For the laminar inflow cases, the computational domain is set to 12.5D0×5D0×5D0 in the x (streamwise), y (spanwise), and z (vertical) directions, respectively, as in the work of . A sensitivity test on the ratio of the rotor-swept area to the domain's cross-section is carried out in Sect. , showing that the effects of blockage are minimal. The rotor center is placed at the origin and is 2.5D0 downstream from the inflow boundary and centered in the yz plane, as indicated by the white line in Fig. b. Within this domain, levels of refined mesh are arranged in a cylindrical shape with diameters of 3D0, 2D0, and 1.44D0 for different refinement levels, as shown in Fig. a. Grids are generated using the application snappyHexMesh, containing 10.6×106 cubic cells. As for the cases with turbulent inflow, the domain extends to 6D0 upstream from the wind turbine rotor, which is the origin, with a refined region at the inflow, resulting in 12.0×106 cells (see Fig. c). This modification accounts for the development of the turbulent structures and helps mitigate undesired pressure fluctuations caused by the synthetic turbulent inlet. For both mesh layouts (Fig. b and c), the grid cell size is Δ/R0=1/5 (i.e., 12.6 m) at level 1. At the most refined level around the rotor, which is level 4, the grid refines to Δ/R0=1/40 (i.e., 1.6 m), which falls within the range suggested by . Note that unless otherwise mentioned, Δ refers to the grid size at level 4 in the remaining part of the current work.

The choice of spatial differencing scheme for the advective term influences energy dissipation levels and numerical stability in the simulation. It is well known that upwind schemes introduce numerical diffusion, whereas central difference schemes can lead to dispersion errors . To balance preserving wake structures and mitigating numerical oscillations, a blended scheme is employed (Gauss fixedBlended) . The composition of the scheme is 95% fourth-order central difference scheme (cubic) and 5% upwind scheme (upwind). The performance of the selected scheme is assessed and compared with several other common schemes in Appendix .

The time step Δt is set at 1.4×10-2 s, corresponding to a one-degree rotor rotation. This ensures that the distance traveled by the rotor tip is below a grid size per time step, which is around 0.7Δ with the current setups. Note that this Δt results in a Courant–Friedrichs–Lewy number of 0.09, which is well below 1. In the simulations, the pressure–velocity coupled system is solved iteratively with PISO (Pressure Implicit with Splitting of Operators) algorithm. The time marching scheme employed is the Crank–Nicolson method with a coefficient of 0.9 (Crank--Nicolson 0.9). The simulations are conducted for 75 and 170 rotor revolutions for the cases subjected to laminar and turbulent inflow conditions, respectively, corresponding to approximately 375 and 850 s. Statistical data are collected after the 70th revolution to eliminate the influence of initial transients. The corresponding sampling windows are five revolutions for the laminar cases and 100 revolutions for the turbulent cases. have demonstrated that these sampling durations are sufficient to achieve convergence of the second-order statistics.

For the cases subjected to laminar inflow, the inflow conditions are specified as Dirichlet, with the velocity set at U∞=11.4 ms-1 and no velocity shear imposed. Slip conditions are applied at the boundaries on all four sides. At the outlet, an advective outflow condition, D/Dt=0, is imposed to ensure mass conservation and to prevent distortion of the flow structures near the outlet .

In addition to the inflow conditions, the cases subjected to turbulent inflow conditions share the same boundary conditions as those with laminar inflow. To introduce inflow turbulence, the divergent-free synthetic eddy method is applied, which is implemented through turbulentDFSEMInlet. The time-averaged streamwise velocity profile is again uniformly set to U∞=11.4 ms-1. To characterize the strength of inflow turbulence intensity TI, it is measured at 2D0 upstream from the rotor, using more than 40 probes. Additionally, the turbulence spectrum obtained is Kolmogorov-like, similar to those reported by . The definition of TI is given in Eq. (), where σu, σv, and σw are the standard deviations of u, v, and w with respect to time. Two turbulence intensity levels are tested, which are a minor level (∼0.5%) and an atmospheric boundary layer level (∼5%). The former represents flow conditions typically encountered in controlled wind or water tunnels , while the latter corresponds to typical inflow conditions for offshore wind farms . 6TI=13(σu2+σv2+σw2)U∞×100%

In addition to closely matching the turbulence intensity typically found in controlled wind or water channel experiments, conditions with TI<1% are also of interest for other reasons. Although these levels are significantly lower than those observed in typical offshore environments, previous numerical studies have shown that even weak turbulence (TI≤0.5%) can trigger instabilities that lead to wake breakdown, which is not observed with perfectly laminar inflow. Motivated by these findings, the present study also investigates whether ambient turbulence at such low levels influences leapfrogging instability and, in turn, affects the wake structure of an asymmetric rotor, with the aim of developing a more comprehensive understanding of the impacts of ambient turbulence.

In addition to the setups described earlier in this section, referred to as the “standard” setups (see Table ), cases with denser grids are also tested, which are the “dense” cases. In general, the setups for the dense cases follow the same structures as for the standard cases, but the grid size Δ is halved. Specifically, Δ=D0/160 (i.e., 0.8 m) is achieved at level 4 in Fig. . This results in the cell count for the laminar and turbulent cases reaching 85.4 and 96.1×106, respectively. Additionally, the time step size Δt for the dense cases is also halved to ensure that the distance that the tip travels per time step is less than one grid. Thus, for the dense case, ΩΔt=0.5°. As one might expect, halving the cell size Δ increases the required CPU hours by approximately 16 times and the required memory by approximately 8 times. Therefore, cases with even finer mesh become unfeasible with the available computational resources.

Running simulations with higher mesh resolution is expected to capture finer details, particularly for modeling the vortex core size rω, which is crucial in terms of vortex dynamics . Indeed, the numerical results of demonstrated that the detailed dynamics of the tip vortices of a two-bladed asymmetric rotor are affected by the vortex core radius when rω/R0 falls between 0.03 and 0.10. Moreover, the overview provided by indicates that rω for rotors used for common industrial applications is around 1×10-2R0 to 5×10-2R0. However, to the authors' best knowledge, the precise vortex core size of a large-scale wind turbine (≥5 MW) is currently not publicly available.

