2. Unsteady potential flow model

To explore low-order models for the LEV generated from a two-dimensional flat plate, this study adopts the unsteady potential flow model developed by Xia & Mohseni (Reference Xia and Mohseni2013), where discrete vortices are introduced to account for the shear layer formation and vortex shedding at the leading and trailing edges. This means that the vortical structures of the LEVs and TEVs are represented by a number of point vortices, as illustrated in figure 1. To establish the flow model, the physical flow in the lab coordinate, denoted $z$ -plane ( $z = x + iy$ ), where the incoming flow with the velocity $U(t)$ impinges on the flat plate at an angle of attack $\alpha (t)$ , can be mapped out to an imaginary flow around a cylinder in the $\zeta$ -plane ( $\zeta = \xi + i\eta$ ) using the Joukowski transformation,

(2.1) \begin{align} z = \zeta + \frac {a^2}{\zeta } + x_0, \end{align}

where the radius $a$ of the cylinder is related to the chord length $c$ of the flat plate as $a = c/4$ ; $x_0$ is the distance from the flat plate’s midpoint to the origin at the rotation centre.

Figure 1.

Unsteady potential flow model around a flat plate with discrete vortices. The physical flow in the $z$ -plane is mapped to that in the $\zeta$ -plane through the Joukowski transformation.

Employing a virtual-reference-frame approach proposed by Minotti (Reference Minotti2002) and applying the circle theorem (Milne-Thomson Reference Milne-Thomson1958), the complex potential of the flow around a rotating flat plate can be expressed in the $\zeta$ -plane as (Xia & Mohseni Reference Xia and Mohseni2013)

(2.2) \begin{align} \begin{split} w_{\zeta }(\zeta ) & = \underbrace {|U|\textrm{e}^{-\textrm{i}\alpha }(\zeta +x_0) + |U|\textrm{e}^{\textrm{i}\alpha } \big(a^2/\zeta +x_0 \big)}_{\text{Translational}} + \underbrace {i\varOmega \frac { x_0^2-2a^2-2 \big(a^2/\zeta + x_0 \big)^2}{2}}_{\text{Rotational}}\\& \quad -\frac {i}{2\pi } \underbrace {\sum _{j=1}^{M} \varGamma _{Lj} \ln \left ( \frac {\zeta - \zeta _{Lj}}{\zeta - a^2/\bar {\zeta }_{Lj}} \right )}_{\text{LEVs}} -\frac {i}{2\pi } \underbrace {\sum _{k=1}^{N} \varGamma _{Tk} \ln \left ( \frac {\zeta - \zeta _{Tk}}{\zeta - a^2/\bar {\zeta }_{Tk}} \right )}_{\text{TEVs}}, \end{split} \end{align}

where $\varOmega (t) = -\dot {\alpha }(t)$ is the angular velocity associated with the flat plate’s rotation. Note that in (2.2), the first two terms represent the translational effect and the third term represents the rotational effect. The last two terms respectively correspond to the induced effects of the point vortices forming the LEVs and TEVs, including the effects of their image vortices inside the cylinder, as shown in figure 1. Here, the subscripts $_{Lj}$ and $_{Tk}$ refer to the $j$ th and $k$ th vortices (i.e. $L_j$ and $T_k$ ) in the LEVs and TEVs, respectively, with the total number of vortices being $M$ for the LEVs and $N$ for the TEVs. Accordingly, $z_{Lj}$ and $z_{Tk}$ are the locations of the point vortices in the $z$ -plane, whereas $\zeta _{Lj}$ and $\zeta _{Tk}$ are their corresponding locations in the $\zeta$ -plane; $\varGamma _{Lj}$ and $\varGamma _{Tk}$ denote their circulations, respectively.

In evaluating (2.2), it is essential to predict circulations and locations for all point vortices. For the former, the vortex circulation is usually determined at the moment when an individual point vortex is generated from the leading or trailing edge, by means of the classical steady-state Kutta condition:

(2.3) \begin{align} \frac {\partial w_{\zeta }(\zeta )}{\partial \zeta } = 0 \quad \text{for} \; \zeta \rightarrow \pm a. \end{align}

This condition is effectively enforcing a stagnation point at the leading or trailing edge in the $\zeta$ -plane, so as to ensure a finite velocity at the flat-plate edges in the $z$ -plane. Note that while such a condition is naturally satisfied at the trailing edge, its applicability at the leading edge is limited to large angle of attack (AoA), where significant reverse flow presents on the windward side (Xia & Mohseni Reference Xia and Mohseni2013). As such, the present work only concerns the large AoA situation. To implement this condition, assuming the vortices $L_m$ and $T_n$ are being generated at the leading and trailing edges, respectively, (2.3) can be further expressed as

(2.4) \begin{align} \frac {\partial w_{\zeta }^{*}(\zeta )}{\partial \zeta } - \frac {i\varGamma _{Lm}}{2\pi } \frac {\partial }{\partial \zeta } \left ( \ln \left ( \frac {\zeta - \zeta _{Lm}}{\zeta - a^2/\bar {\zeta }_{Lm}} \right ) \right ) - \frac {i\varGamma _{Tn}}{2\pi } \frac {\partial }{\partial \zeta } \left ( \ln \left ( \frac {\zeta - \zeta _{Tn}}{\zeta - a^2/\bar {\zeta }_{Tn}} \right ) \right ) = 0 \quad \text{for} \; \zeta \rightarrow \pm a, \end{align}

where $w_{\zeta }^{*}(\zeta )$ is the complex potential without the newly generated vortices. The unknowns here are $\zeta _{Lm}$ , $\zeta _{Tn}$ , $\varGamma _{Lm}$ and $\varGamma _{Tn}$ , which are associated with the newly generated vortices. To solve this problem, I assume that the vorticity feeding into the leading-edge or trailing-edge shear layers occurs in a quasi-steady manner and its rate is determined by the flow advection near the edges. Then, the placement of the two new vortices at each time step follows the $1/3$ -arc rule (Mason Reference Mason2002; Ansari et al. Reference Ansari, Zbikowski and Knowles2006; Xia & Mohseni Reference Xia and Mohseni2013), which requires the new vortex to be placed at the $1/3$ arc position from its vorticity feeding edge to the previously generated vortex. Here, the arc representing the vorticity-feeding shear layer should be tangential to the sharp edge. Combining these conditions yields $z_{Lm}$ and $z_{Tn}$ in the physical plane, which can be plugged into (2.4) to solve for $\varGamma _{Lm}$ and $\varGamma _{Tn}$ .

