An inconsistence-free integral-based dynamic one- and two-parameter mixed model

ABSTRACT

The development of the dynamic procedure as well as the deeper understanding of the link between filtering, modelling and numerics, allowed Large Eddy Simulation (LES) to make great progresses during the last years. Among several modelling approaches, the scale-similar-based modelling is based on the observation that the smallest resolved scales are the most active in the interaction with the unresolved ones. Owing to the low dissipation introduced by the scale-similar models (SSMs), the coupling with the eddy-viscosity model is often used in the so-called mixed models. Dynamic version of mixed models is historically based on the application of the test-filtering on the differential form of the filtered momentum equation. Such an approach is used for both the one and the two-coefficients mixed models. The use of the differential form of the filtered equations produces the well-known mathematical inconsistence caused by the need to extract arbitrarily the model functions out of filtering. It is known that, along with the eddy viscosity assumption, the magnitude of the Germano identity error (GIE) is strongly influenced. The mathematical inconsistence in the extraction of the dynamic eddy viscosity coefficient was recently superseded by using the new integral-based formulation. However, owing to the intrinsic limits of the Smagorinsky model, also in those results, the GIE is still remarkable therefore, the present paper presents a new formulation to the integral-based dynamic procedure for both one and two-coefficients mixed models (IDMM). The original contributions of the present paper can be summarised: (1) A theoretical analysis comparing the spectral errors for the differential and integral-based SSM, assessing that the errors are less relevant for the integral form; (2) The implementation of one and two parameters IDMM for the simulation of turbulence in a plane channel flow, assessing the reduction of the GIE and the good behaviour of the statistics that are compared with those of the other LES codes used in the LESinItaly project.

KEYWORDS:

• Large Eddy simulation
• dynamic mixed model
• integral conservation laws
• finite volume methods
• Germano identity error

AMS SUBJECT CLASSIFICATIONS:

• 65C20
• 65M99
• 76D05

Acknowledgments

The author acknowledges the CASPUR centre (now CINECA) for supporting this study with a grant on its computational cluster MATRIX. The authors also acknowledges the participants of the LESinItaly group for the simulations on the channel flow. The LESinItaly database is available under request to the author.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Filippo Maria Denaro http://orcid.org/0000-0003-4315-0773

Notes

1 In the expression of T the isotropic residual pressure flux is not needed since computed within the total pressure in the elliptic equation.