Abstract
The strength of characteristic waves is related to the local gradient and streamline curvature. This relationship and the equations giving the pressure gradient and streamline curvature are used to determine the relative strengths (the reflection coefficient) of characteristics just downstream of a two-dimensional curved shock wave. It is shown that the characteristics’ strengths are a complex function of the specific heat ratio, the upstream Mach number and shock angle and vary directly with the shock curvature. The reflection coefficient, which is independent of shock curvature, is used to characterise four different types of shock wave whose existence depends on the specific heat ratio of the gas and the upstream flow Mach number. The nature of reflection at the shock’s downstream surface may change up to four times and this is posed as the explanation for the inflected shocks that have been observed both experimentally and computationally. It is concluded that such approximate analytical methods as the Tangent-Wedge should not be used when strong curved shocks are present and that the nature of wave reflections behind a weak shock in air is not properly simulated by tests with helium.
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Abbreviations
- μ± :
-
Mach angle(=±sin(1/M)
- η± :
-
distance measured along characteristics
- S :
-
distance measured along streamline
- n :
-
distance measured normal to streamline
- C± :
-
characteristic lines
- p :
-
static pressure
- ρ :
-
static density
- V :
-
Flow speed
- π± :
-
strengths of C± — characteristics
- P :
-
non-dim. press, gradient along streamline
- D :
-
streamline curvature
- δ :
-
flow deflection
- M :
-
Mach number
- A…C :
-
coefficients in compatibility equations
- A′…C′ :
-
coefficients in compatibility equations
- S a :
-
shock wave curvature measured in a plane which contains both the up-and downstream flow vectors at the shock
- S b :
-
shock wave curvature measured in a plane perpendicular both to the shock and to the plane containing S a
- θ :
-
angle between upstream flow vector and plane of shock
- γ :
-
ratio of specific heats
- λ :
-
reflection coefficient (= π−/π+)
- R :
-
radius of curvature of the shock corresponding to curvature S =−1/R
- 1, 2:
-
up- and downstream sides of the shock wave
- con:
-
conical flow
- *:
-
sonic conditions
- +:
-
refers to the characteristic which is inclined at acute angle +μ to the streamline
- −:
-
refers to the characteristic which is inclined at acute angle −μ to the streamline
- a, b :
-
refer to streamwise and transverse shock curvatures
References
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© 1995 Springer-Verlag Berlin Heidelberg
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Mölder, S. (1995). Strength of Characteristics at a Curved Shock Wave. In: Brun, R., Dumitrescu, L.Z. (eds) Shock Waves @ Marseille I. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-78829-1_14
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DOI: https://doi.org/10.1007/978-3-642-78829-1_14
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