Skip to main content
SpringerLink
Log in

The hypersonic Mach number independence principle in the case of viscous flow

  • Original Article
  • Published:
Shock Waves Aims and scope Submit manuscript

Abstract

The hypersonic Mach number independence principle of Oswatitsch is important for hypersonic vehicle design. It explains why, above a certain flight Mach number (M ∞ ≈ 4−6, depending on the body shape), some aerodynamic properties become independent of the flight Mach number. For ground test facilities this means that it is sufficient for the Mach number in the test section to be high enough, that Mach number independence exists. However, the principle was derived for calorically perfect gas and inviscid flow only. In this paper a theoretical study for blunt bodies in the case of viscous flow is presented. We provide numerical results which give insight into how attached viscous flow behaves at high Mach numbers. The flow past an axisymmetric configuration is analysed by applying a coupled Euler/second-order boundary-layer method. Wall boundaries are treated by assuming an adiabatic or radiation-adiabatic wall for laminar flow. Calorically perfect or equilibrium air is accounted for. Lift, drag, and moment coefficients, and lift-to-drag ratios are given for several combinations of flight Mach number and altitude, i.e. Reynolds number. For blunt bodies considered here, which are pressure dominated, Mach number independence occurs for the adiabatic wall, but not for the radiation-adiabatic wall assumption.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

a :

Speed of sound (m/s)

b :

Shear-stress term [kg/(m2 s2)]

c :

Coefficient

c p :

Specific heat capacity at p = const. [J/(kg K)]

c v :

Specific heat capacity at v = const. [J/(kg K)]

C D :

Drag coefficient

C L :

Lift coefficient

C m :

Moment coefficient

C p :

Pressure coefficient

D:

Drag (N)

h 0 :

Specific total enthalpy (J/kg)

H :

Altitude (km)

L:

Lift (N)

L/D:

Lift-to-drag ratio

M ∞ :

Free-stream Mach number

p :

Static pressure (Pa)

R N :

Nose radius (m)

Re ∞,u :

Unit Reynolds number (m−1)

s :

Specific entropy [J/(kg K)]

T :

Static temperature (K)

u, v, w:

Velocity (m/s)

\({\underline{V}}\) :

Velocity vector (m/s)

x, y, z:

Cartesian coordinates (m)

γ :

Ratio of specific heats

δ :

Hyperboloid asymptotic angle (°)

ε :

Surface radiation emissivity

θ :

Shock angle (°)

μ :

Dynamic viscosity [kg/(m s)]

ρ :

Density (kg/m3)

τ :

Shear-stress (Pa)

KX,MX:

Number of grid points

wall:

At the wall

′:

Dimensionless properties

∞:

Free-stream

References

  1. Oswatitsch, K.: Ähnlichkeitsgesetz für Hyperschallströmung. ZAMP, vol. II, pp. 249–264 (1951). (Similarity Laws for Hypersonic Flow. Royal Institute of Technology. Stockholm. Sweden. KTH-AERO TN 16 (1950))

  2. Oswatitsch, K.: Contributions to the Development of Gasdynamics. In: Schneider, W., Platzer, M. (eds.), pp. 76–88. Vieweg Verlag (1980)

  3. Hayes, W.D., Probstein, R.F.: Hypersonic Flow Theory. In: Inviscid Flows, vol. 1. Academic Press, New York (1966)

  4. Hirschel, E.H.: Basics of Aerothermodynamics. In: Progress in Astronautics and Aeronautics, AIAA, vol. 204. Springer, Berlin (2005)

  5. Schneider W.: Reibungsfreie Hyperschallströmung eines realen Gases um einen angestellten Kreiskegel. Journal de Mécanique 5(1), 45–67 (1966)

    MathSciNet  MATH  Google Scholar 

  6. Oswatitsch, K.: Spezialgebiete der Gasdynamik. Schallnähe, Hyperschall. Tragflächen, Wellenausbreitung, pp. 163–167. Springer, Berlin (1977)

  7. Hirschel, E.H., Weiland, C.: Selected Aerothermodynamic Design Problems of Hypersonic Flight Vehicles. In: Progress in Astronautics and Aeronautics, AIAA, vol. 229. Springer, Berlin (2009)

  8. Mundt, Ch.: Calculation of Hypersonic Viscous Non-Equilibrium Flows Around Reentry Bodies Using a Coupled Boundary Layer/Euler Method. In: AIAA-1992-2856, 27th Thermophysics Conference (1992)

  9. Pfitzner, M., Weiland, C.: 3-D Euler Solution for Hypersonic Mach Numbers. In: AGARD-CP-428, pp. 22-1–22-14 (1987)

  10. Boyce, R.R., Mundt, Ch.: Computational fluid dynamic code validation using a free piston hypervelocity shock tunnel. In: Proceedings of the 18th International Symposium on Shock Waves, Sendai, Japan, 21–26 July (1991)

  11. Boyce, R.R., Houwing, F., Mundt, Ch., Morton, J.: CFD validation of real gas air flows over a blunt body using multiple interferometric views. In: AIAA-1994–2599 (1994)

  12. Mundt, Ch., Boyce, R.R., Hirschel, E.H.: Simulation of hypersonic, reacting flow around reentry bodies—comparisons of numerical and experimental results. Zeitschrift für Flugwissenschaften und Weltraumforschung. Bd. 19, pp. 492–501. Springer, Berlin (1995)

  13. Boyce, R.R., Mundt, Ch., Houwing, F.: Computational Fluid Dynamics Validation using Surface Heat Transfer Measurements of hypersonic Flowfields. In: AIAA-1996-351 (1996)

  14. Mundt, Ch., Keraus, R., Fischer, J.: New, Accurate, Vectorized Approximations of State Surfaces for Thermodynamic and Transport Properties of Equilibrium Air. Zeitschrift für Flugwissenschaften und Weltraumforschung, Bd. 15, pp. 179–184. Springer, Berlin (1991)

  15. Adams, J.C., et al.: Real Gas Scale Effects on Hypersonic Lamiar Boundary-Layer Parameters Including Effects of Entropy-Layer Swallowing. In: AIAA-1976–358 (1976)

Download references

Author information

Authors and Affiliations

  1. IABG mbH, Defence & Security, 85521, Ottobrunn, Germany

    D. Kliche

  2. Institut für Thermodynamik, Universität der Bundeswehr München, 85577, Neubiberg, Germany

    Ch. Mundt

  3. Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, Stuttgart, Germany

    E. H. Hirschel

Authors
  1. D. Kliche
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ch. Mundt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. H. Hirschel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Kliche.

Additional information

Communicated by R. R. Boyce.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kliche, D., Mundt, C. & Hirschel, E.H. The hypersonic Mach number independence principle in the case of viscous flow. Shock Waves 21, 307–314 (2011). https://doi.org/10.1007/s00193-011-0318-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00193-011-0318-y

Keywords

Search

Navigation