Unsteady Loads Research Articles

Accuracy assessment of Beddoes-Leishman and IAG dynamic stall models for wind turbine applications

The study presents a systematic comparison between two of the most-credited dynamic stall models for wind turbine applications: the original Beddoes-Leishman (BL) model and the newly-developed IAG. The scope of such comparison, supported by experimental data, is to shed new light on the actual suitability of current dynamic stall models for their integration into modern wind turbine simulation codes, and on the best practices to calibrate them. Two different strategies are followed for the calibration of the BL model: 1) standard one, compliant with common practices found in the literature; 2) a physics-oriented one, focusing on the constants defining the dynamic stall onset as well as on the parameters governing the duration of the vortex shedding process. The IAG model, initially developed based on the first-order BL formulation and recently improved by reducing the number of constants and removing compressibility effects, is applied instead in its standard form only. The two models are compared across a range of oscillation mean angles, amplitudes, and reduced frequencies. Results demonstrate that the original BL model, although with a challenging calibration process, when properly tuned, can provide a very good description of aerodynamic unsteady loads. While showing consistent results, the IAG formulation appears to be more robust, as it employs fewer constants and extracts most of the needed information directly from the input polar data. The comparison between the calibrated BL and IAG models highlights critical modelling aspects, the computation of drag and determination of the stall onset above all, offering valuable insights for the future development of dynamic stall formulations.

A general framework for the design of efficient passive pitch systems

Mitigating the impact of variable inflow conditions is critical for a wide range of engineering systems such as drones or wind and tidal turbines. Passive control systems are of increasing interest for their inherent reliability, but a mathematical framework to aid the design of such systems is currently lacking. To this end, in this paper a two-dimensional rigid foil that passively pitches in response to changes in the flow velocity is considered. Both an analytical quasi-steady model and a dynamic low-order model are developed to investigate the pivot point position that maximizes unsteady load mitigation. The paper focuses on streamwise gusts, but the proposed methodology would apply equally to any change in the inflow velocity (speed and/or direction). The quasi-steady model shows that the force component in any arbitrary direction can be kept constant if the pivot lies on a particular line, and that the line coordinates depend on the gust and the foil characteristics. The dynamic model reveals that the optimum distance of the pivot location from the foil increases with decreasing inertia. For a foil at small angles of incidence, the optimum pivot point is along the extended chord line. This knowledge provides a methodology to design optimum passively pitching systems for a plethora of applications, including flying and swimming robotic vehicles, and provides new insights into the underlying physics of gust mitigation.

Challenges in Rotor Aerodynamic Modeling for Non-Uniform Inflow Conditions

Within an international collaboration framework, the accuracy of rotor aerodynamic models used for design load calculations of wind turbines is being assessed. Where the use of high-fidelity computation fluid dynamics (CFD) and mid-fidelity free-vortex wake (FVW) models has become commonplace within the wind energy community, these still fail to meet the requirements in terms of execution time and computational cost needed for design load calculations. The fast but engineering fidelity blade-element/momentum (BEM) method can therefore still be considered the industry workhorse for design load simulations.At the same time, upscaling of wind turbine rotors makes inflow non-uniformities (e.g. shear, veer, turbulence) more important. The objective of this work is to assess model accuracy in non-uniform inflow conditions, which violate several BEM assumptions. Thereto a comparison in turbulent inflow has been executed including a wide variety of codes, focusing on the DanAero field measurements, where a 2.3-MW turbine was equipped with, among other sensors, a pressure measurement apparatus. The results indicate that, although average load patterns are in good agreement, this does not hold for the unsteady loads that drive fatigue damage and aero-elastic stability. A simplified comparison round in vertical shear was initiated to investigate the observed differences in a more controlled manner. A consistent offset in load amplitude was observed between CFD and free-vortex codes on the one hand and BEM-type codes on the other hand. To shed more light on the observations, dedicated efforts are ongoing to pinpoint the cause for these differences, in the end leading to guidelines for an improved BEM implementation.

