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Abstract
An onboard LIght Detection And Ranging (LIDAR) sensor designed to track wake vortex created by aircraft in formation flight is presented. It uses short pulses (75 ns) to obtain a spatial resolution of ∼22.5 m required to resolve small-scale structures of vortices and a blind zone of 17.5 m to locate vortices next to the wing tip. Monte Carlo simulations show that vortex centers could be located within ±0.5 m. Flight tests were performed with two aircraft in formation flight configuration. The LIDAR, installed in the following aircraft, was able to measure, in real time (every 6 s), the air flow velocities induced by the vortices created by the leading aircraft. The software was used to determine the vortex centers. These measurements were coupled to global positioning system (GPS) measurements of the two aircraft positions to determine the falling velocity of the vortices and infer their circulations.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Measuring meter-scale air flow velocity structures is important for a wide variety of physics: (1) for meteorology to measure atmospheric turbulence [1–5], its effect on ocean elevation [5] and diffusion of pollutants in cities [6], (2) for airport to detect wind shear [7], vortices and measure their time of life [8–10], (3) for small airship that are sensitive to small-scale air flow velocity structures [11], (4) for wind turbine to optimize them according to the air flow velocity [12], (5) for drone transport, or (6) for airship to perform formation flight [13]. This latter application is the focus of the study presented here. Formation flight is similar to the V formation of birds that uses the wake vortices created by the leading bird to enhance the overall aerodynamic efficiency by reducing the drag that, therefore, increases the flight distance that the bird can travel [14]. Extensive studies have been conducted over the last 10 years to perform similar close formation flight with aircraft as technologies have provided more precise navigation and control systems to maintain formations [13,15]. For commercial flights, close formation flight is not possible due to collision hazards. However, distant formation flight with spacing between 10 to 40 wingspans would still be beneficial because of the persistence of wake vortices created by large commercial aircraft. The induced drag reduction was estimated to be up to 30% [13,15]. As the vortex core radius (that correspond to the distance between the vortex center and the maximum air flow velocity) is typically few meters, positioning the wing tip of the following aircraft where the air flow velocity induced by the vortex is large and pushes the wing upward require a localization of the vortex center within about $\pm 1$ m at 25 m from the aircraft (about half of the wingspan). Typically, the wing tip is positioned 20 m to 30 m above the vortex center and not at the maximum air flow velocity for safety reasons. This needs to be performed in real time to continuously position the following aircraft and avoid variation in drag reduction [Fig. 1(a)]. A recent study [16] have claimed that even though “Physical model can predict location and strength of the vortex […] direct measurement using systems like Doppler LIDAR [..] would be very useful” which correspond to the study presented in this article.
Fig. 1. a) Picture of the aircrafts in formation flight taken during the flight tests perform for the study presented here. b) Schematic of the LIDAR configuration during the flight tests. The LIDAR measure the air flow velocity map induced by the vortices perpendicularly to the vortex axis using a scanner. The vortex centers are determined from the velocity map. The axis directions X, Y, and Z are indicated.
Coherent light detection and ranging (LIDAR) sensors are commonly used to measure air flow velocity remotely. A laser is sent in atmosphere, the signal backscaterred from the aerosol particles (Mie scattering) is collected and its frequency is determined using heterodyne detection. Due to the Doppler effect, the shift in frequency of the backscattered signal is proportional to the particle speed that is similar to the air flow velocity projected on the laser axis. Usually, a pulsed laser is used to obtain spatial resolution where the distance (R) is determined from the duration that the pulse take to go and return to the system ($R=\textrm {c}\Delta \textrm {t}/2$ where $\Delta \textrm {t}$ is the difference between the detection time and the time when the pulse leave the output optic of the LIDAR). In addition, a scanner is typically added to move the LIDAR axis in the vertical direction to obtain the air flow velocity map and localize the vortex centers [Fig. 1(b)].
