Identification Of Coherent Structures Research Articles

Effects of an engineered log jam on spatial variability of the flow field across submergence depths

AbstractEngineered log jams (ELJs) are commonly implemented in rivers to diversify fish habitat, but few studies have verified the hydraulic benefits of ELJs by assessing their effects on the spatial variability and turbulence structure of the flow field. This field and modelling study assessed the effects of an ELJ on various characteristics of the flow field with increasing submergence. A 2‐D hydrodynamic model was applied to simulate flow conditions of the Calapooia River, Oregon, with an ELJ installed at the upstream end of a meander. Output was analyzed using coherent flow structure identification, maps of turbulence measures, and wavelet analysis along linear transects. At the highest discharge (62% ELJ submergence), more and smaller coherent flow structures were identified than in the no ELJ model run, but the magnitudes of turbulence metrics were unaffected. At low flow (1% jam submergence), the presence of the ELJ did not affect the number of coherent flow structures identified and produced only small and localized effects on turbulence measures. Taken together, results indicate that the ELJ primarily affected the flow field by increasing the number of flow structures at higher discharges and submergence depths. However, these changes were small and local to the ELJ. The limited effects of this wood jam over limited spatial scales raise questions about the conditions under which such structures increase hydraulic diversity and highlight the need for more detailed understanding on spatial variability around habitat structures with complex shapes.

On Least Squares Problems with Certain Vandermonde--Khatri--Rao Structure with Applications to DMD

This paper proposes a new computational method for solving the structured least squares problem that arises in the process of identification of coherent structures in dynamic processes, such as, e.g., fluid flows. It is deployed in combination with dynamic mode decomposition (DMD), which provides a nonorthogonal set of modes corresponding to particular temporal frequencies. A selection of these is used to represent time snapshots of the underlying dynamics. The coefficients of the representation are determined from a solution of a structured linear least squares problems with the matrix that involves the Khatri--Rao product of a triangular and a Vandermonde matrix. Such a structure allows for a very efficient normal equation based least squares solution, which is used in state-of-the-art computational fluid dynamics (CFD) tools, such as the sparsity promoting DMD (DMDSP). A new numerical analysis of the normal equations approach provides insights about its applicability and its limitations. Relevant condition numbers that determine numerical robustness are identified and discussed. Further, the paper offers a corrected seminormal solution and the QR factorization based algorithms. It shows how to use the Vandermonde--Khatri--Rao structure to efficiently compute the QR factorization of the least squares coefficient matrix, thus providing a new computational tool for the ill-conditioned cases where the normal equations may fail to compute a sufficiently accurate solution. Altogether, the presented material provides a firm numerical linear algebra framework for a class of structured least squares problems arising in a variety of applications besides the DMD, such as, e.g., multistatic antenna array processing.

Identification of coherent flow structures and experimental analysis of the hydroacoustic emission of a hubless propeller

The progress in drive technology and the increasing needs for compact ship propulsion systems has brought a new class of hubless propellers into the field of operation. So far, no detailed analyses have been done concerning the unsteady flow structures which are important for the hydroacoustic emission. The objective of this paper is to analyze the unsteady flow topology and the related hydroacoustic emission of a hubless propeller with numerical and experimental methods. In order to capture the unsteady wake flow structures, a stress-blended eddy simulation (SBES) has been done. Experimental investigations in a circulating water channel provided the validation data in terms of integral quantities like thrust, wall pressure fluctuation and the hydroacoustic noise emission. The wake flow can be characterized by a multi-helix vortex system which develops from the blade root and tip region as well as from the trailing edge. The transition length of the helixes is larger compared to hub propellers in conjunction with an accelerated dissipation. The application of the proper orthogonal decomposition (POD) onto the unsteady wall pressure fields obtained from the SBES simulation was capable to assign different flow structures to the measured hydroacoustic emissions.

