We perform a new modeling procedure for a 3D turbulent fluid, evolving toward a statistical equilibrium. This will result to add to the equations for the mean field $(v, p)$ the term $-alpha div (ell(\x) D v_t)$, which is of the Kelvin-Voigt form, where the Prandtl mixing length $ell=ell(\x)$ is not constant and vanishes at the solid walls. We get estimates for mean velocity $v$ in $L^infty_t H^1_x cap W^{1,2}_t H^{1/2}_x$, that allow us to prove the existence and uniqueness of a regular-weak solutions $(v, p)$ to the resulting system, for a given fixed eddy viscosity. We then prove a structural compactness result that highlights the robustness of the model. This allows us to consider Reynolds averaged equations and pass to the limit in the quadratic source term in the equation for the turbulent kinetic energy $k$. This yields the existence of a weak solution to the corresponding Navier-Stokes Turbulent Kinetic Energy system satisfied by $(v, p, k)$.

Turbulent flows as generalized Kelvin-Voigt materials: modeling and analysis

Luigi C. Berselli
;
2020-01-01

Abstract

We perform a new modeling procedure for a 3D turbulent fluid, evolving toward a statistical equilibrium. This will result to add to the equations for the mean field $(v, p)$ the term $-alpha div (ell(\x) D v_t)$, which is of the Kelvin-Voigt form, where the Prandtl mixing length $ell=ell(\x)$ is not constant and vanishes at the solid walls. We get estimates for mean velocity $v$ in $L^infty_t H^1_x cap W^{1,2}_t H^{1/2}_x$, that allow us to prove the existence and uniqueness of a regular-weak solutions $(v, p)$ to the resulting system, for a given fixed eddy viscosity. We then prove a structural compactness result that highlights the robustness of the model. This allows us to consider Reynolds averaged equations and pass to the limit in the quadratic source term in the equation for the turbulent kinetic energy $k$. This yields the existence of a weak solution to the corresponding Navier-Stokes Turbulent Kinetic Energy system satisfied by $(v, p, k)$.
2020
Amrouche, Cherif; Berselli, Luigi C.; Lewandowski, Roger; Duong Nguyen, Dinh
File in questo prodotto:
File Dimensione Formato  
Generalized_Voigt-V6.pdf

accesso aperto

Descrizione: Preprin ArXiv
Tipologia: Documento in Pre-print
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 418.33 kB
Formato Adobe PDF
418.33 kB Adobe PDF Visualizza/Apri
Generalized_Voigt-revised.pdf

Open Access dal 01/08/2022

Tipologia: Documento in Post-print
Licenza: Creative commons
Dimensione 431.08 kB
Formato Adobe PDF
431.08 kB Adobe PDF Visualizza/Apri
NA2020-b.pdf

solo utenti autorizzati

Descrizione: Versione editoriale
Tipologia: Versione finale editoriale
Licenza: NON PUBBLICO - Accesso privato/ristretto
Dimensione 943.19 kB
Formato Adobe PDF
943.19 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/999567
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 14
  • ???jsp.display-item.citation.isi??? 13
social impact