1 Introduction

While challenges for flight operations in low visibility and icing conditions are largely overcome, atmospheric turbulence still causes injuries, delays and waste of resources, such as CO2 emissions and excessive fuel consumption [1]. Suppressing atmospheric turbulence in flight carries the potential to reduce CO2 emissions, fuel consumption and flight time by up to 10% for commercial flights [2, 3]. These potentials become even more relevant, as atmospheric turbulence is predicted to increase in response to climate change [4].

In this context, this paper investigates on the turbulence load prediction task, c.f., Fig. 1, which can be seen as a subtask of the turbulence load alleviation objective, also referred to as gust load alleviation [5]. The accurate prediction of disturbance loads caused by atmospheric turbulence subsequently enables the compensation by opposing feedforward deflections of flight control surfaces of an aircraft [6].

Fig. 1
figure 1

Aircraft flying in atmospheric turbulence. The vertical movement of the air is presented according to the colorbar on the right side (yellow for rising air, blue for sinking air). High-dynamic differential pressure sensors in front of the wings provide anticipating measurements of the turbulence field to predict disturbance effects

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Atmospheric turbulence can be modelled making use of spatial power spectral densities (PSD). Examples are the von Kármán [7] and the Dryden [8] wind turbulence field models. Prior approaches to predict turbulence loads include wind LIDAR measurements [9], both for a statistical analysis of the far field to warn the flight crew [10], as well as for prediction of the near field in front of the aircraft for actuation of flight control surfaces [11]. Other approaches include the use of pressure sensors [12] to counteract turbulence effects in wind tunnel tests. Various sensor principles, both anticipating, such as differential pressure sensors [13] and strain gauges [14], as well as reactive measurements, e.g., inertial measurements used for acceleration control [15] are considered. The disadvantage of reactive measurements is that rejection efforts can only be started upon measuring the first negative effects of the disturbance. Thus, only by including anticipating measurements, a theoretically perfect cancellation of disturbances is possible [16]. In contrast to simulative studies of aircraft models [17] and wind tunnel tests [18], literature is lacking research including actual test flight results. After initial test flights with an unmanned system [19], the authors adapted an experimental aircraft to also perform a first test flight in manned size.

The contribution of this paper is the presentation of actual test flight data that are measured with both a UAS test platform as well as with a manned experimental aircraft for various turbulence intensities. The data is analyzed in the time domain, frequency domain, as well as for the statistical distribution. Section 2 presents the approach to model and analyze turbulence based on spectral characteristics. Section 3 states the calculations to transform measured wind quantities into predicted acceleration values. Section 4 describes the set-up of the UAS testbed and the manned experimental aircraft, which allow for anticipating measurements of the airflow in front of the wings. Finally, Sect. 5 presents the test flight data, which is assessed regarding the turbulence load prediction task.

2 Turbulence modelling

For the spatial and temporal analysis of a wind field, which is traversed by an aircraft in atmospheric turbulence, spectral modelling is pursued. According to the Dryden wind turbulence model [8], the PSD of the vertical turbulence component \(w\) can be characterized by

$${\Psi }_{w}\left(\Omega \right)= {\sigma }_{w}^{2}\frac{2{L}_{w}}{\pi }\frac{1+12{\left({L}_{w}\Omega \right)}^{2}}{{\left(1+{4\left({L}_{w}\Omega \right)}^{2}\right)}^{2}} ,$$
(1)

with the spatial frequency \(\Omega ,\) the turbulence intensity \({\sigma }_{w}\), and the turbulence scale length \({L}_{w}\). To generate a representative turbulence field with a PSD according to (1) a suitable transfer function

$${G}_{w}\left(s\right)={\sigma }_{w}\sqrt{\frac{2{L}_{w}}{\pi }}\frac{1+\sqrt{12}{L}_{w}s}{{\left(1+2{L}_{w}s\right)}^{2}}$$
(2)

can be found, which satisfies

$${\Psi }_{w}\left(\Omega \right)= {\left|{G}_{w}\left(j\Omega \right)\right|}^{2}.$$
(3)

Thus, by filtering 2-dimensional, unit-variance, band-limited white noise by (2) representative turbulence fields \(w(x,y)\) can be generated, where \(x\) is the longitudinal coordinate in flight direction and \(y\) is the lateral, spanwise coordinate. In Fig. 1 an exemplary field with scale length \({L}_{w}=\) 3 m is shown, which is the scale length that is observed during test flights with the fixed-wing UAS. In the following, the different effects of spatial variations in x-direction and y-direction of such turbulence field shall be examined.

Spatial variations in x-direction are transformed into time variations as the aircraft flies through the turbulence field. Based on the airspeed \({V}_{a}\) the relation of temporal frequency \(\omega\) and spatial frequency \(\Omega\) can be calculated as

$$\omega = {V}_{a}\Omega .$$
(4)

In consequence, neglecting time change of the turbulence field itself, i.e. assuming a frozen turbulence model [20], a spatial PSD \({\Psi }_{w}\left(\Omega \right)\) can be transformed into a temporal PSD \({\Phi }_{w}\left(\omega \right)\) by

$${\Phi }_{w}\left(\omega \right)= \frac{{\Psi }_{w}\left(\frac{\omega }{{V}_{a}}\right)}{{V}_{a}}.$$
(5)

This implies that \({\Phi }_{w}\) gets broader and smaller for higher airspeeds \({V}_{a}\). For \({L}_{w}=\) 3 m and \({\upsigma }_{w}^{ }=1\frac{m}{{s}^{2}}\), Fig. 2 shows \({\Psi }_{w}\left(\Omega \right)\), \({\left|{G}_{w}\left(j\Omega \right)\right|}^{2},\) as well as \({\Phi }_{w}\left(\omega \right)\) for three different airspeeds \({V}_{a}= 10\frac{m}{s}\), \(30\frac{m}{s}\), \(100\frac{m}{s}\). It can be noticed that the faster the aircraft flies, the stronger the influence of higher temporal frequencies becomes.

Fig. 2
figure 2

Spatial PSD \({\Psi }_{w}\left(\Omega \right)\), transfer function \({\left|{G}_{w}\left(j\Omega \right)\right|}^{2}\) and temporal PSD \({\Phi }_{w}\left(\omega \right)\) for three different airspeeds

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Spatial variations in y-direction, i.e., spanwise variations, determine to which extent various flight quantities, such as vertical acceleration, pitch moment, roll moment, wing bending and higher-order structural dynamics are affected. As an example, symmetric spanwise variations do not cause roll moments as the effects on the left and right wing cancel out.

