Shock Tube Simulation using Converge

The purpose of this project, is to setup and analyze a transient shock tube simulation using Converge Studio. The project will analyze how the interaction of the original shock waves and the reflected waves affect the fluid in the tube, and how the computational mesh has to be set in order to capture these high gradient effects.

Introduction

A shock tube is an instrument widely employed to analyze chemical kinetics. Shock tubes were originally created to replicate and direct blast waves at a sensor or a model in order to simulate actual explosions and their effects, usually on a smaller scale. However, it is currently used for other many applications, such as the analysis of aerodynamic flow under a wide range of temperatures and pressures (difficult to obtain otherwise) and to investigate compressible flow phenomena and gas phase combustion reactions (air-gas ignition delay).

A shock tube consist of two regions, a high-pressure area known as the driver section, and a lower-pressure area known as the driven section. Initially, both zones are separated by a diaphragm to avoid mixture. At a certain moment, the diaphragm bursts (there are several methods to produce this burst) and the fluid from the driver zone rush towards the low-pressure zone. As a result, compression waves are generated towars the low-pressure area, raising both pressure and temperature of the fluid, while expansion waves are generated in the driver zone. But at a certain moment, waves are reflected at the end of the tube, and then the interaction between the initial waves and the reflected waves occurs. 

This experiment is oftenly used to study the ignition delay of a fuel-air mixture, as it relies on the same principle where the generation of shocks waves lead to the increase in temperature and pressure until the ignition starts. 

Problem Setup

The geometry of the tube is imported from a STL file. The geometry consists on a simple 3D rectangular tube, in which the diaphragm is imposed by dividing the geometry into two regions, and allowing them to interact only at a certain starting time (simulating the burst of the diaprhagm).

The high pressure zone is filled with N2 at 6 bar, while the left side contains O2 at atmospheric pressure (and both at atmospheric temperature). Both zones are disconected until the sequential event takes place at 1 ms, where the gas starts flowing from one zone to the other. The boundary conditions are set as follows. Front and back walls are stated as 2D, while the BC for top and bottom walls are No-Slip for the Mometum equation, and Zero Normal Gradient for the Energy equation. The transient simulation is run for 3 ms, and an adaptative mesh refinement (AMR) is used to capture the high gradient phenomena at the fluid front. The mesh refinement is carried out by tracking the change in N2 concentration between cells, and applying a SGS embedding of 0.001. The turbulence model employed is RNG k-eps, and the output files are written every 0.01 ms in order to clearly distinguish the phenomena.

Mesh and Cell Count

As in any transient simulation, the mesh is somehow changed in each time step. In this case, the mesh size is recalculated depending on the gradient of the interest variable within the flow front. These images below show the progression of the mesh within the simulation, due to the AMR explained above.

t=0

t=1.05 ms 

Shortly after the motion starts

t= 1.5 ms

Detail of the mesh at the fluids border. At this time, the shock wave hasn't been reflected yet.

t= 2.5 ms

After the shock reflects, and the two set of waves interact, the problem is on track of reaching a steady state. That is why the refinement of the mesh is not as neccesary as before, as we can see in this picture.

In the following picture, we can see the distribution of the computational cells within the simulation time. As expected, the initial rise is obvisouly once the two regions are set to interact, at 1ms. At this time appears what has been called as a fluid front, this is, the part of the two gases that interacts. Note that this rise is maintained until the shock waves interacts with each other, and the problems tends to a steady situation.

For reference, just be aware that the cell size has been further refined and the boundary conditions have been changed from the model case, that is why an steady state trend happens here, and not in the model case. This decision was based on similar industry papers and researchers. 

Pressure and Temperature Variation at the tube:

Pressure at the regions:

Overall Mean Pressure:

Temperature at the regions:

Overall Mean Temperature:

  • When the simulation starts, there is a 5 bar pressure difference between the two regions. At t=1ms, the diaphragm suddenly bursts, and as a result, a compression wave travels to the low-pressure region, while an expansion wave travels to the high-pressure region. As a result, the pressure difference between the two regions suddenly drops. However, the respective reflected waves cancel the incoming shocks, and thus pressure difference between the two zones rises again due to the recirculation of the gases at the front. This entire phenomena keeps repeating itself, but everytime the magnitude of this pressure difference is going to be lower and lower. As a result, the shocks eventually stabalize, and that is why we can see a clear trend to a steady state value after only a couple cycles.
  • A similar phenomena occurs with the temperature of the gases. At the beggining, the expansion of the driver gas (N2) creates an important decrease in its temperature, and the other way around for the driven gas (O2). After the waves reflection, the recirulation zone creates again a temperature change in the two zones. Let's note that the time for the temperature equilibrium is expected to be longer than the motion equilibrium, as we can see that there is still an important temperature difference between the fluids, where as the pressures are reaching the equlibrium faster.

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