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Normal shock tables

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In aerodynamics, the normal shock tables are a series of tabulated data listing the various properties before and after the occurrence of a normal shock wave. With a given upstream Mach number, the post-shock Mach number can be calculated along with the pressure, density, temperature, and stagnation pressure ratios. Such tables are useful since the equations used to calculate the properties after a normal shock are cumbersome.

The tables below have been calculated using a heat capacity ratio, γ , equal to 1.4. The upstream Mach number, M 1 , begins at 1 and ends at 5. Although the tables could be extended over any range of Mach numbers, stopping at Mach 5 is typical since assuming γ to be 1.4 over the entire Mach number range leads to errors over 10% beyond Mach 5.

Normal shock table equations

Given an upstream Mach number, M 1 , and the ratio of specific heats, γ , the post normal shock Mach number, M 2 , can be calculated using the equation below.

M 2 = M 1 2 ( γ − 1 ) + 2 2 γ M 1 2 − ( γ − 1 )

The next equation shows the relationship between the post normal shock pressure, p 2 , and the upstream ambient pressure, p 1 .

p 2 p 1 = 2 γ M 1 2 γ + 1 − γ − 1 γ + 1

The relationship between the post normal shock density, ρ 2 , and the upstream ambient density, ρ 1 is shown next in the tables.

ρ 2 ρ 1 = ( γ + 1 ) M 1 2 ( γ − 1 ) M 1 2 + 2

Next, the equation below shows the relationship between the post normal shock temperature, T 2 , and the upstream ambient temperature, T 1 .

T 2 T 1 = ( 1 + γ − 1 2 M 1 2 ) ( 2 γ γ − 1 M 1 2 − 1 ) M 1 2 ( 2 γ γ − 1 + γ − 1 2 )

Finally, the ratio of stagnation pressures is shown below where p 01 is the upstream stagnation pressure and p 02 occurs after the normal shock. The ratio of stagnation temperatures remains constant across a normal shock since the process is adiabatic.


p 02 p 01 = ( γ + 1 2 M 1 2 1 + γ − 1 2 M 1 2 ) γ γ − 1 ( 1 2 γ γ + 1 M 1 2 − γ − 1 γ + 1 ) 1 γ − 1

References

Normal shock tables Wikipedia
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