For the existing vortices in the LEVs and TEVs, their circulations can be considered to remain constant following the Helmholtz’s laws of vortex motion. However, their locations need to be updated based on their advective velocities, which can be obtained by taking the derivative of the de-singularised complex potential (Xia & Mohseni Reference Xia and Mohseni2013). For a point vortex $L_v$ in the LEVs, its velocity in the $z$ -plane can be computed as

(2.5) \begin{align} \begin{split} u_{Lv} - iv_{Lv} & = \frac {\zeta _{Lv}^2}{\zeta _{Lv}^2 - a^2}\left [|U| \left (\textrm{e}^{-\textrm{i}\alpha } - \frac {a^2\textrm{e}^{\textrm{i}\alpha }}{\zeta _{Lv}^2} \right ) + i\varOmega \frac {2a^2}{\zeta _{Lv}^2}\left (\frac {a^2}{\zeta _{Lv}}+x_0\right ) \right ] \\[3pt]& \quad - \frac {\zeta _{Lv}^2}{2\pi \big(\zeta _{Lv}^2 - a^2 \big)}\left [ \sum _{j=1,j\neq v}^{M} \frac {i\varGamma _{Lj}}{\zeta _{Lv}-\zeta _{Lj}} - \sum _{j=1}^{M} \frac {i\varGamma _{Lj}}{\zeta _{Lv}-a^2/\bar {\zeta }_{Lj}} \right. \\[3pt] & \quad \left. + \sum _{k=1}^{N} \left (\frac {i\varGamma _{Tk}}{\zeta _{Lv}-\zeta _{Tk}} - \frac {i\varGamma _{Tk}}{\zeta _{Lv}-a^2/\bar {\zeta }_{Tk}}\right )\right ] + \frac {i\varGamma _{Lv}}{\pi }\frac {\zeta _{Lv}a^2}{ \big(a^2 - \zeta _{Lv}^2 \big)^2}, \end{split} \end{align}

where the last term is the classical Routh correction term (Lin Reference Lin1941; Clements Reference Clements1973), which is, in addition to the de-singularised flow field, excluding the effect of the singularity itself (at $\zeta = \zeta _{Lv}$ ), but including its image (at $\zeta = a^2/\bar {\zeta }_{Lv}$ ) induced by the cylinder. Similarly, the velocity of a point vortex in the TEVs can be evaluated by replacing $Lv$ with $Tv$ and excluding the singularity at $\zeta = \zeta _{Tv}$ in (2.5). It should be noted that in applying (2.5), the vortex by default is a free vortex. This is the case for a vortex element (i.e. the discrete point vortex in the present study) or an individual concentrated vortex that is not attached to or has been shed from the flat plate. However, this equation does not hold for an attached vortex, which is connected to the flat plate through a vorticity-feeding shear layer.

Consequently, following Xia & Mohseni (Reference Xia and Mohseni2013), the unsteady aerodynamic force can be obtained by applying the unsteady Bernoulli equation and then integrating the pressure around the flat plate, which yields the expression

(2.6) \begin{align} \begin{split} F_x - iF_y & = 2\pi \rho a^2 ( \underbrace {|\dot {U}| (\textrm{e}^{-\textrm{i} \alpha } - \textrm{e}^{\textrm{i} \alpha }) + 2i\dot {\varOmega }x_0}_{\text{Added Mass}} + \underbrace {2i|U| \textrm{e}^{-\textrm{i} \alpha }\varOmega }_{\text{Rotational}} )\\& \quad - i \rho \sum _{j=1}^{M} \underbrace { \left [\varGamma _{Lj} \frac {\rm d}{\mathrm dt} \left (\bar {\zeta }_{Lj} -\frac {a^2}{\zeta _{Lj}} \right ) \right ]}_{\text{LEVs}} - i \rho \sum _{k=1}^{N} \underbrace { \left [\varGamma _{Tk} \frac {\rm d}{\mathrm dt} \left (\bar {\zeta }_{Tk} - \frac {a^2}{\zeta _{Tk}} \right ) \right ]}_{\text{TEVs}}. \end{split} \end{align}

This formula reveals that a main source of unsteady aerodynamic force is related to the LEVs and TEVs, as a result of the relative advective movement between a vortex and its image.

The previous work (Xia & Mohseni Reference Xia and Mohseni2013) has validated this model by simulating Dickinson & Gotz (Reference Dickinson and Gotz1993)’s classical experiment of a starting flat plate, which has a chord length $c = 5$ cm and an angle of attack $\alpha = 45^\circ$ . The incoming flow is accelerated from rest at a constant rate of $62.5$ cm s⁻ $^{2}$ for $0.16$ s, until it reaches a steady-state velocity of $10$ cm s⁻¹, corresponding to a Reynolds number of $192$ . The simulation was performed for the starting process, covering a plate travel distance of $s = 4c$ . The lift coefficient was computed as $C_L = 2L/(\rho U^2c)$ , where the lift $L = F_y \cos {\alpha }-F_x \sin {\alpha }$ was obtained from (2.6). For time advancement, the previous simulation employed the Euler scheme, which led to convergence issues at later times ( $s/c\gt$ 4). This can be attributed to the emergence of the secondary TEV, which was drawn upstream and perturbed by strong interactions with both the plate and the large LEV.

Figure 2.

Variations of the lift coefficient $C_L$ versus the travel distance $s/c$ predicted by the discrete vortex mode with different time advancement schemes and time steps. The results of Knowles et al. (Reference Knowles, Wilkins, Ansari and Zbikowski2007)’s simulation and discrete-vortex model are plotted for comparison.

