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NASA NTRS1985

Scattering of acoustic waves into Tollmien-Schlichting waves by small streamwise variations in surface geometry

By using the triple-deck scaling of Stewartson (1969) and Messiter (1970) it is shown that small but relatively sudden surface geometry variations that produce only very weak static pressure variations can nevertheless produce strong, i.e. O(1), coupling between an externally imposed acoustic disturbance and a spatially growing Tollmien-Schlichting wave. The analysis provides a qualitative explanation of the Leehey and Shapiro (1979) boundary-layer receptivity measurements and is in good quantitative agreement with the Aizin and Poliakov (1979) experiment. It may also explain why small 'trip wires' can promote early transition.

Goldstein, M. E.↗
NASA NTRS1987

Predicted and hot-film measured Tollmien-Schlichting wave characteristics

The Tollmien-Schlichting (TS) instability is a time-dependence instability which can lead to transition of laminar boundary layers on airfoils. A comparison of theoretical predictions and experimental observations of the TS instability on the NLF(1)-0414F airfoil designed by Viken and Pfenninger. The theoretical predictions were obtained using the SALLY stability code. Test results, from the same hot films that were used to detect transition, revealed that TS waves could be detected by the hot films if the hot-film signal was adequately modified.

Stack, John P.↗
NASA NTRS1988

On the secondary instability of Taylor-Goertler vortices to Tollmien-Schlichting waves in fully developed flows

There are many flows of practical importance where both Tollmien-Schlichting waves and Taylor-Goertler vortices are possible causes of transition to turbulence. The effect of fully nonlinear Taylor-Goertler vortices on the growth of small amplitude Tollmien-Schlichting waves is investigated. The basic state considered is the fully developed flow between concentric cylinders driven by an azimuthal pressure gradient. It is hoped that an investigation of this problem will shed light on the more complicated external boundary layer problem where again both modes of instability exist in the presence of concave curvature. The type of Tollmien-Schlichting waves considered have the asymptotic structure of lower branch modes of plane Poiseuille flow. Whilst instabilities at lower Reynolds number are possible, the latter modes are simpler to analyze and more relevant to the boundary layer problem. The effect of fully nonlinear Taylor-Goertler vortices on both two-dimensional and three-dimensional waves is determined. It is shown that, whilst the maximum growth as a function of frequency is not greatly affected, there is a large destabilizing effect over a large range of frequencies.

Bennett, James↗
NASA NTRS1988

On the interaction of Tollmien-Schlichting waves in axisymmetric supersonic flows

Two-dimensional lower branch Tollmien-Schlichting waves described by triple-deck theory are always stable for planar supersonic flows. The possible occurrence of axisymmetric unstable modes in the supersonic flow around an axisymmetric body is investigated. In particular flows around bodies with typical radii comparable with the thickness of the upper deck are considered. It is shown that such unstable modes exist below a critical nondimensional radius of the body a sub 0. At values of the radius above a sub 0 all the modes are stable while if unstable modes exist they are found to occur in pairs. The interaction of these modes in the nonlinear regime is investigated using a weakly nonlinear approach and it is found that, dependent on the frequencies of the imposed Tollmien-Schlichting waves, either of the modes can be set up.

Duck, P. W.↗
NASA NTRS1989

On the interaction of Tollmien-Schlichting waves in axisymmetric supersonic flows

Two-dimensional lower branch Tollmien-Schlichting waves described by triple-deck theory are always stable for planar supersonic flows. The possible occurrence of axisymmetric unstable modes in the supersonic flow around an axisymmetric body is investigated. In particular flows around bodies with typical radii comparable with the thickness of the upper deck are considered. It is shown that such unstable modes exist below a critical nondimensional radius of the body a sub O. At values of the radius above a sub O all the modes are stable while if unstable modes exist they are found to occur in pairs. The interaction of these modes in the nonlinear regime is investigated using a weakly nonlinear approach and it is found that, dependent on the frequencies of the imposed Tollmien-Schlichting waves, either of the modes can be set up.

Duck, P. W.↗
NASA NTRS1983

The evolution of Tollmien-Schlichting waves near a leading edge

The method of matched asymptotic expansions is used to study the generation of Tollmien-Schlichting waves by free-stream disturbances incident on a flat-plate boundary layer. Near the leading edge, the motion is governed by the unsteady boundary-layer equation, while farther downstream it is governed (to lowest order) by the Orr-Sommerfeld equation with slowly varying coefficients. It is shown that there is an overlap domain where the Tollmien-Schlichting wave solutions to the Orr-Sommerfeld equation and appropriate asymptotic solutions of the unsteady boundary-layer equation match, in the matched-asymptotic-expansion sense. The analysis explains how long-wavelength free-stream disturbances can generate Tollmien-Schlichting waves of much shorter wavelength. It also leads to a set of scaling laws for the asymptotic structure of the unsteady boundary layer.

