Unsteady Compressible Magnetic Laminar Boundary Layers in Hypersonic Flow

  • Paul S. Lykoudis
  • John P. Schmitt

Abstract

The unsteady hypersonic flow of a compressible, viscous, thermally and electrically conducting fluid is considered in the presence of a magnetic field. The class of non-steady flows is studied for which the velocity at the edge of the boundary layer varies hyperbolically with time.

It is shown that under reasonable restrictions the equations of conservation of total mass, energy, and momentum may be brought into similarity form. An approximate transformation is found which reduces their solution to that of the known nonmagnetic steady case. The basic equations are integrated by means of an electronic analogue computer and the results are compared with the approximate solution. Velocity and enthalpy profiles, wall shear, and heat transfer rates are presented for different decelerations and magnetic field intensities. The effects of the magnetic field and the deceleration rate upon the bow shock wave stand-off distance are also evaluated. It is demonstrated that for reasonable decelerations the steady-state solutions are good approximations.

Keywords

Enthalpy Eter Compressibility Bove Compro 

Nomenclature

a

Acoustic velocity

Magnetic induction vector

Bx, By

x and y components of

Cf

Friction factor defined by Eq. (24)

C0

Stagnation point velocity gradient at zero time

Ct

Stagnation point velocity gradient at general time

c

Radius of curvature of bow shock wave

c1

Proportionality factor defined by Eq. (A-1)

c2, c3

Constants appearing in Eq. (A-2)

f

Velocity function defined through Eq. (6)

fo

Value of f corresponding to nonmagnetic case

h

Enthalpy

k1

Proportionality factor defined by Eq. (A-4)

k2

Proportionality factor defined by Eq. (A-3)

k0, kt

Density functions defined in Eqs. (31) and (32)

m

Power appearing in Eq. (21)

Nu

Nusselt number defined by Eq. (27)

Pr

Prandtl number

p

Static pressure

R

Radius of curvature of blunt vehicle nose

Re

Reynolds number defined by Eq. (25)

S

Stagnation enthalpy function defined by Eq. (6)

T

Temperature

t

Time

U

Velocity

U*

Velocity corresponding to absence of magnetic field

u

Longitudinal velocity component

General velocity vector

v

Normal velocity component

X

Transformed longitudinal coordinate:

x

Longitudinal coordinate measured from stagnation point

Y

Transformed normal coordinate:

y

Normal coordinate

Z

Boundary value of f′ defined by Eq. (15)

α

Non-steadiness parameter defined by Eq. (4)

β

Pressure gradient parameter

Δ

Bow shock wave stand-off distance parameter defined by Eq. (29)

ζ

Magnetic parameter defined by Eq. (10)

η

Similarity parameter defined by Eq. (6)

Similarity parameter defined by Eq. (B-3)

ηo

Value of η according to Eq. (A-7)

λ

Transformation variable defined by Eq. (B-2)

μ

Dynamic viscosity

v

Kinematic viscosity

ϱ

Density

σ

Electrical conductivity

τ

Shear stress

Subscripts

(except those defined above)

e

Condition at the edge of the boundary layer

o

Free stream stagnation value behind bow shock wave

s

Stagnation value inside the boundary layer

w

Condition at the wall

Condition ahead of the bow shock wave

Unstetige verdichtbare magnetische Grenzschicht in hypersonischer Strömung

Zusammenfassung

Unstetige hypersonische Strömungen einer verdichtbaren, zähen, thermisch und elektrisch leitfähigen Strömung in Gegenwart eines magnetischen Feldes werden hier untersucht, und zwar insbesondere solche Strömungen, bei denen sich die Geschwindigkeit am Rande der Grenzschicht hyperbolisch mit der Zeit ändert.

Es wird gezeigt, daß unter vernünftigen Bedingungen die Gleichungen der Erhaltung der Gesamtmasse, der Energie und der Bewegungsgröße in eine Ahnlichkeitsform gebracht werden können. Eine Annäherungs-Transformation wurde gefunden, welche ihre Lösung auf die des bekannten unmagnetischen Stetigkeitsfalles reduziert. Die Grundgleichungen werden mit Hilfe einer elektronischen Analogie-Rechenmaschine integriert; die Resultate werden mit den Näherungslösungen verglichen. Geschwindigkeits- und Enthalpieprofile, Wandreibungsschub und Wärmetransport werden für verschiedene Verzögerungen und magnetische Feldstärken angegeben. Für mäßige Verzögerungen erweisen sich die Näherungslösungen des stetigen Zustandes als ausreichend.

