
Full text loading...
The objective of this article is to review some developments in the use of adjoint equations in hydrodynamic stability theory. Adjoint-based sensitivity analysis finds both analytical and numerical applications much beyond those originally imagined. It can be used to identify optimal perturbations, pinpoint the most receptive path to break down, select the most destabilizing base-flow defect in a nominally stable configuration, and map the structural sensitivity of an oscillator. We focus on two flow cases more closely: the noise-amplifying instability of a boundary layer and the global mode occurring in the wake of a cylinder. For both cases, the clever interpretation and use of direct and adjoint modes provide key insight into the process of the transition to turbulence.
Article metrics loading...
Full text loading...
Literature Cited
Data & Media loading...
Download Supplemental Appendix (PDF): The Supplemental Appendix provides an introduction to adjoint methods with a tutorial that deductively leads to the main concepts and their applications. After some historical remarks, adjoint problems are first introduced for simple discrete systems and then gradually expanded to include progressively more complex system, including continuous systems and nonlinear equations. The adjoint of the Navier-Stokes equations and some considerations on adjoint programming conclude the tutorial.
Supplemental Video 1: Spatial evolution of the minimal seed in a boundary layer. The structure evolves with the initial downstream tilting of the low streamwise velocity region (in light blue) by the Orr mechanism, the formation of a Λ structure, and the subsequent creation of a hairpin vortex (visualized in gray through isosurfaces of the Q-criterion). This hairpin can rapidly induce downstream a train of smaller-scale hairpin vortices which display, embedded within, further small minimal seeds; this leads to a repeated sequence of the same events taking place over shorter length scales and timescales, until turbulence (Cherubini et al. 2011, 2012).
Download video file (AVI)
Supplemental Video 2: Structural sensitivity map of the secondary instability of the cylinder wake mode A (Re = 190), calculated as in Giannetti et al. (2010a). The structural sensitivity of this mode is a function of two spatial coordinates and a periodic function of time, just as its base flow is. The sensitivity peak marks the location of the wave maker and its motion during the oscillation period, showing that this structural sensitivity is even more localized (at each instant of time) than the sensitivity of the primary instability (Figure 6 of the article). The three material closed orbits of the primary instability are superimposed on the sensitivity map.
Download video file (AVI)
Supplemental Video 3: Structural sensitivity map of the secondary instability of the cylinder wake mode B (Re = 260), calculated as in Giannetti et al. (2010a). The structural sensitivity of this mode is a function of two spatial coordinates and a periodic function of time, just as its base flow is. The sensitivity peak marks the location of the wave maker and its motion during the oscillation period, showing that this structural sensitivity is even more localized (at each instant of time) than the sensitivity of the primary instability (Figure 6 of the article). The three material closed orbits of the primary instability are superimposed on the sensitivity map
Download video file (AVI)