The onset of shear flow turbulence is characterized by turbulent patches bounded by regions of laminar flow. At low Reynolds numbers localized turbulence relaminarises, raising the question of whether it is transient in nature or it becomes sustained at a critical threshold. We present extensive numerical simulations and a detailed statistical analysis of the data, in order to shed light on the sources of the discrepancies present in the literature. The results are in excellent quantitative agreement with recent experiments and show that the turbulent lifetimes increase super-exponentially with Reynolds number. In addition, we provide evidence for a lower bound below which there are no meta-stable characteristics of the transients, i.e. the relaminarisation process is no longer memoryless.
Linear transient growth analysis is commonly used to suggest the structure of disturbances which are particularly efficient in triggering transition to turbulence in shear flows. We demonstrate that the addition of nonlinearity to the analysis can substantially change the prediction made in pipe flow from simple 2 dimensional streamwise rolls to a spanwise and cross-stream localised 3 dimensional state. This new nonlinear optimal is demonstrably more efficient in triggering turbulence than the linear optimal indicating that there are better ways to design perturbations to achieve transition.
We study the statistics of free-surface turbulence at large Reynolds numbers produced by direct numerical simulations in a fluid layer at different thickness with fixed characteristic forcing scale. We observe the production of a transient inverse cascade, with a duration which depends on the thickness of the layer, followed by a transition to three-dimensional turbulence initially produced close to the bottom, no-slip boundary. By switching off the forcing, we study the decaying turbulent regime and we find that it cannot be described by an exponential law. Our results show that boundary cond…
Turbulence intensity profiles are compared for smooth- and rough-wall pipe flow measurements made in the Princeton Superpipe. The profile development in the transition from hydraulically smooth to fully rough flow displays a propagating sequence from the pipe wall towards the pipe axis. The scaling of turbulence intensity with Reynolds number shows that the smooth- and rough wall level deviates with increasing Reynolds number. We quantify the correspondence between turbulence intensity and the friction factor.
Recent studies have brought into question the view that at sufficiently high Reynolds number turbulence is an asymptotic state. We present the first direct observation of the decay of turbulent states in Taylor-Couette flow with lifetimes spanning five orders of magnitude. We also show that there is a regime where Taylor-Couette flow shares many of the decay characteristics observed in other shear flows, including Poisson statistics and the coexistence of laminar and turbulent patches. Our data suggest that characteristic decay times increase super-exponentially with increasing Reynolds number…
It has been reported that a fully localized turbulent band in channel flow becomes sustained when the Reynolds number is above a threshold. Here we show evidences that turbulent bands are of a transient nature instead. When the band length is controlled to be fixed, lifetimes of turbulent bands appear to be stochastic and exponentially distributed, a sign of a memoryless transient nature. Besides increasing with the Reynolds number, the mean lifetime also strongly increases with the band length. Given that the band length always changes over time in real channel flow, this size dependence may …
SAM, a plant-level system analysis tool for advanced reactors (SFR, LFR, MSR/FHR) is under development at Argonne. As a modern system code, SAM aims to improve the predictions of 3D flows relevant to reactor safety during transient conditions. In order to fulfill this goal, one approach is to implement modeling of turbulent flow in SAM through establishing an embedded surrogate model for Reynolds stress/turbulence viscosity based on machine learning techniques. The proposed approach is based on an assumption that there exists a functional dependency relationship between local flow features and…
Although the equations governing fluid flow are well known, there are no analytical expressions that describe the complexity of turbulent motion. A recent proposition is that in analogy to low dimensional chaotic systems, turbulence is organized around unstable solutions of the governing equations which provide the building blocks of the disordered dynamics. We report the discovery of periodic solutions which just like intermittent turbulence are spatially localized and show that turbulence arises from one such solution branch.
We investigate experimentally the influence of suspended particles on the transition to turbulence. The particles are monodisperse and neutrally-buoyant with the liquid. The role of the particles on the transition depends both upon the pipe to particle diameter ratios and the concentration. For large pipe-to-particle diameter ratios the transition is delayed while it is lowered for small ratios. A scaling is proposed to collapse the departure from the critical Reynolds number for pure fluid as a function of concentration into a single master curve.
We study streamwise turbulence intensity definitions using smooth- and rough-wall pipe flow measurements made in the Princeton Superpipe. Scaling of turbulence intensity with the bulk (and friction) Reynolds number is provided for the definitions. The turbulence intensity scales with the friction factor for both smooth- and rough-wall pipe flow. Turbulence intensity definitions providing the best description of the measurements are identified. A procedure to calculate the turbulence intensity based on the bulk Reynolds number (and the sand-grain roughness for rough-wall pipe flow) is outlined.