For improved accuracy, the fourth-order Runge–Kutta (RK4) scheme can be implemented for time advancement. In addition, the vortex-blob (VB) method (Chorin & Bernard Reference Chorin and Bernard1973; Krasny Reference Krasny1986; Nitsche & Krasny Reference Nitsche and Krasny1994) is adopted to suppress the artificial instability of the vortex sheet induced by numerical discretisation. In the present complex-analysis framework, this is achieved replacing the term $1/(\zeta _{Lv}-\zeta _{Lj})$ in (2.5) with $(\bar {\zeta }_{Lv}-\bar {\zeta }_{Lj})/(|\zeta _{Lv}-\zeta _{Lj}|^2+\sigma ^2)$ , where the smoothing parameter $\sigma$ is set to be a constant value of $0.016c$ . Figure 2 compares the lift coefficient variations predicted by the present model with those from the simulation and discrete-vortex model of Knowles et al. (Reference Knowles, Wilkins, Ansari and Zbikowski2007). It is noted that during the initial acceleration, the lift spike due to the added-mass effect is much weaker in the CFD results, owing to the viscous effect that tends to damp the lift overshoot characteristic of inviscid models during rapid flow transitions. The RK4 and VB predictions agree well with Knowles et al. (Reference Knowles, Wilkins, Ansari and Zbikowski2007)’s discrete-vortex model, and the convergence with decreasing time step show marked improvement over the previous results (Xia & Mohseni Reference Xia and Mohseni2013), demonstrating the effectiveness of the combined RK4 and VB approaches.

Figure 3 further presents a qualitative comparison of the wake evolutions between the experiment and model predictions at four snapshots corresponding to $s/c = 1, 2, 3, 4$ , showing the overall success of the discrete-vortex model in capturing the evolution of the flow pattern and vortical structures. Again, overall good convergence of the RK4 and VB scheme is confirmed by comparing the discrete vortex locations in figure 3(b–e). Notably, better convergence is observed at the early stage ( $s/c=1$ ), when the feeding shear layers in both the LEV and TEV remain stable. Beyond $s/c=2$ , as Kelvin–Helmholtz (K–H) instabilities develop in the feeding shear layers, local vortex roll-up emerges, breaking the shear layers into segments and causing the global vortex distributions to diverge. Furthermore, the LEV exhibits inferior convergence than the TEV during the initial roll-up process, which likely results from the strong LEV–plate interaction, highlighting an intrinsic challenge in LEV modelling. Within the VB framework, convergence can be readily enhanced by increasing the smoothing parameter $\sigma$ ; however, such tuning should be exercised with caution, as it may introduce unphysical viscous damping that suppresses the natural development of the shear-layer instabilities. Overall, the results presented in figures 2 and 3 adequately justify the validity and implementation of the baseline discrete-vortex model.

Figure 3.

Comparisons of (a) the flow visualisations of Dickinson & Gotz (Reference Dickinson and Gotz1993)’s starting flat plate at $\alpha = 45^\circ$ with (b–f) the flow-field evolutions predicted by the discrete-vortex model. Here, panels (b–e) adopt the Runge–Kutta (RK4) and vortex-blob (VB) schemes, with time step $\mathrm dt$ decreasing from 0.008 s to 0.001 s, whereas panel (f) uses the Euler time scheme with $\mathrm dt$ = 0.005 s. For comparison, panels (b–d) and (f) also plot the vortex locations of panel (e) in the background, with the plotting time intervals matching the corresponding time steps.

Although the RK4 and VB scheme demonstrates superior accuracy and convergence, it generally incurs higher computational costs and is more complex to implement. In contrast, the Euler time scheme – while less accurate and subject to deteriorating convergence in long-term simulations – is more efficient and more straightforward to apply. Figure 2 also presents the prediction result obtained using the Euler scheme, showing that the lift closely follows the CFD data. This agreement, of course, is achieved through a judicious choice of time step ( ${\mathrm{d}}{t}=0.005$ s) tailored to this specific problem. Figure 3(f) further shows that the wake vortex pattern compares well with those from both the experiment and the RK4 and VB simulations. Since the present study does not aim to excel in the discrete-vortex model itself, the Euler scheme with ${\mathrm{d}}{t}=0.005$ s is considered sufficient and is adopted for the subsequent studies of low-order models, owing to its efficiency and ease of implementation. In fact, the more accurate RK4 and VB is not essential here, as the core objective is to improve the reduced-order representation of the primary LEV dynamics – particularly during its initial growth upon the onset of plate motion – which is largely independent of the fidelity of the discrete-vortex framework itself.

3. Simple LEV models

The discrete-vortex approach provides relatively accurate predictions of the wake vortex dynamics. Moreover, advances in algorithms – such as the tree code (Lindsay & Krasny Reference Lindsay and Krasny2001) and multipole method (Hussien & Saetti Reference Hussien and Saetti2026) – together with the growing availability of modern computational resources, have rendered the full discrete-vortex method increasingly feasible for simulating the starting flat-plate problem. Nevertheless, it remains desirable to have physically motivated reductions of the vortex system – they can provide useful building blocks for fast prediction or control-oriented models, as they reduce not only the operation count, but also the effective state dimension of the entire system. Therefore, the motivation of the present work extends beyond merely providing a fast reduced simulation framework. A more intriguing question is how to construct a low-order, physically relevant model for the large coherent LEV. To this end, an intuitive thought would be to replace the multitude of point vortices, which emulate the lumped effect of one individual concentrated vortex, with a single point vortex. For a moving flat plate, such models have already been developed by Michelin & Smith (Reference Michelin and Smith2009) and Wang & Eldredge (Reference Wang and Eldredge2013), where each point vortex is adopted to represent an entire LEV or TEV, and the circulation of each vortex is treated as a variable during its growing process. The circulation and location of such growing vortex are determined instantaneously based on the Kutta condition and the Brown-Michael equation (or its modification) (Brown & Michael Reference Brown and Michael1954), which is a physical condition requiring a zero net force is applied on the assumed vortex system consisting of the concentrated vortex and its feeding sheet at the edge.

