Inviscid separation in steady planar flows

Abstract

This paper addresses the unsolved classical problem of the indeterminacy of steady, inviscid, incompressible vortical planar flows due to inviscid separation. We prove that any inviscid steady flow is “totally nonunique” in the following sense: there are a continuum of possible solutions involving recirculation zones that can be always inserted into such a flow satisfying the same boundary conditions. For sufficiently large inflow vorticity (a flow past a smoothing screen gives an example of inflow vorticity generation), the formation of a separation zone is inevitable. Various additional phenomenological conditions to achieve uniqueness and to determine the flow in the separation zone are evaluated, including the frequently used analytic continuation method. We reveal a new ‘measure paradox’ in inviscid hydrodynamics: as inflow of discontinuous vorticity increases, one streamline transforms into a stagnation zone of finite area. We focus on this new steady stagnation zone flow model with continuous velocity distribution. We prove existence and uniqueness theorems for stagnant separation zones and show that the kinetic energy of the entire flow domain is minimized when it contains a separation zone which is stagnant. We consider vortical channel flows with different inflow vorticity distributions and show that a totally quiescent stagnation zone model occurs in the limit as the inflow vorticity becomes discontinuous. For a smooth, but large-gradient vorticity distribution a quasi-stagnant zone occurs. It is shown that the analytic continuation model applied to an irrotational channel flow with a slender vortical core (the planar analog of the Rankine vortex) yields “vorticity breakdown” – a phenomenon somewhat similar to vortex breakdown in swirling flows. Nonuniqueness and hysteresis-related multiple solutions are inherent to those solutions with recirculation zones where very large (physically unrealistic) reversed flow appears. Being free from these drawbacks, the stagnation zone model appears to be superior. A numerical investigation of a vortical flow in a square domain confirms the existence of a critical value of inflow vorticity, leading to steady or permanently unsteady flow regimes.