2.1 Configuration

The high-lift configuration, shown in figure 2, is a modified DLR F15 airfoil equipped with a highly deflected Coanda flap and a droop nose. The details of the airfoil design are described in Jensch et al. (Reference Jensch, Pfingsten, Radespiel, Schuermann, Haupt and Bauss2009) and Burnazzi & Radespiel (Reference Burnazzi and Radespiel2014), where the objective was to accomplish high lift coefficients during take-off and landing. The droop nose design is chosen to satisfy low noise emission restrictions, which exclude other leading edge devices, such as the Krüger slat. The leading edge geometry was reached after an iterative process that improved the airfoil stall behaviour, which is ruled by the suction peak. The highly deflected flap at $65^{\circ }$ has a chord length of $c_{fl}=0.25c$ , where $c$ is the main airfoil chord length with the retracted flap. The flap is specially designed with a round shoulder to enhance the Coanda effect. Henceforth, all physical variables are assumed to be non-dimensionalized with respect to the chord length $c$ , the incoming flow velocity $U_{\infty }$ and the constant density ${\it\rho}$ (constant outside the plenum). The numerical simulations are performed at Mach number $Ma=0.15$ and Reynolds number $Re=U_{\infty }c/{\it\nu}=12\times 10^{6}$ , where ${\it\nu}$ is the kinematic viscosity of the fluid. These flow parameters correspond to the expected conditions during landing.

Figure 2. Part of the numerical mesh surrounding the modified DLR F15 airfoil with a close up of the actuation duct.

Unsteady Coanda blowing is performed through an oscillatory lip motion, simulating the planned experimental device. This yields a velocity fluctuation at the jet exit whose magnitude can be prescribed as

(2.1) $$\begin{eqnarray}\displaystyle \Vert \boldsymbol{U}_{J}\Vert _{2}(t)\approx B_{0}+B_{1}\cos ({\it\omega}^{a}t), & & \displaystyle\end{eqnarray}$$

where ${\it\omega}^{a}$ is the angular actuation frequency, $B_{0}$ is the mean exit velocity, $B_{1}$ is the oscillatory actuation amplitude and $\Vert \cdot \Vert _{2}$ is the Euclidean vector norm. For simplicity, the jet velocity magnitude at the jet exit $\Vert \boldsymbol{U}_{J}\Vert _{2}$ will henceforth be referred to simply as $U_{J}$ . For the current test case, the corresponding actuation velocity parameters are ${\it\omega}^{a}=f^{a}c/U_{\infty }=1.96$ , $B_{0}=4.515$ and $B_{1}=0.07$ . The blowing intensity is characterized by the momentum coefficient,

(2.2) $$\begin{eqnarray}\displaystyle c_{{\it\mu}}(t)=\frac{U_{J}(t){\dot{m}}_{J}(t)}{\frac{1}{2}{\it\rho}U_{\infty }^{2}S_{ref}}, & & \displaystyle\end{eqnarray}$$

where ${\dot{m}}_{J}$ is the jet mass flow rate and $S_{ref}$ is the reference area. Similar to the jet exit velocity given by (2.1), the momentum coefficient can be expressed as $c_{{\it\mu}}(t)=C_{{\it\mu}0}+C_{{\it\mu}1}\cos ({\it\omega}^{a}t)$ , where $C_{{\it\mu}0}$ is the steady mean and $C_{{\it\mu}1}$ is the amplitude of oscillation. This yields a mean momentum coefficient of $C_{{\it\mu}0}=0.0295$ and amplitude $C_{{\it\mu}1}=0.0239$ . The non-dimensional actuation frequency (Seifert, Darabi & Wyganski Reference Seifert, Darabi and Wyganski1996; Seifert et al. Reference Seifert, Greenblatt and Wygnanski2004) can be expressed as,

(2.3) $$\begin{eqnarray}\displaystyle F^{+}=\frac{f^{a}c_{fl}}{U_{\infty }}=\frac{{\it\omega}^{a}c_{fl}}{c}, & & \displaystyle\end{eqnarray}$$

where the flap chord length $c_{fl}$ is used as the length scale, since the flow is only separated over the flap. This definition yields a normalized actuation frequency of $F^{+}=0.49$ for the current test case. All actuation parameters are selected from a previous parametric study, where lift was maximized with respect to the steady blowing momentum coefficient.

2.2 Unsteady Reynolds-averaged Navier–Stokes simulations

The computational fluid dynamics (CFD) solver employed to perform the analysis is the DLR TAU-Code (Kroll et al. Reference Kroll, Rossow, Schwamborn, Becker and Heller2002; Schwamborn, Gerhold & Heinrich Reference Schwamborn, Gerhold and Heinrich2006). The two-dimensional unsteady Reynolds-averaged Navier–Stokes (URANS) equations are solved using a finite volume approach. The discretization schemes are the central scheme and the second-order upwind Roe scheme for the mean-flow inviscid flux and the convective flux of the turbulence transport equation, respectively. The turbulence model is that of Spalart–Allmaras with a curvature correction (Shur et al. Reference Shur, Strelets, Travin and Spalart2000) which allows the one-equation turbulence model to maintain a good accuracy in regions where the streamlines have a high curvature. This characteristic is fundamental for the simulation of the Coanda phenomenon, which is based on the equilibrium between the inertial forces and the momentum transport in the direction normal to the convex surface (Pfingsten et al. Reference Pfingsten, Jensch, Körber and Radespiel2007). The numerical scheme and the turbulence model were previously assessed by comparing the results to wind tunnel experiments (Pfingsten, Cecora & Radespiel Reference Pfingsten, Cecora and Radespiel2009; Pfingsten & Radespiel Reference Pfingsten and Radespiel2009). The lift, drag and pitching moment coefficients are determined by integrating the pressure and shear stress distributions over the airfoil surface. The contribution from the added jet momentum is not included.

The mesh density was determined by means of a mesh convergence exercise based on the Richardson extrapolation (Richardson & Gaunt Reference Richardson and Gaunt1927). This procedure provided an estimation of the space discretization error and of the minimum number of points that produced acceptable accuracy. Three different grid densities with ${\sim}70\,000$ , 230 000 and 920 000 points were tested. The corresponding maximum lift coefficients of 4.410, 4.456 and 4.480 were obtained for the three grids at ${\it\alpha}=3^{\circ }$ angle of attack. Based on these values, the Richardson extrapolation yielded a maximum lift coefficient of 4.496, which is an approximation of using an infinite large number of grid points. This extrapolated value was used as a reference lift coefficient to determine the grid resolution error, which was 1.91 %, 0.89 % and 0.36 % for the three grids, respectively. As such, the medium grid, with ${\sim}230\,000$ points was chosen, which represented a compromise between accuracy and computational cost.

The mesh is composed of a structured and an unstructured region, as the close up in figure 2 shows. The outer unstructured mesh has a C-block topology and extends 50 chord lengths in all directions. The structured grid extends from the airfoil surface outward to cover the region where the main viscous phenomena occur. The viscous sublayer is also resolved with $y_{+}<1$ everywhere over the airfoil surface. An important characteristic of the grid is the high grid density along the pressure side, where the stagnation point can be located, far from the leading edge. The structured region is also extended over a large area behind the highly deflected flap, in order to accurately capture the wake dynamics during the separated non-actuated period. Both the trailing edge and the edge of the slot lip are discretized by means of a local C-block topology.

The lip oscillation movement is numerically simulated by means of grid deformation, as shown in figure 3, where the black/blue lines denote the mesh domain boundaries for the lowest/highest position. To avoid a pressure surge in the plenum and a mesh singularity, no simulations are conducted with a fully closed lip. The total lip height movement is $2h$ ( $3h/4$ below and $5h/4$ above neutral position), where $h=0.0006$ is the lip height at the neutral position. Both, the neutral lip height and the plenum geometry, are based on the results by Jensch et al. (Reference Jensch, Pfingsten, Radespiel, Schuermann, Haupt and Bauss2009), where a range of possible duct geometries was investigated.

Figure 3. Numerically simulated dynamic lip movement. The slot exit mid-point $\boldsymbol{x}_{J}$ is also shown.

