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Workshop Programme

for period 8 - 12 September 2008

Wall Bounded Shear Flows: Transition and Turbulence

8 - 12 September 2008


Monday 8 September
08:30-09:45 Registration
09:45-09:50 David Wallace - welcome
09:50-10:00 Tribute to Professor Phillip Saffman
Chair: R Kerswell
10:00-11:00 Marusic, I (Melbourne)
  Wall-Bounded Turbulence: Introduction Sem 1
11:00-11:30 Coffee
11:30-12:30 Jimenez, J (Universidad Politecnica)
  Introduction - Transition Sem 1
12:30-13:30 Lunch at Wolfson Court
Chair: P Manneville
14:00-14:20 Chernyshenko, S (Imperial College London)
  Ten questions to facilitate discussions during the workshop Sem 1

Turbulence research is distinguished by its complexity making rigorous results scarce. This comparative lack of objectivity is a matter of concern for the turbulence research community, even though it does consist of most brilliant and creative researchers. Logic standards are best maintained in face-to-face discussions, for which the present workshop provides an excellent opportunity. The desire to use it in full motivates this talk. Several questions linking seemingly unrelated results are suggested for discussion below.

14:20-14:40 Nikitin, N (Moscow)
  Universal character of perturbation growth in near-wall turbulence Sem 1

Spatial instability of fully developed turbulent flow in a long straight circular pipe is investigated via DNS. The incompressible Navier-Stokes equations are solved with turbulent inflow velocity field extracted from auxiliary streamwise-periodic simulation which run in parallel with the main spatial simulation. In addition, small perturbations are introduced into the inlet and velocity difference between the flows with and without perturbations is analyzed. It is shown that mean perturbation amplitude $\varepsilon$ increases exponentially with distance downstream until saturating at the level comparable to the level of turbulent fluctuations in the flow. The rate of the exponential growth is found to be constant when normalized by viscous length, $\varepsilon\sim\exp(0.002x^+)$ over the considered Reynolds number range $140\leqslant\Re_\tau \leqslant320$. The universal character of perturbation growth is confirmed also by channel flow simulations.

14:40-15:00 Chen, S (JHU and PKU)
  Reynolds-stress-constrained subgrid-scale stress model for large eddy simulation of wall bounded turbulent flow Sem 1

In the traditional hybrid RANS/LES approaches for the simulation of wall-bounded fluid turbulence, such as the detached eddy simulation (DES), the whole flow domain is divided into the inner layer and outer layer. Typically the Reynolds averaged Navier-Stokes (RANS) equation is used for the inner layer, while the large eddy simulation (LES) is used for the outer layer. The transition from the inner layer solution to the outer layer is often problematic due to lacking small scale dynamics in the RANS region. In this paper we propose to simulate the whole flow region by large eddy simulation while enforcing a RANS Reynolds stress for the inner layer. We verifed our approach by simulating three-dimensional turbulent channel flows. We were able to accurately model the mean velocity and the turbulent stress using the same grid size as DES. The application of the proposed constrained LES to other turbulent flows with separation will also be discussed.

15:00-15:30 Tea
Chair: T Gotoh
15:30-15:50 Vassilicos, JC (Imperial College London)
  The Karman constant is inversely proportional to the number of stagnation points at the upper edge of the buffer layer and is therefore not a universal constant Sem 1

By an adaptation of the Rice theorem to three-dimensional incompressible vector fields we show that the average distance between turbulent velocity stagnation points at a certain distance from the wall is proportional to the Taylor microscale at that distance. Then, by using this result in conjunction with the balance between kinetic energy dissipation and production in the log-layer we calculate the Karman constant as a function of other constants. We show that the Karman constant is inversely proportional to the number of turbulent velocity stagnation points at the edge of the log-layer that is closest to the wall.

We perform three different Direct Numerical Simulations (DNS) of fully developed turbulent channel flows to test our formula. In two of these DNS the flow is forced at the wall in two different ways and in the remaining third simulation it is not. The proportionality between the inverse Karman constant and the number $C_s$ of turbulent velocity stagnation points at the upper edge of the buffer layer is observed in all three simulations even though the values of the Karman constant and $C_s$ are different in each one of these simulations. Our formula is therefore able to predict and explain why the Karman constant is not a universal constant.

15:50-16:10 Nagib, HM (IIT)
  Non-universality of the Von Kármán "constant" Sem 1

The overlap parameters of the logarithmic region in turbulent pipe, channel, and boundary-layer flows are established using a composite profile approach which incorporates the influence of the outer part. The $Re$-specific von Kármán coefficient for channel flows decreases with Reynolds number to a level below the well defined value of $\kappa_{BZ}~=~0.384$ for ZPG TBLs. The proper limiting value of $\kappa_C$ for the channel flow could not be established with a high confidence because of the limited range of available Reynolds numbers, but the best projected value is near $\kappa_C\sim 0.37$. For the pipe flow, reprocessing of the Superpipe data indicates that $\kappa_P\sim0.41$, which on the opposite side of the boundary layer value compared to the channel flow. This collective behavior of $\kappa$ in boundary layers, pipes and channels suggests that the von Kármán coefficient is not universal, and exhibits dependence on not only the pressure gradient but also on the flow geometry, thereby raising fundamental questions regarding turbulence flow theory and modeling for all wall-bounded flows. In contrast, a wide range of data from such canonical flows reveals a universal relation between the overlap parameters; i.e., the von Kármán coefficient and the intercept B.

16:10-16:30 Jimenez, J (Universidad Politecnica)
  Comparison between turbulent boundary layers and channels from direct numerical simulations Sem 1

Preliminary results are presented from a new simulation of the zero-pressure-gradient turbulent boundary layer at Re_\theta=1000-2100, which are compared to simulations of turbulent channels at similar Reynolds numbers. Even the low order statistic differ considerably between the two flows, including for some quantities the values within the buffer layer. In general the pressure and the transverse velocity fluctuations are stronger in boundary layers than in channels, even if the wall-parallel scales that can be derived from the spectra and the two-point correlations are simular in both cases. On the other hand, the streamwise fluctuation intensities are roughly similar in both flows, but their scales are shorter and narrower in boundary layers than in channels. The difference between the two flows is traced to an excess of production of the streamwise turbulent energy in the outer part of the boundary layers compared to channels (by a factor of order two), which is associated with the presence of a stronger wake component of the mean velocity profile. Most of this excess production is compensated by stronger pressure fluctuations and by the pressure-strain term, whicht redistribute the energy to the transverse components. The difference in the statistic can be traced in experimental results at higher Reynolds numbers, although qunatities such as the pressure fluctuations and the energy budgets are not available for them. These results suggest that caution should be used in mixing different flows when document, for example, Reynolds number effects in shear turbulence.

Chair: S Chernyshenko
16:50-17:10 Davidson, P (Cambridge)
  A simple model for the log-law region of a boundary layer Sem 1

We build on the work of Davidson, Krogstad & Nickels (JFM, 2006) and propose an elementary model for the log-law region of a boundary-layer. The model is remarkably simple, contains only one free parameter, and assumes very little about the shape of the boundary-layer eddies. The physical content of the model is simple: we assume that the two-point statistics of the streamwise velocity fluctuations know about the presence of the wall only to the extent that, over a range of eddy sizes, it imposes a kinetic energy scale proportional to the square of the shear velocity. Little else is assumed. Despite its naivety, the model is an excellent fit to experimental data for the k-1 law of the one-dimensional, longitudinal spectrum, F11(k), and also to F11(k), in the inertial range. It is also an excellent fit to experimental data for the real-space analogue of the k-1 law; that is, the logarithmic law for the longitudinal structure-function.

