Summary
For hypersonic flow past an elliptic cone with longitudinal curvature, the outer-expansion analysis of shock layer and the inner-expansion analhsis of vortical layer are two major subjects in this research. The zeroth-order approximation of hypersonic conical flow obtained by nonlinear asymptotic theory is chosen as the basic-cone solution for the outer and inner expansions. In the outer analysis of shock layer, the complicated governing equation of outer flowfield can be simplified by an appropriate approximation scheme, and the first-order approximations of properties are derived. In sequence, to study the phenomenon in vortical layer, the inner expansion based on the outer solutions is proceeded near the cone surface. Thereafter, in accordance with the asymptotic matching principle, the uniformlyvalid approximation can be obtained and expressed in closed form. Moreover, it is verified that the perturbation pressure and azimuthal velocity from outer solutions are actually the perturbation expansion values from inner solutions. However, the analysis of inner solutions illustrates apparently the rapid variations on entropy and density of the vortical layer. These results enable us to understand well about the flow characteristics in the entire shock layer, and may provide an important guide for grid arrangement to accomodate the dramatic adjustment occurring in the vortical layer.
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Abbreviations
- a :
-
speed of sound
- ê r , êθ, êø :
-
unit vectors in spherical coordinates
- G m :
-
shock-perturbation factor
- H :
-
unified supersonic-hypersonic similarity parameter,\(\sqrt {M_\infty ^2 - 1} \delta\)
- h t :
-
total enthalpy
- I n :
-
modified Bessel function of first kind andnth order
- K n :
-
modified Bessel function of second kind andnth order
- K δ :
-
hypersonic similarity parameter where M∞δ
- M:
-
Mach number
- P :
-
pressure
- R, Ψ, Φ:
-
inner variable
- r, θ, ϕ:
-
spherical coordinate system
- s :
-
special entropy
- T :
-
temperature
- u, v, w :
-
velocity components in spherical coordinates
- U, V, W :
-
the first perturbation velocity components
- V :
-
velocity vector
- β:
-
semivertex angle of the unperturbed shock
- δ:
-
semivertex angle of the unperturbed cone
- ε:
-
perturbation parameter (defined extremely small)
- γ:
-
ratio of specific heat (γ=1.4 for air)
- η:
-
(ø0−σ)/(β−σ)
- ø0 :
-
new independent variable, defined in Eq. (4)
- ϱ:
-
density
- σ:
-
β/δ
- ζ:
-
ø0/δ
- ξ0 :
-
ϱ∞/ϱ0(β)
- c :
-
composite solution
- i :
-
inner expansion
- o :
-
outer expansion
- b :
-
cone body surface
- m :
-
mth term in expansion
- o :
-
zeroth-order, unperturbed flow quantities
- s :
-
shock wave
- ∞:
-
freestream condition
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Lin, S.C., Chou, Y.T. Analytical study for hypersonic flow past an elliptic cone with longitudinal curvature. Acta Mechanica 135, 127–151 (1999). https://doi.org/10.1007/BF01305748
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DOI: https://doi.org/10.1007/BF01305748