The boundary-layer equations for the steady laminar flow of a vertical jet, including a buoyancy term caused by temperature differences, are solved by similarity methods. Two-dimensional and axisymmetric jets are treated. Exact solutions in closed form are found for certain values of the Prandtl number, and the velocity and temperature distribution for other Prandtl numbers are found by numerical integration.

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An analytic solution is obtained for the plate-velocity statistics of a turbulent-flow excited, simply supported, rectangular flat plate. The radiated acoustic pressure is neglected as contributing to the plate excitation, leaving only the turbulent-boundary-layer pressure fluctuations as the exciting force. The mathematical model for the turbulent-boundary-layer pressure statistics is based on that of Corcos, which agrees well with experiment. Plate-velocity statistics are expressed in the forms of dimensionless cross-power spectral and power spectral densities. Plate-velocity-spectral and cross-spectral densities were obtained with a digital computer for selected flow and plate parameters. From these computed dimensionless spectra, effects of major parameters on the plate-velocity statistics were determined. A “peak spectrum,” constructed by connecting major spectral peaks of the plate-velocity-spectral density, proved to be a useful engineering concept, inasmuch as knowledge of the “peak spectrum” is equivalent to knowledge of the maximum plate velocity spectral density for a given set of input parameters. The computed dimensionless plate-velocity “peak spectrum” compares well with “peak spectra” constructed from available experimental data.

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The nonlinear equations of motion for a railway vehicle wheelset having curved wheel profiles and wheel-flange/rail contact are presented. The dependence of axle roll and vertical displacement on lateral displacement and yaw is formulated by two holonomic constraint equations. The method of Krylov-Bogoliubov is used to derive expressions for the amplitudes of stationary oscillations. A perturbation analysis is then used to derive conditions for the stability characteristics of the stationary oscillations. The expressions for the amplitude and the stability conditions are shown to have a simple geometrical interpretation which facilitates the evaluation of the effects of design parameters on the motion. It is shown that flange clearance and the nonlinear variation of axle roll with lateral displacement significantly influence the motion of the wheelset. Stationary oscillations may occur at forward speeds both below and above the critical speed at which a linear analysis predicts the onset of instability.