Exemplar-based objective classification of gust-induced loads across multiple flight conditions

# Exemplar-based objective classification of gust-induced loads across multiple flight conditions
Source: [https://arxiv.org/html/2608.12448](https://arxiv.org/html/2608.12448)
Is it possible to find an objective classification criterion that organizes the complexity of gust\-induced loads across many flight conditions? And one that remains as interpretable as a labelling based on coarse parameters, such as the flight attitude? Our approach encodes a large number of experimental observations through a machine\-learned representation and applies a summarization procedure to select a minimal subset of highly significant exemplars\. The exemplars provide a similarity\-based objective classification criterion of all the observations, they can be more conveniently inspected by experts and can become subject of more refined experiments\. We demonstrate the approach on a database of 3480 pressure\-load measurements induced by random gusts on a flying\-wing model across six flight attitudes\. We find nine fundamental response types that recur across multiple attitudes; analysis of a type’s transient response enables physical intuition into the underlying fluid mechanics\.

Paolo OlivucciEmail:[p\.olivucci@tu\-braunschweig\.de](mailto:[email protected])Affiliation:Institute of Fluid Mechanics, TU Braunschweig, Hermann\-Blenk Str\. 37, Braunschweig, 38108, GermanyKowshik SrivatsanEmail:[k\.srivatsan@tu\-braunschweig\.de](mailto:[email protected])Affiliation:Institute of Fluid Mechanics, TU Braunschweig, Hermann\-Blenk Str\. 37, Braunschweig, 38108, GermanyDavid E\. RivalEmail:[david\.rival@tu\-braunschweig\.de](mailto:[email protected])Affiliation:Institute of Fluid Mechanics, TU Braunschweig, Hermann\-Blenk Str\. 37, Braunschweig, 38108, Germany

###### keywords

gust\-wing interaction, unsteady aerodynamics, data summarization, delta wing, machine learning, flight attitude

## 1Introduction

Small uncrewed aerial vehicles \(UAVs\) are increasingly deployed in low\-altitude environments where atmospheric turbulence is spatio\-temporally complex\. In this scenario, the length and velocity scales of unsteady disturbances overlap directly with the vehicle dimensions, which makes gust\-induced load excursions a critical threat to flight stability and structural integrity\([9](https://arxiv.org/html/2608.12448#bib.bib4);[3](https://arxiv.org/html/2608.12448#bib.bib10)\)\. The gust\-wing interaction parameter space is very high\-dimensional, spanning manifold combinations atmospheric conditions, terrain features and vehicle attitude\. Further, the resulting aerodynamic response is typically strongly nonlinear, which contributes to making interactions of gusts with small aircraft poorly understood\([10](https://arxiv.org/html/2608.12448#bib.bib5);[14](https://arxiv.org/html/2608.12448#bib.bib6)\)\.

Traditional gust models, from early deterministic velocity profiles to modern statistical descriptions of atmospheric turbulence, capture only a limited slice of this complexity\([5](https://arxiv.org/html/2608.12448#bib.bib9);[15](https://arxiv.org/html/2608.12448#bib.bib2)\)\. The classical linear framework of indicial response theory provides a closed\-form description of unsteady lift under small\-amplitude disturbances\([13](https://arxiv.org/html/2608.12448#bib.bib15)\)\. However, UAVs in the lower atmospheric boundary\-layer largely operate outside the linear regime\. For instance, the aerodynamics of non\-slender delta wings is dominated by leading\-edge vortex \(LEV\) development and strongly nonlinear separation phenomena\([7](https://arxiv.org/html/2608.12448#bib.bib1);[14](https://arxiv.org/html/2608.12448#bib.bib6)\), under which linear superposition fails\. Identifying an organizing criterion for the complexity of gust\-load responses in this regime, therefore, calls for an approach grounded in empirical data before analytical theory\.

