Real shock tube behaviour and performance

The results in this chapter are derived from ideal gas conditions (i.e., par-ticle interactions do not play a significant role), assuming an instantaneous diaphragm rupture and no dissipative phenomena (e.g. viscocity and heat). In the absence of dissipative phenomena, chemical reactions and particle interac-tions, the shock wave preserves a constant velocity. However, the existence of these phenomena will change the equilibrium in the thermodynamic relations presented herein.

As already pointed out earlier, real gas effects and chemically reacting mixtures of perfect gases are considered as beyond the scope of this thesis. Such effects can be of importance in blast events and close-in detonations, and the reader is referred to the literature (see e.g. [124,125,202]) for a more detailed presentation of these effects. However, as a rule of thumb one may state that shock temperatures larger than 1000 K and pressures stronger than 1 MPa introduces additional degrees of freedom in the air molecules (i.e., air ionizes and dissociates), and the ratio of specific heats is no longer constant and must be modified to allow for these effects. Typically, experiments have shown that for shocks with velocities larger than about 3.5 of the speed of sound (i.e., hydrostatic pressures of about 1.4 MPa), the assumption of a calorically perfect gas leads to an error of more than 1 %. For shocks with velocities greater than about 5.5 of the speed of sound (i.e., hydrostatic pressures of approximately 3.5 MPa), the error is more than 5 % [145].

The state of a shocked gas in a shock tube is therefore not completely described by the idealized theory presented herein. Moreover, even if the real gas proper-ties are known, other phenomena such as diaphragm rupture and boundary (or friction) effects cause deviations from the idealized theory and must be considered when evaluating the shock tube experiments and performance. Ex-periments show that failure of the diaphragm often initiates at the centre and propagates to the edges during tearing and folding of the petals [123]. The gas flow therefore starts as a jet and increases in diameter as the diaphragm opens until the cross-section of the tube is completely filled. The shock wave is formed due to the coalesce and overtaking of compression waves during this opening process. The imperfect burst of a shock tube diaphragm results in multi-dimensional disturbances that can significantly modify the flow field predicted by the idealized 1D theory and a complete opening is seldom observed in experiments. Despite this disturbance in the flow field, experimental work shows that most of the disturbances related to the diaphragm opening process vanish in the distant flow field (i.e., at distances larger than 10×diameter) [123].

Due to the higher temperature and flow velocity in the compressed gas, post-shock waves move faster than the leading post-shock and eventually coalesce into

6.4. General shock tube theory 157

the single shock front in Figure 6.6b (see also Figure 6.3).

Therefore, the assumption of an instantaneous diaphragm burst in analytical and numerical studies does not introduce any relevant error in the prediction of a normal shock wave in the distant flow field. However, the distant flow field is characterized by the occurrence of a normal shock front of constant strength whose intensity is lower than that expected from the 1D theory (i.e., a complete and instantaneous opening of the diaphragm). Thus, when considering shock tube performance at distances larger than 10×diameter downstream the diaphragm, the effect of diaphragm opening may be limited to the reduced strength of the leading shock compared to the expected strength from the 1D theory. Recent numerical and experimental work [119] on shock tube performance using steel diaphragms suggests that the deviation from the idealized theory in terms of loss in intensity of the reflected pressure could be in the range of 50-60 %.

Even though the diaphragm dynamics play a negligible role in the distant flow field, experimental evidence often reveals a shock attenuation (i.e., decreasing velocities) at distances larger than 10×diameter downstream the diaphragm.

Experiments indicate that the shock does not travel at constant velocity in the distant flow field but tends to decelerate. At the same time the contact discontinuity may be accelerated. These effects are due to dissipative phenom-ena introduced by viscous forces resulting from the relative motion between the gas and the interior walls of the tube. A boundary layer is formed with a thickness which varies from zero at the shock front to a maximum at the contact discontinuity. In well-designed shock tubes with limited friction at the side walls, the effects of viscous attenuation is small [124]. The most severe contribution to shock attenuation is usually due to the reflected rarefaction waves catching up with the shock wave due to small driver to driven length ratios (illustrated in Figure 6.6d).

A further description of these real effects are beyond the scope of this thesis.

However, it should be noted that even though the relations in this chapter are derived for ideal gas conditions, the theory and principles presented herein give a fundamental understanding of the underlying physics of the shock tube problem.

