Showing papers in "Journal of Engineering Mathematics in 2007"

An explicit construction of interpolation nodes on the simplex

Abstract: An open question concerns the spatial distribution of nodes that are suitable for high-order Lagrange interpolation on the triangle and tetrahedron. Several current methods used to produce nodal sets with small Lebesgue constants are recalled. A new approach is presented for building nodal distributions of arbitrary order, that is based on curvilinear finite-element techniques. Numerical results are shown which demonstrate that, despite the explicit nature of this construction, the resulting node sets are well suited for interpolation and competitive with existing sets for up to tenth-order polynomial interpolation. Matlab scripts which evaluate the node distributions on the equilateral triangle are included.

A brief historical perspective of the Wiener–Hopf technique

Abstract: It is a little over 75 years since two of the most important mathematicians of the 20th century collaborated on finding the exact solution of a particular equation with semi-infinite convolution type integral operator. The elegance and analytical sophistication of the method, now called the Wiener–Hopf technique, impress all who use it. Its applicability to almost all branches of engineering, mathematical physics and applied mathematics is borne out by the many thousands of papers published on the subject since its conception. The Wiener–Hopf technique remains an extremely important tool for modern scientists, and the areas of application continue to broaden. This special issue of the Journal of Engineering Mathematics is dedicated to the work of Wiener and Hopf, and includes a number of articles which demonstrate the relevance of the technique to a representative range of model problems.

On the accuracy of finite-difference solutions for nonlinear water waves

Abstract: This paper considers the relative accuracy and efficiency of low- and high-order finite-difference discretisations of the exact potential-flow problem for nonlinear water waves. The method developed is an extension of that employed by Li and Fleming (Coastal Engng 30: 235–238, 1997) to allow arbitrary-order finite-difference schemes and a variable grid spacing. Time-integration is performed using a fourth-order Runge–Kutta scheme. The linear accuracy, stability and convergence properties of the method are analysed and high-order schemes with a stretched vertical grid are found to be advantageous relative to second-order schemes on an even grid. Comparison with highly accurate periodic solutions shows that these conclusions carry over to nonlinear problems and that the advantages of high-order schemes improve with both increasing nonlinearity and increasing accuracy tolerance. The combination of non-uniform grid spacing in the vertical and fourth-order schemes is suggested as optimal for engineering purposes.

Penetration of flexural waves through a periodically constrained thin elastic plate in vacuo and floating on water

Abstract: The subject of this paper, the scattering of flexural waves by constrained elastic plates floating on water is relatively new and not an area that Professor Newman has worked in, as far as the authors are aware. However, in two respects there are connections to his own work. The first is the reference to his work with H. Maniar on the exciting forces on the elements of a long line of fixed vertical bottom-mounted cylinders in waves. In their paper (J Fluid Mech 339 (1997) 309–329) they pointed out the remarkable connection between the large forces on cylinders near the centre of the array at frequencies close to certain trapped-mode frequencies, which had been discovered earlier, and showed that there was another type of previously unknown trapped mode, which gave rise to large forces. In Sect. 6 of this paper the ideas described by Maniar and Newman are returned to and it is shown how the phenomenon of large forces is related to trapped, or standing Rayleigh–Bloch waves, in the present context of elastic waves. But there is a more general way in which the paper relates to Professor Newman and that is in the flavour and style of the mathematics that are employed. Thus extensive use has been made of classical mathematical methods including integral-transform techniques, complex-function theory and the use of special functions in a manner which reflects that used by Professor Newman in many of his important papers on ship hydrodynamics and related fields.