Given the above-mentioned motivation, a primary task here is to assess the performance of the LEV models that resolve the LEV based on one main point vortex. Next, four different models are compared by looking at the evolutions of the flow-field pattern, LEV circulation and location, as well as the lift coefficient for a starting flat-plate problem (Dickinson & Gotz Reference Dickinson and Gotz1993). For simplicity, this work concerns the process where only one main concentrated LEV is growing from the leading edge. The baseline model, namely the ‘discrete LEV’, refers to the discrete-vortex model introduced in § 2, which serves as a reference for comparison. The rest of the models are all simple LEV models with one main point vortex, as illustrated in figure 4. Note that all models here adopt the discrete-vortex approach for the TEV; this allows us to focus on the differences resulting merely from the different LEV models. Furthermore, for compatibility with the theoretical framework developed for the ‘discrete LEV’ model, the simple LEV models all consist of two vortices $L_1$ and $L_2$ (i.e. $M \equiv 2$ in (2.2)) so that they allow for the prediction of the LEV circulation growth through the Kutta condition. Specifically, $L_2$ is the vortex being generated from the leading edge; the determination of the vortex location $z_{L2}$ and circulation $\varGamma _{L2}$ follows (2.4) and its subsequent discussion. Here, $L_1$ represents the main LEV and it assimilates the previous $L_2$ at each time step.

Figure 4.

Diagram for the three simple LEV models.

Thus, in the spirit of circulation conservation, the LEV growth can be easily expressed in terms of $\varGamma _{L1}$ and $\varGamma _{L2}$ as

(3.1) \begin{align} \varGamma _{L1}(t+\Delta t) = \varGamma _{L1} + \varGamma _{L2}, \end{align}

where $\Delta t$ is the simulation time step. Note that all quantities by default are estimated at time $t$ . However, it is more challenging to predict $z_{L1}$ , which represents the location of the main LEV, considering that $\varGamma _{L1}$ is time variant. In this regard, three different simple LEV models are next introduced to resolve the evolution of $z_{L1}$ , as explained in figure 4.

The first simple LEV model is the ‘free LEV’ model, which assumes $L_1$ to be a free vortex, the advective motion of which satisfies (2.5). Thus, $z_{L1}$ can be updated as

(3.2) \begin{align} z_{L1}(t+\Delta t) = z_{L1} + V_{L1}\Delta t, \end{align}

where $V_{L1}$ is the advective velocity given by $V_{L1} = u_{L1} + iv_{L1}$ .

The second simple LEV model is the ‘B–M LEV’ model, which decides the LEV centre $z'_{L1}$ based on the Brown–Michael equation (Brown & Michael Reference Brown and Michael1954; Michelin & Smith Reference Michelin and Smith2009),

(3.3) \begin{align} V'_{L1} = u_{L1} + iv_{L1} - \frac {\big(z'_{L1}-z_{L0} \big)}{\varGamma _{L1}} \frac {\varGamma _{L2}}{\Delta t}, \end{align}

where $z_{L0}$ is the leading edge location. Based on $V'_{L1}$ estimated at time $t$ , $z'_{L1}(t+\Delta t)$ can be again determined using a similar equation to (3.2). Apparently, the advective velocities of the LEV given by these two models are different by the last term in (3.3).

The third simple LEV model is the ‘merge LEV’ model, which considers the growth and motion of the LEV as an outcome of a virtual merging process of $L_2$ into $L_1$ . During such a merging process, we may assume momentum conservation of the double-vortex system, meaning that the circulation centroid of the system remains unchanged, which can be expressed as

(3.4) \begin{align} z''_{L1}(t+\Delta t) = \frac {z_{L1}(t+\Delta t) \varGamma _{L1} + z_{L2}(t+\Delta t) \varGamma _{L2}}{\varGamma _{L1} + \varGamma _{L2}}. \end{align}

This gives the LEV location $z''_{L1}(t+\Delta t)$ after merging. Here, $z_{L1}(t+\Delta t)$ and $z_{L2}(t+\Delta t)$ are the vortex locations at $t+\Delta t$ as if they were convected freely, similar to that described by (3.2).

The performance of these models is assessed by simulating the same starting-flat-plate problem. Figure 5 presents the qualitative comparison of the predicted flow field evolutions. The lift predictions of the simple LEV models are compared in figure 6(a), where the ‘free LEV’ appears to be the least accurate model. The undesired prediction of the ‘free LEV’ is also reflected in the LEV core location, $z_{\textit{LEV}}$ , which is the circulation-weighted LEV centroid marked by the $\times$ in figure 5. It is evident from figure 5(b) that the LEV cores are further downstream compared with the benchmark locations in figure 5(a), especially during the early stage at $s/c=1$ and 2. This can be quantitatively shown in figure 6(b), where the evolutions of the LEV core’s streamwise distance, $s_{\textit{LEV}}$ , from the leading edge are compared among the different models. The $s_{\textit{LEV}}$ result not only confirms the large gap between the ‘free LEV’ and the ‘discrete LEV’, but also shows a deflection of the ‘free LEV’, which contrasts with the relatively linear increase of the ‘discrete LEV’. However, both the ‘B–M LEV’ and ‘merge LEV’ yield better predictions of the LEV core location, as seen in figure 5(c,e) and figure 6(b). Specifically, the accuracy of $s_{\textit{LEV}}$ prediction has been significantly improved during $s/c$ from 0 to 3; however, both models exhibit a deflecting trend in figure 6(b), causing notable deviations from the ‘discrete LEV’ after $s/c=3$ .

Figure 5.

Comparisons of the flow field evolutions of a starting flat plate at $\alpha = 45^\circ$ predicted by (a) the ‘discrete LEV’ model, (b, c, e) three simple LEV models, and (d) the classical B–M LEV model. The $\times$ marks the the circulation-weighted centroid $z_{\textit{LEV}}$ of the LEV. The animations of the flow field evolutions (for panels a–e) during $s/c=0$ –4.5 are included in the supplementary movies 1–5 available at https://doi.org/10.1017/jfm.2026.12255.

