scholarly journals Tollmien-Schlichting route to elastoinertial turbulence in channel flow

2021 ◽  
Vol 6 (9) ◽  
Author(s):  
Ashwin Shekar ◽  
Ryan M. McMullen ◽  
Beverley J. McKeon ◽  
Michael D. Graham
Keyword(s):  
Channel Flow ◽  
Tollmien Schlichting

Nonlinear interaction of oblique three-dimensional Tollmien-Schlichting waves and longitudinal vortices, in channel flows and boundary layers

1992 ◽  
Vol 436 (1898) ◽  
pp. 585-602 ◽  
Keyword(s):  
Boundary Layers ◽  
Channel Flow ◽  
Nonlinear Interaction ◽  
Vortex Flow ◽  
Blow Up ◽  
Three Dimensional ◽  
Channel Flows ◽  
Nonlinear Evolution Equations ◽  
Longitudinal Vortex ◽  
Tollmien Schlichting

The present theoretical article considers the nonlinear interaction of oblique three dimensional Tollmien-Schlichting waves and induced or input longitudinal vortex motion, mainly for channel flow at large Reynolds numbers. Both the waves and the vortices are controlled by viscous-inviscid balancing but their respective flow structures are rather different because of the different typical timescales involved. This leads to the vortex-wave interaction being governed by nonlinear evolution equations on the vortex timescale, even though the wave amplitudes are notably small. The analogue in boundary-layer transition, addressed in a previous paper, is also re-considered here. Computational and analytical properties of the interaction equations for both channel flows and boundary layers are investigated, along with certain connections with companion studies of other vortex-wave interactions in channel flow. The nonlinear interactions in channel flow are found to lead to finitetime blow-up in amplitudes or to sustained vortex flow at large scaled times, depending on the input conditions. In particular, increasing the input amplitudes of the vortex or the wave can readily provoke blow-up even in the linearly stable regime; whereas in the case of sustained vortex flow new physical effects come into play on slightly longer timescales. Again, a very interesting feature is that the blowup response is found to be confined to a small range of wave angles near 45° relative to the original flow direction.

Wave Propagation in Flows Across Junctions Between Rigid and Flexible Walls

2002 ◽  
Author(s):  
P. W. Carpenter ◽  
P. K. Sen ◽  
S. Hegde ◽  
C. Davies
Keyword(s):  
Wave Propagation ◽  
Channel Flow ◽  
Incident Wave ◽  
Wave System ◽  
Transmitted Waves ◽  
Generic Problem ◽  
Flexible Walls ◽  
Walled Channel ◽  
Spatially Homogeneous ◽  
Tollmien Schlichting

The generic problem considered is the propagation of vortical waves across junctions between one wave-bearing medium and another. It is assumed that the eigensolutions are known for the corresponding spatially homogeneous problems. The task is how to determine the amplitudes of the reflected and transmitted waves given the amplitude of the incident wave. In general, there may be more than one incident, reflected or transmitted wave. It is shown how this sort of problem may be solved in terms of the homogeneous eigensolutions by drawing an analogy between the junction and a wave-driver. The particular illustrative problem studied is that of a Tollmien-Schlichting wave, propagating along a rigid-walled channel flow, that is incident on a section of the channel where the walls consist of compliant panels. It is shown how the wave system over the compliant panels and the amplitude of the Tollmien-Schlichting wave leaving the compliant section may be determined in terms of the incident wave. The technique developed for this problem is considered to be generic.

Simulations of natural transition in viscoelastic channel flow

2017 ◽  
Vol 820 ◽  
pp. 232-262 ◽  
Author(s):  
Sang Jin Lee ◽  
Tamer A. Zaki
Keyword(s):  
Growth Rate ◽  
Drag Reduction ◽  
Channel Flow ◽  
Secondary Instability ◽  
Transition To Turbulence ◽  
Newtonian Flow ◽  
Final State ◽  
Periodic State ◽  
Nonlinear Amplification ◽  
Tollmien Schlichting

