The inertial regimes in Rayleigh–Bénard convection

Most theoretical studies of Rayleigh–Bénard Convection assume that the velocity boundary layer develops on the whole width or height <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> of the convection cell, or that the development length <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> scales with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> . We argue that it is probably not the case, and examine the consequences of an intermediate asymptotics hypothesis in the sense of Barenblatt [1], the development length <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> scaling with the Nusselt number: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>∝</mml:mo> <mml:mi>H</mml:mi> <mml:msup> <mml:mi mathvariant="normal">Nu</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . This hypothesis is checked through existing experimental data, which show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> can take several different values. The analysis of Grossmann and Lohse [2] is reexamined with this new point of view, stressing on pure scaling regimes.