Eduction Of Coherent Structures Research Articles

Eduction of coherent structures from schlieren images of twin jets using SPOD informed with momentum potential theory in the spectral domain

Abstract This work presents a methodology to extract coherent structures from high-speed schlieren images of turbulent twin jets which are more physically interpretable than those obtained with currently existing techniques. Recently, Prasad and Gaitonde (J Fluid Mech 940:1–11, 2022) introduced an approach which employs the momentum potential theory of Doak (J Sound Vib 131(1):67–90, 1989) to compute potential (acoustic and thermal) energy fluctuations from the schlieren images by solving a Poisson equation, and combines it with spectral proper orthogonal decomposition (SPOD) to educe coherent structures from the momentum potential field instead of the original schlieren field. While the latter field is dominated by a broad range of vortical fluctuations in the turbulent mixing region of unheated high-speed jets, the momentum potential field is governed by fluctuations which are intimately related to acoustic emission, and its spatial structure in the frequency domain is very organized. The proposed methodology in this paper improves the technique of Prasad and Gaitonde (J Fluid Mech 940:1–11, 2022) in three new ways. First, the solution of the Poisson equation is carried out in the frequency-wavenumber domain instead of the time-space domain, which simplifies and integrates the solution of the Poisson equation within the SPOD framework based on momentum potential fluctuations. Second, the issue of solving the Poisson equation on a finite domain with ad hoc boundary conditions is explicitly addressed, identifying and removing those unphysical harmonic components introduced in the solution process. Third, the solution of the SPOD problem in terms of momentum potential fluctuations is used to reconstruct schlieren SPOD fields associated with each mode, allowing the visualization of the obtained coherent structures also in terms of the density gradient. The method is applied here to schlieren images of a twin-jet configuration with a small jet separation at two supersonic operation conditions: a perfectly-expanded and an overexpanded one. The SPOD modes based on momentum potential fluctuations retain the wavepacket structure including the direct Mach-wave radiation, together with upstream- and downstream-traveling acoustic waves, similar to SPOD modes based on the schlieren images. However, for the same dataset, they result in a lower-rank decomposition than schlieren-based SPOD and provide an effective separation of twin-jet fluctuations into independent toroidal and flapping oscillations that are recovered as different SPOD modes. These coherent structures are more consistent with twin-jet wavepacket models available in the literature than those originally obtained with direct schlieren-based SPOD, facilitating their interpretation and comparison against theoretical analyses. Graphical abstract

The effect of flight on a turbulent jet: coherent structure eduction and resolvent analysis

The effect of flight on a turbulent jet: coherent structure eduction and resolvent analysis

Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability

Coherent features of a turbulent Mach 0.9, Reynolds number$10^{6}$jet are educed from a high-fidelity large eddy simulation. Besides the well-known Kelvin–Helmholtz instabilities of the shear layer, a new class of trapped acoustic waves is identified in the potential core. A global linear stability analysis based on the turbulent mean flow is conducted. The trapped acoustic waves form branches of discrete eigenvalues in the global spectrum, and the corresponding global modes accurately match the educed structures. Discrete trapped acoustic modes occur in a hierarchy determined by their radial and axial order. A local dispersion relation is constructed from the global modes and found to agree favourably with an empirical dispersion relation educed from the simulation data. The product between direct and adjoint modes is then used to isolate the trapped waves. Under certain conditions, resonance in the form of a beating occurs between trapped acoustic waves of positive and negative group velocities. This resonance explains why the trapped modes are prominently observed in the simulation and as tones in previous experimental studies. In the past, these tones were attributed to external factors. Here, we show that they are an intrinsic feature of high-subsonic jets that can be unambiguously identified by a global linear stability analysis.

