1. Introduction
Aircraft wake vortices are operationally relevant flow phenomena for all phases of flight. They are unavoidable by-products generated by any flying aircraft (Gerz et al. 2002) and can pose an invisible hazard to the following aircraft landing on that same runway (Zheng and Ash 1996; Holzäpfel and Steen 2007; Stephan et al. 2013; Holzäpfel et al. 2016). Wake vortices may be safety critical during all flight conditions; however, during approach and landing, the risk of a wake vortex encounter is most prominent: Aircraft are closely spaced and share similar glide paths (GPs), and their generated wake vortices interact with the ground, resulting in a boundary layer with detaching secondary vortices (Hallock and Holzäpfel 2018). Additional atmospheric conditions, particularly weak crosswinds, may cause the generated wake vortices of a landing aircraft to stall above the runway (Stephan et al. 2013). Some required pilot reaction times for avoiding fatal incidents at low altitudes are beyond human capability. Wake vortex safety considerations were first implemented in flight operations in the 1970s, when the International Civil Aviation Organization (ICAO) increased the minimum landing separation from the minimum radar separation for certain ICAO leader–follower aircraft pairs (Gerz et al. 2002).
With these separations of landing aircraft, the throughput at airports is limited. Since the introduction of the separation minima, flight demand has continuously increased, and also in the future, air traffic is expected to grow further (EUROCONTROL 2022). Thus, for economical reasons, several new separation programs have been suggested and introduced in the past years. For instance, a time-based separation approach (De Visscher et al. 2020) has been introduced at London Heathrow Airport (Morris et al. 2013), and phase 1 of the European Wake Vortex Recategorisation (RECAT-EU) program (RECAT-1) (Rooseleer and Treve 2018) has been introduced at several airports, such as Paris Charles de Gaulle Airport (Ministère de la Transition Écologique et Solidaire 2018). RECAT-2 considers individual aircraft type pair separations, and RECAT-3 ultimately aims at dynamic pairwise separations considering the effects of the prevailing atmospheric conditions on wake vortex behavior. For realizing these novel separation programs, both theoretical and experimental studies are necessary, evaluating the feasibility, the potential benefit, and maintenance of safety. Equally, wake vortex monitoring plays an important role for urban air mobility (UAM) operations of vertiports in the airport vicinity. In this work, we focus on experimental wake vortex studies, which are commonly performed with light detection and ranging (lidar) instruments in cloud-free conditions (Barbaresco et al. 2015; Oude Nijhuis et al. 2018).
Unlike with field measurements, three-dimensional velocity data as well as all other relevant flow parameters are available when performing wake vortex simulations. Within the wake vortex simulations, virtual lidar can be modeled, providing a comprehensive wake vortex lidar dataset. Simulations of atmospheric measurement instruments in the context of wake vortex detection initiated in the 1970s (Thomson and Meng 1976), branching off to onboard configurations and ground configuration, suiting our purpose. Within this field, studies either focus on fully simulating the operating of a lidar or on simulating wake vortices in full detail. It started with the former where white noise or random backscatter spectra were assumed (Rye 1990). Half a decade later, Salamitou et al. (1995) investigated different atmospheric conditions, altering the backscatter spectrum accordingly, and Frehlich (1997) analyzed the condition’s effect on the lidar performance. The focus on the wake vortex simulation in the context of lidar simulations was, to the authors’ knowledge, first done by Holzäpfel et al. (2003), where Navier–Stokes numerical simulations of a Lamb–Oseen vortex pair (Lamb 1923) and an aircraft generated multivortex system were performed. However, lidar simulations were focused on kinematic wake vortex characteristics and lidar geometry. Frehlich and Sharman (2005) added lidar parameters such as range gate, angle scan rate, wavelength, and pulse duration. Further lidar scanning, processing, and atmospheric scattering parameters were simulated in a large-eddy simulation (LES) by Wassaf et al. (2011) and Jacob et al. (2015). Still today, lidar simulations for wake vortex characterization are performed with various levels of detail. Typically, the simulated lidar is the focus of the studies, as in Lugan (2016), Gao et al. (2018, 2019), Li et al. (2020), and Wei et al. (2024), or the generation of the wake vortices with high-fidelity aircraft simulations (Stephan et al. 2019a). Each approach is typically limited by the incompleteness of the other aspect.
