Investigation of the roughness-induced transition: linear and non-linear optimal perturbations

For small amplitude disturbances and supercritical Reynolds numbers, linear stability theory predicts the slow transition of flat plate boundary layer flows as a result of the generation, amplification and secondary instability of Tollmien-Schlichting (TS) waves. Fransson et al. [1] have shown that the streaks induced by a three-dimensional roughness element placed on the flat plate can stabilize these TS waves. Unfortunately, beyond a given amplitude of the roughness element, the resulting flow can transition right downstream the roughness element. Such transition past cylindrical roughness elements has been investigated by Loiseau et al. [2] in the framework of global stability where it has been shown that transition could be explained by a global instability of the flow. Despite the plausible explanation it provides, this study does not rule out the possibility for the flow to experience bypass transition wherein the flow transitions to turbulence despite all the modes of the linearised Navier-Stokes operator being linearly stable. Such transition is related to the nonnormality of the linearised operator: small disturbances can experience a large transient amplification due to constructive interferences of non-orthogonal stable modes. If the growth is sufficiently large, such disturbances can trigger non-linear effects allowing them to self-sustain, resulting in bypass transition to turbulence. It is the computation and investigation of such perturbations that we are addressing in the present work using the framework of linear and non-linear optimal perturbation analysis. Provided a linearly stable base flow Ub, the aim of this work is to compute the linear and non-linear velocity perturbations u optimizing the objective functional J (u) = 1 2 R V u(T) · u(T) dV. That is, we look for the perturbation at time t = 0 which provides the maximum value of the objective functional at a given target time t = T. The optimization problem is subject to a set of partial differential constraints, i.e. the perturbative (linear or non-linear) Navier-Stokes equations, that must be verified at each time and each point of the computational domain. A constraint on the initial value of the perturbation’s kinetic energy is also imposed, i.e. E(0) = E0. These optimal perturbations are computed using a Lagrange multipliers technique which involves the adjoint Navier-Stokes equations. For the particular flow configuration considered, once spatially discretised, the optimisation problem involves almost 100 millions of degrees of freedom. To solve it, a rotation-update gradient-based method [3] is used. The linearly stable flow investigated is the same as in [1] and [2]. The base flow consists in a horseshoe vortex wrapped around the cylindrical roughness element, whose legs give birth further downstream to streamwise velocity streaks. A central low-speed region is also generated in the wake of the roughness element due to the blockage it induces. Linear optimal perturbation analyses have shown that, despite the flow being linearly stable, infinitesimal perturbations can have their energy amplified by a factor 106 over a time interval T ' 55 before decaying due to the linearly stable nature of the flow. While the optimal perturbation consists in alternated patches of velocity oriented against the base flow shear and localized in the vicinity of the roughness element, the optimal response is a wavepacket of velocity patches, now oriented along the base flow shear, that has travelled further downstream. The underlying amplification mechanisms have been investigated using the Reynolds-Orr equation which governs the transfer of energy between the base flow Ub and the perturbation u. This analysis has revealed that the perturbation bases its transient amplification on two different mechanisms: (i) the Orr mechanism at short times (t < 40), and (ii) the lift-up effect at larger times (t > 40). How nonlinearity influences the optimal perturbation and associated response, as well as the energy extraction process, is currently under investigation with the computation of the corresponding non-linear optimal perturbations.