Low-order Model Research Articles

Modeling probability density functions of instantaneous velocity components at the pedestrian levels of a building array by Gram–Charlier series

Probability density distributions (PDFs) of wind speeds at pedestrian levels are required to predict gusts for wind environment assessments. However, the time-series data are extremely large in data volume for calculating the PDFs. Therefore, we applied the Gram–Charlier series (GCS) to the dataset of the flow fields at the pedestrian level around a simplified block array obtained by a large-eddy simulation to analyze the probability density and low-occurrence strong wind speeds. PDFs are predicted using the GCS-nth model, in which the distribution is modified from the Gaussian distribution by incorporating third to nth-order moments. The GCS can estimate the skewed PDFs of pedestrian winds and more accurately predict the percentile values compared to a Gaussian distribution. In addition, at most locations, the higher-order GCS estimates the probability densities and percentiles more accurately than the lower-order models. In contrast, the accuracy of GCS-6th is not always better than that of GCS-5th or lower models in some locations because of the large values of the coefficient in the polynomial function that modifies a Gaussian distribution. This study proves that using GCS to predict PDFs from statistics is a newly discovered and useful approach for wind environmental assessment.

Compositional noise in nozzles with dissipation

We propose a physical model to predict indirect noise generated by the acceleration of compositional inhomogeneities in nozzles with viscous dissipation (non-isentropic nozzles). First, we derive the quasi-one-dimensional equations from the conservation laws of multicomponent flows. Second, we validate the proposed model with the experimental data available in the literature for binary mixtures of four gases. Third, we calculate the nozzle transfer functions for different Helmholtz numbers and friction factors, in both subsonic and supersonic flows with/without shock waves. We show that friction and dissipation have a significant effect on the generation of indirect noise, for which the physical mechanism is identified and explained. Fourth, we find a semi-analytical solution with path integrals, which provides an asymptotic expansion with respect to the Helmholtz number. Fifth, we introduce the compositional-noise scaling factor, which is applied to quickly estimate compositional noise from the knowledge of only one single-component gas transfer function. The approximation error is less than $1\,\%$ . The proposed low-order model provides accurate estimates of the transfer functions and physical insight into indirect noise for multicomponent gases. This opens up new possibilities to accurately predict, and understand, sound generation in gas turbines.

Fast Tube Model Predictive Control for Driverless Cars Using Linear Data-Driven Models

Model predictive control (MPC) has been widely applied to different aspects of autonomous driving, typically using nonlinear physically derived models for prediction. However, feedback control systems inherently correct for model errors, and thus in many applications it is sufficient to use a linear time-invariant (LTI) model for control design, especially when using robust control methods. This philosophy of approach appears to have been neglected in current driverless car research and is the research gap that we aim to address here. Namely, instead of deriving meticulous nonlinear physical models of vehicle dynamics and solving a correspondingly complex optimal control problem (OCP), we identify a low-order data-driven LTI model and handle its uncertainty via robust linear MPC methods. We develop a two-step control scheme for driverless cars based on tube MPC (TMPC), which introduces structural robustness, ensuring constraint compliance despite modeling error in the data-driven prediction model. Furthermore, we use fast optimization methods designed to exploit the special structure of the linear MPC problem. We evaluate the proposed control scheme using a vehicle model identified from real-world data and simulations in IPGCarmaker, where the model of the vehicle under control is inherently nonlinear and uses detailed 3-D physics. Our results show that an LTI model can be effectively used for the task of lane-keeping, that TMPC can prevent lane departure and possible collisions due to model uncertainty, and that linear models allow for several algorithmic improvements that can decrease computation time by an order of magnitude compared with naive MPC implementations.

Driving Mechanisms in Low-Order Modeling of Longitudinal Combustion Instability

Several test cases in the literature have shown that both transverse and longitudinal high-frequency combustion instability can be driven by the injector dynamics. In these cases, pressure oscillations result in fluctuations in propellant mass flow rate, which yields pulsing heat release. This fundamental mechanism is the focus of the present work, with the aim of including this effect in a quasi-1D nonlinear model of Euler equations suited to studies of longitudinal combustion instability. In particular, the injection dynamics is represented through a simplified formulation, which is the core of the proposed response function. The analysis also addresses the influence of combustion efficiency on the main characteristics of the resulting limit cycle (frequency and amplitude). The obtained model is tested comparing the quasi-1D simulations against the experimental data of the continuously variable resonance combustor available in the literature, considering three different geometrical configurations, with different lengths of the oxidizer post. The proposed formulation is capable of reasonably reproducing the unstable behavior, as well as providing a simple model that explains the mechanism that leads to a low average combustion efficiency during unstable operation.

