2.1. Test geometries

In this study, a symmetric version of the SD7003 airfoil is chosen for the wing geometry so as to eliminate the effects of camber and isolate the effects of the flow disturbances on the aerodynamics of the airfoil. The symmetric airfoil is derived from the SD7003 airfoil by a modification of the airfoil profile to a zero-camber configuration while preserving the airfoil thickness distribution along the chord. This approach maintains the leading-edge curvature of the airfoil, thus nominally maintaining the Reynolds-number-specific critical-LESP value (and thus the LEV-shedding characteristics, as explained later in § 5.1) previously determined by Ramesh et al. (Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014) for the SD7003 airfoil. The wing model used in the experiments is fabricated from plastic (VeroWhite) through additive manufacturing and is reinforced with a spanwise carbon-fibre rod insert. The wing chord length of the model is $c=10.16$ cm.

The cylinder has a diameter, $d$, of 2.54 cm and a length, L, such that the aspect ratio, $L/d$, is 18. The axis of the cylinder is oriented normal to the tunnel walls, parallel to the wing leading edge. Both the wing and the cylinder span the width of the test section to produce a nominally two-dimensional flow field. Experiments were conducted at a free-stream speed of $U_{\infty }=29.6\ \textrm {cm}\ \textrm {s}^{-1}$, corresponding to a diameter-based Reynolds number of $Re_d=7500$ and a chord-based Reynolds number of $Re_c=3 \times 10^{4}$. The wing has a minor gap of ${\sim }0.5$ mm existing at the wing–wall interface at both ends of its span. The cylinder is supported at the tunnel walls via low-profile vertical plate inserts.

The cylinder generates a periodic disturbance to be encountered by the downstream wing in the form of a von Kármán vortex street with a natural shedding frequency, $f$, corresponding to a Strouhal number of $St=fd/U_{\infty }=0.2$. With a chord-to-diameter ratio of $c/d=4$, the resulting street of vortices is comparable in size to LEVs of the wing in dynamic stall. The wing is positioned downstream of the cylinder at a distance of $x/d=8$, measured from the cylinder centre to the wing leading edge. This spacing is selected to ensure that the presence of the wing does not alter the natural shedding dynamics of the cylinder. The set-up allows for various vertical positions of the wing relative to the cylinder centreline. In this paper, we focus on the cases where the airfoil performs pitching motions about a pivot point that coincides with the centreline of $y/d=0$.

2.2. Flow visualization

Two-dimensional time-resolved PIV measurements were performed using the experimental set-up illustrated in figure 2. The water tunnel was seeded with polyamide particles of $60\ \mathrm {\mu }\mathrm {m}$-diameter (LaVision, $\textrm {SG}=1.03$) and illumination was provided by an Nd:YLF laser sheet (Photonics Industries DM50-527, $50\ \textrm {mJ}\ \textrm {pulse}^{-1}$, 10 kHz max) oriented in the streamwise direction and positioned at the three-quarter span of the wing. Images were recorded by high-speed camera (PCO Dimax S4, 4MP, 1279 f.p.s. at max resolution). The velocity vectors were calculated using Fluere version 1.3. Two passes of interrogation areas, with initial and final sizes of $64\ \textrm {px}^{2}$ and $32\ \textrm {px}^{2}$, respectively, were used to determine the particle displacements from single-frame images sampled at 350 Hz. An interrogation area overlap of 50 % was used. The resultant vector fields contained $123 \times 123$ vectors with a resolution of $(x/d, y/d)=(0.074, 0.074)$. Instantaneous vector fields were phase averaged using snapshot proper orthogonal decomposition (POD) (Perrin et al. Reference Perrin, Braza, Cid, Cazin, Barthet, Sevrain, Mockett and Thiele2007) for the cylinder-only cases. For the studies of the airfoil in the cylinder wake, the PIV data were ensemble averaged over consecutive runs of the experiment. A programmed trigger was used to ensure that all the runs were synchronized.

Figure 2.

Experimental set-up: direct force measurement and flow visualization using time-resolved PIV.

