Numerical methods for the weakly compressible Generalized Langevin Model in Eulerian reference frame

Journal Article · Wed Jun 01 04:00:00 EDT 2016 · Journal of Computational Physics

DOI:https://doi.org/10.1016/J.JCP.2016.02.073· OSTI ID:22572306

Azarnykh, Dmitrii; Litvinov, Sergey; Adams, Nikolaus A.

A well established approach for the computation of turbulent flow without resolving all turbulent flow scales is to solve a filtered or averaged set of equations, and to model non-resolved scales by closures derived from transported probability density functions (PDF) for velocity fluctuations. Effective numerical methods for PDF transport employ the equivalence between the Fokker–Planck equation for the PDF and a Generalized Langevin Model (GLM), and compute the PDF by transporting a set of sampling particles by GLM (Pope (1985) [1]). The natural representation of GLM is a system of stochastic differential equations in a Lagrangian reference frame, typically solved by particle methods. A representation in a Eulerian reference frame, however, has the potential to significantly reduce computational effort and to allow for the seamless integration into a Eulerian-frame numerical flow solver. GLM in a Eulerian frame (GLMEF) formally corresponds to the nonlinear fluctuating hydrodynamic equations derived by Nakamura and Yoshimori (2009) [12]. Unlike the more common Landau–Lifshitz Navier–Stokes (LLNS) equations these equations are derived from the underdamped Langevin equation and are not based on a local equilibrium assumption. Similarly to LLNS equations the numerical solution of GLMEF requires special considerations. In this paper we investigate different numerical approaches to solving GLMEF with respect to the correct representation of stochastic properties of the solution. We find that a discretely conservative staggered finite-difference scheme, adapted from a scheme originally proposed for turbulent incompressible flow, in conjunction with a strongly stable (for non-stochastic PDE) Runge–Kutta method performs better for GLMEF than schemes adopted from those proposed previously for the LLNS. We show that equilibrium stochastic fluctuations are correctly reproduced.

• OSTI ID: 22572306
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 314; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
FLUCTUATIONS
FOKKER-PLANCK EQUATION
HYDRODYNAMICS
INCOMPRESSIBLE FLOW
LAGRANGIAN FUNCTION
LANGEVIN EQUATION
NAVIER-STOKES EQUATIONS
NONLINEAR PROBLEMS
PROBABILITY DENSITY FUNCTIONS
RUNGE-KUTTA METHOD
STOCHASTIC PROCESSES
TURBULENT FLOW
VELOCITY