arXiv subjects

Ashley P. Willis

Publications and source records attributed to Ashley P. Willis.

At least 19 recordsLinked to original sources

arXiv2025

Can a convecting magma ocean offer a solution to the puzzling case of core convection in early earth?

Convective flow in Earth's iron-rich liquid core drives self-sustained dynamo action, generating Earth's magnetic field, which is strongest among all terrestrial planets of the solar system. Rock records show that this magnetic field has been operative in Earth for at least 3.4 billion years (b.y). However, advanced high pressure experiments have revised the value of the thermal conductivity of the outer core, which implies an age for the inner core of less than 1 b.y., when compositional convection begins. This creates a puzzle, with a gap between the observations of an early magnetic field on Earth and the young inner core. Previous work has suggested that the pre-inner core dynamo could have been generated in a magma ocean (MO) at the base of the mantle; however, the fluid dynamics of this scenario have received little attention. Here we numerically model the non-magnetic rotating flow in a MO above a convectively stable core in a configuration representing the pre-inner core days of Earth's evolution. Simulations here explore the importance of several dimensionless parameters on coupled core-MO convection -- the Rayleigh number, the ocean/core thermal diffusivity ratio, thermal expansion coefficient ratio, viscosity ratio, and layer thickness ratio. It is found that the MO can easily drive a flow of comparable magnitude in the core, and an approximately linear relationship is observed between the ratio of root-mean-square velocities in the core and the ocean, $(u_c^{RMS}/u_o^{RMS})$, and $(\Nu_o-1)$, where $\Nu_o$ is the Nusselt number for the MO, for the $\Nu_o$ of order 1 to 10 considered. Radial and azimuthal components of the core flow are of similar magnitude, so that, with comparable toroidal and poloidal components, we speculate that the MO-driven core flow could drive an early dynamo.

astro-ph.EP↗

arXiv2025

Longitudinal vortices in unsteady Taylor-Couette flow: solution to a 60-year-old mystery

Applying a sufficiently rapid start-stop to the outer cylinder of the Couette-Taylor system, structures approximately aligned with the axis were recorded in the classic work of Coles (1965). These short-lived rolls are oriented perpendicular to the classic Taylor-vortex rolls. In this work we report numerical observation of this instability, guided by a more recent experimental observation. The instability is shown to be related to an inflection in the azimuthal velocity profile, a finding consistent with the experimental observations of its emergence during the deceleration phase. Despite the transient nature of start-stop experiments, we show that the instability can be linked to that of the oscillating boundary layer problem of Stokes. There are several reasons why the instability may have remained elusive, both for experimental observation and intrinsic to the idealized system. We look in more detail at dependence on the radius ratio for the Taylor-Couette system and find that, in the case where the size of the rolls scales with the gap width, for radius ratios any lower than that used by Coles, R_i/R_o=0.874, the instability is quickly overrun by axisymmetric rolls of Gortler type.

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arXiv2025

Optimal body force for heat transfer in turbulent vertical heated pipe flow

The vertical heated-pipe is widely used in thermal engineering applications, as buoyancy can help drive a flow, but several flow regimes are possible: shear-driven turbulence, laminarised flow, and convective turbulence. Steady velocity fields that maximise heat transfer have previously been calculated for heated pipe flow, but were calculated independently of buoyancy forces, and hence independently of the flow regime and time-dependent dynamics of the flow. In this work, a variational method is applied to find an optimal body force of limited magnitude that maximises heat transfer for the vertical arrangement, with the velocity field constrained by the full governing equations. In our calculations, mostly at Re=3000, it is found that streamwise-independent rolls remain optimal, as in previous steady optimisations, but that the optimal number of rolls and their radial position are dependent on the flow regime. Surprisingly, while it is generally assumed that turbulence enhances heat transfer, for the strongly forced case, time-dependence typically leads to a reduction. Beyond offering potential improvement through the targeting of the roll configuration for this application, wider implications are that optimisations under the steady flow assumption may overestimate improvements in heat transfer, and that strategies that simply aim to induce turbulence may not necessarily be efficient in enhancing heat transfer either. Including time-dependence and the full governing equations in the optimisation is challenging but offers further enhancement and improved reliability in prediction.