We propose a scaling law for the onset of turbulence in pipe flow of neutrally buoyant suspensions. This scaling law, based on a large set of experimental data, relates the amplitude of the particle-induced perturbations ($ε$) to the critical suspension Reynolds number, $Re_{s,c}$. Here $ε$ is a function of the particle-to-pipe diameter ratio and the volume fraction of the suspended particles, $ε= (d/D)^{1/2} ϕ^{1/6}$. $Re_{s,c}$ is found to scale as $ε^{-1}$. Furthermore, the perturbation amplitude allows a distinction between classical, intermediate and particle-induced transition.
As everyone knows who has opened a kitchen faucet, pipe flow is laminar at low flow velocities and turbulent at high flow velocities. At intermediate velocities there is a transition wherein plugs of laminar flow alternate along the pipe with "flashes" of a type of fluctuating, non-laminar flow which remains poorly known. We show experimentally that the fluid friction of flash flow is diagnostic of turbulence. We also show that the statistics of flash flow are in keeping with Kolmogorov's phenomenological theory of turbulence (so that, e.g., the energy spectra of both flash flow and turbulent …
The statistics of the relaminarisation of localised turbulence in a pipe are examined by direct numerical simulation. As in recent experimental data (Peixinho & Mullin, Phys. Rev. Lett. 96, 094501, 2006), the half life for the decaying turbulence is consistent with the scaling (Re_c-Re)^{-1}, indicating a boundary crisis of the localised turbulent state familiar in low-dimensional dynamical systems. The crisis Reynolds number, is estimated as Re_c=1870, a value within 7% of the experimental value 1750. We argue that the frequently-asked question of which Re and initial disturbance are needed t…
In pipes and channels, the onset of turbulence is initially dominated by localized transients, which lead to sustained turbulence through their collective dynamics. In the present work, we study the localized turbulence in pipe flow numerically and elucidate a state space structure that gives rise to transient chaos. Starting from the basin boundary separating laminar and turbulent flow, we identify transverse homoclinic orbits, the presence of which necessitates a homoclinic tangle and chaos. A direct consequence of the homoclinic tangle is the fractal nature of the laminar-turbulent boundary…
Two approaches to the problem of transition to turbulence of shear flows are popular in the literature. The first is the linear one of transient growth which focuses on the likely form of the most 'dangerous' (lowest energy) turbulence-triggering disturbances. The second is the nonlinear calculation of the laminar-turbulent boundary which instead focuses on their typical amplitudes. We look to bridge the gap between these two perspectives by considering the fully nonlinear transient growth problem to estimate both the form and amplitude of the most dangerous disturbance. We thereby discover a …
Transition to turbulence in straight pipes occurs in spite of the linear stability of the laminar Hagen--Poiseuille flow if the amplitude of flow perturbations as well as the Reynolds number exceed a minimum threshold (subcritical transition). As the pipe curvature increases centrifugal effects become important, modifying the basic flow as well as the most unstable linear modes. If the curvature (tube-to-coiling diameter $d/D$) is sufficiently large a Hopf bifurcation (supercritical instability) is encountered before turbulence can be excited (subcritical instability). We trace the instability…
We study the mechanism of energy injection from the mean flow to the fluctuating velocity necessary to maintain wall turbulence. This process is believed to be correctly represented by the linearized Navier--Stokes equations, and three potential linear mechanisms have been considered, namely, modal instability of the streamwise mean cross-flow $U(y,z,t)$, non-modal transient growth, and non-modal transient growth supported by parametric instability. We have designed three numerical experiments of plane turbulent channel flow with additional forcing terms aiming to neutralize one or various lin…
The purpose of this contribution is to summarize and discuss recent advances regarding the onset of turbulence in shear flows. The absence of a clear cut instability mechanism, the spatio-temporal intermittent character and extremely long lived transients are some of the major difficulties encountered in these flows and have hindered progress towards understanding the transition process. We will show for the case of pipe flow that concepts from nonlinear dynamics and statistical physics can help to explain the onset of turbulence. In particular the turbulent structures ('puffs') observed close…
Threshold amplitude of disturbance for transition to turbulence in a pipe Poiseuille flow is investigated. Based on the energy gradient theory, we argued that the transition to turbulence depends on magnitudes of the energy gradient of mean flow and the disturbance energy. Furthermore, the threshold disturbance energy required for turbulence transition is expressed as a function of the complementary angle to the energy angle which also characterizes the behaviour of the energy gradient of mean flow. With some mathematical treatments, the variation of the threshold energy of disturbance versus …
