Euler Solutions Research Articles

Numerical simulation of Mach reflection in steady flows

The structure obtained when two shocks intersect is known to be highly sensitive to various parameters. In the so-called dual solution domain, both regular and Mach reflection patterns are possible, resulting in hysteresis. The phenomenon is important in inlets because of the substantial difference in entropy rise associated with the two manifestations, and the possibility of unstart with Mach reflection. The effect of various numerical and physical parameters on hysteresis are investigated with two-dimensional simulations. The effect of spanwise relief on a three-dimensional situation is also elucidated. It is confirmed that Mach-stem heights determined from inviscid computations are captured relatively accurately by comparison with experimental data and earlier Euler solutions reported in the literature. Near bifurcation points, however, the solution is highly sensitive to the scheme, and the van Leer and Roe schemes can yield converged solutions with different reflection configurations. Viscous terms and downstream conditions are observed to have relatively minor impact on the solution. The three-dimensional simulations reveal that beyond the spanwise limit of the compression surface, the overall shock-structure remains similar in form but the strengths of various shocks are rapidly muted by the expansion from the side surface. Additionally, the flow downstream of the shock that once formed the Mach reflection rapidly becomes supersonic. The Mach-stem height on the symmetry plane and its variation with spanwise position shows reasonable agreement with the experimental data of other investigators.

Surface stress effects on the critical buckling strains of silicon nanowires

The objective of this paper is to quantify how nanoscale surface stresses impact the critical buckling strains of silicon nanowires. These insights are gained by using nonlinear finite element calculations based upon a multiscale, finite deformation constitutive model that incorporates nanoscale surface stress and surface elastic effects to study the buckling behavior of silicon nanowires that have cross sectional dimensions between 10 and 25nm under axial compressive loading. The key finding is that, in contrast to existing surface elasticity solutions, the critical buckling strains are found to show little deviation from the classical bulk Euler solution. The present results suggest that accounting for axial strain relaxation due to surface stresses may be necessary to improve the accuracy and predictive capability of analytic linear surface elastic theories.

Acceleration of 2-D Compressible Flow Solvers with Graphics Processing Unit Clusters

We demonstrate the first use of graphics processing unit clusters for engineering-level compressible flow computations with two separately developed solvers. Each solver adopts the finite-volume approach on multiblock, structured, nonuniform, quadrilateral meshes. We use a second-order accurate integration scheme, with a second-order accurate dual-time integration, using steady-flow acceleration techniques such as local time-stepping and nonlinear multigrid. The first solver is based on an existing multiblock Fortran code, MBFLO, which is augmented with graphics processing unit acceleration for the laminar and unsteady flow subroutines. The second solver is built from scratch for graphics processing unit acceleration, but is limited to the steady Euler solutions over simple geometries. The Euler solver is optimized by reducing the memory bandwidth, recomputing intermediate values, and combining functions to increase producer-consumer locality.These two approaches are used to highlight trade-offs between development effort and performance when accelerating an application with graphics processing units. On test cases with up to 6.4 million grid points, our graphics processing unit solvers outperform their central processing unit counterparts by up to 20×. In our graphics processing unit strong scalability test, 32 graphics processing units scale up to 22× over a single graphics processing unit, which is 16× faster than 32 central processing unit cores, or 500× faster than a single Central Processing Unit core. It is confirmed that graphics processing unit clusters are effective for engineering computational fluid dynamics applications if their order-of-magnitude increase in performance is paired with large problem sizes to amortize the cost of communication.

Determination of the stress-strain state and damages in a rock mass by the displacement measurements on its surface. Part I: Analytical solutions

The article offers two approaches to stress-strain state problem solution: the first is the analytical solution for the elastic deformation of a medium, the second is the numerical solution for the elastic and inelastic deformation. The analytical solution is based on the Kolosov-Muskhelishvili formulas, the numerical solution is analogous to the Euler solution of a regular first-order differential equation. The both methods are compared, and calculation of breaches in half-plane is exemplified.