The unsteady lip movement represents a constantly deforming non-homogeneous boundary condition and introduces additional complexity to any reduced-order model. As simplification, all snapshots are exported, a posteriori, onto a static mesh that is fixed at the neutral position. Critically, this procedure preserves all dynamical interactions between the moving lip wall and the incoming boundary layer over the lip edge along the suction side. All data presented henceforth are those exported on the static mesh. This approximation is deemed acceptable due to the spatially limited extent of the deformed mesh, as can be seen in figure 3. As a result, the unsteady non-homogeneous boundary condition is transferred to the Coanda jet flow across the exit location, which now acts as a boundary with periodic pressure/velocity fluctuation. The implications of such simplification on POD modelling will be discussed in § 3.1.

2.3 Transient flow solutions

Sudden transients between two flow states lead to rich dynamics. The current numerical simulation includes two sudden transients between the unactuated state and an actuated one. The unactuated flow field behind the deflected flap is characterized by massive separation, as seen in figure 4(a). The recirculation region behind the wake is associated with periodic vortex shedding with a dominant Strouhal number of $St_{fl}^{u}=f^{u}c_{fl}/U_{\infty }=0.23$ . With actuation, the effect on the flow field is the near complete attenuation of the natural shedding vortices and the emergence of a new attractor locked-in on the actuation frequency. This observation is also confirmed in the lift coefficient distribution in figure 5. During this state, the flow is attached over the flap length, with the exception of roll-off vortices generated at the jet exit. The averaged effect of this phenomenon is shown in figure 4(b), where a small recirculation region near the trailing edge of the flap can be observed. One can also see the more curved streamlines in comparison with the natural state, which are associated with elevated lift for that case.

Figure 4. Comparison between (a) the unactuated, and (b) the actuated ( $C_{{\it\mu}0}=0.0295$ , $C_{{\it\mu}1}=0.0239$ and $F^{+}=0.49$ ) mean-flow field visualized by streamlines. The black line surrounding the airfoil delimits the structured grid region.

Figure 5. The lift coefficient (black) and the jet exit velocity (grey) from the URANS data: $Re=12\times 10^{6}$ , $Ma=0.15$ , and ${\it\alpha}=0^{\circ }$ . Actuation is turned on at $t_{s}=7.395$ and turned off at $t_{e}=43.843$ .

A more accurate description of the flow behaviour throughout both transients is provided by the lift distribution. Figure 5 shows the lift coefficient (black) and the jet exit velocity (grey) from the URANS data. The simulation starts from the unactuated natural state until $t_{s}=7.395$ , when actuation is turned on. The exit velocity distributions rise sharply with oscillatory modulation. In other words, the lip starts oscillating at the same instant as the total pressure in the plenum is increased. The flow response to the sudden actuation is almost instantaneous, as seen by the quick increase in the lift coefficient. This initial response, hereafter called the off–on transient, is followed by a slow asymptotic convergence towards the new state, which is reached at $t_{c}\approx 39$ . Asymptotic behaviour is assumed reached when one lift coefficient period changes less than 0.1 % than its predecessor. The lift increase through actuation is 80 % ( $C_{l}=1.92$ versus 3.47). Throughout this transition period the flow is locked on the actuation frequency ${\it\omega}^{a}$ , as seen in the $C_{l}$ frequency and in the flow field snapshots (not shown here). After the flow settles on the new attractor for 8 cycles, the actuation is suddenly turned off at $t_{e}=43.843$ and the flow is left to settle towards its natural state. Similar to the first transient, the flow response to the second transient called the on–off transient is immediate, and a sudden drop in the lift coefficient is observed. The flow transition from the actuation frequency to the natural shedding frequency ( ${\it\omega}^{u}=0.919$ ) is also near instantaneous. However, unlike the off–on flow response, the on–off transient is associated with an undershoot and an increased oscillation amplitude. These differences stem from the different damping mechanisms acting on the two transients. The oscillations following the on–off transient decay only through ‘natural’ damping from wall shear stress and the motion of the separation region, whereas those following the off–on transient have the added damping effect of the blown jet that increases the Reynolds stresses (Amitay, Smith & Glezer Reference Amitay, Smith and Glezer1998). Following this transition phase, the flow settles on the unactuated attractor at $t\approx 68$ .

3.1 Actuation modes as lifting method

A Galerkin model of fluid flow is based on an inner product in the observation region. In this study, a circular domain of five chord lengths surrounding the airfoil was chosen,

(3.1) $$\begin{eqnarray}\displaystyle {\it\Omega}:=\{(x,y):(x,y)\not \in \text{airfoil and }x^{2}+y^{2}\leqslant 5^{2}\}. & & \displaystyle\end{eqnarray}$$

This domain is large enough to resolve the main coherent structures and actuation effects, yet small enough to reduce long-term convection effects which are detrimental to the robustness of Galerkin models (Noack et al. Reference Noack, Morzyński and Tadmor2011).

The Galerkin expansion is defined in the space of square-integrable vector functions in the observation domain, $L^{2}({\it\Omega})$ . The corresponding inner product between two velocity fields $\boldsymbol{v}$ and $\boldsymbol{w}$ is defined by

(3.2) $$\begin{eqnarray}\displaystyle (\boldsymbol{v},\boldsymbol{w})_{{\it\Omega}}:=\int _{{\it\Omega}}\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{w}\,\text{d}\boldsymbol{x}, & & \displaystyle\end{eqnarray}$$

where the associated norm reads $\Vert \boldsymbol{v}\Vert _{{\it\Omega}}:=\sqrt{(\boldsymbol{v},\boldsymbol{ v})_{{\it\Omega}}}$ .

In $L^{2}({\it\Omega})$ , there exists a complete countable orthonormal basis $\{\boldsymbol{u}_{i}(\boldsymbol{x})\}_{i=1}^{\infty }$ , such that any velocity field can be arbitrarily closely approximated by a finite Galerkin expansion of $N$ modes. In the following search for the proper Galerkin expansion, we trade completeness for a minimal average residual of given snapshots $\boldsymbol{u}^{m}(\boldsymbol{x})$ , $m=1,\ldots ,M$ . This leads to the proper orthogonal decomposition

(3.3) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(\boldsymbol{x},t)=\boldsymbol{u}_{0}(\boldsymbol{x})+\mathop{\sum }_{i=1}^{N}a_{i}(t)\boldsymbol{u}_{i}(\boldsymbol{x}), & & \displaystyle\end{eqnarray}$$

where $N$ is the number of retained POD modes, $\boldsymbol{u}_{0}$ denotes the mean flow, $\boldsymbol{u}_{i}$ are the spatial POD modes and $a_{i}$ are the temporal mode coefficients (Holmes et al. Reference Holmes, Lumley, Berkooz and Rowley2012).

In the following, two averaging operators are defined. $\langle \cdot \rangle ^{u}$ denotes the time averaging operator of the unactuated flow between $t=0$ and $t_{s}$ , whereas $\langle \cdot \rangle ^{a}$ denotes that of the actuated flow between $t_{c}$ and $t_{e}$ . In this sense, the zeroth mode of (3.3) is defined as the mean base flow of the unactuated state $\boldsymbol{u}_{0}(\boldsymbol{x})=\langle \boldsymbol{u}(\boldsymbol{x},t)\rangle ^{u}$ .

Compared to unactuated flows, Galerkin expansions for actuated flows with a deforming surface face two additional challenges. First, the moving surface (oscillating lip in this case) and thus the domain ${\it\Omega}$ vary with time while the traditional approach of POD–Galerkin projection requires a fixed fluid domain. Second, the actuation does not enter explicitly as a free control command in the Galerkin expansion (3.3). The first challenge is solved by neglecting the small lip oscillations, since they are four orders of magnitude smaller than the chord length. Thus, all operations are performed on the steady domain corresponding to the unactuated state, as detailed in § 2.2, and the tiny lip motion is ignored in this Galerkin expansion. This simplification can be justified following the pioneering work of Bourguet, Braza & Dervieux (Reference Bourguet, Braza and Dervieux2011). In the Hadamard formulation, a solution of a partial differential equation (PDE) in a domain with small time-varying deformations is considered. The effect of these deformations can be approximated by a volume force in the PDE with a steady domain. This approximation leads to an additional linear actuation term in the Galerkin system.

