Mark N. Glauser

The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet

It is shown that the pressure signal measured at the outer edge of a jet mixing layer is entirely hydrodynamic in nature and provides a good measure of the large-scale structure of the turbulent flow. Measurement of the pressure signal provides a unique opportunity to utilize proper orthogonal decomposition (POD) to deduce the streamwise structure. Since pressure is a scalar, a significant reduction in the numerical and experimental complexity inherent in the analysis of velocity vector fields results. The POD streamwise eigenfunctions show that the structure associated with any frequency‐azimuthal mode number combination displays the general characteristics of amplification‐saturation‐decay of an instability wave, all within about three wavelengths. High-frequency components saturate early in x and low-frequency components saturate further downstream, indicative of the inhomogeneous character of the flow in the streamwise direction. Application of the POD technique allows the phase velocity to be determined taking into account the inhomogeneity of the flow in the streamwise direction. The phase velocity of each instability wave (POD eigenvector) is constant and equal to 0.58U j , indicating that the jet structure is non-dispersive. Using the shot-noise decomposition, a characteristic event is constructed. This event is found to contain evidence of both pairings and triplings of vortex structures. The tripling results in a rapid increase in the first asymmetric (m fl 1) component. On average, pairing occurs once every four U j }D while tripling occurs once every 13U j }D. This paper deals with the identification of large-scale structures in turbulent jets through an examination of the near-field pressure signal which is an outgrowth of studies going back more than 20 years (Arndt & George 1974). The measurement procedure is based on the proper orthogonal decomposition (POD) originally proposed by Lumley (1967) for the study of spatial structure associated with inhomogeneous flows. The measurements reported herein are based on work initiated in 1980 to study the influence of coherent structures on jet noise radiation using these earlier concepts. At the time no reliable method for deducing coherent structures in high-Reynolds-number flows in an unbiased manner was available. A review of the literature indicated that coherent structures were readily identified in clean jets at low Mach number and Reynolds numbers less than about 10&. However, subsonic jet noise measurements were made at higher Mach numbers and correspondingly higher Reynolds numbers as reviewed by Long & Arndt (1984). Clearly, there was a need to deduce the presence of

Stochastic estimation and proper orthogonal decomposition: Complementary techniques for identifying structure

The Proper Orthogonal Decomposition (POD) as introduced by Lumley and the Linear Stochastic Estimation (LSE) as introduced by Adrian are used to identify structure in the axisymmetric jet shear layer and the 2-D mixing layer. In this paper we will briefly discuss the application of each method, then focus on a novel technique which employs the strengths of each. This complementary technique consists of projecting the estimated velocity field obtained from application of LSE onto the POD eigenfunctions to obtain estimated random coefficients. These estimated random coefficients are then used in conjunction with the POD eigenfunctions to reconstruct the estimated random velocity field. A qualitative comparison between the first POD mode representation of the estimated random velocity field and that obtained utilizing the original measured field indicates that the two are remarkably similar, in both flows. In order to quantitatively assess the technique, the root mean square (RMS) velocities are computed from the estimated and original velocity fields and comparisons made. In both flows the RMS velocities captured using the first POD mode of the estimated field are very close to those obtained from the first POD mode of the unestimated original field. These results show that the complementary technique, which combines LSE and POD, allows one to obtain time dependent information from the POD while greatly reducing the amount of instantaneous data required. Hence, it may not be necessary to measure the instantaneous velocity field at all points in spacesimultaneously to obtain the phase of the structures, but only at a few select spatial positions. Moreover, this type of an approach can possibly be used to verify or check low dimensional dynamical systems models for the POD coefficients (for the first POD mode) which are currently being developed for both of these flows.

Examination of large-scale structures in a turbulent plane mixing layer. Part 1. Proper orthogonal decomposition

Large-scale structures in a plane turbulent mixing layer are studied through the use of the proper orthogonal decomposition (POD). Extensive experimental measurements are obtained in a turbulent plane mixing layer by means of two cross-wire rakes aligned normal to the direction of the mean shear and perpendicular to the mean flow direction. The measurements are acquired well into the asymptotic region. From the measured velocities the two-point spectral tensor is calculated as a function of separation in the cross-stream direction and spanwise and streamwise wavenumbers. The continuity equation is then used for the calculation of the non-measured components of the tensor. The POD is applied using the cross-spectral tensor as its kernel. This decomposition yields an optimal basis set in the mean square sense. The energy contained in the POD modes converges rapidly with the first mode being dominant (49% of the turbulent kinetic energy). Examination of these modes shows that the first mode contains evidence of both known flow organizations in the mixing layer, i.e. quasi-two-dimensional spanwise structures and streamwise aligned vortices. Using the shot-noise theory the dominant mode of the POD is transformed back into physical space. This structure is also indicative of the known flow organizations.

