The development of three-dimensional wave packets in a boundary layer

1968 ◽  
Vol 32 (1) ◽  
pp. 173-184 ◽  
Author(s):  
M. Gaster
Keyword(s):  
Boundary Layer ◽  
Linear Problem ◽  
Asymptotic Form ◽  
Wave Packets ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Critical Value ◽  
Lateral Spread ◽  
Disturbed Region ◽  
Dimensional Wave

The formation and growth of three-dimensional wave packets in a laminar boundary layer is treated as a linear problem. The asymptotic form of the disturbed region developing from a point source is obtained in terms of parameters describing two-dimensional instabilities of the flow. It is shown that a wave caustic forms and limits the lateral spread of growing disturbances whenever the Reynolds number is √2 times the critical value. The analysis is applied to the boundary layer on a flat plate and shapes of the wave-envelope are calculated for various Reynolds numbers. These show that all growing disturbances are contained within a wedge-shaped region of approximately 10° semi-angle.

Streamwise absolute instability of a three-dimensional boundary layer at high Reynolds numbers

1998 ◽  
Vol 373 ◽  
pp. 111-153 ◽  
Author(s):  
OLEG S. RYZHOV ◽  
EUGENE D. TERENT'EV
Keyword(s):  
Boundary Layer ◽  
Wind Tunnel ◽  
Wave Packets ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Absolute Instability ◽  
Streamwise Direction ◽  
High Reynolds Numbers ◽  
Dimensional Boundary ◽  
High Reynolds

The simplest receptivity problem of linear disturbances artificially excited in a three-dimensional boundary layer adjacent to a solid surface is studied in the framework of the generalized triple-deck theory. In order to provide a mathematical model to be compared with experimental data from wind-tunnel tests we consider the base flow over a swept flat plate. Then crossflow in the near-wall region originates owing to an almost constant pressure gradient induced from outside with a displacement body on top. A pulsed or vibrating ribbon installed on the solid surface serves as an external agency provoking initially weak pulsations. A periodic dependence of the ribbon shape on a coordinate normal to the streamwise direction makes the receptivity problem effectively two-dimensional, thereby allowing a rigorous analysis to be carried out without additional assumptions.The most striking result from the asymptotic theory is the discovery of streamwise absolute instability intrinsic to a three-dimensional boundary layer at high Reynolds numbers. However, due to limitations imposed on the receptivity problem no definite conclusions can be made with regard to possible continued convection of disturbances in the crossflow direction. An investigation of the dispersion-relation roots points to the fact that wave packets of different kinds can be generated by an external source operating in the pulse mode. Rapidly growing wave packets sweep downstream, weaker wave packets move against the oncoming stream. Insofar as the amplitude of all of the modulated signals increases exponentially in time and space, the excitation process gives rise to absolutely unstable disturbances in the streamwise direction. The computation confirms the theoretical prediction about the existence of upstream-advancing wave packets. They can be prevented from being persistently amplified only in a region ahead of the ribbon where nearly critical values of the Reynolds number are attained.The results achieved are shown to be broadly consistent with wind-tunnel measurements. Hence a conjecture is made that the onset of transition is probably associated, under some environmental conditions, with the mechanism of streamwise absolute instability in the supercritical range of the Reynolds numbers.

Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers

2010 ◽  
Vol 650 ◽  
pp. 181-214 ◽  
Author(s):  
ANTONIOS MONOKROUSOS ◽  
ESPEN ÅKERVIK ◽  
LUCA BRANDT ◽  
DAN S. HENNINGSON
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Wave Packets ◽  
Three Dimensional ◽  
Computational Domain ◽  
Initial Condition ◽  
Oblique Wave ◽  
Layer Flow ◽  
Optimal Disturbances ◽  
Matrix Free

The global linear stability of the flat-plate boundary-layer flow to three-dimensional disturbances is studied by means of an optimization technique. We consider both the optimal initial condition leading to the largest growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response. Both optimization problems are solved using a Lagrange multiplier technique, where the objective function is the kinetic energy of the flow perturbations and the constraints involve the linearized Navier–Stokes equations. The approach proposed here is particularly suited to examine convectively unstable flows, where single global eigenmodes of the system do not capture the downstream growth of the disturbances. In addition, the use of matrix-free methods enables us to extend the present framework to any geometrical configuration. The optimal initial condition for spanwise wavelengths of the order of the boundary-layer thickness are finite-length streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths, it is the Orr mechanism combined with the amplification of oblique wave packets that is responsible for the disturbance growth. This mechanism is dominant for the long computational domain and thus for the relatively high Reynolds number considered here. Three-dimensional localized optimal initial conditions are also computed and the corresponding wave packets examined. For short optimization times, the optimal disturbances consist of streaky structures propagating and elongating in the downstream direction without significant spreading in the lateral direction. For long optimization times, we find the optimal disturbances with the largest energy amplification. These are wave packets of Tollmien–Schlichting waves with low streamwise propagation speed and faster spreading in the spanwise direction. The pseudo-spectrum of the system for real frequencies is also computed with matrix-free methods. The spatial structure of the optimal forcing is similar to that of the optimal initial condition, and the largest response to forcing is also associated with the Orr/oblique wave mechanism, however less so than in the case of the optimal initial condition. The lift-up mechanism is most efficient at zero frequency and degrades slowly for increasing frequencies. The response to localized upstream forcing is also discussed.

