2 Governing equations

In the following, the fluid domain is $(x,y\geqslant 0)$ where $y=0$ is ground. The flow is started by a pair of counter-rotating Lamb–Oseen vortices, located at $(x_{c_{0}},y_{c_{0}})=(\pm 0.5b,y_{c_{0}})$ with initial circulation $\unicode[STIX]{x1D6E4}_{0}$. Initially the vortex dispersion radius that defines the vortex velocity profile is set to $a=0.15b$. Length and velocity are normalized on vortex separation $b$ and on the drift velocity of the vortex pair $v^{\ast }=\unicode[STIX]{x1D6E4}_{0}/2\unicode[STIX]{x03C0}b$, yielding a reference time $T^{\ast }=2\unicode[STIX]{x03C0}b^{2}/\unicode[STIX]{x1D6E4}_{0}$ and a circulation-based Reynolds number $Re=\unicode[STIX]{x1D6E4}_{0}/2\unicode[STIX]{x03C0}\unicode[STIX]{x1D708}$ where $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid. The Reynolds number is considered in the range $200\leqslant Re\leqslant 5000$. For the remainder of the paper and unless otherwise stated, all the quantities are normalized upon the aforementioned variables. The flow is governed by the incompressible Navier–Stokes equations,

(2.1)$$\begin{eqnarray}\left.\begin{array}{@{}l@{}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }~\boldsymbol{u}=0,\\ \unicode[STIX]{x2202}_{t}\boldsymbol{u}+(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}=-\unicode[STIX]{x1D735}p+Re^{-1}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u},\\ \boldsymbol{u}(x,y=0,t)=\boldsymbol{u}_{w},\end{array}\right\}\end{eqnarray}$$

where $\boldsymbol{u}_{w}$ is the velocity at the wall, which is zero in the uncontrolled case, and optimized in the controlled case. These equations are solved using the incompressible computational tool Nek5000 (Fischer, Lottes & Kerkemeier Reference Fischer, Lottes and Kerkemeier2008). Numerical simulations have been carried out using parallel computations with 128 cores and solving the set of (2.1) limited to the right half of the domain, denoted $\unicode[STIX]{x1D6FA}$, shown in figure 1 with a mesh containing $143\times 129=18\,447$ elements. Each element contains $8\times 8$ Gauss–Lobatto points, leading to 1.2 M discretization points. Boundary conditions are symmetry conditions at all frontiers except the wall where Dirichlet boundary condition is applied.

Figure 1.

Schematic (not to scale) of the physical set-up. Dimensions are indicated. The right part of the domain is the computational domain and is shaded in grey. The disks represent the two Lamb–Oseen vortices at time $t=0$. Symmetric conditions are applied at the right, upper and left boundaries (respectively denoted $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{2,3,4}$) and Dirichlet is prescribed at the ground denoted $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{1}$.

Numerical validation is carried out using comparisons of vortex trajectories against data from different simulations published by Türk et al. (Reference Türk, Coors and Jacob1999) and Zheng & Ash (Reference Zheng and Ash1996). Usually the vortex centroid $M_{c}=(x_{c},y_{c})$ is obtained by the first moment of the normal to the plane vorticity field $\unicode[STIX]{x1D714}$

(2.2a,b)$$\begin{eqnarray}x_{c}=\displaystyle {\displaystyle \frac{\displaystyle \int _{\unicode[STIX]{x1D6FA}}x\unicode[STIX]{x1D714}\,\text{d}\unicode[STIX]{x1D6FA}}{\displaystyle \int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D714}\,\text{d}\unicode[STIX]{x1D6FA}}};\quad y_{c}=\displaystyle {\displaystyle \frac{\displaystyle \int _{\unicode[STIX]{x1D6FA}}y\unicode[STIX]{x1D714}\,\text{d}\unicode[STIX]{x1D6FA}}{\displaystyle \int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D714}\,\text{d}\unicode[STIX]{x1D6FA}}}.\end{eqnarray}$$

However, in the present configuration, $M_{c}$ does not match the primary vortex locus $M_{pv}=(x_{pv},y_{pv})$ because of the boundary layer and secondary vortices. Instead, the vortex locus $M_{pv}$ is taken as the barycentre of the vorticity field $\unicode[STIX]{x1D714}_{pv}$ given by

(2.3)$$\begin{eqnarray}\unicode[STIX]{x1D714}_{pv}(x,y)=\unicode[STIX]{x1D714}(x,y)\cdot [\unicode[STIX]{x1D714}(x,y)>0]\cdot [Q(x,y)>0],\end{eqnarray}$$

where $[S]$ denotes the Iverson bracket of the statement $S$ (Iverson Reference Iverson1962) and $Q$ is the $Q$-criterion which defines a vortex as a region with a positive second invariant of $\unicode[STIX]{x1D735}u$ (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988; Kolář Reference Kolář2007). The barycentre leading to $M_{pv}$ is then obtained by the same volume integration as in relation (2.2), with $\unicode[STIX]{x1D714}$ replaced by $\unicode[STIX]{x1D714}_{pv}$.

The comparison against Türk et al. (Reference Türk, Coors and Jacob1999) is conducted for $Re_{0}=\unicode[STIX]{x1D6E4}_{0}/\unicode[STIX]{x1D708}=330$ and $3300$ and that against Zheng & Ash (Reference Zheng and Ash1996) for $Re_{0}=\unicode[STIX]{x1D6E4}_{0}/\unicode[STIX]{x1D708}=7650$ (note the use of $Re_{0}=2\unicode[STIX]{x03C0}Re$). The flow is initialized by a pair of Lamb–Oseen vortices localized at $\pm (x_{c_{0}},2x_{c_{0}})$ and $(\pm x_{c_{0}},3.8x_{c_{0}})$ to match the settings of Türk et al. (Reference Türk, Coors and Jacob1999) and Zheng & Ash (Reference Zheng and Ash1996), respectively. The ensemble of trajectories, shown in figure 2, provides comparisons that validate the present numerical approach.