Recent work in data\-driven fluid mechanics show a promising trend in this sense\. Sparse pressure sensing strategies have been developed for real\-time load estimation during gust encounters\([2](https://arxiv.org/html/2608.12448#bib.bib12);[8](https://arxiv.org/html/2608.12448#bib.bib13)\), and deep autoencoders have compressed complex vortical interactions onto low\-dimensional manifolds with physically interpretable coordinates\([4](https://arxiv.org/html/2608.12448#bib.bib11)\)\. Cluster\-based approaches have also shown promise for flow\-state estimation from sparse experimental measurements\([11](https://arxiv.org/html/2608.12448#bib.bib17)\)\. Results from the machine learning literature have demonstrated that data quality often matters more than quantity; predictive models trained on curated, high\-value training sets can outperform those trained on orders of magnitude more unfiltered data\([18](https://arxiv.org/html/2608.12448#bib.bib3);[6](https://arxiv.org/html/2608.12448#bib.bib16)\)\. This motivates a sparse\-in\-the\-data approach: a selected subset of the most informative gust encounters may suffice to capture the essential physics and enable understanding and further analysis\. In this respect we build on previous work111Currently under journal review\.\([16](https://arxiv.org/html/2608.12448#bib.bib14)\), that demonstrated the feasibility of finding a minimal subset, dubbed*textbook*, of a database of gust\-wing encounters at a single angle of incidence with minimal consequences on load prediction accuracy\.

### 1\.1Objectives and approach

The central motivation of this study is the question on what constitutes an objective criterion for classifying gust\-induced loads in a complex flight envelope\. Specifically, we ask whether an interpretable classification criterion such as labeling loads based on the vehicle attitude is adequate, or whether an criterion can be identified from available observations without losing ease of being understood by human experts\.

![Refer to caption](https://arxiv.org/html/2608.12448v1/sketch_labelling.png)

Figure 1:Gust\-induced loads\.\(a\) A flying wing encounters a series of gusts\. As the wing’s attitude varies, each encounter can happen at a different orientation\. \(b\) Classification of gust\-induced load response according to two different criteria: parametric and exemplar\-based\.The problem setting is illustrated in[Fig\.1](https://arxiv.org/html/2608.12448#S1.F1)a\. The case we consider is that of a non\-slender delta wing encountering a number of longitudinal gusts at several incidence configurations\. The choice of a non\-slender delta wing is well suited to investigating complex loads, as its aerodynamics is dominated by the development of unsteady nonlinear phenomena such as leading\-edge vortices \(LEVs\)\.

Our proposed approach is conceptualized in[Fig\.1](https://arxiv.org/html/2608.12448#S1.F1)b and contrasts two classification criteria; the first is based on human\-defined parameters, the other is based on labeling loads according to a set of representative exemplars, called*textbook*examples\.

The flight attitude is an important determinant of the aerodynamic loads acting on a wing under given atmospheric conditions and therefore constitutes a natural candidate as a tentative classification criterion of the gust\-load responses\. It is sensible to ask what extent the flight attitude explains the diversity of the load response and then interrogate the data to find an objective criterion\. Successfully finding such an organizing criterion could serve as the basis for understanding the response phenomenology and for investigating its roots in the fluid dynamics of gust\-wing interaction\.

In practice, our contribution consists of searching for this criterion on a sizable amount of experimental gust\-load measurements across several flight attitudes \(3479 encounters across 6 attitudes\) and extracting a small number of significant exemplars in an objective manner via a learned predictive model\. These*textbook*examples are taken as the canonical gust\-induced load types and used to classify all observations\. This exemplar\-based classification criterion is then compared to the one based on the incidence angles and utilized to guide the search for the fluid mechanical origins of different gust\-load types\.

The article is laid out as follows\. The experimental facility and the aerodynamic load database are described in[Section2](https://arxiv.org/html/2608.12448#S2)and[Section2\.2](https://arxiv.org/html/2608.12448#S2.SS2)\. The data summarization approach is introduced in[Section3](https://arxiv.org/html/2608.12448#S3)and its concrete realization in[Section3\.2](https://arxiv.org/html/2608.12448#S3.SS2)\. The resulting*textbook*examples are in[Section4\.1](https://arxiv.org/html/2608.12448#S4.SS1)and applied to gust\-load classification in[Section4\.2](https://arxiv.org/html/2608.12448#S4.SS2)\. The fluid dynamical characterization of representative cases is considered in[Section4\.3](https://arxiv.org/html/2608.12448#S4.SS3)\.

## 2Experimental setup and database

### 2\.1Random gust generator

The experimental facility, shown in[Fig\.2](https://arxiv.org/html/2608.12448#S2.F2)a, consists of a 9×\\times9 array of computer\-controlled dual tube\-axial DC fans capable of generating unsteady axial inflow across a broad range of conditions\. The facility can be programmed to operate semi\-autonomously, performing a high volume of trials and enabling coverage of a large experimental space spanned by combinations of many forcing parameters\.

