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Because much computing time is required for the refined turbulence simulations, it is imperative to develop efficient acceleration methods. An improved mode multigrid method is proposed to accelera...

Modal analysis of fluid flows is essential for understanding flow physics and fluid–solid interaction mechanisms and for implementing flow control. Unlike unstable flows, the intrinsic stability of subcritical flows has led to failures in experimentally extracting the clear structure of the subcritical flow modes. To this end, this paper proposes an experimental modal testing method for subcritical flows. Using dynamic mode decomposition, dominant modes of flow around a cylinder at subcritical Reynolds numbers are extracted experimentally for the first time. The extracted structure and parameters of the modes are in agreement with the numerical results in the literature. It is found that the first-order mode is the stable von Kármán mode and can be observed at a Reynolds number as low as 19.3, which is nearly identical to the lower boundary of subcritical vortex-induced vibration. This finding provides the first experimental evidence of the correlation between the von Kármán mode and fluid–solid interaction instability in subcritical flows.

We present a numerical analysis of linear multigrid operators for the high-order Flux Reconstruction method. The non-modal analysis is used to assess the short-term numerical dissipation in the context of 1D and 2D linear convection-diffusion. The effect of several parameters, namely the number of coarse-level iterations, the polynomial order and the combination of h- and p-multigrid is explored in an effort to identify the most efficient configurations. V-cycle p-multigrid is shown to have increased efficiency at higher polynomial orders, and the use of W-cycles and/or hp-multigrid appear to offer additional advantages. The effect of high Péclet numbers and high aspect-ratio cells is also explored in 2D, and both factors are shown to decrease the error dissipation. Finally, we relate the non-modal dissipation to the convergence rate of the multigrid in a series of manufactured solutions.

To improve the efficiency of adjoint-based optimization algorithms, dynamic mode decomposition (DMD) technology is used to accelerate the convergence of steady adjoint equations in this paper. During pseudo-time marching, adjoint fields are projected onto the modal space and modal analysis is carried out by DMD. When only first zero frequency mode is reserved, the convergence speed of adjoint equations is significantly enhanced. Through two examples of flow past airfoils in subsonic and transonic flows and a transonic optimization example, the validity of the proposed methodology is verified. Results indicate that the proposed methodology can improve optimization efficiency remarkably, and the iteration step number of adjoint equations is reduced by almost 60%.

We present a framework for airfoil shape optimization to reduce the trailing edge noise for the design of wind turbine blades. Far-field noise is evaluated using Amiet's theory coupled with the TNO-Blake model to calculate the wall pressure spectrum and fast turn-around XFOIL simulations to evaluate the boundary layer parameters. The computational framework is first validated using a NACA0012 airfoil at zero angle of attack. Particle swarm optimization is used to find the optimized airfoil configuration. The multi-objective optimization minimizes the A-weighted overall sound pressure level at various angles of attack, while ensuring enough lift and minimum drag. We compare classic parametrization methods to define the airfoil geometry (i.e., CST) to a machine learning method (i.e., a variational autoencoder). We observe that variational autoencoders can represent a wide variety of geometries, with only four encoded variables, leading to efficient optimizations, which result in improved optimal shapes. When compared to the baseline geometry, a NACA0012, the autoencoder-based optimized airfoil reduces by 3% (1.75 dBA) the overall sound pressure level (with decreased noise across the entire frequency range), while maintaining favorable aerodynamic properties in terms of lift and drag.

This paper presents eigensolution and non-modal analyses for immersed boundary methods (IBMs) based on volume penalization for the linear advection equation. This approach is used to analyze the behavior of flux reconstruction (FR) discretization, including the influence of polynomial order and penalization parameter on numerical errors and stability. Through a semi-discrete analysis, we find that the inclusion of IBM adds additional dissipation without changing significantly the dispersion of the original numerical discretization. This agrees with the physical intuition that in this type of approach, the solid wall is modelled as a porous medium with vanishing viscosity. From a stability point of view, the variation of penalty parameter can be analyzed based on a fully-discrete analysis, which leads to practical guidelines on the selection of penalization parameter. Numerical experiments indicate that the penalization term needs to be increased to damp oscillations inside the solid (i.e. porous region), which leads to undesirable time step restrictions. As an alternative, we propose to include a second-order term in the solid for the no-slip wall boundary condition. Results show that by adding a second-order term we improve the overall accuracy with relaxed time step restriction. This indicates that the optimal value of the penalization parameter and the second-order damping can be carefully chosen to obtain a more accurate scheme. Finally, the approximated relationship between these two parameters is obtained and used as a guideline to select the optimum penalty terms in a Navier-Stokes solver, to simulate flow past a cylinder.

