Wenwen Zhao

Computation of Rarefied Hypersonic Flows Using Modified Form of Conventional Burnett Equations

This paper describes the computations of hypersonic flows in a diatomic gas in rotational nonequilibrium using a newly developed simplified set of Burnett equations designated as simplified conventional Burnett equations. Since the original formulation by Burnett, a number of variations to the original Burnett equations have been proposed, and the differences among these variants and their merits/shortcomings have been described in the literature. A new variant is created based on the conventional Burnett equations for hypersonic flows by neglecting higher-order terms that are inversely proportional to the Mach number. This set of simplified conventional Burnett equations is linearly stable for small disturbances in contrast to the conventional Burnett equations that suffer from Bobylev instability. To simulate the rotational nonequilibrium effect in a diatomic gas, both the Navier–Stokes and the simplified conventional Burnett equations are modified by including a rotational nonequilibrium relaxation mod...

Non-equilibrium Shock Structure Calculation with High-order Modified Navier–Stokes Equations

Abstract High-order modified Navier-Stokes (NS) equations are introduced to describe the transitional flow derived from Burnett equations considering the inaccurate simulation of shock structure using traditional NS continuum methods, particularly at high Mach numbers. This process is performed by defining the effective transportation coefficient, which is associated with Mach number, molecule propriety, and continuum failure criterion. Within this method, the linear Stokes and Fourier relationships are replaced by nonlinear constitutive relationships to describe transitional flow. The traditional NS equation formulation is preserved. The shock structures of both monatomic and diatomic gases are concentrated with different Mach numbers to validate the accuracy and effectiveness of the derived method. The comparison of shock structure with DSMC and Burnett equations shows that the high-order modified NS equations obtain better results than traditional NS equations in terms of shock wave, particularly at high Mach numbers. The modified NS equations also capture the significant features of Burnett equations in shock structure simulation.

Computation of 1-D shock structure using nonlinear coupled constitutive relations and generalized hydrodynamic equations

The moment methods in rarefied gas dynamics could be divided into generalized hydrodynamic equations (GHE) and extended hydrodynamic equations (EHE), e.g., Burnett equations, Grad equations and R-13 equations, theoretically. Eu firstly developed the GHE based on a non-equilibrium canonical distribution function and demonstrated the thermodynamically consistent of this model. Subsequently, nonlinear coupled constitutive relations (NCCR) was proposed by Myong by omitting the product of heat flux and velocity gradient in GHE to reduce the computational complexity. According to the successful application in 1-D shock wave structure and 2-D flat plate flow, the capability of NCCR has already been demonstrated successfully. The motivation of this study was to investigate the different behavior of NCCR and GHE for monatomic and diatomic gases in one-dimensional shock structure problems. Therefore, argon and nitrogen shock structure was calculated using both GHE and NCCR model up to Ma=50. The 3rd order MUSCL sch...

Computation of 1-D shock structure in a gas in rotational non-equilibrium using a new set of simplified Burnett equations

This paper describes the computations of hypersonic shock wave structure in a gas in rotational non-equilibrium using a newly developed simplified set of Burnett equations designated as Simplified Conventional Burnett (SCB) equations. Since the original formulation by Burnett, a number of variations to the original Burnett equations have been proposed and the differences among these variants and their merits/shortcomings have been described in the literature. A new variant is created based on the conventional Burnett equations for hypersonic flows by neglecting terms that are inversely proportional to the Mach number. This simplified set of conventional Burnett equations is linearly stable for small disturbances in contrast to the conventional Burnett equations which suffer from Bobylev instability. To simulate the rotational non-equilibrium effect in a diatomic gas, both the Navier-Stokes (NS) and the SCB equations are modified by including a rotational non-equilibrium relaxation model. The flow variables (density, translational and rotational temperature) in a typical 1-D shock at different Mach numbers (1.2, 5, and 10) in Nitrogen are computed using the SCB and NS equations and are compared with the DSMC results. SCB calculations are in close agreement with the DSMC results at high Mach numbers.

