On the partially reacted boundary layer in rate sticks

Using our temperature dependent reactive flow model (TDRR) to simulate detonation in a rate stick, we observe that a partially reacted layer (PRL) is formed near the boundary. We are not aware that such a PRL has been observed in tests, and this is why we regarded it in the past as a numerical artifact. Assuming that such an artefact may be caused by the finite rise time of the detonation shock, we showed in [1] how it can be eliminated by delaying the outward boundary motion for a length of time comparable with the shock rise time. Here we revisit the PRL problem. We first show that it is not a numerical artifact but a real phenomenon. We do this by repeating the reactive flow run with a finer mesh. By looking at the PRL structure, we see that doubling the resolution affects the PRL only slightly. We then conjecture that the PRL formation has to do with the finite duration of the reaction process (or the finite extent of the reaction zone). By the time the boundary rarefaction reaches a cell near the boundary, it may be only partially reacted, and its reaction may therefore be cut off. To establish our conjecture we show how the PRL structure changes with the reaction duration.


Introduction
When we simulate a rate stick test with our Temperature Dependent Reaction Rate (TDRR) reactive flow model, we usually get that a thin layer near the boundary is only partially reacted. Using a pressure dependent reaction rate like Lee-Tarver Ignition and Growth model [2], we get an even more pronounced partially reacted layer near the boundary. We're not aware that such a partially reacted layer (PRL) has been observed experimentally. We therefore regarded it in the past as a numerical artifact. In [1] we proposed a way to eliminate the PRL from the simulations. We assumed there that the PRL is formed by the finite rise time of the detonation shock. Artificial viscosity usually smears the shock over a length of three mesh cells. By the time the rarefaction wave from the free boundary reaches the first and second cells near the boundary, the shock there has not fully built up yet, and the reaction there is quenched before it has a chance to develop. But in reality, this does not happen, as the shock rise time is of the order of a nanosecond. To eliminate such an artifact we proposed in [1] to artificially delay the boundary motion in the direction normal to itself for the shock rise time duration.
But recently we repeated these rate stick simulations with finer meshes and realized that the PRL changes only slightly, even though the shock rise time was much shorter. We concluded that: 1) the PRL is probably real and not a numerical artifact, and that there is no point in eliminating it artificially; and 2) some other mechanism is responsible for forming the PRL.
In what follows we do rate stick simulations and show that the PRL is mainly an outcome of the reaction zone being of finite length (or finite duration). We also evaluate and show the structure of the PRL.

Simulations
The rate stick in the simulations is made of IHE and has a diameter of 16 mm. We monitor the detonation front at 60 mm from the initiation boundary (almost four stick diameters), where the flow is about stationary. We use our reactive flow model [3] with two mesh resolutions, 10 and 20 cells per mm. The reaction zone length of the explosive is about 1 mm so that the mesh resolutions used have enough cells in the reaction zone to resolve the reaction zone structure. In figures 1 and 2 we show maps with lines of equal W (reaction progress parameter) from runs with the two resolutions, at 10 microseconds after initiation.  From figures 1 and 2 we see that a PRL was formed in both runs. We can also see that for the higher resolution run, the width of the PRL is only slightly narrower.
To get a quantitative perception of the structure of the PRL, we show in figures 3 to 5 the maxima of pressure, reactant temperature and reaction parameter from the two runs, along the radial direction, at 60 mm down the stick. From figures 3 to 5 we see that for a finer mesh, the PRL at the boundary becomes narrower, but only slightly. From this and from additional runs not shown here we conclude that:  The detonation shock rise time is not the main cause for the formation of the PRL.  The main cause is probably the finite duration of the reaction. If the release wave from the boundary arrives at a certain cell before the reaction has time to develop to a significant extent, the reaction there may be quenched down.  As mentioned above, we're not aware that a PRL has been observed in a rate stick test, or in any other detonation configuration. But nevertheless we think that there is indirect evidence of its existence. In rate stick tests, the usual procedure is to photograph the shock front breakout from the far end of the stick, with a smear camera, along a diameter. From such a breakout curve one tries, among other things, to extract the boundary angle, which is important for DSD analysis. But usually, the edge of the breakout curve is somewhat blurred, and the extracted boundary angle has a sizeable uncertainty. As a result, we find in the literature different values for the boundary angle of the same explosive. We suggest here that this uncertainty in the extracted boundary angle has to do with the existence of a PRL near the boundary.
To show that the PRL near the boundary is caused by the finite duration of the reaction, we made additional runs in which we artificially changed the duration of the reaction. The reaction rate in our reactive flow model TDRR is given by [3]: where y(W) is the Burn Topology Function, T s is the reactant temperature, and R T , T * are material parameters calibrated from tests. We changed the reaction duration by changing the coefficient R T . Increasing R T by some factor decreases the reaction duration by the same factor.
In figures 6 and 7 we show the reaction progress parameter maxima along the radial direction near the boundary for three values of R T , and for the two mesh resolutions.
From figures 6 and 7 we see that: 1) The PRL is thicker for longer reaction duration; 2) The relation between the reaction duration and the PRL is nonlinear; 3) When R T decreases by 25% (not shown here), the reaction is extinguished; 4) From the nominal (middle) curve it can be seen that: in the last cell near the boundary there is no reaction, in the second cell from the boundary there is a minimal reaction, and in the third cell from the boundary the extent of reaction is around 10%; 5) The PRL thickness from the run with the finer mesh and with the nominal value of R T is only smaller by about 0.07 mm, which is less than one tenth of the reaction zone length of the explosive. We did not perform a systematical convergence study. We

Summary
When simulating detonation in a rate stick with a reactive flow model, we get a partially reacted layer (PRL) near the boundary. In the past we regarded this PRL as a numerical artefact, and proposed means to eliminate it. Recently we realized that the PRL obtained in simulations is not a numerical artefact, because when performing the same simulations with a finer mesh we get that the PRL changes only slightly. From the simulations we conclude that the PRL near the boundary is caused by the finiteness of the reaction zone, or by the finite duration of the reaction. By the time a cell near the boundary is released by the rarefaction wave from the boundary, the reaction there may not have yet advanced far enough before it is extinguished.