Orderly, or natural, transition to turbulence in dilute polymeric channel flow is studied using direct numerical simulations of a FENE-P fluid. Three Weissenberg numbers are simulated and contrasted to a reference Newtonian configuration. The computations start from infinitesimally small Tollmien–Schlichting (TS) waves and track the development of the instability from the early linear stages through nonlinear amplification, secondary instability and full breakdown to turbulence. At the lowest elasticity, the primary TS wave is more unstable than the Newtonian counterpart, and its secondary instability involves the generation of $\unicode[STIX]{x1D6EC}$-structures which are narrower in the span. These subsequently lead to the formation of hairpin packets and ultimately breakdown to turbulence. Despite the destabilizing influence of weak elasticity, and the resulting early transition to turbulence, the final state is a drag-reduced turbulent flow. At the intermediate elasticity, the growth rate of the primary TS wave matches the Newtonian value. However, unlike the Newtonian instability mode which reaches a saturated equilibrium condition, the instability in the polymeric flow reaches a periodic state where its energy undergoes cyclical amplification and decay. The spanwise size of the secondary instability in this case is commensurate with the Newtonian $\unicode[STIX]{x1D6EC}$-structures, and the extent of drag reduction in the final turbulent state is enhanced relative to the lower elasticity condition. At the highest elasticity, the exponential growth rate of the TS wave is weaker than the Newtonian flow and, as a result, the early linear stage is prolonged. In addition, the magnitude of the saturated TS wave is appreciably lower than the other conditions. The secondary instability is also much wider in the span, with weaker ejection and without hairpin packets. Instead, streamwise-elongated streaks are formed and break down to turbulence via secondary instability. The final state is a high-drag-reduction flow, which approaches the Virk asymptote.

scholarly journals The effect of wall heating on instability of channel flow

2007 ◽  
Vol 577 ◽  
pp. 417-442 ◽  
Author(s):  
A. SAMEEN ◽  
RAMA GOVINDARAJAN
Keyword(s):  
Channel Flow ◽  
Secondary Growth ◽  
Transient Growth ◽  
Wall Heating ◽  
Heat Diffusivity ◽  
Order Of Magnitude ◽  
Tollmien Schlichting ◽  
Prandtl Numbers ◽  
Rayleigh Benard

A comprehensive study of the effect of wall heating or cooling on the linear, transient and secondary growth of instability in channel flow is conducted. The effect of viscosity stratification, heat diffusivity and of buoyancy are estimated separately, with some unexpected results. From linear stability results, it has been accepted that heat diffusivity does not affect stability. However, we show that realistic Prandtl numbers cause a transient growth of disturbances that is an order of magnitude higher than at zero Prandtl number. Buoyancy, even at fairly low levels, gives rise to high levels of subcritical energy growth. Unusually for transient growth, both of these are spanwise-independent and not in the form of streamwise vortices. At moderate Grashof numbers, exponential growth dominates, with distinct Poiseuille–Rayleigh–Bénard and Tollmien–Schlichting modes for Grashof numbers up to ∼ 25 000, which merge thereafter. Wall heating has a converse effect on the secondary instability compared to the primary instability, destabilizing significantly when viscosity decreases towards the wall. It is hoped that the work will motivate experimental and numerical efforts to understand the role of wall heating in the control of channel and pipe flows.

Laminarization mechanisms and extreme-amplitude states in rapidly rotating plane channel flow

2013 ◽  
Vol 730 ◽  
pp. 193-219 ◽  
Author(s):  
Stefan Wallin ◽  
Olof Grundestam ◽  
Arne V. Johansson
Keyword(s):  
Channel Flow ◽  
Plane Channel ◽  
Cross Flow ◽  
Reynolds Numbers ◽  
Spanwise Direction ◽  
Rotation Rates ◽  
Rotation Numbers ◽  
Tollmien Schlichting ◽  
Secondary Instabilities ◽  
Plane Channel Flow