To account for spanwise variations of the turbulence field, a representation of \(w\left(y\right)= w(\bullet ,y)\) by orthonormal polynomial functions is proposed. For this purpose, an inner product of two spanwise distributions \({x}_{1}(y)\) and \({x}_{2}(y)\) can be defined as

$$\left\langle {{x_1},{x_2}} \right\rangle = \frac{1}{b}\mathop \int \limits_{ - \frac{b}{2}}^{\frac{b}{2}} {x_1}\left( y \right){x_2}\left( y \right)dy,$$
(6)

with the span \(b\), and the according induced norm

$$\Vert {x}_{1}\Vert =\sqrt{\langle {x}_{1},{x}_{1}\rangle }.$$
(7)

Therewith, orthonormal polynomial basis functions can be defined by recursively applying the law

$${p}_{i}\left(y\right)=\frac{\left({y}^{i}- \sum_{j=0}^{i-1}\langle {y}^{i},{p}_{j}\left(y\right)\rangle {p}_{j}(y)\right)}{ \Vert \left({y}^{i}- \sum_{j=0}^{i-1}\langle {y}^{i},{p}_{j}\left(y\right)\rangle { p}_{j}(y)\right)\Vert }$$
(8)

for \(i=0,\dots ,\infty\) to fulfil the relations

$$\langle {p}_{i},{p}_{j}\rangle = \left\{\begin{array}{c}1, \quad i=j\\ 0, \quad i \ne j .\end{array}\right.$$
(9)

An arbitrary spanwise wind distribution \(w\left(y\right)\) can then be represented by a coefficient vector \({\varvec{\zeta}}=[{\zeta }_{0}\) \({\zeta }_{1} {\zeta }_{2} \cdots ]\) as

$$w\left(y\right)=\sum_{i=0}^{\infty }{w}_{i}\left(y\right)=\sum_{i=0}^{\infty }{\zeta }_{i}{p}_{i}\left(y\right),$$
(10)

where the coefficients can be calculated as

$${\zeta }_{i}= \langle w,{p}_{i}\rangle .$$
(11)

Figure 3 shows the first three even basis polynomials \({p}_{0}\), \({p}_{2}\), and \({p}_{4},\) as well as the first three uneven basis polynomials \({p}_{1}\), \({p}_{3}\), and \({p}_{5}\) for \(b\)=1.6 m. Additionally, an exemplary distribution \({w}_{0,5}\) acting on an aircraft is illustrated with \({\varvec{\zeta}}=[\text{1 0.5 }- \text{1 }- \text{0.3 0.2 }- \text{0.2 0 0 }\cdots ]\).

Fig. 3
figure 3

Even polynomials \({p}_{0}\), \({p}_{2}\), and \({p}_{4}\), odd polynomials \({p}_{1}\), \({p}_{3}\), and \({p}_{5}\), and distribution \({w}_{0,5}\)

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To quantify the variation of a spanwise wind distribution \(w(y)\), the rooted mean square (RMS) value with (9) and (10) can be determined as

$$RMS\left(w\right)=\Vert w\Vert =\sqrt{\langle w,w\rangle }=\sqrt{\sum_{i=0}^{\infty }{\zeta }_{i}^{2}}={\Vert {\varvec{\zeta}}\Vert }_{2} ,$$
(12)

where \({\Vert \bullet \Vert }_{2}\) denotes the Euclidean norm. Thus, the RMS value of the coefficient vector \({\varvec{\zeta}}\), i.e., \({RMS}({\varvec{\zeta}})={\Vert {\varvec{\zeta}}\Vert }_{2}\), also represents the RMS value of \(w\), where \({\zeta }_{i}\) is the contribution of the i-th component \({w}_{i}={\zeta }_{i}{p}_{i}\).

Assessing the statistical relevance of the i-th component, the ratio of the turbulence scale length \({L}_{w}\) and the span \(b\) of the aircraft is decisive for the expected value \(E({\zeta }_{i}^{2})\). In this regard, Fig. 4 shows \(\sqrt{E({\zeta }_{i}^{2})}\) of the first eight coefficients \({\zeta }_{0}\), \({\zeta }_{1}, \ldots , {\zeta }_{7}\) for different values of \(\frac{{L}_{w}}{b}=0.1, 1, 10\) to be able to assess the expected contribution of the i-th component \({w}_{i}\) to \({RMS}\left(w\right)\).

Fig. 4
figure 4

Expected value \(\sqrt{E}({\upzeta }_{i}^{2})\) of the first eight coefficients \({\upzeta }_{0}\), \({\upzeta }_{1}, \ldots , {\upzeta }_{7}\) depending on \(\frac{{\mathrm{L}}_{\mathrm{w}}}{\mathrm{b}}\)

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For \(\frac{{L}_{w}}{b}=10\) the scale length of the turbulence field is significantly higher than the span, i.e., mainly low frequent spatial variations occur. This relates to higher order coefficients \({\zeta }_{i}\), \(i>2\), being of subordinate importance. For \(\frac{{L}_{w}}{b}=0.1\) the scale length of the turbulence field is significantly lower than the span, i.e., also higher order coefficients need to be included to properly represent the turbulence field.

These considerations need to be taken into account, when discrete measurements shall be performed for reconstruction of the turbulence field. If for example a single sensor is placed at the center of the aircraft, i.e., at \(y=0\), the measured vertical wind according to (10) is

$${w}_{c}=w\left(0\right)=\sum_{i=0}^{\infty }{\zeta }_{i}{p}_{i}\left(0\right)=\sum_{k=0}^{\infty }{\zeta }_{2k}{p}_{2k}\left(0\right),$$
(13)

as \({p}_{i}\left(0\right)=0\) for \(i=1, 3, 5, \cdots\). If now \({w}_{\mathrm{c}}\) is used as estimated 0-th order coefficient \({\widehat{\zeta }}_{0}={w}_{\mathrm{c}}\), i.e., the center measurement is assumed to be valid for the whole span, spatial aliasing occurs leading to the relative error

$$\frac{\hat{\zeta}_0-\zeta_0}{\zeta_0} = \frac{\sum_{k=1}^{\infty}{\zeta }_{2k}{p}_{2k}\left(0\right)}{\zeta_0},$$
(14)

as higher order coefficients are projected into \({\widehat{\zeta }}_{0}.\)

3 Turbulence load prediction

In this section the prediction of disturbances of the vertical acceleration \({a}_{z}\) of an aircraft flying through atmospheric turbulence based on airflow measurements is discussed. The term prediction is used in this context, as by means of differential pressure measurements in front of the wing, c.f., Fig. 1, future values of \({a}_{z}\) are estimated by predicted values \({\widehat{a}}_{z}\) with an anticipation time \({T}_{ant}\), i.e.,

$${a}_{z}\left(t+{T}_{ant}\right)\approx {\widehat{a}}_{z}(t).$$
(15)

The predicted vertical acceleration \({\widehat{a}}_{z}\) is calculated with the objective to minimize the prediction error

$${e}_{{a}_{z}}\left(t\right)={a}_{z}\left(t\right)-{\widehat{a}}_{z}\left(t-{T}_{ant}\right).$$
(16)

For a frozen turbulence field and assuming that the airspeed \({V}_{a}\) stays approximately constant during the comparatively short anticipation time \({T}_{ant}\), with the anticipation distance \({d}_{ant}\) the anticipation time can be calculated as

$${T}_{ant}\left(t\right)=\frac{{d}_{ant} }{{V}_{a}\left(t\right)}.$$
(17)

Thus, the achievable anticipation time for a given aircraft design depends on the aircraft’s airspeed related to the aircraft’s size, as the latter is indicative for realizable anticipation distances. In this context, Table 1 lists typical airspeeds \({V}_{a}\) and spans \(b\) for the UAS of this paper, the ultra-light one-seater Colomban Luciole MC-30, the turbo-prop aircraft Pilatus PC-12, the narrow-body airliner Airbus A320, and the wide-body airliner A380. It is notable that for various aircraft types of different sizes the ratio \({{T}_{b}=b/V}_{a}\) shows similar values in the order of 0.16 s indicating a likewise increase of airspeed with aircraft size for these types. This means that if anticipating measurements are performed at a half-span distance in front of the wings, i.e., \({d}_{ant}=\frac{b}{2}\), an anticipation time in the order of \({T}_{ant}={T}_{b}=\) 0.08 s can be achieved. However, \({T}_{b}=\) 0.16 s shall not be considered as a strict design constant and varies for different aircraft types. As an example, fast subsonic aircraft are limited to around Mach 0.85 and in consequence show to have similar airspeeds despite considerably varying aircraft sizes, as can be seen for example for the A380 with a ratio \({{T}_{b}=b/V}_{a}=\) 0.32 s, i.e., double the value of the A320. Similarly, the anticipation distance \({d}_{ant}\) needs to be increased for fast aircraft designs or may be decreased for particularly slow flying aircraft to obtain similar anticipation times \({T}_{ant}\).

