HPC Tier Module
Rayleigh-Benard Convection
2D Rayleigh-Benard convection with the free-slip onset Ra_c validated against its closed form.
See it run - a worked example, 100% in this browser tab
The problem
Convection studies need a quick way to see roll onset and heat transport, but the critical Rayleigh number depends on the plate boundary condition and the box - a mismatch silently misleads, and full CFD codes are heavy to stand up.
The local-first solution
This solver integrates 2D Boussinesq convection in vorticity-streamfunction form in your browser, measures the Nusselt heat-transport number over the run, and validates the onset against the exact free-slip Ra_c = 27 pi^4/4, always surfacing which Ra_c governs the run. Deterministic f64 math, nothing uploaded.
What it does
Vorticity-transport, temperature, and streamfunction equations on a periodic box
2nd-order centered finite differences with an SOR/Jacobi Poisson solve for psi
Explicit RK2 time march of the Boussinesq system
Nusselt number Nu measured over the run with the late-time steady value reported
Free-slip onset validated against the exact Ra_c = 27 pi^4/4 (~657.51)
GeoNum drift probe on the relative change of Nu to gauge whether the steady state has settled
Honest scope
EXACT: the centered operators, the Poisson solve, the RK2 march, the Nusselt integral, and the closed-form free-slip onset. Ra_c is boundary-condition and box dependent - the rigid-rigid 1707.76 figure is reported for reference but the solver imposes free-slip, and it surfaces which Ra_c it used. NOT modeled: 3D, non-Boussinesq/compressible effects, rotation/magnetic fields, no-slip plates in the nonlinear solver, or high-Ra turbulent scaling; coarse grids under-resolve boundary layers (flagged). A prototype, not a production CFD code.
Authorities cited
- Rayleigh, Lord (1916). On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag. 32, 529-546. (Free-slip onset Ra_c = 27 pi^4/4 ~ 657.51 at k_c = pi/sqrt(2).)
- Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability. Oxford / Dover. Ch. II - Rayleigh-Benard linear stability; rigid-rigid onset Ra_c ~ 1707.762 at k_c ~ 3.117.
- Reid, W. H. & Harris, D. L. (1958). Some further results on the Benard problem. Phys. Fluids 1, 102-110. (Numerical eigenvalue Ra_c = 1707.762 for rigid-rigid plates.)
- Drazin, P. G. & Reid, W. H. (2004). Hydrodynamic Stability, 2nd ed. Cambridge. Ch. 2 - marginal curve Ra(k) = (pi^2 + k^2)^3 / k^2 (free-slip).
- Boussinesq, J. (1903). Theorie analytique de la chaleur, Vol. 2. (Boussinesq approximation: density variation retained only in the buoyancy term.)
- Getling, A. V. (1998). Rayleigh-Benard Convection: Structures and Dynamics. World Scientific. (Nusselt-number definition Nu = 1 + <v theta>; nonlinear roll structure.)
Measure heat transport with a validated onset
Run the convection solve in your browser with nothing uploaded, then route the Nusselt curve and onset check into a Sandbox workspace or attach them to a Worklog case.