On the weakly nonlinear development of Tollmien-Schlichting wavetrains in boundary layers

1996 ◽  
Vol 323 ◽  
pp. 133-171 ◽  
Author(s):  
Xuesong Wu ◽  
Philip A. Stewart ◽  
Stephen J. Cowley
Keyword(s):  
Numerical Solutions ◽  
Asymptotic Methods ◽  
Nonlinear Effects ◽  
Mean Flow ◽  
Flow Distortion ◽  
Weakly Nonlinear ◽  
Nonlinear Development ◽  
Finite Distance ◽  
Tollmien Schlichting ◽  
High Reynolds

The nonlinear development of a weakly modulated Tollmien-Schlichting wavetrain in a boundary layer is studied theoretically using high-Reynolds-number asymptotic methods. The ‘carrier’ wave is taken to be two-dimensional, and the envelope is assumed to be a slowly varying function of time and of the streamwise and spanwise variables. Attention is focused on the scalings appropriate to the so-called ‘upper branch’ and ‘high-frequency lower branch’. The dominant nonlinear effects are found to arise in the critical layer and the surrounding ‘diffusion layer’: nonlinear interactions in these regions can influence the development of the wavetrain by producing a spanwise-dependent mean-flow distortion. The amplitude evolution is governed by an integro-partial-differential equation, whose nonlinear term is history-dependent and involves the highest derivative with respect to the spanwise variable. Numerical solutions show that a localized singularity can develop at a finite distance downstream. This singularity seems consistent with the experimentally observed focusing of vorticity at certain spanwise locations, although quantitative comparisons have not been attempted.

On the nonlinear evolution of a pair of oblique Tollmien–Schlichting waves in boundary layers

1997 ◽  
Vol 340 ◽  
pp. 361-394 ◽  
Author(s):  
XUESONG WU ◽  
S. J. LEIB ◽  
M. E. GOLDSTEIN
Keyword(s):  
Boundary Layer ◽  
Phase Angle ◽  
Spatial Scales ◽  
Asymptotic Methods ◽  
Three Dimensional ◽  
Critical Layer ◽  
Back Reaction ◽  
Finite Distance ◽  
Tollmien Schlichting ◽  
High Reynolds

This paper is concerned with the nonlinear interaction and development of a pair of oblique Tollmien–Schlichting waves which travel with equal but opposite angles to the free stream in a boundary layer. Our approach is based on high-Reynolds-number asymptotic methods. The so-called ‘upper-branch’ scaling is adopted so that there exists a well-defined critical layer, i.e. a thin region surrounding the level at which the basic flow velocity equals the phase velocity of the waves. We show that following the initial linear growth, the disturbance evolves through several distinct nonlinear stages. In the first of these, nonlinearity only affects the phase angle of the amplitude of the disturbance, causing rapid wavelength shortening, while the modulus of the amplitude still grows exponentially as in the linear regime. The second stage starts when the wavelength shortening produces a back reaction on the development of the modulus. The phase angle and the modulus then evolve on different spatial scales, and are governed by two coupled nonlinear equations. The solution to these equations develops a singularity at a finite distance downstream. As a result, the disturbance enters the third stage in which it evolves over a faster spatial scale, and the critical layer becomes both non-equilibrium and viscous in nature, in contrast to the two previous stages, where the critical layer is in equilibrium and purely viscosity dominated. In this stage, the development is governed by an amplitude equation with the same nonlinear term as that derived by Wu, Lee & Cowley (1993) for the interaction between a pair of Rayleigh waves. The solution develops a new singularity, leading to the fourth stage where the flow is governed by the fully nonlinear three-dimensional inviscid triple-deck equations. It is suggested that the stages of evolution revealed here may characterize the so-called ‘oblique breakdown’ in a boundary layer. A discussion of the extension of the analysis to include the resonant-triad interaction is given.

