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We investigate the stability and dynamics of natural convection in twodimensions, subject to inhomogeneous boundary conditions. In particular, weconsider a Rayleigh-B\`enard (RB) cell, where the horizontal top boundarycontains a periodic sequence of alternating thermal insulating and conductingpatches, and we study the effects of the heterogeneous pattern on the globalheat exchange, both at low and high Rayleigh numbers. At low Rayleigh numbers,we determine numerically the transition from a regime characterized by thepresence of small convective cells localized at the inhomogeneous boundary tothe onset of bulk convective rolls spanning the entire domain. Such atransition is also controlled analytically in the limit when the boundarypattern length is small compared with the cell vertical size. At higherRayleigh number, we use numerical simulations based on a lattice Boltzmannmethod to assess the impact of boundary inhomogeneities on the fully turbulentregime up to $Ra \sim 10^{10}$.

Natural convection with mixed insulating and conducting boundary conditions: low and high Rayleigh numbers regimes