This paper, therefore, has the aim to join the two aspects, by implementing virtual lidar instruments including the most crucial lidar parameters for geometry and operation and planting these into high-fidelity wake vortex simulations. This allows us to evaluate the accuracy of the RV method, contextualizing previous studies and further facilitating the development of increasingly advanced data-driven processing algorithms for the characterization of wake vortices. The objective is to develop virtual lidars within high-fidelity hybrid Reynolds-averaged Navier–Stokes (RANS) and LESs of landing aircraft. The therefore called LES lidar simulator (LLS) enables the assessment of virtual lidar scans using the RV method and comparing the vortex characteristic estimates to the simulation truth (ST), which under certain limits we consider to be the ground truth as the three-dimensional field of all relevant flow parameters is available in the simulations for evaluating the position and strength of the wake vortices.
The remainder of this paper is structured as follows: In section 2a, lidar operating for wake vortices and the RV method are outlined; thereafter, section 2b describes the underlying numerical simulations. The LLS is introduced in section 2c. In section 3a, we establish the validity of the numerical simulations. The analysis continues with discussing the LLS measurements in section 3b. Finally, in section 3c, we compute and segment RV method accuracy levels. Note that a precursor of this paper has been presented at the 2023 American Institute of Aeronautics and Astronautics (AIAA) Aviation Forum in San Diego (Wartha et al. 2023).
2. Methods
a. Pulsed coherent Doppler lidar principles
1) Measuring wake vortices
Pulsed coherent Doppler lidars (PCDLs) have become the instrument of choice for measuring wake vortices with a high spatial resolution at airports. PCDLs and lidars are used interchangeably in this paper. Typically, range–height indicator (RHI) scans are conducted perpendicular to the airport runway. Figure 1 illustrates an emitted laser beam from the lidar. Upon interaction with the aerosols (air particles) in the atmosphere, a backscatter is received from which aerosol motion at different ranges (distances) R from the lidar can be detected from the spectrum of Doppler shifts in frequency. Aerosols approaching and moving away from the lidar are associated with negative and positive LOS velocities Vr, respectively. During an RHI scan, the LOS continuously changes its elevation angle φ at a predetermined scanning rate ωS between a minimal and maximal elevation angle φ− and φ+, respectively. Once φ− or φ+ has been reached, ωS is reversed, and a new RHI scan is initiated. RHI lidar scans consist of LOS velocities recorded at (R, φ) in a polar coordinate system. The lidar coordinate system is referred to with a dash (′) in this paper.
Lidar performing RHI scans perpendicular to the runway, with an aircraft flying out of the page, generating a starboard (Str) and a port (Prt) vortex at
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
2) Characterizing wake vortices
The flow field can be decomposed as a sum of the background wind and the turbulent field. Therefore, a radially averaged lidar RHI scan prior to the passage of the aircraft (without wake vortices) is subtracted from the scans. Although a temporal delay between the scans exists, temporally constant atmospheric structures such as crosswinds can be captured, and their influence on the characterization reduced.
b. Numerical wake vortex simulation methods
Four high-fidelity RANS–LES runs of the wake vortices generated by aircraft during final approach and landing under varying meteorological conditions are performed. Below key points of the method by Stephan et al. (2014) are recapitulated, particularly focusing on adaptations of their method.
Fundamental parameters of the wake vortex simulations are summarized in Table 1, detail follows in the below sections. Normalization parameters, the initial/root circulation Γ0, the initial vortex spacing b0, the initial vortex descend velocity w0, the characteristic time t0, and the vorticity unit ω0, assume an elliptic load distribution (Gerz et al. 2002). Parameters normalized throughout this paper are marked with an asterisk (*).
Wake vortex simulation parameters. Aircraft parameters are quoted for the scale 1:1, whereas appropriately, the 1:27 model parameters are given in parentheses. Values used for normalization are marked with an asterisk, and their definition can be found in Gerz et al. (2002). Note that formulas are only given for actively computed parameters in this paper.
1) Governing equations
2) RANS flow field integration
Performed simulations are one-directional RANS simulation and LES couplings, allowing the investigation of all wake vortex evolution phases (see Breitsamter 2007, chapter 2) in ground proximity: near field, extended near field (roll-up phase), mid-/far field (vortex phase), and decay phase. This coupling was first introduced by Misaka et al. (2012), with the addition to simulate landing aircraft by Stephan et al. (2014).
