Nick Moore
Assistant Professor of Mathematics,
Florida State University


Statistical dynamical model to predict extreme events and anomalous features in shallow water waves with abrupt depth change, [PNAS link]
A.J. Majda, M.N.J. Moore, D. Qi
PNAS (2019)

Understanding and predicting extreme events and their anomalous statistics in complex nonlinear systems are a grand challenge in climate, material, and neuroscience as well as for engineering design. Recent laboratory experiments in weakly turbulent shallow water reveal a remarkable transition from Gaussian to anomalous behavior as surface waves cross an abrupt depth change (ADC). Downstream of the ADC, probability density functions of surface displacement exhibit strong positive skewness accompanied by an elevated level of extreme events. Here, we develop a statistical dynamical model to explain and quantitatively predict the above anomalous statistical behavior as experimental control parameters are varied. The first step is to use incoming and outgoing truncated Korteweg–de Vries (TKdV) equations matched in time at the ADC. The TKdV equation is a Hamiltonian system, which induces incoming and outgoing statistical Gibbs invariant measures. The statistical matching of the known nearly Gaussian incoming Gibbs state at the ADC completely determines the predicted anomalous outgoing Gibbs state, which can be calculated by a simple sampling algorithm verified by direct numerical simulations, and successfully captures key features of the experiment. There is even an analytic formula for the anomalous outgoing skewness. The strategy here should be useful for predicting extreme anomalous statistical behavior in other dispersive media.

Anomalous wave statistics induced by abrupt depth change, [PRF link], [arXiv link]
C.T. Bolles, K. Speer, M.N.J. Moore
Physical Review Fluids (Rapid Communications) 4(1), 011801

Laboratory experiments reveal that variations in bottom topography can qualitatively alter the distribution of randomized surface waves. A normally-distributed, unidirectional wave field becomes highly skewed and non-Gaussian upon encountering an abrupt depth transition. A short distance downstream of the transition, wave statistics conform closely to a gamma distribution, affording simple estimates for skewness, kurtosis, and other statistical properties. Importantly, the exponential decay of the gamma distribution is much slower than Gaussian, signifying that extreme events occur more frequently. Under the conditions considered here, the probability of a rogue wave can increase by a factor of 50 or more. We also report on the surface-slope statistics and the spectral content of the waves produced in the experiments.

Boundary integral analysis for non-homogeneous, incompressible Stokes flows, [arXiv link]
L.J. Gray, J. Jakowski, M.N.J. Moore, W. Ye
Advances in Computational Mathematics (2019)

A regular-grid volume-integration algorithm has been previously developed for solving non-homogeneous versions of the Laplace and the elasticity equations. This note demonstrates that the same approach can be successfully adapted to the case of non-homogenous, incompressible Stokes flow. The key observation is that the Stokeslet (Green's function) can be written as the Laplacian of another function that has a simple analytical expression. As a consequence, the volume integral can be reformulated as an easily evaluated boundary integral, together with a remainder domain integral that can be computed using a regular cuboid grid covering the domain.

A minimal model for predicting ventilation rates of subterranean caves, [arXiv link]
K. Khazmutdinova, D. Nof, D. Tremaine, M. Ye, M.N.J. Moore
arXiv and under submission (2018)

The ventilation of carbon dioxide within subterranean caves regulates the growth of speleothems --- mineral deposits found in caves that provide important clues about past climate. While previous studies have used internal measurements to predict ventilation rates, such measurements would not be available for the task of climate reconstruction. Here, we develop a parsimonious model to predict ventilation rates from knowledge of outside temperatures and the cave's physical dimensions only. In the model, ventilation arises from buoyancy-driven flows created in passageways that connect to the outside. A few key simplifications leads to a system amenable to perturbation analysis, resulting in explicit relationships for how ventilation depends on system parameters. We compare these predictions to time-resolved, in-situ measurements of transported cave gases (carbon dioxide and radon). The theory accurately accounts for seasonal and synoptic variations of these gases.

