Order Reduction and UQ of Chaotic Flows with Low-Dimensional Attractors - DO Equations

It is often the case that infinite-dimensional dynamical systems such as Navier-Stokes equations posses global attractors with low dimensionality, i.e. the set that contains all the possible outcomes of the chaotic system "lives" in a low-dimensional space. For such case it is beneficial to perform order-reduction of the original equation and this is usually done by selecting a set of basis elements or fields (usually based on energetic criteria) and perform a Galerkin projection of the dynamics.

In many situations of practical interest this approach gives powerful results. However, in cases where the problem is strongly transient, even if the number of instabilities is small, it is very hard to choose the appropriate set of modes in order to perform effective order-reduction.

In this work we derive an exact, closed set of evolution equations (the Dynamically Orthogonal (DO) equations - read more...) that allow us to simultaneously evolve i) the basis elements that capture the modes of the flow that have important energy and ii) the statistical structure of the coefficients for these modes (i.e. the shape of the attractor). This new probabilistic framework allows for detailed understanding of the dynamical mechanisms (energy transfers, statistics, instabilities) but also for the efficient uncertainty quantification (UQ) of transient flows as long as low-dimensional attractors characterize these.