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Title: Numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows

Abstract

The quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions. Such situations are common in multiscale, intermittent and non-homogeneous fluid and ocean flows. The dynamically orthogonal (DO) field equations provide an adaptive methodology to predict the probability density functions of such flows. The present work derives efficient computational schemes for the DO methodology applied to unsteady stochastic Navier-Stokes and Boussinesq equations, and illustrates and studies the numerical aspects of these schemes. Semi-implicit projection methods are developed for the mean and for the DO modes, and time-marching schemes of first to fourth order are used for the stochastic coefficients. Conservative second-order finite-volumes are employed in physical space with new advection schemes based on total variation diminishing methods. Other results include: (i) the definition of pseudo-stochastic pressures to obtain a number of pressure equations that is linear in the subspace size instead of quadratic; (ii) symmetric advection schemes for the stochastic velocities; (iii) the use of generalized inversion to deal with singular subspace covariances or deterministic modes; and (iv) schemes to maintain orthonormal modes at the numerical level. To verify our implementation andmore » study the properties of our schemes and their variations, a set of stochastic flow benchmarks are defined including asymmetric Dirac and symmetric lock-exchange flows, lid-driven cavity flows, and flows past objects in a confined channel. Different Reynolds number and Grashof number regimes are employed to illustrate robustness. Optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations.« less

Authors:
 [1]
  1. Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Mass. Avenue, Cambridge, MA 02139 (United States)
Publication Date:
OSTI Identifier:
22192349
Resource Type:
Journal Article
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 233; Journal Issue: Complete; Other Information: Copyright (c) 2012 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ADVECTION; ASYMMETRY; BENCHMARKS; BOUNDARY CONDITIONS; COMPUTERIZED SIMULATION; CONVERGENCE; FIELD EQUATIONS; FLUIDS; GRASHOF NUMBER; NAVIER-STOKES EQUATIONS; NONLINEAR PROBLEMS; PROBABILITY DENSITY FUNCTIONS; REYNOLDS NUMBER; SEAS; STOCHASTIC PROCESSES; SYMMETRY; VARIATIONS; VELOCITY

Citation Formats

Ueckermann, M.P., E-mail: mpuecker@mit.edu, Lermusiaux, P. F.J.,, and Sapsis, T.P., E-mail: sapsis@mit.edu. Numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows. United States: N. p., 2013. Web. doi:10.1016/J.JCP.2012.08.041.
Ueckermann, M.P., E-mail: mpuecker@mit.edu, Lermusiaux, P. F.J.,, & Sapsis, T.P., E-mail: sapsis@mit.edu. Numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows. United States. https://doi.org/10.1016/J.JCP.2012.08.041
Ueckermann, M.P., E-mail: mpuecker@mit.edu, Lermusiaux, P. F.J.,, and Sapsis, T.P., E-mail: sapsis@mit.edu. 2013. "Numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows". United States. https://doi.org/10.1016/J.JCP.2012.08.041.
@article{osti_22192349,
title = {Numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows},
author = {Ueckermann, M.P., E-mail: mpuecker@mit.edu and Lermusiaux, P. F.J., and Sapsis, T.P., E-mail: sapsis@mit.edu},
abstractNote = {The quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions. Such situations are common in multiscale, intermittent and non-homogeneous fluid and ocean flows. The dynamically orthogonal (DO) field equations provide an adaptive methodology to predict the probability density functions of such flows. The present work derives efficient computational schemes for the DO methodology applied to unsteady stochastic Navier-Stokes and Boussinesq equations, and illustrates and studies the numerical aspects of these schemes. Semi-implicit projection methods are developed for the mean and for the DO modes, and time-marching schemes of first to fourth order are used for the stochastic coefficients. Conservative second-order finite-volumes are employed in physical space with new advection schemes based on total variation diminishing methods. Other results include: (i) the definition of pseudo-stochastic pressures to obtain a number of pressure equations that is linear in the subspace size instead of quadratic; (ii) symmetric advection schemes for the stochastic velocities; (iii) the use of generalized inversion to deal with singular subspace covariances or deterministic modes; and (iv) schemes to maintain orthonormal modes at the numerical level. To verify our implementation and study the properties of our schemes and their variations, a set of stochastic flow benchmarks are defined including asymmetric Dirac and symmetric lock-exchange flows, lid-driven cavity flows, and flows past objects in a confined channel. Different Reynolds number and Grashof number regimes are employed to illustrate robustness. Optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations.},
doi = {10.1016/J.JCP.2012.08.041},
url = {https://www.osti.gov/biblio/22192349}, journal = {Journal of Computational Physics},
issn = {0021-9991},
number = Complete,
volume = 233,
place = {United States},
year = {Tue Jan 15 00:00:00 EST 2013},
month = {Tue Jan 15 00:00:00 EST 2013}
}