AbstractFully developed plane channel flow rotating in the spanwise direction has been studied analytically and numerically. Linear stability analysis reveals that all cross-flow modes are stable for supercritical rotation numbers, $Ro\gt R{o}_{c} $, where $R{o}_{c} $ will approach 3 from below for increasing Reynolds number. Plane Tollmien–Schlichting (TS) waves are independent of rotation and always linearly unstable for supercritical Reynolds numbers. Direct numerical simulation (DNS) of the laminarization process reveals that the turbulence is damped when $Ro$ approaches $R{o}_{c} $. Hence, the laminarization is dominated by linear mechanisms. The flow becomes periodic for supercritical Reynolds numbers and rotation rates, i.e. when $Ro\gt R{o}_{c} $ and $Re\gt R{e}_{c} $. At such rotation rates, all oblique (cross-flow) modes are damped and when the disturbance amplitude becomes small enough, the TS modes start to grow exponentially. Secondary instabilities are initially blocked by the rotation since all cross-flow modes are linearly stable and the breakdown to turbulence will be strongly delayed. Hence, the TS waves will reach extremely high amplitudes, much higher than for typical turbulent fluctuations. Eventually, the extreme-amplitude state with TS-like waves will break down to turbulence and the flow will laminarize due to the influence of the rapid rotation, thus completing the cycle that will then be repeated. This flow is strongly dominated by linear mechanisms, which is remarkable considering the extremely high amplitudes involved in the processes of laminarization of the turbulence at $Ro\geq R{o}_{c} $ and the growth of the unstable TS waves.

Instabilities in a plane channel flow between compliant walls

1997 ◽  
Vol 352 ◽  
pp. 205-243 ◽  
Author(s):  
CHRISTOPHER DAVIES ◽  
PETER W. CARPENTER
Keyword(s):  
Dispersion Relation ◽  
Shear Layer ◽  
Channel Flow ◽  
Plane Channel ◽  
Mean Flow ◽  
Blasius Flow ◽  
Divergence Instability ◽  
Compliant Walls ◽  
Tollmien Schlichting ◽  
Plane Channel Flow

The stability of plane channel flow between compliant walls is investigated for disturbances which have the same symmetry, with respect to the channel centreline, as the Tollmien–Schlichting mode of instability. The interconnected behaviour of flow-induced surface waves and Tollmien–Schlichting waves is examined both by direct numerical solution of the Orr–Sommerfeld equation and by means of an analytic shear layer theory. We show that when the compliant wall properties are selected so as to give a significant stability effect on Tollmien–Schlichting waves, the onset of divergence instability can be severely disrupted. In addition, travelling wave flutter can interact with the Tollmien–Schlichting mode to generate a powerful instability which replaces the flutter instability identified in studies based on a potential mean-flow model. The behaviour found when the mean-flow shear layer is fully accounted for may be traced to singularities in the wave dispersion relation. These singularities can be attributed to solutions which represent Tollmien–Schlichting waves in rigid-walled channels. Such singularities will also be found in the dispersion relation for the case of Blasius flow. Thus, similar behaviour can be anticipated for Blasius flow, including the disruption of the onset of divergence instability. As a consequence, it seems likely that previous investigations for Blasius flow will have yielded very conservative estimates for the optimal stabilization that can be achieved for Tollmien–Schlichting waves for the purposes of laminar-flow control.

Instabilities in channel flow with system rotation

1989 ◽  
Vol 202 ◽  
pp. 543-557 ◽  
Author(s):  
P. Henrik Alfredsson ◽  
Håkan Persson
Keyword(s):  
Reynolds Number ◽  
Channel Flow ◽  
Large Scale ◽  
Critical Reynolds Number ◽  
Flow Direction ◽  
Reynolds Numbers ◽  
Tollmien Schlichting ◽  
Scale Turbulence ◽  
High Reynolds ◽  
Primary Instability

A flow visualization study of instabilities caused by Coriolis effects in plane rotating Poiseuille flow has been carried out. The primary instability takes the form of regularly spaced roll cells aligned in the flow direction. They may occur at Reynolds numbers as low as 100, i.e. almost two orders of magnitude lower than the critical Reynolds number for Tollmien-Schlichting waves in channel flow without rotation. The development of such roll cells was studied as a function of both the Reynolds number and the rotation rate and their properties compared with results from linear spatial stability theory. The theoretically obtained most unstable wavenumber agrees fairly well with the experimentally observed value. At high Reynolds number a secondary instability sets in, which is seen as a twisting of the roll cells. A wavytype disturbance is also seen at this stage which, if the rotational speed is increased, develops into large-scale ‘turbulence’ containing imbedded roll cells.

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