Low-order Model Research Articles

Point-enhanced convolutional neural network: A novel deep learning method for transonic wall-bounded flows

Low order models can be used to accelerate engineering design processes. Ideally, these surrogates should meet the conflicting requirements of large design space coverage, high accuracy and fast evaluation. Within the context of aerospace applications at transonic conditions, this can be challenging due to the associated non-linearity of the flow regime. Different methods have been investigated in the past to predict the flow-field around shapes such as airfoils or cylinders. However, they usually have reduced spatial resolution, limiting the prediction capabilities within the boundary layer which is of interest for transonic wall-bounded flows. This work proposes a novel Point-Enhanced Convolutional Neural Network (PCNN) method that combines the advantages of the well-established PointNet and convolutional neural network approaches. The PCNN model has relatively low memory requirements in the training process, preserves the spatial correlation in the domain and has the same resolution as a traditional computational method. The architecture is used for the flow-field prediction of civil aero-engine nacelles in which it is demonstrated that the flow features of peak isentropic Mach number (Mis), pre-shock isentropic Mach number and shock location (X/Lnac) are captured within ΔMis = 0.02, ΔMis=0.04, ΔX/Lnac=0.007, respectively. The PCNN model successfully predicts the integral parameters of the boundary layer, in which the incompressible displacement thickness, momentum thickness and shape factor are typically within 5% of the CFD. Overall, the PCNN method is demonstrated for transonic wall-bounded flows for a range of flow physics that include shock waves and shock-induced separation.

Assessment of the performance and robustness of a new PID tuning technique for AVR systems

ABSTRACT In this work, a hybrid PID controller, combining Zeigler–Nichols (ZN) theory and the dominant pole placement, has been developed for the automatic voltage regulator (AVR) systems. Furthermore, the reduced order models (ROMs) of AVR systems, namely, AVR-1, AVR-2, and AVR-3, have been established using Particle Swarm Optimization (PSO), Big Bang Big Crunch (BBBC), and Pade’s approximation, respectively. Then, the proposed PID controller has been applied to both the original and developed lower order models of AVR to check the efficacy and robustness of the developed hybrid controller. It is shown that the proposed hybrid PID controller provides a very small value of the percentage peak overshoot (%M p ) and settling time (t s ). Also, the gain margin (GM) and phase margin (PM) are found to be in the desired range. Finally, the sensitivity analysis has also been performed to check the robustness of the proposed controller. The sensitivity function, complementary sensitivity function, and various performance indices have been evaluated for the proposed controller based on Bode’s Integral (BI), Tyreus–Lyuben (T–L), and Internal Model Control (IMC) approaches. The comparative studies of the results show that the proposed controller has smaller values of performance indices and sensitivity function.

Reduction of vortex-induced vibrations of a cantilevered hydrofoil with passive piezoelectric shunt

This work first investigates the ability of a low order fluid-structure model to fit the vortex-induced vibrations (VIV) observed on a truncated hydrofoil in a hydrodynamic tunnel. A particular VIV area is scrutinized, for which a hydrodynamic excitation mechanism due to a Karman-type vortex wake organization successively locks the first torsional and second bending mode of the cantilevered hydrofoil. Coupling two structure oscillators with a Van der Pol wake oscillator satisfactorily reproduces the amplitude response and the lock-in frequency. In order to build a low order model allowing to optimize control strategy, a fourth degree of freedom corresponding to the electric circuit of a resonant piezoelectric shunt has been added. Composed of an inductance and a resistance connected to a piezoelectric patch, the passive shunt was tuned to minimize the vibration amplitude in the frequency lock-in range. Model predictions are finally compared with experimental results.

Robustness and reliability of state-space, frame-based modeling for thermoacoustics

The Galerkin modal expansion is a well-known method used to develop reduced order models for thermoacoustics. A known issue is the appearance of Gibbs-type oscillations on velocity fluctuations at the interface between subdomains and at boundary conditions. Recent work of Laurent et al. (2019) [20] and Laurent et al. (2021) [23] have shown that it is possible to overcome this issue by using an over-completed frame, instead of a Galerkin modal basis. However, the low-order modeling based on this frame modal expansion may generate spurious modes. In this paper, the origin of these non-physical modes is identified and a method is proposed to automatically remove them from the outcome. By preventing any interaction between the physical and non-physical components, the proposed methodology drastically improves the robustness and reliability of the frame modal expansion modeling for thermoacoustics.

