17 results
Deep reinforcement transfer learning of active control for bluff body flows at high Reynolds number
- Zhicheng Wang, Dixia Fan, Xiaomo Jiang, Michael S. Triantafyllou, George Em Karniadakis
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- Journal:
- Journal of Fluid Mechanics / Volume 973 / 25 October 2023
- Published online by Cambridge University Press:
- 20 October 2023, A32
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We demonstrate how to accelerate the computationally taxing process of deep reinforcement learning (DRL) in numerical simulations for active control of bluff body flows at high Reynolds number ($Re$) using transfer learning. We consider the canonical flow past a circular cylinder whose wake is controlled by two small rotating cylinders. We first pre-train the DRL agent using data from inexpensive simulations at low $Re$, and subsequently we train the agent with small data from the simulation at high $Re$ (up to $Re=1.4\times 10^5$). We apply transfer learning (TL) to three different tasks, the results of which show that TL can greatly reduce the training episodes, while the control method selected by TL is more stable compared with training DRL from scratch. We analyse for the first time the wake flow at $Re=1.4\times 10^5$ in detail and discover that the hydrodynamic forces on the two rotating control cylinders are not symmetric.
Mapping the properties of the vortex-induced vibrations of flexible cylinders in uniform oncoming flow
- Dixia Fan, Zhicheng Wang, Michael S. Triantafyllou, George Em Karniadakis
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- Journal of Fluid Mechanics / Volume 881 / 25 December 2019
- Published online by Cambridge University Press:
- 25 October 2019, pp. 815-858
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Flexible structures placed within an oncoming flow exhibit far more complex vortex-induced dynamics than flexibly mounted rigid cylinders, because they involve the distributed interaction between the structural and wake dynamics along the entire span. Hence, mapping the well-understood properties of rigid cylinder vibrations to those of strings and beams has been elusive. We show here with a combination of experiments, conducted at Reynolds number, $Re$ from 250 to 2300, and computational fluid dynamics that such a mapping is possible for flexible structures in uniform flow undergoing combined cross-flow and in-line oscillations, but only when additional concepts are introduced to model the extended coupling of the flow and the structure. The in-line response consists of largely standing waves that define cells, each cell spanning the distance between adjacent nodes, over which stable vortical patterns form, whose features (‘2S’ versus ‘P$+$S’) depend strongly on the true reduced velocity, $V_{r}=U/f_{y}d$, where $U$ is the inflow velocity, $f_{y}$ is the cross-flow vibration frequency and $d$ is the cylinder diameter, and the phase angle between in-line and cross-flow response; while the cross-flow response may contain travelling waves, breaking the symmetry of the problem. The axial distribution of the highly variable effective added masses in the cross-flow and in-line directions, and the local phase angle between in-line and cross-flow motion determine the single frequency of cross-flow response, while the in-line response vibrates at twice the cross-flow frequency. The cross-flow and in-line lift coefficients in phase with velocity depend strongly on the true reduced velocity but also on the local phase angle between in-line and cross-flow motions. Modal shapes can be defined for in-line and cross-flow, based on the resemblance of the response to conventional modes, which can be in the ratio of either ‘$2n/n$’ or ‘$(2n-1)/n$’, where $n$ is the order of the cross-flow response mode. We use an underwater optical tracking system to reconstruct the sectional fluid forces in a flexible structure and show that, once the cross-flow and in-line motion features are known, employing strip theory and the hydrodynamic coefficients obtained from forced rigid cylinder experiments allows us to predict the distributed forces accurately.
