Theoretical and Computational Fluid Dynamics

, Volume 31, Issue 2, pp 111–126 | Cite as

On the need of mode interpolation for data-driven Galerkin models of a transient flow around a sphere

  • Witold Stankiewicz
  • Marek Morzyński
  • Krzysztof Kotecki
  • Bernd R. Noack
Open Access
Original Article


We present a low-dimensional Galerkin model with state-dependent modes capturing linear and nonlinear dynamics. Departure point is a direct numerical simulation of the three-dimensional incompressible flow around a sphere at Reynolds numbers 400. This solution starts near the unstable steady Navier–Stokes solution and converges to a periodic limit cycle. The investigated Galerkin models are based on the dynamic mode decomposition (DMD) and derive the dynamical system from first principles, the Navier–Stokes equations. A DMD model with training data from the initial linear transient fails to predict the limit cycle. Conversely, a model from limit-cycle data underpredicts the initial growth rate roughly by a factor 5. Key enablers for uniform accuracy throughout the transient are a continuous mode interpolation between both oscillatory fluctuations and the addition of a shift mode. This interpolated model is shown to capture both the transient growth of the oscillation and the limit cycle.


Reduced-order modeling (ROM) Galerkin projection Dynamic mode decomposition (DMD) Continuous mode interpolation CFD 


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Chair of Virtual EngineeringPoznan University of TechnologyPoznańPoland
  2. 2.UPR 3251LIMSI-CNRSOrsay CedexFrance
  3. 3.Institut fr StrömungsmechanikTechnische Universität BraunschweigBraunschweigGermany

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