dt(Eϕ − Zϕ) = −2µ(Eϕ − Zϕ) Eϕ(t) − Zϕ(t) = e−2µt [Eϕ(0) − Zϕ(0)]

Energy-Enstrophy Balance

0.0 0.1 0.2 0.3 0.4 0.5

E

0.0 0.5 1.0

0.40 , 0.00 0.61 , 0.00

µ

Z

• Φ

ϕ(t)

ΦEZ : Eϕ=Zϕ

Eϕ(t) − Zϕ(t) → 0 as t → ∞ ΦEZ = {ϕ ∈ Φ such that Eϕ = Zϕ} ⇒ ΦEZ attracts all initial conditions

Optimization with Constraints

x y Maximize: R[ϕ] ≡

• ϕx ϕy cos x
• |∇ϕ|2

with constraint |∇ϕ|2 = (∇2ϕ)2 Optimal solution R∗ = R[ϕ∗] = 0.3571 ϕ∗(x, y) =

• e i l y ˜

ϕ(x)

• with

l ≈ 0.4166

Energy-Enstrophy (EZ) Stability (ν = 0)

β =

• 0.13

µ2 − µ2 (a = 2.8µ)

0.0 0.2 0.4 0.6 0.8 1 2 3

EZ stability Energy stability Linear stability

B

STABLE

C D

?

UNSTABLE

β µ

Energy-Enstrophy (EZ) Stability (ν = 0)

β =

• 0.13

µ2 − µ2 (a = 2.8µ)

0.0 0.2 0.4 0.6 0.8 1 2 3

EZ stability Energy stability Linear stability

B

STABLE

C D

?

UNSTABLE

β µ

5 10

time

1 2

Eϕ

2 4

time

1

0.0 0.1 0.2 0.3 1.6 1.7

Eϕ

The viscous case: ν > 0

∇2ψt + J(ψ, ∇2ψ) + βψx = −µ∇2ψ + cos x + ν∇2ζ ψL(x) = −a cos(x − xβ) a = 1

• β2 + (µ + ν)2

dEϕ dt = a

• ϕx ϕy cos x
• − 2µEϕ + 2νZϕ

dZϕ dt = a

• ϕx ϕy cos x
• − 2µZϕ + 2νPϕ

Pϕ = 1 2

• |∇(∇2ϕ)|2

Extended Energy-Enstrophy (EEZ) Stability

Consider a family of norm with the parameter α: Q(α) = (1 − α) Eϕ + α Zϕ , 0 α 1 dQ dt = 2

• RQ[ϕ; α, ν, a] − µ
• Q

Global stability : Q(α) → 0 as t → ∞ for some α .

Extended Energy-Enstrophy (EEZ) Stability

0.0 0.2 0.4 0.6 0.8 1 2 3

B STABLE

?

β µ R∗

Q(α2)

R∗

Q(α1)

R∗

Q(α2) < R∗ Q(α1)

Consider a family of norm with the parameter α: Q(α) = (1 − α) Eϕ + α Zϕ , 0 α 1 dQ dt = 2

• RQ[ϕ; α, ν, a] − µ
• Q

Global stability : Q(α) → 0 as t → ∞ for some α . R∗

Q(α, ν, a) = max ϕ∈Φ RQ[ϕ; α, ν, a]

For each α, neutral condition : R∗

Q(α, ν, a) = µ

Extended Energy-Enstrophy (EEZ) Stability

0.0 0.2 0.4 0.6 0.8 1 2 3

B STABLE

?

β µ R∗

Q(α2)

R∗

Q(α1)

R∗

Q(α2) < R∗ Q(α1)

Consider a family of norm with the parameter α: Q(α) = (1 − α) Eϕ + α Zϕ , 0 α 1 dQ dt = 2

• RQ[ϕ; α, ν, a] − µ
• Q

Global stability : Q(α) → 0 as t → ∞ for some α . R∗

Q(α, ν, a) = max ϕ∈Φ RQ[ϕ; α, ν, a]

For each α, neutral condition : R∗

Q(α, ν, a) = µ

"Optimal" neutral condition : min

α

R∗

Q(α, ν, a) = µ

Extended EZ Stability (ν = 10−3)

Q(α) = (1 − α) Eϕ + α Zϕ

0.0 0.2 0.4 0.6 1 2 3

Linear stability EEZ stability Energy stability

B

STABLE ? UNSTABLE

β µ

0.00 0.01 0.02 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018

α∗ ν/µ

0.00 0.01 0.02 0.408 0.410 0.412 0.414 0.416

l∗ ν/µ

Extended EZ Stability for different ν

0.0 0.2 0.4 0.6 µ 1 2 3

• EZ (=0)

=10

5 Energy

=10

5

=10

3 Energy

=10

3 EEZ

=10

1 Energy

=10

1 EEZ

B

STABLE

?

Summary

0.0 0.2 0.4 0.6 1 2 3

Linear stability EEZ stability Energy stability

B STABLE ? UNSTABLE

β µ By incorporating information based on the enstrophy, we develop the EZ and EEZ stability method which allows transient growth in Eϕ(t) ( ϕ(t=0) ΦEZ ) identifies a physically realistic most-unstable disturbance lies closer to the linear stability neutral curve

EZ stability: Tsang & Young, Phys. Fluids 20, 084102 (2008)