NUMERICAL SIMULATIONS OF SHOCK FOCUSING AND SYRINGE STRESSES
5.1 Shock Focusing
5.1.2 Shock Focusing in a Straight Cone
The focusing of shock waves in a straight cone or a wedge has been thoroughly studied in the context of gasdynamics; some important work in this area has been performed here at Caltech, including the work ofBond et al. (2009);Russel (1967);
Setchell, Storm, and Sturtevant (1972). The monograph ofApazidis and Eliasson (2019)should be consulted for a thorough introduction to shock focusing and past
173 research in this field. In most experimental studies, a shock tube (i.e., the working fluid is a gas) was used to create a shock wave upstream of a straight cone or a wedge. The Mach number of the incident pressure wave was typically greater than 1.5, and the shock focusing effect resulted in the shock becoming moderate to strong. Nonlinear effects are important in those studies, and real gas effects were often observed. Most studies have investigated only one half-angle and have focused on understanding the process through which the shock wave is amplified as it propagates toward the apex of the cone rather than performing parametric studies.
One important finding confirmed by past studies is that the amplification mechanism of the pressure on the axis of the cone or wedge is not a continuous process: the shock is alternately strengthened by the repeated crossing of the Mach reflections across the diameter of the cone or wedge, a concept which is clearly discussed by Setchell, Storm, and Sturtevant (1972).
This process is partially illustrated in the right-hand side ofFigure 5.1for acoustic waves. The schematic illustrates the physical situation after the incident wave has entered the straight cone: a corner wave and a diffracted wave have formed. The corner wave results from the reflection of the incident wave on the oblique wall of the cone. Because this is an acoustic problem, the reflection of the incident wave on the wall is regular (Apazidis and Eliasson, 2019;Courant and Friedrichs, 1976): the angle between the wall and the incident wave is the same as the angle between the wall and the corner wave. The diffracted wave forms as a result of the wave turning the corner between the syringe barrel and the cone.
The corner and the diffracted waves converge toward the axis of symmetry of the syringe. Upon reaching the axis of symmetry they reflect and propagate toward the wall of the cone. This cycle is repeated and creates stepwise changes in pressure on the axis of symmetry. The convergence of the waves toward the axis means that some of the energy transported by the waves isfocused on the axis of symmetry;
the energy is concentrated into a volume of decreasing size. Although it is typical to think of pressure as a force per unit area (N/m2), it is also possible to think of pressure as an energy density (J/m3). It then becomes easier to understand the amplification mechanism: the focusing of the energy transported by the pressure waves on the axis of symmetry increases the energy density. This, in turn, translates into a substantial increase in liquid pressure.
There are important differences between the present and previous studies on shock focusing. In the present work, the pressure waves entering the cone are acoustic
(the Mach number is approximately 1) rather than moderate to strong shocks, and the waves propagate in liquid water rather than a gas. The study aims at quantifying the magnitude of pressure wave amplification at the tip of the cone as a function of the half-angle of the cone, the rise time of the pressure pulse, and the acoustic impedance of the cone.
The rise time of the pressure pulse is a parameter which could influence the mag- nitude of amplification within the cone. Pressure pulses with a slow rise time are expected to experience only a small amount of amplification within the cone. The preliminary study ofJohnson et al. (2014)on the focusing of stress waves in a cone has shown that stress waves with an increasing ramp tend to be less amplified than top-hat types of stress waves. However, only one slope for the ramp was investigated.
In the context of autoinjector devices, it is important to quantify the effect of the rise time of the pressure pulse on the focusing effect because the rise time of pulses created in syringes is largely affected by the size of the air gap located between the plunger-stopper and the liquid, as described in Chapters3and4.
Eliasson et al. (2010) and Wang and Eliasson (2012) performed an experimen- tal study on the focusing of shock waves inside structures with varying acoustic impedance. The geometries, a double wedge and a logarithmic spiral, were filled with water. Both studies indicate the focusing effect is affected by the acoustic impedance of the cone or spiral, but the effect of the acoustic impedance on the peak pressure was not quantified. The acoustic impedance of the confining geom- etry affects the proportion of the incident wave which is reflected in the liquid and transmitted in the solid. Physical intuition suggests that cones with rigid walls rather than soft walls could create more amplification of the pressure. In the context of autoinjector devices, this has important implication for glass and plastic syringes, two materials with dissimilar acoustic impedances. The acoustic impedance of glass syringes is approximately 4.7 times larger than the acoustic impedance of plastic syringes, suggesting the focusing effect could be significantly different in glass and plastic syringes.
The shock focusing process in a straight cone filled with water is illustrated in Figure 5.2, a sequence of images obtained using LS-DYNA. The results shown are for a half-angle of 30 degrees, an infinitely sharp pressure wave, and a rigid cone.
The left-hand side is a pseudo-schlieren which makes it possible to track the pressure waves in the cone. The right-hand side is the pressure field. Note that the incident pressure wave, which is initially infinitely sharp, does not appear to be infinitely
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Figure 5.2: Amplification of a sharp pressure wave entering a straight, rigid cone with a half-angle of 30 degrees. The left-hand side of each frame is a pseudo- schlieren, and the right-hand side is the pressure field.
sharp inFigure 5.2, but instead has a finite thickness and rise time. This results from numerical dissipation due to the low order of the numerical methods in LS-DYNA.
Initially, the pressure wave is above the corner located at the junction between the syringe barrel and the cone. The pressure ahead of the pressure wave is 0 MPa, and the pressure behind is 1 MPa. After 5.0 µs, the incident pressure wave enters the cone, and a corner wave and a diffracted wave start forming. The propagation of those waves is not purely in the axial direction. The radial component of motion makes the corner wave and the diffracted wave reverberate in the radial direction within the cone.
0 10 20 30 40 50 60 0
2 4 6 8 10 12
Figure 5.3: Pressure history on the cone axis and the tip wall for a sharp pressure wave entering a straight, rigid cone with a half-angle of 30 degrees.
From 5.0µs to 15.0µs the corner wave and the diffracted wave converge on the axis of symmetry. The magnitude of those waves increases as this happens, resulting in a larger pressure increase upon their passage. After 17.5µs the corner wave and the diffracted wave reach the centerline of the cone and reflect. The corner wave then consists of two approximately straight segments. The radial component of motion of those two segments is in opposite directions. The cone has the effect of "folding"
the incident wave on itself within the cone.
The axial progression of the incident and the corner waves toward the tip of the cone is uninterrupted. The pressure waves are focused into the tip between 27.5 µs and 30 µs, resulting in a peak pressure in the tip that is approximately 12.5 MPa, or 12.5 times the initial magnitude of the incident wave. Following this, the pressure waves still exhibit a radial motion, but the axial direction of travel is reversed. The pressure waves propagate away from the tip, toward the top of the syringe, and the pressure in the tip decays.
The evolution of pressure at 4 stations located on the axis of symmetry is shown in Figure 5.3. The stations are positioned atz =0dtip (at the tip wall) and upstream of the tip wall atz= 2, 4 and 6 tip diameters. The results indicate the pressure increase
177 on the axis of symmetry is not continuous, as expected: a first pressure increase occurs upon arrival of the incident wave, and a second pressure increase occurs when the corner wave converges on the axis of symmetry. This is the same behavior which was observed in previous research studies (Bond et al., 2009;Setchell, Storm, and Sturtevant, 1972). The pressure history also indicates the pressure in the vicinity of the tip does not remain large for a very long time: the pressure is greater than 5 MPa for approximately 5.4µs, and above 1 MPa, the magnitude of the incident wave, for approximately 33 µs.