Spectral analysis of attached and separated turbulent flows over a Gaussian-shaped bump

Abstract

We investigate the broadband turbulent dynamics of attached and separated flows over a Gaussian bump, focusing on the origin of low-frequency coherent structures. The analysis combines time-resolved experimental measurements with physics-based linear models, using mean fields previously assimilated from the same dataset as base flows. Spectral proper orthogonal decomposition reveals coherent dynamics in low- and medium-frequency regimes for both flows, with the low-frequency dynamics being substantially stronger in the separated case. In the separated flow, these dynamics are linked to a three-dimensional zero-frequency modal instability that generates large-scale streaks downstream of the bump. A standing-wave model based on resolvent modes, incorporating sidewall effects, reproduces the experimentally observed spanwise structure of the dynamics and highlights the limitations of simulations with small spanwise extent and periodic boundary conditions. In the attached flow, similar low-frequency streaks are identified. These are weaker, do not form a prominent standing-wave pattern, and cannot be definitively classified as either modal or non-modal. The three-dimensional zero-frequency instability and finite-span standing-wave dynamics, identified as the main drivers of low-frequency coherent structures in the separated flow, offer an explanation for persistent discrepancies between simulations and experiments on the Gaussian bump, and provide guidance on spanwise domain size and boundary conditions for future simulations.

Figures (22)

• Figure 1: Geometry of the bump and measurement locations in the spanwise (left) and streamwise (right) planes, showing mean (blue dots) and instantaneous (purple crosses) pressure measurement positions, as well as PIV measurement regions colored by the streamwise mean velocity component $\bar{u}$ for $\hbox{Re}=2\times10^6$. Instantaneous velocity data are available for the spanwise SPIV window (green border) and the black-bordered streamwise PIV windows labeled "FOV 1" and "FOV 2". The wind tunnel side walls are located at $z=-0.5$ and $z=0.5$.
• Figure 2: PSD of surface pressure measurements in the downstream region of the bump for the attached (a) and separated (b) cases. Tick marks at the top denote the frequency resolution.
• Figure 3: Data-assimilated mean flow used in the approximation of the linear operators for the attached (a) and separated cases (b), with the eddy viscosity field shown as background contours and mean streamlines overlaid in blue. Based on data reported in klopsch_Enabling_2025.
• Figure 4: Validation of the streamwise variation of mean surface pressure for the attached (a) and separated cases (b), with the gray-shaded region indicating measurement uncertainty from the experiment. The pressure coefficient is offset such that $c_p=0$ at $x=-0.4$. Based on data reported in klopsch_Enabling_2025.
• Figure 5: Sketch of the computational domain with gray lines indicating the mesh. The red area denotes regions with PIV data, the blue area the PINN domain, and the cyan area the response domain for RA. Black lines show contour levels of the sponge: 0.1 (solid), 1 (dashed), 10 (dash-dotted), and 50 (dotted). Axis breaks are used to highlight the most relevant part of the domain.