2D Fluid Turbulence (Merz)

8/3/2019 2D Fluid Turbulence (Merz)

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Max−Planck−Institutfür Plasmaphysik

2D Fluid Turbulence

Florian Merz

Seminar on Turbulence, 08.09.05



2D turbulence?

• strictly speaking, there are no two-dimensional flows in nature

• approximately 2D: soap films, stratified fluids, geophysical flows,magnetized plasmas



2D turbulence?

• a ’simplified’ situation (compared to 3D), more accessible to theo-retical, experimental and computational approaches, interesting for

developing and testing general ideas about turbulence

• much easier to visualize than 3D-turbulence

•interesting new phenomena (e.g. dual cascade)



Outline

• Basic equations

• Cascades in 2D

• Coherent structures



Basic equations: velocity

2D-Navier-Stokes equations for an incompressible fluid:

DDtv =

∂ ∂t + v · v = −

1ρ p + f ext + ν 2v

· v = 0

where v is in the (x,y) plane, (v · z = 0).

For viscosity ν = 0 and f ext = 0 these equations are called Euler equations.



Basic equations: vorticity

Taking the curl of the NS equations and discarding the zero x and y

components of the equation gives

DDtω =

∂ ∂t + v ·

ω = g + ν 2ω

for the vorticity ω =

× v· z.

If g = ( × f ext) · z = 0 and ν = 0 (Euler equation), we have DDtω = 0

→ vorticity is conserved



Basic equations: energy and enstrophy

Important quantities:

mean energy E = 12u2

enstrophy Z = 12ω2

(mean square vorticity)

dE dt = −2νZ

dZ dt = −2ν (ω)2

• in 2D with curl free forcing, energy and enstrophy can only decrease

with time, they are conserved in the inviscid case (ν = 0)

• for ν → 0 we get dE dt → 0, energy is a “robust” invariant in 2D

•dZ

dt does not necessarily go to zero for ν → 0, enstrophy is a “fragileinvariant” (dissipation anomaly!)



Dissipation anomaly

• as time evolves, vorticity patches get distorted by background velocity

and generate smaller and smaller filaments

• the vorticity gradient increases and dZ dt = −2ν (ω)2 becomes sizeable

• dissipation even for ν → 0: fragile invariant

• in the enstrophy cascade, the dissipation rate is independent of ν but

depends only on the enstrophy transfer rate

• ν only determines the enstrophy dissipation scale, not the enstrophy

dissipation rate



Exact results

For forced, isotropic and homogenous turbulence:

•for small scales: Corrsin-Yaglom-relation for passive tracers

DL(r)− 2ν dS 2dr

= −4

3ηr

(DL(r) =

(vL(x)

−vL(x + r))(ω(x)

−ω(x + r))2

, S 2(r) =

(ω(x)

−ω(x + r))2), η = ν (ω)2)

• for large scales: 2D-analogue of the Kolmogorov relation (4/5-law)

F 3(r) =3

2r

(F 3(r) = (vL(x)− vL(x + r))3, = ν (ω)2 = 2νZ )

sign of the prefactor reversed!



Cascades



Direction of the energy/enstrophy transfer

In spectral space, the expressions for energy and enstrophy read

E = E (k, t)dk

Z =

k2E (k, t)dk

. . .k2 k3k1

Z 1 Z 3

Z 2

E 1 E 3

. . .

E 2E 2 = E 1 + E 3, Z 2 = Z 1 + Z 3.

