The neutron flux from the nuclear reactors used in experiments involving NPLs varies noticeably at distances that considerably exceed a laser cell’s transverse dimensions [2—14]. Thus, it can be assumed that fission density distribution in a uranium layer, as well as gas temperature and density distribution at any given point within a cylindrical cell, are essentially only dependent upon the radial coordinate, r, and the axial coordinate, z, which coincides with the axis of symmetry.

More precisely, the nonstationary thermal gasdynamic processes taking place in a laser of the type under conditions only permit modeling to be performed using a computer. But even for a geometry that is dependent upon two spatial variables, such calculations prove to be quite difficult and require large outlays of computer time. Therefore, the first investigations of this type were conducted for alternatives involving dependence upon just one spatial variable; the primary objective of these investigations consisted of determining the scale of the inhomogeneities occurring

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S. P. Melnikov et al., Lasers with Nuclear Pumping,

DOI 10.1007/978-3-319-08882-2_8

and establishing the regularities of their dynamics. For simplicity’s sake, it was assumed that all physical parameters, including the distribution of specific energy release sources in the gas and in the uranium layer, are only dependent upon a single radial coordinate, r. Consequently, it was decided that the distribution of the thermal neutron fluxes initiating uranium nucleus fission is not dependent upon coordinates and equals the average value over the length of a cell.

It should be noted that the effect of viscosity forces is negligible. Actually, the order of magnitude of the gas velocity scale comes to u ~ І/Тщ, where І is a typical cell dimension (a transverse dimension of d ~ 1 cm or a longitudinal dimension of L ~ 1 m), and; т1/2 is the excitation pulse duration. The Reynolds number for the gas flow over the course of excitation is then Re = iu /v ~102-107 (here v represents kinematic viscosity, which for rare gas mixtures over the temperature and pressure variation ranges actually used comes to ~10-1 cm2/s), suggesting the negligible effect of viscosity on gas flow.

All the initial calculations were performed using the standard VNIIEF program described in study [15]. In ignoring viscosity, it becomes possible to undertake the numerical solution of the thermal gasdynamic problems described by:

the motion equation

du _ JV-1 dp

dt ds ds = 1pd(rN);

dr

the continuity equation

d (1=p) = Is 1u)’

and the energy equation

Here, E is the internal energy per unit mass of gas; Fp is the internal source power per unit mass; ki is the thermal conductivity coefficient; u is matter velocity; p is pressure; r is the matter particle Eulerian coordinate; s is the matter particle Lagrangian coordinate; T is temperature; and p is density. The N parameter char­acterizes system geometry: N = 1 for a planar geometry, N = 2 for a cylindrical geometry, and N = 3 for a spherical geometry.

Over wide temperature and pressure variation limits, the rare gases that serve as NPL active media can be quite accurately described as ideal. Therefore, a state equation was given in the form

P =(y — 1) cvpT, (8.1)

E = CvT, (8.2)

where у = cp/cv; cp, cv represents specific heat capacities at a constant pressure and a constant volume, respectively.

At the contact boundaries of the gas and the uranium-containing active layer, as well as the active layer and the metal tube (the substrate) to which this layer is applied, the equality of temperatures and heat fluxes was postulated. The substrate’s outer surface was considered to be heat-insulated. The heat capacity and thermal conductivity coefficients of the active layer and the substrate were approximated by a quadratic dependence upon temperature. Based on data [16], the dependences of the gas thermal conductivity coefficients upon temperature were described by a fourth-degree polynomial.

Over the course of irradiation, the gas distribution density in the cell is transformed with the passage of time. Accordingly, because the fragment energy losses per unit range are unequivocally linked to the density of the stopping medium, the specific energy deposition profile along the laser cell cross-section is also deformed. During early calculations, specific source distribution, Fp, in the gas was tied to Lagrangian coordinates in order to take this factor into account, and was given in the form

Fp(s, t) = Fp0(s)q(t).

Here Fp0(s) is the specific power deposition at a point with a Lagrangian coordinate of s in the unperturbed gas (at the reference time) attributed to fission output per uranium layer unit volume, and q(t) is the fission output per unit volume of this layer. The spatial distribution, Fp0(s), was determined using the method described in study [17]. A comparison [18, 19] of this approach to the technique in which the precise dependence of the Fp function on the gas density distribution at any given moment in time is used revealed that discrepancies in the density and temperature calculations using both procedures over typical durations of t1/2 < 3 ms do not exceed 10 % (also see Chap. 7, Sect. 7.2).

After specific energy deposition calculation techniques were worked out for the media with variable densities [19, 20] as mentioned above, the program was modified to make it possible to perform calculations with strict allowance for the effect of the density redistribution on the energy deposition.