laboratoire matière et systèmes complexes

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Bifurcation and Chaos in Rayleigh-Bénard Convection

Mahendra VermaDepartment of Physics

Indian Institute of Technology, Kanpurwith

Pinaki Pal2, Pankaj Wahi2, Supriyo Paul2, Pankaj K. Mishra2 and Krishna Kumar3

2 Department of Mech. Engg., IITK3 Department of Physics and Meteorology, IITKGP

AcknowledgementStephan Fauve, ENS Paris

Friday 10 April 2009

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Objective

Patterns and route to chaos in convection

forZero, Low, and High Prandtl number convection.


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Rayleigh Bénard Convection


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Fluctuating Fields

Velocity field: v(x,y,z,t)

Temperature field: T(x,y,z,t) = T1 +z (ΔT)/d + θ(x,y,z)


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Nondimensional Equations

Rayleigh Number R =αg(T1 - T2)d3/(νκ), Prandtl Number P = ν/κ

For low Prandtl number, we use the units, d for length, d2/ν for time, (△T)ν/κ for temperature :


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Boundary conditions

Free-slip Boundary Conditions


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Prandtl numbers of some important material

Engine oil ≈ 100 - 40000 Water, Earth mantle ≈ 7 Air ≈ 0.7-0.8 Mercury ≈ 0.02 Liquid Sodium ≈ .006

Zero Prandtl limit is smooth. The idealized zero P fluid has essential properties of low Prandtl number fluids.


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Zero Prandtl number equations

where ω=∇×v


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Simulation of zero P equationsHerring tried for the first time to simulate the zero P limit of the convection with free-slip boundary conditions.

He reported divergence of the solutions for a large class of initial conditions.

The nonlinear terms in his zero P equations vanish and the amplitude of these modes grow exponentially.

J. R. Herring, 'Convection at zero Prandtl number' , Woods Hole Oceanogr. Inst. Tech. Rep. WHOI-70-01, (1970).


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Thual's simulation Thual successfully simulated the zero Prandtl number equations in 1992 for the first time.

He reported existence of a relaxation oscillation of square patterns (SQOR) regime at 6.5 % above the onset of convective instability.

He also reported stable square (SQ) convective flows and several chaotic flow regimes at larger Rayleigh number.

However, the complete series of bifurcations connecting these various regimes and the onset was not studied. O. Thual, 'Zero-Prandtl-number convection', J.

Fluid Mech., 240, 229 (1992).Friday 10 April 2009

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Mechanism of selection of square pattern

Pal & Kumar explained the mechanism of selection of square pattern in zero Prandtl number convection using a low dimensional model.

There it was shown that square pattern is selected due to interaction of two mutually perpendicular sets of wavy rolls.

But their model could not capture SQ and SQOR regime. They only observed chaotic squares.P. Pal, and K. Kumar, 'Wavy stripes and squares zero-

Prandtl-number convection', Phys. Rev. E 65, 047302 (2002).


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Our Approach• Direct Numerical Simulations (DNS) are performed in the regime of interest.

• Identify energetic modes from DNS data

• Derive low dimensional model using these modes.

• Perform a detailed study of the low-dimensional model.

•Verify the results of low dimensional model by comparing with DNS.


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Low dimensional model To explore the full bifurcation scenario

near the onset is very costly in DNS. Critical slowing down near onset. Presence of very large number of modes

obscure the actual mechanism. In this situation low dimensional modeling

becomes very useful. However, the modes to be used for the low

dimensional modeling should be chosen judiciously.


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Direct Numerical Simulation We use a pseudospectral code to simulate the flow (free-slip BC). Grid size = 643, time step = 10-3 .

We obtain the SQ and SQOR flow regimes as obtained by Thual.

In addition, we obtain several other interesting behavior explained in the subsequent slides.


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Large scale modes The model is a 13-dimensional coupled ODE for the following Fourier coefficients. q= ky/kx =1.


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Bifurcation diagram

Bifurcation scenario near the onset at r=1


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Stationary Square (SQ)

Isotherm plot at z = 0.5, r = 1.22, r = R/Rc.

Red regions: Hot fluid is going up Blue regions: colder fluid is going down.


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SQ and ASQ regimesStationary Square

r = 1.2471Asymmetric Stationary Square

r=1.1559

r (Model) r( DNS)‏SQ 1.2201 – 1.4373 1.2015 – 1.4297ASQ 1.0703 – 1.2200 1.1407 - 1.2014


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r (Model) : 1.0175 - 1.0702 r (DNS) : 1.0798 - 1.1406

Osc. Assym. Square (OASQ)

Dominantly along x Dominantly along y


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Homoclinic orbit

The limit cycle generated from the Hopf bifurcation keeps increasing in size comes closer to the square fixed point saddle. The exponential growth of the time period also suggest the existence of Homoclinic orbit.