A boundary-integral framework to simulate viscous erosion of a porous medium, [JCP link], [arXiv link]
B.D. Quaife and M.N.J. Moore
Journal of Computational Physics 375, 1-21

We develop numerical methods to simulate the fluid-mechanical erosion of many bodies in two-dimensional Stokes flow. The broad aim is to simulate the erosion of a porous medium (e.g. groundwater flow) with grain-scale resolution. Our fluid solver is based on a second-kind boundary integral formulation of the Stokes equations that is discretized with a spectrally-accurate Nystrom method and solved with fast-multipole-accelerated GMRES. The fluid solver provides the surface shear stress which is used to advance solid boundaries. We regularize interface evolution via curvature penalization using the theta-L formulation, which affords numerically stable treatment of stiff terms and therefore permits large time steps. The overall accuracy of our method is spectral in space and second-order in time. The method is computationally efficient, with the fluid solver requiring O(N) operations per GMRES iteration, a mesh-independent number of GMRES iterations, and a one-time O(N^2) computation to compute the shear stress. We benchmark single-body results against analytical predictions for the limiting morphology and vanishing rate. Multibody simulations reveal the spontaneous formation of channels between bodies of close initial proximity. The channelization is associated with a dramatic reduction in the resistance of the porous medium, much more than would be expected from the reduction in grain size alone.

A fast Chebyshev method for simulating flexible-wing propulsion, [JCP link], [arXiv link]
M.N.J. Moore
Journal of Computational Physics 345, 792-817

We develop a highly efficient numerical method to simulate small-amplitude flapping propulsion by a flexible wing in a nearly inviscid fluid. We allow the wing's elastic modulus and mass density to vary arbitrarily, with an eye towards optimizing these distributions for propulsive performance. The method to determine the wing kinematics is based on Chebyshev collocation of the 1D beam equation as coupled to the surrounding 2D fluid flow. Through small-amplitude analysis of the Euler equations (with trailing-edge vortex shedding), the complete hydrodynamics can be represented by a nonlocal operator that acts on the 1D wing kinematics. A class of semi-analytical solutions permits fast evaluation of this operator with O(N log N) operations, where N is the number of collocation points on the wing. This is in contrast to the minimum N^2 cost of a direct 2D fluid solver. The coupled wing-fluid problem is thus recast as a PDE with nonlocal operator, which we solve using a preconditioned iterative method. These techniques yield a solver of near-optimal complexity, allowing one to rapidly search the infinite-dimensional parameter space of all possible material distributions and even perform optimization over this space.

Riemann-Hilbert problems for the shapes formed by bodies dissolving, melting, and eroding in fluid flows, [preprint], [link]
M.N.J. Moore
Communications on Pure and Applied Mathematics 70(9), 1810-1831

The classical Stefan problem involves the motion of boundaries during phase transition, but this process can be greatly complicated by the presence of a fluid flow. Here, we consider a body undergoing material loss due to either dissolution (from molecular diffusion), melting (from thermodynamic phase change), or erosion (from fluid-mechanical stresses) in a fast-flowing fluid. In each case, the task of finding the shape formed by the shrinking body can be posed as a singular Riemann-Hilbert problem. A class of exact solutions captures the rounded surfaces formed during dissolution/melting, as well as the angular features formed during erosion, thus unifying these different physical processes under a common framework. This study, which merges boundary-layer theory, separated-flow theory, and Riemann-Hilbert analysis, represents a rare instance of an exactly solvable model for high-speed fluid flows with free boundaries.

Torsional spring is the optimal flexibility arrangement for thrust production of a flapping wing, [pdf], [link]
M.N.J. Moore
Physics of Fluids (Letters) 27(9), 091701

While it is understood that flexibility can improve the propulsive performance of flapping wings and fins, the flexibility distribution leading to optimal performance has not been explored. Using 2D small-amplitude theory and a fast Chebyshev method, we examine how thrust depends on the chord-wise distribution of wing stiffness. Through numerical optimization, we find that focusing flexibility at the wing's front, e.g. through a torsional spring, maximizes thrust. A wing with an optimally chosen spring constant typically generates 36% more thrust than a wing of optimal uniform stiffness. These results may relate to material distributions found in nature, such as insect wings, and may apply to the design of biomimetic swimmers and flyers, such as ornithopters.

