Effect of cross-sectional anisotropy on shock train dynamics in supersonic internal flows

TL;DR

This study examines how cross-sectional anisotropy in elliptical ducts, via aspect ratio $AR$, affects shock-train dynamics in supersonic internal flows at $M_ fty=2.1$ using an adaptive mesh refinement embedded-boundary approach. Despite pronounced AR-induced changes in shock-train morphology (fewer shock cells and a weakened central normal stem as $AR$ increases), the overall pressure rise and stagnation-pressure loss across the pseudo-shock remain largely insensitive to $AR$ when the cross-sectional area and upstream blockage are fixed. The results indicate that the global deceleration efficiency is governed primarily by turbulent boundary-layer blockage rather than the detailed local shock-train structure, challenging simple AR-based scaling and highlighting the dominance of boundary-layer effects in pressure recovery. The work demonstrates the capability of AMR-EB methods to resolve complex shock-boundary-layer interactions in anisotropic ducts and suggests avenues for comparing elliptical and rectangular cross-sections to better understand cross-sectional geometry effects on shock trains.

Abstract

This study investigates the effect of duct aspect ratio ($AR$), defined as the ratio of major to minor axis in an elliptical duct, on shock train dynamics for a freestream Mach number of 2.1. The aspect ratio $AR$ is varied from 1.0 to 3.0 while maintaining a constant cross-sectional area and identical upstream conditions, thereby ensuring the same inlet mass flow and nearly constant boundary-layer-induced blockage across all $AR$. This isolates shape-induced confinement effects. Simulations are performed using an embedded-boundary method with adaptive mesh refinement which enabled a finest resolution of 48$μ$m resolving the shocks in the shock train. The results show that increasing $AR$ significantly modifies the shock train morphology. The number of discrete shock cells decreases, and the leading shock front elongates along the major axis while contracting along the minor axis. The normal shock stem prominent in the circular duct (AR=1.0) nearly disappears at AR=3.0. Despite these morphological changes, the wall-pressure trace and stagnation-pressure loss remain largely insensitive to $AR$. These results indicate that while duct cross-section governs the detailed shock train structure, the overall efficiency of flow deceleration and pressure recovery is dictated primarily by the effective blockage imposed by the turbulent boundary layer, rather than the aspect ratio itself for a given mass flow rate and pressure ratio across the pseudo shock.

Figures (12)

• Figure 1: Refinement via AMR shown on (a) an axial plane and (b) a section of the x-plane at $x^* \approx 2.5$ i.e. approximately $2.5R$ upstream of the shock train plotted for $AR=1$. Three levels of refinement can be seen while the coarsest level is not visible in this view.
• Figure 2: Instantaneous Mach number upstream of the shock train plotted at $x^*\approx 1.6$ or at a distance of 0.25 times the shock train length for (a) $AR=1.0$ (b) $AR=2.0$ and (c) $AR=3.0$. Panel (d) overlays the cross-sections for all aspect ratios to highlight the differences in the cross-sections relative to each other.
• Figure 3: (a) Boundary layer thickness upstream of the shock train. (b) Mean velocity profiles plotted in wall-units for different aspect ratios at $2.6R$ upstream of the shock train.
• Figure 4: History of the pressure at the (a,c,e) centerline and (b,d,f) at the wall plotted for (a,b) $AR=1.0$ (c,d) $AR=2.0$ and (e,f) $AR=3.0$.
• Figure 5: Instantaneous three-dimensional structure of the leading shock for (a) $AR=1.0$ (b) $AR=2.0$ and (c) $AR=3.0$.