I am considering the flow field around a hypersonic glide vehicle at $M=7$. As I understand it, air is in chemical equilibrium if the fluid time scale $t_f \ll t_c$, where $t_c$ is the chemical time scale. I can estimate $t_f$ very easily, but I have no clue how to estimate $t_c$.

Could someone tell me how to estimate $t_c$ for my problem ?

  • $\begingroup$ what chemical kinetic reaction is expected to occur in air at Mach 7? In other words is air expected to react and create products at Mach 7? That's the first step: determine the chemical equation in terms of reactants, products and energy. $\endgroup$
    – docscience
    Aug 19, 2015 at 23:39
  • 2
    $\begingroup$ Just to clarify the terminology, the flow is in (or near) equilibrium when the chemical time scale is small compared to the flow time. When the flow time scale is small relative to the chemical time scale, the flow is called frozen. $\endgroup$
    – John1024
    Aug 20, 2015 at 5:21

1 Answer 1


The chemical kinetics of air depend on both how fast you are flying and your altitude. Fortunately, NASA has studied these issues. The figures below are from NASA Report NACA-TN-4359. The predominant chemistry in the stagnation region of an airfoil as a function of flight speed and altitude are shown below:

enter image description here

You say $M=7$. If your vehicle is near sea level, then that is around 6,500 feet per second which, according to the above plot, would mean that vibrational kinetics of the molecules are important but that chemical dissociation reactions are not yet important.

You ask how to estimate chemical reaction time, $t_c$. The same concept can be characterized by chemical reaction length: $d_c = u \times t_c$ where $d_c$ is the characteristic length for the reactions and $u$ is the flow speed. This is handy because it is easy to compare with the dimensions of your vehicle. NASA calculates the characteristic lengths for vibration excitation as follows:

enter image description here

As you can see, for a given flow speed, the characteristic length varies strongly with altitude.

The air is in vibrational equilibrium when the the characteristic length from the plot above is much smaller than the characteristic lengths of your glide vehicle and it's airfoils.

At higher altitudes or higher speeds, molecular dissociation becomes important (see Fig 1 above). NASA calculates the characteristic lengths for oxygen dissociation as follows:

enter image description here

Again, the characteristic length depends strongly on both speed and altitude.

The air will be in (or near) equilibrium for oxygen dissociation when the characteristic length from the plot above is much smaller than the characteristic lengths of your vehicle.


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