arXiv subjects

Edward James Brambley

Publications and source records attributed to Edward James Brambley.

11 recordsLinked to original sources

arXiv2026

A free-surface-only closure model for linear waves on general deep-water flows

We present a novel, spatially two-dimensional (2D) set of equations to study the propagation of linear deep-water surface waves over a general steady three-dimensional (3D) background free surface flow. No assumptions of a flat background free surface, nor of an irrotational background flow, are made. The resulting model is not only a significant theoretical simplification, but also results in orders of magnitude faster computations. The linearized Euler equations are evaluated on the free surface of the base flow, and a closure condition is proposed to account for the vertical derivatives at the free surface. This generalizes a recent result for purely rotating background flows (Zuccoli, Brambley & Barkley, 2025, arXiv:2405.12078). The final model consists of five coupled first order partial differential equations (PDEs) to be solved on the 2D free surface, involving five unknowns: the horizontal and vertical perturbation velocities; the free surface perturbation height; and unexpectedly the gradient of perturbation pressure with depth at the surface. Two test problems are used to validate the model: a uni-directional vortical depth-varying base flow with a flat free surface; and perturbations to a travelling Gerstner wave solution. Eigenvalues and eigenfunctions of the linearized Euler equations are computed and compared with those of the model. Results show remarkable agreement between the two. Our study finds for the first time, to the best of our knowledge, that sufficiently steep two-dimensional Gerstner waves are unstable.

physics.flu-dyn↗

arXiv2026

The Weakly-Nonlinear Admittance at Open Ends of Two- and Three-Dimensional Acoustic Waveguides

We formulate a weakly-nonlinear exit condition for open ends of acoustic waveguides; to our knowledge this is the first time a general acoustic open end has been analysed outside the linear regime. The formulation neglects mean flow, vortex separation, and viscous effects. The resulting admittance boundary condition, and its weakly-nonlinear counterpart, extend recent weakly-nonlinear modelling of curved ducts(Jensen & Brambley 2026, arXiv:2503.11536) to include open ends. We approximate free space by considering the open-ended duct to be enclosed within a much larger hard-walled concentric duct; within the larger duct, the smaller duct exit is modelled as an acoustic discontinuity. Importantly, the superposition principle is unneeded, allowing the model to be applied in the nonlinear regime. The exit condition can be calculated without needing to solve the full problem in either the outer or inner ducts, making it numerically efficient. The exit condition is validated in the linear regime by comparison to Wiener-Hopf solutions of the duct end correction, and a novel nonlinear end correction is proposed; we find that both non-plane waves and nonlinearity cause the end correction to vary significantly from Rayleigh's classical 0.6 radii result. A number of numerical illustrations are then discussed, demonstrating nonlinear effects, sound radiating from curved ducts, sound radiating from an exponential horn (representative of brass instrument bells), and the harmonic effects of the open end on in-duct resonances. The model has potential applications to sound in woodwind and brass instruments. Matlab source code is provided in the supplementary material.

physics.flu-dyn↗

arXiv2026

Modelling Lateral Spread in Wire Flat Rolling

A mathematical model for wire rolling is developed, focusing on predicting the lateral spread. This provides, for the first time, an analytic model of lateral spread without any fitting parameters. The model is derived directly from the governing equations, assuming a rigid, perfectly plastic material and exploiting the thinness of the wire (in thickness and width) relative to the roller size. Results are compared against experiments performed on stainless steel wire using 100mm diameter rolls, demonstrating accurate predictions of lateral spread across a wide range of wire diameters (2.96mm-7.96mm) and reduction ratios (20%-60%), all without the need for fitting parameters. Since the model requires only seconds to compute, the model's valid range is explored for varying roll diameter, wire diameter, and reduction ratio, and their effects on the resulting lateral spread characterized. The model can serve as a robust tool for validating FE results, guiding process design, and laying the foundation for future improved models. Matlab code to evaluate the model is provided in the supplementary material.

physics.class-ph↗

arXiv2025

Through-Thickness Modelling of Metal Rolling using Multiple-Scale Asymptotics

A new semi-analytic model of the metal rolling process is introduced, which, for the first time, is able to predict the through-thickness stress and strain oscillations present in long thin roll-gaps. The model is based on multiple-scales asymptotics, assuming a long thin roll-gap and a comparably small Coulomb friction coefficient. The leading-order solution varies only on a long lengthscale corresponding to the roll-gap length and matches with slab models. The next-order correction varies on both this long lengthscale and a short lengthscale associated with the workpiece thickness, and reveals rapid stress and strain oscillation both in the rolling direction and through the thickness. For this initial derivation, the model assumes a rigid perfectly-plastic material behaviour. Despite these strong assumptions, this model compares well with finite element simulations that employ more realistic material behaviour (including elasticity and strain hardening). These assumptions facilitate the simplest possible model to provide a foundational understanding of the complex through-thickness behaviour observed in the finite element simulations, while requiring an order of only seconds to compute. This model can form the foundation of further improved models with more complicated mechanics in the future. Matlab code for evaluating the model is provided in the supplementary material.