2.3. Control of the disturbance phasing

The focus of this work is the effect of the upstream disturbances generated by the cylinder on the dynamic wing positioned downstream. Critical to this effort is the capability to control the phase of the proximal vortical disturbances relative to the wing's kinematic commencement. Previous works (Okajima, Takata & Asanuma Reference Okajima, Takata and Asanuma1975; Cheng, Liu & Lam Reference Cheng, Liu and Lam2001; Rockwood & Medina Reference Rockwood and Medina2020) have shown that phase control of the periodic wake can be achieved by rotational oscillation of the cylinder about its axis. To this end, a timing belt was integrated flush into one of the vertical plates to provide rotary motion to the cylinder about its respective axial coordinate via a stepper motor, as illustrated in figure 2. The cylinder is driven in a sinusoidal angular profile of rotational amplitude $45^{\circ}$. The angular frequency, $\omega$, corresponds to the natural shedding frequency of the cylinder and thus to a Strouhal number of $St=0.2$.

The commanded cylinder rotation, $\psi _c$, is given by

(2.1)\begin{equation} \psi_c(t) = \psi_{c.max} \sin(\omega t + \varTheta), \end{equation}

in which the cylinder-oscillation phase, $\varTheta$, is set to achieve a desired relative position of the oncoming vortices with respect to the wing leading edge at a particular instant in time. The result is a commanded phase, $\phi$, of the vortical disturbance with respect to the wing's position and kinematic timing. The phasing convention, illustrated in figure 3, corresponds to the relative position of the vortical disturbances with respect to the wing leading edge.

Figure 3.

(a) Definition of the phase $\phi$ based on the position of the oncoming vortical disturbances relative to the airfoil. (b) Demonstration of the control of $\phi$. Lift-coefficient histories of a static airfoil at zero incidence for different values of $\phi$ exhibiting a phase shift equal to $\phi$.

For a static wing, $\phi =0^{\circ }$ is defined as the case when a clockwise vortex is coincident with the streamwise location of the leading edge at a reference point in time. The case where a counter-clockwise vortex is encountered by the leading edge at the same time corresponds to $\phi =180^{\circ }$. Figure 3 also demonstrates the effective phase control enabled by active shedding of the rotating cylinder for a static airfoil placed in the flow centreline at $y/d=0$ at zero incidence. For this configuration, the von Kármán street emanating from the cylinder is bisected by the wing. Lift histories for various values of $\phi$, 90 degrees apart, showcase a phase differential of $90$ degree increments as intended. For a dynamic wing, $\phi$ is similarly defined based on the location of the vortices at the start of the wing's prescribed kinematic profile. Different disturbance configurations (corresponding to different values of $\phi$) can be achieved at the initiation of motion by changing the cylinder-rotation phase $\varTheta$. This leads to different interaction modes between the wing and the oncoming disturbances, as visualized and discussed in detail later in § 5.

Another benefit of using the rotating cylinder is that the rotating cylinder preserves the spatial layout of the wake shed by a static cylinder but reduces spurious cycle-to-cycle variation in the shedding pattern. In cylinder wakes the two most energetic modes are associated with the von Kármán vortices, and when their respective temporal coefficients, denoted by $a_1$ and $a_2$, are normalized by their energy and plotted against each other, they produce a phase plot, an example of which is shown in figure 4(a) for the static cylinder and in figure 4(b) for the rotating cylinder. Using figure 4(a) for the discussion, each point in this figure corresponds to a velocity field at one instant in time. From this figure, a phase angle can easily be determined for each velocity field as $\beta =\tan ^{-1}(a_{2,normalized}/a_{1,normalized})$. In this study the phase plot was divided equally into 36 bins, each 10 degrees wide. A total of 4000 velocity fields were used, which encompassed 55 periods of von Kármán vortex shedding. Each velocity field was placed in the corresponding bin based on its phase angle, and then all velocity fields within each bin were averaged together to form phase-averaged velocity fields. Phase averaging mitigates some of the random errors derived from the PIV velocity-vector calculation. Comparison between static-cylinder and rotating-cylinder POD phase plots of figure 4 reveals significant reduction in hysteresis for the rotating cylinder. This observation translates to a reduction in cycle-to-cycle fluctuations in vortical shedding for the rotating cylinder.