Comparative study on lateral vibration characteristics of the head and tail cars of high-speed trains under unsteady aerodynamic loadings

With the increasing speed of high-speed trains, abnormal vibrations in the tail car pose a threat to its overall stability and running safety. In this study, the aerodynamic and multibody dynamics models for 3-car and 8-car trains are established. The flow field characteristics and dynamic performance around the head and tail cars under both steady and unsteady aerodynamic loadings at a speed of 400 km/h are compared. The results reveal that the surrounding flow field exhibits highly pronounced unsteady characteristics, directly leading to a significant impact of unsteady aerodynamic loadings on the train's dynamic response. Under unsteady loadings, the lateral stability indexes, displacement, velocity, and acceleration of the tail car are all worser compared to those of the head car. Additionally, the larger oscillation amplitudes and frequencies closer to the suspension modal frequencies under unsteady aerodynamic loadings contribute to more pronounced lateral displacements and accelerations, constituting a crucial factor in the observed ‘tail swing' phenomenon for the high-speed train.

A discrete-module-beam hydroelasticity method with finite element theory in analyzing VLFS in different engineering scenarios

A numerical tool is developed in the framework of the Discrete-Module-Beam method to analyze hydroelasticity of very large floating structures (VLFS) such as floating offshore photovoltaic platforms (FOPV). The proposed method utilizes finite element theory to perform structural analysis. The 3D potential flow theory is used to perform a hydrodynamic analysis after macro-submodule division. For the structural analysis part, the platform is divided into macro-submodules and each macro-submodule is represented by a beam, which is then further discretized into micro beam elements with the nodes categorized into three groups, that is, the lumped mass, the boundary nodes and the inner nodes. Derivation of the lumped-mass stiffness matrix and recovery of the displacement distribution are given. Procedures to deal with interconnections, complex boundary conditions and unsteady loads are elaborated. Good agreement is obtained in different loading scenarios to the numerical and experimental results.

Acoustics and Forces from a Subscale Electric Vertical-Takeoff-and-Landing Rotor in Edgewise Flight

The far-field noise and forces of a subscale two-blade rotor were measured in an anechoic wind tunnel for both hover and edgewise flight conditions. The mean thrust and torque coefficients increased with the edgewise advance ratio in relation to the hover coefficients. The tonal and broadband noise contributions were separated and analyzed in both the time and frequency domains. The sound pressure level (SPL) at the blade pass frequency (BPF) had the greatest contribution to the overall SPL, and the dominant broadband noise source changed from high-frequency trailing edge noise to midfrequency blade wake interaction noise with an increasing edgewise advance ratio. The tip Mach number scaling of the noise sources was examined and found to be highly dependent on the oncoming freestream velocity. The BPF SPL directivity showed a dependency on the advance ratio, with a peak magnitude below and upstream of the rotor’s retreating side. The broadband noise had a dipole directivity with a minimum in the rotor plane. The midfrequency broadband noise had signatures indicative of blade wake interaction noise, and the high-frequency noise had signatures indicative of amplitude modulated trailing edge noise. The results demonstrate the importance of unsteady loading noise and broadband noise for subscale urban air mobility rotors.

On the aeroelastic bifurcation of a flexible panel subjected to cavity pressure and inviscid oblique shock

