Stability of developing flow in a pipe: non-axisymmetric disturbances

Spatial stability results for the developing flow in a rigid circular pipe are presented for the velocity profile obtained by the Hornbeck (1963) method and compared with the available temporal stability results for the velocity profile obtained by Sparrow, Lin & Lundgren (1964). The disturbance is taken to be non-axisymmetric, and Gram–Schmidt orthonormalization is used to remove the parasitic errors during numerical integration. It is found that the stability characteristics are very sensitive to the velocity field in the inlet region. At all axial locations investigated the critical frequency and critical wavenumber for the Hornbeck profile are larger than the corresponding values for the Sparrow profile while the critical Reynolds number is smaller. The minimum critical Reynolds number for the Hornbeck profile is only 13250 and occurs at $\overline{X} = 0.0032$ compared with 19780 at $\overline{X} = 0.0049$ for the Sparrow profile. The maximum difference between the two velocity profiles occurs near the boundary-layer edge but is within 5%. Results for the Hornbeck profile are found to be closer to the experimental data of Sarpkaya (1975).