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arXiv2025

Modelling the transition from shear-driven turbulence to convective turbulence in a vertical heated pipe

Heated pipe flow is widely used in thermal engineering applications, but the presence of buoyancy force can cause intermittency, or multiple flow states at the same parameter values. Such changes in the flow lead to substantial changes in its heat transfer properties and thereby significant changes in the axial temperature gradient. We therefore introduce a model that features a time-dependent background axial temperature gradient, and consider two temperature boundary conditions -- fixed temperature difference and fixed boundary heat flux. Direct numerical simulations (DNS) are based on the pseudo-spectral framework, and good agreement is achieved between present numerical results and experimental results. The code extends openpipeflow.org and is available at the website. The effect of the axially periodic domain on flow dynamics and heat transfer is examined, using pipes of length L=5D and L=25D. Provided that the flow is fully turbulent, results show close agreement for the mean flow and temperature profiles, and only slight differences in root-mean-square fluctuations. When the flow shows spatial intermittency, heat transfer tends to be overestimated using a short pipe, as shear turbulence fills the domain. This is particularly important when shear turbulence starts to be suppressed at intermediate buoyancy numbers. Finally, at such intermediate buoyancy numbers, we confirm that the decay of localised shear turbulence in the heated pipe flow follows a memoryless process, similar to that in isothermal flow. While isothermal flow then laminarises, convective turbulence in the heated flow can intermittently trigger bursts of shear-like turbulence.

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arXiv2024

Laminarising turbulent pipe flow by linear and nonlinear optimisation

It has been observed that flattening the mean velocity profile of pipe flow by body force can laminarise turbulence, a promising means to reduce frictional drag substantially. To explore whether there is a more efficient body force to eliminate turbulence, we consider time-independent active body forces with varying spatial dependencies. Results confirm that when using an active force, a flattened forced laminar profile is needed to eliminate turbulence, and that a purely streamwise body force is best for laminarisation. While these results required an expensive nonlinear optimisation, it was also observed that the optimal forced profile exhibits reduced linear transient growth (TG). To determine whether the reduction of linear TG alone is a sufficient target for the laminarisation of turbulence, a linear Lagrange Multiplier technique is used to minimise TG of perturbations to the forced laminar profile. The optimal velocity profiles reveal that TG for each azimuthal wavenumber strongly depends on the radial velocity gradient at some specific radial interval. The optimal velocity profile obtained by minimising TG of perturbations of azimuthal wavenumber m = 1 is shown to be able to eliminate turbulence at Re = 2400, but a more effective reduction of TG, including perturbations of higher m, is needed for laminarisation at higher Reynolds number Re = 3000. The streaks formed with the flattened forced laminar profile reveal the mechanism of laminarisation: it is found that the lift-up mechanism in the more flattened forced laminar profile creates a lower streak (i.e. closer to the wall) that is more stable, so that turbulence is harder to maintain through streak instability. Further numerical experiments by disturbing a periodic orbit validated that the breakdown of turbulence self-sustaining mechanisms is mainly caused by suppression of the formation of streaks.

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arXiv2024

The minimal seed for transition to convective turbulence in heated pipe flow

It is well known that buoyancy suppresses, and can even laminarise turbulence in upward heated pipe flow. Heat transfer seriously deteriorates in this case. Through a new DNS model, we confirm that the deteriorated heat transfer within convective turbulence is related to a lack of near-wall rolls, which leads to a weak mixing between the flow near the wall and centre of pipe. Having surveyed the fundamental properties of the system, we perform a nonlinear nonmodal stability analysis. it is found that, the minimal seed becomes thinner and closer to the wall, with increase of buoyancy number C. Most importantly, we show that the critical initial energy required to trigger shear-driven turbulence keeps increasing, implying that attempts to artificially trigger it may not be an efficient means to improve heat transfer at larger C. The new minimal seed, found at C=6, is localised in streamwise direction and is active in the centre of pipe. To find this branch of optimal, we took advantage of a window of linear stability. While the nonlinear optimal causes transition to convective turbulence directly at this and larger C, transition via the linear instability passes via a travelling wave or periodic orbit solutions. Detailed analysis of the periodic solution reveals three stages: growth of the unstable eigenfunction, the formation of streaks, and the decay of streaks due to suppression of the instability. Flow visualization at C up to 10 also show similar features, suggesting that convective turbulence is sustained by these three typical processes.

