Space-time Velocity Correlations Research Articles

A Kriging-based elliptic extended anisotropic model for the turbulent boundary layer wall pressure spectrum

The present study addresses the computation of the wall pressure spectrum for a turbulent boundary layer flow without pressure gradient, at high Reynolds numbers, using a new model, the Kriging-based elliptic extended anisotropic model (KEEAM). A space–time solution to the Poisson equation for the wall pressure fluctuations is used. Both the turbulence–turbulence and turbulence–mean shear interactions are taken into account. It involves the mean velocity field and space–time velocity correlations which are modelled using Reynolds stresses and velocity correlation coefficients. We propose a new model, referred to as the extended anisotropic model, to evaluate the latter in all regions of the boundary layer. This model is an extension of the simplified anisotropic model of Gavin (PhD thesis, 2002, The Pennsylvania State University, University Park, PA) which was developed for the outer part of the boundary layer. It relies on a new expression for the spatial velocity correlation function and new parameters calibrated using the direct numerical simulation results of Sillero et al. (Phys. Fluids, vol. 26, 2014, 105109). Spatial correlation coefficients are related to space–time coefficients with the elliptic model of He & Zhang (Phys. Rev. E, vol. 73, 2006, 055303). The turbulent quantities necessary for the pressure computation are obtained by Reynolds-averaged Navier–Stokes solutions with a Reynolds stress turbulence model. Then, the pressure correlations are evaluated with a self-adaptive sampling strategy based on Kriging in order to reduce the computation time. The frequency and wavenumber–frequency wall pressure spectra obtained with the KEEAM agree well with empirical models developed for turbulent boundary layer flows without pressure gradient.

Wave-number–frequency spectrum for turbulence from a random sweeping hypothesis with mean flow

We derive the energy spectrum in wave-number-frequency space for turbulent flows based on Kraichnan's idealized random sweeping hypothesis with additional mean flow, which yields the instantaneous energy spectrum multiplied by a Gaussian frequency distribution. The model spectrum has two adjustable parameters, the mean flow velocity and the sweeping velocity, and has the property that the power-law index of the wave-number spectrum translates to the frequency spectrum, invariant for arbitrary choices of the mean velocity and sweeping velocity. The model spectrum incorporates both Taylor's frozen-in flow approximation and the random sweeping approximation in a natural way and can be used to distinguish between these two effects when applied to real time-resolved multipoint turbulence data. Evaluated in real space, its properties with respect to space-time velocity correlations are discussed, and a comparison to the recently introduced elliptic model is drawn.

Validating Large-Eddy Simulation for Jet Aeroacoustics

Engineers charged with making jet aircraft quieter have long dreamed of being able to see exactly how turbulent eddies produce sound, and this dream is now coming true with the advent of large-eddy simulation. Two obvious challenges remain: validating the large-eddy-simulation codes at the resolution required to see the fluid–acoustic coupling, and the interpretation of the massive data sets that are produced. This paper addresses the former, the use of advanced experimental techniques such as particle image velocimetry and Raman and Rayleigh scattering, to validate the computer codes and procedures used to create large-eddy-simulation solutions. This paper argues that the issue of accuracy of the experimental measurements be addressed by cross-facility and cross-disciplinary examination of modern data sets along with increased reporting of internal quality checks in particle image velocimetry analysis. Furthermore, it argues that the appropriate validation metrics for aeroacoustic applications are increasingly complicated statistics that have been shown in aeroacoustic theory to be critical to flow-generated sound, such as two-point space-time velocity correlations. A brief review of data sources available is presented along with examples illustrating cross-facility and internal quality checks required of the data before they should be accepted for validation of large-eddy simulation.

Batchelor’s fourth order velocity correlation.