Over the past twenty years, wake vortex detection at low altitude from the ground has been the goal of many LIDAR studies and developments [8,9,17–27]. High altitude vortices were also measured using on-board LIDAR to detect wake vortices created by a preceding aircraft [10,28,29] or from the ground with a detection range of 900 m [30,31]. In that case, wake vortices were seeded with smoke to insure high LIDAR signal level. In most LIDAR studies, pulse lengths ranging from 400 ns to 800 ns [at full width at half maximum (FWHM)] have been used that resulted in a blind zone of 60 m to 120 m and a spatial resolution of 120 m to 240 m. The blind zone ($\approx \textrm {c}\Delta \textrm {t}_{laser,-30dB}$/2 where $\Delta \textrm {t}_{laser,-30dB}$ is the pulse duration calculated at -30 dB and c is the speed of light) is due to optical reflection of the laser pulse on the LIDAR output optics. The spatial resolution corresponds to the distance over which the signal that is processed to calculate each velocity is coming from ($\approx \textrm {c}\Delta \textrm {t}_{laser,FWHM}$ where $\Delta \textrm {t}_{laser,FWHM}$ is the pulse length at FWHM) [32]. For LIDAR that contain fiber lasers, pulse length down to 250 ns have been used [10] that lead to a blind zone length of 35 m and a spatial resolution of 70 m. However, this is still too long to satisfy the requirements. In our case, the previous developments are not compliant with: (1) the blind zone constrain of $\leq 20$ m and (2) the spatial resolution required to locate vortex centers within $\pm 1$ m.
In this article, a LIDAR designed at ONERA is presented that enable to locate vortex centers within $\pm 0.5$ m at a distance down to 25 m. It uses 75 ns FWHM (120 ns at -30 dB) pulse length that induce a blind zone of $\sim$17.5 m and a spatial resolution of $\sim$22.5 m. It is almost entirely made with fibers to be robust to vibration when used on-board. The optics were chosen to focus the laser at 120 m and obtain an about flat response of the system between 25 m and 200 m to locate vortices in this area. A scanner was added to measure the air flow velocity map between -20$^{\circ }$ and 20$^{\circ }$ in the vertical direction in real time (every 6 s). A postprocessing software was written to precisely determine the positions of the vortex centers from the measured air flow velocity map. The LIDAR and the software were validated using Monte-Carlo simulations. Results show that the vortex center positions could be determined within $\pm$0.5 m. Formation flight tests were carried out where a leading aircraft seeded its wake vortices with smoke to enable the pilot of a second following aircraft to visually locate the vortex and increase the LIDAR signal. Results show that the system was able to measure vortex center positions when they were located at distances between 25 m and 450 m from the following aircraft. Coupled to global positioning system (GPS) measurements of the two aircraft positions, the circulation was inferred within $\pm$4 % by measuring the falling velocity of the vortices and the distance between vortices.
2. LIDAR characteristics
The LIDAR characteristics are summarized in Table 1. The LIDAR uses a fiber laser that emit at 1545 nm followed by a fiber amplifier. It delivers short pulses with narrow spectral width (close to Fourier transform limited) appropriate for heterodyne detection. Their temporal profiles (so called square pulses) were shaped with sharp edges {$\sim 20$ ns [33]} to minimize the blind zone length, optimize the spatial resolution and maximize the energy per pulse (25 $\mu \textrm {J}$, 80 kHz). A scanning mirror was added to give access to a scanning angle ($\theta$) varying from -20$^{\circ }$ to 20$^{\circ }$ and determine the air flow velocity map in the vertical plane perpendicular to the LIDAR axis.
Table 1. LIDAR set up summary.
2.1 Air flow velocity measurement
To determine the air flow velocity, for each time step, the part of the heterodyne current that has a duration of about the FWHM of the laser pulse, centered on the time step, is convolved with a Gaussian function with a FWHM about equal to the FWHM of the laser pulse (apodization function) and the power spectral density is calculated. Each time step is converted into distance using $R=\textrm {c}\Delta \textrm {t}/2$. This leads to a series of power spectral density as a function of distance [Fig. 2(a)]. For each distance, the power spectral density is fitted with a Gaussian function and the frequency that correspond to the maximum of the fit is used to determine the air flow velocity projected on the LIDAR axis [$\textrm {V}_{\perp }=-\lambda /2(\textrm {f}-\textrm {f}_{IF})$ where $\textrm {f}_{IF}$ is the frequency of the laser pulses] [34]. The carrier-to-noise ratio (CNR) is determined integrating the power spectral density over frequencies and dividing it by the integration over frequencies of the power spectral density of the noise [32]. The power spectral density of the noise is obtained by averaging over distance the power spectral densities calculated before the blind zone.
