Local burning velocity in a Bunsen jet flame

A PIV-based system has been set-up for the simultaneous measurement of the local burning velocity of premixed flames and the flame stretch due to the flame front curvature and the incoming flow strain rate. For moderately short jet flames, these measurements allow an indirect determination of the Markstein length, according to Clavin and Joulin (C–J) theory. For tall flames, the flame curvature becomes relatively large in a region around the tip where the C–J theory breaks down. However, our experiments confirm the appearance of a new linear relation between burning velocity and curvature at the flame tip. This relation defines a new proportionality factor which is probably associated to the evolution from rounded tips to slender tips when the jet velocity is increased.

Experimental determination of burning velocity in metal dust explosions

The modeling of metal dust explosion phenomenon is important in order to safeguard industries from potential accidents. A key parameter of these models is the burning velocity, which represents the consumption rate of the reactants by the flame front, during the combustion process. This work is focused on the experimental determination of aluminium burning velocity, through an alternative method, called "Direct method". The study of the methods used and the results obtained is preceded by a general analysis on dust explosion phenomenon, flame propagation phenomenon, characteristics of the metals combustion process and standard methods for determining the burning velocity. The “Direct method” requires a flame propagating through a tube recorded by high-speed cameras. Thus, the flame propagation test is carried out inside a vertical prototype made of glass. The study considers two optical technique: the direct visualization of the light emitted by the flame and the Particle Image Velocimetry (PIV) technique. These techniques were used simultaneously and allow the determination of two velocities: the flame propagation velocity and the flow velocity of the unburnt mixture. Since the burning velocity is defined by these two quantities, its direct determination is done by substracting the flow velocity of the fresh mixture from the flame propagation velocity. The results obtained by this direct determination, are approximated by a linear curve and different non-linear curves, which show a fluctuating behaviour of burning velocity. Furthermore, the burning velocity is strongly affected by turbulence. Turbulence intensity can be evaluated from PIV technique data. A comparison between burning velocity and turbulence intensity highlighted that both have a similar trend.

Numerical investigation of effects of addition of ethanol to gasoline on Laminar Flame Speed (LFS), autoignition, and wall quenching

Since the advent of automobiles, alcohol has been considered a possible engine fuel1,2. With the recent increased concern about the high price of crude oil due to fluctuating supply and demand and environmental issues, interest in alcohol based fuels has increased2,3. However, using pure alcohols or blends with conventional fuels in high percentages requires changes to the engine and fuel system design2. This leads to the need for a simple and accurate conventional fuels-alcohol blends combustion models that can be used in developing parametric burn rate and knock combustion models for designing more efficient Spark Ignited (SI) engines. To contribute to this understanding, numerical simulations were performed to obtain detailed characteristics of Gasoline-Ethanol blends with respect to Laminar Flame Speed (LFS), autoignition and Flame-Wall interactions. The one-dimensional premixed flame code CHEMKIN® was applied to simulate the burning velocity and autoignition characteristics using the freely propagating model and closed homogeneous reactor model respectively. Computational Fluid Dynamics (CFD) was used to obtain detailed flow, temperature, and species fields for Flame-wall interactions. A semi-detailed validated chemical kinetic model for a gasoline surrogate fuel developed by Andrae and Head4 was used for the study of LFS and Autoignition. For the quenching study, a skeletal chemical kinetic mechanism of gasoline surrogate, having 50 species and 174 reactions was used. The surrogate fuel was defined as a mixture of pure n-heptane, isooctane, and toluene. For LFS study, the ethanol volume fraction was varied from 0 to 85%, initial pressure from 4 to 8 bar, initial temperature from 300 to 900K, and dilution from 0 to 32%. Whereas for Autoignition study, the ethanol volume fraction was varied between 0 to 85%, initial pressure was varied between 20 to 60 bar, initial temperature was varied between 800 to 1200K, and the dilution was varied between 0 to 32% at equivalence ratios of 0.5, 1.0 and 1.5 to represent the in-cylinder conditions of a SI engine. For quenching study three Ethanol blends, namely E0, E25 and E85 are described in detail at an initial pressure of 8 atm and 17 atm. Initial wall temperature was taken to be 400 K. Quenching thicknesses and heat fluxes to the wall were computed. The laminar flame speed was found to increase with ethanol concentration and temperature but decrease with pressure and dilution. The autoignition time was found to increase with ethanol concentration at lower temperatures but was found to decrease marginally at higher temperatures. The autoignition time was also found to decrease with pressure and equivalence ratio but increase with dilution. The average quenching thickness was found to decrease with an increase in Ethanol concentration in the blend. Heat flux to the wall increased with increase in ethanol percentage in the blend and at higher initial pressures. Whereas the wall heat flux decreased with an increase in dilution. Unburned Hydrocarbon (UHC) and CO % was also found to decrease with ethanol concentration in the blend.

