Uwe Ehrenstein

Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes

Two-dimensional global eigenmodes are used as a projection basis both for analysing the dynamics and building a reduced model for control in a prototype separated boundary-layer flow. In the presen ...

On the onset of nonlinear oscillations in a separating boundary-layer flow

The stability of a separating boundary-layer flow at the rear of a two-dimensional bump mounted on a flat plate is numerically investigated. Above a critical Reynolds number, the flow field is shown to undergo self-sustained two-dimensional low-frequency fluctuations in the upstream region of the separation bubble, evolving into aperiodic vortex shedding further downstream. The computed steady flow states below the critical Reynolds number are shown to be convectively unstable. On extrapolating the flow field to Reynolds numbers above criticality, some evidence is found that the onset of the oscillatory behaviour coincides with topological flow changes near the reattachment point leading to the rupture of the (elongated) recirculation bubble. The structural changes near reattachment are shown to trigger an abrupt local transition from convective to absolute instability, at low frequencies. On preventing the separation bubble from bursting by reaccelerating the flow by means of a second bump further downstream, the separated flow remains steady for increasing Reynolds numbers, until a local region of absolute instability in the upper part of the geometrically controlled recirculation bubble is produced. The resulting global instability consists of self-sustained nonlinear saturated perturbations oscillating at a well-defined frequency, very distinct from the the low-frequency motion of the elongated recirculation bubble in the single-bump geometry. A frequency selection criterion based on local absolute frequencies, which has been successfully applied to wake flows, is shown to accurately predict the global frequency.

Two-dimensional global low-frequency oscillations in a separating boundary-layer flow

A separated boundary-layer flow at the rear of a bump is considered. Two-dimensional equilibrium stationary states of the Navier–Stokes equations are determined using a nonlinear continuation procedure varying the bump height as well as the Reynolds number. A global instability analysis of the steady states is performed by computing two-dimensional temporal modes. The onset of instability is shown to be characterized by a family of modes with localized structures around the reattachment point becoming almost simultaneously unstable. The optimal perturbation analysis, by projecting the initial disturbance on the set of temporal eigenmodes, reveals that the non-normal modes are able to describe localized initial perturbations associated with the large transient energy growth. At larger time a global low-frequency oscillation is found, accompanied by a periodic regeneration of the flow perturbation inside the bubble, as the consequence of non-normal cancellation of modes. The initial condition provided by the optimal perturbation analysis is applied to Navier–Stokes time integration and is shown to trigger the nonlinear ‘flapping’ typical of separation bubbles. It is possible to follow the stationary equilibrium state on increasing the Reynolds number far beyond instability, ruling out for the present flow case the hypothesis of some authors that topological flow changes are responsible for the ‘flapping’.

Three-dimensional transverse instabilities in detached boundary layers

The direct numerical simulation of the incompressible Navier-Stokes equations of the flow above a bump shows a stationary longitudinal instability at a Reynolds number of Re=400. A three-dimensional global mode linear analysis is used to interpret these results and shows that the most unstable eigenmode is steady and localized in the recirculation bubble, with spanwise wavelength of approximatively ten bump heights. An inviscid geometrical optics analysis along closed streamlines is then proposed and modified accordingly to account for viscous effects. This motivates a final discussion regarding the physical origin of the observed instability.

On the relation between kinetic energy production in adverse-pressure gradient wall turbulence and streak instability

A direct numerical simulation (DNS) of a turbulent channel flow with a lower curved wall is performed at Reynolds number Re τ≃617 at inlet. This adverse-pressure gradient turbulent flow is characterized by strong peaks of turbulent kinetic energy at both walls, as a consequence of the breakdown of more organized flow structures. To elucidate the underlying instability scenario, low-speed streak structures are extracted from the turbulent flow field and base flows formed with conditional streak averages, superimposing the mean streamwise velocity profile, are used for linear stability analyses. The size and shape of the counter-rotating streamwise vortices associated with the instability modes are shown to be reminiscent of the coherent vortices emerging from the streak skeletons in the direct numerical simulation. The distance of the streaks centre from the wall is used as a criterion for the conditional averages and the corresponding streak base flows are characterised by more or less pronounced contour...

