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Description
Background of the study
An expansion tube is one of the ground-based experimental facilities that can generate a high-enthalpy hypersonic flow. A standard expansion tube consists of a compression tube, a shock tube, an acceleration tube, and a test section. In the expansion tube, the acceleration tube at extremely low pressure is connected to the shock tube with a thin diaphragm. The shock wave driven by the shock tube is further accelerated by the acceleration tube, resulting in an extremely high-speed flow. However, measurements in the expansion tube have difficulties since a test time of the generated flow is quite short. Therefore, a numerical simulation of the expansion tube is important to support the measurement [D. E. Gildfind et al. 2018].
The shock wave traveling in a tube is gradually weakened by the interaction with the boundary layer. The shock speed attenuation observed in the expansion tube is quite large [H. Tanno et al. 2016], and it was not reproduced by the numerical simulation [K. Kitazono et al. 2019]. This discrepancy in the shock speed attenuation between the experiment and the numerical simulation has been a problem in recent years since the large shock speed attenuation causes a decrease in the velocity of the test flow, and also undermines the reliability of the numerical simulation used to identify the test time. Recently, a numerical simulation using a turbulence model such as Reynolds-averaged Navier-Stokes model has successfully reproduced the experimentally observed shock speed attenuation [H. Sakamoto et al. 2021]. As a result, it was suggested that the large attenuation of shock speed observed in the expansion tube is caused by the turbulent transition of the boundary layer behind the shock wave. However, the turbulent transition in the actual expansion tube facility is still controversial.
Methodology
In this study, we use an analysis of the linear stability theory to investigate the stability of the boundary layer that develops behind a shock wave propagating inside an expansion tube. Specifically, we consider two base flows for the linear stability analysis: an adiabatic wall condition and an isothermal wall condition. The Chebyshev spectral collocation discretization method was used to discretize the governing equations used in the linear stability analysis. The collocation point number is $N=121$. The solver for the linear stability analysis is an in-house code.
For the base flow used in the linear stability analysis, a self-similar solution of the boundary layer behind the shock wave propagating in the acceleration tube of the HEK-X expansion tube was used. In this study, the analysis was performed in a shock stationary frame (SSF), an inertial system in which the shock wave is almost stationary. This condition simulates the experiment[H. Tanno et al. 2016] conducted in JAXA's free piston type expansion tube HEK-X. Note that in this study, only the airflow behind the shock wave is considered, and the presence of the contact discontinuity is ignored. Under the isothermal wall condition, the heat capacity of the wall is sufficiently large and the wall temperature is assumed to be about room temperature, 300 K.
Results
The stability of the boundary layer flow behind a shock wave propagating inside the acceleration tube of an expansion tube was investigated using linear stability theory. A self-similar solution was used for the base flow in the linear stability analysis, and two conditions, an adiabatic wall condition and an isothermal wall condition, were prepared. The wave number $\alpha$ is normalized by the 99\% velocity boundary layer thickness $y_{99}$, i.e., $\alpha^*y_{99}$. We also used the 99% velocity boundary layer thickness as the characteristic length for the Reynolds number $\operatorname{Re}$.
Under the adiabatic wall condition, an unstable mode was found in which the time growth rate of the disturbance was positive. However, when checking the eigenfunction of this unstable mode, it was found to oscillate violently outside the boundary layer at the grid point interval. This behavior is unlikely to be a physically unstable mode, and it is suggested that the unstable mode found under the adiabatic wall condition is a numerical mode, not a physically unstable mode.
Under the isothermal wall condition, an unstable mode with wave number $\alpha$=5 was found when $x=0.3$ m behind the shock wave. When checking the eigenfunction of this unstable mode, it was found that the amplitude oscillated violently within the boundary layer and gradually decayed outside the boundary layer. The characteristics of this eigenfunction are similar to the supersonic mode, an unstable mode that appears when the wall cooling effect is large[C. P. Knisely & X. Zhong 2019]. The unstable mode was also found to be a high wave number unstable mode, with a frequency equivalent to 3.1 MHz.
The sensitivity of grid resolution on the unstable modes found under the isothermal wall condition was also investigated. The collocation point numbers are $N=241, \,361$. The unstable modes found under the isothermal wall condition shifted to the higher wavenumber side as the grid resolution was increased. In other words, it was found that the unstable modes found under the isothermal wall condition are greatly affected by the grid resolution. It is possible that if the grid resolution is increased sufficiently, the unstable mode will converge at a certain high wavenumber, but such an unstable mode would be on the order of megahertz and is unlikely to physically exist. Therefore, the unstable mode found is considered to be numerical mode under the isothermal wall condition.
In this study, we investigated the stability of the boundary layer behind a shock wave propagating in an expansion tube using the analysis of the linear stability theory. We assumed an ideal gas and investigated the time instability of the wave number in the $x$ direction under the parallel flow approximation. However, it is not appropriate to apply these assumptions to actual flow fields. In particular, the influence of real gas effects is thought to be large, and it is desirable to take these effects into account. Therefore, more advanced analysis is required, as it is possible that the flow field may become unstable due to the influence of the wave number in the spanwise direction, spatial instability, and axisymmetric effects, as well as analyses that take into account real gas effects and stability analyses that do not use parallel flow approximations.
Conclusion
In this study, the analysis of the linear stability theory was used to investigate the stability of the boundary layer behind a shock wave propagating in the expansion tube. Two base flows were prepared for an adiabatic wall and an isothermal wall condition. The analysis was based on the assumption of an ideal gas, and the temporal instability of the wave number in the streamwise direction was investigated.
The unstable modes found under adiabatic wall conditions are likely to be numerical modes since the eigenfunctions had odd shapes from a physical point of view. Under isothermal wall conditions, the unstable mode at high wavenumber was found. The eigenfunction shape of the high-wavenumber unstable mode is similar to the supersonic mode. However, as the grid resolution was increased, the time growth rate became smaller and the unstable mode shifted to higher wavenumber. If the grid resolution was increased sufficiently, the unstable mode may converge at a certain high wavenumber. However, the frequency of the unstable mode would be on the order of megahertz, making it unlikely to be physical.
Therefore, the high-wavenumber unstable mode found under the isothermal wall condition is likely to be a numerical mode, not physical unstable mode.
Reference
D. E. Gildfind et al.: Scramjet Test Flow Reconstruction for a Large-Scale Expansion Tube, Part 2: Axisymmetric CFD Analysis. Shock Waves 28, 899 (2018).
H. Tanno et al.: Basic Characteristics of the Free-Piston Driven Expansion Tube JAXA HEK-X. AIAA Paper 2016-3817 (2016).
K. Kitazono et al.: Numerical Study of Unsteady High Enthalpy Flow in an Expansion Tube. AIAA Paper 2019-1392 (2019).
H. Sakamoto et al.: Numerical Analysis of Shock Speed Attenuation in Expansion Tube. AIAA Paper 2021-0058 (2021).
C. P. Knisely and X. Zhong: Sound Radiation by Supersonic Unstable Modes in Hypersonic Blunt Cone Boundary Layers. I. Linear Stability Theory. Physics of Fluids, 31, 024103, (2019).
Summary
The stability of the boundary layer behind a shock wave in an expansion tube using linear stability analysis was investigated. The analysis considered two types of base flows: an asymmetric flow field along the tube and a symmetric flow field. Wall conditions included an adiabatic wall and an isothermal wall. The results showed that for both asymmetric and symmetric flow fields, the unstable modes found under adiabatic wall conditions were likely numerical modes. Similarly, the high-wavenumber unstable modes found under isothermal wall conditions were also likely numerical modes rather than physically unstable modes.