Sectional Leading Edge Vortex Lift and Drag Coefficients of Autorotating Samaras

Autorotating samaras such as Sycamore seeds are capable of descending at exceptionally slow speeds and the secret behind this characteristic is attributed to a flow mechanism known as the leading edge vortex (LEV). A stable LEV is known to increase the maximum lift coefficient attainable at high angles of attack and recent studies of revolving and flapping wings have proposed suitable lift and drag coefficient models to characterise the aerodynamic forces of the LEV. For the samara, however, little has been explored to properly test the suitability of these low-order lift and drag coefficient models in describing the aerodynamic forces produced by the samara. Thus, in this paper, we aim to analyse the use of two proposed aerodynamic models, namely, the normal force and Polhamus models, in describing the sectional aerodynamic lift of a samara that is producing a LEV. Additionally, we aim to quantify the aerodynamic parameters that can describe the lift and drag of the samara for a range of wind speed conditions. To achieve this, the study first examined the samara flight data available in the literature, and from it, the profiles of the lift coefficient curves were investigated. Subsequently, a numerical Blade Element-Momentum model (BEM) of the autorotating samara encompassing different lift profiles was developed and validated against a comprehensive set of samara flight data, which were measured from wind tunnel experiments conducted at the University of Bristol for three different Sycamores. The results indicated that both the normal force and Polhamus lift models combined with the normal force drag can be used to describe the two-dimensional lift characteristics of a samara exhibiting an LEV. However, the normal force model appeared to be more suitable, since the Polhamus relied on many assumptions. The results also revealed that the aerodynamic force parameters can vary with windspeed and with the samara wing characteristics, as well as along the span of the samara wing. Values of the lift curve slope, zero-lift drag coefficient, and maximum lift coefficient are predicted and presented for different samaras. The study also showed that the low-order BEM model was able to generate a good agreement with the experimental measurements in the prediction of both rotational speed and thrust. Such a validated BEM model can be used for the initial design of bio-inspired rotors for micro-air vehicles.

Experiments and Modeling of the Ceiling Effect with Drone-Scale Propellers

Aircraft designs involving aerodynamic interactions between rotors and other aerodynamic surfaces are becoming increasingly popular. Studying these interactions is key to developing good models for design and aspects of the system such as control. This paper seeks to understand the effect of obstructing the upstream of a propeller at varying distances, also called the ceiling effect, by examining in depth the canonical case of a theoretically infinite obstruction and its interaction with a propeller or actuator disk. Experimental studies of force and pressure are compared to results from computational fluid dynamics (CFD), and good agreement is shown. The CFD results are then compared against the Morillo flowfield model, and a correction is found to match the results. The data indicate that some propellers experienced a nearly twofold increase in thrust. However, a matching force on the surface develops, and the net force drops to nearly zero. The force interaction between the two nearly disappears once the separation exceeds half a propeller diameter. These results are independent of propeller size and pitch. The implemented theoretical model also has low computational cost, and it could be used to improve low-order models such as panel methods or provide a foundation for future rotor–body interaction modeling.

Low-Order Modeling and Sensor-Based Prediction of Stalled Airfoils at Moderate Reynolds Number

