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This paper presents a new formulation and computational solution of an optimal control problem concerning unsteady shock wave attenuation. The adjoint system of equations for the unsteady Euler system in 1D is derived and utilized in an adjoint-based solution procedure for the optimal control. A novel algorithm is used to solve for the optimal control solution that satisfies all necessary first order optimality conditions while locally minimizing an appropriate cost functional. The solution procedure is sufficiently flexible such that it can be used to solve other distributed optimal control problems where the state dynamics are in the form of non-linear hyperbolic systems of partial differential equations and where the initial data is given and the final data is free at a free final time. Distributed control solutions with certain physical constraints are calculated for attenuating blast waves similar to to those generated by Ignition Over Pressure (IOP) from the Shuttle's Solid Rocket Booster during launch. The control solutions give insight to the magnitude and location of energy dissipation necessary to decrease a given blast wave's overpressure to a set target level over a given spatial domain while using only as much control as needed.

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This paper presents the first solution of an optimal control problem concerning unsteady blast wave attenuation where the control takes the form of the initial distribution of liquid water droplets. An appropriate two-phase flow model is adopted for compressible homogeneous two-phase flows. The dynamical system includes an empirical model for water droplet vaporization, the dominant mechanism for attenuating the jump in pressure across the shock front. At the end of the simulation interval, an appropriate target state is defined such that the jump in pressure of the target state is less than that of the simulated blast wave. Given the nature of the non-linear system, the final time must also be a free variable. A novel control algorithm is presented which can satisfy all necessary conditions of the optimal system and avoid taking a variation at the shock front. The adjoint-based method is applied to NASA's problem of Ignition Overpressure blast waves generated during ignition of solid grain rocket segments on launch vehicles. Results are shown for a range of blast waves that are plausible to see in the launch environment of the shuttle. Significant parameters of effective droplet distributions are identified. INTRODUCTIONThere exists a wide range of multi-phase and multi-fluid flow regimes. Much care must be taken in selecting an appropriate model and numeric method. Presently, the flow of interest consists of a gaseous carrier phase with many suspended water droplets. These two fluids' parameters are drastically different and the vast majority of numeric methods in the literature are ill-adapted since they introduce artificial diffusion and mixing at interfaces [1][2][3]. Other methods are poorly suited to mixtures where thermodynamic equilibrium cannot be reasonably assumed except at interfaces. Since the water droplets of interest will be smaller than 1 mm, their properties can be assumed to be homogeneous between computational cells and, therefore, many interfaces will be present in each cell. It will be impractical to use interface tracking methods or solve separate sets of equations for each fluid and to couple their properties at the interface. Instead, the following calculations implement a method proposed by [4] which has the advantage of solving the same set of differential equations throughout the domain. The method uses different equations of state for each phase and allows a difference in temperature at the gas-liquid interfaces. To close the system, a volume fraction variable α is introduced and a non-conservative PDE is appended to the conservative system. The method assumes that both phases are compressible which yields a hyperbolic system. Ignition overpressure (IOP) is a phenomenon present at the start of an ignition sequence in launch vehicles using solid-grain propellants. When the grain is ignited the pressure inside the combustion chamber quickly rises several orders of magnitude. This drives hot combustion products toward the nozzle and out to the open atmosphere at supersonic spee...

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