Table 1 Comparison of span \(b\), typical airspeed \({V}_{a}\), and ratio \({T}_{b}=b/{V}_{a}\) for five differently sized aircraft
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To calculate \({\widehat{a}}_{z}\) based on measurements of the angle of attack (AOA) \({\alpha }\) and the airspeed \({V}_{a}\), a simple lift force model [21] can be written as

$$L= {(C}_{L0}+{C}_{L\alpha }\alpha )\frac{\rho }{2}{V}_{a}^{2} S,$$
(18)

with air density \(\uprho\) and wing area \(\mathrm{S},\) where lifting effects of turn rates and flight surface deflections are neglected. With the aircraft mass \(m\) the corresponding vertical acceleration \({a}_{z}\) results as

$${a}_{z}= \frac{L}{m}= \frac{\rho S}{2 m}( {C}_{L0}+{C}_{L\alpha }\alpha ){V}_{a}^{2}.$$
(19)

With \({C}_{z0}={\frac{\uprho S}{2 m}C}_{L0}\), \({C}_{z\alpha }=\frac{\uprho S}{2 m}{C}_{L\alpha }\) and \(w=\alpha {V}_{a}\) a more concise form is found as

$${a}_{z}= {C}_{z0}{V}_{a}^{2}+{C}_{z\alpha }w{V}_{a}.$$
(20)

Thus, variations of the vertical wind \(w\) have direct effect on the vertical acceleration of the aircraft with the amplification factor \({C}_{z\alpha }{V}_{a}.\) To take into account the spanwise lift distribution, the basic model (20) can be extended by calculating the inner product \(\langle {C}_{z\alpha }(y),w(y)\rangle\) instead of the scalar multiplication \({C}_{z\alpha }w\) resulting in the model

$${a}_{z}= {C}_{z0}{V}_{a}^{2}+\langle {C}_{z\alpha }(y),w(y)\rangle {V}_{a}.$$
(21)

Following the discussions in Sect. 2, (21) can be specialized to a discrete number of measurements of the wind field \(w(y)\) by neglecting higher order polynomial coefficients. As an example, for three measurements, i.e., neglecting polynomial coefficients higher than 2 according to \(w\left(y\right)= {\zeta }_{0}{p}_{0}+{\zeta }_{1}{p}_{1}+{\zeta }_{2}{p}_{2}\), (21) can be evaluated to

$${a}_{z}= {C}_{z0}{V}_{a}^{2}+{{C}_{z{\zeta }_{0}}{\zeta }_{0}V}_{a}+ {C}_{z{\zeta }_{2}}{\zeta }_{2}{V}_{a},$$
(22)

with \({C}_{z{\upzeta }_{0}}=\langle {C}_{z\alpha }(y),{p}_{0}(y)\rangle\), \({C}_{z{\upzeta }_{2}}=\langle {C}_{z\alpha }(y),{p}_{2}(y)\rangle\), and \(\langle {C}_{z\alpha }(y),{p}_{1}(y)\rangle =0\) for symmetry reasons.

To determine estimated values \({\widehat{\zeta }}_{0}\) and \({\widehat{\zeta }}_{2}\) of the coefficients \({\upzeta }_{0}\) and \({\upzeta }_{2}\) based on these measurements, the vertical wind at the three lateral positions \({y}_{L}\), \({y}_{C}\), and \({y}_{R}\), can be written as

$${{\varvec{w}}}_{\mathrm{0,2}}={{\varvec{P}}}_{\mathrm{0,2}}{{\varvec{\zeta}}}_{\mathrm{0,2}},$$
(23)

With \({{\varvec{w}}}_{\mathrm{0,2}}={\left[\begin{array}{ccc}{w}_{L}& {w}_{\mathrm{C}}& {w}_{\mathrm{R}}\end{array}\right]}^{T}\), \({{\varvec{\upzeta}}}_{\mathrm{0,2}}= {\left[\begin{array}{ccc}{\upzeta }_{0}& {\upzeta }_{1}& {\upzeta }_{2}\end{array}\right]}^{\mathrm{T}}\) and the matrix

$${{\varvec{P}}}_{\mathrm{0,2}}= \left[\begin{array}{ccc}{p}_{0}\left({y}_{L}\right)& {p}_{1}\left({y}_{L}\right)& {p}_{2}\left({y}_{L}\right)\\ {p}_{0}\left({y}_{C}\right)& {p}_{1}\left({y}_{C}\right)& {p}_{2}\left({y}_{C}\right)\\ {p}_{0}\left({y}_{R}\right)& {p}_{1}\left({y}_{R}\right)& {p}_{2}\left({y}_{R}\right)\end{array}\right].$$
(24)

For independent measurements, i.e., positions \({y}_{L}, {y}_{C}\), and \({y}_{R}\) are chosen such that \({{\varvec{P}}}_{\mathrm{0,2}}\) is a regular matrix, the estimated polynomial coefficients \({\widehat{{\varvec{\upzeta}}}}_{\mathrm{0,2}}= {\left[\begin{array}{ccc}{\widehat{\upzeta }}_{0}& {\widehat{\upzeta }}_{1}& {\widehat{\upzeta }}_{2}\end{array}\right]}^{\mathrm{T}}\) based on the three measurement \({{\varvec{w}}}_{\mathrm{0,2}}\) are determined as

$${\widehat{{\varvec{\zeta}}}}_{0,2}={{\varvec{P}}}_{0,2}^{-1}{{\varvec{w}}}_{0,2}.$$
(25)

Finally, based on these considerations, for three measurements with anticipation distance \({d}_{ant}\), the predicted acceleration \({\widehat{a}}_{z}\) can be determined as

$${\widehat{a}}_{z}\left(t\right) = {C}_{z0}{{V}_{a}\left(t\right)}^{2}+{{ C}_{z{\zeta }_{0}}{\widehat{\zeta }}_{0}\left(t\right)V}_{a}\left(t\right)+ {C}_{z{\zeta }_{2}}{\widehat{\zeta }}_{2}\left(t\right){V}_{a}\left(t\right).$$
(26)

Figure 1 illustrates this case with the red lines indicating the lateral position of the three probes \({y}_{L}\), \({y}_{C}\), and \({y}_{R}\), where the wind field is sampled at the positions of the white diamonds.

4 Test flight set-up

In order to assess the suitability of differential pressure sensors to predict effects of atmospheric turbulence on the flight dynamics of an aircraft, a fixed-wing UAS test platform and a manned experimental aircraft are equipped with 5-hole probes in front of the wings for anticipating wind field measurements.

