Rotating Navier-Stokes Equations in $${\mathbb R}^{3}_{+}$$ with Initial Data Nondecreasing at Infinity: The Ekman Boundary Layer Problem

The axisymmetric boundary-layer separation of an incompressible impulsively started flow in a wavy-walled tube is analysed at moderate to high values of the Reynolds number. The investigation is carried out by numerical integration of either the Navier–Stokes equations or Prandtl's asymptotic formulation of the boundary-layer problem. The presence of an adverse pressure gradient induces reverse flow at the tube wall independently of the Reynolds number; its occurrence can be predicted by a timescale analysis. Following that, the viscous calculations show different dynamics depending on the Reynolds number. As the Reynolds number increases, the boundary layer has in a well-defined internal structure where longitudinal lengthscales become comparable with the viscous one. Thus the boundary-layer scaling fails locally, with a minimum of pressure inside the boundary layer itself. The formation of the primary recirculation is well captured by the asymptotic model which, however, is not able to describe the roll-up of the vortex structure inside the recirculating region. This inadequacy appears well before the flow evolves to the characteristic terminal singularity usually assumed as foreshadowing the vortex shedding phenomenon. The outcomes are compared with the existing results of analogous problems giving an overall agreement but improving, in some cases, the physical picture.

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Numerical Integration of the Navier-Stokes Equations for a Disk Rotating in a Housing

Zeitschrift für Naturforschung A ◽

10.1515/zna-1985-0802 ◽

1985 ◽

Vol 40 (8) ◽

pp. 789-799 ◽

Cited By ~ 1

Author(s):

A. F. Borghesani

Keyword(s):

Boundary Layer ◽

Cylindrical Cavity ◽

Boundary Layer Thickness ◽

Stokes Equations ◽

Fluid Motion ◽

Rotating Disk ◽

Infinite Medium ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Viscous Torque

The Navier-Stokes equations for the fluid motion induced by a disk rotating inside a cylindrical cavity have been integrated for several values of the boundary layer thickness d. The equivalence of such a device to a rotating disk immersed in an infinite medium has been shown in the limit as d → 0. From that solution and taking into account edge effect corrections an equation for the viscous torque acting on the disk has been derived, which depends only on d. Moreover, these results justify the use of a rotating disk to perform accurate viscosity measurements.

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A simple proof of existence of weak and statistical solutions of Navier–Stokes equations

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1992.0001 ◽

1992 ◽

Vol 436 (1896) ◽

pp. 1-11 ◽

Cited By ~ 12

Keyword(s):

Weak Solution ◽

Initial Data ◽

Natural Number ◽

Simple Proof ◽

Stokes Equations ◽

Galerkin Approximation ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Existence Proofs ◽

Statistical Solutions

The Galerkin approximation to the Navier–Stokes equations in dimension N , where N is an infinite non-standard natural number, is shown to have standard part that is a weak solution. This construction is uniform with respect to non-standard representation of the initial data, and provides easy existence proofs for statistical solutions.

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Direct simulation of transition in an oscillatory boundary layer

Journal of Fluid Mechanics ◽

10.1017/s002211209800216x ◽

1998 ◽

Vol 371 ◽

pp. 207-232 ◽

Cited By ~ 92

Author(s):

G. VITTORI ◽

R. VERZICCO

Keyword(s):

Boundary Layer ◽

Pressure Gradient ◽

Stokes Equations ◽

Navier Stokes ◽

Turbulent Regime ◽

Two Dimensional ◽

Temporal Development ◽

Direct Simulation ◽

Navier Stokes Equations ◽

Oscillatory Boundary Layer

Numerical simulations of Navier–Stokes equations are performed to study the flow originated by an oscillating pressure gradient close to a wall characterized by small imperfections. The scenario of transition from the laminar to the turbulent regime is investigated and the results are interpreted in the light of existing analytical theories. The ‘disturbed-laminar’ and the ‘intermittently turbulent’ regimes detected experimentally are reproduced by the present simulations. Moreover it is found that imperfections of the wall are of fundamental importance in causing the growth of two-dimensional disturbances which in turn trigger turbulence in the Stokes boundary layer. Finally, in the intermittently turbulent regime, a description is given of the temporal development of turbulence characteristics.

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