scholarly journals Local Boundary Regularity for the Navier–Stokes Equations in Non-Endpoint Borderline Lorentz Spaces

2017 ◽  
Vol 224 (3) ◽  
pp. 391-413 ◽  
Author(s):  
T. Barker
Keyword(s):  
Stokes Equations ◽  
Boundary Regularity ◽  
Lorentz Spaces ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Local Boundary

Rayleigh number scaling in numerical convection

1996 ◽  
Vol 310 ◽  
pp. 139-179 ◽  
Author(s):  
Robert M. Kerr
Keyword(s):  
Boundary Layer ◽  
Rayleigh Number ◽  
Large Scale ◽  
Stokes Equations ◽  
Good Evidence ◽  
Navier Stokes ◽  
Fixed Temperature ◽  
Navier Stokes Equations ◽  
Local Boundary ◽  
Velocity Boundary Layer

Using direct simulations of the incompressible Navier-Stokes equations with rigid upper and lower boundaries at fixed temperature and periodic sidewalls, scaling with respect to Rayleigh number is determined. At large aspect ratio (6:6:1) on meshes up to 288 × 288 × 96, a single scaling regime consistent with the properties of ‘hard’ convective turbulence is found for Pr = 0.7 between Ra = 5 × 104 and Ra = 2 × 107. The properties of this regime include Nu ∼ RaβT with βT = 0.28 ≈ 2/7, exponential temperature distributions in the centre of the cell, and velocity and temperature scales consistent with experimental measurements. Two velocity boundary-layer thicknesses are identified, one outside the thermal boundary layer that scales as Ra−1/7 and the other within it that scales as Ra−3/7. Large-scale shears are not observed; instead, strong local boundary-layer shears are observed in regions between incoming plumes and an outgoing network of buoyant sheets. At the highest Rayleigh number, there is a decade where the energy spectra are close to k−5/3 and temperature variance spectra are noticeably less steep. It is argued that taken together this is good evidence for ‘hard’ turbulence, even if individually each of these properties might have alternative explanations.

scholarly journals Stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel

2019 ◽  
Vol 25 ◽  
pp. 66
Author(s):  
Sourav Mitra
Keyword(s):  
Feedback Control ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Boundary Stabilization ◽  
Navier Stokes Equations ◽  
Local Boundary ◽  
Hyperbolic Dynamics ◽  
Finite Dimensional ◽  
Dimensional Range

In this article, we study the local boundary stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel around Poiseuille flow which is a stationary solution for the system under consideration. The feedback control operator we construct has finite dimensional range. The homogeneous Navier–Stokes equations are of parabolic nature and the stabilization result for such system is well studied in the literature. In the present article we prove a stabilization result for non-homogeneous Navier–Stokes equations which involves coupled parabolic and hyperbolic dynamics by using only one boundary control for the parabolic part.

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