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arXiv2021

Suppression of turbulence and travelling waves in a vertical heated pipe

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, which in isothermal pipe flow has recently been linked to complete suppression of turbulence (Kühnen et al. Nat. Phys., 2018), and the flattened laminar base profile has enhanced nonlinear stability (Marensi et al. JFM, 2019). In agreement with these findings, the nonlinear lower-branch travelling-wave solution analysed here, which is believed to mediate transition to turbulence in isothermal pipe flow, is shown to be suppressed by buoyancy. A linear instability of the laminar base flow is responsible for the appearance of the relatively quiescent convection driven state for $C\gtrsim 4$ across the range of Reynolds numbers considered. In the suppression of turbulence, however, i.e. in the transition from turbulence, we find clearer association with the analysis of He et al. (JFM, 2016) than with the above dynamical systems approach, which describes better the transition to turbulence. The laminarisation criterion He et al. propose, based on an apparent Reynolds number of the flow as measured by its driving pressure gradient, is found to capture the critical $C=C_{cr}(Re)$ above which the flow will be laminarised or switch to the convection-driven type. Our analysis suggests that it is the weakened rolls, rather than the streaks, which appear to be critical for laminarisation.

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arXiv2020

Designing a minimal baffle to destabilise turbulence in pipe flows

Motivated by the results of recent experiments (Kühnen et al., Flow Turb. Combust., vol. 100, 2018, pp. 919-943), we consider the problem of designing a baffle (an obstacle to the flow) to relaminarise turbulence in pipe flows. Modelling the baffle as a spatial distribution of linear drag $\mathbf{f}(\mathbf{x},t)=-χ(\mathbf{x})\mathbf{u}_{tot}(\mathbf{x},t)$ within the flow ($\mathbf{u}_{tot}$ is the total velocity field and $χ\ge 0$ a scalar field), two different optimisation problems are considered to design $χ$ at a Reynolds number $Re=3000$. In the first, the smallest baffle defined in terms of a $L_1 $ norm of $χ$ is sought which minimises the viscous dissipation rate of the flow. In the second, a baffle which minimises the total energy consumption of the flow is treated. Both problems indicate that the baffle should be axisymmetric and radially localised near the pipe wall, but struggle to predict the optimal streamwise extent. A manual search finds an optimal baffle one radius long which is then used to study how the amplitude for relaminarisation varies with $Re$ up to $15\,000$. Large stress reduction is found at the pipe wall, but at the expense of an increased pressure drop across the baffle. Estimates are then made of the break even point downstream of the baffle where the stress reduction at the wall due to the relaminarised flow compensates for the extra drag produced by the baffle.

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arXiv2019

Upper edge of chaos and the energetics of transition in pipe flow

In the past two decades, our understanding of the transition to turbulence in shear flows with linearly stable laminar solutions has greatly improved. Regarding the susceptibility of the laminar flow, two concepts have been particularly useful: the edge states and the minimal seeds. In this nonlinear picture of the transition, the basin boundary of turbulence is set by the edge state's stable manifold and this manifold comes closest in energy to the laminar equilibrium at the minimal seed. We begin this paper by presenting numerical experiments in which three-dimensional perturbations are too energetic to trigger turbulence in pipe flow but they do lead to turbulence when their amplitude is reduced. We show that this seemingly counter-intuitive observation is in fact consistent with the fully nonlinear description of the transition mediated by the edge state. In order to understand the physical mechanisms behind this process, we measure the turbulent kinetic energy production and dissipation rates as a function of the radial coordinate. Our main observation is that the transition to turbulence relies on the energy amplification away from the wall, as opposed to the turbulence itself, whose energy is predominantly produced near the wall. This observation is further supported by the similar analyses on the minimal seeds and the edge states. Furthermore, we show that the time-evolution of production-over-dissipation curves provide a clear distinction between the different initial amplification stages of the transition to turbulence from the minimal seed.

