Descent Dynamics

Descent Characteristics

Reynolds Number Regime

Relative Descent Dynamics

Booster Section

Assuming we are falling horizontally with flow conditions above the critical Reynolds number, we expect a drag coefficient (based on diameter) around 1.6. [1] This gives us a C_dS of 0.53 m², given a diameter of 6 in = 0.1524 m and a length of 85.73 in = 2.178 m (this neglects the fins). If we are falling axially, we expect a drag coefficient of around 0.8 [2]. Given a reference area of a circle with diameter 6 in (π*(0.0762m)² = 0.0182 m^₂), we find that the C_dS is 0.0146 m².

It is difficult to predict in which configuration the booster will fall. On the one hand, if it falls horizontally, there will be greater drag-based restoring forces to counteract any perturbations to its configuration. On the other hand, if it falls axially, perturbations will have less of an effect due to smaller moment arms. This analysis is complicated by the fact that we are unsure of the rocket's orientation during deployment and the effect of altitude.

One way to predict things is to look at videos of drogue or tumble recovery experienced by other rockets. We have summarized the findings in the table below:

Drogue

The drogue has a C_dS of (this is at Mach 0.137) 0.662 m².

Mission Package

Assuming that the mission package is falling nose down (it is planned to be stabilized that way via Hermes Payload System), it will have a coefficient of drag of approximately 0.2, given a fineness ratio of approximately 9 [2].Then, taking the same reference area as the axial configuration for the booster, we find a C_dS of 0.00364.

Descent Rates

Drogue

Given the previously mentioned C_dS of 0.662 m² at Mach 0.137 (which corresponds to approximately 150 ft/s at sea level), an air density of 1.225 kg/m³ at sea level, and a dry rocket mass of 89.83 lbs = 40.75kg (taken from the Hermes Mass Budget on 1/31/2018–this includes the drogue parachute itself, but we'll ignore this for now), we can calculate the approximate descent of the rocket under drogue:

$//<![CDATA[ \begin{array}{l}F_{D, dro} = \frac{1}{2} \rho C_{D} S V^2 = m_{dry}*g\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}F_{D, dro} = \frac{1}{2}*1.225 \frac{kg}{m^3}*0.662 m^2 *V^2 = 40.75 kg * 9.81 \frac{m}{s^2}\end{array} //]]>$

V ≈ 31.4 m/s = 103 ft/s

This velocity is well within an acceptable range per Requirement 3.1.1 of the DTEG.

One important caveat to this analysis is that this assumes the drogue will be pulling the entire rocket. As we see above, the C_dS of the drogue and that of the booster section may be comparable, depending on the falling configuration. If this occurs, there will not be tension on the booster webbing, and the drogue therefore will not be "carrying its weight." Hermes' descent under drogue will occur faster. ¹ To take this into account, we can bound the velocity neglecting the weight of the booster section. According to the mass budget, the propulsion system plus fin can weigh 

¹ Admittedly, there is some super weird coupling in this system. If the drogue and mission package start to travel faster downwards than the booster section, eventually the webbing will be fully extended between the mission package and the booster and it will start to pull it down. Then maybe you'd enter some sort of periodic pattern of the webbing jerking the booster down and it falling behind again? I am not sure but I'm going to neglect this from my analysis.

Main

Relative Descent Rates

Drogue to Mission Package

Because the CdS of the drogue is 2 orders of magnitude greater than that on the mission package, we expect there to be a good deal of tension in the line connecting the drogue and the main. At sea level, there will

Resources

[1] DRAG OF CIRCULAR CYLINDERS FOR A WIDE RANGE OF REYNOLDS NUMBERS AND MACH NUMBERS

[2] Aerospaceweb.org, Drag of Cylinders and Cones