Figure 4.

POD phase plot for cylinder wakes: (a) static cylinder, (b) dynamic cylinder.

Figure 5 showcases the typical wake structure behind the rotating cylinder in contrast to the static cylinder. Note that the rotating cylinder also provides for greater vorticity concentration for a given vortical element of the vortex street. This characteristic helps to ensure that a given vortical element maintains coherency downstream at the point of wing interaction.

Figure 5.

Phase-averaged vorticity contours of the wake: (a) static cylinder, (b) rotating cylinder.

3.1. Formulation

The schematic representation in figure 6 shows an airfoil of chord $c$ undergoing arbitrary pitching and heaving kinematics in the presence of an upstream disturbance. The kinematic state of the airfoil is defined by the pitch angle $\theta$, the heave position $h$ and the respective velocities $\dot {\theta }$ and $\dot {h}$. The pivot point, located at a distance $x_p$ aft of the leading edge, denotes the centre of rotation of the airfoil. Also shown is a body-fixed frame $Bxz$, with the origin coinciding with the leading edge of the airfoil and the $x$ and $z$ axes extending in the chord-wise and chord-normal directions, respectively. The upstream disturbance is modelled using a vortical component resembling a von Kármán street superimposed on a mean component referred to here as the ‘uniform-flow speed’, denoted by $U$. Note that this uniform-flow speed, $U$, may be different from the free-stream speed of $U_{\infty }$ for modelling purposes to account for a velocity deficit when the airfoil is in the wake of the cylinder. The combined flow induces fluctuations in the flow velocity and the flow angle experienced by the airfoil. The wake of the airfoil consists of discrete LEVs and trailing-edge vortices (TEVs) shed from either edges of the airfoil in the previous time steps.

Figure 6.

Illustration of the low-order framework showing the oncoming flow disturbances, the vorticity shed from the airfoil and the kinematic variables used in the theoretical formulation. Clockwise vorticity is considered positive.

The solution seeks a velocity potential, ${\varPhi }$, that satisfies the incompressible continuity equation

(3.1)\begin{equation} \nabla^{2}{\varPhi}=0 \end{equation}

subject to the time-dependent zero-normal-flow boundary condition,

(3.2)\begin{equation} (\boldsymbol{\nabla}\boldsymbol{\varPhi}+\boldsymbol{V}_{kin})\boldsymbol{\cdot}\hat{n}=0, \end{equation}

where $\boldsymbol {V}_{kin}$ is the relative fluid velocity experienced by the airfoil due to the airfoil kinematics (pitch and heave), and $\hat {n}$ is the unit normal to the local camberline of the airfoil. The velocity potential, $\varPhi$, consists of the unknown bound potential, $\varPhi _{B}$, the uniform-flow potential, $\varPhi _U$, the potential due to the vortical disturbances, $\varPhi _D$, and the wake potential potential, $\varPhi _{W}$, which in turn can be broken down into the velocity potential due to the discrete LEVs and TEVs

(3.3)\begin{equation} \varPhi=\varPhi_{B}+\varPhi_{U}+\varPhi_{D}+\varPhi_{W}=\varPhi_{B}+\varPhi_{U}+\varPhi_{D}+\varPhi_{LEV}+\varPhi_{TEV}. \end{equation}

The velocity potential, $\varPhi _{D}$, due to the vortical component of the upstream disturbance, is a superposition of the potentials of two infinite rows of alternating-sign vortices of strength $\varGamma _k$, resembling a von Kármán street. Using the boundary condition (3.2), the unknown bound potential can be expressed in terms of the known airfoil kinematics and the free vorticity (comprising upstream vortical disturbances, LEVs and TEVs) as

(3.4)\begin{equation} \frac{\partial\varPhi_{B}}{\partial z}={-}\boldsymbol{V}_{kin}\boldsymbol{\cdot}\hat{n} -\frac{\partial}{\partial z}(\varPhi_{U}+\varPhi_D+\varPhi_{LEV}+\varPhi_{TEV}). \end{equation}