Currently, rω found in the simulations is 7.6×10-2R0 with the standard mesh and 4.4×10-2R0 with the dense mesh (see Fig. ). This indicates that the current setups for standard mesh can marginally meet the resolution requirements to properly capture the vortex cores of an industrial rotor, whereas the cases with the dense mesh are considered to have adequate resolution. However, experimental data from indicate that tip vortex core sizes are around 0.11ctip (ctip is the chord length at the rotor tip) for the Joukowsky rotor (designed to have a constant circulation profile along the entire blade and thus having concentrated tip vortices) they used, and this value is equivalent to 1.4×10-2R0. For the NREL 5 MW turbine used in this study, 0.11ctip corresponds to approximately 1.3×10-3R0 (ctip=0.75 m; ), demonstrating that the vortex core size could be substantially smaller than what can be resolved in the current simulations. Despite this, the numerical results of showed that the predicted leapfrogging distance closely matched experimental observations with a grid size of 4rω. This finding suggests that precisely capturing the vortex core size may not be necessary to predict the leapfrog instability, and this also agrees with the findings of . In the present study, the leapfrogging distances predicted by the standard and dense setups are comparable, and the main conclusions remain consistent despite the predicted values of rω being significantly different (see Figs.  and ). Therefore, this work primarily focuses on results obtained using the standard setup, while results from the dense setup are also frequently included in the discussion to further examine the effects of vortex core size on the detailed tip vortex dynamics.

The 2D point vortex model used in this study is based on the framework established by , which reduces the complex helical tip vortex system of a two-bladed asymmetric rotor to two infinite arrays of point vortices, separated by a radial distance δr. This self-repeating vortex arrangement is schematically illustrated in Fig. , where the x and z axes correspond to the streamwise and radial directions of the helical system, respectively. Initially, the vortices within each array are evenly spaced by a distance of 2h0 to replicate the helical pitch, with δr set equal to ΔR to represent the blade length difference. The two vortex arrays are staggered, with each vortex positioned midway between two adjacent vortices in the opposite array.

The dynamics of the mth vortex, marked as ωm, is represented by its temporal displacement rates in the x and z directions (dxm/dt and dzm/dt, where xm and zm are the x and z positions of ωm), which are determined by the velocities Vx,m and Vz,m. These velocities are the sums of the induction from all other vortices , as formulated in Eq. (). The circulation Γ is assumed to be constant across both vortex arrays, and the index n ranges from -∞ to ∞, as the vortex arrays extend infinitely in the ±x directions. ddtxmzm=Vm,xVm,z7=Γ2π∑n=-∞,n≠m∞zn-zm(xn-xm)2+(zn-zm)2Γ2π∑n=-∞,n≠m∞xn-xm(xn-xm)2+(zn-zm)2

Due to the inversion symmetry and periodicity, the magnitude of the induced velocity is identical for both arrays and is uniform across all vortices, i.e., |Vn,x|=Vx and |Vn,z|=Vz. Specifically, when Γ is positive in the y direction, ωm and ωm+2 in Fig.  travel with +Vx and +Vz, while ωm+1 and ωm+3 travel with -Vx and -Vz. This leads to the pairing motion of ωm and ωm+1. Consequently, the variations of the streamwise and radial separations between a vortex pair, denoted as δh and δr (see Fig. ), change at rates of 2Vx and 2Vz. Furthermore, Vx and Vz can be completely described by δh and δr, as written in Eq. (). Note that the infinite sums can be expressed in closed algebraic expressions, and the detailed derivations can be found in the work of . The main advantage offered by the formulation in Eq. () is that the synchronous displacement across the vortex array ensures identical temporal evolution for any pair, thereby simplifying the study of vortex pairing growth rates to a single representative pair and reducing the system's degrees of freedom. ddtδh(t)δr(t)=2Vx(δh(t),δr(t))2Vz(δh(t),δr(t))=Γ2π∑n=-∞,n∉even∞δr(xn+nh0-δh)2+(δr)2Γ2π∑n=-∞,n∉even∞xn+nh0-δh(xn+nh0-δh)2+(δr)2=Γ2h0sinh⁡πδr/h0cos⁡πδh/h0+cosh⁡πδr/h0sin⁡πδh/h0cos⁡πδh/h0+cosh⁡πδr/h08≡Γ2h0fδh(δh(t),δr(t))fδr(δh(t),δr(t))

To find out how δh and δr evolve in time, the equations of motion for vortex arrays written in Eq. () are integrated in time numerically, as given in Eq. (). It should be noted that with the definitions of δh and δr given in Fig. , dδh/dt and dδr/dt are both 0 when δh=δr=0. Also, the initial conditions are given as δh=0 and δr=ΔR for the cases considered in this work. Notice that because ΔR≠0, dδh/dt≠0. ddtδhδr≡Γ2h0fδh(δh,δr)fδr(δh,δr)⟶Time integrationδh(t)δr(t)9=Γ2h0∫t0tfδh(δh(t′),δr(t′))dt′∫t0tfδr(δh(t′),δr(t′))dt′+δh(t=t0)δr(t=t0)

As derived in Sect. , Eq. () describes the vortex pairing motion due to the misalignment of two vortex arrays of a non-linear system. As detailed in Appendix , by applying a Taylor expansion, the system can be linearized, making it an eigenvalue problem. After solving the eigenvalue problem, an unstable mode with a growth rate of λ can be found, and its mode shape for [δh;δr] is [1;1]. Given the eigenvector obtained for the unstable mode, selecting the L1 norm of δh and δr as the parameter estimating the growth of the unstable mode becomes natural, where the L1 norm is denoted as ||η||1. The definition of ||η||1 is given in Eq. (). Furthermore, the time evolution of the L1 norm, ||η||1, is chosen to be the indicator of the leapfrogging growth rate. Note that by the initial conditions of this work, when ΔR>0, ||η||1/ΔR is expected to grow exponentially in time from unity according to the result after the linearization (see Appendix ), as shown in Eq. (). 10||η||1≡|δh|+|δr|11||η(t)||1/ΔR≃eλtfor larget

However, this linearization may not be valid when δh and δr deviate significantly from the point where the Taylor expansion is carried out, as the system is inherently non-linear. Additionally, Eq. () becomes inaccurate when dδh/dt and/or dδr/dt have large values at t=t0 (see Appendix  for details). To address these limitations, instead of estimating the growth rate using the linearized system, it is derived based on ||η||1 obtained through carrying out time integration in Eq. (), which accounts for non-linear effects. Consequently, the growth rate determined via time integration, denoted as σ2D, is preferred over that determined via the linearized equations, denoted as λ.

