On Low-Dimensional Galerkin Models for Fluid Flow

Rempfer, Dietmar

doi:10.1007/s001620050131

Cited by 161 publications

(100 citation statements)

References 9 publications

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“…Error propagation through the POD mode basis has been explored to some degree as far as implications to reduced-order models (see, e.g., Refs. [23][24][25][26]). The propagation of error through data-driven POD representations of turbulence remains a subject requiring development.…”

Section: Introductionmentioning

confidence: 99%

Anisotropic character of low-order turbulent flow descriptions through the proper orthogonal decomposition

2017

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Proper orthogonal decomposition (POD) is applied to distinct data sets in order to characterize the propagation of error arising from basis truncation in the description of turbulence. Experimental data from stereo particle image velocimetry measurements in a wind turbine array and direct numerical simulation data from a fully developed channel flow are used to illustrate dependence of the anisotropy tensor invariants as a function of POD modes used in low-order descriptions. In all cases, ensembles of snapshots illuminate a variety of anisotropic states of turbulence. In the near wake of a model wind turbine, the turbulence field reflects the periodic interaction between the incoming flow and rotor blade. The far wake of the wind turbine is more homogenous, confirmed by the increased magnitude of the anisotropy factor. By contrast, the channel flow exhibits many anisotropic states of turbulence. In the inner layer of the wall-bounded region, one observes onecomponent turbulence at the wall; immediately above, the turbulence is dominated by two components, with the outer layer showing fully three-dimensional turbulence, conforming to theory for wall-bounded turbulence. The complexity of flow descriptions resulting from truncated POD bases can be greatly mitigated by severe basis truncations. However, the current work demonstrates that such simplification necessarily exaggerates the anisotropy of the modeled flow and, in extreme cases, can lead to the loss of three-dimensionality. Application of simple corrections to the low-order descriptions of the Reynolds stress tensor significantly reduces the residual root-mean-square error. Similar error reduction is seen in the anisotropy tensor invariants. Corrections of this form reintroduce three-dimensionality to severe truncations of POD bases. A threshold for truncating the POD basis based on the equivalent anisotropy factor for each measurement set required many more modes than a threshold based on energy. The mode requirement to reach the anisotropy threshold after correction is reduced by a full order of magnitude for all example data sets, ensuring that economical low-dimensional models account for the isotropic quality of the turbulence field.

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Section: Introductionmentioning

confidence: 99%

Anisotropic character of low-order turbulent flow descriptions through the proper orthogonal decomposition

2017

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“…One cause is that the stability of the Navier-Stokes solution is not preserved under Galerkin projection. In an ingenious example, Rempfer showed how a stable fixed point can become unstable upon Galerkin projection in a lower-dimensional subspace [35]. The authors have identified additional modes for base flow variations as one root cause of fragile ROM [30,32].…”

Section: Introductionmentioning

confidence: 99%

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

Stankiewicz

Morzyński

Kotecki

et al. 2016

Theor. Comput. Fluid Dyn.

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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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“…The possible inconsistency of data set and the reduced-order formulation (neglect or inaccurate treatment of pressure and boundary terms, not veri¦ed incompressibility of the §ow) [43], as well as the structural instability of Galerkin Projection [16,44,45] are other sources of discrepancies between Reduced-Order Galerkin model and the high-¦delity model (like DNS of Navier Stokes equations or Large Eddy Simulation).…”

Section: Flow Controlmentioning

confidence: 99%

Arbitrary Lagrangian-Eulerian approach in reduced order modeling of a flow with a moving boundary

Stankiewicz

Roszak

Morzyński

2013

Progress in Flight Physics

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Flow-induced de §ections of aircraft structures result in oscillations that might turn into such a dangerous phenomena like §utter or bu¨eting. In this paper the design of an aeroelastic system consisting of Reduced Order Model (ROM) of the §ow with a moving boundary is presented. The model is based on Galerkin projection of governing equation onto space spanned by modes obtained from high-¦delity computations. The motion of the boundary and mesh is de¦ned in Arbitrary Lagrangian Eulerian (ALE) approach and results in additional convective term in Galerkin system. The developed system is demonstrated on the example of a §ow around an oscillating wing.

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On Low-Dimensional Galerkin Models for Fluid Flow

Cited by 161 publications

References 9 publications

Anisotropic character of low-order turbulent flow descriptions through the proper orthogonal decomposition

Anisotropic character of low-order turbulent flow descriptions through the proper orthogonal decomposition

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

Arbitrary Lagrangian-Eulerian approach in reduced order modeling of a flow with a moving boundary

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