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WSIMS08, Barcelona, December 1-5, 2008 1

Dynamics of particle trajectories in a Rayleigh–B´ enard problem

Dolors Puigjaner(1) Joan Herrero(2) Carles Sim´

• (3)

Francesc Giralt(2)

(1) Dept. Enginyeria Inform` atica i Matem`

• atiques. Universitat Rovira i Virgili.

(2) Dept. Enginyeria Qu´ ımica. Universitat Rovira i Virgili. (3) Dept. Matem` atica Aplicada i An`

• alisi. Universitat de Barcelona.

Motivation

Motivation and Objectives

• Motivation
• Objectives

Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 2

• Fluid mixing efficiency is a crucial issue in many

engineering applications

• Efficient mixing is usually related to turbulent regimes and

to mechanical devices

• Some industrial applications require an efficient mixing in

the absence of turbulence or high shear stresses

• Rayleigh–B´

enard convection can offer an alternative to the use of mechanical devices

Objectives

Motivation and Objectives

• Motivation
• Objectives

Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 3

The design of reactors in which efficient mixing is achieved without moving parts The application of dynamical systems theory to the analysis of dynamics and mixing properties in flows in- duced by Rayleigh–B´ enard convection inside a cube

• Analyze the rich dynamics of fluid particle trajectories
• Characterize well-mixed regions inside the cube
• Investigate the dependence of mixing properties on the

Rayleigh number

Flow system and equations

Motivation and Objectives Problem description

• Flow system
• Continuation method
• Bifurcation diagram
• Flow patterns
• Symmetries

Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 4 z x y T=T

Ω

h c

T=T

g

L

Ra = gβL3∆T/αν Pr = ν/α β thermal expansion ν kinematic viscosity α thermal diffusivity ∆T = Th − Tc θ=[T−(Th+Tc)/2]/∆T−z

Continuity

∇ · V = 0

Momentum

1 Pr

• ∂

V ∂t +Ra

1 2(

V · ∇) V

• =

∇2 V + Ra

1 2θ

ez −∇p

Energy

∂θ ∂t + Ra

1 2(

V · ∇)θ = ∇2θ + Ra

1 2

V · ez

Boundary conditions

• V = θ = 0 en ∂Ω

Continuation method

Motivation and Objectives Problem description

• Flow system
• Continuation method
• Bifurcation diagram
• Flow patterns
• Symmetries

Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 5

Galerkin spectral method with basis functions

{ Fi(x, y, z)}

satisfying the boundary condi- tions and the continuity equation Solution

V θ

• =
• i

ci(t) Fi

+ Stability Analysis

solution Conductive

b

Ra close to New solution at Ra Bifurcation Ra Rab Main eigenvalues Main eigenvectors

Bifurcation Location Technique Branch Switching Procedure Parameter Continuation

Tracked solution branch

• D. Puigjaner, J. Herrero, C. Sim´
• , F. Giralt, J. Fluid Mechanics, 598, 393–427, (2008)

Bifurcation diagram (Pr = 130)

Motivation and Objectives Problem description

• Flow system
• Continuation method
• Bifurcation diagram
• Flow patterns
• Symmetries

Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 6

Steady solutions that are stable over some Ra range

• 0.4
• 0.2

0.2 20 40 60 80 100 120 140 Nu-0.012Ra1/2 10-3Ra B1 B2 B3 B5

B25

B251 B251

B34

• 0.280
• 0.265

82 92

stable unstable

• steady bifurcation
• Hopf bifurcation

turning point

Flow patterns: λ2 = 0 surfaces

Motivation and Objectives Problem description

• Flow system
• Continuation method
• Bifurcation diagram
• Flow patterns
• Symmetries

Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 7

B2

Initial Ra

6 798

Stability Range

67 730–85 694 Ra=7 000 Ra=51 000 Ra=80 000

B3

Initial Ra

11 612

Stability Range

20 637–79 362 Ra=12 000 Ra=51 000 Ra=80 000

λ2 is the second largest eigenvalue of the tensor S2 + Ω2

Symmetries and invariant planes

Motivation and Objectives Problem description

• Flow system
• Continuation method
• Bifurcation diagram
• Flow patterns
• Symmetries

Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 8

Solution Symmetry Group (generators) Invariant Planes

B2 Sd−, −I x + y = 0 B25 −I

–

B251 Sy, −I y = 0 B3 Sd+, −Sy

• x + y = 0

x − y = 0 Sy

reflection about the plane y = 0

Sd+

reflection about the plane x − y = 0

Sd−

reflection about the plane x + y = 0

−I

simmetry with respect the origin

−Sy

rotation of angle π around the y–axis

Numerical methods

Motivation and Objectives Problem description Dynamical systems approach and results

• Numerical methods

Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 9

Particle trajectories (Negligible diffusivities)