To give an example, Fig.  plots ||η||1 obtained via the time integration method against time t for the case with ΔR/R0=9.8%, where the values of h0 and Γ are based on the un-truncated blade of case Lam10S reported in Table . An exponential growth of ||η||1 is evident, as indicated by the linear region in the semi-logarithmic plot. Specifically, the range 0.6≤t/tHel≤0.8 (where tHel=2h02/Γ) gives the most stable exponential growth rate, and this time window is also applicable to the LES data presented in Sect. . Therefore, the slope of this part is chosen to define the growth rate of the leapfrogging instability, which is denoted as σ2D. In general, σ2D is greater than λ when ΔR>0 (see Fig. ). Furthermore, the leapfrogging time tLF, marked by the vertical dashed line in Fig. , corresponds to the moment when the vortex pair swaps their streamwise positions, which occurs when δh=h0.

In the results section (Sect. ), the growth rate σ2D obtained from the 2D model using the time integration method is compared with the LES results (σLES). This comparison evaluates whether the CFD results align with theoretical predictions. Additionally, following , the characteristic timescale for the helical system of a two-bladed asymmetric rotor, denoted as tHel, is defined as 2h02/Γ. This timescale is commonly used in this study to normalize quantities related to growth rate and time.

As shown in Fig. a, the contours of the instantaneous out-of-plane vorticity field (ωy) reveal that tip vortices shed by a symmetric rotor (ΔR/R0=0%) under laminar inflow conditions are convected downstream in a highly regular pattern. The consecutive tip vortices follow one another with minimal variation in the z direction (radial direction). Beyond approximately 4D0 downstream, individual vortices become indistinguishable due to vortex diffusion, ultimately forming a stable vortex tube. Additionally, a shear layer is formed across this tube and extends beyond x/D0=8 (see Figs. a and a). However, it is important to note that such conditions are unlikely to occur in practical wind farm environments, as perfectly turbulence-free inflow is unrealistic. Indeed, as shown in Fig. , even a minor level of inflow turbulence can trigger the wake breakdown process.

Figure b presents the ωy field for the case with a blade length difference of 9.8% (ΔR/R0≃10%). In this configuration, tip vortices are shed at different radial positions. Owing to the velocity differences that can be deduced from the Biot–Savart law, the outer vortices gradually overtake the inner ones. During this process, the vortices exhibit significant movement in the z direction (radial direction), and the streamwise spacing between consecutive vortices decreases. At approximately x/D0=1.2, the outer vortex of a vortex pair overtakes the inner vortex (corresponding to δh=h0, as defined in Fig. ), marking the event of leapfrogging (further discussed in Sect. ). During this overtaking event, both tip vortices deform from circular to elliptical shapes, which promotes vortex merging shortly after leapfrogging . Following the merging, a secondary helical vortex structure forms and remains stable beyond x/D0=8.

The observed vortex pairing and merging under laminar inflow conditions are consistent with findings reported in the literature . However, these results do not fully align with the predictions of . Under similar conditions (h0/πR0<0.3 and ΔR/R0<0.3), their inviscid vortex model predicts that outer vortices continuously overtake inner vortices without precession motions and merging. This discrepancy can be mainly attributed to the sizes of the vortex cores , which is surveyed and discussed later on in Sects.  and .

For cases with larger blade length differences, the radial separation between the inner and outer vortices naturally increases. When the blade length difference reaches approximately 30% (Fig. d), it is observed that after the first leapfrogging event (defined when δh=h0), the outer vortices continuously overtake the inner vortices without merging. This behavior aligns with the predictions of . On the other hand, for a 20% blade length difference (Fig. c), the tip vortex behavior falls between the cases observed for 10% and 30%. In this intermediate case, the inner tip vortices become sufficiently stretched such that portions of them merge with the outer vortices, while other segments continue to be overtaken within the range of 2≤x/D0≤4. Another noteworthy observation is that larger values of ΔR/R0 cause the outer vortices to overtake the inner vortices at more upstream positions. For instance, overtaking occurs at approximately x/D0≃1.2 for ΔR/R0=10% but shifts upstream to x/D0≃1.0 for ΔR/R0=20% and 30%.

Despite the occurrence of leapfrogging, the vorticity contours in Fig. b–d show that the wake structures remain relatively stable, with no signs of wake breakdown. This finding suggests that, under laminar inflow conditions, triggering the leapfrogging instability is not beneficial for enhancing wake recovery rates. This observation is further supported by the velocity field analyses presented later in Sect. .

In addition to laminar inflow conditions, this study also considers two levels of inflow turbulence intensities, which are TI=0.6% and 5.1%. Simulations are performed for both symmetric rotors and asymmetric rotors with varying blade length differences, as detailed in Table . However, for conciseness, this section focuses on the cases with ΔR/R0=0% and ΔR/R0≃10%, as these configurations are the most representative of leapfrogging behavior, as evidenced by the vorticity contours in Fig. .

Figure a and b show the instantaneous vorticity field ωy for cases with TI=0.6%. For the symmetric rotor case, a comparison between Figs. a and a reveals that even a minor level of turbulence can trigger instabilities, including leapfrogging, ultimately leading to wake breakdown around x/D0=4. This observation is consistent with experimental results from wind and water tunnel studies , where inflow conditions exhibit very low TI but are not perfectly laminar. In contrast, for the asymmetric rotor case with ΔR/R0≃10%, the tip vortex behavior remains largely unchanged between the laminar inflow case (Fig. b) and the case with TI=0.6% (Fig. b) up to approximately x/D0=5. This indicates that the low-level turbulence does not advance the leapfrogging event, a conclusion further supported by the leapfrogging distance analysis in Sect. . Additionally, the coherence of the vortex structures appears to be better maintained in the asymmetric rotor case, with the onset of wake breakdown delayed from approximately x/D0=4 to x/D0=5 compared to the symmetric rotor case. This observation is further validated by the phase-averaged vorticity magnitude |ωy|̃ presented later in Fig. .