Coherent Structure Interaction During Unsteady Separation

Unsteady flow separation in rotationally augmented flow fields plays a significant role in the aerodynamic performance of industrial rotors, including wind turbines. Current computational models underestimate the aerodynamic loads due to the inaccurate prediction of the emergence and severity of unsteady flow separation, especially in response to a sudden change in the effective angle of attack. Through the use of time-resolved particle image velocimetry (PIV), coherent structure formation during the unsteady separation over an experimental wind turbine blade is examined. Time-dependent empirical mode decomposition results during a dynamic pitching cycle give insight into the spatio-temporal scales that influence the transition from attached to separated flow. Empirical mode decomposition (EMD) modes are represented as two-dimensional fields and are analyzed together with Lagrangian coherent structures, the spatial distribution of vortices, the location of the separation point, and velocity contours focusing on the role of vortex shedding and shear-layer perturbation in unsteady flow separation, stall, and reattachment. The combination of these analytical techniques provides experimental evidence that the location of the separation point and the stability of the shear layer are directly influenced by the presence of vortices. Within this paper, each of the scales represented by the EMD are directly connected to the size of the vortices present, from the smallest representing a vortex radius to the largest reaching to two full vortex diameters. The velocity scales and spatial scales provided by the EMD modes are also found to supply valuable inputs into the identification of Lagrangian coherent structures within each of the PIV snapshots. This indicates that the scales captured by the EMD can be used to extract important turbulent scales present at the point of flow separation where the vortices are created, providing relenvant insight into the separation dynamics of the airfoil.

Object‐Oriented Identification of Coherent Structures in Large Eddy Simulations: Importance of Downdrafts in Stratocumulus

AbstractA novel methodology is proposed to characterize coherent structures in large eddy simulations. Based on two passive tracers emitted respectively at the surface and at cloud top, the object‐oriented framework allows individual characterization of coherent tridimensional plumes within the flow. Applying this method in a simulation of the diurnal cycle of a marine stratocumulus‐topped boundary layer shows that coherent updraft and downdraft structures contribute to most of the total transport of heat and moisture, although covering a small part of the domain volume. On average, downdrafts contribute equally compared to updrafts for moisture fluxes and more than updrafts for heat fluxes. The relative contribution of updraft and downdraft objects to heat transport exhibits a large diurnal cycle, which suggests cloud‐turbulence‐radiation interaction. Our results suggest that subgrid downdraft properties within stratocumulus‐topped boundary layers should be represented through nonlocal mass‐flux parameterization in climate models.

Lagrangian Identification of Coherent Structures in Wall-Bounded Flows

Lagrangian Identification of Coherent Structures in Wall-Bounded Flows

On the Application of Proper Orthogonal Decomposition (POD) for In-Cylinder Flow Analysis

Proper orthogonal decomposition (POD) is a coherent structure identification technique based on either measured or computed data sets. Recently, POD has been adopted for the analysis of the in-cylinder flows inside internal combustion engines. In this study, stereoscopic particle image velocimetry (Stereo-PIV) measurements were carried out at the central vertical tumble plane inside an engine cylinder to acquire the velocity vector fields for the in-cylinder flow under different experimental conditions. Afterwards, the POD analysis were performed firstly on synthetic velocity vector fields with known characteristics in order to extract some fundamental properties of the POD technique. These data were used to reveal how the physical properties of coherent structures were captured and distributed among the POD modes, in addition to illustrate the difference between subtracting and non-subtracting the ensemble average prior to conducting POD on datasets. Moreover, two case studies for the in-cylinder flow at different valve lifts and different pressure differences across the air intake valves were presented and discussed as the effect of both valve lifts and pressure difference have not been investigated before using phase-invariant POD analysis. The results demonstrated that for repeatable flow pattern, only the first mode was sufficient to reconstruct the physical properties of the flow. Furthermore, POD analysis confirmed the negligible effect of pressure difference and subsequently the effect of engine speed on flow structures.

