Dynamics of particle trajectories in a Rayleigh–B´ enard problem Dolors Puigjaner (1) Joan Herrero (2) o (3) Francesc Giralt (2) Carles Sim´ (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. WSIMS08, Barcelona, December 1-5, 2008 1
Motivation • Motivation and Fluid mixing efficiency is a crucial issue in many Objectives engineering applications • Motivation • Objectives • Efficient mixing is usually related to turbulent regimes and Problem description to mechanical devices Dynamical systems approach and results Poincar´ e Maps Regular regions and Lyapunov exponents Critical points Streamlines and trajectories Comparison of B 2 and B 3 Conclusions and outlook • 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 WSIMS08, Barcelona, December 1-5, 2008 2
Objectives Motivation and The design of reactors in which efficient mixing is Objectives • Motivation achieved without moving parts • Objectives Problem description Dynamical systems The application of dynamical systems theory to the approach and results analysis of dynamics and mixing properties in flows in- Poincar´ e Maps Regular regions and duced by Rayleigh–B´ enard convection inside a cube Lyapunov exponents Critical points Streamlines and trajectories • Analyze the rich dynamics of fluid particle trajectories Comparison of B 2 and • Characterize well-mixed regions inside the cube B 3 • Conclusions and Investigate the dependence of mixing properties on the outlook Rayleigh number WSIMS08, Barcelona, December 1-5, 2008 3
Flow system and equations T=T Motivation and Continuity c Objectives g ∇ · � V = 0 Problem description • Flow system Ω • Continuation method Momentum • Bifurcation diagram T=T h • Flow patterns � � ∂� 1 V • Symmetries 1 2 ( � V · ∇ ) � L z ∂t + Ra V = y Dynamical systems Pr approach and results x ∇ 2 � 1 Poincar´ e Maps V + Ra 2 θ� e z −∇ p Ra = gβL 3 ∆ T/αν Regular regions and Lyapunov exponents Pr = ν/α Energy Critical points β thermal expansion ∂θ Streamlines and 1 2 ( � ∂t + Ra V · ∇ ) θ = trajectories ν kinematic viscosity Comparison of B 2 and 1 2 � B 3 α thermal diffusivity ∇ 2 θ + Ra V · � e z Conclusions and ∆ T = T h − T c outlook Boundary conditions � V = θ = 0 en ∂ Ω θ =[ T − ( T h + T c ) / 2] / ∆ T − z WSIMS08, Barcelona, December 1-5, 2008 4
Continuation method Motivation and Galerkin spectral method Stability Analysis Objectives + with basis functions Bifurcation Location Problem description { � F i ( x, y, z ) } • Flow system satisfying • Continuation method Bifurcation Ra Ra b Conductive the boundary condi- • Bifurcation diagram Main eigenvalues solution Main eigenvectors • Flow patterns tions and the continuity • Symmetries equation Dynamical systems Branch Switching approach and results Technique Poincar´ e Maps Solution Regular regions and New solution at Ra Lyapunov exponents � � close to Ra b � Critical points V � c i ( t ) � = F i Streamlines and θ trajectories Parameter Continuation i Comparison of B 2 and Procedure B 3 Conclusions and Tracked solution branch outlook D. Puigjaner, J. Herrero, C. Sim´ o, F. Giralt, J. Fluid Mechanics, 598, 393–427, (2008) WSIMS08, Barcelona, December 1-5, 2008 5
Bifurcation diagram ( Pr = 130 ) Steady solutions that are stable over some Ra range Motivation and Objectives Problem description 0.2 • Flow system • Continuation method B 1 • Bifurcation diagram B 34 • Flow patterns 0 • Symmetries Nu-0.012Ra 1/2 -0.265 Dynamical systems B 3 approach and results B 25 B 2 -0.2 Poincar´ e Maps -0.280 82 92 Regular regions and B 5 Lyapunov exponents B 251 Critical points B 251 -0.4 Streamlines and trajectories 20 40 60 80 100 120 140 Comparison of B 2 and B 3 10 -3 Ra Conclusions and stable • steady bifurcation outlook unstable ◦ Hopf bifurcation turning point WSIMS08, Barcelona, December 1-5, 2008 6