The jet exit velocity at the slot exit mid-point $\boldsymbol{x}_{J}$ is selected as the free actuation command. Hence, $U_{J}$ given by (2.1) acts as boundary actuation for ${\it\Omega}$ , and $U_{J}\equiv 0$ represents unactuated flow. The standard lifting method (see § 3.2) constructs an actuation mode $\boldsymbol{u}_{-1}$ which describes the jet blowing such that $U_{J}(t)=a_{-1}(t)$ and $\Vert \boldsymbol{u}_{-1}(\boldsymbol{x}_{J})\Vert _{2}=1$ at the unsteady boundary, and vanishes at all other boundaries. Now, $\boldsymbol{u}_{0}+a_{-1}\boldsymbol{u}_{-1}$ satisfies the unsteady inhomogeneous boundary conditions at the airfoil and the oncoming flow at infinity. The POD decomposition is then applied to the modified snapshots, where $\boldsymbol{u}_{0}+a_{-1}\boldsymbol{u}_{-1}$ is subtracted. Thus, the POD modes – and arbitrary linear combinations thereof – also fulfil homogeneous boundary conditions. In the case of $N_{A}$ actuation modes, the general expression for the instantaneous velocity fields becomes

(3.4) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(\boldsymbol{x},t)=\boldsymbol{u}_{0}(\boldsymbol{x})+\mathop{\sum }_{i=-N_{A}}^{-1}a_{i}^{[N_{A}]}(t)\boldsymbol{u}_{i}(\boldsymbol{x})+\mathop{\sum }_{i=1}^{N}a_{i}^{[N_{A}]}(t)\boldsymbol{u}_{i}^{[N_{A}]}(\boldsymbol{x}), & & \displaystyle\end{eqnarray}$$

where we change the notations of $a_{i}$ and $\boldsymbol{u}_{i}$ to emphasize in superscripts the dependence on the number of actuation modes $N_{A}$ . By construction (see §§ 3.2–3.4), the actuation modes $\boldsymbol{u}_{i}$ , $i=-N_{A},\ldots ,-1$ are independent of the number of actuation modes considered. In addition, the mode amplitude $a_{0}:=1$ has been introduced for convenience following Rempfer & Fasel (Reference Rempfer and Fasel1994). The flow at infinity and through the slot is described by the mean flow $\boldsymbol{u}_{0}$ and the actuation modes $\sum _{i=-N_{A}}^{-1}a_{i}^{[N_{A}]}\boldsymbol{u}_{i}$ , respectively. The POD modes $\boldsymbol{u}_{i}^{[N_{A}]}$ , $i=1,\ldots ,N$ satisfy homogeneous boundary conditions by construction. The actuation mode amplitudes $a_{i}^{[N_{A}]}$ , $i=-N_{A},\ldots ,-1$ represent free actuation commands.

For the purpose of homogenizing the boundary conditions of the POD modes, a single actuation mode $N_{A}=1$ enables a lifting of the boundary actuation into the Galerkin model. This actuation mode, however, lumps the flow response of the steady blowing and the oscillatory component. A refined analysis considers two actuation modes $N_{A}=2$ , one corresponding to steady blowing and one to the unsteady component. However, the convection of high-frequency structures also generates a phase-shifted flow response. Any harmonically oscillating coherent structures requires two modes for an accurate description. These considerations lead to three actuation modes $N_{A}=3$ . For easy reference, the corresponding Galerkin expansions (GE) for $N_{A}=1$ , $N_{A}=2$ and $N_{A}=3$ are termed GE $A$ , $B$ and $C$ , respectively. Figure 6 illustrates the link between these expansions and the decomposition of the actuation command. A preview on the corresponding computational algorithm is offered in figure 7 and will be explained in the following subsections.

Figure 6. Hierarchy of lifting methods visualized by the actuation velocity decomposition. The purpose behind each hierarchy level is explicated on the right-hand side.

Figure 7. Algorithm for computing the three Galerkin expansions. The definition of the mode amplitudes, the actuation modes and the resulting Galerkin expansion is provided on the left, centre and right, respectively.

3.2 Galerkin expansion A: standard lifting method with one actuation mode

The first actuation mode $i=-1$ resolves the difference between the actuated and the unactuated state and is computed in two steps:

(3.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{u}_{-1}^{\ast }(\boldsymbol{x}):=\langle \boldsymbol{u}(\boldsymbol{x},t)\rangle ^{a}-\boldsymbol{u}_{0}(\boldsymbol{x}), & \displaystyle\end{eqnarray}$$

(3.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{u}_{-1}(\boldsymbol{x}):=\frac{\boldsymbol{u}_{-1}^{\ast }(\boldsymbol{x})}{\Vert \boldsymbol{u}_{-1}^{\ast }(\boldsymbol{x}_{J})\Vert _{2}}. & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle \Vert \boldsymbol{u}_{-1}(\boldsymbol{x}_{J})\Vert _{2}=1, & & \displaystyle\end{eqnarray}$$

and the fluctuating velocity at the exit can be expressed as

(3.7) $$\begin{eqnarray}\displaystyle \boldsymbol{U}_{J}(t):=\boldsymbol{u}^{\prime }(\boldsymbol{x}_{J},t)=a_{-1}^{[1]}(t)\boldsymbol{u}_{-1}(\boldsymbol{x}_{J}), & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{u}^{\prime }(\boldsymbol{x},t)=\boldsymbol{u}(\boldsymbol{x},t)-\boldsymbol{u}_{0}(\boldsymbol{x})$ are the fluctuating velocity fields, and where the first actuation mode coefficient is defined as

(3.8) $$\begin{eqnarray}\displaystyle a_{-1}^{[1]}(t)=\left\{\begin{array}{@{}ll@{}}\displaystyle U_{J}(t)=B_{0}+B_{1}\cos ({\it\omega}^{a}t) & \text{during actuation;}\\ \displaystyle 0 & \text{otherwise.}\end{array}\right. & & \displaystyle\end{eqnarray}$$

In this case, $a_{-1}$ encompasses both the steady and the unsteady component of actuation.

The homogenized snapshots are then computed as

(3.9) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{A}^{[N_{A}]}(\boldsymbol{x},t):=\boldsymbol{u}^{\prime }(\boldsymbol{x},t)-a_{-1}^{[N_{A}]}(t)\boldsymbol{u}_{-1}(\boldsymbol{x}), & & \displaystyle\end{eqnarray}$$

with $N_{A}=1$ . By construction, $\boldsymbol{u}_{A}^{[1]}(\boldsymbol{x}_{J},t)\equiv \mathbf{0}$ , i.e. the homogeneous Dirichlet condition is fulfilled. This property carries over to the POD modes which are linear combinations of the snapshots. The Galerkin expansion reads

(3.10) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(\boldsymbol{x},t)=\boldsymbol{u}_{0}(\boldsymbol{x})+a_{-1}^{[1]}(t)\boldsymbol{u}_{-1}+\mathop{\sum }_{i=1}^{N}a_{i}^{[1]}(t)\boldsymbol{u}_{i}^{[1]}(\boldsymbol{x}). & & \displaystyle\end{eqnarray}$$

We would like to emphasize that Galerkin expansion $A$ corresponds to the standard lifting procedure encountered in the literature (Graham et al. Reference Graham, Peraire and Tang1999; Ravindran Reference Ravindran2000). It should be noted that (3.6) and a unit $L^{2}$ -norm are not compatible requirements. The physical interpretation of the mode coefficient is given higher priority over a $L^{2}$ unit norm.

3.3 Galerkin expansion B: two actuation modes for steady and unsteady blowing effects

With the addition of a second actuation mode, it becomes possible to divide the forcing term into its steady and fluctuating components. As such, the first actuation mode corresponds to steady blowing, whereas the second mode to oscillatory modulation. The definition of $\boldsymbol{u}_{-1}$ is the same as in GE $A$ , but with the mode coefficient defined as

(3.11) $$\begin{eqnarray}\displaystyle a_{-1}^{[2]}(t)=\left\{\begin{array}{@{}ll@{}}\displaystyle \langle U_{J}\rangle ^{a} & \text{during actuation;}\\ \displaystyle 0 & \text{otherwise.}\end{array}\right. & & \displaystyle\end{eqnarray}$$

The second actuation mode $i=-2$ is defined as

(3.12) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{-2}^{\ast }(\boldsymbol{x}):=\langle a_{-2}^{[2]}(t)\boldsymbol{u}_{A}^{[2]}(\boldsymbol{x},t)\rangle ^{a}, & & \displaystyle\end{eqnarray}$$

where $a_{-2}^{[2]}$ is the corresponding mode coefficient defined as

(3.13) $$\begin{eqnarray}\displaystyle a_{-2}^{[2]}(t)=\left\{\begin{array}{@{}ll@{}}\displaystyle U_{J}^{\prime }(t)\approx B_{1}\cos ({\it\omega}^{a}t) & \text{during actuation;}\\ \displaystyle 0 & \text{otherwise.}\end{array}\right. & & \displaystyle\end{eqnarray}$$

