[Paper Review] A Lagrangian Dynamic Mode Decomposition

Jörn Sesterhenn, Amir Shahirpour|arXiv (Cornell University)|Mar 8, 2016

Fluid Dynamics and Turbulent Flows5 references3 citations

TL;DR

This paper introduces a Lagrangian Dynamic Mode Decomposition (L-DMD) that transforms data into a moving reference frame aligned with the group velocity of convecting structures, enabling proper low-rank representation via POD or DMD. By doing so, it recovers a few dominant modes for traveling structures—resolving the counterintuitive poor singular value decay—demonstrated on the head vortex of a compressible starting jet.

ABSTRACT

Temporal or spatial structures are readily extracted from complex data by modal decompositions like POD or DMD. Subspaces of that decompositions serve as reduced order models and define spatial structures in time or temporal structures in space. Convecting phenomena pose a major problem to those decompositions. A structure travelling with a certain group velocity will be perceived as a plethora of modes in time or space respectively. This manifests itself for example in poorly decaying Singular Values when using a POD. The poor decay is very counter-intuitive, since we expect a single structure to be represented by a few modes. The intuition proves to be correct and we show that in a properly chosen reference frame along the characteristic defined by the group velocity, a POD or DMD reduces moving structures to a few modes, as expected. Beyond serving as a reduced model, the re- sulting entity can be used to define a constant or minimally changing structure in turbulent flows. This can be interpreted as an empirical counterpart to exact coherent structures. We present the method and its application to the head vortex of a compressible starting jet.

Motivation & Objective

• To address the poor singular value decay in POD and DMD when analyzing convecting structures in fluid flows.
• To resolve the counterintuitive issue where a single traveling structure appears as many modes due to frame misalignment.
• To develop a method that transforms data into a reference frame moving with the group velocity, restoring expected low-rank representation.
• To enable identification of empirical coherent structures in turbulent flows by stabilizing the decomposition in a Lagrangian frame.
• To demonstrate the method’s effectiveness on the head vortex of a compressible starting jet as a real-world application.

Proposed method

• Transform the data into a reference frame moving with the group velocity of the dominant convecting structure, derived from the characteristic direction of propagation.
• Apply Proper Orthogonal Decomposition (POD) or Dynamic Mode Decomposition (DMD) in the moving frame to extract spatial or temporal modes.
• Use the group velocity to define the Lagrangian frame, ensuring that convecting structures appear stationary or slowly varying.
• Leverage the resulting decomposition to identify a minimal set of modes representing the structure, improving mode decay and interpretability.
• Apply the method to the head vortex of a compressible starting jet to validate its effectiveness in capturing coherent dynamics.

Experimental results

Research questions

• RQ1 How does transforming data into a Lagrangian frame aligned with the group velocity affect the mode decomposition of convecting structures in POD and DMD?
• RQ2 Why do convecting structures lead to poor singular value decay in standard POD, and can this be corrected by frame transformation?
• RQ3 Can a single convecting structure be represented by only a few modes when decomposed in a moving reference frame?
• RQ4 To what extent does the Lagrangian DMD method improve the identification of coherent structures in turbulent flows?
• RQ5 How does the method perform on real, complex flows such as the head vortex of a compressible starting jet?

Key findings

• The method successfully restores rapid decay of singular values in POD when applied to convecting structures, confirming that a single structure should be representable by only a few modes.
• In the Lagrangian frame, the head vortex of a compressible starting jet is captured by a minimal number of modes, demonstrating improved low-rank approximation.
• The resulting decomposition reveals a stable, constant, or minimally changing structure in the moving frame, analogous to exact coherent structures.
• The approach provides an empirical counterpart to exact coherent structures by identifying persistent patterns in turbulent flows.
• The application to the compressible starting jet validates the method’s effectiveness in real-world fluid dynamics scenarios with complex convective dynamics.