The RANS simulation is performed for a 1:27 model (reducing computational demand) of an Airbus A340 aircraft in high-lift configuration and a freestream velocity of U∞ = 25 m s−1, with an aspect ratio Λ = 9.3 and a lift coefficient CL = 1.4 using the Deutsches Zentrum für Luft- und Raumfahrt (DLR)’s TAU code (Schwamborn et al. 2006; Keye 2011). To achieve realistic wake vortex lidar scans, the RANS solution dimensions are scaled by a factor of 27, and a realistic flight speed of U∞ = 75 m s−1 (Breitsamter 2007, page 34) is attained by scaling velocities by a factor of 3. This gives a wingspan of B = 60.4 m. The scaling yields a realistic Reynolds number of Re ≈ 30 × 106 and leads to larger vortex core radii, which we assume relatively less significant in comparison to lidars mollifying high velocities and steep gradients of the vortex cores.
The A340 landing is thus realized in two steps. First, wake initialization is performed in a RANS–LES run, where the RANS flow field represents a forcing term in the LES Navier–Stokes equations. Second, the forgoing simulation is temporally extended in pure LES for observing the long-term behavior of wake vortices.
3) Boundary conditions and computational domain
Side boundary conditions of the LES computational domain are periodic, such that vortices can be tracked for great lateral distances and avoid wall effects. The front and back boundary conditions are periodic for achieving an aircraft initialization which adheres to Helmholtz’s vortex theorems (Kundu and Cohen 2002, page 134) as described in Stephan et al. (2014). The bottom and top boundary conditions are nonslip to represent the ground and free-slip, respectively.
The LES computational domain is set to
A fine and uniform grid spacing
In this work, both
For reference, the following notes some key aircraft positions and times. On domain entrance,
4) Ambient turbulent wind
Below an altitude of 1.5b0 (Robins et al. 2001), wake vortices induce vorticity of opposite sign onto the ground, creating a boundary layer (Harvey and Perry 1971). Once the pressure inside the boundary layer is high enough, secondary vortices detach and interact with the primary wake vortices, causing their diverging lateral transport and rebound (Zheng and Ash 1996; Spalart et al. 2001). When a crosswind is present, the lee (downwind) and luff (upwind) secondary vortices are strengthened and attenuated by the wind, respectively. In ground vicinity, a weak crosswind, matching the vortex-induced lateral propagation speed, 0.5w0 = 0.73 m s−1 ≈ 1.4 knots (kt; 1 kt ≈ 0.51 m s−1) (“light air” on the Beaufort wind scale, associative with “still air” conditions at airports) (Holzäpfel and Steen 2007; Stephan et al. 2013; Holzäpfel et al. 2016), can lead to the luff primary wake vortex remaining in the glide path for significant periods of time.
Wake vortex simulations of the landing A340 are performed in a quiescent atmosphere and for turbulent crosswinds at strengths of 0.5w0, 1.0w0, and 2.0w0. In the case without wind, no ambient turbulence is present. Studying a crosswind strength of 1.0 w0 is operationally relevant; London Heathrow Airport records most wake vortex encounters at this crosswind strength (Critchley and Foot 1991). A crosswind strength of 2.0w0 is responsible for enhanced boundary layer turbulence, disturbing the coherency of wake vortices. Beyond 2.0w0 crosswinds, wake vortices advect quickly out of relevant runway areas, inhibiting serious wake vortex encounter threats.
