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Objective functional and scalar products
The objective functional represents a measure of how well the objectives of the optimization procedure have been attained. In our case the functional E T α2 1 T l2 I = ( ) + Z0 E(t)dt + ¿ qw, qw À (2.11) E(0) 2 T 2 is used. The scalar product ¿ ., . À appearing in (2.11) is defined as 1 Z T ¿ q1, q2 À= q1∗M§q2 dt + c.c., (2.12) T 0
where the symbol c.c. denotes the complex conjugate. The scalar product ¿ qw, qw À is then a suitably weighted measure of the wall-blowing energy, as fur-ther discussed below. Similarly, the scalar product Z ∞ [[q1, q2]] = q1∗M⁄q2 dy + c.c. (2.13) 0 is introduced so that the perturbation kinetic energy at time t defined in Table 2.1 may be expressed as E(t) = [[q(y, t), q(y, t)]], i.e. solely in terms of the velocity components u and v.
The linear differential operators M⁄ and M§ appearing in the definitions of the perturbation energy and the wall blowing energy via the scalar products (2.12) and (2.13) arise from the following considerations: by taking advantage of the conti-nuity equation and the boundary conditions (2.10), one finds through successive integrations by parts applied to E(t) defined in Table 2.1 that M⁄ = −k2 •−∂y∂ −1 − k2 ‚ . (2.14) 1 Δ ∂
Gradient of the objective functional
The Gateau differential dL of the Lagrangian evaluated at point Q is defined as d L|Q (δQ) = lim L(Q + εδQ) − L(Q) . (2.20) ε 0 ε → Assuming that L is Fr´echet-differentiable the gradient of the Lagrangian at point Q, denoted rL(Q), is such that for any vector δQ the following expression holds : {rL(Q), δQ} = dL|Q(δQ) , (2.21) with the scalar product introduced in (2.17). The projections of rL(Q) onto the subspaces span{q, 0, 0, 0, 0, 0}, span{0, q0, 0, 0, 0, 0}, etc. are denoted by the more convenient symbols rqL, rq0 L, . . . and referred to as either “ the q, q0, etc. com-ponent of the gradient” or “the gradient with respect to q, q0, etc.” The common procedure in flow control (Gunzburger 1997, Andersson et al. 1999, Corbett & Bottaro 2001a,b and Pralits et al. 2002) is to compute, first, the gradients of the Lagrangian with respect to the adjoint variables q˜, q˜0, q˜w, and, second, the gradient with respect to the flow field q. From these calculations, one can recover the direct and adjoint systems, as well as the direct and adjoint boundary conditions. The gradients of the Lagrangian with respect to the control variables q0 and qw, i.e., the initial perturbation and the wall blowing/suction velocity, are computed last and yield analytic expressions for the gradients of the objective functional with respect to q0 and qw.
Optimal perturbations
In their study of non-modal effects in swept Hiemenz flow, Obrist and Schmid (2003b) present several computations at a Reynolds number Re = 550 and a span-wise wavenumber k = 0.25. Their approach is based on an eigenfunction expansion of the linear initial value problem. With these parameter settings the flow is found to be asymptotically stable but susceptible to short-term energy growth. Compu-tations have been performed at the same parameter settings, but additional results are also presented at a Reynolds number Re = 850 where the flow is linearly un-stable. Even higher Reynolds numbers (Re = 2000) have also been investigated to probe the physical mechanisms responsible for transient energy amplification, which was found to take place throughout the parameter range under consideration (100 ≤ Re ≤ 2500, 0.05 ≤ k ≤ 0.45).
Energy amplification
The optimal perturbation is defined as the initial disturbance exhibiting the largest energy amplification over a given time interval. Minor modifications to the algorithm outlined in Table 2.2 — i.e. setting qw = 0 (no blowing/suction), using q0 as the control variable and rq0 I as the associated objective functional gradient — yield a fast and efficient algorithm to determine both the maximum energy amplification and the initial condition that produces it.