17:10-17:30 Marusic, I (Melbourne)
  Modelling large-scale influences in near-wall turbulence Sem 1

Turbulent boundary layers are traditionally considered in terms of an inner and outer region, not excluding the logarithmic region, which is usually identified with an overlap of the two regions. Recent experiments in high Reynolds number flows have revealed that the key to understanding the scaling behaviour of the turbulence statistics lies in unravelling the influence and interaction of the outer and inner regions. In this talk a model for the influence of the large-scale motions in wall turbulence (termed “superstructures”) on the inner viscous region is presented. The model has implications for predicting wall-shear stress fluctuations based on a filtered velocity signature in the outer part of the boundary layer, as required in high Reynolds number large-eddy simulations.

17:30-18:30 Welcome Wine Reception
18:45-19:30 Dinner at Wolfson Court (Residents Only)
Tuesday 9 September
Chair: D Henningson
09:40-10:00 Eckhardt, B (Philipps-Universitat Marburg)
  Transition to turbulence in pipe flow Sem 1

According to textbook wisdom, flow down a pipe becomes turbulent near a Reynolds number of about 2000. This simple statement misses many subtleties of the transition: the absence of a linear stability of the laminar flow, the sensitive dependence on perturbations that sometimes succeed and sometimes fail to induce turbulence and the unexpected observation that the turbulent state, once achieved, is not persistent but can decay. All these observations are compatible with the formation of a strange saddle in the state space of the system. I will focus on three aspects: on the appearance of 3-d coherent states, on the information contained in lifetime statistics and on results on the boundary between laminar and turbulent regions. They suggest a generic structuring of state space in flows where turbulent and laminar flow coexist.

10:00-10:20 Hof, B (Mathematisches Institut, Göttingen)
  Repeller or attractor? Selecting the dynamical model of shear flow turbulence Sem 1

The collapse of turbulence, observable in shear flows at low Reynolds numbers, raises the question if turbulence is generically of transient nature or becomes sustained at some critical point. Recent data have lead to conflicting views with the majority of studies supporting the model of turbulence turning into an attracting state. Here we present lifetime measurements of turbulence in pipe flow spanning eight orders of magnitude in time, drastically extending all previous investigations. We show that no critical point exists in this regime and that in contrast to the prevailing view the turbulent state remains transient. The observed transient scaling behaviour has been conjectured to occur in turbulent flows more than two decades ago.

10:20-10:40 Bottaro, A (Università di Genova)
  Optimal paths to transition in a duct Sem 1

It is accepted that self sustaining states exist in duct flow past a threshold value of the Reynolds number, which rely on the nonlinear coupling between vortices, streaks and travelling waves. In this work we have devised an optimisation strategy that aims at maximising the energy of the travelling disturbance wave, and thus at enhancing the nonlinear terms that feed back onto vortices and streaks.

10:40-11:00 Pinelli, A (CIEMAT)
  Investigating puffs in square duct flow Sem 1

We are investigating the characteristics of localized disturbances in the flow through a straight duct with a square cross-section. For this purpose, we have performed pseudo-spectral DNS in streamwise periodic domains with a length of 100 duct half-widths. By imposing a body force locally and for a finite temporal interval (similar to the work in a pipe by Willis & Kerswell, Phys. Rev. Lett. 98/100, 2007/2008), a 'puff' is generated when the bulk Reynolds number (based upon the half width) is chosen within the range of approximately 800 to 1000; for lower values the disturbance decays, whereas for higher values the puff expands and ends up filling the entire periodic domain. We have followed the evolution of equilibrium puffs for several thousand bulk time units, enabling us to compute the statistics of the perturbed flow. Phase-locked averaging has been performed in order to reveal the internal structure of the puffs.

11:00-11:30 Coffee
Chair: Z Warhaft
11:30-11:50 Gibson, JF (Georgia Institute of Technology)
  Invariant solutions and visualization of dynamics in plane Couette flow Sem 1

It has recently become possible to compute precise equilibrium, traveling wave, and periodic orbit solutions to pipe and plane Couette flow at moderate Reynolds numbers. These invariant solutions capture the complex dynamics of rolls and streaks (coherent structures) in wall-bounded flows and provide a framework for understanding turbulent flows as dynamical systems. We present (1) a number of weakly unstable equilibria, traveling waves, and periodic orbits of plane Couette flow, (2) visualizations of their physical and state-space dynamics, and (3) statistics on how frequently the solutions appear within turbulent flows. What emerges is a picture of moderate-Reynolds turbulence as a walk among a set of weakly unstable invariant solutions.

11:50-12:10 Cvitanovic, P (Georgia Institute of Technology)
  Symmetry reduced averages over moderately turbulent flows Sem 1

Long-term averages in low-dimensional dynamical systems can be expressed as averages over invariant state-space sets (equilibria, periodic orbits, partially hyperbolic invariant tori). We speculate as to how such theory might work for moderately turbulent flows.

12:10-12:30 Manneville, P (Ecole Polytechnique)
  Transitional plane Couette flow: an alternative to the low dimensional dynamical systems approach Sem 1

The transition from the turbulent state to the laminar regime in plane Couette flow is studied by means of a model focusing on the in-plane (x,z) space dependence of a few velocity amplitudes with reduced wall-normal (y) dependence. The model appears well suited to study the low-R transitional range. I shall present my latest results obtained at very large aspect ratio (typically 768x768 with gap rescaled to 1) and attempt to convince people that the statistical physics of spatiotemporal chaos in extended geometry offers a valuable alternative to the well-accepted interpretation of the relaxation of turbulence in terms of chaotic transients typical of low-dimensional dynamical systems.

12:30-13:30 Lunch at Wolfson Court
Chair: R Moser
14:00-14:20 Adrian, R (Arizona State)
  A mechanism for turbulent drag reduction by polymers Sem 1

Minute quantities of long chained polymers, of order 10 ppm, added to liquids like water or oil are capable of cutting the turbulent wall friction by half. This startling effect- the "Toms phenomenon" -has been known for more than 60 years, but a detailed explanation of how such small amounts of polymer alter the structure of turbulence so dramatically has been lacking. To explore this question, direct numerical simulations have been performed based on a visco-elastic model of the fluid that uses a finite extensible non-linear elastic-Peterlin (FENE-P) constituitive equation. It is found that the stresses due to the polymers circulating around turbulent vortices produce counter-torques that inherently oppose the rotation. Vortices creating the turbulent transport responsible for drag are weakened and the creation of new vortices is inhibited. Thus, both coherent and incoherent turbulent Reynolds stresses are reduced. Interesting, the viscoelastic stresses of the FENE-P model rely upon the vortices being asymmetric and such deviations from axisymmetry occur where the vortices are strained by themselves or by adjacent vortices.

Kim, K, Li, C.-F., Sureshkumar, R., Balachandar, S. and Adrian, R. J., “Effects of polymer stresses on eddy structures in drag-reduced turbulent channel flow,” J. Fluid Mech. 584, 281 (2007).