The recent discovery of unstable travelling waves (TWs) in pipe flow has been hailed as a significant breakthrough with the hope that they populate the turbulent attractor. We confirm the existence of coherent states with internal fast and slow streaks commensurate in both structure and energy with known TWs using numerical simulations in a long pipe. These only occur, however, within less energetic regions of (localized) `puff' turbulence at low Reynolds numbers (Re=2000-2400), and not at all in (homogeneous) `slug' turbulence at Re=2800. This strongly suggests that all currently known TWs si…
Lower-branch traveling waves and equilibria computed in pipe flow and other shear flows appear intermediate between turbulent and laminar motions. We take a step towards connecting these lower-branch solutions to transition by deriving a numerical method for finding certain special disturbances of the laminar flow in a short pipe. These special disturbances cause the disturbed velocity field to approach the lower-branch solution by evolving along its stable manifold. If the disturbance were slightly smaller, the flow would relaminarize, and if slightly larger, it would transition to a turbulen…
Fluid flows in nature and applications are frequently subject to periodic velocity modulations. Surprisingly, even for the generic case of flow through a straight pipe, there is little consensus regarding the influence of pulsation on the transition threshold to turbulence: while most studies predict a monotonically increasing threshold with pulsation frequency (i.e. Womersley number, $α$), others observe a decreasing threshold for identical parameters and only observe an increasing threshold at low $α$. In the present study we apply recent advances in the understanding of transition in steady…
This is a comment on arXiv:physics/0608292, subsequently published in Phys Rev Lett, 98, 014501 (2007), in which Willis and Kerswell study in direct numerical simulations the statistics of turbulent lifetimes in pipe flow. They suggest that their data for six different Reynolds numbers support a divergence of the lifetime near a Reynolds number of about 1870. However, their conclusion is not compelling: a re-analysis of their data shows that it is also compatible with an exponentially increasing lifetime.
We propose a general strategy for determining the minimal finite amplitude isturbance to trigger transition to turbulence in shear flows. This involves constructing a variational problem that searches over all disturbances of fixed initial amplitude, which respect the boundary conditions, incompressibility and the Navier--Stokes equations, to maximise a chosen functional over an asymptotically long time period. The functional must be selected such that it identifies turbulent velocity fields by taking significantly enhanced values compared to those for laminar fields. We illustrate this approa…
This paper presents a new theory of turbulence in time-independent non-Newtonian fluids. The wall layer is modelled in terms of unsteady exchange of viscous momentum between the wall and the main stream, following the classic visualisation of inrush-sweep-ejection/burst. The thickness of the wall layer is found to be the same for Newtonian and purely viscous non-Newtonian fluids, when normalised with the instantaneous wall parameters at the onset of bursting. The results indicate that the mechanisms of turbulence in Newtonian and time-independent fluids are identical when structural similarity…
Despite the nonlinear nature of turbulence, there is evidence that part of the energy-transfer mechanisms sustaining wall turbulence can be ascribed to linear processes. The different scenarios stem from linear stability theory and comprise exponential instabilities, neutral modes, transient growth from non-normal operators, and parametric instabilities from temporal mean-flow variations, among others. These mechanisms, each potentially capable of leading to the observed turbulence structure, are rooted in theoretical and conceptual arguments. Whether the flow follows any or a combination of t…
Despite the nonlinear nature of wall turbulence, there is evidence that the energy-injection mechanisms sustaining wall turbulence can be ascribed to linear processes. The different scenarios stem from linear stability theory and comprise exponential instabilities from mean-flow inflection points, transient growth from non-normal operators, and parametric instabilities from temporal mean-flow variations, among others. These mechanisms, each potentially capable of leading to the observed turbulence structure, are rooted in simplified theories and conceptual arguments. Whether the flow follows a…
We tackle the question of whether the presence of particles in a pipe flow can influence the linear transient growth of infinitesimal perturbations, in view of better understanding the behaviour of particulate pipe flows in regimes of transition to turbulence. The problem is tackled numerically by means of a simple model where particles are modelled as a second fluid, that interacts with the fluid phase through a two-way Stokes drag only. The transient growth is found to be enhanced by the presence of particles, especially so if the particles are localised at a specific radial location of the …
Turbulence in the flow of fluid through a pipe can be suppressed by buoyancy forces. As the suppression of turbulence leads to severe heat transfer deterioration, this is an important and undesirable phenomenon in both heating and cooling applications. Vertical flow is often considered, as the axial buoyancy force can help drive the flow. With heating measured by the buoyancy parameter $C$, our DNS show that shear-driven turbulence may either be completely laminarised or transitions to a relatively quiescent convection-driven state. Buoyancy forces cause a flattening of the base flow profile, …