We now introduce the term downwash, denoted by $W$, as the component of the induced velocity opposite to the normal direction, i.e. along the $-\hat {n}$ direction, due to uniform flow, kinematics, upstream vortical disturbances, LEVs and TEVs. We see that, to satisfy the zero-normal-flow boundary condition along the airfoil chord, $\partial \varPhi _B(x)/\partial z$ should be equal to $W(x)$ everywhere along the chord

(3.5)\begin{align} W(x,t) = \frac{\partial\varPhi_{B}}{\partial z}=\frac{\partial \eta}{\partial x}(U \cos \theta+\dot{h} \sin \theta+u_{ind}) -U \sin \theta -\dot{\theta}(x-x_p) +\dot{h} \cos \theta -w_{ind}, \end{align}

where $\eta (x)$ is the camberline shape of the airfoil, assumed to be small in comparison with the chord length, and $u_{ind}(x)$ and $w_{ind}(x)$ are the chordwise and normal components (along positive $x$ and $z$ axes), respectively, of the velocity induced by all the free vortices along the airfoil chord

(3.6)$$\begin{gather} u_{ind}(x)=\frac{\partial \varPhi_{LEV}}{\partial x}+\frac{\partial \varPhi_{TEV}}{\partial x}+\frac{\partial \varPhi_{D}}{\partial x}, \end{gather}$$

(3.7)$$\begin{gather}w_{ind}(x)=\frac{\partial \varPhi_{LEV}}{\partial z}+\frac{\partial \varPhi_{TEV}}{\partial z}+\frac{\partial \varPhi_{D}}{\partial z}. \end{gather}$$

3.2. Solution of the flow field

Using the Glauert transformation, $x={c}/{2}(1-\cos \nu )$, the chordwise distribution of the bound vortex-sheet strength, $\gamma (x,t)$, which is positive for clockwise direction, can be written in terms of $\nu$ as

(3.8)\begin{equation} \gamma(\nu,t)=2U_{ref}\left[A_{0}(t)\frac{1+\cos \nu}{\sin \nu} + \sum_{n=1}^{\infty}A_{n}(t) \sin(n\nu)\right], \end{equation}

where $U_{ref}$ is the reference velocity used as a scaling parameter, the choice of which is arbitrary. This transformation boils down the solution of the flow field to obtaining the time-dependent Fourier coefficients, $A_{n}(t)$, from the instantaneous downwash, $W(x,t)$

(3.9)$$\begin{gather} A_{0}(t) ={-}\frac{1}{\rm \pi} \displaystyle \int_{0}^{\rm \pi}\frac{W(x,t)}{U_{ref}}\,\textrm{d} \nu , \end{gather}$$

(3.10)$$\begin{gather}A_{n}(t) = \frac{2}{\rm \pi} \displaystyle \int_{0}^{\rm \pi}\frac{W(x,t)}{U_{ref}} \cos (n \nu) \,\textrm{d} \nu,\quad n\geqslant 1. \end{gather}$$

The bound circulation can then be obtained by integrating $\gamma$ as

(3.11)\begin{equation} \varGamma_B(t)= \displaystyle \int_{0}^{c}\gamma(x)\,\textrm{d} \nu={\rm \pi} c U_{{ref}}\left[A_0(t)+\frac{A_1(t)}{2}\right]. \end{equation}

In the time-stepping implementation, the wake vorticity shed from the airfoil edges are represented using discrete vortices. A newly shed vortex is placed at one-third distance between the shedding edge and the previous vortex shed from that edge. The strength of the latest-shed vortices need to be determined to obtain $W$. At every time step a discrete TEV is released, the strength of which is obtained in accordance with Kelvin condition. At the $j\textrm {th}$ time step

(3.12)\begin{equation} {\rm \pi}c U_{{ref}}\left[A_0(t)+\frac{A_1(t)}{2}\right]+\varGamma_{TEV}^{j}=\varGamma_{B}^{j-1}, \end{equation}

where $\varGamma _{TEV}^{j}$ is the unknown strength of the latest TEV and $\varGamma _{B}^{j-1}$ is the bound circulation of the airfoil at the previous time step. After some manipulation, we get