Kim, K,, Adrian, R. J., Balachandar, S. and Sureshkumar, R., “Dynamics of hairpin vortices and polymer-induced turbulent drag reduction,” Phys Rev. Lett. 100 (2008). LJ11563

14:20-14:40 Kim, J (UCLA)
  Physics and control of wall turbulence Sem 1

It has been generally accepted that nonlinearity is an essential characteristic of turbulent flows. Consequently, except for special situations in which a linear mechanism is expected to play a dominant role (e.g., rapidly straining turbulent flows to which the rapid distortion theory can be applied), the role of linear mechanisms in turbulent flows has not received much attention. Even for transitional flows, a common notion is that the most a linear theory can provide is insight into the early stages of transition to turbulence. But several investigators have recently shown that linear mechanisms play an important role even in fully turbulent, and hence fully nonlinear, flows. Examples of such studies include: optimal disturbances in turbulent boundary layers (Butler \& Farrell 1993); transient growth due to non-normality of the Navier-Stokes system (Reddy \& Henningson 1993); applications of a linear control theory to transitional and turbulent channel flows (Joshi \etal 1997); and a numerical experiment (Kim \& Lim 2000) demonstrating that near-wall turbulence could not be maintained in turbulent channel flow when a linear mechanism was artificially suppressed. \medskip Turbulent channel flow is analyzed from a linear system point of view. After recasting the linearized Navier-Stokes equations into a state-space representation, the singular value decomposition (SVD) analysis is applied to the linear system, with and without control input, in order to gain new insight into the mechanism by which various controllers are able to accomplish the viscous drag reduction in turbulent boundary layers. We examine linear-quadratic-regulator (LQR) controllers that we have used, as well as the opposition control of Choi \etal (1994), which has been a benchmark for comparison of various control strategies. The performance of control is examined in terms of the largest singular values, which represent the maximum disturbance energy growth ratio attainable in the linear system under control. The SVD analysis shows a similarity between the trend observed in the SVD analysis (linear) and that observed in direct numerical simulations (nonlinear), thus reaffirming the importance of linear mechanisms in the near-wall dynamics of turbulent boundary layers. It is shown that the SVD analysis of the linearized system can indeed provide useful insight into the performance of linear controllers. Other issues, such as the effect of using the evolving mean flow as control applied to a nonlinear flow system (a.k.a. gain scheduling) and high Reynolds-number limitation, can be also investigated through the SVD analysis. Finally, time permitting, a linear Floquet analysis of a channel flow with periodic control, which had been shown to sustain skin-friction drag below that of a laminar channel, will be discussed to elucidate the drag reducing mechanism.

14:40-15:00 Henningson, D (KTH)
  Input-output analysis, model reduction and flow control applied to the Blasius boundary layer Sem 1

The dynamics and control of disturbances in the spatially evolving boundary layer on a flat-plate are investigated from an input-output viewpoint. From the linearized Navier-Stokes equations with inputs (disturbances and actuators) and outputs (objective function and sensor) controllable, observable and balanced modes are extracted using the snapshot-method and a matrix-free time-stepper approach. A balanced reduced-order model is constructed and shown to capture the input-output behaviour of linearized Navier-Stokes equations. This model is used to design a LQG-feedback controller to suppress the growth of perturbations inside the boundary layer

15:00-15:30 Tea
Chair: A Bottaro
15:30-15:50 Cossu, C (LadHyx)
  Optimal transient growth and very large scale structures in turbulent boundary layers Sem 1

The optimal energy growth of perturbations sustained by a zero pressure gradient turbulent boundary is computed using the eddy viscosity associated with the turbulent mean flow. It is found that even if all the considered turbulent mean profiles are linearly stable, they support transient energy growths. The most amplified perturbations are streamwise uniform and correspond to streamwise streaks originated by streamwise vortices. For sufficiently large Reynolds numbers two distinct peaks of the optimal growth exist respectively scaling in inner and outer units. The optimal structures associated with the peak scaling in inner units correspond well to the most probable streaks and vortices observed in the buffer layer and their moderate energy growth is independent of the Reynolds number. The energy growth associated with the peak scaling in outer units is larger than that of the inner peak and scales linearly with an effective turbulent Reynolds number formed with the maximum eddy viscosity and a modified Rotta-Clauser length based on the momentum thickness. The corresponding optimal perturbations consist in very large scale structures with a spanwise wavelength of the order of 8 $\delta$. The associated optimal streaks scale in outer variables in the outer region and in wall units in the inner region of the boundary layer, there being proportional to the mean flow velocity. These outer streaks protrude far into the near wall region, having still 50% of their maximum amplitude at $y^+=20$. The amplification of very large scale structures appears to be a robust feature of the turbulent boundary layer: Optimal perturbations with spanwise wavelengths ranging from 4 to 15 $\delta$ can all reach 80% of the overall optimal peak growth.

15:50-16:10 Tardu, S (LEGI)
  Genesis of coherent structures through an interactive bypass transition process Sem 1

The interaction between two localized disturbances is analyzed in a subcritical channel flow through direct numerical simulations. The initial perturbations are in the form of two pairs of counter rotating vortices. One of them interacts with the wall normal vorticity layers set-up near the wall, by compressing or stretching locally part of them through the straining motion it induces. The breakdown of spanwise symmetry leads to the rapid development of a new wall normal vorticity patch that is tilted by the shear and rolls up into a new small-scale streamwise vortex. The process results in a localized turbulent spot at later stages of development. A detailed analysis is carried out to determine the role of different parameters entering in the physics of the mechanism. Several critical thresholds that trigger the interactive bypass transition process are found and analyzed. The similitude parameters resulting from the parametric investigation coincide well with those governing the self-sustaining Reynolds shear stress producing eddies in the buffer layer of a fully developed turbulent wall flow. It is suggested that the mechanism we propose may play some role in the regeneration cycle of the near wall turbulence generating structures by precisely bypassing the three-dimensional streak instability mechanism. An active strategy based on this process is applied to the mixing in microsystems. The specific interactions asymmetrically distributed in space lead to a better mixing locally in time and space leading to the possibility of active control of scalar transport in micro devices.

16:10-16:30 Monkewitz, P (EPFL)
  The Reynolds shear stress in zero pressure gradient turbulent boundary layers Sem 1

The Reynolds shear stress (RS) in zero pressure gradient turbulent boundary layers is established using recently developed composite mean velocity profiles based on the “log-law” in the overlap region between inner and outer profiles. The contribution of the normal stress difference is also considered and shown to be relatively small. From this analysis, an asymptotic expansion for the maximum RS and its location is developed. The hypotheses underlying this analysis are discussed and the results are compared with experiments and DNS. Using the friction velocity as scale, the RS determined from composite mean velocity profiles agrees reasonably well with low-Re experimental results. However, when comparing with high-Re experiments and DNS, the agreement is generally limited as near the wall the experimental accuracy and resolution becomes problematic and far from the wall the numerical treatment of the boundary condition is very delicate.

16:30-16:50 Eyink, GL (The John Hopkins University)
  Turbulent flow in pipes and channels as cross-stream inverse cascades of vorticity Sem 1

A commonplace view of pressure-driven turbulence in pipes and channels is as “cascades” of streamwise momentum toward the viscous layer at the wall. We present in this talk an alternative picture of these flows as “inverse cascades” of spanwise vorticity, in the cross-stream direction but away from the viscous sublayer. We show that there is a constant spatial flux of spanwise vorticity, due to vorticity conservation, and that this flux is necessary to produce pressure-drop and energy dissipation. The vorticity transport is shown to be dominated by viscous diffusion at distances closer to the wall than the peak Reynolds stress, well into the classical log-layer. The Perry-Chong model based on “representative" hairpin /horseshoe vortices predicts a single sign of the turbulent vorticity flux over the whole log-layer, whereas the actual flux must change sign at the location of the Reynolds-stress maximum. The Perry-Chong model may be viable at distances beyond the peak. The vortex-cascade picture presented here has a close analogue in the theory of quantum superfluids and superconductors, the “phase slippage” of quantized vortex lines. Most of our results should therefore apply as well to superfluid turbulence in pipes and channels. We also discuss issues about drag-reduction from this perspective.

18:45-19:30 Dinner at Wolfson Court (Residents Only)
Wednesday 10 September
Chair: Y Kaneda
09:40-10:00 Coleman, G (Southampton)
  Implications of Ekman layer DNS for near wall similarity Sem 1
10:00-10:20 Nickels, TB (Cambridge)
  On very long structures in boundary layers Sem 1

The existence, structure and statistics of very long structures in boundary layers will be discussed with particular reference to their three dimensional structure and the extent to which they can be explained as chance concatentations of shorter structures.