Acoustic scattering at a hard-soft lining transition in a flow duct

Abstract: An explicit Wiener-Hopf solution is derived to describe the scattering of sound at a hard-soft wall impedance transition at x = 0, say, in a circular duct with uniform mean flow of Mach number M. A mode, incident from the upstream hard section, scatters at x = 0 into a series of reflected modes and a series of transmitted modes. Of particular interest is the role of a possible instability along the lined wall in combination with the edge singu- larity. If one of the "upstream" running modes is to be interpreted as a downstream-running instability, an extra degree of freedom in the Wiener-Hopf analysis occurs that can be resolved by application of some form of Kutta condition at x = 0, for example a more stringent edge condition where wall streamline deflection h = O(x 3/2 ) at the downstream side. In general, the effect of this Kutta condition is significant, but it is particularly large for the plane wave at low frequencies and should therefore be easily measurable. For small Helmholtz numbers, the reflection coefficient modulus |R001| tends to (1 + M)/(1 − M) without and to 1 with Kutta condition, while the end correction tends to ∞ without and to a finite value with Kutta condition. This is exactly the same behaviour as found for reflection at a pipe exit with flow, irrespective if this is uniform or jet flow. Although the presence of the instability in the model is hardly a question anymore since it has been confirmed numerically, a proper mathematical causality analysis is still not totally watertight. Therefore, the limit of a vortex sheet, separating zero flow from mean flow, approaching the wall has been explored. Indeed, this confirms that the Helmholtz unstable mode of the free vortex sheet transforms into the suspected mode and remains unstable. As the lined-wall vortex-sheet model predicts unstable behaviour for which experimental evidence is at best rare and indirect, the question may be raised if this model is indeed a consistent simplification of reality, doing justice to the double limit of small perturbations and a thin boundary layer. Numerical time-domain methods suffer from this instability and it is very important to decide whether the instability is at least physically genuine. Experiments based on the present problem may provide a handle to resolve this stubborn question.

Nonlinear viscous liquid jets from a rotating orifice

Abstract: A liquid jet follows a curved trajectory when the orifice from which the jet emerges is rotating. Surface-tension-driven instabilities cause the jet to lose coherence and break to form droplets. The sizes of the drops formed from such jets are in general not uniform, ranging from drops with diameters of the order of the jet diameter to droplets with diameters which are several orders of magnitude smaller. This presentation details a theoretical investigation of the effects of changing operating parameters on the break-up of curved liquid jets in stagnant air at room temperature and pressure. The Navier–Stokes equations are solved in this system with the usual viscous free-surface boundary conditions, using an asymptotic method based upon a slender-jet assumption, which is clearly appropriate from experimental observations of the jet. Nonlinear temporal simulations of the break-up of the liquid jets using slender theory are also presented. These simulations based upon both a steady-trajectory assumption, and the more general equations which allow for an unsteady trajectory, show all the break-up modes viewed in experiments. Satellite-droplet formation is also considered.

Regularization of inverse heat conduction by combination of rate sensor analysis and analytic continuation

Abstract: This paper describes some recent observations associated with (1) clarifying and expanding upon the integral relationship between temperature and heat flux in a half-space; (2) offering an analytic-continuation approach for estimating the surface temperature and heat flux in a one-dimensional geometry based on embedded measurements; and, (3) offering a novel digital filter that supports the use of analytic continuation based on a minimal number of embedded sensors. Key to future inverse analysis must be the proper understanding and generation of rate data associated with both the temperature and heat flux at the embedded location. For this paper, some results are presented that are theoretrically motivated but presently adapted to implement digital filtering. A pulsed surface heat flux is reconstructed by way of a single thermocouple sensor located at a well-defined embedded location in a half space. The proposed low-pass, Gaussian digital filter requires the specification of a cut-off frequency that is obtained by viewing the power spectra of the temperature signal as generated by the Discrete Fourier Transform (DFT). With this in hand, and through the use of an integral relationship between the local temperature and heat flux at the embedded location, the embedded heat flux can be accurately estimated. The time derivatives of the filtered temperature and heat flux are approximated by a simple finite-difference method to provide a sufficient number of terms required by the Taylor series for estimating (i.e., the projection) the surface temperature and heat flux. A numerical example demonstrates the accuracy of the proposed scheme. A series of appendices are offered that describe the mathematical details omitted in the body for ease of reading. These appendices contain important and subtle details germane to future studies.