Deep learning of vortex-induced vibrations
- Maziar Raissi, Zhicheng Wang, Michael S. Triantafyllou, George Em Karniadakis
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- Journal:
- Journal of Fluid Mechanics / Volume 861 / 25 February 2019
- Published online by Cambridge University Press:
- 19 December 2018, pp. 119-137
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Vortex-induced vibrations of bluff bodies occur when the vortex shedding frequency is close to the natural frequency of the structure. Of interest is the prediction of the lift and drag forces on the structure given some limited and scattered information on the velocity field. This is an inverse problem that is not straightforward to solve using standard computational fluid dynamics methods, especially since no information is provided for the pressure. An even greater challenge is to infer the lift and drag forces given some dye or smoke visualizations of the flow field. Here we employ deep neural networks that are extended to encode the incompressible Navier–Stokes equations coupled with the structure’s dynamic motion equation. In the first case, given scattered data in space–time on the velocity field and the structure’s motion, we use four coupled deep neural networks to infer very accurately the structural parameters, the entire time-dependent pressure field (with no prior training data), and reconstruct the velocity vector field and the structure’s dynamic motion. In the second case, given scattered data in space–time on a concentration field only, we use five coupled deep neural networks to infer very accurately the vector velocity field and all other quantities of interest as before. This new paradigm of inference in fluid mechanics for coupled multi-physics problems enables velocity and pressure quantification from flow snapshots in small subdomains and can be exploited for flow control applications and also for system identification.
An entropy-viscosity large eddy simulation study of turbulent flow in a flexible pipe
- Zhicheng Wang, Michael S. Triantafyllou, Yiannis Constantinides, George Em Karniadakis
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- Journal:
- Journal of Fluid Mechanics / Volume 859 / 25 January 2019
- Published online by Cambridge University Press:
- 23 November 2018, pp. 691-730
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We present a new approach – the entropy-viscosity method (EVM) – for numerical modelling of high Reynolds number flows and investigate its potential by simulating fully developed incompressible turbulent flow, first in a stationary pipe and subsequently in a flexible pipe. This method, which was first proposed by Guermond et al. (J. Comput. Phys., vol. 230 (11), 2011, pp. 4248–4267), introduces the concept of entropy viscosity, computed based on the nonlinear localized residual obtained from the energy equation. Specifically, this nonlinear viscosity based on the local size of entropy production is added to the spectral element discretization employed in our work for stabilization at insufficient resolution. Unlike its original formulation, which includes an ad hoc tuneable parameter $\unicode[STIX]{x1D6FC}$, here, we determine the value of $\unicode[STIX]{x1D6FC}$ by assuming that the entropy viscosity is analogous to the eddy viscosity of the Smagorinsky model. However, the overall approach has the flavour of the implicit large eddy simulation (ILES) instead of the standard large eddy simulation (LES). Given the empiricism of our approach, we have performed systematic studies of homogeneous isotropic turbulence for validation (see appendix A). We have also carried out a more complete numerical simulation study to investigate incompressible turbulent flow in a stationary pipe at $Re_{D}=5300$ and $Re_{D}=44\,000$, following the work of Wu & Moin (J. Fluid Mech., vol. 608, 2008, pp. 81–112) who performed very accurate direct numerical simulations (DNS) of these two cases. We found that the mean flow, turbulence fluctuations, and two-point correlations of the EVM-based LES are in good agreement with the DNS of Wu & Moin despite the fact that we employed grids with resolution two orders of magnitude smaller. If we instead use the standard Smagorinsky model in our simulations, the computations become unstable due to insufficient resolution of the smaller scales. Another important difference is that the entropy-viscosity model scales with the cube of the distance from the wall and approaches zero at the wall, which is theoretically correct, as shown by our a posteriori tests. Based on the validated EVM approach, we then simulated fully developed turbulent flow at $Re_{D}=5300$ in a flexible pipe subject to prescribed vibrations in the cross-flow plane corresponding to a standing wave of amplitude $A$ and wavelength $\unicode[STIX]{x1D706}=3D$, where $D=2R$ is the pipe diameter and $R$ is the radius. We have simulated 11 cases corresponding to increasing values of wave steepness $s_{o}=2A/\unicode[STIX]{x1D706}$, with $s_{o}\in [0,0.067]$. We found a quadratic dependence of the friction factor on $s_{o}$ with the minimum at approximately $s_{o}\approx 0.01$, so, surprisingly, we have a slight decrease in drag at first and then a substantial increase compared to the stationary pipe. To obtain the turbulence statistics, we averaged the simulated flow over twenty time periods at the nodes and anti-nodes separately. We found substantial changes in the mean velocity profile at distances $(1-r)^{+}>5$ while the peaks of turbulent intensities were amplified by 50 % in the axial direction and by 200 % in the normal and azimuthal directions at $s_{o}=0.067$. The peak shear stress at the node increased by more than 200 % whereas at the anti-node it attained negative values. Turbulent budgets revealed large changes close to the wall at $(1-r)^{+}<50$ while flow visualizations showed that many more strong worm-like vortices were generated in the near-wall regions compared to the stationary pipe. We have also computed various spatio-temporal correlations that show that the pressure fluctuations are very sensitive to the pipe vibration and scale quadratically with $s_{o}$. Both pressure and velocity correlations exhibit cellular patterns consistent with the standing-wave pipe motion.