10:20-10:40 Warhaft, Z (Cornell)
  Lagrangian measurements of inertial particle accelerations in shear, and in grid generated turbulence Sem 1

We describe two dimensional Lagrangian acceleration statistics of inertial particles in a turbulent boundary layer and contrast the results with those observed in decaying grid generated turbulence (Ayyalasomayajula PRL .Vol. 97,144507 (2006)) . The statistics were determined by means of particle tracking techniques using a high speed camera moving along the side of a wind tunnel at the mean flow speed . Water droplets were fed into the flow using two different methods: sprays placed down-stream from an active grid, and from tubes fed into the boundary layer from humidifiers. The boundary layer, which had considerable free stream turbulence, was formed above a flat plate placed horizontally in the tunnel. The flows are described in terms of the Stokes, Froude and Reynolds numbers. For the flow conditions studied, the sprays produced Stokes numbers varying from 0.47 to 1.2, and the humidifiers produced Stokes numbers varying from 0.035 to 0.25, where the low and high values refer to the outer boundary layer edge and the near-wall region, respectively. The Froude number was approximately 1.0 for the sprays and 0.25 for the humidifiers. The boundary layer momentum thickness Reynolds number was approximately 800. The free stream turbulence was varied by operating the grid in the active mode as well as a passive mode (the latter behaves as a conventional grid). The effects of the free stream turbulence on the acceleration statistics were systematically studied. At the outer edge of the boundary layer, where the shear was weak, the acceleration probability density functions were similar to those previously observed in isotropic turbulence for inertial particles. As the boundary layer plate was approached, the tails of the probability density functions narrowed, became negatively skewed, and their peak occurred at negative accelerations (decelerations in the stream-wise direction). The mean deceleration and its rms increased to large values close to the plate. These effects were more pronounced at higher Stokes number. Although there were free stream turbulence effects, and the complex boundary layer structure played an important role, a simple model suggests that the acceleration behavior is dominated by shear and inertia. The results are contrasted with inertial particles in isotropic turbulence and with fluid particle acceleration statistics in a boundary layer. The work was supported by the US National Science Foundation.

10:40-11:00 Cohen, J (Technion)
  Significance of localized vortical disturbances in wall-bounded and free shear flows Sem 1

The resemblance among coherent structures naturally occurring in fully developed bounded turbulent shear flows, transitional flows and free shear layers suggests the existence of a basic mechanism responsible for the formation of the structures under various base flow conditions. The common elements in all such flows are the shear of the base flow and the presence of a localized vortical disturbance within this shear. Due to their localization in space, the surrounding base flow can be assumed to have homogeneous shear to leading order. In this talk we combine analytical, numerical and experimental tools to show that indeed the interaction between a localized dipole-vortical disturbance and the surrounding `simple' laminar shear flow where the velocity vector is (at most) a linear function of the coordinates, can reproduce the generation mechanism and characteristics of the coherent structures that naturally occur in turbulent bounded shear flows (counter-rotating vortex pairs and hairpin vortices) and free shear layers (`rib vortices').

11:00-11:30 Coffee
Chair: J Kim
11:30-11:50 Durbin, R (Iowa State)
  Boundary layer transition by discrete and continuous modes Sem 1

The natural and bypass routes to boundary layer turbulence have traditionally been studied independently. One can characterize our study as an exploration of the situation in which both occur. Experiments indicate that this may be pertinent in certain flow regimes, particularly in adverse pressure gradients. We study this case by direct numerical simulation (DNS). The inflow condition is a superposition of a 3-D continuous mode and a 2-D T-S wave onto a Blasius mean flow. The T-S and continuous modes are obtained by solving the Orr-Sommerfeld and Squire equations by well established numerical methods. The DNS is accomplished with a finite volume, staggered mesh, fractional step algorithm for incompressible Navier-Stokes equations. Either mode, of itself, is unable to provoke transition. With both modes present, transition usually occurred within the computational domain. Transition was preceded by the appearance of Lambda-shaped velocity contours. Although this is reminiscent of secondary instability of T-S waves, the lateral spacing between Lambda's was very much narrower and seemed to be controlled by spanwise wavelength in the continuous mode. However, the spacing and wavelength were not necessarily equal. Two broad classes of behavior were seen, as epitomized by modes 2 and 5. In mode 2 the Lambda's were grouped in staggered rows. The elements of a row are pairs Lambda's. The pairs are aligned in $z$ within the row, which is followed by another row of pairs, shifted horizontally half way between the previous row. Flow visualization will be presented. The lateral spacing between Lambda's within a row is equal to that of the continuous mode --- actually of the perturbation jets spawned by the continuous mode. Mode 5 produced a more irregular pattern, but still arranged in pairs Lambda's. Unlike mode 2, their spanwise spacing differs from that of the continuous mode; it appears to be about three times as wide. Another curious aspect of mode 5 is that a larger streak amplitude can delay transition. Mode 2 shows transition to move upstream as the Klebanoff streaks get stronger. Mode 5 initially promotes transition, then, as its amplitude increases further, it delays transition.

11:50-12:10 Lavioe, P (Toronto)
  Transient growth induced by surface roughness in a Blasius boundary layer Sem 1

Introduction There has been much recent interest in transient growth both theoretically [1-4] and experimentally [2,5]. The present work stems from an interest in model reduction for flow control: starting with a linear physical model representative of near-wall turbulence, the aim of this experiment is to devise a wall-based estimator. The inviscid transient growth instability has an optimal form of spanwise periodic streamwise vortices resulting in high/low – speed streaks. Slow growth and decay in x and spanwise periodicity suggest a reduced requirement for mode observability. Equations (1) and (2) describe the generation of optimal modes through the coupling term, , viz. Experimental Arrangement The Blasius base flow is developed on a vertical cast aluminium plate with a sharp leading edge in a very low-turbulence intensity (around 0.05%) wind tunnel. A periodic array of roughness elements at 200 mm from the leading edge is used to perturb the base state and introduces steady steamwise vortex pairs approximating the optimum mode. The height, k, spanwise separation, Δz and diameter, d, of the roughness elements are all adjustable. For the present work, 0.5 ≤ k ≤ 1.5, 10 ≤ Δz ≤ 30 and 2.5 ≤ d ≤ 5.0 mm and the measurements are conducted over the central six elements. Two hot wires, one normal and one slanted by 45 degrees to the mean flow, are used to obtain u and w in the range 250 ≤ x ≤ 700 mm. Results and Discussion Figure 1 present typical results for these experiments. Figure 1(a) shows the u-disturbance spectra as a function of spanwise-wavenumber. The spectrum is taken at the wall-normal location where the maximum rms of the disturbance occurs below that predicted by theory [3]. For this case, the first harmonic of the disturbance introduced is discernible for the first four streamwise locations. The energy of the disturbance is observed to grow at the first streamwise location and peaks when \beta is equal to 0.45 before it decays monotonically. Figures 1(b) and (c) show the corresponding disturbance contours at x = 300. The double positive peaks in the u-disturbance merge into a single peak further downstream of the roughness elements. The shape of the w-disturbance is more complicated and suggests that the streamwise vortices produced by the roughness elements are very weak by this location. The u-component of the disturbance shows the effects of a spanwise shear at larger x, the shear increasing with streak amplitudes. In common with other experiments [2,5], the evolution of the disturbance shows significant deviation to that predicted by theory, suggesting a suboptimal initial growth. This raises questions regarding the receptivity mechanism of laminar boundary layers subjected to different types of perturbations, such as free stream turbulence or wall roughness [6]. In future work, we intend to investigate this further as well as estimating the v component from continuity. Simultaneous measurements of the two components of surface skin friction will also be made to enable the design of an estimator for control by surface deformation.