Figure 2.

Trajectories of the right vortex for a set of Reynolds numbers. The red, blue and green curves respectively correspond to $Re_{0}=\unicode[STIX]{x1D6E4}_{0}/\unicode[STIX]{x1D708}=330,3300,7650$. The results obtained by Türk et al. (Reference Türk, Coors and Jacob1999) (two cases) and Zheng & Ash (Reference Zheng and Ash1996) are respectively represented with the dashed, dash-dotted and solid lines and the trajectories obtained in the simulations are marked with squares, triangles and circles.

3 Uncontrolled flow

In potential flows, vortices initially located in $(\pm x_{c_{0}},y_{c_{0}})$ follow a hyperbolic trajectory $(x_{c}(t),y_{c}(t))$ described by

(3.1)$$\begin{eqnarray}\frac{1}{x_{c}^{2}(t)}+\frac{1}{y_{c}^{2}(t)}=\frac{1}{x_{c_{0}}^{2}}+\frac{1}{y_{c_{0}}^{2}}.\end{eqnarray}$$

With viscosity, as seen previously in figure 2 and now in figure 3 which plots the vortex trajectories for $Re=500$ and 1000, the dynamics is modified when the vortices near the ground. This so-called rebound results in an oscillation of the altitude of the vortex and a reduction of its sideward motion. It is noteworthy that such an observation is coherent with practical, high Reynolds number cases of aircraft wake vortices impacting ground, as is shown by the lidar (light detection and ranging) measurements also plotted in figure 3. The data come from a set of Doppler lidar measurements performed at Toulouse-Blagnac airport (Hallermeyer Reference Hallermeyer2017). From the simulated data it can be remarked that with a larger Reynolds number the vortex follows the inviscid path up to a larger distance. Rebound yet remains practically unchanged, given that both the current simulations and the real life lidar measurements show identical overall kinematics.

Figure 3.

Trajectories of the right vortex for $Re=1000$ (dashed line), for $Re=5000$ (solid line) and inviscid flow (dotted line). Simulation time is $t=15$. Circle markers correspond to Doppler lidar measurements achieved at Toulouse-Blagnac airport (European Commission 2015; Hallermeyer Reference Hallermeyer2017) during $50$ s (in this case $b\simeq 40~\text{m}$ and circulation is $\unicode[STIX]{x1D6E4}_{0}\simeq 500~\text{m}^{2}~\text{s}^{-1}$). Note the existence of an average cross-wind ($u_{crosswind}\sim -1~\text{m}~\text{s}^{-1}$) in the lidar case, not taken into account in the simulated data.

Rebound is illustrated in figure 4 with plots of the vorticity field at various instants of the vortex evolution. The figure also features a set of velocity profiles in the near wall region and the wall-wise pressure gradient. The velocity profiles show the flow at the wall that is induced by the momentum brought by the vortices. As initially described by Harvey & Perry (Reference Harvey and Perry1971) and due to the large Reynolds number, this subsequently forms a boundary layer. The vortex also imposes a positive pressure gradient at its sides because of the low pressure present in its core. As a consequence of this (adverse) pressure gradient, the boundary layer separates. This occurs when the vortex gets closer to the ground (figure 4b). The separated flow then rolls into opposite sign vorticity, soon embarked around the primary vortex (c). Additional detachments of the separated boundary layer later form other vortices, which themselves induce an opposite boundary layer flow (d).

Figure 4.

Uncontrolled vorticity field computed for $Re=1000$ at $t=0.25,2.5,3.75,5,10,15$. Thick lines represent boundary layer profiles are given at locations $x=0.5,1,1.5,2.0,2.5$. Dashed lines represent the pressure gradient along the wall (at $y=0$). (a) Descending vortex $t=1.25$. (b) Boundary layer and detachment of a secondary vortex $t=2.5$. (c) Rebound of the vortex $t=3.75$. (d) Entrainment of the secondary vortex $t=5$. (e) Entrainment of the secondary vortex $t=10$. (f) Entrainment of the secondary vortex $t=15$.

The question of how the vortex circulation evolves in this context can be analysed using the circulation of the right part of the flow $\unicode[STIX]{x1D6E4}_{tot}=\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D714}\,\text{d}\unicode[STIX]{x1D6FA}$, whose evolution is to be inferred from the equation of the vorticity $\unicode[STIX]{x1D714}$

(3.2)$$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D6E4}_{tot}}{\text{d}t}=\frac{1}{Re}\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D714}\boldsymbol{\cdot }\boldsymbol{n}\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}=-\frac{1}{Re}\left(\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{1}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}}{\unicode[STIX]{x2202}y}\,\text{d}x+\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{4}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}}{\unicode[STIX]{x2202}x}\,\text{d}y\right).\end{eqnarray}$$

Since $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}x>0$ at $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{4}$, the left boundary, and $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}y>0$ at $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{1}$, the bottom boundary, total circulation is expected to decrease. Physically the second and first terms on the right-hand side relate to friction with the left vortex and ground, respectively. It is important to mention that total circulation $\unicode[STIX]{x1D6E4}_{tot}$ is not equal to $\unicode[STIX]{x1D6E4}_{0}$ initially because of the presence of the boundary layer, which features negative vorticity.

