Dynamic feedback control through wall suction in shear flows

Flow control is of interest in many open and wall‐bounded shear flows in order to reduce drag or to avoid sudden large fluctuations that may lead to material failure. An established means of control is the application of suction through a porous wall. Here, we combine suction with a feedback strategy whereby the suction velocity is adjusted in response to either the kinetic energy or the shear stress at the bottom wall. The control procedure is then used in an attempt to stabilize invariant solutions and to carry out direct numerical simulations with a prescribed value of the friction coefficient.


Suction velocity control
We consider a plane Couette setup with a fluid located in the gap between two infinitely extended parallel plates at a distance 2d. The bottom plate is stationary and the fluid is driven by the motion of the top plate with velocity U 0 in x-direction. The flow is assumed to be incompressible and isothermal such that the density can be regarded as constant. For large gap widths and a constant suction velocity −V s in the wall-normal y-direction through the bottom plate, this system is often used to emulate the asymptotic suction boundary layer (ASBL). The aim here is to study the impact of variations in the suction velocity on the global properties of the flow.
The stationary laminar solution of the Navier-Stokes equations is given by in units of U 0 , with ν the kinematic viscosity. The deviations u from the laminar flow then obey the dimensionless equations where p is the pressure divided by the constant density ρ and Re 0 = U 0 d/ν the Reynolds number based on the velocity of the top plate, the half-height of the domain and the kinematic viscosity of the fluid. The feedback control procedure is based on a control variable R, here related to the suction velocity V s by V s /U 0 = R/R 2 0 , and an observable A, e.g. the L 2 -norm u 2 or the friction factor C f = τ w /(ρU 2 0 ), where τ w is the shear stress at the bottom wall. The dynamics of the control variable is given bẏ where (A 0 , R 0 ) are the values of R and A at the desired operating point and γ and µ are free parameters. The first parameter defines the time scale on which the control is adjusted and, and second one regulates the sensitivity of the control variable to changes in the observable. In order to achieve stabilization, the reaction of the control variable has to be faster than the internal dynamics with which the dynamics runs away from the operating point, and the amplitude has to be sufficiently large to switch between stable and unstable regions (details will be given in [4]).

Numerical aspects and simulation data
Equations  Table 1: Simulation parameters and observables, with Vs the suction velocity, C f the friction coefficient, Re0 = U0d/ν the Reynolds number, Vs0 the suction velocity at the operating point, µ and γ the free parameters in Eq. (4) and δu0 a perturbation of the invariant solution u0. Here we choose u 0 to be a travelling wave with one unstable direction in a subspace that enforces the symmetries of the invariant solution [2]. All simulations have been carried out with spanwise, wall-normal and streamwise dimensions L x × L y × L z = 4π × 8 × 2π using N x × N y × N z = 32 × 65 × 32 grid points. Further simulation parameters are summarized in table 1.

Results
Applying a linear feedback control to an invariant solution with one unstable direction can result in stabilisation thereof, if (i) the control overlaps with the unstable direction, and (ii) if the stable directions are not destabilised. Unstable invariant solutions have indeed been stabilised in pipe flow through an instantaneous feedback control strategy where the Reynolds number is adjusted in response to an observable connected with deviations from laminar flow [3]. Here, we attempt to stabilise an unstable travelling wave through the suction control procedure specified in Eqs. (2)-(4). Since the feedback procedure is based on the suction velocity, which is translationally invariant in all directions, perturbations of the mean flow should be easiest to control. The left and middle panels of Fig. 1 show phase-space representations of the controlled dynamics, starting from either a strong perturbation of the mean flow (left panel) or a weak random perturbation (middle panel). One notes that the controlled dynamics approaches the operating point in both examples. However, on longer times the systems slowly drifts away from the target state, due to the adverse effects of the control on the stable directions of the invariant solution [4]. A stable control hence requires a feedback that is orthogonal to the stable directions.
As a second application of the feedback control, we use it to target a global constraint on the flow. Specifically, we pick a target value for the drag that is lower than the average drag C f = 5 × 10 −3 at a Reynolds number of Re 0 = 1600. The right most panel in Fig. 1 shows time series of the controlled for three different values of µ and the free dynamics, where the target value of the shear stress was set to (C f ) 0 = 1.25 × 10 −3 . The uncontrolled dynamics shows large fluctuations. The controlled dynamics shows much smaller fluctuations if µ is large enough, approaches the target value and stays its vicinity for up to 200 time units.