Cubic Spline Collocation Method for the Simulation of Turbulent Thermal Convection in Compressible Fluids

Thesis/Dissertation · Fri Jan 01 04:00:00 EST 1999

DOI:https://doi.org/10.2172/15014452· OSTI ID:15014452

Castillo, Victor Manuel ^[1]

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1. Univ. of California, Davis, CA (United States)

A collocation method using cubic splines is developed and applied to simulate steady and time-dependent, including turbulent, thermally convecting flows for two-dimensional compressible fluids. The state variables and the fluxes of the conserved quantities are approximated by cubic splines in both space direction. This method is shown to be numerically conservative and to have a local truncation error proportional to the fourth power of the grid spacing. A ''dual-staggered'' Cartesian grid, where energy and momentum are updated on one grid and mass density on the other, is used to discretize the flux form of the compressible Navier-Stokes equations. Each grid-line is staggered so that the fluxes, in each direction, are calculated at the grid midpoints. This numerical method is validated by simulating thermally convecting flows, from steady to turbulent, reproducing known results. Once validated, the method is used to investigate many aspects of thermal convection with high numerical accuracy. Simulations demonstrate that multiple steady solutions can coexist at the same Rayleigh number for compressible convection. As a system is driven further from equilibrium, a drop in the time-averaged dimensionless heat flux (and the dimensionless internal entropy production rate) occurs at the transition from laminar-periodic to chaotic flow. This observation is consistent with experiments of real convecting fluids. Near this transition, both harmonic and chaotic solutions may exist for the same Rayleigh number. The chaotic flow loses phase-space information at a greater rate, while the periodic flow transports heat (produces entropy) more effectively. A linear sum of the dimensionless forms of these rates connects the two flow morphologies over the entire range for which they coexist. For simulations of systems with higher Rayleigh numbers, a scaling relation exists relating the dimensionless heat flux to the two-seventh's power of the Rayleigh number, suggesting the existence of ''hard'' turbulence in two-dimensional compressible convection.

• Research Organization: Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
• Sponsoring Organization: USDOE
• DOE Contract Number: W-7405-ENG-48
• OSTI ID: 15014452
• Report Number(s): UCRL-TH--209324
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
97 MATHEMATICS AND COMPUTING
ACCURACY
CONVECTION
ENTROPY
HARMONICS
HEAT FLUX
NAVIER-STOKES EQUATIONS
PRODUCTION
RAYLEIGH NUMBER
SIMULATION
TURBULENCE