MSc (Mathematics) / Department of Mathematics and Applied Mathematics / See the attached abstract below

A nonlinear second order differential equation related to vortex motion is derived. This equation is analyzed using various numerical and analytical techniques including finding approximate solutions using a perturbative approach.

This thesis presents an investigation into the motions of the horseshoe vortices and the passage vortex, within a plane turbine blade cascade. Fluid motion was determined using a tracer gas technique. Ethylene was injected into the pressure-side and suction-side legs of the horseshoe vortex, near the leading edge of the cascade. Ethylene concentrations were determined at two downstream locations using a flame ionization detector. It was found that the pressure-side leg of the horseshoe vortex moved toward the suction side of the passage, starting the formation of the passage vortex, and was distributed throughout the passage vortex. The suction-side leg of the horseshoe vortex convected once around the periphery of the passage vortex before passing the cascade trailing edge. Downstream of the trailing edge, most of the fluid from the suction-side leg diffused into the passage vortex. However, twice as much fluid from the suction-side leg, as opposed to the pressure-side leg, mixed within the blade wake. At a location 40% of the axial chord downstream of the trailing edge, the passage vortex (shown previously to account for 60% of the overall total pressure losses) contained over 65% of the fluids from both legs. / M.S.

A Numerical Solution is presented for the problem of flow past a highly swept, slender wing with sharp leading edges. The lifting surface is modelled as a bound vortex sheet, while the wake is modelled as a force-free vortex sheet. The solution is obtained by the use of a unsteady Vortex-Lattice Method which includes the effect of leading edge separation. Numerical predictions for the aerodynamic loads and pressure distributions are compared with experimental data. A 75° Delta wing and a 60° Delta wing with Leading Edge Vortex flaps in uniform, symmetric and steady flow are studied. Uniform and cosine distributions are used to determine the effect of lattice shape on the solution. The results show that good aerodynamic load predictions are obtained by this Vortex-lattice method. The results also indicated that fewer cosine distribution control points predict pressures as well as the use of a larger number of uniform distribution control points. The numerical results for wings with LEVFs show good agreement with experimental data away from the trailing edge. This may be due to the viscous effects in the experiment not modelled in this method. It is also apparent that the size of the wake, trailing and leading edge wakes, is the important factor effecting computation times. / M.S.

In this dissertation large-eddy simulation(LES) is used to study the transitional and turbulent structures of vortices and free shear layers. The recently developed dynamic model and the basic Smagorinsky model are utilized to model the subgrid-scale(SGS) stress tensor. The dynamic model has many advantages over the existing SGS models. This model has the ability to vary in time and space depending on the local turbulence conditions. This eliminates the need to tune the model constants a priori to suit the flow field being simulated. Three different flow fields are considered. First, the evolution of large-scale turbulent structures in centrifugally unstable vortices is studied. It is found that these structures appear as counter rotating vortex rings encircling the vortex core. The interaction of these structures with the core results in the transfer of angular momentum between the core and the surroundings. The mean tangential velocity decays due to this exchange of angular momentum. Second, the generation and decay of turbulent structures in a vortex with an axial velocity deficit are studied. The presence of a destabilizing wake-like axial velocity field in an otherwise centrifugally stable vortex results in a very complex flow field. The inflectional instability mechanism of the axial velocity deficit amplifies the initial disturbances and results in the generation of large-scale turbulent structures. These structures appear as branches sprouting out of the vortex core. The breakdown of these structures leads to small-scale motions. But the stabilizing effects of the rotational flow field tend to quench the small-scale motions and the vortex returns to its initial laminar state. The mean axial velocity deficit is weakened, but the mean tangential velocity shows no significant decay. Third, a transitional mixing layer calculation is performed.The growth and breakdown to small scales of vortical structures are studied. Emphasis is given to the identification of late transition structures and their subsequent break down. Formation of streamwise vortices in place of the original Kelvin-Helmholtz vortices and the subsequent appearance of hair-pin vortices at the edges of the mixing layer mark the completion of transition. The basic Smagorinsky model is also used in the mixing layer simulations. The performance of the dynamic model is compared with the previous results obtained using the basic Smagorinsky model. As expected, the basic Smagorinsky model is found to be more dissipative. / Ph. D.

The interactions of trailing vortex pairs shed from the tips of two rectangular wings have been studied through helium bubble flow visualizations and extensive hot wire velocity measurements made between 10 and 30 chord lengths downstream. The wings were placed tip to tip at equal and opposite angles of attack, generating pairs of co-rotating and counter rotating vortices. Meaningful hot wire measurements could be made because the vortices were found to be insensitive to probe interference and experienced very small wandering motions. The co-rotating pairs were observed to rotate around each other and merge. Upstream of the merging location, the vortices have approximately elliptical cores. These are surrounded by the two wing wakes which join together around the two cores. Flow in the vicinity of the cores appears fully developed. During the merging process, the cores rotate rapidly about each other, winding the wing wakes into a fine spiral structure. Merger roughly doubles the core size and appears to produce turbulence over abroad range of frequencies. The counter rotating pairs move sideways under their mutual induction and slightly apart; their flow structure changing little with downstream location. These cores remain fairly circular and do not become fully developed within 30 chord lengths of the measurements. / Master of Science