   ˙ x = u(x, y, z) ˙ y = v(x, y, z) ˙ z = w(x, y, z)

Advection equations

• Symmetries and invariant planes
• Poincar´

e sections

• Periodic orbits and their stability
• Size and shape of regular regions
• Maximal Lyapunov exponents and metric entropy
• Critical points in the interior and on the boundary
• Stability analysis
• Poincar´

e–Hopf index theorem

• C. Sim´
• , D. Puigjaner, J. Herrero, F. Giralt, Communications in Nonlinear Science and

Numerical Simulation, doi:10.1016/j.cnsns.2008.07.012. In Press

Poincar´ e Maps I

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps

• Poincar´

e Maps I

• Periodic orbit
• Poincar´

e Maps II Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 10

512 equidistributed initial conditions integrated up to t = 103 B2, z = 0 Ra = 104

Ra = 3.3 × 104

Main stable periodic orbit (fixed elliptic point)

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps

• Poincar´

e Maps I

• Periodic orbit
• Poincar´

e Maps II Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 11

• 2

2 4 20 40 60 80 100 n argument trace 10 (x-coordinate)

y = 2π/k, k = 2, · · · , 7

Poincar´ e Maps II

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps

• Poincar´

e Maps I

• Periodic orbit
• Poincar´

e Maps II Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 12

B2, z = 0 Ra = 6.87099 × 104 Ra = 8.5 × 104

Regular regions

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents

• Regular regions
• Lyapunov exponents I
• LM and Vc
• Metric entropy
• Regular regions I
• Regular regions II

Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 13

Vc= volume occupied by the chaotic zone (points outside

invariant tori) Computation Procedure

• Divide the cavity into n × n × n cubic cells (n = 200)
• Compute trajectories of fluid particles initially located at x0,

for any x0 in the set CI (final time tM = 106)

CI =

• −0.375+ i

8, 0.48, −0.375+ j 8

• , i, j = 0,· · · ,6
• Store the cells visited by one or more trajectories
• Nr(t) = number of cells that at time t have not yet been

visited by any particle trajectory (every ∆t = 200)

• Check that Nr(t) is almost constant in t ∈ [ 3

4tM, tM]

• Points at a distance less than 0.01 from the boundaries are

considered as non-regular

Maximal Lyapunov exponents I

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents

• Regular regions
• Lyapunov exponents I
• LM and Vc
• Metric entropy
• Regular regions I
• Regular regions II

Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 14

LM(x0) = lim

t→∞

1 t log lt l0

• lt = Dxφ(t, x0)l0 where l0 is an arbitrary vector and

φ(t, x0) is a solution of the differential equation with φ(0, x0) = x0

Computation Procedure

• 49 equidistributed initial conditions on the plane y = 0.48
• finite time approximations of LM (final time=105)
• transient values (t ≤ 104)
• calculate log ( lt / l0 ) /t every 103 units of time

after the transient (t > 104)

• average with respect to time and initial conditions

LM and Vc evolution

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents

• Regular regions
• Lyapunov exponents I
• LM and Vc
• Metric entropy
• Regular regions I
• Regular regions II

Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 15

B2

20 40 60 80 100 20 40 60 80 100 120 60 70 80 90 100 10-3Ra, 103LM Size of chaotic region (%) n % Chaotic region Rayleigh number 103 Maximal Lyapunov exponent

Metric entropy, hm

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents

• Regular regions
• Lyapunov exponents I
• LM and Vc
• Metric entropy
• Regular regions I
• Regular regions II

Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 16

hm = LM × Vc B2

0.01 0.02 0.03 0.04 20 40 60 80 100 120 metric entropy n

Shape of regular regions

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents

• Regular regions
• Lyapunov exponents I
• LM and Vc
• Metric entropy
• Regular regions I
• Regular regions II

Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 17

• 0.4-0.2 0 0.2 0.4
• 0.4
• 0.2

0.2 0.4

• 0.4
• 0.2

0.2 0.4

• 0.4-0.2 0 0.2 0.4
• 0.4
• 0.2

0.2 0.4

• 0.4
• 0.2

0.2 0.4

Ra = 104 Ra = 8.5 × 104

• 0.4-0.2 0 0.2 0.4
• 0.4
• 0.2

0.2 0.4

• 0.4
• 0.2

0.2 0.4

Ra = 3.4 × 104

Spherical–like regular regions

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents

• Regular regions
• Lyapunov exponents I
• LM and Vc
• Metric entropy
• Regular regions I
• Regular regions II

Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 18

B2, Ra = 6 × 104

• 0.4-0.2 0 0.2 0.4
• 0.4
• 0.2

0.2 0.4

• 0.4
• 0.2

0.2 0.4

0.2 0.4

• 0.4
• 0.2
• 0.4
• 0.2
• 0.5
• 0.25

0.25 0.5

• 0.5
• 0.25

0.25 0.5 y x

z = 0 z = 0.4

Critical points: interior and boundaries

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points

• Critical points I
• Critical points II
• Poincar´

e–Hopf index

• Critical points III
• Critical points IV
• Critical points V

Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 19

Interior: fixed points of the advection equations (

V = 0)

Boundary: wall x3 = 0

dxi dτ = ∂ui ∂x3 + ∂2ui ∂x3∂x1 (x1−b1)+ ∂2ui ∂x3∂x2 (x2−b2)+ 1 2 ∂2ui ∂x2

3

x3 dx3 dτ = −1 2 ∂2u1 ∂x1∂x3 + ∂2u2 ∂x2∂x3

• x3

τ = x3t rescaled time b = (b1, b2, 0) point on the wall x3 = 0

Trajectories of particles passing very close to the wall are

• btained by taking the limit (x1, x2, x3) → (b1, b2, 0)

dxi dτ = ∂ui ∂x3 i = 1, 2

Critical points: stability

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points

• Critical points I
• Critical points II
• Poincar´

e–Hopf index

• Critical points III
• Critical points IV
• Critical points V

Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 20

λ1, λ2, λ3 eigenvalues associated to the linearization of the

vector velocity field about a critical point xc. Divergence–free flow (volume–preserving flow)

λ1 + λ2 + λ3 = 0

if xc is in the interior of the cube;

1 2(λ1 + λ2) + λ3 = 0

if xc is on a wall of the cube ;

2(λ1 + λ2) + λ3 = 0

if xc is on an edge of the cube; Classification of critical points

• SF: stable focus with a 1D unstable manifold
• UF: unstable focus with a 1D stable manifold
• 2S: saddle with a 2D stable manifold
• 1S: saddle with a 1D stable manifold

Poincar´ e–Hopf index

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points

• Critical points I
• Critical points II
• Poincar´

e–Hopf index

• Critical points III
• Critical points IV
• Critical points V

Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 21

Poincar´ e–Hopf index theorem The sum of the Poincar´ e indexes over all the isolated critical points of a vector field on a compact orientable differentiable manifold is equal to the Euler characteristic of the manifold Change from the cubical domain to the 3D sphere S3

• the interior of the cubical domain is topologically equivalent

to a 3D open ball B3 and its boundary is equivalent to a 2D sphere S2

• deform B3 to get a 3D hemisphere whose equator is S2
• take a symmetric copy of the 3D hemisphere and glue both

hemispheres after identifying the S2 boundaries The Poincar´ e index of a critical point xc satisfying Reλi = 0, is (−1)np, where np = #{λi | Reλi < 0, i = 1, 2, 3}. Interior critical points must be counted twice.

Critical points: bifurcations and Poincar´ e indexes

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points

• Critical points I
• Critical points II
• Poincar´

e–Hopf index

• Critical points III
• Critical points IV
• Critical points V

Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 22

Interior Walls Edges SF UF 2S 1S SF 2S 1S 2S 1S

10−3Ra (2) (-2) (2) (-2) (1) (1) (-1) (1) (-1)

10 2 3 2 4 2 6 2 11 2 3 4 2 2 2 4 28 2 1 4 2 4 41 2 1 4 2 4 58 4 3 4 2 4 60 4 2 1 2 4 2 4 61 4 4 1 4 2 4 63 6 6 1 4 2 4 65 6 6 1 4 2 4 4 68.58 6 6 2 1 4 2 4 4 68.71 6 6 2 1 4 6 8 4 68 6 6 1 8 2 8 4 58 6 6 1 8 6 4 57 6 6 1 4 4 6 4 52 6 6 1 4 2 4

Critical points:bifurcations and invariant planes I

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points

• Critical points I
• Critical points II
• Poincar´

e–Hopf index

• Critical points III
• Critical points IV
• Critical points V

Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 23

Ra = 104 Ra = 5.8 × 104

• 0.5
• 0.25

0.25 0.5

• 0.5
• 0.25

0.25 0.5 z x+y=0

1 3 3 2 2 4 4 1 3 3 2 2 4 4

• 0.5
• 0.25

0.25 0.5 x+y=0

1 4 4 1 4 4

× SF: stable focus + 2S: saddle with two stable directions

UF: unstable focus

Critical points:bifurcations and invariant planes II

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points

• Critical points I
• Critical points II
• Poincar´

e–Hopf index

• Critical points III
• Critical points IV
• Critical points V

Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 24

Ra = 6.3 × 104 Ra = 8.5 × 104

• 0.5
• 0.25

0.25 0.5 x+y=0

1 4 4 6 6 1 4 4 6 6

• 0.5
• 0.25

0.25 0.5

• 0.5
• 0.25

0.25 0.5 z x+y=0

1 4 4 6 6 1 4 4 6 6

×

SF: stable focus

+

2S: saddle with two stable directions UF: unstable focus

Limiting Streamlines (B2)