For the crosswind cases, presimulations are performed, introducing turbulent velocity fluctuations and a physical boundary layer in the LES. The implementation is described in Stephan (2014), and below key steps are recapitulated. A typical half-channel flow is simulated (see e.g., Kundu and Cohen 2002, chapter 13.11) by imposing a pressure gradient dp/dy set such that the mean velocity of the wind at the altitude b0 is the desired multiple χ of w0. In combination with the nonslip bottom boundary, a logarithmic vertical wind profile of υ+(z+) = ln z+/κ + B [κ = 0.4 is the von Kármán constant (Baumert 2013) and B = 5.5] is achieved once characteristic wall vorticity streaks establish and the mean vertical velocity profile
Figure 2 confirms the adherence to the logarithmic turbulent boundary layer described by Moser et al. (1999). The simulated winds contain a marginal underestimation of the average wind speed. The larger variability at low altitudes for the 2.0w0 case is caused by the vorticity streaks detaching due to high pressure gradients at higher wind speeds. Generally, the 2.0w0 case is inherent to higher turbulence as illustrated by the root-mean-square (RMS) of the velocity fluctuations in Fig. 3. The crosswind deviates from the logarithmic trend when approaching the free-slip boundary at the top of the computational domain. The presented wind statistics are satisfactory for the complex flow behind a landing aircraft, particularly when considering the sinking aircraft altitude throughout the simulation and the particularly large domain height—flow effects in this region are negligible for the wake vortex development.
Mean velocity profile in wind direction ⟨υ+⟩ with respect to distance above the ground z+ of fully developed crosswinds for three strengths χw0. The expected profile is given by the dashed line.
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
Mean RMS fluctuations of all velocity components [uu, υυ, ww]+ with respect to distance above the ground z+ of fully developed crosswinds for three strengths χw0. Line colors are analogous to Fig. 2.
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
5) LES vortex tracking
The ground truth position and circulation of the wake vortices in the LESs are obtained via a pressure–vorticity algorithm. The vorticity in flight direction ωx and the pressure p suffice when vortices are strong enough, coherent, and tilting is limited (Hennemann 2010; Stephan 2014). Since RHI lidar scans are limited to flow field slices and also do not consider vortex tilting, the pressure–vorticity algorithm is allowed up until a limited degree of vortex tiling. Above this degree, using only ωx for the characterization underestimates vortex circulation.
c. LLS method
Computationally feasible numerical wake vortex simulations in LES do not offer a sufficient spatial resolution for explicitly performing virtual measurements of scattering along a lidar beam (Robey and Lundquist 2022). Thus, the LLS method focuses on the volume averaging within the scanning volume of the lidar backscatter, by considering the range gates and pulse shape with a range gate weighting function (RWF) analogous to Eq. (8) (Simley et al. 2014; Lundquist et al. 2015; Meyer Forsting et al. 2017; Robey and Lundquist 2022) leading to the low-pass filter smoothing characteristic of a lidar.
This approach is appropriate when the LOS velocity can be estimated accurately via the center of gravity of the Doppler spectrum, which is valid for large signal-to-noise ratio (SNR) values (Banakh and Smalikho 2013, 1997; Frehlich 1997; Robey and Lundquist 2022). On the contrary, when the SNR is small (a great noise component exists), the position of the maximum in the Doppler spectrum allows the most accurate LOS velocity estimation (Stephan et al. 2019b). Although for a PCDL the SNR is typically small, we employ the idealized first approach to circumvent the computation and storage of spectral data. In this manner, we are able to estimate the accuracy of wake vortex processing algorithms, such as the RV method, excluding instrument noise.
1) LOS velocity retrieval
Virtual lidars are installed perpendicular to the runway in the LES computational domain, i.e., y‖y′; thus, the velocity component u can be disregarded for computing Vr. For computational efficiency, a padding method is implemented in less relevant regions of the lidar scans, where LOSs reach beyond the computational domain as sketched in Fig. 4. In field 0, values of V are directly retrieved from the LES. In the horizontal regions outwith of the domain (fields 1 and 2), the z component of the desired measurement point remains unchanged, while the y component is changed to the lowest node in the domain y− and highest node in the domain y+, respectively. This selects V at the domain boundaries to points of the same z. Similarly, field 3 operates in the vertical sense, selecting V from y and z+. Last, fields 4 and 5 select values from (y−, z+) and (y+, z+), respectively. A decreasing weighting factor is applied with increasing distance to the LES domain boundary, avoiding the stimulation of extreme velocities.
Schematic of the lidar and LES coordinate systems. The LES domain is drawn in red, and the lidar measurement window is drawn in black. Field 0 represents the region where velocities can be retrieved from the appropriate location in the LES domain. Fields 1–5 represent regions where flow field padding is applied.