Direct numerical simulations of two-phase flow in an inclined pipe
- Fangfang Xie, Xiaoning Zheng, Michael S. Triantafyllou, Yiannis Constantinides, Yao Zheng, George Em Karniadakis
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- Journal:
- Journal of Fluid Mechanics / Volume 825 / 25 August 2017
- Published online by Cambridge University Press:
- 20 July 2017, pp. 189-207
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We study the instability mechanisms leading to slug flow formation in an inclined pipe subject to gravity forces. We use a phase-field approach, where the Cahn–Hillard model is used to model the interface. At the inlet, a stratified flow is imposed with a specified velocity profile. We validate our numerical results by comparing against previous theoretical models and by predicting the various flow regimes for horizontal and inclined pipes, including stratified flow, slug flow, dispersed bubble flow and annular flow. Subsequently, we focus on slug formation in an inclined pipe and connect its appearance with specific vortical dynamics. A two-dimensional channel geometry is first considered. When the heavy fluid is injected as the top layer, inverted vortex shedding emerges, which periodically impacts on the bottom wall, as it develops further downstream. The accumulation of heavy fluid in the bottom wall causes a back flow that induces rolling waves and interacts with the upstream jet. When the heavy fluid is placed as the bottom layer, the heavy fluid accumulates and initially forms a massive slug at the bottom region, close to the inlet. Subsequently, the heavy fluid slug starts to break into smaller pieces, some of which translate along the pipe. During the accumulation phase, a back flow forms also generating rolling waves. Occasionally, a rolling wave can reach the top of the pipe and form a new slug. To describe the generation of vorticity from the two-phase interface and pipe walls in the slug formation, we study the circulation dynamics and connect it with the resulting two-phase flow patterns. Finally, we conduct three-dimensional (3-D) simulations in a circular pipe and compare the differences between the 3-D flow patterns and its circulation dynamics against the 2-D simulation results.
The flow dynamics of the garden-hose instability
- Fangfang Xie, Xiaoning Zheng, Michael S. Triantafyllou, Yiannis Constantinides, George Em Karniadakis
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- Journal:
- Journal of Fluid Mechanics / Volume 800 / 10 August 2016
- Published online by Cambridge University Press:
- 12 July 2016, pp. 595-612
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We present fully resolved simulations of the flow–structure interaction in a flexible pipe conveying incompressible fluid. It is shown that the Reynolds number plays a significant role in the onset of flutter for a fluid-conveying pipe modelled through the classic garden-hose problem. We investigate the complex interaction between structural and internal flow dynamics and obtain a phase diagram of the transition between states as function of three non-dimensional quantities: the fluid-tension parameter, the dimensionless fluid velocity and the Reynolds number. We find that the flow patterns inside the pipe strongly affect the type of induced motion. For unsteady flow, if there is symmetry along a direction, this leads to in-plane motion whereas breaking of the flow symmetry results in both in-plane and out-of-plane motions. Hence, above a critical Reynolds number, complex flow patterns result for the vibrating pipe as there is continuous generation of new vorticity due to the pipe wall acceleration, which is subsequently shed in the confined space of the interior of the pipe.
