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Aliasing in one-point turbulence measurementsTheory, DNS and hotwire experiments

Aliasing in one-point turbulence measurementsTheory, DNS and hotwire experiments

Vertekening in eenpuntsmetingen aan turbulentieTheorie, DNS en hittedraadproeven

Proefschriftter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnicus K.F. Wakker, in het openbaar te verdedigen ten overstaan van een commissie, door het College voor Promoties aangewezen, op vrijdag 25 juni 1999 te 10.30 uur


Arjan VAN DIJKdoctorandus in de natuurkunde, geboren te Amsterdam

Dit proefschrift is goedgekeurd door de promotor: F.T.M. Nieuwstadt

Samenstelling promotiecommissie: Rector Magnicus, F.T.M. Nieuwstadt, P.G. Bakker, P.M.T. Broersen, H. Hoeijmakers, Prof.dr. J.C.R. Hunt, FRS, A.F.G. Jacobs, Prof.dr. D. Lohse, voorzitter Technische Universiteit Delft, promotor Technische Universiteit Delft Technische Universiteit Delft Universiteit Twente University of Cambridge, Engeland Landbouwuniversiteit Wageningen Universiteit Twente

Copyright c 1999 by A. van Dijk Frederik Hendrikstraat 125 NL - 3583 VJ Utrecht All rights reserved

ISBN 90-9012799-2 NUGI 812, 811 en 815

SummaryAliasing in one-point turbulence measurementsTheory, DNS and hotwire experiments Arjan van DijkThe subject of this thesis is the interpretation of one-point velocity measurements in a turbulent ow eld in terms of three-dimensional spatial energy spectra. Our motivation to start this study is that there is no clear relation between the wavelength in conventional spectra of onepoint measurements and three-dimensional structure size. Conventional spectra of one-point measurements are constructed from time-series of observations. With help of Taylors hypothesis these observations are converted into spatial observations along a line, from which the so called one-point spectrum can be estimated. One studies the one-point spectrum to get information on the spatial structure of turbulence. Unfortunately this one-point spectrum is blurred by aliasing, which means that signal components from small structures are observed as associated with larger structures. Therefore a procedure is needed to un-alias the one-point spectrum such that one can interpret its results in terms of the three-dimensional structure of turbulence. Until now there is not much more to unalias one-point measurements than an old isotropic relation. However, in real situations one observes anisotropic distributions of uctuations. Exchange processes are for example studied in the context of climate research and one-point measurements are a commonly used source of information. Without a proper un-aliasing tool, one cannot estimate the preferential directions of velocity uctuations induced by structures of a certain size. Consequently, without an anisotropic model, one cannot expect to gain much insight into the dynamics of exchange processes from one-point measurements. In this study we have constructed a theoretical model which relates one-point spectra to threedimensional spectra. The resulting relations are assessed by means of a direct numerical simulation (DNS), and are then applied to windtunnel measurements on the upstream stagnation line of grid generated turbulence in stagnant plate ow and the achievements are discussed. The model which we have developed is based on a statistical description of the probability density of the directions in which turbulent structures of a certain wavelength induce a velocity uctuation as a function of the orientation of the wavevector. For isotropic turbulence this probability density function can be regarded as spherical in k-space. Here we make use of ellipsoids, with which we extend the existing theory to incompressible, homogeneous, but anisotropic ow. The three-dimensional spectrum is solved from the model with use of a linear


