Gregory Pujals

A note on optimal transient growth in turbulent channel flows

We compute the optimal transient growth of perturbations sustained by a turbulent channel flow following the same approach recently used by del Alamo and Jimenez [J. Fluid Mech. 559, 205 (2006)]. Contrary to this previous analysis, we use generalized Orr–Sommerfeld and Squire operators consistent with previous investigations of mean flows with variable viscosity. The optimal perturbations are streamwise vortices evolving into streamwise streaks. In accordance with del Alamo and Jimenez, it is found that for very elongated structures and for sufficiently large Reynolds numbers, the optimal energy growth presents a primary peak in the spanwise wavelength, scaling in outer units, and a secondary peak scaling in inner units and corresponding to λz+≈100. Contrary to the previous results, however, it is found that the maximum energy growth associated with the primary peak increases with the Reynolds number. This growth, in a first approximation, scales linearly with an effective Reynolds number based on the cen...

Sensitivity of 2-D turbulent flow past a D-shaped cylinder using global stability

We use adjoint-based gradients to analyze the sensitivity of turbulent wake past a D-shaped cylinder at Re = 13000. We assess the ability of a much smaller control cylinder in altering the shedding frequency, as predicted by the eigenfrequency of the most unstable global mode to the mean flow. This allows performing beforehand identification of the sensitive regions, i.e., without computing the actually controlled states. Our results obtained in the frame of 2-D, unsteady Reynolds-averaged Navier–Stokes compare favorably with experimental data reported by Parezanovic and Cadot [J. Fluid Mech. 693, 115 (2012)] and suggest that the control cylinder acts primarily through a local modification of the mean flow profiles.

Sensitivity of aerodynamic forces in laminar and turbulent flow past a square cylinder

We use adjoint-based gradients to analyze the sensitivity of the drag force on a square cylinder. At Re = 40, the flow settles down to a steady state. The quantity of interest in the adjoint formulation is the steady asymptotic value of drag reached after the initial transient, whose sensitivity is computed solving a steady adjoint problem from knowledge of the stable base solution. At Re = 100, the flow develops to the time-periodic, vortex-shedding state. The quantity of interest is rather the time-averaged mean drag, whose sensitivity is computed integrating backwards in time an unsteady adjoint problem from knowledge of the entire history of the vortex-shedding solution. Such theoretical frameworks allow us to identify the sensitive regions without computing the actually controlled states, and provide a relevant and systematic guideline on where in the flow to insert a secondary control cylinder in the attempt to reduce drag, as established from comparisons with dedicated numerical simulations of the two-cylinder system. For the unsteady case at Re = 100, we also compute an approximation to the mean drag sensitivity solving a steady adjoint problem from knowledge of only the mean flow solution, and show the approach to carry valuable information in view of guiding relevant control strategy, besides reducing tremendously the related numerical effort. An extension of this simplified framework to turbulent flow regime is examined revisiting the widely benchmarked flow at Reynolds number Re = 22 000, the theoretical predictions obtained in the frame of unsteady Reynolds-averaged Navier–Stokes modeling being consistent with experimental data from the literature. Application of the various sensitivity frameworks to alternative control objectives such as increasing the lift and reducing the fluctuating drag and lift is also discussed and illustrated with a few selected examples.

Forcing large-scale coherent streaks in a zero-pressure-gradient turbulent boundary layer

Large-scale coherent streaks are artificially forced in a well-developed turbulent boundary layer at using an array of cylindrical roughness elements. Measures of the velocity field with particle image velocimetry reveal the presence of well-reproducible, streamwise-oriented, steady coherent streaks. We show that the amplitude of these coherent streaks transiently grows in space. The position of the maximum amplitude is proportional to the spanwise wavelength of the streaks and the most amplified spanwise wavelength is of very large scale λ z ≈6δ0. These results are in good agreement with the recent predictions based on the optimal transient growth analysis of turbulent mean flows.

Optimal perturbations of non-parallel wakes and their stabilizing effect on the global instability

We compute the spatial optimal energy amplification of steady inflow perturbations in a non-parallel wake and analyse their stabilizing action on the global mode instability. The optimal inflow perturbations, which are assumed spanwise periodic and varicose, consist in streamwise vortices that induce the downstream spatial transient growth of streamwise streaks. The maximum energy amplification of the streaks increases with the spanwise wavelength of the perturbations, in accordance with previous results obtained for the temporal energy growth supported by parallel wakes. A family of increasingly streaky wakes is obtained by forcing optimal inflow perturbations of increasing amplitude and then solving the nonlinear Navier-Stokes equations. We show that the linear global instability of the wake can be completely suppressed by forcing optimal perturbations of sufficiently large amplitude. The attenuation and suppression of self-sustained oscillations in the wake by optimal 3D perturbations is confirmed by fully nonlinear numerical simulations. We also show that the amplitude of optimal spanwise periodic (3D) perturbations of the basic flow required to stabilize the global instability is much smaller than the one required by spanwise uniform (2D) perturbations despite the fact that the first order sensitivity of the global eigenvalue to basic flow modifications is zero for 3D spanwise periodic modifications and non-zero for 2D modifications. We therefore conclude that first-order sensitivity analyses can be misleading if used far from the instability threshold, where higher order terms are the most relevant