U-shaped fairings suppress vortex-induced vibrations for cylinders in cross-flow
- Fangfang Xie, Yue Yu, Yiannis Constantinides, Michael S. Triantafyllou, George Em Karniadakis
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- Journal of Fluid Mechanics / Volume 782 / 10 November 2015
- Published online by Cambridge University Press:
- 09 October 2015, pp. 300-332
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We employ three-dimensional direct and large-eddy numerical simulations of the vibrations and flow past cylinders fitted with free-to-rotate U-shaped fairings placed in a cross-flow at Reynolds number $100\leqslant \mathit{Re}\leqslant 10\,000$. Such fairings are nearly neutrally buoyant devices fitted along the axis of long circular risers to suppress vortex-induced vibrations (VIVs). We consider three different geometric configurations: a homogeneous fairing, and two configurations (denoted A and AB) involving a gap between adjacent segments. For the latter two cases, we investigate the effect of the gap on the hydrodynamic force coefficients and the translational and rotational motions of the system. For all configurations, as the Reynolds number increases beyond 500, both the lift and drag coefficients decrease. Compared to a plain cylinder, a homogeneous fairing system (no gaps) can help reduce the drag force coefficient by 15 % for reduced velocity $U^{\ast }=4.65$, while a type A gap system can reduce the drag force coefficient by almost 50 % for reduced velocity $U^{\ast }=3.5,4.65,6$, and, correspondingly, the vibration response of the combined system, as well as the fairing rotation amplitude, are substantially reduced. For a homogeneous fairing, the cross-flow amplitude is reduced by about 80 %, whereas for fairings with a gap longer than half a cylinder diameter, VIVs are completely eliminated, resulting in additional reduction in the drag coefficient. We have related such VIV suppression or elimination to the features of the wake flow structure. We find that a gap causes the generation of strong streamwise vorticity in the gap region that interferes destructively with the vorticity generated by the fairings, hence disorganizing the formation of coherent spanwise cortical patterns. We provide visualization of the incoherent wake flow that leads to total elimination of the vibration and rotation of the fairing–cylinder system. Finally, we investigate the effect of the friction coefficient between cylinder and fairing. The effect overall is small, even when the friction coefficients of adjacent segments are different. In some cases the equilibrium positions of the fairings are rotated by a small angle on either side of the centreline, in a symmetry-breaking bifurcation, which depends strongly on Reynolds number.
Distributed lock-in drives broadband vortex-induced vibrations of a long flexible cylinder in shear flow
- Rémi Bourguet, George Em Karniadakis, Michael S. Triantafyllou
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- Journal of Fluid Mechanics / Volume 717 / 25 February 2013
- Published online by Cambridge University Press:
- 01 February 2013, pp. 361-375
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A slender flexible body immersed in sheared cross-flow may exhibit vortex-induced vibrations (VIVs) involving a wide range of excited frequencies and structural wavenumbers. The mechanisms of broadband VIVs of a cylindrical tensioned beam of length-to-diameter aspect ratio 200 placed in shear flow, with an exponentially varying profile along the span, are investigated by means of direct numerical simulation. The Reynolds number is equal to 330 based on the maximum velocity, for comparison with previous work on narrowband vibrations in linear shear flow. The flow is found to excite the structure at a number of different locations under a condition of wake–body synchronization, or lock-in. Broadband responses are associated with a distributed occurrence of the lock-in condition along the span, as opposed to the localized lock-in regions limited to the high inflow velocity zone, reported for narrowband vibrations in sheared current. Despite the instantaneously multi-frequency nature of broadband responses, the lock-in phenomenon remains a locally mono-frequency event, since the vortex formation is generally synchronized with a single vibration frequency at a given location. The spanwise distribution of the excitation zones induces travelling structural waves moving in both directions; this contrasts with the narrowband case where the direction of propagation toward decreasing inflow velocity is preferred. A generalization of the mechanism of phase-locking between the in-line and cross-flow responses is proposed for broadband VIVs under the lock-in condition. A spanwise drift of the in-line/cross-flow phase difference is identified for the high-wavenumber vibration components; this drift is related to the strong travelling wave character of the corresponding structural waves.
