Article

# Analysis of Dynamic Stall Using Dynamic Mode Decomposition Technique

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## Abstract

Dynamic mode decomposition is applied to investigate the unsteady flowfield around a pitching airfoil. The extracted flow structures are termed as dynamic mode decomposition modes. Analyses are performed for both attached flow and dynamic stall cases. Initially, flowfield data generated from numerical simulations are investigated. The effect of exclusion of the flowfield near the surface of the airfoil on the structure of the dynamic mode decomposition modes is examined. This is of vital importance for the experimental measurements, as the flowfield near the airfoil’s surface is difficult to measure using particle image velocimetry. In the latter part of the paper, dynamic mode decomposition modes extracted from the experimental data are analyzed. The structure of these modes are compared with the modes obtained from the proper orthogonal decomposition technique. Finally, effects of phase averaging on the modes are discussed.

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... For example, during dynamic stall, the lift coefficient increases beyond its value found in the static stall condition, due to the formation of a leadingedge vortex (LEV). As the angle of attack increases further, the LEV sheds and the lift coefficient falls [2]. To avoid large, undesirable variations in lift, feedback control may be applied to enhance or regularize unsteady lift production. ...
... In the case of an oscillatory flow, a small number of DMD modes often provides a sufficiently accurate reconstruction of the data [2]. To find those modes and their initial amplitudes, we employ the sparsity promoting DMD algorithm developed by Jovanović et al. [18], which consists of finding the mode amplitudes that minimize the cost function ...
... Note that the test set consists of instantaneous PIV data, which are representative of a case of real-time estimation. POD modes form a more accurate projection of the data; the difference corresponds to using three or four more DMD modes for the same level of accuracy, which is similar to the result obtained in Ref. [2]. The projection error from the POD modes of the training and test data are very close for the first 10 modes, suggesting that they contain similar information. ...
... For example, during dynamic stall, the lift coefficient increases beyond its value found in the static stall condition, due to the formation of a leading edge vortex (LEV). As the angle of attack increases further, the LEV sheds and the lift coefficient falls [2]. To avoid large, undesirable variations in lift, feedback control may be applied to enhance or regularize unsteady lift production. ...
... Note that the test set consists of instantaneous PIV data, representative of a case of real-time estimation. POD modes form a more accurate projection of the data; the difference corresponds to using three or four more DMD modes for the same level of accuracy, similar to the result obtained in [2]. The projection error from the POD modes of the training and test data are very close for the first 10 modes, suggesting that they contain very similar information. ...
... 和圆柱不稳定线性动力学问题的建模过程 [19] 等。 然而，对于某些复杂流动，可能包含某些对流场 动 态 特 性 影 响 很 大 的 低 能 量 特 征 。 近 期 ， Schmid [18] 提出了一种通过动态系统特征值估计流 [17] 结合，通过无限维线性算子 理论使其扩展到非线性流动。DMD在横向射流 [20] ，带襟翼机翼尾流 [21] 和动失速尾流 [22][23] 等复杂 流动现象上有广泛应用。结合优化方法或正则化 理论，研究者也提出一些改进形式的DMD，如 opt-DMD [24] , OMD [25] , SPDMD [26] 等。这些改进形 式对某些特定的问题可能提取出更好的流动模 态。 由于DMD与POD之间存在较多的相关性， 因 此 针 对 某 些 复 杂 流 动 现 象 ， 往 往 通 过 POD 和DMD进行对比研究。Wan等 [20] 通过LES模拟了 横向射流的涡旋流动发展过程，结果表明DMD 得到的中立稳定模态以及主导频率均与LES的计 算结果一致，而由于高阶POD模态包含多种频率 分量，因此不利于流场的动态分析。Mariappan 等 [22] 和Mohan等 [23] 通过POD和DMD方法分别研 究 二 维 翼 型 和 三 维 机 翼 的 动 失 速 现 象 ， 发 现DMD模态得到的流动结构对于描述频域下的 流场和提取流动中的主要不稳定模态具有一定优 势。上述研究中对比了DMD与POD的区别及联 (13) 矩阵  A 的特征分解得到的特征值是 A 的一部 分，即 z 特征值 [17] 。定义第 j 个模态的 Ritz 特 征值为 j ...
... 和圆柱不稳定线性动力学问题的建模过程 [19] 等。 然而，对于某些复杂流动，可能包含某些对流场 动 态 特 性 影 响 很 大 的 低 能 量 特 征 。 近 期 ， Schmid [18] 提出了一种通过动态系统特征值估计流 [17] 结合，通过无限维线性算子 理论使其扩展到非线性流动。DMD在横向射流 [20] ，带襟翼机翼尾流 [21] 和动失速尾流 [22][23] 等复杂 流动现象上有广泛应用。结合优化方法或正则化 理论，研究者也提出一些改进形式的DMD，如 opt-DMD [24] , OMD [25] , SPDMD [26] 等。这些改进形 式对某些特定的问题可能提取出更好的流动模 态。 由于DMD与POD之间存在较多的相关性， 因 此 针 对 某 些 复 杂 流 动 现 象 ， 往 往 通 过 POD 和DMD进行对比研究。Wan等 [20] 通过LES模拟了 横向射流的涡旋流动发展过程，结果表明DMD 得到的中立稳定模态以及主导频率均与LES的计 算结果一致，而由于高阶POD模态包含多种频率 分量，因此不利于流场的动态分析。Mariappan 等 [22] 和Mohan等 [23] 通过POD和DMD方法分别研 究 二 维 翼 型 和 三 维 机 翼 的 动 失 速 现 象 ， 发 现DMD模态得到的流动结构对于描述频域下的 流场和提取流动中的主要不稳定模态具有一定优 势。上述研究中对比了DMD与POD的区别及联 (13) 矩阵  A 的特征分解得到的特征值是 A 的一部 分，即 z 特征值 [17] 。定义第 j 个模态的 Ritz 特 征值为 j ...
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Transonic buffet is due to the self-sustained oscillations of shock wave in the unsteady transonic flow, which induces the forced periodically motion of the structure. For aircrafts in transonic flow, this phenomenon generally happens, leading to negative effects on the structural strength and fatigue life. Analysis based on mode decomposition is an effective tool for developing buffet control design. In this paper, two typical mode analysis methods, i.e., proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD), are utilized for analyzing the transonic buffet of a OAT15A airfoil. Two techniques are compared by studying the frequency of dominant modes, pressure dis-tributions on the surface and the errors of flow construction. Results indicate that because of the consideration of fre-quency contents in DMD, the critical stable characteristics and dominant frequency of transonic buffet are well cap-tured. Besides, DMD techniques accurately mimic the time evolutions of flow variables near the shock wave. Although POD technique provides relatively small errors for flow reconstruction, it performs worse than DMD near the shock wave region, because of the poorer approximates of pressure evolution in time.
... The DMD and POD analyses will be performed on the pressure snapshots to investigate the modal participation in the aerodynamic damping of the dynamic stall phenomenon presented earlier. In earlier DMD analysis of dynamic stall phenomena, either the snapshots of the stream-wise velocity component [7], or the velocity magnitude [23] was used for computing the DMD modes. However, in order to investigate the modal aerodynamic damping, we require the modal pitching moments. ...
... Mesh used for k − ω SST DDES Similar to Ref.[23,7], a reduced computational domain was used for taking the snapshots of the flow. This reduced domain, shown inFig. ...
Preprint
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This study presents an investigation of the aerodynamic damping of light dynamic stall phenomena on a pitching NACA 0012 airfoil via Dynamic Modal Decomposition (DMD) and Proper Orthogonal Decomposition (POD) techniques. DMD analysis predicts a dominant mode having the same frequency of the pitching excitation and contributing all of the aerodynamic damping of the system. The temporal orthogonality of the DMD technique renders all other DMD modes orthogonal to both the dominant DMD mode as well as the pitching motion, resulting in zero aerodynamic work due to these DMD modes. However, the lack of temporal orthogonality of POD modes implies several modes contributing to the aerodynamic damping of the system. The leading-edge suction and the trailing-edge vortex were considered the most energetic features by the two most dominant POD modes and was also observed as the most significant spatial feature in the dominant DMD mode shape. A recently developed Hilbert transform-based methodology was extended here for computing chord-wise intra-cycle aerodynamic damping distribution. This method shows large variations of the intra-cycle aerodynamic damping distribution obtained from DDES surface pressure data over the cycle, which poses a challenge to extract meaningful information for possible flow control applications. However, the dominant DMD modal aerodynamic damping distribution shows no intra-cycle variations and predicts negative damping hot-spots near the leading and trailing edges of the chord. Such conclusions are significantly different than observed for an attached flow case or observed in a related study considering a deep dynamic stall regime at lower Reynolds number.