The aerodynamic model is a non\-slender delta wing with a NACA0012 cross\-section and a mid\-span of chordc=30c=30cm, adapted from previous work\([14](https://arxiv.org/html/2608.12448#bib.bib6);[1](https://arxiv.org/html/2608.12448#bib.bib8)\)\. Four pressure taps are positioned near the leading edge at locations determined by a dedicated optimisation study\([2](https://arxiv.org/html/2608.12448#bib.bib12)\)\. A six\-component force balance mounted on the support sting records the time\-resolved aerodynamic loads\. The sting additionally allows independent adjustment of the static angle of attackα\\alphaand sideslip angleβ\\beta, enabling systematic variation of the wing attitude\. The causal chain linking the experimental parameters to the measured aerodynamic response is illustrated schematically in[Fig\.2](https://arxiv.org/html/2608.12448#S2.F2)b\.

The fan array is driven by three randomized forcing parameters: the base fan velocityπ1\\pi\_\{1\}, the velocity incrementπ2\\pi\_\{2\}, and the forcing interval durationπ3\\pi\_\{3\}\. During each interval, the fans accelerate or decelerate from the base velocity toward the target increment, before the cycle repeats\. The base velocity spans 100 discrete levels, keeping the chord\-based Reynolds number in the range6⋅104<c​U∞ν<3\.5⋅1056\\cdot 10^\{4\}<\\frac\{cU\_\{\\infty\}\}\{\\nu\}<3\.5\\cdot 10^\{5\}, whereU∞U\_\{\\infty\}denotes the mean freestream speed produced by a fan velocity setting away from the wing\. The velocity increment is drawn from a uniform random distribution between 10% and 130% of the base fan velocity, and the non\-dimensional forcing duration is randomized with a minimum value ofG/c=2G/c=2\. This randomized operation ensures unbiased coverage of the gust parameter space across successive trials\.

![Refer to caption](https://arxiv.org/html/2608.12448v1/apparatus_w_sequence.png)Figure 2:Random gust generator and experimental setup\.\(a\) The fan\-array facility with the non\-slender delta wing mounted on its sting, showing pressure taps and 6\-axis balance\. \(b\) Causal chain linking the experimental parameters \(the forcing parametersπk​\(t\)\\pi\_\{k\}\(t\)and the incidence anglesα,β\\alpha,\\beta, highlighted in red\) to the aerodynamic loadsCL​\(t\)C\_\{L\}\(t\)andCY​\(t\)C\_\{Y\}\(t\)\.
### 2\.2Gust\-event database

The experimental database spans six vehicle attitudes, combining two angles of attackα∈\{20∘,30∘\}\\alpha\\in\\\{20^\{\\circ\},30^\{\\circ\}\\\}with three sideslip anglesβ∈\{0∘,15∘,30∘\}\\beta\\in\\\{0^\{\\circ\},15^\{\\circ\},30^\{\\circ\}\\\}, ensuring markedly distinct flow states on the wing\. The attitudeα=30∘,β=0∘\\alpha=30^\{\\circ\},\\beta=0^\{\\circ\}is represented by 100 one\-minute trials, one for every fan speed level\. All remaining attitudes are represented by 50 trials each, yielding a total of 350 one\-minute trials across the full database; an overview is given in[Table1](https://arxiv.org/html/2608.12448#S2.T1)\.

The four pressure tap readings and the full six\-component balance output are recorded continuously throughout each trial, of which the lift coefficientCL​\(t\)C\_\{L\}\(t\)and the side\-force coefficientCY​\(t\)C\_\{Y\}\(t\)capture the two dominant load components under combined angle\-of\-attack and sideslip conditions\. Each trial is a continuous signal that contains the response signatures to many randomly occurring successive gust encounters, which we call “events”\. A time\-series segmentation procedure is therefore applied to each trial to isolate individual gust events as the elementary units of data \(also known as “token”\)\. The automated segmentation procedure is detailed in[SectionA\.1](https://arxiv.org/html/2608.12448#A1.SS1)and is based on the lift coefficient historyCL​\(t\)C\_\{L\}\(t\), which serves as the primary indicator of aerodynamic response\.

Table 1:Overview of the six flight attitudes and the 3479 events in the gust\-load database\.The final database comprises 3479 gust events; some descriptive statistics are summarized in[Table1](https://arxiv.org/html/2608.12448#S2.T1)and the events are visualized in[Fig\.3](https://arxiv.org/html/2608.12448#S2.F3)\. The mean event duration is approximately 115 convective timest∗=t​U∞/ct^\{\*\}=tU\_\{\\infty\}/c, with a standard deviation of approximately 41 convective times\. The database is partitioned into a training set of 2783 events and a test set of 696 events following an 80–20 split\.

