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Analytical linear theory for the shock and re-shock of isotropic density inhomogeneities. (English) Zbl 1248.76106
Summary: We present an analytical model that describes the linear interaction of two successive shocks launched into a non-uniform density field. The re-shock problem is important in different fields, inertial confinement fusion among them, where several shocks are needed to compress the non-uniform target. At first, we present a linear theory model that studies the interaction of two successive shocks with a single-mode density perturbation field ahead of the first shock. The second shock is launched after the sonic waves emitted by the first shock wave have vanished. Therefore, in the case considered in this work, the second shock only interacts with the entropic and vortical perturbations left by the first shock front. The velocity, vorticity and density fields are later obtained in the space behind the second shock. With the results of the single-mode theory, the interaction with a full spectrum of random-isotropic density perturbations is considered by decomposing it into Fourier modes. The model describes in detail how the second shock wave modifies the turbulent field generated by the first shock wave. Averages of the downstream quantities (kinetic energy, vorticity, acoustic flux and density) are easily obtained either for two-dimensional or three-dimensional upstream isotropic spectra. The asymptotic limits of very strong shocks are discussed. The study shown here is an extension of previous works, where the interaction of a planar shock wave with random isotropic vorticity/entropy/acoustic spectra were studied independently. It is also a preliminary step towards the understanding of the re-shock of a fully turbulent flow, where all three of the modes, vortical, entropic and acoustic, might be present.

##### MSC:
 76L05 Shock waves and blast waves in fluid mechanics
##### Keywords:
gas dynamics; shock waves
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##### References:
 [1] DOI: 10.1063/1.3278598 · doi:10.1063/1.3278598 [2] DOI: 10.1063/1.2162803 · doi:10.1063/1.2162803 [3] DOI: 10.1103/PhysRevE.63.056303 · doi:10.1103/PhysRevE.63.056303 [4] DOI: 10.1063/1.2745809 · doi:10.1063/1.2745809 [5] DOI: 10.1103/PhysRevE.85.016301 · doi:10.1103/PhysRevE.85.016301 [6] DOI: 10.1063/1.873986 · doi:10.1063/1.873986 [7] DOI: 10.1017/S0022112010000492 · Zbl 1193.76066 · doi:10.1017/S0022112010000492 [8] DOI: 10.1088/0029-5515/27/3/009 · doi:10.1088/0029-5515/27/3/009 [9] DOI: 10.1063/1.857960 · Zbl 0745.76034 · doi:10.1063/1.857960 [10] DOI: 10.2514/2.397 · doi:10.2514/2.397 [11] DOI: 10.2514/3.9642 · doi:10.2514/3.9642 [12] DOI: 10.1017/S0022112069001121 · Zbl 0164.56002 · doi:10.1017/S0022112069001121 [13] DOI: 10.1063/1.874046 · doi:10.1063/1.874046 [14] DOI: 10.1086/341886 · doi:10.1086/341886 [15] DOI: 10.1017/S0263034604222121 · doi:10.1017/S0263034604222121 [16] DOI: 10.1063/1.874056 · doi:10.1063/1.874056 [17] DOI: 10.1063/1.872690 · doi:10.1063/1.872690 [18] DOI: 10.1017/S0022112097004576 · Zbl 0899.76193 · doi:10.1017/S0022112097004576 [19] DOI: 10.1017/S0022112095003739 · Zbl 0858.76074 · doi:10.1017/S0022112095003739 [20] DOI: 10.1017/S0022112097005107 · Zbl 0899.76194 · doi:10.1017/S0022112097005107 [21] DOI: 10.1017/S0022112093003519 · doi:10.1017/S0022112093003519 [22] DOI: 10.1063/1.3275856 · Zbl 1183.76296 · doi:10.1063/1.3275856 [23] Landau, Fluid Mechanics (1987) [24] DOI: 10.1063/1.2219768 · Zbl 1185.76750 · doi:10.1063/1.2219768 [25] DOI: 10.1063/1.869719 · Zbl 1185.76762 · doi:10.1063/1.869719 [26] DOI: 10.1103/PhysRevLett.74.2961 · doi:10.1103/PhysRevLett.74.2961 [27] DOI: 10.1088/0004-637X/702/1/39 · doi:10.1088/0004-637X/702/1/39 [28] DOI: 10.1007/978-1-4684-9283-5 · doi:10.1007/978-1-4684-9283-5 [29] DOI: 10.1080/14685248.2011.597755 · Zbl 1273.76200 · doi:10.1080/14685248.2011.597755 [30] DOI: 10.1063/1.1927099 · doi:10.1063/1.1927099 [31] DOI: 10.1103/PhysRevE.83.056320 · doi:10.1103/PhysRevE.83.056320 [32] DOI: 10.1016/j.jcp.2004.01.003 · Zbl 1107.76367 · doi:10.1016/j.jcp.2004.01.003 [33] DOI: 10.1088/0031-8949/2010/T142/014022 · doi:10.1088/0031-8949/2010/T142/014022 [34] DOI: 10.1103/PhysRevLett.106.195005 · doi:10.1103/PhysRevLett.106.195005 [35] DOI: 10.1103/PhysRevE.85.026312 · doi:10.1103/PhysRevE.85.026312 [36] DOI: 10.2514/3.13175 · doi:10.2514/3.13175 [37] DOI: 10.1007/s11431-009-0379-7 · doi:10.1007/s11431-009-0379-7 [38] DOI: 10.1063/1.873172 · doi:10.1063/1.873172 [39] DOI: 10.1103/PhysRevE.79.066315 · doi:10.1103/PhysRevE.79.066315 [40] Drake, High-Energy-Density Physics: Fundamentals, Inertial Fusion, and Experimental Astrophysics (2006)
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