Online Dynamic Mode Decomposition Methods for the Investigation of Unsteady Aerodynamics of the DrivAer Model (Second Report)

Transcription

1 Research Paper Online Dynamic Mode Decomposition Methods for the Investigation of Unsteady Aerodynamics of the DrivAer Model (Second Report) - Application on Velocity Fields - Marco Kiewat 1) Daiki Matsumoto 1) Lukas Haag 1) Vincent Zander 2) Thomas Indinger 1) 1) Technical University of Munich, Chair of Aerodynamics and Fluid Mechanics Garching, Germany ( marco.kiewat@tum.de) 2) AUDI AG Ingolstadt, Germany Received on July 6, 2017 ABSTRACT: Temporally resolved flow fields are commonly averaged in time, and mostly the time-averaged flow fields and forces are used for the aerodynamic optimization of road vehicles. Online DMD is found to be well suited for studying transient flow effects and leads to a deeper understanding of the complex flow around the vehicle. The investigated velocity field is computed by a Detached Eddy Simulation of the DrivAer reference body. The CFD setup and key considerations for the application of online DMD on large data sets are outlined, and the most dominant extracted coherent flow structures are analyzed independently. KEY WORDS: heat fluid, computational fluid dynamics, aerodynamic performance, Dynamic Mode Decomposition [D1] 1. Introduction Transient numerical simulations are becoming the industry standard for the optimization of vehicle aerodynamics. Detached Eddy Simulation (DES) has been shown to provide exceptional accuracy when compared with the Reynolds-averaged approaches utilized by Islam et al. (1). Even though the computational cost can be an order of magnitude higher, the tuning of simulation parameters and distributed computations leads to moderate turnaround times. Results from a DES can resolve the flow field in much greater detail and accuracy than commonly used RANS based methods (1). The analysis of CFD results for aerodynamic optimization is usually focused on the temporally averaged flow field and other statistical mean quantities. Visualizations of averaged detachment lines, recirculation regions and pressure fields can be helpful tools to study the effect of a change in geometry on the overall aerodynamic objective quantities, such as the drag and lift coefficient. Studying the temporal evolution of the flow field is usually not feasible due to the complex nature of flow phenomena at high Reynolds numbers and complex geometries. Too many flow field effects overlap in time and it is hard to distinguish dominant effects. Dynamic Mode Decomposition (DMD), first introduced by Schmid in 2010 (2), offers a way of decomposing the flow field into modes of distinct frequency. This method can be applied as a post-processing algorithm and is also referred to as conventional DMD. Using this modal decomposition technique, flow field perturbations at a specific frequency can be identified as coherent spatial structures. By reconstructing modes in time, those structures can be tracked through the flow field and their temporal evolution can be investigated independently. This can be used to draw a strong connection between the geometry and its effects on the temporally averaged flow field. DMD has been gaining popularity among researchers since its introduction in 2010, and many extensions and variants of DMD have been introduced in order to increase its applicability for certain objectives. For vehicle aerodynamics, DMD has been successfully applied to examine CFD data with various objectives. Peichl et al. (3) presents the application of conventional DMD on flow field data from a Detached Eddy Simulation of the DrivAer reference body. They suggest preprocessing the flow field data, which filters out numerical noise and high frequency components, in order to generate a Reduced Order Model (ROM). While they are able to link DMD modes to flow phenomena as the movement of wake vortices and the vortex shedding from the rear wheels, they could not identify a clearly dominant frequency. They also show the limited applicability of this conventional DMD approach (4) for industrial applications. Frank et al. make use of conventional DMD for studying the aeroacoustic behavior of an isolated side mirror. They identify a feedback loop to be responsible for a tonal noise source by focusing their analysis on amplified modes. Matsumoto et al. (5) apply conventional DMD to the surface force vector data of a DES of the DrivAer body. They analyze the most dominant mode for two geometry configurations using a streaming variant of DMD. Matsumoto et al. extend their surface forces study (6), using the recently introduced Incremental Total DMD published by Matsumoto et al. (7). In addition to CFD data, DMD has been found useful for application on time series data from PIV measurements (8,9,10). The second part of this paper outlines the modal analysis method employed and provides a strategy for the proper selection of modes for the creation of a ROM. Important criteria for the selection of the correct sampling interval and required overall timespan are demonstrated using the correlation function and a residual study. Key aspects of the simulation setup are presented and the data reduction by spatial interpolation is discussed. The 72

