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 OJFD  Vol.6 No.4 , December 2016
Laminar-Turbulent Bifurcation Scenario in 3D Rayleigh-Benard Convection Problem
Abstract: We are considering two initial-boundary value problems for Rayleigh-Benard convection in Oberbeck-Boussinesq approximation for incompressible fluid in 3D-rectangular domain with 4:4:1 geometric ratio with periodicity in two directions and cubic domain with 1:1:1 ratio and zero velocity and temperature gradient boundary conditions. For this purpose, we use two numerical method: one is a Pseudo-Spectral-Galerkin method with trigonometric-Chebyshev polynomials and the other is finite element/volume method with WENO interpolation for advection term. Numerical methods are presented shortly and are benchmarked against known DNS data and against one another (for quasi-periodic domain problem). Then we perform stability analysis using analytical expression for main stationary solutions and eigenvalue numerical analysis by applying Implicitly Restarted Arnoldi (IRA) method. The IRA is used to perform linear stability analysis, find bifurcations of stationary points and analyze eigenvalues of monodromy matrices. Thus characteristic exponents of the system for time periodic solutions (limited cycles of various periods and resonance invariant tori) are computed. We show, numerically, the existence of multistable rotes to chaos through chaotic fractal attractors, full Feigenbaum-Sharkovski cascades and multidimensional torus attractors (Landau-Hopf scenario). The existence of these attractors is shown through analysis of phase subspaces projections, Poincare sections and eigenvalue analysis of numerically computed DNS data. These attractors burst into chaos with the increase of Rayleigh number either through resonance and phase-locking or through emergence of singular chaotic attractors.
Cite this paper: Evstigneev, N. (2016) Laminar-Turbulent Bifurcation Scenario in 3D Rayleigh-Benard Convection Problem. Open Journal of Fluid Dynamics, 6, 496-539. doi: 10.4236/ojfd.2016.64035.
References

[1]   Tucker, W. (2002) A Rigorous ODE Solver and Smale’s 14th Problem. Foundations of Computational Mathematics, 2, 53-117. https://doi.org/10.1007/s002080010018

[2]   Lord Rayleigh, O.M.F.R.S. (1916) On Convection Currents in a Horizontal Layer of Fluid, When the Higher Temperature Is on the Under Side. Philosophical Magazine Series 6, 32, 529-546.
https://doi.org/10.1080/14786441608635602

[3]   Getling, A.V. (1998) Rayleigh-Benard Convection: Structures and Dynamics. World Scientific, Singapore. https://doi.org/10.1142/3097

[4]   Gelfgat, A.Y. (1999) Different Modes of Rayleigh-Benard Instability in Two- and Three-Dimensional Rectangular Enclosures. Journal of Computational Physics, 156, 300-324.
https://doi.org/10.1006/jcph.1999.6363

[5]   Gelfgat, A.Y. (2001) Two- and Three-Dimensional Instabilities of Confined Flows: Numerical Study by a Global Galerkin Method. Computational Fluid Dynamics Journal (Special Issue), 9, 437-448.

[6]   Paul, S., Verma, M.K., Wahi, P., Reddy, S.K. and Kumar, K. (2012) Bifurcation Analysis of the Flow Patterns in Two-Dimensional Rayleigh-Benard Convection. International Journal of Bifurcation and Chaos, 22, Article ID: 1230018. https://doi.org/10.1142/S0218127412300182

[7]   Manneville, P. (2010) Rayleigh-Benard Convection: Thirty Years of Experimental, Theoretical, and Modeling Work. Springer Tracts in Modern Physics, 207, 41-65.
https://doi.org/10.1007/978-0-387-25111-0_3

[8]   Evstigneev, N.M. and Magnitskii, N.A. (2013) FSM Scenarios of Laminar-Turbulent Transition in Incompressible Fluids. In: Awrejcewicz, J., Ed., Nonlinearity, Bifurcation and Chaos Theory and Applications, Chapter 10, INTECH, Rijeka, 250-280.