The vorticity fields shown in Fig. c and d show that for cases with TI=5.1%, the wake breakdown process initiates significantly earlier for both ΔR values compared to those with TI=0.6%. In both the symmetric and asymmetric rotor cases, wake breakdown begins around x/D0=1.5. Unlike the cases subjected to laminar and low-turbulence inflow conditions, the ωy contours for these two cases become indistinguishable beyond approximately x/D0=2. This demonstrates that, at this higher turbulence level, wake behavior is more strongly influenced by ambient turbulence rather than rotor asymmetry. This observation is further supported by the integral wake velocity analysis discussed in Sect. . However, in the very near wake region (x/D0<1.5), the effects of ΔR remain identifiable. A visual inspection of the tip vortex patterns reveals that rotor asymmetry still promotes the leapfrogging phenomenon, a conclusion that will be more clearly demonstrated through the phase-averaged field presented in Fig. .

In addition to the instantaneous fields, phase-averaged contours of the out-of-plane vorticity magnitude |ωy|̃ are presented in Fig. . The phase-averaging procedure employed in this study is based on the rotor's rotational period, with the phase of interest corresponding to the position where one of the blades points in the positive z direction. Phase-averaged fields are particularly useful for assessing the coherence of periodically varying flow structures, such as those observed in wind turbine wakes . It is important to note that |ωy|̃ contours for cases with laminar inflow conditions are not shown, as they are effectively identical to the instantaneous fields presented in Fig.  due to the strictly periodic nature of these systems.

Figure a and b present the |ωy|̃ fields for the cases with a symmetric and an asymmetric rotor when TI=0.6%. After phase-averaging, the vortical structures observed in both cases with TI=0.6% become more similar to the ωy fields observed under laminar inflow conditions. The disappearance of the structures having concentrated |ωy|̃ indicates the onset of wake breakdown, as this suggests randomization in the positions of the vortices. A particularly interesting observation is that the vortical structures in the asymmetric rotor case appear slightly more coherent than those in the symmetric rotor case, suggesting that the introduced rotor asymmetries slightly delay the breakdown process. Specifically, clear vortex contours persist up to approximately x/D0≃5 in the asymmetric rotor case, whereas they dissipate around x/D0≃3 in the symmetric case. This enhanced coherence may be attributed to the merging process, which increases the circulation strength of the resulting vortices. However, this increased coherence does not necessarily imply that rotor asymmetry has an adverse effect on wake recovery, as the swirling motions induced by stronger vortices may, in fact, enhance energy entrainment via advection . After all, as shown in later sections, the wake recovery rates for cases with TI=0.6% are largely unaffected by rotor asymmetry.

The contours of |ωy|̃ for the cases subjected to TI=5.1% are shown in Fig. c and d. After phase-averaging, the leapfrogging event in the case with ΔR/R0≃10% becomes more clearly delineated compared to the instantaneous ωy fields presented in Fig. d. Additionally, for the symmetric rotor case, the regular pattern of vortical structures is partially restored through phase-averaging. Similar to the observations at TI=0.6%, the coherence of the vortical structures is enhanced by rotor asymmetry, although to a lesser extent at higher turbulence levels. Furthermore, unlike in the TI=0.6% cases, where elevated values of |ωy|̃ persist beyond x/D0=4.0, the regions of high |ωy|̃ for TI=5.1% diminish rapidly, disappearing before x/D0=1.5. These findings are consistent with the observations from the instantaneous fields and further demonstrate that, at higher turbulence intensities, ambient turbulence dominates the wake aerodynamics, making the effects of rotor asymmetries insignificant.

Contours of the instantaneous ωy fields for the cases with a denser mesh listed in Table  are shown in Fig. . These cases include both symmetric and asymmetric rotor configurations (ΔR/R0=9.8%), subjected to laminar and turbulent inflow conditions (TI=5.9%). Despite the slight difference in TI between the cases subjected to turbulent inflow with two different mesh resolutions (TI=5.9% for the dense resolution and 5.1% for the standard resolution), they are directly compared, as the discrepancy in turbulence levels is considered small.

Focusing on the two cases subjected to laminar inflow (Fig. a and b), their overall features are consistent with those observed in the standard mesh cases (Fig. a and b). In particular, a stable vortex tube forms in the symmetric rotor case, while in the asymmetric rotor case, vortex pairs eventually merge to form a secondary helical structure. However, closer inspection reveals that the vortex cores in the denser mesh case (Lam10D) are smaller than those in the standard mesh case (Lam10S), and the consecutive vortices merge at a significantly later stage in both cases. Furthermore, for the asymmetric rotor configuration, the denser mesh case (Lam10D) exhibits a second leapfrogging event following the first, whereas this is not observed in the standard mesh case (Lam10S). This difference can be attributed to the larger vortex core size rω in Lam10S, which leads to earlier vortex merging , thereby naturally stopping the leapfrogging.

For the two cases subjected to turbulent inflow (Fig. c and d), differences in tip vortex dynamics compared to the standard mesh cases (Fig. c and d) are even less pronounced than those observed under laminar inflow conditions. The main distinctions introduced by the denser mesh are smaller vortex core sizes and more finely resolved turbulent structures. Additionally, the phase-averaged fields |ωy|̃ for the two turbulent cases with the dense mesh (not shown) closely resemble those obtained with the standard mesh (see Fig. c and d). These observations support the idea that the presence of atmospheric-level turbulence diminishes the influence of mesh resolution for the current application.