The Lighthill theory for aerodynamic noise relates far field sound pressure to complex fourth order two point two-time space-time velocity correlations. The Mani–Gliebe–Balsa (MGB) jet noise prediction code follows Lighthill’s theory in the development of its algorithm. Since fourth order turbulence correlations are not modeled in the MGBs flow solver, it is necessary to resort to some form of approximation. The MGB invokes Batchelor’s fourth order decomposition, which is exact for homogeneous and isotropic turbulence. For anistropic flow, like that of a jet, it is not known what the error is in adopting Batchelor’s isotropic decomposition. The question here is can stereo particle image velocimetry (SPIV) provide a means to establish the error incurred with adoption of this decomposition. In an attempt to define the penalty for adopting the Batchelor’s decomposition, a simple 2×94; diameter laboratory jet was operated at ambient temperatures at Mach 0.85. SPIV measurements were acquired of the fourth order two-point space correlation and all of the other second order correlations. A comparison was made to numerical plume simulations with an advanced numerical flow solver running in the RANS mode. This paper provides an explanation for observed variations between measured and predicted jet noise.

On the pseudomomentum and generalized Stokes drift in a spectrum of rotational waves

Two Lagrangian-mean measures crucial to the accurate estimation of mean particle velocities in wavy or turbulent shear flows are considered. The measures are the pseudomomentum and generalized Stokes drift and of particular interest is their expression in terms of quantities directly measurable by fixed instruments. To proceed, the measures are first calculated for broad spectra of progressive symmetric rotational wave pairs of small amplitude. Both discrete and continuous spectra are considered and the waves may grow or decay. The expressions are then cast into a form composed of quantities that are measurable in a fixed reference frame, such as the surface slope spectrum of surface gravity waves or space–time velocity correlations in the interior of wavy shear flows. Finally, an example is given in which the measures are calculated for a plane channel flow subject to a broad spectrum of discrete progressive waves, specifically a numerical simulation of turbulent channel flow. It is seen that while the streamwise component of pseudomomentum is everywhere negative in the flow, the generalized Stokes drift changes sign, giving rise to an enhanced mass transport close to the boundary and a reduction in transport some distance from it. The sign change occurs 12.5 viscous units from the wall, near the centre of a 15 viscous units thick highly sheared layer of Stokes drift.

Space-time velocity correlations in the impeller stream of a Rushton turbine

Simultaneous velocity measurements at two points have been carried out with a two-component laser Doppler velocimeter in order to characterise the turbulence along the impeller stream of a Rushton turbine in a water tank. In addition to mean values of the velocity and its fluctuations, space–time correlations have been studied to enable the determination of the convection velocity as well as of the scales of the turbulent structures and to investigate the validity of Taylor’s hypothesis. A correction to the relationship between space and time correlations is proposed.

A rapidly scanning two-velocity component laser Doppler velocimeter

A two-velocity-component directionally sensitive fringe-type rapidly scanning laser Doppler velocimeter (RSLDV) is described, which scans the measurement volume in a direction perpendicular to the optical axis of the transmitting optics and normal to the test flow floor and measures the velocity components perpendicular to the optical axis. Scan rates of up to 300 Hz are possible over 75 mm using a sinusoidal scanning of the position versus time. The RSLDV was designed to permit space-time velocity correlations along a vertical line scan. To illustrate the use of this instrument, some results are presented from measurements performed in the highly turbulent complex three-dimensional flow in front of a wing-body junction.

Measurements and Axisymmetric Model of Spatial Correlations in Turbulent Pipe Flow

The structure of fully developed turbulent water pipe flow is investigated using data from two hot film x probes to evaluate two-point, two-component (axial and radial) space and space-time velocity correlations. Measurements are made with one probe fixed at F//? = 0.35 from the pipe wall and the other a moveable probe translated, with combined axial and radial separations, to give net probe separations in planes inclined at 0, 10, 20, 40, 65, and 90 deg in both upstream and downstream directions with respect to the mean flow. Departure of these two-point correlations from isotropic behavior is modeled in terms of axisymmetric tensor forms in which the turbulence is characterized by two scalar quantities related to the correlation length scales.

Some two-point statistical properties of a three-dimensional wall jet

The effects of both a straight and a curved wall on the two-point space-time velocity correlations of a turbulent jet are documented. From the correlations, isocorrelation contours are constructed which allow a picture of the various flow patterns to be determined. Autoand cross-correlations of the fluctuating concentration field are also obtained for the different flow configurations. The confining surfaces served to radically change the flow pattern for the freely expanding turbulent jet. The velocity measurements are made with a laser Doppler velocimeter in conjunction with a phase locked-loop processor, while the concentration field is monitored with a laser light-scattering technique.