Floating particles transport through the free surface vortex technique: A novel numerical study to assess the interaction among different scales of vortex structures

Fat, oil, and grease (FOG) floating particles in the sump of sewage pumping stations will accumulate together to form rigid layers, resulting in failure for pump device. To overcome this, the free surface vortex (FSV) technique has been considered and applied to transport floating particles toward the submerged suction pump inlet. This paper investigates the potential of vortices as a means of downward motion of FOG. The entrainment capacity of FSV is investigated by numerical simulations using a coupled level-set and volume-of-fluid method. Two coherent structures are decomposed by proper orthogonal decomposition: FSV represented by the first two orders with high energy content and spiral vortex bands represented by low energy and high order models. The extracted ridges of the finite-time Lyapunov exponent (FTLE) delineate different regions of the flow field and effectively capture the evolution of Lagrangian coherent structures. The floating particles in the sump are first caught by the dividing line formed by the FTLE ridges, mixed in the entrainment zone, and then merged into the vortex. The enstrophy production term dominates the development of vorticity. Subject to the influence of flow velocity gradients, both radial and tangential vortices undergo a transition into axial vortices. This transformation enhances the vortex's capacity to entrain particles within the vortex core area, leading to their rapid inward spiraling toward the vortex center and eventual expulsion due to the vortex's entrainment effect.

Limit cycle oscillations in the zonal-flow-catalyzed interactions of ion-temperature-gradient turbulence

Limit-cycle oscillations are studied for ion temperature gradient turbulence, which, in the absence of large diamagnetic (mean) shear flows, saturates through energy transfer from unstable modes to large-scale stable modes via zonal-flow intermediary modes. Oscillations of zonal flow and turbulence levels are strongly constrained by the reactive, largely non-dissipative character of the zonal flows. Since existing predator–prey models for observed oscillations in experiments do not include energy transfer through zonal flows to stable modes, low-order fluid models with this physics are constructed and investigated. A simple three-wave truncation produces low-amplitude zonal flows that slowly oscillate around a zero mean, with turbulence oscillations between coupled wavenumbers that exceed linear frequencies by orders of magnitude. This inconsistency with experimental observations is caused by the weak non-linear drive of zonal flows in three-wave systems and the lack of multiple-wavenumber turbulent interactions. A more comprehensive model that preserves multiple wavenumber interactions within the context of conservative zonal-flow-mediated energy transfer to stable modes accurately reflects observed dynamics when the phase between stable and unstable modes is occasionally randomized.

An Image Compensation-Based Range–Doppler Model for SAR High-Precision Positioning

The range–Doppler (R–D) model is extensively employed for the geometric processing of synthetic aperture radar (SAR) images. Refining the sensor motion state and imaging parameters is the most common method for achieving high-precision geometric processing using the R–D model, comprising a process that involves numerous parameters and complex computations. In order to reduce the specialization and complexity of parameter optimization in the classic R–D model, we introduced a novel approach called ICRD (image compensation-based range–Doppler) to improve the positioning accuracy of the R–D model, implementing a low-order polynomial to compensate for the original imaging errors without altering the initial positioning parameters. We also designed low-order polynomial compensation models with different parameters. The models were evaluated on various SAR images from different platforms and bands, including spaceborne TerraSAR-X and Gaofen3-C images, manned airborne SAR-X images, and unmanned aerial vehicle-mounted miniSAR-Ku images. Furthermore, image positioning experiments involving the use of different polynomial compensation models and various numbers and distributions of ground control points (GCPs) were conducted. The experimental results demonstrate that geometric processing accuracy comparable to that of the classical rigorous positioning method can be achieved, even when applying only an affine transformation model to the images. Compared to classical refinement models, however, the proposed image-compensated R–D model is much simpler and easy to implement. Thus, this study provides a convenient, robust, and widely applicable method for the geometric-positioning processing of SAR images, offering a potential approach for the joint-positioning processing of multi-source SAR images.