4.1 Airflow measurements

The airflow measurements are conducted by means of 5-hole probes with a geometry as shown in Fig. 5. The probes are 3D printed making use of resin-based stereolithography (SLA), which allows for fine resolutions as low as 47 μm laterally and 20 μm vertically. The pressure port \({p}_{1}\) is used to determine the local airspeed \({V}_{a}\), while pressure ports \({p}_{2}\) and \({p}_{3}\) are used to determine the local AOA \({\alpha}\) of the respective probe. The pressure ports \({p}_{4}\) and \({p}_{5}\) could be used to determine the sideslip angle, however, are not connected and sealed rearwards, as lateral dynamics are not in the focus of the current investigations.

Fig. 5
figure 5

Geometry of the 5-hole probes to measure the airflow in front of the aircraft in mm. The probes are 3D printed by resin-based stereolithography. Pressure port \({p}_{1}\) is used for airspeed estimation, pressure ports \({p}_{2}\) and \({p}_{3}\) are used for angle of attack estimation. Pressure ports \({p}_{4}\) and \({p}_{5}\) are not connected and sealed rearwards

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The difference of \({p}_{1}\) and static pressure \({p}_{s}\) is measured as

$${\Delta {p}_{{V}_{a}}=p}_{1}-{p}_{s}\approx \stackrel{-}{q}=\frac{\rho }{2} {V}_{a}^{2},$$
(27)

with the dynamic pressure \(\stackrel{-}{q}\) and the air density \(\rho\) to determine the airspeed according to

$${V}_{a}\approx \sqrt{\frac{2\Delta {p}_{{V}_{a}}}{\rho }}.$$
(28)

The difference of \({p}_{2}\) and \({p}_{3}\) is measured as

$${\Delta {p}_{\alpha }= p}_{2}-{p}_{3}\approx {c}_{p,\alpha }\alpha \stackrel{-}{q},$$
(29)

with a constant coefficient \({c}_{p,\alpha }\) to determine the AOA according to

$$\alpha =\frac{\Delta {p}_{\alpha }}{{c}_{p,\alpha } \stackrel{-}{q}} \approx \frac{1}{{c}_{p,\alpha }} \frac{\Delta {p}_{\alpha }}{ \Delta {p}_{{V}_{a}}}.$$
(30)

Additionally, correction factors are implemented to correct quadratic measurement errors of the airspeed at higher AOA values [22]. The peak-to-peak noise measured in calm air is 0.19 m/s for \({V}_{a}\) and 0.17° for \(\alpha\), which corresponds to noise-related errors of the load prediction \({\widehat{a}}_{z}\) in the order of 0.2 m/s2 according to (20) with the calculated parameters according to Sect. 5. Bearing in mind that for the related turbulence load alleviation objective the predicted load \({\widehat{a}}_{z}\) shall be used for opposing flap deflections, errors in \({\widehat{a}}_{z}\) would lead to erroneous compensation actions. Thus, the noise level needs to be considered as a limiting factor for turbulence load alleviation.

To measure \(\Delta {p}_{{V}_{a}}\) and \(\Delta {p}_{\alpha}\), differential pressure sensors (Sensirion SDP33) are used. As the ambient offset pressure of approximately 1 bar = 105 Pa is several orders of magnitude higher than the aerodynamic pressure changes in the order of \(\stackrel{-}{q}\)=100 Pa, measuring differential pressures instead of subtracting absolute pressure measurements is pursued for improved accuracy. The measuring range of the differential pressure sensors of 1500 Pa still allows airspeed measurements \({V}_{a}\) up to 50 m/s.

4.2 Unmanned aircraft

The UAS is based on the unmanned aircraft Volantex Ranger 1600 with a span of \(b\) = 1.6 m, c.f., Fig. 6. At a distance \({d}_{x,CG}=\frac{b}{2}\) = 0.8 m in front of the aircraft’s center of gravity (CG) airflow measurements are conducted at three different spanwise positions \({y}_{L}=-{d}_{y,CG}=-\text{0.5 m}\), \({y}_{C}= \text{ 0 m}\), \({y}_{R}={d}_{y,CG}=\text{0.5 m}\). With three independent measurements, the first three coefficients \({\zeta }_{0}\), \({\zeta }_{1}, {\zeta }_{2}\) are determined according to Sect. 3 with the matrix (24) resulting as

Fig. 6
figure 6

Scheme of UAS testbed equipped with an air data boom for airflow measurements at three points

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$${{\varvec{P}}}_{0,2}= \left[\begin{array}{ccc}{1} & \text{-1.05} & \text{ 0.12}\\ {1} & {0} & \text{-1.12}\\ {1} & \text{ 1.05} & \text{ 0.12}\end{array}\right].$$
(31)

The coefficient \({\zeta }_{1}\) is not used in this paper, however, may be used for future research on lateral dynamics. It is worth noting, that the third measurement is either way necessary for determining \({\zeta }_{2}\), as for two measurements only, parts of \({\zeta }_{0}\) or \({\zeta }_{1}\) would be projected into \({\widehat{\zeta }}_{2}\), analogously to (14).

As the probes are positioned \({d}_{x,CG}\) in front of the CG and the left and right probe are positioned \({d}_{y,CG}\) to the side of the CG, c.f., Fig. 6, a roll rate \(p\) and a pitch rate \(q\) of the aircraft cause local perpendicular airflow, which making use of small angle approximations can be corrected by

$$\begin{aligned} \alpha _{{L,CG}} = \alpha _{L} + \frac{{d_{{x,CG}} q}}{{V_{a} }} + \frac{{d_{{y,CG}} p}}{{V_{a} }}, \hfill \\ \alpha _{{C,CG}} = \alpha _{C} + \frac{{d_{{x,CG}} q}}{{V_{a} }}, \hfill \\ \alpha _{{R,CG}} = \alpha _{R} + \frac{{d_{{x,CG}} q}}{{V_{a} }} - \frac{{d_{{y,CG}} p}}{{V_{a} }}. \hfill \\ \end{aligned}$$
(32)

Furthermore, for each of the considered AOA \({\alpha }_{\chi }\) the corresponding local vertical wind \({w}_{\chi }\) can be calculated according to a small angle approximation

$${w}_{\chi }={\alpha }_{\chi }{V}_{a}.$$
(33)

For the unmanned test flights, static pressure \({p}_{s}\) is taken from the fuselage and provided to the sensors in front of the aircraft by means of one common static pressure line. From the three airspeed measurements \({V}_{a,L}\), \({V}_{a,C}\), and \({V}_{a,R}\), c.f., Fig. 6, the mean airspeed.

$${V}_{a}=\frac{1}{3}\left({V}_{a,L}+{V}_{a,C}+{V}_{a,R}\right)$$
(34)

is determined, which is used as airspeed \({V}_{a}\).

Figure 7 shows the UAS before take-off, where the pneumatic tubing from the 5-hole probes to the SDP33 sensors, which are placed at the center of the air data boom (ADB), can be seen. Furthermore, a mast is used together with cables to reduce vertical and torsional motions of the ADB relatively to the fuselage by pretensioning. The battery is placed right before the empennage to balance the CG, as the ADB shifts the CG forward, which otherwise would lead to reduced maneuverability.