Coherent Structures in the Actively Controlled Wake of a High-Lift Configuration

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 by employing the recently introduced method of spectral proper orthogonal decomposition, allowing for an identification of coherent flow structures in the Eulerian specification that can be robustly related to specific frequencies. It is shown that 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, whereas the greater actuation intensity is sufficient to ensure a stable prevention of turbulent flow separation.

An extension of Oberbeck–Boussinesq approximation for thermal convection problems

An extension of the Oberbeck–Boussinesq approximation (OB) with expressive thermal effects is investigated to model specific mass with a temperature-dependent approach. An expansion of OB is proposed using a mathematical formulation with null velocity divergence in the continuity equation and variable specific mass in the momentum and energy equations (NOB). The NOB method has been previously tested in the literature, and the present paper aims to carry out a quantitative analysis between NOB and OB. The specific mass is calculated based on the temperature field and a divergence-free velocity is imposed on the NOB due to the small effects of variations of the specific mass in the continuity equation, as previously demonstrated by the OB. The purpose of the NOB is to overcome the OB’s restriction to single-phase flows, as well as to model the effects of specific mass variations more accurately, especially in problems with prominent thermal transfer effects, as that occur in the turbulent regime. Since the effects of a variable specific mass in the momentum and energy equations were taken into account by the NOB, it was expected that the thermal transfer results of the NOB would be improved compared with those of the OB. Then, three-dimensional natural convection simulations for a large range of Rayleigh numbers were performed using OB and NOB in a cubic cavity. Turbulence modeling was performed using Large Eddy Simulation (LES). Numerical results from both approaches were validated and all the results presented good quality. Thermal transfer was evaluated by the calculation of the Nusselt number at the heated wall and compared to experimental data from the literature. The OB and NOB provided adequate thermal transfer rate results; however, the differences between the literature and OB were higher than for NOB. Therefore, NOB is presented as a useful mathematical formulation for modeling incompressible flows with a temperature-dependent specific mass approach in single-phase or multiphase problems instead of solving the full compressible mathematical formulation. In addition, the differences between the literature and OB results increased as the Rayleigh number was raised, especially in the turbulent regime. The results of the NOB were closer to the literature than were those of the OB for the entire range of Rayleigh numbers tested. Therefore, the numerical results confirmed the higher accuracy of NOB compared to OB despite the NOB’s still being an approximate model. Lastly, flow visualization allowed the identification of coherent turbulent structures near the cavity walls in the simulation with Rayleigh number $$10^{10}$$ . The presence of hairpin and Tollmien–Schlichting instabilities revealed the importance of modeling the three-dimensional effects of natural convection in the turbulent regime.

Description and detection of burst events in turbulent flows

A mathematical and computational framework is developed for the detection and identification of coherent structures in turbulent wall-bounded shear flows. In a first step, this data-based technique will use an embedding methodology to formulate the fluid motion as a phase-space trajectory, from which state-transition probabilities can be computed. Within this formalism, a second step then applies repeated clustering and graph-community techniques to determine a hierarchy of coherent structures ranked by their persistencies. This latter information will be used to detect highly transitory states that act as precursors to violent and intermittent events in turbulent fluid motion (e.g., bursts). Used as an analysis tool, this technique allows the objective identification of intermittent (but important) events in turbulent fluid motion; however, it also lays the foundation for advanced control strategies for their manipulation. The techniques are applied to low-dimensional model equations for turbulent transport, such as the self-sustaining process (SSP), for varying levels of complexity.