A nonlinear magnetoelastic tube under extension and inflation in an axial magnetic field: numerical solution

Abstract: In the context of nonlinear magnetoelasticity theory very few boundary-value problems have been solved. The main problem that arises when a magnetic field is present, as compared with the purely elastic situation, is the difficulty of meeting the magnetic boundary conditions for bodies with finite geometry. In general, the extent of the edge effects is unknown a priori, and this makes it difficult to interpret experimental results in relation to the theory. However, it is important to make the connection between theory and experiment in order to develop forms of the magnetoelastic constitutive law that are capable of correlating with the data and can be used for making quantitative predictions. In this paper the basic problem of a circular cylindrical tube of finite length that is deformed by a combination of axial compression (or extension) and radial expansion (or contraction) and then subjected to an axial magnetic field is examined. Such a field cannot be uniform throughout, since the boundary conditions on the ends and the lateral surfaces of the tube would be incompatible in such circumstances. The resulting axisymmetric boundary-value problem is formulated and then solved numerically for the case (for simplicity of illustration) in which the deformation is not altered by the application of the magnetic field. The distribution of the magnetic-field components throughout the body and the surrounding space is determined in order to quantify the extent of the edge effects for both extension and compression of the tube.

Middle-field formulation for the computation of wave-drift loads

Abstract: New formulations of second-order wave loads contributed by a first-order wave field are developed by applying two variants of Stokes’s theorem and Gauss’s theorem to a formulation consisting of direct pressure integrations on a body’s hull which is called the near-field formulation. In addition to this direct formulation and the formulation derived from the momentum theorem called the far-field formulation, for the computation of drift (surge/sway) forces in horizontal directions and drift (yaw) moment around the vertical axis, one of new formulations is defined on the control surfaces surrounding the body and called the middle-field formulation. After a brief summary of both pressure-integration (near-field) and momentum (far-field) formulations, the development of the middle-field formulation involving control surfaces is described and complemented in detail in the appendices. The application of the new formulation shows that the near-field and far-field formulations are mathematically equivalent for wall-sided, as well as non-wall-sided bodies and under the condition that the mean yaw moments are expressed with respect to a space-fixed reference point. It is shown that the middle-field formulation is as robust as the far-field formulation and as general as the near-field formulation of second-order loads on a single body as well as on multiple bodies. Furthermore, the extension to the computation of a second-order oscillatory load, which is so far accessed only by the near-field formulation, is envisioned.

Second-order Wagner theory of wave impact

Abstract: The paper deals with the two-dimensional unsteady problem of the impact of a liquid parabola onto a rigid flat plate at a constant velocity. The liquid is assumed ideal and incompressible and its flow potential. The initial stage of the impact is the main concern in this study. The non-dimensional half-width of the contact region between the impacting liquid and the plate plays the role of a small parameter in this problem. The flow region is subdivided into four parts: (i) the main flow region, the dimension of which is of the order of the contact-region width, (ii) the jet-root region, where the curvature of the free surface is very high and the flow is strongly nonlinear, (iii) the jet region, where the flow is approximately one-dimensional, (iv) the far-field region, where the flow is approximately uniform at the initial stage of impact. A second-order solution in the main flow region has been derived and matched to the first-order inner solution in the jet-root region. The matching conditions provide an estimate of the dimension of the contact region for small time. Pressure distributions in both the main flow region and the inner region are derived. The accuracy of the obtained asymptotic formulae is estimated. The second-order hydrodynamic force acting on the plate is obtained and compared with available experimental data. A fairly good agreement is reported.

Nodal DG-FEM solution of high-order Boussinesq-type equations

Abstract: A discontinuous Galerkin finite-element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one horizontal dimension is presented. The continuous equations are discretized using nodal polynomial basis functions of arbitrary order in space on each element of an unstructured computational domain. A fourth-order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy and convergence of the model with both h (grid size) and p (order) refinement are confirmed for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar, and reflection of a steep solitary wave from a vertical wall. Test cases for two horizontal dimensions will be considered in future work.