2 results section shows the DMD results via mode spectrum, eigenvalue and mode distribution plots. Modes of vortex shedding, and locally fixed recirculation structures are identified Simulation Setup 2. Methods The CFD simulations are performed using DDES turbulence modelling with OpenFOAM (11). The turbulent length scale and the turbulent viscosity in the near wall regions are evaluated using the Spalart-Allmaras RANS turbulence model. In cells away from the wall, scale resolving LES turbulence modelling is employed using the Smagorinsky sub-grid scale viscosity model. The exceptional accuracy of this method compared to pure, RANSbased vehicle aerodynamics simulations is shown by Islam et al. (1). Spatial discretization is implemented on a finite volume computational domain of 157 million hexahedral cells. In order to match the experimental wind tunnel test results, the computational domain includes main features of the large model scale wind tunnel A (WTA) at the Technical University of Munich, as depicted in Fig. 1. The wind tunnel nozzle is explicitly modelled in the simulation to capture the effects of nozzle blockage, shear layer instabilities and jet expansion. Nozzle effects influence the effective velocity around the vehicle and have been found to be of crucial importance when comparing CFD simulations with wind tunnel measurements by Collin et al. (12). Furthermore, the simulated geometry includes the single belt rolling road system, model support via top sting as well as wheel arms. The vehicle geometry is the DrivAer geometry in notchback configuration, which includes the engine bay flow and structured underbody, as investigated by Haag et al. (13). The model geometry has a wheel base length of L = m and the flow velocity is U = 45 m s Spatial Data Reduction As previously described by Peichl et al. (3), singular value truncation and temporal low-pass filtering can help to reduce the amount of erroneous noise from CFD data significantly. In a typical CFD setup for industrial aerodynamics applications, large parts of the turbulence effects are modelled. In addition, due to the need for small turn-around times, simulation setups are usually trimmed for stability at maximum Courant numbers above 1, as explained in Islam et al. (1). Both measures increase numerical noise, which can influence the results of modal analysis significantly, similar to the analysis of experimental data. Due to the incremental nature of the modal analysis method in this work, temporal low-pass filtering without phase shift cannot be carried out using the same approach. Instead, mapping the simulation data to a coarser equidistant grid on a subdomain in the area of interest is used. The cell-volume-weighted mapping acts as a conservative spatial filter for the simulation data and reduces high-frequency oscillations. Spatial filtering not only smoothens out the obtained structure shapes, but also effects the frequency spectrum, minimizing high-frequency oscillations. The outline of the interpolation mesh enclosing the vehicle geometry is visualized in Fig. 2. The interpolation grid consists of 2.6 million equidistant cells with an edge length of x = 7.5mm. Fig. 2: Subdomain for DMD analysis. The number of cells is reduced from 157 million cells in the CFD domain to 2.6 million cells on the equidistant sampling grid for the snapshots built from the 3 velocity components Temporal Sampling Strategy Fig. 1: Simulation domain resembling the model scale wind tunnel setup of the Technical University of Munich. The Reynolds number based on the wheel base as the characteristic length scale is Re L = , and a total of 7s physical simulation time is computed. The flow field is initialized by a potential flow solution. The rotation of the wheels is included using the sliding mesh approach with a cylinder, annulus-shaped mesh interface region enclosing the non-symmetric part of the 5- spoke DrivAer rims. In order to obtain meaningful insights into the temporal dynamics of the flow, choosing a good sampling interval and a long enough sampling time span is of crucial importance. This work focuses on the globally most energetic perturbations of the flow. The frequency range of interest has to be known beforehand and requires knowledge about the timescales involved. With a similar Reynolds number and geometry, similar frequency ranges can be expected in bluff-body aerodynamics. This way, only a few a priori studies are required if modal analysis is to be conducted on slightly modified geometries. A common tool for analyzing a data time series are correlation functions. They can be used as an indicator for how similar two data sets recorded at different time steps are. In statistics, correlation is used to separate random noise from recurring phenomena in data series. The normalized correlation of snapshot x t and snapshot x t+τ can be expressed as 73

