Mixing enhancement via the release of strongly nonlinear longitudinal Görtler vortices and their secondary instabilities into the mixing region

Abstract

Mixing enhancement in a mixing layer is considered in terms of a ‘vortex generator’ that uses fluid dynamically generated counter-rotating longitudinal vortices rather than explicit winglets or similar devices. This view is reached through considering the centrifugal instability of weak initial Görtler vortices on a slightly concave wall that are allowed to develop to their various nonlinear stages through selecting the cutoff lengths of the trailing edge prior to their release into the mixing region. These vortices are released from one side of the (say, upper) stream in the present work. The quantitative entrainment properties of the longitudinal vortices are studied to select an optimal trailing-edge cutoff for fixed upstream conditions. As the vortices develop along the wall, they are intensified because of the centrifugal instability mechanism and because of the work done by the Reynolds stress of the vortices against the local mean flow rate of strain; simultaneously, the region of strong streamwise vorticity moves away from the wall. This selection process is explained through a balance between the vorticity strength and proximity to the lower stream when the trailing edge is cut off: it is shown, therefore, that vortices of relatively modest strength and kinetic energy that are close to the interface separating the two streams provide mixing properties superior to stronger vortices located too far from the interface. Energy-balancing mechanisms and the stretching of the initial interface are studied, as are the effects of the velocity ratio and the spanwise wavelengths other than the fundamental. In order further to enhance mixing by exploiting the inherent secondary instability of primary steady longitudinal vortices, the most amplified secondary instability of the optimal-trailing-edge cutoff situation, which is the sinuous mode, is studied in detail in terms of the nonlinear development and modification of the steady vortical flow. Local energy-exchange mechanisms are studied, as are the mixing properties of the modified steady flow, which are shown to be significantly improved compared to the unmodified steady flow. Though the initiation of steady longitudinal vortices relies on centrifugal instability upstream, such vortices are able to develop self-sustaining and amplifying properties through the Reynolds stresses in the mixing region even without centrifugal instability reinforcement. The secondary instability is initiated and sustained entirely through its own three-dimensional Reynolds stress properties, which work against the three-dimensional rates of strain in the entire steady flow. This contrasts with initially generated potential-like vortices that decay downstream in the presence of dissipative mechanisms without the production mechanisms due to the Reynolds stresses.