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JOURNALS // Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki // Archive

Zh. Vychisl. Mat. Mat. Fiz., 2013 Volume 53, Number 11, Pages 1869–1893 (Mi zvmmf9948)

On the structure of steady axisymmetric Navier-Stokes flows with a stream function having multiple local extrema in its definite-sign domains

B. V. Pal'tsev, M. B. Solov'ev, I. I. Chechel'

Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

Abstract: A new high-order accurate method and a corresponding computer program developed previously by the first and third authors for the numerical solution of the axisymmetric stationary Dirichlet boundary value problem for the Navier–Stokes equations in spherical layers at low Reynolds numbers were used to reliably study the structure of certain flows with a stream function in a meridional plane having multiple local extrema in its positive-sign domains. Regimes of rotation of the boundary spheres were detected that ensure this flow pattern: the inner sphere rotates at a constant angular velocity, while the outer sphere rotates at zenith-angle-dependent angular velocities. To describe the structure of these flows, the domain where the stream function is positive was partitioned into subdomains (circulation zones) by the separatrices of the saddle points of the stream function, which generate manifolds of unstable initial points of trajectories. Unexpected phenomena in the circulation of such flows were discovered. Examples were presented that illustrate the behavior of fluid particle trajectories. The computed trajectories were shown to be of high accuracy even on long time intervals.

Key words: incompressible Navier–Stokes equations, axisymmetric steady flows in spherical layers, numerical method with splitting of boundary conditions, stream function in a meridional plane, local extrema, saddle points, separatrices of saddle points, flow structures, fluid particle trajectories.

UDC: 519.634

Received: 20.05.2013

DOI: 10.7868/S0044466913110124


 English version:
Computational Mathematics and Mathematical Physics, 2013, 53:11, 1696–1719

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© Steklov Math. Inst. of RAS, 2025