We present analysis from planar time-resolved particle image velocimetry fields in the streamwise surface-normal plane of turbulent flow surrounding NACA 0012 and NACA 65-410 airfoils at focusing on stall phenomena at intermediate to high angles of attack . Dominant flow frequencies, identified from the lift spectra obtained from a load cell mounted to the foils, highlight the presence of bluff-body shedding [ of , where is the frequency, the airfoil chord, and the freestream velocity] for all cases and prominent low frequencies [ of ] for cases in transient stall (TS). A data-driven modeling framework via the proper orthogonal decomposition (POD) reveals that the low frequencies are associated to flow separation and reattachment driven by underlying expansion and contraction normal to the suction surface. Further, the ability to predict the low-order features from (pseudo) pressure probes at the leading, midchord, and trailing edges for both TS and deep stall (DS) cases is quantified using linear stochastic estimation (LSE). The framework pinpoints the centroid of the region of reverse flow with error on the order of 5 and 20% for DS and TS regimes, respectively. Notably, it is found that LSE coefficients governing the low-order POD correlations do not strongly depend on the airfoil geometry. This is demonstrated by the comparative performance of training the LSE using the probes of the NACA 0012 cases to predict the NACA 65-410 velocity fields and vice versa. This work demonstrates the insight afforded by the POD on the low-order features of turbulent stalled airfoils as well as the similarity of such features across geometries toward predicting flow features for potentially any airfoil geometry without the need to retrain an LSE library.

Impact of spray dynamics on combustion instabilities investigated by changing the atomizer recess in a swirl combustor

The present article is aimed at identifying the impact of spray-wall interaction on combustion instabilities. This is achieved by systematically modifying the liquid fuel atomizer location with respect to the injector outlet in experiments carried out on a single-injector swirling combustor. The injector is characterized by a high swirl number and a relatively high pressure drop, and features a pressure atomizer that delivers liquid heptane fuel in the form of a fine spray. In the first set of experiments, longitudinal self-sustained oscillations (SSOs) are examined by varying the atomizer recess distance and using three combustion chamber lengths. Three distinct instability regions characterized by jumps in amplitude and frequency are observed as a function of the atomizer recess. At lower SSO frequencies (obtained for longer chambers), the pressure fluctuation amplitude in the chamber decreases as the atomizer recess distance is reduced, tending towards a stable operation. An opposite behavior is observed at comparatively higher SSO frequencies (obtained with a shorter chamber), where the system becomes less unstable at higher atomizer recess distances. The oscillation frequency is also found to change and increases as the recess distance is reduced. Three recess distances corresponding to each zone of operation are selected to further analyze the flow and flame structures and dynamics. It is found that the mean velocity profiles only reveal moderate differences in the air flow field, but there are significant changes in the fuel spray distribution in the neighborhood of the injector axis. Laser sheet images of the spray reveal two patterns—one where the spray predominantly interacts with the injector end piece when the atomizer is recessed, and the other where this interaction is minimal. In the latter case, corresponding to a small recess, the fuel spray is directly conveyed into the chamber. It is next found that the spray distribution affects the flame pattern defining two distinct configurations. Finally, the flame response to external disturbances is characterized by flame describing functions (FDFs) that allow carrying out a stability analysis. FDF results obtained with two independent methods indicate that the gain and phase at the three recess positions vary substantially and that this can be used in combination with a low-order model to interpret the instability behavior of the system. This study might serve to guide the modeling of combustion dynamics and help design injectors that are less sensitive to instabilities.

High-Fidelity Aerostructural Optimization with a Geometrically Nonlinear Flutter Constraint

When designing aircraft, avoiding dynamic aeroelastic instabilities such as flutter is a key requirement. One way to meet this requirement is to use a multidisciplinary design optimization subject to a flutter constraint. Flutter-constrained design optimizations have used geometrically linear detailed models, which do not accurately predict flutter for very flexible aircraft, or geometrically nonlinear low-order models, which do not accurately trade off cruise range and structural mass. This paper presents a framework for integrating a geometrically nonlinear, subsonic flutter constraint that captures the large in-flight deflections of very flexible aircraft into high-fidelity gradient-based aerostructural optimization. The cruise range and stress constraints are computed accurately with detailed aerostructural analyses, which use a built-up finite element model coupled to Reynolds-averaged Navier–Stokes computational fluid dynamics. The detailed model is condensed to a low-order aeroelastic model to compute the geometrically nonlinear flutter constraint and its adjoint derivatives with computational cost and robustness suitable for optimization. The framework is demonstrated by maximizing the cruise range of a subsonic high-aspect-ratio wing with respect to panel thicknesses, sweep, and span. The impact of the geometrically nonlinear flutter constraint highly depends on the in-flight deformation level. At low deflection levels, sweeping the wing backward is the most effective flutter prevention method, and the flutter constraint has little impact on the achievable cruise range. At high deflection levels, shortening the wing span is necessary to suppress flutter, reducing the achievable cruise range by up to 5.9%. This work is a step toward making multidisciplinary design optimization a practical tool for designing the energy-efficient, very flexible aircraft of the future, which require geometrically nonlinear flutter analyses early in the design cycle to prevent flutter.