Investigation of coherent structures in a turbulent channel with built-in longitudinal vortex generators

The present study discusses the results pertaining to eduction of coherent structures and assesses their contribution in promoting mixing between low and high momentum fluid streams to enhance thermal convection. The analysis has been carried out numerically in a periodic parallel-plate-channel containing built-in longitudinal vortex generators on the bottom wall. The Reynolds number of the investigation is 6000 based on the channel height and bulk mean velocity. The computational geometry resembles heat exchanger surfaces on which the vortex generators create strong longitudinal vortices that enhance heat transfer on the plane channel wall. At high Reynolds numbers, the enhancement of convective heat transfer near the wall largely depends on the large scale turbulent structures (coherent structures) that disappear and reappear frequently and exhibit typical correlated scales in time or space. Proper orthogonal decomposition (POD) is an efficient tool to identify these coherent turbulent structures from instantaneous flow field data. Despite geometrical simplicity, the complex swirling motion behind the vortex generators demands highly accurate computational method, like the large eddy simulation (LES) of turbulence for capturing the nuances of flow. The instantaneous velocity data at a cross-flow plane have been assembled over a long time-window and analyzed deploying POD technique to extract the large- and small-scale structures based on the turbulent kinetic energy of the structures. Reconstructions of the velocity field using different combinations of eigenmodes have yielded different patterns of the flow. Large-scale structures have been visualized using first few energetic modes, while remaining less energetic modes have provided small-scale structures. The technique enables us to identify coherent structures hidden in the random flow field by filtering the small scale effects. The analysis yields as large as 15 dominant POD modes to cumulatively capture as low as 25% of the total turbulent kinetic energy of the flow, indicating broad range of energy spectrum that shares a large number of POD modes and thereby concludes highly complex turbulent fields with numerous spatial and temporal turbulent scales. Reconstructed velocity fields considering the leading dominant modes reveal a cut-off of 43% turbulent kinetic energy to clearly expose the coherent structures, which are consistent with published literature on similar turbulent channel flows. Additionally, quadrant analysis on the reconstructed velocity fields has clearly revealed that sweep and ejection modes are the only dominating Reynolds-stress-producing events for the current flow of interest, where the vortices lift the low-momentum fluid away from wall through the ejection mechanism. The high-momentum fluid impinges on the wall in sweep mode.

A simple technique for the visualisation of eddy kinematics in turbulent flows

We developed and tested a simple technique to predict, for flow visualisation purposes only, the evolution of coherent structures in between two given realisations of a turbulent flow. Classic coherent-structure eduction methods are adopted, such as the Q-criterion, pressure fluctuations and contours of velocity fluctuations. The kinematics of the evolving structures are reconstructed by means of an advection-based reconstruction technique and captured in a movie. The resulting quality of the animations has been assessed via the Structural Similarity Index (SSIM). The sensitivity to increasing spacing in time of the available flow realisations has been tested and several improvements implemented. The abrupt transition from reconstructed frames of the animation to the available realisations results in a noticeable lack of smoothness. The replacement of the available realisations with a similar advection-based average increases the perceived smoothness of the movies. This is confirmed by the reduced total variation of the SSIM index. The residual minor periodic variations of accuracy have been reduced by introducing a stochastic weighting function. The sensitivity of the results to changes in Reynolds number, resolution and structure representation methods has been tested.

Techniques for the Eduction of Coherent Structures from Flow Measurements in the Atmospheric Boundary Layer

Two empirical methods to detect coherent motions embedded in the flow field have been compared, namely the variable interval time average (VITA) method and a wavelet-based technique, both with artificial signals as well as velocity measurements from the atmospheric boundary layer over a forest canopy. It has been found that the wavelet method is slightly better than the VITA approach in coherent structure eduction, even if the results of both techniques are comparable. Also the application of the present approach to simultaneous conditionally sampled wind data has highlighted some important features of coherent structures and gust generation in canopy flows.

Research Progress of Wavelet Analysis in Flow Structure Characterization

The wavelet analysis is a time-frequency analysis method, which plays an important role in processing non-stationary signals obtained by Particle Image Velocimetry (PIV) to measure turbulent flow fields. Current research status for wavelet analysis in the flow field structure was reviewed: eduction of coherent structures and multi-scale structures(the large-, intermediate- and relatively small-scale structures)by wavelet transform and combine with the maximum energy criterion, extraction of coherent vortical structures and analysis of multi-scale turbulent structures information from the time–frequency diagram. Application of wavelet analysis in the flow structures was also prospected.