Vortex-induced vibrations of a long flexible cylinder in shear flow
- REMI BOURGUET, GEORGE E. KARNIADAKIS, MICHAEL S. TRIANTAFYLLOU
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- Journal of Fluid Mechanics / Volume 677 / 25 June 2011
- Published online by Cambridge University Press:
- 27 April 2011, pp. 342-382
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We investigate the in-line and cross-flow vortex-induced vibrations of a long cylindrical tensioned beam, with length to diameter ratio L/D = 200, placed within a linearly sheared oncoming flow, using three-dimensional direct numerical simulation. The study is conducted at three Reynolds numbers, from 110 to 1100 based on maximum velocity, so as to include the transition to turbulence in the wake. The selected tension and bending stiffness lead to high-wavenumber vibrations, similar to those encountered in long ocean structures. The resulting vortex-induced vibrations consist of a mixture of standing and travelling wave patterns in both the in-line and cross-flow directions; the travelling wave component is preferentially oriented from high to low velocity regions. The in-line and cross-flow vibrations have a frequency ratio approximately equal to 2. Lock-in, the phenomenon of self-excited vibrations accompanied by synchronization between the vortex shedding and cross-flow vibration frequencies, occurs in the high-velocity region, extending across 30% or more of the beam length. The occurrence of lock-in disrupts the spanwise regularity of the cellular patterns observed in the wake of stationary cylinders in shear flow. The wake exhibits an oblique vortex shedding pattern, inclined in the direction of the travelling wave component of the cylinder vibrations. Vortex splittings occur between spanwise cells of constant vortex shedding frequency. The flow excites the cylinder under the lock-in condition with a preferential in-line versus cross-flow motion phase difference corresponding to counter-clockwise, figure-eight orbits; but it damps cylinder vibrations in the non-lock-in region. Both mono-frequency and multi-frequency responses may be excited. In the case of multi-frequency response and within the lock-in region, the wake can lock in to different frequencies at various spanwise locations; however, lock-in is a locally mono-frequency event, and hence the flow supplies energy to the structure mainly at the local lock-in frequency.
Three-dimensionality effects in flow around two tandem cylinders
- GEORGIOS V. PAPAIOANNOU, DICK K. P. YUE, MICHAEL S. TRIANTAFYLLOU, GEORGE E. KARNIADAKIS
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- Journal:
- Journal of Fluid Mechanics / Volume 558 / 10 July 2006
- Published online by Cambridge University Press:
- 04 July 2006, pp. 387-413
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The flow around two stationary cylinders in tandem arrangement at the laminar and early turbulent regime, ($\hbox{\it Re}\,{=}\,10^2$–$10^3$), is studied using two- and three-dimensional direct numerical simulations. A range of spacings between the cylinders from 1.1 to 5.0 diameters is considered with emphasis on identifying the effects of three-dimensionality and cylinder spacing as well as their coupling. To achieve this, we compare the two-dimensional with corresponding three-dimensional results as well as the tandem cylinder system results with those of a single cylinder. The critical spacing for vortex formation and shedding in the gap region depends on the Reynolds number. This dependence is associated with the formation length and base pressure suction variations of a single cylinder with Reynolds number. This association is useful in explaining some of the discrepancies between the two-dimensional and three-dimensional results. A major effect of three-dimensionality is in the exact value of the critical spacing, resulting in deviations from the two-dimensional predictions for the vorticity fields, the forces on the downstream cylinder, and the shedding frequency of the tandem system. Two-dimensional simulations under-predict the critical spacing, leading to erroneous results for the forces and shedding frequencies over a range of spacings where the flow is qualitatively different. To quantify the three-dimensional effects we first employ enstrophy, decomposed into a primary and a secondary component. The primary component involves the vorticity parallel to the cylinder axis, while the secondary component incorporates the streamwise and transverse components of the vorticity vector. Comparison with the single cylinder case reveals that the presence of the downstream cylinder at spacings lower than the critical value has a stabilizing effect on both the primary and secondary enstrophy. Systematic quantification of three-dimensionalities involves finding measures for the intensity of the spanwise fluctuations of the forces. This also verifies the stabilization scenario, suggesting that when the second cylinder is placed at a distance smaller than the critical one, three-dimensional effects are suppressed compared to the single-cylinder case. However, when the spacing exceeds the critical value, the upstream cylinder tends to behave like a single cylinder, but three-dimensionality in the flow generally increases.