Coherent structure dynamics and identification during the multistage transitions of polymeric turbulent channel flow

Drag reduction induced by polymer additives in wall-bounded turbulence has been studied for decades. A small dosage of polymer additives can drastically reduce the energy dissipation in turbulent flows and alter the flow structures at the same time. As the polymer-induced fluid elasticity increases, drag reduction goes through several stages of transition with drastically different flow statistics. While much attention in the area of polymer-turbulence interactions has been focused on the onset and the asymptotic stage of maximum drag reduction, the transition between the two intermediate stages – low-extent drag reduction (LDR) and high-extent drag reduction (HDR) – likely reflects a qualitative change in the underlying vortex dynamics according to our recent study [1]. In particular, we proposed that polymers start to suppress the lift-up and bursting of vortices at HDR, leading to the localization of turbulent structures. To test our hypothesis, a statistically robust conditional sampling algorithm, based on Jenong and Hussain [2]’s work, was adopted in this study. The comparison of conditional eddies between the Newtonian and the highly elastic turbulence shows that (i) the lifting “strength” of vortices is suppressed by polymers as reflected by the decreasing lifting angle of the conditional eddy and (ii) the curvature of vortices is also eliminated as the orientation of the head of the conditional eddy changes. In summary, the results of conditional sampling support our hypothesis of polymer-turbulence interactions during the LDR-HDR transition.

Wall-based identification of coherent structures in wall-bounded turbulence

During the last decades, a number of reduced order models based on coherent structures have been proposed to describe wall-bounded turbulence. Many of these models emphasize the importance of coherent wall-normal velocity eddies (ν-eddies), which drive the generation of the very long streamwise velocity structures observed in the logarithmic and outer region. In order to use these models to improve our ability to control wall-bounded turbulence in realistic applications, these ν-eddies need to be identified from the wall in a non-intrusive way. In this paper, the possibility of using the pressure signal at the wall to identify these ν-eddies is explored, analyzing the cross-correlation between the wall-normal velocity component and the pressure fluctuations at the wall in a DNS of a turbulent channel flow at Reτ = 939. The results show that the cross-correlation has a region of negative correlation upstream, and a region of positive correlation backwards. In the spanwise direction the correlation decays monotonously, except very close to the wall where a change of sign of the correlation coefficient is observed. Moreover, filtering the pressure fluctuations at the wall in space results in an increase of the region where the cross-correlation is strong, both for the positively and the negatively correlated regions. The use of a time filter for the pressure fluctuations at the wall yields different results, displacing the regions of strong correlation without changing much their sizes. The results suggest that space-filtering the pressure at the wall is a feasible way to identify ν-eddies of different sizes, which could be used to trigger turbulent control strategies.

Study on turbulence characteristics and sensitivity of quadrant analysis to threshold level in Lake Taihu

The identification of coherent structures is very important in investigating the sediment transport mechanism and controlling the eutrophication in shallow lakes. This study analyzed the turbulence characteristics and the sensitivity of quadrant analysis to threshold level. Simultaneous in situ measurements of velocities and suspended sediment concentration (SSC) were conducted in Lake Taihu with acoustic Doppler velocimeter (ADV) and optical backscatter sensor (OBS) instruments. The results show that the increase in hole size makes the difference between dominant and non-dominant events more distinct. Wind velocity determines the frequency of occurrence of sweep and ejection events, which provide dominant contributions to the Reynolds stress. The increase of wind velocity enlarges the magnitude of coherent events but has little impact on the events frequency with the same hole size. The events occurring within short periods provide large contributions to the momentum flux. Transportation and diffusion of sediment are in control of the intermittent coherent events to a large extent.

Quasi-bivariate variational mode decomposition as a tool of scale analysis in wall-bounded turbulence