The aeroelasticity of a panel in the presence of a shock is a fundamental issue of great significance in the development of hypersonic vehicles. In practical engineering, cavity pressure emerges as a crucial factor that influences the nonlinear dynamical characteristics of the panel. This study focuses on the aeroelastic bifurcation of a flexible panel subjected to both cavity pressure and oblique shock. To this end, a computational method is devised, coupling a high-fidelity reduced-order model for unsteady aerodynamic loads with nonlinear structural equations. The solution is meticulously tracked by continuous calculations. The obtained results indicate that cavity pressure plays a pivotal role in determining the bifurcation and stability characteristics of the system. First, the system exhibits hysteresis behaviour in response to the ascending and descending dynamic pressures. The evolution of hysteresis behaviour originates from the phenomenon of cusp catastrophe. Second, variations in cavity pressure induce three types of bifurcation phenomena, exhibiting characteristics akin to supercritical Hopf bifurcation, subcritical Hopf bifurcation and saddle-node bifurcation of cycles. The system's response at the critical points of these bifurcations manifests as long-period asymptotic flutter or explosive flutter. Lastly, the evolution of the dynamical system among these three types of bifurcations is an important factor contributing to the discrepancies observed in certain research results. This study enhances the understanding of the nonlinear dynamical behaviour of panel aeroelasticity in complex practical environments and provides new explanations for the discrepancies observed in certain research results.

Actuator Surface Modeling of Rotors at the Ship–Helicopter Dynamic Interface

The aerodynamics of the ship–helicopter dynamic interface play an important role in determining ship–helicopter operating limits. A review of prior studies, in conjunction with experimental work, indicates that a time-accurate rotor model and a physics-based turbulence-resolving flow solver with two-way coupling between these components are required to accurately predict rotor–airwake and rotor–rotor interactions. Here, a recently developed actuator surface model is validated against two further multirotor cases and employed for two large-scale ship–helicopter dynamic interfaces with up to three concurrent rotorcraft and five rotors. Simulations of the Sikorsky X2 rotor with significant rotor–rotor interaction match resolved unsteady rotor loading well using 1.2% of the computational effort of resolved blade calculations. Computations of the outwash from a Boeing CH-47D in hover gave results within experimental error bars for most azimuths, where discrepancies may be attributed to the estimated trim and division of rotor loading. Simulations of the dynamic interface between the Simple Frigate Shape 1 and a hovering UH-60a were presented for three wind-over-deck angles and three positions for 40 knots of wind speed. The computed thrust and root mean square thrust coefficients in the closed-loop pilot control band are presented and demonstrated to have an to have an expected dependence on immersion in the ship airwake. Finally, a large-scale dynamic interface computation was performed with two CH-47Ds and one UH-60a operating concurrently over the deck of the landing helicopter assault ship at 40 knots of wind speed and 0° and 30° angle. Results highlighted the importance of understanding rotor-wake/airwake interactions dependent on rotorcraft type and position and wind-over-deck. The computation provided unsteady thrust and torque for each of the five rotors from 60 s of data using just 11,424 central processing unit hours on a grid of 19.5×106 points. These simulations show that hybrid Navier–Stokes simulations employing actuator line or surface methods are capable of providing high-fidelity, time-accurate predictions of rotor loads at the ship–helicopter dynamic interface at substantially lower cost.

Efficient prediction of tidal turbine fatigue loading using turbulent onset flow from Large Eddy Simulations

To maximise the availability of power extraction from a tidal stream site, tidal turbines need to be able to operate reliably when located within arrays. This requires a thorough understanding of the operating conditions, which include turbulence, velocity shear due to bed proximity and roughness, ocean waves and due to upstream turbine wakes, over the range of flow speeds that contribute to the loading experienced by the devices. High-fidelity models such as Large Eddy Simulation (LES) can be used to represent these complex flow conditions and turbine device models can be embedded to predict loading. However, to inform micro-siting of multiple turbines with an array, the computational cost of performing multiple simulations of this type is impractical. Unsteady onset conditions can be generated from the LES to be used in an offline coupling fashion as input to lower-fidelity load prediction models to enable computationally efficient array design. In this study, an in-house Blade Element Momentum (BEM) method is assessed for prediction of the unsteady loads on the turbines of a floating tidal device with unsteady inflow developed with the in-house LES solver DOFAS. Load predictions are compared to those obtained using the same unsteady inflow to the commercial tool Tidal Bladed and from an Actuator Line Model (ALM) embedded in the LES solver. Estimates of fatigue loads differ by up to 3% for mean thrust and 11% for blade root bending moment for a turbine subject to a turbulent channel flow. When subjected to more complex flows typical of a turbine wake, the predictions of rotor thrust fatigue differ by up to 10%, with loads reduced by the inclusion of a pitch controller.