Criteria for Ballast Flight Initiation Induced by High Speed Trains

One of the phenomena that limit the velocity of trains in high speed lines is the so- called “ballast pick-up”. It is a ballast train-induced-wind erosion (or BATIWE) that can produce damage to the train under body and the infrastructure surrounding the tracks. The analysis of the measurements taken during several passes of the train allows for a criterion of ballast flight initiation to be obtained. The first rotation of a ballast stone occurs when the impulse given to the stone (arising from the aerodynamic loading produced by the wind gust genera ted by the passing train) overpasses a critical impulse. This impulse depends on the physical properties of the stone (mass, shape, moment of inertia, etc. ...) and its posture on the track bed. The aim of this paper is to report on the experimental results obtained in the ADIF’S Brihuega (Guadalajara) test station, in the Madrid to Barcelona high speed line, and the way they can be used to support the feasibility of the definition of a criterion to evaluate the BA TIWE capability of trains. The results obtained show the feasibility of the proposed method, and contribute to a method of BATIWE characterization, which can be relevant for the development of train interoperability standardization.

A model for turbulent combustion simulation of large scale hydrogen explosions

El agotamiento, la ausencia o, simplemente, la incertidumbre sobre la cantidad de las reservas de combustibles fósiles se añaden a la variabilidad de los precios y a la creciente inestabilidad en la cadena de aprovisionamiento para crear fuertes incentivos para el desarrollo de fuentes y vectores energéticos alternativos. El atractivo de hidrógeno como vector energético es muy alto en un contexto que abarca, además, fuertes inquietudes por parte de la población sobre la contaminación y las emisiones de gases de efecto invernadero. Debido a su excelente impacto ambiental, la aceptación pública del nuevo vector energético dependería, a priori, del control de los riesgos asociados su manipulación y almacenamiento. Entre estos, la existencia de un innegable riesgo de explosión aparece como el principal inconveniente de este combustible alternativo. Esta tesis investiga la modelización numérica de explosiones en grandes volúmenes, centrándose en la simulación de la combustión turbulenta en grandes dominios de cálculo en los que la resolución que es alcanzable está fuertemente limitada. En la introducción, se aborda una descripción general de los procesos de explosión. Se concluye que las restricciones en la resolución de los cálculos hacen necesario el modelado de los procesos de turbulencia y de combustión. Posteriormente, se realiza una revisión crítica de las metodologías disponibles tanto para turbulencia como para combustión, que se lleva a cabo señalando las fortalezas, deficiencias e idoneidad de cada una de las metodologías. Como conclusión de esta investigación, se obtiene que la única estrategia viable para el modelado de la combustión, teniendo en cuenta las limitaciones existentes, es la utilización de una expresión que describa la velocidad de combustión turbulenta en función de distintos parámetros. Este tipo de modelos se denominan Modelos de velocidad de llama turbulenta y permiten cerrar una ecuación de balance para la variable de progreso de combustión. Como conclusión también se ha obtenido, que la solución más adecuada para la simulación de la turbulencia es la utilización de diferentes metodologías para la simulación de la turbulencia, LES o RANS, en función de la geometría y de las restricciones en la resolución de cada problema particular. Sobre la base de estos hallazgos, el crea de un modelo de combustión en el marco de los modelos de velocidad de la llama turbulenta. La metodología propuesta es capaz de superar las deficiencias existentes en los modelos disponibles para aquellos problemas en los que se precisa realizar cálculos con una resolución moderada o baja. Particularmente, el modelo utiliza un algoritmo heurístico para impedir el crecimiento del espesor de la llama, una deficiencia que lastraba el célebre modelo de Zimont. Bajo este enfoque, el énfasis del análisis se centra en la determinación de la velocidad de combustión, tanto laminar como turbulenta. La velocidad de combustión laminar se determina a través de una nueva formulación capaz de tener en cuenta la influencia simultánea en la velocidad de combustión laminar de la relación de equivalencia, la temperatura, la presión y la dilución con vapor de agua. La formulación obtenida es válida para un dominio de temperaturas, presiones y dilución con vapor de agua más extenso de cualquiera de las formulaciones previamente disponibles. Por otra parte, el cálculo de la velocidad de combustión turbulenta puede ser abordado mediante el uso de correlaciones que permiten el la determinación de esta magnitud en función de distintos parámetros. Con el objetivo de seleccionar la formulación más adecuada, se ha realizado una comparación entre los resultados obtenidos con diversas expresiones y los resultados obtenidos en los experimentos. Se concluye que la ecuación debida a Schmidt es la más adecuada teniendo en cuenta las condiciones del estudio. A continuación, se analiza la importancia de las inestabilidades de la llama en la propagación de los frentes de combustión. Su relevancia resulta significativa para mezclas pobres en combustible en las que la intensidad de la turbulencia permanece moderada. Estas condiciones son importantes dado que son habituales en los accidentes que ocurren en las centrales nucleares. Por ello, se lleva a cabo la creación de un modelo que permita estimar el efecto de las inestabilidades, y en concreto de la inestabilidad acústica-paramétrica, en la velocidad de propagación de llama. El modelado incluye la derivación matemática de la formulación heurística de Bauwebs et al. para el cálculo de la incremento de la velocidad de combustión debido a las inestabilidades de la llama, así como el análisis de la estabilidad de las llamas con respecto a una perturbación cíclica. Por último, los resultados se combinan para concluir el modelado de la inestabilidad acústica-paramétrica. Tras finalizar esta fase, la investigación se centro en la aplicación del modelo desarrollado en varios problemas de importancia para la seguridad industrial y el posterior análisis de los resultados y la comparación de los mismos con los datos