In an electric circuit analogy, $a_{-1}^{[2]}$ and $a_{-2}^{[2]}$ denote the direct current (DC) and alternating current (AC) components, respectively. The DC and AC components for GE $B$ are shown in figure 6. Similar to $\boldsymbol{u}_{-1}$ , the second actuation mode is normalized by its value at the slot exit,

(3.14) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{-2}(\boldsymbol{x}):=\frac{\boldsymbol{u}_{-2}^{\ast }(\boldsymbol{x})}{\Vert \boldsymbol{u}_{-2}^{\ast }(\boldsymbol{x}_{J})\Vert _{2}}. & & \displaystyle\end{eqnarray}$$

Thus, the fluctuating jet velocity reads,

(3.15) $$\begin{eqnarray}\displaystyle \boldsymbol{U}_{J}(t)=\boldsymbol{u}^{\prime }(\boldsymbol{x}_{J},t)=a_{-1}^{[2]}(t)\boldsymbol{u}_{-1}(\boldsymbol{x}_{J})+a_{-2}^{[2]}(t)\boldsymbol{u}_{-2}(\boldsymbol{x}_{J}). & & \displaystyle\end{eqnarray}$$

The POD is then computed from the homogenized snapshots,

(3.16) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{B}^{[2]}(\boldsymbol{x},t):=\boldsymbol{u}^{\prime }(\boldsymbol{x},t)-a_{-1}^{[2]}(t)\boldsymbol{u}_{-1}(\boldsymbol{x})-a_{-2}^{[2]}(t)\boldsymbol{u}_{-2}(\boldsymbol{x}) & & \displaystyle\end{eqnarray}$$

and the Galerkin expansion is expressed as,

(3.17) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(\boldsymbol{x},t)=\boldsymbol{u}_{0}(\boldsymbol{x})+\mathop{\sum }_{i=-2}^{-1}a_{i}^{[2]}(t)\boldsymbol{u}_{i}(\boldsymbol{x})+\mathop{\sum }_{i=1}^{N}a_{i}^{[2]}(t)\boldsymbol{u}_{i}^{[2]}(\boldsymbol{x}). & & \displaystyle\end{eqnarray}$$

As discussed in § 3.2, an $L^{2}$ normalization of the actuation modes would obstruct the physical meaning of $a_{-1}^{[N_{A}]}$ and $a_{-2}^{[N_{A}]}$ . Following Kasnakoğlu et al. (Reference Kasnakoğlu, Serrani and Efe2008), one could envision, at minimum, an orthogonalization of the actuation mode with respect to each other and to the POD modes. This would simplify the resulting Galerkin system but has severe disadvantages. First, in Kasnakoğlu et al. (Reference Kasnakoğlu, Serrani and Efe2008), this orthonormalization is independent of the POD modes since the authors consider unactuated snapshots for determining the POD modes. In our approach, the POD modes are affected by the actuation modes. Second, we prefer to have identical actuation modes $\boldsymbol{u}_{-i}$ ( $i=1,2,3$ ) in Galerkin expansions $A$ , $B$ and $C$ . If the actuation modes are orthogonalized with respect to the POD modes, this independence is lost. Third, after the orthogonalization, the actuation modes would lose their physical meaning as correlated flow responses to the DC or AC actuation component.

3.4 Galerkin expansion C: three actuation modes for steady, unsteady and time-delayed blowing effects

The third actuation mode $i=-3$ is an additional unsteady mode that is time shifted with respect to the main unsteady actuation term. It is defined as,

(3.18) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{-3}^{\ast }(\boldsymbol{x}):=\langle a_{-3}^{[3]}(t)\boldsymbol{u}_{B}^{[2]}(\boldsymbol{x},t)\rangle ^{a}, & & \displaystyle\end{eqnarray}$$

where

(3.19) $$\begin{eqnarray}\displaystyle a_{-3}^{[3]}(t)=\left\{\begin{array}{@{}ll@{}}\displaystyle U_{J}^{\prime }(t-T/4)\approx B_{1}\sin ({\it\omega}^{a}t) & \text{during actuation;}\\ \displaystyle 0 & \text{otherwise,}\end{array}\right. & & \displaystyle\end{eqnarray}$$

and $T$ is one period of the actuated shedding cycle. The normalization of $\boldsymbol{u}_{-3}$ is less restricted than that of $\boldsymbol{u}_{-2}$ and $\boldsymbol{u}_{-1}$ , since the spatial evolution of the control is already represented by the first two actuation modes. Hence, $\boldsymbol{u}_{-3}$ must vanish at the exit (i.e. $\boldsymbol{u}_{-3}(\boldsymbol{x}_{J})=\mathbf{0}$ ), and the fluctuating jet velocity reads,

(3.20) $$\begin{eqnarray}\displaystyle \boldsymbol{U}_{J}(t):=\boldsymbol{u}^{\prime }(\boldsymbol{x}_{J},t)=a_{-1}^{[3]}(t)\boldsymbol{u}_{-1}(\boldsymbol{x}_{J})+a_{-2}^{[3]}(t)\boldsymbol{u}_{-2}(\boldsymbol{x}_{J}), & & \displaystyle\end{eqnarray}$$

with $a_{-1}^{[3]}:=a_{-1}^{[2]}$ and $a_{-2}^{[3]}:=a_{-2}^{[2]}$ . $\boldsymbol{u}_{-3}$ is simply normalized by the same factor as the second actuation mode, i.e.

(3.21) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{-3}(\boldsymbol{x}):=\frac{\boldsymbol{u}_{-3}^{\ast }(\boldsymbol{x})}{\Vert \boldsymbol{u}_{-2}^{\ast }(\boldsymbol{x}_{J})\Vert _{2}}. & & \displaystyle\end{eqnarray}$$

This choice makes the fluctuation levels of the second and third actuation mode as an oscillatory flow response comparable.

The homogenized snapshots are determined as

(3.22) $$\begin{eqnarray}\displaystyle \boldsymbol{u}_{C}^{[3]}(\boldsymbol{x},t):=\boldsymbol{u}^{\prime }(\boldsymbol{x},t)-a_{-1}^{[3]}(t)\boldsymbol{u}_{-1}(\boldsymbol{x})-a_{-2}^{[3]}(t)\boldsymbol{u}_{-2}(\boldsymbol{x})-a_{-3}^{[3]}(t)\boldsymbol{u}_{-3}(\boldsymbol{x}) & & \displaystyle\end{eqnarray}$$

and the Galerkin expansion is expressed as

(3.23) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(\boldsymbol{x},t)=\boldsymbol{u}_{0}(\boldsymbol{x})+\mathop{\sum }_{i=-3}^{-1}a_{i}^{[3]}(t)\boldsymbol{u}_{i}(\boldsymbol{x})+\mathop{\sum }_{i=1}^{N}a_{i}^{[3]}(t)\boldsymbol{u}_{i}^{[3]}(\boldsymbol{x}). & & \displaystyle\end{eqnarray}$$

For highly convective flows, the first POD modes are the most energetic. For the current flow, the first four POD modes contain 78 % of the total turbulent kinetic energy, and are deemed sufficient to represent the dynamics. The mean base flow, the first four POD modes and all three actuation modes for Galerkin expansion GE $C$ are shown in figure 8. By construction, the actuation modes are independent of $N_{A}$ , i.e. they are the same for GE $A$ and $B$ and therefore are not shown separately. All modes can be physically interpreted. Mode $\boldsymbol{u}_{0}$ represents the unactuated low-lift base flow. The first actuation mode $\boldsymbol{u}_{-1}$ displays the difference to the high-lift actuated equilibrium. The nearly circular streamlines clearly indicate the enhanced lift. The higher-order actuation modes $\boldsymbol{u}_{-2}$ and $\boldsymbol{u}_{-3}$ resolve the oscillatory flow response to the blowing modulation, as intended by construction.

Figure 8. The mean base flow, the first four POD modes and all three actuation modes for Galerkin expansion GE $C$ . The results are visualized by the vorticity fields and vortex lines.

The first two POD modes $\boldsymbol{u}_{1}^{[3]}$ and $\boldsymbol{u}_{2}^{[3]}$ represent von Kármán vortex shedding. The third and fourth mode $\boldsymbol{u}_{3}^{[3]}$ and $\boldsymbol{u}_{4}^{[3]}$ resolve slow transient base flow changes; $\boldsymbol{u}_{3}^{[3]}$ resolves global flow changes, while $\boldsymbol{u}_{4}^{[3]}$ is concentrated in the near wake. Their coefficients get non-vanishing amplitudes during change of the actuation and vanish with time. Thus, these modes can be considered as transient modes. Their impact on the Galerkin system dynamics will be elaborated in §§ 4 and 5.

