Shock Tube simulation in Converge CFD

Objective:

To set up a transient shock tube simulation in Converge CFD and to post-process the results in Paraview.

Theory- Shock tube

A shock tube is a device used to generate shocks. This device is used in experiments for determining the ignition delay of fuel when subjected to the auto-ignition temperature. The generated shock travels back and forth raising the temperature of the pre-mixed fuel. When the temperature reaches the auto-ignition temperature of the fuel, the time that the fuel uses to undergo combustion is measured.

The shock tube consists of two regions, namely, the high-pressure region and the low-pressure region. The high-pressure region contains the driver gas and the low-pressure region contains the test gas (premixed fuel). These two regions are separated by a diaphragm with the required tensile strength. On the high-pressure side, the pressure is steadily increased until it reaches the required pressure at which point the diaphragm breaks and the generated shock travels back and forth inside the shock-tube raising the temperature of the fuel to the auto-ignition temperature.

The experimental setup is as shown in the picture below;

Problem Definition

It is required to simulate the shock wave that is generated inside the shock-tube. The high-pressure chamber contains nitrogen gas. The low-pressure chamber contains oxygen at 1 atm. The diaphragm breaks when the pressure inside the high-pressure chamber reaches 6 bar.

Since it is difficult to simulate the actual breaking of the diaphragm, we are going to directly assign the required pressure (at which diaphragm breaks) to the high-pressure region and connect both regions only after a certain time with the \'Events\' feature in Converge Studio in order to simulate the breaking diaphragm.

Case setup

Geometry

The geometry is a cylinder with two chambers for high-pressure and low-pressure as shown in the picture below;

Mesh

The grid size is chosen as 0.0005 m with the total number of cells for 2D geometry as 8060. A species (N2)-based Adaptive Mesh Refinement feature is enabled with a maximum embed level of 3 and with the SGS parameter as 0.001. The below video shows how the mesh is generated for each time step.

The region with a high intensity of cells inside the shock-tube in the above animation represents a steep change in the amount of Nitrogen. In other words, this region is where the mixing of N2 and O2 is taking place.

Initial and Boundary Conditions

There are two regions. The high-pressure region containing N2 is initialized with a pressure of 6 bar and the low-pressure region containing O2 is initialized with the atmospheric pressure. The temperature in both regions is 300 K.

All the boundaries except the front and back ones are specified as \'wall\' with the \'law of wall\' boundary condition. The front and back boundaries are specified as 2D since this is a 2D simulation.

End time

The end time for this transient simulation is chosen as 0.003 s.

Turbulence model

We\'re using the RNG K-Epsilon model with the default parameters.

Simulation Results

The following video shows the distribution of Nitrogen as well as x_component of velocity inside the shock tube with time;

In the simulation that shows the variation of Velocity_x with time, the shock wave can be clearly seen propagating to and fro inside the shock tube. The other simulation that shows the distribution of Nitrogen over time shows the degree of mixing taking place inside the shock tube. The temperature and pressure variation with time inside the shock tube is shown in the animation below;

The variation in the flow variables with time can be presented in the form of graphs for a clearer understanding. 

Variation of Pressure with time

The above graph shows the variation of pressure with time in both regions as well as the entire domain as a whole. It can be noted that there is no change occurring in pressure until 0.001 s because both the high-pressure and low-pressure regions remain disconnected until 0.001 s. After 0.001 s, it appears that the pressure at the HP region is decreasing, but in the LP region, it is increasing. This is because the high-pressure and low-pressure regions are connected after 0.001 s and the large difference in their pressure creates a shockwave in the low-pressure region moving away from the high-pressure region. Therefore, the high-pressure region encounters expansion while the low-pressure region encounters compression.

After some time, however, the trend reverses, the high-pressure region increases in pressure while the low-pressure region decreases. This trend again reverses after some time. This is because the shock wave is getting reflected back and forth inside the shock tube.

It can also be noticed that as the shock wave is reflected back and forth, the amplitude of pressure variation decreases with time. This is because the shock tube is closed on all sides and there are no outside forces acting on it. Therefore, with time, a steady-state (equilibrium) is reached inside the shock tube.

Variation of Temperature with time

The above graph shows the variation of temperature with time in both regions as well as the entire domain as a whole. It can be noted that there is a decrease in temperature in the high-pressure region and an increase in the low-pressure region. This is because, as the shock wave is moving away from the HP region towards the LP region, there is an expansion of nitrogen occurring inside the HP region and compression of Oxygen in the LP region. And since the shock wave is moving back and forth, the trend is alternating and with time will reach an equilibrium.

Cell count variation with time

As it can be observed in the graph above, the total cell count is changing with time. This is due to the Adaptive Mesh Refinement feature in Converge. Due to the occurrence of steep gradients in the shock wave regions, more cells of smaller sizes are embedded to improve the resolution in those regions. This improves the accuracy in locating the shock.