Flow field around a vibrating cantilever: coherent structure eduction by continuous wavelet transform and proper orthogonal decomposition

The velocity field around a vibrating cantilever plate was experimentally investigated using phase-locked particle image velocimetry. Experiments were performed at Reynolds numbers of Reh = 101, 126 and 146 based on the tip amplitude and the speed of the cantilever. The averaged vector fields indicate a pseudo-jet flow, which is dominated by vortical structures. These vortical structures are identified and characterized using the continuous wavelet transform. Three-dimensional flow features are also clearly revealed by this technique. Furthermore, proper orthogonal decomposition was used to investigate regions of vortex production and breakdown. The results show clearly that the investigation of phase-averaged data hides several key flow features. Careful data post-processing is therefore necessary to investigate the flow around the vibrating cantilever and similar highly transient periodic flows.

Coherent structure eduction from PIV data of an electromagnetically forced separated flow

Abstract Periodic addition of momentum by wall parallel electromagnetic forces has a strong influence on the separated region of a stalled airfoil. As in the case of periodic blowing and suction, actuation frequency and momentum input are the main factors of influence. However, the control authority depends as well on the actuation wave form. This latter aspect is investigated in the present paper by means of time resolved particle image velocimetry data from the suction side of an inclined flat plate. The control effect is rated by the size of the remaining recirculation region in the time averaged flow fields. Typically, the controlled flow possesses a small number of relatively large vortices, which are assumed to be related to the control mechanism. Consequently, the time resolved flow fields have been analysed by proper orthogonal decomposition and continuous wavelet transform to extract dominant features of the flow.

Coherent Structure Eduction and Convection Velocity in Compressible Blunt-Body Wakes

An iterative eduction procedure has been developed to study coherent structures in blunt-base cylinder wakes at 0 and 10 deg angles of attack to an M = 2.46 freestream flow. The eduction procedure uses an initial wavelet pattern to find candidate coherent structures of a particular scale and then uses the average intensity pattern of the educed structures to create an updated pattern. This iterative procedure continues until the average structure converges, indicating that a self-consistent intensity pattern has been described. Using this eduction procedure, the average appearance of the structures is described. Moreover, a convection velocity measurement is made by directly tracking individual coherent structures, unlike conventional imaging-based convection velocity measurements, which are not conditioned on coherent structures. The evolutionary trends of the structures are also quantified. The coherent structure averages show many of the same trends found in more conventional autocorrelation analysis. For example, the structures are inclined toward the freestream flow direction, and the angle of this inclination is affected by the recompression and reattachment processes. Convection velocity measurements show that the convection velocity is dependent on the transverse position of the structure in the shear layer, particularly for small structures. Finally, the coherent structures evolve in such a way that they become poorly representative of the average coherent structure pattern within 5-15 μs after being first identified.

Dynamic eduction of coherent structures in turbulent jet flow imagery by wavelet techniques: part I

The use of two-dimensional wavelet techniques enables us to discern, at appropriate wavelet scales, certain features of a coherent structure from a single flow visualization image. By analysing a large number of planar laser-induced fluorescence images of chiefly the diametral section of a turbulent jet, with some supplementary axial sections, we show that it is possible to describe the passage of whole coherent structures through a specified section of the flow, and thereby acquire useful information on their life cycle. In particular it is found that a typical cycle is characterized by a lobed ring-like structure about 8% of the time, a dye-filled core a similar fraction of the time, with a mixed regime in between. Based on comparisons with dye-concentration data and DNS results, it is suggested that the lobed ring seen in the wavelet transform of the raw image may represent a vortex ring exhibiting the Widnall instability.

Coherent Structures in a Jet Indicating Edgetone Oscillation (2nd Report, Eduction of Coherent Structures by Using Low Pressure Region and Velocity Gradient Tensor)

Coherent vortical structures in a high-Reynolds number turbulent jet indicating edgetone oscillations were extracted from velocity and pressure data. Three quantities were chosen as candidate indicators of the coherent structures : (i) phase-averaged vorticity, (ii) phase-averaged pressure, and (iii) the phase-averaged velocity gradient tensor. All these quantities worked well in extracting coherent structures in a test case of a low-Reynolds number wake of a cylinder. However, when applied to the high-Reynolds number edgetone flow, (ii) and (iii) indicated much better performance than (i), since (i) was affected by the background time-averaged vorticity field. The advection of the coherent structures was well captured by using (ii) and (iii).

