Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection

Area of Science:

• Fluid dynamics
• Nonlinear dynamical systems
• Computational mathematics

Background:

• Variational problems arise in analyzing forced-dissipative infinite-dimensional nonlinear dynamical systems.
• Rigorous upper-bound analysis is crucial for understanding phenomena like convection.
• Existing numerical methods often require accurate initial conditions and numerical continuation.

Purpose of the Study:

• To introduce and analyze an alternative computational procedure for solving variational problems.
• To apply this method to Rayleigh-Bénard convection and determine optimal bounds on heat transport.
• To prove the algorithm's convergence to the global optimum for specific flow configurations.

Main Methods:

• Development of an alternative computational procedure for variational problems.
• Application to Rayleigh-Bénard convection with stress-free isothermal boundaries.
• Analysis of convergence to the global optimum for three canonical flow configurations.

Main Results:

• The algorithm converges to the global optimum of the variational problem.
• Optimal bounds for heat transport (Nusselt number, Nu) were determined as a function of Rayleigh number (Ra), Prandtl number (Pr), and aspect ratio (L).
• For Ra≤10(10) and fixed L=2√[2], Nu≤0.106Pr(0)Ra(5/12), indicating molecular transport is significant in the high-Ra regime.

Conclusions:

• The novel computational method is easy to implement and avoids numerical continuation.
• The findings provide a precise scaling relation for heat transport in Rayleigh-Bénard convection.
• Molecular transport cannot be neglected in the ultimate high-Rayleigh number regime.