The Symmetry Group of the Non-Isothermal Navier–Stokes Equations and Turbulence Modelling

Abstract

In this work, the non-isothermal Navier-Stokes equations are studied from the group theory point of view. The symmetry group of the equations is presented and discussed. Some standard turbulence models are analyzed with the symmetries of the equations. A class of turbulence models which preserve the physical properties contained in the symmetry group is built. The proposed turbulence models are applied to an illustrative example of natural convection in a differentially heated cavity, and the results are presented.

Lie groups and continuum mechanics: where do we stand today?

“…An approach developped in [120][121][122][123] enables to build a class of turbulence models which preserve the Lie symmetry group of the Navier-Stokes equations. The works have latter been extended to the anisothermal case [108,[124][125][126]. This approach will be recalled in Section 3.3.…”

“…An approach in [108,122,126] suggests to construct T s and h s such that the symmetry groups of equations ( 1), summarized in Table 2, are also symmetry groups of equations (1). With this approach, the resolved variables have the same properties as v , p and θ, regarding the consequences of symmetries (self-similar solutions, conservation laws, .…”

Group theoretic approach and analytical solutions of the compressible Navier–Stokes equations

“…The group theoretic approach helped to analyze turbulence ( [30]) and turbulence models in the framework of large-eddy simulation [19,23]. Symmetry preservation even served as a guide to develop turbulence models in [22,26,2,24]. Lastly, note that Lie symmetry group theory was used to design robust numerical schemes in [11,7,6,8].…”

“…In ( 16) and ( 17), the sums are over all dependent variables q = u, v, w, p, ρ, and over all independent variables r, s = t, x, y, z. The coefficients of X (1) and X (2) are linked to those of X by the relations:…”

A review of some geometric integrators

“…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]. It is then important that numerical schemes do not break symmetries if one wishes to reproduce numerically the cited model solutions and properties.…”