Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken. We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator. This generalizes DMD to a larger class of datasets, including nonsequential time series. We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively. We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples. Such computations are not considered in the existing literature, but can be understood using our more general framework. In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory. It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science. We show that under certain conditions, DMD is equivalent to LIM.

On Dynamic Mode Decomposition: Theory and Applications

Shock wave/boundary layer interactions occur in a wide range of supersonic internal and external flows, and often these interactions are associated with turbulent boundary layer separation. The resulting separated flow is associated with large-scale, low-frequency unsteadiness whose cause has been the subject of much attention and debate. In particular, some researchers have concluded that the source of low-frequency motions is in the upstream boundary layer, whereas others have argued for a downstream instability as the driving mechanism. Owing to substantial recent activity, we are close to developing a comprehensive understanding, albeit only in simplified flow configurations. A plausible model is that the interaction responds as a dynamical system that is forced by external disturbances. The low-frequency dynamics seem to be adequately described by a recently proposed shear layer entrainment-recharge mechanism. Upstream boundary layer fluctuations seem to be an important source of disturbances, but the evidence suggests that their impact is reduced with increasing size of the separated flow.

Low-Frequency Unsteadiness of Shock Wave/Turbulent Boundary Layer Interactions

The need for better understanding of the low-frequency unsteadiness observed in shock wave/turbulent boundary layer interactions has been driving research in this area for several decades. We present here a large-eddy simulation investigation of the interaction between an impinging oblique shock and a Mach 2.3 turbulent boundary layer. Contrary to past large-eddy simulation investigations on shock/turbulent boundary layer interactions, we have used an inflow technique which does not introduce any energetically significant low frequencies into the domain, hence avoiding possible interference with the shock/boundary layer interaction system. The large-eddy simulation has been run for much longer times than previous computational studies making a Fourier analysis of the low frequency possible. The broadband and energetic low-frequency component found in the interaction is in excellent agreement with the experimental findings. Furthermore, a linear stability analysis of the mean flow was performed and a stationary unstable global mode was found. The long-run large-eddy simulation data were analyzed and a phase change in the wall pressure fluctuations was related to the global-mode structure, leading to a possible driving mechanism for the observed low-frequency motions.

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Large-eddy simulation of low-frequency unsteadiness in a turbulent shock-induced separation bubble

This paper addresses one of the most persistent errors in wall-modeled large eddy simulation: the inevitable presence of numerical and subgrid modeling errors in the first few grid points off the wall, which leads to the so-called “log-layer mismatch” with its associated 10-15% error in the predicted skin friction. By considering the behavior of turbulence length scales near a wall, the source of these errors is analyzed, and a method that allows for the log-layer mismatch to be removed, thereby yielding accurately predicted skin friction, is proposed.

Wall-modeling in large eddy simulation: Length scales, grid resolution, and accuracy

Progress in shock wave/boundary layer interactions

A model to explain the low-frequency unsteadiness found in shock-induced separation is proposed for cases in which the flow is reattaching downstream. It is based on the properties of fluid entrainment in the mixing layer generated downstream of the separation shock whose low-frequency motions are related to successive contractions and dilatations of the separated bubble. The main aerodynamic parameters on which the process depends are presented. This model is consistent with experimental observations obtained by particle image velocimetry (PIV) in a Mach 2.3 oblique shock wave/turbulent boundary layer interaction, as well as with several different configurations reported in the literature for Mach numbers ranging from 0 to 5.

A simple model for low-frequency unsteadiness in shock-induced separation

Unsteadiness in shock wave boundary layer interactions with separation

The interaction of an oblique shock wave impinging on a turbulent boundary layer at Mach number 2.3 is experimentally investigated for a wide range of shock intensities. Characteristic time and length scales of the unsteady reflected shock and inside the downstream interaction region are obtained and compared with already existing results obtained in compression ramp experiments as well as in subsonic detached flows. Dimensionless characteristic frequencies are highlighted to characterize low-frequency shock unsteadiness as well as the different large scales which develop inside the initial part of the interaction. The possibility of describing the spatial development of the large scales inside the interaction zone using a mixing-layer scheme including compressibility effects is tested for a wide range of Mach numbers, shock intensities and geometrical configurations. Moreover, strong evidence of a statistical link between low-frequency shock movements and the downstream interaction is given. Finally, the downstream evolution of the structures shed into the boundary layer is characterized and shows features specific of our configuration.

Space and time organization in a shock-induced separated boundary layer

The effect of the interaction strength on the unsteady behavior of a planar shock wave impinging on a low Reynolds turbulent boundary layer is investigated. This is achieved by means of a variation in incident shock angle under otherwise constant flow conditions. In addition, the effect of an order-of-magnitude variation in the Reynolds number is considered. This has been done for equivalent interaction strength, based on a similar probability of occurrence of instantaneous flow separations. The measurement technique employed is two-component planar particle image velocimetry. Common mechanisms for the large-scale reflected-shock unsteadiness are deduced by means of conditional statistics based on the separation bubble height. The results indicate that both upstream and downstream mechanisms are at work, the dominant mechanism depending on the interaction strength. No significant dependence on the Reynolds number was observed for interactions with a similar probability of instantaneous flow separations.

Effect of Interaction Strength on Unsteadiness in Shock-Wave-Induced Separations

The organization and length scales of turbulent structures and unsteadiness generated in a shock-wave-induced separation at Mach number of 2.3 are investigated experimentally using particle image velocimetry. Processing of the velocity fields displays and demonstrates the existence of structures in the mixing layer developed in the separation bubble. Moreover, we show in evidence a link between the reflected shock excursions and the size of the separated flow. This overview of the spatial organization of the interaction provides a more comprehensive picture of the flow. Nomenclature C f = friction coefficient L = length of the interaction M = Mach number Re = Reynolds number based on the momentum thickness S L = Strouhal number T t = stagnation temperature U 0 = upstream external velocity U  = V=u u = friction velocity V = Van Driest transformed velocity X = x X 0 =L X 0 = mean position of the reflected shock x = longitudinal coordinate Y = y= 0 y = normal to the wall coordinate y  = yu = 0 = upstream boundary-layer thickness = incidence angle of the shock generator = kinematic viscosity

Jean-François Debiève

Papers

A simple model for low-frequency unsteadiness in shock-induced separation

Unsteadiness in shock wave boundary layer interactions with separation

Space and time organization in a shock-induced separated boundary layer

Effect of Interaction Strength on Unsteadiness in Shock-Wave-Induced Separations

Investigation by Particle Image Velocimetry Measurements of Oblique Shock Reflection with Separation