Aerodynamic data can be obtained from different sources, which vary in fidelity, availability and cost. As the fidelity of data increases, the cost of data acquisition usually becomes higher. Therefore, to obtain accurate unsteady aerodynamic model with very low cost and the desired level of accuracy, this paper proposes an unsteady multi-fidelity aerodynamic modeling framework. The approach integrates ideas from data fusion, multi-fidelity modeling, nonlinear system identification and machine learning. Data fusion reduces the total cost of data generation for model construction, while multi-fidelity modeling with a nonlinear autoregressive with exogenous input (NARX) description provides a general framework for unsteady aerodynamics. The correction term from the low-fidelity model to the high-fidelity result is then identified by a machine learning approach, i.e., a multi-kernel neural network. To validate the proposed method, unsteady aerodynamics of a NACA0012 airfoil pitching at Mach number 0.8 is modeled. The high-fidelity data is obtained from a Navier–Stokes-equation-based solver, while the low-fidelity solution is taken from an Euler-equation-based flow solver. The main difference between two types of data is that the high-fidelity solution takes into account the viscous effect, while the low-fidelity solution is based the invisicid flow assumption. Besides, to mimic the practical situation where high-fidelity data are limited in amount and diversity due to high cost (e.g., the experimental condition), only three high-fidelity unsteady aerodynamic solutions from harmonic motion are available. After performing a multi-fidelity analysis on a typical harmonic motion, the model is applied to the prediction of aerodynamic loads from either new harmonic motions or random motions. The multi-fidelity model shows a good agreement with the high-fidelity solution, indicating that by using only a few high-fidelity data and a low-fidelity model, high-fidelity results can be accurately reproduced. Furthermore, the model convergence with respect to increasing training data, and the comparison with a single high-fidelity reduced-order model (ROM) are also studied. The proposed approach becomes more accurate as the number of high-fidelity samples increases, and outperforms a single aerodynamic ROM in most of test cases. Compared with ROM method, additional computational cost for the proposed approach is small, therefore the total time cost of model training is still low.

The wind-tunnel experiment plays a critical role in the design and development phases of modern aircraft, which is limited by prohibitive cost. In contrast, numerical simulation, as an important alternative paradigm, mimics complex flow behaviors but is less accurate compared to experiment. This leads to the recent development and emerging interest in applying data fusion for aerodynamic prediction. In particular, the accurate prediction of aerodynamic with lower computational cost can be achieved by fusing experimental (high-fidelity) and computational (low-fidelity) aerodynamic data. Currently, existing works on aerodynamic model using data fusion mainly concern integral data (lift, drag, etc.). In this paper, a multi-fidelity aerodynamic model based on Deep Neural Network (DNN), where both numerical and experimental data are introduced in the loss function with proper weighting factor to balance the overall accuracy, is developed for aerodynamic distribution over the wing surface. Specially, the proposed approach is illustrated by modeling the surface pressure distribution of ONERA M6 wing in transonic flow, including different secnarios where the flow condition varies or there is less high-fidelity data. The results demonstrate that the proposed approach with decent number of low-fidelity data and a few high-fidelity data can accurately predict the surface pressure distribution on transonic wing. The outperformance of the proposed model over other DNNs from only high fidelity or low-fidelity, has been reported.

This paper proposes a mode multigrid (MMG) method, and applies it to accelerate the convergence of the steady state flow on unstructured grids. The dynamic mode decomposition (DMD) method is used to analyze the convergence process of steady flow field according to the solution vectors from the previous time steps. Unlike the traditional multigrid method, we project the flowfield solutions from the physical space into the modal space, and truncate all the high-frequency modes but only the first-order mode are retained based on the DMD analysis. The real solutions in the physical space can be obtained simply by the inverse transformation from the modal space. The developed MMG method ingeniously avoids the complicated process of coarsening computational mesh, and does not need to make any change for the grid in physical space. Therefore, it is very convenient to be applied to any numerical schemes with just a little change for the flow solver, which is also suitable for unstructured grids and easy for parallel computing. Several typical test cases have been used to verify the effectiveness of the proposed method, which demonstrates that the MMG can dramatically reduce the number of iterative steps for the different mesh types, different accuracy of spatial discretization and different time-marching schemes. The method is 3 to 6 times faster than the baseline method while ensuring the computational accuracy.

