Generally in a shock tube facility, the shock wave intensity depends mainly on the ratio of driver/driven gas pressure and on the used gas. In ideal conditions, the shock wave propagates at a constant velocity and the flow quantities are constant behind it. The Hugoniot`s relations describe the flow parameters can easily be found in the literature {1}. But in the sixties, many authors have shown that this shock wave velocity decreases along the tube and the useful test time doesn’t vary linearly with the tube length when, for a given tube diameter, the driven gas pressure is low enough {2}. The main reason is the interaction of the wall boundary layer and the core flow which leads to a mass loss term behind the shock and the contact surface. This behavior is unsteady but it is possible to study it at limit regime in a steady fixed shock coordinate system as proposed by Mirels {3}. In the last decade, due to the development and design of certain micro-electromechanical system (MEMS), analytical numerical and experimental works {4}, {5} have been published which concern the shock wave propagation in a tube with very small diameter (milli- or micrometer dimension). In these tubes, the viscous effects play an important role compared to the convective ones and the shock wave propagation is strongly affected. This can lead to a transformation of the shock wave into a compression wave after a certain distance along the tube {5}.
From a dimensionless analysis of the viscous flow conservation equations, Brouillette {4} brings out a scaling ratio Sc=Re1DH/4L, where Re1 is the sonic Reynolds number based on the driven gas conditions, DH is the hydraulic diameter and L is a reference length. This sonic Reynolds number can be written as function of the quantity P1DH for a given initial temperature of the driven gas. It was also pointed that the lower this scaling ratio is, the stronger boundary layer effects are {4}, {5}. Consequently the intensity of the shock wave and the flow behavior are impacted along the tube through more or less shock wave attenuation with non-constant hot flow parameters behind the shock wave.
In the same way, the flow regime in the tube can be laminar or turbulent and the development of the boundary layer behind the shock wave will depend on this regime, just as the mass loss term through boundary layer thickness and the skin friction term. The limit application of laminar theory is a value of P1DH < 0.5 [cmHg.inch] and P1DH > 5 for turbulent theory.
The scaling ratio Sc merges the sonic Reynolds number Re1 and the shape ratio DH/L, but to study the propagation of the shock wave and its attenuation along the tube, it is interesting to build a local scaling ratio from the shock wave local position s, written as Scx=Re1DH/ 4xs. The aim of this work is to revisit the previous numerical and experimental results {6}, {7} and {8} giving the shock wave propagation in millimeter or micrometer diameter tubes and to study their behavior as function of this local scaling ratio Scx. Two power-law correlations will be proposed and discussed for laminar and turbulent flow regimes.
References
1. Glass I.I., Sislian J.P, Nonstationary flows and shock waves. Oxford Sciences Publications,(1994)
2. Duff R.E., Shock tube performance at initial low pressure. Phys. of Fluids, 4, 2, 207-216(1959)
3. Mirels H., Test time in low pressure shock tubes. Phys. of Fluids, 6, 9, 1201-1214 (1963)
4. Brouillette M., Shock waves at microscales. Shock Wave Journal, 13, 1, 3-12 (2003)
5. Zeitoun D.E., Burtschell Y., Navier Stokes computations in micro shock tubes. Shock Waves 15,3-4, 241-246 (2006)
6. Austin J.M., Bodony D.J., Wave propagation in gaseous small-scale channel flows. Shock Waves, 21, 6, 547-557 (2011)
7. Mirshekari G., Brouillette M., Giordano J., Hebert C., Parisse J.D, Perrier P., Shock waves in microchannels. J. Fluid Mechanics, 724, 259-283 (2013)
8. Zeitoun D.E., Microsize and initial pressure effects on shock wave propagation in a tube, Shock Waves, DOI: 10.1007/s00193-014-0512-9 (2014)