Shape dynamics and scaling laws for a body dissolving in fluid flow, [pdf], [link]
J.M. Huang, M.N.J. Moore, L. Ristroph
Journal of Fluid Mechanics (Rapids) 765, R3

While fluid flows are known to promote dissolution of materials, such processes are poorly understood due to the coupled dynamics of the flow and the receding surface. We study this moving boundary problem through experiments in which hard candy bodies dissolve in laminar high-speed water flows. We find that different initial geometries are sculpted into a similar terminal form before ultimately vanishing, suggesting convergence to a stable shape-flow state. A model linking the flow and solute concentration shows how uniform boundary-layer thickness leads to uniform dissolution, allowing us to obtain an analytical expression for the terminal geometry. Newly derived scaling laws predict that the dissolution rate increases with the square root of the flow speed and that the body volume vanishes quadratically in time, both of which are confirmed by experimental measurements.

Analytical results on the role of flexibility in flapping propulsion, [pdf], [link]
M.N.J. Moore,
Journal of Fluid Mechanics 757, 599

Wing or fin flexibility can dramatically affect the performance of flying and swimming animals. Both laboratory experiments and numerical simulations have been used to study these effects, but analytical results are notably lacking. Here, we develop small-amplitude theory to model a flapping wing that pitches passively due to a combination of wing compliance, inertia, and fluid forces. Remarkably, we obtain a class of exact solutions describing the wing's emergent pitching motions, along with expressions for how thrust and efficiency are modified by compliance. The solutions recover a range of realistic behaviors and shed new light on how flexibility can aid performance, the importance of resonance, and the separate roles played by wing and fluid inertia. The simple, robust estimates afforded by our theory will likely be valuable even in situations where details of the flapping motion and wing geometry differ.

Self-similar evolution of a body eroding in a fluid flow, [pdf], [link]
M.N.J. Moore, L. Ristroph, S. Childress, J. Zhang, and M.J. Shelley
Physics of Fluids 25(11), 116602

Erosion of solid material by flowing fluids plays an important role in shaping landforms, and in this natural context is often dictated by processes of high complexity. Here, we examine the coupled evolution of solid shape and fluid flow within the idealized setting of a cylindrical body held against a fast, unidirectional flow, and eroding under the action of fluid shear stress. Experiments and simulations both show self-similar evolution of the body, with an emerging quasi-triangular geometry that is an attractor of the shape dynamics. Our fluid erosion model, based on Prandtl boundary layer theory, yields a scaling law that accurately predicts the body's vanishing rate. Further, a class of exact solutions provides a partial prediction for the body's terminal form as one with a leading surface of uniform shear stress. Our simulations show this predicted geometry to emerge robustly from a range of different initial conditions, and allow us to explore its local stability. The sharp, faceted features of the terminal geometry defy the intuition of erosion as a globally smoothing process.

Sculpting of an erodible body by flowing water, [pdf], [link]
L. Ristroph, M.N.J. Moore, S. Childress, M.J. Shelley, and J. Zhang
Proceedings of the National Academy of Sciences 109(48), 19606

Erosion by flowing fluids carves striking landforms on Earth and also provides important clues to the past and present environments of other worlds. In these processes, solid boundaries both influence and are shaped by the surrounding fluid, but the emergence of morphology as a result of this interaction is not well understood. We study the coevolution of shape and flow in the context of erodible bodies molded from clay and immersed in a fast, unidirec- tional water flow. Although commonly viewed as a smoothing process, we find that erosion sculpts pointed and cornerlike features that persist as the solid shrinks. We explain these observations using flow visualization and a fluid mechanical model in which the surface shear stress dictates the rate of material removal. Experiments and simulations show that this interaction ultimately leads to self- similarly receding boundaries and a unique front surface character- ized by nearly uniform shear stress. This tendency toward confor- mity of stress offers a principle for understanding erosion in more complex geometries and flows, such as those present in nature.