Two point correlations of jet noise

A large body of careful experimental measurements of two-point broad band correlations of far-field jet noise has been carried out and was briefly reported recently by Lucio Maestrello in NASA TM X-72835. The rather sharp directional lobes and marked absence of axisymmetry were striking and motivated the present effort to bring theory to bear. The model of jet-noise generation is an approximate version of an earlier work of Ribner, based on the foundations of Lighthill. The model incorporates isotropic turbulence superimposed on a specified mean shear flow, with assumed space-time velocity correlations, but with source convection neglected. The particular vehicle is the Proudman format, and the previous work (mean-square pressure) is extended to display the two-point space-time correlations of pressure. The shape of polar plots of correlation is found to derive from two main factors: (1) the non-compactness of the source region, which allows differences in travel times to the two microphones—the dominant effect—and (2) the directivities of the constituent quadrupoles —a weak effect. The non-compactness effect causes the directional lobes in a polar plot to have pointed tips (cusps) and to be especially narrow in the plane of the jet axis. In these respects, and in the quantitative shapes of the normalized correlation curves, results of the theory show generally good agreement with Maestrello's experimental measurements.

Intermittency of coherent structures in the core region of fully developed turbulent pipe flow

The present investigation is oriented towards a better understanding of the turbulent structure in the core region of fully developed and completely wall-bounded flows. In view of the already existing results concerning the bursting process in boundary layers (which are semi-bounded flows), an amplitude analysis of the Reynolds shear stress fluctuation u1u2, sorted into four quadrants of the u1, u2 plane, was carried out in a turbulent pipe flow. For the wall side of the core region, in which the correlation coefficient u1u2/u’1u’2 does not change appreciably with the distance from the wall, the structure of the Reynolds stress is found to be similar to that obtained in boundary layers: bursts, i.e. ejections of low speed fluid, make the dominant contribution to the Reynolds stress; the regions of violent Reynolds stress are small fractions of the overall flow; and the mean time interval between bursts is found to be almost constant across the flow. For the core region, the large cross-stream evolution of the correlation coefficient u1u2/u’1u’2 is associated with a new structure of the Reynolds stress induced by the completely wall-bounded nature of the flow. Very large amplitudes of u1u2 are still observed, but two distinct burst-like patterns are now identified and related to ejections originating from the two opposite halves of the flow. In addition to this interaction, a focusing effect caused by the circular section of the pipe is observed. As a result of these two effects, the mean time interval between the bursts decreases significantly in the core region and reaches a minimum on the pipe axis. Investigation of specific space-time velocity correlations reveals the possible existence of rotating structures similar to those observed at the outer edge of turbulent boundary layers. These coherent motions are found to have a scale noticeably larger than that of the bursts.

Two-point correlations of sound pressure in the far-field of a jet: theory

A large body of careful experimental measurement of two-point jet noise correlation have been carried out and are reported in a companion paper by Lucio Maestrello. The result were not altogether as expected and motivated the present effort to bring theory to bear. The model of jet noise generation is that of Ribner (1963, 1964, 1969), which is development of the Lighthill for malism. It postulates isotropic turbulence superimposed on a specified mean shear flow, with assumed space-time velocity correlations. The particular vehicle is the Proudman format, which is extended from the 1969 work (mean square pressure) to display the two-point space-time correlations of pressure. Here, again, the self-noise (from turbulence alone) is accompanied by shear noise (from turbulence-mean flow interaction), with distinct and different contributions. The shape of polar plots of correlation derives from two main factors: (1) the derivatives of the constituent quadrupoles and (2) the noncompactness of the source region which allows differences in travel time to the two microphones. Comparisons are made with the results of Maestrello's measurements.

Relation between Time Symmetry and Reflection Symmetry of Turbulent Fluids

An assumption of statistical reflection symmetry is proposed for certain turbulent flows. On the basis of this assumption it is shown that some space-time velocity correlations characterizing the flow are even functions of the time delay between the correlation measurements. In particular the second- and one of the fourth-order velocity correlations are symmetric in this relative time.