3 c(τ) = (x t x mean ) (x t+τ x mean ) (x t x mean ) 2. (1) The expression c(τ) is an even function so that c(τ) = c( τ), i.e. the correlation between subsequent time steps in either direction is qualitatively consistent. Fig. 3 shows the normalized correlation function for the correlation between 200 simulation time steps spaced at t sim = s. Fig. 4: Convergence of the DMD residual norm divided by the number of snapshots. As presented in the research of Alenius (15), the convergence of the DMD mode residual can then be visualized as shown in Fig. 4. Here, the last time step is held constant so that a continuous residual decrease is observed. A total time span of 2.5s physical time is sampled for the DMD analysis in this research Streaming-Total DMD Fig. 3: Correlation function over time. The time increment for the DMD analysis used in this work is chosen as 25 simulation time steps t DMD = s, visualized as the red dot in Fig. 3 with a correlation value of Using this method, the modal analysis can be directed toward correctly resolving the most relevant perturbations, neglecting high-frequency, low energy content. In contrast, the methods and guidelines described by Duke et al. (14) make it possible to focus the DMD analysis on a specific frequency range, rather than simply resolving energy-containing frequencies, as done in this work. While the correlation function can be used as an indicator for the sampling frequency, the lower frequency limit is set by the total time span analyzed and the number of snapshots. As described in Schmid et al. (2), a representative data basis is required for a sensible evaluation of DMD modes. If the number of snapshots is large enough, a time step x m can be approximated using previous time steps x 1 to x m 1 as x m = a 1 x 1 + a 2 x a m 1 x m 1 + r. (2) The residual vector r becomes smaller for increasing the number of snapshots and saturates when the columns of X become linearly dependent. DMD is a data-driven method that processes snapshots of observables in time and can be used to extract coherent structures that oscillate at distinct frequencies. The observables in this work are the 3 velocity components, but could be any other flow quantity of interest, e.g. surface force vectors on the geometry, as in Matsumoto et al. (5). The data is ordered in column vectors of length n. Using m such snapshots for DMD analysis, the snapshot matrix can be built as X = [ x 1 x 2 x m ]. (3) The DMD operator A describes the temporal evolution of the data series. AX 1 m 1 = X 2 m (4) It is a linear mapping operator with eigenvalues and eigenvectors providing information about the dynamics of the system. Instead of computing the pseudo inverse of X 1 m 1, singular value decomposition X 1 m 1 = UΣV T (5) is commonly used to compute the projected DMD operator A. A = U T AU = U T X 2 m VΣ 1 (6) The method of choice in this work, Streaming-Total DMD (STDMD), published in 2016 by Hemati et al. (16), combines the ideas of Total DMD (17) and Streaming DMD (18). STDMD offers a low-memory alternative to conventional DMD to create a ROM of modes at distinct frequencies. Instead of batch processing all flow field data at once, STDMD incrementally builds an 74

4 orthonormal basis that is used to compute the DMD modes. Whereas, in conventional DMD, all time steps need to be in memory at once, STDMD accepts an incremental feed of time step information and allows the orthonormal basis to be compressed upon reaching a user-defined limit. Due to an erroneous equation part in the original publication (equation (8)), a short description of STDMD is shown below. The projected STDMD operator writes procedures, which might be prohibitive in an industrial context in which quick turn-around times are needed. In addition, STDMD does not provide the required inputs for such an amplitude optimization procedure. An alternative procedure for ordering the modes by relevance as a post-processing routine has been recently proposed by Kou et al. (19). A mode ordering parameter I i for each mode is evaluated using the temporal evolution of each mode s amplitude as A = Q X T AQ X = Q X T [0 I]Q Z G Z Q Z T [ I 0 ] Q XG X +, (7) m I i = α i (λ i ) j 1 2 φ i F t. (13) j=1 where Q Z is an incrementally updated orthonormal basis. Q X is the orthonormal matrix from QR decomposition of the upper half of Q Z Q X R X = [I 0]Q Z. (8) The square matrices G Z and G X contain products of the upper triangular matrices R Z and R X. POD compression can be carried out upon reaching a user-defined rank limit by computing the eigenvalue decomposition of G Z, removing the eigenvectors associated with the smallest eigenvalues and projecting the reduced eigenvectors back onto the orthonormal basis. This way, it is possible to process large data sets such as those from vehicle aerodynamics simulations without surpassing the given memory limitations as well as preventing expensive hard drive write and read operations, as the incremental feed of snapshots can be carried out in memory during the simulation. After the projected DMD operator is evaluated, the eigenvalues and eigenvectors are obtained from eigenvalue decomposition Finally, the DMD modes can be evaluated as with respective DMD mode frequencies A y i = λ i y i. (9) φ i = Q X y i, (10) f i = Im[logλ i ]. 