Guderley–Landau–Stanyukovich problem

Guderley–Landau–Stanyukovich problem describes the time evolution of converging shock waves.

The problem was discussed by G. Guderley in 1942[1] and independently by Lev Landau and K. P. Stanyukovich in 1944, where the later authors' analysis was published in 1955.

[2] Consider a spherically converging shock wave that was initiated by some means at a radial location

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 self-similar solution to be described corresponds to the region

, that is to say, the shock wave has travelled enough to forget about the initial condition.

is very large in comparison with the pressure ahead of the wave

, which in turn removes the velocity scale by setting

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.

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

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

; 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

such that the converging shock wave reaches the origin at time

, the reflected shock wave emerges from the origin.

The reflected shock emerges with the same similarity index.

is determined from the condition that a self-similar solution exists, whereas the constant

and therefore can be determined only when the entire region of the flow is solved.

For Taylor–von Neumann–Sedov blast wave, dimensional arguments can be used to obtain

immediately behind the strong shock front, for an ideal gas are given by These will serve as the boundary conditions for the flow behind the shock front.

To obtain the self-similar equations, we introduce[3][4][5] Note that since both

At the moment of collapse, the flow variables at any distance from the origin must be finite, that is to say,

This is possible only if Substituting the self-similar variables into the governing equations lead to From here, we can easily solve for

This additional condition can be satisfied not for any arbitrary value of

as shown in the figure as a solid curve.

is the initial condition for the differential equation, i.e.,

appears implying that this ratio vanishes at point

can vanish, indicating that the aforementioned functions have extrema.