Although the two-dimensional planes presented in Figs.  and  are able to effectively illustrate the vortical structures of an asymmetric rotor, some of their important three-dimensional traits are not clearly depicted. To better visualize the vorticity structures associated with the asymmetric rotor studied in this work, three-dimensional iso-surfaces of the instantaneous vorticity magnitude |ω| from case Lam10D are presented in Fig. . This case is highlighted because it is the only one that clearly exhibits two leapfrogging events, making it particularly illustrative for demonstrating the leapfrogging phenomenon. It should be noted that the case with the standard mesh (Lam10S) has very similar three-dimensional structures but with the merging happening earlier (see the supplementary animations ).

In Fig. , tip vortices shed by different blades are color-coded differently to enhance visualization. The interlaced helical structures are clearly visible, and the leapfrogging event can be easily identified as the tip vortices from different blades exchange their streamwise positions. Additionally, the figure shows that the radii of the helices vary as they undergo the leapfrogging process, which are features that may not be readily apparent from two-dimensional contour plots. Furthermore, Fig.  straightforwardly demonstrates that only global pairing modes are observed with the absence of local pairing modes, which is consistent with the findings of previous studies conducted under similar setups . Note that the absence of a local pairing mode holds for all cases subjected to laminar inflow conditions, as can be confirmed with the supplementary animations .

This part demonstrates how the positions and trajectories of the vortices are identified and tracked, forming the basis for determining the leapfrogging growth rate and leapfrogging distance. For brevity, only cases Lam10S and Lam10D from Table  are demonstrated, with a focus on illustrating the methodology for extracting leapfrogging-related quantities and highlighting the influence of mesh resolution.

To track the positions of the vortices, vortex centroids are defined at locations where local maxima of |ωy| are detected at the initial time step, following the approach described by . Examples of detected vortex centroids are shown by the markers in Fig. a. The vortex trajectories are then obtained by tracking the temporal evolution of these centroids' positions using a technique similar to that employed in particle tracking velocimetry . This method assumes that each vortex travels as a coherent structure and its centroid can always be identified. Starting from the initial positions of the most recently shed vortices, their motion is tracked by searching for the nearest neighbor within a defined radius around the predicted position in the subsequent time frame. This process is repeated until the vortex exits the domain of interest. The resulting trajectories for cases Lam10S and Lam10D (laminar inflow, ΔR/R0≃10%, standard and dense mesh) are shown in Fig. b and d. These trajectories capture the spatiotemporal development of the vortex pairs, revealing their precessional (leapfrogging) motion.

The obtained trajectories (Fig. a and c) show good agreement with the corresponding ωy contours presented in Figs. b and b. In both cases, the outer vortex (indicated by blue pluses) overtakes the inner vortex (indicated by green crosses) at approximately x/D0=1.2. As previously noted, in the standard mesh case (Lam10S), the vortex pairs rapidly merge following their first leapfrogging event. In contrast, for the dense mesh case (Lam10D), the two vortices remain clearly distinguishable, and the onset of a second leapfrogging event is observed.

After capturing the vortex trajectories, key quantities, including the temporal evolution of the vortex pair's separation vector, can be extracted. These measurements are subsequently used to determine the leapfrogging distance xLF, leapfrogging time tLF, and leapfrogging growth rate σLES in the following sections.

Before analyzing the leapfrogging instability, two characteristic parameters, which are the initial streamwise spacing between consecutive vortices, h0, and the circulation strength, Γ (see Fig. ), are measured and reported in Table , along with the characteristic helical timescale tHel≡2h02/Γ. Note that the vortex core size rω is also reported for completeness, which is determined based on the tangential velocity profile around the vortex centroid (see Sect.  for details). For consistency, all LESs in this study use the same values of h0, Γ, and tHel, which are based on those of case Lam10S (ΔR/R0≃10%, standard mesh, laminar inflow). That is, despite differences in rotor asymmetry, inflow conditions, and mesh resolutions, all cases share these characteristic parameters. Unless otherwise specified, h0 and Γ are measured from the tip vortices shed by the un-truncated blade at the moment when the vortex age is half a rotor rotational period, which are those enclosed by the magenta circles in Fig. . In this work, h0 is determined as the streamwise distance between that vortex centroid and the rotor plane, while Γ is obtained by evaluating a circular line integral having a diameter of h0, centered on that vortex centroid in the y plane . Also, note that torsional effects due to helical pitch are not accounted for in these measurements.

In Table , Γ and rω of the truncated blade are also measured at the moment when its vortex age is half a rotor rotational period. The results show that the relative difference in circulation strength between the truncated and un-truncated blades is within 1% for both mesh resolutions, confirming that the assumption Γtruncated=Γun-truncated holds fairly well. Additionally, relative differences between rω,truncated and rω,un-truncated are also merely around 3%, further justifying the assumption that the tip vortices shed by the truncated and un-truncated blades are identical. Furthermore, the relative differences in h0, Γ, and tHel between different mesh resolutions are approximately 2%, 3%, and 1%, respectively. These small differences indicate that treating these parameters as constant across different mesh resolutions is well justified. On the other hand, a discussion on the differences in rω found in different mesh resolutions is provided in Sect. .

Following the notations defined in the 2D vortex model (see Fig. ), the evolutions of the vortex pair for cases Lam10S and Lam10D are presented in Fig. . The angle between the vortex pair and the streamwise direction β, the separation distance b, the streamwise separation h0-δh, and the radial separation δr are plotted against the streamwise location of the vortex pair's centroid (averaging the centroids of the two vortices).

In Fig. , both β and δh are observed to increase after the vortices are released from the rotor at x/D0=0, indicating that the outer vortex of the pair gradually overtakes the inner vortex. When δh=h0 and β=90°, the vortex pair becomes aligned in the radial direction, and the corresponding streamwise location is defined as the leapfrogging distance in this study, which is denoted as xLF. Additionally, δr increases with x and reaches its maximum near xLF. The decreasing trend of b before the leapfrogging event demonstrates that the vortices move closer together. At the moment when only a single vorticity peak is detected within the search radius, b is set to 0, indicating that the vortices have merged. Notably, before merging, only one leapfrogging event is identified in the case with the standard mesh, whereas two leapfrogging events are found in the case with the dense mesh, as indicated by β exceeding 270° prior to merging. This behavior is consistent with the contour plots shown previously (Figs. b and b) as well as the previous findings , which show that the detail tip vortex dynamics after the first leapfrogging event, including the merging process, are sensitive to the vortex core size rω and fluid diffusivity.