Goldstein, M. E.↗
NASA NTRS1986

On the secondary instability of Taylor-Goertler vortices to Tollmien-Schlichting waves in fully-developed flows

There are many flows of practical importance where both Tollmien-Schlichting waves and Taylor-Goertler vortices are possible causes of transition to turbulence. The effect of fully nonlinear Taylor-Goertler vortices on the growth of small amplitude Tollmien-Schlichting waves is investigated. The basic state considered is the fully developed flow between concentric cylinders driven by an azimuthal pressure gradient. It is hoped that an investigation of this problem will shed light on the more complicated external boundary layer problem where again both modes of instability exist in the presence of concave curvature. The type of Tollmein-Schlichting waves considered have the asymptotic structure of lower branch modes of plane Poisseulle flow. Whilst instabilities at lower Reynolds number are possible, the latter modes are simpler to analyze and more relevant to the boundary layer problem. The effect of fully nonlinear Taylor-Goertler vortices on both two-dimensional and three-dimensional waves is determined. It is shown that, whilst the maximum growth as a function of frequency is not greatly affected, there is a large destabilizing effect over a large range of frequencies.

Bennett, James↗
NASA NTRS1983

Generation of Tollmien-Schlichting waves by free-stream disturbances at low Mach numbers

The method of matched asymptotic expansions is used to study the generation of Tollmien-Schlichting waves by free stream disturbances incident on a flat plate boundary layer. Near the leading edge, the motion is governed by the unsteady boundary layer equation, while farther downstream it is governed (to lowest order) by the Orr-Sommerfeld equation with slowly varying coefficients. It is shown that there is an overlap domain where the Tollmien-Schlichting wave solutions to the Orr-Sommerfeld equation and an appropriate asymptotic solution of the unsteady boundary layer equation match, in the matched asymptotic expansion sense. The analysis leads to a set of scaling laws for the asymptotic structure of the unsteady boundary layer.

Goldstein, M. E.↗
NASA NTRS1979

Effect of streamwise vortices on Tollmien-Schlichting waves

The method of multiple scales is used to determine a first order uniform expansion for the effect of counter rotating steady streamwise vortices in growing boundary layers on Tollmien-Schlichting waves. The results show that such vortices have a strong tendency to amplify three dimensional Tollmien-Schlichting waves having a spanwise wavelength that is twice the wavelength of the vortices. An analytical expression is derived for the growth rates of these waves. These growth rates increase linearly with increasing amplitudes of the vortices.

Nayfeh, A. H.↗
NASA NTRS1981

Effect of streamwise vortices on Tollmien-Schlichting waves

The method of multiple scales is used to determine a first-order uniform expansion for the effect of counter-rotating steady streamwise vortices in growing boundary layers on oblique Tollmien-Schlichting waves. The results show that such vortices have a strong tendency to amplify oblique Tollmien-Schlichting waves having a spanwise wavelength that is twice the wavelength of the vortices. An analytical expression is derived for the growth rates of these waves. These exponential growth rates increase linearly with increasing amplitudes of the vortices. Numerical results are presented. They suggest that this mechanism may dominate the instability.

Nayfeh, A. H.↗
NASA NTRS1983

The evolution of Tollmien-Schlichting waves near a leading edge. II - Numerical determination of amplitudes

In the first part of this investigation, Goldstein (1983) has shown that the amplitude of the spatially growing Tollmien-Schlichting wave generated by a time-harmonic free-stream disturbance is related to the coefficient multiplying the lowest-order asymptotic eigensolution of the unsteady boundary-layer equation. In the present study, a numerical solution of the unsteady boundary-layer equation is used to relate the amplitude of the asymptotic eigensolution, and consequently of the Tollmien-Schlichting wave, to that of the imposed free-stream disturbance for the special case of a uniformly pulsating stream. It is pointed out that the ideas of this study can be extended to other, more complex bodies and free-stream oscillations.

