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.
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.
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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Abbreviations
- 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
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Lykoudis, P.S., Schmitt, J.P. (1960). Unsteady Compressible Magnetic Laminar Boundary Layers in Hypersonic Flow. In: Hecht, F. (eds) Xth International Astronautical Congress London 1959 / X. Internationaler Astronautischer Kongress / Xe Congrès International d’Astronautique. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-39914-9_51
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