Transition In Plane Poiseuille Flow Research Articles

Subcritical transition scenarios via linear and nonlinear localized optimal perturbations in plane Poiseuille flow

Subcritical transition in plane Poiseuille flow is investigated by means of a Lagrange-multiplier direct-adjoint optimization procedure with the aim of finding localized three-dimensional perturbations optimally growing in a given time interval (target time). Space localization of these optimal perturbations (OPs) is achieved by choosing as objective function either a p-norm (with ) of the perturbation energy density in a linear framework; or the classical (1-norm) perturbation energy, including nonlinear effects. This work aims at analyzing the structure of linear and nonlinear localized OPs for Poiseuille flow, and comparing their transition thresholds and scenarios. The nonlinear optimization approach provides three types of solutions: a weakly nonlinear, a hairpin-like and a highly nonlinear optimal perturbation, depending on the value of the initial energy and the target time. The former shows localization only in the wall-normal direction, whereas the latter appears much more localized and breaks the spanwise symmetry found at lower target times. Both solutions show spanwise inclined vortices and large values of the streamwise component of velocity already at the initial time. On the other hand, p-norm optimal perturbations, although being strongly localized in space, keep a shape similar to linear 1-norm optimal perturbations, showing streamwise-aligned vortices characterized by low values of the streamwise velocity component. When used for initializing direct numerical simulations, in most of the cases nonlinear OPs provide the most efficient route to transition in terms of time to transition and initial energy, even when they are less localized in space than the p-norm OP. The p-norm OP follows a transition path similar to the oblique transition scenario, with slightly oscillating streaks which saturate and eventually experience secondary instability. On the other hand, the nonlinear OP rapidly forms large-amplitude bent streaks and skips the phases of streak saturation, providing a contemporary growth of all of the velocity components due to strong nonlinear coupling.

Subcritical transition in plane Poiseuille flow as a linear instability process

In this work, a transition scenario is demonstrated, in which most of the stages are followed analytically. The transition is initiated by the linear transient growth mechanism in plane Poiseuille flow subjected to an infinitesimally small secondary disturbance. A novel analytical approximation of the linear transient growth mechanism enables us to perform a secondary linear stability analysis of the modified base-flow. Two possible routes to transition are highlighted here, both correspond to a small secondary disturbance superimposed on a linear transient growth. The first scenario is initiated by four decaying odd normal modes which form a counter-rotating vortex pair; the second is initiated by five even decaying modes which form a pair of counter-rotating pairs. The approximation of the linear transient growth stage by a combination of minimal number of modes allows us to follow the transition stages analytically by employing the multiple time scale method. In particular, the secondary instability stage is followed analytically using linear tools, and is verified by obtaining transition in a direct numerical simulation initiated by conditions dictated by the transient growth analytical expressions. Very good agreement is observed, verifying the theoretical model. The similarities between the two transition routes are discussed and the results are compared with similar results obtained for plane Couette flow.

Modeling and Distributed Control of Transition in Plane Poiseuille Flow

Recently a mathematical model for flow transition control in 2-D channels has been proposed that is in interconnected systems representation. The model is valid for all spatial frequencies and can be used for the synthesis of controllers for flow transition control problems in channels of infinite length. In this brief the model is validated against known features of plane Poiseuille flow and against the response of a nonlinear simulation at Reynolds number 2000. Based on this model a distributed control scheme is then synthesized for Reynolds number 10 000 using a recently proposed approach for interconnected systems. The controller is designed to stabilize the otherwise spatially unstable flow. This instability causes the transition from laminar to turbulence. Furthermore, a desired closed loop frequency response is obtained by tuning the weighting filters. The designed controller is then tested in a nonlinear simulation and closed-loop results are presented. The approach proposed here does not make any assumption on the periodicity of the channel, as is the case in most of previously published work. The control scheme can be implemented on microelectro-mechanical systems (MEMS)-based control systems. Such systems are spatially discrete in nature which is taken into consideration in the controller synthesis phase.

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

The failure of normal-mode linear stability analysis to predict a transition Reynolds number (Retr) 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 Retr of approximately 2000 associated with the bifurcation between two deviation solutions.

Influence of small imperfections on the stability of plane Poiseuille flow and the limitation of Squire’s theorem

In a recent paper [Luo and Wu, Phys. Fluids 16, 2852 (2004)], a theoretical model was proposed to illustrate the key role played by small imperfections in the subcritical transition of plane Poiseuille flow. In that model, the imperfections are represented by a steady but spatially periodic two-dimensional surface roughness, and a small oscillatory pressure gradient representing the inevitable background fluctuations. The present paper extends the previous work by considering an improved model, where the wall roughness is three-dimensional. A steady perturbation in the form of spatially periodic suction is also considered. In both cases, the resulting steady and unsteady components interact to produce a forcing that is in near resonance with a pair of oblique Tollmien-Schlichting (T-S) waves. The latter are generated as a result, and grow in proportion to the time during the initial stage. This excitation process and the subsequent nonlinear development of the T-S waves are studied by a weakly nonlinear analysis. A forced nonlinear (and nonlocal) amplitude equation is derived to provide a simple mathematical framework to link the critical Reynolds number for nonlinear instability to the level of imperfections. The theoretical result demonstrates that the onset critical Reynolds number is lower for three-dimensional than for two-dimensional disturbances if the imperfection exceeds a critical level, which turns out to be remarkably small. This offers a possible explanation as to why Squire’s theorem is applicable only when external perturbations are suppressed to an extremely low level. Further analysis of the nonlinear interactions indicates that the much reduced critical Reynolds number for three-dimensional external perturbations can be attributed to the fact that the excited pair of oblique T-S waves interact to generate extremely strong streaks, which then result in a very strong nonlinear destabilizing effect.