Aerodynamic reduced-order model (ROM) is a useful tool to predict nonlinear unsteady aerodynamics with reasonable accuracy and very low computational cost. The efficacy of this method has been validated by many recent studies. However, the generalization capability of aerodynamic ROMs with respect to different flow conditions and different aeroelastic parameters should be further improved. In order to enhance the predicting capability of ROM for varying operating conditions, this paper presents an unsteady aerodynamic model based on long short-term memory (LSTM) network from deep learning theory for large training dataset and sampling space. This type of network has attractive potential in modeling temporal sequence data, which is well suited for capturing the time-delayed effects of unsteady aerodynamics. Different from traditional reduced-order models, the current model based on LSTM network does not require the selection of delay orders. The performance of the proposed model is evaluated by a NACA 64A010 airfoil pitching and plunging in the transonic flow across multiple Mach numbers. It is demonstrated that the model can accurately capture the dynamic characteristics of aerodynamic and aeroelastic systems for varying flow and structural parameters.

Galloping is a type of fluid-elastic instability phenomenon characterized by large-amplitude low-frequency oscillations of the structure. The aim of the present study is to reveal the underlying mechanisms of galloping of a square cylinder at low Reynolds numbers ($Re$) via linear stability analysis (LSA) and direct numerical simulations. The LSA model is constructed by coupling a reduced-order fluid model with the structure motion equation. The relevant unstable modes are first yielded by LSA, and then the development and evolution of these modes are investigated using direct numerical simulations. It is found that, for certain combinations of$Re$and mass ratio ($m^{\ast }$), the structure mode (SM) becomes unstable beyond a critical reduced velocity$U_{c}^{\ast }$due to the fluid–structure coupling effect. The galloping oscillation frequency matches exactly the eigenfrequency of the SM, suggesting that the instability of the SM is the primary cause of galloping phenomenon. Nevertheless, the$U_{c}^{\ast }$predicted by LSA is significantly lower than the galloping onset$U_{g}^{\ast }$obtained from numerical simulations. Further analysis indicates that the discrepancy is caused by the nonlinear competition between the leading fluid mode (FM) and the SM. In the pre-galloping region$U_{c}^{\ast }, the FM quickly reaches the nonlinear saturation state and then inhibits the development of the SM, thus postponing the occurrence of galloping. When$U^{\ast }>U_{g}^{\ast }$, mode competition is weakened because of the large difference in mode frequencies, and thereby no mode lock-in can happen. Consequently, galloping occurs, with the responses determined by the joint action of SM and FM. The unstable SM leads to the low-frequency large-amplitude vibration of the cylinder, while the unstable FM results in the high-frequency vortex shedding in the wake. The dynamic mode decomposition (DMD) technique is successfully applied to extract the coherent flow structures corresponding to SM and FM, which we refer to as the galloping mode and the von Kármán mode, respectively. In addition, we show that, due to the mode competition mechanism, the galloping-type oscillation completely disappears below a critical mass ratio. From these results, we conclude that transverse galloping of a square cylinder at low$Re$is essentially a kind of single-degree-of-freedom (SDOF) flutter, superimposed by a forced vibration induced by the natural vortex shedding. Mode competition between SM and FM in the nonlinear stage can put off the onset of galloping, and can completely suppress the galloping phenomenon at relatively low$Re$and low$m^{\ast }$conditions.

