Optimization and control of a separated boundary-layer flow Journal: 2011 Hawaii Summer Conferences Manuscript ID: Draft lumeetingid: 2225 Date Submitted by the Author: n/a Contact Author: PASSAGGIA, Pierre-Yves
Page 1 of 6 2011 Hawaii Summer Conferences Optimization and control of a separated boundary-layer flow P.-Y. Passaggia and U. Ehrenstein Institut de Recherche sur les Phénomènes Hors Equilibre (IRPHE), CNRS / Aix-Marseille Université, F-13384 Marseille, France In order to investigate the possibilities of controlling wall-bounded flows, optimal control of the nonlinear dynamics of a separated boundary-layer flow is studied using a localized blowing-suction at the wall. Control laws are computed using a formulation based on the augmented Lagrangian approach using the full knowledge of the flow dynamics. The case study consists of a laminar separated boundary-layer flow over a bump. I. Introduction Flow separation is ubiquitous to wall bounded flows when subject to an adverse pressure gradient or geometrical devices. The resulting recirculation bubble are source of instability phenomena leading to a loss of aerodynamic performances. It is essential to propose control solutions capable of minimizing there impact on the entire flow. In this paper, an adjoint-based optimization technique is presented in order to control the flow behind an obstacle. The feedback control of such dynamics has been recently investigated with success in the case of a flow over a shallow cavity 2 using model reduction based on the linearized perturbation dynamics. 6 However, model reduction based on the linear dynamics is likely to be not reliable as soon as absolute/convective instabilities coexist in the nonlinear regime. In order to assess the possibility of feedback control of such flow dynamics, the augmented Lagrangian approach has been implemented. This formulation takes into account the entire flow dynamics and was for instance shown to relaminarize a turbulent channel flow. 4 This formulation can also be adapted to the computation of optimal perturbations which are known to play an important role in the transient dynamics of open flows. In the present investigation, the flow is a laminar boundary layer where flow separation is triggered by a bump. This configuration produces a long recirculation bubble which becomes unstable beyond a critical Reynolds number. 5 This flow has been studied numerically by Marquillie & Ehrenstein 7 and their algorithm has been adapted to the present study. For the optimization problem with constraints, the gradient method based on a formulation of the adjoint Navier-Stokes system has been implemented. The control problem of the fully developed nonlinear dynamics, detailed in section 2, is formulated using a cost function based on either the integral or the terminal time kinetic energy of the perturbation. The effect of nonlinearities on the optimal initial disturbance and the control of the fully developed dynamics is presented in section 3. Finally the results are discussed in section 4. II. Formulation and numerical procedure The flow is decomposed into a baseflow and a perturbation and the Navier-Stokes system is written t u + (U )u + (u )U + (u )u + p (1/Re) u = 0, u = 0 (1) where U is the velocity of the baseflow and u, p are the velocity and pressure pertubation, respectively. The Reynolds number PhD Student, IRPHE, 49 rue F. Joliot Curie, B.P. 146, F-13384 Marseille Cedex 13, France Professor, IRPHE, Aix-Marseille Université 1 of 6
Page 2 of 6 Re = δ U (2) ν is based on the displacement thickness δ of the Blasius profile imposed at the entrance of the domain. Here, the numerical procedure of Marquillie & Ehrenstein 7 has been adapted for the optimization formulation. The bump is taken into account by use of a mapping which transforms the physical domain into a cartesian one. The streamwise direction x is discretized using finite differences whereas y, the wall-normal direction, is discretized using Chebyshev-collocation. The time integration is performed using a second order semiimplicit scheme. The difference between the present formulation and the one described in Marquillie & Ehrenstein comes from the algorithm used to solve the pressure. In the optimization formulation, the choice of outflow boundary condition is essential. In order to ensure the compatibility between the Navier-Stokes system and its adjoint, stress-free outflow boundary condition are to be implemented which is achieved using the influence matrix technique which couples pressure and velocity 8 II.A. Unstable baseflow At a supercritical Reynolds number of 650 used in the presnet investigation, a large domain with length L = 250 and a height H = 100 has been considered to minimize finite boundary effects. The first step of the analysis consists in generating a baseflow and the method of Åkervik et al. 1 ( called selective frequency damping) has been implemented in order to converge towards a stationary solution of the Navier-Stokes system. The method consists in solving the coupled system { q = f(q, Re) χ(q q) q = (q q) (3) Λ where f represents the Navier-Stokes system, q contains the velocity U and the pressure P of the baseflow. Here Λ is a cutoff frequency and χ is a damping coefficient. In the present case, the value of Λ corresponds to the dominant frequency measured using a probe localized inside the shear layer. A good estimation of the parameter χ is given by the growth rate of the linear instability, provided by the global stability analysis of the flow. 5 For Re = 650, the parameters used are = 6.37, χ=0.03 and figure 1 shows the resulting baseflow. Figure 1. Baseflow for Re = 650 II.B. Augmented Lagrangian and gradient In the case of the control, the aim is to minimize the kinetic energy in the entire domain and the cost function is either J 1,int (φ, u) = 1 2 when energy integral is considered or J 1,term (φ, u) = 1 2 T1 Ω o u u dx dt + γ Ω o u(t 1 ) u(t 1 ) dx dt + γ T1 T1 Γ c gφ gφ ds dt (4) Γ c gφ gφ ds dt (5) for energy optimization at time T 1. The portion of the boundary where the control is applied is denoted by Γ c, whereas g(x) defines the spatial distribution (taken as uniform in the present case) of the blowing-suction. The control has been applied at three different locations (each one of length 5) in the vicinity of the bump (cf. figure 2). The function φ(t) is the control law and γ refers to the cost of the control. The optimization 2 of 6
Page 3 of 6 2011 Hawaii Summer Conferences time interval is [, T 1 ]. The aim is to minimize the functional J 1, u being the solution of the Navier-Stokes system written as f(u, u, p) = 0, g(u, φ) = 0 with g(u, φ) = u Γc gφ. (6) In the case of the optimal perturbation analysis, the objective function becomes J 2 (φ, u) = 1 u(t 1 ) u(t 1 ) dx (7) 2 Ω the unknown being now the initial condition u 0, that is u is now solution of f(u, u, p) = 0, g(u, u 0 ) = u(x, 0) u 0 = 0, h(u 0, ɛ) = u 0 u 0 dx ɛ = 0 (8) where ɛ is the amplitude of the initial condition u 0. Lagrange multipliers u +, p +, φ + are introduced and in Ω Figure 2. Sketch of the domain and of the control setup order to solve the optimization problem, the Lagrangian L L(u, u +, p, p +, φ, φ +, ɛ, ɛ + ) = J 1,int (φ, u) < f(u, u, p), (u +, p + ) > < g(u, φ), φ + > (9) is to be rendered stationary. Cancelling the Fréchet derivatives with respect to the Lagrange multipliers consist in imposing the Navier-Stokes constraints. The derivatives with respect to the state variables (u, p) writes T1 T1 D u,p L (û, ˆp) = u û dxdt D u,p < f(u, p), (u +, p + ) > (û, ˆp) û gφ + ds dt (10) Ω Γ c The expression of D u,p < f(u, p), (u +, p + ) > (û, ˆp) is detailed for example in Bewley et al. 3 the above expression is equivalent to solve the adjoint system and canceling t u + (U )u + + ( U) t u + (u )u + + ( u) t u + + p + (1/Re) u + = u, u + = 0. (11) The right hand side of the adjoint momentum equation is zero in the case of a terminal time optimization. The adjoint system is integrated backward in time from T 1 to, starting with u + (T 1 ) = 0 in the intgral time formulation whereas for terminal time u + (T 1 ) = u(t 1 ). The integrations by part necessary to obtain the adjoint system introduce the integral on the boundary Ω T1 Ω [ 1 Re u+ ( û) n 1 Re ( u+ n)û (U n)u + û (u n)u + û + (p + n)û u + (ˆp n)] ds dt. (12) The stress-free conditions 1 Re u n pn = 0 being imposed at the outflow of the domain Ω S, the outflow condition for the adjoint system (implemented using the influence matrix) 1 Re ( u+ )n ΩS (U n)u + ΩS (u n)u + ΩS + p + n ΩS = 0, u + Ω ΩS = 0 (13) cancels this integral on Ω Γ c. Finally, the contribution on Γ c is cancelled taking gφ + = 1 Re u+ n + p + n (14) 3 of 6
