Solution of the Noh problem with an arbitrary equation of state

The classic self-similar solutions of the nonstationary compressible Euler equations obtained for a blast-wave propagation (Sedov, Taylor, and von Neumann), a shock-wave implosion (Guderley, Landau, and Stanyukovich), or an impulsive loading of a planar target (von Hoerner, Häfele, and Zel’dovich) have all been derived for a polytropic ideal gas. None of them can be generalized for a fluid with an arbitrary equation of state (EOS), such as the van der Waals EOS of a non-ideal-gas or a three-term EOS of a condensed material. We demonstrate here that the Noh accretion-shock problem is an exception. Its self-similar solutions exist in cylindrical and spherical geometry for fluids and materials with an arbitrary EOS. Such solutions for finite accretion-shock strength and nonuniform inflow velocity are constructed semianalytically with a model three-term equation of state that includes cold, thermal ion (lattice), and thermal electron contributions to the pressure and internal energy. Examples are presented for aluminum and copper. Other material- and EOS-specific semianalytic solutions of the Noh problem can be easily constructed using the same method for any material that in the pressure range of interest can be approximated as a dissipation-free fluid with an arbitrary equation of state.

Figure 1

(a) Grüneisen coefficient vs density compression as given by Eq. (7) for Al and Cu (lines) and inferred experimental values (diamonds for Al, triangles for Cu) from Ref. [62]. (b) and (c) Cold compression curve and Hugoniot curves for Al and Cu, respectively, starting from the zero-temperature isotherm-isentrope, the preshock density being ${\rho }_1/{\rho }_{0a}=1$, 1.1, 1.2, and 1.4. Empty circles represent the cold compression curve calculations from Ref. [62]. Solid circles, triangles, and diamonds represent the experimental data for the principal Hugoniots of aluminum and copper taken from Refs. [62, 72], and [73] (Al only), respectively.

Figure 2

Self-similar profiles of density compression (a), pressure (b), and radial velocity (c) for solutions of the spherical Noh problem for Al. Shock parameters for each case are listed in Table 1. Lines with box symbols represent the constant-velocity approximate solution: the density profile is given by Eq. (31), the preshock pressure is zero, and the preshock velocity is equal to $-u_0$.

Figure 3

Self-similar profiles of density compression (a), pressure (b), and radial velocity (c) for solutions of the cylindrical Noh problem for Cu. Shock parameters for each case are listed in Table 1. Lines with box symbols represent the constant-velocity approximate solution: the density profile is given by Eq. (31), the preshock pressure is zero, and the preshock velocity is equal to $-u_0$.