In this paper, a fuzzy scalar radial basis function neural network is proposed, in order to overcome the limitation of traditional aerodynamic reduced-order models having difficulty in adapting to input variables with different orders of magnitude. This network is a combination of fuzzy rules and standard radial basis function neural network, and all the basis functions are defined as scalar basis functions. The use of scalar basis function will increase the flexibility of the model, thus enhancing the generalization capability on complex dynamic behaviors. Particle swarm optimization algorithm is used to find the optimal width of the scalar basis function. The constructed reduced-order models are used to model the unsteady aerodynamics of an airfoil in transonic flow. Results indicate that the proposed reduced-order models can capture the dynamic characteristics of lift coefficients at different reduced frequencies and amplitudes very accurately. Compared with the conventional reduced-order model based on recursive radial basis function neural network, the fuzzy scalar radial basis function neural network shows better generalization capability for different test cases with multiple normalization methods.

This paper develops a hybrid and parallel-structured reduced-order framework for modeling unsteady aerodynamics, which incorporates both linear and nonlinear system identification methods. To reflect unsteady flow physics, the hybrid model introduces time-delayed output feedback to both linear and nonlinear subsystems. The linear output and nonlinear residual are identified by the autoregressive with exogenous input model and the multi-kernel neural network, respectively. The proposed approach is illustrated here with the reduction of computational-fluid-dynamics-based aeroelastic analysis of a NACA0012 airfoil oscillating in transonic and viscous flows. In particular, we exploit the potential of this model in analyzing complex aeroelastic phenomena including limit-cycle oscillations, the beat phenomenon at high reduced velocities, and nodal-shaped oscillations induced by the interaction between buffet and flutter. Results demonstrate that the proposed approach approximates the dynamically linear and nonlinear aerodynamic characteristics obtained from high-fidelity time-marching methods with a high level of accuracy. This framework can be used as a general reduced-order modeling strategy to represent dynamic systems exhibiting both linear and nonlinear characteristics.

Ice accretion on wind turbine blades and wings changes the effective shape of the airfoil and considerably deteriorates the aerodynamic performance. However, the unsteady performance of iced airfoil is often difficult to predict. In this study, the unsteady aerodynamic performance of iced airfoil is simulated under different pitching amplitudes and reduced frequencies. In order to efficiently predict aerodynamic performance under icing conditions, a multi-fidelity reduced-order model based on multi-task learning is proposed. The model is implemented using lift and moment coefficient of clean airfoil as low-fidelity data. Through using few aerodynamic data from iced airfoils as high-fidelity data, the model can achieve aerodynamic prediction for different ice shapes and pitching motions. The results indicate that, compared with single-fidelity and single-task modeling, the proposed model can achieve better accuracy and generalization capability. At the same time, the model can be generalized to different ice shapes, which can effectively improve the unsteady prediction efficiency.

High-order (HO) methods are of concerted academic and industrial interest in recent years due to their improved accuracy and their capability to deal with complex geometries [1]. Of particular note is the flux reconstruction method [2], which unifies several existing HO schemes into a simpler and computationally efficient approach that has been shown to work on all element types (including simplices) in two and three dimensions. There is considerable interest to apply HO methods to industrially relevant problems. At the same time, accurate and robust turbulence modeling techniques are essential for reliable results. As outlined in the National Aeronautics and Space Administration's CFD vision 2030 study, Large Eddy Simulation (LES) still remains impractical for industrial cases therefore, Reynolds-Averaged Navier Stokes (RANS) and hybrid RANS-LES methods hold high significance in the near future [4]. Achieving fastest convergence to steady-state is important in the context of RANS simulations, for which several convergence acceleration techniques are being investigated. Multigrid methods are an industry standard in Finite Volume (FV) type schemes and are increasingly being applied to HO methods in the form of p-multigrid [22]. They exploit the polynomial hierarchy of the solution space to represent errors on a coarser resolution. A natural extension of this idea is hp-multigrid, where we can augument the classical h-multigrid to the polynomial hierarchy [23]. In this paper we illustrate the application of high-order flux reconstruction methods to simulate compressible, turbulent flows on body-fitted meshes. The case in point is the turbulent flow over a flat plate [24]. Turbulence is modeled through the RANS approach using the one-equation Spalart-Allmaras model. Grid-coarsening for the h-levels is performed by removing every other line in each direction from the original mesh. The system is driven to a steady-state solution using hp-multigrid convergence acceleration with local time-stepping using an explicit Runge-Kutta time-marcher. We show that the augumented h-multigrid is highly effective with a 10X to 24X drop in convergence time.

