Tracking stages of transition in Couette flow analytically

2014 ◽  
Vol 748 ◽  
pp. 896-931 ◽  
Author(s):  
Michael Karp ◽  
Jacob Cohen
Keyword(s):  
Growth Mechanism ◽  
Inflection Point ◽  
Three Dimensional ◽  
Linear Process ◽  
Base Flow ◽  
Transition To Turbulence ◽  
Initial Structure ◽  
Transient Growth ◽  
Flow Vorticity ◽  
Transition Mechanism

AbstractThe current study focuses on a transition scenario in which the linear transient growth mechanism is initiated by four decaying normal modes. It is shown that the four modes, the initial structure of which corresponds to counter-rotating vortex pairs, are sufficient to capture the transient growth mechanism. More importantly, it is demonstrated that the kinetic energy growth of the initial disturbance is not the key parameter in this transition mechanism. Rather, it is the ability of the transient growth process to generate an inflection point in the wall-normal direction and consequently to make the flow susceptible to a three-dimensional disturbance leading to transition to turbulence. Because of the minimal number of modes participating in the transition process, it is possible to follow its earlier key stages analytically and to compare them with the results of direct numerical simulation. This procedure reveals the role of various flow parameters during the transition, such as the difference between symmetric and antisymmetric transient growth scenarios. Moreover, it is shown that the resulting modified base flow of the linear process is not sufficient to produce a significant localized maximum of the base-flow vorticity (i.e. a ‘strong’ inflection point), and it is only due to nonlinear effects that the base flow becomes unstable with respect to an infinitesimal three-dimensional disturbance. Finally, the physical mechanism during key stages of transition is well captured by the analytical expressions. Furthermore, the vortex dynamics during these stages is very similar to the model proposed by Cohen, Karp & Mehta (J. Fluid Mech., vol. 747, 2014, pp. 30–43) according to which streamwise variation of the initial counter-rotating vortex pair is required to generate concentrated spanwise vorticity, which together with the lift-up by the induced velocity and shear of the base flow generates packets of hairpins.

Aspects of linear and nonlinear instabilities leading to transition in pipe and channel flows

2008 ◽  
Vol 367 (1888) ◽  
pp. 509-527 ◽  
Author(s):  
Jacob Cohen ◽  
Jimmy Philip ◽  
Guy Ben-Dov
Keyword(s):  
Growth Mechanism ◽  
Pipe Flow ◽  
Scaling Laws ◽  
Scaling Law ◽  
Base Flow ◽  
Initial Disturbance ◽  
Transition To Turbulence ◽  
Transient Growth ◽  
Hairpin Vortices ◽  
Linear And Nonlinear

The failure of normal-mode linear stability analysis to predict a transition Reynolds number ( Re tr ) in pipe flow and subcritical transition in plane Poiseuille flow (PPF) has led to the search of other scenarios to explain transition to turbulence in both flows. In this work, various results associated with linear and nonlinear mechanisms of both flows are presented. The results that combine analytical and experimental approaches indicate the strong link between the mechanisms governing the transition of both flows. It is demonstrated that the linear transient growth mechanism is based on the existence of a pair of least stable nearly parallel modes (having opposite phases and almost identical amplitude distributions). The analysis that has been applied previously to pipe flow is extended here to a fully developed channel flow predicting the shape of the optimized initial disturbance (a pair of counter-rotating vortices, CVP), time for maximum energy amplification and the dependence of the latter on Re . The results agree with previous predictions based on many modes. Furthermore, the shape of the optimized initial disturbance is similar in both flows and has been visualized experimentally. The analysis reveals that in pipe flow, the transient growth is a consequence of two opposite running modes decaying with an equal decay rate whereas in PPF it is due to two stationary modes decaying with different decay rates. In the first nonlinear scenario, the breakdown of the CVPs (produced by the linear transient growth mechanism) into hairpin vortices is followed experimentally. The associated scaling laws, relating the minimal disturbance amplitude required for the initiation of hairpins and the Re , are found experimentally for both PPF and pipe flow. The scaling law associated with PPF agrees well with the previous predictions of Chapman, whereas the scaling of the pipe flow is the same as the one previously obtained by Hof et al ., indicating transition to a turbulent state. In the second nonlinear scenario, the base flow of pipe when it is mildly deviated from the Poiseuille profile by an axisymmetric distortion is examined. The nonlinear features reveal a Re tr of approximately 2000 associated with the bifurcation between two deviation solutions.

