Reflection of a rarefaction wave from the center of symmetry: theoretical analysis of the flow features and calculation by the method of characteristics

For an unsteady spherical rarefaction wave in an ideal (inviscid and non-heat-conducting) gas, the features of the flow near the reflection of the first characteristic from the center of symmetry are investigated. Computations performed by the method of characteristics on nearly uniform grids usually used in such problems reveal sawtooth irregularities in the parameter distributions near the reflection point, whereas similar irregularities in the cylindrical and plane cases are absent. The amplitudes of the irregularities and the sizes of the domains where they are observed remain nearly unchanged when the number of points of the characteristic grid is increased by many times. Away from the reflection point in both time and space, the numerical solution completely “forgets” about the irregularities. This finding explains why these irregularities were ignored or overlooked earlier, but the nature of this phenomenon remains an open question. The present study has established that the spherical rarefaction flow near the reflection point differs fundamentally in structure from its plane and cylindrical counterparts. In the spherical case, the rarefaction flow near the reflection point was found to be nearly conical (entirely conical in the linear approximation). Allowance for this feature in the method of characteristics led to continuous regular distributions of the parameters. The performed analysis and computations revealed that a spherical rarefaction wave is strengthened (cumulates) theoretically unlimitedly in a small neighborhood of the reflection point (center of symmetry) of the first characteristic. Moreover, the claim that a gradient catastrophe occurs in this neighborhood was found to be untenable.