Complex ray theory successfully describes temporal and spatial development of a two-dimensional wave packet into steady and semi-infinite distribution of very large disturbances in the wake accompanied by a region of reverse flows. Final formation of the steady state is caused by the existence, on integral path of propagation equations, of a logarithmic singularity, at which the group velocity and wave-number variation both vanish and the temporal growth rate is positive. Nearly singular frequencies around the complex frequency defined at the singularity pass through a close vicinity of the singular point while spending a very large time and then compose the distributed disturbances at the limit of large time. Propagation of the nearly singular frequencies can be pursued by an appropriate application of the complex-time method of integral to the region away from the singularity and the complex-coordinate method of integral to its neighborhood.

Instability of the flow on a rotating disk is governed by linearized disturbance equations of the partial differential with respect to the radial distance from the rotation axis and the normal distance from the disk surface. Applying uniform suc-tion from the surface brings a small parameter associated with displacement thickness of the circumferential velocity pro-file into a dimensionless form of the equation system. Two kinds of series solutions expanded by the powers of this parameter are obtained to describe the cross-flow and centrifugal instabilities of the flow having a twisted velocity profile. The leading terms of the series solutions are determined from two eigenvalue problems of slightly different ordinary differ-ential equations, and the superposition of those equations leads to an eigenvalue problem applicable to multiple-instability characteristics of such three-dimensional boundary layers.

The frequency of acoustic sound emanating from the trailing edge of a two-dimensional airfoil is known to exhibit a ladder-like variation, displaying discontinuous jumps between discretely identifiable states as the free stream velocity varies. In order to reveal the underlying causes for this behavior, a two-dimensional jet issuing into still air with no aerodynamic sound emission is used as a model platform to study this phenomenon, because prescribed aero-acoustic sound may be readily introduced into the flow at the jet exit. When unstable disturbances growing in the shear layer of the jet are excited by a loudspeaker, an acoustic feedback loop automatically selects one frequency from the unstable frequencies present in the shear layer, and the resulting ladder-like variations are found to be similar to those present in airfoil trailing-edge noise. In addition, the observed slope of each rung of the ladder in the selected frequency behavior, and the observed jump frequency between ladder steps, show good agreement with existing empirical models. It is also discovered that when the remainder of the distance between the speaker and jet divided by the wavelength of the selected acoustic sound is equivalent to one-half wavelength of the accepted sound, the selected frequency jumps to another state.

Exact partial differential equations are derived to describe Gortler instability, caused by a weakly concave wall, of axisymmetric boundary layers with similar velocity profiles that are decomposed into a sequence of ordinary differential systems on the assumption that the solution can be expanded into inverse powers of local Reynolds number. The leading terms of the series solution are determined by solving a non-parallel version of Gortler's eigen-value problem and lead to a neutral stability curve and finite values of critical Gortler number and wave number for stationary and longitudinal vortices. Higher-order terms of the series solution indicate Reynolds-number dependence of Gortler instability and a limited validity of Gortler's approximation based on the leading terms only. The present formulation is simply applicable to two-dimensional boundary layers of similar profiles, and critical Gortler number and wave number of the Blasius boundary layer on a flat plate are given by G(2c) = 1.23 and beta(2c) = 0.288, respectively, if the momentum thickness is chosen as the reference length.

The frequency of instability waves in a wake flow is uniquely determined by a logarithmic singularity of complex ray trajectories describing the propagation of a two-dimensional wave packet. Conditions for the singularity are given by simultaneous equations indicating that the group velocity and X-derivative of the complex dispersion relation for a given flow field are both equal to zero, where X is the downstream coordinate and the dispersion relation defines the complex frequency as a function of the complex wave number and X. Simple mathematical models are introduced to simulate spatial variations of the wake behind a moderately thin flat plate. Stability calculations of the model flow indicate that the logarithmic singularity is located in the vicinity of the real axis of the complex coordinate X.

