Entropy Conditions Involved in the Nonlinear Coupled Constitutive Method for Solving Continuum and Rarefied Gas Flows
Abstract
:1. Introduction
2. Boltzmann Equations and Entropy Transport Equation
2.1. Boltzmann Equations and the Feature of Irreversibility
2.2. H Theorem and Entropy Balance Equation
3. Nonlinear Coupled Constitutive Method
3.1. Conservation Laws in the NCCM
3.2. Evolution Equations of the Nonconservation Variables in NCCM
3.3. Treatment of the Distribution Function in the Boltzmann Equation
- (1)
- The definition of the distribution function from nonequilibrium to equilibrium is weak in form and dynamic. In other words, the distribution function has no exact and specific expression in the process of nonequilibrium.
- (2)
- The distribution function should fulfil not only the conservations of mass, momentum, and energy, but also the H theorem and positive entropy generation.
3.4. Treatment of the Collision Term in the Boltzmann Equation
3.5. Nonlinear Coupled Constitutive Method
3.6. Comparisons among the Viscous Stress in the NCCM, the Grad Moment Method, and the Burnett Equation
- (1)
- The NCCM and the Burnett and Grad moment methods show nonlinear constitutive relations which is quite different from the NSF equation. However, the NCCM and the Burnett and Grad moment methods have the same linear relations as those of Newton’s laws near the equilibrium state. In other words, the NSF equation can be regarded as the low-order approximation of NCCM, Burnett, and Grad. Further discussion can be found in [4,22].
- (2)
- All three of these nonlinear relationships are consistent in the near-equilibrium region. Because both of the latter methods can solve the problem of near-equilibrium gas flows, the NCCM should be correct in the near-equilibrium region.
- (3)
- The NCCM has a different nonlinear trend than those of the Burnett equation and the Grad moment method in the far-from-equilibrium region. Note that both the Burnett equation and the Grad moment method cannot be used in far-from-equilibrium states. In contrast, the NCCM shows the opposite trends as those of the Grad moment and Burnett methods. This finding has been validated by the DSMC method, as shown in Figure 1. This feature indicates that the NCCM can be used to describe the gas flow in the far-from-equilibrium state.
4. Results
4.1. Verification and Validation
4.2. Cavity Flow
4.3. Hypersonic Gas Flow in a Low Kn Number
4.4. Hypersonic Gas Flow at a High Kn Number
5. Discussion
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A. The Derivation of the Nonlinear Parameter in the NCCM
Appendix B. Positive Entropy Production in Eu Moment Equations
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Case | 20 × 20 | 40 × 40 | 80 × 80 | 100 × 100 |
---|---|---|---|---|
CD | 0.4101 | 0.3951 | 0.3901 | 0.3902 |
CL | 9.0 × 10−3 | 1.23 × 10−4 | 1.02 × 10−6 | 1.11 × 10−6 |
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Tang, K.; Xiao, H. Entropy Conditions Involved in the Nonlinear Coupled Constitutive Method for Solving Continuum and Rarefied Gas Flows. Entropy 2017, 19, 683. https://doi.org/10.3390/e19120683
Tang K, Xiao H. Entropy Conditions Involved in the Nonlinear Coupled Constitutive Method for Solving Continuum and Rarefied Gas Flows. Entropy. 2017; 19(12):683. https://doi.org/10.3390/e19120683
Chicago/Turabian StyleTang, Ke, and Hong Xiao. 2017. "Entropy Conditions Involved in the Nonlinear Coupled Constitutive Method for Solving Continuum and Rarefied Gas Flows" Entropy 19, no. 12: 683. https://doi.org/10.3390/e19120683
APA StyleTang, K., & Xiao, H. (2017). Entropy Conditions Involved in the Nonlinear Coupled Constitutive Method for Solving Continuum and Rarefied Gas Flows. Entropy, 19(12), 683. https://doi.org/10.3390/e19120683