The Big Three: Geometric Unification of the Noh, Sedov, and Guderley Problems

The idealized stagnation shock (formulated by W. Noh in 1983), point blast wave (formulated independently by G. Taylor, L. Sedov, and J. von Neumann in the early 1940s), and symmetric implosion (formulated by K. Guderley in 1942) problems are perhaps the three most famous self-similar shock wave solutions of the gas dynamics equations. These venerable solutions remain indispensable within an incredibly diverse set of physics modeling efforts including supernovae, inertial confinement fusion, or even the ultrasonic destruction of kidney stones. The applicability of these solutions across so large a range of physical scales is itself a natural consequence of the invariance of the gas dynamics equations under scaling transformations, as established classically using the Buckingham-Pi Theorem. In turn, by the 1950s, scale-invariance concepts were successfully integrated within S. Lie's broader group-theory (i.e., geometric, or symmetry) interpretation of differential equations. Accordingly, self-similar solutions of the gas dynamics equations have long since been understood to encode specific realizations of a broader set of underlying symmetries – namely, certain invariant scaling transformations. To this point, this work demonstrates how Lie's symmetry analysis formalism may be employed to rigorously identify both the common basis and distinguishing invariant scaling transformations that give rise to the classical Noh, Sedov-Taylor-von Neumann, and Guderley solutions. In addition to providing a unification framework for these solutions, the symmetry interpretation may also be leveraged to provide a differential geometric view of self-similar shock wave propagation.