On certain aspects of three-dimensional instability of parallel flows

1983 ◽  
Vol 385 (1788) ◽  
pp. 53-84 ◽  
Keyword(s):  
Three Dimensional ◽  
Periodic Variation ◽  
Finite Amplitude ◽  
Basic Flow ◽  
Spanwise Direction ◽  
Equilibrium Solutions ◽  
Wave Disturbances ◽  
Parallel Flows ◽  
Tollmien Schlichting ◽  
Dimensional Instability

The effect of small imperfections in shear flows on the development of finite-amplitude three-dimensional disturbances in the flow is examined. A model problem is studied, one in which the basic flow is plane Poiseuille flow in a channel and the small imperfection in the form of spanwise periodic variation of the basic flow is introduced from the channel boundaries. It is shown that this has a positive effect on the growth of larger Tollmien-Schlichting wave disturbances, which are in the form of standing waves in the spanwise direction. Equations for the amplitudes of these disturbances, based on Stuart-Watson-Eckhaus theory, are derived and over a range of Reynolds number, the regions in the wavenumber plane over which equilibrium solutions are possible are identified. The possibility of three-dimensional disturbances that are oscillatory in the streamwise direction but that may be growing exponentially in a spanwise direction is examined.

Passive boundary layer control of oblique disturbances by finite-amplitude streaks

2014 ◽  
Vol 749 ◽  
pp. 1-36 ◽  
Author(s):  
Shahab Shahinfar ◽  
Sohrab S. Sattarzadeh ◽  
Jens H. M. Fransson
Keyword(s):  
High Speed ◽  
Passive Control ◽  
Control Method ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Transition To Turbulence ◽  
Spanwise Direction ◽  
Passive Flow Control ◽  
Tollmien Schlichting ◽  
Streamwise Streaks

AbstractRecent experimental results on the attenuation of two-dimensional Tollmien–Schlichting wave (TSW) disturbances by means of passive miniature vortex generators (MVGs) have shed new light on the possibility of delaying transition to turbulence and hence accomplishing skin-friction drag reduction. A recurrent concern has been whether this passive flow control strategy would work for other types of disturbances than plane TSWs in an experimental configuration where the incoming disturbance is allowed to fully interact with the MVG array. In the present experimental investigation we show that not only TSW disturbances are attenuated, but also three-dimensional single oblique wave (SOW) and pair of oblique waves (POW) disturbances are quenched in the presence of MVGs, and that transition delay can be obtained successfully. For the SOW disturbance an unusual interaction between the wave and the MVGs occurs, leading to a split of the wave with one part travelling with a ‘mirrored’ phase angle with respect to the spanwise direction on one side of the MVG centreline. This gives rise to $\Lambda $-vortices on the centreline, which force a low-speed streak on the centreline, strong enough to overcome the high-speed streak generated by the MVGs themselves. Both these streaky boundary layers seem to act stabilizing on unsteady perturbations. The challenge in a passive control method making use of a non-modal type of disturbances to attenuate modal disturbances lies in generating stable streamwise streaks which do not themselves break down to turbulence.

Finite-amplitude three-dimensional instability of inviscid laminar boundary layers

1995 ◽  
Vol 290 ◽  
pp. 203-212
Author(s):  
Melvin E. Stern
Keyword(s):  
Three Dimensional ◽  
Streamwise Velocity ◽  
Finite Amplitude ◽  
Transition To Turbulence ◽  
Stream Velocity ◽  
Normal Velocity ◽  
Laminar Boundary Layers ◽  
Vertical Thickness ◽  
Layer Flow ◽  
Dimensional Instability

An inviscid laminar boundary layer flow Û(ŷ) with vertical thickness λy, and free stream velocity U is disturbed at time $\tcirc$ = 0 by a normal velocity $\vcirc$ and by a spanwise velocity ŵ([xcirc ],ŷ, $\zcirc$, 0) of finite amplitude αU, with spanwise ($\zcirc$) scale λz, and streamwise ([xcirc ]) scale λx = λz/α; the streamwise velocity û([xcirc ],ŷ,$\zcirc$,$\tcirc$) is initially undisturbed. A long wave λy/λz → 0) expansion of the Euler equations for fixed α and time scale $\tcirc$s = U−1λz/α results in a hyperbolic equation for Lagrangian displacements ŷ. Within the interval $\tcirc$ > $\tcirc$s of asymptotic validity, finite parcel displacements (O(λy)) with finite (O(U)) û fluctuations occur, independent of α no matter how small; the basic flow Û is therefore said to be unstable to streaky (λx [Gt ] λz) spanwise perturbations. The temporal development of the ('spot’) region in the (x,z) plane wherein inflected û profiles appear is computed and qualitatively related to observations of ‘breakdown’ and transition to turbulence in the flow over a flat plate. The maximum $\vcirc$([xcirc ],ŷ,$\zcirc$,$\tcirc$) increases monotonically to infinity as $\tcirc$ → $\tcirc$s.

