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Shock Wave Reflection Phenomena pp 38–174Cite as

Shock Wave Reflections in Pseudo-Steady Flows

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Abstract

Shock waves having constant velocities with respect to an inertial frame of reference can be investigated using steady flow concepts by attaching a frame of reference to the shock wave. In such a frame of reference, the shock wave is stationary, and the entire flow field is known as either pseudo-stationary or pseudo-steady. The transformation, known as a Galilean transformation, is shown schematically in figure 2.1. In figure 2.1a a constant velocity shock wave, having a velocity of Vs, is propagating from left to right, towards a flow having a velocity Vj, and inducing behind it a flow velocity Vj. The velocities with respect to a frame of reference attached to the shock wave are shown in figure 2.1b. In this frame of reference the flow in state (i) propagates towards the stationary shock wave with the velocity Uj = Vs - Vj. Upon passing through the shock wave its velocity is reduced to u: = Vs - Vj. The velocity field shown in figure 2.1b is actually obtained from the velocity field shown in figure 2.1a by superimposing on it a velocity equal to the shock wave velocity but in the opposite direction. The flow field of figure 2.1b is pseudo-steady, and hence can be treated using the steady flow theory for oblique shock waves.

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Abbreviations

ai :

local speed of sound in state (i)

Aij :

ai/aj

Cp :

specific heat capacity at constant pressure

Cv :

specific heat capacity at constant volume

hi :

enthalpy in state (i)

k:

thermal conductivity

ld :

dissociational relaxation length

lv :

vibrational relaxation length

L:

Lr/LT

LK :

horizontal distance of the kink from the leading edge of the reflecting wedge

Lr :

length of the reflected shock wave in TMR and DMR

LT :

horizontal distance of the triple point from the leading edge of the reflecting wedge

M ji :

flow Mach number in state (i) with respect to point (j)

pi :

static pressure in state (i)

Pr :

Prandtl number

Pij :

pi/pj

t:

time

Ti :

flow temperature in state (i)

ui :

flow velocity in state (i) with respect to R in RR or T in MR

u’i :

flow velocity in state (i) with respect to K in TMR or T in DMR

u’ix :

x component of u’i

uiy :

y component of u’i

V ba :

velocity of point “a” with respect to point “b”

V bax :

x component of V ba

V bay :

y component of V ba

Vij :

Vi/aj

Vi :

flow velocity in state (i) in a laboratory frame of reference

Vs :

incident shock wave velocity

x:

coordinate

Xchar :

characteristic length

y:

coordinate

γ:

specific heat capacities ratio

δ:

kinematic boundary layer thickness

γT :

thermal boundary layer thickness

γ*:

boundary layer displacement thickness

Δt :

time interval

ε:

roughness

ζ:

boundary layer displaced angle

η:

θ trw (ε)/θ trw (0)

θi :

deflection angle of the flow while passing across an oblique shock wave into state (i) with respect to R in RR or T in MR

θ’i :

deflection angle of the flow while passing across an oblique shock wave into state (i) with respect to K in TMR or T in DMR

θw :

reflecting wedge angle

θ Cw :

complementary wedge angle

θ trw :

transition wedge angle

θ trw (ε):

transition wedge angle over a surface having a roughness ε

λ:

mean free path

μ:

dynamic viscosity

ζ:

inverse pressure ratio across the incident shock wave (= P01)

ρi :

flow density in state (i)

Φi :

angle of incidence between the flow and the oblique shock wave across which the flow enters state (i) with respect to RinRRorTinMR

Φ’i :

angle of incidence between the flow and the oblique shock wave across which the flow enters state (i) with respect to K in TMR or T in DMR

χ:

first triple point trajectory angle in MR

χ’:

kink trajectory angle in TMR or second triple point trajectory angle in DMR

χg :

first triple point trajectory angle in MR at glancing incidence

ωij :

angle between the discontinuities i and j

0:

flow state ahead of the incident shock wave, i.

1:

flow state behind the incident shock wave, i.

2:

flow state behind the reflected shock wave, r.

3:

flow state behind the Mach stem, m.

4:

flow state behind the secondary Mach stem, m’.

5:

flow state behind the secondary reflected shock wave, r’

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Authors and Affiliations

  1. Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    Gabi Ben-Dor

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  1. Gabi Ben-Dor
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© 1992 Springer Science+Business Media New York

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Ben-Dor, G. (1992). Shock Wave Reflections in Pseudo-Steady Flows. In: Shock Wave Reflection Phenomena. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4279-4_2

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  • DOI: https://doi.org/10.1007/978-1-4757-4279-4_2

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  • Print ISBN: 978-1-4757-4281-7

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