Stability of three-dimensional supersonic boundary layers
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A rotating cone that is located in a supersonic free stream at zero angle of attack is used as a model to investigate the stability of three-dimensional supersonic boundary layers. The boundary-layer profiles on the surface are calculated using the CebeciKeller box scheme. The stability equations are solved to determine the eigenvalues using a two-point fourth-order finite-difference scheme [Malik etal., Z. Angew. Math. Phys. 33, 189 (1982)]. The results show that the amplification rate of the first mode is increased by a factor of 2 to 4 due to the cross-flow, compared with a two-dimensional flow with the same streamwise profile. This increase decreases with increasing Mach number. The instability with cross-flow covers a wide range of unstable frequencies (including zero) and wave numbers. The results also show that the second mode in a three-dimensional boundary layer is oblique whereas the second mode in a two-dimensional boundary layer is two dimensional. The maximum amplification rate of the second mode decreases more slowly with increasing wave angle in a three-dimensional boundary layer than in a two-dimensional boundary layer. It is concluded that the cross-flow instability becomes important for cross-flow Reynolds number on the order of 50 for low Mach numbers and 100 for high Mach numbers, this Reynolds number range corresponds to a maximum cross-flow velocity of about 4%.
