Guderley–Landau–Stanyukovich problem describes the time evolution of converging shock waves. The problem was discussed by G. Guderley in 1942 and independently by Lev Landau and K. P. Stanyukovich in 1944, where the later authors' analysis was published in 1955.
Mathematical description Consider a spherically converging shock wave that was initiated by some means at a radial location r = R 0 {\displaystyle r=R_{0}} and directed towards the center. As the shock wave travels towards the origin, its strength increases since the shock wave compresses lesser and lesser amount of mass as it propagates. The shock wave location r = R ( t ) {\displaystyle r=R(t)} thus varies with time. The self-similar solution to be described corresponds to the region r ∼ R ≪ R 0 {\displaystyle r\sim R\ll R_{0}} , that is to say, the shock wave has travelled enough to forget about the initial condition. Since the shock wave in the self-similar region is strong, the pressure behind the wave p 1 {\displaystyle p_{1}} is very large in comparison with the pressure ahead of the wave p 0 {\displaystyle p_{0}} . According to Rankine–Hugoniot conditions, for strong waves, although p 1 ≫ p 0 {\displaystyle p_{1}\gg p_{0}} , ρ 1 ∼ ρ 0 {\displaystyle \rho _{1}\sim \rho _{0}} , where ρ {\displaystyle \rho } represents gas density; in other words, the density jump across the shock wave is finite. For the analysis, one can thus assume p 0 = 0 {\displaystyle p_{0}=0} and ρ 0 ≠ 0 {\displaystyle \rho _{0}\neq 0} , which in turn removes the velocity scale by setting c 0 = 0 {\displaystyle c_{0}=0} since c 0 2 = γ p 0 / ρ 0 {\displaystyle c_{0}^{2}=\gamma p_{0}/\rho _{0}} . At this point, it is worth noting that the analogous problem in which a strong shock wave propagating outwards is known to be described by the Taylor–von Neumann–Sedov blast wave. The description for Taylor–von Neumann–Sedov blast wave utilizes ρ 0 {\displaystyle \rho _{0}} and the total energy content of the flow to develop a self-similar solution. Unlike this problem, the imploding shock wave is not self-similar throughout the entire region (the flow field near r = R 0 {\displaystyle r=R_{0}} depends on the manner in which the shock wave is generated) and thus the Guderley–Landau–Stanyukovich problem attempts to describe in a self-similar manner, the flow field only for r ∼ R ≪ R 0 {\displaystyle r\sim R\ll R_{0}} ; in this self-similar region, energy is not constant and in fact, will be shown to decrease with time (the total energy of the entire region is still constant). Since the self-similar region is small in comparison with the initial size of the shock wave region, only a small fraction of the total energy is accumulated in the self-similar region. The problem thus contains no length scale to use dimensional arguments to find out the self-similar description i.e., the dependence of R ( t ) {\displaystyle R(t)} on t {\displaystyle t} cannot be determined by dimensional arguments alone. The problems of these kind are described by the self-similar solution of the second kind. For convenience, measure the time t {\displaystyle t} such that the converging shock wave reaches the origin at time t = 0 {\displaystyle t=0} . For t < 0 {\displaystyle t<0} , the converging shock approaches the origin and for t > 0 {\displaystyle t>0} , the reflected shock wave emerges from the origin. The location of shock wave r = R ( t ) {\displaystyle r=R(t)} is assumed to be described by the function
R ( t ) = A ( − t ) α {\displaystyle R(t)=A(-t)^{\alpha }}
… excerpt ends here. Continue reading the full article.
