Shock Tube Simulation

Shock Tube Simulation:

A shock tube simulation case is setup with a high pressure region containing Nitrogen and a low pressure region containing oxygen.

Shock Tube:

The shock tube is a device in which a plane propagating shock is produced in a long tube by
the abrupt rupturing of the diaphragm separating one section of the tube at higher pressure than that in the other section.

The shock tube is a long tube of uniform cross section and with uniform internal dimensions. The diaphragm separates the high pressure driver section from the low pressure driven or test section. The material of the diaphragm and its thickness are dictated by the pressure ratio between the sections. On abrupt rupturing of the diaphragm, pressure waves emanating from the diaphragm station coalesce to form the shock front which propagates in to the low pressure section. This shock wave increases the temperature and pressure of the driven gas and induces a flow in the direction of the shock wave but at lower velocity than the lead wave. Simultaneously, a rarefraction wave often reffered to as Prandtl-Meyer expansion wave travels back into the driver gas. Reflected shocks cancle the motion of the driven fluid initiated by the primary shock thus, providing more resistance against mixing.

Following is the diagram explaining differnet regions in the shock tube after the diaphram bursts.

1,2,3,4 numbers in the above image describe the diferent regions in the shock tube. Region 1 represents the initial conditions in the driven section. The Region 2 corresponds to the shocked gas conditions. The region between the contact surface and the expansion fan is referred to as 3. Region 4 represents the conditions of the initial high pressure driver gas.

Follwion is the geometry of the shock tube.

The triangles are colored by regions, the green represents the high pressure region and the brown represents the low pressure region.

Boundary and Initial Condition:

There is no inflow or out-flow boundary condition. The side wall are defined as 2-D and the upper and lower boundary were given wall boundary condition.

The pressure difference of 6 bar is set and initial temperature of 300k is set for both the rgions.

In simulation we don\'t have a real diaphram nor can we pop the diaphram hence we creat events for this purpose we define the time at which both the high and low pressure region are closed and when they are open.

Following is the image of the events parameters:

Both the regions are closed initially and then they are open at time = 0.001 seconds and stay open for rest of the time.

Case setup Paramaters:

• A transient simulation is being run this setup.
• The time for simulation is 0.005 seconds.
• Base Grid size of 0.001 mm is used.
• A species based adaptive mesh refinement is used in this setup for both regions. N2 is used as the species for this with a sgs of 0.001 and the type of AMR is permanent.
• For turbulance modelling RNG k-Epsilon is used.

Results:

Following are the plots comparing the pressure, temperature and density in the high and low pressure region after the regions are open.

Pressure plot:

Temperature Plot:

Density Plot:

Observation:

• As soon as the regions are opened there is a sudden increase in the low pressure region and a sudden decrease in the high pressure region.
• After some time the pressure rises for the high pressure region and decreases for the low pressure region.
• It can be seen that there is this continious rise and fall in pressure in low pressure region and a continious fall and rise in the high pressure region.
• Pressure differences are the driving force for the formation of shocks, so as the pressure difference reduces, the magnitude of the wave front generated will be low, thus leading to the gradual reduction in the amplitude of the fluctuations as time progresses.
• As the time pregresses the amplitude of the of the change pressure wave decreases and if the simultion is ran for a longer period of time it can be shown that the pressure wave would eventualy dies down.
• The above mentioned observation is made upon the following idea:

In the form of waves, energy gets transferred with the goal of attaining equilibrium. Shocks are formed due to the coalesce of the waves during the process of attaining equilibrium.

• The same observations can be seen for the temperature and density and these observations are accurate as accoring to the ideal gas law pressure, temperature and density are all praportional to each other. Hence, there is a similarity in the increase and decrease of the graphs of these.
• As pressure decreases for the high pressure region so does the temperature and density.

Cell Count:

Following is the image for the cell count for this simulation.

Cell count starts to increase after the regions are open and the AMR is set into play.

The sub-grid criterion value of 0.001 is the mimimum mass fraction of the selected species that will trigger the AMR. So the Species based AMR refines the mesh in the N2 filled region at the places where sudden change in the properties exist.

It can be clearly seen in the following animation created. Showing the mass fraction of N2.

The following is the animation with grids for the mass fraction of N2 and the increase in cell count can be clearly seen w.r.t species mass fraction of N2 due to species based AMR.

The process of generation of shock and expansion waves continues till the pressure difference in the domain dies out, therefore AMR option keeps on refining the mesh in the new regions of wave front generation, which results in increasing no. of cells in the domain.

Following is the animation for Temperature.

Animation for Velocity:

Animation for pressure:

As we can see from the mass fraction animation that due to the generation and reflection of the shocks from the walls creats a recirculation region and that prevents the mixing of the two species.