The left column of[Fig\.3](https://arxiv.org/html/2608.12448#S2.F3)a shows the time series of all 2783 training events for each channel including the four pressure readingsCp​1​\(t\)C\_\{p1\}\(t\)throughCp​4​\(t\)C\_\{p4\}\(t\)and the two load outputsCL​\(t\)C\_\{L\}\(t\)andCY​\(t\)C\_\{Y\}\(t\)overlaid across all the attitudes\. The projections of each channel againstCLC\_\{L\}reveal that the sensed pressure and the aerodynamic loads have a non\-linear dependence, with a non\-trivial data geometry which reflects the complexity of the underlying gust\-wing interactions\.

![Refer to caption](https://arxiv.org/html/2608.12448v1/database_attitudes.png)Figure 3:Gust\-load event database\.\(a\) event time\-series \(left\-hand column\) and their projection on each coordinate plane of the input\-response space \(right\-hand column\)\. Time is expressed in convective unitst∗=t​U∞/ct^\{\*\}=tU\_\{\\infty\}/c\. \(b\) Distribution of the 2783 training gust events on theCLC\_\{L\}–CYC\_\{Y\}response plane, colored by flight attitude\. Large symbols mark the centroids of each event and symbol shape distinguishes the two values ofα\\alpha\.[Fig\.3](https://arxiv.org/html/2608.12448#S2.F3)b provides a scaled\-up view of the projection of the data on theCL−CYC\_\{L\}\-C\_\{Y\}response plane and also labels events by the flight attitude they were recorded at\. The six attitudes occupy partially overlapping regions of the response space\. Increasing the sideslip angleβ\\betaproduces a systematic shift toward more negative side\-force valuesCYC\_\{Y\}across both angles of attack, while the higher angle of attackα=30∘\\alpha=30^\{\\circ\}extends to larger lift values\. The data geometry in the response space raises the question of whether the attitude is a fundamental organizing parameter of gust\-induced load response or whether objective response patterns that transcend attitude boundaries can be identified\.

## 3Data\-driven response modelling and summarization

The size and complexity of the database supplied by the experimental facility raises the issue of extracting the information contained therein in a form accessible to a human expert\. Compressing a large experimental sample to a concise, maximally informative subset of examples would bridge the gap between large\-scale automated data collection and physical understanding\. Borrowing from a previous study, we refer to such a subset as a*textbook*\([16](https://arxiv.org/html/2608.12448#bib.bib14)\)\.

### 3\.1*textbook*datasets

A*textbook*is understood as a small subsetDtxt⊂DD\_\{\\text\{txt\}\}\\subset Dof a large databaseDDthat is accurate, in that it guarantees high generalization accuracy on unseen data; and parsimonious, in that it is as small as possible while remaining accurate\. Formally, the first property can be defined as follows\. Given a predictive model trained on subsetsDm⊂DD\_\{m\}\\subset Dof cardinality\|Dm\|=m\|D\_\{m\}\|=mwithm≪\|D\|=m∞m\\ll\|D\|=m\_\{\\infty\}and evaluated on a held\-out test set, the*textbook*of sizemmis the subset that minimizes:

Dtxt​\(m\)=arg⁡minDm⊂D,\|Dm\|=m​ε​\(Dm\)D\_\{\\text\{txt\}\}\(m\)=\\underset\{D\_\{m\}\\subset D,\\,\|D\_\{m\}\|=m\}\{\\arg\\min\}\\,\\varepsilon\(D\_\{m\}\)\(1\)
whereε⁡\(Dm\)\\varepsilon\(D\_\{m\}\)denote the test error of the model trained onDmD\_\{m\}\.[Eq\.1](https://arxiv.org/html/2608.12448#S3.E1)defines a combinatorial search problem over all\(\|D\|m\)\\binom\{\|D\|\}\{m\}possible subsets, which is computationally intractable for any dataset of practical size\. Surrogate approaches are therefore needed\.

The computer science literature \(whereDtxtD\_\{\\text\{txt\}\}are usually called*core\-sets*\) identifies two tractable ways of constructing approximations toDtxtD\_\{\\text\{txt\}\}: one is based on selecting individual samples of high value or “difficulty” for the model to learn\([17](https://arxiv.org/html/2608.12448#bib.bib7);[12](https://arxiv.org/html/2608.12448#bib.bib18)\); the other is to aim at good coverage of the sample diversity through clustering\-like procedures\([18](https://arxiv.org/html/2608.12448#bib.bib3)\)\. Here we take the second path, which is empirically known to be superior for smaller subsets sizes\.