Secondary instability and subcritical transition of the leading-edge boundary layer

2016 ◽  
Vol 792 ◽  
pp. 682-711 ◽  
Author(s):  
Michael O. John ◽  
Dominik Obrist ◽  
Leonhard Kleiser
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
High Speed ◽  
Three Dimensional ◽  
Leading Edge ◽  
Reynolds Numbers ◽  
Secondary Instability ◽  
Bypass Transition ◽  
Wall Suction ◽  
Transition Mechanism

The leading-edge boundary layer (LEBL) in the front part of swept airplane wings is prone to three-dimensional subcritical instability, which may lead to bypass transition. The resulting increase of airplane drag and fuel consumption implies a negative environmental impact. In the present paper, we present a temporal biglobal secondary stability analysis (SSA) and direct numerical simulations (DNS) of this flow to investigate a subcritical transition mechanism. The LEBL is modelled by the swept Hiemenz boundary layer (SHBL), with and without wall suction. We introduce a pair of steady, counter-rotating, streamwise vortices next to the attachment line as a generic primary disturbance. This generates a high-speed streak, which evolves slowly in the streamwise direction. The SSA predicts that this flow is unstable to secondary, time-dependent perturbations. We report the upper branch of the secondary neutral curve and describe numerous eigenmodes located inside the shear layers surrounding the primary high-speed streak and the vortices. We find secondary flow instability at Reynolds numbers as low as$Re\approx 175$, i.e. far below the linear critical Reynolds number$Re_{crit}\approx 583$of the SHBL. This secondary modal instability is confirmed by our three-dimensional DNS. Furthermore, these simulations show that the modes may grow until nonlinear processes lead to breakdown to turbulent flow for Reynolds numbers above$Re_{tr}\approx 250$. The three-dimensional mode shapes, growth rates, and the frequency dependence of the secondary eigenmodes found by SSA and the DNS results are in close agreement with each other. The transition Reynolds number$Re_{tr}\approx 250$at zero suction and its increase with wall suction closely coincide with experimental and numerical results from the literature. We conclude that the secondary instability and the transition scenario presented in this paper may serve as a possible explanation for the well-known subcritical transition observed in the leading-edge boundary layer.

Three-Dimensional Boundary-Layer Flow About an Ablating Slender Cone

1969 ◽  
Vol 91 (4) ◽  
pp. 632-648 ◽  
Author(s):  
T. K. Fannelop ◽  
P. C. Smith
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Cone Angle ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Low Reynolds Numbers ◽  
Boundary Layer Equations ◽  
Transverse Curvature ◽  
Dimensional Boundary ◽  
Layer Flow

A theoretical analysis is presented for three-dimensional laminar boundary-layer flow about slender conical vehicles including the effect of transverse surface curvature. The boundary-layer equations are solved by standard finite difference techniques. Numerical results are presented for hypersonic flow about a slender blunted cone. The influences of Reynolds number, cone angle, and mass transfer are studied for both symmetric flight and at angle-of-attack. The effects of transverse curvature are substantial at the low Reynolds numbers considered and are enhanced by blowing. The crossflow wall shear is largely unaffected by transverse curvature although the peak velocity is reduced. A simplified “channel flow” analogy is suggested for the crossflow near the wall.

The three-dimensional nature of boundary-layer instability

1962 ◽  
Vol 12 (1) ◽  
pp. 1-34 ◽  
Author(s):  
P. S. Klebanoff ◽  
K. D. Tidstrom ◽  
L. M. Sargent
Keyword(s):  
Boundary Layer ◽  
Wave Motion ◽  
Three Dimensional ◽  
Mean Flow ◽  
Linear Effect ◽  
Non Linear Effect ◽  
Theoretical Approaches ◽  
Boundary Layer Instability ◽  
Non Linear ◽  
Dimensional Wave

An experimental investigation is described in which principal emphasis is given to revealing the nature of the motions in the non-linear range of boundary-layer instability and the onset of turbulence. It has as its central purpose the evaluation of existing theoretical considerations and the provision of a sound physical model which can be taken as a basis for a theoretical approach. The experimental method consisted of introducing, in a two-dimensional boundary layer on a flat plate at ‘incompressible’ speeds, three-dimensional disturbances under controlled conditions using the vibrating-ribbon technique, and studying their growth and evolution using hot-wire methods. It has been definitely established that longitudinal vortices are associated with the non-linear three-dimensional wave motions. Sufficient data were obtained for an evaluation of existing theoretical approaches. Those which have been considered are the generation of higher harmonics, the interaction of the mean flow and the Reynold stress, the concave streamline curvature associated with the wave motion, the vortex loop and the non-linear effect of a three-dimensional perturbation. It appears that except for the latter they do not adequately describe the observed phenomena. It is not that they are incorrect or may not play a role in some aspect of the local behaviour, but from the over-all point of view the results suggest that it is the non-linear effect of a three-dimensional perturbation which dominates the behaviour. A principal conclusion to be drawn is that a new perspective, one that takes three-dimensionality into account, is required in connexion with boundary-layer instability. It is demonstrated that the actual breakdown of the wave motion into turbulence is a consequence of a new instability which arises in the aforementioned three-dimensional wave motion. This instability involves the generation of ‘hairpin’ eddies and is remarkably similar in behaviour to ‘inflexional’ instability. It is also shown that the results obtained from the study of controlled disturbances are equally applicable to ‘natural’ transition.

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