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories

• Limiting Streamlines
• Projected trajectories

Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 25

z = −0.5

• 0.5
• 0.25

0.25 0.5

• 0.5
• 0.25

0.25 0.5 y x 1S 2S 1S 2S

• 0.5
• 0.25

0.25 0.5

• 0.5
• 0.25

0.25 0.5 y x 1S

• 0.5
• 0.25

0.25 0.5

• 0.5
• 0.25

0.25 0.5 y x 1S

x = −0.5

• 0.5
• 0.25

0.25 0.5

• 0.5
• 0.25

0.25 0.5 z y SF 1S SF 1S

• 0.5
• 0.25

0.25 0.5

• 0.5
• 0.25

0.25 0.5 z y 2S 2S

• 0.5
• 0.25

0.25 0.5

• 0.5
• 0.25

0.25 0.5 z y SF SF

104 6 × 104 8.5 × 104

Projected trajectories (B2)

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories

• Limiting Streamlines
• Projected trajectories

Comparison of B2 and

B3

Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 26

• 0.5
• 0.25

0.25 0.5

• 0.5
• 0.25

0.25 0.5 y x

• 0.5
• 0.25

0.25 0.5

• 0.5
• 0.25

0.25 0.5 y x

4 × 104 8.5 × 104

• 0.5
• 0.25

0.25 0.5

• 0.5
• 0.25

0.25 0.5 y x

• 0.5
• 0.25

0.25 0.5

• 0.5
• 0.25

0.25 0.5 y x

8.5 × 104 8.5 × 104

Vc, LM and hm

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

• Vc, LM and hm
• Poincar´

e sections Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 27

20 40 60 80 100 10 20 30 40 50 60 70 80 Volume of chaotic region (%) Ra/1000 B2 B3 82 100 50 70 0.01 0.02 0.03 0.04 10 20 30 40 50 60 70 80 Metric entropy Ra/1000 B3 B2 0.01 0.02 0.03 0.04 0.05 10 20 30 40 50 60 70 80 LM Ra/1000 B3 B2

Poincar´ e sections

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

• Vc, LM and hm
• Poincar´

e sections Conclusions and

• utlook

WSIMS08, Barcelona, December 1-5, 2008 28

B2 2 × 104 5.1 × 104 8 × 104 B3

Conclusions

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook
• Conclusions
• Outlook

WSIMS08, Barcelona, December 1-5, 2008 29

• The dynamics are characterized by regions with regular

motion surrounded by regions of chaotic motion

• Changes on the topology and on the chaotic level of the

flows are related to bifurcations of critical points

• The detailed knowledge of the flow provided by the

dynamical systems approach can be relevant in selecting the parameter ranges and flow patterns at which more efficient mixing is achieved

Future work

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook
• Conclusions
• Outlook

WSIMS08, Barcelona, December 1-5, 2008 30

• Study the relative position of the most relevant invariant

manifolds of the fixed points and hyperbolic periodic orbits

• Study the effect of a non–negligible molecular diffusion on

the dynamics of particle trajectories

• Extend the study to non-stationary flows

Basis Functions I

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

Basis Functions

• Basis Functions I
• Basis Functions II

WSIMS08, Barcelona, December 1-5, 2008 31

V θ

• =

Nx

• i=1

Ny

• j=1

Nz

• k=1
• a(1)

ijkG(1) ijk+a(2) ijkG(2) ijk+a(3) ijkG(3) ijk+a(4) ijkG(4) ijk

• G(1)

ijk =

     −gifjf ′

k

1 h2 gif ′

jfk

     , G(2)

ijk =

     −figjf ′

k

1 h1 f ′

igjfk

     , G(3)

ijk =

       − 1 h2 fif ′

jh′ k

1 h1 f ′

ifjh′ k

       , G(4)

ijk =

    gigjgk     N = 4 × 8 × Nx × Ny × Nz terms

Basis Functions II

Motivation and Objectives Problem description Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B2 and

B3

Conclusions and

• utlook

Basis Functions

• Basis Functions I
• Basis Functions II

WSIMS08, Barcelona, December 1-5, 2008 32

fk(x) : Ck(x) = cosh(λkx) cosh(λk/2) − cos(λkx) cos(λk/2); Sk(x) = sinh(µkx) sinh(µk/2) − sin(µkx) sin(µk/2) gk(x) : cos((2k − 1) π x) ; sin(2k π x) λk y µk are the positive solutions of tanh(λk/2) + tan(λk/2) = coth(µk/2) − cot(µk/2) =