3.1. Direct numerical simulations

We study 3-D flows over wings by numerically solving the incompressible Navier–Stokes equations

(3.1a)$$\begin{gather} \frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u} ={-}\boldsymbol{\nabla} p +{\frac{1}{Re_c}} \nabla^2 \boldsymbol{u} + \boldsymbol{e}, \end{gather}$$

(3.1b)$$\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = 0, \end{gather}$$

where $\boldsymbol {u} = (u_x,u_y,u_z)$ is the velocity vector, $p$ is the pressure and $\boldsymbol {e}$ is external forcing. The latter term is modelled as a harmonic body force using the spatial–temporal characteristics of the triglobal forcing modes obtained from resolvent analysis (§ 3.2). The set of equations (3.1) is solved using Cliff, the incompressible flow solver from the CharLES package, developed by the Cascade Technologies, Inc. This solver uses collocated node-based second-order-accurate finite volume formulation to spatially discretize mass and momentum equations and a fractional step scheme for time integration (Ham & Iaccarino Reference Ham and Iaccarino2004; Ham, Mattsson & Iaccarino Reference Ham, Mattsson and Iaccarino2006).

With the origin of the Cartesian coordinate system placed at the leading edge of the wing root $(x,y,z)/c = (0,0,0)$, the computational domain extends over approximately $(x,y,z)/c \in [-20,25] \times [-20,20] \times [0,20]$. The present computational grids were also used in Ribeiro et al. (Reference Ribeiro, Neal, Burtsev, Amitay, Theofilis and Taira2023a). A symmetry boundary condition is imposed at the root. At the inlet, we prescribe a free-stream velocity vector $\boldsymbol {u} = (U_\infty,0,0)$. At the outlet, we specify the convective boundary condition. A slip boundary condition is set on all other far-field boundaries. Lastly, a no-slip wall boundary condition is enforced on the wing surface.

We perform DNS for baseline ($\boldsymbol {e} = 0$ in (3.1a)) and controlled flows ($\boldsymbol {e}$ modelled with harmonic forcing modes). For both cases, simulations are initiated from uniform flow with no external forcing, being performed with a constant Courant–Friedrichs–Lewy (CFL) number of 1 until transients are washed out of the computational domain, which takes approximately $t = 90$. After the transient flow features are washed out of the domain, flows are simulated with and without external forcing using a constant time step defined such that the CFL number is smaller than one. Statistics are collected for approximately $t = 100$ to ensure statistical convergence.

3.2. Resolvent analysis

Resolvent analysis can provide valuable insights for the design of active flow control strategies. Let us consider the Reynolds decomposition of state variable $\boldsymbol {q} = \bar {\boldsymbol {q}} + \boldsymbol {q}^\prime$, where $\bar {\boldsymbol {q}}$ is the time-averaged flow and $\boldsymbol {q}'$ is the statistically stationary fluctuation component. In this study, the base flows $\bar {\boldsymbol {q}}$ used for resolvent analysis are derived from compressible flow simulations with a free-stream Mach number $M_\infty \equiv U_\infty / a_\infty = 0.1$, where $a_\infty$ represents the free-stream speed of sound. The spatial distribution of the velocity components for both incompressible and compressible time-averaged flows exhibits similar patterns, as depicted in figure 3. For compressible flows, the major variations in thermodynamic quantities occur closer to the wing surface and remain within a $1\,\%$ difference from the free-stream values. Resolvent modes can be derived from both compressible and incompressible base flows. In the present work, a compressible resolvent approach is employed to build on our previous studies (Ribeiro et al. Reference Ribeiro, Yeh and Taira2023b; Ribeiro & Taira Reference Ribeiro and Taira2023).

Figure 3. Time-averaged incompressible baseline flows from DNS (dashed, red) and the compressible base flows used for resolvent analysis (solid, black) shown as an example for the flow over a tapered swept wing with $\lambda = 0.27$ and $\varLambda = 30^\circ$ at $\alpha = 22^\circ$. Isocontour lines of (a) $\overline {u_x} = [-0.3,0.3]$, (b) $\overline {u_y} = [-0.3,0.7]$ and (c) $\overline {u_z} = [-0.25,0.55]$ shown in 2-D slices ($(x,y)$ plane) at $z/c = 0.5$ (top) and $1.2$ (bottom).

With the compressible base flow, the aforementioned Reynolds decomposition is used to linearize the compressible Navier–Stokes equations about $\bar {\boldsymbol {q}}$ to yield

(3.2)\begin{equation} \frac{\partial \boldsymbol{q}^\prime}{\partial t} = \boldsymbol{\mathsf{L}}_{\bar{\boldsymbol{q}}} \boldsymbol{q}^\prime + \boldsymbol{f}^\prime, \end{equation}

where $\boldsymbol{\mathsf{L}}_{\bar {\boldsymbol {q}}}$ is the discrete form of the linearized Navier–Stokes operator and $\boldsymbol {f}^\prime$ encompasses the nonlinear and external forcing terms (McKeon & Sharma Reference McKeon and Sharma2010). By considering the Fourier representation for $\boldsymbol {q}^\prime$ and $\boldsymbol {f}^\prime$, we have

(3.3)\begin{equation} [\boldsymbol{q}^\prime(\boldsymbol{x},t), \boldsymbol{f}^\prime(\boldsymbol{x},t)] = \int_{-\infty}^{\infty} [\hat{\boldsymbol{q}}_{\omega}(\boldsymbol{x}), \hat{\boldsymbol{f}}_{\omega}(\boldsymbol{x})] \exp({-{\rm i} \omega t})\,{\text d} \omega, \end{equation}

which transforms (3.2) to read

(3.4)\begin{equation} {-}{\rm i} \omega \hat{\boldsymbol{q}}_{\omega} = \boldsymbol{\mathsf{L}}_{\bar{\boldsymbol{q}}} \hat{\boldsymbol{q}}_{\omega} + \hat{\boldsymbol{f}}_{\omega}, \end{equation}

where $\boldsymbol {x} = (x,y,z)$ and the triglobal response and forcing modes are $\hat {\boldsymbol {q}}_{\omega }$ and $\hat {\boldsymbol {f}}_{\omega }$, respectively, for a temporal frequency $\omega$. This expression leads to

(3.5)\begin{equation} \hat{\boldsymbol{q}}_{\omega} = \boldsymbol{\mathsf{H}}_{\bar{\boldsymbol{q}},\omega} \hat{\boldsymbol{f}}_{\omega}, \end{equation}

with the resolvent operator $\boldsymbol{\mathsf{H}}_{\bar {\boldsymbol {q}},\omega } \in \mathbb {C}^{m \times m}$. The operator size $m$ is defined by the product of the number of state variables and the number of spatial grid points. Herein, the linear operators have size $m$ of approximately $5 \times 10^6$. We analyse the resolvent operator through the singular value decomposition

(3.6)\begin{equation} \boldsymbol{\mathsf{H}}_{\bar{\boldsymbol{q}}} = [-{\rm i} \omega \boldsymbol{\mathsf{I}} - \boldsymbol{\mathsf{L}}_{\bar{\boldsymbol{q}}}]^{{-}1} = \boldsymbol{\mathsf{Q}} \boldsymbol{\varSigma} \boldsymbol{\mathsf{F}}^*, \end{equation}

where $\boldsymbol{\mathsf{F}} = [\hat {\boldsymbol {f}}_1,\hat {\boldsymbol {f}}_2,\ldots,\hat {\boldsymbol {f}}_m]$ is an orthonormal matrix comprised of forcing modes, the diagonal matrix $\boldsymbol {\varSigma }=\text { diag}[\sigma _1,\sigma _2,\ldots,\sigma _m]$ holds the singular values (gain) in descending order and $\boldsymbol{\mathsf{Q}} = [\hat {\boldsymbol {q}}_1,\hat {\boldsymbol {q}}_2,\ldots,\hat {\boldsymbol {q}}_m]$ is an orthonormal matrix comprised of response modes (Trefethen et al. Reference Trefethen, Trefethen, Reddy and Driscoll1993; Jovanović & Bamieh Reference Jovanović and Bamieh2005).

