{"id":2139,"date":"2012-05-29T12:57:10","date_gmt":"2012-05-29T16:57:10","guid":{"rendered":"http:\/\/mseas.mit.edu\/?p=2139"},"modified":"2013-03-21T09:55:43","modified_gmt":"2013-03-21T13:55:43","slug":"numerical-schemes-for-dynamically-orthogonal-equations-of-stochastic-fluid-and-ocean-flows-2","status":"publish","type":"post","link":"https:\/\/mseas.mit.edu\/?p=2139","title":{"rendered":"Numerical Schemes and Studies for Dynamically Orthogonal Equations of Stochastic Fluid and Ocean Flows"},"content":{"rendered":"The quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their\r\ngoverning equations or are constrained by limited knowledge of initial and boundary conditions. Such\r\nsituations are common in multi-scale, intermittent and non-homogeneous fluid and ocean flows, and other\r\nnon-linear dynamical systems. The Dynamically Orthogonal (DO) fi\feld equations provide an e\u000efficient time-\r\ndependent adaptive methodology to predict the probability density functions of such dynamics. The present\r\nwork derives e\u000efficient computational schemes for the DO methodology applied to unsteady stochastic Navier-\r\nStokes and Boussinesq equations, and illustrates and studies the numerical aspects of these schemes. Semi-\r\nimplicit projection methods are developed for the mean and for the orthonormal modes that defi\fne a basis\r\nfor the evolving DO subspace, and time-marching schemes of \ffirst to fourth order are used for the stochastic\r\ncoe\u000efficients. Conservative second-order \fnite-volumes are employed in physical space with new advection\r\nschemes based on Total Variation Diminishing methods. Other results speci\ffic to the DO equations include:\r\n(i) the definition of pseudo-stochastic pressures to obtain a number of pressure equations that is linear in\r\nthe subspace size instead of quadratic; (ii) symmetric advection schemes for the stochastic velocities; (iii)\r\nthe use of generalized inversion to deal with singular subspace covariances or deterministic modes; and (iv)\r\nschemes to maintain orthonormal modes at the numerical level. While (i) and (ii) are specifi\fc to fluid flows,\r\n(iii) and (iv) are important for any system of equations discretized using the DO methodology. To verify\r\nthe correctness of our implementation and study the properties of our schemes and their variations, a set\r\nof stochastic flow benchmarks are defi\fned including asymmetric Dirac and symmetric lock-exchange flows,\r\nlid-driven cavity flows, and flows past objects in a confi\fned channel. Di\u000bfferent Reynolds number and Grashof\r\nnumber regimes are employed to illustrate robustness. Optimal convergence under both time and space\r\nrefi\fnements is shown as well as the convergence of the probability density functions with the number of\r\nstochastic realizations.","protected":false},"excerpt":{"rendered":"<p>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 multi-scale, intermittent and non-homogeneous fluid and ocean flows, and other non-linear dynamical systems. The Dynamically Orthogonal (DO) fi\feld equations provide an [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[87],"tags":[],"class_list":["post-2139","post","type-post","status-publish","format-standard","hentry","category-mseas-reports"],"_links":{"self":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/2139","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2139"}],"version-history":[{"count":11,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/2139\/revisions"}],"predecessor-version":[{"id":2528,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/2139\/revisions\/2528"}],"wp:attachment":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2139"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2139"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2139"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}