# Entropy Conditions Involved in the Nonlinear Coupled Constitutive Method for Solving Continuum and Rarefied Gas Flows

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## Abstract

**:**

## 1. Introduction

## 2. Boltzmann Equations and Entropy Transport Equation

#### 2.1. Boltzmann Equations and the Feature of Irreversibility

_{i}denotes the probability of finding a particle of species $i$ in the range of ${v}_{i}+d{v}_{i}$ and ${r}_{i}+d{r}_{i}$ at time $t$. The Boltzmann equation yields the evolution equation for ${f}_{i}$,

#### 2.2. H Theorem and Entropy Balance Equation

## 3. Nonlinear Coupled Constitutive Method

#### 3.1. Conservation Laws in the NCCM

**P**is the stress tensor and is defined by $P=pI+\Delta \mathrm{I}+\u041f$. ${F}_{i}$ is the external force per unit mass. In the gas flows, the external force is usually omitted. Thus, the conservation laws are simplified into [22,23,24,25,26]:

**P**and

**Q**are nonconservation variables with molecular expressions that do not yield a collisional invariant.

#### 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

**NCCM**at steady state for the Grad moment method, the second-order Burnett equation, and the third-order Burnett equation. The following observations are made:

- (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

^{−6}.

#### 4.1. Verification and Validation

#### 4.2. Cavity Flow

_{w}= 290 K. The Knudsen number is defined as the ratio of the mean free path to the length of the cavity’s side wall, and the mean free path of gas is evaluated for a hard sphere model. Similar to the previous work [28], the wall velocity is ${U}_{\infty}=300\mathrm{m}/\mathrm{s}$ for cases at Kn = 0.671 and 6.712. The solutions are compared with the DSMC ones. The computational domain is discretized using a mesh of 100 × 100 cells in physical space. The results of the present study and the DSMC at Kn = 0.671 and 6.712 are presented in Figure 3 and Figure 4, where the temperature contours are shown. Excellent agreements are obtained.

#### 4.3. Hypersonic Gas Flow in a Low Kn Number

_{outer}= 30 R. The cell size in unstructured mesh is 200 × 120. In other words, there are 200 points on the cylinder surface and 120 points in the radial direction of the computational domain. The linear element (P

^{1}) and the quadratic element (P

^{2}) are initially tested for the 2D case; because their numerical results are not distinguishable, the linear element is chosen for all 2D simulations.

#### 4.4. Hypersonic Gas Flow at a High Kn Number

_{outer}= 30 R. In this equation, R is the characteristic size of the vehicles. The input parameters for the first case are Ma = 20.0, Kn = 2.531, and Pr = 0.707, and the working gas is air. Good agreement between the results of the NCCM and the DSMC method was observed. This case is of a high Kn number and in the region of far-equilibrium states.

## 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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**Figure 1.**Comparisons of the normal viscous stress in the nonlinear coupled constitutive method (NCCM) and the Grad, Burnett, and Navier–Stokes–Fourier (NSF) methods.

**Figure 5.**The contours of Ma number, density, and Q

_{x}(heat flux in the × direction) of High Mach Number gas flow around a Cylinder at Ma = 5.48 and Kn = 0.001. NSF (Top), NCCM (Bottom).

**Figure 6.**Normalized Heat Flux around the Solid Surface of High Mach Number gas flow around a Cylinder. (

**a**) Ma = 20, Kn = 0.01; (

**b**) Ma = 20, Kn = 1.0.

Case | 20 × 20 | 40 × 40 | 80 × 80 | 100 × 100 |
---|---|---|---|---|

C_{D} | 0.4101 | 0.3951 | 0.3901 | 0.3902 |

C_{L} | 9.0 × 10^{−}^{3} | 1.23 × 10^{−}^{4} | 1.02 × 10^{−}^{6} | 1.11 × 10^{−}^{6} |

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Tang, 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