experimentales correspondientes. Concretamente, se abordo la simulación de explosiones en túneles y en contenedores, con y sin gradiente de concentración y ventilación. Como resultados generales, se logra validar el modelo confirmando su idoneidad para estos problemas. Como última tarea, se ha realizado un analisis en profundidad de la catástrofe de Fukushima-Daiichi. El objetivo del análisis es determinar la cantidad de hidrógeno que explotó en el reactor número uno, en contraste con los otros estudios sobre el tema que se han centrado en la determinación de la cantidad de hidrógeno generado durante el accidente. Como resultado de la investigación, se determinó que la cantidad más probable de hidrogeno que fue consumida durante la explosión fue de 130 kg. Es un hecho notable el que la combustión de una relativamente pequeña cantidad de hidrogeno pueda causar un daño tan significativo. Esta es una muestra de la importancia de este tipo de investigaciones. Las ramas de la industria para las que el modelo desarrollado será de interés abarca la totalidad de la futura economía de hidrógeno (pilas de combustible, vehículos, almacenamiento energético, etc) con un impacto especial en los sectores del transporte y la energía nuclear, tanto para las tecnologías de fisión y fusión. ABSTRACT The exhaustion, absolute absence or simply the uncertainty on the amount of the reserves of fossil fuels sources added to the variability of their prices and the increasing instability and difficulties on the supply chain are strong incentives for the development of alternative energy sources and carriers. The attractiveness of hydrogen in a context that additionally comprehends concerns on pollution and emissions is very high. Due to its excellent environmental impact, the public acceptance of the new energetic vector will depend on the risk associated to its handling and storage. Fromthese, the danger of a severe explosion appears as the major drawback of this alternative fuel. This thesis investigates the numerical modeling of large scale explosions, focusing on the simulation of turbulent combustion in large domains where the resolution achievable is forcefully limited. In the introduction, a general description of explosion process is undertaken. It is concluded that the restrictions of resolution makes necessary the modeling of the turbulence and combustion processes. Subsequently, a critical review of the available methodologies for both turbulence and combustion is carried out pointing out their strengths and deficiencies. As a conclusion of this investigation, it appears clear that the only viable methodology for combustion modeling is the utilization of an expression for the turbulent burning velocity to close a balance equation for the combustion progress variable, a model of the Turbulent flame velocity kind. Also, that depending on the particular resolution restriction of each problem and on its geometry the utilization of different simulation methodologies, LES or RANS, is the most adequate solution for modeling the turbulence. Based on these findings, the candidate undertakes the creation of a combustion model in the framework of turbulent flame speed methodology which is able to overcome the deficiencies of the available ones for low resolution problems. Particularly, the model utilizes a heuristic algorithm to maintain the thickness of the flame brush under control, a serious deficiency of the Zimont model. Under the approach utilized by the candidate, the emphasis of the analysis lays on the accurate determination of the burning velocity, both laminar and turbulent. On one side, the laminar burning velocity is determined through a newly developed correlation which is able to describe the simultaneous influence of the equivalence ratio, temperature, steam dilution and pressure on the laminar burning velocity. The formulation obtained is valid for a larger domain of temperature, steam dilution and pressure than any of the previously available formulations. On the other side, a certain number of turbulent burning velocity correlations are available in the literature. For the selection of the most suitable, they have been compared with experiments and ranked, with the outcome that the formulation due to Schmidt was the most adequate for the conditions studied. Subsequently, the role of the flame instabilities on the development of explosions is assessed. Their significance appears to be of importance for lean mixtures in which the turbulence intensity remains moderate. These are important conditions which are typical for accidents on Nuclear Power Plants. Therefore, the creation of a model to account for the instabilities, and concretely, the acoustic parametric instability is undertaken. This encloses the mathematical derivation of the heuristic formulation of Bauwebs et al. for the calculation of the burning velocity enhancement due to flame instabilities as well as the analysis of the stability of flames with respect to a cyclic velocity perturbation. The results are combined to build a model of the acoustic-parametric instability. The following task in this research has been to apply the model developed to several problems significant for the industrial safety and the subsequent analysis of the results and comparison with the corresponding experimental data was performed. As a part of such task simulations of explosions in a tunnel and explosions in large containers, with and without gradient of concentration and venting have been carried out. As a general outcome, the validation of the model is achieved, confirming its suitability for the problems addressed. As a last and final undertaking, a thorough study of the Fukushima-Daiichi catastrophe has been carried out. The analysis performed aims at the determination of the amount of hydrogen participating on the explosion that happened in the reactor one, in contrast with other analysis centered on the amount of hydrogen generated during the accident. As an outcome of the research, it was determined that the most probable amount of hydrogen exploding during the catastrophe was 130 kg. It is remarkable that the combustion of such a small quantity of material can cause tremendous damage. This is an indication of the importance of these types of investigations. The industrial branches that can benefit from the applications of the model developed in this thesis include the whole future hydrogen economy, as well as nuclear safety both in fusion and fission technology.