The Galerkin expansion of the high-lift configuration by Luchtenburg et al. (Reference Luchtenburg, Günther, Noack, King and Tadmor2009) constitutes a good basis for comparison with the current study. In both models, the first POD mode pair resolves unactuated vortex shedding. Luchtenburg et al. (Reference Luchtenburg, Günther, Noack, King and Tadmor2009) incorporate the high-frequency flow response in modes 3 and 4. These modes resolve the forcing of the zero-net-mass-flux actuator and their amplitudes are determined by the Galerkin system. Base-flow changes in the Galerkin system are not kinematically resolved. In the present study, actuation modes $i=-2$ and $i=-3$ take on this role commanded by control. The response to steady and unsteady is kinematically and dynamically resolved with additional modes corresponding to the indices $i=-1$ , $-2$ and $-3$ .

The first four POD modes $\boldsymbol{u}_{i}$ ( $i=1,\ldots ,4$ ) depend hardly on the number of actuation modes considered for their determination. For this reason, we remove hereafter in our notation the explicit dependence of $N_{A}$ for the spatial POD modes $\boldsymbol{u}_{i}$ and the temporal mode coefficients $a_{i}$ .

4.1 Derivation of the mean-field model

The following consideration contains two key components of mean-field theory (Stuart Reference Stuart1971), namely frequency decomposition and slaving. The resulting Galerkin expansion is based on a standard inner product of square-integrable fields in steady observation domain ${\it\Omega}$ as introduced in (3.2). We shall not pause to include a bifurcation expansion as this is irrelevant for a turbulent flow.

The mean-field model will be derived in four steps. First, a suitable mean-field expansion is developed (§ 4.1.1). Second, the dynamics of the natural frequency is derived (§ 4.1.2). Third, the dynamics of the forcing frequency is considered (§ 4.1.3). Finally, the base flow dynamics is investigated (§ 4.1.4).

4.1.1 Mean-field expansion

The considered flow has three frequency components: a slowly varying mean-field $\boldsymbol{u}^{b}$ (superscript ‘ $b$ ’ for base flow), a natural shedding component $\boldsymbol{u}^{u}$ (superscript ‘ $u$ ’ for unforced) and a forced high-frequency component $\boldsymbol{u}^{a}$ (superscript ‘ $a$ ’ for actuated):

(4.1) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(\boldsymbol{x},t)=\boldsymbol{u}^{b}(\boldsymbol{x},t)+\boldsymbol{u}^{u}(\boldsymbol{x},t)+\boldsymbol{u}^{a}(\boldsymbol{x},t). & & \displaystyle\end{eqnarray}$$

Upon short-term averaging the terms with natural and high frequency vanish. The harmonic contribution at natural shedding frequency may be composed of two modes $\boldsymbol{u}_{1}$ , $\boldsymbol{u}_{2}$ ,

(4.2) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{u}(\boldsymbol{x},t)=a_{1}(t)\boldsymbol{u}_{1}(\boldsymbol{x})+a_{2}(t)\boldsymbol{u}_{2}(\boldsymbol{x}). & & \displaystyle\end{eqnarray}$$

In what follows, POD modes 1 and 2 of figure 8 are taken as $\boldsymbol{u}_{i}$ , $i=1,2$ . These two modes may also be obtained from the real and imaginary part of the dominant Fourier mode, from the first two POD modes of natural shedding or from any other suitable filter. Higher harmonics tend to be small and are neglected. The corresponding mode amplitudes describe in good approximation a limit cycle

(4.3a-c ) $$\begin{eqnarray}\displaystyle a_{1}(t)=r\cos ({\it\phi}),\quad a_{2}(t)=r\sin ({\it\phi}),\quad \frac{\text{d}{\it\phi}}{\text{d}t}={\it\omega} & & \displaystyle\end{eqnarray}$$

were ${\it\phi}$ is the phase, ${\it\omega}$ a constant or slowly varying angular frequency and $r$ a constant or slowly varying amplitude. Without loss of generality, $a_{1}$ follows the cosine function. Otherwise, the origin of the time is shifted. Similarly, $a_{2}$ follows the sinus. Otherwise, the second mode $\boldsymbol{u}_{2}$ needs to be adjusted.

In completely analogous manner, the high-frequency contribution is characterized by

(4.4) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{a}(\boldsymbol{x},t)=a_{-2}(t)\boldsymbol{u}_{-2}(\boldsymbol{x})+a_{-3}(t)\boldsymbol{u}_{-3}(\boldsymbol{x}), & & \displaystyle\end{eqnarray}$$

and

(4.5a,b ) $$\begin{eqnarray}\displaystyle a_{-2}(t)=s\cos ({\it\omega}^{a}t),\quad a_{-3}(t)=s\sin ({\it\omega}^{a}t). & & \displaystyle\end{eqnarray}$$

In the choice of symbols for the modes and their amplitudes we have anticipated the use of actuations modes $-2$ and $-3$ of figure 8 for the flow response at forcing frequency.

The base flow is conceptualized as the sum of the steady Navier–Stokes solution $\boldsymbol{u}_{s}$ and a constant or slowly varying base-flow deformation $\boldsymbol{u}^{{\it\Delta}}$ , following mean-field theory and departing from the POD expansion of § 3,

(4.6) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{b}(\boldsymbol{x},t)=\boldsymbol{u}_{s}(\boldsymbol{x})+\boldsymbol{u}^{{\it\Delta}}(\boldsymbol{x},t). & & \displaystyle\end{eqnarray}$$

The unforced base-flow deformation may be characterized by a single mode $\boldsymbol{u}_{3}$ pointing from the steady solution to the unforced mean flow $\langle \boldsymbol{u}\rangle ^{u}$ , often called the shift mode (Noack et al. Reference Noack, Afanasiev, Morzyński, Tadmor and Thiele2003; Tadmor et al. Reference Tadmor, Lehmann, Noack and Morzyński2010). The corresponding shift mode amplitude $a_{3}$ is driven by the fluctuation of the vortex shedding and considered as a slowly varying quantity. Changes by steady blowing may be incorporated in a single actuation mode $\boldsymbol{u}_{-1}$ pointing from the unforced averaged velocity to the averaged actuated one $\langle \boldsymbol{u}\rangle ^{a}$ (Weller, Lombardi & Iollo Reference Weller, Lombardi and Iollo2009). The amplitude $a_{-1}$ determines the strength of blowing in the high-lift configuration. An additional mode $\boldsymbol{u}_{4}$ may be needed to capture transient effects parameterized by the amplitude $a_{4}$ . Summarizing, the mean-field deformation reads

(4.7) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{{\it\Delta}}(\boldsymbol{x},t)=a_{-1}(t)\boldsymbol{u}_{-1}(\boldsymbol{x})+a_{3}(t)\boldsymbol{u}_{3}(\boldsymbol{x})+a_{4}(t)\boldsymbol{u}_{4}(\boldsymbol{x}) & & \displaystyle\end{eqnarray}$$

and (4.1) becomes

(4.8) $$\begin{eqnarray}\displaystyle \boldsymbol{u}(\boldsymbol{x},t)=\boldsymbol{u}_{s}(\boldsymbol{x})+\mathop{\sum }_{i=-3}^{4}a_{i}(t)\boldsymbol{u}_{i}(\boldsymbol{x}). & & \displaystyle\end{eqnarray}$$

The resulting components of the mean-field expansion are catalogued in table 1.

Table 1. Components of the mean-field expansion. The table comprises unforced (top) and forced components (bottom) contributing to the base flow $\boldsymbol{u}^{b}$ (left), the vortex shedding $\boldsymbol{u}^{u}$ (middle) and the high-frequency actuation $\boldsymbol{u}^{a}$ (right).