Constructing nonlinear data-driven models from pitching wing experiments using multisine excitation signals

Accurate modelling of unsteady nonlinear aerodynamic loads is crucial for the effective design and control of aerodynamic systems. Data-driven modelling approaches have emerged as powerful tools for capturing intricate system dynamics without prior knowledge of the underlying physical principles. This paper focuses on the construction of nonlinear data-driven models for pitching wing systems using multisine excitation signals and their relative merits compared to swept sine signals. The polynomial nonlinear state-space (PNLSS) modelling framework is used, which has been proven in the system identification community to offer flexibility and richness in representing diverse nonlinear phenomena. Data are gathered with a dedicated experimental wind tunnel setup. A NACA 0018 wing is made to oscillate in pitch according to a swept sine and a multisine with a frequency up to 2 Hz. The experiments are repeated to cover the attached flow, light stall, and deep stall regimes. PNLSS models are trained on multisines and benchmarked against models trained on swept sines to assess their comparative performance. Then, all models are successfully cross-validated using single sine data at different angles of attack. We conclude that one single data-driven model can accurately represent the lift force across different linear (attached flow) and nonlinear (light and deep stall) operating regimes.

Aerodynamic interference and unsteady loads for a trimming intermeshing rotor compound propeller helicopter at level forward flight

Traditional trim methods for isolated rotor helicopters are not suitable for intermeshing rotor compound propeller helicopters (IRcPHs) due to inherent structural differences. This study aims to establish a novel trim method for an intermeshing rotor with an auxiliary propeller. First, the relationship between the rotor cyclic pitch and the flapping response in the fully articulated rotor system with hinged extension is derived. The study then formulates the equilibrium state equations and the corresponding flight conditions for trimming the IRcPH in level forward flight. In addition, the unsteady vortex lattice method is employed to calculate the periodic averaging states on the rotor disk and correct for induced losses in flight dynamics. Finally, the paper focuses on the aerodynamic interference, blade–vortex interaction, and harmonic load patterns of the IRcPH, serving as a guide for rotor active control systems. Results show that the positions of the mutual aerodynamic interference occur constantly at 160° and 340° azimuths on the intermeshing rotor disk, serving as the phase basis for suppressing vibration. Moreover, the propeller enters the comfort zone as the forward flight speed increases, and the loads only have frequency orders of (2/nw)/rev with respect to the main frequency of the intermeshing rotor.

Conformally Decambered Natural Laminar Flow Blades for Vertical-Axis Wind Turbines

A two-element natural laminar flow airfoil—commonly used on medium-altitude, long-endurance, unmanned air vehicles—was conformally decambered for application on a two-bladed, H-rotor, vertical-axis wind turbine. Blade kinematics were used to determine the virtual camber line, which was then used to conformally map the original profile onto the chord line. Both decambered and original blade profiles were evaluated experimentally using large chord-radius ratios (0.6 and 0.75) that exploited dynamic stall to produce the driving torque. Decambered blades showed substantially greater power and torque coefficients than the original blades, up to 60 and 27%, respectively, which represents the first experimental validation of conformal decambering. Relatively large peak power coefficients of 0.28 were attained, despite maximum chord-based Reynolds numbers being less than 2×105. Depending upon the chord-radius ratio, either light or deep dynamic stall occurred in the second upstream quadrant, and the flap flow remained attached virtually throughout. In contrast, on the original profiles, massive separation was observed on the blades, and the flap flow remained separated due to assumed outer surface flow separation. Future research should consider surface pressure and flowfield measurements, significantly higher Reynolds numbers, and variable intracycle flap deflection mechanisms to optimize performance and minimize unsteady loads.