Energy and enstrophy conservation for three Fourier modes k1, k2 = 2k1,

k3 = 3k1




δE 1 + δE 2 + δE 3 = 0

k21δE 1 + k2

2δE 2 + k23δE 3 = 0

with δE i = E (ki, t2)− E (ki, t1). Combining the equations gives

δE 1 = −5

8δE 2 δE 3 = −3

8δE 2

k2

1δE

1=−

5

32k2

2δE

2k2

3δE

3=−

27

32k2

2δE

2

→ enstrophy goes to higher k (direct enstrophy cascade),

energy goes to lower k (inverse energy cascade)




Alternatively:

E =

E (k, t)dk

Z =

k2E (k, t)dk

an evolution of E (k, t) to larger k conflicts with the boundedness of en-

strophy

→ E (k, t) must evolve towards small wave numbers / large

scales



KBL theory: dual cascade

Kraichnan, Batchelor, Leith proposed the existence of a dual cascade insteady state turbulence (∼1968):

• energy/enstrophy is constantly injected at some intermediate ki

• direct enstrophy cascade to higher k → dissipation scale kd = (β/ν 3)1/6

(equivalent to Kolmogorov microscale in 3D)

• inverse energy cascade to lower k → condenses in the lowest mode

(for bounded domain) / is stopped by Ekman friction (

−µv-term in

NS-equation) at (kE = µ3/)1/2

•E (k) is stationary, the transfer rates of energy and enstrophy β far

from the dissipation scales are independent of k (self-similarity)



KBL theory: inertial ranges

• inertial range of the energy cascade: for kE k ki, the energy

spectrum can only depend on .

Dimensional analysis:

k = [L]−1; E (k) = [L]3[T ]−2; = [L]2[T ]−3 → E (k) = C2/3k−5/3

• inertial range of the enstrophy cascade (ki k kd): the energy

spectrum can only depend on β .Dimensional analysis:

k = [L]−1; E (k) = [L]3[T ]−2; β = [T ]−3 → E (k) = C β 2/3k−3

C, C constant and dimensionless.

• zero enstrophy transfer in the energy inertial range, zero energy trans-

fer in the enstrophy inertial range



KBL: Inertial ranges in steady state turbulence



Experiments: soap films

• vertically flowing soap films are approximately 2D (thickness variations

of about 10% - condition of incompressibility is slightly violated)

• turbulence is generated by grids/combs inserted in the flow



Experiments: soap films

Energy spectrum in soap film



Correction to the energy spectrum

• the k−3-spectrum of the enstrophy cascade gives rise to inconsistency:

infrared divergence of for the (k-dependent) enstrophy transfer rate

Λ(k) for ki → 0

• reason: contributions of larger structures to the shear on smaller struc-

tures (nonlocality in k-space).

• Kraichnan (TFM-closure approximation): logarithmic correction to re-

store constant transfer rate

E (k) = C β 2/3k−3 ln

k

ki

−1/3



Correction to the energy spectrum

• attempts to measure the corrections are being made: k3E (k) and

(enstrophy flux)/β for various Reynolds numbers (direct numerical si-

mulation [DNS] results)



Cascades: 2D vs. 3D

Vorticity equation in 3 dimensions (analogous to MHD kinematic equation

for ω →

B

):

D

Dt ω = ( ω · )v + g + ν 2 ω

the additional vortex-stretching term changes the behaviour significantly:

• gradients in the velocity field stretch embedded vortex tubes

• as the cross section decreases, the vorticity increases



Cascades: 2D vs. 3D

• enstrophy in 3D is not conserved even for ν = 0 but increases with

time (in 2D: fragile invariant) → no enstrophy cascade in 3D!

• energy in 3D follows dE dt = −2νZ and is a fragile invariant (in 2D:

robust invariant)

• energy cascades to smaller scales in 3D (direct cascade), in 2D to

large scales (inverse cascade)

• E (k) has the same k−5/3-dependence in the inertial range of the ener-

gy cascade



Cascades: 2D vs. 3D

experimental results for grid turbulence in 3D/2D



Coherent structures



Coherent structures

• physical and numerical experiments: long lived vortical structures (li-

fetime turnover time) spontaneously emerging from the turbulent

background

• these “coherent” structures alter the cascading behavior

• especially important in freely decaying turbulence (they can be inhi-

bited / destroyed in forced systems)

• clear definition/identification of coherent structures difficult, several

competing methods: e.g. simple threshold criteria, Weiss criterion,

wavelet decomposition..