For both cases, the initial value of δr is expected to match the imposed radial offset ΔR, which is 10% of R0, for the cases shown in Fig. . However, a discrepancy of approximately 0.04R0 between the initial δr and the prescribed ΔR is observed. This deviation can likely be attributed to wake expansion. Specifically, by the time the subsequent vortex is shed from the un-truncated blade, the previously shed vortex from the truncated blade has already moved outward, reducing the effective radial separation between them. Additionally, the growth rate of δh is found to be nearly twice that of δr, deviating from the eigenvector prediction of [1;1] derived through linearization in Appendix . This difference is likely due to the influence of initial conditions and the presence of non-linear effects in the vortex dynamics.

As defined in Sect. , the instability growth rate σLES is determined based on the L1 norm of the measured δh and δr, denoted as ||η||1. Figure  presents a semi-log plot of ||η||1 vs. time, with data from case Lam10S, case Lam10D, and the 2D vortex model. Overall, the σLES values obtained from the LES data show good agreement with the predictions of the 2D vortex model prior to the leapfrogging event. Moreover, the ||η||1 curve for the case with the denser mesh (Lam10D) aligns more closely with the 2D vortex model results. This improved agreement is likely due to the smaller grid size Δ and reduced smoothing factor ϵ, which lead to smaller vortex core sizes rω and cause the vortices to behave more like idealized point vortices. Additionally, the smaller grid size reduces the subgrid-scale viscosity νsgs in Eq. (), making the simulated flow more closely resemble the inviscid conditions, which is also a key assumption imposed by the 2D vortex model.

A monotonic increase in ||η||1 is observed before the leapfrogging time tLF for both Lam10S and Lam10D. Furthermore, consistent with the behavior predicted by the 2D vortex model, distinct linear regions can be identified, indicating that ||η||1 obtained from the LESs also grows exponentially with time. These linear segments enable the effective determination of the leapfrogging instability growth rate based on the slopes obtained from the LES data, denoted as σLES. Additionally, the curves in Fig.  show a transient phase that precedes the linear growth region, followed by a decrease in slope after the leapfrogging event. This behavior is in good agreement with the evolution of streamwise vortex separation documented in previous studies by and . For consistency, both σ2D and σLES are determined based on the time interval 0.6≤t/tHel≤0.8, during which the ||η||1 curves exhibit the most stable exponential growth phase.

The growth rates σLES obtained under different inflow conditions and blade length differences (ΔR) are plotted in Fig. , accompanied by the σ2D predicted by the 2D vortex model. For the turbulent inflow cases, the evaluation is based on 90–100 vortex pairs. The averages and standard errors of σLES are calculated, with the resulting 95% confidence intervals being less than ±0.1% for TI=0.6% and up to ±9.7% for TI=5.1% relative to their respective means. It should be noted that leapfrogging events are not always detected for every vortex pair under turbulent inflow conditions. Specifically, for cases with ΔR/R0≃10%, the detection rate of leapfrogging is approximately 97% when TI=0.6% and around 75% when TI=5.1%. In the instances where leapfrogging is not detected, the corresponding data are still included in the statistical analysis of σLES, but they are excluded from the calculations of leapfrogging time tLF and leapfrogging distance xLF.

Under laminar inflow conditions, the normalized growth rates σLEStHel (represented by red crosses in Fig. ) are observed to decrease with increasing ΔR, from 1.71 to 1.22. This trend aligns well with the predictions of the 2D vortex model, which drops from 1.53 to 1.01, demonstrating that the LES results are consistent with the theoretical framework. However, systematic deviations between σLES and σ2D are evident. These discrepancies are expected, given that the 2D vortex model incorporates several simplifying assumptions, such as neglecting the three-dimensional effects, presence of the hub vortices, spatial wake development, and finite vortex core size. In particular, the difference between the LES-derived growth rates and the model predictions is approximately 10% at small ΔR. Specifically, the approximately 10% higher σLES at smaller ΔR can be quantitatively attributed, where the three-dimensional effects contribute around 3% (as shown in Fig. ), while the omission of hub vortices accounts for another 5% (based on the analysis by ).

Figure  also shows that inflow turbulence intensity has a limited effect on σLES. For cases with TI=0.6%, the average values of σLES differ by no more than 3% compared to the corresponding laminar cases. In cases with TI=5.1%, the average values of σLES differ by up to 7%, but the associated uncertainties are notably larger.

Lastly, the results indicate that variations in mesh resolution do not significantly impact the obtained growth rates. Specifically, the relative differences in σLES between the two mesh resolutions are 6.2% for the cases subjected to laminar inflow conditions (cases Lam10S and Lam10D) and only 1.5% for the cases with turbulent inflow conditions (cases T10S and T10D).

To the author's best knowledge, this is the first study to demonstrate that the leapfrogging growth rate is largely insensitive to ambient turbulence. Given that perturbations introduced by inflow turbulence are known to trigger leapfrogging instability , one might have expected these fluctuations to significantly amplify the leapfrogging growth rate. However, the present results show that the growth rate remains nearly the same as that observed in cases with laminar inflow. A possible explanation is that the vortex dynamics in the very near wake of the wind turbine (x/D0<1.0) is relatively unaffected by inflow turbulence, as evidenced by the phase-averaged vorticity fields presented in Fig. . To further elucidate the interactions between the leapfrogging modes, mean flow, and stochastic fluctuations, future studies involving modal analyses may offer valuable insights .

The leapfrogging distance, xLF, is defined as the distance from the rotor plane to the first downstream location where the vortex pairs exchange their streamwise positions. In the LESs, these distances are determined based on the tracked positions of the vortex centroids, as described in Sect. .