To characterize the vortex circulation, $\unicode[STIX]{x1D6E4}_{pv}$ is introduced, which is defined as

(3.3)$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{pv}=\int _{\unicode[STIX]{x1D6FA}_{pv}}\unicode[STIX]{x1D714}\,\text{d}\unicode[STIX]{x1D6FA}_{pv}.\end{eqnarray}$$

The configuration imposes $\unicode[STIX]{x1D6E4}_{pv}=\unicode[STIX]{x1D6E4}_{0}$ initially. The zone $\unicode[STIX]{x1D6FA}_{pv}$ defines the spatial extent of the primary vortex. Here, to discriminate it from the several vortical zones featured by the flow, $\unicode[STIX]{x1D6FA}_{pv}$ is defined as the zone of positive vorticity about the primary vortex locus $M_{pv}=(x_{pv},y_{pv})$. Note that the simple integration of $\unicode[STIX]{x1D714}_{pv}$ over the half-domain $\unicode[STIX]{x1D6FA}$, as used to define $M_{pv}$ in (2.3), does not yield the vortex circulation as desired, as, due to the $Q$-criterion constraint, it misses some part of the vortical area associated with the vortex. The zone $\unicode[STIX]{x1D6FA}_{pv}$ cannot be easily described and therefore a specific procedure is employed to automatically track its shape in the snapshots of the flow. The first step is to locate the primary vortex centre using procedure (2.3). Once the vortex centre is known, the method consists in using a cylindrical coordinate system about $M_{pv}$ and to locate at all azimuthal positions $\unicode[STIX]{x1D703}$ the radial extent $R$ where $\unicode[STIX]{x1D714}_{pv}(R,\unicode[STIX]{x1D703})=0$. In practice, an ensemble of $N_{\unicode[STIX]{x1D703}}=128$ azimuthal positions $\unicode[STIX]{x1D703}_{i}$ is considered, and the radial direction is discretized into $R_{i}$ with a spacing of $\unicode[STIX]{x1D6FF}_{r}=0.01$. The ensemble of $(\unicode[STIX]{x1D703}_{i},R_{i})$ forms a polygonal area that is taken as $\unicode[STIX]{x1D6FA}_{pv}$.

Figure 5.

Evolution of $\unicode[STIX]{x1D6E4}_{pv}$ and $\unicode[STIX]{x1D6E4}_{tot}$ at $Re=200,1000$ and $T=15$ for the uncontrolled case. Vertical dotted lines correspond to times $t=1.25,2.5,3.75,5,10$ referring to figure 4.

Figure 5 shows the evolution of $\unicode[STIX]{x1D6E4}_{tot}$ and $\unicode[STIX]{x1D6E4}_{pv}$ as a function of time for the two Reynolds numbers $Re=200$ and 1000. Total circulation $\unicode[STIX]{x1D6E4}_{tot}$ is seen to decrease significantly up to $t=4$ and to level off after that. This can be analysed in the light of figure 4, which corresponds to the case $Re=1000$. Indeed the development of the boundary layer, its separation and the formation of the secondary vortex (see figure 4a–c) explain the initial strong decrease of $\unicode[STIX]{x1D6E4}_{tot}$ in figure 5 while the mild decrease of total circulation after $t=4$ stems from the end of these processes and corresponds to the slow viscous interaction between the different vorticity zones.

It is observed that the evolution of $\unicode[STIX]{x1D6E4}_{tot}$ hardly changes between $Re=200$ and 1000, suggesting a weak Reynolds number dependency of this quantity. The formulation of circulation written below

(3.4)$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{tot}=\oint _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}\boldsymbol{u}\boldsymbol{\cdot }\text{d}\boldsymbol{l},\end{eqnarray}$$

can help understand why this is so. In (3.4) $\text{d}\boldsymbol{l}$ is the tangent vector along the closed contour $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$. In the no-slip case, the contour integration reduces to the symmetry boundary $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{4}$. Indeed, the velocities at the top and right boundaries are very small and their contribution can thus be neglected. Equation (3.4) can hence be rewritten as

(3.5)$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{tot}=-\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{4}}v(0,y)\,\text{d}y,\end{eqnarray}$$

where $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{4}$ is oriented downward. The vertical velocity at the symmetry plane $v(0,y)$ essentially evolves as a function of the lateral distance of the vortex and one expects little influence of the no-slip boundary condition applied at the ground. As a check, we introduce $\unicode[STIX]{x1D6E4}_{inv}=-\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{4}}v_{inv}(0,y)\,\text{d}y$, the contribution of the left symmetry frontier to the total circulation in the inviscid case. The velocity $v_{inv}$ results from the Biot–Savart law applied to the point vortex along the hyperbolic trajectory and its images. This circulation can be calculated following

(3.6)$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{inv}=\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{4}}{\displaystyle \frac{\unicode[STIX]{x1D6E4}_{0}}{2\unicode[STIX]{x03C0}}}\left\{{\displaystyle \frac{2x_{c}}{\sqrt{x_{c}^{2}+(y-y_{c})^{2}}}}-{\displaystyle \frac{2x_{c}}{\sqrt{x_{c}^{2}+(y+y_{c})^{2}}}}\right\}\,\text{d}y.\end{eqnarray}$$

Crossed markers in figure 5 show the time evolution of total circulation using (3.6). The evolution of $\unicode[STIX]{x1D6E4}_{inv}$ follows closely that of $\unicode[STIX]{x1D6E4}_{tot}$ provided by (3.2) up to $t=3$. Departure from the inviscid evolution appears afterwards, indicating the impact of no slip at the ground on the lateral position of the vortex. However the difference between $\unicode[STIX]{x1D6E4}_{inv}$ and $\unicode[STIX]{x1D6E4}_{tot}$ remains small, which concludes on the weak influence of Reynolds number on $\unicode[STIX]{x1D6E4}_{tot}$.