Three-dimensional dynamics and transition to turbulence in the wake of bluff objects
- George Em Karniadakis, George S. Triantafyllou
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- Journal of Fluid Mechanics / Volume 238 / May 1992
- Published online by Cambridge University Press:
- 26 April 2006, pp. 1-30
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The wakes of bluff objects and in particular of circular cylinders are known to undergo a ‘fast’ transition, from a laminar two-dimensional state at Reynolds number 200 to a turbulent state at Reynolds number 400. The process has been documented in several experimental investigations, but the underlying physical mechanisms have remained largely unknown so far. In this paper, the transition process is investigated numerically, through direct simulation of the Navier—Stokes equations at representative Reynolds numbers, up to 500. A high-order time-accurate, mixed spectral/spectral element technique is used. It is shown that the wake first becomes three-dimensional, as a result of a secondary instability of the two-dimensional vortex street. This secondary instability appears at a Reynolds number close to 200. For slightly supercritical Reynolds numbers, a harmonic state develops, in which the flow oscillates at its fundamental frequency (Strouhal number) around a spanwise modulated time-average flow. In the near wake the modulation wavelength of the time-average flow is half of the spanwise wavelength of the perturbation flow, consistently with linear instability theory. The vortex filaments have a spanwise wavy shape in the near wake, and form rib-like structures further downstream. At higher Reynolds numbers the three-dimensional flow oscillation undergoes a period-doubling bifurcation, in which the flow alternates between two different states. Phase-space analysis of the flow shows that the basic limit cycle has branched into two connected limit cycles. In physical space the period doubling appears as the shedding of two distinct types of vortex filaments.
Further increases of the Reynolds number result in a cascade of period-doubling bifurcations, which create a chaotic state in the flow at a Reynolds number of about 500. The flow is characterized by broadband power spectra, and the appearance of intermittent phenomena. It is concluded that the wake undergoes transition to turbulence following the period-doubling route.
Frequency selection and asymptotic states in laminar wakes
- George Em Karniadakis, George S. Triantafyllou
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- Journal of Fluid Mechanics / Volume 199 / February 1989
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- 26 April 2006, pp. 441-469
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A better understanding of the transition process in open flows can be obtained through identification of the possible asymptotic response states in the flow. In the present work, the asymptotic states in laminar wakes behind circular cylinders at low supercritical Reynolds numbers are investigated. Direct numerical simulation of the flow is performed, using spectral-element techniques. Naturally produced wakes, and periodically forced wakes are considered separately.
It is shown that, in the absence of external forcing, a periodic state is obtained, the frequency of which is selected by the absolute instability of the time-average flow. The non-dimensional frequency of the vortex street (Strouhal number) is a continuous function of the Reynolds number. In periodically forced wakes, however, non-periodic states are also possible, resulting from the bifurcation of the natural periodic state. The response of forced wakes can be characterized as: (i) lock-in, if the dominant frequency in the wake equals the excitation frequency, or (ii) non-lock-in, when the dominant frequency in the wake equals the Strouhal frequency. Both types of response can be periodic or quasi-periodic, depending on the combination of the amplitude and frequency of the forcing. At the boundary separating the two types of response transitional states develop, which are found to exhibit a low-order chaotic behaviour. Finally, all states resulting from the bifurcation of the natural state can be represented in a two-parameter space inside ‘resonant horn’ type of regions.