We report the results of three-dimensional direct numerical simulations for incompressible viscous fluid in a circular pipe flow with a sudden expansion. At the inlet, a parabolic velocity profile is applied together with a finite amplitude perturbation in the form of a vortex with its axis parallel to the axis of the pipe. At sufficiently high Reynolds numbers the recirculation region breaks into a turbulent patch that changes position axially depending on the strength of the perturbation. This vortex perturbation is believed to produce a less abrupt transition than in previous studies with a…
Modelling the sudden depressurisation of superheated liquids through nozzles is a challenge because the pressure drop causes rapid flash boiling of the liquid. The resulting jet usually demonstrates a wide range of structures, including ligaments and droplets, due to both mechanical and thermodynamic effects. As the simulation comprises increasingly numerous phenomena, the computational cost begins to increase. One way to moderate the additional cost is to use machine learning surrogacy for specific elements of the calculations. The present study presents a machine learning-assisted computatio…
In linearly stable shear flows at moderate Re, turbulence spontaneously decays despite the existence of a codimension-one manifold, termed the edge of chaos, which separates decaying perturbations from those triggering turbulence. We statistically analyse the decay in plane Couette flow, quantify the breaking of self-sustaining feedback loops and demonstrate the existence of a whole continuum of possible decay paths. Drawing parallels with low-dimensional models and monitoring the location of the edge relative to decaying trajectories we provide evidence, that the edge of chaos separates state…
Advanced nuclear reactors often exhibit complex thermal-fluid phenomena during transients. To accurately capture such phenomena, a coarse-mesh three-dimensional (3-D) modeling capability is desired for modern nuclear-system code. In the coarse-mesh 3-D modeling of advanced-reactor transients that involve flow and heat transfer, accurately predicting the turbulent viscosity is a challenging task that requires an accurate and computationally efficient model to capture the unresolved fine-scale turbulence. In this paper, we propose a data-driven coarse-mesh turbulence model based on local flow fe…
The identification of stream in the straight pipe as a flexible rod has allowed to present the criterion expression for determination of transition of the laminar flow regime to the turbulent as a loss of stability of the rectilinear static structure of translational motion of stream in pipe and its transition to the flexural-vortical dynamic structure of translational motion, just as a flexible rod buckling. The introduced criterion allows to take into account an influencing of the inlet geometry, the pipe length, the flow velocity, and also of any physical factors on stability of the rectili…
The laminar-turbulent boundary S is the set separating initial conditions which relaminarise uneventfully from those which become turbulent. Phase space trajectories on this hypersurface in cylindrical pipe flow look to be chaotic and show recurring evidence of coherent structures. A general numerical technique is developed for recognising approaches to these structures and then for identifying the exact coherent solutions themselves. Numerical evidence is presented which suggests that trajectories on S are organised around only a few travelling waves and their heteroclinic connections. If the…
We use laser-Doppler velocimetry (LDV) experiments and Reynolds-averaged Navier--Stokes (RANS) simulations to study the characteristic flow patterns downstream of a standardized clockwise swirl disturbance generator. After quantifying the impact of the mesh size, we evaluate the potential of various eddy-viscosity turbulence models in providing reasonable approximations with respect to the experimental reference. The choice of turbulent models reflects current industry practice. Our results suggest that models from the $k$-$ε$ family are more accurate in predicting swirling flows than models f…
Transitional pipe flow is modeled as a one-dimensional excitable and bistable medium. Models are presented in two variables, turbulence intensity and mean shear, that evolve according to established properties of transitional turbulence. A continuous model captures the essence of the puff-slug transition as a change from excitability to bistability. A discrete model, that additionally incorporates turbulence locally as a chaotic repeller, reproduces almost all large-scale features of transitional pipe flow. In particular it captures metastable localized puffs, puff splitting, slugs, a continuo…
In this Letter we show that a bifurcation cascade and fully sustained turbulence can share the phase space of a fluid flow system, resulting in the presence of competing stable attractors. We analyse the toroidal pipe flow, which undergoes subcritical transition to turbulence at low pipe curvatures and supercritical transition at high curvatures, as was previously documented. We provide decisive evidence that the nature of the supercritical transition is of Ruelle--Takens type and that, in a narrow range of intermediate curvatures, its dynamics competes with that of sustained turbulence emergi…
Influence of statistically stationary, homogeneous, and isotropic turbulence on the mean area of a passive self-propagating front and, hence, on the rate of fluid consumption by the front is analysed in the case of asymptotically high turbulent Reynolds number $Re_L$ and asymptotically high ratio of the Kolmogorov velocity to a constant speed $u_0$ of the front. By considering an early stage of the front evolution, the mean (over the studied early stage) front area and consumption velocity $\bar{u}_T$ are analytically determined. The analysis shows that the mean $\bar{u}_T$ is proportional to …