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
2) Application of the RWF
3) LLS specifications
The LLSs in this study aim to model idealized lidars in the style of the Vaisala (formerly Leosphere) WindCube 200S (λ = 1.54 μm). The Vienna International Airport campaign (Holzäpfel et al. 2021) employed WindCubes at five locations along the glide path. Lidar parameters in Table 2 in front of the semicolon indicate virtual lidars modeling the aforementioned campaign. Virtual lidars after the semicolon aim to generalize the LLS analysis further.
LLS parameters. The
In Vienna, all lidars were placed on the starboard side of the glide path. We additionally position LLSs on the port side. By mirroring scans from port side LLSs about the glide path axis and swapping starboard and port vortices (assuming mirror symmetry), we obtain LLS scans from the starboard side with both port and starboard crosswinds.
3. Results
a. Numerical wake vortex simulation results
The wake vortex simulations capture all phases of wake vortex evolution in ground proximity well. In the near field, “A” in Fig. 5, the wake behind an aircraft is highly complex—multiple vortices detach from the aircraft geometry. In high-lift configuration, two vortex types dominate in the extended near field (see “B”): the flap-tip and the wing-tip vortices. Figure 5 shows that the flap-tip vortices dominate, until the merging of the two main corotating vortices on each side of the aircraft, resulting in their superposition and a single vortex (see “C”).
Airbus A340 landing at
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
High vorticity is visible on the ground as the aircraft reduces its altitude, eventually leading to detaching secondary vortices from the ground (Fig. 6), with positive streamwise vorticity in comparison to negative streamwise vorticity by the primary wake vortices (see red arrow). These secondary vortices then interact with the primary wake vortices (red arrow in Fig. 7), leading to their decay.
Detaching secondary vortices from the ground at
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
Interacting secondary vortices and wake vortices at
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
Wake vortex trajectories in quiescent atmosphere are plotted in Figs. 8 and 9. The simulation neatly reproduces trends from Stephan et al. (2014). Prior to touchdown (
Temporal Str and Prt vortex development, in terms of circulation
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
Longitudinal Str and Prt vortex development, in terms of circulation
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
Limitations of the numerical simulations and the associated vortex analysis are twofold. First, the fluctuation of strength and position tends to increase with time, for instance, consider the
Similar trends hold for the simulated crosswind aircraft landings (see Figs. 10 and 11). With an increase in crosswind strength, the asymmetry between the counterrotating vortices becomes more prominent. It was expected that a crosswind strength of 0.5 w0 halts the lateral movement of the luff vortex; however, due to descend of the aircraft and the logarithmic boundary layer, only at an altitude of b0, this condition can be expected. As the altitude plot of Fig. 10a shows, most wake vortices (with the exception of later rebound) are located below b0, thus experiencing lower crosswind speeds and higher mutual velocity induction with the vorticity layer establishing at the ground surface. Nonetheless, the blue case, i.e., vortices at
Temporal Str and Prt vortex development, in terms of circulation
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
Longitudinal Str and Prt vortex development, in terms of circulation
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
The vortex-induced lateral propagation speed is matched more frequently when the crosswind strength of 1.0 w0 is set at the altitude of b0. In the
b. LLS results
Virtual lidar measurements within the wake vortex simulations are subject to typical lidar measurement restrictions: 1) the smearing out of the flow in the atmosphere due to the RWF, 2) the temporal delay in recording an RHI scan, and 3) the RHI lidar scan geometry. For investigating the impact of the RWF, we assume that point measurements are possible and solely compute Eq. (14) at the desired range gate centers along the LOS, without the application of the RWF shown in Fig. 12 (right). Four exemplary LLS scans, produced in challenging scenarios, are given in Fig. 12 (left), where the RWF has been applied. As expected, we observe high velocities and steep gradients being filtered out by the RWF scans; scans without the RWF feature at least one further velocity layer. On top of this, a smearing out of the velocity field results, leading to an increased vortex core. Both effects may have an effect on the vortex strength evaluation.