The influence of gravity on the performance of planing vessels in calm water

Abstract: Usually gravity can be neglected for planing vessels at very high planing speed. However, if the planing speed becomes lower, the influence of gravity must be considered. A 2D+t theory with gravity effects is applied to study the steady performance of planing vessels at moderate planing speeds. In the framework of potential theory, a computer program based on a boundary-element method (BEM) in two dimensions is first developed, in which a new numerical model for the jet flow is introduced. The spray evolving from the free surface is cut to avoid the plunging breaker to impact on the underlying water. Further, flow separation along a chine line can be simulated. The BEM program is verified by comparing with similarity solutions and validated by comparing with drop tests of V-shaped cylinders. Then the steady motion of prismatic planing vessels is studied by using the 2D+t theory. The numerical results are compared with the results by Savitsky’s empirical formula and the experiments by Troesch. Significant nonlinearities in the restoring force coefficients can be seen from the results. Three-dimensional effects are discussed to explain the difference between the numerical results and the experimental results. Finally, in the comparison of results at high planing speed and moderate planing speed, it is shown that the gravity not only affects the free-surface profile around the hull, but also influences the hydrodynamic force on the hull surface.

On Helmholtz and higher-order resonance of twin floating bodies

Abstract: The Helmholtz mode and other symmetric modes of resonance of a moonpool between two heaving rectangular floating cylinders are investigated. The hydrodynamic behavior around these resonant modes is examined together with the associated mode shapes in the moonpool region. It is observed that near each of the resonance frequencies, the damping coefficient can vanish. The Helmholtz mode is characterized by a region of modest variation of added-mass value from negative to positive near the Helmholtz frequency. The peaks are, however, bounded with the cross-over point in sign corresponding to a bounded spike in damping. The higher-order resonant modes are characterized by the presence of standing waves in the moonpool, which leads to large spikes in the hydrodynamic behavior near the resonance frequencies. The Helmholtz frequency has a distinct value, while the higher-order resonances occur at fairly regular intervals of the frequency parameter, σ2(w − b)/g, where w − b is the moonpool gap. The parametric dependence of the hydrodynamic behavior on frequency and geometry is discussed. With best wishes to my colleague and good friend, Nick Newman, on the occasion of his 70th birthday. A leader and staunch supporter of marine hydrodynamics, Nick has expanded the reach and influence of this field through his insights and publications. His contributions have been wide-ranged and his graciousness to young researchers is exemplary. May he enjoy the best of health in the years to come. R.W. Yeung

Hydroelastic analysis of plates and some approximations

Abstract: The paper concerns the modelling of very large pontoon-type floating structures by thin beams and plates of shallow draft, excited by regular waves. It is shown how the classical theory of hydroelasticity, involving the concepts of added mass and damping associated with the structural responses, may be reconciled with more recent formulations. In the latter, coupled equations for displacement and total hydrodynamic pressure are solved directly, without the breakdown into diffraction and radiation problems. A numerical model is adopted based on a Galerkin approach, and the nature of the various components of hydrodynamic loading on a shallow draft beam is investigated. The approach is then extended to the case of thin plate in waves, where the hydrodynamic effects are fully three dimensional.

A 3D numerical model for computing non-breaking wave forces on slender piles

Abstract: In this paper a numerical model for water-wave-body interaction is validated by comparing the numerical results with laboratory data. The numerical model is based on Euler’s equation without considering the effects of energy dissipation. The Euler equations are solved by a two-step projection finite-volume scheme and the free-surface displacements are tracked by the volume-of-fluid method. The numerical model is used to simulate solitary waves as well as periodic waves and their interaction with a vertical slender pile. A very good agreement between the experimental data and numerical results is observed for the time history of free-surface displacement, fluid-particle velocity, and dynamic pressure on the pile.

Numerical simulation of the head-on collision of two equal-sized drops with van der Waals forces

Abstract: The head-on collision of two equal-sized drops in a hyperbolic flow is investigated numerically. An axisymmetric volume-of-fluid (VOF) method is used to simulate the motion of each drop toward a symmetry plane where it interacts and possibly coalesces with its mirror image. The volume-fraction boundary condition on the symmetry plane is manipulated to numerically control coalescence. Two new numerical methods have been developed to incorporate the van der Waals forces in the Navier–Stokes equations. One method employs a body force computed as the negative gradient of the van der Waals potential. The second method employs the van der Waals forces in terms of a disjoining pressure in the film depending on the film thickness. Results are compared to theory of thin-film rupture. Comparisons of the results obtained by the two methods at various values of the Hamaker constant show that the van der Waals forces calculated from the two methods have qualitatively similar effects on coalescence. A study of the influence of the van der Waals forces on the evolution and rupture of the film separating the drops reveals that the film thins faster under stronger van der Waals forces. Strong van der Waals forces lead to nose rupture, and small van der Waals forces lead to rim rupture. Increasing the Reynolds number causes a greater drop deformation and faster film drainage. Increasing the viscosity ratio slows film drainage, although the effect is small for small viscosity ratio.