Fig. 7
figure 7

UAS testbed with three 5-hole probes connected to high-dynamic differential pressure sensors

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The flight controller Pixhawk4 with customized firmware of the flight stack PX4 is positioned inside the fuselage close to the CG. The vertical acceleration \({a}_{z}\) is measured by the on-board inertial measurement units. The cycle rate of the custom flight code of 500 Hz is much higher than the investigated frequencies, allowing for quasi-continuous time considerations.

4.3 Manned aircraft

For manned test flights the aircraft Colomban Luciole MC-30 with a span \(b\)=6.9 m is equipped with the same type of flight controller, differential pressure sensors and 5-hole probes as used for the unmanned test flights. The airflow measurements are conducted at a distance \({d}_{x,CG} =2.65\mathrm{m}\) in front of the aircraft’s center of gravity (CG) at the spanwise positions \({y}_{L}=-2.5{\text{m}}\) and \({y}_{R}={d}_{y,CG}=\text{2.5m}\), c.f., Fig. 8. Due to the propeller position at the nose of the aircraft, a center probe like for the unmanned aircraft is not feasible. With the two independent measurements, the first two coefficients \({\zeta }_{0}\), \({\zeta }_{1}\) can be determined. For future research further measurement points on the wings, e.g., at the wing root and wing tip, can be considered to also determine higher order coefficients \({\zeta }_{2},\) \({\zeta }_{3}\), \({\zeta }_{4}\), and \({\zeta }_{5}\). To determine the airspeed \({V}_{a}\) of the manned aircraft the differential pressure of the conventional total pressure and static pressure lines is measured.

Fig. 8
figure 8

Scheme of the manned experimental aircraft equipped with one air data boom for each wing

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Unlike for the unmanned aircraft, a mast construction and pretensioning of the ADBs, is not possible, as it would require major adaptions of the aircraft structure. Instead, the sensor boards at the ADB tips are additionally equipped with inertial measurement units (IMU) to determine the motion of the 5-hole probes and correct for resulting local airflow variations, c.f., Fig. 9. To this end, the measured AOA at the tip of the ADB \({\alpha }_{ADB}\) results as the AOA close to the CG \({\alpha }_{CG}\), i.e., the quantity of interest for turbulence load prediction, and the superimposed local AOA \({\alpha }_{loc}\) due to translational and rotational motion of the ADB tip relative to the motion of the CG of the aircraft, i.e.,

Fig. 9
figure 9

Manned experimental aircraft with 5-hole probe and differential pressure sensors in front of the wing

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$${\alpha }_{ADB}= {\alpha }_{CG}+ {\alpha }_{loc}.$$
(35)

The local AOA results as sum of the local pitch rotation of the probe \({\theta }_{y,loc}\) and the local airflow due to vertical motion of the probe \({w}_{z,loc}\) as

$${\alpha }_{loc}={\theta }_{y,loc}+ \frac{{w}_{z,loc}}{{V}_{a}}.$$
(36)

As the IMUs measure accelerations and angular rates, the time derivatives \({\dot{\theta }}_{y,loc}={q}_{loc}\) and \({\dot{w}}_{z,loc}={a}_{z,loc}\) can be directly determined, by considering the difference of the measurements \({a}_{z,ADB}\), \({q}_{ADB}\) at the ADB and \({a}_{z,CG},\) \({q}_{CG}\) at the CG of the aircraft, i.e.,

$${q}_{loc}= {q}_{ADB}-{q}_{CG} {a}_{z,loc}= {a}_{z,ADB}-{a}_{z,CG}$$
(37)

Assuming that the dynamics of airspeed changes are much slower than the ADB motion, i.e., \({\dot{V}}_{a}\approx 0\), with (36) the time derivate of the local AOA is found as

$${\dot{\alpha }}_{loc}={q}_{loc}+ \frac{{a}_{z,loc}}{{V}_{a}}.$$
(38)

Finally, by numerical integration of \({\dot{\alpha }}_{loc}\) and high-pass filtering with order 3 and cut-off frequency 0.5 Hz to avoid integration errors due to sensor offsets, the local AOA \({\alpha }_{loc}\) is calculated by the flight controller to correct the ADB measurements according to.

$${\alpha }_{CG}= {\alpha }_{ADB}- {\alpha }_{loc}$$
(39)

With these corrections and \({\alpha }_{CG,L}\) and \({\alpha }_{CG,R}\) being the corrected AOAs for the left and the right ADB, \({\zeta }_{0}\) can be found in accordance with (24) as

$${\zeta }_{0}=\frac{{\alpha }_{L,CG}+{\alpha }_{R,CG}}{2}{V}_{a}.$$
(40)

5 Test flights

To investigate on the possibilities to predict the vertical acceleration of an aircraft in atmospheric turbulence by differential pressure measurements in front of the wings, test flights with a UAS testbed and a manned experimental aircraft, c.f., Sect. 4, are performed. The flights are conducted in different intensities of atmospheric turbulence from light to moderate turbulence with g-load variations of \({a}_{z}\) in the order of 0.3 to 0.5 g up to severe turbulence with variations of the g-load of more than 3 g, with the gravitational acceleration 1 g = 9.81 m/s2.

5.1 Unmanned test flights

First the spectral properties of the measured turbulence in unmanned test flights are assessed by comparing the PSD of the vertical wind measured with the three probes \({w}_{L,CG}= {\alpha}_{L,CG}{V}_{a},\) \({w}_{C,CG}= {\alpha }_{C,CG}{V}_{a}, {w}_{R,CG}= {\alpha }_{R,CG}{V}_{a}\) to the turbulence model (1) with the temporal PSD \({\Phi }_{w}\left(\omega \right)\), which is calculated making use of transformation (5). Figure 10 shows the result for \({L}_{w}=\) 3 m, \({\sigma }_{w}=\) 0.6 m/s and \({V}_{a}=\) 13.4 m/s, where \({V}_{a}\) and \({\sigma }_{w}\) are the mean values of the measurements in flight and \({L}_{w}\) is the fitted parameter. A very good compliance of the measured turbulence field with \({\Phi }_{w}\left(\omega \right)\) can be observed. The measurements of the three probes show similar PSD magnitudes. By the correction of turn rates (32) measurement errors due to the short period mode oscillation [21] are corrected, while the uncorrected phugoid mode with a time constant \({T}_{ph}\approx 8s\) seems to affect measurements in the region of \({\omega }_{ph}\approx \frac{2\pi }{8s}=0.78\frac{rad}{s}\).

Fig. 10
figure 10

Comparison of the PSDs of \({w}_ {{L},{CG}}\), \({w}_{{C},{CG}}\), \({w}_{{R},{CG}}\) with the PSD \({\Phi }_{w}\left(\upomega \right)\) of turbulence model (1), \({L}_{w}=\) 3 m

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To determine the parameters \({c}_{z0}\), \({c}_{z{\upzeta }_{0}}\), \({c}_{z{\upzeta }_{2}}\) of (26) a least squares optimization problem is solved to minimize the prediction error \({e}_{{a}_{z}}\) of the recorded flight data. Additionally, it showed to be beneficial to also introduce a fourth parameter \({c}_{z{V}_{a}}\) to account for linear effects of \({V}_{a}\), which leads to the predicted acceleration.

$${\widehat{a}}_{z}= {c}_{z0}{{V}_{a}}^{2}+{c}_{z{V}_{a}}{V}_{a}+{{c}_{z{\zeta }_{0}}{\zeta }_{0}V}_{a}+ {c}_{z{\zeta }_{2}}{\zeta }_{2}{V}_{a}$$
(41)

The optimal parameters result as \({c}_{z0}= -0.017\), \({c}_{z{V}_{a}}= 0.565\), \({c}_{z{\upzeta }_{0}}= 0.618\), and \({c}_{z{\upzeta }_{2}}= -0.148\) based on the test flight data for different turbulence intensities.

