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Shock Wave Reflections in Pseudo-Steady Flows

  • Gabi Ben-Dor

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.

Keywords

Shock Wave Triple Point Transition Line Wedge Angle Incident Shock Wave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of Symbols

Latin Letters

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

Mij

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

Vab

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

Vaxb

x component of V a b

Vayb

y component of V a b

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

Greek Letters

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 w tr (ε)/θ w tr (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

θwC

complementary wedge angle

θwtr

transition wedge angle

θwtr(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

Subscripts

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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Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Gabi Ben-Dor
    • 1
  1. 1.Department of Mechanical EngineeringBen-Gurion University of the NegevBeer-ShevaIsrael

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