LLS scans in challenging scenarios (left) with and (right) without RWF applied for various crosswinds and directions. Wake vortex location characterizations (markers) and strength characterizations (see legends) of the ST and the RV method are given. Gray markers represent the ST, which are often overshadowed by the pink and violet markers which arise from the RV method with and without the RWF, respectively. The wake vortex age
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
Vortices close to the ground appear to have a strong increase in velocity in their lower half compared to their upper half. Axisymmetric vortices are typically expected; however, the ground represents a wall along which secondary vorticity is produced. The effect of this vorticity can be modeled and understood with the concept of imaginary mirror vortices of opposite vorticity below the computational domain that interacts with the flow field. Their presence is confirmed by vertical velocity contours of the velocity layers close to the ground. So, the vortex half closer to the ground is strengthened, while the upper vortex-half is attenuated. With no RWF applied, lidar scans also contain far more detail and secondary vortices in comparison to LLS scans, where differentiating between wing-tip and flap-tip vortices is not possible. Even before the merging of corotating vortices, we must regard the vortices as a whole overlapping vortex.
Since RHI lidar scans are made up of multiple LOSs, a time delay of several seconds exists within each scan. Depending on whether the scanning rate ωS is positive or negative, during the recording of an RHI lidar scan, the elevation angle φ continuously increases
Another downside of lidar recordings is their scan geometry. It is not always possible to include the entire wake of an aircraft in lidar scans as vortices may advect out of the measurement window. Particularly when vortices are partly visible in the scans, such as in Fig. 12b, lidar characterization algorithms may struggle to correctly identify them and strength under- or overestimations may result.
A limitation exclusive to LLS scans concerns the padding method, which is responsible for velocities outwith the computational domain, the black rectangle. Generally, it extends the flow field without unphysical gradients, for example, in Fig. 12b. However, in few cases, such as Figs. 12a and 12c, gradients are excessively extended from within the computational domain. In other cases, new velocity gradients emerge due to padding being performed in the Cartesian coordinate system. For instance, Fig. 12d introduces new gradients in the padding region, caused by velocities outwith the computational domain targeted to approach zero. The padding is most suitable for a quiescent atmosphere in comparison to a crosswind case. Minimal impact on the success of wake vortex characterization is expected, as they are rarely located in the vicinity of the padded scan areas.
c. RV method assessment
1) Dataset
A subset of obtained LLS scans is characterized by the RV method for its evaluation. Selecting a representative subset follows two steps. First, an accurate ground truth is computed via the pressure–vorticity algorithm from section 2b. By implementing both quantitative and qualitative checks, primarily relying on following the vortex trajectories and strength trends, a carefully curated dataset is obtained. Quantitative checks include circulation sense, strength drop, and extreme strength as well as vortex position. Nonetheless, outliers cannot be ruled out completely, only minimized, due to the turbulent nature of the flow. The subset is given in Table 3. Cases where the RV method fails completely occur when vortex centers are too close to the edges of the scan, or when vortices possess minimal coherency. The latter leads to the lower number of LLS scans measured in the 2.0w0 crosswind strength simulation.
Dataset split by crosswind strength χw0 and wake vortex type.
2) Quantitative accuracy
The characterization accuracy of the investigated RV method version is given in Table 4. The medians of the error values (omitting outliers) of the polar coordinates to a vortex center
The medians of the error values E of the polar coordinates for defining a vortex center position (RC, φC), the circulation
Two-norm error for the vortex center (left)
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
Errors in the circulation characterization of the RV method are twofold. An inaccurate vortex localization significantly disrupts an accurate circulation characterization, which itself has multiple sources of error. Combining this leads to a systematic error drawn in Fig. 14 in the full range of circulation magnitude. Unsurprisingly, the error spread is higher for weak vortices as their coherency is lower if generated by the same aircraft type.
Correlation of the circulation
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
In Table 5, the RV method accuracy is given for increasing turbulent crosswind strengths. A significant detriment in accuracy occurs with a crosswind strength of 2.0w0, and both the median accuracy and its standard deviation increase. If we define a circulation characterization accuracy range as above, for the 2.0w0 crosswind case, a vortex circulation uncertainty ranging from −15.3 to +68.2 m2 s−1 must be considered.
The medians of the error values for the circulation
Table 6 splits the vortex characterizations by luff and lee vortices for the LLS scans performed under crosswind conditions. Only a marginal difference is found for the vortex strength characterization accuracy; however, the localization error of lee vortices is nearly twice that of the luff vortices. Luff vortices are expected to be coherent for longer periods of time; thus, already at younger vortex age, the characterization of lee vortices is more challenging. At the same vortex age, lee vortices are also weaker; hence, the impact of this discrepancy is not hugely consequential for operational purposes.