Donaldson's single-point second-order model [13] is used to close the Reynolds stress transport equations in cylindrical coordinates. A reduced set of equations are then solved for the decay of axisymmetric vortices and jets. A self-similar solution to the axisymmetric vortices is obtained numerically. The characteristics of the mean flow variables as well as the Reynolds stresses in this solution are discussed. Comparisons of the current results with Donaldson[13J and Donaldson and Sullivan[16] are also presented. The results show that the vortex core is free from turbulent shear stresses. The turbulent kinetic energy is also found to be relatively weak within the core region. The overshoot of the circulation is found to be 5% of the circulation at infinity over a wide range of Reynolds numbers. The effects of Reynolds number on the decay of the vortices are computed and discussed. Some of the quantities, such as mean flow circulation and turbulent kinetic energy, are found to be sensitive to the Reynolds number. However, the overshoot is found to be insensitive to the Reynolds number but its location does. A set of suitable model constants for the axisymmetric jets is also found and a self similar solution for the jet case is obtained. Comparisons of the computed results with some recent experimental data are presented. / Master of Science

Using a general, unsteady, nonlinear vortex lattice method, the aerodynamic loads have been found on a 75° delta wing with and without leading edge vortex flaps. The flap had an area approximately 26 percent of the wing area with a constant chord of 6.7 percent of the wing mean aerodynamic chord and was deflected at 30°. Results for lift, drag, axial force, and pitching moment coefficients are compared with experimental data and show very good agreement. Individual pressure difference coefficients along the wing and flap are also presented and compared with experimental data. Overall, the method shows the leading edge vortex flap to be very effective in reducing drag while maintaining lift comparable to that of the plain wing. / Master of Science

A pressure-driven three-dimensional turbulent boundary layer flow upstream and around a junction vortex was experimentally studied and is offered for use as a benchmark flow for testing and evaluating the predictive ability of state-of-the-art three-dimensional turbulent boundary layer codes. The pressure-driven flow and junction vortex system was generated by a streamlined cylinder placed normal to a flat surface. Measurements of wall static pressure, wall shear stress, mean velocity, and Reynolds stress tensor field are reported at several stations in the three-dimensional turbulent boundary layer region. Documentation of the flow edge conditions is provided as well as upstream initial conditions along a plane with measured mean velocity and Reynolds stress tensor to permit the testing of intermediate and higher order turbulence models. Measurements of wall shear stress magnitude were made with a Preston tube and the wall shear stress directions were taken from an oil streak flow visualization. These results are compared with earlier direct force wall shear measurements of both magnitude and direction. Mean velocity magnitude and direction were measured with a single hot film probe. Measurements of the complete Reynolds stress tensor were carried out with three hot film x-array probes. Supporting work includes a wind tunnel calibration which examined the sensitivity and effects of spanwise nonuniformities and a two-dimensional momentum integral calculation along the tunnel center plane; the development of a calibration technique to determine individual sensor yaw characteristics in more complex probe geometries; and a generalized response analysis for a sensor with arbitrary orientation to the flow which allows for the use of an arbitrary yaw cooling law, allows for modest amounts of probe misalignment and yields a precise definition of matched sensors, geometric guidelines for constructing x-array probes, and a general mean velocity correction for turbulence where several existing formulas are compared. In addition, two popular cooling laws are studied, comparisons are made with other response equations, and an extensive discussion of the errors associated with the matched sensor approximations is given. Comparisons are made of several mean velocity measurements using different probes and redundant normal and shear stresses measured by the different x-array film probes, a single wire, and single film probe are compared. / Ph. D.

An experimental investigation of an incompressible, turbulent junction vortex formed at the base of a streamlined cylinder with a circular leading edge placed normal to a flat surface is documented. The flow environment is characterized by a body Reynolds number of 183,000, based on the diameter of the model. The investigation centered around measurements of mean velocity vectors at over 1,700 locations on the plane of symmetry. In addition, extensive turbulence measurements in regions of interest on the plane of symmetry were also made. All of the measurements are performed with a two-color, two-component, frequency shifted laser Doppler velocimeter. The mean flow field shows the presence of a single vortex on the plane of symmetry and a singular separation point upstream of the vortex. From the measured data, the locations of the vortex center and the singular separation point were determined. The mean velocity field correlates well with previously published surface pressure data and surface flow visualization results. Extensive comparisons were made with earlier Conrad and five-hole pressure probe measurements and hot film measurements. Comparisons with the Conrad probe results were made in a two-dimensional turbulent boundary layer, and with the five-hole probe results for the plane of symmetry junction vortex flow at six stations in the upstream region and four stations inside the separated region. Hot film results were compared at two stations in the upstream region. Excellent agreement was found in the two-dimensional turbulent boundary layer. On the plane of symmetry junction vortex flow, agreement among the three instruments was good, especially for data taken in the upstream region and away from the floor. Turbulence data were also obtained with the laser Doppler velocimeter in conjunction with a minicomputer. Results indicate the turbulence level inside the separated region is significantly higher than in the upstream region. Highest levels of turbulence were found near the singular separation point and near the center of the vortex. Doppler spectra variations in these regions were also shown. In addition, Doppler histograms obtained by a PDP-11 minicomputer were compared to Doppler spectra obtained using an FFT. Excellent agreement was found between the Doppler spectra and the histograms obtained from the two different instruments. / Ph. D.