Goldstein, M. E.↗
NASA NTRS1988

The nonlinear interaction of Tollmien-Schlichting waves and Taylor-Goertler vortices in curved channel flows

The development of Tollmien-Schlichting waves (TSWs) and Taylor-Goertler vortices (TGVs) in fully developed viscous curved-channel flows is investigated analytically, with a focus on their nonlinear interactions. Two types of interactions are identified, depending on the amplitude of the initial disturbances. In the low-amplitude type, two TSWs and one TGV interact, and the scaled amplitudes go to infinity on a finite time scale; in the higher-amplitude type, which can also occur in a straight channel, the same singularity occurs if the angle between the TSW wavefront and the TGV is greater than 41.6 deg, but the breakdown is exponential and takes an infinite time if the angle is smaller. The implications of these findings for external flow problems such as the design of laminar-flow wings are indicated. It is concluded that longitudinal vortices like those observed in the initial stages of the transition to turbulence can be produced unless the present interaction mechanism is destroyed by boundary-layer growth.

Hall, P.↗
NASA NTRS1987

Interaction of Tollmien-Schlichting waves and Goertler vortices

There are many fluid flows of practical interest where transition can be caused by competing hydrodynamic instabilities. Thus in three-dimensional boundary-layer flows over curved walls, instability might be caused by Tollmien-Schlichting waves, Goertler vortices or crossflow vortices. If a particular type of instability is suppressed by some means, there is the possibility that another one might be stimulated. Hence it is important to understand the mechanisms by which these different instabilities interact. Some properties of the interaction which can take place between Goertler vortices and Tollmien-Schlichting waves are discussed.

Hall, Phil↗
NASA NTRS1987

A note on the generation of Tollmien-Schlichting waves by sudden surface-curvature change

This note is primarily concerned with the generation of spatially growing Tollmien-Schlichting waves by the interaction of very long-wavelength free-stream disturbances with a discontinuity in the curvature of a bounding surface (whose slope may or may not be continuous). The theory is combined with a numerical solution of the local Orr-Sommerfeld equation, and the result is used to predict the Tollmien-Schlichting amplitude in a relevant experiment carried out by Leehey and Shapiro (1980). The calculated results are in satisfactory agreement with their observations.

Goldstein, M. E.↗
NASA NTRS1985

Taylor-Goertler instabilities of Tollmien-Schlichting waves and other flows governed by the interactive boundary layer equations

The Taylor-Gortler vortex instability equations are formulated for steady and unsteady interacting boundary layer flows of the type which arise in triple-deck theory. The effective Gortler number is shown to be a function of the all shape in the boundary layer and the possibility of both steady and unsteady Taylor-Gortler modes exists. As an example the steady flow in a symmetrically constricted channel is considered and it is shown that unstable Gortler vortices exist before the boundary layers at the wall develop the Goldstein singularity. As an example of an unsteady spatially varying basic state the instability of high frequency large amplitude Tollmien-Schlichting waves in a curved channel were considered. It is shown that they are unstable in the first Stokes layer stage of the hierarchy of nonlinear states. The Tollmien-Schlichting waves are shown to be unstable in the presence of both convex and concave curvature.

Hall, P.↗
NASA NTRS1986

Taylor-Goertler instabilities of Tollmien-Schlichting waves and other flows governed by the interactive boundary-layer equations

The Taylor-Goertler vortex instability equations are formulated for steady and unsteady interacting boundary-layer flows. The effective Goertler number is shown to be a function of the wall shape in the boundary layer and the possibility of both steady and unsteady Taylor-Goertler modes exists. As an example the steady flow in a symmetrically constricted channel is considered and it is shown that unstable Goertler vortices exist before the boundary layers at the wall develop the Goldstein singularity discussed by Smith and Daniels (1981). As an example of an unsteady spatially varying basic state, it is considered the instability of high-frequency large-amplitude two- and three-dimensional Tollmien-Schlichting waves in a curved channel. It is shown that they are unstable in the first 'Stokes-layer stage' of the hierarchy of nonlinear states discussed by Smith and Burggraf (1985). This instability of Tollmien-Schlichting waves in an internal flow can occur in the presence of either convex or concave curvature. Some discussion of this instability in external flows is given.

Hall, Philip↗
NASA NTRS1990

Boundary-layer receptivity due to a wall suction and control of Tollmien-Schlichting waves

A numerical study of the generation of Tollmien-Schlichting (T-S) waves due to the interaction between a small free-stream disturbance and a small localized suction slot on an otherwise flat surface was carried out using finite difference methods. The nonlinear steady flow is of the viscous-inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier-Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of T-S waves generated by the interaction between the free-stream disturbance and the suction slot, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T-S waves and the demonstration of the possible active control of the growth of T-S waves.

Bodonyi, R. J.↗