The circulation of the primary vortex $\unicode[STIX]{x1D6E4}_{pv}$ shows a different evolution than $\unicode[STIX]{x1D6E4}_{tot}$, with three distinct phases and a net effect of Reynolds number. The first phase, that corresponds to the initial descent of the vortices, up to $t=2.5$, preserves circulation, i.e. $\unicode[STIX]{x1D6E4}_{pv}\simeq \unicode[STIX]{x1D6E4}_{0}$. The second phase, up to $t=5$, features a sudden drop of circulation and coincides with the rebound of the vortex. In this step the vortex experiences strong friction at the wall because $\unicode[STIX]{x1D735}\unicode[STIX]{x1D714}\boldsymbol{\cdot }\boldsymbol{n}$ is increased by the presence of opposite sign vorticity at the wall (boundary layer). The decay rate then slightly levels off, corresponding to the third phase of evolution. Vortex decay in the two last phases shows a clear dependence upon the Reynolds number. The start of the third phase ($t\geqslant 5$) corresponds to the slow decay phase of the total circulation $\unicode[STIX]{x1D6E4}_{tot}$, and to the quasi-disappearance of the pressure gradient at the wall, which goes along with the quasi-suppression of the boundary layer, see figure 4. The vortex system altitude is now large due to the previous rebound and the friction at the wall is not as significant.

4 Optimal control approach

We consider the effect of either blowing or suction applied at the ground to counteract the stagnation of the vortex system and promote increased lateral displacement. Practically, a modification of the boundary condition at $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{1}$ in (2.1) is applied with $\boldsymbol{u}_{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{1}}=\boldsymbol{u}_{\boldsymbol{w}}(x,t)$ where $\boldsymbol{u}_{\boldsymbol{w}}(x,t)$ is free to vary in intensity and orientation. The control is restricted to a zone of finite extent $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{1,w}=\{0\leqslant x\leqslant 10\}$. Figure 6 provides a schematic of the control set-up.

Figure 6.

Schematic of the control scheme applied along the portion $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{1,w}$ of the wall. $u_{w}$ and $v_{w}$ denote the flow velocity imposed at the wall.

The optimal control strategy requires an appropriate objective function ${\mathcal{J}}$ to optimize $\boldsymbol{u}_{\boldsymbol{w}}$ and maximize the lateral position of the vortex system at a given horizon time $T$. A choice could have been the first moment of vorticity $\int _{\unicode[STIX]{x1D6FA}}x\unicode[STIX]{x1D714}\,\text{d}\unicode[STIX]{x1D6FA}$ as it evaluates, once divided by $\unicode[STIX]{x1D6E4}_{tot}$ (see (2.2)), the horizontal component of the barycentre of vorticity.

However, because of the different vorticity areas, which yield opposite signs, this optimization would lead to maximizing the lateral position of the positive vorticity (primary vortex) and minimizing that of the negative vorticity (separated boundary layer). Therefore, the maximization of the lateral position of the vortex seems not to be guaranteed. A way to shape the objective function to our goal is to consider instead the square of vorticity $\unicode[STIX]{x1D714}^{2}$. The benefit is that it mechanically embarks the vorticity of the primary vortex and the opposite vorticity coming from the separated boundary layer alike.

The implementation hence proceeds from an evaluation of the horizontal position of the vortex based on the horizontal component of the barycentre of vorticity squared, that is

(4.1)$$\begin{eqnarray}{\mathcal{J}}(\boldsymbol{u};\boldsymbol{u}_{\boldsymbol{w}};T,l)={\mathcal{J}}_{1}(\boldsymbol{u};\boldsymbol{u}_{\boldsymbol{w}};T,l)-l{\mathcal{J}}_{2}(\boldsymbol{u}_{\boldsymbol{w}};T,l),\end{eqnarray}$$

with

(4.2)$$\begin{eqnarray}{\mathcal{J}}_{1}(\boldsymbol{u};\boldsymbol{u}_{\boldsymbol{w}};T,l)=\displaystyle {\displaystyle \frac{\displaystyle \int _{\unicode[STIX]{x1D6FA}}x\unicode[STIX]{x1D714}^{2}(T)\,\text{d}\unicode[STIX]{x1D6FA}}{\displaystyle \int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D714}^{2}(T)\,\text{d}\unicode[STIX]{x1D6FA}}},\end{eqnarray}$$

and

(4.3)$$\begin{eqnarray}\displaystyle {\mathcal{J}}_{2}(\boldsymbol{u}_{\boldsymbol{w}};T,l)=\displaystyle {\displaystyle \frac{1}{2}}\int _{0}^{T}\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{1}}\boldsymbol{u}_{\boldsymbol{w}}^{2}\,\text{d}x\,\text{d}t.\end{eqnarray}$$

The term ${\mathcal{J}}_{2}$ (spatio-temporal norm of the control) penalizes ${\mathcal{J}}_{1}$ by acting as a cost function for control. The mathematical parameter $l>0$ (which holds no physical interpretation) prevents $\boldsymbol{u}_{\boldsymbol{w}}$ from reaching an unphysically large amplitude and increases the convergence of the optimization algorithm. As such, it favours the existence and unicity of a solution as well as eases convergence of the optimization loops. The case $l\rightarrow \infty$ amounts to the uncontrolled case i.e. ${\mathcal{J}}_{m}(\cdot ,\infty )={\mathcal{J}}_{1}(\cdot ,\infty )$ since $\lim _{l\rightarrow \infty }{\mathcal{J}}_{2}=0$ (a validation is provided further down, see § 5.1).

The maximum ${\mathcal{J}}_{m}(T,l)$ of ${\mathcal{J}}$ in the space of $\boldsymbol{u}_{w}$ for a given $T$ and $l$ is found using the Lagrange multiplier method in which the constrained optimization problem is transformed into an unconstrained one. To this end, the cost functional and the constraints are assembled into the Lagrangian function ${\mathcal{L}}$, in the form

