Buoyancy-Driven Entrainment in Dry Thermals. (arXiv:1906.07224v1 [physics.flu-dyn])
<a href="http://arxiv.org/find/physics/1/au:+McKim_B/0/1/0/all/0/1">Brett McKim</a>, <a href="http://arxiv.org/find/physics/1/au:+Jeevanjee_N/0/1/0/all/0/1">Nadir Jeevanjee</a>, <a href="http://arxiv.org/find/physics/1/au:+Lecoanet_D/0/1/0/all/0/1">Daniel Lecoanet</a>
Turner (1957) proposed that dry thermals entrain because of buoyancy (via a
constraint which requires an increase in the radius $a$). This however, runs
counter to the scaling arguments commonly used to derive the entrainment rate,
which rely on either the self-similarity of Scorer (1957) or the turbulent
entrainment hypothesis of Morton et al (1956). The assumption of
turbulence-driven entrainment was investigated by Lecoanet and Jeevanjee
(2018), who found that the entrainment efficiency $e$ varies by less than
$20%$ between laminar (Re = 630) and turbulent (Re = 6300) thermals. This
motivated us to utilize Turner’s argument of buoyancy-controlled entrainment in
addition to the thermal’s vertical momentum equation to build a model for
thermal dynamics which does not invoke turbulence or self-similarity. We derive
simple expressions for the thermals’ kinematic properties and their fractional
entrainment rate $epsilon$ and find close quantitative agreement with the
values in direct numerical simulations. We then directly validate the role of
buoyancy-driven entrainment by running simulations where gravity is turned off
midway through a thermal’s rise. The entrainment efficiency $e$ is observed to
drop to less than 1/3 of its original value in both the laminar and turbulent
cases when $g=0$, affirming the central role of buoyancy in entrainment in dry
thermals.
Turner (1957) proposed that dry thermals entrain because of buoyancy (via a
constraint which requires an increase in the radius $a$). This however, runs
counter to the scaling arguments commonly used to derive the entrainment rate,
which rely on either the self-similarity of Scorer (1957) or the turbulent
entrainment hypothesis of Morton et al (1956). The assumption of
turbulence-driven entrainment was investigated by Lecoanet and Jeevanjee
(2018), who found that the entrainment efficiency $e$ varies by less than
$20%$ between laminar (Re = 630) and turbulent (Re = 6300) thermals. This
motivated us to utilize Turner’s argument of buoyancy-controlled entrainment in
addition to the thermal’s vertical momentum equation to build a model for
thermal dynamics which does not invoke turbulence or self-similarity. We derive
simple expressions for the thermals’ kinematic properties and their fractional
entrainment rate $epsilon$ and find close quantitative agreement with the
values in direct numerical simulations. We then directly validate the role of
buoyancy-driven entrainment by running simulations where gravity is turned off
midway through a thermal’s rise. The entrainment efficiency $e$ is observed to
drop to less than 1/3 of its original value in both the laminar and turbulent
cases when $g=0$, affirming the central role of buoyancy in entrainment in dry
thermals.
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