... The DMD and POD analyses will be performed on the pressure snapshots to investigate the modal participation in the aerodynamic damping of the dynamic stall phenomenon presented earlier. In earlier DMD analysis of dynamic stall phenomena, either the snapshots of the stream-wise velocity component [7], or the velocity magnitude [23] was used for computing the DMD modes. However, in order to investigate the modal aerodynamic damping, we require the modal pitching moments. ...
... Mesh used for k − ω SST DDES Similar to Ref.[23,7], a reduced computational domain was used for taking the snapshots of the flow. This reduced domain, shown inFig. ...
Article
Full-text available
This study presents an investigation of the aerodynamic damping of light dynamic stall phenomena on a pitching NACA 0012 airfoil via Dynamic Modal Decomposition (DMD) and Proper Orthogonal Decomposition (POD) techniques. DMD analysis predicts a dominant mode having the same frequency of the pitching excitation and contributing all of the aerodynamic damping of the system. The temporal orthogonality of the DMD technique renders all other DMD modes orthogonal to both the dominant DMD mode as well as the pitching motion, resulting in zero aerodynamic work due to these DMD modes. However, the lack of temporal orthogonality of POD modes implies several modes contributing to the aerodynamic damping of the system. The leading-edge suction and the trailing-edge vortex were considered the most energetic features by the two most dominant POD modes and was also observed as the most significant spatial feature in the dominant DMD mode shape. A recently developed Hilbert transform-based methodology was extended here for computing chord-wise intra-cycle aerodynamic damping distribution. This method shows large variations of the intra-cycle aerodynamic damping distribution obtained from DDES surface pressure data over the cycle, which poses a challenge to extract meaningful information for possible flow control applications. However, the dominant DMD modal aerodynamic damping distribution shows no intra-cycle variations and predicts negative damping hot-spots near the leading and trailing edges of the chord. Such conclusions are significantly different than observed for an attached flow case or observed in a related study considering a deep dynamic stall regime at lower Reynolds number.
... The difference between respective eigenvalues is less than 0.3 % of the standard deviation, σ(λ i ), after the 6th cycle. This result is in accordance with other studies which found convergence after five cycles for quasi-2D periodic inflow conditions 34 . An analysis of the available 13 cycles is thus sufficient to capture the periodicity of the present flow by means of POD. ...
Preprint
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Rotor blades of wind turbines in the atmospheric boundary layer regularly experience the aerodynamic phenomenon of dynamic stall consisting of a temporary overshoot of lift and detrimental fatigue loads. Particularly the formation of dynamic stall under three-dimensional inflow conditions raises open questions. Aerodynamic behavior of a DU 91-W2-250 wind profile undergoing light dynamic stall is thus analyzed in a wind tunnel. Effects of a gust with streamwise and spanwise periodic variation are investigated by comparing total and local lift generation with flow formation above the airfoil. The observed stall cycle is divided into five stages of which one reveals lift overshoot of up to 16 %. The aerodynamic response of the airfoil shows a delay of about 1/8 period between evolution of local angle of attack and lift giving a counterclockwise dynamic polar. A proper orthogonal decomposition (POD) analysis of the flow field contributes to understand aerodynamic consequences of the three-dimensional gust. Local inflow, total lift as well as certain lift events are captured by one POD eigenmode, respectively. Obtained results lead to the conclusion that the flow and particularly the stalled wake of an airfoil facing a three-dimensional gust are strongly coupled in the spanwise direction. This yields to flow stabilization, inhibition of stall, and in turn, counterclockwise dynamic polar along with augmented total lift.
... The POD modes show the spatial structures of the high energy flow features. Both the POD and DMDmodes were weighted with the the grid cell area as mentioned in Ref[36] and it was seen that the grid weighting did not significantly affect the mode spatial structure and properties compared to the non-weighted results. This is most likely due to the fact that the mesh used in this case is extremely smooth with minimal skewness. ...
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The ability to generate massive amounts of high-resolution data, both experimentally and computationally, has led to a surge of interest in mathematical model reduction using modal decomposition algorithms. When applied to complex unsteady flows, different techniques highlight different flow dynamics which are difficult to extract directly from large datasets. A widely used technique is Proper Orthogonal Decomposition (POD) which ranks modes by their relative energy content. A newer method, Dynamic Mode Decomposition (DMD), focuses on rates of growth of different modes, thus retaining temporal information pertinent to dynamic events which dominate highly transient flows. In this work, we employ these techniques, with emphasis on DMD, to analyze flow past an SD7003 airfoil undergoing periodic plunging motion representative of small unmanned vehicles. Snapshots are obtained from a high-fidelity, experimentally validated Large-Eddy Simulation (LES). The stability characteristics of DMD modes show that the flow structure associated with the leading edge vortex (LEV) in dynamic stall is unstable. The influence of DMD modes in local regions of the flow provide insight into which flow frequencies may be targeted by leading edge actuators to have maximum impact in controlling the unstable flow structures inducing stall. Dominant POD modes, each of which can have multiple frequency components, are shown to be comprised primarily of the dominant DMD mode contributions. A practical framework is introduced to identify components of a global flow structure across different velocity components. The method is shown to successfully reproduce the global flow structure. A few dominant DMD modes are used to reconstruct the flow near the leading edge, where rapid changes occur during parts of the plunging cycle and the ability of individual probe to provide insight into global phenomena is assessed. Finally, a stability analysis of each mode is performed to identify flow instabilities near the leading edge.
... DMD has already been applied to different flows such as swirling jet 46 and flows around cylinders of different diameters 47 or around airfoils. 48,49 A brief description of the algorithm will be presented below, but for a more rigorous treatment, we refer the reader to the original works of Schmid 44,45 and the analysis of Jovanović et al. 50 The review paper of Bagheri 51 provides a more general discussion on model reduction methods. ...
Article
Direct numerical simulations of the flow field around a NACA 0012 airfoil at Reynolds number 50 000 and angle of attack 5° with 3 different trailing edge shapes (straight, blunt, and serrated) have been performed. Both time-averaged flow characteristics and the most dominant flow structures and their frequencies are investigated using the dynamic mode decomposition method. It is shown that for the straight trailing edge airfoil, this method can capture the fundamental as well as the subharmonic of theKelvin-Helmholtz instability that develops naturally in the separating shear layer. The fundamental frequency matches well with relevant data in the literature. The blunt trailing edge results in periodic vortex shedding, with frequency close to the subharmonic of the natural shear layer frequency. The shedding, resulting from a global instability, has an upstream effect and forces the separating shear layer. Due to forcing, the shear layer frequency locks onto the shedding frequency while the natural frequency (and its subharmonic) is suppressed. The presence of serrations in the trailing edge creates a spanwise pressure gradient, which is responsible for the development of a secondary flowpattern in the spanwise direction. This pattern affects the mean flow in the near wake. It can explain an unexpected observation, namely, that the velocity deficit downstream of a trough is smaller than the deficit after a protrusion. Furthermore, the insertion of serrations attenuates the energy of vortex shedding by de-correlating the spanwise coherence of the vortices. This results in weaker forcing of the separating shear layer, and both the subharmonics of the natural frequency and the shedding frequency appear in the spectra.
... Although there is widespread interest in applying DMD to practical problems, its use in the analysis of stall is relatively limited in literature. DMD was used to analyze the dynamic stall of a pitching two-dimensional airfoil [41], where the effect of neglecting the boundary layer in particle image velocimetry measurements was studied. Mohan et al. [42] studied the dynamic stall of a plunging airfoil to isolate flow frequencies associated with stall. ...
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Airfoil stall is a major inhibitor of aircraft performance. Many methods have been successfully shown to inhibit or eliminate stall. However, the underlying dynamics of control remains relatively obscure because of the highly nonlinear nature of turbulence. A major challenge is the a priori estimation of actuation parameters without resorting to trial and error. Such estimation of parameters requires a physics-based understanding of the flowfield and its response to actuation. To study this, high-fidelity large-eddy simulations are employed to analyze active control by exploring the key dynamic modes, without and with control, through dynamic mode decomposition. A NACA 0015 stalled airfoil is considered at a Reynolds number of 100,000 and a 15 deg angle of attack. The results suggest that the dominant mode representing stall has an effective Strouhal number of two. Simulations are then performed by modeling control at St=2 using a nanosecond-pulsed dielectric barrier discharge near the leading edge. Intro...
... DMD has already been applied to different flows such as swirling jet 46 and flows around cylinders of different diameters 47 or around airfoils. 48,49 A brief description of the algorithm will be presented below, but for a more rigorous treatment, we refer the reader to the original works of Schmid 44,45 and the analysis of Jovanović et al. 50 The review paper of Bagheri 51 provides a more general discussion on model reduction methods. ...
... To do so, velocity data of the flow field on the suction side of a T∕E flap, subject to a varied actuation intensity, are obtained through time-resolved particle image velocimetry (PIV). Before this, the velocity fields are assessed with the help of the recently introduced method of spectral proper orthogonal decomposition (SPOD) [13], extending the classical snapshot POD approach that is well established in terms of studying flow separation on airfoils [14][15][16]. Thus, coherent structures in Eulerian specification as well as related time scales can be identified. ...