Perturbation dynamics in unsteady pipe flows

2007 ◽  
Vol 570 ◽  
pp. 129-154 ◽  
Author(s):  
M. ZHAO ◽  
M. S. GHIDAOUI ◽  
A. A. KOLYSHKIN
Keyword(s):  
Growth Mechanism ◽  
Flow Instability ◽  
Initial Conditions ◽  
Laminar Flows ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Transient Growth ◽  
Energy Growth ◽  
Long Time ◽  
Short Time

This paper deals with perturbed unsteady laminar flows in a pipe. Three types of flows are considered: a flow accelerated from rest; a flow in a pipe generated by the controlled motion of a piston; and a water hammer flow where the transient is generated by the instantaneous closure of a valve. Methods of linear stability theory are used to analyse the behaviour of small perturbations in the flow. Since the base flow is unsteady, the linearized problem is formulated as an initial-value problem. This allows us to consider arbitrary initial conditions and describe both short-time and long-time evolution of the flow. The role of initial conditions on short-time transients is investigated. It is shown that the phenomenon of transient growth is not associated with a certain type of initial conditions. Perturbation dynamics is also studied for long times. In addition, optimal perturbations, i.e. initial perturbations that maximize the energy growth, are determined for all three types of flow discussed. Despite the fact that these optimal perturbations, most probably, will not occur in practice, they do provide an upper bound for energy growth and can be used as a point of reference. Results of numerical simulation are compared with previous experimental data. The comparison with data for accelerated flows shows that the instability cannot be explained by long-time asymptotics. In particular, the method of normal modes applied with the quasi-steady assumption will fail to predict the flow instability. In contrast, the transient growth mechanism may be used to explain transition since experimental transition time is found to be in the interval where the energy of perturbation experiences substantial growth. Instability of rapidly decelerated flows is found to be associated with asymptotic growth mechanism. Energy growth of perturbations is used in an attempt to explain previous experimental results. Numerical results show satisfactory agreement with the experimental features such as the wavelength of the most unstable mode and the structure of the most unstable disturbance. The validity of the quasi-steady assumption for stability studies of unsteady non-periodic laminar flows is discussed.

scholarly journals Convective instability and transient growth in flow over a backward-facing step

2008 ◽  
Vol 603 ◽  
pp. 271-304 ◽  
Author(s):  
H. M. BLACKBURN ◽  
D. BARKLEY ◽  
S. J. SHERWIN
Keyword(s):  
Convective Instability ◽  
Three Dimensional ◽  
Nonlinear Effects ◽  
Expansion Ratio ◽  
Base Flow ◽  
Time Interval ◽  
Transient Growth ◽  
Two Dimensional ◽  
Backward Facing Step ◽  
Transient Energy

Transient energy growths of two- and three-dimensional optimal linear perturbations to two-dimensional flow in a rectangular backward-facing-step geometry with expansion ratio two are presented. Reynolds numbers based on the step height and peak inflow speed are considered in the range 0–500, which is below the value for the onset of three-dimensional asymptotic instability. As is well known, the flow has a strong local convective instability, and the maximum linear transient energy growth values computed here are of order 80×103 at Re = 500. The critical Reynolds number below which there is no growth over any time interval is determined to be Re = 57.7 in the two-dimensional case. The centroidal location of the energy distribution for maximum transient growth is typically downstream of all the stagnation/reattachment points of the steady base flow. Sub-optimal transient modes are also computed and discussed. A direct study of weakly nonlinear effects demonstrates that nonlinearity is stablizing at Re = 500. The optimal three-dimensional disturbances have spanwise wavelength of order ten step heights. Though they have slightly larger growths than two-dimensional cases, they are broadly similar in character. When the inflow of the full nonlinear system is perturbed with white noise, narrowband random velocity perturbations are observed in the downstream channel at locations corresponding to maximum linear transient growth. The centre frequency of this response matches that computed from the streamwise wavelength and mean advection speed of the predicted optimal disturbance. Linkage between the response of the driven flow and the optimal disturbance is further demonstrated by a partition of response energy into velocity components.

Control of the secondary cross-flow instability using localized suction

2012 ◽  
Vol 706 ◽  
pp. 470-495 ◽  
Author(s):  
Tillmann Friederich ◽  
Markus J. Kloker
Keyword(s):  
Flow Instability ◽  
Three Dimensional ◽  
Cross Flow ◽  
Base Flow ◽  
Transition To Turbulence ◽  
Significant Delay ◽  
Transition Control ◽  
Dimensional Boundary ◽  
Nonlinear Disturbance ◽  
Layer Flow

AbstractTransition control by suction in a three-dimensional boundary-layer flow subject to cross-flow instability is investigated using direct numerical simulation. Whereas the classical application of (homogeneous) suction at the wall is aimed at modifying the quasi-two-dimensional base flow to weaken primary cross-flow instability, here the three-dimensional nonlinear disturbance state with large-amplitude steady cross-flow vortices (CFVs) is controlled. Strong, localized ‘pinpoint’ suction is shown to be suitable for altering the CFVs and the associated flow field such that secondary instability is weakened or even completely suppressed. Thus significant delay of transition to turbulence can be achieved.