cond-mat.mtrl-sci↗

arXiv2025

Multimodal nonlinear acoustics in two- and three-dimensional curved ducts

We develop a weakly nonlinear model of duct acoustics in two and three dimensions (without flow). The work extends the previous work of McTavish & Brambley (2019, J. Fluid Mech. 875, pp. 411-447) to three dimensions and significantly improves the numerical efficiency. The model allows for general curvature and width variation in two-dimensional ducts, and general curvature and torsion with radial width variation in three-dimensional ducts. The equations of gas dynamics are perturbed and expanded to second order, allowing for wave steepening and the formation of weak shocks. The resulting equations are then expanded temporally in a Fourier series and spatially in terms of straight duct modes, and a multi-modal method is applied, resulting in an infinite set of coupled ODEs for the modal coefficients. A linear matrix admittance and its weakly-nonlinear generalization to a tensor convolution are first solved throughout the duct, and then used to solve for the acoustic pressures and velocities. The admittance is useful in its own right, as it encodes the acoustic and weakly-nonlinear properties of the duct independently from the specific wave source used. After validation, a number of numerical examples are presented that compare two- and three-dimensional results, the effects of torsion, curvature and width variation, acoustic leakage due to curvature and nonlinearity, and the variation in effective duct length of a curved duct due to varying the acoustic amplitude. The model has potential future applications to sound in brass instruments. Matlab source code is provided in the supplementary material.

physics.flu-dyn↗

arXiv2025

A deep-water closure model for surface waves on axisymmetric swirling flows

We consider the propagation of linear gravity waves on the free surface of steady, axisymmetric flows with purely azimuthal velocity. We propose a two-dimensional set of governing equations for surface waves valid in the deep-water limit. These equations come from a closure condition at the free surface that reduces the three-dimensional Euler equations in the bulk of the fluid to a set of two-dimensional equations applied only at the free surface. Since the closure condition is not obtained rigorously, it is validated numerically through comparisons with full three-dimensional calculations for vortex flows, including for a Lamb-Oseen vortex. The model presented here overcomes three limitations of existing models, namely: it is not restricted to potential base flows; it does not assume the base flow to have a flat free surface; and it does not require the use of infinite-order differential operators (such as $\tanh(\nabla)$) in the governing equations. The model can be applied in the case of rapid swirl (large Froude number) where the base free surface is substantially deformed. Since the model contains only derivatives of finite order, it is readily amenable to standard numerical study.

physics.flu-dyn↗

arXiv2024

Careful finite element simulations of cold rolling with accurate through-thickness resolution and prediction of residual stress

Metal rolling is a widespread and well-studied process, and many finite-element (FE) rolling simulations can be found in the scientific literature. However, these FE simulations are typically limited in their resolution of through-thickness variations. In this paper, we carefully assess the accuracy of a number of FE approaches, and find that at least 60 elements through-thickness are needed to properly resolve through-thickness variation; this is significantly more than is used elsewhere in the metal rolling literature. In doing so, we reveal an oscillatory stress pattern, which is not usually observed in simulations but which we can validate by comparison with recent analytical work, and which is completely deterministic, not arising from numerical noise or error. We show that these oscillations contribute to the formation of residual stress and may help predict curvature in asymmetric rolled sheets, a phenomenon which is currently not well understood. Accurate through-thickness variation of stress and strain would also have implications for modelling microstructure evolution, damage, and surface finish.

cond-mat.mtrl-sci↗

arXiv2024

Trapped Free Surface Waves for a Lamb-Oseen Vortex Flow

Trapped surface waves have been observed in a swimming pool trapped by, and rotating around, the cores of vortices. To investigate this effect, we have numerically studied the free-surface response of a Lamb--Oseen vortex to small perturbations. The fluid has finite depth but is laterally unbounded. The numerical method used is spectrally accurate, and uses a novel non-reflecting buffer region to simulate a laterally unbounded fluid. While a variety of linear waves can arise in this flow, we focus here on surface gravity waves. We investigate the linear modes of the vortex as a function of the perturbation azimuthal mode number and the vortex rotation rate. We find that at low rotation rates, linear modes decay by radiating energy to the far field, while at higher rotation rates modes become nearly neutrally stable and trapped in the vicinity of the vortex. While trapped modes have previously been seen in shallow water surface waves due to small perturbations of a bathtub vortex, the situation considered here is qualitatively different owing to the lack of an inward flow and the dispersive nature of non-shallow-water waves. We also find that for slow vortex rotation rates, trapped waves propagate in the opposite direction to the vortex rotation, whereas, above a threshold rotation rate, waves co-rotate with the flow.