Computing the zeros, maxima and inflection points of Chebyshev, Legendre and Fourier series: solving transcendental equations by spectral interpolation and polynomial rootfinding

Abstract: Recently, both companion-matrix methods and subdivision algorithms have been developed for finding the zeros of a truncated spectral series. Since the Chebyshev or Legendre coefficients of deriv- atives of a function f (x) can be computed by trivial recurrences from those of the function itself, it follows that finding the maxima, minima and inflection points of a truncated Chebyshev or Fourier series fN(x) is also a problem of finding the zeros of a polynomial when written in truncated Chebyshev series form, or computing the roots of a trigonometric polynomial. Widely scattered results are reviewed and a few previously unpublished ideas sprinkled in. There are now robust zerofinders for all species of spectral series. A transcendental function f (x) can be approximated arbitrarily well on a real interval by a truncated Chebyshev series fN(x) of sufficiently high degree N. It follows that through Chebyshev interpolation and Chebyshev rootfinders, it is now possible to easily find all the real roots on an interval for any smooth transcendental function.

Experimental and numerical analyses of sloshing flows

Abstract: Recently the demand for sloshing analyses is rising because of the construction of large LNG carriers and LNG platforms. This study considers the experimental and numerical observations of strongly nonlinear sloshing flows in ship cargo and their coupling effects with ship motion. Violent sloshing flows in experiments are observed, and two different numerical methods, the finite-difference method and smoothed-particle-hydrodynamics (SPH) method, are applied for the simulation of violent sloshing flows. Several physical issues are introduced in the analysis of sloshing flows, and the corresponding numerical models are described. This study demonstrates that physics-based numerical schemes are essential in the prediction of violent sloshing flows and sloshing-induced impact pressure. To study the sloshing effects on ship-motion, a ship-motion program based on an impulsive response function (IRF) is coupled with the developed numerical models for sloshing analysis. The results show that the nonlinearity of sloshing-induced forces and moments plays a critical role in the coupling effects.

Weak or strong nonlinearity: the vital issue

Abstract: Marine hydrodynamics is characterised by both weak nonlinearities, as seen for example in drift forces, and strong nonlinearities, as seen for example in wave breaking. In many cases their relative importance is still a controversial matter. The phenomenon of particle escape, seen in linear theory, appears to offer a guide to when strongly nonlinear effects will start to become important, and what will happen when they do. In the case of the “ringing” of vertical cylinders in steep waves, particle escape is shown to correspond approximately to local wave breaking, which leads to the cavitation responsible for “ringing”. Another example is rogue waves, where recent results from weakly nonlinear theory are disappointing, and also fail to explain the rogue waves seen in relatively shallow water, as in the data from the Draupner and Gorm platforms. Recent laboratory experiments, too, show wave crests continuing to grow in height after all frequency components have come into phase, which is inconsistent with weakly nonlinear theory. Particle escape, which is more frequent in shallow water, offers a simple alternative explanation for these observations, as well as for the violent motion at the wave crests, which often confuses rogue-wave data. Extreme wave crests have long been known to be strongly nonlinear, so it appears possible that rogue waves are primarily a strongly nonlinear phenomenon. Fully nonlinear computations of two interacting regular waves are presented, to explore further the connection between particle escape and wave breaking. They are combined with Monte-Carlo simulations of particle escape in hurricane conditions, and the very few measurements of large breaking waves during hurricanes. It is concluded that large breaking waves will have occurred about once per hour, and once per 100 h, respectively, in the recent hurricanes LILI and IVAN. These findings call into question the use of non-breaking wave models in the design codes for fixed steel offshore structures.

