Interfacial dynamics and energy cascade in immiscible Rayleigh–Taylor turbulence

We investigate interfacial dynamics and multiscale energy transfer in immiscible Rayleigh–Taylor turbulence using numerical simulations with varying surface-tension coefficients sigma <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:miσ</mml:mi> </mml:math> $\sigma$ . Capillarity is shown to control characteristic length scales, interfacial area, and global energy and enstrophy budgets. The flow exhibits self-similar evolution with respect to surface tension, with the maximum kinetic energy scaling as sigma Superscript 1 divided by 2 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:msup mml:miσ</mml:mi> mml:mrow mml:mn1</mml:mn> mml:mrow mml:mo/</mml:mo> </mml:mrow> mml:mn2</mml:mn> </mml:mrow> </mml:msup> </mml:math> $\sigma ^{1/2}$ and the flow duration as sigma Superscript negative 1 divided by 4 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:msup mml:miσ</mml:mi> mml:mrow mml:mo−</mml:mo> mml:mn1</mml:mn> mml:mrow mml:mo/</mml:mo> </mml:mrow> mml:mn4</mml:mn> </mml:mrow> </mml:msup> </mml:math> $\sigma ^{-1/4}$ . A scale-by-scale budget shows that surface tension removes kinetic energy at large scales while injecting it at small scales, with the crossover occurring near the Hinze scale. We further recast and verify a local kinematic relationship between surface-tension power and interface stretching, up to conservative transport, bold italic f Superscript sigma Baseline bold dot bold italic u equals minus sigma script upper S StartAbsoluteValue bold nabla c EndAbsoluteValue plus normal t normal r normal a normal n normal s normal p normal o normal r normal t <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:msup <mml:mi mathvariant="bold-italic">f</mml:mi> mml:miσ</mml:mi> </mml:msup> <mml:mo mathvariant="bold">⋅</mml:mo> <mml:mi mathvariant="bold-italic">u</mml:mi> mml:mo=</mml:mo> mml:mo−</mml:mo> mml:miσ</mml:mi> mml:mrow <mml:mi mathvariant="script" class="MJX-tex-caligraphic">S</mml:mi> </mml:mrow> <mml:mspace width="thinmathspace"/> mml:mrow <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi mathvariant="bold">∇</mml:mi> mml:mic</mml:mi> mml:mrow <mml:mo stretchy="false">|</mml:mo> </mml:mrow> mml:mo+</mml:mo> mml:mrow <mml:mi mathvariant="normal">t</mml:mi> <mml:mi mathvariant="normal">r</mml:mi> <mml:mi mathvariant="normal">a</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> <mml:mi mathvariant="normal">o</mml:mi> <mml:mi mathvariant="normal">r</mml:mi> <mml:mi mathvariant="normal">t</mml:mi> </mml:mrow> </mml:math> $\boldsymbol{f}^\sigma \boldsymbol{\cdot }\boldsymbol{u} = - \sigma \mathcal{S}\,|\boldsymbol{\nabla }c| + \mathrm{transport}$ , where bold italic f Superscript sigma <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:msup <mml:mi mathvariant="bold-italic">f</mml:mi> mml:mrow mml:miσ</mml:mi> </mml:mrow> </mml:msup> </mml:math> $\boldsymbol{f}^{\sigma }$ is the surface-tension force, bold italic u <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi mathvariant="bold-italic">u</mml:mi> </mml:math>