On the flow characteristics in the shock formation region due to the diaphragm opening process in a shock tube

TL;DR

The paper addresses how diaphragm rupture dynamics modify shock formation in shock tubes, challenging predictions based on ideal one-dimensional theory. It advances a combined experimental–numerical approach, using high-speed imaging and 2D CFD (CONVERGE) with experimentally measured diaphragm-opening profiles to capture early-time flow features and validate predictions. Key contributions include new correlations for the shock-deceleration time $t_{dec}$ and minimum Mach number $M_{s,min}$, a mass-flow–based model for the peak Mach number $M_{s,peak}$ that accounts for diaphragm opening, and a set of relationships linking the shock-formation distance $x_f$ and time $t_f$ to opening times and driving pressures. The work delivers a unified framework to predict shock evolution in the formation region, improving shock-tube design, diagnostics, and interpretation across a range of operating conditions.

Abstract

The shock formation process in shock tubes has been extensively studied; however, significant gaps remain in understanding the effects of the diaphragm rupture process on the resulting flow non-uniformities. Existing models predicting the shock attenuation and propagation dynamics overlook critical diaphragm mechanics and their impact on shock behavior. Addressing this gap is vital for improving predictive capabilities and optimizing shock tube designs for applications in combustion kinetics, aerodynamics, and high-speed diagnostics. This study investigates the shock wave formation and propagation through combined experimental and numerical approaches over a range of driver-to-driven pressure ratios (Driver pressure: 9.4 - 24.2 bar of helium; Driven pressure: 100 Torr of argon). High-speed imaging captures the diaphragm opening dynamics, while pressure and shock velocity measurements along the entire driven section of the shock tube provide key validation data for CFD. Two-dimensional numerical simulations incorporate experimentally measured diaphragm opening profiles, offering detailed insights into flow features and thermodynamic gradients behind the moving shock front. Key parameters, including deceleration and acceleration phases within the shock formation region, shock formation distances, and times, have been quantified. A novel theoretical framework is introduced to correlate these parameters, enabling accurate predictions of shock Mach number evolution under varying conditions. This unified methodology bridges theoretical and experimental gaps, providing a robust foundation for advancing shock tube research and design.

Figures (19)

• Figure 1: Schematic of the high-pressure shock tube facility at KAUST. A light source placed at driven end illuminates the diaphragm and a high-speed camera placed at the endwall captures the images of the diaphragm. (a) A zoomed cross-sectional view of the area change near the diaphragm station. (b) Cross-sectional cut at the center-line of 1.6 mm thick aluminum diaphragm highlighting the effective thickness of the diaphragm after an 'X' shaped indentation.
• Figure 2: The diaphragm opening process as visualized using a high-speed camera placed at the end wall for case S2. The line plot shows the opened area as a function of time. The time zero indicates the instant when the diaphragm stretching began. Scatter points indicate the opened aperture area measured after 5% of opening. Multimedia view shows the diaphragm opening as a function of time for all the seven cases. (Multimedia available online)
• Figure 3: Experimentally measured values of (a) $t_{op,5}$, (b) $t_{op,5 \rightarrow 100}$, and (c) $t_{op,100}$ along with their respective fitted curves, and (d) Diaphragm opening profile measurements with a sigmoid fit (\ref{['eq:sigmoid']}) to represent the opening behavior.
• Figure 4: Schematic describing the numerical domain for the shock tube including the incorporation of diaphragm opening in CFD.
• Figure 5: Grid convergence study comparing the pressure measurements at two locations (1.9 and 6.6 X/D) for minimum grid sizes from 1 - 0.125 mm for case S1. The recommended grid sizes based on this study is 0.25 mm