Local Instability in Curved Flow

Expressions are obtained for the rates of change of the vorticity components about the tangent, principal normal, and binormal of inviscid flow of arbitrary configuration with curvature and torsion of the particle paths, and non-uniform density. If a steady flow pattern exists the vorticity changes are exactly those required to carry the fluid particles through it. A small rotational displacement of a fluid element about each of these directions is then considered separately and if the additional rate of change of vorticity has the same sign and order of magnitude as the displacement the motion is unstable locally regardless of the motion elsewhere. The equations depend on four quantities: the curvature, torsion, velocity and vorticity at the point where stability is being investigated. These four quantities define a helical vortex, with the same stability properties, in which the flow is equivalent. There is therefore special merit in studying this case. It is found that a vortex with helical particle paths is unstable for rotation of the fluid elements about directions which lie between the vorticity vector and the direction of the axis of the motion, when the density is uniform. More generally, the motion is unstable for these disturbances if the stagnation pressure decreases radially outwards. A gradient of axial velocity component always causes some local instability, and the most likely (fastest growing) disturbance is one in which rotations occur around a line everywhere bisecting the angle between the vorticity vector and the direction of the axis. The analysis shows that in two dimensional circular flow the most unstable disturbances are toroidal and in general curved flow in two dimensions (without torsion) the most likely disturbances are rotations around the tangent, i.e. longitudinal rolls. The criteria obtained for local instability do not agree with a Richardson-type criterion for local stability, but both show the destabilizing effect of a radiating gradient of axial velocity.