##### Author

Newman, Andrew

##### Other Contributors

Drew, Donald A. (Donald Allen), 1945-; Castillo, Luciano; Schwendeman, Donald W.; Kapila, Ashwani K.;

##### Date Issued

2013-12

##### Subject

Mathematics

##### Degree

PhD;

##### Terms of Use

This electronic version is a licensed copy owned by Rensselaer Polytechnic Institute, Troy, NY. Copyright of original work retained by author.;

##### Abstract

In order to address the question of characteristic motions which were responsible for the bulk of the vertical entrainment of mean kinetic energy due to the Reynolds shear stress, Proper Orthogonal Decomposition was selected as an analysis tool. Due to the nature of the experimental data, it was desired to derive a methodology which would allow POD to be applied to data which possessed a time shift. Two approaches were taken to do this. The first was to use POD modes computed from sub domains which did not have a time shift to create expansions for full domain POD modes where the data had a time shift. This approach lead to a non-linear system whose solution provided the necessary coefficients to construct POD modes for data with a time shift out of the smaller portions of data which did not have a time shift. It was proven that this could be done and that further, the Fourier coefficients computed by projecting the full domain modes onto the sub domain modes satisfied the non-linear system. The second approach was to prove that the data with a time shift could still be used to construct modal expansions of the Reynolds shear stress and thus interrogate the characteristic motions this way.; After establishing the rates of development for the layers it was of interest to use the experimental data to analyze how the various terms in the transport equation of mean kinetic energy compare with one another. It was found that the dominant terms were the horizontal advection by the mean streamwise velocity and the vertical entrainment by the Reynolds shear stress. Furthermore, this analysis revealed that the vertical entrainment by the Reynolds shear stress contributes to the potential power of the turbines not only from above the array but also from below the rotors. Furthermore, it continues to do so at the deepest point measured in the experiment indicating that this is a non negligible contribution for a significant portion of the WTBL.; This work is primarily concerned with analyzing the wind turbine boundary layer (WTBL) in order to further understand the physics involved in the energy transfer process from the atmospheric boundary layer to the array. In order to do this much understanding of the WTBL and new mathematical tools for analysis are needed. First to better understand the physics within the wind turbine boundary layer, experimental data from a scaled 3 x 5 wind turbine array in a wind tunnel was collected using the technique of particle image velocimetry. Doing so revealed three distinct layers within the WTBL: the above rotor region, rotor swept region and the below rotor region. Rates of streamwise development in these regions were then computed using an appropriately defined difference norm. Upon doing so it was discovered that the mean streamwise velocity develops much faster than all other turbulent statistics in the WTBL. Thereby it is shown that studies which used this quantity as the definition of a developed WTBL may have overlooked continuing development in important higher order statistics such as the vertical entrainment of mean kinetic energy by the Reynolds shear stress.; Having discovered this, the technique to expand the Reynolds shear stress was used to interrogate the scale but not details of the motions which supplied much of the mean kinetic energy to the array by vertical entrainment. Doing so showed the relatively few modes contributed to 75% of the total entrainment and that these modes had characteristic wavelengths larger than D, the rotor diameter. Furthermore, it was shown that two types of modes exist: idiosyncratic and asymptotic. The idiosyncratic modes were those which had large wavelengths, made significant contributions to the vertical entrainment of mean kinetic energy and whose wavelengths did not follow a prescribed decay. Alternatively, the asymptotic modes were those which had small characteristic wavelengths, made little contribution to the vertical entrainment and whose wavelength followed a decay of n-1/2 where n is mode number. Comparing the results with those from a boundary layer without turbines and those from smaller domains where not only the size but the details of the characteristic motions could be investigated revealed that low order modes with large characteristic wavelengths are associated with large scale coherent motions in the velocity field, thus revealing the importance of such large scale structures in the energy transfer process in the wind turbine boundary layer.; Examples were then considered to show the mechanics of using the non-linear system to compute the coefficients. Experimental data was collected so that full domain and sub domain modes could be computed, and expansions of the full domain modes in terms of the sub domain modes constructed. It was found that such expansions could be constructed from the sub domain modes though large sums of modes were needed to achieve reasonable accuracy. It was then discovered that the Fourier coefficients computed from the experimental data were not in fact solutions to the derived non-linear system. The reason for this was then shown to be due to the difference between time correlations computed using members of one ensemble (i.e. one run of an experiment) and members of two different ensembles (two separate runs of an experiment).;

##### Description

December 2013; School of Science

##### Department

Dept. of Mathematical Sciences;

##### Publisher

Rensselaer Polytechnic Institute, Troy, NY

##### Relationships

Rensselaer Theses and Dissertations Online Collection;

##### Access

Restricted to current Rensselaer faculty, staff and students. Access inquiries may be directed to the Rensselaer Libraries.;