The $\boldsymbol{\mathsf{H}}_{\bar {\boldsymbol {q}},\omega }$ operators are discretized over 3-D structured grids with the leading edge at the root positioned at $(x,y,z)/c = (0,0,0)$, extending over $(x, y, z)/c \in [-10,15] \times [-10,10] \times [0,10]$. The computational grids used for resolvent analysis are smaller in size than those used for DNS, we perform linear interpolation of the flow field between DNS and resolvent grids. Homogeneous Neumann boundary conditions are prescribed for $T'$ and homogeneous Dirichlet boundary conditions are set for the fluctuating variables $\rho '$ and $u'$ along the far field, airfoil surface and outlet. Far from the airfoil and in conjunction with the boundary conditions, sponges are applied (Freund Reference Freund1997).

The resolvent modes are computed using the randomized resolvent analysis algorithm (Ribeiro, Yeh & Taira Reference Ribeiro, Yeh and Taira2020b) and the direct and adjoint linear systems were directly solved using the MUMPS (multifrontal massively parallel sparse direct solver) package (Amestoy et al. Reference Amestoy, Duff, L'Excellent and Koster2001). The adjoint-based sensitivity analysis was used to interpolate the resolvent norm over frequencies $\omega$ (Schmid & Brandt Reference Schmid and Brandt2014; Fosas de Pando & Schmid Reference Fosas de Pando and Schmid2017). The codes used to compute the resolvent modes are part of the linear analysis package made available by Skene, Ribeiro & Taira (Reference Skene, Ribeiro and Taira2022).

4.1. Time-averaged characteristics

To achieve the control objectives in this study, we first comprehensively examine the wake patterns and dynamics of the baseline flows. Specifically, for the control goal of reducing the separation bubble size, focus is placed on investigating key features of the reversed-flow bubble 3-D topology through the time-averaged flow field. The reversed-flow bubble has a non-homogeneous spatial distribution, significantly influenced by both the wing planform geometry and the angle of attack, as illustrated in figure 4. For visualization, we present the full span wing mirrored with respect to the wing root. The time-averaged reversed flow is depicted with light blue and purple isosurfaces of $\overline {u_x} = 0$ and $-0.1$, respectively.

Figure 4. Time-averaged reversed flows over wings visualized with light blue and dark purple isosurfaces of $\overline {u_x} = 0$ and $-0.1$, respectively. Flow fields around wings with different $\varLambda$ and $\lambda$ combinations at (a–c) $\alpha = 14^\circ$ and (d–f) $22^\circ$ are presented. Wings are plotted with full span for visualization purposes. The time-averaged lift and lift-to-drag ratio, $\overline {C_L}$ and $\overline {C_L/C_D}$, respectively, are provided for each wing.

Additionally, we present the time-averaged aerodynamic forces, reported as non-dimensional drag and lift coefficients, ${C_D}$ and ${C_L}$, respectively, defined as

(4.1a,b)\begin{equation} C_D = \frac{F_x}{\dfrac{1}{2}U_\infty^2 b c} \quad \text{and} \quad C_L = \frac{F_y}{\dfrac{1}{2}U_\infty^2 b c}, \end{equation}

where $F_x$ and $F_y$ are the $x$ and $y$ force components, respectively. For low-Reynolds-number flows, the time-averaged lift-to-drag ratio ($\overline {C_L/C_D}$), used throughout this work to quantify the aerodynamic performance of the wing, typically remains around $O(1)$ and reaches its peak at high angles of attack (Taira & Colonius Reference Taira and Colonius2009b; Zhang et al. Reference Zhang, Hayostek, Amitay, He, Theofilis and Taira2020b). Lift increases within the depicted $\alpha$ range, as seen in the $\overline {C_L}$ values in figure 4(a–c) for $\alpha = 14^\circ$ and figure 4(d–f) for $\alpha = 22^\circ$. Notwithstanding, the lift-to-drag ratio decreases with $\alpha$, indicating loss of aerodynamic performance as the angle of attack increases. This decline in aerodynamic efficiency at $\alpha = 22^\circ$, compared with $\alpha = 14^\circ$, is linked to a post-stall flow condition. Consequently, a flow control implementation becomes essential to assist the wing in maintaining aerodynamic efficiency.

Wing sweep and taper also influence aerodynamic loads. Overall, wing sweep is associated with a reduction in lift for untapered wings (Zhang et al. Reference Zhang, Hayostek, Amitay, Burstev, Theofilis and Taira2020a). The combination of a high leading edge sweep and wing taper proves advantageous for the aerodynamic performance of the wing, especially for post-stall flows (Ribeiro et al. Reference Ribeiro, Neal, Burtsev, Amitay, Theofilis and Taira2023a). Achieving a high lift-to-drag ratio is possible for steady wakes at lower angles of attack. However, for enhanced aerodynamic performance, the role of flow unsteadiness must be considered (Zhang et al. Reference Zhang, Hayostek, Amitay, He, Theofilis and Taira2020b). To take advantage of the inherent unsteadiness, we consider resolvent-analysis-based actuation that can excite unsteady structures in the wake. For realizing effective flow modification, we first investigate the characteristics of the reversed-flow bubble around the wings shown in figure 4.

For laminar flows over wings at high incidence angles, the separation bubble becomes massive, beyond the size of the wing. As shown in figure 4, the size of the reversed-flow bubble is more pronounced for flows over wings at $\alpha = 22^\circ$ and the intensity of the reversed flow within the separation bubble increases with the angle of attack, as evident from the larger $\overline {u_x} = -0.1$ isosurfaces. Additionally, the non-homogeneity of the reversed flow is observed in the spanwise direction, with the bubble expanding and contracting over the wingspan. For example, for untapered unswept wings $(\lambda,\varLambda = 1,0^\circ )$, the bubble shrinks near the wing tip while expanding in the inboard region.

The wingspan location where the separation bubble reaches its maximum sectional area in the $(x,y)$ plane is a key aspect of the reversed-flow topology. For control, this location is important as a direct wake modification implemented over this region can significantly reduce the reversed-flow volume. For untapered unswept wings at lower $\alpha$, the reversed-flow bubble is marked by the contraction of the reversed-flow region near the midspan, as depicted in figure 4(a). Zhu et al. (Reference Zhu, Wang, Xu, Qu and Long2023) noted that similar reversed-flow bubble shapes resemble the tail of a swallow and termed this reversed-flow topology as a swallow-tailed structure. With an increase in the angle of attack, they noted that the bubble transforms into a single-tailed formation, where the tail refers to a portion of the reversed-flow bubble with larger extension downstream of the wing in the streamwise direction. A single-tailed reversed-flow structure is analogous to the structure presented in figure 4(d). Such wake formation is commonly observed for 3-D flows over untapered unswept low-aspect-ratio wings at high angles of attack (Chen et al. Reference Chen, Bai and Wang2016).

In contrast to the flows over untapered unswept wings, for swept wings $(\varLambda = 30^\circ )$, the flow is attached over the midspan. For untapered swept wings $(\lambda,\varLambda = 1,30^\circ )$, shown in figure 4(b,e), the larger portion of the reversed-flow bubble emerges over the outboard side of the wing, closer to the tip. The characteristics of the reversed-flow bubble are further influenced by wing taper, as shown in figure 4(c,f). For tapered swept wings $(\lambda,\varLambda = 0.27,30^\circ )$, the larger portion of the reversed-flow structure appears over the inboard section of the wing. To gain further insights into the wakes, we examine the vortex dynamics of the wakes in the following section.