physics.flu-dyn↗

arXiv2022

Analysis of a Rigid Cylinder Rolling over a Linear Elastic Half-space in the Full-slip Regime

This paper provides an analytical solution for the deformation of an elastic half-space caused by a cylindrical roller. The roller is considered rigid, and is forced into the half space and rolls across its surface, with contact modelled by Coulomb friction. In general, portions of the surface of the roller in contact with the half space may slip across the surface of the half space, or may stick to it. In this paper, we consider only the regime where all of the rollers contact surface is slipping. This results in a mixed boundary value problem, which is formulated as a $2\times 2$ matrix Wiener-Hopf problem. The exponential factors in the Wiener-Hopf matrix allows a solution by following the iterative method of Priddin, Kisil, and Ayton (Phil. Trans. Roy. Soc. A 378, p. 20190241, 2020) which is implemented numerically by computing Cauchy transforms using a spectral method following Slevinsky and Olver (J. Comput. Phys. 332, pp. 290-315, 2017). The limits of the contact region are located a posteriori by applying an optimisation method. The solution is illustrated with several examples, and numerical code to compute the solutions in general is included in the supplementary material.

physics.class-ph↗

arXiv2019

Optimized Runge-Kutta (LDDRK) timestepping schemes for non-constant-amplitude oscillations

Finite differences and Runge-Kutta time stepping schemes used in Computational AeroAcoustics simulations are often optimized for low dispersion and dissipation (e.g. DRP or LDDRK schemes) when applied to linear problems in order to accurately simulate waves with the least computational cost. Here, the performance of optimized Runge-Kutta time stepping schemes for linear time-invariant problems with non-constant-amplitude oscillations is considered. This is in part motivated by the recent suggestion that optimized spatial derivatives perform poorly for growing and decaying waves, as their optimization implicitly assumes real wavenumbers. To our knowledge, this is the first time the time-stepping of non-constant-amplitude oscillations has been considered. It is found that current optimized Runge-Kutta schemes perform poorly in comparison with their maximal order equivalents for non-constant-amplitude oscillations. Moreover, significantly more accurate results can be achieved for the same computation cost by replacing a two-step scheme such as LDDRK56 with a single step higher-order scheme with a longer time step. Attempts are made at finding optimized schemes that perform well for non-constant-amplitude oscillations, and three such examples are provided. However, the traditional maximal order Runge-Kutta time stepping schemes are still found to be preferable for general problems with broadband excitation. These theoretical predictions are illustrated using a realistic 1D wave-propagation example.

math.NA↗

arXiv2016

Viscous effects on the acoustics and stability of a shear layer over an impedance wall

The effect of viscosity and thermal conduction on the acoustics in a shear layer above an impedance wall is investigated numerically and asymptotically by solving the compressible linearised Navier-Stokes equations. It is found that viscothermal effects can be as important as shear, and therefore including shear while neglecting viscothermal effects by solving the linearised Euler equations is questionable. In particular, the damping rate of upstream propagating waves is found to be dramatically under-predicted by the LEE in certain instances. The effects of viscosity on stability are also found to be important. Short wavelength disturbances are stabilised by viscosity, greatly altering the characteristic wavelength and maximum growth rate of instability. For the parameters typical of aeroacoustic simulations considered here, the Reynolds number below which the flow stabilizes ranges from $10^5$ to $10^7$. By assuming a thin but nonzero-thickness boundary layer, asymptotic analysis leads to a system of boundary layer governing equations for the acoustics. This system may be solved numerically to produce an effective impedance boundary condition, applicable at the wall of a uniform inviscid flow, that accounts for both the shear and viscosity within the boundary layer. An alternative asymptotic analysis in the high frequency limit yields a different set of equations with analytic solutions. The acoustic mode shapes and axial wavenumbers from both asymptotic analyses compare well with numerical solutions of the full LNSE. A closed-form effective impedance boundary condition is derived from the high-frequency asymptotics, suitable for application in frequency-domain numerical simulations. Finally, surface waves are considered, and it is shown that a viscous flow over an impedance lining supports a greater number of surface wave modes than an inviscid flow.

physics.flu-dyn↗