Moreover, the shock attenuation observed in experiments are most evident when predicting the shock parameters based on the initial conditions using Eqs. (6.79) and (6.80). The shock attenuation effects are not that important when measuring the velocity in the vicinity of the plate, and it is possible to calculate the shock parameters with high accuracy by using the Mach number Msand Eqs. (6.45), (6.73) and Eqs. (6.91) to (6.93). This is useful since this velocity can easily be measured in the experiment.

7

The SIMLab Shock Tube Fa-cility

Chapter 6 introduced the shock tube as an alternative to explosive detonations when studying the dynamic response of flexible structures and fluid-structure interaction effects in blast environments. Shock tubes produce shock waves under controlled laboratory conditions, where the shock strength is determined by the initial conditions. It was therefore decided to establish such a test facility at SIMLab, NTNU. This chapter presents the premises and design of the SIMLab Shock Tube Facility (SSTF). The shock tube presented herein is developed for blast applications where the properties of a planar shock wave acting on a structure may be studied by placing a test object inside or at the end of the tube. Finally, two different camera models to be used in the 3D-DIC analyses are evaluated to ensure an accurate calibration of the mathematical relation between the target and image coordinates. The shock tube design presented in this chapter is also presented in the second paper published in International Journal of Protective Structures [209].

7.1 Introduction

Shock tube designs are typically specialized according to the application, and the literature reveals a rather widespread use of the shock tube as a research tool (see e.g. [123]). Tube length and internal cross-section shape and area are therefore determined by the particular application and, of course, the funds available. The test time, driving method and temporal distribution of the pressure are dependent on the driver and driven section lengths (see Section 6.4.1). Choice of tube internal cross-section geometry is typically influenced by the desired flow conditions, dimensions of the test specimens, and the type of instrumentation to be used for flow measurement.

Use of optical techniques (such as high-speed cameras and schlieren photog-raphy) is simplified with rectangular or square tube cross-sections which can

accommodate plane parallel windows. However, rectangular and square cross-sections are inferior to the circular cross-section when considering structural strength and general ease of construction and sealing [123]. The circular cross-section is preferable from a structural viewpoint, particularly for the driver section of the tube. This may be combined with a square driven section using a transition from a circular to a square shape at or downstream the diaphragm(s). For high-pressure applications, the structural limitations of the square cross-section may then be overcome by encasing it in a pipe with a practically incompressible material.

The European [210] or ASME standards for Unfired Pressure Vessels and traditional design methods for material strength ensure sufficient static strength of the tube sections. However, the most severe stress conditions arise from transient loadings during the operation of the tube. High-pressure loading of the low-pressure section(s) typically results from reflection of the incident shock wave (see Sections 6.4.1 and 6.4.4). Eq. (6.88) shows that the (initial) reflected overpressure to incident overpressure ratio increases steadily with shock strength tending asymptotically to the value 2 + (γ+ 1)/(γ−1) for very strong shocks. For air (γ= 1.4) the upper limit is 8, whereas for monatomic gases such as helium (γ = 5/3) the upper limit is 6. However, depending on the properties of the remaining right-running flow (illustrated in Figure 6.13a) additional compression and heating of the reflected shock wave may result in a further pressure increase of the order of 20 (see e.g. [123–125]). The transient stressing of the material usually reduces to an estimation of what is considered an adequate safety factor over the expected maximum static loadings. The influence of the recoil induced in the gas during diaphragm burst and momentum changes in the gas (when the gas comes to rest during reflections at the closed ends) depend on the tube diameter. Impulsive loads in the tube and supports which restrain axial movement may be neglected for small diameters, while limited recoil and axial movement may be desirable to avoid excessively transient stressing for larger diameters and high driver pressures. More detailed information on shock tube design may be found in Ref. [123].

Based on the general design and construction aspects discussed hitherto, the fol-lowing premises were established for the SIMLab Shock Tube Facility (SSTF):

• The overall design was limited by the dimensions of the location where the shock tube was installed. The height, width and length of the room are 3.0 m, 4.0 m and 23.5 m, respectively.

• The shock tube diameter depends on the dimension of the test specimens.

The design should enable mounting of test specimens either inside or at the end of the tube. A literature review [44,65,103,113,114,116,117,211,212]

7.2. Shock tube design 161