The effect of different LEV models on the TEV dynamics is also evident in figure 5. Again, the TEV locations predicted by the ‘free LEV’ differ significantly from the other models that the downstream advection of the primary TEV seems to be much slower. Furthermore, the shedding of the primary TEV and the subsequent formation of the secondary TEV occur at a much earlier stage prior to $s/c=2$ , as opposed to $s/c=2$ –3 for the other models. The TEV behaviours predicted by the ‘B–M LEV’ and ‘merge LEV’ bear closer resemblances to the ‘discrete LEV’, although the primary TEVs still slightly lag behind the benchmark result. Better agreements can be observed in the formation of the secondary TEV at $s/c=3$ . However, at $s/c=4$ , the secondary TEV tends to experience excessive growth towards to the leading edge for all three simple LEV models. This possibly results from the failure of the over-simplified LEV in resolving vortex interaction, as the secondary TEV grows under the influence of the large LEV.

The different performances of lift prediction in figure 6(a) can be understood from a fundamental perspective of vortex force. Specifically, the ‘LEVs’ and ‘TEVs’ terms in (2.6) indicate that a major lift contribution originates from the relative movement between the wake vortices and their image counterparts. Since the image vortices effectively correspond to the bound circulation around the flat plate, this unsteady lift is actually associated with the vortex movement relative to the flat plate itself. It can be further deduced that a downstream advection of the LEV causes negative lift, whereas the same motion of the TEV generates positive lift; similar interpretations have been previously discussed by Yu et al. (Reference Yu, Tong and Ma2003) and Xia & Mohseni (Reference Xia and Mohseni2013). Therefore, compared with the benchmark result of the ‘discrete LEV’, the significantly reduced lift generation during $s/c=0$ –2 for the ‘free LEV’ can be attributed to the faster advection of the LEV and the slower advection of the TEV away from the flat plate, which respectively result in greater negative lift and less positive lift. After $s/c=2$ , the dramatically increased lift can be explained by the lingering tendency of the LEV, as is evident from the notable deflection of the solid-green curve in figure 6(b) – the slower downstream advection of the LEV leads to significantly reduced generation of negative lift. Similarly, for both the ‘B–M LEV’ and ‘merge LEV’, their lift discrepancies from the ‘discrete LEV’ can also be attributed to the mispredictions of the LEV motion; nevertheless, their performances are notably improved over the ‘free LEV’, as a result of the improvements in capturing the motions of both the LEV and TEV.

Figure 6.

Evolutions of (a) the lift coefficient $C_L$ and (b) the LEV core’s streamwise distance $s_{\textit{LEV}}$ from the leading edge.

Figure 7.

Evolutions of (a) the non-dimensional total circulation of the LEV and (b) the non-dimensional growth rate of the LEV circulation. The absolute values are adopted here as $\varGamma _{\textit{LEV}}$ has a negative sign.

The vortex circulation is another key parameter affecting the lift generation. The growth of the LEV circulation, $\varGamma _{\textit{LEV}}$ , is compared among the same models in figure 7(a), whereas their corresponding rates of circulation growth, $\dot {\varGamma }_{\textit{LEV}}$ , are plotted in figure 7(b). The three simple LEV models are able to capture the overall trends of $\varGamma _{\textit{LEV}}$ and $\dot {\varGamma }_{\textit{LEV}}$ for $s/c\lt 3$ , despite some degrees of overestimation; and the ‘free LEV’ again shows a larger gap from the benchmark result of the ‘discrete LEV’ than the other two models. Especially, the performance of the ‘free LEV’ further deteriorates after $s/c=3$ such that the over-prediction of $\varGamma _{\textit{LEV}}$ rises to be as high as 100 %. Evidently, this is associated with an unrealistic surge of $\dot {\varGamma }_{\textit{LEV}}$ , which could arise from the unphysical growth of the secondary TEV towards the leading edge, as shown in figure 5(b), thereby exerting a strong influence on the Kutta condition at the leading edge. As a result, this can also have a partial contribution to the significantly overestimated $C_L$ of the ‘free LEV’ in figure 6(a). For the ‘B–M LEV’ and ‘merge LEV’, the secondary TEV is less eccentric and farther away from the leading edge. Therefore, the surge of $\dot {\varGamma }_{\textit{LEV}}$ after $s/c=3$ is weaker in figure 7(b).

Interestingly, ‘B–M LEV’ and ‘merge LEV’ offer highly similar results in figures 5–7, which is not surprising as the two models share the same fundamental principle of momentum conservation. However, Wang & Eldredge (Reference Wang and Eldredge2013) showed that, by accounting for the impulse of both the main vortex and its image, their modified Brown–Michael formula, i.e. impulse matching, yielded notably different results from the classical B–M method. Here, I also present the results of the classical B–M, denoted ‘B–M LEV (classic)’, in figures 5–7 for side-by-side comparison with the two-vortex ‘B–M LEV’ model. Note that the ‘B–M LEV (classic)’ does not include the feeding vortex $L_2$ and applying the Kutta condition leads to a direct change in $\varGamma _{L1}$ . Basically, the two B–M models show close agreements in both their vortex evolutions and lift variations when $s/c\lt 3$ . Their gap expands after $s/c=3$ when the secondary TEVs show distinct roll-up behaviours, as is evident from figures 5(c) and 5(d). This suggests that the different B–M implementations yield largely consistent predictions of the LEV growth rate and movement during its initial formation stage; however, the prediction accuracy is sensitive to the change in the trailing-edge Kutta condition, which can be altered dramatically during the formation of a secondary TEV.

It should be mentioned that Sheng et al. (Reference Sheng, Ysasi, Kolomenskiy, Kanso, Nitsche and Schneider2012) reported good agreement between the discrete-vortex method and the B–M method in simulating the wake behind a flapping plate. However, their model considered vortex shedding only from the trailing edge, without accounting for the LEV formation. In this regard, the present work and the study of Sheng et al. together suggest that the Brown–Michael approach offers better accuracy in predicting the dynamics of TEVs than that of LEVs. This can be partially attributed to the fact that the LEV forms in the near field of the plate, where the point-vortex approximation may introduce considerable errors in computing the induced velocity associated with the leading-edge Kutta condition. Furthermore, the matured LEV prior to shedding can significantly influence the formation of the secondary TEV by modifying the trailing-edge Kutta condition, which would further amplify errors from the LEV modelling.