The coverage approach necessitates two main elements\. The first is a representation functionF:D→ℝdF:D\\rightarrow\\mathbb\{R\}^\{d\}\(an embedding\) that maps each event in the database to a fixed\-length descriptor vectorξ\\xiof dimensiondd\. The second is a diversity score𝒱ξ​\(Dm\)\\mathcal\{V\}\_\{\\xi\}\(D\_\{m\}\)that measures the internal diversity of subsetDmD\_\{m\}by comparing its elements inξ\\xi\-space\. The surrogate*textbook*of[Eq\.1](https://arxiv.org/html/2608.12448#S3.E1)is then found as the subset that maximizes

Dtxt​\(m\)≈arg⁡maxDm⊂D,\|Dm\|=m​𝒱ξ​\(Dm\),D\_\{\\text\{txt\}\}\(m\)\\approx\\underset\{D\_\{m\}\\subset D,\\,\|D\_\{m\}\|=m\}\{\\arg\\max\}\\,\\mathcal\{V\}\_\{\\xi\}\(D\_\{m\}\),\(2\)
rewarding coverage of the full dataset while penalizing redundancy among selected events\. Crucially, the information captured by[Eq\.2](https://arxiv.org/html/2608.12448#S3.E2)reflects the choice of embedding coordinatesξ\\xiand diversity score𝒱\\mathcal\{V\}\. An ideal*textbook*would therefore use a representation that reflects the underlying functional relation \(in the present case, the pressure\-load response\)\.

Finally, the choice of*textbook*sizemmis based on a trade\-off with predictive accuracy, in analogy with lossy compression\. Ifε⁡\(D\)\\varepsilon\(D\)is the baseline test error when training on the full database, the optimal*textbook*sizem∗m^\{\*\}is the smallest subset that achieves performanceε⁡\(Dtxt​\(m\)\)≥δ⋅ε⁡\(D\)\\varepsilon\\\!\\left\(D\_\{\\text\{txt\}\}\(m\)\\right\)\\geq\\delta\\cdot\\varepsilon\(D\)within a desired toleranceδ∈\(0,1\)\\delta\\in\(0,1\)of the full\-database limit\.

The specific choice of predictive model, representationξ\\xi, diversity score𝒱\\mathcal\{V\}and toleranceδ\\deltaare described in[Section3\.2](https://arxiv.org/html/2608.12448#S3.SS2)\.

### 3\.2Data summarization and*textbook*selection

The aerodynamic load prediction model used to validate*textbook*quality is a Multi\-Layer Perceptron \(MLP\) which takes the four instantaneous pressure readingsCp​1C\_\{p1\}throughCp​4C\_\{p4\}as inputs and predictingCLC\_\{L\}andCYC\_\{Y\}simultaneously as outputs\. The model has two hidden layers with 32 ReLU units each\. The MLP is trained using the combined mean squared error \(MSE\) loss onCLC\_\{L\}andCYC\_\{Y\}as the loss metricε\\varepsilon\. First, training is performed on the entire database to determine the baseline accuracy, then it is repeated from scratch for each of the subsets of increasing size \(selected as explained below\), resulting in a different set of trained weights for each subset\. Each model is evaluated on the same held\-out test set using the same MSE loss metric\.

A schematic view of the*textbook*\-finding procedure is in[Fig\.4](https://arxiv.org/html/2608.12448#S3.F4)a\. As the embedding coordinatesξ\\xiwe take the output of the trained MLP truncated before its final \(linear\) layer\. This constitutes a set of learned coordinates that embed the network inputs into a linear manifold of dimension equal to the layer width \(d=32d=32in the present case\)\. L2 distances between data point embeddings i\.e\.\(ξ,CL,CY\)\(\\xi,C\_\{L\},C\_\{Y\}\)are thus a natural candidate for an objective, data\-driven pairwise diversity criterion\.

The*textbook*subsets are identified through a data summarization procedure based onkk\-medoids clustering in embedding space \([Fig\.4](https://arxiv.org/html/2608.12448#S3.F4)a bottom row\), in which themmcluster centers constitute the*textbook*events\. Formally, this amounts to solving[Eq\.2](https://arxiv.org/html/2608.12448#S3.E2)with𝒱⁡\(Dm\)\\mathcal\{V\}\(D\_\{m\}\)defined as the sum of the average distances from cluster centers\([19](https://arxiv.org/html/2608.12448#bib.bib19)\)\. The pairwise similarity between events is the L2 distance between their barycenters inξ\\xi\-space\.