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Pulsed blowing is applied to prevent turbulent flow separation on the trailing edge flap of a two-element high-lift wing configuration that is based on a state-of-the-art transonic profile shape. While ensuring experimental conditions relevant to application on aircraft level, the flow field along the suction side of the flap is investigated by means of time-resolved particle image velocimetry. The obtained velocity fields are investigated employing the recently introduced method of Spectral Proper Orthogonal Decomposition, allowing for an identification of coherent flow structures in Eulerian specification that can be robustly related to specific frequencies. As we are able to show, the spectral content of the observed modes can be employed as an indicator for the flow control authority. The approach is supported by the computation of finite-time Lyapunov exponent fields, containing ridges that indicate Lagrangian coherence. In the context of this study, they correspond to separation profiles and thus allow for further assessment of the actuation effectivity. The baseline configuration is shown to be characterized by two distinct material lines connecting downstream of the trailing edge flap and enclosing a recirculation zone of significant size. In the case of actuation with the lower studied momentum input, the recirculation zone is intermittently suppressed while the greater actuation intensity is sufficient to ensure a stable prevention of turbulent flow separation.
... Bekemeyer (Bekemeyer and Timme, 2017) discussed the ROMs of large civil aircraft under gust excitation in transonic flight for routine gust load analysis. For the POD method, which projects the flow snapshots into several dominant low-order vector bases, has been successfully applied to aeroelastic analysis in many fields of engineering, such as turbine blades (Cizmas and Palacios, 2003;Clark et al., 2012), helicopter rotor blade (Mariappan et al., 2014), wings (Demasi and Palacios, 2010;Stanford and Beran, 2013;Hesse and Palacios, 2014) and complete aircraft configurations Amsallem et al., 2007). Until now, CFD-based POD/ROM has been exercised for active aeroelastic control , LCO control in transonic flow (Chen et al., 2013), gust response analysis (Zhou et al., 2016a;Zhou et al., 2017), and transonic flutter suppression with control delay (Zhou et al., 2016b). ...
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Current and future trends in the aerospace industry leverage on the potential benefits provided by lightweight materials that can be tailored to realize desired mechanical characteristics when loaded. For aircraft design, the deployment of aeroelastic tailoring is hindered by the need to re-compute, for any possible modification of the structure, the dependence of the aerodynamic field on the underlying structural properties. To make progress in this direction, the work presents a rapid computational fluid dynamics based aeroelastic tool which is built around a reduced order model for the aerodynamics that is updated for any modification of the structure by using the structural dynamics reanalysis method. The aeroelastic tailoring tool is demonstrated in transonic flow for the AGARD 445.6 wing, suitably modified with composite materials. It was found that the proposed method provides accurate engineering predictions for the aeroelastic response and stability when the structure is modified from the baseline model.
... Kou [14] developed a hybrid ROM for linear and nonlinear aerodynamics, through constructing mappings between input and output data calculated from the CFD solver and its application to NACA0012 aeroelastic analysis in transonic and viscous flows. The POD method, in particular, have been successfully applied in some coupling systems to expedite the aeroelastic analysis, such as turbine blades [15,16], helicopter rotor blade [17], wings [18][19][20] and complete aircraft configurations [21,22]. In our research group, https://doi.org/10.1016/j.ast.2019.105354 ...
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... Later investigations showed that while the analysis of the POD mode time development coefficients was useful, the analysis of the POD mode shapes was only of limited use, and dynamic mode decomposition (DMD) was unsuitable for the analysis of dynamic stall [88]. However the correlations in flows between the sizes of the flow structures and the energy contained in the modes means that reconstruction of a flow excluding higher POD modes can be used effectively as a filter of fine structures, see Fig. 27. ...
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The helicopter group at the DLR in Göttingen has been actively involved in the development of measurement techniques for unsteady flows, particularly as they apply to the problems found in unsteady rotor blade aerodynamics. This includes the development and validation of new techniques for the detection of dynamically moving boundary layer transition, and for the detection of dynamic stall and other transient flow separation events. These new techniques include pressure sensor analysis, differential infrared thermography, local infrared thermography and the automated analysis of hot-film data. Particle image velocimetry (PIV) and background oriented schlieren (BOS) have been used for the analysis of the unsteady off-body flow, and synchronised PIV-BOS-pressure measurements have allowed direct comparisons between different methods. The Lagrangian volumetric PIV variant, shake-the-box, has been used to analyse secondary vortex structures in the vortex wake. This review article will give an overview of the advances in that group, as well as placing their activities in the context of international advances in these areas.
... The plethora of DMD variants existing not only illustrates the flexibility of the method, but also indicate its wide applications on multiple circumstance, such as back step flow [50,61,62], cavity flow [15,60,63,64], cylinder flow [16,46,65,66], flow around the sphere [67,68], surface flow on airfoils [69][70][71], jets [15,17,51,72,73], combustion [74][75][76][77], wall-bounded flow [51,78,79], laminarturbulent transition [80][81][82], and even in epidemiology [83], neuroscience [84], financial trading [85]. ...
Thesis
In this thesis, we develop several techniques for Dynamic Mode Decomposition (DMD) method to deal with the problem of large flow database analysis. The techniques applied can be classified into three categories: (i) composite DMD to help extract DMD modes from composite variables that are closely related to the physics of interest, (ii) agglomeration strategies applied with DMD that highly compress the large databases while retaining the accuracy of feature detection, and (iii) a new θ-DMD to deal with non-uniformly sampled databases from numerical or experimental results. The first category of the techniques introduces multiple variables into the snapshots sequence and proposes a weighted β factor to classify the modes of close relevance with weighted variable, which highly reduce the number of modes that needed to reconstruct the flow filed. In this work, we have considered combining skin friction (C_f(t_j)) and either the streamwise perturbation velocity field (u’(x, t_j)) or the Reynolds shear stress field (u’v‘(x, t_j)). From the application of composite DMD to turbulent channel flow databases at ReT~200 and ReT~932, we find that a dozen modes suffice to reconstruct the Reynolds stresses profiles. The second category of the techniques considers the compression of the databases along their temporal dimension and the agglomeration of the spatial degrees of freedom prior to the application of the DMD analysis. An overall experiment on a toy model indicates the advantages of spatial reduction over temporal decimation. Thus, in order to explore the effects of different clustering algorithms combined with DMD on flow cases, twelve different clustering algorithms/methods on three test cases have been carefully compared. Three flow cases encompass different flow regimes: a synthetic flow field, a ReD = 60 flow around a cylinder cross section, and a ReT~200 turbulent channel flow. The third category of the techniques proposes a new DMD based strategy which is capable of handling snapshots that are not equally separated in time. The θ-DMD departs from a θ-like linear discretization of non-linear system governing equation, showing accuracy and robustness on three test cases which outperforms other two methods (DMD and Non-Uniform DMD). Finally, even though the main focus of this work is on the compression and feature extraction of large fluid flow problems, the methods developed here can also be applied together with other mode decomposition methods.
... And Φ i is the flow field at a moment of a single frequency after decomposition. The detailed solution method can be seen in references [30,31]. ...
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Corner separation is a vital unsteady flow phenomenon in a compressor and plays an essential role in flow field instability. Moreover, corner separation under high subsonic Mach number conditions is a scientific problem with practical engineering significance. The prime motivation of this work is using the most cost-effective method to more accurately resolve the corner separation phenomenon in a high-speed compressor cascade and to determine its inherent unsteady behavior under high-incidence conditions. Two configurations of the cascade are investigated, an incidence of 5.0° (stable) and 7.5° (unstable, near-stall), using a detached eddy simulation (DES) set to "Re=7.5×10^5 and Ma = 0.6 at the inlet boundary. The flow structures are analyzed after verifying the accuracy of the simulation. The dynamic mode decomposition (DMD) method is applied to analyze the main sources of instability in the flow field. By comparing the stable and unstable conditions, the suction surface separation vortex (SSV) found in the recirculation region is the key factor for flow instability. The mixture of the SSV and corner vortex (CV) is the main reason for the flow field becoming unstable and eventually causing a high-incidence stall. The formation and unsteady characteristics of the SSV at the near-stall condition are discussed. Suggestions for flow control and stall warning are also given. Thus, the results can provide a theoretical basis for the flow control of a high-speed compressor blade.
... The DMD snapshots then constructed from the relatively fixed domain. This method is implemented for the analysis of the dynamic stall phenomenon in the work of Mariappan et al., 24 where the snapshots were extracted from a region of the upper side of the airfoil. Moreover, data interpolation is a feasible way to overcome the DMD limitation. ...