A CFD-Based Data-Driven Reduced Order Modeling Method for Damaged Ship Motion in Waves

A simple CFD-based data-driven reduced order modeling method was proposed for the study of damaged ship motion in waves. It consists of low-order modeling of the whole concerned parameter range and high-order modeling for selected key scenarios identified with the help of low-order results. The difference between the low and high-order results for the whole parameter range, where the main trend of the physics behind the problem is expected to be captured, is then modeled by some commonly used machine learning or data regression methods based on the data from key scenarios which is chosen as Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) in this study. The final prediction is obtained by adding the results from the low-order model and the difference. The low and high-order modeling were conducted through computational fluid dynamics (CFD) simulations with coarse and refined meshes. Taking the roll Response Amplitude Operator (RAO) of a DTMB-5415 ship model with a damaged cabin as an example, the proposed physics-informed data-driven model was shown to have the same level of accuracy as pure high-order modeling, whilst the computational time can be reduced by 22~55% for the studied cases. This simple reduced order modeling approach is also expected to be applicable to other ship hydrodynamic problems.

The validity of a general factor of emotional intelligence in the South African context

Emotional intelligence (EI) plays an important role in the prediction of important work-related outcomes, such as work performance. Southern African scholars frequently deploy total scores of EI without considering its hierarchical structure. This study investigated the presence of a general factor, as manifested among the subscales of the EQ-i 2.0, using an archival dataset of 16 581 employees in Southern Africa. Orthogonal first-order, single-factor, higher-order, oblique lower-order and bifactor models were specified to investigate the hierarchical structure of EI. The evidence supports the notion that a total score could be calculated for EI based on the EQ-i 2.0. A total EI score also appears to be predictive of employees’ individual work performance, as measured by their managers. It might, therefore, be practically meaningful for practitioners to calculate or use a total score when making selection decisions about employees based on the EQ-i. 2.0.Contribution: The findings of the present study offer insights into the theoretical and empirical structure of EI based on statistical techniques that have not been used on the construct in the Southern African context. Concurrent validity evidence further provides additional support that an overall quantitative score, based on the EQ-i. 2.0, has utility in hiring practices, where the aim is to predict future work performance.

Model and experimental analysis of a rotor rig dynamics with time-varying characteristics

This work is motivated by the rapid developments of a broad range of electrically powered vertical take-off and landing vehicles which frequently feature two-bladed thrust generating rotor propulsion units. The interaction between a two-bladed rotor and the surrounding structure causes an emergence of the complex dynamics specific to the systems with the time-varying characteristics. This paper introduces a new rotor-structure test rig aimed to support the focused analyses of such systems. The test rig is realized as an idealised configuration consisting of a flexible slender cantilevered beam-like support structure which accommodates a single brushless electrical motor and a two-bladed propeller at its free end. Whilst retaining relative simplicity and scalability, this system allows analysis of complex flexible structure-propeller coupled phenomena, some of which are evidenced in this study. To form a compact and fully defined low-order model suitable for the computationally efficient in-vacuo analysis presented here, a Lagrange-based energy formulation is used to derive the equations of motion. The frequency and time domain techniques, including the complex modal analysis of the linear time-periodic systems, frequency response function and Udwadia-Kalaba method, are used to describe and explain the observed dynamics. The predicted dynamic changes, their impact on the modal interactions and the validity of the finite-sized time-invariant approximating system are successfully demonstrated through correlation with the experiment in the frequency range which encompasses the first four modal families and the rotor speed range spanning up to the first observed experimental instability. Specifically, the study confirms the occurrence of the modal veering and lock-in phenomena, modal splits as well as the strong dominance of the 0th order frequency modulation clusters in the structural response. The results presented in this work show that the proposed experimental platform represents a useful minimal dynamic system with the behavioral characteristics of high significance for the design of novel electrically powered and propeller-driven aircrafts.

