A Micro State Dynamic Model and Compatibility Conditions for the One Dimensional Non-Stationary Compressible Flow in Pressure Wave Machines

Shock tube like applications such as the pressure wave machines are subjected to the possible build up of moving discontinuities in the material, temperature and pressure. The method of characteristics offered a suitable platform to take these aspects into consideration. Jenny [1,2,3] has applied this method to obtain the direction and the corresponding compatibility conditions for the isentropic non-stationary one dimensional gas flow. Following Jenny’s derivation, it was possible to derive the general compatibility conditions [4–8] along the corresponding characteristic directions. The variable grid method [4–8] used a second order approximation of the direction and compatibility conditions. A system of non-linear equations [4–8] for the local states and the grid positions was obtained. In spite of the accurate results achieved by the variable grid method, the obligation to grid reorganization -while numerically tractable- have drawn the attention to the possibility that there may be persistent and inherent analytical/physical inadequacy with respect to the resulting compatibility equations that are numerically reflected by inhomogeneous build up of the variable grid. This paper, therefore, revisits the formulation of the dynamics of the one dimensional non stationary compressible flow and develops a new micro state dynamic model. The micro state equations of continuity, momentum and energy are derived. A micro state variable such as Θ or P represents the micro state dynamics, while the corresponding flow variables of temperature T or density ρ are to be regarded as observed measurement variables. Further, the micro state compatibility conditions are derived. The resulting micro state total differentials dS₀ & dS_± along the particle path & sound propagation ± Lines are linear functions of the micro state differentials. It is shown that dS₀ is the total differential of the micro state energy equation along the particle path. It is proportional to the “Clausius”-entropy differential ds through the gas constant R and to the differential of the number of micro states dΩ* in “Boltzmann”-relation by the Avogadro number N_A. For isentropic flow, dS_± is proportional — through κ and the sound speed a — to the transformed micro state “Riemann”-differential dR_±. Otherwise, the relation ±Δ^b_aR_± = [Δ,^b_aS_± + Δ^b_aS₀] holds along the ± Lines where ±Δ^b_aR_± & Δ^b_aS_± are the weighted change of the micro state differential of Riemann-Invariants and the change of dS_± along the ± Lines respectively. δ ^b_aS₀ is the “jump” along the ± Lines of dS₀. Beside providing novel micro state compatibility conditions for the variable grid method, the presented formulation clears the existence as well as the sign of entropy change (reversibility/irreversibility) within the second law of thermodynamics.