Page 4 of 6 in (10). The derivative of the Lagrangian with respect to the control variable φ becomes D φ L ˆφ T1 = γ gφ g ˆφ T1 ds dt ( g ˆφ) gφ + ds dt (15) Γ c Γ c and taking into account (14), the gradient for the control is φ J 1 (φ) = γφ g g ds + ( 1 Γ c Γ c Re u+ n + p + n) g ds. (16) The same procedure is used for the optimal perturbation, the expression of the gradient now being { u0 J 2 (u 0 ) = u + 0 + 2ɛ+ u 0 with u + 0 = u+ (x, ) ɛ + = Ω u+ ( ) u + ( ) dx/4ɛ (17) An iterative gradient algorithm φ k+1 = φ k α k φ J (φ k, u +,k ). is used to converge towards the optimal solution. A main difficulty is to appropriately choose the step α k and this parameter is optimized using a classical line search procedure. Once the optimal control is computed over the first time interval, the time marching is insured using a recieding-predictive control algorithm described in 4 with a time shift T a = T/2 equal to half the time of optimization in our case (see figure 3). Figure 3. Schematic of the recieding-predictive algorithm III. Optimal growth and control The linear optimal perturbation analysis 5 showed that perturbations localized near the separation point are responsible for the largest possible transient growth of the system. In order to examine the effect of Figure 4. a) Evolution of the structure of the optimal perturbation as a function of the initial amplitude, b) transient growth of optimal perturbations as a function of initial amplitude for T 1 = 150. 4 of 6
Page 5 of 6 2011 Hawaii Summer Conferences nonlinearities on these optimal perturbations, the initial amplitude of the disturbance is first set to small values in order to maintain the flow close to the linear regime. Then the amplitude of the initial disturbance is increased from 10 8 to 10 4 and successive optimization times were considered between T1 = 50 and T1 = 150. Figure 4a shows the shape of the optimal initial disturbance for three amplitudes considered and its structure is seen to spread in the shear layer, as well as upstream the bump when the ampliude is increased. This clearly shows that the nonlinear flow is not only sensitive to actuators located on the separation point, but also to regions located inside the recirculation bubble as well as upstream the bump. Figure 4b shows the energy evolution for the time horizon T1 = 150. The flow perturbation starts to saturate at T = 120 when the highest initial perturbation amplitude is considered. The possibilities of controlling the fully developed dynamics is then investigated. This unstable dynamics is characterized by a low frequency oscillations of the recirculation bubble5. The uncontrolled flow energy is shown as the solid line in figure 5 and the flapping frequency of Λ 200 is clearly visible. Figure 5. a) Energy of the perturbation for the uncontrolled case ( ), control using the integral time formulation with a single actuator (...), control with 3 actuators (.. ), control with 3 actuators ( ) using the terminal time formulation and b) associated control laws φ for the integral time formulation. Ideally one would like to take arbitrarily long time horizons for optimization. However, the direct solution in time u is input of the adjoint system and memory constraints put a bound on the maximum time T1 which can be considered. Here we have chosen T1 = 300 which is larger than the flapping period. The optimization procedure implies 5 to 7 computations of the gradient whereas 4 to 7 evaluations of the objective function are necessary to evaluate the optimal step α. A first actuator is introduced at the separation point shown as 1 in figure 2, where the nonlinear optimal perturbation dominates. The problem is then solved in time iterating Figure 6. Time-averaged streamlines for the