Two-dimensional nonlinear plane Poiseuille–Couette flow homotopy revisited

A recent paper by Rincon [Phys. Fluids 19, 074105 (2007)] readdressed the question of the existence of two-dimensional steady nonlinear states in plane Couette flow, coming to the conclusion that it is not possible to obtain the nonlinear plane Couette flow solutions reported by Cherhabili and Ehrenstein [Eur. J. Mech. B/Fluids 14, 667 (1995)] using their Poiseuille–Couette homotopy. Exploring the multiparameter space by performing several consecutive and distinct continuations, we show that it is possible to find a complex numerical path from plane Poiseuille streamwise periodic waves to two-dimensional nonlinear steady states for the plane Couette flow limit. The nonlinear Couette flow states are retrieved using three independent solution procedures and the disturbance flow structure is shown to be localized in the streamwise periodic box. Numerical evidence is provided that the width of the nonlinear plane Couette flow disturbance decreases with increasing resolution at fixed boxlength. This singular-t...

Skin friction on a flapping plate in uniform flow.

To calculate the energy costs of swimming or flying, it is crucial to evaluate the drag force originating from skin friction. This topic seems not to have received a definite answer, given the difficulty in measuring accurately the friction drag along objects in movement. The incoming flow along a flat plate in a flapping normal motion has been considered, as limit case of a yawed cylinder in uniform flow, and applying the laminar boundary layer assumption it is demonstrated that the longitudinal drag scales as the square root of the normal velocity component. This lends credit to the assumption that a swimming-like motion may induce a drag increase because of the compression of the boundary layer, which is known as the ‘Bone–Lighthill boundary-layer thinning hypothesis’. The boundary-layer model however cannot predict the genuine three-dimensional flow dynamics and in particular the friction at the leeward side of the plate. A multi-domain, parallel, compact finite-differences Navier–Stokes solution procedure is considered, capable of solving the full problem. The time-dependent flow dynamics is analysed and the general trends predicted by the simplified model are confirmed, with however differences in the magnitude of the friction coefficient. A tentative skin friction formula is proposed for flow states along a plate moving at steady as well as periodic normal velocities.

Open-loop control of noise amplification in a separated boundary layer flow

Linear optimal gains are computed for the subcritical two-dimensional separated boundary-layer flow past a bump. Very large optimal gain values are found, making it possible for small-amplitude noise to be strongly amplified and to destabilize the flow. The optimal forcing is located close to the summit of the bump, while the optimal response is the largest in the shear layer. The largest amplification occurs at frequencies corresponding to eigenvalues which first become unstable at higher Reynolds number. Nonlinear direct numerical simulations show that a low level of noise is indeed sufficient to trigger random flow unsteadiness, characterized here by large-scale vortex shedding. Next, a variational technique is used to compute efficiently the sensitivity of optimal gains to steady control (through source of momentum in the flow, or blowing/suction at the wall). A systematic analysis at several frequencies identifies the bump summit as the most sensitive region for control with wall actuation. Based on ...

Model Reduction and Control of a Cavity-Driven Separated Boundary Layer

The control of a globally unstable boundary-layer flow along a two-dimensional cavity is considered. When perturbed by the worst-case initial condition, the flow exhibits a large transient growth associated with the development of a wave packet along the cavity shear layer followed by a global cycle related to the least stable global eigenmodes. The flow simulation procedure is coupled to a measurement feedback controller, which senses the wall shear stress at the downstream lip of the cavity and actuates at the upstream lip. A reduced model for the control optimization is obtained by a projection on the least stable global eigenmodes. The LQG controller is run in parallel to the Navier-Stokes time integration. It is shown that the controller is able to damp out the global oscillations.

Optimal control of a separated boundary-layer flow over a bump

The optimal control of a globally unstable two-dimensional separated boundary layer over a bump is considered using augmented Lagrangian optimization procedures. The present strategy allows of controlling the flow from a fully developed nonlinear state back to the steady state using a single actuator. The method makes use of a decomposition between the slow dynamics associated with the baseflow modification, and the fast dynamics characterized by a large scale oscillation of the recirculation region, known as flapping. Starting from a steady state forced by a suction actuator located near the separation point, the baseflow modification is shown to be controlled by a vanishing suction strategy. For weakly unstable flow regimes, this control law can be further optimized by means of direct-adjoint iterations of the nonlinear Navier-Stokes equations. In the absence of external noise, this novel approach proves to be capable of controlling the transient dynamics and the baseflow modification simultaneously.