Energy dissipation in body-forced plane shear flow

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作者：

DOERING, C. R.，ECKHARDT, B.，SCHUMACHER, J.

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摘要：

We study the problem of body-force driven shear flows in a plane channel of width l with free-slip boundaries. A mini-max variational problem for upper bounds on the bulk time averaged energy dissipation rate epsilon is derived from the incompressible Navier-Stokes equations with no secondary assumptions. This produces rigorous limits on the power consumption that are valid for laminar or turbulent solutions. The mini-max problem is solved exactly at high Reynolds numbers Re = U*l/nu, where U is the rms velocity and nu is the kinematic viscosity, yielding an explicit bound on the dimensionless asymptotic dissipation factor beta=epsilon*l/U^3 that depends only on the ``shape'' of the shearing body force. For a simple half-cosine force profile, for example, the high Reynolds number bound is beta <= pi^2/sqrt{216} = .6715... . We also report extensive direct numerical simulations for this particular force shape up to Re approximately 400; the observed dissipation rates are about a factor of three below the rigorous high-Re bound. Interestingly, the high-Re optimal solution of the variational problem bears some qualitative resemblence to the observed mean flow profiles in the simulations. These results extend and refine the recent analysis for body-forced turbulence in J. Fluid Mech. 467, 289-306 (2002).Dring, C; Eckhardt, B; Schumacher, J

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关键词：

Theoretical or Mathematical/ channel flow compressible flow flow simulation kinematics laminar flow Navier-Stokes equations shear flow turbulence variational techniques/ energy dissipation body-forced plane shear flow body-force-driven shear flows plane channel free-slip boundaries mini-max variational problem bulk time-averaged energy dissipation rate incompressible Navier-Stokes equations power consumption limits high Reynolds number rms velocity kinematic viscosity dimensionless asymptotic dissipation factor shearing body force half-cosine force profile direct numerical simulations body-forced turbulence/ A4760 Flows in ducts, channels, and conduits A4740 Compressible flows shock and detonation phenomena A4715 Laminar flows A4725 Turbulent flows, convection, and heat transfer A4710 General fluid dynamics theory, simulation and other computational methods