scholarly journals Vanishing viscosity limit for global attractors for the damped Navier–Stokes system with stress free boundary conditions

2018 ◽  
Vol 376-377 ◽  
pp. 31-38 ◽  
Author(s):  
Vladimir Chepyzhov ◽  
Alexei Ilyin ◽  
Sergey Zelik
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Stokes System ◽  
Global Attractors ◽  
Navier Stokes ◽  
Vanishing Viscosity ◽  
Vanishing Viscosity Limit ◽  
Free Boundary Conditions ◽  
Viscosity Limit ◽  
Stress Free Boundary

scholarly journals Transition to geostrophic convection: the role of the boundary conditions

2016 ◽  
Vol 799 ◽  
pp. 413-432 ◽  
Author(s):  
Rudie P. J. Kunnen ◽  
Rodolfo Ostilla-Mónico ◽  
Erwin P. van der Poel ◽  
Roberto Verzicco ◽  
Detlef Lohse
Keyword(s):  
Boundary Layer ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Large Scale ◽  
Scaling Laws ◽  
Stokes Equations ◽  
Fluid Layer ◽  
Navier Stokes ◽  
Free Boundary Conditions ◽  
Stress Free Boundary

Rotating Rayleigh–Bénard convection, the flow in a rotating fluid layer heated from below and cooled from above, is used to analyse the transition to the geostrophic regime of thermal convection. In the geostrophic regime, which is of direct relevance to most geo- and astrophysical flows, the system is strongly rotating while maintaining a sufficiently large thermal driving to generate turbulence. We directly simulate the Navier–Stokes equations for two values of the thermal forcing, i.e. $Ra=10^{10}$ and $Ra=5\times 10^{10}$, at constant Prandtl number $Pr=1$, and vary the Ekman number in the range $Ek=1.3\times 10^{-7}$ to $Ek=2\times 10^{-6}$, which satisfies both requirements of supercriticality and strong rotation. We focus on the differences between the application of no-slip versus stress-free boundary conditions on the horizontal plates. The transition is found at roughly the same parameter values for both boundary conditions, i.e. at $Ek\approx 9\times 10^{-7}$ for $Ra=1\times 10^{10}$ and at $Ek\approx 3\times 10^{-7}$ for $Ra=5\times 10^{10}$. However, the transition is gradual and it does not exactly coincide in $Ek$ for different flow indicators. In particular, we report the characteristics of the transitions in the heat-transfer scaling laws, the boundary-layer thicknesses, the bulk/boundary-layer distribution of dissipations and the mean temperature gradient in the bulk. The flow phenomenology in the geostrophic regime evolves differently for no-slip and stress-free plates. For stress-free conditions, the formation of a large-scale barotropic vortex with associated inverse energy cascade is apparent. For no-slip plates, a turbulent state without large-scale coherent structures is found; the absence of large-scale structure formation is reflected in the energy transfer in the sense that the inverse cascade, present for stress-free boundary conditions, vanishes.

scholarly journals Vanishing viscosity limit for the compressible Navier–Stokes system via measure-valued solutions

2020 ◽  
Vol 27 (6) ◽  
Author(s):  
Danica Basarić
Keyword(s):  
Strong Convergence ◽  
Strong Solution ◽  
Stokes System ◽  
Spatial Domain ◽  
Euler System ◽  
Navier Stokes ◽  
Strong Uniqueness ◽  
Vanishing Viscosity ◽  
Vanishing Viscosity Limit ◽  
Viscosity Limit

AbstractWe identify a class of measure-valued solutions of the barotropic Euler system on a general (unbounded) spatial domain as a vanishing viscosity limit for the compressible Navier–Stokes system. Then we establish the weak (measure-valued)–strong uniqueness principle, and, as a corollary, we obtain strong convergence to the Euler system on the lifespan of the strong solution.

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