Computational Flow Physics

A stochastic data-driven reduced-order model applicable to a wide range of turbulent natural and engineering flows is presented. Combining ideas from Koopman theory and spectral model order reduction, the stochastic low-dimensional inflated convolutional Koopman model accurately forecasts short-time transient dynamics while preserving long-term statistical properties. A discrete Koopman operator is used to evolve convolutional coordinates that govern the temporal dynamics of spectral orthogonal modes, which, in turn, represent the energetically most salient large-scale coherent flow structures. Turbulence closure is achieved in two steps: first, by inflating the convolutional coordinates to incorporate nonlinear interactions between different scales, and second, by modelling the residual error as a stochastic source. An empirical dewhitening filter informed by the data is used to maintain the second-order flow statistics. The model uncertainty is quantified through either Monte–Carlo simulation or by directly propagating the model covariance matrix. The model is demonstrated on the Ginzburg–Landau equations, large-eddy simulation data of a turbulent jet, and particle image velocimetry data of the flow over an open cavity. In all cases, the model is predictive over time horizons indicated by a detailed error analysis and integrates stably over arbitrary time horizons, generating realistic surrogate data.

Most model reduction methods reduce the state dimension and then temporally evolve a set of coefficients that encode the state in the reduced representation. In this paper, we instead employ an efficient representation of the entire trajectory of the state over some time interval of interest and then solve for the static coefficients that encode the trajectory on the interval. We use spectral proper orthogonal decomposition (SPOD) modes, which are provably optimal for representing long trajectories and substantially outperform any representation of the trajectory in a purely spatial basis (e.g., POD). We develop a method to solve for the SPOD coefficients that encode the trajectories for forced linear dynamical systems given the forcing and initial condition, thereby obtaining the accurate prediction of the dynamics afforded by the SPOD representation of the trajectory. The method, which we refer to as spectral solution operator projection (SSOP), is derived by projecting the general time-domain solution for a linear time-invariant system onto the SPOD modes. We demonstrate the new method using two examples: a linearized Ginzburg-Landau equation and an advection-diffusion problem. In both cases, the error of the proposed method is orders of magnitude lower than that of POD-Galerkin projection and balanced truncation. The method is also fast, with CPU time comparable to or lower than both benchmarks in our examples. Finally, we describe a data-free space-time method that is a derivative of the proposed method and show that it is also more accurate than balanced truncation in most cases.

Experimental measurements often present corrupted data and outliers that can strongly affect the main coherent structures extracted with the classical modal analysis techniques. This effect is amplified at high frequencies, whose corresponding modes are more susceptible to contamination from measurement noise and uncertainties. Such limitations are overcome by a novel approach proposed here, the robust spectral proper orthogonal decomposition (robust SPOD), which implements the robust principal component analysis within the SPOD technique. The new technique is firstly presented with details on its algorithm, and its effectiveness is tested on two different fluid dynamics problems: the subsonic jet flow field numerically simulated, and the flow within an open cavity experimentally analyzed in [48]. The analysis of the turbulent jet data, corrupted both with salt and pepper and Gaussian noise, shows how the robust SPOD produces more converged and physically interpretable modes than the classical SPOD; moreover, the use of the robust SPOD as a tool for de-noising data, based on the signal reconstruction from de-noised modes, is also presented. Applying robust SPOD to the open cavity flow has revealed that it yields smoother spatial distributions of modes, particularly at high frequencies and when considering higher-order modes, compared to standard SPOD.