Fig. 2. a) Example of power spectral densities calculated on LIDAR data acquired during the flight tests described in section 4. The dashed line corresponds to the positions of the maxima of the gaussian fits of the spectral densities. b) Measured pulse shape. c) Power spectral densities calculated around the blind zone. The dashed line correspond to the measured blind zone distance. In a) and c), the power spectral densities of the signal are normalized to be consistent with simulation. This is performed by subtracting the power spectral density of the noise and dividing by the power spectral density of the noise.
2.2 Measurement of the pulse shape and the blind zone
The laser pulse shape was measured using a reflection of the laser on a solid target. The pulse shape ($\textrm {I}_{laser}$) is proportional to the envelope of the heterodyne current ($\textrm {I}_{het}$) that is deduced from the measured current:
To characterize accurately the blind zone length, the power spectral densities of the heterodyne current were calculated around the blind zone [Fig. 2(c)]. We obtained a minimal distance where the power spectral densities were lower than the standard deviation of the power spectral density of the noise of 17.5 m.
2.3 Measurement of the CNR profile
The diameter, focal length of the exit lens and the laser beam diameter were chosen to obtain an about flat response between 25 m and 200 m. Figure 3(a) shows measurements of the CNR profiles performed on the ground (where the amount of particles is high) over different days. To obtain the day to day variation of the CNR, each curve corresponds to day averaged data. We can notice that the absolute value of the CNR varied over 10 dB between the different days but the shape remained about constant. This is due to the fact that the concentration of particles varied from day to day but was about spatially homogeneous. The small variation of the CNR shape was likely due to air flow turbulence [35].
Fig. 3. a) Comparison of the CNR measured with the LIDAR on different days. Two simulations are shown (dashed curves) for $\beta =1*10^{-6} \textrm {sr}^{-1}\textrm {m}^{-1}$(upper curve) and $\beta =1.5*10^{-7} \textrm {sr}^{-1}\textrm {m}^{-1}$ (lower curve). b) Example of CNR map obtained during the flight tests where only the closer vortex was seeded with smoke (see section 4). c) CNR averaged over boxes with edges of 15 m centered at the vortex center (shown in b) measured over different shots performed the same day as a function of the distance between the vortex center and the LIDAR. This is compared with a simulation of the CNR using a backscattering coefficient $\beta =1.6*10^{-5} \textrm {sr}^{-1}\textrm {m}^{-1}$.
A second measurement of the CNR [Fig. 3(b) and 3(c)] was performed during the flight tests (presented in section 4). Figure 3(b) shows an example of spatially resolved CNR map calculated on a LIDAR scan. The CNR was large around the vortex because it was seeded with smoke. On Fig. 3(c), the CNR averaged over 15-m squares centered at the vortex center is plotted, for a series of scan, as a function of the distance between the vortex center and the LIDAR. The points are compared with the CNR profile simulated at 1545 nm using a backscattering coefficient of $\beta =1.6*10^{-5} \textrm {sr}^{-1}\textrm {m}^{-1}$ that was chosen to best match the data. The good agreement indicate that simulations well reproduce the experimental CNR profile and give an estimate of the backscattering coefficient induced by the smoke used during the campaign.
3. LIDAR signal processing
To measure the position of vortex centers, the air flow velocity map measured by the LIDAR over the scanning angles was analysed with a postprocessing software. In this section, the software is presented and benchmarked using Monte-Carlo simulations.
3.1 Measurement of vortex center positions
A simulation was performed with two vortices located at distances of 40 m and 102.6 m and an altitude of -10 m from the LIDAR with a core radius of 3 m and a circulation of 630 $m^2/s$. The air flow velocity field induced by the vortices was simulated using a Hallock-Burnham model [36]. A backscattering coefficient of $\beta =5*10^{-6} \textrm {sr}^{-1}\textrm {m}^{-1}$ was used.
