The Symmetry Group of the Non-Isothermal Navier–Stokes Equations and Turbulence Modelling
Abstract
:1. Introduction
2. The Symmetry Group of the Equations
- the group of time translations, corresponding to :
- the group of pressure translations, corresponding to :
- the group of pressure-temperature translations, corresponding to :
- the group of horizontal rotations, corresponding to :
- the (three-parameter) group of generalized Galilean transformations, spanned by the ’s, :
- the group of the first scaling transformations generated by :
- and the group of the second scaling transformations corresponding to :
- the reflections which are discrete symmetries:
- and the material indifference in the limit of a 2D horizontal flow in a simply connected domain [17] which is a time-dependent rotation:
- –
- the time, the pressure and the generalized Galilean translations,
- –
- the pressure-temperature translations,
- –
- the reflections, the horizontal (constant or time-dependent) rotations,
- –
- the scaling transformations.
3. Model Analysis
3.1. Subgrid Models
- The most widely used model is the Smagorinsky model, which was derived by adopting the concept of turbulent viscosity for and an analogy for :
- Using Germano–Lilly procedure ([18]), the model constants can be calculated in a dynamic way to give more flexibility to the model. This leads to the dynamic model:The tilde symbolizes a test filtering, with a width .
- Another model, which introduces the buoyancy term, is the Eidson model ([19]):
- To avoid having a negative radicand, Peng and Davidson ([20]) propose a modified version of the Eidson model:
- Another model, based on the scale-similarity hypothesis, is the scale-similarity model, adapted from Bardina model to the non-isothermal case:
- The scale similarity hypothesis can be used to obtain others models which are combined with the Smagorinsky model to give a mixed model in the following form ([21,22,23,24]):From the point of view of the symmetries, these models behave generally in the same way; so, we study only the following generic model:
3.2. Time, Pressure and Galilean Translations
3.3. Pressure-Temperature Translations
- Next,The scale-similarity model (38) is then invariant.
- For the dynamic model (34), , and are unchanged. And since , remains also unchanged. The model is then invariant.
- At last, the invariance of the Smagorinsky and the scale-similarity models leads to the invariance of the mixed model (40) under the pressure-temperature translations.
3.4. Reflections and Rotations
- For this reflection, we have:These relations imply the invariance of the Smagorinsky model (31).
- In addition,
- For the scale-similarity model (38),This model is then invariant.
- Next, for the dynamic model (34),Hence, and . It follows that the dynamic model is invariant under the third reflection.
- Lastly, the invariance of the mixed model (40) under the third reflection follows from the invariance of the Smagorinsky and the scale-similarity models.
3.5. Scaling Transformations
- Since and , we have, for the Smagorinsky model (31):The model verifies (58) neither when nor when . Thus, it is invariant neither under the first nor under the second scaling transformations.
- The scale-similarity model (38) is invariant under the two types of scaling transformations becauseCondition (58) is verified.
- For the dynamic model (34), we have:This implies thatThus,The dynamic model is invariant.
- At last, the mixed model (40) is not invariant under the scaling transformations because of the Smagorinsky part. However, the (Leonard) terms which correspond to the scale-similarity model are invariant.
4. Symmetry-Invariant LES Models
5. Model Simplification
5.1. Strongly Coupled Model
5.2. Decoupled Model
5.3. Linear Model
6. Numerical Example
7. Conclusions
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Time, pressure, Galilean | Pressure-temperature | Rotation, reflection, material indifference | Scaling | |
---|---|---|---|---|
Smagorinsky | invariant | invariant | invariant | non-invariant |
Dynamic | invariant | invariant | invariant | invariant |
Eidson | invariant | invariant | invariant | non-invariant |
Modified Eidson | invariant | invariant | invariant | non-invariant |
Similarity | invariant | invariant | invariant | invariant |
Mixed | invariant | invariant | invariant | non-invariant |
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Al Sayed, N.; Hamdouni, A.; Liberge, E.; Razafindralandy, D. The Symmetry Group of the Non-Isothermal Navier–Stokes Equations and Turbulence Modelling. Symmetry 2010, 2, 848-867. https://doi.org/10.3390/sym2020848
Al Sayed N, Hamdouni A, Liberge E, Razafindralandy D. The Symmetry Group of the Non-Isothermal Navier–Stokes Equations and Turbulence Modelling. Symmetry. 2010; 2(2):848-867. https://doi.org/10.3390/sym2020848
Chicago/Turabian StyleAl Sayed, Nazir, Aziz Hamdouni, Erwan Liberge, and Dina Razafindralandy. 2010. "The Symmetry Group of the Non-Isothermal Navier–Stokes Equations and Turbulence Modelling" Symmetry 2, no. 2: 848-867. https://doi.org/10.3390/sym2020848