Shock Structures Using the OBurnett Equations in Combination with the Holian Conjecture

Ravi Sudam Jadhav,Amit Agrawal

doi:10.3390/fluids6120427

Ravi Sudam Jadhav, Amit Agrawal

Open Access

https://doi.org/10.3390/fluids6120427

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Abstract

In the present work, we study the normal shock wave flow problem using a combination of the OBurnett equations and the Holian conjecture. The numerical results of the OBurnett equations for normal shocks established several fundamental aspects of the equations such as the thermodynamic consistency of the equations, and the existence of the heteroclinic trajectory and smooth shock structures at all Mach numbers. The shock profiles for the hydrodynamic field variables were found to be in quantitative agreement with the direct simulation Monte Carlo (DSMC) results in the upstream region, whereas further improvement was desirable in the downstream region of the shock. For the discrepancy in the downstream region, we conjecture that the viscosity–temperature relation (μ∝Tφ) needs to be modified in order to achieve increased dissipation and thereby achieve better agreement with the benchmark results in the downstream region. In this respect, we examine the Holian conjecture (HC), wherein transport coefficients (absolute viscosity and thermal conductivity) are evaluated using the temperature in the direction of shock propagation rather than the average temperature. The results of the modified theory (OBurnett + HC) are compared against the benchmark results and we find that the modified theory improves upon the OBurnett results, especially in the case of the heat flux shock profile. We find that the accuracy gain is marginal at lower Mach numbers, while the shock profiles are described better using the modified theory for the case of strong shocks.

Highlights

In the continuum regime for vanishing Knudsen numbers (Kn(= λ/L) → 0), the linear constitutive laws (Newton’s law of viscosity and Fourier’s law of heat conduction) employed in the Navier-Stokes equations are sufficient to discern the flow physics
To generate the direct simulation Monte Carlo (DSMC) results, Bird’s DSMC code [9] for the normal shock wave flow problem was implemented; details are provided in our earlier work [43]
By employing the combination of the OBurnett equations and the Holian conjecture, we first verified the thermodynamic consistency of the modified theory and the existence of the heteroclinic trajectory

Summary

Introduction

In the continuum regime for vanishing Knudsen numbers (Kn(= λ/L) → 0), the linear constitutive laws (Newton’s law of viscosity and Fourier’s law of heat conduction) employed in the Navier-Stokes equations are sufficient to discern the flow physics. Several non-equilibrium flow phenomena [6,7,8] start arising in the flow that cannot be described using the Navier-Stokes equations For such flows, it becomes necessary to employ particle methods such as the direct simulation Monte Carlo (DSMC) technique [9,10] or formulate higher-order continuum theories starting from the Boltzmann kinetic equation, for example, conventional Burnett equations [11,12], Grad 13 moment equations [13,14], Onsager-Burnett equations [15], and Onsager 13 moment equations [16].

OBurnett Equations

Significance of the OBurnett Equations

Problem Definition

Modified Theory

Results

Conclusions