Data-driven filtered reduced order modeling of fluid flows

Xuping Xie, Muhammad Mohebujjaman, Leo Rebholz, Traian Iliescu

Abstract:

We propose a data-driven filtered reduced order model (DDF-ROM) framework for
the numerical simulation of fluid flows. The novel DDF-ROM framework consists of two steps: (i)
In the first step, we use ROM projection to filter the nonlinear PDE and construct a filtered ROM.
This filtered ROM is low-dimensional but is not closed (because of the nonlinearity in the given
PDE). (ii) In the second step, we use data-driven modeling to close the filtered ROM, i.e., to model
the interaction between the resolved and unresolved modes. To this end, we use a quadratic ansatz
to model this interaction and close the filtered ROM. To find the new coefficients in the closed
filtered ROM, we solve an optimization problem that minimizes the difference between the full order
model data and our ansatz. We emphasize that the new DDF-ROM is built on general ideas of
spatial filtering and optimization and is independent of (restrictive) phenomenological arguments.
We investigate the DDF-ROM in the numerical simulation of a 2D channel flow past a circular
cylinder at Reynolds numbers Re = 100, Re = 500, and Re = 1000. The DDF-ROM is significantly
more accurate than the standard projection ROM. Furthermore, the computational costs of the
DDF-ROM and the standard projection ROM are similar, both costs being orders of magnitude
lower than the computational cost of the full order model. We also compare the new DDF-ROM
with modern ROM closure models in the numerical simulation of the 1D Burgers equation. For this
simplified computational setting, the DDF-ROM is more accurate and significantly more efficient
than these ROM closure models.