The performance of*textbook*s selected according to this procedure is benchmarked against random subsets of equivalent size, which serve as a reference for the expected accuracy of unguided subset selection\. The resulting learning curves are discussed in[Section4\.1](https://arxiv.org/html/2608.12448#S4.SS1)\.

The optimal*textbook*sizem∗m^\{\*\}is selected using a tolerance identified from the learning curve as the point of diminishing returns, beyond which additional*textbook*events yield negligible improvement in test accuracy and is reported in[Section4\.1](https://arxiv.org/html/2608.12448#S4.SS1)\.

Figure 4:*textbook*datasets\.a\) Data subset selection procedure\. Symbols are as defined in[Section3\.1](https://arxiv.org/html/2608.12448#S3.SS1)and the procedure explained in[Section3\.2](https://arxiv.org/html/2608.12448#S3.SS2)\. b\) Learning curves\. Test error of models trained on increasingly larger subsets of the full dataset, selected according to several criteria: randomly \(black solid\),*textbook*subsets selected without attitude labels \(red empty\), and including attitude labels \(blue solid\)\. The dotted line marks the baseline accuracy achieved by the full database\.

## 4Results

### 4\.1*textbook*selection

The learning curves resulting from the selection procedure for random subsets, labeled*textbook*s, and unlabeled*textbook*s are shown in[Fig\.4](https://arxiv.org/html/2608.12448#S3.F4)b\. Across all training set sizes, both*textbook*conditions achieve substantially lower test error than random subsets of equivalent size, confirming that the summarization procedure successfully identifies high\-information events\. At the chosen*textbook*size ofm∗=9m^\{\*\}=9, the*textbook*achieves a test mean\-squared error of 0\.161, compared to 0\.473 for a random subset of the same size which is a reduction of approximately 66%, while the full\-database baseline stands at 0\.099\. This corresponds to a compression ratio ofm∗/m∞≈0\.3%m^\{\*\}/m\_\{\\infty\}\\approx 0\.3\\%, representing a reduction of the full training set size by over two orders of magnitude, and about15%15\\%when compared to random subsets of size6060that yield comparable test accuracy\.

In order to assess to which extent the attitude provides a sensible classification criterion for gust\-load response,*textbook*selection is repeated with the inclusion of attitude labels for each event\. In this setting, the subset selection algorithm will select that diverse attitude make\-up in terms of the attitude\. If the two subset selection setting yield*textbook*s of comparable generalization accuracy, it follows that attitude labeling is superfluous for the purpose of capturing the essential diversity of the data\.

It is apparent in[Fig\.4](https://arxiv.org/html/2608.12448#S3.F4)b that the learning curves for attitude\-labeled and unlabeled*textbook*s do not show diverging trends to a significant level\. Atm∗=9m^\{\*\}=9the unlabeled condition marginally outperforms the labeled one, while atm=13m=13andm=20m=20the two conditions converge to within 5% of each other\. The absence of any significant effect from attitude labeling indicates that the summarization procedure already incorporates the essential data diversity, including in the attitude\.

### 4\.2Cross\-attitude response

Each of them∗=9m^\{\*\}=9*textbook*events serves as a representative exemplar of a response type within the full gust\-load database, providing the objective labeling criterion theorized in[Fig\.1](https://arxiv.org/html/2608.12448#S1.F1)b\. All the 3479 gust\-load events are assigned to their nearest neighboring*textbook*event through the same similarity criterion used during selection\. The resulting clustering is visualized in[Fig\.5](https://arxiv.org/html/2608.12448#S4.F5)a, which shows the sameCL−CYC\_\{L\}\-C\_\{Y\}response space as[Fig\.3](https://arxiv.org/html/2608.12448#S2.F3)b but now with each event colored by assigned response type rather than flight attitude\.

The nine clusters occupy distinct, geometrically coherent regions of theCL−CYC\_\{L\}\-C\_\{Y\}response plane\. Events at high lift with near\-zero side force are captured by clusters concentrated in the upper\-right region of the plane, while the strongly asymmetric lateral\-load events populate the lower portion\. The intermediate clusters span the curved central band, capturing transitional types between these regimes\. The*textbook*centroids, marked by large symbols, sit near the geometric centers of each cluster\. This is consistent with each*textbook*event being representative of its assigned group rather than an outlier\. Taken together, the nine clusters suggest that the gust\-load response space is structured into physically distinct regimes that the*textbook*has successfully identified\.