A weak-coupling expansion for viscoelastic fluids applied to dynamic settling of a body, [pdf], [link]
M.N.J. Moore and M.J. Shelley
Journal of Non-Newtonian Fluid Mechanics 183, 25

The flow of viscoelastic fluids is an area in which analytical results are difficult to attain, yet can provide invaluable information. We develop a weak-coupling expansion that allows for semi-analytical computations of viscoelastic fluid flows coupled to immersed structures. In our method, a leading-order poly- meric stress evolves according to a Newtonian velocity field, and this stress is used to correct the motion of bodies. We apply the method to the transient problem of a sphere settling through a viscoelas- tic fluid using the Oldroyd-B model, and recover previous results and observed behavior. The theory pre- sented here is in contrast to the retarded-motion, or low-Weissenberg-number, expansions that have seen much application. One advantage of the weak-coupling method is that it offers information for Weissenberg numbers larger than one. The expansion's limit of validity is closely related to the diluteness criterion of a Boger fluid. We extend the classical settling problem to include an oscillatory body-force, and show how the introduction of a second time-scale modifies the body-dynamics.

Stratified flows with vertical layering of density: experimental and theoretical study of flow configurations and their stability, [pdf], [link]
R. Camassa, R.M. McLaughlin, M.N.J. Moore, and K. Yu
Journal of Fluid Mechanics 690, 571

The vertical motion of bodies through a density-stratified fluid is a problem with important applications in the ocean and atmosphere. Here we create a "stripped-down" flow configuration in order to isolate one feature of vertical motion through stratification - namely the creation of vertical density layers by viscous entrainment. This flow configuration is created in a table-top experiment by towing a fiber vertically through a stratified corn syrup solution and lubrication theory is used to model the dynamic flow configuration. We find long-wave instabilities along the fluid interface for layers that are larger than a critical length-scale.

Brachistochrones in potential flow and the connection to Darwin's theorem, [pdf], [link]
R. Camassa, R.M. McLaughlin, M.N.J. Moore, and A. Vaidya
Physics Letters A 372(45), 6742

Here we characterize the brachistochrone path, or path of shortest time, in potential flow past a body, and we apply our theory to obtain new results related to Darwin's drift volume. Drift volume is related to a body's added mass and has important practical applications. Our theory extends previous infinite and semi-infinite results to finite distances and times, which is an important extension in the context of laboratory experiments. We also discuss an application to a rigid sphere sedimenting through a sharply stratified fluid in order to account for an experimentally observed levitation phenomenon. Further theoretical work related to the potential energy carried by the drift volume is underway.


A weak-coupling expansion applied to the motion of a settling rod in a viscoelastic fluid, [pdf]
Dustin J. Hill,
Senior honors thesis (2013)

Viscoelastic fluids are fluids that, in addition to having the property of viscosity, also have the property of elasticity. This is often the result of the introduction of a polymeric material to a non-viscoelastic fluid - the presence of these materials greatly affects the behavior of the fluid by introducing additional internal stresses. An example of this - and the focus of this paper - is the dynamics of a settling rod in a viscoelastic fluid. In a Newtonian fluid, a rod settling under gravity adopts a horizontal orientation; whereas, in a viscoelastic fluid, the same rod would adopt a vertical orientation. In this paper, we discuss a number of theoretical and computational models toward the goal of demonstrating the stability of the vertical orientation of the rod in a viscoelastic fluid. The most important of these techniques is the weak- coupling expansion. In this technique, we consider the elastic stresses and Newtonian stresses in the fluid to be coupled by a parameter. Considering this to be small, we expand the equations of motion for the rod in terms of it, allowing for an efficient solution of these equations.