$ M_- $ | 2 | 5 | 10 | 20 | 40 | 60 |
$ \max\epsilon $ | 0.2786 | 0.5506 | 0.6137 | 0.6387 | 0.6478 | 0.6499 |
$ \mathrm{Pr}_* $ | 0.5184 | 0.5454 | 0.5539 | 0.5576 | 0.5590 | 0.5593 |
$ (3/2)\mathrm{Pr}_* $ | 0.7776 | 0.8181 | 0.8309 | 0.8364 | 0.8385 | 0.8390 |
Entropic property of the Ellipsoidal Statistical model with the Prandtl number Pr below 2/3 is discussed. Although 2/3 is the lower bound of Pr for the H theorem to hold unconditionally, it is shown that the theorem still holds even for $ \mathrm{Pr}<2/3 $, provided that anisotropy of stress tensor satisfies a certain criterion. The practical tolerance of that criterion is assessed numerically by the strong normal shock wave and the Couette flow problems. A couple of moving plate tests are also conducted.
Citation: |
Figure 1. Structure of the shock wave for typical Mach numbers $ M_- $ in the case of $ \mathrm{Pr} = 2/3 $. (a) $ M_- = 2 $, (b) $ M_- = 5 $, (c) $ M_- = 20 $, and (d) $ M_- = 40 $. Here, $ \ell_- $ is the mean free path of molecules in the equilibrium state at rest with density $ \rho_- $ and temperature $ T_- $. In each panel, solid lines indicate $ \rho_* = (\rho-\rho_-)/(\rho_+-\rho_-) $, $ u_* = (v_1-u_+)/(u_–u_+) $, and $ T_* = (T-T_-)/(T_+-T_-) $ respectively, while a dash-dotted line indicates $ \epsilon $
Figure 2. The maximum value of $ \epsilon $ vs. $ (3/2)\mathrm{Pr} $ for various upstream Mach numbers $ M_- $. Symbols indicate the results of computations. The results of common $ M_- $ are connected by solid lines in the acceptable range of $ \mathrm{Pr} $ and by dashed lines in the range where condition (8) is violated
Figure 4. The function $ \epsilon_P $ and the dimensionless density $ \hat{\rho} $ in the range $ -5<\hat{U}<5 $. (a) $ \epsilon_P $, (b) $ \hat{\rho} $. In (a), the values of $ \mathcal{S}(\mathrm{Pr}) $ for $ (3/2)\mathrm{Pr} = 0.76,0.8,0.84,\dots,0.96 $ are also indicated by dash-dotted lines for reference
Table 1.
The maximum
$ M_- $ | 2 | 5 | 10 | 20 | 40 | 60 |
$ \max\epsilon $ | 0.2786 | 0.5506 | 0.6137 | 0.6387 | 0.6478 | 0.6499 |
$ \mathrm{Pr}_* $ | 0.5184 | 0.5454 | 0.5539 | 0.5576 | 0.5590 | 0.5593 |
$ (3/2)\mathrm{Pr}_* $ | 0.7776 | 0.8181 | 0.8309 | 0.8364 | 0.8385 | 0.8390 |
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Structure of the shock wave for typical Mach numbers
The maximum value of
The functions
The function