Reduced-order models are used widely to make fluid flow simulation computationally tractable for example for the purpose of real-time control of fluid systems, like wind-turbine wakes. Reduced-order models are created by first performing (offline) expensive high-fidelity simulations, and then extracting the most important features in order to construct the reduced model. An important ongoing issue in current reduced-order models is their lack of stability.

In this work we take a new look at the stability of reduced-order models for incompressible flow. For incompressible flow, a stability bound can be obtained by considering the kinetic energy of the flow, which is bounded by

$$\frac{d K}{d t} = -\nu \| \nabla u \|^2. % -\nu \int_{\Omega} \nabla \vt{u} : \nabla \vt{u} \, \rd \Omega.$$

The novelty is that we have derived a POD-Galerkin method that satisfies a similar energy equation on the level of the reduced-order model:

$$\frac{K_{r}^{n+1} – K_{r}^{n}}{\Delta t} = -\nu \| Q_{h} \Phi a^{n+1/2} \|^2.$$

As a consequence, the reduced-order model is stable, independent of the number of modes used, grid, or time-step. The two essential elements to non-linear stability in our method are:

  1. first discretise the full order model, and then project the discretised equations,
  2. use spatial and temporal discretisation schemes that are globally energy-conserving (in the limit of vanishing viscosity). For this purpose, as full order model a staggered-grid finite volume method in conjunction with an implicit Runge-Kutta method is employed.

In addition, a new constrained singular value decomposition is proposed which enforces global momentum conservation. The resulting ROM is thus globally conserving mass, momentum and kinetic energy.

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