Experimental characterization of hydrogen containing flames

The current environmental and socio-economic situation promotes the development of carbon-neutral and sustainable solutions for energy supply. In this framework, the use of hydrogen has been largely indicated as a promising alternative. However, safety aspects are of concern for storage and transportation technologies. Indeed, the current know-how promotes its transportation via pipeline as compressed gas. However, the peculiar properties of hydrogen make the selection of suitable materials challenging. For these reasons, dilution with less reactive species has been considered a short and medium solution. As a way of example, methane-hydrogen mixtures are currently transported via pipelines. In this case, the hydrogen content is limited to 20% in volume, thus keeping the dependence on natural gas sources. On the contrary, hydrogen can be conveniently transported by mixing it with carbon dioxide deriving from carbon capture and storage technologies. In this sense, the interactions between hydrogen and carbon dioxide have been poorly studied. In particular, the effects of composition and operative conditions in the case of accidental release or for direct use in the energy supply chain are unknown. For these reasons, the present work was devoted to the characterization of the chemical phenomena ruling the system. To this aim, laminar flames containing hydrogen and carbon dioxide in the air were investigated experimentally and numerically. Different detailed kinetic mechanisms largely validated were considered at this stage. Significant discrepancies were observed among numerical and experimental data, especially once a fuel consisting of 40%v of hydrogen was studied. This deviation was attributed to the formation of a cellular flame increasing the overall reactivity. Hence, this observation suggests the need for combined models accounting for peculiar physical phenomena and detailed kinetic mechanisms characterizing the hydrogen-containing flames.