12:10-12:30 Govindarajan, R (JNCASR)
  Instabilities in variable-property flows, and the continuous spectrum Sem 1

The stability work to be presented here is motivated by our ongoing numerical study of vortex merger in the presence of density-stratification. We find that very strong density stratification (even ``stable'' stratification) can (a) prevent the merger and (b) cause the vortices to break up. At low diffusivity levels the latter results from an inviscid instability due to alternately-signed density jumps packed in a known pattern. The analytical solution yields several co-existing unstable modes, and transient growth can add significantly to the linear growth at moderate times, thus speeding up the break-up of a single vortex. A related problem of Couette-Poiseuille flow and its continuous spectrum in the presence of low levels of base-flow vorticity will be discussed.

12:30-13:30 Lunch at Wolfson Court
Chair: V Zeitlin
14:00-14:20 Tuckerman, L (PMMH-ESPCI)
  Turbulent-laminar patterns in plane Couette flow Sem 1

Plane Couette flow near transition displays steady periodic oblique bands of alternating turbulent and laminar flow. Numerical simulations of the Navier-Stokes equations in a tilted domain show a rich variety of such patterns, including spatio-temporal intermittency, branching and travelling states, and localized states analogous to spots. Quantitative analysis of the Reynolds-averaged equations reveals that both the mean flow and the turbulent force are centrosymmetric and can be described by only three trigonometric functions, leading to a model of 6 ODEs. The transition is best described as a bifurcation in the probability distribution function of the first Fourier component.

14:20-14:40 Barkley, D (Warwick)
  Turbulent-laminar patterns II Sem 1

In this talk extensions of the earlier work on turbulent-laminar patterns will be presented. In particular large-scale computations of turbulent-laminar coexistence in pipe flow will be presented for pipes up to 400 diameters in length.

14:40-15:00 Willis, A (Bristol)
  Turbulent dynamics of pipe flows captured in a reduced model Sem 1

Significant success has been achieved in recent years by a dynamical systems approach to low Reynolds number turbulence, particularly regarding transition and their relation to coherent structures. Establishing links with experiments and simulations of turbulent flows, however, has been stifled by the vastness of the number of degrees of freedom in real flows. This both serves to mask the underlying dynamics and renders the simulations expensive. In plane Couette flows the Minimal Flow Unit was introduced as a testing ground for probing transitional dynamics. For pipe flow we introduce a 2.5-dimensional model which preserves the key spatio-temporal features -- localised puffs, expanding slugs, and long-term transients. Simulations within this model are already proving fruitful in directing parallel simulations in 3-dimensions, which would otherwise prove costly to run without a clear target.

15:00-15:30 Tea
Chair: J Jimenez
15:30-15:50 Duggleby, A (Texas A & M)
  Structure and dynamics of turbulent pipe flow Sem 1

Results of a proper orthogonal decomposition of a turbulent pipe flow generated by direct numerical simulation are presented. The flow field is decomposed into its optimal basis functions as found by solving a Fredholm integral whose kernel is the two-point velocity autocorrelation tensor. The energy, structure, and dynamics of the basis functions are examined. The basis functions are categorised into two classes and six subclasses based on their wavenumber and coherent vorticity structure, and compared to similar results found in turbulent channel flow. The turbulent pipe flow is generated by a direct numerical simulation of the Navier-Stokes equations using a spectral element algorithm at a Reynolds number Re_tau=150 and a domain length of ten diameters (L/D=10). Evidence of very-large-scale motions will also be presented, as well as preliminary results from a corresponding investigation using a new large-domain (L/D=100) simulation.

15:50-16:10 Toh, S (Kyoto)
  Dynamical characterization of large scale structures in channel flow turbulence Sem 1

We obtained unstable periodic solutions up to at Re=6000 and travelling wave solution up to at Re=12,000 of streamwise-minimal channel flow by he shooting method. These solutions are on the basin boundary of the turbulent attractor and have characteristics of large scale motions observed in ully-developed channel flow turbulence. We show that these unstable solutions well describe the temporal evolution of the large scale motion embeded in turbulence that is defined by the local minima of streamwise velocity averaged over the streamwise direction.

16:10-16:30 Duguet, Y (KTH)
  Lower-branch travelling waves and transition to turbulence in pipe flow Sem 1

Transition to turbulence in cylindrical pipe flow is still not a well understood phenomenon. However, Direct numerical simulation in pipes with periodic boundary conditions shares a lot of common features with recent experiments. Here we are interested in the structure of the laminar-turbulent boundary, the manifold in phase-space separating the initial conditions leading to turbulence from those who lead to quick relaminarisation. Depending on the symmetries of the problem, phase-space trajectories on this manifold approach transiently various travelling wave solutions sitting on it. We can study the implications of this on the transition problem for both short and long pipes.

Chair: B Eckhardt
16:50-17:10 Mellibovsky, F (Universitat Politècnica de Catalunya)
  Underlying large-scale structures in transitional pipe flow Sem 1

Pipe flow undergoes transition to turbulence despite the linear stability of its basic laminar solution. Finite amplitude solutions in the form of travelling waves (H. Faisst and B. Eckhardt, Phys. Rev. Lett. 91(22), 224502 (2003)), coexisting with the basic flow, have been identified in the last few years. While they have been proved to play a certain role in the turbulent dynamics (B. Hof et al., Science 305, 1594 (2004)), their involvement in the transition process seems to be simply ungrounded. Furthermore, some recent experimental results point at a transitory nature of turbulence (B. Hof et al., Nature 443(7107), 59--62 (2006)), thus questioning the mere existence of a well defined critical threshold. The region of phase space dominated by turbulent dynamics would then be constituted by a surging amount of bifurcating complex solutions as the Reynolds Number is increased, acting as an attractor most of the time, but always retaining some probability that any trajectory finds its way back to laminarity. However transient may turbulence be, the notion of a threshold separating initial conditions that lead to transition from others that end up decaying still applies. It suffices to define the threshold as the point where the perturbation lifetime seems to diverge, possibly not to infinity if turbulence is a transient phenomenon, but still abruptly. Then, the threshold regains interest, and the question can be asked of how a solution wandering about criticality (T. Schneider et al., Phys. Rev. Lett. 99(3), 034502 (2007)) would look like. Starting from different initial conditions, and through accurate refinements, trajectories on the edge between turbulence and laminarity can then be analysed to elucidate which properties of a solution determine whether it belongs to the laminar or the turbulent basin of attraction. We analyse these trajectories to try and understand transition. Using an adapted Newton method we systematically search for travelling wave solutions underlying the dynamics of these critical trajectories.

17:10-17:30 Lebovitz, N (Chicago)
  Subcritical instability in shear flows: the shape of the basin boundary Sem 1

The boundary of the basin of attraction of the stable, laminar point is investigated for several of the dynamical systems modeling subcritical instability. In the cases thus far considered, this boundary contains a linearly unstable structure (equilibrium point or periodic orbit). The stable manifold of this unstable structure coincides at least locally with the basin boundary. The unstable structure plays a decisive role in mediating the transition in that transition orbits cluster tightly around its (one-dimensional) unstable manifold, illustrating a scenario proposed by Waleffe. The picture that emerges augments the bypass scenario for transition and reconciles it with Waleffe's scenario. We consider a model proposed by Waleffe (W97) for which an unstable equilibrium point U lies on the basin boundary. We find numerically that all orbits starting near U decay to the origin, whereas 'half' of them should remain permanently bounded away from the origin. We offer an interpretation of this tendency toward decay based on the structure of the basin boundary.

19:30-23:00 Conference dinner at Emmanuel College
Thursday 11 September
Chair: S Kida
09:40-10:00 Mullin, T (Manchester Centre for Nonlinear Dynamics)
  Transition to turbulence in a pipe Sem 1

The puzzle of why fluid motion along a pipe is observed to become turbulent as the flow rate is increased remains the outstanding challenge of hydrodynamic stability theory, despite more than a century of research. The issue is both of deep scientific and engineering interest since most pipe flows are turbulent in practice even at modest flow rates. All theoretical work indicates that the flow is linearly stable i.e. infinitesimal disturbances decay as they propagate along the pipe and the flow will remain laminar. Finite amplitude perturbations are responsible for triggering turbulence and these become more important as the non-dimensional flow rate, the Reynolds number Re, increases. Transition is generally abrupt and elucidating the details is difficult in practice. Here we report new experimental results on the appearance of periodic states which arise below the transition threshold. They are in accord with recent numerical results and their role in the transition process will be discussed.