One-degree-of-freedom galloping instability of a 3D bluff body pendulum at high Reynolds number

The cross-flow swinging dynamics of a cube pendulum is studied experimentally in a flow at high Reynolds numbers (Re∼105) with a low free-stream turbulence intensity. A galloping instability is observed and results in the exponential growth of the swinging motion. The onset of galloping is found to be very sensitive to the static yaw angle of the cube. Despite the 3D geometry of the cube, flow mechanisms similar to the case of a square cylinder appear to govern the onset of the instability. A quasi-steady linear model of the motion is assessed to predict the stability of the pendulum.For the lowest reduced velocity investigated (U∗=18.5), unsteady phenomena arise during the saturation phase of the pendulum oscillations. From the analysis of the unsteady loads and the pressure distribution on the faces of the cube, an unsteady phase delay between the wake and the pendulum dynamics is identified. It produces an energy loss in the pendulum motion which favors its saturation.

Turbulent Airwake Estimation from Helicopter–Ship Wind-Tunnel Data

This paper presents a stochastic approach for modeling the turbulent airwake suitable for real-time simulation of the helicopter–ship dynamic interface. This approach relies on the measurements of unsteady loads collected during a wind-tunnel test campaign with a scaled helicopter operating over the deck of simple frigate shape 1. Power spectral densities of the measured aerodynamic loads combined with the estimated frequency response functions are used to find, through an optimization algorithm, a model of airwake spectra over the range of frequencies which mainly affects the pilot workload during shipboard operations. Then, a set of autoregressive filters is designed for every particular rotor position and wind-over-deck condition, so that when driven by white noise, the spectrum of the output will reproduce those obtained from the optimization. This approach is applied to three different tested wind directions and three rotor positions by implementing the autoregressive filters into the multibody model of the experimental rotor. Frequency response analysis of the aerodynamic loads demonstrates that the turbulent airwake model obtained from the experimental data can predict the unsteadiness of loads comparable to those measured in the wind tunnel across the bandwidth of interest for pilot activities. The identified airwake models could be applied to a full-scale model to simulate the unsteady loads effectively experienced by the helicopter during a ship landing flight.

Modeling the along-wind loading on a high-rise building considering the turbulence scale effects in the wind tunnel tests

The three-dimensional (3D) stretch and distortion of the approaching turbulent flow will cause significant unsteady wind loads on line-like structures, such as high-rise buildings (HRBs). This phenomenon becomes more prominent when the scale ratio Lu/D (Lu is turbulence scale, D is body size) decreases, which is defined as the turbulence scale effect (TSE). Consequently, the traditional aerodynamic admittance (AAF) fails to estimate the along-wind loading (AWL) on HRBs, especially in wind tunnel tests. This paper studies the TSE on the AAF of an HRB with a 2:1 rectangular cross-section based on the pressure measurement of a segmental model. A 3D spectral method is presented to decompose the traditional AAF of AWL into the two-dimensional AAF (2D-AAF) in fully coherent gusts and the 3D correction factor to consider the 3D effect. The results indicate that the reduction of Lu/D will amplify the high-frequency decay slope of 2D-AAF and attenuate the low-frequency value of traditional AAF, determined by the distorted turbulence and the 3D effect, respectively. Notably, the presented 2D-AAF model based on the experimental data from grid-generated turbulence can be adopted to predict the unsteady AWL on a 2:1 HRB in the simulated boundary layer (ABL) flow accurately, confirming the uniqueness of the proposed model.