In the last decade, there has been a lot of interest in developing high-order methods as viable option for unsteady scale-resolving-simulations which are increasingly important in the industrial design process. High-order methods offer the advantage of low numerical dissipation, high efficiency on modern architectures and quasi mesh-independence. Despite significant advances in high-order solution methods, the general CFD workflow (geometry, CAD preparation, meshing, solution, post-processing) has largely remained unchanged, with mesh generation being a significant bottleneck and often determining the overall quality of the solution. In this work, we aim to combine the numerical advantages of the high-order Flux-Reconstruction (FR) method and the simplicity of the mesh generation (or lack thereof) of the Immersed Boundary Method (IBM) for steady and unsteady problems over moving geometries. The volume-penalization (penalty-IBM) method is selected for its ease of implementation and robustness. Detailed discussions about numerical implementation, including the boundary representation, mask function, data reconstruction, and selection of the penalization parameter are given. Advantages of combining volume penalization in the high-order framework are shown by various numerical test cases. The approach is firstly demonstrated for the linear advection-diffusion equation by investigating the numerical convergence for the coupled FR-IBM approach. Thereafter, the accuracy of the approach is demonstrated for canonical (static) test cases in 2D and 3D when compared to a standard body-fitted unstructured simulation. Finally, the efficiency of the method to handle moving geometries is demonstrated for the flow around an airfoil with pitching and plunging motions., This paper has been accepted by Journal of Computational Physics and can be accessed through: https://doi.org/10.1016/j.jcp.2021.110721

We propose a strategy to estimate the maximum stable time-steps for explicit time-stepping methods for hyperbolic systems in a high-order flux reconstruction framework. The strategy is derived through a von-Neumann analysis (VNA) framework for the advection-diffusion equation on skewed two- and three-dimensional meshes. It directly incorporates the spatial polynomial- and mesh-discretization in estimating the convective and diffusive length-scales. The strategy is extended to the density-based Navier-Stokes system of equations, taking into account the omnidirectionality of the speed of sound. We compare the performance of this strategy with three other popular choices of length-scales across a wide range of polynomial-orders, meshes of drastically varying cell-quality, and flow-physics. The proposed strategy shows robust behavior across all test-scenarios with limited variation of the maximum stable CFL-number (0.1 to 1) for polynomial-orders 1 through 10, unlike other strategies where the CFL-number varies sharply. Finally, we show the advantage of the proposed methodology for local-timestepping for p-multigrid through a RANS-modeled steady-state turbulent flow case, on a mesh with large disparity of mesh elements and aspect ratios.

Numerical simulation results in recent years show that vortex-induced vibration (VIV) can occur at a subcritical Reynolds number. And the VIV has been observed numerically at Reynolds numbers as low as Re = 20. The current study presents an experimental evidence for the subcritical VIV of a cylinder. We designed and built a rotating channel that makes it possible to perform VIV experiments at subcritical Reynolds numbers. Based on the rotating channel, two sets of tests were carried out for fixed natural frequency with variable incoming flow speed and fixed incoming flow speed with variable natural frequency. In both sets of experiments, subcritical VIV were observed and the VIV can be observed at a Reynolds number as low as 23, which is close to the numerical results of Mittal.

In the last decade, high-order methods for Computational Fluid Dynamics (CFD) are becoming attractive for unsteady scale-resolving-simulations in industrial CFD applications, due to their advantages of low numerical dissipation, high efficiency on modern architectures and quasi mesh-independence. However, the generation of body-fitted mesh for high-order methods is still a significant bottleneck and often determines the overall quality of the solution. To avoid the complexity of mesh generation, the present work combines the numerical advantages of the high-order Flux Reconstruction (FR) method and the simplicity of the mesh generation based on Immersed Boundary Method (IBM) that allows solving flow past obstacles on a non body-fitted mesh. The volume penalization method is selected for its ease of implementation and robustness. The proposed method is validated by several test cases, including flow past a cylinder and NACA0012 airfoil for static and moving boundaries. Good agreement with body-fitted simulation is reported.

