Reinterpreting shock wave structure predictions using the Navier–Stokes equations

Abstract

Classical Navier-Stokes equations fail to predict shock wave profiles accurately. In this paper, the Navier-Stokes system is fully transformed using a velocity variable transformation. The transformed equations termed the recast Navier-Stokes equations display physics not initially included in the classical form of the equations. We then analyze the stationary shock structure problem in a monatomic gas by solving both the classical and the recast Navier-Stokes equations numerically using a finite difference global solution (FDGS) scheme. The numerical results are presented for different upstream Mach numbers ranging from supersonic to hypersonic flows. We found that the recast Navier-Stokes equations show better agreement with the experimentally measured density and reciprocal shock thickness profiles.

Accurate Constitutive Relations for Shock Wave Structures in Gases

“…These previous works paid less attention to temperature profile description across the shock layer. Recently, the authors reinterpreted shock structure predictions of the classical Navier-Stokes equations using a change of velocity variable [43]. The results on the shock density profiles and shock thicknesses better agreed with the experimental data.…”

“…To justify the non-negativity of the entropy production rate for the modified continuum equations one observes the following in Eq. (43). Across the shock layer, the gradient of velocity is always negative as the velocity progressively decreases from supersonic/hypersonic upstream to subsonic downstream.…”

“…where m 0 , p 0 and h 0 are integration constants which represents the mass flow rate, the stagnation pressure and the stagnation specific enthalpy, respectively, and their values can be obtained using the well-known Rankine-Hugoniot conditions (14). In order to solve the eqns (17) and (18), it is convenient to work with its dimensionless form. We use the following set of dimensionless variables based on the upstream reference states (denoted with subscript 1) as in Reese et al [5], Reddy and Dadzie [17], and Dadzie and Reddy [22]:…”

“…In order to solve the eqns (17) and (18), it is convenient to work with its dimensionless form. We use the following set of dimensionless variables based on the upstream reference states (denoted with subscript 1) as in Reese et al [5], Reddy and Dadzie [17], and Dadzie and Reddy [22]:…”

“…Recent notable works on improving shock structure prediction results over the classical Navier-Stokes may include: a second-order continuum theory of Paolucci and Paolucci [41], a linear irreversible thermodynamic model of Velasco and Uribe [42], recast Navier-Stokes of Reddy and Dadzie [43], Onsager-Burnett equations of Jadhav and Agrawal [44]. These previous works paid less attention to temperature profile description across the shock layer.…”