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Several aspects of the subharmonic instability of boundary layers are studied. First, the subharmonic instability of incompressible flows over a flat plate is investigated using the resonant triad model and the Floquet model. The primary wave is taken in the form of a two-dimensional (2-0) Tollmien-Schlichting (T-S) wave. The subharmonic wave is taken in the form of a three-dimensional (3-D) wave. Results from both models are presented and compared with the experimental data and numerical simulation. It is found that the results of the Floquet model are in good agreement with the experimental data and numerical simulation, whereas the results of the resonant triad model agree only qualitatively with the experimental data. Second, the subharmonic instability of incompressible flows over a 2-0 hump is studied using the Floquet model. The mean flow over the hump is calculated by using interacting boundary layers, thereby accounting for viscid/inviscid interactions. The results show that increasing the hump height results in an increase in the amplification factors of the primary and subharmonic waves. When the hump causes separation, the growth rates of both the primary and subharmonic waves are considerably larger than those obtained in the case of no separation. Third, the subharmonic instability of compressible boundary layers over a flat plate is studied using the Floquet model. Results are presented for adiabatic wall boundary conditions and subsonic, transonic, and supersonic flows. For supersonic flows results are presented for first- and second-mode primary waves. The effect of Mach number, spanwise wavenumber, primary-wave amplitude, Reynolds number, and frequency are studied. Fourth, results for the effect of heat transfer on the subharmonic instability of a two-dimensional compressible boundary layer over a flat plate are presented for different Mach numbers. For supersonic flows results are presented for first- and second-mode waves. The effect of different levels of heat transfer on changing the features of the subharmonic compressible instability is evaluated. Fifth, results for the effect of suction on the subharmonic instability of a two-dimensional compressible boundary layer over a flat plate are presented. It is found that when the primary wave is a first-mode merging with a second-mode, the subharmonic wave is strongly destabilized by suction. Sixth, the effect of a bulge on the subharmonic instability of compressible boundary layers is studied. It is found that the effect of compressibility on reducing the growth rate of the disturbances weakens as the hump height increases. / Ph. D.

Instability waves in a free shear layer formed by two parallei compressibie streams are analyzed using the linear spatial stability theory. Both viscous and inviscid disturbances are considered. The basic state is obtained by solving the compressibie laminar boundary-layer equations or is specified by the hyperbolic tangent velocity profile. The effects of viscosity, Mach number, the velocity and temperature ratios on the growth rate are determined. Unlike the boundary layer flow, viscosity has a stabilizing effect on the mixing layer flow. Increasing the temperature ratio produces a strong stabilizing effect on the growth of the mixing flow; this stabilization does not, however, persist at higher Mach numbers. Whereas the maximum growth rate of the Incompressible mixing layer varies linearly with the velocity ratio, the maximum growth rate of the compressible mixing flow varies nonlinearly with the velocity ratio. The numerical results substantiate the fact that the convective Mach number Is the appropriate parameter for correlating the compressibility effects on the spreading rate of the mixing layer. The ratio of the spreading rate of a compressible layer to that of an incompressible layer at the same velocity and density ratios depends primarily on the convective Mach number. Three-dimensional waves become important when the convective Mach number is greater than 0.6. The influence of nonparallelism on the spatial growth rate of two-dimensional disturbances is evaluated and is found to be negligible. Linear subharmonic Instabilities of a compressible mixing layer, which Is spatially periodic in a translating frame of reference, are analyzed by using Floquet theory. The basic state is obtained by the linear superposition ofa steady mean flow, which is given by a solution to the compressible boundary-layer equations or by a hyperbolic tangent velocity profile approximation, and the neutral primary wave of that mean flow. The results show that the growth rates of two-dimensional subharmonic instabilities (pairing mode) increase with increasing amplitude of the periodicity but decrease with increasing the convective iVIach number. In the incompressible flow case, the most amplified subharmonic wave is a two-dimensional mode, which is in agreement with the published results. For subsonic convective Mach numbers, the presence of the periodicity enhances the growth rates of three-dimensional subharmonic waves over a wide range of spanwise wavenumber which shows a preferred band over which the growth rate is maximum. However, when the convective IVIach number is greater than one, the interaction between the subharmonic wave and the primary wave marginally increases the maximum growth rate of the subharmonic. Nevertheless, that interaction dramatically increases the range of amplified spanwise wave numbers. Fourth-order compact finite-difference codes are developed for solving the compressible boundary-layer equations and investigating their primary and subharmonic instabilities. The codes proved to be very accurate and versatile. / Ph. D.