It is also worth discussing the zoomed-in streamlines passing by the leading edge, for which representative snapshots at $s/c=1$ are plotted in figure 5. We can see that the streamlines closely follow the feeding vortex sheet represented by the discrete vortices in figure 5(a) and the stagnation streamline is tangential to the leading edge, agreeing well with the requirement by the Kutta condition. However, the leading-edge streamlines for the three simple LEV models in figure 5(b,c,e) are less physical that they tend to turn around the leading edge, although the Kutta condition is still satisfied in these cases. Interestingly, the ‘B–M LEV (classic)’ in figure 5(d) also generates smooth, tangential streamlines at the leading edge; yet, the streamline angle off the edge and the location of the windward stagnation point still differ profoundly from the ‘discrete LEV’ model. This underscores the necessity of including the feeding vortex sheet to correctly resolve the leading-edge streamlines.

4. Multi-vortices LEV models

The previous section has exposed the limitations of the simple LEV models, especially in predicting the LEV’s advection and growth during the late stages. Nevertheless, the results of the ‘B–M LEV’ and ‘merge LEV’ models demonstrate notable improvements over the ‘free LEV’ model, especially during the LEV’s initial growing stage when $s/c\lt 0.5$ or so. Essentially, this indicates the relevance of incorporating the dynamics of the leading-edge feeding vortex $L_2$ in considering the dynamics of the main LEV $L_1$ . In reality, vorticity feeding to the main LEV is achieved through a shear layer (or vortex sheet) emanating from the leading edge. And a tractable way of representing the feeding sheet is readily available, i.e. through an array of discrete vortices.

In this work, such a model is proposed as the ‘feeding-sheet LEV’ model, where the LEV core is still accounted for by a single point vortex, whereas multiple vortices are introduced to emulate the feeding vortex sheet. This idea somewhat resembles the hybrid model implemented by Darakananda & Eldredge (Reference Darakananda and Eldredge2019) and Dumoulin et al. (Reference Dumoulin, Eldredge and Chatelain2023), which can be traced to Pullin (Reference Pullin1978)’s early work. As depicted in figure 8(a), this model can be readily extended from the ‘merge LEV’ model, with $L_1$ being the main LEV plus a series of feeding vortices from $L_2$ to $L_M$ . Similar to the ‘merge LEV’ model, $L_1$ and $L_2$ merge during each time step to form the new $L_1$ of the next time step, following the merging principles described in (3.1) and (3.4). In addition, $L_3$ – $L_M$ are treated as free vortices, the motions of which can be computed based on (3.2). Furthermore, the generation of the new vortex $L_M$ , including its placement $z_{LM}$ and circulation $\varGamma _{LM}$ , is again determined by the Kutta condition specified in (2.4). This way, the ‘merge LEV’ model in § 3 is simply the ‘feeding-sheet LEV’ model with $M = 2$ .

The ‘feeding-sheet LEV’ model offers a reasonable representation of both the LEV core and its vorticity-feeding shear layer; however, the length of the shear layer depends on the number ( $M-1$ ) of its constituting vortices, which is more or less arbitrary. Basically, the LEV core is formed as the result of the shear layer roll-up, which occurs as the local fluid is subject to a sufficiently strong rotational effect. Given this understanding, it would be ideal to determine the roll-up onset physically, based on the local rotation strength of the shear layer. Such a criterion can be readily adopted from the Rortex strength (Gao & Liu Reference Gao and Liu2018) in the form,

(4.1) \begin{align} R_s(z) = |\omega (z)| - \omega _0(z). \end{align}

For a given point $z$ , $R_s \gt 0$ predicts that $z$ is within the rolled-up vortex core region. Here, $\omega$ is the vorticity defined as $\omega = \partial v/\partial x - \partial u/\partial y$ and $\omega _0$ is the non-rotational vorticity expressed as

(4.2) \begin{align} \omega _0 = \sqrt {\left ( \frac {\partial v}{\partial y} - \frac {\partial u}{\partial x} \right )^2 + \left ( \frac {\partial v}{\partial x} + \frac {\partial u}{\partial y} \right )^2}, \end{align}

where the velocity gradients can be estimated with a central difference approach. Here, the velocity components $u$ and $v$ are related to the complex potential $w_{\zeta }(\zeta )$ in (2.2) as

(4.3) \begin{align} u - iv = \frac {\zeta ^2}{\zeta ^2-a^2} \frac {\partial w_{\zeta }}{\partial \zeta }. \end{align}

Figure 8.

Diagram for the two multi-vortices LEV models.

However, in the present potential flow, $\omega$ is zero everywhere, which gives $R_s \leqslant 0$ except for all the discrete vortices, rendering (4.1) ineffective in differentiating the LEV core from the feeding shear layer. This paradox results from the discretisation of the vorticity field in the framework of potential flow. To remedy this difficulty, this work adopts the classical vortex blob method to approximate $\omega (z)$ in (4.1) as

(4.4) \begin{align} \omega (z) = \frac {\sum \varGamma _{Lr}}{\pi r_o^2} e^{-r^2/r_o^2}, \end{align}

which assumes a Gaussian vorticity distribution in the vortex core. Here, $\varGamma _{Lr}$ represent the circulations of all point vortices within a circular region, which shares the common centre, $z_o$ , of the vortex core of interest; the radius of this circle is given by $r=|z-z_o|$ so that $z$ is on its boundary. Additionally, $r_o$ is the size of the vortex core or vortex blob, which can be estimated as $r_o = C_o \sqrt {\nu t_o}$ , where $t_o$ is the characteristic growth time of the vortex core. In this work, $t_o$ is simply the existing time of the point vortex that is identified to be the vortex core and $C_o$ is a prefactor for adjusting the rate of diffusion, with $C_o = 2$ corresponding to the classical Lamb–Oseen vortex.

