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 Howard ( J. Fluid Mech. , vol. 17, issue 3, 1963, pp. 405–432) states that the dimensionless heat flux $\textit {Nu}$ carried out by convection is such that $\textit {Nu} < (3/64 \ Ra)^{1/2}$ 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 $\textit {Nu} -1 < 0.02634 \ Ra^{1/2}$ by Plasting & Kerswell ( J. Fluid Mech. , vol. 477, 2003, pp. 363–379). 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 an 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 $\mathcal {D}$ and on the equation of state. For monatomic gases and large Rayleigh numbers, the bound is $\textit {Nu} < 25.8\, Ra^{{1}/{2}} / (1-\mathcal {D}/2 )^{{5}/{2}}$ .