10:00-10:20 Pringle, C (Bristol)
  Highly-symmetric travelling waves in pipe flow Sem 1

The recent theoretical discovery of finite-amplitude travelling waves in pipe flow has re-ignited interest in the transitional phenomena that Osborne Reynolds studied 125 years ago. Despite all being unstable, these waves are providing fresh insight into the flow dynamics. We describe two new classes of travelling wave which while possessing more restrictive symmetries than the previously found travelling waves of Faist & Eckhardt (2003) and Wedin & Kerswell (2004) seem to be more fundamental to the hierarchy of exact solutions. They exhibit much higher wall-shear stresses and appear at notably lower Reynolds numbers. The first M-class comprises of the various discrete-rotationally symmetric analogues of the mirror-symmetric wave found in Pringle & Kerswell (2007) and have a distinctive double layer structure of fast and slow streaks across the pipe radius. The second N-class has the more familiar separation of fast streaks to the exterior and slow streaks to the interior and looks like the precursor to the class of non-mirror-symmetric waves already known.

10:20-10:40 Viswanath, D (Michigan)
  Computational methods for finding exact solutions of shear flows Sem 1

Direct numerical solution begins with an initial velocity field and uses the incompressible Navier-Stokes equation to evolve that field forward in time. It has been a huge success and has provided theoretical support for a large number of experiments and natural phenomena. To find steady solutions and traveling waves, one must solve for velocity fields that satisfy certain nonlinear requirements. To find periodic or relative periodic solutions, one must solve for an initial velocity field that evolves in time over a single period to reach a final state that is equal to the initial state modulo certain symmetries. This talk will describe the use of direct numerical solution, Krylov subspace methods, and the hookstep technique from nonlinear optimization to find such solutions.

10:40-11:00 Daviaud, F (CEA/Saclay)
  Transition to turbulence and turbulent bifurcation in a von Karman flow Sem 1

We study the transition from laminar flow to fully developed turbulence in a von Karman flow (Re from 50 to 106). The flow undergoes a classical succession of bifurcations driven by the destabilization of the azimuthal shear-layer. We observe that the transition to turbulence is globally supercritical: the kinetic energy of the velocity fluctuations can be used as a single order parameter to characterize the transition. We also measure the dissipation through the torque injected in the flow. For high Reynolds numbers, the mean flow presents multiple solutions: the canonical symmetric solution becomes marginally unstable towards a flow which breaks the basic Rπ-symmetry. The global bifurcation between these states is highly subcritical and the system thus keeps a memory of its history. The transition recalls low-dimension dynamical system transitions and exhibits slow dynamics and peculiar statistics. References F. Ravelet et al., J. Fluid Mech. 601, 339 (2008) F. Ravelet et al., Phys. Rev. Lett. 93, 164501 (2004)

11:00-11:30 Coffee
Chair: A Leonard
11:30-11:50 Kawahara, G (Osaka)
  The role of coherent structures in low-Reynolds-number turbulent wall flows Sem 1

Coherent states found numerically in plane Couette turbulence (Kawahara & Kida, J. Fluid Mech. 449, 2001; Kawahara, Phys. Fluids 17, 2005) and square duct turbulence (Uhlmann, Pinelli, Kawahara & Sekimoto, J. Fluid Mech. 588, 2007) are reviewed to discuss the role of coherent structures in wall-bounded turbulent flows at low Reynolds numbers. The coherent states in Couette flow are represented by unstable periodic solutions to the incompressible Navier-Stokes equation. The periodic solution, which exhibits a near-wall regeneration cycle of streamwise vortices and streaks, is shown to satisfy Prandtl's wall law, implying that the regeneration cycle has relevance to the statistical law of near-wall turbulent flow. The coherent states in marginally turbulent square duct flow are characterized by a four-vortex secondary flow and associated streaks. It is shown that at low Reynolds numbers streamwise vortices play a crucial role in the appearance of secondary flow of Prandtl's second kind.

11:50-12:10 Kida, S (Kyoto)
  Statistics of passive vectors in an unstable periodic flow of Couette system Sem 1

The statistics of orientation and location of passive vectors in an unstable periodic flow of Couette system are investigated numerically with special attention to the correlation with the coherent structures in the flow. The flow domain is divided into many small cubic regions and the distribution of directions of many passive vectors in each cube is calculated at several temporal phases of the unstable periodic flow. It is found that in most of cubes the passive vectors align either in a single direction (linear region) or on a plane (planar region). The unstable periodic flow exhibits the characteristic features of Couette turbulence having the streamwise vortices and the low-speed streaks. The linear regions are observed in streamwise vortices and low-speed streaks (or the ejection region near the wall), the planar regions in the periphery of vortices and in the sweep region near the wall. The dispersion in direction of passive vectors depends on their near-past history over the characteristic times of the unstable periodic flow.

12:10-12:30 Lee, C (Peking)
  Transition in wall-bounded flows Sem 1

In this article we present direct comparisons of experimental results on transition in wall-bounded flows obtained by flow visualizations, hot-film measurement, and particle-image velocimetry (PIV), along with a brief mention of relevant theoretical progresses, based on a critical review of about 120 selected publications. Despite somewhat different initial disturbance conditions used in experiments, the flow structures were found to be practically the same. The following observed flow structures are considered to be of fundamental importance in understanding transitional wall-bounded flows: The three-dimensional nonlinear wave packets called solitons-like coherent structures (SCS) in boundary layer and pipe flows; the ƒ©-vortex; the secondary vortex loops; and the chain of ring vortices. The dynamic processes of the formation of these structures and transition as newly discovered by recent experiments include, among others: (1) The sequential interaction processes between the ƒ©-vortex and the secondary vortex loops, which controls the manner by which the chain of ring vortices is periodically introduced from the wall region into the outer region of the boundary layerG (2) The generation of high-frequency vortices, which is one of the key issues for understanding both transitional and developed turbulent boundary layers (as well as other flows), of which several explanations have been proposed but a particularly clear interpretation can be provided by the experimental discovery of secondary vortex loops. The ignorance of secondary vortex loops would make the dynamic processes and flow structures in a transitional boundary layer inconsistent with previous discoveries; and (3) The dominant role of SCS in all turbulent bursting, which is considered as the key mechanism of turbulent production in a low Reynolds-number turbulent boundary layer. Of direct relevance to bursting is the low-speed streaks, whose formation mechanism and link to the flow structures in wall-bounded flows can be answered more clearly than before in terms of the SCS dynamics. We combine these newly observed structures and processes to those well-known ones to form a more integrated physical picture of the transitional dynamics. This not only enables revisiting the classic story of wall-bounded flow transition, but also opens a new avenue to reconstruct the possible universal scenario for wall bounded flow transition.

12:30-13:30 Lunch at Wolfson Court
Chair: J Klewicki
14:00-14:20 Hutchins, N (Melbourne)
  The existence of very-large scale motions or `superstructures' in wall-bounded turbulent flows Sem 1

Recent experiments in the logarithmic and wake regions of wall-bounded turbulent flows have revealed the existence of very large-scale motions (with instantaneous length-scales up to 20 boundary layer thicknesses). This regime of very long meandering positive and negative streamwise velocity fluctuations (and associated roll-modes) appear to be universal features of all wall-bounded turbulent flows, having been recently documented in turbulent boundary layers, pipes, channels and atmospheric surface layers. Importantly, these structures appear to maintain a presence or footprint in the near-wall region, seeming to modulate or influence the near-wall cycle. As Reynolds number increases, not only is there an increase in scale-separation between these large-scale motions and the near-wall cycle, but also an increase in the overall energy of these long meandering features as compared to the viscous scales. This would seem to imply that these features will play an increasingly dominant role in high Reynolds number wall-bounded turbulent flows.