Article
Dynamic Mode Decomposition (DMD) is a data-driven reduced order method, which is known for its power to capture the basic features of dynamical systems. In fluid dynamics, modal analysis of unsteady fluid flows over moving structures is significant in terms of state estimation and control. However, the underlying algorithm of the DMD requires a fixed spatial domain, which is an obstacle for applying the DMD on the numerically investigated problems using dynamic meshes. In this study, a hybrid method called Hybrid Dynamic Mode Decomposition (HDMD) is presented for analysis of unsteady fluid flows over moving structures based on the DMD and machine learning. According to the assessment of several data interpolation methods, the K-nearest neighbor algorithm is employed for the interpolation of the numerical data from dynamic meshes at each time step to a single stationary grid. Three different case studies (rotating cylinder, oscillating airfoil, and Savonius wind turbine) are assessed to ensure the validity of the proposed method. Minimum mean R² equal to 0.92 has been obtained for all of the mentioned cases, indicating the robustness of the HDMD algorithm for a variety of fluid flow simulations.
... Some turbulent flow problems have been analyzed based on the DMD method. [26][27][28] For example, Schmid 29 decomposed the experimental data of a slow jet entering a quiescent fluid to extract the jet flow dynamics. Liu and Zhang 30 analyzed the dynamic characteristics of the separated flow around a finite blunt plate based on the DMD modes and the corresponding energy spectra. ...
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To understand the flow dynamic characteristics of a centrifugal compressor, the dynamic mode decomposition (DMD) method is introduced to decompose the complex three-dimensional flow field. Three operating conditions, peak efficiency (OP1), peak pressure ratio (OP2), and small mass flow rate (near stall, OP3) conditions, are analyzed. First, the physical interpretations of main dynamic modes at OP1 are identified. As a result, the dynamic structures captured by DMD method are closely associated with the flow characteristics. In detail, the BPF/2BPF (blade passing frequency) corresponds to the impeller–diffuser interaction, the rotor frequency (RF) represents the tip leakage flow (TLF) from leading edge, and the 4RF is related to the interaction among the downstream TLF, the secondary flow, and the wake vortex. Then, the evolution of the dynamic structures is discussed when the compressor mass flow rate consistently declines. In the impeller, the tip leakage vortex near leading edge gradually breaks down due to the high backpressure, resulting in multi-frequency vortices. The broken vortices further propagate downstream along streamwise direction and then interact with the flow structures of 4RF. As a result, the 8RF mode can be observed in the whole impeller, this mode is transformed from upstream RF and 4RF modes, respectively. On the other hand, the broken vortices show broadband peak spectrum, which is correlated to the stall inception. Therefore, the sudden boost of energy ratio of 14RF mode could be regarded as a type of earlier signal for compressor instability. In the diffuser, the flow structures are affected by the perturbation from the impeller. However, the flow in diffuser is more stable than that in impeller at OP1–OP3, since the leading modes are stable patterns of BPF/2BPF.
... Dynamic mode decomposition (DMD) [10] is one of the most commonly used modal analysis methods along with proper orthogonal decomposition (POD) [11]. DMD has been applied in various studies [2,[12][13][14][15][16] due to its advantages in extracting both spatial modes and their associated temporal behavior. ...
Preprint
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In this study, we proposed the truncated total least squares dynamic mode decomposition (T-TLS DMD) algorithm, which can perform DMD analysis of noisy data. By adding truncation regularization to the conventional TLS DMD algorithm, T-TLS DMD improves the stability of the computation while maintaining the accuracy of TLS DMD. The effectiveness of the proposed method was evaluated by the analysis of the wake behind a cylinder and pressure-sensitive paint (PSP) data for the buffet cell phenomenon. The results showed the importance of regularization in the DMD algorithm. With respect to the eigenvalues, T-TLS DMD was less affected by noise, and accurate eigenvalues could be obtained stably, whereas the eigenvalues of TLS and subspace DMD varied greatly due to noise. It was also observed that the eigenvalues of the standard and exact DMD had the problem of shifting to the damping side, as reported in previous studies. With respect to eigenvectors, T-TLS and exact DMD captured the characteristic flow patterns clearly even in the presence of noise, whereas TLS and subspace DMD were not able to capture them clearly due to noise.
... DMD was initially employed as a spectral decomposition method for complex fluid flows [37]. More recently, it has proved successful in a wide range of settings such as background/foreground separation in real-time video [38], characterization of dynamic stall [39], and analysis of the propagation of infectious diseases [40]. DMD hinges on the theory of Koopman operators [41], which allows to represent the flow of a nonlinear dynamical system via an infinite-dimensional linear operator on the space of measurement functions. ...
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High-resolution simulations of particle-based kinetic plasma models typically require a high number of particles and thus often become computationally intractable. This is exacerbated in multi-query simulations, where the problem depends on a set of parameters. In this work, we derive reduced-order models for the semi-discrete Hamiltonian system resulting from a geometric particle-in-cell approximation of the parametric Vlasov-Poisson equations. Since the problem's non-dissipative and highly nonlinear nature makes it reducible only locally in time, we adopt a nonlinear reduced basis approach where the reduced phase space evolves in time. This strategy allows a significant reduction in the number of simulated particles, but the evaluation of the nonlinear operators associated with the Vlasov-Poisson coupling remains computationally expensive. We propose a novel reduction of the nonlinear terms that combines adaptive parameter sampling and hyper-reduction techniques to address this. The proposed approach allows decoupling the operations having a cost dependent on the number of particles from those that depend on the instances of the required parameters. In particular, in each time step, the electric potential is approximated via dynamic mode decomposition (DMD) and the particle-to-grid map via a discrete empirical interpolation method (DEIM). These approximations are constructed from data obtained from a past temporal window at a few selected values of the parameters to guarantee a computationally efficient adaptation. The resulting DMD-DEIM reduced dynamical system retains the Hamiltonian structure of the full model, provides good approximations of the solution, and can be solved at a reduced computational cost.
... The eigendecomposition of the Koopman operator can provide an in-depth insight into the system behavior. The applications of Kooman theoretic formalism can be classified into three major categories, (i) system identification, (ii) state estimation and prediction, (iii) control [27], [28]. ...
Preprint
This paper explores a novel data-driven approach based on recent developments in Koopman operator theory and dynamic mode decomposition (DMD) for modeling signalized intersections. Vehicular flow and queue formation on signalized intersections have complex nonlinear dynamics, making system identification, modeling, and controller design tasks challenging. We employ a Koopman theoretic approach to transform the original nonlinear dynamics into locally linear infinite-dimensional dynamics. The data-driven approach relies entirely on spatio-temporal snapshots of the traffic data. We investigate several key aspects of the approach and provide insights into the usage of DMD-type algorithms for application in adaptive signalized intersections. To demonstrate the utility of the obtained linearized dynamics, we perform prediction of the queue lengths at the intersection; and compare the results with the state-of-the-art long short term memory (LSTM) method. The case study involves the morning peak vehicle movements and queue lengths at two Orlando area signalized intersections. It is observed that DMD-based algorithms are able to capture complex dynamics with a linear approximation to a reasonable extent.
... This technique has been applied on a variety of fluid mechanic problems, and its advantages over the POD method have been illustrated. [23][24][25][26] Muld et al. 27 showed that the advection of flow structures, represented by two POD modes, can be summarized in a single DMD mode. However, some researchers expressed their doubt on whether this method is suitable for the extraction of coherent structures in highly turbulent flows. ...
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Surface pressure measurement is a general tool for evaluating wind flow qualitatively and quantitatively. Due to its complex temporal and spatial features, modal analysis is an interesting tool to be used for interpretation and discussion. The most common technique for modal representation is proper orthogonal decomposition (POD), also referred to as principal component analysis. However, it is believed that POD sometimes fails to extract meaningful features of the pressure field. To remove the non-physical POD modes and provide a closer physical description of the pressure field, an advanced method independent component analysis (ICA) is applied. Furthermore, these two methods are generalized in the frequency domain, called dynamic POD and dynamic ICA, to provide the temporal evolutions of coherent structures over the spatial domain. Modal analysis is used to isolate the different coherent structures in tornado-like vortices, e.g., wandering, vortex breakdown, and two-cell structure, and find the spectral characteristic of each phenomenon. Moreover, a comparison of modal analysis between the current paper and the previous paper on the velocity field {see Karami et al., [“Coherent structures in tornado-like vortices,” Phys. Fluids 31, 085118 (2019)]} presents new insight into the pressure–velocity correlation of the POD modes.
