Unsteady Metric-Based Adaptation via Koopman Expansion

In this study, we investigated the stability of dynamic mode decomposition\n(DMD) algorithms to noisy data. To achieve a stable DMD algorithm, we applied\nthe truncated total least squares (T-TLS) regression and optimal truncation\nlevel selection to the TLS DMD algorithm. By adding truncation regularization\nto the TLS DMD algorithm, T-TLS DMD improves the stability of the computation\nwhile maintaining the accuracy of TLS DMD. The effectiveness of the T-TLS DMD\nwas evaluated by the analysis of the wake behind a cylinder and practical\npressure-sensitive paint (PSP) data for the buffet cell phenomenon. The results\nshowed the importance of regularization in the DMD algorithm. With respect to\nthe eigenvalues, T-TLS DMD was less affected by noise, and accurate eigenvalues\ncould be obtained stably, whereas the eigenvalues of TLS and subspace DMD\nvaried greatly due to noise. It was also observed that the eigenvalues of the\nstandard and exact DMD had the problem of shifting to the damping side, as\nreported in previous studies. With respect to eigenvectors, T-TLS and exact DMD\ncaptured the characteristic flow patterns clearly even in the presence of\nnoise, whereas TLS and subspace DMD were not able to capture them clearly due\nto noise.\n

Dynamic characteristics and thermodynamic energy dissipations are vital for comprehending unsteady flow behaviors, while their interrelationship remains underexplored. Based on entropy production theory and dynamic mode decomposition, a general theoretical model is derived to reveal the intrinsic connection between dynamic mode characteristics and entropy production rate (EPR). Two unsteady simple flow cases, i.e., the flow around a cylinder and the flow around an airfoil, are employed to verify this theoretical model. It is found that the model enables the quantitative assessment of the contributions of individual modes to irreversible losses, providing an efficient tool to identify the dominant modes associated with energy losses in flow fields. The local amplitude of a mode and the amplitude of its time coefficient directly determine the magnitude of the EPR generated by a single mode. For the simple flows, the dynamic modes with larger local amplitudes typically have larger time coefficient amplitudes as well, and these modes are often associated with regions in the flow field where significant irreversible dissipation occurs. The proposed model is applicable as long as the dominant frequencies extracted from the mode decomposition of the partial derivatives of the velocity components field are consistent with those obtained from the original EPR field. This model highlights the critical role of specific modes in EPR, providing a valuable theoretical framework for exploring the interplay between flow dynamics and thermodynamics in unsteady flow fields.

Dynamic mode decomposition (DMD) is performed to investigate the unsteady flow field in the tip region of transonic axial compressor NASA Rotor35 at 100% design rotating speed. With DMD technology, the complex flow field containing spatial and temporal information can be decomposed hierarchically. The results show that DMD obtain the oscillation frequency 5.60 kHz of the unsteady flow field in the tip region at low mass flow condition, and the corresponding modes with large scale oscillation structure at blade pressure side show the flow field fluctuating nature. In addition, flow field reconstruction is conducted by the second DMD mode and the oscillations alternating positive and negative is the main unsteady flow source for the compressor. The oscillation in the tip region conducts as like a standing wave.

The analysis of the unsteady flow field in axial compressor cascade is conducted using methods such as proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). Data on the unsteady flow field of the Stage-35 compressor cascade are acquired via computational fluid dynamics (CFD) simulations and subsequently processed using POD and DMD for dimensionality reduction. Using singular value decomposition, the POD technique identifies the dominant modes, showing that the first nine modes account for 99% of the energy in the flow field, thus highlighting the primary flow structures. On the other hand, the DMD approach isolates the periodic and high-frequency dynamics within the flow field by decomposing the dynamic modes, effectively identifying fine variations in the unsteady flow. The study examines the flow field at three distinct moments within an unsteady cycle, specifically at 1/4T, 1/2T, and 3/4T, reconstructing the flow field at each instance and performing root mean square error analysis. Reconstruction results and error analysis demonstrate that the POD method excels at reconstructing low-frequency features, whereas the DMD method accurately identifies the unsteady dynamic aspects of the flow field, excelling in resolving high-frequency details. Both methods demonstrate high feasibility regarding the accuracy and efficiency of flow field reconstruction.

Traditional flow field modeling methods are limited by high computational complexity, making them difficult to apply in practical engineering. This study applies the Dynamic Mode Decomposition (DMD) method to perform reduced-order modeling of unsteady flow fields over an airfoil and a compressor cascade. As a data-driven modal decomposition technique, DMD extracts low-dimensional modes from high-dimensional spatiotemporal data, preserving key dynamic characteristics and significantly reducing computational costs. Numerical simulations were conducted to generate time snapshots, forming matrices of pressure and Mach number snapshots. DMD analysis identified a few dominant modes and their eigenvalues, capturing the primary dynamic behavior of the flow field. The results demonstrate that these modes effectively reconstruct the system’s main characteristics, reducing the need for extensive computational resources and time. The DMD method not only improves modeling efficiency, but also accurately reconstructs complex flow structures. This study validates the feasibility and effectiveness of DMD in reduced-order modeling for unsteady flow fields and includes error analysis for further evaluation.