Entropy Production Analysis of Burnett Equations Using Classical Thermodynamics with Gibbs Equations

An expression describing the entropy production in the Burnett equations is constructed from the classical thermodynamic theory using the continuum conservation relations for mass, momentum and energy in conjunction with Gibbs equation in both oneand three-dimensions. The entropy production is transformed into a polynomial in Mach number and Knudsen number which is employed to analyze the thermodynamic properties of both expanding and compressed gases. Each term in the entropy production polynomial in three-dimensions is studied separately due to the complexity of the polynomial expression in three-dimensions. The results indicate that the irreversible entropy production in conventional Burnett equations, augmented Burnett equations and simplified conventional Burnett equations is not positive semi-definite for an expanding but is positive semi-definite for a gas under compression. Thus these equations may violate the second law of thermodynamics in a flow undergoing both compression and expansion. This result may explain the numerical instability encountered in the numerical solution of the Burnett equations.

One-dimensional non-equilibrium shock wave structure analysis by simplified conventional Burnett equations

The inner shock wave structure with non-equilibrium effect is difficult to be accurately simulated due to the great gradient of density and temperature. In this paper, simplified conventional Burnett (SCB) equations were formulated for the study of hypersonic shock wave structure in continuum-transition regime. The conventional Burnett equations were derived by using the second-order Chapman–Enskog expansion of the velocity distribution function in Boltzmann equation. By neglecting conventional Burnett terms which are inversely proportional to Mach number, the constitutive relations in SCB equations were simplified specifically for hypersonic flow. The rotational and vibrational energy balance equations were also introduced into the governing equations to study the non-equilibrium relaxation processes inside shock waves. Meanwhile, generalized Rankine–Hugoniot relations were established to obtain the post-shock flow parameters in non-equilibrium flow. The numerical methods included three-order Runge-Kutta...

Simulation of Couette flow using conventional Burnett equations with modified slip boundary conditions

Gas or liquid flow through small channels has become more and more popular due to the micro-electro-mechanical systems (MEMS) fabrication technologies such as micro-motors, electrostatic comb-drive, micro-chromatographs, micro-actuators, micro-turbines and micro-pumps, etc. The flow conditions in and around these systems are always recognized as typical transitional regimes. Under these conditions, the mean free path of gas molecules approaches the characteristic scale of the micro-devices itself, and due to the little collisions the heat and momentum cannot equilibrate between the wall and fluids quickly. Couette flow is a simple and critical model in fluid dynamics which focuses on the mechanism of the heat transfer in shear-driven micro-cavities or micro-channels. Despite numerous work on the numerical solutions of the Couette flow, how to propose stable and accurate slip boundary conditions in rarefied flow conditions still remains to be elucidated. In this paper, converged solutions for steady-state ...

3D linearized stability analysis of various forms of Burnett equations

Burnett equations were originally derived in 1935 by Burnett by employing the Chapman-Enskog expansion to Classical Boltzmann equation to second order in Knudsen number Kn. Since then several variants of these equations have been proposed in the literature; these variants have differing physical and numerical properties. In this paper, we consider three such variants which are known in the literature as ‘the Original Burnett (OB) equations’, the Conventional Burnett (CB) equations’ and the recently formulated by the authors ‘the Simplified Conventional (SCB) equations.’ One of the most important issues in obtaining numerical solutions of the Burnett equations is their stability under small perturbations. In this paper, we perform the linearized stability (known as the Bobylev Stability) analysis of three-dimensional Burnett equations for all the three variants (OB, CB, and SCB) for the first time in the literature on this subject. By introducing small perturbations in the steady state flow field, the trajectory curve and the variation in attenuation coefficient with wave frequency of the characteristic equation are obtained for all the three variants of Burnett equations to determine their stability. The results show that the Simplified Conventional Burnett (SCB) equations are unconditionally stable under small wavelength perturbations. However, the Original Burnett (OB) and the Conventional Burnett (CB) equations are unstable when the Knudsen number becomes greater than a critical value and the stability condition worsens in 3D when compared to the stability condition for 1-D and 2-D equations. The critical Knudsen number for 3-D OB and CB equations is 0.061 and 0.287 respectively.