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Turbulent Flow Heat Transfer Enhancement
Published in Je-Chin Han, Lesley M. Wright, Analytical Heat Transfer, 2022
For turbulent air flow in rectangular channels, Nusselt number can be predicted by a given Reynolds number. In addition to Reynolds number, Rotation number and Buoyancy number are important parameters under rotation conditions. The following is a brief summary of these parameters. ReD=ρVDhμ=VDhν=10,000to60,000foraircraftturbinesNu0=hDhk=0.023ReD0.8Pr0.4forsmoothchannelheattransfercorrelationRo=Rotationnumber=CoriolisforceInertiaforce=ΩDhV=0.2~0.5Ω=RPMrotationalspeedDh=coolingchannelhydraulicdiameter
Ordinary differential equations defined by a trigonometric polynomial field: behaviour of the solutions
Published in Dynamical Systems, 2023
In this article, we study the asymptotic behaviour of solutions for ordinary differential equations (ODE) defined by a trigonometric polynomial field. The idea comes from the scalar case, where in this case H. Poincaré defined the rotation number for circle homeomorphisms [7]. The simple example is a scalar differential equation where is lipschitz, 1-periodic and is the state of the system. There exists a rotation number for which the function is bounded (periodic). We know that any non-autonomous system can be written as an autonomous system. Our result is a generalization of this asymptotic behaviour to any dimension. In this case, λ is a vector and called a rotation vector or rotation set as it is defined in [5]. Under some assumptions of stability [8], [2] proved the existence of the rotation vector. Some biological works use the ODE defined by a trigonometric polynomial field and study the rotation vector components as in [1], [3], [4], [10]. Our contribution to this biological works has two key points, the mathematical proof of existence of the rotation vector and the study of the behaviour of solutions.
Breakup of transport barriers in plasmas with flow described by symplectic maps
Published in Radiation Effects and Defects in Solids, 2022
Carolina A. Tafoya, Julio J. Martinell
Depending on the nature of the map the transport barriers have different properties. For non-degenerate maps (when the rotation number has a non-vanishing derivative), also known as twist maps, the last KAM surface to break up is the one with the most stable irrational rotation number (usually the golden mean). For non-twist maps which violate the non-degeneracy condition the most robust torus is the shearless surface. When there are no background plasma flows, transport barriers are not a characteristic feature of the map, although in (6) they were characterized for the kicked Harper model. However, the presence of flows produces clear transport barriers since the flow lines give rise to KAM torii that are resilient to chaos.
Linearization of a quasi-periodically forced flow on 𝕋m under Brjuno–Rüssmann non-resonant condition
Published in Applicable Analysis, 2018
For the map T, one can define the rotation number , which measures the average speed of rotation of orbits around the circle. Poincaré [1] proved that this was the same for all orbits and hence was an invariant of T. This leads to Poincar’s celebrated classification of the dynamics of circle homeomorphisms: if the rotation number is rational then T has a periodic orbit of period q; if the rotation number is irrational, then f is semi-conjugate to the irrational rotation The picture in the irrational case was further completed by Denjoy [2], where he showed that if the rotation number of T is irrational and the derivative of T has bounded variation, then the semi-conjugacy is a homeomorphism. The properties of the map T decide the general character of the behaviour of trajectories of the differential Equation (1.1). H. Poincaré and A. Denjoy’s results gave a strong motivation and stimulus for later researches on trajectories on a tours. In 1961, Bolyai [3] researched firstly the question, whether the differential Equation (1.1) can be reduced, by a suitable quasi-periodic transformation to where is a constant vector. In higher dimensions tori, a pioneering work should due to Arnold [4], where he proved that if the rotation vector satisfies Diophantine condition, then there exists such that for every real analytic vector field on there exists a vector for which the differential equations