Coherent structures in plumes with and without off-source heating using wavelet analysis of flow imagery

Laser flow visualization using fluorescent dye provides instantaneous snapshots of dye concentration across selected sections of the flow. We demonstrate the potential power of wavelet analysis in revealing the nature of organization present in such pictures of turbulent shear flows, in a case where it is difficult to discern any such organization in the raw image. The technique consists of examining the contours of the transform coefficients using two-dimensional wavelets on the raw image. It is applied here to diametral cross sections of a plume with and without off-source heating. It is found that in the unheated plume there is definite evidence for a sinuous large-scale ring-like organization, which, however, is absent when the off-source heating is switched on. The proposed technique not only enables eduction of coherent structures in flow-visualization pictures, but can also provide a wealth of scale-specific information on flow structure.

A study on spanwise heat transfer in a turbulent channel flow – eduction of coherent structures by a conditional sampling technique

Coherent structures of temperature fields in the near-wall turbulence ( y +<60) were educed by a conditional sampling technique from the DNS data. Treated situation is the fully developed turbulent channel flow for two kinds of temperature fields: (a) the uniform heat flux assigned on two walls of the channel (UHF) and (b) the constant spanwise mean temperature gradient imposed on the flow over the whole channel width (STG). Following conclusions are obtained. (1) Flow and temperature fields around the quasi-streamwise vortex are roughly uniform in the streamwise direction. (2) Temperature distribution in/around the vortex for UHF and that for STG are similar in shape with azimuthal phase difference of about 90°. (3) Mechanisms for destruction of turbulent heat flux are similar between UHF case and STG case.

Wavelet Multiresolution Analysis Applied to Coherent Structure Eduction of a Turbulent Jet.

To evaluate coherent structures in the dimension of time and scale, a definition of wavelet multiresolution autocorrelation based on the discrete wavelet transform is first developed. Then a new identification technique that combines the wavelet multiresolution analysis combined with the wavelet multiresolution autocorrelation analysis is proposed. By analyzing u- and v-components of fluctuating velocity, the coherent structure and its scales can be identified when larger local amplitude fluctuation and stronger autocorrelation appear at the same wavelet level. For a turbulent jet at a downstream distance of x/d=6, the coherent structures with frequency 39Hz can be deduced about times 0.29, 0.53, 0.6, and 0.67s. This also represents the passing of eddies through the shear layer and concentration of the energy of the flow at these instants.

A Study on Three-Dimensionality of Turbulent Heat Transfer in a Channel. Eduction of Coherent Structures by a Conditional Sampling Technique.

Coherent structures near the wall (y+<60)in a turbulent channel flow were educed by a conditional sampling technique from the DNS data for two kinds of temperature fields : (a) the uniform heat flux assigned on the two walls (UHF) and (b) the constant spanwise mean temperature gradient imposed on the working fluid (STG). Attention was paid for roles of the quasi-streamwise vortex in heat transfer mechanisms. Flow and temperature fields around the quasi-streamwise vortex have basically two-dimensional structure. In the case of UHF, temperature fluctuation is high at right-and left-sides of the vortex core where the rotating flow motion due to the vortex has a wallnormal component. In STG, large value of the temperature fluctuation was observed at the top-and bottom-sides of the vortical structure where flow in the spanwise direction is conspicuous. In both cases, gradients of the time-mean temperature and the temperature fluctuation in the same direction have opposite signs since excessive temperature is transported by the vortical flow motion in the azimuthal direction with respect to the vortex axis. Therefore, temperature distribution for UHF and that for STG have azimuthally phase difference of about 90 degree, and they are almost similar in shape. Destruction (correlation between pressure and temperature gradient) of wall-normal turbulent heat flux and that of spanwise component are based on similar mechanisms. They are caused by the rotating motion of the vortex by which, near the center axis of the vortex, pressure fluctuation takes negative value and the gradient of temperature fluctuation becomes inverted for that of the mean temperature.