Nonlinear interaction of shear flow with a free surface
- Athanassios A. Dimas, George S. Triantafyllou
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- Journal of Fluid Mechanics / Volume 260 / 10 February 1994
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- 26 April 2006, pp. 211-246
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In this paper the nonlinear evolution of two-dimensional shear-flow instabilities near the ocean surface is studied. The approach is numerical, through direct simulation of the incompressible Euler equations subject to the dynamic and kinematic boundary conditions at the free surface. The problem is formulated using boundary-fitted coordinates, and for the numerical simulation a spectral spatial discretization method is used involving Fourier modes in the streamwise direction and Chebyshev polynomials along the depth. An explicit integration is performed in time using a splitting scheme. The initial state of the flow is assumed to be a known parallel shear flow with a flat free surface. A perturbation having the form of the fastest growing linear instability mode of the shear flow is then introduced, and its subsequent evolution is followed numerically. According to linear theory, a shear flow with a free surface has two linear instability modes, corresponding to different branches of the dispersion relation: Branch I, at low wavenumbers; and Branch II, at high wavenumbers for low Froude numbers, and low wavenumbers for high Froude numbers. Our simulations show that the two branches have a distinctly different nonlinear evolution.
Branch I: At low Froude numbers, Branch I instability waves develop strong oval-shaped vortices immediately below the ocean surface. The induced velocity field presents a very sharp shear near the crest of the free-surface elevation in the horizontal direction. As a result, the free-surface wave acquires steep slopes, while its amplitude remains very small, and eventually the computer code crashes suggesting that the wave will break.
Branch II: At low Froude numbers, Branch II instability waves develop weak vortices with dimensions considerably smaller than their distance from the ocean surface. The induced velocity field at the ocean surface varies smoothly in space, and the free-surface elevation takes the form of a propagating wave. At high Froude numbers, however, the growing rates of the Branch II instability waves increase, resulting in the formation of strong vortices. The free surface reaches a large amplitude, and strong vertical velocity shear develops at the free surface. The computer code eventually crashes suggesting that the wave will break. This behaviour of the ocean surface persists even in the infinite-Froude-number limit.
It is concluded that the free-surface manifestation of shear-flow instabilities acquires the form of a propagating water wave only if the induced velocity field at the ocean surface varies smoothly along the direction of propagation.
On the formation of vortex streets behind stationary cylinders
- George S. Triantafyllou, Michael S. Triantafyllou, C. Chryssostomidis
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- Journal of Fluid Mechanics / Volume 170 / September 1986
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- 21 April 2006, pp. 461-477
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The formation of vortex streets behind stationary cylinders is found to be caused by an absolute instability in the wake immediately behind the cylinder. The inviscid Orr–Sommerfeld equation is used together with measured profiles at Reynolds numbers of (a) Re = 56 when the absolute instability provides a Strouhal number of 0.13; and (b) Re = 140000 providing a Strouhal number of 0.21, both in agreement with experimental values. At the subcritical Re = 34 the instability is of the convective type; i.e. the disturbance decays, being convected away once the external disturbance is removed, in agreement with experimental observations. Finally, the instability of the mode which causes a symmetric array of vortices is shown to be always of the convective type.
Effect of surfactants on free-surface turbulent flows
- LIAN SHEN, DICK K. P. YUE, GEORGE S. TRIANTAFYLLOU
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- Journal of Fluid Mechanics / Volume 506 / 10 May 2004
- Published online by Cambridge University Press:
- 28 April 2004, pp. 79-115
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In two earlier papers, we studied the statistical and mechanistic structure of the turbulent boundary layer under a stress-free (clean) free surface. Findings there, such as the presence of inner and outer surface layers, are very much the direct result of the absence of shear stresses at the surface. The latter condition is easily lost when the surface is contaminated and surface elasticity varies with space and time. In this paper we consider the effect of surfactant on features of the free-surface turbulent flow. We perform direct numerical simulations of the Navier–Stokes equations subject to surfactant-laden free-surface boundary conditions for varying Reynolds and Marangoni numbers and low Froude numbers. As expected, the Marangoni effect decreases the horizontal turbulence intensity and normal vorticity at the surface. The direct effect on the turbulent kinetic energy is an increase in the dissipation and viscous diffusion and a decrease in the production near the surface relative to the clean case. The most prominent effect of the presence (of even a small amount) of surfactant is the drastic reduction in the surface divergence and the associated sharp decrease of up- and downwelling at the surface which has direct implications to near-surface turbulent transport. The observed surfactant effects on turbulent kinetic energy budget can be attributed to the generation of Marangoni vorticity at the free surface by approaching hairpin vortices. The Marangoni effect has also a direct effect on the boundary-layer structure, causing an increase of the thickness of the boundary layer and in the maxima of the mean shear near the surface. For moderate values of the Marangoni number, up-/downwelling effectively vanishes and the flow approaches a state independent of the Marangoni number. Guided by these results and to obtain theoretical insight, we develop a similarity solution for the mean flow. The analytic solution agrees well with the numerical data and provides precise measures for the multi-layer structure of the boundary layer. Based on the theoretical model, we derive scaling laws for the thickness of the inner and the outer boundary layers, which are also confirmed by numerical simulations.