Academician V.N. Boltinsky’s Scientific article Legacy in Agricultural Engineering Education and Science (Commemorating the 120th Anniversary of the Birth)

The article delves into the scientific contributions and advancements of the eminent Russian scientist in agricultural tractor construction, Vasily Nikolaevich Boltinsky. (Research purpose) The paper aims to shape and consolidate historical, scientific, engineering and technical concepts regarding the establishment and development of scientific and pedagogical institutions focused on the study of tractor engine operation and machine-tractor units in unsteady modes. It also seeks to underscore the contributions made by Doctor of Technical Sciences, Professor, and Member of VASKHNIL, V.N. Boltinsky. (Materials and methods) The study employs a chronological approach, utilizing historical, scientific, and informational resources, including publications by prominent scientists and original works by the academician himself. The paper also analyzes and summarizes historical facts detailing the role of V.N. Boltinsky in enhancing the design and efficiency of Soviet tractors and machinery units. (Results and discussions) Throughout his career, from assistant to associate professor to vice-president of VASKHNIL, V.N. Boltinsky dedicated himself to developing and testing new models of both foreign and domestic tractor equipment. His extensive experience and years of research culminated in the publication of the monograph «Operation of a tractor engine under unsteady load», which extensively explores the impact of varying loads on engine performance. V.N. Boltinsky successfully solved problems related to enhancing the operational speed of machine and tractor units, as well as innovating the design of agricultural machinery. (Conclusions) The life and professional endeavors of V.N. Boltinsky serve as a shining example of dedication to science, education, and the professional advancement of engineering and scientific professionals. To honor his legacy, an annual scientific and practical seminar entitled «Academician V.N. Boltinsky Readings» is held in Russia, ensuring his contributions continue to be celebrated and remembered.

Wind turbine rotors in surge motion: new insights into unsteady aerodynamics of floating offshore wind turbines (FOWTs) from experiments and simulations

Abstract. An accurate prediction of the unsteady loads acting on floating offshore wind turbines (FOWTs) under consideration of wave excitation is crucial for a resource-efficient turbine design. Despite a considerable number of simulation studies in this area, it is still not fully understood which unsteady aerodynamic phenomena have a notable influence on the loads acting on a wind turbine rotor in motion. In the present study, investigations are carried out to evaluate the most relevant unsteady aerodynamic phenomena for a wind turbine rotor in surge motion. As a result, inflow conditions are determined for which a significant influence of these phenomena on the rotor loads can be expected. The experimental and numerical investigations are conducted on a two-bladed wind turbine rotor subjected to a tower-top surge motion. A specialised wind tunnel test rig has been developed to measure the aerodynamic torque response of the rotor subjected to surge motions with moderate frequencies. The torque measurements are compared to two free-vortex-wake (FVW) methods, namely a panel method and a lifting-line method. Unsteady contributions that cannot be captured using quasi-steady modelling have not been detected in either the measurements or the simulations in the covered region of motions ranging from a rotor reduced frequency of 0.55 to 1.09 and with motion velocity amplitudes of up to 9 % of the wind speed. The surge motion frequencies were limited to a moderate range (5 to 10 Hz) due to vibrations occurring in the experiments. Therefore, a numerical study with an extended range of motion frequencies using the panel and the lifting-line method was performed. The results from both FVW methods reveal significant unsteady contributions of the surge motions to the torque and thrust response that have not been reported in the recent literature. Furthermore, the results show the presence of the returning wake effect, which is known from helicopter aerodynamics. Additional simulations of the UNAFLOW scale model and the IEA 15 MW rotor demonstrate that the occurrence of the returning wake effect is independent from the turbine but determined by the ratio of 3P and surge motion frequency. In the case of the IEA 15 MW rotor, a notable impact of the returning wake effect was found at surge motion frequencies in the range of typical wave periods. Finally, a comparison with OpenFAST simulations reveals notable differences in the modelling of the unsteady aerodynamic behaviour in comparison to the FVW methods.