A replacement model to simulate the nonlinear dynamics of electro-responsive liquid crystal coatings

An electric circuit replacement model is proposed to simulate the key nonlinear dynamics of electro-responsive liquid crystal polymer networks (LCNs). LCNs are known for having great potential to be integrated into smart functional surfaces due to their ability to generate various surface patterns. However, due to their complex molecular dynamics, low-order dynamic models that can accurately describe and predict their dynamic behavior are still lacking. In light of this research gap, we develop a lumped-parameter replacement model based on the observed dynamics in the experimental data and the physics of LCN dielectric properties. The unique assembly of lumped parameters in its simplest form describes the transformation of a high-frequency input voltage to a relatively slow increase in the local height of the LCN coating in between the electrodes, serving as an excitation mechanism to induce height change. The nonlinear dynamics of this height increase, as a function of both excitation frequency and voltage, is described by the proposed model. Furthermore, the comparison of the simulation results with the experimental data from LCN shows that key LCN response characteristics are captured well by the model. This model makes it possible to accurately predict and control the response of the electro-responsive LCN surfaces to obtain a predefined desired deformation pattern, which is a vital requirement for integrating them in haptic and smart surface devices.

Latent-Variable-Modeling-Based Fault Diagnosis for Unknown Nonlinear Distributed Parameter Systems with Application to Snap Curing Oven

A novel fault detection method based on time/space separation and latent variable model is proposed for unknown nonlinear distributed parameter systems in sensor-constrained environments. By performing time/space separation, the augmented matrix formed by the spatio-temporal distribution data of the DPSs can be split into a spatial basis function (SBFs) and a time series model, and the dimensionality reduction capability of the SBF is further utilized to obtain a low-order temporal model. Then, the temporal model is further extracted by using a dynamic latent variable modeling method to obtain the dominant time components and establish the corresponding monitoring statistics. Utilizing the appropriate kernel density function, confidence bounds are selected for the monitoring statistics when the system is normal. As a data-based fault diagnosis method, it requires only the past data records of the system and no reliance on complex mathematical models. Two sets of experiments performed on a snap curing oven verified the effectiveness of the proposed method.

Drop Interactions with the Conical Shock Structure Generated by a Mach 4.5 Projectile

This work presents measurements of liquid drop deformation and breakup time behind approximately conical shock waves and evaluates the predictive capabilities of low-order models and correlations developed using planar shock experiments. A conical shock was approximated by firing a bullet at Mach 4.5 past a vertical column of water drops with a mean initial diameter of . The time-resolved drop position and maximum transverse dimension were characterized using backlit stereo images taken at 500 kHz. The gas density and velocity fields experienced by the drops were estimated using a Reynolds-averaged Navier–Stokes simulation of the bullet. Classical correlations predict drop breakup times and deformation in error by a factor of 3 or more. The Taylor analogy breakup (TAB) model predicts deformed drop diameters that agree within the confidence bounds of the ensemble-averaged experimental values using a dimensionless constant compared to the accepted value . Results demonstrate existing correlations are inadequate for predicting the drop response to the three-dimensional relaxation of the flowfield downstream of a conical-like shock and suggest the TAB model results represent a path toward improved predictions.

Low-order dynamical model and distributed coordinated model predictive control for multi-stage belt conveyor systems

This work studies a distributed economic model predictive control (DEMPC) approach for multi-stage belt conveyor (BC) systems based on a novel low-order model. First, the existing BC finite element dynamic model is reduced to a low-order model with five micro-element segments. In describing the speed and acceleration range of the BC, the low-order model is consistent with the original model. Then, a DEMPC method is proposed for multi-stage BC systems. In this method, the state information of the adjacent BCs is used to express the collaborative cost function for the local optimization control problem. By using the obtained method, the economic performance is improved, and the energy consumption is reduced. Finally, hardware-in-the-loop simulation tests are utilized to demonstrate the benefits of the results gained.

