Degeneracy of turbulent states in two-dimensional channel flow

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Searching turbulence for periodic orbits with dynamic mode decomposition

We present a new method for generating robust guesses for unstable periodic orbits (UPOs) by post-processing turbulent data using dynamic mode decomposition (DMD). The approach relies on the identification of near-neutral, repeated harmonics in the DMD eigenvalue spectrum from which both an estimate for the period of a nearby UPO and a guess for the velocity field can be constructed. In this way, the signature of a UPO can be identified in a short time series without the need for a near recurrence to occur, which is a considerable drawback to recurrent flow analysis, the current state-of-the-art. We first demonstrate the method by applying it to a known (simple) UPO and find that the period can be reliably extracted even for time windows of length one quarter of the full period. We then turn to a long turbulent trajectory, sliding an observation window through the time series and performing many DMD computations. Our approach yields many more converged periodic orbits (including multiple new solutions) than a standard recurrent flow analysis of the same data. Furthermore, it also yields converged UPOs at points where the recurrent flow analysis flagged a near recurrence but the Newton solver did not converge, suggesting that the new approach can be used alongside the old to generate improved initial guesses. Finally, we discuss some heuristics on what constitutes a "good" time window for the DMD to identify a UPO

Can libration maintain Enceladus's ocean?

The process by which the subsurface ocean on Enceladus is heated remains a puzzle. Tidal interaction with Saturn and Dione is the leading candidate but whether the dominant heating occurs in the solid core, ice crust or in the ocean itself is an outstanding question. Here we consider the driving e ect of the longitudinal libration of the ice crust on the subsurface ocean and argue that the ow response should be turbulent even in the most benign situation of a smooth spherical ice shell when the motion of the boundary is only transmitted viscously. A rigorous upper bound on the turbulent viscous dissipation rate is then derived and used to argue that libration should be potent enough to explain the observed heat ux emanating out of Enceladus when the e ects of tidal distortion and roughness of the ice crust are included

Koopman analysis of Burgers equation

The emergence of Dynamic Mode Decomposition (DMD) as a practical way to attempt a Koopman mode decomposition of a nonlinear PDE presents exciting prospects for identifying invariant sets and slowly decaying transient structures buried in the PDE dynamics. However, there are many subtleties in connecting DMD to Koopman analysis and it remains unclear how realistic Koopman analysis is for complex systems such as the Navier-Stokes equations. With this as motivation, we present here a full Koopman decomposition for the velocity field in Burgers equation by deriving explicit expressions for the Koopman modes and eigenfunctions - the first time this has been done for a nonlinear PDE. The decomposition highlights the fact that different observables can require different subsets of Koopman eigenfunctions to express them and presents a nice example where: (i) the Koopman modes are linearly dependent and so cannot be fit a posteriori to snapshots of the flow without knowledge of the Koopman eigenfunctions; and (ii) the Koopman eigenvalues are highly degenerate which means that computed Koopman modes become initial-condition dependent. As way of illustration, we discuss the form of the Koopman expansion with various initial conditions and assess the capability of DMD to extract the decaying nonlinear coherent structures in run-down simulations.EPSR

Stabilisation and drag reduction of pipe flows by flattening the base profile

Recent experimental observations (Kuehnen et al., 2018) have shown that flattening a turbulent streamwise velocity profile in pipe flow destabilises the turbulence so that the flow relaminarises. We show that a similar phenomenon exists for laminar pipe flow profiles in the sense that the nonlinear stability of the laminar state is enhanced as the profile becomes more flattened. Significant drag reduction is also observed for the turbulent flow when triggered by sufficiently large disturbances. The flattening is produced by an artificial body force designed to mimick a baffle used in the experiments of Kuehnen et al. (2018) and the nonlinear stability measured by the size of the energy of the initial perturbations needed to trigger transition. In order to make the latter computation more efficient, we examine how indicative the minimal seed for transition is in measuring transition thresholds. We first show that the minimal seed is relatively robust to base profile changes and spectral filtering. We then compare the (unforced) transition behaviour of the minimal seed with several forms of randomised initial conditions in the range of Reynolds numbers Re=2400 to 10000 and find that the energy of the minimal seed after the Orr and oblique phases of its evolution is close to that of a localised random disturbance. In this sense, the minimal seed at the end of the oblique phase can be regarded as a good proxy for typical disturbances (here taken to be the localised random ones) and is thus used as initial condition in the simulations with the body force. The enhanced nonlinear stability and drag reduction predicted in the present study are an encouraging first step in modelling the experiments of Kuehnen et al. and should motivate future developments to fully exploit the benefits of this promising direction for flow control