Transverse instability of gravity–capillary solitary waves

Abstract: Gravity–capillary solitary waves of depression that bifurcate at the minimum phase speed on water of finite or infinite depth, while stable to perturbations along the propagation direction, are found to be unstable to transverse perturbations on the basis of a long-wave stability analysis. This suggests a possible generation mechanism of the new class of gravity–capillary lumps recently shown to also bifurcate at the minimum phase speed.

Computing elliptic membrane high frequencies by Mathieu and Galerkin methods

Abstract: Resonant modes of an elliptic membrane are computed for a wide range of frequencies using a Galerkin formulation. Results are confirmed using Mathieu functions and finite-element methods. Algorithms and their implementations are described to handle Dirichlet or Neumann boundary conditions and draw animations or contour plots of the modal surfaces. The methods agree to four or more digit accuracy for the first one hundred modes. The effects of high function order and high frequency parameter upon the convergence of the modified Mathieu function series are discussed and quantified. The Galerkin method is conceptually simple and requires only an eigenvalue solver without the need of special functions.

Sloshing in a vertical circular cylindrical tank with an annular baffle. Part 2. Nonlinear resonant waves

Abstract: Weakly nonlinear resonant sloshing in a circular cylindrical baffled tank with a fairly deep fluid depth (depth/radius ratio ≥ 1) is examined by using an asymptotic modal method, which is based on the Moiseev asymptotic ordering. The method generates a nonlinear asymptotic modal system coupling the time-dependent displacements of the linear natural modes. Emphasis is placed on quantifying the effective frequency domains of the steady-state resonant waves occurring due to lateral harmonic excitations, versus the size and the location of the baffle. The forthcoming Part 3 will focus on the vorticity stress at the sharp baffle edge and related generalisations of the present nonlinear modal system.

Global stability of the rotating-disk boundary layer

Abstract: The global stability of the von KAirmAin boundary layer on the rotating disk is reviewed. For the genuine, radially inhomogeneous base flow, linearized numerical simulations indicate that convectively propagating forms of disturbance are predominant at all radii. The presence of absolute instability does not lead to the formation of any unstable linear global mode, even though the temporal growth rate of the absolute instability increases along the radial direction. Analogous behaviour can be found in the impulse solutions of a model amplitude equation, namely the linearized complex Ginzburg–Landau equation. These solutions show that, depending on the precise balance between spatial variations in the temporal growth rate and the corresponding shifts in the temporal frequency, globally stable behaviour can be obtained even in the presence of a strengthening absolute instability. The radial dependency of the absolute temporal frequency is sufficient to detune the disturbance oscillations at different radial positions, thus overcoming the radially increasing absolute growth, thereby giving rise to a stable global response. The origin of this form of behaviour can be traced to the fact that the cylindrical geometry of the rotating-disk flow dictates a choice of a globally valid time non-dimensionalization that, when properly employed, leads to a significant radial variation in the frequency for the absolute instability.

An anisotropic model of the Mullins effect

Abstract: The Mullins effect in rubber-like materials is inherently anisotropic. However, most constitutive models developed in the past are isotropic. These models cannot describe the anisotropic stress-softening effect, often called the Mullins effect. In this paper a phenomenological three-dimensional anisotropic model for the Mullins effect in incompressible rubber-like materials is developed. The terms, damage function and damage point, are introduced to facilitate the analysis of anisotropic stress softening in rubber-like materials. A material parametric energy function which depends on the right stretch tensor and written explicitly in terms of principal stretches and directions is postulated. The material parameters in the energy function are symmetric second-order damage and shear-history tensors. A class of energy functions and a specific form for the constitutive equation are proposed which appear to simplify both the analysis of the three-dimensional model and the calculation of material constants from experimental data. The behaviour of tensional and compressive ground-state Young’s moduli in uniaxial deformations is discussed. To further justify our model we show that the proposed model produces a transversely anisotropic non-virgin material in a stress-free state after a simple tension deformation. The proposed anisotropic theory is applied to several types of homogenous deformations and the theoretical results obtained are consistent with expected behaviour and compare well with several experimental data.

