Nobutake Itoh

Observation of traveling waves in the three-dimensional boundary layer along a yawed cylinder

Experimental investigations are made of crossflow instability of the three-dimensional boundary layer along a yawed circular cylinder. To explore the effect of environmental disturbances on transition, comparative experiments are performed in high- and low-turbulence wind tunnels. Independent of turbulence level, unsteady disturbances with definite frequencies instead of stationary vortices are observed in hot-wire surveys of the unstable flow filed. Detailed measurements with the aid of hot-wire arrays have confirmed that the unsteady disturbances are identical with the so-called crossflow traveling waves predicted by linear stability theory as the most unstable disturbances, although some discrepancies still remain between experimental results and theoretical predictions.

Simple cases of the streamline-curvature instability in three-dimensional boundary layers

A new instability of the centrifugal type due to the curvature of external streamlines was theoretically predicted in a recent study on boundary layers along a swept wing. It is, however, not clear how this instability relates to already-known instability phenomena in various three-dimensional flows. So the basic idea developed in the analysis of boundary layers is applied to the simpler problems of the flow on a rotating disk and along the leading edge of a yawed circular cylinder, and the resulting eigenvalue problems are numerically solved to show multiple stability characteristics of the flows. Computational results confirm that the streamline-curvature instability does appear in the rotating-disk flow and that it is in fact identical with the instability called the ‘parallel’ or ‘type 2’ mode in the atmospheric literature. This instability is also found to occur in the steady flow near the attachment line and to give the lowest values of the critical Reynolds number except for a very narrow region close to the attachment line, where the viscous and cross-flow instabilities are dominant. These facts provide evidence to show that the same mode of instability as the classical one observed in rotating flows can appear in general three-dimensional boundary layers without rotation.

Instability of three-dimensional boundary layers due to streamline curvature

Investigated are the fundamental roles of the curvature of external streamlines in the stability of three-dimensional boundary layers, the partial differential equations governing small disturbances superimposed on the basic flows are modeled by a simple system of ordinary differential equations, which include a nondimensional parameter denoting magnitude of the streamline curvature. The eigenvalue problem posed by the model equations is numerically solved to evaluate effects of this parameter on critical Reynolds numbers of the Falkner–Skan–Cooke family of three-dimensional velocity profiles. Computational results predict the possibility of a new instability, essentially due to the streamline curvature, of the centrifugal type similar to the Taylor–Gortler instability caused by a concave curvature of the wall.

Development of wedge-shaped disturbances originating from a point source in a three-dimensional boundary layer

A recent study predicted possibility of existence of a new instability due to the curvature of external streamlines in three-dimensional boundary layers, besides the familiar cross-flow instability, but no reliable evidence of this phenomenon has yet been obtained in experiments. In expectation of dispersive development of the two instabilities, the present study deals with small disturbances induced by continuous forcing from a point source in the boundary layer along a yawed circular cylinder, and attempts to describe their spatial development into wedge-shaped distribution with a linear stability theory, which is applicable to both of the above instabilities. Unlike plane-wave disturbances, the point-source disturbances have an important peculiarity that their propagation is governed by a complex group velocity, and a new method based on the complex property of the group velocity is presented to predict the paths of propagation along which growth rates of disturbances are integrated. Results of this stability calculation clarify important differences in development between the cross-flow disturbances and the streamline-curvature disturbances. These differences will make it possible to observe the new mode of disturbances separately from the other in experiments.