Based on the above-mentioned $R_s$ criterion, a ‘physical-merge LEV’ model is proposed, as shown in figure 8(b), which also has a multi-vortices framework similar to the ‘feeding-sheet LEV’ model, i.e. the main LEV is represented by the vortex $L_1$ and the feeding shear layer by the vortices $L_2{-}L_M$ . The distinction here is that, during each time step, the $R_s$ criterion is computed at $z_{Lj}$ ( $2 \leqslant j \leqslant M$ ) to dynamically decide whether each shear-layer vortex $L_j$ should remain in the shear layer or become a part of the core vortex $L_1$ . To estimate $\omega$ in (4.4), the circular region for each $L_j$ is centred at $z_{L1}$ , with the radius given by the distance between $L_1$ and $L_j$ as $r = |z_{Lj}-z_{L1}|$ . Then, all vortices $L_R$ with $R_s \gt 0$ are identified and merged with $L_1$ according to

(4.5) \begin{align} \begin{split} &\varGamma _{L1}(t+\Delta t) = \varGamma _{L1} + \sum \varGamma _{LR},\\ &z''_{L1}(t+\Delta t) = \frac {z_{L1}(t+\Delta t) \varGamma _{L1} + \sum z_{LR}(t+\Delta t) \varGamma _{LR}}{\varGamma _{L1} + \sum \varGamma _{LR}}, \end{split} \end{align}

where $z_{L1}(t+\Delta t)$ and $z_{LR}(t+\Delta t)$ are again the vortex locations advanced using the free vortex velocity in (3.2), and $z''_{L1}(t+\Delta t)$ is $L_1$ ’s location after merging. Equation (4.5) is basically the circulation and momentum conservations generalised from (3.1) to (3.4) that apply to the double-vortex merge process. In practice, to improve the merge smoothness and dynamics differentiability, abrupt change in the vortex system should be avoided. Here, only a maximum of three shear-layer vortices that are of the highest $R_s$ values are allowed to merge with $L_1$ during each merge event.

It should be noted that the previous framework can be readily extended to consider the formation of secondary LEVs subsequent to the shedding of the primary LEV. The key issue is to identify the emergence of a new vortex core near the leading edge. To this end, at each time step, all point vortices $L_2{-}L_M$ formed after the primary core are treated as new vortex core candidates. Then, by setting $z_o = z_{L_o}$ for each candidate vortex core $L_o$ , the Rortex strength $R_s$ is evaluated at the other point vortices. In this way, a new vortex core is identified if a threshold number ( $M_c$ ) of point vortices $L_R$ satisfy $R_s \gt 0$ . After that, $L_R$ are merged with $L_o$ to form the merged vortex core, based on the conservation laws similar to (4.5). Here, $M_c$ serves as a parameter for tuning the core identification sensitivity – we may capture only the main LEV cores by setting a sufficiently large $M_c$ , whereas a smaller $M_c$ could pick out local vortex rolls, e.g. those arise from the K–H instability of the feeding shear layer. In the present work, the main focus is on evaluating the reduced-order model for the growing process of the first LEV – the model performance for the later vortex shedding process depends on proper handling the strong TEV–plate and LEV–TEV interactions, which will be investigated in future work. Nevertheless, supplementary movie 8 shows a longer-term simulation ( $s/c$ up to 7.5), demonstrating the identification and reduction of the second LEV (with $M_c = 40$ ).

To evaluate the performances of the multi-vortices LEV models, the same starting flat-plate problem in § 3 is simulated. For the ‘feeding-sheet LEV’ model, two different settings are adopted, $M=11$ and $M=51$ , respectively corresponding to the total number of point vortices being 10 and 50 in the feeding vortex sheet of the LEV. The ‘physical-merge LEV’ model also takes two different settings with $C_o$ being 2.0 and 1.5, respectively. The former can be treated as a standard setting based on the diffusion model of the Lamb–Oseen vortex, while the latter gives relatively smaller values of $\omega$ and $R_s$ , corresponding to a lower tendency of the peripheral vortices to merge with the vortex core. The flow field evolution results are compared with the benchmark of the ‘discrete LEV’ in figure 9. In general, all multi-vortices LEV models exhibit overall qualitative agreements with the ‘discrete LEV’, including the configurations and locations of the wake vortices, especially during $s/c \leqslant 2$ . In particular, the behaviours of the primary TEV are rather consistent among different models for $s/c$ up to 4, showing a significant improvement over the TEV predictions of the simple LEV models in figure 5.

Figure 9.

Comparison of the flow-field evolutions predicted by (a) the ‘discrete LEV’ model and (b–e) the multi-vortices LEV models. The $\times$ marks the the circulation-weighted centroid $z_{\textit{LEV}}$ of the LEV, which differs from the LEV core at $z_{L1}$ for the multi-vortices LEV models. The flow-field animations of the multi-vortices LEV models during $s/c=0$ –4.5 are included in supplementary movies 6–9.

Further quantitative analysis reveals substantially different performances of the multi-vortices LEV models. The lift coefficient predictions are presented in figure 10(a). Compared with the ‘merge LEV’ in figure 6(a), the ‘feeding-sheet LEV’ ( $M=11$ ) has a largely reduced gap from the benchmark result during the early stage of $s/c \leqslant 2$ , while this gap for the $M=51$ model further diminishes. Similar to the analysis in § 3, this improvement in lift prediction can also be understood in terms of the vortex force pertaining to the streamwise LEV movement $s_{\textit{LEV}}$ , which is plotted in figure 10(b). Specifically, increasing $M$ from 11 to 51 yields a more accurate prediction of $s_{\textit{LEV}}$ during $s/c\lt 1$ , thereby decreasing the initial gap in the lift calculation. However, the ‘feeding-sheet LEV’ models still suffer from significant over-predictions of lift at the later stage of $s/c\gt 2$ . This again can be attributed to the deflection of $s_{\textit{LEV}}$ in figure 10(b) – the slower LEV advection generates less negative lift, which amounts to a higher overall lift; this issue has also been discussed for the simple LEV models in figure 6.