Figure 4(a) shows the power spectral densities of the synthetic LIDAR heterodyne current obtained at $\theta =0^{\circ }$ as a function of distance. The air flow velocity map determined by analyzing the power spectral densities simulated for a scanning angle varying between $-20^{\circ }$ and $20^{\circ }$ is shown in Fig. 4(b). For angles located above and below the vortex centers, at distances that correspond to the distances of the vortex center, we observe local velocity maxima and minima. As the LIDAR measure the velocity projected on its axis, the maximum or minimum velocity occurs where the velocity is about parallel to the LIDAR axis. Due to the circular symmetry of the air flow velocity induced by the vortex, this happens at a distance that corresponds about to the distance of the vortex centers. In addition, the vortex induce a maximum air flow velocity at its core radius ($\textrm {r}_{c}$). Therefore, the maximum and minimum velocities are located at angle differences with the vortex center angle of $\pm \Delta \theta _{m}(\textrm {R})=\tan ^{-1}(\frac {\textrm {r}_{c}}{\textrm {R}})$ (about $\pm 4.3^{\circ }$ for the closer vortex and $\pm 1.7^{\circ }$ for the further vortex).
Fig. 4. a) Simulated LIDAR spectral densities at an angle $\theta =0^{\circ }$. b) Synthetic velocity field measured by the LIDAR over a scan. c) Difference between the air flow velocity map and the air flow velocity map angularly shifted by shifted by $2*\Delta \theta _{m}(\textrm {R})$. The rectangles ($24$ m$*12\Delta \theta _{m}(\textrm {R})$) in b) are centered at the first estimate of the vortex center deduced from the maximum and minimum velocity difference in c). A more precise determination of the vortex centers are performed in these rectangles and are indicated by black crosses. d) Comparison of the angularly averaged velocity square profiles calculated for the closer vortex (blue curve) with its Gaussian fit (dashed red curve). e) Schematic of the positions of the maximum and minimum velocity measured by the LIDAR in a vortex which induce a correction in radii. f) Comparison of the evolution of the standard deviation along R of the velocity as a function of scanning angle with its polynomial 2 fit. The curves in d) and f) were calculated using the air flow velocities inside the rectangle centered at the closer vortex center in b).
Figure 4(c) show the difference between the air flow velocity map and the air flow velocity map angularly shifted by $2*\Delta \theta _{m}(\textrm {R})$. This allows the local maximum and minimum located above the vortex centers to be superimposed with the local maximum and minimum located below the vortex centers. This leads to a maximum of $\sim 2\max (\textrm {V}_{\perp })$ and a minimum of $\sim -2\max (\textrm {V}_{\perp })$ in the map difference located at an angle $\Delta \theta _{m}(\textrm {R})$ above the two vortex centers. These are used to perform a first estimate of the vortex center positions.
A more precise localization is performed in a second step. For each vortex, a rectangle of $24$ m$*12\Delta \theta _{m}(\textrm {R})$ centered at the first estimate is defined [Fig. 4(b)]. The size of the rectangle is chosen to isolate the large air flow velocities induced by the vortex. To determine the distance between the LIDAR and the vortex center, on this rectangle, the velocity map is interpolated to change the R axis to $\textrm {R}'=\textrm {R} \cos {\theta }$ [Fig. 4(d)]. This correction in radii accounts for the fact that, for each angle, the exact position where the vortex-induced air flow velocity is parallel to the LIDAR axis occurs at $\textrm {R}'=\textrm {R} \cos {\theta }$ [Fig. 4(e)]. Then, the interpolated velocities are squared and averaged over $\theta$ for each $\textrm {R}'$. The resulting function is fitted with a Gaussian function and the distance between the vortex-center and the LIDAR corresponds to the maximum of the fit [Fig. 4(d)]. The vortex center angle is obtained by calculating the standard deviation of the velocity as a function of the scanning angle and determining its minimum using a polynomial 2 fitting function [Fig. 4(f)]. Along the axis that go through the vortex center angle, the air flow induced by the vortex is perpendicular to the LIDAR axis resulting in a constant velocity (that corresponds to the background velocity) that minimize the standard deviation parameter. This method is different from the technique described in [25,26]. In particular, it does not require measuring the maximum and minimum peaks of the velocity or the value of the background velocity. Therefore, it is well adapted to the seeded vortices that are analyzed during the flight tests where the CNR is large only next to the vortex center (so the background velocity is difficult to evaluate) and the velocity maximum and minimum are not necessarily observable if the vortex is on the side of the LIDAR field of view (so called truncated vortices).