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Phase plot projections in model & DNS

Model DNS


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Flow oscillations in SQOR at* Time Dependent * Oscillates between perpendicular sets of wavy rolls.

r=1.065


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Phase plot projections in model & DNS ...

Model DNS

Model(a) : r= 1.0041(c) : r=1.0038(e) : r=1.0030

DNS(b): r=1.0045(d): r=1.0032(f): r=1.0023


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Important Flow Regimes

SQ : Stationary Square regimeASQ : Asymmetric Square RegimeOASQ : Oscillatory Asymmetric Square RegimeSQOR : Relaxation Oscillation squares


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Model vs. DNS (P=0)


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Low P Convection

The relevant equations for this regime are

• Various patterns predicted by Busse, Thual [92]• Fauve, Coullet, Perrin, ... • Low-dimensional models of Kumar, Fauve, ..• Here we compare with zero-P calculations.


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Large scale modes (30)

11 v3 modes, 7 vertical vorticity modes


Bifurcation Diagram (30-mode model)P=0.02


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Phase plot projections in model & DNSP=0.02

Model DNS


SR : Stationary Roll ASQ : Stationary Asymmetric SquareOASQ : Oscillatory Asymmetric Square SQOR : Square Relaxation OscillatorySQ: Stationary Square

Unfortunately no chaos in the model.Chaos seen in 42 mode model.Crucial mode for chaos: W111, θ111. 30

Comparison: Model vs DNSP=0.02


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High P convection

We use the units, d for length, d2/κ for time, (△T) for temperature :

• Various patterns predicted by Busse, Thual [92]• Fauve, Coullet, Perrin, ... • Tuckerman and Barkley [88]: saddle-node (no-slip BC, cylinder)• Preliminary bifurcation study of ours.


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30-mode 2D Model (P=6.8)


Bifurcation Diagram (2D RBC)

Superharmonic frequency

Hopf at around r=27


Time series of dual soln (r=40)

Superharmonic frequency

Hopf Bifurcation


Quasiperiodic State for 2D RBC (r=43)


Chaotic State for 2D RBC (r=45)


DNS result 2D RBC: Timeseries

3.9 4−150

0

!(w

101),

"(w

101) (d)

3.5 4−200

0(e)2 3

0

40(a)

2.8 30

60(b)

2.8 3−100

100

0

(c)

0.5 0.950

350(g) 0 5

−300

200(f)

0.995 110.13

10.45(h)

time (d2/#)0.791 0.796

12.5

13.5

(i)

"(w101)

!(w101)r=70 "(w101)

!(w101)

"(w101)

r=400 r=700

r=830

"(w101)r=1500

!(w101) r=7000"(w101)

!(w101)r=10000

!(w101)

"(w101)"(w101)

!(w101)

r=100!(w101) !(w101) r=140

!(w101)

"(w101)

"(w101)


Chaotic travelling waves &wind reversal

• free-slip condition• Possibly similar to the wind reversal observed in experiments• Periodic BC along horizontal crucial.•http://home.iitk.ac.in/~mkv/Turbulence/Convection.html (movie)


http://home.iitk.ac.in/~mkv/Turbulence/Convection.html




Chaotic travelling waves &wind reversal

No-slip conditionsimulation by A. De

•http://home.iitk.ac.in/~mkv/Turbulence/Convection.html (movie)

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3D RBC: Bifurcation Diagram (7-mode model)

Modes: 101, 011, 112, 002


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Conclusions1. A possible bifurcation scenarios near the onset of convection

2. A systematic methodology to derive low dimensional models for study of Rayleigh Benard conevection.

3. Low dimensional models give very good qualitative behavior of the flow near the onset of convection.

4. Low-P and Zero-P bifurcation diagrams are consistent with each other.

5. Extension for larger r is required.

6. Large-P route to chaos through quasiperiodicity.

7. Need to extend to q ≠ 1, and for no-slip boundary condition.


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References S. Chandrasekhar, 'Hydrodynamic and hydromagnetic

stability', Oxford, (1961). E. A. Spiegel, 'Thermal turbulence in a very small Prandtl

number fluid', J. Geophys. Res., 67, 3063 (1962). F. H. Busse, 'The oscillatory instability of convection rolls in

a low Prandtl number fluid', J. Fluid Mech., 52, 97 (1972). R. M. Clever and F. H. Busse, 'Convection at very low Prandtl

numbers', Phys. Fluids A2, 334 (1990). O. Thual, 'Zero-Prandtl-number convection', J. Fluid Mech.,

240, 229 (1992). K. Kumar, S. Fauve, and O. Thual, 'Critical self-tuning: the

example of zero Prandtl number convection', J. Phys. II (France) 6, 945 (1996).

P. Pal, and K. Kumar, 'Wavy stripes and squares in zero-Prandtl-number convection', Phys. Rev. E 65, 047302 (2002).


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Thank You!


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