Extreme multiplicity in cylindrical Rayleigh-Bénard convection. II. Bifurcation diagram and symmetry classification

Abstract

A large number of flows with distinctive patterns have been observed in experiments and simulations of Rayleigh-Bénard convection in a water-filled cylinder whose radius is twice the height. We have adapted a time-dependent pseudospectral code, first, to carry out Newton's method and branch continuation and, second, to carry out the exponential power method and Arnoldi iteration to calculate leading eigenpairs and determine the stability of the steady states. The resulting bifurcation diagram represents a compromise between the tendency in the bulk toward parallel rolls and the requirement imposed by the boundary conditions that primary bifurcations be toward states whose azimuthal dependence is trigonometric. The diagram contains 17 branches of stable and unstable steady states. These can be classified geometrically as roll states containing two, three, and four rolls; axisymmetric patterns with one or two tori; threefold-symmetric patterns called Mercedes, Mitsubishi, marigold, and cloverleaf; trigonometric patterns called dipole and pizza; and less symmetric patterns called CO and asymmetric three rolls. The convective branches are connected to the conductive state and to each other by 16 primary and secondary pitchfork bifurcations and turning points. In order to better understand this complicated bifurcation diagram, we have partitioned it according to azimuthal symmetry. We have been able to determine the bifurcation-theoretic origin from the conductive state of all the branches observed at high Rayleigh number.

Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation

“…Consequently L −1 plays the role of a preconditioner for N u +L, and G u −I is essentially a preconditioned version of N u +L [15,19,134]. This method has been used to calculate bifurcation diagrams in many physical systems, such as spherical Couette flow [73], cylindrical Rayleigh-Bénard convection [14,133], Bose-Einstein condensation [57], and binary fluid convection [1,3]. Far away from u=0, the linear solver may fail to converge, and other preconditioners of N u +L may be sought.…”

Order-of-Magnitude Speedup for Steady States and Traveling Waves via Stokes Preconditioning in Channelflow and Openpipeflow

“…This allowed the linear systems whose solution is required by Newton's method to be solved economically by matrix-free iterative methods, which in turn allowed steady states in large systems (O(10 5 )-O(10 7 ) degrees of freedom) to be computed. This method, called Stokes preconditioning, was applied to calculate bifurcation diagrams in spherical Couette flow by Mamun & Tuckerman [2], convection in Cartesian [3][4][5][6][7][8][9][10][11][12], cylindrical [13][14][15][16][17][18][19] and spherical [20][21][22][23][24] geometries by researchers such as Xin, Chenier, Henry, and Feudel, to von Kármán flow [25][26][27] by Daube, Nore, and Le Quéré and to Bose-Einstein condensation [28,29] by Huepe, Brachet and co-workers…”

“…by adding C∂U/∂ x to the operator N and imposing an additional condition to fix the spatio-temporal phase of the traveling wave. The bottleneck for Newton's method is the solution of the linear system (17). Assuming that the dimension of G U is too large for (17) to be solved directly via Gaussian elimination, it must be solved iteratively using a matrix-free approach that avoids explicit construction of the Jacobian.…”

“…The consequences of this are particularly severe when GMRES is used, since part of its algorithm requires a time which is quadratic in K. We can propose several remedies for this, in increasing order of difficulty and effectiveness. First, K may be reduced by relaxing the convergence criterion for solving the linear equation (17). This has the counterbalancing effect of increasing the number of Newton steps required, but some improvement is possible.…”

Bifurcation analysis of steady natural convection in a tilted cubical cavity with adiabatic sidewalls

“…This method has become a well-established numerical technique used in the context of three-dimensional convection by several authors, e.g. Bergeon & Knobloch (2002) and more recently Borońska & Tuckerman (2010), Mercader, Batiste & Alonso (2010), Beaume, Bergeon & Knobloch (2013), and Lo Jacono, . In a previous study (Torres et al 2013), we used this method to study convection of a pure fluid (Pr = 1) confined in a tilted truncated square duct (aspect ratio, A = 2), focusing on the effect of the tilt around the first instability thresholds.…”