4.2. Vortex dynamics

Let us continue to study the features of flows over low-aspect-ratio wings by examining their spectral characteristics and instantaneous wakes illustrated in figure 5. In figure 5(a,b) we present the power spectral density (PSD) of $C_L$, where frequencies are normalized as

(4.2)\begin{equation} St \equiv \frac{\omega}{2{\rm \pi}} \frac{c \sin \alpha}{ U_\infty \cos \varLambda}, \end{equation}

which is a modified Fage–Johansen Strouhal number (Fage & Johansen Reference Fage and Johansen1927) with a $1/\cos \varLambda$ scaling to account for the sweep angle. The lift spectra reveal peaks associated with vortex shedding. At $\alpha = 14^\circ$, the shedding structures are predominantly two dimensional near the wing root, resulting in smoother lift PSD curves when compared with those obtained at $\alpha = 22^\circ$, where shedding vortices are three dimensional. Despite the differences in wake shedding, lift peaks are found between $0.13 \le St \le 0.16$ for all cases shown. Within this frequency range, untapered unswept wings $(\lambda,\varLambda = 1,0^\circ )$ peak around the lower frequencies $St \approx 0.14$ while tapered swept wings $(\lambda,\varLambda = 0.27,30^\circ )$ peak around the higher portion of the frequency range $St \approx 0.16$. In figure 5(c,f) we examine the time-varying flows over low-aspect-ratio untapered unswept wings $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 14^\circ$ and $22^\circ$. Notably, the global unsteady characteristics of the flow share similarities for both angles of attack. Near the root, unsteadiness is predominantly marked by the emergence of vortices aligned in the spanwise direction. At the wing tip, a steady flow displays a tip vortex oriented along the streamwise direction. The streamwise length of this tip vortex is more pronounced for $\alpha = 22^\circ$.

Figure 5. (a,b) The PSD of $C_L$ and (c–h) instantaneous flow fields around tapered swept wings visualized with isosurfaces of $Q = 1$ coloured by the streamwise velocity $u_x$. Along the root plane, spanwise vorticity $\omega _z$ contours are shown. We present flow fields around wings with different $\varLambda$ and $\lambda$ combinations at (c–e) $\alpha = 14^\circ$ and (f–h) $22^\circ$.

Wing sweep plays a stabilizing role in attenuating wake oscillations around 3-D wings (Zhang et al. Reference Zhang, Hayostek, Amitay, Burstev, Theofilis and Taira2020a; Burtsev et al. Reference Burtsev, He, Hayostek, Zhang, Theofilis, Taira and Amitay2022), as shown in figure 5(d,g). At the lower $\alpha$, unsteadiness is entirely suppressed. However, at $\alpha =22^\circ$, the flow exhibits unsteadiness near the wing tip, while remaining steady near the wing root. For the higher $\alpha$, the unsteady flow structures develop farther away from the wing.

Figure 5(e,h) reveals that the wake patterns of tapered swept wings $(\lambda,\varLambda = 0.27,30^\circ )$ exhibit notable similarities with those around untapered unswept wings $(\lambda,\varLambda = 1,0^\circ )$. However, a distinctive feature is the significant shortening of the tip vortex in tapered swept wings. Despite the pronounced leading edge sweep angle, the wakes of tapered swept wings are unsteady. Vortex shedding structures feature spanwise rolls, akin to the flows around untapered unswept wings shown in figure 5(c,f). The shift in flow unsteadiness along the spanwise direction, particularly toward the quarter-span (located halfway between the wing root and tip), coincides with the larger expansion of the reversed-flow bubble size over the wingspan, as depicted in figure 4(e,h).

6.1. Implementation of resolvent-analysis-based actuation

Resolvent analysis finds the optimal perturbation that can be amplified in the flow field. However, the receptivity of low-Reynolds-number separated 3-D flows to external perturbations remains elusive. To address this issue, forcing modes obtained from resolvent analysis are introduced as a harmonic external body force $\boldsymbol {e}$ in (3.1a). The body force is used in the numerical simulation to model actuators employed in practical flow control applications. For instance, the control effects of dielectric barrier discharge plasma actuators have been studied numerically using body forces (Shyy, Jayaraman & Andersson Reference Shyy, Jayaraman and Andersson2002; Mullenix, Gaitonde & Visbal Reference Mullenix, Gaitonde and Visbal2013; Waindim & Gaitonde Reference Waindim and Gaitonde2016) and surface heat actuators (Yeh, Munday & Taira Reference Yeh, Munday and Taira2017; Prasad & Unnikrishnan Reference Prasad and Unnikrishnan2023). For this reason, even though this study is a fundamental analysis of the 3-D post-stall flow control, the spatio-temporal characteristics of the body forces presented here can be approximated in practical flow control applications.

The spatial distribution of the forcing modes and their frequency $\omega$ is used within the definition of the body force $\boldsymbol {e}$ expressed as

(6.1)\begin{equation} \boldsymbol{e}(\boldsymbol{x},t) = A [\text{Re}(\,\hat{\boldsymbol{f}}(\boldsymbol{x}))\sin(\omega t + \phi) + \text{Im}(\,\hat{\boldsymbol{f}}(\boldsymbol{x}))\cos(\omega t + \phi)], \end{equation}

where $\hat {\boldsymbol {f}}(\boldsymbol {x})$ is the spatial forcing mode, with real $\text {Re}({\cdot })$ and imaginary $\text {Im}({\cdot })$ parts, $A$ is the amplitude and $\phi$ is the phase. For all controlled flows considered herein, $\phi$ is set to initiate the body force actuation at the time of maximum $C_L$. The resolvent modes $\hat {\boldsymbol {f}}$ and $\hat {\boldsymbol {q}}$ are weakly compressible, being derived from a base flow characterized by a free-stream Mach number of $M_\infty = 0.1$ (Ribeiro & Taira Reference Ribeiro and Taira2023). Only velocity components of $\hat {\boldsymbol {f}}$ are used in (6.1).

Forcing modes are spatially global, extending over the entire computational domain $(\boldsymbol {x})$, with higher spatial support upstream of the wing, as depicted in figure 7(a). To provide fundamental insights into separation control, we narrow our focus to actuation over a specific subset of $\boldsymbol {x}$. We term this approach as spatially confined actuation, in which body forces are introduced within a volume of actuation ($V_{act}$) and its corresponding actuation surface ($S_{act}$), defined as the outer surface of the actuation volume. Our procedure to define $V_{act}$ will be discussed in the following section. It is noteworthy that global and localized actuation lead to distinct control outcomes, as depicted in figure 7(b,c).

Figure 7. Global and confined forcing-mode actuation for the flow control over a untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 14^\circ$. (a) Isosurfaces of $St = 0.14$ forcing $\hat {\boldsymbol {f}}_{uy} = \pm 0.5$ (bottom) and response $\hat {\boldsymbol {f}}_{uy} = \pm 0.2$ (top). (b) Plot of $C_L$ versus time for global and confined controlled flows at two actuation frequencies. (c) Global and confined controlled flows using $St = 0.14$ forcing mode, visualized with $Q = 0.3$, coloured by $u_x$. Arrows show how upstream perturbation emerges in the forcing mode and how it appears in the flow actuated using global forcing.

This study adopts a confined actuation approach for two primary reasons. First, global forcing modes emerge upstream far from the wing, making them unrealistic for practical aircraft actuation and flow control. Secondly, confined actuation enables us to discern the effects of control over specific regions of the wingspan. The spanwise support of forcing modes changes as the frequency of actuation varies (§ 5). Let us take the example of a global forcing actuation using forcing modes at $St = 0.14$ and $0.26$ using the same body force amplitude $A$. Notably, the forcing modes at $St = 0.14$ have stronger spatial support over the wing root, while those at $St = 0.26$ are more pronounced at the wing tip. Nevertheless, global actuation yields similar lift over time for both frequencies, as seen in figure 7(b). Additionally, global actuation with the $St = 0.14$ forcing mode alters the wake across the entire wingspan, perturbing the tip vortex upstream in the wake, as observed in figure 7(c). In contrast, the control effects of the confined actuation in the wake are concentrated in the root region.

