Computing optimal flow disturbances

We outline methodologies for computation of the spatial distributions of energy-optimal linear initial and boundary disturbances to incompressible flows. The theory presented here is based in techniques developed for constrained optimisation, but we show that there are equivalent eigenvalue interpretations. As a result the computations may be carried out either by optimisation or eigensystem methods, leading to the same outcomes though typically the eigensystem approaches converge more rapidly for optimal initial condition calculations. We show how the methods have been applied to example open flows. Introduction Methods for computing linear large-time stability and optimal initial condition disturbances for transient growth in general incompressible flows are now well established (see e.g. [1, 15]). For open flows (ones with an inflow and an outflow) one may well be more interested in finding an inflow disturbance which can lead to energetic disturbances further downstream than in optimal initial disturbances: it is rather difficult to conceive how an initial disturbance could actually be created within an open flow. We here discuss how to compute optimal inflow boundary condition perturbations, those that produce an optimal gain, i.e. kinetic energy in the domain at a given time horizon normalised by a measure of time-integrated energy on the inflow boundary segment. The conceptual setting for these discussions lies in the arena of optimal flow control, as described e.g. in [8, 13]. From consideration of suitable constrained optimisation problems, we show that one can compute either optimal initial or boundary condition perturbations to prescribed base flows (U ,P) using iterative gradient-based methods. However, further consideration also shows that in each case there is an equivalent eigenvalue problem which delivers the same perturbation. We examine the relative computational efficiencies of these two alternative approaches. We demonstrate that, similarly to the optimal initial condition problem, the gain can be interpreted as the leading singular value of the forward linearized operator that evolves the boundary conditions to the final state at a fixed time. In this investigation we restrict our attention to problems where the temporal profile of the perturbations examined is a product of a Gaussian bell and a sinusoid, whose frequency is selected to excite axial wavelengths similar to those of the optimal initial perturbations in the same geometry. Comparison of the final state induced by the optimal boundary perturbation with that induced by the optimal initial condition demonstrates a close agreement for the selected problem. Previous works dealing with optimal boundary perturbation, e.g. [7] considered a prescribed spatial structure and computed an optimal temporal variation of a walln 3 Figure 1: Schematic representation of a spatial flow domain Ω, boundary ∂Ω and unit outward normal vector n. normal velocity component, whereas here we consider perturbations with a prescribed temporal structure and compute the optimal spatial variation of velocity boundary conditions over a one-dimensional inflow boundary segment. The methodology is capable of finding optimal boundary condition perturbations in general non-parallel twoand three-dimensional flows. The gradient-based optimisation approach to these problems has been described in an earlier paper [11], but here more consideration is given to comparing optimisation and eigensystem methodologies. The optimal boundary perturbation problem is, in the flows examined to date at least, closely linked to the optimal initial condition problem in that ultimately the same physics are excited and the wave-packet nature of the optimal inflow perturbation is closely linked to that for the optimal initial disturbance and outcome. Problem definition Working from the incompressible Navier–Stokes equations ∂tu =−u ·∇u−∇p+Re−1∇2u, with ∇ · u = 0, where p is the modified or kinematic pressure, u is the velocity vector, while the Reynolds number Re = UD/ν where U and D are convenient velocity and length scales and ν is kinematic viscosity. Decomposing the flow field as the sum of a base flow and a perturbation i.e. (u, p) = (U ,P)+ (u′, p′) and omitting the interaction of perturbations, we obtain the linearized Navier–Stokes (LNS) equations, which govern the evolution of infinitesimal perturbations, as ∂tu=−U ·∇u′−(∇U )T ·u′−∇p′+Re−1∇2u′, with ∇ ·u′= 0, or more compactly, considering that pressure is a dependent variable in an incompressible flow, ∂tu−L(u) = 0. (1) Flow is considered within a spatial domain Ω which has a boundary surface ∂Ω and unit outward normal n, as indicated in figure 1. Flow is taken to evolve over the time interval [0,τ], so the space-time domain considered is Ω× [0,τ]. We introduce scalar products defined on spatial domain Ω and its boundary ∂Ω