In Figure 2.2a the temporal energy evolution of such initial perturbations for the case of a linearly stable (solid line) or unstable (dashed line) basic flow is displayed. In the linearly stable case (Re = 550, k = 0.25), transient energy growth amounts to 123 times the initial energy before perturbations eventually decay. In the unstable case (Re = 850, k = 0.25), high energy levels may be reached significantly earlier than would be possible by a purely exponential growth of the unstable eigenmode only. The time required to amplify the initial energy by a factor 220 is only 15 time units which should be compared to the 200 time units required to amplify the energy of the most unstable mode by the same amount. Figure 2.2b illustrates how the optimization time Tp influences the energy evo-lution of the optimal perturbations. At a Reynolds number Re = 550 and spanwise wavenumber k = 0.25, the maximum energy amplification can be achieved by setting the optimization time to Tp = 14.3 (solid line). The energy at t = Tp is then 123 times the initial energy. Optimal perturbations for shorter optimization times, e.g. Tp = 5, are slightly more amplified initially. Their overall amplification, however, is lower than for the case Tp = 14.3. Beyond a specific value of the optimization time, the least stable mode (or the most unstable mode in the linearly unstable parameter r´egime) prevails. Any optimization time larger than Tp = 25 results in a very similar optimal initial condition — the one that excites the least stable mode most efficiently during the early stages of the energy amplification.
Table of contents :
1 Introduction
1.1 A general introduction to viscous flow and boundary-layer instabilities
1.1.1 Flying in a viscous fluid
1.1.2 On the stability of shear flows
1.1.3 Controlling shear flows
1.2 Disturbances and control of the attachment-line boundary layer
1.2.1 Flow configuration at the leading-edge of swept wings
1.2.2 Previous experimental and theoretical results
1.2.3 Control strategies
1.3 Outline
2 Optimal perturbations and optimal control in the G¨ortler-H¨ammerlin framework
2.1 Introduction
2.2 Linear perturbation model
2.3 Elements of optimization theory
2.3.1 Objective functional and scalar products
2.3.2 Lagrangian formulation
2.3.3 Gradient of the objective functional
2.3.4 Optimization procedure
2.4 Application to swept Hiemenz flow and numerical implementation
2.5 Optimal perturbations
2.5.1 Energy amplification
2.5.2 Parameter study
2.5.3 Energy transfer analysis
2.5.4 Physical mechanisms leading to transient energy growth
2.6 Optimal control
2.6.1 Control of optimal perturbations
2.6.2 Physical mechanisms
2.6.3 Constant gain feedback control
2.7 Concluding remarks
3 Optimal temporal disturbances of arbitrary shape
3.1 Introduction
3.2 Direct numerical simulation
3.2.1 Direct perturbation equations
3.2.2 Adjoint perturbation equations
3.2.3 Temporal scheme
3.2.4 Spatial discretization
3.3 Optimization techniques in swept Hiemenz flow
3.3.1 Definition of the energy
3.3.2 Adjoint-based optimization and the fringe region technique
3.4 Optimal disturbances in swept Hiemenz flow
3.4.1 Introduction
3.4.2 Flow configuration and numerical techniques
3.4.3 Three-dimensional optimal disturbances
3.4.4 Discussion
4 Optimal spatial perturbations
4.1 Introduction
4.2 Spatial energy amplification and the Ginzburg-Landau equation
4.2.1 Parabolized versus non-parabolized spatially developing perturbations
4.2.2 Conclusion
4.3 Parabolized spatial equations and numerical techniques
4.3.1 Direct and adjoint parabolized equations in swept Hiemenz flow107
4.3.2 Numerical techniques
4.4 Validation of the numerical scheme
4.5 Optimal spatial perturbations in swept Hiemenz flow
4.5.1 Flow parameters in the spatial framework
4.5.2 Optimal spatial energy amplification
4.5.3 Optimal spatial disturbances
4.5.4 Discussion
5 Conclusion : research on the attachment-line boundary layer in a historical perspective
Appendix
