# Bifurcation analysis of two-dimensional Rayleigh-Bénard convection using deflation

@article{Boulle2022BifurcationAO, title={Bifurcation analysis of two-dimensional Rayleigh-B{\'e}nard convection using deflation}, author={Nicolas Boull'e and Vassilios Dallas and Patrick E. Farrell}, journal={Physical review. E}, year={2022}, volume={105 5-2}, pages={ 055106 } }

We perform a bifurcation analysis of the steady states of Rayleigh-Bénard convection with no-slip boundary conditions in two dimensions using a numerical method called deflated continuation. By combining this method with an initialization strategy based on the eigenmodes of the conducting state, we are able to discover multiple solutions to this nonlinear problem, including disconnected branches of the bifurcation diagram, without the need for any prior knowledge of the solutions. One of the…

## Figures and Tables from this paper

## 2 Citations

### Optimal control of Hopf bifurcations

- MathematicsArXiv
- 2022

We introduce a numerical technique for controlling the location and stability properties of Hopf bifurcations in dynamical systems. The algorithm consists of solving an optimization problem…

### Control of bifurcation structures using shape optimization

- MathematicsSIAM J. Sci. Comput.
- 2022

A numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain, which is able to delay or advance a given bIfurcation point to a given parameter value, often to within machine precision.

## References

SHOWING 1-10 OF 56 REFERENCES

### Bifurcations in two-dimensional Rayleigh-Bénard convection

- Physics
- 1998

Two-dimensional bouyancy-driven convection in a horizontal fluid layer with stress-free boundary conditions at top and bottom and periodic boundary conditions in the horizontal direction is…

### Multiplicity of steady states in cylindrical Rayleigh-Bénard convection.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2006

Three-dimensional steady Rayleigh-Bénard convection in a vertical cylinder is investigated by numerical simulation and bifurcation analysis and the coexistence of multiple stable states is observed near the threshold of the first bIfurcation.

### Bifurcation analysis of multiple steady flow patterns for Rayleigh-Bénard convection in a cubical cavity at Pr = 130.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2006

The stability analysis predicted that six flow patterns were stable and that two, three, or even four of these patterns coexisted over certain ranges of Ra in the studied domain, in agreement with flow visualizations previously reported in the literature.

### Zonal flow reversals in two-dimensional Rayleigh-Bénard convection

- Physics, Environmental SciencePhysical Review Fluids
- 2020

We analyse the nonlinear dynamics of the large scale flow in Rayleigh-Benard convection in a two-dimensional, rectangular geometry of aspect ratio $\Gamma$. We impose periodic and free-slip boundary…

### Asymmetry and Hopf bifurcation in spherical Couette flow

- Mathematics
- 1995

Spherical Couette flow is studied with a view to elucidating the transitions between various axisymmetric steady‐state flow configurations. A stable, equatorially asymmetric state discovered by…

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

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2010

A time-dependent pseudospectral code is adapted to carry out Newton's method and branch continuation and Arnoldi iteration to calculate leading eigenpairs and determine the stability of the steady states of the bifurcation diagram.

### Bifurcation Analysis of the Flow Patterns in Two-Dimensional Rayleigh-BéNard convection

- PhysicsInt. J. Bifurc. Chaos
- 2012

This work investigates the origin of various convective patterns for Prandtl number P = 6.8 using bifurcation diagrams that are constructed using direct numerical simulations (DNS) of Rayleigh–Benard convection (RBC).

### Extreme multiplicity in cylindrical Rayleigh-Bénard convection. I. Time dependence and oscillations.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2010

Simulation of Rayleigh-Bénard convection in a cylindrical container at the same Prandtl number, that of water, and a radius-to-height aspect ratio of two yields a wide variety of coexisting steady and time-dependent flows.

### Bifurcation Analysis for Timesteppers

- Computer Science
- 2000

It is shown that the implicit linear step of a time-stepping code can serve as a highly effective preconditioner for solving linear systems involving the full Jacobian via conjugate gradient iteration.