Linear temporal stability analysis on cross sheared flow: The stabilization effects via cross shear

Abstract

The stability features of the inviscid, homogenous, and free cross sheared flow, with base flow velocities U=tanh(z) in the primary direction, which is unstable, and V=ξz3 in the orthogonal direction, which is stable, are thoroughly examined with linear temporal stability analysis, where z is the transverse coordinate perpendicular to the flow and ξ is the cross shear ratio which is the ratio of the characteristic magnitudes of V to U. The map of the unstable regions directly related to χ=ξ(β/α) is obtained, where (β/α) is the ratio between the orthogonal and primary wavenumbers. Further examination of the eigenfunctions shows that the eigenfunction structures divide into the orthogonal wavenumber (OW) mode where (β/α) dominates and the cross shear (CS) mode where ξ dominates. The cross shear is found necessary for stabilization in spite of different fashions for the OW and CS modes. The transition from the OW mode to the CS mode shows that the developments of the two modes inherently compete with each other, so that when ψ=(β/α)/ξ decreases the enhanced cross shear needs to deteriorate the OW mode before it helps the growth of the CS mode. Based on the magnitudes of the associated eigenfunctions in the enstrophy budget, the map of the OW, CS, and hybrid modes, which includes the mixed features of both the OW and CS modes, is produced and discussed.