TY - JOUR

T1 - Numerical methods for the weakly compressible Generalized Langevin Model in Eulerian reference frame

AU - Azarnykh, Dmitrii

AU - Litvinov, Sergey

AU - Adams, Nikolaus A.

N1 - Publisher Copyright:
© 2016 Elsevier Inc..

PY - 2016/6/1

Y1 - 2016/6/1

N2 - 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.

AB - 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.

KW - Eulerian reference frame

KW - Fluctuating hydrodynamics

KW - Generalized Langevin Model

KW - Nonlinear stochastic partial differential equations

KW - Turbulent flow

UR - http://www.scopus.com/inward/record.url?scp=84960441382&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2016.02.073

DO - 10.1016/j.jcp.2016.02.073

M3 - Article

AN - SCOPUS:84960441382

SN - 0021-9991

VL - 314

SP - 93

EP - 106

JO - Journal of Computational Physics

JF - Journal of Computational Physics

ER -