On the generalized eigenvalue problem for the Rossby wave vertical velocity in the presence of mean flow and topography

In a series of papers, Killworth and Blundell have proposed to study the effects of a background mean flow and topography on Rossby wave propagation by means of a generalized eigenvalue problem formulated in terms of the vertical velocity, obtained from a linearization of the primitive equations of motion. However, it has been known for a number of years that this eigenvalue problem contains an error, which Killworth was prevented from correcting himself by his unfortunate passing and whose correction is therefore taken up in this note. Here, the author shows in the context of quasigeostrophic (QG) theory that the error can ulti- mately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a non- linear one (the eigenvalue appears both in the numerator and denominator), unlike that for the pressure. The reason that this nonlinear term is lacking in the Killworth and Blundell theory comes from neglecting the depth dependence of a depth-dependent term. This nonlinear term is shown on idealized examples to alter significantly the Rossby wave dispersion relation in the high-wavenumber regime but is otherwise irrelevant in the long-wave limit, in which case the eigenvalue problems for the vertical velocity and pressure are both linear. In the general dispersive case, however, one should first solve the generalized eigenvalue problem for the pressure vertical structure and, if needed, diagnose the vertical velocity vertical structure from the latter.

A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Age-related differences in time-limit performance and force platform-based balance measures during one-leg stance

Poor posture control has been associated with an increased risk of falls and mobility disability among older adults. This study was conducted to assess the test-retest reliability and sensitivity to group differences regarding the time-limit (TLimit) of one-leg standing and selected balance parameters obtained with a force platform in older and young adults. A secondary purpose was to assess the relationship between TLimit and these balance parameters. Twenty-eight healthy older adults (age: 69±5years) and thirty young adults (age: 21±4years) participated in this study. Two one-leg stance tasks were performed: (1) three trials of 30s maximum and (2) one TLimit trial. The following balance parameters were computed: center of pressure area, RMS sway amplitude, and mean velocity and mean frequency in both the anterio-posterior and medio-lateral directions. All balance parameters obtained with the force platform as well as the TLimit variable were sensitive to differences in balance performance between older and young adults. The test-retest reliability of these measures was found to be acceptable (ICC: 0.40-0.85), with better ICC scores observed for mean velocity and mean frequency in the older group. Pearson correlations coefficients (r) between balance parameters and TLimit ranged from -0.16 to -0.54. These results add to the current literature that can be used in the development of measurement tools for evaluating balance in older and young adults. © 2013 Elsevier Ltd.

Beyond the diffraction limit of optical/IR interferometers I. Angular diameter and rotation parameters of Achernar from differential phases

Context. Spectrally resolved long-baseline optical/IR interferometry of rotating stars opens perspectives to investigate their fundamental parameters and the physical mechanisms that govern their interior, photosphere, and circumstellar envelope structures. Aims. Based on the signatures of stellar rotation on observed interferometric wavelength-differential phases, we aim to measure angular diameters, rotation velocities, and orientation of stellar rotation axes. Methods. We used the AMBER focal instrument at ESO-VLTI in its high-spectral resolution mode to record interferometric data on the fast rotator Achernar. Differential phases centered on the hydrogen Br gamma line (K band) were obtained during four almost consecutive nights with a continuous Earth-rotation synthesis during similar to 5h/night, corresponding to similar to 60 degrees position angle coverage per baseline. These observations were interpreted with our numerical code dedicated to long-baseline interferometry of rotating stars. Results. By fitting our model to Achernar's differential phases from AMBER, we could measure its equatorial radius R-eq = 11.6 +/- 0.3 R-circle dot, equatorial rotation velocity V-eq = 298 +/- 9 km s(-1), rotation axis inclination angle i = 101.5 +/- 5.2 degrees, and rotation axis position angle (from North to East) PA(rot) = 34.9 +/- 1.6 degrees. From these parameters and the stellar distance, the equatorial angular diameter circle divide(eq) of Achernar is found to be 2.45 +/- 0.09 mas, which is compatible with previous values derived from the commonly used visibility amplitude. In particular, circle divide(eq) and PA(rot) measured in this work with VLTI/AMBER are compatible with the values previously obtained with VLTI/VINCI. Conclusions. The present paper, based on real data, demonstrates the super-resolution potential of differential interferometry for measuring sizes, rotation velocities, and orientation of rotating stars in cases where visibility amplitudes are unavailable and/or when the star is partially or poorly resolved. In particular, we showed that differential phases allow the measurement of sizes up to similar to 4 times smaller than the diffraction-limited angular resolution of the interferometer.

