Hopfions

2018 ◽  
Vol 30 (08) ◽  
pp. 1840017
Author(s):  
Paul Sutcliffe
Keyword(s):  
Three Dimensional ◽  
Original Model ◽  
Frustrated Magnets ◽  
Numerical Computations ◽  
Topological Solitons ◽  
Hopf Invariant ◽  
Similarities And Differences ◽  
Ferromagnetic Fluids

More than 40 years ago, Faddeev proposed the existence of three-dimensional topological solitons classified by the integer-valued Hopf invariant. These solitons are now known as hopfions and have been investigated in a range of systems, including the original model suggested by Faddeev, where a variety of stable knot and link solutions have been computed numerically. Very recently, numerical computations have predicted the existence of nanoscale hopfions in frustrated magnets and experiments have realized micrometer-sized hopfions in chiral ferromagnetic fluids. All these examples of hopfions will be described and their similarities and differences discussed.

Standing and travelling oscillatory blob convection

1995 ◽  
Vol 297 ◽  
pp. 255-273 ◽  
Author(s):  
R. M. Clever ◽  
F. H. Busse
Keyword(s):  
Rayleigh Number ◽  
Heat Transport ◽  
Three Dimensional ◽  
Horizontal Layer ◽  
Time Dependent ◽  
Stable Form ◽  
Two Dimensional ◽  
Mean Flow ◽  
Numerical Computations

Results of numerical computations are presented of time-dependent three-dimensional convection flows in a horizontal layer heated from below which evolve from the oscillatory blob instability of steady two-dimensional rolls. It is shown that the heat transport is typically increased in the transition to blob convection. Oscillatory blob convection exists in the forms of standing or travelling blob convection. The latter type of solution represents the stable form bifurcating supercritically at the Rayleigh number RII for the onset of the oscillatory blob instability. In contrast to standing blob convection travelling blob convection exhibits a mean flow.

scholarly journals Design Optimization of a Transonic-Fan Rotor Using Numerical Computations of the Full Compressible Navier-Stokes Equations and Simplex Algorithm

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
M. A. Aziz ◽  
Farouk M. Owis ◽  
M. M. Abdelrahman
Keyword(s):  
Stokes Equations ◽  
Pressure Ratio ◽  
Three Dimensional ◽  
Simplex Algorithm ◽  
Navier Stokes ◽  
Numerical Computations ◽  
Navier Stokes Equations ◽  
Pressure Losses ◽  
Optimization Parameters ◽  
Transonic Fan

The design of a transonic-fan rotor is optimized using numerical computations of the full three-dimensional Navier-Stokes equations. The CFDRC-ACE multiphysics module, which is a pressure-based solver, is used for the numerical simulation. The code is coupled with simplex optimization algorithm. The optimization process is started from a suitable design point obtained using low fidelity analytical methods that is based on experimental correlations for the pressure losses and blade deviation angle. The fan blade shape is defined by its stacking line and airfoil shape which are considered the optimization parameters. The stacking line is defined by lean, sweep, and skews, while blade airfoil shape is modified considering the thickness and camber distributions. The optimization has been performed to maximize the rotor total pressure ratio while keeping the rotor efficiency and surge margin above certain required values. The results obtained are verified with the experimental data of Rotor 67. In addition, the results of the optimized fan indicate that the optimum design is found to be leaned in the direction of rotation and has a forward sweep from the hub to mean section and backward sweep to the tip. The pressure ratio increases from 1.427 to 1.627 at the design speed and mass flow rate.

Divergent expansion, Borel summability and three-dimensional Navier–Stokes equation

2008 ◽  
Vol 366 (1876) ◽  
pp. 2775-2788 ◽  
Author(s):  
Ovidiu Costin ◽  
Guo Luo ◽  
Saleh Tanveer
Keyword(s):  
Nonlinear Partial Differential Equations ◽  
Local Existence ◽  
Three Dimensional ◽  
Nonlinear Pdes ◽  
Navier Stokes ◽  
Borel Summability ◽  
Numerical Computations ◽  
Navier Stokes Equation ◽  
Existence Proofs ◽  
General Data

We describe how the Borel summability of a divergent asymptotic expansion can be expanded and applied to nonlinear partial differential equations (PDEs). While Borel summation does not apply for non-analytic initial data, the present approach generates an integral equation (IE) applicable to much more general data. We apply these concepts to the three-dimensional Navier–Stokes (NS) system and show how the IE approach can give rise to local existence proofs. In this approach, the global existence problem in three-dimensional NS systems, for specific initial condition and viscosity, becomes a problem of asymptotics in the variable p (dual to 1/ t or some positive power of 1/ t ). Furthermore, the errors in numerical computations in the associated IE can be controlled rigorously, which is very important for nonlinear PDEs such as NS when solutions are not known to exist globally. Moreover, computation of the solution of the IE over an interval [0, p 0 ] provides sharper control of its p →∞ behaviour. Preliminary numerical computations give encouraging results.

scholarly journals A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations

2007 ◽  
Vol 48 (6) ◽  
pp. 065204 ◽  
Author(s):  
Sergei I. Chernyshenko ◽  
Peter Constantin ◽  
James C. Robinson ◽  
Edriss S. Titi
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
A Posteriori ◽  
Numerical Computations ◽  
Navier Stokes Equations
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