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Thesis
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Investigations into dynamic stall and dynamic stall control on airfoils are detailed using pitching airfoil experiments and numerical investigations at Mach 0.3, 0.4 and 0.5 on the airfoils EDI-M109, EDI-M112 and OA209. Two-dimensional dynamic stall was investigated, and the effects of the wind tunnel interference, rotation and the finite wing on the three-dimensional stall process were detailed. The curvature of the stall vortex and its effect in reducing the strength of the dynamic stall compared to a two-dimensional treatment was investigated using CFD and experiments with high-speed pressure sensitive paint (PSP) and pressure transducers. Stall control was designed based on its ability to increase the lift in CFD simulations of static stall and implemented by using constant and pulsed blowing with high-pressure air jets in the vertical direction, and the stall was demonstrated to be significantly reduced at all Mach numbers investigated. Optimal mass flux and jet spacing were found for Mach 0.3 and Mach 0.5, and depended on the test case investigated. Optimums for deep stall were around Cμ =0.12 for M=0.3 and Cμ =0.02 for M=0.5. Pulsed blowing was found to be at best as effective as constant blowing with the same mass flux, for the jet configuration and test cases investigated. Flow control by blowing reduced drag for separated flow, but the energy required in compressed air to achieve this was more than the savings in drag, and no cases were found in which flow control resulted in a reduction in total power used.
Conference Paper
View Video Presentation: https://doi.org/10.2514/6.2021-2520.vid We evaluate different approaches to characterize the onset of leading-edge type dynamic stall in pitching airfoils for incompressible flows. The first approach is by calculating the time variation of two flow parameters, namely, the Leading Edge Suction Parameter (LESP) and the Boundary Enstrophy Flux (BEF), both of which reach a critical value in the vicinity of stall onset. The alternate approaches include the use of Dynamic Mode Decomposition (DMD) and Wavelet Transform (WT) to identify the occurrence of critical flow states. Using wall-resolved LES results, we found that both LESP and BEF were effective in indicating stall onset, with the critical value of the BEF preceding that of the LESP. However, we were not able to identify any distinguishing behavior from DMD or WT that clearly indicates stall onset. DMD yielded unstable eigenvalues both within and outside of the stall onset regime. WT indicated the presence of energetic small-scale structures, whose time of incidence varied relative to the stall onset regime for different cases with no observable trend. The novel element in the current work is the use of CFD data with fine spatial and temporal resolution within the stall onset regime, to provide a composite picture of the stall onset process using different types of analyses.
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The evolution of vortex structures over flapping NACA0012 foils in shear flows and the corresponding aerodynamic performance are numerically studied using a two dimensional (2D) high-order accurate spectral difference Navier–Stokes flow solver, and further analyzed using the dynamic mode decomposition (DMD) method and vortex theory. Several types of vortex structures over pitching or plunging foils are simulated and analyzed to answer the following questions: (1) how mean flow shear affects the evolution of vortex structures, including both leading and trailing edge vortices, over flapping foils; and (2) how mean flow shear affects the aerodynamic performance under different kinematics. A temporal DMD method is used to analyze vortex structures. It is found that mean flow shear does not modify the dominant temporal frequencies in flow fields, but strong mean flow shear can significantly alter the growth rate, amplitude, and spatial patterns of coherent structures. From simulation results, it is observed that mean flow shear can affect evolution as well as interaction among leading and trailing edge vortices, thus altering the direction of wakes behind flapping foils. The mechanism of shear-induced deflective wakes is explained via qualitative analysis of evolution of simplified vortex street models. Finally, the effects of mean flow shear on aerodynamic performances of flapping foils with different kinematics are studied. By comparing the practical aerodynamic performances with those predicted by the steady aerodynamic theory, it is shown that flapping motion can significantly promote unsteady lift generation in mean flow shear. Furthermore, compared with flapping foils with positive mean angles of attack in a uniform incoming flow, the lift over flapping foils in flows with negative mean flow shear is enhanced without compromising thrust generation.
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Due to the damage caused by stall flutter, the investigation and modeling of the flow over a wind turbine airfoil at high angles of attack are essential. Dynamic mode decomposition (DMD) and dynamic mode decomposition with control (DMDc) are used to analyze unsteady flow and identify the intrinsic dynamics. The DMDc algorithm is found to have an identification problem when the spatial dimension of the training data is larger than the number of snapshots. IDMDc, a variant algorithm based on reduced dimension data, is introduced to identify the precise intrinsic dynamics. DMD, DMDc and IDMDc are all used to decompose the data for unsteady flow over the S809 airfoil that are obtained by numerical simulations. The DMD results show that the dominant feature of a static airfoil is the adjacent shedding vortices in the wake. For an oscillating airfoil, the DMDc results may fail to consider the effect of the input and have an identification problem. IDMDc can alleviate this problem. The dominant IDMDc modes show that the intrinsic flow for the oscillating case is similar to the unsteady flow over the static airfoil. Moreover, the input–output model identified by IDMDc can give better predictions for different oscillating cases than the identified DMDc model.
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The time-resolved velocity field (2D2C high-speed PIV) and surface pressure (pressure taps) of a stalled NACA0012 airfoil have been simultaneously measured. These measurements enable cross-evaluation of the velocity field and pressure time series. The present paper evaluates the flow field and surface pressure using POD, cross-correlation and conditional averaging. First, the flow and surface pressure are analysed independently. Both the time-average and the dynamic structures of the flow are presented. Distinct frequency bands are observed in the energy spectra of the surface-pressure signal. It is shown that a high-pressure event at the foremost pressure port (at x/c = 0.34) is followed by high-pressure events at the other pressure ports (x/c = 0.51-0.93), indicating a decaying pressure wave that travels over the surface with roughly half the free-stream velocity. Next, the link between the flow field and the surface pressure is explored. Using cross-correlation, it is shown that the pressure signal correlates with specific regions in the flow. Conditional averages of the flow fields, conditioned to high-pressure events at the surface, indicate that these pressure events are caused by coherent structures in the flow. These structures consist of a train of alternating vortices that induce velocity components toward and away from the surface, causing high-and low-pressure events at this surface. By filtering the pressure signal with different frequency bands, the flow structures responsible for different peaks in the energy spectra are recovered. A POD analysis indicates that the energy of the coherent flow structures leading to surface-pressure fluctuations only contain a fraction of the total energy of the flow. While POD and cross-correlation are applied to a full time-series, the conditional averages show that coherent structures in the flow can be identified in real-time using the surface pressure. Identification of coherent structures in the flow using only the surface pressure signal enables real-time control of such structures.
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Inlet guide vanes (IGVs) affect compressor performance by characterizing the upstream flow of the rotor blade row. After operating a long period, the IGV may suffer from blade deformation and consequently influence the downstream flow fields. The deformation of IGV always occurs locally, and the deformed structure can be regarded as a partially-flapped variable-camber IGV. In the current study, six typical configurations, with different span ranges of variable-camber IGVs, were selected. The impact on the compressor performance was discussed through numerical simulations. The dynamic mode decomposition (DMD) approach was applied over the computational domain for investigating the dynamic features. Results show that the stall margin and the total pressure rise of the compressor rotor are sensitive to the span range of the variable-camber IGV. For the unsteady feature in the flow field, the variation of IGV has a limited effect on the IGV itself while significantly changes the dynamic characteristics over the rotor domain. Compared to the case with conservative non-variable IGV, the extension in the flapped span yields a stall margin ranging from −4.1% to 8.1% as well as a monotone drop in the total pressure ratio. The mechanism behind the effect of IGV on the stall margin and the total pressure rise is dependent on the particular range of the modified IGV span.
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Aerodynamic shape refinement optimization for passenger aircraft is difficult and requires a significant workload. The adjoint-based gradient optimization method can quickly find local optimal solutions based on the initial shape in these types of problems. The optimization model of the common research model for the drag coefficient minimization and wing thickness constraints with a large-scale grid is established, and the drag coefficient is reduced by 10.2 counts while maintaining the lift coefficient. The stress-blended eddy simulation is used for unsteady simulations, and the optimized configuration can effectively eliminate oscillations in the middle of the upper wing surface. The spanwise flow is reduced and the pressure response on the wing surface is due primarily to shock chordal motion. For aerodynamic analyses with similar shapes, the dynamic mode decomposition (DMD) analysis shows that the upper wing surface mode amplitudes and spanwise instability modes of the optimized design are weaker, and the fluctuations of the pressure are more stable. Therefore, DMD is suitable for refined shape optimization analyses.
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In this study the pure effect of raindrops on dynamic stall of a pitching airfoil has been investigated. The simulation was performed at Reynolds number of $${10}^{6}$$ 10 6 with raindrop diameter equal to $${10}^{-5}\text{m}$$ 10 - 5 m . A couple of multiphase models based on Eulerian and Lagrangian frames of reference have been implemented to simulate the raindrops. In the first step the accuracy of each multiphase model has been appraised. As a result, the Lagrangian multiphase model, which is called Discrete Phase Model, has been proven to be of better accuracy. It has been concluded that in general raindrops has negative effects on the lift coefficient of the pitching airfoil. In addition, a lead in aerodynamic phenomena has been observed due to the presence of water drops. This phenomenon has also been observed in the formation and separation of Leading Edge and Trailing Edge vortices which come to existence in advance of the dry case. Finally, it has been illustrated that the main effect of raindrops is on the phase of force oscillation rather than the force amplitude.