Axisymmetric stagnation-point flow, if it is along a concave wall, is susceptible to the Gortler instability, which is governed by partial differential equations with respect to both the normal-to-wall coordinate and the local Reynolds number. A series solution of the exact disturbance equations is obtained in the form expanded into inverse powers of the Reynolds number. The lowest-order equations concerning the leading terms of the series construct a non-parallel version of Gortler's eigenvalue problem and present a neutral stability curve with the critical point in a finite range of wave numbers for disturbances of the steady longitudinal-vortex type. Higher-order terms of the series solution indicate the dependence of the Gortler instability on the local Reynolds number and suggest a quantitative limit of validity of the Gortler approximation based on the lowest-order equation system alone. In the case of the concave wall that slightly rotates around the axis of symmetry, the instability induces oblique traveling waves instead of steady longitudinal vortices.

Three-dimensional laminar boundary layers are susceptible to cross-flow instability and streamline-curvature instability, both of which lead to growth of longitudinal vortices. It is practically difficult to distinguish one instability mode from the other in the natural process of laminar-turbulent transition. Unlike plane-wave disturbances, however, the point-source disturbances evolve into dispersive development of their different components, which will result in separate appearance of two instability modes downstream of a point source. Continuous excitation from a small hole is applied to the boundary layer on a yawed circular cylinder evolving into a wedge-shaped pattern downstream of the hole, while a pulsed jet through a tiny hole is used to generate disturbances of a wave-packet type in the flow on a rotating-disk. Spatial development of localized disturbances corresponding to these experimental configurations can be described by linear stability analysis based on the complex ray theory. Comparison between experimental results and theoretical calculations shows qualitative and even quantitative agreement for either case.

An experimental study is done to confirm the existence of a new instability due to the curvature of external streamlines in a three-dimensional boundary layer. Monochromatic-wave excitation from a tiny hole near the attachment line of a yawed circular cylinder is used to separate unsteady disturbances due to the streamline-curvature in-stability from traveling waves of the cross-flow instability. Experimen-tal results show that a point-source disturbance evolves into a wedge-shaped distribution and that amplitude and phase distributions in the spanwise direction definitely include both modes arising from the two instabilities. Observed characteristics and behavior of those disturbances are shown to be in excellent agreement with the latest results of a linear stability theory based on the complex characteristic method.

Development of localized disturbances generated by an oscillating point source in compressible boundary layers with a zero pressure gradient at Mach numbers from 0.2 to 2.0 is studied theoretically on the basis of the linear stability theory. The method of complex characteristics recently proposed by Itch as an extension of Whitham's kinematic wave theory, is applied to describe wave propagation from the oscillating source. The analysis demonstrates distinct differences in the development of localized disturbances between the subsonic and supersonic boundary layers. Importantly, maximum growth occurs away from the midspan in supersonic boundary layers, while it occurs at the midspan in subsonic boundary layers.

Small disturbances superimposed on the growing boundary-layer flow along a long swept wing are governed by partial differential equations with respect to the local chordwise Reynolds number and the nondimensional vertical coordinate. For a simple and widely applicable method of stability estimation, however, it is desirable to reduce the exact disturbance equations to an eigenvalue problem of the corresponding ordinary differential equations, as in the stability analysis of two-dimensional parallel flows. This paper proposes such a simple model of the ordinary differential system that includes the most important terms of boundary-layer nonparallelism and wall curvature. Numerical computations show that the eigensolutions can properly describe multi-instability characteristics of the three-dimensional boundary layers near the attachment line of a long swept wing.

Multi-instability characteristics of the three-dimensional boundary layer around a yawed circular cylinder are investigated by an e(N) method involving the effects of wall curvature and nonparallelism. Velocity profiles of the boundary layer are approximated by members of Falkner-Skan-Cooke similarity solutions and the local dispersion relation of each member is determined from the nonparallel eigenvalue problem proposed in part 1 of this study. A complex ray theory is adopted in the integration procedure for N factor, and a numerical estimation of N is made to describe wedge-shaped disturbances originating from a point source. The analysis using these methods shows that the influence of wall curvature and nonparallelism stabilizes and destabilizes the flow, respectively, though their quantitative effects on the N factor depend on kinds of instability, range of frequency, and values of flow parameters. It is also found that the destabilizing, effect of nonparallelism increases with the increase of sweep angle.