As first consideration, before presenting time and frequency analysis of \({\widehat{a}}_{z}\), the anticipation distance \({d}_{ant}\) shall be validated by calculating \({\widehat{a}}_{z}\) according to (41) and varying \({d}_{ant}\) in the range of \({d}_{x,CG}\pm \text{0.2}\) m, i.e., from \(\text{0.6}\) m to \({1}\) m for moderate turbulence. To this end, Fig. 11 shows the RMS value of the prediction error \({e}_{{a}_{z}}\), which becomes minimal for \({d}_{ant} =\) 0.821 m. As only a small deviation of 0.021 m from \({d}_{x,CG}=\) 0.8 m is observed, which increases the RMS(\({e}_{{a}_{z}}\)) by less than 1%, the geometry based anticipation distance \({d}_{ant}={d}_{x,CG}=\frac{b}{2}=\) 0.8 m is kept for the following investigations. It shall be emphasized, that while \({d}_{ant}\) is constant, according to (17) the anticipation time \({T}_{ant}\) varies depending on the airspeed from 0.1 s for low airspeeds \({V}_{a}=\) 8 m/s to 0.05 s for high airspeeds \({V}_{a}=\) 16 m/s.

Fig. 11
figure 11

Analysis how the anticipation distance \({\mathrm{d}}_{\mathrm{ant}}\) affects \({\mathrm{RMS}(\mathrm{e}}_{{\mathrm{a}}_{\mathrm{z}}})\) in the range of \({{\mathrm{d}}_{\mathrm{ant}}=\mathrm{d}}_{\mathrm{x},\mathrm{CG}}\pm \text{0.2m}\)

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To assess the ability of \({\widehat{a}}_{z}\) to predict the time behavior of \({a}_{\mathrm{z}}\) in moderate turbulence, Fig. 12 presents the time signal of \({a}_{\mathrm{z}}\), \({\widehat{a}}_{z}\), and the prediction error \({e}_{{a}_{z}}\). An accurate prediction of the time behavior can be observed, where \({e}_{{a}_{z}}\) for the most part stays below 1 m/s2, while \({a}_{\mathrm{z}}\) varies from 6 m/s2 up to 16 m/s2.

Fig. 12
figure 12

Time signal of acceleration \({a}_{z}\), predicted acceleration \({\widehat{a}}_{z}\), and prediction error \({e}_{{a}_{z}}\) measured during a test-flight with the UAS in moderate turbulence

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To allow for a more detailed examination of the predictive character of \({\widehat{a}}_{z}\), Fig. 13 shows a 2 s time interval from 649 to 651 s of Fig. 12. The predicted acceleration \({\widehat{a}}_{z}\) appears shifted by the anticipation time \({T}_{ant}\approx 0.1s\) relatively to \({a}_{\mathrm{z}}\), which is consistent with the flown airspeed \({V}_{a}\approx 8\) m/s during this time interval.

Fig. 13
figure 13

Detail view of Fig. 12, where the predicted acceleration \({\widehat{a}}_{z}\) appears shifted relatively to \({a}_{z}\) by the anticipation time \({T}_{ant}\approx 0.1s\)

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To assess the frequency behavior, the PSD of \({a}_{\mathrm{z}}\), \({\widehat{a}}_{z}\), and \({e}_{{a}_{z}}\) for moderate turbulence are presented in Fig. 14. For frequencies below 2 Hz the PSD of the prediction error PSD(\({e}_{{a}_{z}}\)) is more than 10 times lower than PSD(\({a}_{\mathrm{z}}\)). Above 2 Hz, PSD(\({e}_{{a}_{z}}\)) is noticeably increasing relatively to PSD(\({a}_{\mathrm{z}}\)), up to reaching similar values at 8 Hz. At 15 Hz, which is both in the dynamic range of the wing bending mode and the ADB bending mode, a pronounced peak of PSD(\({a}_{\mathrm{z}}\)) is visible. Further investigations and design improvements are planned to investigate on this resonance phenomenon.

Fig. 14
figure 14

PSD of acceleration \({a}_{z}\), predicted acceleration \({\widehat{a}}_{z}\), and prediction error \({e}_{{a}_{z}}\) for moderate turbulence

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To assess the ability of \({\widehat{a}}_{z}\) to predict the time behavior of \({a}_{\mathrm{z}}\) also in severe turbulence, Fig. 15 presents the time signal of \({a}_{\mathrm{z}}\), \({\widehat{a}}_{z}\), and the prediction error \({e}_{{a}_{z}}\). A mostly accurate prediction of the time behavior can be observed, where \({e}_{{a}_{z}}\) for the most part stays below 2 m/s2, while \({a}_{z}\) varies from −1 m/s2 up to 35 m/s2. An error of over 5 m/s2 can be observed at 829 s when the acceleration peak of 35 m/s2 is reached. As the high acceleration value correlates to AOA \({\alpha}\) of 15° the wings at this point most probably already show significant airflow detachment, such that the lift model (18) would need to be extended by nonlinear terms of \({\alpha}\) for more accurate tracking of \({a}_{z}.\)

Fig. 15
figure 15

Time signal of acceleration \({a}_{z}\), predicted acceleration \({\widehat{a}}_{z}\), and prediction error \({e}_{{a}_{z}}\) measured during a test-flight with the UAS in severe turbulence

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The difference of flying in moderate turbulence and severe turbulence is additionally illustrated by Fig. 16 and Fig. 17, which show the empirical probabilities Pr(\({a}_{\mathrm{z}})\) and Pr(\({e}_{{a}_{z}})\) of \({a}_{\mathrm{z}}\) and \({e}_{{a}_{z}}\) with a bin width of 0.2 m/s2. All empirical probabilities show distributions approximately according to Gaussian curves. For both turbulence intensities Pr(\({a}_{\mathrm{z}})\) shows a mean value around the trim load \({a}_{z0}\)=1 g = 9.81 m/s2 of straight and level flight. Regarding the variation, as can be expected, for severe turbulence the values of \({a}_{\mathrm{z}}\) vary more intensely leading to a broader distribution Pr(\({a}_{\mathrm{z}})\). The distributions of Pr(\({e}_{{a}_{z}})\) for both Figures show a mean value of approximately 0 and are much narrower than Pr(\({a}_{\mathrm{z}})\), being indicative for a good prediction accuracy.