2π t (11) Using the formulation in equation (13), modes with large damping rates are penalized. For the creation of a ROM, this leads to preferential selection of stable modes over damped modes. Compared to the selection of modes by amplitude only, the eigenvalue weighted formulation by Kou leads to superior ROM performance and allows for the reduction to less modes while keeping the same ROM accuracy. This tendency has been shown by Kou et al. (19), in which a ROM performance similar to the computationally expensive SPDMD approach is obtained. Due to its performance and small computational effort of mode ordering by eigenvalue-weighted amplitudes, this method is chosen for extracting the dominant modes in this work and proofs to be a valuable tool for judging the dominance of modes Simulation Results 3. Results The simulation results are compared to wind tunnel measurement results by Haag et al. (13). Table 1 shows the timeaveraged integrated drag values for the body and wheels separately as well as the total drag value. Table 1: Wind tunnel experiment vs. CFD simulation drag values published by Haag et al. (13). Wind tunnel experiment C D,body C D,wheels C D,total DMD Mode Selection CFD Simulation In order to generate a ROM that approximates the original flow field as good as possible and to identify the most dominant modes, DMD mode amplitudes need to be calculated. The most common way of evaluating mode amplitudes α is the method of first snapshot x 1 = Φα. (12) Other ways of computing mode amplitudes have been developed, including widely used Sparsity Promoting DMD (SPDMD) as proposed Jovanovic et al. (8). While this method claims to find the optimal amplitudes that best reproduce the original flow field, it uses computationally expensive optimization The slight discrepancies between experimental values from the wind tunnel measurement and CFD simulation in the drag value of the body can partially be explained by the missing collector in the simulation setup. It has been shown by Collin et al. (12) that the steep pressure gradient in front of the collector of the WTA increases the base pressure behind the DrivAer body significantly when compared to measurements of the same 1:2.5 scale model, measured in the full-scale aeroacoustic wind tunnel at Audi (AAWK). This effect leads to an under-prediction of drag values in WTA measurements. The investigations by Collin et al. show a difference in drag values of C D = between the wind tunnels and an almost exact match between CFD simulations and AAWK measurements. Difficulties in the evaluation of 75

5 rolling resistance in the wind tunnel measurement also lead to uncertainty in the calculation of the drag values of the wheels. Since the rolling resistance is included in the forces measured by the balance of the wheel arms, it needs to be subtracted for the evaluation of aerodynamic drag, which is flawed due to uncertainties in predicting the rolling resistance DMD Analysis This section presents the main flow field processes involved in the most dominant STDMD modes. STDMD has been validated against conventional DMD in terms of its ability to generate ROMs for the flow around a single rotating wheel at a Reynolds number of Re D = in a study by Kiewat et al. (20). To the authors knowledge, this is the first time that STDMD is applied on a vehicle simulation velocity field. The most dominant frequencies are found to reach up to 35Hz, as can be seen in the frequency spectrum in Fig. 5. Fig. 6: FFT frequency spectrum of the integrated drag force coefficient C D. Fig. 7: Mode 2 at 9.6Hz. Iso-surfaces of U x = ±29.1 m s. Fig. 5: STDMD relative eigenvalue-weighted amplitude vs. frequency. The most dominant mode extracted in this modal analysis fluctuates at f 2 = 9.6Hz. It shows a perturbation below the mirror support travelling along the vehicle surface at the height of the door handles. Transported downstream, the structure initiates a large separation along the vertical line downstream of the door handles. This process is more pronounced on the left side of the vehicle, but can also be observed on the right with slight differences. The perturbation on the right side travels slightly above the door handles and ends up initiating large scale separation on the right end of the trunk lid, above the separation location of the structure on the left. The structures from both sides detach almost in anti-phase. In addition, the rear wheels are responsible for a large amount of the oscillations in the wake. The frequency spectrum of the integrated forces also shows a peak around 9Hz (see Fig. 6); the extracted structures of mode 2 are likely to be the dominant source