The origin and subsequent development in space of Tollmien-Schlichting waves in a boundary layer

Two kinds of small-parameter expansions are used to derive a series of ordinary differential equations governing spatially growing or decaying disturbances of the travelling-wave type in non-parallel laminar boundary layers. Eigensolutions of the lowest-order equation are shown to tend to resemble eigensolutions of the unsteady boundary-layer equation as the Reynolds number Re decreases, and to the Tollmien-Schlichting wave solutions of the Orr-Sommerfeld equation as Re increases. Examination of the next-order approximation reveals that the unsteady boundary-layer equation has eigensolutions only if Re exceeds a critical value depending on the frequency of disturbances, while it remains a partial differential equation for lower Reynolds numbers. Thus the Tollmien-Schlichting wave may be considered to originate at the above critical value of Re. The subsequent development of the wave with fixed frequency can be followed by integrating the spatial growth rate of the eigensolutions with respect to the downstream distance. The amplitude is at first damped but grows even higher than its original value farther downstream, provided the frequency is less than a certain value.

Dispersive evolution of crossflow disturbances excited by an airjet column in a three-dimensional boundary layer

An experimental study is made to show that dispersive properties of disturbances originating from a point source lead to the separate appearance of stationary vortices and travelling waves of the crossflow instability in a three-dimensional boundary layer. The spatial development of disturbances induced by a weak airjet issuing from a small surface hole near the attachment line of a yawed circular cylinder is observed at several downstream stations. Experimental results show that the disturbances evolve into a wedge-shaped distribution, which includes both stationary and nonstationary modes of the crossflow instability. The observed behavior of those disturbances is shown to be in fairly good agreement with recent results of a linear stability theory based on the method of complex characteristics, although there are some incongruities in frequency and growth rate of the most unstable disturbances. The experiment also shows the existence of a new instability due to the curvature of external streamlines, which was simultaneously predicted by the theory.

Stability Calculations of the Three-Dimensional Boundary Layer Flow on a Rotating Disk

The Orr-Sommerfeld eigenvalue problem is solved to establish the neutral stability curve on the wavenumber plane for each of supercritical Reynolds numbers. Then the weakly nonlinear stability theory of Stuart and Watson is formally applied to obtain numerical values of the Landau coefficient, which are found to be negative on most part of the neutral curve, suggesting the existence of the supercritical stable equilibrium amplitude. It is however shown that the theory is inapplicable to a certain region of the wavenumbers, where an alternative analysis is made by the use of the resonant instability theory developed by Craik.

Experiments on Streamline-Curvature Instability in Boundary Layers on a Yawed Cylinder

Instability of the three-dimensional boundary layer on a yawed circular cylinder placed in a uniform flow is investigated experimentally by introducing acoustic disturbances from a point near the attachment line. To exemplify the flow dominated by streamline-curvature instability rather than crossflow instability, which has been often observed in many swept-wing flows, is the aim here. In upstream regions of the disturbance wedge originating from the point source, both streamline-curvature and crossflow disturbances are superposed on each other and yield complicated amplitude distributions. A newly proposed method enables the decomposition of the distorted amplitude distribution into contributions from the two instability modes. Detailed observations, however, show that the crossflow mode decays with the distance from the source much faster than the streamline-curvature mode and allows the latter to be dominant in a region further downstream. A fundamental characteristic of the streamline-curvature instability wave is confirmed by examining its phase distribution in the spanwise and normal directions.

A non-parallel theory for Görtler instability of Falkner–Skan boundary layers

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 Gortlers 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 Gortler instability.

Development of an axisymmetric wave packet in rotating-disk flow

The three-dimensional flow on a rotating disk admits two kinds of wavy disturbances induced by the cross-flow instability and the streamline-curvature instability. Their separate appearances by some artificial forcing are very desirable for experimental investigation of differences in dynamics and structure between the two instabilities. Thus the present study considers an instantaneous excitation of the boundary-layer flow through an annular slit on the disk surface and describes temporal and spatial development of the excited wavy disturbances by applying the method of complex characteristics to eigensolutions of approximate stability equations. Computational results show that observation at a downstream station will first catch the streamline-curvature disturbances with smaller circumferential wavenumbers and then the familiar cross-flow ones, because of a definite difference in propagation speed of the two modes. This study provides evidence to show that the method of complex characteristics is a very useful tool for theoretical prediction of dispersive properties of such instability waves.