The identification and separation of multi-scale coherent structures is a critical task for the study of scale interaction in wall-bounded turbulence. Here, we propose a quasi-bivariate variational mode decomposition (QB-VMD) method to extract structures with various scales from instantaneous two-dimensional (2D) velocity field which has only one primary dimension. This method is developed from the one-dimensional VMD algorithm proposed by Dragomiretskiy and Zosso (IEEE Trans Signal Process 62:531–544, 2014) to cope with a quasi-2D scenario. It poses the feature of length-scale bandwidth constraint along the decomposed dimension, together with the central frequency re-balancing along the non-decomposed dimension. The feasibility of this method is tested on both a synthetic flow field and a turbulent boundary layer at moderate Reynolds number ( $$Re_{\tau }$$ = 3458) measured by 2D particle image velocimetry (PIV). Some other popular scale separation tools, including pseudo-bi-dimensional empirical mode decomposition (PB-EMD), bi-dimensional EMD (B-EMD) and proper orthogonal decomposition (POD), are also tested for comparison. Among all these methods, QB-VMD shows advantages in both scale characterization and energy recovery. More importantly, the mode mixing problem, which degrades the performance of EMD-based methods, is avoided or minimized in QB-VMD. Finally, QB-VMD analysis of the wall-parallel plane in the log layer (at $$y/\delta$$ = 0.12) of the studied turbulent boundary layer shows the coexistence of large- or very large-scale motions (LSMs or VLSMs) and inner-scaled structures, which can be fully decomposed in both physical and spectral domains.

Identification of coherent structures in the flow past a NACA0012 airfoil via proper orthogonal decomposition

Proper orthogonal decomposition, POD, is employed to identify coherent structures in the turbulent flow past a NACA0012 airfoil. An analysis of different POD techniques is presented including the standard snapshot method, the spectral POD (SPOD) method, and a Fourier-POD implementation combined with the SPOD. The latter technique is viable once the present turbulent flow has a homogeneous direction along the airfoil span. An assessment of different vector norms and filter functions employed in the POD reconstruction is presented. The evaluation of the several POD techniques allows the investigation of the physics of the current turbulent flow including the observation of coherent structures and their relation to the airfoil noise generation mechanisms. The SPOD technique is applied assuming periodic and non-periodic temporal signals in the construction of the covariance matrix. Although it slightly alters the spectrum of the POD singular values, the periodic SPOD considerably reduces the noise observed in the singular vectors. The application of the Gaussian filtering allows an enhanced control in the response of the SPOD when compared with a square-box filter. The present POD reconstructions employ norms based on kinetic energy and pressure. For both norms, two-dimensional coherent structures are observed along the turbulent boundary layer and wake regions for the first pair of modes. These structures have a spectral content at the same frequency of the tonal noise radiation by the airfoil. The POD analysis also allows the identification of further coherent structures in the present flow. The kinetic energy norm is able to reconstruct low-frequency structures in the flow field for higher POD modes, while the pressure norm reconstructs high-frequency structures. This behavior is related to the physics captured by each POD norm in the present turbulent flow.

Coherent structure colouring: identification of coherent structures from sparse data using graph theory

We present a frame-invariant method for detecting coherent structures from Lagrangian flow trajectories that can be sparse in number, as is the case in many fluid mechanics applications of practical interest. The method, based on principles used in graph colouring and spectral graph drawing algorithms, examines a measure of the kinematic dissimilarity of all pairs of fluid trajectories, measured either experimentally, e.g. using particle tracking velocimetry, or numerically, by advecting fluid particles in the Eulerian velocity field. Coherence is assigned to groups of particles whose kinematics remain similar throughout the time interval for which trajectory data are available, regardless of their physical proximity to one another. Through the use of several analytical and experimental validation cases, this algorithm is shown to robustly detect coherent structures using significantly less flow data than are required by existing spectral graph theory methods.

Identification and characterization of coherent structures in gasoline injector nozzle flow using proper orthogonal decomposition

Proper orthogonal decomposition (POD) methodology is used to identify and characterize coherent structures in the gasoline nozzle flow. A correlation matrix is generated from the fluctuating velocity data snapshots produced by in-house Large Eddy simulations (LES) of an internal flow of a gasoline nozzle at a pressure of 1.1 MPa (11 bar). The modes obtained from the solution of correlation matrix identify the most energetic and dominant recirculation zones along with strong axial and radial pulsations. Initial modes show big vortical structures and inter-mode coupling but later modes identify less energetic, smaller and uncoupled structures. The dominant modes demonstrate the key role of highly energetic large coherent structures in the flow organization at the exit of the injector. The frequencies obtained from Fast Fourier transform (FFT) analysis of axial and radial fluctuating velocities give an additional insight to the characteristics of important coherent structures that govern the nozzle flow.