There has been an increasing interest in developing efficient immersed boundary method (IBM) based on Cartesian grids, recently in the context of high-order methods. IBM based on volume penalization is a robust and easy to implement method to avoid body-fitted meshes and has been recently adapted to high order discretisations (Kou et al., 2021). This work proposes an improvement over the classic penalty formulation for flux reconstruction high order solvers. We include a selective frequency damping (SFD) approach (Aakervik et al., 2006) acting only inside solid body defined through the immersed boundary masking, to damp spurious oscillations. An encapsulated formulation for the SFD method is implemented, which can be used as a wrapper around an existing time-stepping code. The numerical properties have been studied through eigensolution analysis based on the advection equation. These studies not only show the advantages of using the SFD method as an alternative of the traditional volume penalization, but also show the favorable properties of combining both approaches. This new approach is then applied to the Navier-Stokes equation to simulate steady flow past an airfoil and unsteady flow past a circular cylinder. The advantages of the SFD method in providing improved accuracy are reported.

We propose a data-driven method to perform eigensolution analyses and quantify numerical errors in a non-intrusive manner. In classic eigensolution analysis methods, explicit matrices need to be constructed, whilst in our approach only solution snapshots from numerical simulations are required to quantify the numerical errors (dispersion and diffusion) in time and/or space. This new approach is based on a recent data-driven method: the Spatio-Temporal Koopman Decomposition (STKD), that approximates spatio-temporal data as a linear combination of standing or travelling waves growing or decaying exponentially in time and/or space. We validate our approach with classic matrix-based approaches, where accurate predictions of the dispersion-dissipation behaviour for both temporal and spatial eigensolution analyses are reported.

Computational-fluid-dynamics-based prediction of unsteady aerodynamics is an essential research topic in the design of aircraft, which usually requires very high computational cost. Therefore, developing efficient and accurate reduced-order models (ROMs) for unsteady aerodynamics is important. With recent progress in modeling unsteady aerodynamics for a fixed configuration, there is a need of developing reduced-order models across multiple geometrical shapes. In this work, the long short-term memory (LSTM) network, a traditional model in machine learning for modeling time-dependent dynamics, is utilized to construct the unsteady aerodynamic reduced-order model, which takes into account the variation of airfoil geometry. The proposed approach is illustrated by some NACA 6-series airfoils under pitching and plunging motions in the transonic flow, where the geometrical parametrization is based on the maximum camber and thickness. Results show that the model can accurately capture the unsteady aerodynamics for varying shape parameters. Finally, as a proof of concept, the constructed ROM is applied to uncertainty and sensitivity analysis with respect to the variation of geometry parameters.

Aerodynamic modeling plays an important role in multiphysics and design problems, in addition to experiment and numerical simulation, due to its low-dimensional representation of unsteady aerodynamics. However, in the traditional study of aerodynamics, developing aerodynamic and flow models relies on classical theoretical (potential flow) and empirical investigation, which limits the accuracy and extensibility. Recently, with significant progress in high-fidelity computational fluid dynamic simulation and advanced experimental techniques, very large and diverse fluid data becomes available. This rapid growth of data leads to the development of data-driven aerodynamic and flow modeling. Through advanced mathematical methods from control theory, data science and machine learning, a lot of data-driven aerodynamic models have been proposed. These models are not only more accurate than theoretical models, but also require very low computational cost compared with numerical simulation. At the same time, they help to gain physical insights on flow mechanism, and have shown great potential in engineering applications like flow control, aeroelasticity and optimization. In this review paper, we introduce three typical data-driven methods, including system identification, feature extraction and data fusion. In particular, main approaches to improve the performance of data-driven models in accuracy, stability and generalization capability are reported. The efficacy of data-driven methods in modeling unsteady aerodynamics is described by several benchmark cases in fluid mechanics and aeroelasticity. Finally, future development and potential applications in related areas are concluded.