Large eddy simulation of highly underexpanded sonic jets from elliptical nozzles

The injection of high-pressure jets into quiescent air poses significant challenges in fluid dynamics, pertinent to various industrial engineering applications. This study used large eddy simulations on a massively parallel computational framework, employing a grid of over 600 million nodes, to investigate the behavior of highly underexpanded sonic jets from elliptical nozzles at a nozzle pressure ratio of 15. Three elliptical nozzles, with aspect ratios of 1.5, 2.2, and 3.0, each having a sectional area equivalent to that of a circular jet with a diameter of D=1 mm, were analyzed. The aim was to clarify the gasdynamic and mixing characteristics of these jets to guide the design of next-generation injectors. A detailed analysis of the flow provided insights into the mechanisms of turbulence generation and Reynolds stress anisotropy. This was achieved using the componentality contour approach and a modified barycentric color mapping scheme, offering valuable reference data for developing lower-order models. The results indicate a non-axisymmetric radial expansion of the jet boundary in all elliptical injectors, leading to an axis switch phenomenon. The use of elliptical orifices was found to reduce jet penetration, mitigating issues such as fuel impingement in small engine combustion chambers and promoting improved air–fuel mixing quality.

The effect of hydrogen enrichment on the conical premixed methane–air flame response and thermoacoustic modes coupling

This paper investigates the thermoacoustic dynamic responses of hydrogen-enriched laminar premixed conical methane–air flames under dual-mode coupling. Utilizing the level set method and the G-equation model, one meticulously derives the flame describing function (FDF) in relation to variations in hydrogen enrichment levels (ηH). This precisely derived FDF is then integrated into the low-order network model of the Rijke tube to analyze the thermoacoustic unstable modes of the system. Finally, dual-frequencies incoming flow velocity perturbations are reintroduced as inputs to obtain the flame response under thermoacoustic modes coupling. Results show that while keeping the unstretched steady flame aspect ratio constant, an increase in ηH not only raises the FDF’s cut-off frequency but also enhances the FDF gain within the nonlinear frequency region, leading to more unstable modes, especially high frequency modes within the Rijke tube system. Furthermore, the flame response is further altered under the excitation of dual unstable modes. When both modes are within the linear frequency region of the FDF, the flame response is co-controlled by the two modes, predominantly by the mode with a larger velocity perturbation amplitude, with weaker modes coupling leading to the additional frequency perturbation having a suppressive effect on the flame response at the original frequency. Conversely, when one or two modes are within the nonlinear frequency region of the FDF, the flame response is dominated by the lower-frequency mode, with higher nonlinear modes coupling allowing the additional frequency perturbation to both promote and suppress the response at the original frequency and also couple to produce a significant difference frequency response. Novelty and significance The novelty of this paper lies in the determination of the flame describing function (FDF) with varying hydrogen enrichment levels and the stability of thermoacoustic systems, and more significantly, conducting an in-depth and comprehensive investigation into the nonlinear dynamic responses of hydrogen-enriched flames to different types of dual-mode coupling perturbations. The dual-mode perturbations result in either attenuation or amplification of the flame response, depending on whether the corresponding frequencies lie within the linear or nonlinear frequency regions of the FDF, which innovatively incorporates the influence of the FDF phase and examines the responses at the difference frequencies generated by the coupling. This lays a foundation for further exploration into the mechanisms of modes coupling in hydrogen-enriched and other thermoacoustic systems.

Sensitivity analysis of a multi-scale biofuel primary atomization simulation tool

The purpose is to produce a benchmarked commercial Volume of Fluid (VOF)-to-Discrete Phase Modeling (DPM) numerical algorithm using an experimental study of high-viscosity jet fuel blend C-3. Where the mesh resolution was too coarse to resolve sufficient gas-liquid interfacial curvature, the solver automatically converted VOF biofuel phase mass, momentum, and energy into the equivalent DPM biofuel phase at the sub-grid scale level. Particular attention herein was given to the sensitivity of the results to various numerical degrees of freedom related to the local criteria for conversion of a biofuel from VOF to DPM. The proliferation of commercial simulation tools mandates detailed parametric evaluations. It was found that, while instantaneous conversion rates were extremely sensitive to conversion criteria, atomization physics were not sensitive to the conversion rate effects. Also, this flow system exhibited an outstanding dependence on simulated azimuthal angle, rendering lower order models ineffective. Data sampling methods and durations, along with mesh resolution, were explored such that a particular set of parameters provided a validated model.

