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]

Mathematical description

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Consider a spherically converging shock wave that was initiated by some means at a radial location   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   thus varies with time. 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.

Since the shock wave in the self-similar region is strong, the pressure behind the wave   is very large in comparison with the pressure ahead of the wave  . According to Rankine–Hugoniot conditions, for strong waves, although  ,  , where   represents gas density; in other words, the density jump across the shock wave is finite. For the analysis, one can thus assume   and  , which in turn removes the velocity scale by setting   since  .

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   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   on   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   such that the converging shock wave reaches the origin at time  . For  , the converging shock approaches the origin and for  , the reflected shock wave emerges from the origin. The location of shock wave   is assumed to be described by the function

 

where   is the similarity index and   is a constant. The reflected shock emerges with the same similarity index. The value of   is determined from the condition that a self-similar solution exists, whereas the constant   cannot be described from the self-similar analysis; the constant   contains information from the region   and therefore can be determined only when the entire region of the flow is solved. The dimension of   will be found only after solving for  . For Taylor–von Neumann–Sedov blast wave, dimensional arguments can be used to obtain  

The shock-wave velocity is given by

 

According to Rankine–Hugoniot conditions the gas velocity  , pressure   and density   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.

Self-similar solution

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The governing equations are

 

where   is the density,   is the pressure,   is the entropy and   is the radial velocity. In place of the pressure  , we can use the sound speed   using the relation  .

To obtain the self-similar equations, we introduce[3][4][5]

 

Note that since both   and   are negative,  . Formally the solution has to be found for the range  . The boundary conditions at   are given by

 

The boundary conditions at   can be derived from the observation at the time of collapse  , wherein   becomes infinite. At the moment of collapse, the flow variables at any distance from the origin must be finite, that is to say,   and   must be finite for  . This is possible only if

 

Substituting the self-similar variables into the governing equations lead to

 

From here, we can easily solve for   and   (or,  ) to find two equations. As a third equation, we could two of the equations by eliminating the variable  . The resultant equations are

 

where   and  . It can be easily seen once the third equation is solved for  , the first two equations can be integrated using simple quadratures.

 
Guderley-Landau-Stanyukovich integral curve for  

The third equation is first-order differential equation for the function   with the boundary condition   pertaining to the condition behind the shock front. But there is another boundary condition that needs to be satisfied, i.e.,   pertaining to the condition found at  . This additional condition can be satisfied not for any arbitrary value of  , but there exists only one value of   for which the second condition can be satisfied. Thus   is obtained as an eigenvalue. This eigenvalue can be obtained numerically.

The condition that determines   can be explained by plotting the integral curve   as shown in the figure as a solid curve. The point   is the initial condition for the differential equation, i.e.,  . The integral curve must end at the point  . In the same figure, the parabola   corresponding to the condition   is also plotted as a dotted curve. It can be easily shown than the point   always lies above this parabola. This means that the integral curve   must intersect the parabola to reach the point  . In all the three differential equation, the ratio   appears implying that this ratio vanishes at point   where the integral curve intersects the parabola. The physical requirement for the functions   and   is that they must be single-valued functions of   to get a unique solution. This means that the functions   and   cannot have extrema anywhere inside the domain. But at the point  ,   can vanish, indicating that the aforementioned functions have extrema. The only way to avoid this situation is to make the ratio   at   finite. That is to say, as   becomes zero, we require   also to be zero in such a manner to obtain  . At  ,

 

Numerical integrations of the third equation provide   for   and   for  . These values for   may be compared with an approximate formula  , derived by Landau and Stanyukovich. It can be established that as  ,  . In general, the similarity index   is an irrational number.

See also

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References

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  1. ^ Guderley, K. G. (1942). Starke kugelige und zylindrische verdichtungsstosse in der nahe des kugelmitterpunktes bnw. der zylinderachse. Luftfahrtforschung, 19, 302.
  2. ^ Stanyukovich, K. P. (2016). Unsteady motion of continuous media. Elsevier.
  3. ^ Landau, L. D., & Lifshitz, E. M. (2000). Fluid Mechanics (Course of Theoretical Physics, Volume 6). Reed Educational and Professional Publishing Ltd,.
  4. ^ Zeldovich, Y. B., Raizer, Y. P., Hayes, W. D., & Probstein, R. F. (1967). Physics of shock waves and high-temperature hydrodynamic phenomena. Vol. 2 (pp. 685-784). New York: Academic Press.
  5. ^ Sedov, L. I., & Volkovets, A. G. (2018). Similarity and dimensional methods in mechanics. CRC press.