Transition to turbulence over 2D and 3D periodic large-scale roughnesses

2016 ◽  
Vol 804 ◽  
Author(s):  
A. M. Hamed ◽  
M. Sadowski ◽  
Z. Zhang ◽  
L. P. Chamorro
Keyword(s):  
Large Scale ◽  
Inflection Point ◽  
Three Dimensional ◽  
Transition To Turbulence ◽  
Transitional Region ◽  
Mean Velocity ◽  
Distinctive Features ◽  
Laminar To Turbulent Transition ◽  
Index Matching ◽  
Refractive Index Matching

A laboratory investigation was performed to study distinctive features of the laminar-to-turbulent transition over distributed roughness characterized by two-dimensional (2D) and three-dimensional (3D) periodic, low-order topographies at roughness Reynolds number $Re_{k}\approx 300$. Systematic experiments were performed using high-spatial-resolution planar particle image velocimetry (PIV) in a refractive-index-matching (RIM) channel, where the roughness covered the entire length of the test section. The results show that the flow over the 2D roughness becomes turbulent much sooner than its 3D counterpart ($Re_{x}=50\,000$ versus 120 000). This is attributed to the presence of a velocity inflection point resulting from flow separation within the troughs of the 2D roughness. In the transitional region, unsteady disturbances above the two roughnesses appear upstream of near-roughness disturbances. The above-roughness disturbances are associated with the inflection point in the vertically displaced boundary layer for the 2D case, and with the mean velocity deficit resulting from the interaction of the wakes of upstream elements for the 3D case. The near-roughness fluctuations are associated with the shear layer present behind the crests of both roughnesses. The transitional region is characterized by the interaction between above- and near-roughness disturbances, which merge, leading to a rapid vertical growth of the turbulent fluctuations.

scholarly journals Subcritical transition to turbulence in accretion disc boundary layer

2018 ◽  
Vol 619 ◽  
pp. A44
Author(s):  
V. V. Zhuravlev ◽  
D. N. Razdoburdin
Keyword(s):  
Boundary Layer ◽  
Shear Rate ◽  
Three Dimensional ◽  
Rotational Frequency ◽  
Accretion Disc ◽  
Transition To Turbulence ◽  
Linear Feedback ◽  
Transient Growth ◽  
Rayleigh Line ◽  
Non Linear

Context. Enhanced angular momentum transfer through the boundary layer near the surface of weakly magnetised accreting star is required in order to explain the observed accretion timescales in low-mass X-ray binaries, cataclysmic variables, or young stars with massive protoplanetary discs. The accretion disc boundary layer is locally represented by incompressible homogeneous and boundless flow of the cyclonic type, which is linearly stable. Its non-linear instability at the shear rates of the order of the rotational frequency remains an issue. Aims. We put forward a conjecture that hydrodynamical subcritical turbulence in such a flow is sustained by the non-linear feedback from essentially three-dimensional vortices, which are generated by quasi-two-dimensional trailing shearing spirals grown to high amplitude via the swing amplification. We refer to those three-dimensional vortices as cross-rolls, since they are aligned in the shearwise direction in contrast to streamwise rolls generated by the anti-lift-up mechanism in rotating shear flow on the Rayleigh line. Methods. Transient growth of cross-rolls is studied analytically and further confronted with direct numerical simulations (DNS) of the dynamics of non-linear perturbations in the shearing box approximation. Results. A substantial decrease of transition Reynolds number RT is revealed as one changes a cubic box to a tall box. DNS performed in a tall box show that RT as a function of shear rate accords with the line of constant maximum transient growth of cross-rolls. The transition in the tall box has been observed until the shear rate is three times higher than the rotational frequency, when RT ∼ 50 000. Conclusions. Assuming that the cross-rolls are also responsible for turbulence in the Keplerian flow, we estimate R T ≲ 108 in this case. Our results imply that non-linear stability of Keplerian flow should be verified by extending turbulent solutions found in the cyclonic regime across the solid-body line rather than entering a quasi-Keplerian regime from the side of the Rayleigh line. The most favourable shear rate to test the existence of turbulence in the quasi-Keplerian regime may be sub-Keplerian and equal approximately to 1/2.

scholarly journals Non-Modal Three-Dimensional Optimal Perturbation Growth in Thermally Stratified Mixing Layers

