High-Fidelity Model

A.**Basic model:**
For SRBs with high length-to-diameter ratios a very good approximation to the internal ballistics can be obtained using a modified model based on stochastic partial differential equations (SPDEs) for momentum, energy, and mass of combustion products averaged over a cross-section area

that takes into account fluctuations in gas dynamics arising mainly due to the propellant density variations, the burning rate, a graded propellant performance, and external noise,

These equations for continuity of the mass, momentum and energy of the gas flow in the combustion chamber have to be extended by including equations for dynamics of the burn distance, nozzle ablation, and melting, flame erosion, and burning of the metal walls. In the simplest case of the propellant burning law and Bartz. approximation for the heat transfer between the gas flow and walls of the nozzle and the hole in the metal case we have the following equations for the burn distance

nozzle throat ablation

and melting of the metal walls of the hole in the rocket case.

It is assumed that the equation of state for an ideal gas holds in the combustion chamber,

To incorporate real propellant geometry the following key assumption is introduced: at every moment of time the burning area Sb is determined by the burn distance Rp and the corresponding design curve

The system (1)-(7) is completed by adding equation for the nozzle exit ablation

and equation for the nozzle FN and case breach fault-induced Fh thrusts

The equations above represent a basic model of the case breach fault that incorporates essential dynamical features of the fault-induced changes in the internal ballistics of the gas flow. The value is not controlled during a flight of the rocket. Therefore, we have to solve an inverse problem: to find the hidden variable using data of sensors of head gas pressure and temperature in combustion chamber and to predict its probable dynamics. We solve this inverse problem based on the Bayesian inferential framework.

**B. Validation of the basic model using FLUENT simulations**
To validate the model of the case breach fault (see equations above) introduced above we build a FLUENT model of the case breach (see Figure 5). Next we notice that of the propellant surface, metal flame erosion, and nozzle ablation are relatively slow processes as compared to the time scale for redistribution of gas parameters along the combustion chamber (typically trel . L/c . 10 msec). Under these conditions it becomes possible to find stationary solutions of the Eqs. (1) analytically in the combustion chamber

Figure 5. (left) Velocity distribution obtained using FLUENT simulations after 0.14 sec. The geometry of the model surfaces is shown in the figure. Note that the hole wall, propellant surface wall, and the nozzle wall are deforming according to the equations (3)-(5), (8). (right) Velocity distribution generated by the FLUENT model for t = 5.64 sec. Note the changes in the geometry of the rocket walls and the corresponding changes in the velocity distribution.

and in the nozzle area

where M(SUB>0 is given by the solution of the nozzle equation

The results of the comparison of the analytical distributions obtained with the axial velocity and pressure distributions obtained using FLUENT simulations are shown in the Figure 6 and Figure 7. It can be seen from these figures that the model provides a very good approximation to the results of FLUENT simulations.

*Click to enlarge images above*

Figure 6 Axial velocity (left) and pressure (right) profiles generated by the FLUENT model for t = 0.05 sec (black solid line) as compared to the analytical solutions (red dashed lines) given by the equations above.

Note that the difference in the time scales for dynamics of burn distance, metal flame erosion, and nozzle ablation as compared to the characteristic relaxation time of the distributions to their quasi-stationary values trel allows us to integrate equations (1)-(13) in quasi-stationary approximation. As a result we obtain analytical solution for the quasi-stationary dynamics of the axial distributions of the gas parameters in the combustion chamber and in the nozzle area. The comparison of this analytical solution with the results of FLUENT simulations also demonstrates good agreement between the theory and numerical solution of the high-fidelity model.

Figure 7. (left) Results of the FLUENT calculation of the pressure distribution in the nozzle under conditions adopted for the HFPM. (right) The comparison of the axial values of the nozzle pressure (red line) calculated using FLUENT and the analytical results (black line) of this section.

A further validation and verification of the case breach model (1)-(10) can be obtained using results of the ground firing test as will be described in details elsewhere. Here we will discuss necessary modifications of the basic case breach model that have to be introduced to explain experimentally observed deviations of the case breach dynamics from the predictions based on the model (1)-(10).