arXiv subjects

Nicholas J. Moore

Publications and source records attributed to Nicholas J. Moore.

10 recordsLinked to original sources

arXiv2025

Optimal velocity fields for instantaneous magnetic field growth

We consider a variant of the kinematic dynamo problem. Rather than prescribing a velocity field and searching for high-growth magnetic fields via an eigenvalue problem, we treat the seed magnetic-field structure as given and ask which velocity field maximally enhances its instantaneous growth. We show this second problem has an elegant formulation in terms of variational calculus. Upon simultaneously constraining the velocity's kinetic energy and enstrophy, the Euler-Lagrange equation leads to a forced Helmholtz partial differential equation for the optimal velocity field. For the special case of fixed kinetic energy and unconstrained enstrophy, the optimal velocity field everywhere opposes the divergence-free projection of the Lorentz force. In the more general setting, the optimal velocity field can be found through numerical solution of the forced Helmholtz equation. We construct 2.5-dimensional numerical examples to support the theoretical findings, and then leverage the newly found optimal velocity fields to accelerate numerical optimization of the magnetic-velocity field pair for maximal instantaneous growth rate.

physics.flu-dyn↗

arXiv2024

Low-order reaction-diffusion system approximates heat transfer and flow structure in annular convection

Heat transfer in a fluid can be greatly enhanced by natural convection, giving rise to the nuanced relationship between the Nusselt number and Rayleigh number that has been a focus of modern fluid dynamics. Our work explores convection in an annular domain, where the geometry reinforces the large-scale circulatory flow pattern that is characteristic of natural convection. The flow must match the no-slip condition at the boundary, leading to a thin boundary layer where both the flow velocity and the temperature vary rapidly. To understand the system's heat transfer characteristics, we derive a reduced model from the Navier-Stokes-Boussinesq equations, whereby the equations of flow and heat are transformed to a system of low-order partial differential equations (PDEs) that take the form of a reaction-diffusion system. Solutions to the reaction-diffusion system, though they fail to predict dynamic events, preserve the same boundary-layer structure seen in the direct numerical simulation (DNS). By matching the solutions inside and outside the boundary layer, asymptotic analysis predicts a power-law relationship Nu $\propto$ Ra$^{1/4}$. Though difficult to distinguish from an exponent of 2/7, the predicted power law agrees well with measurements from DNS over several decades of the Rayleigh number. Considering the model's deficiencies in describing turbulent fluctuations and reversal events, the agreement regarding heat transfer characteristics is encouraging and suggests that the methodology of systematically deriving low-order PDEs from the governing equations may provide a useful complement to existing theories.

physics.flu-dyn↗

arXiv2024

In search of rogue waves: a novel proposal distribution for parallelized rejection sampling of the truncated KdV Gibbs measure

The Gibbs ensemble of the truncated KdV (TKdV) equation has been shown to accurately describe the anomalous wave statistics observed in laboratory experiments, in particular the emergence of extreme events. Here, we introduce a novel proposal distribution that facilitates efficient rejection sampling of the TKdV Gibbs measure. Within parameter regimes accessible to laboratory experiments and capable of producing extreme events, the proposal distribution generates 1-6 orders of magnitude more accepted samples than does a naive, uniform distribution. When equipped with the new proposal distribution, a simple rejection algorithm enjoys key advantages over a Markov chain Monte Carlo algorithm, include better parallelization properties and generation of uncorrelated samples.