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An experimental investigation of the non-normal nature of thermoacoustic interactions in an electrically heated horizontal Rijke tube is performed. Since non-normality and the associated transient growth are linear phenomena, the experiments have to be confined to the linear regime. The bifurcation diagram for the subcritical Hopf bifurcation into a limit cycle behavior has been determined, after which the amplitude levels, for which the system acts linearly, have been identified for different power inputs to the heater. There are two main objectives for this experimental investigation. The first one deals with the extraction of the linear eigenmodes associated with the acoustic pressure from experimental data. This is accomplished by the Dynamic Mode Decomposition (DMD) technique applied in the linear regime. The nonorthogonality between the eigenmodes is determined for various values of heater power. The second objective is to identify evidence of transient perturbation growth in the system. The total acoustic energy in the duct has been monitored as the thermoacoustic system evolves from its initial condition. Transient growth, on the order of previous theoretical studies, has been found, and its parameteric dependence on amplitude ratio and phase angle of the initial eigenmode components has been determined. This study represents the first experimental confirmation of non-normality in thermoacoustic systems.
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The flow in a cylindrical cavity driven by a rotating lid undergoes a sequence of bifurcations as the Reynolds number is increased. It is a well-known and well-studied fluid system and a common benchmark for the identification of emerging coherent structures and for the quantification of bifurcation points. Time-resolved particle-image velocimetry (PIV) data have been taken in a cross-sectional plane, and a sequence of snapshots has then been processed by two algorithms: the Dynamic Mode Decomposition (DMD) and the Proper Orthogonal Decomposition (POD).
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For feedback control using low-dimensional proper orthogonal decomposition (POD) models, the mode amplitudes of the POD mode coefficients need to be estimated based on sensor readings. This paper is aimed at suppressing the von Kairman vortex street in the wake of a circular cylinder using a low-dimensional approach based on POD. We compare sensor placement methods based on the spatial distribution of the POD modes to arbitrary ad hoc methods. Flow field data were obtained from Navier-Stokes simulation as well as particle image velocimetry (PIV) measurements. A low-dimensional POD was applied to the snapshot ensembles from the experiment and simulation. Linear stochastic estimation was used to map the sensor readings of the velocity field on the POD mode coefficients. We studied 53 sensor placement configurations, 32 of which were based on POD eigenfunctions and the others using ad hoc methods. The effectiveness of the sensor configurations was investigated at Re = 100 for the computational fluid dynamic data, and for a Reynolds number range of 82-99 for the water tunnel PIV data. Results show that a five-sensor configuration can keep the root mean square estimation error, for the amplitudes of the first two modes to within 4% for simulation data and within 10% for the PIV data. This level of error is acceptable for a moderately robust controller The POD-based design was found to be simpler. more effective, and robust compared to the ad hoc methods examined.
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We demonstrate a three-step method for estimating time-resolved velocity fields using time-resolved point measurements and non-time-resolved particle image velocimetry data. A variant of linear stochastic estimation is used to obtain an initial set of time-resolved estimates of the flow field. These estimates are then used to identify a linear model of the flow dynamics. The model is incorporated into a Kalman smoother, which provides an improved set of estimates. We verify this method with an experimental study of the wake behind an elliptical-leading-edge flat plate at a thickness Reynolds number of 3,600. We find that, for this particular flow, the Kalman smoother estimates are more accurate and more robust to noise than the initial, stochastic estimates. Consequently, dynamic mode decomposition more accurately identifies coherent structures in the flow when applied to the Kalman smoother estimates. Causal implementations of the estimators, which are necessary for flow control, are also investigated. Similar outcomes are observed, with model-based estimation outperforming stochastic estimation, though the advantages are less pronounced.
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An experimental study has been conducted on a transitional water jet at a Reynolds number of Re = 5,000. Flow fields have been obtained by means of time-resolved tomographic particle image velocimetry capturing all relevant spatial and temporal scales. The measured three-dimensional flow fields have then been postprocessed by the dynamic mode decomposition which identifies coherent structures that contribute significantly to the dynamics of the jet. Both temporal and spatial analyses have been performed. Where the jet exhibits a primary axisymmetric instability followed by a pairing of the vortex rings, dominant dynamic modes have been extracted together with their amplitude distribution. These modes represent a basis for the low-dimensional description of the dominant flow features.
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The extraction of dynamically relevant structures from time-resolved flow data has commonly be restricted to numerically generated flow fields. Equivalent structures, however, could not be obtained from experimental measurements, since the commonly used mathematical techniques required the explicit or implicit availability of an underlying model equation. A numerical scheme based on a Krylov subspace method for the extraction of dynamic modes directly from flow fields --- without the need to resort to a model equation --- will be introduced. This technique can be applied equally to numerically generated or experimental data and thus provides a means to decompose time-resolved measurements into dynamically dominant structures. The treatment of subdomains, spatially evolving flows, PIV data and simple flow visualizations will be demonstrated; a connection to the proper orthogonal decomposition (POD) technique, which is a byproduct of the dynamic mode decomposition, will be pointed out.
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A parametric study was undertaken to investigate the effect of periodic excitation (with zero net mass flux) on a NACA 0015 airfoil undergoing pitch oscillations at rotorcraft reduced frequencies under incompressible conditions. The primary objective of the study was to maximize airfoil performance while limiting moment excursions to typical prestalled conditions. The incidence angle excursions were limited to +/-5 deg, and a wide range of reduced excitation frequencies and amplitudes were considered for 0.3 x 10(6) less than or equal to Re less than or equal to 0.9 x 10(6) with various flap deflections and excitation locations. Significant increases in maximum lift and reductions in drag were attained while containing the moment excursions, Oscillatory excitation was found to be far superior to steady blowing, which was even detrimental under certain conditions, and flap-shoulder excitation was found to be superior to leading-edge excitation.
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Turbulentfluidhasoftenbeenconceptualizedasatransientthermodynamicphase. Here, a finite-time thermodynamics (FTT) formalism is proposed to compute mean flow and fluctuation levels of unsteady incompressible flows. The pro- posed formalism builds upon the Galerkin model framework, which simplifies a continuum 3D fluid motion into a finite-dimensional phase-space dynamics and, subsequently, into a thermodynamics energy problem. The Galerkin model consists of a velocity field expansion in terms of flow configuration dependent modes and of a dynamical system describing the temporal evolution of the mode coefficients. Each mode is treated as one thermodynamic degree of freedom, characterized by an energy level. The dynamical system approaches local ther- mal equilibrium (LTE) where each mode has the same energy if it is governed only by internal (triadic) mode interactions. However, in the generic case of un- steady flows, the full system approaches only partial LTE with unequal energy levels due to strongly mode-dependent external interactions. The FTT model is
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The dynamic stall process of a NACA 0012 airfoil undergoing a constant-rate pitching up motion is studied experimentally in a water towing tank facility. This study focuses on the detailed measurement of the unsteady separated flow in the vicinity of the leading and trailing edges of the airfoil. The measurements are carried out using the Particle Image Velocimity (PIV) technique. This technique provides the two-dimensional velocity and associated vorticity fields, at various instants in time, in the mid-span of the airfoil. Near the leading edge, large vortical structures emerge as a consequence of Van Dommelen and Shen type separation and a local vorticity accumulation. The interaction of these vortices with the reversing boundary layer vorticity initiates a secondary flow separation and the formation of a secondary vortex. The mutual induction of this counter-rotating vortex pair eventually leads to the ejection process of the dynamic stall vortex from the leading edge region.
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We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an infinite-dimensional linear operator associated with the full nonlinear system. These modes, referred to as Koopman modes, are associated with a particular observable, and may be determined directly from data (either numerical or experimental) using a variant of a standard Arnoldi method. They have an associated temporal frequency and growth rate and may be viewed as a nonlinear generalization of global eigenmodes of a linearized system. They provide an alternative to proper orthogonal decomposition, and in the case of periodic data the Koopman modes reduce to a discrete temporal Fourier transform. The Arnoldi method used for computations is identical to the dynamic mode decomposition recently proposed by Schmid & Sesterhenn (Sixty-First Annual Meeting of the APS Division of Fluid Dynamics, 2008), so dynamic mode decomposition can be thought of as an algorithm for finding Koopman modes. We illustrate the method on an example of a jet in crossflow, and show that the method captures the dominant frequencies and elucidates the associated spatial structures.
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The unsteady flow field above a NACA 0012 airfoil pitching under deep dynamic stall conditions has been investigated in a low-speed wind tunnel by means of particle image velocimetry. The measurements of the instantaneous flow velocity field show the characteristic features of the dynamic stall process: formation and development of an organized vortex structure for increasing incidences and the subsequent separation. Vorticity and divergence estimated from the measured data give a good insight into the complex flow behaviour during the downstroke motion. Furthermore, small-scale structures could be observed in the separated flow field and even within the dynamic stall vortex.