4.1.2 Dynamics of the natural frequency

The dynamics of the mean-field expansion may be inferred from the non-dimensionalized Navier–Stokes equation,

(4.9) $$\begin{eqnarray}\displaystyle \partial _{t}\boldsymbol{u}=\frac{1}{Re}{\rm\Delta}\boldsymbol{u}-\boldsymbol{{\rm\nabla}}\boldsymbol{u}\otimes \boldsymbol{u}-\boldsymbol{{\rm\nabla}}p. & & \displaystyle\end{eqnarray}$$

Here $\partial _{t}$ , $\boldsymbol{{\rm\nabla}}$ and ${\rm\Delta}$ denote the partial temporal derivative, the nabla operator and Laplace operator, respectively. The product sign $\otimes$ denotes an outer tensor product, also called dyadic vector product. We shall not pause to discuss the pressure term as it will vanish under Galerkin projection for sufficiently large domains (Deane et al. Reference Deane, Kevrekidis, Karniadakis and Orszag1991; Noack, Papas & Monkewitz Reference Noack, Papas and Monkewitz2005).

The natural frequency component of (4.9) is described by

(4.10) $$\begin{eqnarray}\displaystyle \partial _{t}\boldsymbol{u}^{u}=\frac{1}{Re}{\rm\Delta}\boldsymbol{u}^{u}-\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{b}\otimes \boldsymbol{u}^{u}-\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{u}\otimes \boldsymbol{u}^{b}-\boldsymbol{{\rm\nabla}}p^{u}. & & \displaystyle\end{eqnarray}$$

All other terms of (4.8) vanish upon filtering at frequency ${\it\omega}$ . For the following, we assume that $\boldsymbol{u}_{1}$ and $\boldsymbol{u}_{2}$ are orthogonal, $(\boldsymbol{u}_{1},\boldsymbol{u}_{2})_{{\it\Omega}}=0$ . Otherwise, the modes may be orthogonalized. From (4.6) and (4.2), Galerkin projection on $\boldsymbol{u}_{i}$ and $i=1,2$ yields

(4.11) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}t}\left(\begin{array}{@{}c@{}}a_{1}\\ \displaystyle a_{2}\end{array}\right)=\unicode[STIX]{x1D63C}\left(\begin{array}{@{}c@{}}a_{1}\\ \displaystyle a_{2}\end{array}\right)\quad \text{where }\unicode[STIX]{x1D63C}=\unicode[STIX]{x1D63C}_{s}+a_{-1}\unicode[STIX]{x1D63C}_{-1}+a_{3}\unicode[STIX]{x1D63C}_{3}+a_{4}\unicode[STIX]{x1D63C}_{4}. & & \displaystyle\end{eqnarray}$$

Here, the system matrix $\unicode[STIX]{x1D63C}=\big(\begin{smallmatrix}\unicode[STIX]{x1D608}_{11} & \unicode[STIX]{x1D608}_{12}\\ \unicode[STIX]{x1D608}_{21} & \unicode[STIX]{x1D608}_{22}\end{smallmatrix}\big)$ depends affinely on $a_{-1}$ , $a_{3}$ and $a_{4}$ . $\unicode[STIX]{x1D63C}_{s}$ corresponds to the $2\times 2$ matrix from a linearization around the fixed point $\boldsymbol{u}_{s}$ , $\unicode[STIX]{x1D63C}_{3}$ characterises a change of the dynamics from an unforced mean-field deformation, and $\unicode[STIX]{x1D63C}_{-1}$ and $\unicode[STIX]{x1D63C}_{4}$ the corresponding change from actuation. The assumed ansatz (4.3) further restricts the system matrix to

(4.12a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D608}_{11}=\unicode[STIX]{x1D608}_{22}={\it\sigma}_{1}-{\it\beta}_{-1}a_{-1}-{\it\beta}_{3}a_{3}-{\it\beta}_{4}a_{4}, & \displaystyle\end{eqnarray}$$

(4.12b ) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D608}_{12}=\unicode[STIX]{x1D608}_{21}={\it\omega}_{1}+{\it\gamma}_{-1}a_{-1}+{\it\gamma}_{3}a_{3}+{\it\gamma}_{4}a_{4}. & \displaystyle\end{eqnarray}$$

The first equation (4.12a ) describes the growth rate ${\it\sigma}_{1}$ near the fixed point and its changes from mean-flow deformation parameterised by $a_{-1}$ , $a_{3}$ and $a_{4}$ . The second equation (4.12b ) is the analogue for the frequency.

4.1.3 Dynamics of the forcing frequency

In this study, the flow component at forcing frequency is kinematically imposed by the actuation command $a_{-2}$ and $a_{-3}$ and the corresponding actuation modes $\boldsymbol{u}_{-2}$ and $\boldsymbol{u}_{-3}$ . This is the most simple representation of the actuation component.

In Luchtenburg et al. (Reference Luchtenburg, Günther, Noack, King and Tadmor2009), the actuated flow follows an oscillatory dynamics described by a stable forced oscillator

(4.13a ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{d}a_{3}/\text{d}t={\it\sigma}^{a}a_{3}-{\it\omega}^{a}a_{4}+g_{3}(t), & \displaystyle\end{eqnarray}$$

(4.13b ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{d}a_{4}/\text{d}t={\it\sigma}^{a}a_{4}+{\it\omega}^{a}a_{3}+g_{4}(t), & \displaystyle\end{eqnarray}$$

where ${\it\sigma}^{a}$ and ${\it\omega}^{a}$ are the growth rate and frequency of a stable oscillator and $g_{3}$ and $g_{4}$ represent ${\it\omega}^{a}$ -periodic forcing. This refined model accounts for time delays from forcing to flow response. In this study, we choose the more simple kinematic representation because the high-frequency component was found to be small and the time delays are not critical.

4.1.4 Dynamics of the base flow

The slowly varying base flow near the fixed point is described by the linearized Reynolds equation,

(4.14) $$\begin{eqnarray}\displaystyle \partial _{t}\boldsymbol{u}^{{\it\Delta}}=\frac{1}{Re}{\rm\Delta}\boldsymbol{u}^{{\it\Delta}}-\boldsymbol{{\rm\nabla}}\boldsymbol{u}_{s}\otimes \boldsymbol{u}^{{\it\Delta}}-\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{{\it\Delta}}\otimes \boldsymbol{u}_{s}-\boldsymbol{{\rm\nabla}}\langle \boldsymbol{u}^{u}\otimes \boldsymbol{u}^{u}\rangle -\boldsymbol{{\rm\nabla}}\langle \boldsymbol{u}^{a}\otimes \boldsymbol{u}^{a}\rangle -\boldsymbol{{\rm\nabla}}p^{b}.\quad & & \displaystyle\end{eqnarray}$$

This equation is obtained by substituting the mean-field expansion (4.8) in the Navier–Stokes equation (4.9), subtracting the steady Navier–Stokes equation for $\boldsymbol{u}_{s}$ , filtering out the natural and forced frequency terms and neglecting the second-order term $-\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{{\it\Delta}}\otimes \boldsymbol{u}^{{\it\Delta}}$ . The first three terms on the right-hand side of (4.14) denote a linear operator in $\boldsymbol{u}^{{\it\Delta}}$ , the fourth one the Reynolds stress at the natural frequency and the fifth the analogue for the forcing frequency. The brackets denote short-time averages.

For simplicity, we assume that $\boldsymbol{u}_{3}$ and $\boldsymbol{u}_{4}$ are orthogonal to each other. A Galerkin projection of (4.14) on both shift modes yields

(4.15a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{3}}{\text{d}t}=l_{3,-1}a_{-1}+l_{33}a_{3}+l_{34}a_{4}+{\it\alpha}_{3r}r^{2}+{\it\alpha}_{3s}s^{2}, & \displaystyle\end{eqnarray}$$

(4.15b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{4}}{\text{d}t}=l_{4,-1}a_{-1}+l_{43}a_{3}+l_{44}a_{4}+{\it\alpha}_{4r}r^{2}+{\it\alpha}_{4s}s^{2}. & \displaystyle\end{eqnarray}$$

The linear terms $l_{ij}$ parameterise the linear operator for $\boldsymbol{u}^{{\it\Delta}}$ of (4.14). The coefficients ${\it\alpha}_{\star }$ represent the gain of the Reynolds stresses at natural and forced frequency on the shift-mode amplitudes. We did not include an evolution equation for $a_{-1}$ as the blowing is imposed in our high-lift configuration.