6.2. Volume of actuation and momentum coefficient

The objective of control is to achieve notable flow modifications using a small forcing input. To quantify the magnitude of the control input, we consider the momentum coefficient $C_\mu$ defined as

(6.2)\begin{equation} C_\mu \equiv \frac{\rho\dfrac{S_{act}}{V_{act}}\displaystyle\int_{V_{act}} ({u_x^\prime}^2 + {u_y^\prime}^2 + {u_z^\prime}^2)\,\text{d}V}{\dfrac{1}{2} \rho U_\infty^2 bc}, \end{equation}

where ${u_x^\prime }$, ${u_y^\prime }$ and ${u_z^\prime }$ are the velocity components induced by the actuation body force $\boldsymbol {e}$ defined in $V_{act}$. The induced velocity is measured in a calibration simulation in which the free-stream flow is turned off. The determination of an actuation volume involves the analysis of the magnitude of $\hat {\boldsymbol {f}}$:

(6.3)\begin{equation} K_{\hat{\boldsymbol{f}}}(\boldsymbol{x},\omega) \equiv | \,\hat{\boldsymbol{f}}_{u_x}^*\hat{\boldsymbol{f}}_{u_x} + \hat{\boldsymbol{f}}_{u_y}^*\hat{\boldsymbol{f}}_{u_y} + \hat{\boldsymbol{f}}_{u_z}^*\hat{\boldsymbol{f}}_{u_z} |. \end{equation}

This scalar variable is defined across the spatial domain for each forcing mode and serves as a velocity-based metric depicting the spatial support of the forcing modes.

The actuation volume, denoted as $V_{act}$, is specified within the region where

(6.4)\begin{equation} K_{\hat{\boldsymbol{f}}}(\boldsymbol{x},\omega) \ge 0.5 \max_{\boldsymbol{x}}(K_{\hat{\boldsymbol{f}}}(\boldsymbol{x},\omega)). \end{equation}

This threshold is selected for two main reasons: (i) $V_{act}$ is limited to a region closer to the wing surface for all cases studied herein, and (ii) the region of spanwise support for the forcing modes can be clearly distinguished for modes at different frequencies (see figure 6). A yellow isosurface illustrates the definition of $V_{act}$, as seen in figure 8, along with red and blue isosurfaces representing the global modes with $\hat {\boldsymbol {f}}_{ux} / \| \hat {\boldsymbol {f}}_{ux} \|_\infty = \pm 0.1$.

Figure 8. Confined forcing and amplitude definition for untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 14^\circ$. (a) Actuation volume $V_{act}$ (yellow) and forcing modes visualized with isosurfaces of $\hat {\boldsymbol {f}}_{ux}/\| \hat {\boldsymbol {f}}_{ux} \|_\infty = \pm 0.1$ (blue-red). (b) Actuation in quiescent flow visualized with isosurfaces of $Q = 0.1$ coloured by $u_x$ for $St = 0.14$ forcing modes. (c) Lift coefficient $C_L$ over time for baseline (right axis) and actuated quiescent flows (left axis). (d) Amplitude of body force actuation $A$ for estimated momentum coefficients of $0.01 \le C_\mu \le 0.05$.

Following the definition of $V_{act}$, simulations are conducted under quiescent free-stream flow with actuation to estimate the momentum coefficient $C_\mu$. The velocity components of the forcing mode, denoted as $\hat {\boldsymbol {f}} = [\hat {\boldsymbol {f}}_{ux},\hat {\boldsymbol {f}}_{uy},\hat {\boldsymbol {f}}_{uz}]$, each having unit magnitude ($\| \hat {\boldsymbol {f}} \|_2 = 1$), are introduced to the momentum equation (3.1a), as illustrated in figure 8(b), with $Q = 0.1$. To enlarge the flow structures for visualization purposes, the chosen value of $Q$ in this context is $10$ times lower than that used throughout the paper. As shown in figure 8(c), lift fluctuation induced by actuation under quiescent free-stream flow conditions remains under approximately $1\,\%$ of the baseline $C_L$ for $C_\mu \le 0.05$. Through simulations of quiescent free-stream flows, we obtain $u_x^\prime$, $u_y^\prime$ and $u_z^\prime$ for (6.2) to be evaluated over $V_{act}$, revealing a linear relation between $C_\mu$ and the actuation amplitude $A$, as illustrated in figure 8(d). For $C_\mu = 0.05$, the value of $A$ is of order $O(1)$, exhibiting a magnitude similar to previous studies implementing body force actuation in comparable flow conditions (Edstrand et al. Reference Edstrand, Sun, Schmid, Taira and Cattafesta2018b).

In our analysis we maintain a constant $C_\mu = 0.05$. It is noteworthy that variations in $C_\mu$ may induce changes in the controlled wake, as illustrated in figure 9. This figure depicts the impact of different $C_\mu$ values on flow control over an untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 22^\circ$. Focusing on the controlled flows with a forcing mode at $St = 0.40$, situated near the wing tip, the effect of $C_\mu$ on the controlled tip vortex length is relatively minor, as demonstrated in figure 9(a). Notably, an approximately $60\,\%$ reduction in the tip vortex length is achieved for all considered $C_\mu$ values.

Figure 9. Effect of momentum coefficient $C_\mu$ on controlled flows over an untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 22^\circ$ using (a) $St = 0.40$ (wing tip) and (b) $St = 0.14$ (reversed flow) forcing modes. (a) Tip vortex attenuation visualized with time-averaged isosurfaces of $\bar {Q} = 1$ coloured in light grey for baseline, blue for $C_\mu = 0.01$, red for $C_\mu = 0.02$ and dark grey for $C_\mu = 0.05$. (b) Plots of ${C_L}$ and ${C_L/C_D}$ versus time for controlled flows with distinct input $C_\mu$.

When examining controlled flows with a $St = 0.14$ forcing mode located inboard over the wing, an increase in $C_\mu$ results in higher time-averaged and root-mean-squared (RMS) values for $C_L$ and $C_L/C_D$, as presented in figure 9(b). We note that a smaller $C_\mu$ could be used to reduce RMS levels and its impact on structural vibration of the wing. Given our objective to reduce the separation bubble size and enhance time-averaged lift and the lift-to-drag ratio, we specifically select $C_\mu = 0.05$ as it yields superior improvements in aerodynamic performance. Importantly, even at lower $C_\mu$ values, there is a notable increase in both $\overline {C_L}$ and $\overline {C_L/C_D}$, such as a $4.73\,\%$ increase in $\overline {C_L}$ for $C_\mu = 0.05$ compared with $C_\mu = 0.01$. Similarly, $C_{L_{RMS}}$ is $0.067$ for $C\mu = 0.01$, while it increases to $0.103$ for $C_\mu = 0.05$.

6.3. A priori assessment and guidance for flow control

With the forcing-mode-based actuation established, let us consider an a priori assessment of the control design and its impact on the wake. To this end, we quantify mixing introduced by the modal structures using the modal streamwise, transverse and spanwise Reynolds stresses (Luhar et al. Reference Luhar, Sharma and McKeon2014; Yeh & Taira Reference Yeh and Taira2019) respectively defined as

(6.5a–c)\begin{equation} \hat{R}_x(\boldsymbol{x},\omega) = \text{Re}(\hat{\boldsymbol{q}}_{u_y}^*\hat{\boldsymbol{q}}_{u_z}), \quad \hat{R}_y(\boldsymbol{x},\omega) = \text{Re}(\hat{\boldsymbol{q}}_{u_z}^*\hat{\boldsymbol{q}}_{u_x}), \quad \hat{R}_z(\boldsymbol{x},\omega) = \text{Re}(\hat{\boldsymbol{q}}_{u_x}^*\hat{\boldsymbol{q}}_{u_y}), \end{equation}

where $\text {Re}({\cdot })$ denotes the real part of the complex-valued variable. To assess the modal capability to perturb a flow structure, we employ the following mixing metric as

(6.6)\begin{equation} M(\omega) = \int_{V_{resp}} [\sigma_1^2 (\hat{R}_x(\boldsymbol{x},\omega) + \hat{R}_y(\boldsymbol{x},\omega) + \hat{R}_z(\boldsymbol{x},\omega)]^{1/2}\,\text{d}V, \end{equation}

where $V_{resp}$ denotes a specific region in space of the response modes over which the integral is computed. This region is chosen based on the flow structure targeted for modification through flow control.

