Modal and non-modal stabilities of flow around a stack of plates

R. Theobald, Xuerui Mao, A.J. Jaworski, A. Berson
2015 European journal of mechanics. B, Fluids  
Modal and non-modal stabilities of flow around a stack of flat plates are investigated by means of asymptotic stability and transient growth analyses respectively. It is observed that over the parameters considered, both the base flow and the stabilities vary as a function of ReW 2 /(W − 1) 2 , i.e. the product of the Reynolds number and the square of the expansion ratio of the stack. The most unstable modes are found to be located downstream of the recirculation bubble while the global optimal
more » ... initial perturbations (resulting in maximum energy growth over the entire domain) and the weighted optimal initial perturbations (resulting in maximum energy growth in the close downstream region of the stack) concentrate around the stack end owing to the Orr mechanism. In direct numerical simulations (DNS) of the base flow initially perturbed by the modes, it is noticed that the weighted optimal initial perturbation induces periodic vortex shedding downstream of the stack much faster than the most unstable mode. This observation suggests that the widely reported vortex shedding in flow around a stack of plates, e.g. in thermoacoustic devices, is associated with perturbations around the stack end. (X. Mao). shedding occur was calculated. For flow at Reynolds number well above the critical value, three-dimensional stability of the wake flow around a thin plate has been studied [12] . The literature about flow around a stack of plates (or more general bluff bodies) has focused on either the flow instabilities, which can be interpreted as the onset of vortex shedding, or the fully developed vortex shedding state. The gap between the two states, i.e. the route from initial infinitesimal perturbations to periodic vortex shedding, is still an open question. The current study will target this gap and focus on identifying the origin of vortex shedding and how the infinitesimal perturbation to a steady base flow develops into vortex shedding in flow past a periodic plate array. In the rest of this work, the methodology to calculate perturbations relevant to vortex shedding, e.g. modal stability (asymptotic stability) theory to calculate the modal modes (most unstable modes) and nonmodal stability (non-normality, transient growth) theory to calculate the nonmodal modes (optimal initial perturbations) are presented, followed by the numerical setup, and then the modal and nonmodal modes as well as their nonlinear developments to vortex shedding are discussed. Methodology Both modal and nonmodal studies performed in this work involve the linearization of the incompressible Navier-Stokes (NS) http://dx.
doi:10.1016/j.euromechflu.2015.04.005 fatcat:x56qql3jivc2tglym2q6xndcne