Espen Åkervik

Steady solutions of the Navier-Stokes equations by selective frequency damping

A new method, enabling the computation of steady solutions of the Navier-Stokes equations in globally unstable configurations, is presented. We show that it is possible to reach a steady state by damping the unstable (temporal) frequencies. This is achieved by adding a dissipative relaxation term proportional to the high-frequency content of the velocity fluctuations. Results are presented for cavity-driven boundary-layer separation and a separation bubble induced by an external pressure gradient.

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 ...

Matrix-Free Methods for the Stability and Control of Boundary Layers

This paper presents matrix-free methods for the stability analysis and control design of high-dimensional systems arising from the discretized linearized Navier-Stokes equations. The methods are ap ...

Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers

Transition to turbulence and flow control are studied by means of numerical simulations for different simple shear flows. Linear and non-linear optimisation methods using the Lagrange multiplier technique are employed. In the linear framework as objective function the standard disturbance kinetic energy is chosen and the constraints involve the linearised Navier–Stokes equations. We consider both the optimal initial condition leading to the largest disturbance energy growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response for the case of the flat plate boundary layer excluding the leading edge. The optimal disturbances for spanwise wavelengths of the order of the boundary layer thickness are streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths it is the Orr mechanism combined with the amplification of oblique wave packets that is responsible for the disturbance growth. Also linear optimal disturbances are computed around a leading edge and the effect of the geometry is considered. It is found that two-dimentional disturbances originating upstream, relative to the leading edge of the plate are inefficient at generating a viable disturbance, while three dimentional disturbances are more amplified. In the non-linear framework a new approach using ideas from non-equilibrium thermodynamics is developed. We determine the initial condition on the laminar/turbulent boundary closest to the laminar state. Starting from the general evolution criterion of non-equilibrium systems we propose a method to optimise the route to the statistically steady turbulent state, i.e. the state characterised by the largest entropy production. This is the first time information from the fully turbulent state is included in the optimisation procedure. The method is applied to plane Couette flow. We show that the optimal initial condition is localised in space for realistic flow domains, while the disturbance visits bent streaks before breakdown. Feedback control is applied to the bypass-transition scenario with high levels of free-stream turbulence. The flow is the flat-plate boundary layer. In this scenario low frequency perturbations enter the boundary layer and streamwise elongated disturbances emerge due to non-modal growth. The so-called streaky structures are growing in amplitude until they reach high enough energy levels and break down into turbulent spots via their secondary instability. When control is applied in the form of wall blowing and suction, the growth of the streaks is delayed, which implies a delay of the whole transition process. Additionally, a comparison with experimental work is performed demonstrating a remarkable agreement in the disturbance attenuation once the differences between the numerical and experimental setup are reduced. Open-loop control with wall travelling waves by means of blowing and suction is applied to a separating boundary layer. For downstream travelling waves we obtain a mitigation of the separation of the boundary layer while for upstream travelling waves a significant delay in the transition location accompanied by a modest reduction of the separated region.

Linear feedback control and estimation applied to instabilities in spatially developing boundary layers

This paper presents the application of feedback control to spatially developing boundary layers. It is the natural follow-up of Hogberg & Henningson (J. Fluid Mech. vol. 470, 2002, p. 151), whe ...

The use of global modes to understand transition and perform flow control

The stability of nonparallel flows is considered using superposition of global modes. When perturbed by the worst case initial condition, these flows often exhibit a large transient growth associated with the development of wave packets. The global modes of the systems also provide a good starting point for the design of reduced order models used to control the growing disturbances. Three recent investigations are reviewed. The first example is the growth of a wave packet on a falling liquid sheet. The optimal perturbation analysis shows that the worst case initial condition is a localized disturbance that creates a propagating wave packet that hits the downstream end, regenerating a wave packet upstream through a global pressure pulse. Second, we consider two-dimensional disturbances in the Blasius boundary layer. It is found that a wave packet is optimally excited by an initial condition consisting of localized backward leaning Orr structures. Finally, the control of a globally unstable boundary-layer f...

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 disturbances with iterative methods

The flat-plate boundary layer is a classic example of convectively unstable flows; these behave as broadband amplifiers of incoming disturbances. As a consequence, a global stability analysis based on the asymptotic behaviour of single eigenmodes of the system do not capture the relevant dynamics. From this global perspective all the eigenmodes are damped, and one has to resort to an input/output formulation in order to obtain the initial conditions yielding the largest possible disturbance growth at any given time and the optimal harmonic forcing. To this end, an optimisation procedure is adopted. We consider both the optimal initial condition leading to the largest growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response.

FEEDBACK CONTROL IN SPATIALLY GROWING BOUNDARY LAYERS

Linear feedback control has been applied to transitional boundary layer flows. Information from wall-mounted sensors is used to estimate the flow state. The estimated state is then used to compute the optimal feedback control which is applied as blowing and suction with zero net mass-flux through the wall. The performance of the controller is tested in direct numerical simulations of a spatially growing Falkner-Skan- Cooke boundary layer where an inflectional instability is triggered. The extension to spatial boundary layer flows is an important step towards real applications.