Linear temporal stability analysis on non-parallel free cross sheared flow with a primary hyperbolic velocity and an orthogonal Bickley jet velocity

Abstract

A linear temporal stability analysis is carried out on non-parallel free cross sheared flow consisting of the unstable hyperbolic velocity U=tanh(z) in the primary direction and the comparable unstable Bickley jet velocity V=sech2(z) in the orthogonal direction, where z is the coordinate perpendicular to the plane of the primary and orthogonal directions. The cross sheared flow involves the non-parallel effects such as twisted flow and cross flow. The linearized perturbation equations are derived which are subsequently used to examine the stability features. It is found that the instability associated with U=tanh(z) and V=sech2(z) transitions asymptotically to each other as the combined factor χ=ξ(β/α) varies, where ξ=||V||/||U|| is the cross shear ratio and β/α is the ratio between the orthogonal and primary wavenumbers with || || representing the characteristic magnitudes of the velocity components. In addition to the hyperbolic flow (HF) mode and the Bickley jet flow (BJF) mode associated with U=tanh(z) and V=sech2(z), respectively, the orthogonal wavenumber (OW) mode where β/α dominates and the cross shear (CS) mode where ξ dominates are found in the eigenfunction structures. The physical mechanisms of the four eigenfunctions modes are examined with the kinetic energy and enstrophy budgets. The map of the unstable regions influenced and dominated by the OW, CS, HF, and BJF eigenfunction modes is also obtained and discussed.