Neural network-based multi-point, multi-objective optimisation for transonic applications

In the context of aircraft applications, the overall design process can be challenging due to the different aerodynamic requirements at several operating conditions and the total associated computational overhead. For this reason, the use of low order models for the optimisation of complex non-linear problems is sometimes used. This paper addresses the challenge of transonic aerodynamic design optimisation through the integration of a set of neural networks for the prediction of integral values, the classification of flow features and the estimation of flow field characteristics. The design method improves the computational efficiency relative to an expensive design process driven by Computational Fluid Dynamics (CFD) evaluations. The approach is used for the multi-point, multi-objective optimisation of a compact aero-engine nacelle in which the design outcomes are validated using a CFD in-the-loop optimisation strategy. It is demonstrated that the method based on the neural network capability identifies similar nacelle designs at a 75% reduction in the overall computational cost, a drag uncertainty prediction within 2.8%, and a predictive accuracy for the classification metric of 98%. For downselected configurations, the main flow characteristics in terms of peak Mach number, pre-shock Mach number and shock location are well predicted by the neural network models compared with the CFD-based evaluations.

Minimum environmental load extension through compressed air extraction: Numerical analysis of a dry low NOx combustor

The operational flexibility of gas turbine (GT) engines is a key requirement to coexist alongside increasing renewable energy that is often intermittent. One of the GT flexibility criteria is the Minimum Environmental Load (MEL). This is the lowest load the engine can be operated, without infringing on emissions limits (particularly CO) and is relevant to periods when there is a priority to renewable generation or low power demand. This study along with a series of related works of the authors proposes compressor air extraction for MEL extension. Here, a stand-alone three-dimensional numerical dry low NOx combustor demonstrates the technical viability concerning combustor performance and emissions. In addition, supplemented with low-order models for durability and stability evaluations. For the first time, there is evidence to show that the combustor can handle the 18% compressed air extraction to sustain a new MEL. This operation is characterised by a 12.3% reduction in CO through an increase of the fuel split ratio by 2% after design exploration cases. However, at the expense of a smaller overall rise in NO emissions by 5%. The durability analysis focused on the wall liner temperature assessments, which show no unusually high temperature rise for the new MEL. Similarly, the thermoacoustic instability frequencies and gains are around the normal operation mode. When benchmarked against previous related engine-level analysis, the evidence shows that the new MEL is a 7% points reduction of load.

Development of a novel low-order model for atrial function and a study of atrial mechano-electric feedback

Numerical models of the cardiovascular system have largely focused on the function of the ventricles, with atrial function often neglected. Furthermore, the time-varying elastance method that prescribes the pressure–volume relationship rather than calculating it consistently is frequently used for the ventricles and atrium. This method has yet to be validated however, so its applicability for cardiac modelling is frequently questioned. To overcome this challenge, we propose a synergistic model of left atrium (LA) and left ventricle (LV) by self-consistently integrating various feedback mechanisms among the electro-mechanical and chemical functions of the micro-scale myofiber, the macro-scale dynamics of the LA and LV, the atrioventricular node (AV), and circulation. The model is tested and shown to reproduce the essential features of the atrium cycling, such as the characteristic figure of eight pressure–volume loops. Our model is further developed to investigate the effect of dysfunctions of the mechanical-electric feedback (MEF) in the atrium. Our model not only successfully reproduces key experimental MEF observations such as prolonged action-potential and increases in action-potential magnitude induced by atrial stretch but also shows how MEF and arrhythmia of the atrium lead to a degradation of cardiac output and pumping power with significant consequences. In particular, MEF reproduces arrhythmia such as ectopic and erratic cycling, missed heart beats and restricted function.

Single and multi-row turbine performance in bounded shear flow

Predicting the performance of large turbine arrays requires the understanding of many physical factors, such as array geometry, turbine operation, inflow conditions and turbulent wake mixing. Due to the large parameter space that an array may be optimised over, low-order models with low computational cost are often employed. This paper extends one of these models, the inviscid–viscous coupled model, for multi-row turbine modelling. Firstly, an extension to the inviscid actuator disc theory is presented by removing the limit on the number of discrete streamtubes computed. The extended model allows for the quantification of the impact of shear in the bypass and core flows separately. In particular, it is shown that averaging a sheared bypass flow profile can result in a substantial over-prediction of the power of a turbine in a laterally bounded flow as the effective blockage of the flow increases. The model is also used to confirm that an approximation using a limited number of streamtubes in some previous applications of the inviscid–viscous approach has a negligible impact on the results. Secondly, we explore the performance of a multi-row array with either uniform or varying turbine resistance across different rows. Results suggest that by varying resistance across rows, the array may outperform the uniform resistance case. The performance gain is dependent, however, on the arrangement and inter-turbine spacing both in the spanwise and streamwise directions.