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

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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.


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Correspondence to Witold Stankiewicz.

Additional information

Communicated by Dr. Oleg V. Vasilyev.

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Stankiewicz, W., Morzyński, M., Kotecki, K. et al. On the need of mode interpolation for data-driven Galerkin models of a transient flow around a sphere. Theor. Comput. Fluid Dyn. 31, 111–126 (2017). https://doi.org/10.1007/s00162-016-0408-7

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