Vortices and invariant surfaces generated by symmetries for the 3D Navier–Stokes equations

Grassi, Viviane; Leo, R. A.; Soliani, G.; Tempesta, Piergiulio

doi:10.1016/s0378-4371(00)00223-5

Cited by 29 publications

(30 citation statements)

References 18 publications

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“…It is suitable to recall than an approach of group-theoretical type to the equations for the fluid dynamics is not new and has been carried out by many authors (see, for example, [11,12,[24][25][26][27]). On the other hand, a full and exhaustive classification of Lie subgroups of the symmetry group of MHD equations in (3 + 1) dimensions was never been established before [12], and therefore allows us for a systematic approach to the task of constructing both invariant and partially invariant solutions of Eqs.…”

Section: Discussionmentioning

confidence: 99%

Some exact solutions of the ideal MHD equations through symmetry reduction method

Picard

2008

Journal of Mathematical Analysis and Applications

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We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, the so-called invariant solutions, of the ideal magnetohydrodynamic equations in (3 + 1) dimensions. In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1 r 4) was already known. We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. Here, some examples of these solutions are presented with a brief physical interpretation.

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Section: Discussionmentioning

confidence: 99%

Some exact solutions of the ideal MHD equations through symmetry reduction method

Picard

2008

Journal of Mathematical Analysis and Applications

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“…A dilation ( D ) invariant solution to this system was found in Wang P and Wang X . Section 4.5 in Fushchych and Popowych is devoted to solutions of this system. The case of W = α = β =0 yields a planar flow.…”

Section: Semi‐invariant Manifolds In Cylindrical Coordinatesmentioning

confidence: 95%

Semi‐invariant solutions of the Navier‐Stokes equations

Shankar

Squellati

2018

Math Methods in App Sciences

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We present new exact solutions and reduced differential systems of the Navier‐Stokes equations of incompressible viscous fluid flow. We apply the method of semi‐invariant manifolds, introduced earlier as a modification of the Lie invariance method. We show that many known solutions of the Navier‐Stokes equations are, in fact, semi‐invariant and that the reduced differential systems we derive using semi‐invariant manifolds generalize previously obtained results that used ad hoc methods. Many of our semi‐invariant solutions solve decoupled systems in triangular form that are effectively linear. We also obtain several new reductions of Navier‐Stokes to a single nonlinear partial differential equation. In some cases, we can solve reduced systems and generate new analytic solutions of the Navier‐Stokes equations or find their approximations, and physical interpretation.

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“…Yet, many of them have important invariance properties, called symmetries [54], under some transformations. Symmetries play a fundamental role since they may encode exact model solutions (self-similar, vortex, shock solutions, …), conservation laws via Nother's theorem or physical principles (Galilean invariance, scale invariance, …) [55][56][57][58][59][60][61][62][63][64][65][66][67][68]. They have also been used for modelling purposes such as the establishment of wall laws in turbulent flows or the development of turbulence models [69][70][71][72][73][74][75][76].…”

Section: Introductionmentioning

confidence: 99%

A review of some geometric integrators

Razafindralandy

Hamdouni

Chhay

2018

Adv. Model. and Simul. in Eng. Sci.

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Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler-Lagrange equations) and the conservation laws (Noether's theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

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Vortices and invariant surfaces generated by symmetries for the 3D Navier–Stokes equations

Cited by 29 publications

References 18 publications

Some exact solutions of the ideal MHD equations through symmetry reduction method

Some exact solutions of the ideal MHD equations through symmetry reduction method

Semi‐invariant solutions of the Navier‐Stokes equations

A review of some geometric integrators

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