Regarding the ‘physical-merge LEV’, it is evident from figure 10(a) that both settings offer improved lift predictions over the ‘feeding-sheet LEV’. Again, these improvements are consistent with their LEV core movements shown in figure 10(b). For the model with $C_o=2.0$ , the $s_{\textit{LEV}}$ curve displays a correction from the deflected ones of the ‘feeding-sheet LEV’ during $s/c\gt 3$ , which recovers the LEV’s normal advection speed, thereby suppressing further decline of the negative lift. The $s_{\textit{LEV}}$ evolution corresponding to $C_o=1.5$ shows a slightly better matching with the ‘discrete LEV’, which is responsible for the best agreement of lift prediction with the benchmark model.

Figure 10.

Evolutions of (a) the lift coefficient $C_L$ and (b) the LEV core’s streamwise distance $s_{\textit{LEV}}$ from the leading edge for the multi-vortices LEV models.

The evolutions of the LEV circulation $\varGamma _{\textit{LEV}}$ and its growth rate $\dot {\varGamma }_{\textit{LEV}}$ for the multi-vortices models are presented in figure 11, showing considerable improvements over those of the simple LEV models in figure 7. Similar to the results of $C_L$ and $s_{\textit{LEV}}$ , the ‘physical-merge LEV’ models also render better performances than the ‘feeding-sheet LEV’ models in calculating $\varGamma _{\textit{LEV}}$ and $\dot {\varGamma }_{\textit{LEV}}$ . Furthermore, the $M=51$ setting of the ‘feeding-sheet LEV’ offers more accurate predictions of $\varGamma _{\textit{LEV}}$ than the $M=11$ , while the ‘ $C_o=1.5$ ’ setting of the ‘physical-merge LEV’ yields the best matching with the ‘discrete LEV’. It is worth noting that the surge of $\dot {\varGamma }_{\textit{LEV}}$ after $s/c\gt 3$ in figure 7 has been effectively mitigated for the ‘feeding-sheet LEV’ and has been completely rectified for the ‘physical-merge LEV’. On the one hand, for the ‘physical-merge LEV’, the improvements can be attributed to the success in resolving the development of the secondary TEV, which is evident through the comparison among figure 9(a,d,e). On the other hand, for the ‘feeding-sheet LEV’, the secondary TEV still exhibits an unphysical growth extending to the leading edge, as depicted in figure 9(b,c). Nevertheless, the improved $\dot {\varGamma }_{\textit{LEV}}$ predictions over the simple LEV models indicate a suppression of the undesired impact of the secondary TEV on the leading-edge Kutta condition. This can be understood by that the added feeding vortex sheet is closer to the leading edge than the eccentric secondary TEV, thereby playing a crucial role in offsetting the influence of the latter.

Figure 11.

Evolutions of (a) the non-dimensional total circulation of the LEV and (b) the non-dimensional growth rate of the LEV circulation for the multi-vortices LEV models.

It is worth summarising the key messages. Although the present models are evaluated based on a LEV growing process with a finite distance of travel and limited wake vortices, they nonetheless shed light on the broader applications of the low-order vortex models. A primary finding is that the ‘free LEV’ model, albeit being prevalently used in the literature, gives significant error in modelling the LEV advection and growth rate, which is responsible for the significant mispredictions of the lift as well as the behaviours of the TEV. This is because the LEV is essentially an attached vortex with varying circulation, the dynamics of which is strongly affected by vorticity feeding from the leading edge. Therefore, the ‘B–M LEV’ and ‘merge LEV’ models are introduced to account for the influence of vorticity feeding based on momentum conservation, which indeed improves the predictions of the LEV movement and circulation growth, yielding more accurate modelling of the TEV dynamics and calculation of the unsteady lift. However, the vorticity feeding in these two models are still considered in a primitive way such that it lacks necessary detail of the feeding shear layer attached to the leading edge. By adding an array of point vortices to represent the feeding vortex sheet, the ‘feeding-sheet LEV’ model is able to offer further improvements in resolving the LEV and TEV dynamics. However, issues remain as the vorticity feeding from the vortex sheet to the LEV core is determined by the total number of vortices in the feeding sheet, which is still arbitrary. To this end, the newly proposed ‘physical-merge LEV’ has demonstrated effectiveness in physically deciding the annexation between the feeding sheet and the central vortex core. In fact, by setting a merge criterion based on the concept of Rortex (Gao & Liu Reference Gao and Liu2018), this model preserves not only the feeding sheet, but also the LEV part with relatively low rotation strength, as is evident in figure 9(d,e); the level of detail of the low-rotation part, including the feeding sheet, can be adjusted through the diffusion factor $C_o$ . This leads to significantly improved accuracy in almost all aspects of the simulation, especially during the later stage when the secondary TEV forms. The success of the ‘physical-merge LEV’ model hints at a more general guiding principle in the lower-order aerodynamic modelling that the reduction to a point vortex should be targeted at the concentrated vortex core, which has high rotation strength; however, details of the feeding shear layer and low-rotation part of the main vortex should be sufficiently resolved, as they could exert non-negligible influences on the vorticity generation from the vortex-shedding edges.

Last, the limitations and contributions of this work are clarified. Admittedly, the proposed merging scheme in the multi-vortices LEV models requires further mathematical justification – issues such as well-posedness and differentiability warrant more rigorous examination in future studies. Here, for a preliminary check, the Appendix includes the ‘physical-merge’ simulation results of different time steps based on the RK4 and VB scheme, showing overall good convergence. Regarding its potential application to control, the merging operation may introduce event-like changes into the dynamical system, which could restrict its applicability to control formulations that demand smooth differentiability. Nevertheless, as has been discussed, this can be improved by limiting the number of vortices being merged during each merge event. It is worth emphasising that the primary value of this work lies more in offering physical insight into low-order LEV model than in furnishing a reduction algorithm per se. In particular, the ‘physical-merge LEV’ model is developed mainly to demonstrate the necessity of physically resolving the feeding shear layer, rather than to establish a full-blown reduced-order framework for vortex shedding simulation. Accordingly, reducing the computational cost is not a central concern here; the real significance resides in the physically reduced LEV model itself, as physical interpretability and inferential power remain always desirable in model reduction.