[9]   Peyret, R. (2002) Spectral Methods for Incompressible Viscous Flow. Applied Mathematical Sciences, Vol. 148, Springer-Verlag, New York.

[10]   Langtangen, H.P., Mardal, K.-A. and Winther, R. (2002) Numerical Methods for Incompressible Viscous Flow. Advances in Water Resources, 25, 1125-1146.
https://doi.org/10.1016/S0309-1708(02)00052-0

[11]   Sigal, G., Shu, C.-W. and Tadmor, E. (2001) Strong Stability-Preserving High-Order Time Discretization Methods. SIAM Review, 43, 89-112.
https://doi.org/10.1137/S003614450036757X

[12]   Lehoucq, R.B. and Scott, J.A. (1997) Implicitly Restarted Arnoldi Methods and Eigenvalues of the Discretized Navier Stokes Equations. SIAM Journal on Matrix Analysis and Applications, 23, 551-562. https://doi.org/10.1137/S0895479899358595

[13]   Nebauer, J.R.A. and Blackburn, H.M. (2012) Floquet Stability of Time Periodic Pipe Flow. Proceedings of International Conference on CFD in the Minerals and Process Industries CSIRO, Melbourne, 10-12 December 2012, 1-6.

[14]   Abdessemed, N., Sherwin, S.J. and Theofilis, V. (2005) Floquet Stability Analysis of Periodic States in Low Pressure Turbine Flows at Moderate Reynolds Numbers. Congreso de Metodos Numericos en Ingenieria, Granada, 4 a 7 de julio, Espana.

[15]   Burroughs, E.A., Romero, L.A., Lehoucq, R.B. and Salinger, A.G. (2001) Large Scale Eigenvalue Calculations for Computing the Stability of Buoyancy Driven Flows. Sandia Report SAND2001-0113 Unlimited Release.

[16]   Saad, E. (2011) Numerical Methods for Large Eigenvalue Problems. 2nd Edition, Society for Industrial and Applied Mathematics, Twin Cities. https://doi.org/10.1137/1.9781611970739

[17]   Evstigneev, N.M. (2009) Numerical Integration of Poisson’s Equation Using a Graphics Processing Unit with CUDA-Technology. Online Journal Numerical Method and Programing, 10, 268-274.

[18]   Geuzaine, C. and Remacle, J.-F. (2009) GMSH: A Three-Dimensional Finite Element Mesh Generator with Built-In Pre- and Post-Processing Facilities. International Journal for Numerical Methods in Engineering, 79, 1309-1331. https://doi.org/10.1002/nme.2579

[19]   http://iketwww1.fzk.de/iket/turbit/anlagensicherheit_und_systemsimulation/fluid_dynamics/
simulation/e_index.html


[20]   Worner, M., Schmidt, M. and Grotzbach, G. (1997) Direct Numerical Simulation of Turbulence in an Internally Heated Convective Fluid Layer and Implications for Statistical Modelling. Journal of Hydraulic Research, 35, 773-798. https://doi.org/10.1080/00221689709498388

[21]   Kuznetsov, Y.A. (1998) Elements of Applied Bifurcation Theory. 2nd Edition, Springer, New York.

[22]   Ryabkov, O.I. (2013) On Polymodal Maps and Their Application to Chaotic Dynamics of Differential Equations. Proceedings of ISA RAN, 63, 70-84. (In Russian)

[23]   Dijkstra, H.A., Wubs, F.W., Cliffe, A.K., et al. (2014) Numerical Bifurcation Methods and Their Application to Fluid Dynamics: Analysis beyond Simulation. Communications in Computational Physics, 15, 1-45. https://doi.org/10.4208/cicp.240912.180613a

[24]   Ashwin, P. and Podvigina, O. (2003) Hopf Bifurcation with Cubic Symmetry and Instability of ABC Flow. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 459, 1801-1827. https://doi.org/10.1098/rspa.2002.1090

 
 
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