1. Introduction

Various mathematical theories and simulation methods were developed in the past for describing gas flows in nonequilibrium, in particular, hypersonic rarefied regime. They range from the mesoscale models like the Boltzmann equation [1–6], the direct simulation Monte Carlo methods [7], and the high order hydrodynamic equations [1–6, 8–20]. Among these models, the kinetic Boltzmann equation plays a central role in the hierarchy of PDE-based mathematical models for gas kinetic theory. The kinetic Boltzmann equation can be transformed into the

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

moment equations by introducing the statistical average in velocity space. Based on the Maxwell's equation of change and the so-called method of moments, Grad [8] in 1949 derived the constitutive equations of viscous shear stresses and heat fluxes from the kinetic Boltzmann equation of the distribution of monatomic gas particles. However, it was found by Grad [9] himself that, within the framework of his constitutive equations, there is a critical Mach number (1.65) beyond which no continuous shock wave solution in high compressive regime is possible.

2. The second-order constitutive model of the Boltzmann equation

The Boltzmann equation plays a central role in the hierarchy of mathematical models for gas kinetic theory. It was derived as an evolution equation for the singlet distribution function of a gas by considering the collision dynamics of two particles and combining it with a statistical molecular chaos assumption. Since the molecular chaos assumption is not of a mechanical nature, that is, the Boltzmann equation is based on the assumptions made to "arrive at it" from the reversible Liouville equations of motion, the Boltzmann equation should be regarded as a fundamental kinetic equation at the mesoscopic level of description of macroscopic processes. Thus, it is a postulate for dynamic evolution of singlet distribution functions f(t,r,v) in the

Numerical Simulation of Hypersonic Rarefied Flows Using the Second-Order Constitutive Model…

� �fð Þ¼ <sup>v</sup>;r; <sup>t</sup> C f ; <sup>f</sup> <sup>2</sup>

which cannot be derived from the pure mechanical deterministic consideration. Although it is a first-order partial differential equation in space and time, its solution becomes very complicated because it is nonlinear owing to the collision integral C[ f,f2], which is made up of

The moment equations can be obtained by differentiating the statistical definition of the variable in question with time and later combining with the Boltzmann equation [2–5, 8]; it

The symbols c,u,〈〉,Λ(n) denote the peculiar velocity, the average bulk velocity, the integral in

The conservation laws of mass, momentum, and total energy can be derived directly from the kinetic Boltzmann equation. For example, in the case of momentum conservation law, differentiating the statistical definition of the momentum with time and combining with the

¼ �h i mð Þ v � ∇f v þ mvC f ; f <sup>2</sup>

�h i <sup>m</sup>ð Þ <sup>v</sup> � <sup>∇</sup><sup>f</sup> <sup>v</sup> ¼ �<sup>∇</sup> � h i <sup>m</sup>vv<sup>f</sup> ¼ �<sup>∇</sup> � <sup>ρ</sup>uu <sup>þ</sup> h i <sup>m</sup>cc<sup>f</sup> � �: (4)

d dthð Þ <sup>n</sup> � � � �, (1)

http://dx.doi.org/10.5772/intechopen.70657

5

D E � � � �: (2)

� � � � : (3)

� <sup>f</sup> <sup>c</sup>�∇hð Þ <sup>n</sup> D E¼Λð Þ <sup>n</sup> � <sup>h</sup>ð Þ <sup>n</sup> C f ;<sup>f</sup> <sup>2</sup>

2.1. The kinetic Boltzmann equation and the method of moments

∂ ∂t

yields for molecular expressions of general moment h(n)

D E<sup>þ</sup> <sup>c</sup>hð Þ <sup>n</sup> <sup>f</sup> � � D E � <sup>f</sup>

2.2. Exact derivation of the conservation laws

∂ ∂t

Then the first term on the right-hand side becomes

h i mvf ¼ mv

velocity space, and the dissipation (or production) terms, respectively.

∂f ∂t � �

þ v � ∇

phase space (time, position, velocity),

products of distribution functions.

D Eþ∇� <sup>u</sup> <sup>h</sup>ð Þ <sup>n</sup> <sup>f</sup>

Boltzmann equation yield

∂ ∂t hð Þ <sup>n</sup> f

After Grad's pioneering work in developing gas kinetic theory and subsequent failure of his 13-moment method in describing hypersonic shock structure, there have been enormous efforts to resolve the problem from various perspectives, not only by physicists and mathematicians, but engineers and also chemists. Among such efforts, Eu's works [2–5] to develop the gas kinetic theory consistent with the second law of thermodynamics beyond the linear irreversible thermodynamics stand out. By recognizing the logarithmic form of the nonequilibrium entropy production, Eu [2] in 1980 proposed a canonical distribution function in the exponential form, instead of Grad's polynomial form. He also generalized the equilibrium Gibbs ensemble theory—providing the relationship between thermodynamic variables and the partition functions—to nonequilibrium processes. It turns out that such canonical exponential form assures the nonnegativity of the distribution function and satisfies the second law of thermodynamics in rigorous way, regardless of the level of approximations.

Recently, Myong [15] in 2014 developed a new closure theory which plays a critical role in the development of gas kinetic theory. The new closure was derived from a keen observation of the fact that, when closing open terms in the moment equations derived from the Boltzmann kinetic equation, the number of places to be closed is two (movement and interaction), rather than one (movement only) misled by the Maxwellian molecule assumption in previous theory. Therefore, the order of approximations in handling the two terms—kinematic (movement) and dissipation (interaction) terms—must be the same; for instance, second-order for both terms, leading to the name of the new closure as the balanced closure. Then, after applying the Eu's cumulant expansion based on the canonical distribution function to the explicit calculation of the dissipation term and the aforementioned new closure, Myong [15] derived the secondorder constitutive models from the Boltzmann kinetic equation and proved that the new models indeed remove the high Mach number shock structure singularity completely, which had remained unsolved for decades.

On the basis of these new theories, this chapter will first describe a recent development in theoretical models for numerical simulation of hypersonic rarefied flows from the viewpoint of the method of moments. It will focus on the detailed derivation of the second-order constitutive model from the original Boltzmann equation and the development of associated computational models for numerical simulation of hypersonic rarefied gas flows in simple geometry as well as complicated real vehicles. Finally, some practical applications of the second-order constitutive model to hypersonic rarefied flows are summarized.