In Fig. a, it can be observed that under laminar inflow conditions (marked by red crosses), the leapfrogging distance xLF decreases asymptotically from approximately 2D0 and levels off around 1D0 as ΔR/R0 increases from 2.5% to 19.5%. This trend compares well with the experimental observations reported by . This asymptotic behavior can be explained by Eq. (), where dδh/dt increases with increasing δr, eventually saturating at Γ/2h0 as δr≫h0 (note that dδr/dt→0 as δr≫h0). Additionally, the results indicate that leapfrogging occurs earlier (i.e., at a shorter xLF) with greater rotor asymmetry, despite the fact that the growth rate σLES decreases with increasing ΔR. This seemingly contradictory relationship between xLF and σLES can largely be attributed to the initial conditions. As noted earlier, both ||η||1 and dδh/dt at t=t0 are higher for cases with larger initial δr (i.e., larger ΔR), making the early system evolution more dependent on the initial conditions rather than on the intrinsic instability growth rate.

Similar to σLES, both inflow turbulence intensity TI and mesh resolution are found to have minimal effects on the leapfrogging distance xLF. Particularly focusing on when TI differs, the leapfrogging distances when TI=0.6% (green circles in Fig. a) are very close to those obtained with laminar inflow conditions, while when TI=5.1% (blue triangles in Fig. a), the data become more scattered, with mean values of xLF being less than 6% lower than those subjected to laminar conditions. To conclude, the minimal deviation in xLF and σLES between laminar and turbulent inflow conditions suggests that inflow turbulence has minor effects on the near-wake tip vortex behavior for the asymmetric rotor studied in this work, particularly regarding the leapfrogging phenomenon. However, it should be noted that inflow turbulence still plays an important role in triggering secondary instabilities of the vortex helix, as widely reported in the literature , and this can also be identified in the animations provided in the accompanied data repository .

Figure b plotted the leapfrogging time tLF obtained through different methods against ΔR. The results from the 2D vortex model show good agreement with the LESs under laminar inflow conditions, with elative deviations of no more than 7% across the range of ΔR considered. A similar level of agreement has also been reported in . Furthermore, analogous to the behavior of xLF, the value of tLF approaches an asymptotic value as ΔR surpasses 0.5h0. This asymptotic value corresponds to Γ/2h02, which can be explained by Eq. (), where dδh/dt approaches with a velocity of Γ/2h0 as δr≫h0. Dividing h0 by this velocity yields 2h02/Γ, which is the time needed to complete the first leapfrogging event. This relationship was previously highlighted by and forms the basis for using tHel≡2h02/Γ as the characteristic timescale in this study. As shown in Fig. b, tHel indeed serves as the asymptotic value for tLF as ΔR increases for both the LESs and the 2D vortex model. This agreement further demonstrates that the methodologies and analysis techniques employed in the current work are well-grounded in established theoretical frameworks.

In addition to the numerical results and theoretical predictions, two data points from wind tunnel experiments are also provided for comparison in Fig. b. The experiments were conducted in the low-speed wind tunnel at Delft University of Technology (W-tunnel), which was configured as an open jet with a cross-sectional area of 60cm×60cm at the outlet. The tested inflow velocity was approximately 5.4 ms-1, with a turbulence intensity below 2%. Particle image velocimetry (PIV) was employed to capture the flow field. The rotor model used in the experiments was a four-bladed configuration with a diameter of 30 cm, operating at a tip speed ratio of 3.5. This setup results in a helix geometry comparable to that of the LESs, with both configurations featuring h0/D0≃0.2. The trajectories of the tip vortices in the experiments were tracked using the same method applied to the LESs (see Sect. ). The measured circulation strength of the tip vortices in the experiments was approximately 0.08 m2s-1.

As shown in Fig. b, the experimental data align remarkably well with the predictions of the 2D vortex model and the results of the LESs, despite the Reynolds numbers based on the diameter (U∞D0/ν) and the circulation (Γ/ν) being several orders different. This strong consistency between the theoretical predictions, numerical results, and experimental measurements further reinforces the validity of the findings of the present study.

Downstream of a symmetric rotor subjected to laminar inflow conditions (Fig. a), a clear region of significant velocity deficit, known as the wake, is observed. After wake expansion ceases around x/D0=1, the shear layer remains stable further downstream, consistent with the ωy fields shown in Fig. a, where undisturbed tip vortices form a stable vortex tube. This observation aligns well with the findings of and . When a blade length difference of ΔR/R0≃10% is introduced (Fig. b), the overall characteristics of the u‾ field remain similar to those of the symmetric rotor case, suggesting that the wake recovery rate is not significantly affected by this level of rotor asymmetry.

For cases subjected to TI=0.6%, the shear layer exhibits continuous downstream spreading for both symmetric and asymmetric rotors, as shown in Fig. c and d. When the turbulence intensity increases to TI=5.1%, the spreading becomes even more pronounced, as observed in Fig. e and f. Overall, it is evident that the extent of the wake deficit is more strongly influenced by TI than by ΔR. In particular, higher turbulence intensity leads to more rapid shear layer spreading, while the impact of rotor asymmetry remains minimal. As illustrated in Fig. , ambient turbulence disrupts the tip vortex helices, which typically act to suppress wake breakdown . The breakdown of these vortex helices enhances momentum exchange between the wake and the surrounding flow, promoting lateral spreading of the velocity deficit and reducing the peak momentum deficit along the wake centerline.

To further investigate the recovery of momentum deficit, velocity profiles at x/De=2, 5, and 8 downstream are presented in Fig. , where De denotes the effective diameter (see Table ).

At x/De=2 (Fig. a), all cases exhibit the characteristic near-wake velocity profile of a wind turbine, with a pronounced velocity deficit around 0.65U∞ at the center and higher velocities near the hub region. Strong velocity shear is present in both the tip and hub regions, indicated by sharp velocity gradients. Under laminar inflow conditions (black lines) and at TI=0.5% (red lines), the velocity profiles show intricate but noticeable sensitivity to rotor asymmetry, with stronger shear at the tips for the symmetric rotor. This is attributed to periodic loading variations near the tip caused by blade length differences, which reduce the sharpness of the time-averaged loading gradient in those asymmetric cases. However, as the turbulence intensity increases to TI=5.1% (blue lines), the influence of rotor asymmetry becomes unidentifiable. For both the wakes of a symmetric and an asymmetric rotor, the velocity gradients become more gentle, and the velocity deficits become milder with stronger ambient turbulence, indicating enhanced momentum diffusion and accelerated wake recovery.