14:20-14:40 Tutkun, M (Norwegian Defense Research Establishment)
  Large scale structures of high Reynolds number turbulent boundary layers Sem 1

Time-resolved measurements from the European Wallturb consortium experiment conducted using the large Laboratoire de Mecanique de Lille (LML) wind tunnel at Lille, France are reported. Data were taken at Re_theta = 9,800 and 19,100. Data were obtained using a rake of 143 single hot-wires which were spaced logarithmically across the flow in the spanwise and vertical directions over a distance approximately equal to the boundary layer thickness of 0.3 m. The wires were calibrated in situ using data from the simultaneous PIV experiment in a plane just upstream of the wire. Other PIV planes were also recorded in synch with the hot-wire data acquisition. This paper reports the cross-correlations and spectra generated among the wires themselves and those generated the upstream PIV plane immediately upstream of the rake. The existence of very long elongated structures are also shown by the space-time correlations obtained using the hot-wire rake data. Proper orthogonal decomposition (POD) applied on the same how-wire rake data are found to be very efficient in term of energy such that only a few spanwise wavenumber modes are necessary to capture 90% of the energy, and the details POD results like eigenspectra and eigenfunctions are discussed in the paper.

14:40-15:00 Monty, J (New Mexico State)
  Large scale features in turbulent pipe and channel flows Sem 1

In recent years there has been significant progress made towards understanding t he large-scale structure of wall-bounded shear flows. Most of this work has been conducted with turbulent boundary layers leaving scope for further work in pipe s and channels. Here, the structure of fully-developed turbulent pipe and channel flow has been studied using custom-made arrays of hot-wire probes. R esults reveal long, meandering structures of length up to 25 pipe radii or chann el half-heights. These appear to be qualitatively similar to those reported in t he log region of a turbulent boundary layer. However, for the channel case, larg e-scale coherence persists further from the wall than in boundary layers. Further comparison of the three turbulent flows sh ows that the characteristic structure width in the logarithmic region of a bound ary layer is at least 1.6 times smaller than that in a pipe or channel.

15:00-15:30 Tea
Chair: R Adrian
15:30-15:50 Ganapathisubramani, B (Imperial College London)
  Large-scale motions in supersonic turbulent boundary layers Sem 1

Wide-field and high-speed Particle Image Velocimetry measurements were performed in a Mach 2 supersonic turbulent boundary layer to characterise the structure of large-scale coherence. Instantaneous velocity fields in the logarithmic region reveal the presence of elongated uniform low- and high-speed regions. These elongated regions exhibit strong similarities with the large-scale motions found in incompressible boundary layers. Application of Taylor's hypothesis together with high-speed PIV data indicates that these large-scale structures could extend over 30-40 boundary layer thickness in length. Alternately, these can interpreted as 5-10 boundary layer thickness long structures lasting over several integral time scales. Regardless of interpretation, these elongated (spatio-temporal) structures appear to be play a significant role in the dynamics of shock-induced turbulent boundary layer separation. A possible mechanism that relates the passage of these elongated structures to the frequency of unsteadiness of shock-induced separation will be presented.

15:50-16:10 Narasimha, R (Indian Institute of Science)
  An event-based description of the heat-flux time series in an atmospheric boundary layer Sem 1

A new method of describing the turbulent momentum flux time series (which we call the ‘extended point-process’ method) has recently been proposed (Narasimha+ 2007 Phil. Trans A). In this method ‘flux events’ are identified in the associated time series, and their characteristics defined in terms of their contributions to the mean flux (so events can be of either sign). We extend the method here to provide a similar description of the eddy heat flux time series, and show the enormous influence exerted by the state of stability of the atmospheric boundary layer on flux-event statistics.

16:10-16:30 Gungor, A (Gatech)
  Multiscale simulation of wall-bounded flows Sem 1
Chair: B Sawford
16:50-17:10 Klewicki, J (New Hampshire)
  On describing mean flow dynamics in wall turbulence Sem 1

The study of wall-flow dynamics and their scaling behaviors with increasing Reynolds number warrants considerable attention. Attempts to date, however, have primarily focused on questions relating to what scaling behaviors occur, rather than the dynamical reasons why they occur. Given these considerations, the present talk is organized in three parts. In the first part it is shown that the predominant methodology for discerning the dominant mechanisms associated with the mean flow dynamics is problematic, and can lead to erroneous conclusions. In the second part we examine the Millikan-Izakson (inner/outer/overlap) arguments that underpin the widely accepted derivation for a logarithmic mean profile. Existing rigorous results from the theory of functions are outlined. They reveal that the Millikan-Izakson arguments constitute something very close to a tautology and embody little physics specific to turbulent wall-flows. The first two parts establish the context for the third. The presentation concludes with a physical interpretation of the mathematical conditions necessary for a logarithmic (or nearly logarithmic) mean profile. The basis for this interpretation is the analysis of Fife et al., (2005 JFM 532}, 165) which reveals that the mean differential statement of Newton’s second law rigorously admits a hierarchy of physical layers each having their own characteristic length. These analyses show that the condition for exact logarithmic dependence exists when the normalized equations of motion (normalized using the local characteristic length) attain a self-similar structure, and physically indicate that the leading coefficient in the logarithmic law (von Karman constant) will only be truly constant when an exact self-similar structure in the gradient of the turbulent force is attained across a range of layers of the hierarchy. These results are discussed relative to the physics of boundary layer Reynolds number dependence and recent data indicating that the von Karman constant varies for vary ing mean momentum balance.

17:10-17:30 Morrison, J (Imperial College London)
  Inner-outer interaction in turbulent wall layers Sem 1

A physical, but non-taxonomical, description of wall turbulence is presented. The requirements of self-similarity are discussed and, in particular, the effects of inner-outer interaction are explored. This interaction may be described as "bottom-up" (e.g. surface roughness) or "top-down" (e.g. high Reynolds number) and the consequences of each are discussed. It is shown that, while a weak interaction is, in fact, a statement of Townsend's ideas of ‘inactive’ motion, it is only a linear first approximation of a nonlinear process, which occurs primarily through the wall-normal component of velocity and the static pressure. It is shown that the interaction leads to a lack of similarity in terms of the structure and the second-order statistics. Using observations from a discrete wavelet analysis, a heuristic model of wall turbulence is described in which it is shown that the motion is driven by pressure-gradient fluctuations arising from the dominance of quasi-streamwise vortices. For linear control, the need for model reduction therefore suggests transient growth. Early results from measurements of roughness-induced transient growth in a Blasius base flow are described: issues concerning receptivity (to the initial disturbance) and observability are discussed. Some results from associated control experiments and simulations involving surface deformation are also presented.

18:00-19:30 Cambridge University Press - Wine Reception and Book Display
Friday 12 September
Chair: T Mullin
09:20-09:40 Burguete, J (Navarra)
  Slow temporal scales in the dynamics of vortices at large Reynolds numbers in a cylindrical experiment Sem 1

We present results concerning an experimental setup where a fluid is stirred in a cylindrical cavity up to a Reynolds number of $10^6$. We show that the averaged velocity field of the turbulent flow bifurcates subcritically breaking some symmetries of the problem and becomes time-dependent because of equatorial vortices moving with a precession movement. This subcriticality produces a bistable regime, whose main characteristics are successfully reproduced using a three well potential model with additive noise. We characterized the hysteresis region, not previously observed, in this bifurcation. This hysteresis appears only for a extremely small range of parameters. Three different time-scales are relevant to the dynamics, two of them very slow compared to the impeller frequency.

