A transported PDF modeling method applied to one-dimensional stochastic Burgers equation

Abstract

In this paper we apply the transported probability density function (PDF) method to model the Burgers turbulence governed by the one-dimensional Burgers equation with a stochastic external force and random initial data. The PDF modeling method can provide the mean and high order moments of the random solution variable which are useful for the characterization of Burgers turbulence. Firstly, an exact one-point PDF transport equation is derived from the Burgers equation with a stochastic external force, and then the conditional advection and diffusion terms are modeled for closure by using two respective models. Secondly, three numerical methods are employed to solve the modeled PDF transport equation: the mesh-based Lagrangian particle Monte Carlo (MC) method, the original quadrature method of moments (QMOM), and the linear QMOM (LQMOM) as modified in this work. The numerical tests verify the viability of using the PDF modeling method to study Burgers turbulence and show that the MC method, LQMOM and QMOM can yield results comparable to the DNS solution, and the LQMOM is more efficient than the QMOM and MC method. However, the MC method has less dissipation for turbulence in high wavenumber regimes.