The cluster membership per attitude is shown in[Fig\.5](https://arxiv.org/html/2608.12448#S4.F5)b and tabulated in[Table2](https://arxiv.org/html/2608.12448#S4.T2)\. Types 1, 9, 3 and 7 are exclusive to the two zero\-sideslip attitudes \(β\\beta=0∘0^\{\\circ\}\) and occur at both angles of attack\. Types 2, 5, 6, 8, 9 are associated with non\-zero sideslip conditions, with 2 and 5 being the only types that only occur at a single attitude\. The zero sideslip angleβ=0\\beta=0is a strong qualitative differentiator of the response as it impliesCY=0C\_\{Y\}=0, a property which is correctly detected by the objective summarization procedure\. This is consistent with the finding of[Section4\.1](https://arxiv.org/html/2608.12448#S4.SS1)that attitude labeling does not affect*textbook*quality\. Types 8 and 9 occur at four angles of attack and represent the most commonly occurring gust\-load types\. The presence of cross\-attitude clusters confirms that a subset of fundamental response types recurs independently of the flight configuration\. The exemplar\-based classification exposes structure of the response space that a parametric criterion based on the attitude would obscure\.

![Refer to caption](https://arxiv.org/html/2608.12448v1/manifold_barchart.png)Figure 5:Objective classification of gust\-load events\.\(a\) Distribution of all events in response space, colored by type as identified through clustering around the*textbook*events\. Large symbols represent*textbook*event centroids as in[Fig\.3](https://arxiv.org/html/2608.12448#S2.F3)b\. \(b\) Cluster membership share per attitude, showing which response types are attitude\-specific\.Table 2:Event type occurrence across flight attitudes\. A solid square indicates the cluster contributes more than 5% of events at that attitude\.
### 4\.3Characterization of*textbook*events

The*textbook*exemplars identified above organize the diversity of the wing’s gust\-load response in terms of objective response types\. A physically interesting question is investigating the nature of the flow phenomena that generate a certain response type\. This is especially true for explaining how similar response types arise at different attitudes, which would be valuable physical insight into the complexity of the wing’s unsteady aerodynamics\.

As no direct measurements of the flow field was attempted, we resort to the closest available proxy, i\.e\. the time histories of pressure and loading transients\. As the position of the taps on the wing is fixed, the pressure time histories provide a sparse view of the time\-varying pressure field on the wing surface\. The force time histories \(surface integrals of the pressure\) subsume global pressure information\.

As a study case, we consider responses of type 7 \(highlighted in green in[Fig\.5](https://arxiv.org/html/2608.12448#S4.F5)\), which give rise to the simultaneously strongestCL−CYC\_\{L\}\-C\_\{Y\}responses and belong to two different attitudes, namely those where the sideslip angleβ=0\\beta=0\. The time histories of type\-7 events is visually inspected in[Fig\.6](https://arxiv.org/html/2608.12448#S4.F6)alongside the type exemplar, which belongs toα=30\\alpha=30\.

We concentrate the analysis on two aspects: first, we recall from[Fig\.5](https://arxiv.org/html/2608.12448#S4.F5)that around90%90\\%of type 7 events occurs atα=30\\alpha=30; second, we notice from[Fig\.6](https://arxiv.org/html/2608.12448#S4.F6)that the time histories of type\-7 events share attitude\-wise qualitative commonalities\. Specifically, type 7 load histories fromα=30,β=0\\alpha=30,\\,\\beta=0are qualitatively well represented by the type exemplar, being generally single\-peaked and with a median duration of108​t∗108t^\{\*\}and median peak time at53​t∗53t^\{\*\}\. The other attitudeα=20,β=0\\alpha=20,\\,\\beta=0show generally longer median durations of106​t∗106t^\{\*\}and predominantly multi\-peaked or sawtooth\-shaped histories\.49​t∗49t^\{\*\}These facts hint to two different mechanisms leading to type\-7 responses depending on the attack angle\.

To this end, we consider existing PIV measurements of the flow around the same delta wing model taken from a previous study\([14](https://arxiv.org/html/2608.12448#bib.bib6)\), which were collected at an attitude ofα=30,β=0\\alpha=30,\\,\\beta=0whilst the wing was subjected to a impulsive forward acceleration simulating an encounter with a longitudinal gust\. The PIVs exhibited leading\-edge vortex \(LEV\) formation, followed by pinch\-off which causes the flow to reattach and the lift force to revert to the baseline\. These time\-dependent features can be larger or smaller scale, depending on the attitude and the magnitude of the acceleration, which results on average in similar instantaneous response\.