Coherent structures: axisymmetrisation

•experiments: elliptical structures are not stable but become circular

• exact theoretical result: circular patches of uniform vorticity are non-

linearly stable



Coherent structures: vortex merging

• experiments: if like sign vortices of comparable strength get too close,they merge (axisymmetrization)

• theory: analytical solution for the merging of two identical vortices



Coherent structures: vortex break-up

• experiments: the strain caused by the stronger vorices distorts weaker

vortices up to destruction → vorticity adds to background vorticity

C h t t t ti l ti



Coherent structures: time evolution

• by vortex merging and break-up, the coherent structures in a freely

decaying system become fewer and larger

• observables: evolution of vortex density ρ, typical radius a, intervortical

distance r, extremal vorticity ωext

’U i l d th ’ f dil t t



’Universal decay theory’ for dilute vortex gas

empirical approach by Carnevale et al.assumption: two invariants

•E ∼

ρω2

exta4, contributions outside vortices negligible

• vorticity extremum ωext of the system (observation)

→length scale l =√

E/ωext, time scale τ = 1/ωext

Dimensional reasoning gives ρ = l−2g(t/τ ). Assumption g(t/τ ) = (t/τ )−ξ

gives (ξ is to be measured)

ρ

∼l−2(t/τ )−ξ, a

∼l(t/τ )ξ/4, Z

∼τ −2(t/τ )−ξ/2

r ∼ l(t/τ )ξ/2, v ∼ √E,

’Uni ersal deca theor ’ comparison with DNS



’Universal decay theory’: comparison with DNS

• left: decay law for the number of vortices

•right: inverse vortex density, intervortical distance, size, extremal vor-

ticity (lines for ξ = 0.75)

Decay of vortex populations



Decay of vortex populations

• vortex merging and vortex break-up lead to ever larger and fewer

coherent structures

• the system evolves towards a final dipole

Intermittency in 2D



Intermittency in 2D

• experimental results: no intermittency in 2D turbulence

• no theoretical explanation yet

PDFs for longitudinal, transverse velocity increment (energy cascade) and

the vorticity increment (enstrophy cascade) for several scales

. δv = vL(x)−vL(x+r) δv⊥ = vT (x)−vT (x+r) δω = ω(x)−ω(x+r)

Intermittency in 2D



Intermittency in 2D

hyperflatness H 2n(l) = F 2n(l)F 2(l)n

, F n(l) = δv(l)n (energy cascade)

and structure functions of vorticity S n(l) = δω(l)n (enstrophy cascade)

→ no intermittency, slight deviations from gaussianity are assumed to

stem from coherent structures

Summary



Summary

• the existence of a dual enstrophy-energy cascade (KBL-theory) is

experimentally confirmed

• coherent structures play an important role (especially in decaying tur-

bulence) and modify the energy spectrum predicted by KBL-theory

• there is no intermittency found in experiments

• several systems of interest (e.g. geophysical flows, magnetized plas-mas) are approximately 2-dimensional - results of 2D fluid turbulence

are applicable

Further reading



Further reading

General 2D turbulence:

• P.A. Davidson, Turbulence, Oxford University Press (2004)

• M. Lesieur, Turbulence in Fluids, Kluwer (1997)

• U. Frisch, Turbulence, Cambridge University Press (1995)

• P. Tabelling, Two-dimensional turbulence: a physicist approach, Phys. Rep. 362,1-62 (2002)

Cascade classics

• Kraichnan, Inertial Ranges in Two-Dimensional Turbulence, Phys. Fluids 10, 1417(1967)

• Leith, Diffusion Approximation for Two-Dimensional Turbulence, Phys. Fluids 11,1612 (1968)

• Batchelor, Computation of the Energy Spectrum in Homogenous Two-Dimensional

Turbulence, Phys. Fluids 12, II-233 (1969)

2D Fluid Turbulence (Merz)

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