4.2 Dynamics of the mean-field model

Summarizing (4.11), (4.12) and (4.15), the resulting mean-field system reads

(4.16a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{1}}{\text{d}t}={\it\sigma}a_{1}-{\it\omega}a_{2}, & \displaystyle\end{eqnarray}$$

(4.16b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{2}}{\text{d}t}={\it\sigma}a_{2}+{\it\omega}a_{1}, & \displaystyle\end{eqnarray}$$

(4.16c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{3}}{\text{d}t}=l_{3,-1}a_{-1}+l_{33}a_{3}+l_{34}a_{4}+{\it\alpha}_{3r}r^{2}+{\it\alpha}_{3s}s^{2}, & \displaystyle\end{eqnarray}$$

(4.16d ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{4}}{\text{d}t}=l_{4,-1}a_{-1}+l_{43}a_{3}+l_{44}a_{4}+{\it\alpha}_{4r}r^{2}+{\it\alpha}_{4s}s^{2}, & \displaystyle\end{eqnarray}$$

(4.16e ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\sigma}={\it\sigma}_{1}-{\it\beta}_{-1}a_{-1}-{\it\beta}_{3}a_{3}-{\it\beta}_{4}a_{4}, & \displaystyle\end{eqnarray}$$

(4.16f ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\omega}={\it\omega}_{1}+{\it\gamma}_{-1}a_{-1}+{\it\gamma}_{3}a_{3}+{\it\gamma}_{4}a_{4}, & \displaystyle\end{eqnarray}$$

(4.16g ) $$\begin{eqnarray}\displaystyle & \displaystyle r=\sqrt{a_{1}^{2}+a_{2}^{2}}, & \displaystyle\end{eqnarray}$$

(4.16h ) $$\begin{eqnarray}\displaystyle & \displaystyle s=\sqrt{a_{-2}^{2}+a_{-3}^{2}}. & \displaystyle\end{eqnarray}$$

$a_{1}$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{-1}$

$a_{-2}$

$a_{-3}$

Equation (4.16) describes a self-amplified, amplitude-limited oscillation which is mitigated by steady- or high-frequency actuation. In the following, we consider special cases which illuminate the role of the coefficients, the states and their dynamic interaction.

(1) Unforced equilibrium. By construction, i.e. the Galerkin expansion (4.8), the unforced fixed point $\boldsymbol{u}_{s}$ is a trivial solution of (4.16). This implies $a_{1}=a_{2}=a_{3}=a_{4}=0$ and the absense of actuation $a_{-1}=a_{-2}=a_{-3}=0$ . This is readily seen to satisfy the mean-field system.

(2) Linear unforced dynamics $(a_{-1}=a_{-2}=a_{-3}=a_{4}=0)$ . Linearizing the unforced dynamics around the fixed point yields an oscillator

(4.17a-c ) $$\begin{eqnarray}\displaystyle \frac{\text{d}a_{1}}{\text{d}t}={\it\sigma}_{1}a_{1}-{\it\omega}_{1}a_{2},\quad \frac{\text{d}a_{2}}{\text{d}t}={\it\sigma}_{1}a_{2}+{\it\omega}_{1}a_{1},\quad a_{3}=0. & & \displaystyle\end{eqnarray}$$

The assumed self-amplification requires a positive growth rate ${\it\sigma}_{1}>0$ . The shift-mode amplitude should be linearly stable $l_{33}<0$ as it is only driven by the Reynolds stresses.

(3) Unforced periodic dynamics $(a_{-1}=a_{-2}=a_{-3}=a_{4}=0)$ . The initial oscillations converge to a periodic limit cycle implying a vanishing growth rate ${\it\sigma}=0$ , converged shift-mode amplitude $\text{d}a_{3}/\text{d}t=0$ and ${\it\beta}_{3}>0$ . The resulting oscillation and shift-mode amplitude read

(4.18a,b ) $$\begin{eqnarray}\displaystyle r=\sqrt{\frac{-{\it\sigma}_{1}l_{33}}{{\it\alpha}_{3r}{\it\beta}_{3}}},\quad a_{3}=\frac{{\it\sigma}_{1}}{{\it\beta}_{3}}. & & \displaystyle\end{eqnarray}$$

(4) Unforced transient dynamics $(a_{-1}=a_{-2}=a_{-3}=a_{4}=0)$ . Inserting vanishing forcing in (4.16) yields a mean-field Galerkin model (Noack et al. Reference Noack, Afanasiev, Morzyński, Tadmor and Thiele2003):

(4.19a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{1}}{\text{d}t}=({\it\sigma}_{1}-{\it\beta}_{3}a_{3})a_{1}-({\it\omega}_{1}+{\it\gamma}_{3}a_{3})a_{2}, & \displaystyle\end{eqnarray}$$

(4.19b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{2}}{\text{d}t}=({\it\sigma}_{1}-{\it\beta}_{3}a_{3})a_{2}+({\it\omega}_{1}+{\it\gamma}_{3}a_{3})a_{1}, & \displaystyle\end{eqnarray}$$

(4.19c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{3}}{\text{d}t}=l_{33}a_{3}+{\it\alpha}_{3r}(a_{1}^{2}+a_{2}^{2}). & \displaystyle\end{eqnarray}$$

$\text{d}a_{3}/\text{d}t=0$

(4.20a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{1}}{\text{d}t}=({\it\sigma}_{1}-{\it\beta}_{r}r^{2})a_{1}-({\it\omega}_{1}+{\it\gamma}_{r}r^{2})a_{2}, & \displaystyle\end{eqnarray}$$

(4.20b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{2}}{\text{d}t}=({\it\sigma}_{1}-{\it\beta}_{r}r^{2})a_{2}+({\it\omega}_{1}+{\it\gamma}_{r}r^{2})a_{1}, & \displaystyle\end{eqnarray}$$

(4.20c ) $$\begin{eqnarray}\displaystyle & \displaystyle a_{3}={\it\alpha}_{r}r^{2}, & \displaystyle\end{eqnarray}$$

${\it\alpha}_{r}=-{\it\alpha}_{3r}/l_{33}>0$

${\it\beta}_{r}={\it\alpha}_{r}{\it\beta}_{3}>0$

${\it\gamma}_{r}={\it\alpha}_{r}{\it\gamma}_{3}$

(4.21) $$\begin{eqnarray}\displaystyle \frac{\text{d}r}{\text{d}t}={\it\sigma}_{1}r-{\it\beta}_{r}r^{3}. & & \displaystyle\end{eqnarray}$$

The slaving assumption is found to be valid for the laminar cylinder wake at distinctly supercritical Reynolds numbers from direct numerical simulations (Noack et al. Reference Noack, Afanasiev, Morzyński, Tadmor and Thiele2003) and from the identified parameters $|l_{33}|\gg {\it\sigma}_{1}$ (Tadmor & Noack Reference Tadmor and Noack2004). Experiments with turbulent wakes also corroborate the existence of a mean-field paraboloid (Bourgeois, Martinuzzi & Noack Reference Bourgeois, Martinuzzi and Noack2013; Hosseini, Martinuzzi & Noack Reference Hosseini, Martinuzzi and Noack2015). More complex mean-field Galerkin models with cubic terms have already been proposed by Aubry et al. (Reference Aubry, Holmes, Lumley and Stone1988) and elaborated by Podvin (Reference Podvin2009).

(5) Shedding suppression by steady actuation. The high-lift configuration is designed to stabilise vortex shedding with steady Coanda blowing $a_{-1}>0$ , $a_{-2}=a_{-3}=a_{4}=0$ . In the mean-field model (4.16), this stabilisation implies ${\it\sigma}<0$ for $a_{3}=a_{4}=0$ . For blowing $a_{-1}>0$ , negative growth rates can only be achieved with ${\it\beta}_{-1}>0$ . Summarizing, (4.16e ) determines the minimal blow rate for complete stabilisation:

(4.22) $$\begin{eqnarray}\displaystyle {\it\sigma}={\it\sigma}_{1}-{\it\beta}_{-1}a_{-1}<0\Rightarrow a_{-1}>\frac{{\it\sigma}_{1}}{{\it\beta}_{-1}}. & & \displaystyle\end{eqnarray}$$

The blow rate increases with natural growth rate ${\it\sigma}_{1}$ and decreases with the actuation gain ${\it\beta}_{-1}$ .