(4.4)$$\begin{eqnarray}\displaystyle \displaystyle {\mathcal{L}}(\boldsymbol{u},\boldsymbol{u}^{+},p,p^{+},\boldsymbol{u}_{\boldsymbol{w}},\boldsymbol{u}_{\boldsymbol{w}}^{+}) & = & \displaystyle {\mathcal{J}}(\boldsymbol{u}_{\boldsymbol{w}})-\left[\unicode[STIX]{x2202}_{t}\boldsymbol{u}+(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}+\unicode[STIX]{x1D735}p-{\displaystyle \frac{1}{Re}}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u},\boldsymbol{u}^{+}\right]\nonumber\\ \displaystyle & & \displaystyle -\,[\unicode[STIX]{x1D735}\boldsymbol{\cdot }~\boldsymbol{u},p^{+}]-(\boldsymbol{u}-\boldsymbol{u}_{\boldsymbol{w}},\boldsymbol{u}_{\boldsymbol{w}}^{+}),\end{eqnarray}$$

where $[\cdot ,\cdot ]$ and $(\cdot ,\cdot )$, denote the following inner products

(4.5)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}[\boldsymbol{a}(\boldsymbol{x},t),\boldsymbol{b}(\boldsymbol{x},t)]=\displaystyle \int _{0}^{T}\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{a}(\boldsymbol{x},t)\boldsymbol{\cdot }\boldsymbol{b}(\boldsymbol{x},t)\,\text{d}\unicode[STIX]{x1D6FA}\text{d}t,\\[12.0pt] (\boldsymbol{c}(x),\boldsymbol{d}(x))=\displaystyle \int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{1}}\boldsymbol{c}(x)\boldsymbol{\cdot }\boldsymbol{d}(x)\,\text{d}x,\end{array}\right\}\end{eqnarray}$$

and $\boldsymbol{a},\boldsymbol{b}$ are functions of space and time and $\boldsymbol{c},\boldsymbol{d}$ are functions of space alone.

The Lagrangian function ${\mathcal{L}}$ depends on the direct $(\boldsymbol{u},p,\boldsymbol{u}_{\boldsymbol{w}})$ variables and Lagrange multipliers $(\boldsymbol{u}^{+},p^{+},\boldsymbol{u}_{\boldsymbol{w}}^{+})$, also known as adjoint variables. The optimal control is found by cancelling the first variation of the Lagrangian with respect to its variables. For a complete derivations of these equations the reader is referred to Choi et al. (Reference Choi, Hinze and Kunisch1999). Here, only the most important relations are provided. The first variation of ${\mathcal{L}}$ with respect to the adjoint variables yields the equations of the constraints and the first variation with respect to the direct variables provides a set of equations for the adjoint variables,

(4.6)$$\begin{eqnarray}\left.\begin{array}{@{}l@{}}\unicode[STIX]{x1D735}\boldsymbol{\cdot }~\boldsymbol{u}^{+}=0,\\ -\unicode[STIX]{x2202}_{t}\boldsymbol{u}^{+}-(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}^{+}+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\intercal }\boldsymbol{u}^{+}=-\unicode[STIX]{x1D735}p^{+}+{\displaystyle \frac{1}{Re}}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}^{+},\end{array}\right\}\end{eqnarray}$$

where $^{\intercal }$ denotes the transpose operator. The minus sign in front of the time partial derivative can be interpreted as a change of variable $t\longrightarrow T-t$. As a consequence, the adjoint equation is solved backward in time from $t=T$ to $t=0$. By matching terms of the integration by part of the partial time derivative, one obtains the following compatibility equation (which is derived in appendix A) for $t=T$

(4.7)$$\begin{eqnarray}\boldsymbol{u}^{+}(\boldsymbol{x},T)={\displaystyle \frac{2}{I_{0}(T)}}(x\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D714}(T)\boldsymbol{e}_{z}+\unicode[STIX]{x1D735}x\times \unicode[STIX]{x1D714}(T)\boldsymbol{e}_{z})-{\displaystyle \frac{2I_{1}(T)}{I_{0}(T)^{2}}}(\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D714}(T)\boldsymbol{e}_{z}),\end{eqnarray}$$

where

(4.8)$$\begin{eqnarray}\displaystyle I_{0}(t)=\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D714}^{2}(t)\,\text{d}\unicode[STIX]{x1D6FA},\end{eqnarray}$$

and

(4.9)$$\begin{eqnarray}\displaystyle I_{1}(t)=\int _{\unicode[STIX]{x1D6FA}}x\unicode[STIX]{x1D714}^{2}(t)\,\text{d}\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

Relation (4.7) initializes the adjoint variable $\boldsymbol{u}^{+}(\boldsymbol{x},T)$. The gradient of the Lagrangian with respect to $\boldsymbol{u}_{\boldsymbol{w}}$

(4.10)$$\begin{eqnarray}\unicode[STIX]{x1D735}_{\boldsymbol{u}_{\boldsymbol{w}}}{\mathcal{L}}=-l\boldsymbol{u}_{\boldsymbol{w}}+\boldsymbol{u}_{\boldsymbol{w}}^{+}=-l\boldsymbol{u}_{\boldsymbol{ w}}+\left(p^{+}(x,y=0,t)-{\displaystyle \frac{1}{Re}}{\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}^{+}}{\unicode[STIX]{x2202}y}}(x,y=0,t)\right)\boldsymbol{e}_{\boldsymbol{y}},\end{eqnarray}$$

eventually provides the direction to follow in order to update the control variable.

This optimal system is based on a set of nonlinear partial differential equations. While this nonlinearity could lead to difficulties in converging an optimal solution, in the present case the dynamics of the vortex to be controlled follows a deterministic evolution, so that similarly to Önder & Meyers (Reference Önder and Meyers2016), a steepest descent algorithm with a generic backtracking line search is successfully used in order to perform the optimization. The whole procedure is sketched in figure 7. Starting from an initial $\boldsymbol{u}_{w,0}=0$, the algorithm follows two loops. The main loop goes towards the optimal velocity at the wall by computing the gradient and updating $\boldsymbol{u}_{w}$ with a factor $\unicode[STIX]{x1D6FC}_{k}$ of the gradient. The second loop concerns the line search to adapt the value of $\unicode[STIX]{x1D6FC}_{k}$ when the chosen value does not optimize. The starting value is 0.5. Whenever the chosen value does not lead to optimization, $\unicode[STIX]{x1D6FC}_{k}$ is halved and the direct simulation is repeated until optimization is achieved. The iterative algorithm is eventually stopped when $({\mathcal{J}}_{k+1}-{\mathcal{J}}_{k})/{\mathcal{J}}_{k}\leqslant \unicode[STIX]{x1D700}=10^{-4}$.