For reducing the size of the reversed-flow bubble, let us define $V_{resp}$ where $\overline {u_x} \le 0$, as shown by the blue isosurfaces over both wings in figure 10. For the untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$, note that the peak of the blue curve in figure 10(a) suggests that the forcing modes at $St \approx 0.14$ have higher potential to induce momentum mixing within the separation bubble, consequently achieving the control objective of reducing the reversed-flow bubble size. Conversely, when analysing $M$ over the wing tip vortex, we define $V_{resp}$ to be the region for $\overline {\omega _x} \le -0.5$ and $z/c \ge 1.65$, as shown in figure 10(a). In this case, the control objective shifts to perturbing and mitigating the tip vortex (represented in red). The red curve peaks around $St \approx 0.26$, suggesting that forcing modes at this frequency can augment momentum mixing within the tip vortex, potentially weakening its core and reducing its length.

Figure 10. Assessment of control effects from response mode Reynolds stress metric $M$ over $St$ for an (a) untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 14^\circ$, (b) untapered swept wing $(\lambda,\varLambda = 1,30^\circ )$ at $\alpha = 14^\circ$ and a (c) tapered swept wing $(\lambda,\varLambda = 0.27,30^\circ )$ at $\alpha = 22^\circ$. Grey arrows indicate frequencies in which controlled flows yielded maximum contraction of reversed-flow (RF) bubble, maximum tip vortex (TV) attenuation and maximum aerodynamic performance.

Turning our attention to figure 10(b), we focus on the untapered swept wing $(\lambda,\varLambda = 1,30^\circ )$ scenario. Here, the peak of $M$ for the blue curve suggests that a $St \approx 0.10$ forcing mode is likely to yield a larger contraction of the reversed-flow bubble. For the untapered swept wing, utilizing the tip-dominant $St = 0.10$ forcing mode substantially modifies the flow around the tip. Nevertheless, this actuation may not impact the entire reversed-flow bubble, as the upstream portion of the separated flow is located near the wing root. To address this issue, we pursue higher-frequency forcing modes that emerge near the wing root. To identify the forcing modes that enhance momentum mixing and modify the reversed-flow bubble near the wing root (highlighted in yellow), we define $V_{resp}$ as the region where $|\overline {\omega _z}| \ge 5$ and $z/c \le 0.5$. This span subset confines the analysis to the root region while also encompassing the upstream portion of the reversed-flow bubble. The $|\overline {\omega _z}|$ threshold only affects the magnitude of $M$ and not its distribution over $St$. We note that earlier discussions about an untapered unswept wing have excluded the wing-root-based analysis because it closely aligns with the blue curve for the reversed-flow-based analysis shown in figure 10(a).

For the untapered swept wing, the distribution of the wing-root-based $M$ suggests the utilization of a root-dominant $St = 0.18$ forcing mode for actuation. For frequencies $St \ge 0.20$, both wing-root-based and reversed-flow-based $M$ decrease. A rapid decline in the reversed-flow-based $M$ implies that the actuation with $St \ge 0.20$ forcing modes increases momentum mixing near the root while minimally affecting the reversed-flow bubble. Distinguishing the effects of reversed-flow- and wing-root-based metrics is challenging for tapered swept wings $(\lambda,\varLambda = 0.27,30^\circ )$ because the separation bubble over this wing is centred at quarter-span, as depicted in figure 10(c). For a tapered swept wing at $\alpha = 22^\circ$, the reversed-flow-based $M$ peaks at $St = 0.14$, suggesting this mode for an actuation to reduce the reversed-flow bubble size. The wing-root-based metric exhibits two peaks: the first at $St = 0.14$ as the wing-root-based $V_{resp}$ encompasses part of the reversed-flow bubble. The second peak of the wing-root-based $M$ appears at $St = 0.20$ for a root-based actuation, while the reversed-flow-based metric $M$ remains high. This result suggests that actuation using $St = 0.20$ forcing mode can yield significant effects over the reversed-flow region.

It is important to note that analysing $M$ alone does not guarantee improvements in lift and lift-to-drag ratio for the controlled flows. Such enhancements need to be confirmed through experiments and numerical simulations, as demonstrated in the subsequent sections. Furthermore, we show how the behaviour of controlled flows concurs with predictions made by the Reynolds stresses using the response modes. For the remainder of this paper, we narrow our discussion to the three cases depicted in figure 10, as they present distinct and challenging flow features for control, including a variety of separation bubble spatial characteristics and the emergence of wing tip vortices.

6.4. Control of flow separation around untapered unswept wings

Let us study the wake modification achieved by the present resolvent-analysis-based active flow control approach for an untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 14^\circ$. For reference, the baseline flow is visualized using grey-coloured isosurfaces of $Q$, as shown in figure 11(a). Also shown over the wing is a blue isosurface representing the reversed-flow bubble, where $\overline {u_x} \le 0$. For this flow, we observe spanwise-aligned vortex shedding over the majority of the wing and a tip vortex near the free end.

Figure 11. Assessment of flow modification using forcing-mode actuation for an untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 14^\circ$. (a) Baseline flow. (b) Percentage of reversed-flow (RF) volume contraction, tip vortex (TV) length reduction and aerodynamic forces modification for controlled flows compared with baseline. (c–e) On the left, isosurfaces of forcing $\hat {\boldsymbol {f}}_{uy} = \pm 1$ and response modes $\hat {\boldsymbol {q}}_{uy} = \pm 0.5$ at (c) $St = 0.06$, (d) $St = 0.14$ and (e) $St = 0.28$. On the right, controlled flows. All flow fields are visualized with grey-coloured isosurfaces of $Q= 1$ and blue-coloured isosurfaces of $\overline {u_x} = 0$.

The DNS of controlled flows shows substantial differences compared with baseline flows. A quantitative assessment of the control effect is shown in figure 11(b). Here, the percentage difference ($\varDelta$) is used to illustrate variations in flow features between the baseline and controlled flows. In particular, let us analyse the blue curve that represents the contraction (negative) and expansion (positive) effect of flow control on the reversed-flow bubble, which is one of the objectives of the present study. The minimum reversed-flow bubble volume is achieved when the forcing mode at $St = 0.14$ is used for actuation. At this frequency, the reversed-flow bubble undergoes a $50\,\%$ volume reduction, as seen in figure 11(d).

The notable reduction in the reversed-flow bubble size at $St = 0.14$ leverages the forcing-response mode pair at this frequency. In particular, the response mode emerges over the baseline separation bubble, indicating that the forcing-mode actuation excites structures within the reversed-flow bubble. Indeed, along the spanwise-aligned root vortices, the controlled flow using forcing modes at $St = 0.14$ induces spatial oscillations in the wake that are absent in the baseline flow, as seen in figure 11(d). These structures perturb the baseline reversed-flow bubble and result in its volumetric contraction.

Concurrently, the controlled flow with $St = 0.14$ forcing-mode actuation exhibits the highest increase in lift and lift-to-drag ratio, shown by the percentage difference in $\overline {C_L}$ and $\overline {C_L/C_D}$ between the baseline and controlled flows (grey curves). We note that improvements in the aerodynamic performance are achieved because the amount of lift increase is substantially greater than that of the drag. The concomitant $\overline {C_L/C_D}$ increase and reversed-flow bubble contraction shown in figure 11(b) indicate that diminishing the reversed-flow size enhances the overall aerodynamic performance of the wing. As the actuation frequency increases, the higher spatial support of the forcing-mode pairs shifts toward the wing tip and the modes emerge outside the reversed-flow region. As a result, the effect of contracting the reversed-flow bubble is minimized for high-frequency actuation, reaching a plateau for $St \ge 0.26$.

