arXiv subjects

Jacob Page

Publications and source records attributed to Jacob Page.

25 records · Page 2Linked to original sources

arXiv2021

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, $β$. In the dilute limit ($β\rightarrow 1$) with $L_{max} =O(100)$, the linear instability can brought down to more physically-relevant $Wi\gtrsim 110$ at $β=0.98$, compared with the threshold $Wi=O(10^3)$ at $β=0.994$ 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 $Wi$.

physics.flu-dyn↗

arXiv2020

Revealing the state space of turbulence using machine learning

Despite the apparent complexity of turbulent flow, identifying a simpler description of the underlying dynamical system remains a fundamental challenge. Capturing how the turbulent flow meanders amongst unstable states (simple invariant solutions) in phase space, as envisaged by Hopf in 1948, using some efficient representation offers the best hope of doing this, despite the inherent difficulty in identifying these states. Here, we make a significant step towards this goal by demonstrating that deep convolutional autoencoders can identify low-dimensional representations of two-dimensional turbulence which are closely associated with the simple invariant solutions characterizing the turbulent attractor. To establish this, we develop latent Fourier analysis that decomposes the flow embedding into a set of orthogonal latent Fourier modes which decode into physically meaningful patterns resembling simple invariant solutions. The utility of this approach is highlighted by analysing turbulent Kolmogorov flow (flow on a 2D torus forced at large scale) at $Re=40$ where, in between intermittent bursts, the flow resides in the neighbourhood of an unstable state and is very low dimensional. Projections onto individual latent Fourier wavenumbers reveal the simple invariant solutions organising both the quiescent and bursting dynamics in a systematic way inaccessible to previous approaches.

physics.flu-dyn↗

arXiv2020

Exact travelling wave solutions in viscoelastic channel flow

Elasto-inertial turbulence (EIT) is a new, two-dimensional chaotic flow state observed in polymer solutions with possible connections to inertialess elastic turbulence and drag-reduced Newtonian turbulence. In this Letter, we argue that the origins of EIT are fundamentally different from Newtonian turbulence by finding a dynamical connection between EIT and an elasto-inertial linear instability recently found at high Weissenberg numbers (Garg et al. Phys. Rev. Lett. 121, 024502, 2018). This link is established by isolating the first known exact coherent structures in viscoelastic parallel flows - nonlinear elasto-inertial travelling waves (TWs) - borne at the linear instability and tracking them down to substantially lower Weissenberg numbers where EIT exists. These TWs have a distinctive ``arrowhead'' structure in the polymer stretch field and can be clearly recognised, albeit transiently, in EIT, as well as being attractors for EIT dynamics if the Weissenberg number is sufficiently large. Our findings suggest that the dynamical systems picture in which Newtonian turbulence is built around the co-existence of many (unstable) simple invariant solutions populating phase space carries over to EIT, though these solutions rely on elasticity to exist.

physics.flu-dyn↗

arXiv2019

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.

physics.flu-dyn↗

arXiv2019

Koopman analysis of isolated fronts and solitons

A Koopman decomposition of a complex system leads to a representation in which nonlinear dynamics appear to be linear. The existence of a linear framework with which to analyse nonlinear dynamical systems brings new strategies for prediction and control, while the approach is straight-forward to apply to large datasets owing to a connection with dynamic mode decomposition (DMD). However, it can be challenging to connect the output of DMD to a Koopman analysis since there are relatively few analytical results available, while the DMD algorithm itself is known to struggle in situations involving the propagation of a localised structure through the domain. Motivated by these issues, we derive a series of Koopman decompositions for localised, finite-amplitude solutions of classical nonlinear PDEs. We first demonstrate that nonlinear travelling wave solutions to both the Burgers and KdV equations have two Koopman decompositions; one of which converges upstream and another which converges the other downstream of the soliton or front. We then use the inverse scattering transform to derive a full Koopman decomposition for (pure soliton) solutions to the KdV equation, identifying Koopman eigenvalues, eigenfunctions and modes. Our analysis indicates that there are many possible Koopman decompositions when the solution involves the interaction of multiple solitons. The existence of multiple expansions in space and time has a critical impact on the ability of DMD to extract Koopman eigenvalues and modes - which must be performed within a temporally and spatially localised window to correctly identify the separate expansions. In addition, we provide evidence that these features may be generic for isolated nonlinear structures by applying DMD to a moving breather solution of the sine-Gordon equation.

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arXiv2018

Koopman mode expansions between simple invariant solutions

A Koopman decomposition is a powerful method of analysis for fluid flows leading to an apparently linear description of nonlinear dynamics in which the flow is expressed as a superposition of fixed spatial structures with exponential time dependence. Attempting a Koopman decomposition is simple in practice due to a connection with Dynamic Mode Decomposition (DMD). However, there are non-trivial requirements for the Koopman decomposition and DMD to overlap which mean it is often difficult to establish whether the latter is truly approximating the former. Here, we focus on nonlinear systems containing multiple simple invariant solutions where it is unclear how to construct a consistent Koopman decomposition, or how DMD might be applied to locate these solutions. First, we derive a Koopman decomposition for a heteroclinic connection in a Stuart-Landau equation revealing two possible expansions. The expansions are centred about the two fixed points of the equation and extend beyond their linear subspaces before breaking down at a crossover point in state space. Well-designed DMD can extract the two expansions provided that the time window does not contain this crossover point. We then apply DMD to the Navier-Stokes equations near to a heteroclinic connection in low-Reynolds number ($Re=O(100)$) plane Couette flow where there are multiple simple invariant solutions beyond the constant shear basic state. This reveals as many different Koopman decompositions as simple invariant solutions present and again indicates the existence of crossover points between the expansions in state space. Again, DMD can extract these expansions only if it does not include a crossover point.

physics.flu-dyn↗

arXiv2018

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.

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