scholarly journals Precessional instability of a fluid cylinder

2011 ◽  
Vol 666 ◽  
pp. 104-145 ◽  
Author(s):  
ROMAIN LAGRANGE ◽  
PATRICE MEUNIER ◽  
FRANÇOIS NADAL ◽  
CHRISTOPHE ELOY
Keyword(s):  
Reynolds Number ◽  
Nonlinear Effects ◽  
Intermittent Flow ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Aspect Ratios ◽  
Image Velocimetry ◽  
Precessional Motion ◽  
Weakly Nonlinear Model

In this paper, the instability of a fluid inside a precessing cylinder is addressed theoretically and experimentally. The precessional motion forces Kelvin modes in the cylinder, which can become resonant for given precessional frequencies and cylinder aspect ratios. When the Reynolds number is large enough, these forced resonant Kelvin modes eventually become unstable. A linear stability analysis based on a triadic resonance between a forced Kelvin mode and two additional free Kelvin modes is carried out. This analysis allows us to predict the spatial structure of the instability and its threshold. These predictions are compared to the vorticity field measured by particle image velocimetry with an excellent agreement. When the Reynolds number is further increased, nonlinear effects appear. A weakly nonlinear theory is developed semi-empirically by introducing a geostrophic mode, which is triggered by the nonlinear interaction of a free Kelvin mode with itself in the presence of viscosity. Amplitude equations are obtained coupling the forced Kelvin mode, the two free Kelvin modes and the geostrophic mode. They show that the instability saturates to a fixed point just above threshold. Increasing the Reynolds number leads to a transition from a steady saturated regime to an intermittent flow in good agreement with experiments. Surprisingly, this weakly nonlinear model still gives a correct estimate of the mean flow inside the cylinder even far from the threshold when the flow is turbulent.

scholarly journals Stability of zero-pressure-gradient boundary layer distorted by unsteady Klebanoff streaks

2011 ◽  
Vol 681 ◽  
pp. 116-153 ◽  
Author(s):  
NICHOLAS J. VAUGHAN ◽  
TAMER A. ZAKI
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Low Frequency ◽  
Finite Amplitude ◽  
Base Flow ◽  
Streamwise Vorticity ◽  
Mean Flow ◽  
S Wave ◽  
Flow Distortion ◽  
Tollmien Schlichting

The secondary instability of a zero-pressure-gradient boundary layer, distorted by unsteady Klebanoff streaks, is investigated. The base profiles for the analysis are computed using direct numerical simulation (DNS) of the boundary-layer response to forcing by individual free-stream modes, which are low frequency and dominated by streamwise vorticity. Therefore, the base profiles take into account the nonlinear development of the streaks and mean flow distortion, upstream of the location chosen for the stability analyses. The two most unstable modes were classified as an inner and an outer instability, with reference to the position of their respective critical layers inside the boundary layer. Their growth rates were reported for a range of frequencies and amplitudes of the base streaks. The inner mode has a connection to the Tollmien–Schlichting (T–S) wave in the limit of vanishing streak amplitude. It is stabilized by the mean flow distortion, but its growth rate is enhanced with increasing amplitude and frequency of the base streaks. The outer mode only exists in the presence of finite amplitude streaks. The analysis of the outer instability extends the results of Andersson et al. (J. Fluid Mech. vol. 428, 2001, p. 29) to unsteady base streaks. It is shown that base-flow unsteadiness promotes instability and, as a result, leads to a lower critical streak amplitude. The results of linear theory are complemented by DNS of the evolution of the inner and outer instabilities in a zero-pressure-gradient boundary layer. Both instabilities lead to breakdown to turbulence and, in the case of the inner mode, transition proceeds via the formation of wave packets with similar structure and wave speeds to those reported by Nagarajan, Lele & Ferziger (J. Fluid Mech., vol. 572, 2007, p. 471).