uncontrolled flow a), and for the controlled flow b) using the integral formulation and 3 actuators. back and forth the direct and the adjoint system to achieve convergence for each time interval. Using this single actuator, the control is capable of diminishing the energy to half of the uncontrolled value without however achieving definite control, as seen in figure 5. In the attempt to improve the control, two additional actuators shown as 2 and 3 in figure 2 have been added. The corresponding energy evolutions are depicted in figure 5: although the energy is decreased further for T > 500 the perturbation energy cannot be reduced to zero. As shown in Marquillie & Ehrenstein,7 in the perturbed nonlinear regime, the mean recirculation 5 of 6
Page 6 of 6 Figure 7. Instantenous snapshot of the vorticity at time T = 600 for the uncontrolled flow a), for the controlled flow b) using the integral formulation and 3 actuators. length diminishes when compared with that of the baseflow. Therefore the recirculation length is likely to provide informations about the quality of the control. Figure 6 shows an increase from approximately 48.5 for the uncontrolled case to 63.4 when the control is applied. Note that the mean values have been computed for the interval 450 t 900. A snapshot of the vorticity at time t = 600 is depicted in figure 7. The use of terminal time formulation in the objective function has been advocated by ref 3, which in the case of the turbulent channel flow was capable to achieve a relaminarization. Here we imposed finite time energy optimization starting at time 350, however for the present flow dynamics the results are comparable with the integral time formulation, as can be seen in figure5. IV. Discussion The optimal growth and control of a separated laminar boundary layer flow over a bump is investigated numerically using an adjoint-based augmented Lagrangian approach. In this configuration, stress-free outflow boundary conditions compatible between the Navier-Stokes system and its adjoint were implemented, using the influence-matrix approach. The problem of optimal perturbations in the nonlinear regime has been studied, highlighting the regions of the flow which are the most sensitive to actuator devices. It is shown that in the nonlinear regime the flow dynamics becomes sensitive to perturbations aligned along the whole shearlayer, in contrast to the linear regime which is only sensitive to upstream perturbations. The control problem using blowing-suction at the wall has been addressed as well, using a recieding-predictive algorithm for time marching. The unstable dynamics related to the flapping of the recirculation bubble is clearly attenuated but a full control of the nonlinear dynamics could not be achieved and this point is still under investigations. For instance to take larger time horizons and/or multiple actuators is likely to improve the control capabilities. Acknowledgements The authors gratefully acknowledge the financial support of the Agence Nationale pour la Recherche (ref. ANR-09-SYSC-011). References 1 E. Åkervik, L. Brandt, D. Henningson, J. Hœpffner, O. Marxen, and P. Schlatter. Steady solutions of the navier-stokes equations by selective frequency damping. Phys. of Fluids, 18:068102, 2006. 2 E. Åkervik, J. Hœpffner, U. Ehrenstein, and D. Henningson. Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech., 579:305 314, 2007. 3 T.R. Bewley. Flow control: new challenges for a new renaissance. Prog. in Aero. Sci., 37:21 58, 2001. 4 T.R. Bewley, P. Moin, and R. Temam. Dns-based predictive control of turbulence: an optimal benchmark of feedback algortihms. J. Fluid Mech., 447:179 225, 2001. 5 U. Ehrenstein and F. Gallaire. Two-dimensional global low-frequency oscillations in a separating boundary-layer flow. J. Fluid Mech., 614:315 327, 2008. 6 J. Kim and T. R. Bewley. A linear systems approach fo flow control. Annu. Rev. Fluid Mech., 39:383 417, 2007. 7 M. Marquillie and U. Ehrenstein. On the onset of nonlinear oscillations in a separating boundary-layer flow. J. Fluid Mech., 490:169 188, 2003. 8 R. Peyret. Spectral Methods for Incompressible Flows. Springer, 2002. 6 of 6