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No AccessFluid Dynamics

Method for Optimally Controlling Unsteady Shock Strength in One Dimension

Published Online:https://doi.org/10.2514/1.J051924

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 one dimension is derived and used in an adjoint-based solution procedure for the optimal control. A novel algorithm is used to satisfy all necessary first-order optimality conditions while locally minimizing an appropriate cost functional. Distributed control solutions with certain physical constraints are calculated for attenuating blast waves similar to those generated by ignition overpressure from the shuttle’s solid rocket booster during launch. Results are presented for attenuating shocks traveling at Mach 1.5 and 3.5 down to 85%, 80%, and 75% of the uncontrolled wave’s driving pressure. The control solutions give insight into 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. The solution procedure is sufficiently flexible such that it can be used to solve other optimal control problems constrained by partial differential equations that admit discontinuities and have fixed initial data and free final data at a free final time.

References

  • [1] Sritharan S. S. (ed.), Optimal Control of Viscous Flow, SIAM Books, Philadelphia, 1998, pp. 1–42. CrossrefGoogle Scholar

  • [2] Gunzburger M. D., Perspectives in Flow Control and Optimization, SIAM Books, Philadelphia, 2003, pp. 75–94. Google Scholar

  • [3] Gad-el-Hak M., Flow Control: Passive, Active and Reactive Flow Management, Cambridge Univ. Press, New York, 2000, pp. 150–204. CrossrefGoogle Scholar

  • [4] Huan J. C. and Modi V., “Optimum Design of Minimum Drag Bodies in Incompressible Laminar Flow Using a Control Theory Approach,” Inverse Problems in Engineering, Vol. 1, No. 1, 1994, pp. 1–25.doi:https://doi.org/10.1080/174159794088027570 IPENEK 1068-2767 CrossrefGoogle Scholar

  • [5] Jameson A., “Optimum Aerodynamic Design Using CFD and Control Theory,” AIAA Paper  95-1729-CP, 1995. LinkGoogle Scholar

  • [6] Jameson A., “Optimum Aerodynamic Design Using Control Theory,” Computational Fluid Dynamics Review, AIAA Paper  1995-1729, 1995, pp. 495–528. LinkGoogle Scholar

  • [7] Nadarajah S. and Jameson A., “Optimum Shape Design for Unsteady Flows with Time-Accurate Continuous and Discrete Adjoint Methods,” AIAA Journal, Vol. 45, No. 7, 2007, pp. 1478–1491. doi:https://doi.org/10.2514/1.24332 AIAJAH 0001-1452 LinkGoogle Scholar

  • [8] Jameson A., Martinelli L. and Pierce N. A., “Optimum Aerodynamic Design Using the Navier–Stokes Equations,” Theoretical and Computational Fluid Dynamics, Vol. 10, Nos. 1–4, 1998, pp. 213–237. doi:https://doi.org/10.1007/s001620050060 TCFDEP 0935-4964 CrossrefGoogle Scholar

  • [9] Srinath D. N. and Mittla S., “An Adjoint Method for Shape Optimization in Unsteady Viscous Flows,” Journal of Computational Physics, Vol. 229, No. 6, 2010, pp. 1994−2008. doi:https://doi.org/10.1016/j.jcp.2009.11.019 JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [10] Bressan A. and Marson A., “A Maximum Principle for Optimally Controlled Systems of Conservation Laws,” Rendiconti del Seminario Matematico della Universita di Padova, Vol. 94, 1995, pp. 79–94. Google Scholar

  • [11] Fursikov A. V., Optimal Control of Distributed Systems: Theory and Applications, American Mathematical Soc., Providence, RI, 2000, pp. 193–232. Google Scholar

  • [12] Walsh E. J. and Hart P. M., “Flight-Measured Lift-off Ignition Overpressure—A Correlation with Subscale Model Tests,” AIAA Paper  1981-2458, Nov. 1981. Google Scholar

  • [13] Canabal F., “Suppression of the Ignition Overpressure Generated by Launch Vehicles,” Ph.D. Dissertation, Mechanical and Aerospace Engineering Dept., Univ. of Alabama, Huntsville, AL, 2004. LinkGoogle Scholar

  • [14] Jourdan G., Biamino L., Mariani C., Blanchot C., Daniel E., Massoni J., Houas L., Tosello R. and Praguine D., “Attenuation of a Shock Wave Passing Through a Cloud of Water Droplets,” Shock Waves, Vol. 20, No. 4, 2010, pp. 285–296.doi:https://doi.org/10.1007/s00193-010-0251-5 SHWAEN 0938-1287 CrossrefGoogle Scholar

