The problem

The FNGUN1D software employs a one-dimensional, two-phase flow, finite difference solver, coupled to a numerical representation of a grain's geometry throughout burning. These features permit physical effects to be modelled more rapidly than full 2D codes. FNGUN2D is developed and maintained by Frazer-Nash in conjunction with QinetiQ. It incorporates QinetiQ's QIMIBS 2D internal ballistics solver which has been extensively validated against a wide range of internal ballistics problems [1, 2].

In both codes, the internal ballistics cycle is represented by a two-phase flow comprising of a gas phase and a solid phase, representing the propellant. The combustion of propellant is represented by a mass transfer between the solid and gas phases. In principle there can be many propellants and gas species present in the internal ballistics cycle each represented by its own phase and conservation equations. However, here the project only considered a single gas and a single propellant phase in order to simplify matters.

The objective of this work was to analyse the two codes FNGUN1D and FNGUN2D to understand mathematically their relative benefits, their reliability and the differences between them.

The approach

Linear stability analysis was used to find the susceptibility of the 1D solutions to perturbations that break the axisymmetry. However, the equations solved in FNGUN1D and FNGUN2D differed by more than just the addition of an extra spatial dimension which meant the solutions obtained in FNGUN1D were not exact solutions to the reduced 2D equations. Therefore any insight into the stability of the 1D solution could only be taken as an indication of the true behaviour of a 1D solution to the 2D equations.

Whilst the 1D solution may be technically unstable, it could still be modelled accurately by a 1D solver. It was important to study the directions which each instability grew to see if the instability caused a significant radial variation. The 1D radial symmetry can be broken through the radial velocity differing from zero. Therefore, to asses the stability of the 1D solutions, small perturbation were introduced which reduced the original nonlinear two-phase flow equations to a system of linear PDEs for the unknown perturbed variables, ignoring the spatial dependence of the instability.

Multiple copies of the system of linear PDEs were solved at each grid point in space and time of the original 1D simulation. At each point, solutions which lead to an eigenvalue problem were sought, specifically, eigenvalues with real positive parts which correspond to an instability. The solution of the eigenvalue problem was a 9-dimensional eigenvalue problem and could be solved at run time once the solution to the PDE had been obtained at the corresponding grid point. Hence each grid point could be coloured according to an appropriate measure of the growth rate of the radially-significant eigenmodes.

Figure 1 shows the results to a test run. The top panel shows a colour map of the radial instabilities. Here, blue represents weak instabilities and warm colours represent strong instabilities. White regions represent no radial instability. These data are compared in the two lower panels of Figure 1 with the radial velocity components of the corresponding 2D simulations. In this test case, the radial instabilities remain relatively small and there is little radial velocity in the 2D simulations.

Figure 2 shows the results to a second test run which has the same simple rectangular geometry and differs only in the spatial location of the initial propellant. In this case there are more ``warm'' regions in the stability diagram and the maximum eigenvalue is 3 times bigger. The 2D simulations can be seen to generate significant ``hotspots'' of high radial velocity, which convect down the domain. When the time-varying pressure profiles of the 1D and 2D simulations are compared for this test case the differences are apparent. The overall pressure is predicted well by the 1D code in both cases. However, significant differences arise for case 2 where the 1D code generally speaking over-predicts the differential pressures.

Figure 1: Regions of weak stability only give rise to very small radial instabilities that do not appreciably affect the flow. 1D and 2D models give good agreement.	Figure 2: Large regions with rapid instability growth give rise to large radial velocities and appreciable difference between 1D and 2D models.

References

[1] Woodley, C., Finbow, D., Titarev, V. and Toro, E. Two-Dimensional Modelling of Mortar Internal Ballistics, 22nd International Symposium on Ballistics, 2005.
[2] Woodley, C. and Fuller, S. Two-dimensional modelling of modular charge gun firings, 24th International Symposium on Ballistics, Sept 2008.

related resources:

	Comparison of the complexity, fidelity and cost of internal ballistics models
»	Technical summary

other projects: