A Higher Order Ensemble Simulation Algorithm for Fluid Flows


This report presents an efficient, higher order method for fast calculation of an ensemble of solutions of the Navier-Stokes equations. We give a complete stability and convergence analysis of the method for laminar flows and an extension to turbulent flows. For high Reynolds number flows, we propose and analyze an eddy viscosity model with a recent reparameterization of the mixing length. This turbulence model depends on an ensemble mean compatible with the higher order method. We show the turbulence model has superior stability, also demonstrated in numerical tests. We also give tests showing the potential of the new method for exploring flow problems to compute turbulence intensities, effective Lyapunov exponents, windows of predictability and to verify the selective decay principle.


Mathematics and Statistics


The research of the author described herein was partially supported by NSF Grant DMS 1216465 and Air Force Grant FA 9550-12-1-0191.

Keywords and Phrases

Algorithms; Laminar flow; Lyapunov methods; Reynolds number; Turbulence models; Viscosity; BDF2AB2; Eddy viscosity model; Ensemble simulation; High Reynolds number flows; Higher-order methods; Stokes equations; Turbulence intensity; Uncertainty quantifications; Navier Stokes equations; Ensemble calculation; Navier–Stokes equations

International Standard Serial Number (ISSN)

0885-7474; 1573-7691

Document Type

Article - Journal

Document Version


File Type





© 2015 Springer Verlag, All rights reserved.

Publication Date

01 Jul 2015