of low pressure regions in the vehicle wake due to centrifugal forces of the vortex structures. Due to the alternating detachment of large-scale structures from the left and right rear-end, turbulent mixing in the y-direction increases and low pressure regions are generated on the rear surface. Mode 4 is dominated by bluff-body vortex shedding from the rear wheels in the x and y components. Large structures are excited close to the street on both sides of the rear wheels and transported downstream with small z-direction fluctuation components, similar to vortex shedding from a wall-mounted cube or cylinder. The inbound vortex structures move towards the centerline and seem to support another pair of separations from the underbody in the area of the underbody diffusor, as depicted in Fig. 8. Fig. 8: Mode 4 at 25.7Hz. Iso-surfaces of U x = ±5.8 m s. Parts of the excited structures are exactly in phase, which puts into question whether the bluff-body vortex shedding is connected to the rotation of the wheels and the initialization of the rim rotation in the simulation setup. Another dominant process in mode 4 is the vortex shedding of the side mirrors, as can be seen in Fig. 9. Structures detach from the upper and lower part almost in phase. These structures are transported downstream with similar speed. The lower structure initiates a large scale separation above the mechanism from mode 2, while the upper side mirror structure crosses the C-pillar towards the centerline of the vehicle. 76

6 This process can be observed on both sides of the geometry, as in mode 2. lower frequency components of the flow might be important and longer sampling time spans are appropriate for full vehicle simulation, especially due to the large amplitude of these low frequency modes. Mode 5, visualized in Fig. 12, shows only minor vortex shedding processes of the wheels and the side mirrors. The large oscillations in the wake of the vehicle are rather fixed in space and can be seen as recirculation regions. Fig. 9: Mode 4 at 25.7Hz. Iso-surfaces of U x = ±5.8 m s. In addition to the structures from the mirror, which appear on both sides of the vehicle, a small oscillation of the otherwise spatially fixed A-pillar vortex is observed on the right. It reaches the trunk lid where it interacts with pressure-induced separations from the rear windshield. These structures then discharge into the vehicle wake, creating large scale structures that oscillate in z- direction, as visualized in Fig. 11. Due to their initiation very close to the vehicle surface, a large influence on the fluctuations of the lift forces is self-explanatory. As for the connection of mode 2 with the integrated drag values, the frequency of mode 4 can be found in the frequency spectrum of the rear-wheel lift forces shown in Fig. 10. Even though the connection is not as obvious, it is still obvious that mode 4 plays a major role in the oscillations of the lift forces. Fig. 10: FFT frequency spectrum of the integrated lift-force coefficient on the rear wheels C L,rearWheels. Fig. 11: Mode 4 at 25.7Hz. Iso-surfaces of U z = ±5.8 m s. STDMD mode 5 is the lowest frequency of all investigated modes. It can still be resolved due to the long total sampling time span chosen for DMD analysis. Mode 3, on the other hand, coincides with the mean flow mode frequency. This hints that even Fig. 12: Mode 5 at 0.8Hz. Iso-surfaces of U x = ±16.3 m s. 4. Conclusion The current work aims to overcome the limitations of applying conventional DMD, regarding hard drive disk and memory requirements, by using an incremental DMD variant plus data reduction by spatial, cell-volume-weighted mapping. The interpolation to a coarser grid removes high-frequency numerical noise components and purges the need for temporal low-pass filtering of CFD results in order to compute unbiased DMD modes. In addition, a new mode-ordering technique is employed to define the relevance of a DMD mode with respect to the entire time series and to be able to extract the most dominant modes for the creation of a ROM. A quantitative method for defining the sampling frequency using the correlation function is found useful for reducing the amount of sampled time steps to the bare minimum required to resolve physically relevant phenomena. The residual study presented here shows reasonable convergence for the total amount of time steps used, but additional investigations using longer sampling time spans are required to validate the current modal analysis results, especially for the lowest frequency modes. The most dominant STDMD modes lead to a deeper insight into the flow processes involved in the generation of oscillations. The most dominant fluctuating mode is found to coincide with the dominant frequency of the integrated drag forces spectrum, which hints at a strong connection between large-scale separation processes that are partially triggered by smaller perturbations upstream and the resulting vortex-induced drag. Bluff-body vortex shedding from the rear wheels is shown to be another dominant source of oscillations in the wake of the vehicle. No dominant mode with an obvious connection to the wheel-rotation frequency is found. Furthermore, it can be seen that structures that detach from the side mirror and their convection path downstream seem to have a big influence on the location and the phase of large-scale separations on the vehicle s rear end. The problem of degenerate 77