Transition to turbulence in rotating-disk boundary layers—convective and absolute instabilities

Abstract: This work is an experimental study of mechanisms for transition to turbulence in the boundary layer on a rotating disk. In one case, the focus was on a triad resonance between pairs of traveling cross-flow modes and a stationary cross-flow mode. The other was on the temporal growth of traveling modes through a linear absolute instability mechanism first discovered by Lingwood (1995, J Fluid Mech 314:373–405). Both research directions made use of methods for introducing controlled initial disturbances. One used a distributed array of ink dots placed on the disk surface to enhance a narrow band of azimuthal and radial wave numbers of both stationary and traveling modes. The size of the dots was small so that the disturbances they produce were linear. Another approach introduced temporal disturbances by a short-duration air pulse from a hypodermic tube located above the disk and outside the boundary layer. Hot-wire sensors primarily sensitive to the azimuthal velocity component, were positioned at different spatial (r,θ) locations on the disk to document the growth of disturbances. Spatial correlation measurements were used with two simultaneous sensors to obtain wavenumber vectors. Cross-bicoherence was used to identify three-frequency phase locking. Ensemble averages conditioned on the air pulses revealed wave packets that evolved in time and space. The space–time evolution of the leading and trailing edges of the wave packets were followed past the critical radius for the absolute instability, r c A . With documented linear amplitudes, the spreading of the disturbance wave packets did not continue to grow in time as r c A was approached. Rather, the spreading of the trailing edge of the wave packet decelerated and asymptotically approached a constant. This result supports the linear DNS simulations of Davies and Carpenter (2003, J Fluid Mech 486:287–329) who concluded that the absolute instability mechanism does not result in a global mode, and that linear-disturbance wave packets are dominated by the convective instability. In contrast, wave-number matching between traveling cross-flow modes confirmed a triad resonance that lead to the growth of a low azimuthal number (n = 4) stationary mode. At transition, this mode had the largest amplitude. Signs of this mechanism can be found in past flow visualization of transition to turbulence in rotating disk flows.

Aliasing errors due to quadratic nonlinearities on triangular spectral /hp element discretisations

Abstract: In this paper, consideration is given to how aliasing errors, introduced when evaluating nonlinear products, inexactly affect the solution of Galerkin spectral/hp element polynomial discretisations on triangles. A theoretical discussion is presented of how aliasing errors are introduced by a collocation projection onto a set of quadrature points insufficient for exact integration, and consider interpolation projections to geometrically symmetric ollocation points. The discussion is corroborated by numerica examples that elucidate the key features. The study is first motivated with a review of aliasing errors introduced in one-dimensional spectral-element methods (these results extend naturally to tensor-product quadrilaterals and hexahedra.) Within triangular domains two commonly used expansions are a hierarchical, or modal, expansion based on a rotationally non-symmetric collapsed-coordinate system, and a Lagrange expansion based on a set of rotationally symmetric nodal points. Whilst both expansions span the same polynomial space, the construction of the two bases numerically motivates a different set of collocation points for use in the collocation projection of a nonlinear product. The purpose of this paper is to compare these two collocation projections. The analysis and results show that aliasing errors produced using a collocation projection on the rotationally non-symmetric, collapsed-coordinate system are significantly smaller than those for a collocation projection using the rotationally symmetric nodal points. In the case of the collapsed coordinate projection, if the Gaussian quadrature order employed is less than half the polynomial order of the integrand, then it is possible for the aliasing error to modify the constant mode of the expansion and therefore affect the conservation property of the approximation. However, the use of a collocation projection onto a polynomial expansion associated with a set of rotationally symmetric nodal points within the triangle is always observed to be non-conservative. Nevertheless, the rotationally symmetric collocation will maintain the overall symmetry of the triangular region, which is not typically the case when a collapsed coordinate quadrature projection is used.

