Lawrence Ukeiley

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.

Suppression of Pressure Loads in Cavity Flows

The need to suppress dynamic-pressure loads in open cavities represents an important problem in many aeronautical applications. Many studies have been conducted using passive and active control techniques at the leading and trailing edges of cavities that have shown some success at reducing the dynamic-pressure levels in simulated weapons bays. In this work a leading-edge fence along with a cylindrical rod, suspended in the approaching boundary layer parallel to the leading edge of the cavity, was examined. The overall pressure levels along with the spectral distribution of the surface pressure in the cavity have been shown to he altered in a favorable manner by both of these devices. Suppressing the dynamic-pressure levels in the cavity was also found to alter the correlation between sensors along the floor cavity. Although it was found that both leading-edge devices lift the shear layer away from the cavity, the manner in which it is lifted appears to play an important role in the level of surface-pressure suppression.

Low-dimensional characteristics of a transonic jet. Part 2. Estimate and far-field prediction

Complementary low-dimensional techniques are modified to estimate the most energetic turbulent features of a Mach 0.85 axisymmetric jet in the flows near-field regions via spectral linear stochastic estimation. This model estimate is three-dimensional, comprises all three components of the velocity field and is time resolved. The technique employs the pressure field as the unconditional input, measured within the hydrodynamic periphery of the jet flow where signatures (pressure) are known to comprise a reasonable footprint of the turbulent large-scale structure. Spectral estimation coefficients are derived from the joint second-order statistics between coefficients that are representative of the low-order pressure field (Fourier-azimuthal decomposition) and of the low-order velocity field (proper orthogonal decomposition). A bursting-like event is observed in the low-dimensional estimate and is similar to what was found in the low-speed jet studies of others. A number of low-dimensional estimates are created using different velocity-pressure mode combinations from which predictions of the far-field acoustics are invoked using Lighthills analogy. The overall sound pressure level (OASPL) directivity is determined from the far-field prediction, which comprises qualitatively similar trends when compared to direct measurements at r/D=75. Retarded time topologies of the predicted field at 90° and 30° are also shown to manifest, respectively, high- and low-frequency wave-like motions when using a combination of only the low-order velocity modes (m = 0, 1, 2). This work thus constitutes a first step in developing low-dimensional and dynamical system models from hydrodynamic pressure signatures for estimating and predicting the behaviour of the energy-containing events that govern many of the physical constituents of turbulent flows.

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.

On the near field pressure of a transonic axisymmetric jet

The near field fluctuating pressure of a cold subsonic jet (Mach 0.85) issuing from a contoured convergent nozzle was studied. Conventional time series analysis (i.e., correlation and spectral analysis) as well as a wavelet analysis were used to characterize time-dependent events thereby revealing features consistent with current understandings of jet noise while also uncovering modal features consistent with previous measurements of the jet flow field itself. Specifically, the measurements reveal extensive sources of multiple frequencies in the streamwise direction. The near field information of the present investigation should help to bridge our understanding of the relationship between the velocity field and the far field acoustic emission associated with subsonic jet noise as well as supply a data set that can be used for verification of computational efforts.

Properties of subsonic open cavity flow fields

Flow over an open cavity was studied for several different subsonic free stream Mach numbers ranging from 0.19 to nearly 0.73. Velocity field information was acquired through an application of particle image velocimetry, while the fluctuating surface pressure was acquired through a linear array of surface pressure sensors. These data were acquired on the centerline of the cavity which had a length to depth ratio of 6 and a turbulent boundary layer upstream of its leading edge. Over the range of free stream Mach numbers the fluctuating surface pressure spectra in the cavity exhibited different behavior ranging from no apparent resonance to resonance being dominated by the second or third Rossiter modes. The broadband levels of surface pressure spectra with strong resonant tones collapse with scaling by the flow dynamic pressure. Velocity measurements reveal that the center of circulation of the flow within the cavity moves from the aft wall towards the center of the cavity with increasing Mach number. The ...

Control of Pressure Loads in Geometrically Complex Cavities

The need to reduce the fluctuating surface pressure loads in realistic three-dimensional cavity configurations is clear for many applications. In this paper, we describe the results of an experimental study that examined the properties of flow over a highly three-dimensional cavity, which included angled side walls and a sloped floor. The unsteady pressure measurements revealed that the primary spectral properties, such as the frequencies of the cavity tones, are very similar to those of simpler, rectangular cavities. The study also explored the effects of different active fluidic injection methods, at the cavity leading edge, on the unsteady loads generated in the cavity. Specifically, the two active suppression concepts examined were microjets and rectangular slots at the leading edge. Both concepts showed significant reductions in the fluctuating surface pressures, upwards of 50% on the cavity aft wall, with very modest amounts of mass flowing through the injectors. When appropriately scaled for full-scale applications, the actuator mass flux required falls well within the practical range for most aircraft. Different angles for the fluidic Injection were also examined and maximum reductions were observed when injection was perpendicular to the approaching freestream flow. Additionally, the best blowing configurations were found when the injectors did not fully span the leading edge of the cavity. The reductions observed in the fluctuating surface pressure levels resulted from decreases in both the broadband and resonant features of the surface pressures. By conducting these experiments at two different facilities and over a range offreestream dynamic pressures and temperatures, this study also demonstrated that appropriate scaling of the spectral features can be achieved. This allows for the expansion of the results presented here to larger (and different) scale studies and ultimately to full-scale applications.

Modified quadratic stochastic estimation of resonating subsonic cavity flow

There is an obvious need for the development of reduced-order time-resolved models of the off-body physics of wall-bounded shear flows. This work represents a preliminary step toward producing such a model for the resonating subsonic cavity flow by presenting the modified quadratic stochastic estimation (mQSE) as a means to estimate flow field quantities using surface pressure measurements. The mQSE estimates these flow field quantities using statistics calculated between surface-pressure fluctuations and the flow fields POD expansion coefficients. In the current work, the mQSE is applied separately to both the velocity and density fields for subsonic flow over a resonating cavity. The velocity data are obtained using particle image velocimetry, while the density gradients are obtained using spark Schlieren photography. The results demonstrate the ability of the mQSE to estimate the structure of the complex cavity flow field using a few wall-pressure measurements. The results also show an interaction bet...

Spatial Correlations in a Transonic Jet

properties. The data are presented from two separate experiments; one with the light sheet orientated in the streamwise direction (r–x plane) and one with the light sheet perpendicular to the flow direction (r–� plane). The instrument’s characteristics allow for the calculation and subsequent analysis of the two-point spatial correlations which are known to be relevant to the source terms in acoustics analogies where sound production is concerned. An examination of the spatial correlations demonstrates the averaged spatial evolution of the jet’s large-scale turbulent structures throughout the noise producing region. In particular, the (r–x) spatial dependence of the axial and azimuthal normal stresses manifest a oblique structure in the mixing layer regions of the flow, whereas the radial normalstressesevolvemoreuniformlytowardtheendofthepotentialcore.Quadrupolesourcetermsrelevanttothe soundproduction mechanismsare alsocalculated from which their spatial distributions areanalyzed with zerotime delay.Theanalysisofthesesourcetermsatx=D � 4showhowthepeakenergyfortheshear-noisecomponentresides on the high-speed side of the shear layer around r=D � 0:33, whereas the self-noise terms peak along the lip-line at r=D � 0:5,andaremostenergeticforthestreamwiseandazimuthalcomponentsofthevelocity.Tofullyevaluatethe quadrupole sources of noise, the space-time correlations of the full three-dimensional turbulent flowfield are required which are currently not available from experiments.