(3.13)\begin{equation} \varGamma_{TEV}^{j}=\frac{\varGamma_{B}^{j-1}+c\displaystyle \int_{0}^{\rm \pi} W^{0}(1-\cos\nu)\,\textrm{d} \nu} {1-c\displaystyle \int_{0}^{\rm \pi} W^{\prime}_{TEV}(1-\cos\nu)\,\textrm{d} \nu}, \end{equation}

where $W^{\prime }_{TEV}(x)$ is the downwash distribution induced by a unit-strength discrete vortex placed at the location of the latest TEV and $W^{0}$ is the downwash due to all the other discrete vortices present in the flow field as well as the kinematic state of the airfoil. Ramesh et al. (Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014) presented a criterion to predict the initiation and termination of LEV shedding based on the maximum allowable value of suction that can be supported at the leading edge. The non-dimensional measure of the chord-parallel leading-edge suction force, called the LESP (denoted by $ {\mathcal {L}}$ henceforth), was shown to be associated with the leading Fourier-coefficient $A_0$

(3.14)\begin{equation} {\mathcal{L}}(t)=A_0(t) . \end{equation}

In LDVM, discrete LEVs are shed when $ {\mathcal {L}}$ exceeds a critical value denoted by $ {\mathcal {L}}_{crit}$. The value of $ {\mathcal {L}}_{crit}$ for a given airfoil is largely independent of motion kinematics at a given $Re$. Once the value of $ {\mathcal {L}}_{crit}$ for a given airfoil at a given $Re$ is determined using experimental or CFD data, LDVM can predict the LEV shedding characteristics of that geometry for any arbitrary motion kinematics at that $Re$. The strengths of the LEVs are determined so as to maintain the $ {\mathcal {L}}$ at $ {\mathcal {L}}_{crit}$. Hence, when LEV shedding is active, the strengths of the LEV and TEV shed at every time step have to be determined to satisfy the Kelvin condition as well as to maintain $ {\mathcal {L}}$ at $ {\mathcal {L}}_{crit}$. This leads to a simultaneous two-variable problem in the form

(3.15)\begin{equation} [A] \begin{Bmatrix} \varGamma_{TEV} \\ \varGamma_{LEV} \end{Bmatrix} =\{B\} \end{equation}

where,

(3.16)\begin{equation} [A]= \begin{bmatrix} 1-c\displaystyle \int_{0}^{\rm \pi} W^{\prime}_{TEV}(1-\cos\nu)\,\textrm{d} \nu & 1-c \displaystyle \int_{0}^{\rm \pi} W^{\prime}_{LEV}(1-\cos\nu)\,\textrm{d} \nu \\ \displaystyle \int_{0}^{\rm \pi} W^{\prime}_{TEV}\,\textrm{d} \nu & \displaystyle \int_{0}^{\rm \pi} W^{\prime}_{LEV}\,\textrm{d} \nu \end{bmatrix} \end{equation}

and

(3.17)\begin{equation} \{B\}= \begin{Bmatrix} \varGamma_{B}^{j-1}+c\displaystyle \int_{0}^{\rm \pi} W^{0}(1-\cos\nu)\,\textrm{d} \nu\\ \displaystyle -{\rm \pi} U_{ref} {\mathcal{L}}_{crit}-\int_{0}^{\rm \pi} W^{0} \,\textrm{d} \nu \end{Bmatrix}, \end{equation}

where $W^{\prime }_{LEV}(x)$ is the downwash distribution induced by a unit-strength discrete vortex placed at the location of the latest-shed LEV. As noted by Faure, Dumas & Montagnier (Reference Faure, Dumas and Montagnier2020), this approach eliminates the Newton-iteration procedure suggested by Katz & Plotkin (Reference Katz and Plotkin2000) and implemented previously by Ramesh et al. (Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014), and easily extends to multiple lifting surfaces. Once the strengths of the latest discrete vortices are determined, the solution of the flow field can be completed by computing the Fourier coefficients and subsequently obtaining the bound vortex distribution that satisfies the zero-normal-flow boundary condition.