As the flow progresses from x/De=2 to 5 and then to 8, the spatial evolution of the wake can be further examined through the velocity profiles.

When subjected to laminar inflow conditions, noticeable changes when moving downstream primarily occur near the hub region for both symmetric and asymmetric rotors, where the initially sharp velocity gradients gradually smooth out with increasing downstream distance. However, the overall structure of the wake remains largely unchanged, as clearly demonstrated by the velocity contour plots in Fig. a and b, which show minimal wake development and recovery under laminar inflow conditions.

When TI=0.5%, the wake progressively evolves toward the well-known self-similar profiles described in the literature , as clearly observed in the u‾ profiles at x/De=5 and x/De=8. The concave velocity deficit near mid-span flattened due to enhanced turbulent mixing, with the symmetric rotor case showing slightly more advanced development. At TI=5.1%, the effects of turbulent mixing become even more pronounced. The inflection points in the velocity profiles vanish as early as x/De=5, and the velocity deficit spreads further outward in the radial direction, indicating a stronger mixing process and accelerated wake recovery.

In this study, the disk-averaged mean streamwise velocity, 〈u‾〉disk, is employed as an integral metric to evaluate wake recovery. It is calculated by averaging the mean streamwise velocity, u‾, over a circular disk with an effective diameter De. This metric provides an integrated measure of the overall streamwise velocity and serves as an estimator for the wind resource available to a downstream turbine. A more rapid increase in 〈u‾〉disk along the streamwise direction can be interpreted as stronger and faster wake recovery.

Figure  presents the evolution of 〈u‾〉disk at various downstream locations for cases with different rotor asymmetries, inflow turbulence intensities, and mesh resolutions. It is important to note that the characteristic length scale used is the effective diameter, De, which varies slightly between cases depending on rotor asymmetry (see Table ). For all cases, 〈u‾〉disk initially drops to approximately 0.7U∞ just behind the rotor, reflecting the energy extraction by the turbine. Under laminar inflow conditions, little to no increase in 〈u‾〉disk is observed for both symmetric and asymmetric rotors, indicating limited momentum recovery. When low-level inflow turbulence (TI=0.6%) is introduced, noticeable recovery is evident. At TI≃5%, the increase in 〈u‾〉disk becomes even more pronounced. However, across all turbulence levels, no significant differences are observed between symmetric and asymmetric rotors, suggesting that rotor asymmetry has minimal influence on wake recovery in the presence of turbulence.

The trend observed in 〈u‾〉disk is predominantly influenced by the turbulence intensity (TI) rather than rotor asymmetry. As TI increases, the velocity recovers more rapidly, indicating more effective wake recovery. This observation is consistent with previous studies in the literature . When these findings are combined with observations with the ωy contours (Figs.  and ) and phase-averaged vorticity magnitude fields |ωy|̃ (Fig. ), a clear relationship emerges between the leapfrogging and the wake recovery process. Specifically, the rotor asymmetries considered in this study trigger the leapfrogging phenomenon but do not promote wake breakdown. Instead, new stable helices formed from paired tip vortices are established in the wake of an asymmetric rotor, resembling the helical structures seen in symmetric rotor cases. These stable structures (structures in equilibrium, to be precise) again delay mixing between the wake and the surrounding freestream. As a result, with the absence of vortex breakdown, which can be induced by ambient turbulence or other mechanisms, the wake recovery rate is not significantly enhanced by rotor asymmetries alone.

The findings presented in this study appear to contrast with those reported by , who conducted experiments using a two-bladed wind turbine rotor with uneven blade pitch angles to trigger leapfrogging. In their work, they concluded that leapfrogging accelerates wake recovery by promoting wake breakdown. However, the cases with laminar inflow presented in this study indicate that leapfrogging alone does not necessarily accelerate wake recovery, which is supported by the profiles of 〈u‾〉disk (Fig. ) and the vorticity contours (Fig. ). This discrepancy can likely be attributed to the presence of other secondary instabilities in the experiments conducted by , such as the local vortex pairing observed by . These secondary instabilities can arise from unavoidable experimental perturbations, including structural vibrations, turbulence originating from blade boundary layers, and rotor imperfections. As these instabilities develop, they disrupt the coherence and symmetry of the tip vortex helices, potentially serving as the primary drivers of wake breakdown . Therefore, the sudden momentum ingestion observed by may not be entirely attributed to leapfrogging but may instead be only correlated with it. This interpretation is further supported by the current simulation results. When under laminar inflow conditions, 〈u‾〉disk in Fig.  remains relatively constant, while in cases with TI=0.6%, it begins to increase after xLF. Therefore, combining this observation and the vorticity contours provided in Figs.  and , it can be concluded that the driver of the wake breakdown process is turbulence-induced instabilities, rather than leapfrogging alone.

The present findings also diverge from those of , who, using vortex modeling, reported an 11% increase in available power (area-averaged u‾3 over an area of πR02) for an asymmetric rotor (ΔR/R0≃2%) compared to a symmetric rotor at a downstream distance of 7D0 when subjected to laminar inflow conditions. Several factors are likely to contribute to this discrepancy. Notably, their study employs a three-bladed rotor and includes the floor in the simulation setup, whereas the current work uses a two-bladed rotor and excludes the floor. These differences are known to influence the vortex dynamics in wind turbine wakes, as they affect rotor loading, the pitch between successive tip vortices (i.e., the equivalent h0), and the overall symmetry of the system . Additionally, the numerical frameworks used in the two studies differ substantially. It is worth noting that both the LES approach with actuator lines used in the present study and the vortex model with lifting lines employed in their work are relatively sensitive to the choice of model parameters . These include spatial and temporal discretization, vortex core treatment, and diffusion modeling. All of the factors discussed in this paragraph may each partially contribute to the differences in the reported outcomes, and a precise identification of their individual effects is left for future work.