Fig. 16
figure 16

Empirical probability Pr(\({a}_{z})\) and Pr(\({e}_{{a}_{z}})\) with bin width 0.2 m/s2 for moderate turbulence

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Fig. 17
figure 17

Empirical probability Pr(\({a}_{z})\) and Pr(\({e}_{{a}_{z}})\) with bin width 0.2 m/s2 for severe turbulence

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Assessing the results regarding the objective of turbulence load alleviation, low-dynamic load variations are of less importance, as they are sufficiently rejected by feedback control action of the pilot or conventional autopilots. One approach to account for this fact is to assess the load deviation

$${\Delta a}_{z}= {a}_{z}-{a}_{z0}$$
(42)

from a steady-state trim load \({a}_{z0}\) rather than \({a}_{z}\) itself. For an assumed perfect compensation action based on the predicted load deviation \({\Delta \widehat{a}}_{z}= {\widehat{a}}_{z}-{a}_{z0}\), the residual load results to \({a}_{z}-{\Delta \widehat{a}}_{z}={a}_{z0}+{e}_{{a}_{z}}\). In the context of turbulence load alleviation, this means, that perfect compensating control actions based on erroneous predicted loads \({\widehat{a}}_{z}\) would reduce the load deviation from the trim load \({a}_{z0}\) from \({\Delta a}_{z}\) to the load prediction error \({e}_{{a}_{z}}\). Another approach, which can be specialized to the application, is to evaluate PSD values of a specific frequency band of interest, e.g., for perfect compensating control action \({\Delta a}_{z}\) would be reduced to under 10% from 0.3 Hz to 3 Hz in Fig. 15.

Finally, the impact of \({\widehat{\zeta }}_{0}\) and \({\widehat{\zeta }}_{2}\) as well as the use of different probe configurations to determine these values is assessed for moderate and severe turbulence. To this end, for seven different cases Table 2 states \(RMS({e}_{{a}_{z}})\) and the relative error of the load deviation \({\Delta a}_{z}\)

$${\varepsilon }_{{\Delta a}_{z}}=\frac{RMS\left({\Delta a}_{z}-\Delta {\widehat{a}}_{z}\right)}{RMS\left({\Delta a}_{z}\right)}=\frac{RMS\left({e}_{{a}_{z}}\right)}{RMS\left({\Delta a}_{z}\right)},$$
(43)
Table 2 Comparison of \(RMS({e}_{{a}_{z}})\) and relative errors \({\varepsilon }_{{a}_{z}}\) and \({\varepsilon }_{{\Delta a}_{z}}\) for different cases of \({\widehat{\zeta }}_{0}\) for severe turbulence with \(RMS\left({\Delta a}_{z}\right)=\) 4.80 m/s2 and moderate turbulence with \(RMS\left({\Delta a}_{z}\right)=\) 2.26 m/s2
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being related to the reference value \(\mathrm{RMS}\left({\Delta a}_{z}\right)=\) 2.26 m/s2 for moderate turbulence and \(\mathrm{RMS}\left({\Delta a}_{z}\right)=\) 4.80 m/s2 for severe turbulence. The parameters \({c}_{z0}\) , \({c}_{z{V}_{a}}\) , \({c}_{z{\upzeta }_{0}}\) , and \({c}_{z{\upzeta }_{2}}\) are calculated by least squares optimization for each case individually to obtain a fair comparison of the achievable prediction error \({e}_{{a}_{z}}\) for each case.

For the first six cases \({c}_{z{\upzeta }_{2}}=0\), i.e., assuming only a 0-th order field \({w}_{0}\left(y\right)={\widehat{\zeta }}_{0}{p}_{0}\left(y\right)\), c.f., Sect. 2, while for the last case also the estimated 2-nd order coefficient \({\widehat{\zeta }}_{2}\) is included. The lowest prediction accuracy is obtained for \({{\widehat{\zeta }}_{0}={\alpha }}_{L}{V}_{a}\) and \({{\hat \zeta }_0} = {\alpha _C}{V_a}\), i.e., single measurements without turn rate compensations (32). The turn rate compensations included in \({{\hat \zeta }_0} = {\alpha _{L,CG}}{V_a}\) and \({{\hat \zeta }_0} = {\alpha _{R,CG}}{V_a}\;\) noticeably improve the prediction accuracy, e.g., for moderate turbulence \({\varepsilon }_{{\Delta a}_{z}}\) is reduced from 64.97% to 40.15% for the left probe and 55.14% to 34.65% for the center probe. Comparing the result for the left probe with the center probe, it can be noted that the center probe shows better performance. That an off-center probe performs worse than the center probe may be explained, as for the off-center probe also odd order fields \({\upzeta }_{1}\),\({\upzeta }_{3}\),… are projected into \({\widehat{\zeta }}_{0}\) increasing spatial aliasing effects and, additionally, torsional movements of the air data boom cause off-center errors only.

The case \({{\hat \zeta }_0} = {\alpha _{LR}}{V_a}\;\) includes two measurements, namely of the left and the right probe with the mean AOA \({\alpha _{LR}} = 0.5\left( {{\alpha _{L,CG}} + {\alpha _{R,CG}}} \right)\), where \(RMS({e}_{{a}_{z}})\) and \({\varepsilon }_{{a}_{z}}\) are further reduced, e.g., to 30.88% for moderate turbulence. Finally, the cases (\({\widehat{\zeta }}_{0}\)) and \(({\widehat{\zeta }}_{0}, {\widehat{\zeta }}_{2})\) include all three measurements according to (25), where (\({\widehat{\zeta }}_{0}\)) only takes the 0-th order coefficient and \(({\widehat{\zeta }}_{0}, {\widehat{\zeta }}_{2})\) also includes the 2-nd order coefficient, what becomes apparent by the non-zero parameter \({c}_{z{\upzeta }_{2}}\).

As expected, by taking all three measurements into account the prediction error is further reduced. Also including \({\widehat{\zeta }}_{2}\) results in a slightly better performance, than for \({\widehat{\zeta }}_{0}\) only, e.g., for moderate turbulence \({\varepsilon }_{{\Delta a}_{z}}\) is reduced from 29.81% to 28.81%, i.e., a prediction accuracy of 71.19%.

5.2 First manned test flight

To investigate the possibility to predict turbulence effects also in manned sized aircraft a first test flight with the experimental aircraft, c.f. Section 4, is performed. The aircraft flies with three different airspeeds \({V}_{a}=\) 23 m/s, \({V}_{a}=\) 30 m/s, and \({V}_{a}=\) 38 m/s, at constant altitude and circling with approximately double rate, i.e., one 360° turn per minute, to stay within the same region of turbulence.

First, the frequency characteristics of the vertical acceleration \({a}_{z}\) measured at the CG of the experimental aircraft are analyzed. For this purpose, Fig. 18 shows the PSD of \({a}_{z}\) for \({V}_{a}=\) 23 m/s, \({V}_{a}=\) 30 m/s, and \({V}_{a}=\) 38 m/s. For low frequencies a similar value PSD\(({a}_{z})\) = 0.8 m/s2/√Hz can be observed for all airspeeds. This can be expected, as the impact of the vertical wind \(w\) on \({a}_{z}\) increases linearly with \({V}_{a}\) according to (20), while the PSD of \(w\) decreases with \(1/{V}_{a}\) according to (5), c.f. Figure 2. That the overall disturbance impact of \(w\) on \({a}_{z}\) still increases for higher airspeeds, in the frequency domain is reflected by the frequency shift of the low pass characteristic. While for \({V}_{a}=\) 23 m/s the cut-off frequency can be observed at around 0.7 Hz, it increases to 0.9 Hz for \({V}_{a}=\) 30 m/s and to 1.1 Hz for \({V}_{a}=\) 38 m/s. This corresponds to the assumed frozen turbulence model which implies relation (5), i.e., a broader disturbance spectrum for higher airspeeds. In the region from 1 to 10 Hz the PSDs show a -2 slope according to the von Kármán turbulence model (1). Around 10 Hz some smaller deviations can be noted, which might relate to structural modes such as the wing bending mode and wing torsional mode. Finally, very pronounced peaks can be observed above 20 Hz related to vibrations of the engine and propeller. The engine directly drives the propeller which turns at around 2500 rpm = 42 Hz for \({V}_{a}=\) 23 m/s, 3000 rpm = 50 Hz for \({V}_{a}=\) 30 m/s and 3500 rpm = 58 Hz for \({V}_{a}=\) 38 m/s. This results in corresponding vibration peaks depending on the airspeed between 42 and 58 Hz, as well as at half-frequencies between 21 and 29 Hz. To reduce the impact of engine vibrations and higher order structural modes the signals of the manned test flight are filtered with a 3rd order low-pass with a cut-off frequency of 12 Hz.