Time-dependent modelling and asymptotic analysis of electrochemical cells

Abstract: A (time-dependent) model for an electrochemical cell, comprising a dilute binary electrolytic solution between two flat electrodes, is formulated. The method of matched asymptotic expansions (taking the ratio of the Debye length to the cell width as the small asymptotic parameter) is used to derive simplified models of the cell in two distinguished limits and to systematically derive the Butler–Volmer boundary conditions. The first limit corresponds to a diffusion-limited reaction and the second to a capacitance-limited reaction. Additionally, for sufficiently small current flow/large diffusion, a simplified (lumped-parameter) model is derived which describes the long-time behaviour of the cell as the electrolyte is depleted. The limitations of the dilute model are identified, namely that for sufficiently large half-electrode potentials it predicts unfeasibly large concentrations of the ion species in the immediate vicinity of the electrodes. This motivates the formulation of a second model, for a concentrated electrolyte. Matched asymptotic analyses of this new model are conducted, in distinguished limits corresponding to a diffusion-limited reaction and a capacitance-limited reaction. These lead to simplified models in both of which a system of PDEs, in the outer region (the bulk of the electrolyte), matches to systems of ODEs, in inner regions about the electrodes. Example (steady-state) numerical solutions of the inner equations are presented.

Strip theory for underwater vehicles in water of finite depth

Abstract: This paper addresses the need to know the unsteady forces and moments on an underwater vehicle in finite-depth water, at small enough submergences for it to be influenced by sea waves. The forces are those due to the waves themselves, as well as the radiation forces due to unsteady vehicle motions. Knowledge of these forces and the mass distribution of the vehicle allow solution of the equations of motion at a single-frequency. Since the theory is linear, any incident wave field can be decomposed into the sum of many individual single-frequency sinusoidal waves. The motions due to each frequency component can then be added together to obtain the total predicted vehicle motions. The wave forces are due to the undisturbed sea wave plus those due to the diffracted wave necessary to satisfy boundary conditions on the vehicle. The long-used strip theory for ships, with the inviscid-flow approximation, is modified for finite depth and inclusion of lift forces on the vehicle fins. The two-dimensional solutions for the forces on each strip are found by a different method than is commonly used for strip theory. This form of the theory is easier to deal with and requires much less computing time than a fully three-dimensional approach. Experiments are conducted and their results are compared with the theory. Excellent agreement is found between the theoretical and experimental wave forces, including the diffracted wave. It is shown that inclusion of the forces on the fins not only improves the theoretical wave forces, but also brings the results of theory for the radiation forces and moments due to vehicle motions much closer to the experimental values that the theory without inclusion of fin lift forces.

Motion trapping structures in the three-dimensional water-wave problem

Abstract: Trapped modes in the linearized water-wave problem are free oscillations of finite energy in an unbounded fluid with a free surface. It has been known for some time that such modes are supported by certain structures when held fixed, but recently it has been demonstrated that in two dimensions trapped modes are also possible for freely floating structures that are able to respond to the hydrodynamic forces acting upon them. For a freely floating structure such a mode is a coupled oscillation of the fluid and the structure that, in the absence of viscosity, persists for all time. Here previous work on the two-dimensional problem is extended to give motion trapping structures in the three-dimensional water-wave problem that have a vertical axis of symmetry.

Primary crossflow vortices, secondary absolute instabilities and their control in the rotating-disk boundary layer

Abstract: The three-dimensional boundary layer produced by a disk rotating in otherwise still fluid is analytically investigated and its stability properties are systematically established. Using a local parallel flow approximation, finite-amplitude primary travelling vortices governed by a nonlinear dispersion relation are obtained. A secondary stability analysis yields the secondary linear dispersion relation and the secondary absolute growth rate, which determines the long-term stability of the primary nonlinear vortex-trains. By using these local characteristics, spatially developing global patterns of crossflow vortices are derived by employing asymptotic techniques. This approach accounts for both the self-sustained behaviour, exhibiting a sharp transition from laminar to turbulent flow, and the spatial response to external harmonic forcing, for which onset of nonlinearity and transition both depend on the forcing parameters. Based on these results, an open-loop control method is described in detail. Its aim is not to suppress the primary fluctuations but rather to enhance them and to tune them to externally imposed frequency and modenumber, and thereby to delay onset of secondary absolute instability and transition. It is shown that transition can be delayed by more than 100 boundary-layer units.