Spurious Far-Field-Boundary Induced Drag in Two-Dimensional Flow Simulations

A hypothesis is made and evidence is given of a spurious far-field-boundary-induced drag component in twodimensional flow numerical solutions, which is of a different nature from irreversible spurious drag. In the case of solutions to the Euler equations, analysis is carried out of its behavior with far-field boundary condition, lift coefficient, computational domain size, uniform grid refinement, far-field grid refinement, and flow regime (subcritical/transonic). Phenomenological breakdown of this drag component is derived through Taylor series expansion. Finally, the concept of far-field-boundary-induced drag is applied to the interpretation of some of the data contained inVassberg and Jameson’s results [Vassberg, J. C., and Jameson, A., “In Pursuit of GridConvergence for Two-Dimensional Euler Solutions,” Journal of Aircraft, Vol. 474, July–Aug. 2010, pp. 1152–1166]. Estimates of this component based on the aforementioned analysis are found to be coherentwith, and to provide insight into, Vassberg and Jameson’s results.

Investigating Negative Drag in Grid Convergence for Two-Dimensional Euler Solutions

Investigating Negative Drag in Grid Convergence for Two-Dimensional Euler Solutions

Families of symmetric relative periodic orbits originating from the circular Euler solution in the isosceles three-body problem

We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches. We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge to “previously unknown” periodic orbits in the planar problem.

Zero-electron-mass limit of hydrodynamic models for plasmas

This paper is concerned with the limit of the vanishing ratio of the electron mass to the ion mass in the hydrodynamic models for plasmas in critical Besov spaces. We give a new construction of approximation solutions and show that periodic initial-value problems of certain scaled hydrodynamic models have smooth solutions in a (finite) time interval where the Euler solution is known to exist. Furthermore, it is justified that, as the electron mass tends to zero, the smooth solutions converge rigorously to solutions of the incompressible Euler equations, and the definite convergence orders are also obtained.

Application of frequency-domain linearized Euler solutions to the prediction of aft fan tones and comparison with experimental measurements on model scale turbofan exhaust nozzles

Although it is widely accepted that aircraft noise needs to be further reduced, there is an equally important, on-going requirement to accurately predict the strengths of all the different aircraft noise sources, not only to ensure that a new aircraft is certifiable and can meet the ever more stringent local airport noise rules but also to prioritize and apply appropriate noise source reduction technologies at the design stage. As the bypass ratio of aircraft engines is increased – in order to reduce fuel consumption, emissions and jet mixing noise – the fan noise that radiates from the bypass exhaust nozzle is becoming one of the loudest engine sources, despite the large areas of acoustically absorptive treatment in the bypass duct. This paper addresses this ‘aft fan’ noise source, in particular the prediction of the propagation of fan noise through the bypass exhaust nozzle/jet exhaust flow and radiation out to the far-field observer. The proposed prediction method is equally applicable to fan tone and fan broadband noise (and also turbine and core noise) but here the method is validated with measured test data using simulated fan tones. The measured data had been previously acquired on two model scale turbofan engine exhausts with bypass and heated core flows typical of those found in a modern high bypass engine, but under static conditions (i.e. no flight simulation). The prediction method is based on frequency-domain solutions of the linearized Euler equations in conjunction with perfectly matched layer equations at the inlet and far-field boundaries using high-order finite differences. The discrete system of equations is inverted by the parallel sparse solver MUMPS. Far-field predictions are carried out by integrating Kirchhoff's formula in frequency domain. In addition to the acoustic modes excited and radiated, some non-acoustic waves within the cold stream-ambient shear layer are also captured by the computations at some flow and excitation frequencies. By extracting phase speed information from the near-field pressure solution, these non-acoustic waves are shown to be convective Kelvin–Helmholtz instability waves. Strouhal numbers computed along the shear layer, based on the local momentum thickness also confirm this in accordance with Michalke's instability criterion for incompressible round jets with a similar shear layer profile. Comparisons of the computed far-field results with the measured acoustic data reveal that, in general, the solver predicts the peak sound levels well when the farfield is dominated by the in-duct target mode (the target mode being the one specified to the in-duct mode generator). Calculations also show that the agreement can be considerably improved when the non-target modes are also included, despite their low in-duct levels. This is due to the fact that each duct mode has its own distinct directionality and a non-target low level mode may become dominant at angles where the higher-level target mode is directionally weak. The overall agreement between the computations and experiment strongly suggests that, at least for the range of mean flows and acoustic conditions considered, the physical aeroacoustic radiation processes are fully captured through the frequency-domain solutions to the linearized Euler equations and hence this could form the basis of a reliable aircraft noise prediction method.

Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations

The generalized Riemann problem (GRP) scheme for the Euler equations and gas-kinetic scheme (GKS) for the Boltzmann equation are two high resolution shock capturing schemes for fluid simulations. The difference is that one is based on the characteristics of the inviscid Euler equations and their wave interactions, and the other is based on the particle transport and collisions. The similarity between them is that both methods can use identical MUSCL-type initial reconstructions around a cell interface, and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time-dependent flux function. Although both methods have been applied successfully to the inviscid compressible flow computations, their performances have never been compared. Since both methods use the same initial reconstruction, any difference is solely coming from different underlying mechanism in their flux evaluation. Therefore, such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances. Since GRP is so faithfully solving the inviscid Euler equations, the comparison can be also used to show the validity of solving the Euler equations itself. The numerical comparison shows that the GRP exhibits a slightly better computational efficiency, and has comparable accuracy with GKS for the Euler solutions in 1D case, but the GKS is more robust than GRP. For the 2D high Mach number flow simulations, the GKS is absent from the shock instability and converges to the steady state solutions faster than the GRP. The GRP has carbuncle phenomena, likes a cloud hanging over exact Riemann solvers. The GRP and GKS use different physical processes to describe the flow motion starting from a discontinuity. One is based on the assumption of equilibrium state with infinite number of particle collisions, and the other starts from the non-equilibrium free transport process to evolve into an equilibrium one through particle collisions. The different mechanism in the flux evaluation deviates their numerical performance. Through this study, we may conclude scientifically that it may NOT be valid to use the Euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution. To adapt the Navier–Stokes (NS) equations is NOT valid either because the NS equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity. This fact alludes to the consistency of the Euler and Navier–Stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high Mach number flows. The development of numerical algorithm is similar to the modeling process in deriving the governing equations, but the control volume here cannot be shrunk to zero.

Aerospectrometric and aeromagnetic characteristics associated Nb–Ta–Sn mineralization, G. Nuweibi albite granite Central Eastern Desert, Egypt

A geophysical signature associated with Nb–Ta–Sn mineralization of G. (G. : abbreviation to word Gebel which means mountain in Arabic) Nuweibi area, located the Central Eastern Desert of Egypt is presented. This signature was established by an integration of airborne gamma ray spectrometric and magnetic data. Variations seen in the gamma ray spectrometric data are used as a base to study the three granitic suites: younger-, albite-, and older granites in G. Nuweibi area. Graphical techniques such as frequency histograms and box-plots are used to visualize the shape of the distribution and determine the anomaly thresholds of the three radioelements eU, eTh, and K% data in these granitic suites. The box-plot graphical representations and calculations made on data sets indicate that no samples have eU values above the thresholds, i.e., no outliers representing values of the box-plots. Nuweibi albite granite is associated with a gamma ray response that includes the strongest eU, eTh, K%, and eTh/K ratio anomalies in the study area. K–eTh plot shows that the albite granite has a higher eTh concentration than the older and younger granites. The increase in K concentration and raise in Th/K ratio of Nuweibi albite granite points to unusual geological processes leading to mineralization and reflects the highly fractionated nature of the magma which results in thorium enrichment. This also reflects that K alteration associated with Nb–Ta–Sn mineralization is both poorly focused spatially and very much weaker than observed in any other mineralizing districts. The distribution of magnetic sources and their locations and depths in the study region are determined by Euler deconvolution and analytical signal techniques. Good clustering of Euler solutions were obtained using SI = 0.5 and SI = 1.0 for most of the features in the area under consideration. The solutions obtained have shown magnetic sources which can be related to the impact structure whose depths varies between ground surface to 1.66 km. The analytical signal revealed that the metamorphosed basic rocks (mainly olivine metagabbro), serpentinite and dyke bodies are the main sources of high magnetic anomalies, particularly within the area east G. Nuweibi region.

