A model for frequency scaling of flow oscillations in high-speed double cones

Abstract: Coherent small-amplitude unsteadiness of the shock wave and the separation region over a canonical double cone flow, termed in literature as oscillation-type unsteadiness, is experimentally studied at Mach 6. The double cone model is defined by three non-dimensional geometric parameters: fore- and aft-cone angles ($\theta_1$ and $\theta_2$), and ratio of the conical slant lengths ($\Lambda$). Previous studies of oscillations have been qualitative in nature, and mostly restricted to a special case of the cone model with fixed $\theta_1 = 0^\circ$ and $\theta_2 = 90^\circ$ (referred to as the spike-cylinder model), where $\Lambda$ becomes the sole governing parameter. In the present effort we investigate the self-sustained flow oscillations in the $\theta_1$-$\Lambda$ parameter space for fixed $\theta_2 = 90^\circ$ using time-resolved schlieren visualization. The experiments reveal two distinct sub-types of oscillations, characterized by the motion (or lack thereof) of the separation point on the fore-cone surface. The global time scale associated with flow oscillation is extracted using spectral proper orthogonal decomposition. The non-dimensional frequency (Strouhal number) of oscillation is seen to exhibit distinct scaling for the two oscillation sub-types. The relationship observed between the local flow properties, instability of the shear layer, and geometric constraints on the flow suggests that an aeroacoustic feedback mechanism sustains the oscillations. Based on this insight, a simple model with no empiricism is developed for the Strouhal number. The model predictions are found to match well with experimental measurements. The model provides helpful physical insight into the nature of the self-sustained flow oscillations over a double cone at high-speeds.

• The paper develops a scalable aeroacoustic model to predict the frequency scaling of flow oscillations in high-speed double cones by adapting Rossiter's formula.
• Experimental analysis at Mach 6 identified two distinct oscillation behaviors, free and anchored, with spectral proper orthogonal decomposition revealing different Strouhal number dependencies on geometry.
• The theoretical model successfully aligns with experimental data, providing valuable insights into shock wave and boundary layer interactions crucial for designing stable aerospace structures.

An Assessment of Flow Oscillations in High-Speed Double Cones

The study titled "A model for frequency scaling of flow oscillations in high-speed double cones" provides a detailed examination of frequency scaling related to flow oscillations in hypersonic double-cone configurations. The authors, Kumar et al., explore the nature of self-sustained oscillatory behaviors, focusing on experimental investigations conducted at Mach 6 on varying double cone geometries. The study stands out by transitioning from qualitative observations often seen in prior analyses to a model-based quantitative understanding.

High-speed double cones present a significant area of inquiry in the field of aerodynamics due to their complex shock wave configurations and resulting flow separations. In this paper, the authors consider a parametrization of the double cone geometries using fore-cone angle, aft-cone angle, and the relative lengths of the cone's slant surfaces. By varying these parameters, they aim to capture the regimes under which distinct oscillatory flow behaviors manifest.

A key innovation of their work is the identification of two sub-types of oscillatory behaviors, namely "free-oscillations" and "anchored-oscillations." In free-oscillations, the separation point exhibits notable movement, contributing to fluctuations in shock wave locations and flow structures. In contrast, anchored-oscillations are characterized by a fixed separation point and pronounced transverse wave movement, resulting in different dynamic characteristics.

The methodology employs time-resolved schlieren imaging to visualize these flow phenomena, and spectral proper orthogonal decomposition (SPOD) is used to extract dominant temporal scales from experimental data. This approach allows for the capture of non-dimensional frequency metrics (Strouhal numbers), which assert a dependency on geometric parameters for free-oscillations, but remain largely invariant for anchored-oscillations. This suggests that anchored-oscillations may be governed by inherent flow constraints that differ significantly from those influencing free-oscillations.

A major contribution of the paper lies in the development of an aeroacoustic model that provides a theoretical framework for predicting the Strouhal number of oscillations based on measurable parameters. This model adapts Rossiter's formula, commonly associated with cavity flow dynamics, and considers the downstream propagation of shear layer disturbances coupled with upstream acoustic feedback—a mechanism analogous to that in compressible open cavity flows. This model successfully reconciles observed experimental data, providing theoretical grounding to the oscillatory behaviors detected.

The implications of this research are manifold. Practically, the findings can inform the design of aerospace structures exposed to high-speed flow conditions, where understanding the oscillatory regimes is crucial for stability and performance. Theoretically, the work adds depth to aeroacoustic studies concerning shock wave and boundary layer interactions, potentially guiding future computational and experimental designs. The scalable aeroacoustic model developed offers pathways for more advanced explorations into complex oscillatory flow phenomena in various geometries beyond the canonical double-cone configuration.

Speculating on future developments, this integration of experimental insight with theoretical modeling may pave the way for adaptive control strategies in aerospace designs, where active or passive flow control can be employed to mitigate adverse effects of oscillatory flow dynamics. As computational capabilities expand, coupling this model with high-fidelity simulations of turbulent flows and shock interactions could lead to a more comprehensive understanding and control of flow-induced vibrational stresses in real-world applications.

In conclusion, this work by Kumar et al. significantly advances the quantitative understanding of oscillation dynamics in hypersonic flow over double cones, offering a robust model linked to measurable parameters and setting a foundational framework for future studies in related aeroacoustic phenomena.