Session AO: General Stability

Momentum transport and dissipation scaling in turbulent shear flows

We expose analogies between turbulence in a fluid heated from below (Rayleigh-B{e}nard flow), in a fluid between rotating cylinders (Taylor-Couette flow) and in pressure driven flow down a pipe. The analogy is based on the identification of heat and momentum fluxes, respectively, and on corresponding dissipation rates. With this identification it becomes possible to transfer the scaling and modelling ideas of Grossmann and Lohse from Rayleigh-Benard to shear flows. The model is consistent with data for turbulent Taylor-Couette and pipe flow, and gives testable predictions for the dependence on gap widths.   

Finite turbulence lifetimes in pipe flow

It is generally thought that turbulence in pipe flow is a sustained flow state once the Reynolds number (Re) has passed a critical value. Numerical simulations showed that in the transition region there is a distribution of lifetimes and indicated that the median lifetime diverges near a Reynolds number of about 2250. This would indicate a transition to an attractor in phase space that describes the turbulent flow state. Re-analysis of the original numerical results shows that when an initial formation period of the disturbance is omitted, the lifetime variation with Re is better described by an exponential increase, not a divergence. To distinguish between the two types of behaviour it is necessary to collect lifetime statistics spanning a wide range of lifetimes, extending beyond 2000 D. A pipe flow facility was constructed with a length of 7500 D and lifetime statistics for fixed pipe length as a function of Re were obtained. The shape of the probability distributions P(t,Re) to stay turbulent for a time t at Reynolds number Re is not compatible with a transition to an attractor at finite Re, and supports an exponential increase with Re. This implies that an infinite lifetime is only reached in the limit of infinite Re. The same scaling was observed in extended numerical simulations, and in a re-analysis of plane Couette flow data. The results therefore imply that turbulence in pipes is only a transient event.   

Edge States in Transitional Pipe Flow

We study the boundary of the laminar region near the onset of turbulence. Approaching the boundary from the laminar side, the lifetime of perturbations increases, diverges when the boundary is reached, and varies chaotically for larger amplitudes. In the chaotic region, lifetimes vary sensitively with amplitude, consistent with the strange saddle picture of the turbulence proposed earlier. The trajectory on the edge between the laminar and chaotic regions is asymptotic to a single well defined state, essentially independent of the type of perturbation. The edge then becomes the stable manifold of this structure. In the case of a model shear flow, the edge states are simple or period doubled or chaotic trajectories. In the case of pipe flow the edge state seems to remain close to a state with simple vortical structure. \newline Edge of Chaos in a Parallel Shear Flow, Joseph D. Skufca, James A. Yorke, and Bruno Eckhardt, Phys. Rev. Lett. 96, 174101 (2006)   

Threshold exponents of streamwise transition in pipe flows

Subcritical transition in pipe flow is explored for a wide range of Reynolds numbers within the interval ${\rm Re} \in [2.5\times10^3,1.26\times10^4]$ by means of a spectral method that resolves the transitional dynamics with nearly $3.5 \times 10^4$ degrees of freedom on a medium aspect-ratio domain of length $32 \pi /5$. The aim of this exploration is to provide a characterization of the basin of attraction of the basic regime by measuring the minimal amplitude of an initial global perturbation leading to transition. The analysis is based on a particular theoretical scenario that considers streamwise-independent finite amplitude initial vortical perturbations that trigger global transition via optimal inflectional instabilities of streamwise-dependent modes with selected axial wavenumbers. Disturbances consisting of $1, 2$ and $3$ pairs of vortices are investigated. Long lasting turbulent regimes and relaminarized flows are distinguished by means of time integrations of suitable length between $T_{\rm min}=600$ and $T_{\rm max}=1000$ advective time units. Some transitional runs are specifically analized to exemplify the transition scenario under investigation and its independence of pipe length is verified with a few computations on a longer pipe of length $32 \pi$ ($1.4 \times 10^5$ degrees of freedom). For large values of the Reynolds number, a theoretical scaling law for the threshold amplitude of a perturbation required to trigger transition is provided. Different types of perturbations seem to respond to different scaling laws.   

Localized impulsive perturbations in pipe flow transition.

A numerical study of the destabilizing effects of localized impulsive perturbations in pressure-driven pipe flow is presented. The numerics intend to ellucidate the intrinsic mechanisms of subcritical transition to turbulence in pipe flow by reproducing recent experimental explorations carried out by Hof, Juel and Mullin (Phys. Rev. Lett. {\bf 91(24)}, 244502-4, 2003), concluding that the minimum amplitude of a perturbation required to cause transition scales as the inverse of the Reynolds number, i.e., $\mathcal{O} (\mathrm{Re}^{-1})$. A comprehensive exploration of the critical amplitudes that trigger transition as a function of the injection duration is carried out, concluding that injections lasting longer than $24$ advective time units do not remarkably decrease the critical amplitude of transition. The critical threshold is then tracked for long enough injections and up to Re = 14000 with reasonably good agreement for $\mathrm{Re} > 4000$. The apparent disagreement at low $\mathrm{Re}$ is explained in terms of the differences between constant mass-flux and pressure-driven pipe flows. finally, relaminarizing puffs at very low $\mathrm{Re}$ are analysed in search for coherent structures (travelling waves). Traces of these structures, numerically found, have been experimentally observed recently.   