expansion of the anisotropy around isotropy. We found a set of six relations in closed form, which give the symmetric part of the three-dimensional spectrum as a function of the symmetric part of the one-point spectrum. This work can be considered as the rst theory to exploit all independent information in the symmetric part of one-point turbulence spectra to estimate the three-dimensional spectrum. Our unaliasing relations depend on the third derivative of the one-point spectrum. Therefore we require an estimator for the one-point spectrum, which allows for differentiation. We have constructed two different methods to estimate these spectral derivatives. The rst method is based on logarithmic smoothing of the one-point spectrum, estimated via Fast Fourier Transform (FFT), followed by curve tting of the double logarithmic spectrum. The resulting tcurve is then differentiated analytically. The second method uses autoregressive modelling (AR) to nd an analytic expression for the spectrum, which then is differentiated to nd the spectral derivatives. To eliminate insignicant wiggles due to noise from the estimated functions, the function is smoothed via logarithmic convolution. We have a preference for the FFT-based method to the AR-based method, because the latter method is simply incapable of estimating spectra for large structures and for the small scales we have not managed to eliminate the inuence of noise amplication by differentiation. In the DNS we have simulated grid generated turbulence. We have made a comparison between direct estimates for the three-dimensional spectrum via the Fourier transform of three-dimensional velocity elds and indirect estimates via application of our new unaliasing relations to simulated one-point measurements. The comparison has not been successful. There was no correspondence between the two estimates but the scatter in the estimates was also very large. We have many suggestions to improve this test. Most of them are related to the spatial resolution of the simulation and consequently to computer power. In this direction further study is advisable. Our next step to verify our theory has been to collect time-series of turbulent velocity elds. For this we have constructed a four-hotwire probe and a setup for three dimensional calibration. We have examined which hot-wire calibration model gives the best estimates for both the velocity vector and for the anisotropy of a ow. In an investigation of many calibration methods we found that the best method is to model the responses of the wires as functions of the velocity eld and not vice versa. To estimate the velocity vector from a set of four wire readings one has to solve the four response equations simultaneously. A fast method has been developed to make a temperature calibration of the probe. With our hot-wire probe we have measured the velocity vector in grid generated turbulent ow on the stagnation line upstream of a vertical at plate. The plate was implemented to break the initial axisymmetry of the grid turbulence, to create a situation in which our unaliasing theory could show its power to extend the reconstruction of the three-dimensional spectrum from one-point observations to anisotropic turbulence. Moreover the ow geometry presented the possibility to verify the Rapid Distortion theories of Batchelor and Proudman (1954) and of Hunt (1973). In the one-point spectra we observed that when the plate is approached, all velocity components of the large scales are amplied and all velocity components of the small scales are attenuated. This observation is at variance with the asymptotic Rapid Distortion theories by Hunt and by Batchelor and Proudman. The relative contribution the three velocity components to the kinetic energy is found to react to the approach of the plate as follows. For all wavelengths


the reaction to the plate is the same. In longitudinal (streamwise) direction nothing happened and the relative gain of the vertical uctuations was made at the cost of the relative contribution to the energy by the lateral uctuations. For the lateral and vertical uctuations this re-ordering of uctuations is supported by small scale RDT-asymptotics. The indifference of the longitudinal uctuations could not be explained. Nor could we explain why the redistribution of the uctuations over the three dimensions did not depend on the structure size. Our unaliasing relations were used to estimate the three-dimensional spectrum from the one-point spectrum. The AR-based implementation resulted in more or less constant functions as estimates for the three-dimensional spectrum. We ascribe this non-realistic behaviour to the earlier mentioned problems with AR. The FFT-based method resulted in a spectrum which satised two of our expectations: from the one-point spectrum to the three-dimensional spectrum the longitudinal uctuations are amplied and the lateral uctuations are attenuated. Moreover we observed that characteristics were shifted from the large scales to the smaller scales. We conclude that we have succeeded in developing a set of practically applicable relations, with which for the rst time one can interpret anisotropic one-point spectra in terms of threedimensional spectra.



SamenvattingVertekening in eenpuntsmetingen aan turbulentieTheorie, DNS en hittedraadproeven Arjan van DijkDit proefschrift behandelt de uitleg van eenpunts snelheidsmetingen aan een turbulente stroming in termen van driedimensionale energiespectra. We hebben dit onderzoek uitgevoerd omdat er geen goede relatie bestaat tussen de golengte in spectra van eenpuntsmetingen en ruimtelijke structuurgrootten. Conventionele spectra worden gemaakt van tijdreeksen van waarnemingen. Met behulp van Taylors hypothese worden deze waarnemingen vertaald naar ruimtelijke waarnemingen langs een lijn, waaruit het zogenaamde eenpuntsspectrum wordt geschat. Het