CFD analysis of a Shock Tube using CONVERGE
Shock Tube Analysis
Objective: To carry out a CFD analysis of the aerodynamics of the shockwaves inside a shock tube.
Figure 1: Shock Tube
Background: Compressed gas driven shock tubes were first introduced by French scientist Paul Vielle in 1899, yet it was not till 1940s that the apparatus was referred to as Shock Tube. Compressed-gas driven shock tubes range from rectangular to circular cross-sections in the range of 0.6 to 2 m and lengths between 3 to 15m. Compressed-gas driven shock tubes have lower peak pressures as opposed to blast-driven shock tubes and are thus more robustly used in laboratories.
Compressed-gas shock tubes have two sections separated by a diaphragm, a compressed higher-pressure driver section and a lower pressure driven section. When the diaphragm is ruptured or burst due the pressure differential or even ignited due to combustion gases, a shock is sent down the driven section in the form of shock waver while a rarefaction wave travels up the driver section. Thus, flow propagates from driver to driven section and shock and expansion waves oscillate between the tube, reflected off the walls, until ambient pressure steadies out in the tube and flow seizes. The shock tube herein consists of high-pressure nitrogen in the driver section and low-pressure oxygen in the driven section, separated by a diaphragm.
Shock tubes are utilized to study high velocity compressible flow phenomena of compressed-gas driven flows and even combustion kinematics of air-fuel mixtures. A shockwave inside a shock tube may be generated by the build of high pressure which may cause the dividing diaphragm to rupture and a shockwave to propagate down the shock-tube.
Data and Analysis
Nitrogen Concentration:
The animation below shows the nitrogen species concentration in the tube as the diaphragm ruptures and flow persists.
From the animation it can be observed that the breaking of the diaphragm corresponds to the propagation of a shock wave down the driven section. As flow persists, Nitrogen species begins to fill out the driven section. However, the propagated shockwave reflected off the wall interferes with the flow of nitrogen and prevents further mixing of species in what can be observed as recirculation zones. As rarefaction waves propagate up driver section, regions of lower pressure persist while regions of higher-pressure manifest in the shockwave driven section. This causes an adverse pressure gradient which creates flow reversal. The back-and-forth behavior persists, as shock and expansions waves reflect back-and-forth, and the oscillating pattern is observed.
Pressure:
The animation above is characteristic of the pressure profile inside a shock-tube and replicates pressure profile as expansion and shock waves oscillate back-and-forth within the tube. The diaphragm event is accompanied by propagation of shock wave down the driven section while rarefaction wave persists up the driven section. Thus, it can be observed that the pressure increases as the shock wave travels to the right and further superimposed when the wave reflects back. Towards the left the Prandtl Meyer wave causes the pressure to drop and further decline as the expansion waves reflects back. The two waves cross at some point in the driven section, and the pattern reverses. The reflected expansion waves, now propagating to the right, cause regions of reduced pressures and the reflected shockwaves propagating to the left cause regions of increased pressure in the driver section. The superimposition reoccurs as the waves reflect off the walls and once again switch propagation direction, and cross at some point in the driver section. Thereafter, the oscillating profile is set-up back-and-forth whilst the pressure-gradient persist, and the pressure oscillates between higher and lower magnitudes in both the driver and driven sections.
The plot below showcases the oscillation in pressure as regions of low pressure and high pressure develop in regions of oscillating expansion and shock waves, respectively.
Figure 2: Pressure Profile in the Shock Tube
Temperature:
Since shock waves are characterized by jump in temperatures and expansion waves are characterized by drop in temperature, a very similar profile to the pressure profile in observed in the temperature contours.
The animation depicts the same oscillating profile with reflected shock and expansions waves. The propagating shock waves in the driver section cause regions of increases temperatures. The wave reflects off the wall and superimpose to create regions of even higher temperatures in the driven section. Meanwhile, the propagating expansion waves creates regions of reduced temperature as it moves to the left, reflects of the wall and further reduces the temperature in the driven section. The two reflected waves cross-over and now switch propagation direction. Thus, regions of higher temperature now manifest in the driver section whereas the reflected expansion waves, now persistent to the right, create regions of reduced temperatures in the driven section. Thereby, the oscillating pattern in temperature is manifested and persists while driving pressure-gradient persists.
The plot below showcases the oscillation in temperature as regions of low temperature and high temperature develop in regions of oscillating expansion and shock waves, respectively.
Figure 3: Temperature Profile in Shock Tube
Velocity:
As depicted in the animations, we observe regions of oscillating high and low velocities. Initially as flow sets-up post diaphragm event, we observe increase in velocity in either direction. To the left we observe an increment in velocity wave post shock and to the right we observe decremental velocity wave or the shock wave. The shock wave reflects and causes region of reduced velocity to manifest in the driven section. The expansion wave reflects and causes region of high velocity to manifest in the driver section. The two reflected waves cross-over, and the pattern switches with the reflected shockwave now creating regions of low velocity to the left, while reflected expansions waves create regions of high velocity to the right. Thus, the oscillating velocity contour manifest as the two sets of waves reflect back and forth.
Adaptive Mesh Refinement and Cell Count:
Once the diaphragm is compromised, flow persists from driver to driven section causing the effective mixing of species. We utilize a CFD technique of adaptive mesh refinement in regions where a differential change in the species of nitrogen is detected. This creates a transient system, which auto refines the mesh and thus the cell count varies with time. The to fold transient change in refinement can be observed in the animation below.
Figure 4: Cell Count Over Time
Conclusion
We observe that within a shock-tube, once the diaphragm is compromised, both compressive shock waves and expansive Prandtl Meyer waves manifest. One in the flow-ward direction and the other up the driver section, respectively. The sets of waves reflect back and forth and associated characteristics of pressure, velocity, density, and temperature begin to manifest and oscillate, driven by the back-and-forth movement of the reflected waves.