Jacques Lewalle

ON THE STRUCTURE OF A TWO-DIMENSIONAL WAKE BEHIND A PAIR OF FLAT PLATES

Unsteady and asymmetric vortex shedding was investigated experimentally in the wake of two flat plates placed side by side normal to the flow. The experiment was conducted in water at Re=1500 based on an individual plate width. The spacing between the plates was varied from one plate width to two plate widths. For a small spacing between the plates, the deflection of the jet passing through the gap resulted in a larger wake with mostly symmetric vortex shedding on one side and a smaller wake with asymmetric shedding on the other. The intermediate wake for the small plate spacing resembled that of a single bluff body, whereas a more intermittent structure was observed at the wider spacing. The velocity time traces were obtained by laser Doppler velocimetry (LDV) at several locations in the wake. The Fourier spectra of the velocity data at certain positions indicated the presence of more than one dominating frequency. Wavelet analysis mapped the intermittent near‐periodic structures, and quantified their mean energy spectrum in relation to the ensemble mean. These results are interpreted by comparison with flow visualization data.

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.

Properties of the far-field pressure signatures of individual jet noise sources

The detection of intermittent near-field noise sources is achieved by continuous wavelet transform of far-field acoustic data. The similarities between signals from three far-field microphones in the annular cone of acoustic radiation from coherent sources in a Ma = 0.6 cold jet are used to identify some of the loudest events. Cross-correlation of relevant band-passed excerpts identifies the most reliable matches, which occur at the rate of one every 20 dimensionless time units on average, and are intermittent in both time and frequency with no obvious pattern. Processing our database produced a catalog of over 9500 individual events with quantitative properties including the magnitude, frequency and time of detection at each microphone. The differences in detection times (lags) are distributed smoothly around the peak of cross-correlations for the raw data. We show how source magnitude and frequency depends on lags, and relate the lags to their relative source location.

Correlations and wavelet based analysis of near-field and far-field pressure of a controlled high speed jet

Discussed within are the experimental results obtained via simultaneous near-field and far-field measurements of a controlled jet. A large database of both open-loop and closedloop control cases are carried out. The two most interesting investigations from the openloop (OL) and closed-loop (CL) studies are highlighted here and compared to the uncontrolled jet. For the open-loop forcing, the shear layer is seeded with an axisymmetric sinusoidal input with a frequency of 1700 Hz. For the closed-loop case, the shear layer is seeded with a sinusoidal forcing amplitude modulated with the mode-filtered near-field pressure sampled from the stream-wise location at x/D = 6. Data analysis looks at both averaged and instantaneous quantities. In an averaged sense the introduction of control significantly modifies the characteristics of the developing shear layer. Open-loop control is shown to modify the phase lag between both near-field stations with a penalty in an increase in the overall sound pressure level. The closed-loop control imparts more subtle changes to the flow field but yields a slight reduction in the acoustic pressure at the microphone closest to the jet axis. Wavelet based filtering exposes the evolution of the broadband intermittent events seen at both stations.

Wavelet transforms of some equations of fluid mechanics

SummaryThis paper explores the application of wavelet transforms to equations rather than to data sets. An entire class of wavelets, obtained from recursive shifts and changes in scale of Gaussian filters, transforms Laplacians into first order derivatives in the scale factor. As a result, parabolic and elliptic equations are transformed into first-order wave equations or into ordinary differential equations. Examples are given for the diffusion, Burgers, Poisson and Navier-Stokes equations, which are formally integrated by the method of characteristics. It is also shown that the even-indexed Gaussian wavelets decompose a function into the local spectral contributions to its amplitude as well as to its variance. This gives a simpler inversion formula and a new form of the convolution of wavelet transforms.

Decay of velocity and temperature fluctuations in grid turbulence

The dependence of the velocity and temperature fluctuation decay exponents on the initial conditions is resolved on the basis of the cascade model for isotropic turbulence. In the large-Reynolds-number limit, self-preserving solutions are obtained for the nondissipative eddies. For those solutions, the energy decay exponent n and the small-wave-number exponent s for the three-dimensional energy spectrum obey the relation n =2(s + !)/(? + 3). For temperature fluctuations, the decay exponent m and the small-wave-number spectrum exponent q satisfy m = 2(q + l)/(s + 3). Thus, the ratio r m/n of the thermal-to-mechanical time scales is completely determined by the initial spectral exponents for very large eddies, if the initial spectra are quasi-selfpreserving. These conclusions are shown to be independent of the model. The model equations are solved numerically. The ratio of thermal to mechanical integral scales is shown to depend on both n and m. The conclusions have implications in the context of exact flow simulations.

Formal improvements in the solution of the wavelet-transformed Poisson and diffusion equations

Hermitian wavelets’ relation to the Laplace operator leads to a natural measure of the scale factor that emphasizes the largest component wave number. For the Poisson equation (e.g. the pressure equation in Navier–Stokes turbulence), the wavelet transform of the solution at a given location and scale depends on the wavelet transform of the source field at the same location and at nearby and larger scales. For the diffusion problem, the Hamiltonian formulation is simplified through a canonical transformation.

Coherent Structures: Past, Present and Future

It is now commonplace to recognize that coherent structures (OS’s), beautifully illustrated in the art of many cultures, but long held to be marginal (transitional, low-Reynolds-number) in the science and engineering of fluid turbulence, were brought back to the core of the field by the work of Townsend [31] and the explosion of literature following the paper of Brown & Roshko [10] and e.g Hussain [20].

Identifying coherent structures in a 3-stream supersonic jet flow using time-resolved schlieren imaging

We analyze time-resolved schlieren images of the near-field of a 3-stream supersonic jet. The primary jet operates in the vicinity of Ma = 1.6, and the images are collected at the rate of 50 to 400 kfps. We analyze transverse-axial images by constructing time series from more than 400 points selected for their possible significance, based on a qualitative analysis of the schlieren images. The points are grouped along the various shear layers and in the near-field outside the jet. We examine in turn the power spectra and cross-correlations between points. Overall qualitative and quantitative trends in both spectra and correlation are noted, revealing a strong dependence of both on transverse and axial location in the flow field. Defining features in the spectra give insight into the frequency bands which will be more closely analyzed in future phases of this study. The results from this preliminary study point to the validity of using time-resolved schlieren imaging as a non-intrusive experimental method to generate time series, to which a range of analysis methods is applicable.

A cascade model for neutrally buoyant dispersed two-phase homogeneous turbulence—I. Model formulation

Abstract This work proposes a cascade model approach to describe two-phase turbulent flows of neutrally buoyant liquid dispersions for high dispersed phase fractions. A continuum framework is selected for the representation of the mean flow momentum and continuity equations. The cascade model of Desnyansky & Novikov is extended to describe the energy spectrum and eddy intermittency in the presence of a dispersed phase. The Reynolds stress in the mean balance equation is related to the cascade variables by a spectral eddy viscosity model. A population balance is formulated for each eddy size. This approach permits the specification of the immediate environment of the drop processes modeled in the population balance. Specific drop-eddy events such as grazing collisions, drop entrapment and eddy shattering are suggested and their effects on the turbulent spectrum, eddy intermittencies and drop size distributions are examined. The energy, intermittency and population balance equations modified to include the effects of drop-eddy interactions form the proposed two-phase cascade model for homogeneous, neutrally buoyant, turbulent dispersions.