Direct Numerical Simulations of Thermal Convection over a Wavy Surface

Abstract

The influence of a wavy surface on thermal convection of Rayleigh-Bénard type in a Boussinesq fluid is investigated by direct numerical simulations. The surface height varies sinusoidally in one direction. The wave amplitude amounts up to 10% of the fluid layer height and the wavelength equals about the critical wavelength of Rayleigh-Bénard convection. The horizontal size of the computational domain equals this wavelength. For isothermal no-slip boundaries, two-dimensional convection sets in at subcritical Rayleigh numbers in close agreement with linear theory. The heat-transfer rate grows almost with the square of the surface-wave amplitude. Convection in a fluid layer over a no-slip surface with prescribed heat flux and an adiabatic free-slip boundary at the top is investigated for supercritical Rayleigh numbers and a Prandtl number of 0.7 in two and three dimensions. Two-dimensional simulations show oscillatory roll convection which becomes almost stationary if the Rayleigh number is of order 7000 or less. The two-dimensional convection is unstable with respect to three-dimensional disturbances and a cross-role pattern evolves even over a surface which is undulated in one direction only. For Rayleigh numbers exceeding about 15 000, the flow becomes turbulent. The results exhibit little sensitivity of the convection to the wavy surface for a 10% amplitude.