Forcing-mode actuation for $St \ge 0.26$ is also important for flows over untapered unswept wings $(\lambda,\varLambda = 1,0^\circ )$ due to their ability to control the tip vortex while moderately increasing lift and lift-to-drag ratio, as seen in figure 11(b). The attenuation and control of tip vortices has been extensively studied (Gursul et al. Reference Gursul, Vardaki, Margaris and Wang2007; Greenblatt Reference Greenblatt2012; Gursul & Wang Reference Gursul and Wang2018) due to its adverse effects on wing aerodynamics (Francis & Kennedy Reference Francis and Kennedy1979; Katz & Bueno Galdo Reference Katz and Bueno Galdo1989; Green & Acosta Reference Green and Acosta1991; Devenport et al. Reference Devenport, Rife, Liapis and Follin1996) and its negative impact in air transportation (Spalart Reference Spalart1998). In prior studies involving laminar steady flow around an untapered unswept wing, a $21\,\%$ reduction was achieved using instability analysis modes as body forces (Edstrand et al. Reference Edstrand, Schmid, Taira and Cattafesta III2018a,Reference Edstrand, Sun, Schmid, Taira and Cattafestab). For this approach, a vortex instability was taken advantage of to increase dissipation of the tip vortex.

While the $St = 0.14$ forcing mode proves highly effective in diminishing the volume of the separation bubble, its impact on the reduction of the tip vortex length is relatively modest, as seen in the red curve of figure 11(b). Here, the control effect on the tip vortex is measured through the percentage difference between the baseline and the controlled tip vortex lengths. For figure 11(b), the tip vortex length is estimated using the distance in the streamwise direction between the trailing edge and the tip of the $\overline {\omega _x} = -1$ isosurface. While the chosen value of the $\overline {\omega _x}$ isosurface is arbitrary, it does not affect the findings.

A decrease in the tip vortex length is achieved for the control using $St \le 0.10$ forcing modes, reaching a local minimum at $St = 0.06$. The forcing-response mode pair at $St = 0.06$ and the corresponding controlled flow are depicted in figure 11(c). This actuation induces prominent streamwise vortices over the inboard region, perturbing the reversed-flow bubble and diminishing its volume. These structures also interact with the downstream portion of the tip vortex, resulting in an overall reduction in its length.

To achieve a more substantial weakening of the tip vortex, forcing modes at higher frequencies are employed for direct wake modification. The controlled flows with $St \ge 0.26$ forcing modes achieve a higher level of reduction in the tip vortex length, while preserving the baseline reversed-flow bubble, as depicted in figure 11(e) for the controlled flow with the $St = 0.28$ forcing mode near the wing tip, structures excited by the forcing mode emerge predominantly from the pressure side of the wing tip and induce a helical vortical structure formation that is associated with a significant reduction of the tip vortex length. Moreover, the weakening of the tip vortex alleviates the inboard downwash, resulting in a $6\,\%$ overall lift increase.

Figure 12(a) shows the sectional lift ($C_l$) over the wingspan for baseline and controlled flows at different actuation frequencies. The $z/c$ region where $C_l$ increases, compared with the baseline flow, coincides with the spanwise region where the forcing-response mode pairs have higher spatial support. For instance, for the $St = 0.10$ forcing-mode actuation, the most pronounced increase in $C_l$ is observed between $0.5 \le z/c \le 1.2$, contributing to a $16\,\%$ overall lift increase. Similarly, for the controlled flow with $St = 0.14$ forcing mode, a larger rise in $C_l$ is evident over a broader portion of the wingspan between $0 \le z/c \le 1.5$, resulting in $31\,\%$ total lift increase. For the controlled flow with $St = 0.18$ forcing mode, the greatest increase in lift occurs over a narrower wingspan region compared with the control with $St = 0.14$. For the control with $St = 0.18$ forcing mode, lift increases especially closer to the wing tip between $0.7 \le z/c \le 1.7$, resulting in a $22\,\%$ increase in the overall lift.

Figure 12. (a) Sectional lift $C_l$ distribution over span for an untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 14^\circ$. (b) Time-averaged skin friction lines over the suction side of the wing.

The $C_l$ increase results from vortical lift from near-wake structures excited by the actuation. The signature of these vortical structures advecting over the wing marks the skin friction field over their suction side, as seen in figure 12(b), showing that separation is not suppressed. Using force element analysis (details in Appendix A), we reveal how such near-wake structures contribute to lift in figure 13. Drag elements distribution, qualitatively similar to lift elements, are not shown for brevity. Lift elements show that the major lift contribution (positive, red) comes from the flow structures located outside of the reversed-flow region over a significant portion of the wingspan.

Figure 13. Lift elements around an untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 14^\circ$, (a) baseline, (b) controlled flows with $St = 0.14$ and (c) $St = 0.18$ forcing modes. Leftmost figures show 3-D time-averaged lift elements with isosurfaces of $\overline {(\boldsymbol {\omega } \times \boldsymbol {u}) \boldsymbol {\cdot } \boldsymbol {\nabla } \phi _y} = \pm 1$. To the right of the 3-D view, slices at $z/c = 0.5$, $0.8$ and $1.2$ show the red and blue contours of lift elements and a black line contour of $\overline {u_x} = 0$.

For force element analysis, the volume force elements are identified by the dot product of the Lamb vector ($\boldsymbol {\omega } \times \boldsymbol {u}$) and the gradient of the auxiliary potential ($\boldsymbol {\nabla } \phi _i$). The auxiliary potential field decays rapidly in magnitude far from the wing surface. The $\boldsymbol {\nabla } \phi _i$ characteristics are only affected by the wing planform, being consistent for baseline and controlled flows. For this reason, a strategic wake modification that reduces the reversed-flow bubble size has the potential to bring vortices closer to the wing where $\boldsymbol {\nabla } \phi _i$ has a higher magnitude. Consequently, flow structures emerging in controlled flows near the wing have higher contribution to the overall vortical lift. It is important to note that the higher magnitude of $\boldsymbol {\nabla } \phi _i$ consistently appears near the wing surface for all wing planforms and local lift enhancements result from the strengthening of lift elements caused by the flow field. For instance, for flows over untapered and tapered swept wings, Zhang & Taira (Reference Zhang and Taira2022) and Ribeiro et al. (Reference Ribeiro, Neal, Burtsev, Amitay, Theofilis and Taira2023a) have shown that major increases in root contribution to the overall lift results from the emergence of root-based vortices.

As shown in figure 13(a,b) for the baseline and controlled flow with the $St = 0.14$ forcing mode, the reversed-flow bubble size significantly decreases at $z/c = 0.5$ and $0.8$. This causes lift elements to emerge nearer to the wing compared with the baseline. Consequently, the highest lift increase occurs near the inboard quarter-span of the wing, as depicted by the red curve in figure 12(a). For the controlled flow with $St = 0.18$, the reversed-flow bubble reduces significantly over the outboard portion of the wingspan at $z/c = 1.2$ when compared with the baseline case, as depicted in figure 13(a,c). This reversed-flow structure contraction at $z/c = 1.2$ causes lift elements to emerge over this region and closer to the wing. As a result, this flow modification leads to a substantial lift increase over the outboard quarter-span of the wing, as seen in the yellow curve in figure 12(a). The control effects resulting from actuation at the shedding frequency over an untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 14^\circ$ are analogous to those observed at $\alpha = 22^\circ$ (discussions are omitted for brevity).

6.5. Control of flow separation around untapered swept wings

For separated flows over untapered swept wings $(\lambda,\varLambda = 1,30^\circ )$, our objective is to directly modify the wake and control the size of the reversed-flow bubble. Unlike the findings presented in § 6.4, achieving the maximum reduction in the volume of the reversed-flow bubble around swept wings may not necessarily lead to optimal lift and lift-to-drag ratio enhancements. To illustrate the control effects, let us examine the laminar separated flow around a swept wing with $\varLambda = 30^\circ$ at $\alpha = 14^\circ$. The steady baseline flow is depicted in the insert visualization of figure 14(a). The reversed-flow region is highlighted with a blue isosurface at $\overline {u_x} = 0$. The application of resolvent-analysis-based actuation results in unsteady wakes that modify the 3-D dynamics about the reversed-flow bubble, contracting its volume.

