William L. Keith

Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation

This paper concerns the quadratically-damped Mathieu equation: 3 x +( � + � cos t)x +˙ x| ˙ x| =0 : Numerical integration shows the existence of a secondary bifurcation in which a pair of limit cycles come together and disappear (a saddle-node of limit cycles). In � –� parameter space, this secondary bifurcation appears as a curve which emanates from one of the transition curves of the linear Mathieu equation for � ≈ 1:5. The bifurcation point along with an approximation for the bifurcation curve is obtainedby a perturbation methodwhich uses Mathieu functions rather than the usual sines andcosines. Publishedby Elsevier Ltd .

Momentum Thickness Measurements for Thick Axisymmetric Turbulent Boundary Layers

Experimental measurements of the mean wall shear stress and boundary layer momentum thickness on long, thin cylindrical bodies are presented. To date, the spatial growth of the boundary layer and the related boundary layer parameters have not been measured for cases where δ/a (a =cylinder radius) is much greater than one. Moderate Reynolds numbers (10 4 <RE θ <10 5 ) encountered in hydrodynamic applications are considered. Tow tests of cylinders with diameters of 0.61 0.89, and 2.5 mm and lengths ranging from approximately 30 meters to 150 meters were performed. The total drag (axial force) was measured at low speeds up to 17.4 m/sec. These data were used to determine the tangential drag coefficients on each test specimen, which were found to be two to three times greater than the values for the corresponding hypothetical flat-plate cases. Using the drag measurements, the turbulent boundary layer momentum thickness at the downstream end of the cylindrical bodies is determined, using a control volume analysis

Wavenumber-frequency analysis of turbulent wall pressure fluctuations over a wide Reynolds number range of Turbulent Pipe Flows

Measurements of the autospectra and coherence of turbulent wall pressure fluctuations were made in the circular test section of the Quiet Water Tunnel Facility at the Naval Undersea Warfare Center in Newport, Rhode Island. The pipe diameter Reynolds numbers varied from 2.09 × 105 to 1.85 × 106. The coherence measurements are shown to collapse well with the similarity scaling over the entire range of Reynolds numbers. Wavenumber-frequency spectra are estimated by computing the spatial Fourier transform of the measured coherence, using the model of Corcos. The results are shown to accurately represent the convective ridge portion of the wavenumber-frequency spectra where the dominant energy exists.

Experimental investigation of axisymmetric turbulent boundary layers

Measurements of the turbulent boundary layer over an experimental towed array were made at the David Taylor Model Basin, Naval Surface Warfare Center Carderock Division using a stationary stereo- particle image velocimetry (SPIV) system. Data were collected at discrete transverse planes along the length of the array at tow speeds between 6.2 and 15.4 m/s. The corresponding momentum thickness Reynolds numbers Reθ varied from 4.8 × 105 to 1.1 × 106. This range is representative of values occurring in Navy applications and is much greater than those typically achieved in laboratory or with computational investigations. Instantaneous three-dimensional velocity fields and profiles have been published by the authors. However, the streamwise growth of the boundary layer and the variation with speed were not determined. The goal of this analysis is to provide additional insight into the development of axisymmetric boundary layers over long lengths and at moderate to high Reynolds numbers.

SPIV measurements of axisymmetric turbulent boundary layers

This paper presents a description of turbulent boundary layer velocity measurements made on an experimental towed array during testing at the David Taylor Model Basin, Naval Surface Warfare Center Carderock Division in June 2007. The experimental array had an aspect ratio L/a = 7 × 103 and was towed at Reynolds numbers Reθ varying from 4.6 × 105 to 8.9 × 105. This range falls well outside that which has been investigated to date in laboratories or with computational fluid dynamics. Previous lake tests of this array were performed and documented in Cipolla and Keith [1]. However, details of the high Reynolds turbulent boundary layer were not obtained during these tests. The goal of the follow-on tow tank testing was to obtain measurements of the mean and turbulent flow field which are not feasible in lake or sea trial testing. A stationary stereo-particle image velocimetry (SPIV) system was used to obtain three-dimensional velocity measurements and evaluate the boundary layer flow development along full-scale fleet towed array modules. Measurements were collected at discrete transverse planes along the length at tow speeds between 6.2 and 15.4 m/s. Algorithms for image pre-processing and filtering were applied to enhance the instantaneous images and mask the array and its shadow. The data will be analyzed to extract mean velocity profiles and compared with wind tunnel measurements on cylinders [2]. Further, relevant boundary layer parameters will be used to refine the scaling of the wall pressure measurements obtained simultaneously as reported by [3]. Independent load cell measurements of the total drag on the towed model provided the momentum thickness at the end of the model and the spatially-averaged friction velocity uτ. These data supplement the SPIV data near the array wall, completing the velocity profile over the entire boundary layer. The load cell also provided a highly accurate value of the mean wall shear stress which is traditionally very difficult to obtain. The velocity profiles can be compared with existing models for the mean velocity which include the velocity defect law and Clausers log law. In particular, the velocity defect law is expected to provide the best collapse of the data in the outer region of the boundary layer, while the log law relation is expected to provide a good collapse very close to the surface of the towed array (near wall region). Trends in the data with Reynolds number will be evaluated. In addition, the boundary layer thickness and mean wall shear stress at particular streamwise locations along the array will be quantified. The growth of the turbulent boundary layer over the length of the array is an important metric with regard to estimating the maximum turbulent boundary layer thickness which exists over a fleet towed array. The underlying structure of the axisymmetric boundary layer, which leads to significant increases in wall shear stress with respect to flat plate cases, is of primary importance. These new insights will facilitate efforts toward towed array reliability and an accurate prediction of drag and flow noise for any towed array application.