  • [15] Sutton G. and Biblarz O., “Solid Propellant Rocket Fundamentals,” Rocket Propulsion Elements, 8th ed., Wiley, Hoboken, NJ, 2010, pp. 437–438. Google Scholar

  • [16] Gordon S. and McBride B.Software Engineering Associates, Inc., “Cequel Chemical Equilibrium in Excel,” http://cearun.grc.nasa.gov/ [retrieved 7 Nov. 2012]. Google Scholar

  • [17] ESI Group, “CFD-FASTRAN-Manual,” http://www.esigroup.com/ [retrieved 7 Nov. 2012]. Google Scholar

  • [18] Van Leer B., “Flux Vector Splitting for the Euler Equations,” Lecture Notes in Physics, Vol. 170, 1982, pp. 507–512.doi:https://doi.org/10.1007/3-540-11948-5_66 CrossrefGoogle Scholar

  • [19] Barth T. J. and Jespersen D. C., “The Design and Application of Upwind Schemes on Unstructured Meshes,” AIAA Paper  89-0366, Jan. 1989. LinkGoogle Scholar

  • [20] Schwer D. and Kailasanath K., “Blast Mitigation by Water Mist 2, Shock Wave Mitigation Using Glass Particles and Water Droplets,” NRL/MR 6410-03-8658, Jan. 2003. Google Scholar

  • [21] Peng D. Y. and Robinson D. B., “A New Two-Constant Equation of State,” Industrial and Engineering Chemistry Fundamentals, Vol. 15, No. 1, 1976, pp. 59–64. doi:https://doi.org/10.1021/i160057a011 IECFA7 0196-4313 CrossrefGoogle Scholar

  • [22] Kurganov A. and Tadmor E., “New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection-Diffusion Equations,” Journal of Computational Physics, Vol. 160, No. 1, 2000, pp. 241–282. doi:https://doi.org/10.1006/jcph.2000.6459 JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [23] Godunov S. K., “A Finite Difference Method for the Numerical Computation of Discontinuous Solutions of the Equations of Hydrodynamics,” Matematicheskii Sbornik, Vol. 47, No. 3, 1959, pp. 271–306. MATSAB Google Scholar

  • [24] Van Leer B., “Toward the Ultimate Conservative Difference Scheme. V. A Second-Order Sequel to Godunov’s Method,” Journal of Computational Physics, Vol. 32, No. 1, July 1979, pp. 101–136. doi:https://doi.org/10.1016/0021-9991(79)90145-1 JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [25] Eidelman S., Colella P. and Shreeve R. P., “Application of the Godunov Method and Its Second-Order Extension to Cascade Flow Modeling,” AIAA Journal, Vol. 22, No. 11, Nov. 1984, pp. 1609–1615. doi:https://doi.org/10.2514/3.8825 AIAJAH 0001-1452 LinkGoogle Scholar

  • [26] Chen H., “Two-Dimensional Simulation of Stripping Breakup of a Water Droplet,” AIAA Journal, Vol. 46, No. 5, May 2008, pp. 1135–1143. doi:https://doi.org/10.2514/1.31286 AIAJAH 0001-1452 LinkGoogle Scholar

  • [27] Saurel R. and Abgrall R., “A Multiphase Godunov Method for Compressible Multi-Fluid and Multiphase Flows,” Journal of Computational Physics, Vol. 150, No. 2, 1999, pp. 425–467. doi:https://doi.org/10.1006/jcph.1999.6187 JCTPAH 0021-9991 CrossrefGoogle Scholar

  • [28] Ulbrich S., “Adjoint-Based Derivative Computations for the Optimal Control of Discontinuous Solutions of Hyperbolic Conservation Laws,” Systems and Control Letters, Vol. 48, Nos. 3–4, 2003, pp. 313–328.doi:https://doi.org/10.1016/S0167-6911(02)00275-X SCLEDC 0167-6911 CrossrefGoogle Scholar

  • [29] Giles M. B., “Non-Reflecting Boundary Conditions for Euler Equation Calculations,” AIAA Paper  89-1942-CP, 1989. LinkGoogle Scholar

  • [30] Alekseev A. K. and Navon I. M., “On Adjoint Variables for Discontinuous Flow,” Systems and Control Letters (submitted for publication). Google Scholar

  • [31] Moshman N. D., “Optimal Control of Unsteady Shock Wave Attenuation in Single- and Two-Phase Flow with Application to Ignition Overpressure in Launch Vehicles,” Ph.D. Dissertation, Mechanical and Astronautical Engineering Dept., Naval Postgraduate School, Monterey, CA, 2011. Google Scholar

  • [32] Jacket—The GPU Acceleration Engine for Matlab,” 2011, http://www.accelereyes.com [retrieved 7 Nov. 2012]. Google Scholar