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arXiv2019

Equilibria, periodic orbits and computing them

In this short exposition, we describe equilibria and periodic orbits in terms of the flow map, Φ, and discuss the essentials of the Jacobian-free Newton-Krylov (JFNK) method that can be used to find them. This method requires little more than calls to an existing time stepping code, which Φ can be considered to represent. Fortran90 / MATLAB code is available to try it out for yourself, where, in the template/example the method is applied to the Lorenz system. This code is problem-independent and can be applied to large systems, having initially been developed to find periodic orbits in simulations of pipe flow.

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arXiv2018

The Self-Sustaining Process in Taylor-Couette Flow

The transition from Tayor vortex flow to wavy-vortex flow is revisited. The Self-Sustaining Process (SSP) of Waleffe [Phys. Fluids 9, 883-900 (1997)] proposes that a key ingredient in transition to turbulence in wall-bounded shear flows is a three-step process involving rolls advecting streamwise velocity, leading to streaks which become unstable to a wavy perturbation whose nonlinear interaction with itself feeds the rolls. We investigate this process in Taylor-Couette flow. The instability of Taylor-vortex flow to wavy-vortex flow, a process which is the inspiration for the second phase of the SSP, is shown to be caused by the streaks, with the rolls playing a negligible role, as predicted by Jones [J. Fluid Mech. 157, 135-162 (1985)] and demonstrated by Martinand et al. [Phys. Fluids 26, 094102 (2014)]. In the third phase of the SSP, the nonlinear interaction of the waves with themselves reinforces the rolls. We show this both quantitatively and qualitatively, identifying physical regions in which this reinforcement is strongest, and also demonstrate that this nonlinear interaction depletes the streaks.

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arXiv2017

Dynamo Action in a Quasi-Keplerian Taylor-Couette Flow

We numerically compute the flow of an electrically conducting fluid in a Taylor-Couette geometry where the rotation rates of the inner and outer cylinders satisfy $Ω_o/Ω_i=(r_o/r_i)^{-3/2}$. In this quasi-Keplerian regime a non-magnetic system would be Rayleigh-stable for all Reynolds numbers $Re$, and the resulting purely azimuthal flow incapable of kinematic dynamo action for all magnetic Reynolds numbers $Rm$. For $Re=10^4$ and $Rm=10^5$ we demonstrate the existence of a finite-amplitude dynamo, whereby a suitable initial condition yields mutually sustaining turbulence and magnetic fields, even though neither could exist without the other. This dynamo solution results in significantly increased outward angular momentum transport, with the bulk of the transport being by Maxwell rather than Reynolds stresses.

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arXiv2017

Transport properties of the Azimuthal Magnetorotational Instability

The magnetorotational instability (MRI) is thought to be a powerful source of turbulence in Keplerian accretion disks. Motivated by recent laboratory experiments, we study the MRI driven by an azimuthal magnetic field in an electrically conducting fluid sheared between two concentric rotating cylinders. By adjusting the rotation rates of the cylinders, we approximate angular velocity profiles $ω\propto r^{q}$. We perform direct numerical simulations of a steep profile close to the Rayleigh line $q \gtrsim -2 $ and a quasi-Keplerian profile $q \approx -3/2$ and cover wide ranges of Reynolds ($Re\le 4\cdot10^4$) and magnetic Prandtl numbers ($0\le Pm \le 1$). In the quasi-Keplerian case, the onset of instability depends on the magnetic Reynolds number, with $Rm_c \approx 50$, and angular momentum transport scales as $\sqrt{Pm} Re^2$ in the turbulent regime. The ratio of Maxwell to Reynolds stresses is set by $Rm$. At the onset of instability both stresses have similar magnitude, whereas the Reynolds stress vanishes or becomes even negative as $Rm$ increases. For the profile close to the Rayleigh line, the instability shares these properties as long as $Pm\gtrsim0.1$, but exhibits a markedly different character if $Pm\rightarrow 0$, where the onset of instability is governed by the Reynolds number, with $Re_c \approx 1250$, transport is via Reynolds stresses and scales as $Re^2$. At intermediate $Pm=0.01$ we observe a continuous transition from one regime to the other, with a crossover at $Rm=\mathcal{O}(100)$. Our results give a comprehensive picture of angular momentum transport of the MRI with an imposed azimuthal field.