09:40-10:00 Meseguer, A (Universitat Politècnica de Catalunya)
  Subcritical equilibria in counter-rotating Taylor-Couette flow Sem 1

We provide a numerical exploration of the spiral regimes that appear in small-gap Taylor-Couette flow of radius ratio 0.883 and for high counter-rotating Reynolds numbers. In particular, the exporation is carried out for a fixed outer rotation Reynolds number value of R_o=-1200. The spiral regimes found beyond the linear stability critical values are tracked back as a function of the inner rotation Reynolds number R_i with an arch-length continuation scheme via Newton-Krylov algorithms suitably tailored for axially-moving and/or azimuthally rotating reference frames. The spirals found have been found to be subcritical. This study is the first step taken in order to provide the essential inner structure of the skeleton of equilibria that may be responsible for the subcritical transition and hysteretic phenomena that has been reported by many experimentalists in the past.

10:00-10:20 Nagata, M (Kyoto)
  The sliding Couette flow problem Sem 1

The sliding Couette flow, categorised by Joseph (1976), is a flow between concentric cylinders of radii, a and b ( > a), where the inner cylinder is pulled with an axial speed, U , relative to the stationary outer cylinder. It is known that the linear critical Reynolds number based on the speed U and the gap width, b-a, is infinite, at least when the radius ratio is not very small, so that secondary flows, if they exist, must bifurcate abruptly from the laminar state. The absence of linear instabilities occurs similarly in the problems, such as plane Couette flow, pipe Poiseuille flow and flow in a square duct, which have been extensively explored with success in recent years. As far as the author knows finite amplitude solutions in the sliding Couette flow have not yet been found. In this short paper we analyse both linear and nonlinear instabilities of the sliding Couette flow in the limit of narrow gap. Following Masuda, Fukuda & Nagata (2008) we apply a uniform rotation.O, in the streamwise direction in order to provoke rotational instabilities. The idea is to see whether bifurcated flows developed with increasing O may be sustained in the subcritical region and even exist as O is reduced back to zero. We show numerically that the critical Reynolds number approaches the global stability limit determined by energy theory in the limit of large rotation rate. A nonlinear analysis indicates that secondary flows bifurcating at a moderate rotation rate are characterized by three-dimensional spiral vortex structures. Attempted continuation of the secondary flow branch to the zero rotational rate will be discussed. References [1] Joseph,D. D. (1976) Stability of Fluid Motions I, Springer-Verlag. [2] Masuda, S., Fukuda, S. & Nagata, M. (2008) 'Instabilities of plane Poiseuille flow with a streamwise system rotation', J. Fluid Mech., 603, 189-206.

10:20-10:40 Morozov, A (Edinburgh)
  Exact solutions in the 2-dimensional viscoelastic channel flow Sem 1

Recently, it has been discovered that flows of polymer solutions can become unstable and exhibit turbulent-like behaviour at very small Reynolds numbers. As a rule, viscoelastic flows with curved streamlines are linearly unstable, while parallel shear flows are believed to exhibit a subcritical transition to a turbulent state. In the absence of inertia, these instabilities are driven by anisotropic elastic stresses.

Here I try to identify exact solutions in the 2D viscoelastic channel flow. Starting from the exact solutions of the Navier-Stokes equation found by Th. Herbert, solutions for the Oldroyd-B viscoelastic model are obtained by analytic continuation from the Newtonian case. It is found that these solutions persist at relatively small Reynolds numbers if the normal-stress difference is large enough. Nevertheless, so far I was unsuccessful in tracking these solutions down to the Re=0 limit. Other types of analytic continuation will be discussed as well.

10:40-11:00 Sameen, A (ICTP)
  Stability in a plane channel flow with viscosity stratification Sem 1

We discuss the stability of a channel flow with viscosity stratification. Three mechanisms of route to transition, namely linear, transient and secondary instability growths are investigated here. How does a viscosity-stratification alter the stability behaviour of these mechanisms in a channel flow? A temperature dependent viscosity is used obtain the viscosity gradients. It is concluded that all three routes to transition has different stability characteristics for any given kind of stratification. 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 Rayleigh-Benard and Poiseuille modes for Grashof numbers upto $\sim 25000$, which merge thereafter. We conclude that a knowledge of transition mechanism and type of stratification is necessary for viscosity variation to be exploited as an effective flow control mechanism.

11:00-11:30 Coffee
Chair: M Nagata
11:30-11:50 Chomaz, JM (CBRS-Ecole Polytechnique)
  Lift-up and convective nonnormalities : the dynamics of a recirculation bubble Sem 1

The stability of the recirculation bubble behind a smoothed backward-facing step is numerically computed. Destabilization occurs through a stationary three-dimensional mode. Analysis of the direct global mode shows that the instability corresponds to a deformation of the recirculation bubble in which streamwise vortices induce low and high speed streaks as in the classical lift-up mechanism. Formulation of the adjoint problem and computation of the adjoint global mode show that both the lift-up mechanism associated to the transport of the base flow by the perturbation and the convective nonnormality associated to the transport of the perturbation by the base flow explain the properties of the flow. The lift-up nonnormality differentiates the direct and adjoint modes by their component: the direct is dominated by the streamwise component and the adjoint by the cross-stream component. The convective nonnormality results in a different localization of the direct and adjoint global modes, respectively downstream and upstream. Implication of these properties on the linear and nonlinear dynamics will be discussed. co-workers: O. MARQUET, M. LOMBARDI , D. SIPP AND L. JACQUIN Departement d'Aerodynamique Fondamentale et Experimentale, ONERA, 92190 Meudon, France

11:50-12:10 Mittal, S (Indian Institute of Technology)
  Global analysis of convective instabilities in nonparallel flows Sem 1

A new scheme for the global analysis of convective instabilities in nonparallel flows is proposed. The linearized perturbation equations for an incompressible flow are written in a moving frame of reference that travels with the perturbation. In the moving frame, the base flow varies with time. However, at t=0, it is same as the one in stationary frame. Therefore, this analysis, for determining the global convective instability, is valid in an instantaneous sense. A stabilized finite element method is utilized to discretize these equations. A sub-space iteration procedure is utilized to solve the resulting generalized eigenvalue problem. Unlike local analysis, the proposed method gives the global eigenmode and the corresponding growth rate. The scheme is applied to assess the stability of uniform flow past bluff bodies. For the flow past a circular cylinder the critical Re for the onset of convective instability is found to be 4, approximately. The critical Re for the onset of the shear layer instability has been a point of contention. Various estimates have been proposed ranging from Re_c=350 to 2600. The proposed method is applied to find Re_c. To suppress the wake mode, that leads to Karman vortex shedding, flow past one half of the cylinder is studied. The Re_c is found to be ~54. The wake and shear layer modes for a full cylinder are compared to bring out the differences between the two. Also, the connection of the instability at low Re to the shear layer modes at higher Re (=500) is presented. The results are compared with earlier work from local stability analysis. Results are also computed for flow past a flat plate normal to the flow. All the results are in excellent agreement with the direct numerical simulation of the linearized flows equations.

12:10-12:30 Seddon, J (Manchester)
  Bifurcation phenomena in the flow through a sudden expansion in a circular pipe Sem 1

We report the results of an experimental investigation of laminar and time-dependent flows through a sudden expansion in a circular pipe. The flow state was investigated using high resolution MRI imaging techniques which have allowed us to settle a long standing debate on the first instability that occurs. As Re is increased, the flow passes through a steady symmetry breaking bifurcation such that the position of the recirculating eddy becomes asymmetric within the pipe. This in turn gives way to simple periodic motion via a Hopf bifurcation with further increase in Re.

12:30-13:30 Lunch at Wolfson Court
Chair: H Nagib
  Summary and discussion Sem 1
15:00-15:30 Tea
  Summary and discussion continued Sem 1
16:20-16:40 Closing remarks
18:45-19:30 Dinner at Wolfson Court (Residents Only)

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