## Abstract

Large-scale dissipation mechanisms have been routinely employed in

numerical simulations of two-dimensional turbulence to absorb energy

at large scales, presumably mimicking the quasi-steady picture of

Kraichnan in an unbounded fluid. Here, ``side effects'' of such a

mechanism---mechanical friction---on the small-scale dynamics of

forced two-dimensional Navier--Stokes turbulence are elaborated by

both theoretical and numerical analysis. Given a positive friction

coefficient $\alpha$, viscous dissipation of enstrophy has been known

to vanish in the inviscid limit $\nu\to0$. This effectively renders

the scale-neutral friction the only mechanism responsible for

enstrophy dissipation in that limit. The resulting dynamical picture

is that the classical enstrophy inertial range becomes a dissipation

range, in which the dissipation of enstrophy by friction mainly occurs.

For each $\alpha>0$, there exists a critical viscosity $\nu_c$, which

depends on physical parameters, separating the regimes of predominant

viscous and frictional dissipation of enstrophy. It is found that

$\nu_c=[{\eta'}^{1/3}/(Ck_f^2)]\exp\{-{\eta'}^{1/3}/(C\alpha)\}$, where

$\eta'$ is half the enstrophy injection rate, $k_f$ is the forcing wave

number, and $C$ is a nondimensional constant (the Kraichnan--Batchelor

constant). The implication of the present results is that mechanical

friction is a poor choice in numerical attempts to address fundamental

issues concerning the direct enstrophy transfer.

numerical simulations of two-dimensional turbulence to absorb energy

at large scales, presumably mimicking the quasi-steady picture of

Kraichnan in an unbounded fluid. Here, ``side effects'' of such a

mechanism---mechanical friction---on the small-scale dynamics of

forced two-dimensional Navier--Stokes turbulence are elaborated by

both theoretical and numerical analysis. Given a positive friction

coefficient $\alpha$, viscous dissipation of enstrophy has been known

to vanish in the inviscid limit $\nu\to0$. This effectively renders

the scale-neutral friction the only mechanism responsible for

enstrophy dissipation in that limit. The resulting dynamical picture

is that the classical enstrophy inertial range becomes a dissipation

range, in which the dissipation of enstrophy by friction mainly occurs.

For each $\alpha>0$, there exists a critical viscosity $\nu_c$, which

depends on physical parameters, separating the regimes of predominant

viscous and frictional dissipation of enstrophy. It is found that

$\nu_c=[{\eta'}^{1/3}/(Ck_f^2)]\exp\{-{\eta'}^{1/3}/(C\alpha)\}$, where

$\eta'$ is half the enstrophy injection rate, $k_f$ is the forcing wave

number, and $C$ is a nondimensional constant (the Kraichnan--Batchelor

constant). The implication of the present results is that mechanical

friction is a poor choice in numerical attempts to address fundamental

issues concerning the direct enstrophy transfer.

Original language | English |
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Article number | 046322 |

Number of pages | 7 |

Journal | Physical Review. E, Statistical, nonlinear, and soft matter physics |

Volume | 84 |

Issue number | 4 |

DOIs | |

Publication status | Published - 28 Oct 2011 |