The role of nonlinear critical layers in boundary layer transition

1995 ◽  
Vol 352 (1700) ◽  
pp. 425-442 ◽  
Keyword(s):  
Asymptotic Methods ◽  
Nonlinear Effects ◽  
Boundary Layer Transition ◽  
Critical Layer ◽  
Mean Flow ◽  
Layer Transition ◽  
Instability Waves ◽  
Laminar Boundary Layers ◽  
Instability Modes ◽  
Critical Layers

Asymptotic methods are used to describe the nonlinear self-interaction between pairs of oblique instability modes that eventually develops when initially linear spatially growing instability waves evolve downstream in nominally two-dimensional laminar boundary layers. The first nonlinear reaction takes place locally within a so-called ‘critical layer’, with the flow outside this layer consisting of a locally parallel mean flow plus a pair of oblique instability waves - which may or may not be accompanied by an associated plane wave. The amplitudes of these waves, which are completely determined by nonlinear effects within the critical layer, satisfy either a single integro-differential equation or a pair of integro-differential equations with quadratic to quartic-type nonlinearities. The physical implications of these equations are discussed.

The stability of spiral flow between rotating cylinders

1968 ◽  
Vol 263 (1135) ◽  
pp. 57-91 ◽  
Keyword(s):  
Asymptotic Methods ◽  
Stability Boundary ◽  
Nonlinear Effects ◽  
Present Theory ◽  
Rotating Cylinders ◽  
Axial Pressure Gradient ◽  
Tollmien Schlichting ◽  
The Stability ◽  
Sommerfeld Equation ◽  
Axisymmetric Disturbances

The effect of an axial pressure gradient on the stability of viscous flow between rotating cylinders is discussed on the basis of the narrow gap approximation, the assumption of axisymmetric disturbances, and the assumption that the cylinders rotate in the same direction. The onset of instability then depends on both the Taylor number ( T ) and the axial Reynolds number (R). For large values of R, the dominant mechanism of instability is of the Tollmien-Schlichting type and the present theory is based therefore on a generalization of the asymptotic methods of analysis that have been developed for the Orr-Sommerfeld equation. The present results, when combined with previous results for small values of R, give the complete stability boundary in the -plane. Only limited agreement is found with existing experimental data and it is suggested therefore that it may be necessary to consider either non-axisymmetric disturbances or nonlinear effects.

Nonlinear temporal-spatial modulation of near-planar Rayleigh waves in shear flows: formation of streamwise vortices

1993 ◽  
Vol 256 ◽  
pp. 685-719 ◽  
Author(s):  
Xuesong Wu
Keyword(s):  
Shear Flows ◽  
Numerical Solutions ◽  
Broad Class ◽  
Nonlinear Effects ◽  
Amplitude Equation ◽  
Spatial Modulation ◽  
Linear Growth Rate ◽  
Instability Wave ◽  
Streamwise Vortices ◽  
Mean Flow

The nonlinear temporal-spatial modulation of a near-planar Rayleigh instability wave is studied. The amplitude of the wave is allowed to be a slowly varying function of spanwise position as well as of time (or streamwise variable in the spatial evolution case). It is shown that the development of the disturbance is controlled by critical-layer nonlinear effects when the linear growth rate decreases to O(ε⅖), where ε is the magnitude of the disturbance. Nonlinear interactions influence the evolution by producing spanwise dependent mean-flow distortions. The evolution is governed by an integro-partial-differential equation containing history-dependent nonlinear terms of Hickernell (1984) type. A notable feature of the amplitude equation is that the highest derivative with respect to spanwise position appears in the nonlinear terms. These terms are associated with three-dimensionality. The possible properties of the amplitude equation are discussed. Numerical solutions show that a disturbance initially centred at a spanwise position can propagate laterally to form concentrated, quasi-periodic streamwise vortices. This qualitatively captures the phenomena observed in experiments. The focusing of vorticity may be associated with a localized singularity which can occur at a finite distance downstream or within a finite time. It is noted that the amplitude equation is rather generic and applies to a broad class of shear flows which is inviscidly unstable.