astro-ph.SR↗

arXiv2017

Surfing the edge: using feedback control to find nonlinear solutions

Many transitional wall-bounded shear flows are characterised by the coexistence in state-space of laminar and turbulent regimes. Probing the edge boundary between the two attractors has led in the last decade to the numerical discovery of new (unstable) solutions to the incompressible Navier-Stokes equations. However, the iterative bisection method used to achieve this can become prohibitively costly for large systems. Here we suggest a simple feedback control strategy to stabilise edge states, hence accelerating their numerical identification by several orders of magnitude. The method is illustrated for several configurations of cylindrical pipe flow. Travelling waves solutions are identified as edge states, and can be isolated rapidly in only one short numerical run. A new branch of solutions is also identified. When the edge state is a periodic orbit or chaotic state, the feedback control does not converge precisely to solutions of the uncontrolled system, but nevertheless brings the dynamics very close to the original edge manifold in a single run. We discuss the opportunities offered by the speed and simplicity of this new method to probe the structure of both state space and parameter space.

physics.flu-dyn↗

arXiv2017

Relative periodic orbits form the backbone of turbulent pipe flow

Chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this also be the case for the infinite-dimensional dynamics of Navier--Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics---specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier--Stokes equations as they do in lower-dimensional systems---is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.

physics.flu-dyn↗

arXiv2017

The Openpipeflow Navier--Stokes Solver

Pipelines are used in a huge range of industrial processes involving fluids, and the ability to accurately predict properties of the flow through a pipe is of fundamental engineering importance. Armed with parallel MPI, Arnoldi and Newton--Krylov solvers, the Openpipeflow code can be used in a range of settings, from large-scale simulation of highly turbulent flow, to the detailed analysis of nonlinear invariant solutions (equilibria and periodic orbits) and their influence on the dynamics of the flow.

physics.flu-dyn↗

arXiv2017

Symmetry reduction in high dimensions, illustrated in a turbulent pipe

Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynamical system. However, equilibria are atypical for systems with continuous symmetries, i.e. for systems with homogeneous spatial dimensions, whereas relative equilibria (traveling waves) are generic. In order to visualize the unstable manifolds of such solutions, a practical symmetry reduction method is required that converts relative equilibria into equilibria, and relative periodic orbits into periodic orbits. In this article we extend the fixed Fourier mode slice approach, previously applied 1-dimensional PDEs, to a spatially 3-dimensional fluid flow, and show that is substantially more effective than our previous approach to slicing. Application of this method to a minimal flow unit pipe leads to the discovery of many relative periodic orbits that appear to fill out the turbulent regions of state space. We further demonstrate the value of this approach to symmetry reduction through projections (projections only possible in the symmetry-reduced space) that reveal the interrelations between these relative periodic orbits and the ways in which they shape the geometry of the turbulent attractor.

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arXiv2016

Flow in a Circular Expansion Pipe Flow: Effect of a Vortex Perturbation on Localized Turbulence

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 tilt perturbation, as the localized turbulence is observed via the formation of a wavy structure at a low order azimuthal mode, which resembles an optimally amplified perturbation. For higher amplitude, the localized turbulence remains at a constant axial position. It is further investigated using proper orthogonal decomposition, which indicates that the centre region close to the expansion is highly energetic.

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