Figure 14. Assessment of flow modification using forcing-mode actuation for an untapered swept wing $(\lambda,\varLambda = 1,30^\circ )$. Control with forcing modes located over the reversed-flow (RF) bubble and wing root (WR) are covered in detail. (a) Reversed-flow volume contraction and aerodynamic changes for controlled flows. Baseline flow on the bottom-right corner. (b,c) On top, isosurfaces of forcing $\hat {\boldsymbol {f}}_{uy} = \pm 1$ and response modes $\hat {\boldsymbol {q}}_{uy} = \pm 0.5$ at (b) $St = 0.10$ and (c) $St = 0.18$. On the bottom, instantaneous controlled flows over half-span. All flows are visualized with grey-coloured isosurfaces of $Q= 1$ and blue-coloured isosurfaces of $\overline {u_x} = 0$. (d) Plot of $C_l$ versus wingspan, (e) lift PSD and (f) probed $u_y$ for controlled flows.

The baseline reversed-flow bubble is larger near the tip. Consequently, to achieve the most significant reduction in the reversed-flow bubble volume, direct wake modification is applied near the wing tip. In this regard, the flow field is actuated using a forcing mode in the frequency range of $0.10 \le St \le 0.14$, whose spatial support is predominantly concentrated near the wing tip. As shown in figure 14(b), the controlled flow at these frequencies induces a local shedding close to the tip, exciting structures that perturb and modify the reversed-flow bubble. Given the actuation near the wing tip, the inboard flow near the wing root remains undisturbed.

The reduction of the reversed-flow bubble size, through actuation with $0.10 \le St \le 0.14$ forcing mode, results in a significant improvement in lift and lift-to-drag ratio. The most pronounced aerodynamic enhancements in $\overline {C_L}$ and $\overline {C_L/C_D}$ are achieved when promoting structures near the wing root, where the baseline flow is attached to the wing surface. To actuate in this region, we employ the forcing mode at $St = 0.18$, which exhibits larger spatial support near the wing root, as depicted in figure 14(c). This actuation substantially increases the root contribution to the overall lift, as seen in figure 14(d). As discussed for controlled flows over the untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ in § 6.4, the strengthening of near-surface vortices over a section of the wingspan yields sectional lift enhancements. The coherent structures promoted by the actuation mechanism produce the $C_l$ increase near the tip for the controlled flow with the tip-based forcing mode at $St = 0.10$. In contrast, the actuation with the root-based forcing mode at $St = 0.18$ produces a higher increase in $C_l$ near the root.

The lift and lift-to-drag ratio enhancements achieved through control using the $St = 0.18$ forcing mode surpass those obtained with other actuation frequencies. In fact, controlled flows with actuation frequencies $St \ge 0.20$ experience a significant decline in lift and lift-to-drag ratio improvements, despite utilizing a spatial support over the wing root similar to that of the $St=0.18$ forcing mode. Moreover, as depicted in the blue curve of figure 14(a), controlled flows using $St \ge 0.24$ forcing modes increase the size of the reversed-flow bubble. As illustrated in figure 15(a–c), which compares controlled flows using $St=0.12$, $0.18$ and $0.24$, only actuation using the $St = 0.18$ forcing mode induces a global wake modification that alters flow dynamics across the entire wingspan. Both lower- and higher-frequency forcing modes primarily alter flow in specific regions where actuation occurs, near the wing tip for $St=0.12$ and near the root for $St=0.24$. Controlled flows with the $St \ge 0.24$ forcing modes yield minor changes on the reversed flow because their spatial support is concentrated near the wing root region and do not merge into the separated flow bubble. The aerodynamic modifications achieved with $St \ge 0.24$ are not related to any changes in the reversed-flow bubble but are mainly due to the emergence of near-surface vortices excited by the actuation mechanism. As shown in figure 15(d), for the controlled flow with $St =0.18$, the global wake modification is initiated by the perturbation, arising near the wing root and rapidly evolving downstream into hairpin-like vortices that reshape the overall wake dynamics. Consequently, the controlled flow using the $St=0.18$ forcing mode benefits from a local lift increase around the root-based actuation and a subsequent reduction in the reversed-flow bubble, leading to significant aerodynamic improvements.

Figure 15. Instantaneous visualizations of flow modification for controlled flows with (a) $St = 0.12$, (b) $St = 0.18$ and (c) $St = 0.24$ forcing-mode actuation for an untapered swept wing with $\varLambda = 30^\circ$ at $\alpha = 14^\circ$. Inserts of percentual difference ($\varDelta$) of time-averaged reversed-flow (RF) volume, $\overline {C_L}$ and $\overline {C_L/C_D}$ with respect to baseline flow. Panel (d) shows the initial evolution of the unsteady controlled flow at $St = 0.18$ for $1.8 \le t \le 4.5$. Flow structures are visualized with isosurfaces of $Q = 2$ coloured with ${\omega _z}$.

The periodic controlled flows synchronize with the actuation frequency, as shown by the PSD of $C_L$ in figure 14(e). Here, the lift spectra peaks at the input frequencies and its harmonics for both $St=0.10$ (blue) and $St = 0.18$ (yellow) controlled flows. The actuation frequency affects the size of flow structures, as the higher-frequency input yields elongated spanwise-aligned vortices, as shown by the probed $u_y$ along $(x,y)/c = (3,-0.5)$ over time in figure 14(f). Additionally, the higher-frequency actuation yields a lower lift-to-drag RMS. While $(C_L/C_D)_{RMS} = 0.148$ for the controlled flow at $St = 0.10$, the root-based controlled flow at $St = 0.18$ results in $(C_L/C_D)_{RMS} = 0.079$. For brevity, only the control effects for the untapered swept wing $(\lambda,\varLambda = 1,30^\circ )$ at $\alpha = 14^\circ$ are presented here. We note that the control effects are analogous for both angles of attack. At $\alpha = 22^\circ$, the control using a lower-frequency mode near the wing tip is also less effective in improving lift and aerodynamic performance than a higher-frequency forcing mode located near the wing root (see figure 1).

6.6. Control of flow separation around tapered swept wings

For post-stall flows over tapered swept wings, our aim is to control the size of the reversed-flow bubble. Flow control effects over tapered swept wings are consistent with those observed for untapered swept wings, as discussed in § 6.5. For a tapered swept wing $(\lambda,\varLambda = 0.27,30^\circ )$ at $\alpha = 22^\circ$, the actuation frequency that maximally reduces the reversed-flow bubble size does not coincide with the frequency leading to the highest aerodynamic improvements in $\overline {C_L}$ and $\overline {C_L/C_D}$.

The baseline flow exhibits high levels of wake fluctuations across a significant portion of the wingspan, as shown in figure 16(a). The spatial support of the resolvent modes employed for flow control is also distributed over the wingspan. Forcing-response mode pairs are primarily concentrated over the quarter-span ($St \le 0.18$), the wing root ($0.18 < St < 0.28$) and the wing tip ($St \ge 0.28$). Actuation using forcing modes at frequencies $St \ge 0.28$ introduces flow perturbations near the tip region but proves ineffective in modifying the wake or enhancing the overall aerodynamic performance. This is attributed to the absence of a noticeable tip vortex structure in the baseline flow.

Figure 16. Assessment of flow modification using forcing-mode actuation for a tapered swept wing $(\lambda,\varLambda = 0.27,30^\circ )$ at $\alpha = 22^\circ$. (a) Baseline flow. (b,c) On top, isosurfaces of forcing $\hat {\boldsymbol {f}}_{uy} = \pm 1$ and response modes $\hat {\boldsymbol {q}}_{uy} = \pm 0.5$ at (b) $St = 0.10$ and (c) $St = 0.18$. On the bottom, instantaneous controlled flows over half-span. All flows are visualized with grey-coloured isosurfaces of $Q= 1$ and blue-coloured isosurfaces of $\overline {u_x} = 0$. Improvements in $\overline {C_L}$ and $\overline {C_L/C_D}$ are shown. (d) Plot of $C_l$ versus wingspan. (e) Temporal behaviour of probed $u_y$ for controlled flows.