Low-order-model-based design of additive-manufactured heat exchanger

Abstract This paper discusses a comprehensive approach to the design of an additive-manufactured heat exchanger for a high-temperature application. This manufacturing route is attractive because of the flexibility afforded in design, but there is a requirement for calibration of internal friction factors and heat transfer coefficients because of uncertainty regarding the characteristics of the manufactured internal surfaces. The design was informed by physics-based low-order models (total pressure loss model; conjugate heat transfer model) calibrated against experimental data. To extract heat transfer coefficients we use transient experiments performed over timescales for which the surfaces can be regarded as semi-infinite. The calibrated models are used to predict accurate performance characteristics for the heat exchanger. Experimental results for a prototype design were found to be in excellent agreement with the model over a wide operational range, validating both the design methodology and the utility of the particular design for the given application. Our process, though not unique, has wide applicability, has been conducted with considerable care, and will be of interest to those engaged in low-order-model-based heat exchanger design.

A review of resistance–capacitance thermal network model in urban building energy simulations

With the increasing population and urbanization promotion, energy consumption and carbon emissions have increased, and concern for inefficient energy use and deteriorating urban environments is growing. The prediction of building energy consumption and environmental evaluation, especially at large scales, are considered to be a major challenge confronting the research community. An outstanding strategy for mitigating energy consumption and carbon emissions resides in the field of energy modeling. As a simplified building energy modeling model, resistance–capacitance (RC) network model has the applications in fast predicting building energy consumption. Notably, in recent years, there has been an evident absence of thorough review endeavors related to RC model for urban building energy loads and climate. This review systematically explores the application of low-order models and reduction methods for urban or regional energy simulation starting from the typical RC model for building. In summation, the challenges associated with employing this model for urban building energy loads and climate can be succinctly summarized as follows: the absence of a unified platform for RC modeling; insufficient readily applicable urban datasets; the need for modeling tools that facilitate cross-platform analyses in conjunction with existing Geographic Information Systems (GIS) and urban thermal environment simulation research platforms, and the essential requirement for seamless integration with other complementary modules.

Identification of flame transfer function and mechanism analysis of the influence of boundary impedance characteristics on thermoacoustic instability

Thermoacoustic instability is a common problem in the operation of modern gas turbines. The prediction of thermoacoustic instability and the clarification of its mechanism are the research focus and difficulty in the gas turbine industry. As a response function of flame to acoustic disturbance, flame transfer function is a key parameter in the study of thermoacoustic instability. In this paper, based on the scaled adaptive simulation (SAS) model combined with the eddy dissipation concept (EDC) combustion model, the time-domain flow field data are processed by the system identification method, and the results of flame transfer function extraction are in good agreement with the experimental values. Then, the detailed derivation process of the low-order thermoacoustic network model (LOTAN) is given to capture the behavior characteristics of the acoustic wave in the thermoacoustic system. On this basis, the effects of acoustic boundary conditions and hysteresis time on the thermoacoustic instability of the combustion system are analyzed, and the relationship between the mode shape, pressure, vibration velocity phase, and thermoacoustic instability is explored. It is found that the phase relationship between pressure and vibration mode can be used to determine the thermoacoustic instability of the system. This is of great practical significance for determining the thermoacoustic instability of the system and clarifying the internal mechanism of its generation and provides theoretical support for the subsequent thermoacoustic instability control.

Optimal Agile Satellite Target Scheduling with Learned Dynamics

Target sequencing is an important aspect of agile Earth-observing satellite scheduling. The objective of the problem is to find a feasible imaging sequence that maximizes the cumulative value of unique heterogeneously valued requests; no expendable resource constraints (power and data storage) are considered. Two challenges are encountered: transition times between targets are dictated by dynamics of an arbitrary controller, and the solution space is combinatorial with respect to request count. To find dynamically accurate time-dependent attitude maneuver transition times, a neural network is trained on simulated slew data; this allows for the generation of more physically accurate, and thus better performing, schedules than when using the traditional approach of lower-order models for slew feasibility. Next, slews between requests are represented by a sparsified graph, and a mixed-integer linear program is formulated to solve them; the sparse formulation keeps the problem tractable over longer planning horizons and larger numbers of requests when compared to the standard approach. The solutions are optimal up to the quality of the transition time estimator and a time discretization. Finally, the solutions are verified in a high-fidelity simulation, demonstrating validity when deployed on a lifelike system. Time-dependent request values and multiple satellites are also considered.

