Shock Waves

, Volume 29, Issue 2, pp 297–306 | Cite as

In-pipe aerodynamic characteristics of a projectile in comparison with free flight for transonic Mach numbers

  • R. HruschkaEmail author
  • D. Klatt
Original Article


The transient shock dynamics and drag characteristics of a projectile flying through a pipe 3.55 times larger than its diameter at transonic speed are analyzed by means of time-of-flight and pipe wall pressure measurements as well as computational fluid dynamics (CFD). In addition, free-flight drag of the 4.5-mm-pellet-type projectile was also measured in a Mach number range between 0.5 and 1.5, providing a means for comparison against in-pipe data and CFD. The flow is categorized into five typical regimes the in-pipe projectile experiences. When projectile speed and hence compressibility effects are low, the presence of the pipe has little influence on the drag. Between Mach 0.5 and 0.8, there is a strong drag increase due to the presence of the pipe, however, up to a value of about two times the free-flight drag. This is exactly where the nose-to-base pressure ratio of the projectile becomes critical for locally sonic speed, allowing the drag to be estimated by equations describing choked flow through a converging–diverging nozzle. For even higher projectile Mach numbers, the drag coefficient decreases again, to a value slightly below the free-flight drag at Mach 1.5. This behavior is explained by a velocity-independent base pressure coefficient in the pipe, as opposed to base pressure decreasing with velocity in free flight. The drag calculated by CFD simulations agreed largely with the measurements within their experimental uncertainty, with some discrepancies remaining for free-flying projectiles at supersonic speed. Wall pressure measurements as well as measured speeds of both leading and trailing shocks caused by the projectile in the pipe also agreed well with CFD.


Projectile Pipe Transonic 

List of symbols

\(\rho \)


\(\gamma \)

Ratio of specific heat capacities

\(A_{{\mathrm {d}}}\)

Area of pipe cross section

\(A_{{\mathrm {p}}}\)

Projectile projected area


Area in smallest cross section \(\left( A_{{\mathrm {d}}}-A_{{\mathrm {p}}}\right) \)

\(a_{{\mathrm {p}}}\)

Projectile acceleration

\(C_{{\mathrm {d}}}\)

Drag coefficient

\(C_{{\mathrm {d,n}}}\)

Nose drag coefficient

\(C_{{\mathrm {d,b}}}\)

Base drag coefficient


Diameter of the projectile


Mach number

\(M_{{\mathrm {p}}}\)

Projectile Mach number

\(m_{{\mathrm {p}}}\)

Projectile mass


Mass flux rate

\(\dot{m}_{{\mathrm {n}}}\)

Mass flux rate through smallest cross section


Static pressure

\(p_{{\mathrm {t}}}\)

Total pressure


Specific gas constant




Static temperature

\(T_{{\mathrm {t}}}\)

Total temperature



\(v_{{\mathrm {p}}}\)

Projectile velocity

\(v_{{\mathrm {s,l}}}\)

Leading shock velocity

\(v_{{\mathrm {s,t}}}\)

Trailing shock velocity


Projectile position


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.French-German Research Institute of Saint LouisSaint-Louis CedexFrance

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