평행 Poiseuille, 평행 Couette, Blasius Flow의 준안정 해석

We investigate the spectra and the pseudospectra in plane Poiseuille flow, plane Couette flow and Blasius flow. At subcritical Reynolds number, the spectra arc lied strictly inside the stable complex half-plane, but the pseudospectra are lied in the unstable half-plane, reflecting the large linear transient growth that certain perturbations may excite. It means that the smooth flows may become to turbulent even though all the eigenmodes decay monotonicalh We found that pseudospectra is one reason that causes subcrilical transition in plane Poiseuille flow and plane Couette flow and bypass transition in Blasius flow.<br/>

The effect of a small initial distortion of the basic flow on the subcritical transition in plane Poiseuille flow

This paper presents an instability theory in which a mean flow and multiple wave interactions in the Poiseuille flow transition process are studied. It is shown that not only can this mean flow term come as the result of the Fourier decomposition of a general disturbance, it can also come as an exact solution to the unsteady Navier-Stokes equations. The presence of this term, though small, can produce totally different linear and nonlinear stability behavior for the flow at subcritical Reynolds numbers. In the linear stability case, with the presence of this mean flow perturbation term, the instabilities are obtained well below the critical value of 5772. When this mean flow term is introduced into the interactions with other harmonic perturbation waves, for the plane Poiseuille flow case with Reynolds numbers around 1200, the nonlinear interactions rapidly modify the total mean flow profile toward the mean flow profile observed in turbulence while the other two- and three-dimensional waves remain small. The initial energies needed to trigger the instabilities are much smaller than those reported by previous investigators. The intermittent character of the disturbance observed in transition experiments is also captured.

An experimental study of oblique transition in plane Poiseuille flow

Interactions of oblique waves have recently been investigated theoretically and numerically and found to give rise to rapid transition in flows subcritical to linear wave disturbances. The transition scenario consists of the formation and transient growth of streamwise streaks of high and low velocity and later a rapid growth of high-frequency disturbances leading to breakdown. The present study is the first extensive experimental investigation of oblique transition. The experiments were carried out in a plane Poiseuille flow air channel in which the oblique waves were generated, one at each wall, by vibrating ribbons and the development of the flow was mapped with hot-wire anemometry. The experiments consist both of low- and high-amplitude wave disturbances; in both cases streaky structures are created. For the low-amplitude case these structures decay, whereas for the high amplitude the flow goes towards breakdown. This study has confirmed and extended previous theoretical and numerical results showing that oblique transition may be an important transition scenario.

A nonlinear receptivity process generating Tollmien-Schlichting waves

Our ability to predict the boundary layer transition relies heavily on our understanding of receptivity, the process through which external disturbances excite Tollmien–Schlichting (T–S) waves. In this paper, a nonlinear process generating T–S waves, found in our experiment on the ribbon-induced transition in plane Poiseuille flow, is reported and examined through a direct numerical simulation. The results show that external forcing of frequency f can generate a T–S wave of frequency 2f (as well as a T–S wave of f) through the receptivity process if frequency f is much below the lower branch of the neutral stability curve.

Transition theory and experimental comparisons on (I) amplification into streets and (II) a strongly nonlinear break-up criterion

The two stages I, II are studied by using recent nonlinear theory and then compared with the experiments of Nishioka et al . (1979) on the transition of plane Poiseuille flow. The first stage I starts at low amplitude from warped input, which is deformed through weakly nonlinear interaction into a blow-up in amplitude and phase accompanied by spanwise focusing into streets. This leads into the strongly nonlinear stage II. It holds for a broad range of interactive boundary layers and related flows, to all of which the nonlinear break-up criterion applies. The experimental comparisons on I, II for channel flow overall show encouraging quantitative agreement, supporting recent comparisons (in the boundary-layer setting) of the description of stage I in Stewart &amp; Smith (1992) with the experiments of Klebanoff &amp; Tidstrom (1959) and of the break-up criterion of Smith (1988 a ) with the computations of Peridier et al (1991 a , b ).

Numerical investigation of instability and transition in rotating plane Poiseuille flow

Instability and transition in plane Poiseuille flow with spanwise system rotation is studied via direct numerical simulations with initial small-amplitude random disturbances. The results confirm and clarify the recent experimental findings of Alfredsson and Persson [J. Fluid Mech. 202, 543 (1989)]. A large computational domain is used to allow natural selection of the dominant spanwise wavelength of the observed streamwise vortices, and also the streamwise wavelength of the dominant secondary modes which subsequently develop in the presence of the streamwise vortices. Temporal evolutions of these vortices for several different cases are studied in detail. It is found that the spanwise wavelength of the streamwise vortices can be significantly different from that obtained from linear stability analysis. In all cases examined, the unstable primary three-dimensional waves, which develop during the linear stage, are still persistent in the nonlinear stage. It is also found that for the cases in which the primary three-dimensional waves decay, the finite-amplitude streamwise vortices can become unstable to two different three-dimensional secondary modes of instability, depending on the range of Reynolds number (Re) and rotation number (Ro). The first mode has a streamwise wavelength an order of magnitude longer than the spanwise wavelength; the other, referred to by Alfredsson and Persson as the ‘‘twisting mode’’ because of its visual appearance, has a streamwise wavelength comparable to the spanwise wavelength, and is shown to consist of counter-rotating streamwise vortices whose centers oscillate mostly in the vertical direction. It is shown that these modes are nondispersive traveling waves.

Randomization Process of Fluctuations in the Ribbon-Induced Transition of Plane Poiseuille Flow

Randomization Process of Fluctuations in the Ribbon-Induced Transition of Plane Poiseuille Flow

Experimental investigation of transition in plane Poiseuille flow. 1st Report. Newtonian fluids.

Transition in plane Poiseuille flow was investigated by means of flow visualization and hot film velocimeters. The intermittency factor attains the 1 at a Reynolds number of about 2 OOO in natural transition at a minimum critical Reynolds number. But the profiles of turbulent spots are still observable. The flow is fully developed and turbulent for Reynolds numbers greater than 3 OOO. Another experiment was made on spots artificially created in laminar flow over the Reynolds number range 2 000<Re<3 000. The spreading half angle increased from 2° to 10° with increasing Reynolds number from 2 000 to 3 000. The propagation velocity of the leading edge also increased from 0.5umax to 0.6umax.

Experimental investigation of transition in plane poiseuille flow. 2nd report. Dilute polymer solutions.

Transition in dilute polymer solutions was investigated in the same manner as in the first report. At first we investigated natumal transition at a minimum critical Reynolds number. The result suggested that the structural change of a turbulent spot resulted in polymer drag reduction. Subsequently, we maintained the flow laminar above the minimum critical Reynolds number and visualized the evolving process of spots artificially created in the laminar flow. The result showed that polymer additives reduced the spreading half angle. The striped streak lines of ink were observed around the spots. Jhis phenomenon is supposedly caused by the instability of polymer solutions.

Secondary Instability in Transition of Plane Poiseuille Flow (“Flowers of the Sea”)

Secondary Instability in Transition of Plane Poiseuille Flow (“Flowers of the Sea”)

Some observations of the subcritical transition in plane Poiseuille flow

The subcritical transition in plane Poiseuille flow (generated in a long wind channel of rectangular cross-section) was studied experimentally. Cylinder-generated vortical disturbances were introduced into the parabolic flow (in the test section) or the inlet flow. The parabolic flow was also disturbed by a controlled periodic jet from a wall orifice. On the basis of the three kinds of observations, we come to the conclusion that the minimum transition Reynolds number is about 1000 and the related threshold intensity (of the external disturbance triggering the transition) is comparable to the maximum intensity of u-fluctuations in fully turbulent channel flows.

Hydrodynamic Stability

Theoretical and experimental developments for the stability and transition of plane Poiseuille flow and for Couette flow between rotating concentric cylinders are reviewed. The paper concludes with brief comments on the stability of Hagen-Poiseuille flow in a pipe and brief comments on the stability of slowly varying flows.

A flow-visualization study of transition in plane Poiseuille flow

Flow visualization of artificially triggered transition in plane Poiseuille flow in a water channel by means of 10–20 μm diameter tihnium-dioxide-coated mica particles revealed some striking features of turbulent spots. Strong oblique waves were observed both at the front of the arrowhead-shaped spot as well as trailing from the rear tips. Both natural and artificially triggered transition were observed to occur for Reynolds numbers slightly greater than 1000, above which the flow became fully turbulent. The front of the spot moves with a convection speed of about two-thirds of the centreline velocity, while the rear portion moves at about . The spot expands into the flow with a spreading half-angle of about 8°. After growing to a size of some 36 times h (the channel depth) at a downstream distance x/h of about 130, the spot began to split into two spots, accompanied by strong wave activity. The spot(s) was followed visually downstream of its origin a distance x/h of about 300. These results indicate that wave propagation and breakdown play a crucial role in transition to turbulence in Poiseuille flow.

Closure to “Discussion of ‘An Experimental Investigation of Transition of Plane Poiseuille Flow’” (1975, ASME J. Fluids Eng., 97, pp. 274–275)

Closure to “Discussion of ‘An Experimental Investigation of Transition of Plane Poiseuille Flow’” (1975, ASME J. Fluids Eng., 97, pp. 274–275)