Figure 7.

Sketch of the optimization algorithm. The line search parameter is $\unicode[STIX]{x1D6FC}_{k}$.

In the current algorithm the cost of storing the control variable when integrating the adjoint is decreased by using linear interpolation over uniform time intervals $[t_{i},t_{i+1}]$ of the horizon time $T$

(4.11)$$\begin{eqnarray}\boldsymbol{u}_{\boldsymbol{w}}(t_{k})=\unicode[STIX]{x1D6FD}_{k}\boldsymbol{u}_{\boldsymbol{w}}(t_{i})+(1-\unicode[STIX]{x1D6FD}_{k})\boldsymbol{u}_{\boldsymbol{w}}(t_{i+1}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}_{k}=(t_{i+1}-t_{k})/\unicode[STIX]{x0394}t$ and $\unicode[STIX]{x1D6FF}t_{i}=t_{i+1}-t_{i}$ is chosen such that $\unicode[STIX]{x1D6FF}t_{i}\leqslant 5\,\%$. The convergence of the results is provided in appendix B.1 and convergence in $\unicode[STIX]{x1D6FF}t_{i}$ is shown in appendix B.2.

5 Controlled flow

5.1 Optimal parameters

Optimizations have been carried out for $T\in [2,10]$ and $l\in [1,100]$, and Reynolds numbers of 200 and 1000.

Figure 8.

(a) Evolution of the cost of control ${\mathcal{J}}_{2}$ as a function of $l$ for Reynolds numbers 200 and 1000, $T=5$, showing that the optimal control amounts to the uncontrolled case for increasing values of $l$. (b) Ratio ${\mathcal{J}}_{1}(5,l)/{\mathcal{J}}_{1}(5,\infty )$ as a function of the penalization coefficient for computations carried at $Re=200$ (dashed line and crosses) and $Re=1000$ (solid line and squared markers) for $T=5$.

In figure 8(a) the cost of control ${\mathcal{J}}_{2}$ is plotted as a function of $l$. It shows, as expected, that growing $l$ returns to the uncontrolled case ${\mathcal{J}}_{2}\rightarrow 0$. Low values of $l$ have been tested and have shown that when $l<1$ the optimization breaks down: the intensity of $\boldsymbol{u}_{w}$ becomes too high for the capability of the mesh and time step that are used (which correspond to the initial simulation of the uncontrolled case). Figure 8(b) shows the optimized vortex barycentre ${\mathcal{J}}_{1}$ normalized upon its value for infinite $l$ (no control) as a function of $l$ for $Re=200$ and 1000. It shows that the gain increases as $l$ is lowered, reaching a maximum of 1.5 and 2 for $l=1$ at the two Reynolds numbers, respectively. Increasing Reynolds number yields a favourable effect for augmenting control capacity. Plots of the vorticity field at final time $t=T=5$ for selected values of $l\in [1,\infty ]$ and the two values of Reynolds number are displayed in figure 9. They show the effect of control on the separation pattern and timing of the boundary layer and the intensity, location and number of the secondary vortices that rotate around the primary one.

Figure 9.

Vorticity field at horizon time $T=5$ for $Re=200$ (a–d) and 1000 (e–h) for increasing $l$. Dashed lines show the locations of the vorticity-squared barycentres, ${\mathcal{J}}_{1}(5,l)$.

The effect of horizon time $T$ is shown in figure 10 for $Re=200$ and $Re=1000$ at $l=1$, the best value within the range under consideration. Figure 10(a) plots the absolute position of the vortex ${\mathcal{J}}_{1}(T,l)$ for the uncontrolled and controlled cases. The increase in lateral distance provided by control is apparent from the offset between the two curves. Above $T=5$, the lateral position of the vortex shows a limit at approximately $x=3$ and no decisive increase beyond that time. Since $l=1$ provides the strongest control effect, this shows that optimization at a larger time is not required: $T=5$ appears to be quasi-optimal for the present parameter space.

For $Re=200$, it is found that optimization cannot be carried out beyond $T=5$. Indeed, the routine produces irregular vorticity at the end of the domain $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{1,w}$, which is attributed to the finite length of the control area at the ground. Coincidentally, the vortex is also sucked by the control at the wall and disappears. Calculations beyond $T=5$ are not conducted for $Re=200$.

Figure 10(b) shows the amplification of the full functional ${\mathcal{J}}_{m}$ between the controlled and uncontrolled cases. It shows that the optimal for $T=5$ is also the time for which the dynamics can be optimally balanced between the possibility of increased lateral displacement and cost of control. This shows that an optimal time exists over the parameter range investigated here, at approximately 5.

Figure 10.

Effect of horizon time. (a) Evolution of the vortex position ${\mathcal{J}}_{1}(T,l)$ for $Re=200,1000$, respectively in dashed and dotted lines, for the uncontrolled case $l=\infty$ (black) and for $l=1$ (red). (b) Evolution of the ratio ${\mathcal{J}}_{m}(T,1)/{\mathcal{J}}_{m}(T,\infty )$, computed for $Re=200$ (dashed line), $Re=1000$ (solid lines) and $l=1$. For this value of $l$ and $Re=200$, increasing the horizon time, beyond $T\geqslant 5$, leads to the complete absorption of the primary vortex. No data are consequently available for larger time.

