The proper orthogonal decomposition identifies basis functions or modes which optimally capture the average energy content from numerical or experimental data. By projecting the Navier-Stokes equations onto these modes and truncating, one can obtain low-dimensional ordinary differential equation models for fluid flows. In this paper we present a tutorial on the construction of such models. In addition to providing a general overview of the procedure, we describe two different ways to numerically calculate the modes, show how symmetry considerations can be exploited to simplify and understand them, comment on how parameter variations are captured naturally in such models, and describe a generalization of the procedure involving projection onto uncoupled modes that allow streamwise and cross-stream components to evolve independently. We illustrate for the example of plane Couette flow in a minimal flow unit -a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence.
We model turbulent plane Couette flow in the minimal flow unit (MFU) -a domain whose spanwise and streamwise extent is just sufficient to maintain turbulenceby expanding the velocity field as a sum of optimal modes calculated via proper orthogonal decomposition from numerical data. Ordinary differential equations are obtained by Galerkin projection of the Navier-Stokes equations onto these modes. We first consider a 6-mode (11-dimensional) model and study the effects of including losses to neglected modes. Ignoring these, the model reproduces turbulent statistics acceptably, but fails to reproduce dynamics; including them, we find a stable periodic orbit that captures the regeneration cycle dynamics and agrees well with direct numerical simulations. However, restriction to as few as six modes artificially constrains the relative magnitudes of streamwise vortices and streaks and so cannot reproduce stability of the laminar state or properly account for bifurcations to turbulence as Reynolds number increases. To address this issue, we develop a second class of models based on 'uncoupled' eigenfunctions that allow independence among streamwise and cross-stream velocity components. A 9-mode (31-dimensional) model produces bifurcation diagrams for steady and periodic states in qualitative agreement with numerical Navier-Stokes solutions, while preserving the regeneration cycle dynamics. Together, the models provide empirical evidence that the 'backbone' for MFU turbulence is a periodic orbit, and support the roll-streak-breakdown-roll reformation picture of shear-driven turbulence.
We model turbulent plane Couette flow ͑PCF͒ by expanding the velocity field as a sum of optimal modes calculated via the proper orthogonal decomposition from numerical data. Ordinary differential equations are obtained by Galerkin projection of the Navier-Stokes equations onto these modes. For a minimal truncation including only the most energetic modes having no streamwise variation, we show under quite general conditions the existence of linearly stable nontrivial fixed points, corresponding to a state in which the mean flow is coupled to streamwise vortices and their associated streaks. When the two next most energetic modes, still lacking streamwise variations, are included, chaos and heteroclinic cycles associated with the fixed points are found. The attractors involve repeated visits near unstable fixed points and periodic orbits corresponding to steady and periodically varying vortices, and account for a self-sustaining process in which vortices interact with the mean flow. The models considered in this paper can also serve as a foundation for more sophisticated ordinary differential equation models for turbulent PCF, including those which include modes with streamwise variations.
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