The observation that the higher angle of attack is responsible for most type\-7 events agrees with the physical intuition that higher mean lift must be more common; analogously, formation and pinch\-off of isolated LEVs is more likely at higherα\\alpha\. On the other hand, atα=20\\alpha=20only a minority of the encountered gusts have sufficient intensity to produce similarly high baseline lift; flow detachment appears to persist for longer and produce a series of weaker LEVs which form and pinch off intermittently\.

A conclusive investigation of the fluid mechanics responsible for the observed response features would necessitate repeating the experiment and perform flow\-field measurements\. However, these more sophisticated and costlier experiments would be focused on the exemplar cases rather than repeated for many different combinations of attitude and fan forcing\. This possibility highlights another practical advantage of the proposed approach\.

![Refer to caption](https://arxiv.org/html/2608.12448v1/timeseries_7.png)Figure 6:*textbook*event signatures\.Pressure\-load histories for response type 7 and their signatures in input\-response space\. Different colors indicate the original attitude each event was recorded at\.

## 5Conclusions

We investigate the pressure and load transient events induced on a non\-slender delta wing by 3479 random gust encounters across six angles of incidence, measured thanks to a partially automated facility\. The instantaneous pressure\-load response is complex and not trivially separated by the incidence angles of the wing with respect to the flow, motivating the search for an objective classification criterion\.

We contribute with three points of methodological novelty\.

First, building on the concept of*textbook*\([16](https://arxiv.org/html/2608.12448#bib.bib14)\), we are able to isolate of nine essential gust\-load response types by means of a purely data\-driven, objective procedure\. The procedure leverages an accurate response model learned from the full database to distill the database diversity into a few high\-quality exemplars\. This is the first application of such data\-driven techniques to experimental aerodynamics that we are aware of and represents a principled improvement on the previous iteration\. We also notice that quality and number of the*textbook*exemplars are insensitive to whether attitude labeling is included or not into the selection process, hinting that the intuitive attitude\-based classification does not provide any information on the load response that is not detected by the data\-based procedure\.

Second, we construct the desired objective classification criterion by labeling observations according to their closest*textbook*exemplar\. We find that most fundamental gust\-load types are not confined to one attitude, confirming that it is not the most compressive organizing parameter\. While a zero sideslip angle segregates load types into two non\-overlapping groups, loads of the same type occur at up to four different combinations of sideslip and angle of attack\.

Third, the manageable number of just a few response types is advantageous for mapping the unsteady flow physics responsible for the aerodynamic response variety\. A campaign of high\-quality flow\-field measurements could now be focused on the representative cases rather than distributed across the experimental parameter range following a partial or intuitive understanding of the response dependence\. We examine the transient load histories of type 7, which reveal both common and exclusive characteristics depending on the angle of attack\. Existing flow\-field data on the same wing model suggest the observed patterns are consistent with the dynamics of leading\-edge vortex formation and pinch off\.

In conclusion, this study provides a demonstration of a data\-based framework for unfolding complex aerodynamic phenomena which goes beyond sheer predictive power and emphasizes simplicity and human\-centric understanding\. We believe this and related approaches have the potential to benefit unsteady aerodynamics, as well as other areas of fluid mechanics where complex flow configurations are a primary concern\.

#### Acknowledgements

## Declarations

Funding:

Conflict of interest:The authors declare no conflict of interest\.

Ethics approval:Not applicable\.

Consent for publication:Not applicable\.

Data availability:Upon reasonable request\.

Code availability:The code used for the data analysis will be available on GitHub upon publication of the manuscript\.

Author contribution:P\.O\. and K\.S\. contributed to the data analysis methodology\. D\.E\.R\. conceived the project and acquired funding\. All authors contributed to the manuscript\.

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## Appendix AAppendix

### A\.1Trial segmentation

The procedure to segment continuous trials into discrete events is as follows\. The mode ofCLC\_\{L\}across each trial is computed as a proxy for the baseline flow state, and candidate event boundaries are identified at points whereCLC\_\{L\}reverses to this baseline\. Each candidate segment is subsequently assessed against four quality thresholds on its maximum and minimum duration, noise\-to\-signal ratio and monotonicity, and discarded if it fails to satisfy any of them\.

![Refer to caption](https://arxiv.org/html/2608.12448v1/figures/segmentation_14.png)Figure 7:Segmentation of a representative one\-minute trial ofCL​\(t\)C\_\{L\}\(t\)into individual gust events\. Valid segments are shown against a colored background; red dots mark event boundaries\.