Couches limites laminaires en écoulement hypersonique non stationnaire en présence d’un champ magnétique

Résumé

L’article étudie la classe d’écoulements non stationnaires pour lesquels la vitesse à la limite de la couche a une variation hyperbolique avec le temps.

Sous des hypothèses restrictives raisonnables, les équations de conservation peuvent être mises sous une forme à similitude. Une transformation approchée réduit leur solution à celle du cas connu: permanent et sans champ magnétique. Les équations de base sont intégrées sur machine électronique analogue et les résultats sont comparés avec la solution approchée. Pour diverses valeurs de la décélération et du champ, des résultats sont donnés pour les profils de vitesse, de tensions tangentielles d’enthalpie et de taux de transfert de chaleur. Pour des valeurs raisonnables de la décélération les solutions permanentes constituent une bonne approximation.

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References

  1. 1.
    H. J. Allen and A. J. Eggers, A Study of the Motion and Aerodynamic Heating of Missiles Entering the Earth’s Atmosphere at High Supersonic Speeds. NACA Technical Note 4047, October 1957.Google Scholar
  2. 2.
    W. B. Bush, Magnetohydrodynamic-Hypersonic Flow Past a Blunt Body. J. Aero/Space Sci. 25, 685 (1958).CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    C. B. Cohen and E. Reshotko, Similar Solutions for the Compressible Laminar Boundary Layer with Heat Transfer and Pressure Gradient. NACA Report 1293, 1956.Google Scholar
  4. 4.
    S. Feldman, Hypersonic Gas Dynamic Charts for Equilibrium Air. AVCO Research Laboratory, January 1957.Google Scholar
  5. 5.
    N. H. Kemp, On Hypersonic Stagnation-Point Flow with a Magnetic Field. J. Aeronaut. Sci. (Readers’ Forum) 25, 405 (1958).MATHMathSciNetGoogle Scholar
  6. 6.
    T. Y. Li and R. E. Geiger, Stagnation Point of a Blunt Body in Hypersonic Flow. J. Aeronaut. Sci. 24, 25 (1957).CrossRefMATHGoogle Scholar
  7. 7.
    P. S. Lykoudis, On a Class of Compressible Laminar Boundary Layers with Pressure Gradient for an Electrically Conducting Fluid in the Presence of a Magnetic Field. Proceedings of the IXth International Astronautical Congress, Amsterdam 1958, p. 168. Wien: Springer, 1959.Google Scholar
  8. 8.
    P. S. Lykoudis, The Matching of the Viscid and Inviscid Regions for the Stagnation Magnetic Flow. J. Aero/Space Sci. (Readers’ Forum) 26, 315 (1959).CrossRefGoogle Scholar
  9. 9.
    J. L. Neuringer and W. Mcilaoy, Incompressible Two-Dimensional Stagnation-Point Flow of an Electrically Conducting Viscous Fluid in the Presence of a Magnetic Field. J. Aeronaut. Sci. 25, 194 (1958).MATHGoogle Scholar
  10. 10.
    V. J. Rossow, Magnetohydrodynamic Analysis of Heat Transfer near a Stagnation Point. J. Aeronaut. Sci. (Readers’ Forum) 25, 334 (1958).Google Scholar
  11. 11.
    K. T. Yang, Unsteady Laminar Boundary Layers in an Incompressible Stagnation Flow. American Society of Mechanical Engineers paper no. 58-A-3, presented at ASME Annual Meeting, New York, December 1958.Google Scholar
  12. 12.
    R. W. Ziemer, Experimental Magneto-Aerodynamics. American Rocket Society paper no. 707–58, presented at ARS 13th Annual Meeting, New York, November 1958.Google Scholar

Copyright information

© Springer-Verlag Wien 1960

Authors and Affiliations

  • Paul S. Lykoudis
    • 1
    • 2
  • John P. Schmitt
    • 1
  1. 1.Allison DivisionGeneral Motors CorporationIndianapolisUSA
  2. 2.Purdue UniversityLafayetteUSA

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