5.2 Physical mechanisms

The physics behind the action of control is now analysed. The time sequence displayed in figure 11, which is obtained for $T=5$ and corresponds to the same Reynolds number ($Re=1000$) as the uncontrolled one shown in figure 4, is used in preamble. Some of the time indices used in figure 4 are also repeated in figure 11, to ease the comparison. The main effect of control which can be seen in this time sequence is the delay in boundary layer separation, which is only apparent near the final time $t>4$, while in the uncontrolled case (figure 4) it is already effective at $t=2.5$. The doubling of the lateral distance, presented in the previous section, is also directly visible here, changing from $x\simeq 1.2$ (see figure 4d) to $x\simeq 3$ when control is active. Interestingly, the vortex kinematics under the effect of control bears a strong similarity to the vortex evolution obtained with the $v$-control scheme developed by Choi et al. (Reference Choi, Moin and Kim1994) (figure 22b).

Figure 11.

Vorticity field in the controlled case, computed for $(Re,T,l)=(1000,5,10)$, showing the sequence up to the optimization time $t=5$. Thicker borders refer to the times used in figure 4 showing the uncontrolled case at the same Reynolds number, with the same iso-contours. Note however the different spatial extensions of the plots.

The evolution of the velocity at the wall imposed by control is displayed in figure 12 with some of the same time stamps as in figure 11. It is observed that control initially applies blowing at the centre of the two vortex system $x\in [0,1]$. After $t=2$, suction starts to act, essentially below the vortex, with increasing strength up to $t=3$. At this time, the primary vortex has reached its lowest altitude (see figure 11e). Subsequent control essentially reduces to slight blowing at the left of the primary vortex. After $t=2$, suction starts to act, essentially below the vortex, with increasing strength up to $t=3$. At this time, the primary vortex has reached its lowest altitude (see figure 11e). Subsequent control essentially reduces to slight blowing at the left of the primary vortex. Figure 13 shows the time trace of the control at $x=1$, that is, under the descent path of the vortex. The suction then blowing strategy of the control described above is grasped immediately using this plot, along with the dominance of the vertical velocity component upon the lateral component.

Figure 12.

Evolution of the optimal control $\boldsymbol{u}_{w}$ along the control area for different time steps. The thicker borders refer to the same times as those used in figure 4. Solid (respectively dashed) lines represent $v_{w}$ (respectively $u_{w}$).

Figure 13.

Evolution of the optimal control velocity at the wall as a function of time at $x=1$. Solid (respectively dashed) lines represent $v_{w}$ (respectively $u_{w}$).

Two main factors have been identified to explain how control physically achieves increased lateral displacement. One is related to (i) delayed separation, which reduces the vortex updraft by secondary vorticity, hence maintaining lower vortex altitude and increased induction effect by image vorticity about the ground, and the other one to (ii) reduced effective friction at the ground to decrease the opposition to the vortex lateral displacement.

The factor (i) relates to the suction that is applied by control in the region $x\in [1,1.5]$, slightly to the right of the primary vortex. Suction is a generic principle to delay boundary layer separation (Lachmann Reference Lachmann2014). Its interest in the present configuration can be grasped by comparing figure 4, without control, for which the boundary layer separates in the region $x\in [1,1.5]$ around time $t=2.5$, with figure 11, that yields control and which, as a result of suction, allows a delayed separation, up to time $t>4$.

The second factor (ii) is vortex induction. It can be identified using figure 11(d,e), which indicates that a specific vorticity pattern is formed at the wall by control between $t=2$ and 3, this pattern being absent in the flow at the same time when control is absent (figure 4b). The field of $Q$-criterion between the uncontrolled and controlled cases at $t=2$ and focused in the region between the primary vortex and the wall is shown in figures 14(a) and 14(b), respectively, to highlight the associated structure. The plot shows that a rotational area is brought about by control which, referring to figure 11(e), amounts to negative vorticity. Albeit the zone does not present a pressure minimum (at least not apparently), for simplicity’s sake we coin it as a vortex. Considering Biot–Savart induction, it can be stated that the effect of this vortex upon the primary vortex favours the motion to the right of the latter, and effectively pushes it along the wall. In the following, this vortex is conveniently referred to as the pusher vortex.

The formation of this pusher vortex can be directly linked to the velocity imposed by the control at the wall. Indeed figure 12 shows that the vertical velocity component generates a strong gradient $\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}x$ below the primary vortex for time $t=2$ to $3$ (figure 12d,e) and this is to be analysed as creating the $\unicode[STIX]{x1D714}$ associated with the pusher vortex. In effect $\unicode[STIX]{x1D714}\simeq \unicode[STIX]{x2202}v/\unicode[STIX]{x2202}x$ since $|u_{w}|\ll |v_{w}|$ and characteristic length scales for $u_{w}$ and $v_{w}$ are similar.

Figure 14.

Plot of the $Q$-criterion at time $t=2$ for (a) $(1000,5,\infty )$ and (b) $(1000,5,1)$. Creation of a rotational area in the boundary layer (positive values of the $Q$-criterion) that corresponds to the pusher vortex.

An interesting consequence of control concerns circulation, which is found to be significantly diminished with the application of control. The behaviours of $\unicode[STIX]{x1D6E4}_{tot}$ and $\unicode[STIX]{x1D6E4}_{pv}$ are shown in figure 15(a) for $Re=200$ and 1000, $T=5$ and $l=1$. Comparison can be made with the uncontrolled case in figure 5. For $Re=200$, the circulation drops by more than 60 % at $t=5$ compared to 50 % after 15 time units in the uncontrolled case. For $Re=1000$, the decrease is approximately 15 % compared to less than 10 % in the uncontrolled case, also at $t=5$.

Figure 15(b) sums up the previous analysis by showing the trajectory of the controlled vortex against the uncontrolled and inviscid trajectories, with equal time stamps to enable the comparison of the vortex progress in each case. The controlled vortex is initially moved above the uncontrolled trajectory as a consequence of the initial blowing imposed by control. Then, as the uncontrolled vortex rebounds, it moves laterally and close to the ground, reaching an altitude lower than the inviscid hyperbolic path. At the final time, the controlled vortex achieves a more outward position than the inviscid vortex, concluding the efficiency of the blowing/suction strategy.