Barriers to front propagation in ordered and disordered vortex flows

We present experiments on reactive front propagation in a two-dimensional (2D) vortex chain flow (both time-independent and time-periodic) and a 2D spatially disordered (time-independent) vortex-dominated flow. The flows are generated using magnetohydrodynamic forcing techniques, and the fronts are produced using the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. In both of these flows, front propagation is dominated by the presence of burning invariant manifolds (BIMs) that act as barriers, similar to invariant manifolds that dominate the transport of passive impurities. Convergence of the fronts onto these BIMs is shown experimentally for all of the flows studied. The BIMs are also shown to collapse onto the invariant manifolds for passive transport in the limit of large flow velocities. For the disordered flow, the measured BIMs are compared to those predicted using a measured velocity field and a three-dimensional set of ordinary differential equations that describe the dynamics of front propagation in advection-reaction-diffusion systems.

Anomalous Velocity Distributions in Active Brownian Suspensions

Large-scale simulations and analytical theory have been combined to obtain the nonequilibrium velocity distribution, f(v), of randomly accelerated particles in suspension. The simulations are based on an event-driven algorithm, generalized to include friction. They reveal strongly anomalous but largely universal distributions, which are independent of volume fraction and collision processes, which suggests a one-particle model should capture all the essential features. We have formulated this one-particle model and solved it analytically in the limit of strong damping, where we find that f (v) decays as 1/v for multiple decades, eventually crossing over to a Gaussian decay for the largest velocities. Many particle simulations and numerical solution of the one-particle model agree for all values of the damping.

Measurement of the neutrino velocity with the OPERA detector in the CNGS beam using the 2012 dedicated data

In spring 2012 CERN provided two weeks of a short bunch proton beam dedicated to the neutrino velocity measurement over a distance of 730 km. The OPERA neutrino experiment at the underground Gran Sasso Laboratory used an upgraded setup compared to the 2011 measurements, improving the measurement time accuracy. An independent timing system based on the Resistive Plate Chambers was exploited providing a time accuracy of ∼1 ns. Neutrino and anti-neutrino contributions were separated using the information provided by the OPERA magnetic spectrometers. The new analysis profited from the precision geodesy measurements of the neutrino baseline and of the CNGS/LNGS clock synchronization. The neutrino arrival time with respect to the one computed assuming the speed of light in vacuum is found to be δtν≡TOFc−TOFν=(0.6±0.4 (stat.)±3.0 (syst.)) ns and δtν¯≡TOFc−TOFν¯=(1.7±1.4 (stat.)±3.1 (syst.)) ns for νμ and ν¯μ, respectively. This corresponds to a limit on the muon neutrino velocity with respect to the speed of light of −1.8×10−6<(vν−c)/c<2.3×10−6 at 90% C.L. This new measurement confirms with higher accuracy the revised OPERA result.

Did they ‘Read before burning’? The Coalition and quangos

Attitudes to quangos are paradoxical. On the one hand they are perceived to be undemocratic, unaccountable organisations, while on the other they are seen to improve effectiveness, limit political interference and increase public confidence in government. This paradox is reflected in the behaviour of political parties, which generally adopt a harsh line towards quangos in opposition, but come to rely on these bodies in office. Ahead of the 2010 general election it was, however, noticeable that the Conservative party rejected this dynamic by promising to pursue ‘a more sophisticated approach’. This article explores the Coalition government's subsequent ‘public bodies reform programme’, assessing its progress against recommendations contained within the Institute for Government's Read before Burning report of July 2010. It concludes that while the Coalition has addressed long-standing concerns about the day-to-day governance of public bodies, it has failed to resolve a set of broader and strategic (metagovernance) issues.

Stokes waves revisited:exact solutions in the asymptotic limit

The Stokes perturbative solution of the nonlinear (boundary value dependent) surface gravity wave problem is known to provide results of reasonable accuracy to engineers in estimating the phase speed and amplitudes of such nonlinear waves. The weakling in this structure though is the presence of aperiodic “secular variation” in the solution that does not agree with the known periodic propagation of surface waves. This has historically necessitated increasingly higher-ordered (perturbative) approximations in the representation of the velocity profile. The present article ameliorates this long-standing theoretical insufficiency by invoking a compact exact n-ordered solution in the asymptotic infinite depth limit, primarily based on a representation structured around the third-ordered perturbative solution, that leads to a seamless extension to higher-order (e.g., fifth-order) forms existing in the literature. The result from this study is expected to improve phenomenological engineering estimates, now that any desired higher-ordered expansion may be compacted within the same representation, but without any aperiodicity in the spectral pattern of the wave guides.