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The dynamic mode decomposition (DMD) is a data-decomposition technique that allows the extraction of dynamically relevant flow features from time-resolved experimental (or numerical) data. It is based on a sequence of snapshots from measurements that are subsequently processed by an iterative Krylov technique. The eigenvalues and eigenvectors of a low-dimensional representation of an approximate inter-snapshot map then produce flow information that describes the dynamic processes contained in the data sequence. This decomposition technique applies equally to particle-image velocimetry data and image-based flow visualizations and is demonstrated on data from a numerical simulation of a flame based on a variable-density jet and on experimental data from a laminar axisymmetric water jet. In both cases, the dominant frequencies are detected and the associated spatial structures are identified.
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The decomposition of experimental data into dynamic modes using a data-based algorithm is applied to Schlieren snapshots of a helium jet and to time-resolved PIV-measurements of an unforced and harmonically forced jet. The algorithm relies on the reconstruction of a low-dimensional inter-snapshot map from the available flow field data. The spectral decomposition of this map results in an eigenvalue and eigenvector representation (referred to as dynamic modes) of the underlying fluid behavior contained in the processed flow fields. This dynamic mode decomposition allows the breakdown of a fluid process into dynamically revelant and coherent structures and thus aids in the characterization and quantification of physical mechanisms in fluid flow. KeywordsDynamic mode decomposition–Arnoldi method–Iterative techniques–Experimental fluid dynamics
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Passive low-aspect ratio cylinders mounted near the leading edge of an airfoil have been found to significantly improve the performance under dynamic stall conditions. In order to clarify the principle of operation these cylindrical disturbance generators (DG), high speed PIV and simultaneous pressure measurements have been performed on the pitching rotary aircraft wing profile OA209. Time resolved information for the flow field at mid chord and the pressure distribution was taken. In addition to the best configuration found in previous experiments, various geometries and sizes of these devices have been investigated. The experiments showed the effectiveness of the disturbance generators in reducing the negative pitching moment peak and hysteresis effects. Moreover, the development of stall compared to the clean case has been investigated and it was found that the disturbance generators reduce the strength and the distance normal to the chord of the vortices.
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The paper will discuss the latest results of a comprehensive validation activity on the numerical methods TAU and elsA, conducted with respect to dynamic stall applications. The work has been performed in the joint German/French project SIMCOS (Advanced Simulation and Control of Dynamic Stall) in the frame of the DLR/ONERA cooperation on helicopter technologies. The validation was conducted by two-dimensional unsteady RANS simulations on two dynamic stall test cases for the rotor blade airfoil OA209. Test case DS1 measured in the French ONERA-F2 wind tunnel represents a low speed deep dynamic stall case at M = 0.16, Re = 1.8e6, α = 13° ± 5°, and ω* = 0.1 (ω* = 2πfc/v<sub>∞</sub>). Test case DS2 measured in the German DNW-TWG wind tunnel represents a high speed deep dynamic stall case at M = 0.31, Re = 1.16e6, α = 13° ± 7°, and ω* = 0.1. The numerical methods used were the ONERA in-house flow solver elsA and the DLR in-house flow solver TAU. While the elsA code is a structured solver, the TAU code is an unstructured solver using hybrid grids. Simulations were performed with respect to the investigation of the influence of temporal resolution, grid characteristics, turbulence modeling, and transition prediction on the results of the codes. The results of the individual investigations will be discussed and the numerical methods, as well as turbulence and transition modeling from TAU and elsA will be compared.
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Dynamic stall on a helicopter rotor blade comprises a series of complex aerodynamic phenomena in response to the unsteady change of the blade’s angle of attack. It is accompanied by a lift overshoot and delayed massive flow separation with respect to static stall. The classical hallmark of the dynamic stall phenomenon is the dynamic stall vortex. The flow over an oscillating OA209 airfoil under dynamic stall conditions was investigated by means of unsteady surface pressure measurements and time–resolved particle image velocimetry. The characteristic features of the unsteady flow field were identified and analysed utilising different coherent structure identification methods. An Eulerian and a Lagrangian procedure were adopted to locate the axes of vortices and the edges of Lagrangian coherent structures, respectively; a proper orthogonal decomposition of the velocity field revealed the energetically dominant coherent flow patterns and their temporal evolution. Based on the complementary information obtained by these methods the dynamics and interaction of vortical structures were analysed within a single dynamic stall life cycle leading to a classification of the unsteady flow development into five successive stages: the attached flow stage; the stall development stage; stall onset; the stalled stage; and flow reattachment. The onset of dynamic stall was specified here based on a characteristic mode of the proper orthogonal decomposition of the velocity field. Variations in the flow field topology that accompany the stall onset were verified by the Lagrangian coherent structure analysis. Furthermore, the orientation of the trajectories of vortices that originated at the very leading edge shortly before and after stall onset was observed to be altered. The mechanism that results in the detachment of the dynamic stall vortex from the airfoil was identified as vortex induced separation caused by strong viscous interactions. Finally, a revised criterion to discern between light and deep dynamic stall was formulated.
Conference Paper
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Chapter
The stalling of helicopter rotor blades occurs on the retreating side of the rotor disk. The lower velocity on the retreating side (typically 0.3 to 0.5 Mach number) is compensated for by higher lift coefficients, which require higher angles of attack. When stall occurs, the blade dynamic and elastic properties become important in determining the subsequent changes in angle of attack. The aerodynamics of dynamic stall govern the aeroelastic response and can lead to a dynamic response known as stall flutter. In describing the fluid mechanics involved in dynamic stall, the blade motions must be taken into account. The three basic structural modes that are excited by dynamic stall are the blade torsional mode, the bending (normal to the chord) mode, and the flapping mode. The excitation of these three modes is illustrated in Fig. 1 from Crimi’s aeroelastic analysis of a two-dimensional section of a helicopter blade in [5.83] and [5.84]. The three basic structural modes of motion, at three different frequencies, contribute to the angle of attack. The mixture of modes causes the sequence of stall and unstall occurrences to be at irregular time intervals. The structural response most important to the angle of attack is the pitching-angle displacement. The sharp spikes in the aerodynamic pitching moment act as a series of impulses to cause pitching oscillations.
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The term dynamic stall refers to unsteady flow separation occurring on aerodynamic bodies, such as airfoils and wings, which execute an unsteady motion. The prediction of dynamic stall is important for flight vehicle, turbomachinery, and wind turbine applications. Due to the complicated flow physics of the dynamic stall phenomenon the industry has been forced to use empirical methods for its prediction. However, recent progress in computational methods and the tremendous increase in computing power has made possible the use of the full fluid dynamic governing equations for dynamic stall investigation and prediction in the design process. It is the objective of this review to present the major approaches and results obtained in recent years and to point out existing deficiencies and possibilities for improvements. To this end, potential flow, boundary layer, viscous–inviscid interaction, and Navier–Stokes methods are described. The most commonly used numerical schemes for their solution are briefly described. Turbulence models used for the computation of high Reynolds number turbulent flows, which are of primary interest to industry, are presented. The impact of transition from laminar to turbulent flow on the dynamic stall phenomenon is discussed and currently available methods for its prediction are summarized. The main computational results obtained for airfoil and wing dynamic stall and comparisons with available experimental measurements are presented. The review concludes with a discussion of existing deficiencies and possibilities for future improvements.
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A mathematical model of an unsteady separated flow around an oscillating airfoil is considered. This model is based on a viscid-inviscid approach. The points of separation and the intensity of vorticity displaced into the external flow are determined using boundary-layer equations in an integral form. Dynamic stall on an oscillating airfoil is studied. The mechanism and nature of antidamping are discovered.
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INSECTS cannot fly, according to the conventional laws of aerodynamics: during flapping flight, their wings produce more lift than during steady motion at the same velocities and angles of attack1–5. Measured instantaneous lift forces also show qualitative and quantitative disagreement with the forces predicted by conventional aerodynamic theories6–9. The importance of high-life aerodynamic mechanisms is now widely recognized but, except for the specialized fling mechanism used by some insect species1,10–13, the source of extra lift remains unknown. We have now visualized the airflow around the wings of the hawkmoth Manduca sexta and a 'hovering' large mechanical model—the flapper. An intense leading-edge vortex was found on the down-stroke, of sufficient strength to explain the high-lift forces. The vortex is created by dynamic stall, and not by the rotational lift mechanisms that have been postulated for insect flight14–16. The vortex spirals out towards the wingtip with a spanwise velocity comparable to the flapping velocity. The three-dimensional flow is similar to the conical leading-edge vortex found on delta wings, with the spanwise flow stabilizing the vortex.