math.NA↗

arXiv2023

A convective fluid pendulum revealing states of order and chaos

We examine thermal convection in a two-dimensional annulus using fully resolved direct numerical simulation (DNS) in conjunction with a low-dimensional model deriving from Galerkin truncation of the governing Navier-Stokes Boussinesq (NSB) equations. The DNS is based on fast and accurate pseudo-spectral discretization of the full NSB system with implicit-explicit time stepping. Inspired by the numerical results, we propose a reduced model that is based on a Fourier-Laurent truncation of the NSB system and can generalize to any degree of accuracy. We demonstrate that the lowest-order model capable of satisfying all boundary conditions on the annulus successfully captures reversals of the large-scale circulation (LSC) in certain regimes. Based on both the DNS and stability analysis of the reduced model, we identify a sequence of transitions between (i) a motionless conductive state, (ii) a state of steady circulation, (iii) non-periodic dynamics and chaotic reversals of the LSC, (iv) a high Rayleigh-number state in which LSC reversals are periodic despite turbulent fluctuations at the small scale. The reduced model reveals a link to a damped pendulum system with a particular form of external forcing. The oscillatory pendulum motion provides an accurate prediction for the LSC reversal frequency in the high Rayleigh-number regime.

physics.flu-dyn↗

arXiv2023

Large-scale circulation reversals explained by pendulum correspondence

We introduce a low-order dynamical system to describe thermal convection in an annular domain. The model derives systematically from a Fourier-Laurent truncation of the governing Navier-Stokes Boussinesq equations and accounts for spatial dependence of the flow and temperature fields. Comparison with fully-resolved direct numerical simulations (DNS) shows that the model captures parameter bifurcations and reversals of the large-scale circulation (LSC), including states of (i) steady circulating flow, (ii) chaotic LSC reversals, and (iii) periodic LSC reversals. Casting the system in terms of the fluid's angular momentum and center of mass (CoM) reveals equivalence to a damped pendulum with forcing that raises the CoM above the fulcrum. This formulation offers a transparent mechanism for LSC reversals, namely the inertial overshoot of a forced pendulum, and it yields an explicit formula for the frequency $f^*$ of regular LSC reversals in the high Rayleigh-number limit. This formula is shown to be in excellent agreement with DNS and produces the scaling law $f^* \sim Ra^{0.5}$.

physics.flu-dyn↗

arXiv2022

Predicting convection configurations in coupled fluid-porous systems

A ubiquitous arrangement in nature is a free-flowing fluid coupled to a porous medium, for example a river or lake lying above a porous bed. Depending on the environmental conditions, thermal convection can occur and may be confined to the clear fluid region, forming shallow convection cells, or it can penetrate into the porous medium, forming deep cells. Here, we combine three complementary approaches -- linear stability analysis, fully nonlinear numerical simulations, and a coarse-grained model -- to determine the circumstances that lead to each configuration. The coarse-grained model yields an explicit formula for the transition between deep and shallow convection in the physically relevant limit of small Darcy number. Near the onset of convection, all three of the approaches agree, validating the predictive capability of the explicit formula. The numerical simulations extend these results into the strongly nonlinear regime, revealing novel hybrid configurations in which the flow exhibits a dynamic shift from shallow to deep convection. This hybrid shallow-to-deep convection begins with small, random initial data, progresses through a metastable shallow state, and arrives at the preferred steady-state of deep convection. We construct a phase diagram that incorporates information from all three approaches and depicts the regions in parameter space that give rise to each convective state.

physics.flu-dyn↗

arXiv2022

How fluid-mechanical erosion creates anisotropic porous media

Using a Cauchy integral formulation of the boundary integral equations, we simulate the erosion a porous medium comprised of up to 100 solid bodies embedded in a Stokes flow. The grains of the medium are resolved individually and erode under the action of surface shear stress. Through nonlinear feedback with the surrounding flow fields, microscopic changes in grain morphology give way to larger-scale features in the medium such as channelization. The Cauchy-integral formulation and associated quadrature formulas enable us to resolve dense configurations of nearly contacting bodies. We observe substantial anisotropy to develop over the course of erosion; that is, the configurations that result from erosion generally permit flow in the longitudinal direction more easily than in the transverse direction by up to a factor of six. These results suggest that the erosion of solid material from groundwater flows may contribute to previously observed anisotropy of natural porous media.

