Exploratory Data Analysis of The KelvinHelmholtz instability in Jets

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.

💡 Research Summary

The paper investigates the use of two data‑driven modal decomposition techniques—Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD)—to identify and analyse Kelvin‑Helmholtz (KH) instability structures in a two‑dimensional co‑axial jet at Reynolds number 10 000. The authors generate large‑eddy simulation (LES) data on a uniform 256 × 256 grid and extract 30 snapshots at a temporal spacing of five time steps. Two jet configurations are examined: a “close‑jet” case (CASE = L/10) where the two jets interact early (step 40) and a “far‑jet” case (CASE = L/5) with later interaction (step 100). The jet inlet parameters (Umax = 0.1, Vmax = Umax/30, shear‑layer thickness B = 10.5, jet diameter ro = L/20) are clearly defined.

Methodology
POD: The mean field is subtracted from each snapshot to form a fluctuation matrix U. The covariance matrix C = UᵀU is assembled, and its eigen‑decomposition yields eigenvalues λi (ordered descending) and eigenvectors Ai. POD modes φi are reconstructed as linear combinations of the snapshots weighted by Ai. Energy content is assessed via λi, with the first few modes typically capturing >95 % of the total kinetic energy.

DMD: The snapshot matrix V₁ (v₁ … v_{N‑1}) and its one‑step‑ahead counterpart V₂ (v₂ … v_N) are formed. An economy‑size QR factorisation of V₁ produces Q and R, from which the low‑dimensional companion matrix S = R⁻¹Q*V₂ is computed. Eigenvalues D and eigenvectors X of S are obtained; continuous‑time eigenvalues λj = log(D_jj)/Δt give growth rates and frequencies. Dynamic modes DM_j = V₁X(:,j) are then reconstructed. This procedure bypasses the need for an explicit system matrix A and works with both structured and unstructured data.

Results
Figures (not reproduced here) display the first five POD and DMD modes for both jet cases. POD modes 1–2 reveal large‑scale symmetric vortex structures associated with the inverse KH instability, while modes 3–4 capture smaller‑scale features near the nozzle exit. The time coefficients of POD modes 3 and 4 exhibit a phase shift of five time steps, indicating strong correlation and suggesting the presence of helical or columnar instability components. DMD modes highlight the roll‑up of the shear layer: the second DMD mode shows a clear vortex sheet roll‑up pattern, and higher‑order DMD modes resolve finer structures. The DMD eigenvalue spectrum (Fig. 6) is plotted on the complex unit disk; eigenvalues outside the unit circle correspond to unstable growth, those on the circle to neutral oscillations, and those inside to decay. A logarithmic mapping converts these to continuous‑time growth rates, with positive real parts indicating instability.

Discussion
The authors argue that POD excels at rapidly converging to the dominant energetic structures, making it suitable for reduced‑order modelling and isolating large‑scale dynamics. DMD, by contrast, provides a temporally orthogonal basis that directly encodes growth/decay rates and frequencies, offering clearer insight into the dynamics of coherent structures. The paper notes that DMD’s ability to handle arbitrary spatial sampling and unstructured data is advantageous for sub‑domain analysis. However, several limitations are acknowledged: the relatively small number of snapshots (30) and fixed Δt may limit spectral resolution; the assumption of uncorrelated snapshots in POD is not verified; and quantitative validation against experimental PIV or alternative numerical methods is absent.

Conclusions and Future Work
Both POD and DMD are shown to be powerful tools for analysing LES data of subsonic jets and for extracting KH instability features. The authors propose future investigations into the effect of the time window Δt on DMD accuracy, sensitivity analysis of POD modes with respect to snapshot count, and comparisons with nonlinear dimensionality‑reduction techniques (e.g., kernel POD, autoencoders). They also suggest cross‑validation with experimental measurements to strengthen confidence in the extracted modes.

Overall, the study provides a clear, if preliminary, demonstration of how linear data‑driven decompositions can complement each other in revealing both energetic and dynamic aspects of shear‑layer instabilities in high‑Reynolds‑number jet flows.