We note that the recent work by Narsipur et al. (Reference Narsipur, Hosangadi, Gopalarathnam and Edwards2020) has proposed an updated definition for the LESP to take into account the effect of pitch rate on the reference velocity, $U_{ref}$. Their work considered the critical LESP at the initiation of LEV formation for pitching airfoils with a range of pivot points from leading edge to trailing edge, a range of non-dimensional pitch rates from $K = 0.005$ to 0.6, and two Reynolds numbers of 30 000 and 3 million. It was shown that, while the updated definition of LESP significantly improved the collapse of the critical LESP values for all cases at the higher Reynolds number, there was little to no improvement between the original and updated LESP definitions in the critical LESP values at LEV initiation at the lower Reynolds number, especially for the low-pitch-rate cases. Because the current work considers a low-Reynolds-number and low-pitch-rate motion, in this work we use the original definition of LESP in which $U_{ref}$ is set to a constant value of the free-stream velocity, $U_\infty$.

3.3. Calculation of the unsteady loads

The pressure distribution over the airfoil is given by the unsteady Bernoulli equation

(3.18)\begin{equation} \Delta p(x)=\rho\left[\left( U\cos \theta+\dot{h}\sin \theta +u_{ind}(x) \right)\gamma(x) +\frac{\partial}{\partial t} \int_{x'=0}^{x} \gamma(x')\,{\textrm{d} x}' + \dot{\varGamma}_{lev} \right] . \end{equation}

The last term in (3.18) explicitly takes into account the effect of circulation production due to LEV shedding on the unsteady loads of the airfoil (SureshBabu, Ramesh & Gopalarathnam Reference SureshBabu, Ramesh and Gopalarathnam2019). The forces and moment on the airfoil are calculated by the integration of the pressure distribution over the chord. The normal force per unit span of the airfoil can be obtained in terms of the Fourier coefficients using (3.8) as

(3.19)\begin{align} F_N &= \rho {\rm \pi}cU_{ref} \left[(U\cos \theta +\dot{h}\sin \theta)\left(A_{0}(t)+\frac{1}{2}A_{1}(t)\right)\right.\notag\\ &\quad + \left. c \left(\frac{3}{4}\dot{A_{0}}(t)+\frac{1}{4}\dot{A_{1}}(t)+\frac{1}{8}\dot{A_{2} }(t)\right)\right] \nonumber\\ & \quad + \rho \int_{0}^{c}u_{ind}(x)\gamma(x,t)\,{\textrm{d} x} +\rho c \dot{\varGamma}_{lev}, \end{align}

and the suction force per unit span of the airfoil is given by

(3.20)\begin{equation} F_S=\rho {\rm \pi}c U_{ref}^{2} A_{0}^{2}. \end{equation}

The normal and suction force coefficients $C_N$ and $C_S$ can be obtained through non-dimensionalization of the forces using the quantity $(\rho U_{ref}^{2}c)/{2}$. The lift and drag coefficients can then be evaluated as

(3.21)$$\begin{gather} C_L=C_N\cos\theta+C_S\sin\theta, \end{gather}$$

(3.22)$$\begin{gather}C_D=C_N\sin\theta-C_S\cos\theta. \end{gather}$$

The detailed derivation of these expressions is given by Ramesh et al. (Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014) and SureshBabu et al. (Reference SureshBabu, Ramesh and Gopalarathnam2019).

