Numerical Schemes and Computational Studies for Dynamically Orthogonal Equations (Multidisciplinary Simulation, Estimation, and Assimilation Systems: Reports in Ocean Science and Engineering)
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 efficient time-dependent adaptive methodology to predict the probability density functions of such dynamics. This work derives efficient computational schemes for the DO methodology applied to unsteady stochastic Navier-Stokes and Boussinesq equations. Semi-implicit projection methods are developed for the mean and for the orthonormal modes that define a basis for the evolving DO subspace, 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 specific to the DO equations 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. While (i) and (ii) are specific to fluid flows, (iii) and (iv) are important for any system of equations discretized using the DO methodology. To verify the correctness of 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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 01, 2011
- Accession Number
- ADA568415
Entities
People
- Mattheus P. Ueckermann
- Pierre F. J. Lermusiaux
- Themis P. Sapsis
Organizations
- Massachusetts Institute of Technology