(6) Shedding suppression by high-frequency actuation. As a last case, we mention that (4.16) incorporates – in principle – the stabilizing effect of ‘pure’ high-frequency forcing (Glezer, Amitay & Honohan Reference Glezer, Amitay and Honohan2005; Luchtenburg et al. Reference Luchtenburg, Günther, Noack, King and Tadmor2009), i.e.  $a_{-1}\equiv 0$ and $s^{2}=a_{-2}^{2}+a_{-3}^{2}>0$ . High-frequency forcing introduces a Reynolds stress which changes the corresponding shift-mode amplitude $a_{4}$ as portrayed in (4.16d ). Without loss of generality, we can assume $a_{4}>0$ or change the sign of $\boldsymbol{u}_{4}$ . This base flow change affects the stability of vortex shedding (4.16e ). A minimal base-flow change for complete stabilisation is given by

(4.23) $$\begin{eqnarray}\displaystyle {\it\sigma}={\it\sigma}_{1}-{\it\beta}_{4}a_{4}<0\Rightarrow a_{4}>\frac{{\it\sigma}_{1}}{{\it\beta}_{4}} & & \displaystyle\end{eqnarray}$$

implying ${\it\beta}_{4}>0$ . This minimal mean-field deformation translates into a minimal actuation fluctuation level which can be derived from (4.16d ). This equation simplifies for stabilization, $r=a_{3}=0$ and equilibrium conditions, $\text{d}a_{4}/\text{d}t=0$ , to

(4.24) $$\begin{eqnarray}\displaystyle l_{44}a_{4}+{\it\alpha}_{4s}s^{2}=0, & & \displaystyle\end{eqnarray}$$

which defines another mean-field paraboloid for the forced dynamics.

We do not claim that pure high-frequency forcing will mitigate vortex shedding for the studied high-lift configuration – unlike another actuation study by Luchtenburg et al. (Reference Luchtenburg, Günther, Noack, King and Tadmor2009). But the mean-field model structure should also include this mechanism as the derivation assumes nothing about the particular form of actuation.

4.3 The need for a cubic term

The mean-field model (4.16) is based on a shift mode $\boldsymbol{u}_{3}$ for unforced dynamics. The construction of the shift mode requires the knowledge of the unstable solution which is an art in itself (Åkervic et al. Reference Åkervic, Brandt, Henningson, Hœpffner, Marxen and Schlatter2006). In this study, the steady solution and hence the shift mode are not available, but accounted for by mean-field paraboloid (4.20c ). The result is a three-dimensional model:

(4.25a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{1}}{\text{d}t}={\it\sigma}a_{1}-{\it\omega}a_{2}, & \displaystyle\end{eqnarray}$$

(4.25b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{2}}{\text{d}t}={\it\sigma}a_{2}+{\it\omega}a_{1}, & \displaystyle\end{eqnarray}$$

(4.25c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{4}}{\text{d}t}=l_{4,-1}a_{-1}+l_{44}a_{4}+{\it\alpha}_{4r}r^{2}+{\it\alpha}_{4s}s^{2}, & \displaystyle\end{eqnarray}$$

(4.25d ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\sigma}={\it\sigma}_{1}-{\it\beta}_{-1}a_{-1}-{\it\beta}_{4}a_{4}-{\it\beta}_{r}r^{2}, & \displaystyle\end{eqnarray}$$

(4.25e ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\omega}={\it\omega}_{1}+{\it\gamma}_{-1}a_{-1}+{\it\gamma}_{4}a_{4}+{\it\gamma}_{r}r^{2}. & \displaystyle\end{eqnarray}$$

$-{\it\beta}_{r}r^{2}a_{1}$

$-{\it\beta}_{3}a_{3}a_{1}$

$l_{43}a_{3}$

${\it\alpha}_{4r}$

$l_{43}a_{3}+{\it\alpha}_{4r}r^{2}=(l_{43}{\it\alpha}_{r}+{\it\alpha}_{4r})r^{2}$

$a_{3}$

4.4 The need for a noise term

The mean-field model (4.16) elucidates how actuation suppresses vortex shedding. This includes the off–on transient from natural shedding to forced suppression. The on–off transient from suppression to natural shedding is more challenging. Once, vortex shedding is suppressed, $r=0$ , the equations have no resurrection mechanism and $a_{1},a_{2}$ will remain zero. Even if $r>0$ , the transient period would strongly depend on the numerical noise. A transient starting with $r=10^{-50}$ needs much more time to rich the limit cycle than one starting with $r=10^{-10}$ although both are effectively at the origin. This is an unphysical feature. In an experiment, turbulent fluctuations excite vortex shedding again. These fluctuations represent a well-defined noise level with well-defined transient times. Hence, we introduce a noise term in the first two equations of (4.16),

(4.26a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{1}}{\text{d}t}={\it\sigma}a_{1}-{\it\omega}a_{2}+{\it\kappa}{\it\xi}_{1}(t), & \displaystyle\end{eqnarray}$$

(4.26b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}a_{2}}{\text{d}t}={\it\sigma}a_{2}+{\it\omega}a_{1}+{\it\kappa}{\it\xi}_{2}(t). & \displaystyle\end{eqnarray}$$

${\it\xi}_{i}(t)$

${\it\kappa}$

The need for a noise term was postulated already in the first POD model of Aubry et al. (Reference Aubry, Holmes, Lumley and Stone1988). Bourgeois et al. (Reference Bourgeois, Martinuzzi and Noack2013) have implemented a noise term in a least-order Galerkin model to feature slow wake modulations.

The evolution equation (4.26) augmented by the shift-mode equation (4.25c ) and parameters (4.25d ), (4.25e ) can be considered as a physics-based, deterministic stochastic, least-order model which describes the off–on–off transient of § 2.

4.5 Galerkin projection

Section 4.4 provides the description of a robust control-oriented low-order Galerkin model, in which each state and each term has a well-defined effect. In this section, we pursue an alternative path starting with the Galerkin expansion (3.23). A Galerkin projection of the Navier–Stokes equations onto the mentioned subspace yields a Galerkin system of the form (Graham et al. Reference Graham, Peraire and Tang1999; Luchtenburg et al. Reference Luchtenburg, Günther, Noack, King and Tadmor2009; Noack et al. Reference Noack, Morzyński and Tadmor2011),

(4.27) $$\begin{eqnarray}\displaystyle \frac{\text{d}a_{i}}{\text{d}t}={\it\nu}\mathop{\sum }_{j=-N_{A}}^{N}l_{ij}^{{\it\nu}}a_{j}+\mathop{\sum }_{j,k=-N_{A}}^{N}q_{ijk}^{c}a_{j}a_{k}+\mathop{\sum }_{j=-N_{A}}^{-1}m_{ij}\frac{\text{d}a_{j}}{\text{d}t}. & & \displaystyle\end{eqnarray}$$

Here, we follow again the elegant compact nomenclature of Rempfer & Fasel (Reference Rempfer and Fasel1994) in which the basic mode is included as zeroth mode with $a_{0}\equiv 1$ . In (4.27), each actuation mode gives rise to an additional acceleration term $m_{ij}\,\text{d}a_{j}/\text{d}t$ , with $m_{ij}=(\boldsymbol{u}_{i},\boldsymbol{u}_{j})_{{\it\Omega}}$ , $j=-N_{A},\ldots ,-1$ . The dissipation term is characterized by the inverse Reynolds number ${\it\nu}=1/Re$ and $l_{ij}^{{\it\nu}}=(\boldsymbol{u}_{i},{\rm\Delta}\boldsymbol{u}_{j})_{{\it\Omega}}$ , while the convective term is determined from $q_{ijk}^{c}=(\boldsymbol{u}_{i},\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }[\boldsymbol{u}_{j}\otimes \boldsymbol{u}_{k}])_{{\it\Omega}}$ . The pressure term vanishes for many closed flows and is negligible for open flows with large observation domains. For smaller domains, the inclusion of the pressure term in the Galerkin method just modifies the coefficients of (4.27) (Noack et al. Reference Noack, Papas and Monkewitz2005). In many cases, a calibrated linear term offers a good approximation of the Galerkin pressure term representation (Galletti et al. Reference Galletti, Bruneau, Zannetti and Iollo2004).

The Galerkin system (4.27) allows us to track the influence of viscous and convective terms on the dynamics. The mean-field system of § 4.4 lumps both terms but distils the dynamic significance of each frequency component. By construction, the mean-field system lives on a manifold defined by the Reynolds equation. This gives rise to cubic terms which are not present in (4.27). Secondly, the mean-field system contains a noise term to account for high-frequency background turbulence. In principle, similar noise terms could be added to the Galerkin system. Equation (4.27) includes an actuation-induced acceleration term $m_{ij}\,\text{d}a_{j}/\text{d}t$ . This term vanishes for no or steady blowing. Periodic actuations lead to additional terms which may be lumped into the terms $q_{i0k}a_{k}$ , $k<0$ . Drastic changes like sudden blowing or reversal thereof are ignored by mean-field considerations. This will be reflected in the hierarchy of Galerkin models introduced in § 5.