The medians of the error values for the circulation
To this day, only theoretical accuracy estimations by Smalikho et al. (2015) and numerical experiments by Smalikho (2019) have existed for rating the RV method. Both these investigations do not account for the complex flow behind an aircraft, employ vortex models, and assume a quiescent atmosphere. Table 7 compares these theoretical RMS error (RMSE) values for the elevation angle to the vortex center φC, the range to the vortex center RC, and the vortex strength Γ5–15 to the RMSE values from our study. In practice, one can expect RMSE values of just over triple the theoretical RMSE (without considering the error spread). Primary reasons for the underestimation of the error by Smalikho et al. (2015) and Smalikho (2019) can be associated with the lack of both incoherence and unambiguousness of vortices analyzed, in comparison to field measurements.
LLS-based and theoretical [by Smalikho et al. (2015), Smalikho (2019)] RMSE values of the polar coordinates for defining a vortex center position (RC, φC), the circulation Γ, and the RMSE of the two-norm error for the vortex center
3) Discussion
Returning to the spatial averaging caused by the RWF as discussed in section 3b. The legends of Fig. 12 as well as markers within the lidar scans allow a second look at the LLS scans in comparison to simulated lidar scans without the RWF applied. The ground truth is given by gray markers. The pink and violet markers give the RV method estimations with and without the RWF applied, respectively. Note that the model function
Sample LLS scan geometry of [R−, R+] = [80, 530] and [φ−, φ+] = [0, 20] with vortex characterization. Estimates are divided into Str and Prt vortex groups. Markers of individual wake vortex characterizations are colored by the error between the RV method and the ground truth. The ± represent an over- and underestimation by the RV method in comparison to the ground truth, respectively.
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
Vortices at low altitudes are subject to larger inaccuracies. Similar observations have been made in field measurements (F. Holzäpfel 2024, personal communication). The vortex altitude with respect to the circulation error is plotted in Fig. 16. In section 3b, the presence and effect of mirror vortices were discussed. The strength of vortices close to the ground is overestimated as a result. Depending on the method for computing the vortex circulation, the impact on the estimate differs. Equation (1) showed two ways for computing the vortex circulation, either via the vorticity or via the tangential velocity along a line (other methods exist, but these play no role in this paper) (Holzäpfel et al. 2003). The ground truth algorithm makes use of the vorticity. Instead, the RV method fits Burnham–Hallock vortex models via few tangential velocities to the primary vortex pair in Eq. (10). No mirror vortices are considered by this RV method version. It assumes the superposed velocities only result from the primary vortices, and thus, their strengths are overestimated. Moreover, when the vorticity is employed for circulation computation, it allows the vortex to be incoherent. The RV method assumes axisymmetric coherent vortices and, therefore, interprets the additional velocity from the mirror vortices as genuine primary vortices. Note that the free-slip boundary at the top of the computational domain can also be modeled by mirror vortices (Doligalski et al. 1994), and an analogous physical mechanism can be expected in the simulation. In reality, such effects would not exist. With a computational domain height of 2.9b0, the minimum error is expected in the vertical center at 1.45b0—and indeed at this altitude, the vortex circulation is just not overestimated on average (see Fig. 16). Figure 11 in Holzäpfel et al. (2021) reports a vortex strength underestimation on the order of 10% in field measurements instead. Potentially even at the vertical center of our computational domain, mirror vortices still impact the strength estimation, overshadowing the inherent underestimation of the method [the field measurements by Holzäpfel et al. (2021) generally feature vortices at higher altitudes, reducing the mirror vortex effect]. With regards to Fig. 16, for
Circulation error
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
In section 3b, vortex stretching and compression was discussed in relation to the vortex movement and the lidar scanning rate. A representative number of vortex characterizations are plotted in Fig. 17, comparing the simulation truth and RV method. The impact of vortex stretching and compressing appears minimal. Still, the strength overestimation of stretched vortices appears more pronounced, confirming the idealized study for strong vortex descend by Smalikho (2019).