Proportional closed-loop feedback control of flow separation

The aim of this experimental study is the implementation of a practical and efficient closed-loop feedback control of the turbulent flow over a NACA-4412 airfoil equipped with leading-edge zero-net-mass-flux actuators. By using prior computation of correlations between particle image velocimetry data and multiple surface pressure measurements, real-time instantaneous low-dimensional estimates of the velocity field over the wing are then computed from the unsteady surface pressure. From such estimates, a direct knowledge of the state of the flow above the airfoil is obtained (i.e., attached, incipient separation, or fully separated flow). We first show the effectiveness of the low-dimensional modeling approach in extracting and estimating the underlying large-scale structures in a turbulent flow, using the proper orthogonal decomposition and the modified linear/quadratic stochastic measurements. We then show how such an approach is used successfully in a simple, but practical, proportional feedback loop to delay the separation of the flow over the wing at high angles of attack. The benefits of closed-loop vs open-loop control are then discussed. These fundamental results validate the use of low-dimensional modeling techniques for further, more sophisticated, closed-loop feedback control algorithms.

Coherent structures in the axisymmetric turbulent jet mixing layer.

In 1967 Lumley proposed two different, but complimentary approaches to the objective determination of coherent structures. The first uses an orthogonal decomposition to extract eigenvectors from two point velocity measurements, the lowest order eigenvector representing the largest structure. Where there are partial homogeneities, or when the flow is stationary, the eigenfunctions are the harmonic ones and the coherent features are impossible to identify. To organize these fluctuating Fourier modes into coherent features, a second decomposition is used, the shot-noise decomposition.

An application of the stochastic estimation to the jet mixing layer

The linear stochastic estimation is a powerful technique that provides a means of estimating conditional eddies given unconditional two‐point correlation data. This procedure was used to reconstruct estimates of multipoint conditional averages of the dominant structures in the jet mixing layer of Glauser and George (Proceedings of the Sixth Symposium on Turbulent Shear Flows, Toulouse, France, 1987). The pseudodynamic evolution of these conditional eddies was systematically compared to the instantaneous velocity fields and the results were quantified in terms of the percentage of the energy captured by the multipoint stochastic estimates. It was found that the single‐point estimates do not yield adequate representations of the instantaneous velocity field, but that two reference points located on opposite sides of the shear layer yield realistic estimates, with little gained by adding more reference points.

Orthogonal Decomposition of the Axisymmetric Jet Mixing Layer Including Azimuthal Dependence

In 1967 LUMLEY [4] proposed an approach to the objective determination of coherent structures. The method uses an orthogonal decomposition to extract eigenvectors from two point velocity measurements, the lowest order eigenvector representing the largest structure. If the flow is homogenous, stationary or periodic the orthogonal decomposition reduces to a harmonic decomposition. These directions therefore are fit with the harmonic eigenfunctions before applying Lumley’s orthogonal decomposition to the inhomogeneous directions.

Examination of large-scale structures in a turbulent plane mixing layer. Part 2. Dynamical systems model

The temporal dynamics of large-scale structures in a plane turbulent mixing layer are studied through the development of a low-order dynamical system of ordinary differential equations (ODEs). This model is derived by projecting Navier–Stokes equations onto an empirical basis set from the proper orthogonal decomposition (POD) using a Galerkin method. To obtain this low-dimensional set of equations, a truncation is performed that only includes the first POD mode for selected streamwise/spanwise ( k 1 / k 3 ) modes. The initial truncations are for k 3 = 0; however, once these truncations are evaluated, non-zero spanwise wavenumbers are added. These truncated systems of equations are then examined in the pseudo-Fourier space in which they are solved and by reconstructing the velocity field. Two different methods for closing the mean streamwise velocity are evaluated that show the importance of introducing, into the low-order dynamical system, a term allowing feedback between the turbulent and mean flows. The results of the numerical simulations show a strongly periodic flow indicative of the spanwise vorticity. The simulated flow had the correct energy distributions in the cross-stream direction. These models also indicated that the events associated with the centre of the mixing layer lead the temporal dynamics. For truncations involving both spanwise and streamwise wavenumbers, the reconstructed velocity field exhibits the main spanwise and streamwise vortical structures known to exist in this flow. The streamwise aligned vorticity is shown to connect spanwise vortex tubes.

Application of multipoint measurements for flow characterization

Abstract Some of the possibilities for inferring the structure of complicated flows from simultaneous measurements at many points are reviewed. Conditional sampling, pseudo-flow visualization, stochastic estimation, and the proper orthogonal decomposition are briefly reviewed and illustrated by example. Resolution criteria for multipoint spatial arrays are proposed that minimize the possibilities for misinterpreting the data.

Eddy Structure Identification in Free Turbulent Shear Flows

This book is a unique opportunity to present in a single volume information that is needed for both experimentalists, theoreticians and computationalists for the detection, analysis, prediction and control of eddy structures in turbulent shear flows. Major identification techniques of Eddy Structures in Turbulent Shear Flows are presented together with applications to vortex dynamics, turbulence management and flow control, for experimental and numerical applications with new prediction methods: Eduction Schemes, Proper Orthogonal Decomposition, Stochastic Estimation, Pattern Recognition Analysis, Wavelet Transform. Illustrations of the use of the different methods are given.