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An experimental investigation was conducted to examine the dynamic stall characteristics of a NACA 23012 aerofoil section at a Reynolds number of 1.5 million. Time-dependent data were obtained from thirty miniature pressure transducers and three hot film gauges situated at the mid-span of the wing. The static stall mechanism of the NACA 23012 was determined to be via abrupt upstream movement of trailing edge separation. Under dynamic conditions, stall was found to occur via leading edge separation, followed by a strong suction wave that moved across the aerofoil. This suction wave is characteristic of a strong moving vortex disturbance. Evidence of strong secondary vortex shedding was also found to occur, and this appears symptomatic of dynamic stall only at low Mach numbers. Some evidence of flow reversals over the trailing edge of the aerofoil were indicated prior to the development of leading edge separation and dynamic stall.
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A joint comprehensive validation activity on the structured numerical method elsA and the hybrid numerical method TAU was conducted with respect to dynamic stall applications. In order to improve two-dimensional prediction, the influence of several factors on the dynamic stall prediction were investigated. The validation was performed for three deep dynamic stall test cases of the rotor blade airfoil OA209 against experimental data from two-dimensional pitching airfoil experiments, covering low speed and high speed conditions. The requirements for spatial discretization and for temporal resolution in elsA and TAU are shown. The impact of turbulence modeling is discussed for a variety of turbulence models ranging from one-equation Spalart-Allmaras-type models to state-of-the-art seven-equation Reynolds stress models. The influence of the prediction of laminar/turbulent boundary layer transition on the numerical dynamic stall simulation is described. Results of both numerical methods are compared to allow conclusions to be drawn with respect to an improved prediction of dynamic stall.
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The effectiveness of a sensor configuration for feedback flow control on the wake of a circular cylinder is investigated in both direct numerical simulation as well as in a water tunnel experiment. The research program is aimed at suppressing the von Kármán vortex street in the wake of a cylinder at a Reynolds number of 100. The design of sensor number and placement was based on data from a laminar two-dimensional simulation of the Navier–Stokes equations for the unforced condition. A low-dimensional proper orthogonal decomposition (POD) was applied to the vorticity calculated from the flow field and sensor placement was based on the intensity of the resulting spatial eigenfunctions. The numerically generated data was comprised of 70 snapshots taken over three cycles from the steady state regime. A linear stochastic estimator (LSE) was employed to map the velocity data to the temporal coefficients of the reduced order model. The capability of the sensor configuration to provide accurate estimates of the four low-dimensional states was validated experimentally in a water tunnel at a Reynolds number of 108. For the experimental wake, a sample of 200 particle image velocimetry (PIV) measurements was used. Results show that for experimental data, the root mean square estimation error of the estimates of the first two modes was within 6% of the desired values and for the next two modes was within 20% of the desired values. This level of error is acceptable for a moderately robust controller.
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Multi-POD is a new proper orthogonal decomposition (POD) based reduced order modeling (ROM) technique for modeling flows on deforming grids. Presented is the application of multi-POD to flow about a pitching and plunging airfoil. The multi-POD technique expands the parameter space in which POD is applied through selection of the best available ROM for a given set of grid deformations. For application to unconstrained pitching and plunging motion of an airfoil, multi-POD is shown to be effective when trained for forced grid motion, reducing the training requirements significantly. A three-orders of magnitude reduction in the number of degrees of freedom is also shown in the use of POD/ROM for the aeroelastic problem.
Article
Dynamic stall and unsteady boundary layer separation have been studied in incompressible flow at moderately large Reynolds numbers. By varying the leading-edge geometry of an NACA 0012 airfoil, three different types of stall were produced. For most of the configurations studied, including the basic NACA 0012 profile, dynamic stall was found not to originate with the bursting of a leading-edge laminar separation bubble, as is commonly believed. Instead, the vortex shedding phenomenon, which is the predominant feature of dynamic stall, appears to be fed its vorticity by the breakdown of the turbulent boundary layer.
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The results of an investigation of boundary layers close to the stagnation point of an oscillating airfoil are reported. Two procedures for generating initial conditions, the characteristics box scheme and a quasi-static approach, were investigated, and the quasi-static approach was shown to be appropriate provided the initial region was far from any flow separation. With initial conditions generated in this way, the unsteady boundary layer equations were solved for the flow in the leading edge region of a NACA 0012 airfoil oscillating from 0 to 5 deg. Results were obtained for both laminar and turbulent flow, and, in the latter case, the effect of transition was assessed by specifying its occurrence at different locations. The results demonstrate the validity of the numerical scheme and suggest that the procedures should be applied to calculation of the entire flow around oscillating airfoils.
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A comprehensive review of research in dynamic stall is presented with applications for helicopters, fighter aircraft, and wind turbines illustrated and evaluated. Emphasis is placed on review of research contributing to better understanding of the dynamic stall mechanism, including influence of type of motion, Mach number, Reynolds number, and three-dimensional effects.
Article
Phenomena that control the flow during the stall portion of a dynamic stall cycle are analyzed, and their effect on blade motion is outlined. Four mechanisms by which dynamic stall may be initiated are identified: (1) bursting of the separation bubble, (2) flow reversal in the turbulent boundary layer on the airfoil upper surface, (3) shock wave-boundary layer interaction behind the airfoil crest, and (4) acoustic wave propagation below the airfoil. The fluid mechanics that contribute to the identified flow phenomena are summarized, and the usefulness of a model that incorporates the required fluid mechanics mechanisms is discussed.
Article
Dynamic stall is a term often used to describe the complex series of events that result in the dynamic delay of stall, on airfoils and wings experiencing unsteady motion, to angles sigificantly beyond the static-stall angle. This delay of stall, usually followed by large excursions in lift and pitching moment, has challenged aerodynamicists for many years. This paper presents some history of dynamic-stall research and a detailed description of the events associated with dynamic stall. The subjects discussed include: dynamic-stall motions, Reynonlds number effects, 3D effects, dynamic-stall flow and load models and control of dynamic stall.
Article
The effects of blade and root-flexure elasticity and dynamic stall on the stability of hingeless rotor blades are investigated. The dynamic stall description is based on the ONERA models of lift, drag, and pitching moment. The structural analysis is based on three blade models that range from a rigid flap-lag model to two elastic flap-lag-torsion models, which differ in representing root-flexure elasticity. The predictions are correlated with the measured lag damping of an experimental isolated three-blade rotor; the correlation covers rotor operations from near-zero-thrust conditions in hover to highly stalled, high-thrust conditions in foward flight. That correlation shows sensitivity of lag-damping predictions to structural refinements in blade and root-flexure modeling. Moreover, this sensitivity increases with increasing control pitch angle and advance ratio. For high-advance-ratio and high-thrust conditions, inclusion of dynamic stall generally improves the correlation.
Article
this paper (Part I) we discuss this process. In the companion paper (Part II), we describe the application of the resulting low order models to the control problem. The test case we choose is the two-dimensional flow past a circular cylinder, with control action achieved via cylinder rotation. For Reynolds numbers above 50 (approximately) this example exhibits vortex shedding, and is generally regarded as a canonical unsteady, separated flow. We shall consider the case with Reynolds number 100 where, in the light of the above discussion, a single degree-of-freedom control action cannot be expected to suppress the shedding entirely, and we must instead concentrate on ameliorating its effects. The flow data used to generate the low order model are provided by a numerical simulation, described in Sec. 2. The application of the POD to this data is then discussed in Sec. 3. In order to impose control action boundary conditions two extensions to the basis function derivation/Galerkin projection procedure are proposed in Sec. 4. The range of applicability of the resulting low order model is found to depend critically on the ability of the basis functions to describe sufficiently general solutions. The generation of suitable basis functions is addressed in Sec. 5, and the corresponding low order models are assessed by means of two "open-loop" test cases. The implications of the results here are then discussed in the light of the control to be implemented in Part II. 2. Numerical Simulation of the Cylinder Flow
Analysis of the Development of Dynamic Stall Based on Oscillating Airfoil Experiments
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Increase in the Maximum Lift of an Airplane Wing Due to a Sudden Increase in the Effective Angle of Attack Resulting From a Gust
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14 Variation of E DMD av and E POD av between the phase-averaged and reconstructed velocity field with the number of DMD and POD modes used for the reconstruction
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Fig. 14 Variation of E DMD av and E POD av between the phase-averaged and reconstructed velocity field with the number of DMD and POD modes used for the reconstruction. 2438 MARIAPPAN ET AL.
14 Variation of E DMD av and E POD av between the phase-averaged and reconstructed velocity field with the number of DMD and POD modes used for the reconstruction Fluid Mechanic Mechanisms in the Stall Process of Airfoils for Helicopters
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Fig. 14 Variation of E DMD av and E POD av between the phase-averaged and reconstructed velocity field with the number of DMD and POD modes used for the reconstruction. [5] Young, W. H. J., " Fluid Mechanic Mechanisms in the Stall Process of Airfoils for Helicopters, " NASA Technical Memorandum TR-81956, 1981.
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Development of the DLR TAU-Code for Aerospace Applications
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