The evolution of a localized disturbance in a laminar boundary layer. Part 2. Strong disturbances

1990 ◽  
Vol 220 ◽  
pp. 595-621 ◽  
Author(s):  
Kenneth S. Breuer ◽  
Marten T. Landahl
Keyword(s):  
Boundary Layer ◽  
Plate Boundary ◽  
Nonlinear Effects ◽  
Initial Distribution ◽  
Initial Disturbance ◽  
Vertical Shear ◽  
Navier Stokes ◽  
Mean Velocity ◽  
Weakly Nonlinear ◽  
Tollmien Schlichting

Navier–Stokes calculations were performed to simulate the evolution of a moderate-amplitude localized disturbance in a laminar flat-plate boundary layer. It was found that, in accordance with previous results for linear and weakly nonlinear disturbances, the evolving disturbance consists of two parts: an advective, or transient portion which travels at approximately the local mean velocity, and a dispersive wave portion which grows or decays according to Tollmien–Schlichting instability theory. The advective portion grows much more rapidly than the wave portion, initially linearly in time and, in contrast to the weak-disturbance case, gives rise to two distinct nonlinear effects. The first is a streamwise growth of the disturbed region producing a low-speed streak, bounded in the vertical and spanwise directions by intense shear layers. The second nonlinear effect is the onset of a secondary instability on the vertical shear layer formed as a result of spanwise stretching of the mean vorticity and giving rise to oscillations in the v- and w-components with a substantially smaller spatial scale than that of the initial disturbance. The effect of initial spanwise scale is assessed by calculating the disturbance for three different cases in which the spanwise scale and the initial disturbance amplitude were varied. It was found that the resulting perturbation depends primarily on the initial distribution of v in each plane z = const., but is approximately independent of the spanwise scale.

Anelastic Internal Wave Packet Evolution and Stability

2011 ◽  
Vol 68 (12) ◽  
pp. 2844-2859 ◽  
Author(s):  
Hayley V. Dosser ◽  
Bruce R. Sutherland
Keyword(s):  
Nonlinear Dynamics ◽  
Growth Rate ◽  
Wave Packet ◽  
Internal Wave ◽  
Internal Gravity Wave ◽  
Nonlinear Effects ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Stability And Instability ◽  
Anelastic Equations

Abstract As upward-propagating anelastic internal gravity wave packets grow in amplitude, nonlinear effects develop as a result of interactions with the horizontal mean flow that they induce. This qualitatively alters the structure of the wave packet. The weakly nonlinear dynamics are well captured by the nonlinear Schrödinger equation, which is derived here for anelastic waves. In particular, this predicts that strongly nonhydrostatic waves are modulationally unstable and so the wave packet narrows and grows more rapidly in amplitude than the exponential anelastic growth rate. More hydrostatic waves are modulationally stable and so their amplitude grows less rapidly. The marginal case between stability and instability occurs for waves propagating at the fastest vertical group velocity. Extrapolating these results to waves propagating to higher altitudes (hence attaining larger amplitudes), it is anticipated that modulationally unstable waves should break at lower altitudes and modulationally stable waves should break at higher altitudes than predicted by linear theory. This prediction is borne out by fully nonlinear numerical simulations of the anelastic equations. A range of simulations is performed to quantify where overturning actually occurs.

scholarly journals Interaction of oblique instability waves with weak streamwise vortices

1995 ◽  
Vol 284 ◽  
pp. 377-407 ◽  
Author(s):  
M. E. Goldstein ◽  
David W. Wundrow
Keyword(s):  
Streamwise Vortices ◽  
Rayleigh Instability ◽  
Mean Flow ◽  
Flow Distortion ◽  
Instability Waves ◽  
Instability Modes ◽  
Blasius Boundary Layer ◽  
The Common ◽  
Tollmien Schlichting ◽  
Streamwise Distance

This paper is concerned with the effect of a weak spanwise-variable mean-flow distortion on the growth of oblique instability waves in a Blasius boundary layer. The streamwise component of the distortion velocity initially grows linearly with increasing streamwise distance, reaches a maximum, and eventually decays through the action of viscosity. This decay occurs slowly and allows the distortion to destabilize the Blasius flow over a relatively large streamwise region. It is shown that even relatively weak distortions can cause certain oblique Rayleigh instability waves to grow much faster than the usual two-dimensional Tollmien–Schlichting waves that would be the dominant instability modes in the absence of the distortion. The oblique instability waves can then become large enough to interact nonlinearly within a common critical layer. It is shown that the common amplitude of the interacting oblique waves is governed by the amplitude evolution equation derived in Goldstein & Choi (1989). The implications of these results for Klebanoff-type transition are discussed.

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