Transonic buffet is a phenomenon of aerodynamic instability with shock wave motions which occurs at certain combinations of Mach number and mean angle of attack, and which limits the aircraft flight envelope. The objective of this study is to develop a modelling method for unstable flow with oscillating shock waves and moving boundaries, and to perform model-based feedback control of the two-dimensional buffet flow by means of trailing-edge flap oscillations. System identification based on the ARX algorithm is first used to derive a linear model of the input–output dynamics between the flap rotation (the control input) and the lift and pitching moment coefficients (system outputs). The model features a pair of unstable complex-conjugate poles at the characteristic buffet frequency. An appropriate reduced-order model (ROM) with a lower dimension is further obtained by a balanced truncation method that keeps the pair of unstable poles in the unstable subspace but truncates the dynamics in the stable subspace. Based on this balanced ROM, two kinds of feedback control are designed by pole assignment and linear quadratic methods respectively. These independent designs, however, result in similar suboptimal static output feedback control laws. When introduced in numerical simulations, they are both able to completely suppress the buffet instability. Furthermore, the resulting controllers are even able to stabilize buffet flows with nonlinear disturbances and in off-design flow conditions, thus implying their robustness. The analysis of the feedback control laws indicates that parameters (frequency and phase) corresponding to the ‘anti-resonance’ of the linear input–output model are vital for optimal control. The best performance is obtained when the control operates close to the ‘anti-resonance’, which is supported by the optimal frequency and the phase of the open-loop control as well as by the optimal phase of the closed-loop control.

Dynamic mode decomposition (DMD) has been extensively utilized to analyze the coherent structures in many complex flows. Although specific flow patterns with dominant frequency and growth rate can be captured, extracting dominant DMD modes for flow reconstruction and dynamic modeling still needs a priori knowledge on flow physics, especially for some transient states of unstable flows. In this paper, a criterion to select dominant modes from DMD technique is developed. The unsteady flow can be described by the superposition of each normalized DMD mode multiplied by its time coefficient. The dominance of each DMD mode can be ordered by time integration of its time coefficient. Compared with standard DMD approach, which decides the dominance of DMD modes from the order of amplitude or mode norm, this criterion considers the evolution of each mode within the whole sampling space, and ranks them according to their contribution to all samples. The proposed mode selection strategy is evaluated by test cases including both equilibrium and transient states of a cylinder at Reynolds number of 60 and a transient state of a NACA0012 airfoil buffeting in transonic flow. Results indicate that using this criterion, dominant DMD modes can be identified and flow dynamics in unstable or transient systems can be reconstructed accurately with fewer modes. Besides, this approach has better convergence against mode number and lower sensitivity to the initial condition than standard DMD method.

This paper presents a novel reduced-order model (ROM) of thrust for an efficiently flapping airfoil using system identification method. A NACA0012 airfoil pitching and plunging at a low Reynolds number of 40,000 is used to test the ROM. Unlike conventional aerodynamic models which introduce the airfoil displacements directly as inputs, this study utilizes the quadratic-terms of displacements as inputs to overcome the frequency-doubling effect of propulsion forces over the oscillation frequency. The autoregressive with exogenous input (ARX) model is adopted to construct mappings between the input and output data. Meanwhile, a heuristic searching strategy is applied for sensitivity analysis of the input variables and the optimal input-vector is determined. The ROMs are then validated in the time domain by comparing their predicted thrust responses with those of CFD simulations under either harmonic or random excitations. Results show that the proposed ROMs can accurately predict the thrust responses of a flapping airfoil with arbitrary motions from moderate to small oscillation amplitudes where a leading-edge vortex does not develop, while the computational cost can be reduced by nearly 2 orders of magnitude compared to the high-fidelity CFD simulation method. Finally, the validity of ROMs is mostly clearly shown by using them for propulsive characteristic analysis of a flapping airfoil. Excellent qualities of the ROMs indicate that they can be used for flapping mode optimization and flapping flight control in future research.

A layered reduced-order modeling approach for nonlinear unsteady aerodynamics comprising both linear and nonlinear characteristics is developed. The constructed reduced-order model (ROM) with a parallel connection structure is composed of a linear surrogate model and a nonlinear one. The linear autoregressive with exogenous input (ARX) model is constructed by a training case at small amplitude, while the nonlinear radial basis function neural network (RBFNN) model is identified to model the residual nonlinear responses from the second training case under large amplitude motions. Outputs from both models are superimposed to obtain the total aerodynamic coefficients. Either quasi-steady or unsteady layered model can be constructed by introducing only input delay or considering both input and output delayed effects. The present approach is tested by predicting the unsteady aerodynamic forces, limit cycle oscillations and flutter behaviors of a transonic NACA 64A010 airfoil. Results show that both the quasi-steady and the unsteady models agree well with those from high fidelity simulations within a wide range of amplitude and frequency, and the LCO trends are reasonable after the models are coupled with the structural equations of motion. A better agreement is achieved by the unsteady layered model, even though this model may sometimes become unstable.

