[Paper Review] Exploratory Data Analysis of The KelvinHelmholtz instability in Jets

Santosh Tirunagari|arXiv (Cornell University)|Mar 21, 2015

Fluid Dynamics and Turbulent Flows14 references3 citations

TL;DR

This study applies Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) to 2D direct numerical simulation (DNS) data of co-axial jets at Re = 10,000 to analyze Kelvin–Helmholtz (KH) instabilities. It demonstrates that DMD excels in capturing temporal dynamics and coherent structures near shear layers, while POD effectively isolates large-scale energetic modes, with both methods revealing phase-locked instabilities and vortex roll-up patterns in different jet configurations.

ABSTRACT

The KelvinHelmholtz (KH) instability is a fundamental wave instability that is frequently observed in all kinds of shear layer (jets, wakes, atmospheric air currents etc). The study of KH-instability, coherent flow structures has a major impact in understanding the fundamentals of fluid dynamics. Therefore there is a need for methods that can identify and analyse these structures. In this Final assignment, we use machine-learning methods such as Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) to analyse the coherent flow structures. We used a 2D co-axial jet as our data, with Reynolds number corresponding to Re: 10,000. Results for POD modes and DMD modes are discussed and compared.

Motivation & Objective

• To investigate the effectiveness of POD and DMD in identifying coherent flow structures in turbulent jet flows.
• To compare the performance of POD (spatially orthogonal) and DMD (temporally orthogonal) in resolving Kelvin–Helmholtz instabilities.
• To analyze the impact of jet proximity (CASE = L/10 vs. L/5) on instability development and mode dynamics.
• To evaluate the utility of these methods for reduced-order modeling and time-resolved fluid dynamics analysis.
• To provide a foundation for using machine learning techniques in experimental and simulation-based fluid dynamics research.

Proposed method

• POD is applied using the snapshot method to compute spatial modes from time-resolved LES data, minimizing the mean squared error in representing the flow field.
• The method involves computing the auto-covariance matrix from snapshots and solving the eigenvalue problem to extract POD modes and their associated time coefficients.
• DMD is implemented by forming a snapshot matrix V₁ᴺ and approximating the linear mapping A such that vₙ₊₁ ≈ A vₙ, using an Arnoldi iteration-based algorithm.
• DMD computes eigenvalues and modes that represent dynamically relevant structures, with eigenvalues mapped logarithmically to interpret growth/decay rates.
• The analysis uses 2D co-axial jet simulations with Re = 10,000, jet diameter ro = L/20, and a steepness parameter B = 10.5.
• Two cases are studied: jets close together (L/10) and far apart (L/5), with flow fields analyzed via passive scalar (PS), U, and V velocity components.

Experimental results

Research questions

• RQ1 How do POD and DMD compare in resolving large-scale coherent structures in KH-instability dominated jet flows?
• RQ2 What is the role of jet proximity (L/10 vs. L/5) in the development and interaction of Kelvin–Helmholtz vortices?
• RQ3 How do the time coefficients of POD modes reflect the dynamic behavior and phase relationships of instabilities?
• RQ4 To what extent do DMD eigenvalues capture the temporal dynamics and stability characteristics of the flow?
• RQ5 Can POD and DMD effectively separate large-scale coherent motions from turbulent fluctuations in LES data?

Key findings

• POD modes capture 95% of total intensity fluctuations using only the first few modes, effectively isolating large-scale coherent structures.
• In case 1 (jets close), the first few POD modes show symmetric inverse KH instability structures due to jet interaction, with vortex roll-up visible in the second DMD mode.
• DMD modes clearly resolve dynamic patterns near the shear layer, with the second mode exhibiting a characteristic roll-up of the vortex sheet.
• The time coefficients of POD modes 3 and 4 in case 1 show a 5-step phase difference, indicating strong correlation and potential helical or columnar instability features.
• DMD eigenvalues for case 1 show unstable modes (|λ| > 1) outside the unit disk, with logarithmic mapping revealing positive real parts for unstable dynamics.
• Both POD and DMD successfully identify and differentiate KH instability mechanisms, with DMD offering superior temporal resolution of dynamic behavior.