Figure 15.

(a) Evolution of primary vortex and total circulation for $(200,5,1)$ (black) and $(1000,5,1)$ (red). For times beyond $t=5$, control has been turned off. (b) Evolution of the centroid trajectories for $0\leqslant t\leqslant 5$ for inviscid, controlled $(1000,5,1)$ and uncontrolled flow $(1000,5,\infty )$. Circled markers correspond to the centroid position at $t=0,1,2,3,4,5$.

5.3 Control robustness

In this subsection, we discuss the robustness of the optimal control approach with respect to the initial height, radius and activation time of the control law. It is noteworthy that assessing the robustness of the method with respect to the initial vortex circulation would imply only varying the circulation-based Reynolds number. Previous results comparing $Re=200$ and 1000 show no major modification of the present strategy with Reynolds number. Three distinct simulations H (for height), R (for radius) and DAT (for delayed activation time) have been carried out where we modify the initial height, dispersion radius and activation time separately. In simulation H, all parameters are kept the same, except for the initial height $y_{c_{0}}=5$ (instead of $2.5$) and the horizon time ($T=10$ instead of 5). Note that doubling the descending distance justifies doubling of the horizon time in order to compare with the case at $(1000,5,1)$ and $y_{c_{0}}=2.5$. Simulation R keeps all the same parameters as for the case at $(1000,5,1)$ except for the initial dispersion radius $a=0.1$ (instead of 0.15). In simulation DAT, we use the optimal control law of the optimal case $(1000,5,1)$ for the situation when the vortex initially departs from $y_{c_{0}}=5$. Control is then only active between $t_{a}=2.5$ and $t=7.5$. Table 1 summarizes the objective function results and compares them to the reference case $(1000,5,1)$, $y_{c_{0}}=2.5$ and $a=0.15$. The performance of the control, when looking at the value of ${\mathcal{J}}_{2}$, is moderately affected for cases H and R, with a relative 18 % decrease, and is unchanged for case DAT. This suggests that the control does not suffer much from changes in the parameters of the present model. The final vorticity fields and time trace of the optimal control at $x=1$ are shown in figure 16(a,b) for cases H and R. The case DAT leads to negligible changes compared to the reference in figures 11(h) and 13 and is thus not shown in figure 16. Similar behaviours and amplitudes of the cost function are observed between the optimal reference case $(1000,5,1)$ and cases H, R and DAT. In particular, the time traces shown in figure 16 resemble those in figure 13(c,d).

Table 1.

Effect of the initial height, radius and activation time on the objective function results. Left column corresponds to the reference optimal case $(1000,5,1)$ already shown in figure 11. The second, third and fourth columns correspond to simulations H, R and DAT respectively, which yield similar results.

Figure 16.

(a,b) Final vorticity field and (c,d) time trace of the control at $x=1$ achieved by the control. Panels (a,c) (respectively b,d) refer to simulation H (respectively R). To be compared to the optimal case $(1000,5,1)$, see figures 11 and 12.

5.4 Vorticity flux at the wall

The physical analysis can be backed by looking theoretically at the evolution of $I_{1}(t)$, the numerator of ${\mathcal{J}}_{1}(t)$, introduced in § 4 (see (4.8)). Following the calculations given in appendix C, the time derivative of $I_{1}$ yields

(5.1)$$\begin{eqnarray}{\dot{I}}_{1}=-{\displaystyle \frac{2}{Re}}\int _{\unicode[STIX]{x1D6FA}}x|\unicode[STIX]{x1D735}\unicode[STIX]{x1D714}|^{2}\,\text{d}\unicode[STIX]{x1D6FA}+{\displaystyle \frac{1}{Re}}\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{1}}x(\unicode[STIX]{x1D735}\unicode[STIX]{x1D714}^{2})\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}x+\int _{\unicode[STIX]{x1D6FA}}u\unicode[STIX]{x1D714}^{2}\,\text{d}\unicode[STIX]{x1D6FA}={\dot{I}}_{11}+{\dot{I}}_{12}+{\dot{I}}_{13}.\end{eqnarray}$$

This equation shows that $I_{1}$ is reduced by the dissipative term (${\dot{I}}_{11}$), and increased by the vorticity flux at the bottom boundary (${\dot{I}}_{12}$) which also can be written as

(5.2)$$\begin{eqnarray}{\dot{I}}_{12}=-\frac{2}{Re}\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{1}}x\unicode[STIX]{x1D714}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}}{\unicode[STIX]{x2202}y}\,\text{d}x,\end{eqnarray}$$

and by the advection of the vorticity field (${\dot{I}}_{13}$).

Figure 17 plots the evolution of ${\dot{I}}_{11,12,13}$ for the controlled and uncontrolled cases at $Re=1000$. The term ${\dot{I}}_{13}$ is almost unaffected by control, unlike the terms ${\dot{I}}_{11}$ and ${\dot{I}}_{12}$, which are increased by nearly 3 orders of magnitude when reaching their maximum values (shortly after $t=2$). The term ${\dot{I}}_{12}$ becomes larger than ${\dot{I}}_{11}$, which implies the net increase of $I_{1}$ and also means that the vorticity flux provoked by control at the wall, which results in the negative pusher vortex being present there, is the dominant mechanism at $Re=1000$. As explained previously, this pusher vortex creates a horizontal induction to the primary vortex which minimizes the impact of zero velocity at the wall. This way, it allows the primary vortex to slide along the wall and move to larger lateral distances.

Figure 17.

Time evolution $-{\dot{I}}_{11},{\dot{I}}_{12},{\dot{I}}_{13}$ from left to right for the uncontrolled case (dashed lines) and the controlled case (solid lines) for $(Re,T,l)=(1000,4,10)$.