Flow patterns: λ 2 = 0 surfaces Motivation and Objectives B 2 Problem description • Flow system Initial Ra • Continuation method 6 798 • Bifurcation diagram • Flow patterns Stability Range • Symmetries 67 730 – 85 694 Dynamical systems Ra =7 000 Ra =51 000 Ra =80 000 approach and results Poincar´ e Maps Regular regions and B 3 Lyapunov exponents Initial Ra Critical points Streamlines and 11 612 trajectories Stability Range Comparison of B 2 and B 3 20 637 – 79 362 Conclusions and outlook Ra =12 000 Ra =51 000 Ra =80 000 λ 2 is the second largest eigenvalue of the tensor S 2 + Ω 2 WSIMS08, Barcelona, December 1-5, 2008 7
Symmetries and invariant planes Motivation and Symmetry Group Invariant Objectives Solution Problem description (generators) Planes • Flow system • Continuation method B 2 S d − , − I x + y = 0 • Bifurcation diagram • Flow patterns B 25 − I – • Symmetries Dynamical systems B 251 S y , − I y = 0 approach and results Poincar´ e Maps � x + y = 0 Regular regions and B 3 S d + , − S y Lyapunov exponents x − y = 0 Critical points Streamlines and trajectories S y reflection about the plane y = 0 Comparison of B 2 and B 3 S d + reflection about the plane x − y = 0 Conclusions and outlook S d − reflection about the plane x + y = 0 − I simmetry with respect the origin − S y rotation of angle π around the y –axis WSIMS08, Barcelona, December 1-5, 2008 8
Numerical methods Motivation and Particle trajectories (Negligible diffusivities) Objectives Problem description x = u ( x, y, z ) ˙ Dynamical systems y = v ( x, y, z ) ˙ Advection equations approach and results • Numerical methods z = w ( x, y, z ) ˙ Poincar´ e Maps • Symmetries and invariant planes Regular regions and Lyapunov exponents • Poincar´ e sections Critical points • Periodic orbits and their stability Streamlines and trajectories • Size and shape of regular regions Comparison of B 2 and B 3 • Maximal Lyapunov exponents and metric entropy Conclusions and • Critical points in the interior and on the boundary outlook ◦ Stability analysis ◦ Poincar´ e–Hopf index theorem C. Sim´ o, D. Puigjaner, J. Herrero, F. Giralt, Communications in Nonlinear Science and Numerical Simulation, doi:10.1016/j.cnsns.2008.07.012. In Press WSIMS08, Barcelona, December 1-5, 2008 9
Poincar´ e Maps I Motivation and 512 equidistributed initial conditions integrated up to t = 10 3 Objectives Problem description B 2 , z = 0 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 B 2 and B 3 Conclusions and outlook Ra = 10 4 Ra = 3 . 3 × 10 4 WSIMS08, Barcelona, December 1-5, 2008 10
Main stable periodic orbit (fixed elliptic point) Motivation and Objectives Problem description argument trace Dynamical systems 4 approach and results 10 (x-coordinate) Poincar´ e Maps • Poincar´ e Maps I • Periodic orbit • Poincar´ e Maps II 2 Regular regions and Lyapunov exponents Critical points Streamlines and 0 trajectories Comparison of B 2 and B 3 Conclusions and outlook -2 20 40 60 80 100 n y = 2 π/k, k = 2 , · · · , 7 WSIMS08, Barcelona, December 1-5, 2008 11
Poincar´ e Maps II B 2 , z = 0 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 B 2 and B 3 Conclusions and outlook Ra = 6 . 87099 × 10 4 Ra = 8 . 5 × 10 4 WSIMS08, Barcelona, December 1-5, 2008 12
Regular regions V c = volume occupied by the chaotic zone (points outside Motivation and Objectives invariant tori) Problem description Dynamical systems approach and results Computation Procedure Poincar´ e Maps Divide the cavity into n × n × n cubic cells ( n = 200 ) • Regular regions and Lyapunov exponents Compute trajectories of fluid particles initially located at x 0 , • • Regular regions • Lyapunov exponents I for any x 0 in the set C I (final time t M = 10 6 ) • L M and V c � � � � − 0 . 375+ i 8 , 0 . 48 , − 0 . 375+ j • Metric entropy C I = , i, j = 0 , · · · , 6 • Regular regions I 8 • Regular regions II • Store the cells visited by one or more trajectories Critical points N r ( t ) = number of cells that at time t have not yet been • Streamlines and trajectories visited by any particle trajectory (every ∆ t = 200 ) Comparison of B 2 and Check that N r ( t ) is almost constant in t ∈ [ 3 B 3 4 t M , t M ] • Conclusions and Points at a distance less than 0 . 01 from the boundaries are • outlook considered as non-regular WSIMS08, Barcelona, December 1-5, 2008 13
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