Aeroelastic tailoring of stiffened cantilever plate using composites and structural layouts: A parametric study

AbstractThis paper studies the aeroelasticity of a stiffened cantilever plate using composite material and novel structural layouts. A comprehensive parametric study is conducted to determine the influence of different design parameters on the aeroelastic boundaries. Design parameters include plate sweep angle, ply orientation, stringer cross-section and stringer sweep angle. Nastran is used to run the aeroelastic analysis, and the process is automated using ${\rm{Matla}}{{\rm{b}}^{}}$ . The structure of the plate is modelled using laminate elements whereas the stringers are modelled using the Euler-Bernoulli beam elements. The unsteady aerodynamic loads are modelled using doublet-lattice method (DLM) and the structural and aerodynamic meshes are connected using an infinite plate surface (IPS) spline. A mesh sensitivity analysis is conducted to ensure fine meshes for the aerodynamics and structure. The study’s findings demonstrate the benefits of employing forward swept (Fw) stringers since it increases flutter speed by almost 38% compared to the unswept stringers case and prevents divergence. Moreover, the static aeroelastic analysis illustrates that the utilisation of Fw swept stringers can reduce the average tip displacement and tip twist effectively. T-shaped stringers are recommended to stiffen the plate due to their lower impact on the total mass of the plate. In some configurations, the structural layout has a much higher effect on the aeroelastic instabilities when compared to the material effect (ply orientation). However, results suggest combining both for some cases to get balanced washin and washout effects.

Analyzing Unsteady Turbomachinery Flow Simulations With Mixing Entropy

Abstract The prediction of unsteady aerodynamic loads is a central problem during the design of turbomachinery. Over the last 20 years, harmonic balance methods have been shown to be highly efficient for this task. A CPU-cost optimal setup of a harmonic balance simulation, however, requires knowledge of relevant harmonics. In the case of a single blade row with a periodic disturbance this question amounts to the classical problem of harmonic convergence, a problem which is solely due to the nonlinearity of the unsteady flow physics. In contrast, for multi-stage configurations, the choice of harmonics is further complicated by the fact that the interactions of disturbances with blade rows may give rise to a vast spectrum of harmonics that possibly have important modal content, e.g., Tyler–Sofrin modes. The aim of this paper is to show that the mixing entropy attributed to circumferential modes of a given harmonic can serve as a disturbance metric on the basis of which a criterion could be derived whether a certain harmonic should be included or not. The idea is based on the observation that the entropy due to the temporal and circumferential mixing of the flow at a blade row interface may be decomposed, up to third-order terms, into independent contributions from different frequencies and mode orders. For a given harmonic balance (and steady) flow result, the mixing entropy attributed to modes that are simply mixed out, rather than resolved in the neighboring row, is shown to be a natural indicator of a potential inaccuracy. We present important features of the mixing entropy for unsteady disturbances, in particular a close relationship to sound power for acoustic modes. The problem of mode selection in a 1.5-stage compressor configuration serves as a practical example to illustrate our findings.

Modelling the unsteady lift of a pitching NACA 0018 aerofoil using state-space neural networks

The development of simple, low-order and accurate unsteady aerodynamic models represents a crucial challenge for the design optimisation and control of fluid dynamical systems. In this work, wind tunnel experiments of a pitching NACA 0018 aerofoil conducted at a Reynolds number $Re = 2.8 \times 10^5$ and at different free-stream turbulence intensities are used to identify data-driven nonlinear state-space models relating the time-varying angle of attack of the aerofoil to the lift coefficient. The proposed state-space neural network (SS-NN) modelling technique explores an innovative methodology, which brings the flexibility of artificial neural networks into a classical state-space representation and offers new insights into the construction of reduced-order unsteady aerodynamic models. The work demonstrates that this technique provides accurate predictions of the nonlinear unsteady aerodynamic loads of a pitching aerofoil for a wide variety of angle-of-attack ranges and frequencies of oscillation. Results are compared with a modified version of the Goman–Khrabrov dynamic stall model. It is shown that the SS-NN methodology outperforms the classical semi-empirical dynamic stall models in terms of accuracy, while retaining a fast evaluation time. Additionally, the proposed models are robust to noisy measurements and do not require any pre-processing of the data, thus involving only a limited user interaction. Overall, these features make the SS-NN technique an excellent candidate for the construction of accurate data-driven models from experimental fluid dynamics data, and pave the way for their adoption in applications entailing design optimisation and real-time control of systems involving lift.