For the tapered swept wing $(\lambda,\varLambda = 0.27,30^\circ )$, based on the characteristics of the modal structures and insights gained from flow control discussions in §§ 6.4 and 6.5, we assess the control effects at only two frequencies, $St = 0.14$ and $0.20$. We recall that the largest $(x,y)$ sectional area of the baseline separation bubble for this wing is located at the quarter-span (see § 4.1). For the first actuation frequency using the $St = 0.14$ forcing mode, whose spatial support is present near the quarter-span and within the reversed-flow bubble, perturbations are excited within the reversed-flow bubble, modifying the near wake and reducing the reversed-flow volume, as seen in figure 16(b).

The second frequency of actuation involves wake modification near the wing root region, aiming to assess whether a more significant enhancement in $\overline {C_L}$ and $\overline {C_L/C_D}$ can be achieved by perturbing this area, akin to the effects observed for swept wings. The forcing mode at $St = 0.20$ exhibits higher spatial support over the root region, as seen in figure 16(c). The DNS of the controlled flows at this frequency promotes structures near the wing root, leading to an increased root contribution to the overall lift, as depicted in figure 16(d). Additionally, this actuation results in a greater increase in lift and lift-to-drag ratio compared with the lower-frequency actuation at $St = 0.14$.

The differences in the wake patterns between baseline and controlled flows are not readily apparent from the 3-D depiction of coherent structures in figure 16(b,c). Additionally, controlled flows with $St = 0.14$ and $0.20$ forcing modes achieve approximately $38\,\%$ of contraction of the reversed-flow volume. To delve into the distinctions between baseline and controlled flows, we examine the probed $u_y$ at $(x,y)/c = (3,-0.5)$ over time and frequency in figure 16(e). While omitted here for brevity, we note that the baseline and actuated wakes using the forcing mode at $St = 0.14$ exhibit minimal differences in both temporal and spectral analyses of the probed data.

Notable differences emerge when comparing the probed data of controlled flows at $St = 0.14$ and $0.20$ as the peak frequencies of the controlled flow with the $St = 0.20$ forcing mode shift to a higher $St$ range. Wake vortices exhibit a spanwise split into two shedding modes: near-root vortices synchronized with the actuation frequency and quarter-span vortices with a lower frequency unsynchronized with the actuation frequency. This behaviour induces a low-frequency beating in the lift spectrum. This phenomenon, known as vortex dislocation, has been observed in the wakes of high-aspect-ratio blunt bodies and controlled flows (Eisenlohr & Eckelmann Reference Eisenlohr and Eckelmann1989; Williamson Reference Williamson1989; Zhang et al. Reference Zhang, Hayostek, Amitay, He, Theofilis and Taira2020b). Despite the lower aspect ratio in the present controlled flow, a similar wake pattern emerges due to the interplay between near-tip and near-root shedding structures. Lastly, we note that both $St = 0.14$ and $St = 0.20$ forcing-mode actuation strategies increase lift-to-drag RMS, which is originally $(C_L/C_D)_{RMS} = 0.017$ for the baseline flow. For the controlled flow with the $St = 0.14$ forcing mode, $(C_L/C_D)_{RMS}$ increases to $0.098$, while the root-based actuation at $St = 0.20$ yields a lower $(C_L/C_D)_{RMS} = 0.059$. Again, we note that the control effects for the tapered swept wing at $\alpha = 14^\circ$ and $22^\circ$ are analogous and we have omitted the results for the lower incidence angle for brevity.

6.7. Control of tip vortex around untapered unswept wings

To understand the mechanisms that lead to the tip vortex attenuation, we focus our discussion on the flow over an untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 14^\circ$. We compute the streamwise circulation $\varGamma _x = \int _{C} = \boldsymbol {u} \boldsymbol {\cdot } \text {d}\boldsymbol {l}$, where $C$ is the isocontour of $\overline {\omega _x} = -1$ on the $(y,z)$ plane at varied $x$. To compute $\varGamma _x$, only the isocontours of $\overline {\omega _x} = -1$ aligned at the wing tip at $z/c \approx 2$ are considered. In this manner, we limit the inboard vorticity influence on $\varGamma _x$, caused by the interplay of the tip vortex and the root shedding structures. Our discussions focus on wake modifications relative to the baseline and are not significantly influenced by the specific value of the $\overline {\omega _x}$ isocontour used to compute the circulation.

Let us analyse the specifics of the $St = 0.06$ forcing-mode actuation. This actuation results in $8\,\%$ reduction in tip vortex length, as shown in figure 17(c), compared with the baseline length shown in figure 17(b). As seen in the vorticity contours at $x/c = 3$ in figure 17(e), this low-frequency actuation has minimal impact on the core of the tip vortex near the wing because its perturbations arise over the inboard region of the flow, increasing unsteadiness locally. This inboard wake interacts with the tip vortex downstream increasing its dissipation, as shown by the higher decay rate of circulation $\mathrm {d}|\varGamma _x|/\mathrm {d}\kern0.06em x$, resulting in an overall reduction in the tip vortex length.

Figure 17. Tip vortex attenuation for optimal forcing-mode actuation at specific frequencies for an untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 14^\circ$. Throughout the figure, black is used for the baseline case, blue is used for $St = 0.06$, red is used for $St = 0.26$ and yellow for $St = 0.32$ forcing-mode actuation. (a) Tip vortex streamwise circulation $|\varGamma _x|$ over $x/c$. (b–d) Side view of the time-averaged flow structures using isosurfaces of $\bar {Q} = 1$. (e) Streamwise vorticity contours shown in 2-D slices ($(y,z)$ plane) at $x/c = 3$.

When employing tip-based actuation at $St \ge 0.22$, the resulting flows exhibit a substantial attenuation of the tip vortex. The controlled flow with the $St = 0.26$ forcing mode achieves a remarkable reduction in tip vortex length, reaching approximately $42\,\%$, as depicted in figure 17(d). Examining the vorticity contours at $x/c=3$ in figure 17(e), we reveal that the actuation associated with the $St=0.26$ forcing mode effectively modifies the vortex core near the wing. This modification is predominantly observed in the lower portion of the vortex core between $-0.5 \le y/c \le -0.25$, as a consequence of this forcing-mode actuation being localized on the pressure side of the wing.

For controlled flows with $St = 0.32$, circulation near the wing is reduced, as illustrated by the vorticity contours at $x/c=3$. However, this forcing-mode actuation is suboptimal for reducing the tip vortex length downstream in the wake. The structures excited by the $St = 0.32$ actuation remain coherent downstream, resulting in a lower circulation decay rate. It is noteworthy that the resolvent-analysis-based control achieves substantial tip vortex attenuation for $St \ge 0.26$ while also maintaining the increases in $\overline {C_L}$, $\overline {C_D}$ and $\overline {C_L/C_D}$ within $6\,\%$ difference from the baseline flow, as demonstrated in table 1.

Table 1. Effect of tip vortex control using $St \ge 0.26$ forcing modes on time-averaged and RMS of $C_L$, $C_D$ and $C_L/C_D$.

Similar results were obtained when investigating tip vortex attenuation for an untapered unswept wing $(\lambda,\varLambda = 1,0^\circ )$ at $\alpha = 22^\circ$. For this case, the optimal actuation for tip vortex attenuation is identified for the tip-dominant forcing mode at $St = 0.40$, leading to a sizeable reduction in the tip chord length of approximately $60\,\%$. In the Reynolds number regime considered herein, the analysis of tip vortex attenuation is conducted for untapered unswept wings, given that tapered swept wings $(\lambda,\varLambda = 0.27,30^\circ )$ lack a distinct streamwise vortex core at the wing tip, as shown in figure 5.