3.4. Low-order model of the oncoming disturbance

The oncoming vortical disturbance is modelled in LDVM as clusters of discrete vortices with alternating sense as shown in figure 7. The clusters are introduced into the flow field at a shedding location $15c$ upstream of the airfoil to minimize any sudden disturbances to the flow field. The experimentally observed vortex-shedding frequency $f$ is expressed in terms of the diameter-based Strouhal number, $St=fd/U_\infty =0.2$. This corresponds to a chord-based non-dimensional shedding interval of $T^{*}=c/(0.8U_\infty )$. In LDVM, two clusters of opposite strength are simultaneously released at this shedding interval (instead of releasing one cluster at every $0.5T^{*}$) to maintain zero net circulation in the flow domain. Based on the suggestion by Ramesh et al. (Reference Ramesh, Gopalarathnam, Granlund, Ol and Edwards2014), a non-dimensional time step of $\delta t^{*}=\delta tU_\infty /c=0.014$ is used in LDVM. Thus, a pair of clusters with opposite strength is introduced at every 89 non-dimensional time steps in LDVM. Each cluster is initially introduced in the form of a circular ring containing 50 equally spaced discrete vortices. One cluster is released at the shedding location while the other one is released at $0.5UT^{*}$ downstream of the shedding location to account for convection. From the PIV data of the rotating-cylinder wake, the distance from the wake centreline to the centre of a given vortex at $x/d=8$ in figure 5 was measured to be $y/d=0.6$ using the Galilean invariant $\varGamma _2$ criterion of Graftieaux, Michard & Grosjean (Reference Graftieaux, Michard and Grosjean2001). Based on this information, the clusters are introduced with a vertical offset of $1.2d$ between their centres. The predictions of the model is in best agreement with experimental observations when using more than 30 vortices per cluster. Meanwhile, using more than 50 vortices is not seen to change the low-order predictions and also results in large computation times. Using 30 to 50 vortices per cluster is recommended to strike a balance between capturing the flow-interaction phenomena accurately and doing so with reasonable computational expense. The diameter of the rings was set to $0.6d$. The evolution of the vortex street is not seen to be sensitive to this parameter. Other initial configurations such as a random distribution of vortices in a circular area are also seen to result in von Kármán streets similar to those produced with the ring shape for the initial configuration.

Figure 7.

Discrete-vortex rings introduced into the flow field at the upstream shedding location forming a vortex street as they convect downstream towards the airfoil location.

Once released into the flow domain, the clusters convect downstream with the uniform-flow speed $U$ and under mutual influence. As they move downstream, the rings deform to form patches and stabilize into two rows of alternating-sign vortex clusters, representing a von Kármán street. The vertical separation between the centroids of the two rows is not seen to vary much during the downstream convection. The airfoil is held static for a duration of $t^{*}=5$ after the first cluster reaches the leading edge. This allows the airfoil–vortex-street interactions to reach a steady state before data collection. For unsteady simulations, the pitch motion is initiated when the centroid of a cluster aligns with the airfoil leading edge.

The aggregate strength of a vortex cluster in LDVM and the uniform-flow speed $U$ experienced by the airfoil are estimated based on the experimentally observed lift fluctuations of a static airfoil placed in the cylinder wake. For this portion of the experimental study, the static airfoil is placed at a pitch angle of $\theta =6^{\circ }$ in the wake of the cylinder at a streamwise position of $x/d=8$ aft of the longitudinal axis of the cylinder and at a centreline vertical position. In the absence of any oncoming disturbance, a lift coefficient of $0.59$ is observed (blue line in figure 8a). Under the influence of the vortical and velocity-deficit effects of the disturbances generated by the cylinder, the airfoil exhibits a fluctuating lift history with a reduction in mean lift. Previous studies have reported a reduction in mean lift when a static wing is placed at pre-stall angles near the centreline of a bluff-body wake (Lefebvre & Jones Reference Lefebvre and Jones2019; Zhang, Wang & Gursul Reference Zhang, Wang and Gursul2020). Ten typical cycles of lift oscillations are shown using the black curve in figure 8a). The average peak-to-peak amplitude of $C_L$ fluctuations is 0.52 and the mean $C_L$ is 0.39 (red line), corresponding to a $35\,\%$ decrease in mean $C_L$. In LDVM, the velocity components induced by the vortex clusters cause the airfoil to experience velocity- and flow-angle fluctuations and thus result in $C_L$ oscillations of the static airfoil. The amplitude of the oscillations is seen to be largely determined by the strength of the vortex clusters, while the mean of the lift fluctuations are primarily dependent on the uniform-flow speed. Through an error minimization study, a value of $\varGamma /(cU_\infty )=0.3$ of the non-dimensional strength of the vortex clusters and a uniform-flow speed of $0.85U_\infty$ were found to result in $C_L$ oscillations with similar amplitude and percentage reduction in mean. The lift history predicted by the low-order model for the airfoil at $\theta =6^{\circ }$ and located at the centreline is shown in figure 8(b). Figure 8(b) shows a $35\,\%$ reduction in mean $C_L$ (from 0.66 to 0.43) and a lift amplitude of $0.55$ using these values in the LDVM simulations. In previous work (Medina et al. Reference Medina, SureshBabu, Rockwood, Gopalarathnam and Anwar2019), we presented the studies on an airfoil positioned outside the von Kármán street at an offset location of $y/d=-1.5$. The low-order model of the vortex street with the same value of non-dimensional strength and $U= 0.92U_\infty$ was obtained using the same approach and the predicted flow-interaction phenomena were in good agreement with experimental results. All the studies presented in this paper are for the airfoil positioned at the centreline, and hence the value of 0.3 for the non-dimensional strength of the vortex clusters and a value of $0.85U_\infty$ for the uniform-flow speed are used in all the LDVM simulations presented in § 5.