Correlation of the circulation
Citation: Journal of Atmospheric and Oceanic Technology 42, 3; 10.1175/JTECH-D-24-0093.1
An additional limitation of the RV method concerns the requirement of setting a core radius for the Burnham–Hallock vortex model being fit to the vortices. A fixed core radius of 2 m is allocated for wake vortices generated by the A340 aircraft, whereas throughout the lifetime of a wake vortex, its core size may vary substantially. When the core radius is larger or smaller than 2 m, an underestimation or overestimation of the vortex circulation may result.
The localization accuracy of the RV method can be considered satisfactory for tasks such as a ground-based warning system, with the standard deviation of the error constituting the primary downside. The circulation characterization accuracy is more critical, with a characterization window of 66.2 m2 s−1 (two standard deviations). The narrower error spread in Fig. 14 implies that young and strong vortices are characterized with an error close to the medians of the error values presented in Table 4. Thus, the accuracy of the analyzed RV method is, particularly at high vortex altitudes, a satisfactory algorithm for characterizing wake vortices.
4. Conclusions
A new method of assessing the accuracy of wake vortex characterization methods based on lidar measurements has been developed. High-fidelity simulations of the wake vortices generated by a landing Airbus A340 aircraft have been performed under various turbulent crosswind strengths relevant for airport operations. These allow the study of wake vortices from their generation until decay, while their position and strength are known throughout via methods unavailable for field measurements. Virtually installed lidars, so-called LES lidar simulators (LLSs), realistically model the lidar operating, with the primary focus on the volume averaging performed for obtaining line-of-sight (LOS) velocities. A range gate weighting function (RWF) based on the pulse shape and range gate pattern has shown to provide realistic LOS velocities of range–height indicator lidar scans. In general, lidars mollify high velocities and steep gradients due to the convolution with the RWF.
With the aid of a wide range of LLS scans and the available knowledge of the vortices within, we are able to evaluate characterization algorithms for lidar scans used in field measurements. Here, the radial velocity (RV) method, a state-of-the-art wake vortex characterization algorithm for lidar scans, was evaluated. To date, only theoretical accuracy estimates and an experimental numerical extension were available. Particularly under turbulent atmospheric conditions, the performance was unknown. It was found that the overall RMSE values are approximately three times higher than those estimated with analytical means (Smalikho et al. 2015; Smalikho 2019). The medians of the error values for both the position and strength characterization can be considered small for most applications, and the localization of vortices is within instrument accuracy. However, the error spread unveils a high uncertainty in the characterizations. Particularly under highly turbulent conditions, the RV method struggles to provide a narrow error bound. Careful segmentation of the results allows distinguishing several origins of the errors. These arise from simulation assumptions, lidar operation aspects, but crucially also the RV method itself. Most notably, the investigated RV method version does not consider mirror vortices when characterizing the strength of vortices in LLS scans in ground proximity. At altitudes superseding one initial vortex separation above ground, the systematic error of strength characterization by the RV method appears very low. However, serious vortex strength overestimation exists for vortices near the ground. Overall, an overestimation in vortex strength is found; given that most vortices in this study are close to the ground, the impact of the lack of mirror vortices seems critical for the RV method.
Future studies can greatly benefit from the variety in wake vortex simulations and LLSs placed in the LES, providing an extensive and labeled dataset of high-fidelity simulated lidar scans. We aim to evaluate further lidar characterization methods with this dataset, allowing more reliable judgment of past and also future studies involving lidar measurements. The availability of this labeled dataset is also a valuable asset for machine learning (ML). Training ML models with LLS scans has the potential to process wake vortex lidar data both in real time and with an accuracy exceeding traditional processing algorithms as the RV method. This in turn may advance the development and implementation of systems to safely decrease aircraft separations in the terminal environment.
Acknowledgments.
Funding was received from the German Federal Ministry for Digital and Transport within the mFUND project “KIWI” and from the German Aerospace Center (DLR) undertaking “Wetter und Disruptive Ereignisse.” We thank DLR’s Institute of Aerodynamics and Flow Technology for providing the results of the RANS simulations from the EU-funded AWIATOR project and Airbus for the allowance to use them. Furthermore, we appreciate the kind provision of the original MGLET code by the Technical University of Munich, Hydromechanics. Finally, the authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC-NG at the Leibniz Supercomputing Centre (www.lrz.de). The authors declare that they have no conflicts of interest.
Data availability statement.
The generated LES lidar simulator dataset can be found via Wartha et al. (2024).
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