Three-dimensional instabilities in steady and unsteady non-parallel boundary layers, including effects of Tollmien-Schlichting disturbances and cross flow

1987 ◽  
Vol 409 (1837) ◽  
pp. 229-248 ◽  
Keyword(s):  
Boundary Layers ◽  
Large Scale ◽  
Three Dimensional ◽  
Steady Motion ◽  
Cross Flow ◽  
Parallel Flows ◽  
Roughness Elements ◽  
Tollmien Schlichting ◽  
Flow Effects ◽  
Secondary Instabilities

Three-dimensional (3D) linear stability properties are considered for steady and unsteady 2D or 3D boundary layers with significant non-parallelism present. Two main examples of such non-parallel flows whose stability is of interest are, firstly, steady motion, over roughness elements, in cross flow, or in large-scale separation and, secondly, unsteady 2D Tollmien-Schlichting (TS) motion, with its associated question of secondary instabilities. A high-frequency stability analysis is presented here. It is found that, for 2DTS or steady boundary layers, there is a swing in the direction of maximum TS spatial growth rate, from 0° for parallel flow towards 64.68° away from the free-stream direction, as the nonparallel flow effects increase. These effects then depend principally on, and indeed are proportional to, the local slope of the boundary-layer displacement. Cross flow can also have a profound impact on TS instabilities. Further implications for higher-amplitude and/or fasterscale disturbances, their secondary instability, and nonlinear interactions, are also discussed.

scholarly journals Transition to turbulence when the Tollmien–Schlichting and bypass routes coexist

2019 ◽  
Vol 880 ◽  
Author(s):  
Stefan Zammert ◽  
Bruno Eckhardt
Keyword(s):  
Initial Conditions ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Transition To Turbulence ◽  
Stable Region ◽  
Bypass Transition ◽  
Parallel Plates ◽  
Tollmien Schlichting

Plane Poiseuille flow, the pressure-driven flow between parallel plates, shows a route to turbulence connected with a linear instability to Tollmien–Schlichting (TS) waves, and another route, the bypass transition, that can be triggered with finite-amplitude perturbation. We use direct numerical simulations to explore the arrangement of the different routes to turbulence among the set of initial conditions. For plates that are a distance $2H$ apart, and in a domain of width $2\unicode[STIX]{x03C0}H$ and length $2\unicode[STIX]{x03C0}H$, the subcritical instability to TS waves sets in at $Re_{c}=5815$ and extends down to $Re_{TS}\approx 4884$. The bypass route becomes available above $Re_{E}=459$ with the appearance of three-dimensional, finite-amplitude travelling waves. Below $Re_{c}$, TS transition appears for a tiny region of initial conditions that grows with increasing Reynolds number. Above $Re_{c}$, the previously stable region becomes unstable via TS waves, but a sharp transition to the bypass route can still be identified. Both routes lead to the same turbulent state in the final stage of the transition, but on different time scales. Similar phenomena can be expected in other flows where two or more routes to turbulence compete.

On the development of large-sized short-scaled disturbances in boundary layers

1985 ◽  
Vol 399 (1816) ◽  
pp. 25-55 ◽  
Keyword(s):  
Boundary Layers ◽  
Practical Importance ◽  
Three Dimensional ◽  
Viscous Sublayer ◽  
Parallel Flow ◽  
Basic Flow ◽  
Weakly Nonlinear ◽  
Second Stage ◽  
Tollmien Schlichting ◽  
Flow Effects

For high Reynolds numbers the planar nonlinear stability properties of boundary layers, as governed first by the triple-deck interaction, are discussed theoretically. When a disturbance of relatively high frequency is introduced or as the distance downstream increases in the Blasius case the characteristic length scale shortens. An account is given of the possible progress of such a disturbance, from an initially small Tollmien-Schlichting state, through a weakly nonlinear first stage that does not interrupt the amplitude growth, and thence into a fully nonlinear second stage where the Benjamin-Ono equation comes into force. Still higher frequencies point to the full Euler equations holding, although significant non-parallel flow effects then emerge. In the Euler stage, however, and in the previous second stage, bursts of vorticity from the viscous sublayer closest to the solid surface are almost certain to occur. The relations with other basic flow problems, and with turbulence-modelling, are noted, as well as the need to consider three-dimensional motion. Finally, an Appendix deals with certain issues of practical importance arising with respect to non-parallel flow effects, for example in breakaway separations or flow over surface-mounted obstacles, and points out that the slope of the local displacement in the basic flow is then a dominant controlling feature of the short-scale instabilities.

scholarly journals Localized vortex/Tollmien–Schlichting wave interaction states in plane Poiseuille flow

2016 ◽  
Vol 791 ◽  
pp. 97-121 ◽  
Author(s):  
L. J. Dempsey ◽  
K. Deguchi ◽  
P. Hall ◽  
A. G. Walton
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Wave Interaction ◽  
Travelling Wave ◽  
Navier Stokes ◽  
Spanwise Direction ◽  
Plane Poiseuille Flow ◽  
Tollmien Schlichting

Strongly nonlinear three-dimensional interactions between a roll–streak structure and a Tollmien–Schlichting wave in plane Poiseuille flow are considered in this study. Equations governing the interaction at high Reynolds number originally derived by Bennett et al. (J. Fluid Mech., vol. 223, 1991, pp. 475–495) are solved numerically. Travelling wave states bifurcating from the lower branch linear neutral point are tracked to finite amplitudes, where they are observed to localize in the spanwise direction. The nature of the localization is analysed in detail near the relevant spanwise locations, revealing the presence of a singularity which slowly develops in the governing interaction equations as the amplitude of the motion is increased. Comparisons with the full Navier–Stokes equations demonstrate that the finite Reynolds number solutions gradually approach the numerical asymptotic solutions with increasing Reynolds number.

scholarly journals Three-Dimensional Instability of Finite-Amplitude Water Waves

1981 ◽  
Vol 46 (13) ◽  
pp. 817-820 ◽  
Author(s):  
J. W. McLean ◽  
Y. C. Ma ◽  
D. U. Martin ◽  
P. G. Saffman ◽  
H. C. Yuen
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Dimensional Instability
Sign in / Sign up

Export Citation Format

Share Document