Fig. 18
figure 18

PSD of vertical acceleration \({a}_{z}\) of the manned aircraft during light turbulence for three different airspeeds

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To correct for the expected ADB motion relative to the aircraft’s rigid body motion, the local AOA \({\alpha }_{loc}\) is calculated by the flight controller according to Sect. 4.3. By this means the measured AOA at the ADB tip \({\alpha }_{ADB}\) can be corrected according to (39) to obtain a better estimate of \({\alpha }_{CG}\) which causes the actual disturbance effect on the aircraft. To evaluate the necessity and effectiveness of this correction, Fig. 19 shows the PSD of \({\alpha }_{ADB}\), \({\alpha }_{loc}\), and \({\alpha }_{CG}\). A pronounced resonance peak of \({\alpha }_{loc}\) at 4.2 Hz can be observed, which according to ground tests and simulations can be related to the first bending mode of the ADB. The resonance of the second bending mode becomes apparent at 14.2 Hz. While the effect of the second resonance of \({\alpha }_{loc}\) appears negligible, significant errors with an amplification of \({\alpha }_{ADB}\) by almost a factor of 10 result due to the resonance of \({\alpha }_{loc}\) at 4.2 Hz. The correction of \({\alpha }_{ADB}\) by removing \({\alpha }_{loc}\) according to (39) is assessed to be effective, as \({\alpha }_{CG}\) indeed shows the expected behavior of the AOA without the resonance peak of the ADB.

Fig. 19
figure 19

PSD of measured AOA at the ADB tip \({\alpha }_{ADB}\), calculated local AOA \({\alpha }_{loc}\), and corrected AOA \({\alpha }_{CG}\)

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Finally, the predicted acceleration \({\widehat{a}}_{z}\) is calculated by the flight controller analogously to the unmanned test flights according to (41) with \({c}_{z{\upzeta }_{2}}=0\), as \({\upzeta }_{2}\) cannot be determined with only 2 probes, and \({\upzeta }_{0}\) according to (40). By solving a least squares optimization problem to minimize the prediction error \({e}_{{a}_{z}}\) the optimal parameters for the manned aircraft result as \({c}_{z0}= 0.0012\), \({c}_{z{V}_{a}}= 0.1747\), \({c}_{z{\upzeta }_{0}}= 0.0680\). For RMS(\({\Delta a}_{z}\)) = 0.9603 m/s2 a prediction error of RMS(\({e}_{{a}_{z}})\)=0.3652 m/s2 is observed corresponding to \({\varepsilon }_{{\Delta a}_{z}}\)=38.03%, i.e., a prediction accuracy of 61.97%.

Evaluating the anticipation distance similarly to Fig. 11 the value \({d}_{ant}\) = 2.65 m can be confirmed, which corresponds to an anticipation time of \({T}_{ant}=\) 115.2 ms for \({V}_{a}=\) 23 m/s, \({T}_{ant}=\) 88.3 ms for \({V}_{a}=\) 30 m/s, and \({T}_{ant}=\) 69.7 ms for \({V}_{a}=\) 38 m/s.

Finally, Fig. 20 presents the time signal of \({a}_{\mathrm{z}}\), \({\widehat{a}}_{z}\), and \({e}_{{a}_{z}}\) for a segment of the test flight with the manned experimental aircraft with \({V}_{a}=\) 30 m/s, where \({a}_{\mathrm{z}}\) shows variations of about 3 m/s2. Observing the signals of the predicted acceleration \({\widehat{a}}_{z}\) and actual acceleration \({a}_{z}\), the anticipating character of \({\widehat{a}}_{z}\) with a time lead of \({T}_{ant}=\) 88.3 ms becomes apparent. Paying attention to the good correlation of \({\widehat{a}}_{z}\) and \({a}_{z}\) it is worth pointing out, that \({a}_{z}\) is the sensor output of an accelerometer, whereas \({\widehat{a}}_{z}\) is based on a completely different sensor principle measuring differential pressure in front of the wings, c.f., Sect. 4.

Fig. 20
figure 20

Time signal of acceleration \({a}_{z}\), predicted acceleration \({\widehat{{a}}_{z}}\), and prediction error \({\mathrm{e}}_{{\mathrm{a}}_{\mathrm{z}}}\) of the manned experimental aircraft flying with an airspeed of \({\mathrm{V}}_{\mathrm{a}}=\) 30 m/s in light to moderate turbulence

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5.3 Discussion

To further improve the prediction accuracy, especially for higher disturbance frequencies, c.f., Fig. 15, further research on the following error sources may be conducted:

  • the time evolution of the turbulence field itself, i.e., the turbulence field may not be able to be assumed frozen,

  • flight dynamics such as forces and moments due to turn rates and control surface actuation,

  • spatial aliasing, as higher order coefficients \({\upzeta }_{3},\) \({\upzeta }_{4}, \dots\) are neglected,

  • measurement errors such as miscalibration, limited bandwidth and measurement noise,

  • structural modes of the ADB and the aircraft,

  • aerodynamic transients causing lags of lift generation.

In summary, by analyzing time, frequency, and statistical characteristics of the predicted acceleration \({\widehat{a}}_{z}\), it can be concluded, that the use of anticipating high-dynamic differential pressure measurements is a very promising approach for turbulence load prediction. The measured load prediction accuracy of over 70% will allow to advantageously use the predicted acceleration \({\widehat{a}}_{z}\) for feedforward turbulence load alleviation in future work, especially bearing in mind the anticipation time of up to 0.1 s, which allows for data processing and compensation of limited actuator dynamics.

6 Conclusion and outlook

In this paper the prediction of the vertical acceleration of an aircraft in atmospheric turbulence by means of high-dynamic differential pressure sensors is investigated. A spatial and temporal turbulence model is presented to develop a turbulence prediction formulation which is validated by actual test flights with an UAS platform and a manned experimental aircraft in different turbulence intensities. By determining the airflow in front of the wings, an anticipation time of the predicted acceleration of up to 0.1 s is obtained, which can be used to compensate for time delays and low-pass behavior of actuators and control algorithms. For the unmanned test flights, the prediction accuracy is assessed to be 71.19% for moderate turbulence and 71.05% for severe turbulence, where vertical acceleration disturbances higher than 30 m/s2 are measured. The first manned test flight in light to moderate turbulence revealed a prediction accuracy of 61.97%.

By deflecting control surfaces according to the predicted disturbances, a significant reduction of turbulence effects on the flight dynamics of an aircraft is expected in future work, which is aimed at improving energy efficiency, safety, and passenger comfort of manned aviation.