Turbulent diffusion near a free surface
- LIAN SHEN, GEORGE S. TRIANTAFYLLOU, DICK K. P. YUE
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- Journal of Fluid Mechanics / Volume 407 / 25 March 2000
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- 25 March 2000, pp. 145-166
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We study numerically and analytically the turbulent diffusion characteristics in a low-Froude-number turbulent shear flow beneath a free surface. In the numerical study, the Navier–Stokes equations are solved directly subject to viscous boundary conditions at the free surface. From an ensemble of such simulations, we find that a boundary layer develops at the free surface characterized by a fast reduction in the value of the eddy viscosity. As the free surface is approached, the magnitude of the mean shear initially increases over the boundary (outer) layer, reaches a maximum and then drops to zero inside a much thinner inner layer. To understand and model this behaviour, we derive an analytical similarity solution for the mean flow. This solution predicts well the shape and the time-scaling behaviour of the mean flow obtained in the direct simulations. The theoretical solution is then used to derive scaling relations for the thickness of the inner and outer layers. Based on this similarity solution, we propose a free-surface function model for large-eddy simulations of free-surface turbulence. This new model correctly accounts for the variations of the Smagorinsky coefficient over the free-surface boundary layer and is validated in both a priori and a posteriori tests.
The surface layer for free-surface turbulent flows
- LIAN SHEN, XIANG ZHANG, DICK K. P. YUE, GEORGE S. TRIANTAFYLLOU
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- Journal:
- Journal of Fluid Mechanics / Volume 386 / 10 May 1999
- Published online by Cambridge University Press:
- 10 May 1999, pp. 167-212
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Direct numerical simulation (DNS) is used to examine low Froude number free-surface turbulence (FST) over a two-dimensional mean shear flow. The Navier–Stokes equations are solved using a finite-difference scheme with a grid resolution of 1283. Twenty separate simulations are conducted to calculate the statistics of the flow. Based on the velocity deficit and the vertical extent of the shear of the mean flow, the Reynolds number is 1000 and the Froude number is 0.7. We identify conceptually and numerically the surface layer, which is a thin region adjacent to the free surface characterized by fast variations of the horizontal vorticity components. This surface layer is caused by the dynamic zero-stress boundary conditions at the free surface and lies inside a thicker blockage (or ‘source’) layer, which is due to the kinematic boundary condition at the free surface. The importance of the outer blockage layer is manifested mainly in the redistribution of the turbulence intensity, i.e. in the increase of the horizontal velocity fluctuations at the expense of the vertical velocity fluctuation. A prominent feature of FST is vortex connections to the free surface which occur inside the surface layer. It is found that as hairpin-shaped vortex structures approach the free surface, their ‘head’ part is dissipated quickly in the surface layer, while the two ‘legs’ connect almost perpendicularly to the free surface. Analysis of the evolution of surface-normal vorticity based on vortex surface-inclination angle shows that both dissipation and stretching decrease dramatically after connection. As a result, vortex structures connected to the free surface are persistent and decay slowly relative to non-connected vorticities. The effects of surface and blockage layers on the turbulence statistics of length scales, Reynolds-stress balance, and enstrophy dynamics are examined, which elucidate clearly the different turbulence mechanisms operating in the respective near-surface scales. Finally we investigate the effect of non-zero Froude number on the turbulence statistics. We show that the most significant effect of the presence of the free surface is a considerable reduction of the pressure–strain correlation at this surface, compared to that at a free-slip at plate. This reduction is finite even for very low values of the Froude number.