Figure 8.

Static airfoil at $6^{\circ }$ incidence in the presence of the disturbance: (a) experimentally observed $C_L$, (b) $C_L$ predicted by LDVM, (c) discrete-vortex plots from LDVM compared with experimental vorticity contours at four time instants in a cycle of lift oscillation.

Vorticity-contour images from experiment and discrete-vortex plots from LDVM corresponding to four instants of time during a lift-oscillation cycle are presented in figure 8(c). The discrete-vortex plots show the clockwise (CW) vortices in blue and counter-CW vortices (CCW) in red. The flow images can be used to infer the association between the lift fluctuations and the spatio-temporal features of the oncoming disturbance. From figure 8(c), a local lift peak can be associated with the presence of a CW vortex near the mid-chord on the upper surface and a CCW vortex near the leading edge on the lower surface ($t=mT+0.5$, where $m$ is an integer to denote the start of a period in the lift oscillation). It is known from the vortex-force-map study of Li & Wu (Reference Li and Wu2015) that a CW vortical structure on the upper surface of the airfoil is lift enhancing while it is in the vicinity of the leading edge and becomes lift reducing as the vortex convects downstream towards the trailing edge. A CCW structure convecting downstream below the airfoil has a similar effect on lift. Thus, the lift peak occurs when the airfoil experiences the best combined lift enhancement due to the positions of both vortical structures. From an effective-angle-of-attack perspective, a CW vortex located above the airfoil induces maximum upwash at the leading edge and thus the largest effective angle of attack when it is slightly downstream of the leading edge. Also, a CCW vortex convecting downstream on the bottom surface of the airfoil induces the largest upwash when it is slightly upstream of the leading edge. The lift-peak configuration thus corresponds to the situation where the net upwash induced by the two vortex structures is maximized resulting in the largest effective angle of attack. Based on similar arguments, a local valley in lift can be associated with the presence of an approaching CW vortex at the leading edge on the upper surface ($t=mT+0.0$).

3.5. Numerical aspects of the discrete-vortex method

The discrete vortices are represented as vortex blobs using the vortex-core model proposed by Vatistas, Kozel & Mih (Reference Vatistas, Kozel and Mih1991), which is a close approximation of the Lamb–Oseen vortex. The components of velocity induced at a point $(x,z)$ by the $j\textrm {th}$ discrete vortex are given by,

(3.23)$$\begin{gather} u=\frac{-\varGamma_j}{2{\rm \pi}}\frac{z_j-z}{\sqrt{\left[(x_j-x)^{2}+(z_j-z)^{2}\right]^{2}+r^{4}_{core}}}, \end{gather}$$

(3.24)$$\begin{gather}w=\frac{\varGamma_j}{2{\rm \pi}}\frac{x_j-x}{\sqrt{\left[(x_j-x)^{2}+(z_j-z)^{2}\right]^{2}+r^{4}_{core}}}. \end{gather}$$

A core radius $r_{core}$ of $0.02c$ is used, which is approximately $1.3$ times the average spacing between vortices according to the suggestion by Leonard (Reference Leonard1980). The discrete vortices are convected using the total velocity induced at their respective locations by the uniform-flow speed, the bound vorticity and the other discrete vortices in the flow field. Thus, the current discrete-vortex approach simulates the wake rollup in the flow field and accommodates non-planar wakes.