Fluids ◽  
2021 ◽  
Vol 6 (1) ◽  
pp. 37
Author(s):  
Helena Vitoshkin ◽  
Alexander Gelfgat
Keyword(s):  
Mixing Layer ◽  
Three Dimensional ◽  
Stable Stratification ◽  
Base Flow ◽  
Transient Growth ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Dimensional Growth ◽  
Layer Flow ◽  
Perturbation Growth

A non-modal transient disturbances growth in a stably stratified mixing layer flow is studied numerically. The model accounts for a density gradient within a shear region, implying a heavier layer at the bottom. Numerical analysis of non-modal stability is followed by a full three-dimensional direct numerical simulation (DNS) with the optimally perturbed base flow. It is found that the transient growth of two-dimensional disturbances diminishes with the strengthening of stratification, while three-dimensional disturbances cause significant non-modal growth, even for a strong, stable stratification. This non-modal growth is governed mainly by the Holmboe modes and does not necessarily weaken with the increase of the Richardson number. The optimal perturbation consists of two waves traveling in opposite directions. Compared to the two-dimensional transient growth, the three-dimensional growth is found to be larger, taking place at shorter times. The non-modal growth is observed in linearly stable regimes and, in slightly linearly supercritical regimes, is steeper than that defined by the most unstable eigenmode. The DNS analysis confirms the presence of the structures determined by the transient growth analysis.

EXPERIMENTAL INVESTIGATION OF FLOW RESPONSE TO STATIONARY DISTURBANCE IN ROTATING-DISK FLOW

2019 ◽  
Vol XVI (2) ◽  
pp. 13-22
Author(s):  
Muhammad Ehtisham Siddiqui
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Three Dimensional ◽  
Rotating Disk ◽  
Careful Analysis ◽  
Laboratory Frame ◽  
Transition To Turbulence ◽  
Turbulent Regime ◽  
Mean Velocity ◽  
Layer Flow

Three-dimensional boundary-layer flow is well known for its abrupt and sharp transition from laminar to turbulent regime. The presented study is a first attempt to achieve the target of delaying the natural transition to turbulence. The behaviour of two different shaped and sized stationary disturbances (in the laboratory frame) on the rotating-disk boundary layer flow is investigated. These disturbances are placed at dimensionless radial location (Rf = 340) which lies within the convectively unstable zone over a rotating-disk. Mean velocity profiles were measured using constant-temperature hot-wire anemometry. By careful analysis of experimental data, the instability of these disturbance wakes and its estimated orientation within the boundary-layer were investigated.

scholarly journals Specific sagittal alignment patterns are already present in mild adolescent idiopathic scoliosis

2021 ◽  
Author(s):  
Tom P. C. Schlösser ◽  
René M. Castelein ◽  
Pierre Grobost ◽  
Suken A. Shah ◽  
Kariman Abelin-Genevois
Keyword(s):  
Adolescent Idiopathic Scoliosis ◽  
Idiopathic Scoliosis ◽  
Inflection Point ◽  
Thoracic Kyphosis ◽  
Sagittal Alignment ◽  
Three Dimensional ◽  
Sagittal Profile ◽  
Thoracolumbar Kyphosis ◽  
C7 Slope ◽  
Sagittal Malalignment

Abstract Purpose The complex three-dimensional spinal deformity in AIS consists of rotated, lordotic apical areas and neutral junctional zones that modify the spine’s sagittal profile. Recently, three specific patterns of thoracic sagittal ‘malalignment’ were described for severe AIS. The aim of this study is to define whether specific patterns of pathological sagittal alignment are already present in mild AIS. Methods Lateral spinal radiographs of 192 mild (10°–20°) and 253 severe (> 45°) AIS patients and 156 controls were derived from an international consortium. Kyphosis characteristics (T4–T12 thoracic kyphosis, T10–L2 angle, C7 slope, location of the apex of kyphosis and of the inflection point) and sagittal curve types according to Abelin-Genevois were systematically compared between the three cohorts. Results Even in mild thoracic AIS, already 49% of the curves presented sagittal malalignment, mostly thoracic hypokyphosis, whereas only 13% of the (thoraco) lumbar curves and 6% of the nonscoliosis adolescents were hypokyphotic. In severe AIS, 63% had a sagittal malalignment. Hypokyphosis + thoracolumbar kyphosis occurred more frequently in high-PI and primary lumbar curves, whereas cervicothoracic kyphosis occurred more in double thoracic curves. Conclusions Pathological sagittal patterns are often already present in curves 10°–20°, whereas those are rare in non-scoliotic adolescents. This suggests that sagittal ‘malalignment’ patterns are an integral part of the early pathogenesis of AIS.

Sign in / Sign up

Export Citation Format

Share Document