The self-excited pressure and velocity oscillations caused by the downstream back pressure in a hypersonic inlet/isolator directly affect the operational stability of a scramjet. In this work, we conduct the two-dimensional unsteady RANS (Reynolds averaged Navier–Stokes) simulation to predict the flow field in the Mach 7 inlet/isolator with 147 times the freestream static pressure. With the model validated, the time evolution process of the instantaneous flow fields in the isolator is analyzed first. It is shown that the structure of the shock train exhibits an up-down asymmetry. The flow oscillations as observed near the upper wall are more complex than those near the lower wall. Furthermore, the dominant frequency of the wall pressure oscillations is approximately 520 Hz. However, the self-excited oscillations in the isolator flow are dominated with approximately 510 Hz mode but coupled with multiple harmonic higher frequencies. To identify and decompose the modes of these oscillations in the unsteady flow of the isolator, different mode decomposition technologies are applied to further examine the mechanism of such oscillations. By conducting Proper Orthogonal Decomposition (POD) analysis on the x-axis velocity and Mach number fields, it is shown that the first two modes consist of more than 77% of the total oscillations’ kinetic energy. Additionally, the oscillations of the quadrilateral separation zone and low-speed zone near the upper wall largely determine the dominant frequency of the overall unsteady flow field. As Dynamic Mode Decomposition (DMD) investigation is conducted on the x-axis velocity and Mach number fields, it is found that the identified first mode is almost the same as that identified by using POD. It is also found that both DMD and POD can well predict the viscous flows such as the separation zone, shear layer, and low-speed zone. Finally, the dominant frequencies predicted by the POD and DMD are revealed to be very close to the dominant frequency of the wall static pressure oscillations. The movements of multiple shock waves near the upper wall are the major incentive for the complicated change in the entropy generation rate. In general, the self-excited thermodynamic oscillations phenomenon as observed in the hypersonic inlet/isolator in a scramjet could be well analyzed by decomposing and identifying the dominant modes, which contribute mostly to the total energy of such oscillations.

The Coanda effect nozzle is a fluid thrust vectoring technology that uses the Coanda effect to control jet vector deflection. The jumping phenomenon often occurs in the process of controlling jet vector deflection. This phenomenon leads to the nonlinearity of thrust vector control. It destroys the control performance of the aircraft and brings potential dangers to the safety of the aircraft. The jumping phenomenon occurs in an unsteady flow field different from the traditional flow phenomenon. The flow structure in an unsteady flow field changes with time, so it is not easy to control by the traditional active flow control method. This paper explains the reasons for the jumping phenomenon from two aspects: flow field stability and flow structure. Secondly, the unsteady flow field with the jumping phenomenon is studied and analyzed by a flow visualization experiment and dynamic force measurement. Furthermore, the dynamic modal decomposition (DMD) method is used to extract the characteristic frequencies of the critical vortices causing jets to jump in unsteady flow fields. Finally, a pulsed jet with the same characteristic frequency is used to control the varying vortices in the unsteady flow field. The experimental results show that the active flow control method, which extracts the characteristic frequency of the critical flow field structure by DMD, effectively suppresses the jumping phenomenon in the unsteady flow field. It also linearizes the process of jet nonlinear vector deflection.

Source term estimation in the unsteady flow with dynamic mode decomposition

Vehicle acceleration typically occurs at traffic lights, intersections, or congested sections within urban streets, where high densities of pedestrians and vehicles pose a direct threat to respiratory health due to PM2.5 dispersion. Computational Fluid Dynamics (CFD) simulations, combined with the Dynamic Mode Decomposition (DMD) method, are used to analyze the dynamic characteristics of PM2.5 dispersion during vehicle acceleration. The DMD method can effectively analyze the dynamic change in pollutant concentration in an unsteady flow field and clarify the influence mechanism of vehicle acceleration on pollutant dispersion. The results indicate that PM2.5 dispersion during the initial stage of acceleration is primarily influenced by low-frequency and large-scale flows, such as exhaust emissions, natural wind, and trailing vortices. In the middle stage, PM2.5 dispersion tends to stabilize, while in the final stage, high-frequency modes dominate, and intense flow field fluctuations significantly enhance PM2.5 dispersion. Furthermore, the analysis reveals the critical role of upward and downward airflow phenomena around the vehicle in driving PM2.5 dispersion. This study offers a new perspective on the dispersion characteristics of PM2.5 under unsteady flow conditions in urban streets and provides a scientific basis for developing speed management strategies to mitigate the impact of pollutant dispersion.