With this postprocessing software, vortex center positions of $X=39.6$ m and $Z=-9.9$ m (respectively $X=102.6$ m and $Z=-9.9$ m) were obtained for the closer (respectively the further) vortex center. This correspond to an error smaller than $0.5$ m.
3.2 Validation of the software using Monte-Carlo simulations
The postprocessing software was validated using Monte-Carlo simulations (Fig. 5). 100 simulations were performed varying the distance of the closer vortex between 20 m and 190 m from the LIDAR, the distance between the two vortices from 50 m to 65 m, their altitude relative to the plane between -20 m and 20 m, their circulation between 350 $m^2/s$ and 650 $m^2/s$ and using a backscattering coefficient of $\beta =5*10^{-6}$ $\textrm {sr}^{-1}\textrm {m}^{-1}$ that slightly underestimate the seeded vortex conditions.
Fig. 5. Differences obtained during the Monte-Carlo simulations between the positions of the vortex centers along X (a) and Z (b) determined by the software on the synthetic LIDAR velocity map and the corresponding positions of the simulated vortex center. This is performed for the closer vortex (circles) and the further vortex (squares).
Results show that the vortex center positions were localized between 25 m and 200 m with a maximum error of $\pm 0.5$ m along X and $\pm 0.3$ m along Z [Fig. 5]. It results in a total error of about $\pm 0.5$ m. A systematic shift of about $-0.4$ m and $+0.2$ m can be observed along X before 50 m and after 150 m respectively. These shifts disappear when simulations are performed with a constant CNR profile showing that they are due to the variation of the CNR along the beam axis. This is one of the reason why the LIDAR was designed to maximize the flatness of the CNR profile (see section 2.3). Beyond 200 m, the error increases due to the reduction of the CNR at these distances.
4. Flight tests
Flight tests were performed with a leading A380 aircraft followed by a A350 aircraft in formation flight configuration [Fig. 1(a)]. The LIDAR was installed in the following aircraft and measure the air flow velocity map in the plane perpendicular to the LIDAR axis [Fig. 1(b)]. The optical head that include the laser and all the optical components was inserted in a $73~\textrm {cm} \times 44~\textrm {cm} \times 18~\textrm {cm}$ box suitable for flight certification [Fig. 6(a)]. Laser electrical power and control, scanner power and control, a rack-mountable computer dedicated to signal acquisition, real time signal processing and display were set in the flight test bay. Five flights took place, the LIDAR was operated during 12 hours and approximately 11000 scans were recorded that contained 5000 detectable vortices. During the flights, the leading aircraft seeded the vortex with smoke which increases the LIDAR signal [blue plume in Fig. 1(a)]. The distance between the closer vortex and the LIDAR was varied between 25 m and more than 500 m. The estimated vortex center altitude with respect to the aircraft wing (vertical axis) varied from $+20$ m to $-20$ m. Real time display of the CNR map and of the velocity map were available for the engineer of the flight tests [Fig. 6(b) and 6(c)]. This real time display enabled to give directions to the pilot so that he could precisely position the aircraft wing tip relative to the vortex position.
Fig. 6. a) LIDAR installed in the aircraft. Real time display of the velocity map (b) and CNR map (c) shown inside the aircraft. The map are shown for two consecutive LIDAR scans.
4.1 Vortex localization between 25 m and 450 m
Figure 7 shows examples of localization of the two vortices performed during the flight tests at short [Fig. 7(a) and 7(b)], medium [Fig. 7(c) and 7(d)], and long distance [Fig. 7(e) and 7(f)]. The zeros distance of the data was measured by fitting the blind zone signal (using the same treatment of the heterodyne current as in Eq. (1)) with the laser pulse shape. It corresponds about to the position of the output optic of the LIDAR.
