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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.

Experiments in Fluids 17 (i994) 307 314 © Springer-Verlag 1994 decomposition: ComplementaryAbstract 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 space simultaneously 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.

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The true global source of the farfield sound radiated from a subsonic jet is the entire dynamic found within the confines of the hydrodynamic field, a dynamic which comprises an unsteady compressive excitation of the medium resulting from turbulent mixing and unsteady temperature fluctuations, these being driven by a wide range of turbulence scales. Improved understanding, and subsequent modelling of the space-time character of the global source term has been acheived in the past through study of the different physical mechanisms implicated in its dynamic. The phenomena which have to date been accepted as important in terms of the radiated sound are (i) turbulent mixing and shear, (ii) fluctuating entropy, (iii) convective amplification and (iv) refraction and scattering of sound by the mean and turbulent components of the velocity field. Once identified as important, specific modelling strategies can and have been developed in order to deal with these phenomena. The existence of coherent structures in turbulent jets was identified as important in the 1970's and their role in the production of sound has received considerable attention in more recent years. However, direct identification of the causal link between this component of the turbulence and its sound field is a delicate matter due to the virtual impossibility of directly measuring the source dynamic of the flow, this being due to an overwhelming dominance of hydrodynamic energy in the source region, and a total absence of any hydrodynamic signature in the linear, acoustic region. As a result, modelling strategies where this component of the source term is concerned have remained, at best, highly empirical.An interesting region of the flow where our understanding of the relationship between the hydrodynamic cause and its acoustic effect is concerned is the near pressure field, found just outside the rotational region of the flow. In this region the signature of the sound production mechanism and its resultant sound field are both present, coincident in space and in time.

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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 (k1/k3) modes. The initial truncations are for k3 = 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.