Transition process of a supersonic streamwise vortex at M = 2.5; the vortical structure is described by an isosurface of the second invariant of the velocity gradient tensor.
Comparison of theoretical model profiles with experimental measurements 21–26 in supersonic streamwise vortices for (a) axial velocity, (b) azimuthal velocity, and (c) normalized circulation profiles.
(a) Stagnation temperature and (b) entropy profiles of supersonic streamwise vortices for the measurement profiles 3,22 at M = 3.0 (note that (a) ΔS = 0, the supposed stagnation temperature based on q = 0.2, μ = 0.29 fitted with the measurements).
Entropy and entropy gradient distributions for streamwise vortices q = 0.32, μ = 0.5 at M = 0.01, 2.5, and 5.0, using Eq. (17) , given that stagnation temperature is uniform.
Spatial growth rates −α i as a function of frequency ω for M = 0.01, 2.5, and 5.0.
Growth properties of the streamwise vortices obtained by making a comparison between the isentropic flow, the density and pressure constant, the stagnation temperature uniform, and the stagnation temperature deficit of 8%; for Φ = 0.32 (q = 0.16, μ = 0.5), m = −2.
Amplitude profiles of disturbances; axial vorticity |Ω x |, azimuthal vorticity |Ωθ|, and entropy |S †| for m = −2 at M = 0.01, 2.5, and 5.0.
Conceptual diagram of instability features for Φ vs. M.
Eigenfunction profiles of the disturbance; density R, velocities U, V, W, and pressure P, for M = 2.5, m = −2, Φ = 0.64 (q = 0.32, μ = 0.5), (a) and (b) amplitude, (c) and (d) phase.
Source terms to the perturbation total energy, , , and .
Normalized maximum growth rates as a function of the ratio Φ for various Mach numbers.
Normalized maximum growth rates as a function of convective Mach number M c ; (a) the eigenvalues of supersonic streamwise vortices with the uniform stagnation temperature, (b) the comparison of a single curve organized the vortices (a) with measurements of supersonic mixing layers 1,41,42 (- - - broken line, linear stability analysis of a mixing layer; 44 - - - - dashed line, Langley experimental curve 40 ) and compressible swirling flows. 4
Validation of eigenvalues for stability of the Batchelor vortex.
Parameters of supersonic streamwise vortices.
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