physics.flu-dyn↗

arXiv2022

Rigorous criteria for anomalous waves induced by abrupt depth change using truncated KdV statistical mechanics

The truncated Korteweg-De Vries (TKdV) system, a shallow-water wave model with Hamiltonian structure that exhibits weakly turbulent dynamics, has been found to accurately predict the anomalous wave statistics observed in recent laboratory experiments. Majda et al. (2019) developed a TKdV statistical mechanics framework based on a mixed Gibbs measure that is supported on a surface of fixed energy (microcanonical) and takes the usual macroconical form in the Hamiltonian. This paper reports two rigorous results regarding the surface-displacement distributions implied by this ensemble, both in the limit of the cutoff wavenumber $Λ$ growing large. First, we prove that if the inverse temperature vanishes, microstate statistics converge to Gaussian as $Λ\to \infty$. Second, we prove that if nonlinearity is absent and the inverse-temperature satisfies a certain physically-motivated scaling law, then microstate statistics converge to Gaussian as $Λ\to \infty$. When the scaling law is not satisfied, simple numerical examples demonstrate symmetric, yet highly non-Gaussian, displacement statistics to emerge in the linear system, illustrating that nonlinearity is not a strict requirement for non-normality in the fixed-energy ensemble. The new results, taken together, imply necessary conditions for the anomalous wave statistics observed in previous numerical studies. In particular, non-vanishing inverse temperature and either the presence of nonlinearity or the violation of the scaling law are required for displacement statistics to deviate from Gaussian. The proof of this second theorem involves the construction of an approximating measure, which we find also elucidates the peculiar spectral decay observed in numerical studies and may open the door for improved sampling algorithms.

physics.flu-dyn↗

arXiv2022

Morphological attractors in natural convective dissolution

Recent experiments demonstrate how a soluble body placed in a fluid spontaneously forms a dissolution pinnacle -- a slender, upward pointing shape that resembles naturally occurring karst pinnacles found in stone forests. This unique shape results from the interplay between interface motion and the natural convective flows driven by the descent of relatively heavy solute. Previous investigations suggest these structures to be associated with shock-formation in the underlying evolution equations, with the regularizing Gibbs-Thomson effect required for finite tip curvature. Here, we find a class of exact solutions that act as attractors for the shape dynamics in two and three dimensions. Intriguingly, the solutions exhibit large but finite tip curvature without any regularization, and they agree remarkably well with experimental measurements. The relationship between the dimensions of the initial shape and the final state of dissolution may offer a principle for estimating the age and environmental conditions of geological structures.

physics.flu-dyn↗

arXiv2021

Generalization of waving-plate theory to multiple interacting swimmers

Early research in aerodynamics and biological propulsion was dramatically advanced by the analytical solutions of Theodorsen, von K\'{a}rm\'{a}n, Wu and others. While these classical solutions apply only to isolated swimmers, the flow interactions between multiple swimmers are relevant to many practical applications, including the schooling and flocking of animal collectives. In this work, we derive a class of solutions that describe the hydrodynamic interactions between an arbitrary number of swimmers in a two-dimensional inviscid fluid. Our approach is rooted in multiply-connected complex analysis and exploits several recent results. Specifically, the transcendental (Schottky-Klein) prime function serves as the basic building block to construct the appropriate conformal maps and leading-edge-suction functions, which allows us to solve the modified Schwarz problem that arises. As such, our solutions generalize classical thin aerofoil theory, specifically Wu's waving-plate analysis, to the case of multiple swimmers. For the case of a pair of interacting swimmers, we develop an efficient numerical implementation that allows rapid computations of the forces on each swimmer. We investigate flow-mediated equilibria and find excellent agreement between our new solutions and previously reported experimental results. Our solutions recover and unify disparate results in the literature, thereby opening the door for future studies into the interactions between multiple swimmers.

physics.flu-dyn↗