Measurements of surface shear stress magnitude and direction are reported for a three-dimensional, pressure driven, turbulent boundary layer around a wing body junction. Measurements were made using a dual-beam oil film laser interferometer at 56 locations. An iterative procedure was developed which increased the precision of the data extracted from the data records. Skin friction directions computed using a least square error fit were compared to angles obtained from surface oil flows, hot wire anemometry, and LDV measurements. Also, the magnitude of the skin friction coefficients were compared to independently obtained skin friction coefficients. The data agreed to within experimental error outside the effects from the vortex legs present along the side of the wing-body. No accurate data was available for quantitative comparison under the effects of the vortex, but the magnitudes followed the qualitative trends expected. This method failed badly in the region of large three-dimensional effects and requires further study in this area of application. / Master of Science

A three dimensional, pressure driven turbulent boundary layer created by an idealized wing-body junction flow is experimentally studied. The body used is a 3 : 2 elliptical nosed NACA 0020 tailed symmetric profile which has a chord length of 30.5 cm (12 inches), maximum thickness of 7.17 cm (2.824 inches) , height of 22.9 cm (9.016 inches). The body was sitting on a flat plate. The nominal reference velocity of the flow is 27 m/sec and the Reynolds number based on the momentum thickness at 0.75 chord upstream of the body on the centerline of the tunnel is ≃ 5936. The data presented include time-mean static pressure, skin friction magnitude and direction on the wall, as well as the mean velocity and all Reynolds stresses at several stations on a line determined with the mean velocity vector component parallel to the wall in the layer where the u²¯ normal stress is maximum. The mean velocity and stress data were obtained both with hot-wire ( HW ) and laser-Doppler-velocimeter ( LDV ) techniques. The LDV measurements were taken twice due to the differences observed between the HW and LDV data, which is also shown with the present study. This gave a chance to study the uncertainties on the mean velocity and the stresses extensively. Pressure distributions on the wing and the on the bottom plate were obtained with a Scanivalve and an inclined manometer. Skin friction vectors at several locations on the wall were measured in another study done by Allinger ( 1990 ) with a laser interferometer technique. The data show that the eddy viscosity of the flow is not isotropic, but the ratio of eddy viscosities perpendicular and parallel to the direction of the mean velocity vector component parallel to the wall at the point in the layer where u²¯ is maximum is close to unity, and the shear-stress vector direction in the flow lags behind the flow gradient vector direction. A₁, Townsend's structural parameter is not a constant of 0.15 as expected. The production of the turbulent kinetic energy and shear stresses are important below the logarithmic regions of the U axial velocity component profiles. The skin friction velocity is not the scale of the turbulence in such a flow. Further, a collection of 3-D turbulent boundary layer data including the present study is used to investigate the concept of the Law of the Wall velocity profile and the limitations of eddy-viscosity turbulence models in 3-D flows. For this purpose, several Law-of-the-Wall velocity profile models and eddy-viscosity models were tested. Johnston's Law-of-the-Wall relation and, for the pressure-driven flows the Johnson-King eddy-viscosity model and for the shear-driven flows Patel's eddy-viscosity model are most promising. / Ph. D.

Investigations of the three-dimensional flow around a 6:1 prolate spheroid model 1.37 m long were conducted in the separation and near wake regions along the leeward side. Mean velocity flow field measurements, at α = 10∘ and 15∘ , and at Re = 1.3 x 10⁶ (U_re= 15.2 mls) and 4.0x 10⁶ (Ure=45.7 m/s), were obtained at four axial locations along the afterbody. Boundary layer profiles and Reynolds shear stress measurements were obtained at two axial locations, with a = 100 and Re=4.0 X 10⁶. Results of the flow field measurements indicate vortical flow along the surface of the body, growing in strength with increasing Reynolds number and increasing angle of attack. Skewing of the three-dimensional boundary layer is seen in the boundary layer profiles, with the surface shear stress direction lagging the local free-stream velocity direction. Growth of the boundary layer is evident circumferentially and axially along the body. Results of the turbulence measurements show that the distribution of Reynolds stress quantities is different from that of a two-dimensional flow over a flat plate, due to the three-dimensional flow and separation that is present. Estimates of x and z eddy viscosities show that the eddy viscosity is not isotropic. Estimates of the mixing length compared to values for a two-dimensional flow model indicate that the model predicts high values for the mixing length. Comparisons made with results obtained at DFVLR in West Germany show good agreement for the mean velocity and Reynolds normal stress values; however, the agreement of the Reynolds shear stresses is not as good. / Master of Science