Creating analytically divergence-free velocity fields from grid-based data

We present a method, based on B-splines, to calculate a C2 continuous analytic vector potential from discrete 3D velocity data on a regular grid. A continuous analytically divergence-free velocity field can then be obtained from the curl of the potential. This field can be used to robustly and accurately integrate particle trajectories in incompressible flow fields. Based on the method of Finn and Chacon (2005) [10] this new method ensures that the analytic velocity field matches the grid values almost everywhere, with errors that are two to four orders of magnitude lower than those of existing methods. We demonstrate its application to three different problems (each in a different coordinate system) and provide details of the specifics required in each case. We show how the additional accuracy of the method results in qualitatively and quantitatively superior trajectories that results in more accurate identification of Lagrangian coherent structures.

Free-stream turbulence effects on the boundary layer of a high-lift low-pressure-turbine blade

The suction side boundary layer evolution of a high-lift low-pressure turbine cascade has been experimentally investigated at low and high free-stream turbulence intensity conditions. Measurements have been carried out in order to analyze the boundary layer transition and separation processes at a low Reynolds number, under both steady and unsteady inflows. Static pressure distributions along the blade surfaces as well as total pressure distributions in a downstream tangential plane have been measured to evaluate the overall aerodynamic efficiency of the blade for the different conditions. Particle Image Velocimetry has been adopted to analyze the time-mean and time-varying velocity fields. The flow field has been surveyed in two orthogonal planes (a blade-to-blade plane and a wall-parallel one). These measurements allow the identification of the Kelvin-Helmholtz large scale coherent structures shed as a consequence of the boundary layer laminar separation under steady inflow, as well as the investigation of the three-dimensional effects induced by the intermittent passage of low and high speed streaks. A close inspection of the time-mean velocity profiles as well as of the boundary layer integral parameters helps to characterize the suction side boundary layer state, thus justifying the influence of free-stream turbulence intensity on the blade aerodynamic losses measured under steady and unsteady inflows.

Spectral proper orthogonal decomposition

The identification of coherent structures from experimental or numerical data is an essential task when conducting research in fluid dynamics. This typically involves the construction of an empirical mode base that appropriately captures the dominant flow structures. The most prominent candidates are the energy-ranked proper orthogonal decomposition (POD) and the frequency-ranked Fourier decomposition and dynamic mode decomposition (DMD). However, these methods are not suitable when the relevant coherent structures occur at low energies or at multiple frequencies, which is often the case. To overcome the deficit of these ‘rigid’ approaches, we propose a new method termed spectral proper orthogonal decomposition (SPOD). It is based on classical POD and it can be applied to spatially and temporally resolved data. The new method involves an additional temporal constraint that enables a clear separation of phenomena that occur at multiple frequencies and energies. SPOD allows for a continuous shifting from the energetically optimal POD to the spectrally pure Fourier decomposition by changing a single parameter. In this article, SPOD is motivated from phenomenological considerations of the POD autocorrelation matrix and justified from dynamical systems theory. The new method is further applied to three sets of PIV measurements of flows from very different engineering problems. We consider the flow of a swirl-stabilized combustor, the wake of an airfoil with a Gurney flap and the flow field of the sweeping jet behind a fluidic oscillator. For these examples, the commonly used methods fail to assign the relevant coherent structures to single modes. The SPOD, however, achieves a proper separation of spatially and temporally coherent structures, which are either hidden in stochastic turbulent fluctuations or spread over a wide frequency range. The SPOD requires only one additional parameter, which can be estimated from the basic time scales of the flow. In spite of all these benefits, the algorithmic complexity and computational cost of the SPOD are only marginally greater than those of the snapshot POD.