Worked Example

Credits: F-22 Demonstration Team

The mighty F-22 raptor is traveling at its supercruise speed of Mach 1.8 when it turns up at an angle of 7 degrees. The airfoil can be modeled as a symmetrical rhombus with 5 degree half angles as pictured below. Calculate the Mach numbers on surface 1, 2, 3, and 4.

Solution

From the figure we come to the conclusion that we will have Prandtl-Meyer expansion fans from the freestream to surface 1, and from surface 1 to surface 2. Using the functions prandtl_meyer_angle_from_mach and prandtl_meyer_mach_from_angle, we can get the Mach number information pretty quickly.

>>> import gas_dynamics as gd
>>> mach_initial = 1.8
>>> angle_init = gd.prandtl_meyer_angle_from_mach(mach_initial)
>>> angle_init
20.72506424776447
>>> angle_surface_1 = angle_init + 2
>>> mach_surface_1 = gd.prandtl_meyer_mach_from_angle(angle_surface_1)
>>> mach_surface_1
1.869672760055039
>>> angle_surface_2 = angle_surface_1 + 10
>>> mach_surface_2 = gd.prandtl_meyer_mach_from_angle(angle_surface_2)
>>> mach_surface_2
2.2385201020582
>>>

We can also get the corresponding pressures on these surfaces using pressure_from_mach_ratio

>>> pressure_initial = 1
>>> pressure_surface_1 = gd.pressure_from_mach_ratio(pressure_initial=pressure_initial, mach_initial=mach_initial, mach_final=mach_surface_1)
>>> pressure_surface_1
0.8985710141625418
>>> pressure_surface_2 = gd.pressure_from_mach_ratio(pressure_initial=pressure_surface_1, mach_initial=mach_surface_1, mach_final=mach_surface_2)
>>> pressure_surface_2
0.5059159131613694
>>>

For surfaces 3 and 4, we notice that we will first have a shock as the flow is turning into itself, followed by another Prandtl-Meyer expansion from surface 3 to 4. Before proceeding with shock case, lets import the sine function that works in degrees from the “extra” module.

>>> from gas_dynamics.extra import sind
>>> shock_angle = gd.shock_angle(mach=mach_initial, flow_deflection=12)
>>> shock_angle
[46.685991854146, 80.21459032863389]
>>> mach_initial_normal = mach_initial * sind(shock_angle[0])
>>> mach_initial_normal
1.3096891104605881
>>> mach_surface_3_normal = gd.shock_mach(mach_initial_normal)
>>> mach_surface_3_normal
0.7810840690884703
>>> mach_surface_3 = mach_surface_3_normal / sind(shock_angle[0] - 12)
>>> mach_surface_3
1.3725418664628979

We conclude with the final Prandtl-Meyer expansion from surface 3 to 4.

>>> angle_surface_3 = gd.prandtl_meyer_angle_from_mach(mach_surface_3)
>>> angle_surface_4 = angle_surface_3 + 10
>>> mach_4 = gd.prandtl_meyer_mach_from_angle(angle_surface_4)
>>> mach_4
1.7132972931071317
>>>