Parameter-elimination method

The calculation of d(k)_{requires inverting a matrix with size equal to the number of}

constraints. When the number of constraints is large, this can become prohibitive. If during the optimisation procedure the magnitude of d(k) _{vanishes and the}

conjugate gradient method falls back on the steepest descent method (β = 0 in (2.13)), an optimum is reached (Chong and Zak, 1996). In this case, the gradient of the cost functional is linearly dependent on the gradients of the constraints, implying that the cost functional can only further improve if a constraint is violated. The KKT conditions (2.47-2.48) are then satisfied (see Chong and Zak, 1996). The projected gradient of the steepest descent direction is known to be a descent direction (Hinze et al., 2009; Brezillon and Gauger, 2004). In Hinze et al. (2009) it is however noted that this is not the case for the projection of any search direction, an example is given where the projection of the Newton search direction is not a descent direction (Hinze et al., 2009). In this work the search direction g(k) _is

the conjugate-gradient direction. It is difficult to prove that the projection of the conjugate gradient is a descent direction, or d(k)_.G(k) _{< 0 with g}(k)_{in (2.53) given}

by (2.13). Equation (2.13) however shows that if the step lengths decrease, for example near the optimum, the gradient G(k−1)_{and G}(k)_{become equal, and β goes}

to zero. The conjugate gradient method is then reduced to the steepest descent direction for which it is known that the projection is still a descent direction. If

β is non negligible and the projection would result in an ascent direction, this is

not a problem as the line-search algorithm mnbrack and Brent (Press et al., 1996, and section 2.3.2) are robust enough to handle this situation. In that case, the line search searches for negative step lengths.

The conjugate gradient method combined with the gradient projection method shows similarities with line search algorithms on manifolds (see, e.g., Absil, Mahony, and Sepulchre, 2008) where the parameter surface is related to the manifold, and the αproj correction step to the retraction. Optimisation on

manifolds is used in the framework of matrix optimisation problems (Absil et al., 2008; Vandereycken and Vandewalle, 2010).

2.5.3

Parameter-elimination method

The last method used is the parameter-elimination method, which reduces the parameter set to a parameter set that can be freely optimised without constraints. It is a very convenient method for linear constraints and might for some cases also

be used for non linear constraints (see, for more details Nocedal and Wright, 2006). The method reduces the parameters to a subset of linear independent parameters. Instead of formulating the iterative optimisation algorithm (cf. section 2.3) in terms of φ, it is directly formulated using the reduced parameters. The application of the parameter-elimination method to the envisaged optimisation problem is discussed in section 3.3.1.

Numerical approach

This chapter starts with the specification of the optimisation problem for the temporal mixing layer (section 3.1). Subsequently the computational set-up, discretisation of the flow, and adjoint simulations are discussed in section 3.2. Finally, the application of the constraint methods on the envisaged optimisation problem is detailed in section 3.3.

3.1

Optimisation problem

In Chapter 2 the flow optimisation techniques were introduced for a generic cost functional and constraints on the parameters. Here, the techniques are applied to the temporal mixing layer, introduced in section 1.2.

The perturbations φ(x) on the initial mean flow field of the temporal mixing layer are optimised such that an optimal volume-averaged property is achieved after a selected simulation time, called time horizon T . For the initial mean velocity field a hyperbolic-tangent in the streamwise direction is taken. The dimensionless initial velocity perturbations φ, which have no mean velocity component, are added to this dimensionless mean velocity field (see figure 3.1):

u(x, 0, φ) = tanh(x3) e1+ φ(x), (3.1)

with e1the unit vector in the streamwise direction x1(see figure 3.1). This velocity

field was made dimensionless according to section 2.1.1 with as reference velocity

uref= ∆U/2 [m/s], and as reference length Lref= δω/2 [m], with ∆U the velocity

X3 X 2

X1

Figure 3.1: Schematic illustration of a temporal mixing layer. Mean velocity profile (—) and the perturbations (grey arrows).

difference between the upper and lower flow, and δωthe initial vorticity thickness:

δω= ∆U  ∂hu(x, 0, φ)i ∂x3   max . (3.2)

The ensuing evolution of the flow is governed by the non-dimensional incompress- ible Navier–Stokes equations (2.11-2.12).

The temporal mixing-layer case is used as a substitute case to study how long flow control can affect the evolution of the mixing-layer solution. Several cost functionals are formulated that measure volume-averaged properties at the time horizon T , such that J (Eq. (2.14)) simplifies to JT.

The optimisation will minimise the cost functional subject to two constraints on

φ(x), i.e. min φ JT(q(x, T, φ)) subject to ( ∇ · φ(x) = 0 ∀x ∈ Ω (3.3) 1 2 1 Ω Z Ω φ· φ dx = E0 1 2 (3.4)

The first constraint (3.3) follows from the requirement that the initial velocity field

u(x, 0, φ) should satisfy the continuity equation. When the system is discretised, this constraint has to be satisfied in every point on the grid. This leads to an equation per grid point expressing the continuity of the discretised φ. The enforcement of these constraints is essential, since it is required for a physically relevant solution of the Navier–Stokes equations (2.11–2.12).

The second constraint (3.4) keeps the dimensionless total energy per unit volume of the perturbations fixed at a constant level E0 times the initial dimensionless

mean-flow energy per unit volume (1/2 for a box infinite in the normal direction). From a practical point of view, the kinetic energy of the controls (i.e. the initial perturbations of the temporal mixing layer) should be low compared to the mean- flow energy, since, if we would consider an actual implementation of the control in a spatial framework, this energy would be roughly related to the amount of work required from an upstream actuator array on a splitter plate. However, in the current work, it is not the intention to elaborate on the connection between the optimised perturbations in the temporal framework and the actuations in the spatial framework. It is obvious that controls, i.e. the perturbations, with a high level of energy can have a large impact on the mean-flow statistics; they correspond to a brute-force approach to the mixing layer flow control problem. The effectiveness of low-energy controls on the mean-flow evolution over long time horizons seems less obvious. Therefore, this work focusses on low-energy controls, and strictly enforces a selected constant energy level of the controls as a constraint during the optimisation.