Decomposition of Mixing Layer Turbulence into Coherent Structures and Background Fluctuations

The eduction of coherent structures from cross-wire rake data in a fully turbulent incompressible mixing layer confirms the feasibility of a decomposition of a turbulent flow field, first suggested by Farge, as non-periodic non-equilibrium coherent structures interacting with a ‘thermalized’ broad-band turbulence. A simple wavelet coefficient decimation algorithm and orthogonalization yields non-periodic dominant flow structures and a background field that has a Gaussian distribution of velocities at the centerline. The coherent structures are classified in terms of their topology. The non-coherent background field has flat energy spectra and normal distribution of velocity components. Most background field statistics depend only weakly on the type of structure on which they are superposed. It may be possible to adapt existing subgrid scale models to this decomposition.

Collaborative testing of eddy structure identification methods in free turbulent shear flows

The thrust of this paper is to validate, test and compare several Coherent Structure eduction methods utilizing the same data base. The flow chosen was that of an experimental study of a plane, incompressible, fully developed turbulent two-stream mixing layer. The mixing layer was chosen as the data base because it has been studied extensively from a coherent structures point of view. In addition, its characteristics (similarity, convection velocities, etc.) are well documented. There are also no wall effects so that comparisons between techniques are simplified. The data was collected from hot wire rakes with good spatial resolution thus allowing the contributors to apply and test different structure eduction techniques. The techniques chosen for discussion and used here have found wide utilization over the past decade, and all hold forth the promise of extensive application in the future. These include: Conditional Sampling (Vorticity-based and other methods); Wavelets; Pattern Recognition Analysis; Proper Orthogonal Decomposition; Stochastic Estimation; Topological Concept-based methods; Full Field Methods (e.g., pseudo flow visualization). All are illustrated by application to the mixing layer data base, and comparisons made between the results. This common study has shown that direct comparisons between results of several methods are now possible. Good quantitive and qualitative agreement between the different methods have been observed as well as some differences noted. As an example, the size of the averaged structures computed from the various methods compare to within 6 percent.

On the decomposition of turbulent flow fields for the analysis of coherent structures

The decomposition of the turbulent velocity, pressure, and vorticity fields for the analysis of coherent structures is examined from a fundamental theoretical standpoint. It is shown that the commonly used double and triple decompositions yield coherent and incoherent parts of the turbulence that are not Galilean invariant and, consequently, severe doubts are raised concerning their general usefulness for the eduction of coherent structures. Alternative triple decompositions are proposed which are invariant under an arbitrary change of observer. Applications of this triple decomposition to the construction of helicity fluctuations which are more suitable for the study of coherent structures are also discussed.

Coherent structures—reality and myth

The nature and significance of large-scale coherent structures in turblent shear flows are addressed. A definition for the coherent structure is proposed and its implications discussed. The characteristic coherent structure properties are identified and the analytical and experimental constraints in the eduction of coherent structures are examined. Following a few comments on coherent motions in wall layers, the accumulated knowledge from a number of recent and ongoing coherent structure investigations in excited and unexcited free shear flows in the author’s laboratory is reviewed. Also briefly addressed are effects of initial conditions, the role of coherent structures in jet noise production and broadband noise amplification, the feedback effect of coherent structures, the use of the Taylor hypothesis in coherent structure description, negative production, turbulence suppression via excitation, validity of the Reynolds number similarity hypothesis, etc. From the detailed quantitative results, a picture of the state of the art in coherent structure studies emerges. While coherent structures are highly interesting characteristic features of (perhaps all) turbulent shear flows, it is argued that their dynamical significance has been overemphasized. These are predominant only in their early stages of formation following instability, or in resonant situations and excited flows, or in regions adjacent to a wall of a turbulent boundary layer. The coherent Reynolds stress, vorticity, and production are comparable to (and not an order of magnitude larger than) the time-average Reynolds stress, vorticity, and production, respectively, in fully developed states of turbulent shear flows, where incoherent turbulence is also important and cannot be ignored. The concept and importance of coherent structures are here to stay; understanding and modeling of turbulent shear flows will be incomplete without them; but they are not all that matter in turbulent shear flows.