7 eigenvalues as well as further automated mode selection and analysis methods need to be developed for sensible application in an industrial context. Acknowledgements We would like to express our gratitude to our industry partner AUDI AG for their financial support and thank Dr. M. Islam for the opportunity to carry out research in this exciting field. We thank M. Peichl, V. Pasquariello, C.W. Rowley, R. Grigoriev and I. Mezić for the valuable discussions on the application of DMD methods for vehicle aerodynamics. This publication is supported by TUM Graduate School s Faculty Graduate Centre of Mechanical Engineering. This research is supported in part by the National Science Foundation under Grant No. NSF PHY This paper is written based on a proceeding presented at JSAE 2017 Annual Congress. References (1) M. Islam, F. Decker, E. de Villiers, A. Jackson, J. Gines, T. Grahs, A. Gitt-Gehrke and J. Comas i Font, "Application of Detached-Eddy Simulation for Automotive Aerodynamics Development", SAE Paper SAE (2009). (2) P. J. Schmid, "Dynamic mode decomposition of numerical and experimental data", Journal of Fluid Mechanics, Vol. 656, p (2010). (3) M. Peichl, S. Mack, T. Indinger and F. Decker, "Numerical Investigation of the Flow Around a Generic Car using Dynamic Mode Decomposition", Proceedings of ASME Fluids Engineering Division Summer Meeting FEDSM (2014). (4) H. M. Frank and C.-D. Munz, "Direct aeroacoustic simulation of acoustic feedback phenomena on a side-view mirror", Journal of Sound and Vibration, Vol. 371, p (2016). (5) D. Matsumoto, L. Haag and T. Indinger, "Investigation of the unsteady external and underhood air flow of the DrivAer model by Dynamic Mode Decomposition methods", International Journal of Automotive Engineering, Vol.8, No.2, p (2017). (6) D. Matsumoto, M. Kiewat and T. Indinger, "Online Dynamic Mode Decomposition Methods for the Investigation of Unsteady Aerodynamics of the DrivAer Model (First Report) - Validation of DMD Methods for Surface Forces" International Journal of Automotive Engineering, Vol.9, No.2, p (2018). (7) D. Matsumoto and T. Indinger, "On-the-fly algorithm for Dynamic Mode Decomposition using Incremental Singular Value Decomposition and Total Least Squares", arxiv: [physics.flu-dyn] (2017). (8) M. R. Jovanovic, P. J. Schmid and J. W. Nichols, "Sparsity-promoting dynamic mode decomposition", Physics of Fluids, Vol. 26, pp p (2014). (9) A. Wynn, D. S. Pearson, B. Ganapathisubramani and P. J. Goulart, "Optimal mode decomposition for unsteady flows", Journal of Fluid Mechanics, Vol. 733, pp (2013). (10) J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton and N. J. Kutz, "On Dynamic Mode Decomposition: Theory and Applications", Journal of Computational Dynamics, pp (2014). (11) H. G. Weller, G. Tabor, H. Jasak and C. Fureby, "A tensorial approach to computational continuum mechanics using object-oriented techniques", American Institute of Physics (1998). (12) C. Collin, S. Mack, T. Indinger and J. Mueller, "A Numerical and Experimental Evaluation of Open Jet Wind Tunnel Interferences using the DrivAer Reference Model", SAE Paper (2016). (13) L. Haag, M. Kiewat, T. Indinger and T. Blacha, "Numerical and Experimental Investigations of Rotating Passenger Car Wheel Aerodynamics on the DrivAer Model with Engine Bay Flow", Proceedings of ASME Fluids Engineering Division Summer Meeting FEDSM (2017). (14) D. Duke, J. Soria and D. Honnery, "An error analysis of the dynamic mode decomposition", Experiments in Fluids, Vol. 52, p (2011). (15) E. Alenius, "Flow Duct Acoustics - An LES Approach", Stockholm: KTH (2012). (16) M. S. Hemati, E. A. Deem, M. O. Williams, C. W. Rowley and L. N. Cattafesta, "Improving Separation Control with Noise-Robust Variants of Dynamic Mode Decomposition", Proceedings of 54th AIAA Aerospace Sciences Meeting AIAA (2016). (17) M. S. Hemati, C. W. Rowley, E. A. Deem and L. N. Cattafesta, "De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets", in Theoretical and Computational Fluid Dynamics, Berlin Heidelberg, Springer-Verlag, p (2017). (18) M. S. Hemati, M. O. Williams and C. W. Rowley, "Dynamic mode decomposition for large and streaming datasets", Physics of Fluids, Vol. 26, p. 1-6 (2014). (19) J. Kou and W. Zhang, "An improved criterion to select dominant modes from dynamic mode decomposition", European Journal of Mechnics B/Fluids, p (2016). (20) M. Kiewat, L. Haag, T. Indinger and V. Zander, "Low Memory Reduced Order Modelling with Dynamic Mode Decomposition Applied on Unsteady Wheel Aerodynamics", Proceedings of ASME Fluids Engineering Division Summer Meeting FEDSM (2017). 78