Reduced-order models such as dynamic mode decomposition (DMD) and proper orthogonal decomposition (POD) have been extensively utilized to model unsteady flow. Although the major flow patterns can be captured by DMD and POD, due to the linear assumption, the modeling accuracy is low for complex and strongly nonlinear flow structures such as shock wave and vortex. To improve the accuracy and robustness of predicting unsteady flow, this work proposes a novel modeling method based on a hybrid reduced-order model. Since the flow can be regarded as a fusion of the main flow and the residual flow from a modeling perspective, the hybrid reduced-order model is constructed by DMD and POD, which are, respectively, used to obtain different flow properties. First, DMD is applied in describing the main flow, which contains the dominant modes determining most properties of the flow. Then, POD combining the long short-term memory is conceived to model the residual flow that the DMD cannot capture, to further enhance the modeling accuracy. The proposed method is validated by modeling two unsteady flows, which are the flow past a two-dimensional circular cylinder at Reynolds number 100 and the forced oscillation of an airfoil at transonic speed. The results indicate that the proposed method with proper modeling efficiency gains better accuracy and robustness than the existing methods. In particular, this approach has better forecasting accuracy of shock wave and vortex.

Visualization and selection of Dynamic Mode Decomposition components for unsteady flow

With the development of offshore oil fields, gas–liquid multiphase mixed transport technology has attracted more attention by various countries for its remarkable economic benefits. Due to its capability to pump out large gas content, compact structure, and insensitivity to solid particles, so the multiphase rotodynamic pump (MPP) is operating in many gas–oil fields. By the mean of dynamic mode decomposition method, the unsteady flow field and mechanism of pressure pulsation are analyzed, the main flow field structures are extracted, and its complex flow field is decomposed into flow field characteristics in the second compression cell with different energies and frequencies, including basic mode characteristics, dynamic mode flow field characteristics of rotor–stator interaction, and its high-order harmonic behavior. In different locations of second stage, intricate pressure pulsation characteristics occur and harm the MPP operation and ruled by various frequencies. The mutual matching of different rotor and stator blade numbers resulting in a rotor–stator interaction frequency in impeller and diffuser, which will excite higher harmonics of the impeller blade frequency. At low inlet gas volume fraction (IGVF i.e., 10%), there are low-and high-pressure pulsation in the impeller and diffuser flow passages. However, under high IGVF (i.e., 20%), the pressure pulsation is characterized by low- and high-pressure pulsation in the impeller passage, while in the both diffuser conditions there are groups of flow passages of low- and high-pressure pulsation.

A novel wavelet-based adaptive delayed detached eddy simulation (W-DDES) approach for simulations of wall-bounded compressible turbulent flows is proposed. The new approach utilizes anisotropic wavelet-based mesh refinement and its effectiveness is demonstrated for flow simulations using the Spalart–Allmaras DDES model. A variable wavelet thresholding strategy blending two distinct thresholds for the Reynolds-averaged Navier–Stokes (RANS) and large-eddy simulation (LES) regimes is used. A novel mesh adaptation on mean and fluctuating quantities with different wavelet threshold levels is proposed. The new strategy is more accurate and efficient compared to the adaptation on instantaneous quantities using a priori defined uniform thresholds. The effectiveness of the W-DDES method is demonstrated by comparing the results of the W-DDES simulations with results already available in the literature. Supersonic plane channel flow for two different configurations is tested as benchmark wall-bounded flows. Both the accuracy indicated by the threshold and efficiency in terms of degrees of freedom for the novel adaptation strategy are successfully gained compared with the wavelet-based adaptive LES method. Moreover, the newly proposed W-DDES resolves the typical log-layer match issue encountered in the conventional non-adaptive DDES method mainly due to the use of wavelet-based adaptive mesh refinement. The W-DDES capability for simulations of complex turbulent flows is validated by two other flow configurations – a subsonic channel flow with periodic hill constrictions and a supersonic flow over a compression ramp inducing the shock wave–turbulent boundary layer interaction. The current study serves as a crucial step towards construction of a unified wavelet-based adaptive hierarchical RANS/LES modelling framework, capable of performing simulations of varying fidelities from no-modelling direct numerical simulations to full-modelling RANS simulations.

A novel approach is proposed for defining the amplitudes of modes extracted by dynamic mode decomposition based on the amplitudes and coefficients derived from proper orthogonal decomposition. To verify this methodology, unsteady flow fields are numerically simulated using an in-house flow solver and subsequently decomposed. By employing the amplitudes defined through our proposed framework, the dominant mode derived from dynamic mode decomposition can be readily identified. The frequency of this dominant mode aligns closely with the dominant frequency observed in the aerodynamic force spectrum. Furthermore, the coherent structures associated with the dominant mode exhibit clear and well-organised spatial patterns. These results demonstrate that the dominant mode can be effectively distinguished from other modes extracted by dynamic mode decomposition using the amplitudes derived from the proposed definition. The introduced methodology provides researchers with a valuable tool to efficiently identify dominant coherent structures in unsteady flow fields and elucidate flow mechanisms.

Spatio–temporal dynamic mode decomposition in a shear layer flow