A novel modeling approach for nonlinear aerodynamic reduced-order models (ROMs) is developed to enhance the generalization capability of current ROMs. The proposed method is called the “modeling with validation case” approach. Instead of the conventional modeling process through training and test cases, three types of cases, namely, training, validation, and test cases, are introduced. The validation case is used to find the best parameters (widths of hidden neurons) of the model, in order to enhance the generalization capability of the nonlinear aerodynamic ROMs. Searching for optimal parameters is accomplished by the particle swarm optimization (PSO) algorithm, obtaining the minimal mean squared error of the validation case. The approach is applied to a recursive radial basis function neural network aerodynamic model. Two examples of a NACA0012 airfoil pitching in transonic flow are presented to compare the proposed approach with the conventional modeling process. The aerodynamic model with the proposed approach shows high accuracy from small to large pitching amplitudes in the time domain. In the frequency domain, comparisons of the first-order Fourier series indicate that the dynamic characteristics at different reduced frequencies and amplitudes are well captured. The conventional modeling process shows equivalent accuracy at large amplitudes but fails to predict the dynamic linear behavior at small amplitudes. Compared with the conventional modeling process, the proposed approach can capture both linear and nonlinear characteristics.

This paper proposes a nonlinear aerodynamic reduced-order model robust to different Mach numbers based on a recursive neural network. To model the nonlinear features with varying flow param...

In recent years, the data-driven turbulence model has attracted widespread concern in fluid mechanics. The existing approaches modify or supplement the original turbulence model by machine learning based on the experimental/numerical data, in order to augment the capability of the present turbulence models. Different from the previous researches, this paper directly reconstructs a mapping function between the turbulent eddy viscosity and the mean flow variables by neural networks and completely replaces the original partial differential equation model. On the other hand, compared with the machine learning models for the low Reynolds (Re) number flows based on direct numerical simulation data, high Reynolds number flows around airfoils present the apparent scaling effects and strong anisotropy, which induce large challenges in accuracy and generalization capability for the machine learning algorithm. We mainly concentrate on the high Reynolds number turbulent flows around the airfoils and take the results calculated by the computational fluid dynamics solver with the Spallart-Allmaras (SA) model as training data to construct a high-dimensional data-driven network model based on machine learning. The radial basis function neural network and the auxiliary optimization methods are adopted, and the individual models are built separately for the flow fields of the near-wall region, wake region, and far-field region. The training data in this paper is extracted from only three subsonic flow fields of NACA0012 airfoil. The data-driven turbulence model can be applied to various airfoils and flow states, and the predicted eddy viscosity, lift/drag coefficients, and skin friction distributions are all in good agreement with the results of the original SA model. This research demonstrates the promising prospect of machine learning methods in future studies about turbulence modeling.

This study proposes an improvement in the performance of reduced-order models (ROMs) based on dynamic mode decomposition to model the flow dynamics of the attractor from a transient solution. By combining higher order dynamic mode decomposition (HODMD) with an efficient mode selection criterion, the HODMD with criterion (HODMDc) ROM is able to identify dominant flow patterns with high accuracy. This helps us to develop a more parsimonious ROM structure, allowing better predictions of the attractor dynamics. The method is tested in the solution of a NACA0012 airfoil buffeting in a transonic flow, and its good performance in both the reconstruction of the original solution and the prediction of the permanent dynamics is shown. In addition, the robustness of the method has been successfully tested using different types of parameters, indicating that the proposed ROM approach is a tool promising for using in both numerical simulations and experimental data.

It is well known that the occurrence of vortex-induced vibration (VIV) at a subcritical Reynolds number, which is lower than 47, is induced by fluid-structure interaction. However, for the free flow, this phenomenon disappears at a Reynolds number about 20. The current study provides an explanation to the disappearance of the VIV by capturing the evolution of the purely fluid modes at Reynolds numbers between 12 and 55. To ensure accurate mode extraction, the dynamic mode decomposition technique is utilized. Results show that the stable von Karman vortex shedding mode exists in a subcritical flow regime but becomes less distinct as the Reynolds number decreases. When the Reynolds number is lower than 18, this mode nearly vanishes, characterizing the lower boundary of fluid-structure instability.