Pipe flow instability for the `critical' Reynolds number in transition to turbulence

Experimental results obtained over a century have shown that the flow in a circular pipe becomes `naturally' turbulent at a `critical' Reynolds number of $\sim 2000$. In this work a theoretical explanation, based on the minimum energy of an axisymmetric deviation, is suggested for this `critical' value. For this purpose a temporal stability analysis is preformed for the pipe Poiseuille flow, which has been modified by a primary axially-independent axisymmetric and finite amplitude deviation (following the method by Gavarini, Bottaro and Nieustadt, {\it J. Fluid Mech.} {\bf 517}, 131, 2004, who analyzed the spatial case). The optimal modification is defined as the primary base-flow deviation, with a given additional energy density, that yields the maximum growth rate to the secondary disturbances. Optimal modifications are computed by a variational technique. It is found, that above the Reynolds number of $\sim 2000$ the minimum energy of the deviation, which is concentrated at the central part of the pipe, becomes a global minimum for triggering secondary instabilities. Below this critical number the minimum energy deviation, which is concentrated next to the pipe wall, is the one having a global minimum for instability. Previous experimental observations support these results.   

Stability of flows in periodicaly varying smooth channels.

Previous studies investigated 2D and 3D flows in between two patterned surfaces focusing on either sinuous or grooved patterns. We wish to consider the possibility of non stationary disturbances in 3D textured surface. This type of geometrically forced instability are of fundamental importance in the design of efficient passive micromixers and heat exchangers. Here, we use a numerical continuation technique to investigate the three dimensional stability of flows driven by a constant pressure gradient in between two periodic patterned surfaces.   

Jeffery-Hamel flow: an experimantal study of instabilities, bifurcations and multiple solutions

We present an experimental realization of the classical Jeffery-Hamel flows inside a wedge-shaped channel. We measure the velocity fields in the nearly-two-dimensional flow produced in our experimental apparatus, and compare the data with the predictions of Jeffery-Hamel theory. The flow is governed by two dimensionless parameters, the Reynolds number $R$ and the wedge angle $\alpha$. In the plane of these parameters, we perform a detailed study of bifurcation diagrams for the solutions that reveals the absolute stability of the pure outflow solution and an interesting hysteretic structure for bifurcations. We also observe a multiple-vortex flow regime predicted earlier numerically and analytically.   

Impact of Noise on the Onset of Vortex Breakdown

The effect of noise on the dynamical properties of a fluid flow in a bounded container is studied. The flow is modelled by the Navier-Stokes equations. The noise, which is defined by a random process with predefined statistical properties, is introduced in a physically relevant manner, via the boundary conditions. The stability of critical states of the deterministic system is analyzed via the simultaneous linearization of the system in physical and probability spaces. The resulting equations form a stochastic system, which depend on the deterministic critical state and on a Wiener process whose auto-correlation function is directly connected to the type of noise as well as on the critical state itself. The behavior of the stochastically forced system around critical states is explored numerically. This work represents a continuation of the preliminary results presented at the APS-DFD annual meeting in 2005 in Chicago.   

Relaminarisation of localised turbulence in pipe flow

The transition to turbulence in pipe flow is a longstanding problem of classical fluid mechanics. The basic laminar state is believed always to be linearly stable, however, finite amplitude disturbances lead to localised turbulent `puffs' clearly seen in experiments. Recent experimental work (Peixinho \& Mullin, 2006) has indicated that these puffs can become sustained in time, rather than suddenly decaying, beyond a threshold Reynolds number of $1750\pm 10$. Their median survival times were found to scale with $(Re_c-Re)^{-1}$. Other experimental results for $Re_c$ vary upward to 2000 (Wygnanski \& Champagne, 1973) and recently (Hof {\it et al.}, 2006) no critical Reynolds number was found with median times increasing exponentially. Given the range of results, we investigate the transition through direct numerical simulation and consider the implications for the threshold to sustained turbulence.   

Numerical Study of Solitary Waves in a Channel

We investigate the dynamics of large amplitude internal solitary wave solutions of Euler equations in a channel using a vortex sheet model discretized by a point vortex method. The initial conditions are taken to be traveling solitary wave solutions of a strongly nonlinear long-wave model studied by Jo and Choi [Stud. Appl. Math. 109 (2002) 205--227]. The approach uses a boundary integral representation in which the fluid interface is modeled by a free vortex sheet and the channel walls by bound vortex sheets. We validate numerical results first by considering the case when channel is flat and horizontal. We simulate the interaction of two solitary waves and also study the deformation of a solitary wave propagating over non-uniform topography. The goal is to compare our numerical results using the vortex sheet model to those obtained using the long-wave model cited above.