Upper bound of heat flux in an anelastic model for Rayleigh-Bénard convection

Bounds on heat transfer have been the subject of previous studies concerning convection in the Boussinesq approximation: in the Rayleigh-Bénard configuration, the first result obtained by states that Nu < (3/64 Ra)^1/2 for large values of the Rayleigh number Ra, independently of the Prandtl number Pr. This is still the best known upper bound, only with the prefactor improved to Nu < 1/6 Ra^1/2 by . In the present paper, this result is extended to compressible convection. An upper bound is obtained for the anelastic liquid approximation, which is similar to the anelastic model used in astrophysics based on a turbulent diffusivity for entropy. The anelastic bound is still scaling as Ra^1/2, independently of Pr, but depends on the dissipation number 𝒟 and on the equation of state. For monatomic gases and large Rayleigh numbers, the bound is Nu < 146 Ra^1/2 / (2-𝒟 )^5/2.