Table 2.
Spectral slope and the inferred Mach number for our different shock tube models.
| Flat | Turb | Flat | Turb | ||
|---|---|---|---|---|---|
| Theory | Initial ℳ | 2.00 | 2.00 | 3.00 | 3.00 |
| Expected slope | –3.33 | –3.33 | –2.5 | –2.5 | |
| Low momenta | Measured slope | -3.36 | –2.73 | –2.49 | -2.42 |
| Implied ℳ | 1.98 | 2.54 | 3.03 | 3.25 | |
| High momenta | Measured slope | –4.28 | –3.50 | –3.45 | -3.35 |
| Implied ℳ | 2.03 | 3.01 | 3.15 | 3.54 | |
Notes. Top two rows: Initial Mach number for each simulation and the expected slope calculated using Eq. (15). Middle two rows: Measured slope at low (10 < p < 103) momenta and the Mach number that one would infer using Eq. (14). Bottom two rows: As previous but for high momenta (104 < p < 105) and making the typical assumption that this slope is αe − 1. The tail of the Mach number distribution produces shallower slopes and breaks the αe − 1 assumption. These effects both lead to an overestimation of the peak Mach number in the shock.
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