Centrifugal instability of two-dimensional similar boundary layers along a concave wall is governed by partial differential equations with respect to the normal-to-wall coordinate in a nondimensional form and the Reynolds number based on local velocity of external stream and a boundary-layer thickness. In a particular case of the stagnation-point flow with the Falkner–Skan parameter m=1, however, the exact equations admit a series solution expanded in inverse powers of the Reynolds number and its coefficients can be obtained by solving a sequence of ordinary differential systems. Of particular importance is that the leading term is determined from an eigenvalue problem more involved than Görtler's parallel-flow approximation. Numerical evaluation of the series solution thus obtained shades light on fundamental effects of the boundary-layer nonparallelism and finite Reynolds numbers on theoretical prediction of Görtler instability.

The stability of compressible three-dimensional boundary layers to stationary disturbances is examined on the basis of the linear stability theory. Comparisons of stability characteristics are made between the subsonic and supersonic boundary layers at the edge Mach numbers 0.2 and 2.0, respectively. The result shows that the boundary layer becomes unstable to stationary three-dimensional modes when the cross-flow velocity exceeds a rather small threshold of less than 1% of the external flow velocity. Important to note that the critical Reynolds number for stationary modes does not strongly depend on the Mach number. It is also found that the wavelength of the most amplified stationary three-dimensional mode is four or five times the boundary-layer thickness, not depending on the magnitude of crossflow velocity both for the subsonic and supersonic flows.

The behaviors and the structures of the instability waves originating from a point-source in a 50-degrees-yawed cylinder boundary layer are investigated by computing incompressible Navier-Stokes equations. A principal attention is paid to the two different modes, namely the well-known cross-flow instability mode and the streamline-curvature instability mode which derives from the curvature of the external streamlines. It is shown that both, the streamline-curvature instability wave that travels to the cross-flow direction, and the cross-flow instability wave that travels to the opposite direction, are co-existing in the wedge region behind the point-source. It is also shown that two instability waves are made of two completely different vortical structures. Streamline-curvature instability wave mainly consists of vortical structures with their vorticity fluctuation vectors pointing normal to the wall, while in the cross-flow instability mode, the vorticity fluctuation vectors pointing parallel to the wall become important.

Linear stability of a similar boundary-layer flow to oblique-wave disturbances is governed by partial differential equations with respect to nondimensional vertical coordinate and the chordwise Reynolds number R1. In the case of swept Hiemenz flow, the equations admit a series solution expanded into inverse powers of R 1 2 and are then decomposed into an infinite sequence of ordinary differential systems with the leading one posing an eigenvalue problem to determine the first approximation to the complex dispersion relation. Numerical estimation of the series solution indicates a much lower critical Reynolds number of the oblique-wave instability than R3 = 583 of the spanwise-traveling T-S instability.

An experimental investigation of rotating-disk flow was made to separate the cross-flow traveling mode from the streamline-curvature mode by introducing a point-source disturbance through a hole on the disk. A glue-on hotwire probe was successfully applied to discriminate between the introduced traveling modes and the stationary modes, which contaminate the measurement using a conventional probe fixed in the laboratory frame, because of no sensitivity to cross-flow stationary vortices. The experimental results show a definite separation of the two modes appearing in wedge-shaped region downstream of the point source in accordance with the dispersion relation. The characteristic features, such as wavenumber, wave-crest inclination, and phase velocity of the observed disturbances are in good agreement with the linear stability theory making use of the method of complex characteristics, although there is slight disagreement in the spatial growth rate of cross-flow mode.

Instability of a non-parallel similar-boundary-layer flow to small and wavy disturbances is governed by partial differential equations with respect to the non-dimensional vertical coordinate ζ and the local Reynolds number R1 based on chordwise velocity of external stream and a boundary-layer thickness. In the particular case of swept Hiemenz flow, the equations admit a series solution expanded in inverse powers of R12 and then are decomposed into an infinite sequence of ordinary differential systems with the leading one posing an eigenvalue problem to determine the first approximation to the complex dispersion relation. Numerical estimation of the series solution indicates a much lower critical Reynolds number of the so-called oblique-wave instability than the classical value Rc = 583 of the spanwise-traveling Tollmien–Schlichting instability. Extension of the formulation to general Falkner–Skan–Cooke boundary layers is proposed in the form of a double power series with respect to 1/R12 and a small parameter ε denoting the difference of the Falkner–Skan parameter m from the attachment-line value m = 1.