 The main reason for the low pressure inside the hyperloop tubes is to increase efficiency by lowering aerodynamic drag. This can be achieved since drag scales linearly with air density. Thus having 0.1% atmosphere will result in 0.1% aerodynamic drag compared to the drag one would experience at sea level. However, this only holds if the volume around a vehicle can be assumed to be infinitely large, which is a perfectly valid assumption for aircraft. A hyperloop pod violates this assumption by having the walls of the tube closely around the vehicle. In this way, a hyperloop pod somewhat resembles a supersonic wind tunnel with choked flow and shockwaves as a result. This article provides insight into the complex aerodynamics that is involved in a hyperloop pod at full speed.

The drag equation describes the total drag experienced by a body submerged in a gas and is dependent on the dynamics pressure (0.5 ρ V2), the drag coefficient (CL) and the surface of the body (S). The aerodynamic drag increases with the square of the velocity in the subsonic (M<1) region according to the drag equation: $D= 0.5\rho \cdot V^2 \cdot C_L \cdot$ $D= 0.5\rho \cdot V^2 \cdot C_L \cdot$

The upper and lower side around a hyperloop pod act as contracting-expending nozzles which are scenarios that have been studied thoroughly for wind-tunnels and supersonic jet inlets. The flow around the pod accelerates as the bypass area decreases to a maximum of M=1 at the so-called throat (the location where the bypass area is minimum). The airflow will become choked if the speed of the pod increases beyond this point. This causes a large increase in pressure drag and shock waves will start to appear at the tail. This is known as the Kantrowitz limit but is sometimes referred to in the literature as the isentropic contraction limit.

In order to analyse this Kantrowitz limit and its effects, one more concept needs to be introduced, the blockage ratio. This parameter is defined as the ratio of the frontal area of the pod with the total area of the tube. A high blockage ratio means that the tube is close to the side of a hyperloop pod and a low blockage ratio means that there is a lot of space between the hyperloop pod and the tube. The combination between blockage ratio and velocity of the pod determines if the Kantrowitz limit is reached, as visualized in Figure 1. In this figure, the Mach number is used instead of velocity. The Mach number is simply defined as the ratio between the velocity and the local speed of sound (where M=1 equals the speed of sound). All on the right side of the blue line in Figure 1 will result in reaching this limit. This means that for speeds close to the speed of sound, a very low blockage ratio is required to prevent reaching the Kantrowitz limit.

Figure 1: Kantrowitz limit as a function of blockage ratio and Mach number

As said before, the speed of the airflow increases from the free stream velocity in front of the pod to exactly the speed of sound at the throat, at which the flow becomes choked. At this point, the mass flow of air is maximal. That is, if the pod would continue to accelerate, no more additional air would be able to flow alongside the pod increasing the pressure in front of the pod. Simply said; not enough air can flow around the pod so the pod will need to push a pocket of air in front of it.

Another inherent effect of the choked flow is the creation of shockwaves at the tail. The effect on the drag is simulated by using analytical equations which were later verified with Computational Fluid Dynamics (CFD).  A drag profile is shown in Figure 2. This was simulated with a blockage ratio of 0.71, a tube pressure of 100 Pa, pod length of 30m and a temperature of 288.15 K, which is a realistic scenario for the hyperloop pod designed by Delft Hyperloop (see Designing the Delft Hyperloop Passenger Pod). The flow becomes choked once the pod reaches 200 km/h resulting in a sudden drag increase from 40N to 690N.

Elon Musk’s original idea was to put a huge compressor in front of the pod that would reduce this pressure build-up. However, this would require a compression ratio several times higher than current compressors used in jet engines which are already pushing the limit. Besides that, to cope with the heat development inside the compressor, a special cooling system would need to be installed which complicates things even more. A more realistic approach towards designing for the high-speed aerodynamics is to cope with the increased drag by designing the propulsion system accordingly. The shape of the hyperloop pod can be optimized for aerodynamic performance using CFD with the knowledge of the aerodynamic effects explained in this article.

By Delft Hyperloop, March 2019

## 3 Comments

• ### Rafael

I’m curious about the settings you used to do those simulations, specially the boundary conditions you imposed on the left and right ends of the piece of tube

• ### Delft Hyperloop

Thank you for the question! The simulations were done using analytical equations in MATLAB. A mass flow with a constant speed is introduced at the beginning of the tube. For a more detailed explanation, I would like to refer you to the report: The Future of Hyperloop.

• ### Jerry Roane

TriTrack has a measured wind tunnel Cd of 0.07 in open air on an elevated guideway. It also owns the IP for a circular cross section vehicle so many of the Hyperloop designs violate existing patents. The large cabins shown in the press releases are not optimized for moving the human form so the unused standing room shown is a total waste of energy in a system that is claiming to be energy superior. It would be nice if a Hyperloop advocate could document in a wind tunnel the Cd of their particular vehicle in a specific proposed vacuum level like .1 atmosphere or some measurement system for vacuum. It would also be good to know the tube diameter versus the pod diameter or pod rectangle dimensions.