Fast and accurate modelling of twin-screw compressors: A generalised low-order approach

Twin-screw compressors (TSC) are commonly used in heat pump processes due to their robustness and flexibility. They exhibit two core properties, i.e. the swept volume and the built-in volume ratio (BVR), which heavily influence their capacity limits and off-design efficiency. This work presents a new low-order model (i.e. a polynomial model), which can accurately predict a TSC’s behaviour. The model uses the external pressure ratio and volumetric compressor inlet flow rate to calculate isentropic efficiency and compressor speed. The input parameters are normalised with a reference flow rate (calculated from the swept volume) and the BVR, respectively. This results in a generalised model of low numerical cost, which can be used for explorative studies independent of the specific machine size and BVR. A gain in computational speed by a factor of 375 is achieved compared to a semi-empiric reference model. The model displays very good predictive accuracy when used to predict the performance of machines with similar BVRs, but different sizes. A mean deviation from the manufacturer data of 4.29%, 0.88°C and 1.38% for the shaft power, the outlet temperature and the compressor speed can be observed, respectively. When there is a difference in size and BVR, the prediction accuracy is still reasonable but significantly declines for small and very large pressure ratios. Nevertheless, the proposed new approach extends the state-of-the-art by introducing a low-order model, which combines the advantages of low computational cost, high accuracy, physically correct predictions over a wide operational range and scalability to different machine capacities and BVRs. The validation for different fluids indicates a good general prediction accuracy relatively independent of the used fluid.

Model-Based Inference of Flame Transfer Matrices from Acoustic Measurements in an Aero-Engine Test Rig

Abstract Flame dynamics, represented as a flame transfer matrix (FTM), is not directly measurable in test rigs and must be deduced from transfer matrix measurements of the combustion system. The burner-flame transfer matrix (BFTM) approach for FTM estimation is based on local pressure signals from microphones located upstream and downstream of the combustor. It combines acoustic measurements in non-reacting and reacting conditions, with the latter implicitly including flame dynamics. A simple matrix operation yields the FTM. However, this approach assumes loss-free wave propagation at constant speed of sound with no change in cross-sectional area between the microphones and the burner/flame. The present work demonstrates the limitations of these assumptions when applied to a test rig with effusion cooling, bypass annulus, and end contraction. This work proposes a method to infer the FTM for complex combustors by combining reactive transfer matrix measurements of the entire combustor with an accurate low-order model (LOM) of the test rig. This generalized method reduces to the BFTM approach as a special case. The Rolls-Royce SCARLET test rig, operating under realistic engine conditions, is used to analyze the capabilities of the proposed model-based inference method and the limitations of the BFTM approach. First, a LOM based on SCARLET's geometry and operating point is formulated using a generic FTM. This model visualizes the limitations of the BFTM approach concerning various physical and geometrical parameters. Finally, experimental data is used to infer the FTM of SCARLET using the proposed approach.

The role of time-varying external factors in the intensification of tropical cyclones

Abstract. The role of time-varying external parameters in tropical cyclone (TC) dynamics is explored through a low-order conceptual box model. Specifically, we look at stable-to-stable state transitions which may be linked to tropical cyclone intensification, dissipation, or eyewall replacement cycles (ERCs). To this end, we identify two parameters of interest: the exponent of radial decline and sea-surface temperature. We examine how variations in these parameters affect the stable states of the model and consider the behaviour of the system under different time-dependent forcing profiles for the parameters. By externally forcing the exponent of radial decline and sea-surface temperature, we show the existence of rate-dependent behaviour in the model. These findings are brought together in a case study of Hurricane Irma (2017). The results highlight the role of the radial vorticity gradient in behaviour such as rate-induced tipping and overshoot recovery. They also show that a simple model can be used to explore relatively complex tropical cyclone dynamics.

Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model

Ocean circulation plays an important role in the formation and occurrence of extreme climate events. The study shows that the periodic variation of ocean circulation under strong wind stress is closely related to climate oscillation. Ocean circulation is a nonlinear dynamic system, which shows complex nonlinear characteristics, so the essence behind ocean circulation has not been clearly explained. Therefore, the response and evolution of the circulation system to wind stress are studied based on the bifurcation and catastrophe theories in nonlinear dynamics. First, the quasi-geostrophic gyre equation and the normalized gravity model are introduced and developed to study ocean circulation driven by wind stress, and solved using the Galerkin method. Then, the bifurcation and catastrophe behaviors of the system governed by the low-order ocean circulation model during the change in wind stress intensity and the coexistence of multiple equilibria in the circulation system are studied in detail. The results show that saddle and unstable nodes appear in the system after a cusp catastrophe. With the change in model parameters, the unstable node becomes the unstable focus, and then the subcritical Hopf bifurcation occurs. The system forms a bistable interval when the system undergoes a catastrophe twice, and the system shows hysteresis. In addition, multiple equilibrium states are coexisting in the circulating system, and the unstable equilibrium state always changes into a stable equilibrium state through vortex movement. Therefore, there is a route for the system to induce short-term climate oscillation, that is, in the multi-stable equilibrium state of the system, the vortex oscillates after being subject to small disturbances, and then triggers climate oscillation.

Conversion of a Coaxial Rotorcraft to a UAV—Lessons Learned

A coaxial helicopter with a maximum take-off weight of 600 kg was converted to an unmanned aerial vehicle. A minimally invasive robotic actuator system was developed, which can be retrofitted onto the copilot seat of the rotorcraft in a short period of time to enable automatic flight. The automatic flight control robot includes electromechanical actuators, which are connected to the cockpit inceptors and control the helicopter. Most of the sensors and avionic components were integrated into the modular robotic system for faster integration into the rotorcraft. The mechanical design of the control system, the development of the robot control software, and the control system architecture are described in this paper. Furthermore, the multi-body simulation of the robotic system and the estimation of the linear low-order actuator models from hover-frame flight test data are discussed. The developed technologies in this study are not specific to a coaxial helicopter and can be applied to the conversion of any crewed flight vehicle with mechanical controls to unmanned or fly-by-wire. This agile development of a full-size flying test-bed can accelerate the testing of advanced flight control laws, as well as advanced air mobility-related functions.

Forced response of transverse reacting jet in vitiated crossflow to harmonic velocity disturbances

The orthogonally oriented injector configuration typically used in axial fuel staging-capable gas turbine combustors is known to produce complicated thermoacoustic interactions between two distinct reaction zones. Despite the fundamental importance of transfer functions of transverse reacting jets in vitiated crossflow, however, their distinctive properties remain unknown, and associated low-order modeling combined with two different flame transfer functions remains entirely undemonstrated. From detailed measurements of transfer functions of lean-premixed primary and secondary flames in response to harmonic velocity disturbances, here we show that the transfer function of a transverse reacting jet in high-temperature crossflow is described as having relatively larger gain without undulating patterns, near-linear slow decay in magnitude with respect to the forcing frequency, and remarkably shorter duration response time. By leveraging a reduced-order modeling approach to account for the finite time delay between the two distinct transfer functions and by validating the calculation results against velocity and pressure measurement data, we demonstrate that self-induced instabilities in an axial fuel-staged system can be simultaneously driven by the dynamics of primary and secondary flames, and that the triggering of second-stage-induced instability is characteristically connected to higher-order acoustic modes. Such non-axisymmetric intra-combustor interactions generate a cascade of spectral peaks, including the first and third longitudinal modes and their respective higher harmonics, and additional peaks at the sum and difference of two individual frequencies. While the primary flame's dynamics are capable of selectively exciting the first mode, the secondary jet flame exhibits more complicated modal dynamics, manifested as the L3 mode-coupled jet merging-related flame surface annihilation and the L1 mode-coupled lateral movements of the transverse jet column.