Designing a more nonlinearly stable laminar flow via boundary manipulation

AbstractWe show how a fully nonlinear variational method can be used to design a more nonlinearly stable laminar shear flow by quantifying the effect of manipulating the boundary conditions of the flow. Using the example of plane Couette flow, we demonstrate that by forcing the boundaries to undergo spanwise oscillations in a certain way, it is possible to increase the critical disturbance energy for the onset of turbulence by 41 %. If this is sufficient to ensure laminar flow (i.e. ambient noise does not exceed this increased threshold), nearly four times less energy is consumed than in the turbulent flow which exists in the absence of imposed spanwise oscillations.This work is supported by the Engineering and Physical Sciences Research Council [grant number EP/H050310/1].This is the author accepted manuscript. The final version is available from Cambridge University Press via https://doi.org/10.1017/jfm.2013.60

Weakly nonlinear analysis of the viscoelastic instability in channel flow for finite and vanishing Reynolds numbers

The recently-discovered centre-mode instability of rectilinear viscoelastic shear flow (Garg et al. Phy. Rev. Lett. 121, 024502, 2018) has offered an explanation for the origin of elasto-inertial turbulence (EIT) which occurs at lower Weissenberg ($Wi$) numbers. In support of this, we show using weakly nonlinear analysis that the subcriticality found in Page et al. (Phys. Rev. Lett. 125, 154501, 2020) is generic across the neutral curve with the instability only becoming supercritical at low Reynolds ($Re$) numbers and high $Wi$. We demonstrate that the instability can be viewed as purely elastic in origin even for $Re=O(10^3)$, rather than `elasto-inertial', as the underlying shear does not energise the instability. It is also found that the introduction of a realistic maximum polymer extension length, $L_{max}$, in the FENE-P model moves the neutral curve closer to the inertialess $Re=0$ limit at a fixed ratio of solvent-to-solution viscosities, $\beta$. In the dilute limit ($\beta \rightarrow 1$) with $L_{max} =O(100)$, the linear instability can brought down to more physically-relevant $Wi\gtrsim 110$ at $\beta=0.98$, compared with the threshold $Wi=O(10^3)$ at $\beta=0.994$ reported recently by Khalid et al. (arXiv: 2103.06794) for an Oldroyd-B fluid. Again the instability is subcritical implying that inertialess rectilinear viscoelastic shear flow is nonlinearly unstable - i.e. unstable to finite amplitude disturbances - for even lower $Wi$

Kelvin-Helmholtz billows above Richardson number 1/4

We study the dynamical system of a forced stratified mixing layer at finite Reynolds number $Re$, and Prandtl number $Pr=1$. We consider a hyperbolic tangent background velocity profile in the two cases of hyperbolic tangent and uniform background buoyancy stratifications. The system is forced in such a way that these background profiles are a steady solution of the governing equations. As is well-known, if the minimum gradient Richardson number of the flow, $Ri_m$, is less than a certain critical value $Ri_c$, the flow is linearly unstable to Kelvin-Helmholtz instability in both cases. Using Newton-Krylov iteration, we find steady, two-dimensional, finite amplitude elliptical vortex structures, i.e. `Kelvin-Helmholtz billows', existing above $Ri_c$. Bifurcation diagrams are produced using branch continuation, and we explore how these diagrams change with varying $Re$. In particular, when $Re$ is sufficiently high we find that finite amplitude Kelvin-Helmholtz billows exist at $Ri_m>1/4$, where the flow is linearly stable by the Miles-Howard theorem. For the uniform background stratification, we give a simple explanation of the dynamical system, showing the dynamics can be understood on a two-dimensional manifold embedded in state space, and demonstrate the cases in which the system is bistable. In the case of a hyperbolic tangent stratification, we also describe a new, slow-growing, linear instability of the background profiles at finite $Re$, which complicates the dynamics

Layer formation and relaminarisation in plane Couette flow with spanwise stratification

Recent research has shed light on the role of coherent structures in forming layers when stably stratified turbulence is forced with horizontal shear (Lucas, Caulfield & Kerswell, J. Fluid Mech., vol. 832, 2017, pp. 409-437). Here we extend our previous work to investigate the effect of rigid boundaries on the dynamics by studying stably-stratified plane Couette flow with gravity oriented in the spanwise direction. We observe near-wall layering and associated new mean flows in the form of large scale spanwise-flattened streamwise rolls. The layers exhibit the expected buoyancy scaling $l_z\sim U/N$ where $U$ is a typical horizontal velocity scale and $N$ the buoyancy frequency. We associate the new coherent structures with a stratified modification of the well-known large scale secondary flow in plane Couette and find that the possibility of the transition to sustained turbulence is controlled by the relative size of this buoyancy scale to the spanwise spacing of the streaks. We also investigate the influence on the transition to turbulence of the newly discovered linear instability in this system (Facchini et. al. 2018 arXiv:1711.11312).EPSR

The effects of Prandtl number on the nonlinear dynamics of Kelvin-Helmholtz instability in two dimensions

Abstract EPSRC DT