Fig. 7. Localization of vortex centers from experimental data for close vortices (a,b), medium-range vortices (c, d) and long-range vortices (e, f). Velocity maps are displayed in (R, $\theta$) (a, c, e) and in (X, Z) (b, d, f). The vortex centers determined by the software are shown (black crosses).
We can notice that the calculation of the power spectral densities of the heterodyne current was faster than the acquisition of the data so that the duration of the measurements was limited by the duration of the scan. The shorter scanning duration used in the campaign was 6 s that corresponded to 4 s of acquisition time followed by 2 s for the scanner to go back to its initial position. This allowed the air flow velocity map to be determined every 6 s.
In Fig. 8(a), the measured positions of the vortices are compared with the positions of the leading plane (measured with a GPS) over multiple shots. An excellent agreement is observed between the middle points of the two vortex centers and the positions of the plane along Y. The standard deviation of the difference between the two is larger than the accuracy in the measurement of the positions of the vortex centers determined with Monte-Carlo simulations. It can be due to uncertainties in the GPS position or motion along Y, due to turbulence, of the vortices from their creation to their measurement.
Fig. 8. Comparison of the measured position of the vortex centers (red circles correspond to the closer vortex and green circles to the further vortex) with the position of the leading plane (blue curve). b) Measured distance between vortices. The averaged distance over all shots is shown (dashed red line).
The averaged distance between the vortex centers ($\textrm {b}_{0}$) was deduced within $\pm 3.3 \%$ from these data [in Fig. 8(b), the average is performed over 4500 second]. The variation of the distance from scan to scan is larger than the accuracy of the vortex center measurement. It can also be due to motion along Y, due to turbulence, of the vortices from their creation to their measurement.
4.2 Measurement of the circulation
The simultaneous measurement of the position of the leading plane relative to the following plane with GPS and of the vortex centers relative to the following plane with the LIDAR allowed the position of the vortex centers relative to the leading plane to be determined. In particular the distances over which the vortices have fallen between their creation and their measurement were inferred (falling distance in Fig. 9). To determine accurately these distances, the plane angles were accounted for (their roll, side slip and pitch). Figure 9(a) shows a good agreement between the evolution of the falling distance and the distance between planes along X determined with the GPS.
Fig. 9. a) Comparison of the falling distance of the closer vortex (red circles) and the further vortex (blue circles) with the distance between the two planes (dashed black curve). b) Evolution of the falling distance of the closer vortex (red circles) and the further vortex (blue circles) as a function of the falling time measured over 3 hours. The linear fit of the data (using a $\chi ^2$ analysis) is shown (dashed black line).
This results in a falling distance about proportional to the falling time [Fig. 9(b), the falling time is obtained from the distance between the two planes measured with GPS and the second plane velocity]. The falling velocity (within $\pm 2.7 \%$) and the initial vortex position were deduced from the linear fit. The error were determined at 90% chance using a $\chi ^2$ analysis. With these results the circulation was calculated using $\Gamma =\textrm {v}_{fall}*2*\pi *\textrm {b}_0$ within $\pm 4.5 \%$ where the error was deduced from the errors in $\textrm {v}_{fall}$ and $\textrm {b}_0$.
5. Conclusions
In this article, a LIDAR developed at ONERA is presented where short pulses were used to obtain a small spatial resolution ($\sim$22.5 m) and a small blind zone length ($\sim$18 m). It allowed measuring the position of vortex centers within $\pm 0.5$ m (using an adapted postprocessing software) from an aircraft satisfying the requirements to perform formation flight. During flight tests, air flow velocity maps were measured in real time (every 6 s). Vortex centers were located at distances between 25 m and 450 m from the LIDAR. These precise localizations coupled to GPS measurements of the two aircraft positions allowed determining the falling velocity of the vortices, the distance between vortices, and the vortex circulation with an accuracy of $\pm$ 4.5%.
Funding
Airbus; Office national d'études et de recherches aérospatiales.
Acknowledgments
The authors wish to acknowledge R. Domel and J. Cammas for their support of this study.
Disclosures
The authors declare no conflicts of interest.
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