Extending Lundgren’s Transformation to Construct Stretched Vortex Solutions of the 3D Navier-Stokes and Euler Equations

2000 ◽  
pp. 271-282
Author(s):  
J. D. Gibbon
Keyword(s):  
Euler Equations ◽  
Navier Stokes ◽  
Vortex Solutions

scholarly journals Circulation and Energy Theorem Preserving Stochastic Fluids

2019 ◽  
Vol 150 (6) ◽  
pp. 2776-2814 ◽  
Author(s):  
Theodore D. Drivas ◽  
Darryl D. Holm
Keyword(s):  
Variational Principle ◽  
Energy Conservation ◽  
Euler Equations ◽  
Navier Stokes ◽  
Fluid Models ◽  
Smooth Solutions ◽  
Stochastic Fluid Models ◽  
Natural Extensions ◽  
Stochastic Fluid ◽  
Incompressible Euler

AbstractSmooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincaré and stochastic Navier–Stokes–Poincaré equations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.

scholarly journals The Global Well-Posedness for Large Amplitude Smooth Solutions for 3D Incompressible Navier–Stokes and Euler Equations Based on a Class of Variant Spherical Coordinates

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1195
Author(s):  
Shu Wang ◽  
Yongxin Wang
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Large Amplitude ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Well Posedness ◽  
Spherical Coordinates ◽  
Smooth Solutions ◽  
Euler Systems

This paper investigates the globally dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solutions with the finite energy to the three-dimensional (3D) incompressible Navier–Stokes (NS) and Euler systems. The global well-posedness for large amplitude smooth solutions to the Cauchy problem for 3D incompressible NS and Euler equations based on a class of variant spherical coordinates is obtained, where smooth initial data is not axi-symmetric with respect to any coordinate axis in Cartesian coordinate system. Furthermore, we establish the existence, uniqueness and exponentially decay rate in time of the global strong solution to the initial boundary value problem for 3D incompressible NS equations for a class of the smooth large initial data and a class of the special bounded domain described by variant spherical coordinates.

An unsteady two-cell vortex solution of the Navier—Stokes equations

1970 ◽  
Vol 41 (3) ◽  
pp. 673-687 ◽  
Author(s):  
P. G. Bellamy-Knights
Keyword(s):  
Radial Flow ◽  
Stokes Equations ◽  
Circumferential Velocity ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Radial Flux ◽  
Vortex Solution ◽  
Flow Systems ◽  
Vortex Solutions

The steady two-cell viscous vortex solution of Sullivan (1959) is extended to yield unsteady two-cell viscous vortex solutions which behave asymptotically as certain analogous unsteady one-cell solutions of Rott (1958). The radial flux is a parameter of the solution, and the effect of the radial flow on the circumferential velocity, is analyzed. The work suggests an explanation for the eventual dissipation of meteorological flow systems such as tornadoes.

Toward a Global Model for FSI in Tube Bundles

2008 ◽  
Author(s):  
Daniel Broc ◽  
Marion Duclercq
Keyword(s):  
Fluid Flow ◽  
Euler Equations ◽  
Dynamic Behaviour ◽  
Navier Stokes ◽  
Inertial Effects ◽  
Navier Stokes Equation ◽  
Dissipative Effects ◽  
Tube Bundles ◽  
Homogenization Methods ◽  
Physical Phenomena

It is well known that a fluid may strongly influence the dynamic behaviour of a structure. Many different physical phenomena may take place, depending on the conditions: fluid at rest, fluid flow, little or high displacements of the structure. Inertial effects can take place, with lower vibration frequencies, dissipative effects also, with damping, instabilities due to the fluid flow (Fluid Induced Vibration). In this last case the structure is excited by the fluid. The paper deals with the vibration of tube bundles in a fluid, under a seismic excitation or an impact. In this case the structure moves under an external excitation, and the movement is influenced by the fluid. The main point in such system is that the geometry is complex, and could lead to very huge sizes for a numerical analysis. Many works has been made in the last years to develop homogenization methods for the dynamic behaviour of tube bundles (/2/ and /3/). The size of the problem is reduced, and it is possible to make numerical simulations on wide tubes bundles with reasonable computer times. These homogenization methods are valid for “little displacements” of the structure (the tubes), in a fluid at rest. The fluid movement is governed by the Euler equations. In this case, only “inertial effects” will take place, with globally lower frequencies. It is well known that dissipative effects due to the fluid may take place, even if the displacements of the tube are no so high, or if the fluid is not still (/4/, /5/, /6/ and /8/). Such effects may be described in the homogenized models by using a Rayleigh damping, but the basic assumption of the model remains the “perfect fluid” hypothesis. It seem necessary, in order to get a best description of the physical phenomena, to build a more general model, based on the general Navier Stokes equation for the fluid. The homogenization of such system will be much more complex than for the Euler equations. The paper doesn’t pretend to give a general solution of the problem, but only points out the most important key points to build such homogenized model for the dynamic behaviour of tubes bundles in a fluid.

Amplitude–phase decompositions and the growth and decay of solutions of the incompressible Navier–Stokes and Euler equations

2014 ◽  
Vol 24 (05) ◽  
pp. 1017-1035
Author(s):  
Thomas J. R. Hughes
Keyword(s):  
Euler Equations ◽  
Viscous Dissipation ◽  
Finite Time ◽  
Blow Up ◽  
Navier Stokes ◽  
Decay Of Solutions ◽  
New Criteria

We introduce the concept of amplitude–phase decompositions and apply it to the study of growth and decay of solutions of the incompressible Navier–Stokes and Euler equations. Amplitudes coincide with functionals of physical and mathematical interest. We are able to explicitly solve for the amplitudes in terms of the phases. The results obtained provide insights into the growth and decay of enstrophy and viscous dissipation, and identify new criteria for solutions to remain smooth for all time or blow-up in finite time.

Sign in / Sign up

Export Citation Format

Share Document