Scaling in thermal convection: A unifying theory

Scaling in thermal convection: A unifying theory

February 5, 2008 | Siegfried Grossmann and Detlef Lohse
A systematic theory for the scaling of the Nusselt number (Nu) and Reynolds number (Re) in strong Rayleigh-Benard (RB) convection is proposed and shown to be compatible with recent experiments. The theory assumes a coherent large-scale convection roll ("wind of turbulence") and is based on the dynamical equations in the bulk and boundary layers. Several regimes are identified in the Ra-Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation. The crossover between regimes is calculated. In the regime most frequently studied experimentally (Ra ≲ 10¹¹), the leading terms are Nu ∼ Ra¹/⁴Pr¹/⁸, Re ∼ Ra¹/²Pr⁻³/⁴ for Pr ≲ 1, and Nu ∼ Ra¹/⁴Pr⁻¹/¹², Re ∼ Ra¹/²Pr⁻⁵/⁶ for Pr ≳ 1. In most measurements, these laws are modified by additive corrections from neighboring regimes, leading to an apparent slightly larger effective Nu vs Ra scaling exponent. The most important neighboring regimes include the Kraichnan regime (Nu ∼ Ra¹/²Pr¹/², Re ∼ Ra¹/²Pr⁻¹/²), the small Pr regime (Nu ∼ Ra¹/⁵Pr¹/⁵, Re ∼ Ra²/⁵Pr⁻³/⁵), the larger Pr regime (Nu ∼ Ra¹/³, Re ∼ Ra⁴/⁹Pr⁻²/³), and the even larger Pr regime (Nu ∼ Ra³/⁷Pr⁻¹/⁷, Re ∼ Ra⁴/⁷Pr⁻⁶/⁷). A linear combination of the 1/4 and 1/3 power laws for Nu with Ra, Nu = 0.27Ra¹/⁴ + 0.038Ra¹/³, mimics a 2/7 power law exponent in a regime as large as ten decades. For very large Ra, the laminar shear boundary layer is speculated to break down through nonnormal-nonlinear transition to turbulence, leading to another regime. The theory is summarized in the phase diagram figure 1. The theory is validated by comparing the scaling exponents with experimental data, showing good agreement. The scaling exponents for Nu and Re are found to depend on Ra and Pr, with the Nu scaling exponent being approximately 2/7 for large Ra. The theory accounts for the observed crossover between different regimes and provides a comprehensive framework for understanding the scaling behavior in RB convection.A systematic theory for the scaling of the Nusselt number (Nu) and Reynolds number (Re) in strong Rayleigh-Benard (RB) convection is proposed and shown to be compatible with recent experiments. The theory assumes a coherent large-scale convection roll ("wind of turbulence") and is based on the dynamical equations in the bulk and boundary layers. Several regimes are identified in the Ra-Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation. The crossover between regimes is calculated. In the regime most frequently studied experimentally (Ra ≲ 10¹¹), the leading terms are Nu ∼ Ra¹/⁴Pr¹/⁸, Re ∼ Ra¹/²Pr⁻³/⁴ for Pr ≲ 1, and Nu ∼ Ra¹/⁴Pr⁻¹/¹², Re ∼ Ra¹/²Pr⁻⁵/⁶ for Pr ≳ 1. In most measurements, these laws are modified by additive corrections from neighboring regimes, leading to an apparent slightly larger effective Nu vs Ra scaling exponent. The most important neighboring regimes include the Kraichnan regime (Nu ∼ Ra¹/²Pr¹/², Re ∼ Ra¹/²Pr⁻¹/²), the small Pr regime (Nu ∼ Ra¹/⁵Pr¹/⁵, Re ∼ Ra²/⁵Pr⁻³/⁵), the larger Pr regime (Nu ∼ Ra¹/³, Re ∼ Ra⁴/⁹Pr⁻²/³), and the even larger Pr regime (Nu ∼ Ra³/⁷Pr⁻¹/⁷, Re ∼ Ra⁴/⁷Pr⁻⁶/⁷). A linear combination of the 1/4 and 1/3 power laws for Nu with Ra, Nu = 0.27Ra¹/⁴ + 0.038Ra¹/³, mimics a 2/7 power law exponent in a regime as large as ten decades. For very large Ra, the laminar shear boundary layer is speculated to break down through nonnormal-nonlinear transition to turbulence, leading to another regime. The theory is summarized in the phase diagram figure 1. The theory is validated by comparing the scaling exponents with experimental data, showing good agreement. The scaling exponents for Nu and Re are found to depend on Ra and Pr, with the Nu scaling exponent being approximately 2/7 for large Ra. The theory accounts for the observed crossover between different regimes and provides a comprehensive framework for understanding the scaling behavior in RB convection.
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Understanding Scaling in thermal convection%3A a unifying theory