A vorticity surge event that could be a paradigm for a wide class of bursting events in turbulence is studied to examine how the energy cascade is established and how this event could serve as a new test of LES turbulence models. This vorticity surge event is tied to the formation of the energy cascade in a direct numerical simulation by the traditional signatures of a turbulent energy cascade such as spectra approaching -5/3 and strongly Beltramized vortex tubes. A coherent mechanism is suggested by the nearly simultaneous development of a maximum of the peak vorticity $\|ω\|_\infty$, growth …
In this paper it is shown that the location of the characteristic overshoot of the Richardson's annular effect changes with the Kinematic Reynolds number $ω^*=ωr_0^2 /ν$ in the range of frequencies within the laminar regime. From the study of the Richardson's overshoot at different times it was identified the existence of aparent transverse damped waves similar to those ones observed in the famous Stokes second problem, the physical analysis of this waves was used for the establishment of a semi-empirical law that gives the functional relation of the mean overshoot maximum with the kinematic r…
A turbulent mean profile for pipe flow is prescribed which closely matches experimental observations. The nature of perturbations superimposed upon this profile is then considered. Optimal growth calculations predict two distinct classes of structures, clearly associated with near-wall and large-scale structures. Quantitative correspondence of the spanwise wavelength of wall-structures with experimental observations is very good. The response to harmonic forcing is also considered, and the linear growth tested with direct numerical simulation of forced turbulence. Despite the very simple eddy …
We analyze a possible extension of Gioia and Chakraborty's momentum transfer model of friction in steady turbulent pipe flows (Phys. Rev. Lett. 96, 044502 (2006)) to the case of time and/or space dependent turbulent flows. The end result is an expression for the stress at the wall as the sum of an steady and a dynamic component. The steady part is obtained by using the instantaneous velocity in the expression for the stress at the wall of a stationary flow. The unsteady part is a weighted average over the history of the flow acceleration, with a weighting function similar to that proposed by V…
Turbulence locally created in laminar pipe flows shows sudden decay or splitting after a stochastic waiting time. In laboratory experiments, the mean waiting time was observed to increase double-exponentially as the Reynolds number (Re) approaches its critical value. To understand the origin of this double-exponential increase, we perform many pipe flow direct numerical simulations of the Navier-Stokes equations, and measure the cumulative histogram of the maximum axial vorticity field over the pipe (turbulence intensity). In the domain where the turbulence intensity is not small, we observe t…
A dynamical model is proposed for isotropic turbulence driven by steady forcing that yields a viscosity independent dynamics for the small-scale (inertial) regime. This reproduces the Kolmogorov spectrum for the two-point velocity correlation function in the fully developed (stationary) stage, while predicting intermittency corrections for higher order moments. The model also yields a transient stage with a power-law time evolution. The crossover time to fully developed turbulence scales with the turbulent system size as $\sim L^{11/3}$. The physical origin of the transient behavior is explain…
Polymer additives can substantially reduce the drag of turbulent flows and the upper limit, the so called "maximum drag reduction" (MDR) asymptote is universal, i.e. independent of the type of polymer and solvent used. Until recently, the consensus was that, in this limit, flows are in a marginal state where only a minimal level of turbulence activity persists. Observations in direct numerical simulations using minimal sized channels appeared to support this view and reported long "hibernation" periods where turbulence is marginalized. In simulations of pipe flow we find that, indeed, with inc…
Reynolds proposed that after sufficiently long times, the flow in a pipe should settle to a steady condition: below a critical Reynolds number, flows should (regardless of initial conditions) always return to laminar, while above, eddying motion should persist. As shown, even in pipes several thousand diameters long, the spatio-temporal intermittent flow patterns observed at the end of the pipe strongly depend on the initial conditions, with no indication of an approach to a (statistical) steady state. Exploiting the fact that turbulent puffs do not age, we continuously recreate the puff seque…
The experiments by Darbyshire and Mullin (J. Fluid Mech. 289, 83 (1995)) on the transition to turbulence in pipe flow show that there is no sharp border between initial conditions that trigger turbulence and those that do not. We here relate this behaviour to the possibility that the transition to turbulence is connected with the formation of a chaotic saddle in the phase space of the system. We quantify a sensitive dependence on initial conditions and find in a statistical analysis that in the transition region the distribution of turbulent lifetimes follows an exponential law. The characteri…