Stability of columns with semi-rigid connections including shear effects using Engesser, Haringx and Euler approaches

A complete column classification and the corresponding stability equations that can be used to evaluate the elastic critical axial loads of prismatic columns with sidesway uninhibited, partially inhibited, and totally inhibited including the simultaneous effects of bending and shear deformations and semi-rigid connections are presented and derived using three different approaches. The first two approaches are those by Engesser and Haringx that include the shear component of the applied axial force proportional to the total slope ( d y / d x ) and to the angle of rotation of the cross-section ( ψ ) along the member, respectively. The third approach is a simplified formulation based on the classical Euler theory that includes the effects of shear deformations but neglects the shear component of the applied axial force along the member. Comparisons among the critical loads obtained using these three approaches and those using the classical Euler solutions for classical column cases are presented. Examples that demonstrate the effectiveness and accuracy of the proposed stability equations and the importance of shear deformations and shear component of the applied axial force are presented in detail. These effects must be taken into account particularly in structures made of columns with relatively low effective shear areas A s (like laced columns, columns with batten plates or with perforated cover plates, and columns with open webs) or with low shear stiffness GA s of the same order of magnitude as π 2 E I / h 2 (like short columns made of laminated composites with low shear modulus G when compared to their elastic modulus E ). These effects become even more significant when the external supports are not perfectly clamped. The effects of semi-rigid connections and those due to shear forces and deformations are condensed into the proposed characteristic equations.

Least action principle and the incompressible Euler equations with variable density

In this article we study a variational formulation, proposed by V. I. Arnold and by Y. Brenier, for the incompressible Euler equations with variable density. We consider the problem of minimizing an action functional, the integral in time of the kinetic energy, among fluid motions considered as trajectories in the group of volume-preserving diffeomorphisms beginning at the identity and ending at some fixed diffeomorphism at a given time. We show that a relaxed version of this variational problem always has a solution, and we derive an Euler-Lagrange system for the relaxed minimization problem which we call the relaxed Euler equations. Finally, we prove consistency between the relaxed Euler equations and the classical Euler system, showing that weak solutions of the relaxed Euler equations with the appropriate geometric structure give rise to classical Euler solutions and that classical solutions of the Euler system induce weak solutions of the relaxed Euler equations. The first consistency result is new even in the constant density case. The remainder of our analysis is an extension of the work of Y. Brenier (1999) to the variable density case.

Adaptive modeling of turbulent flow with residual based turbulent kinetic energy dissipation

In this paper we first review our recent work on a new framework for adaptive turbulence simulation: we model turbulence by weak solutions to the Navier–Stokes equations that are wellposed with respect to mean value output in the form of functionals, and we use an adaptive finite element method to compute approximations with a posteriori error control based on the error in the functional output. We then derive a local energy estimate for a particular finite element method, which we connect to related work on dissipative weak Euler solutions with kinetic energy dissipation due to lack of local smoothness of the weak solutions. The ideas are illustrated by numerical results, where we observe a law of finite dissipation with respect to a decreasing mesh size.

Convergence of the semi-implicit Euler method for neutral stochastic delay differential equations with phase semi-Markovian switching

Recently, in the numerical analysis for stochastic differential equations (SDEs), it is a new topic to study the numerical schemes of neutral stochastic functional differential equations (NSFDEs) (see Wu and Mao [1]). Especially when Markovian switchings are taken into consideration, these problems will be more complicated. Although Zhou and Wu [2] develop a numerical scheme to neutral stochastic delay differential equations with Markovian switching (short for NSDDEwMSs), their method belongs to explicit Euler–Maruyama methods which are in general much less accurate in approximation than their implicit or semi-implicit counterparts. Therefore, to propose an implicit method becomes imperative to fill the gap. In this paper we will extend Zhou and Wu [2] to the case of the semi-implicit Euler–Maruyama methods and equations with phase semi-Markovian switching rather than Markovian switching. The employment of phase semi-Markovian chains can avoid the restriction of the negative exponential distribution of the sojourn time at a state. We prove the semi-implicit Euler solution will converge to the exact solution to NSDDEwMS under local Lipschitz condition. More precise inequalities and new techniques are put forward to overcome the difficulties for the existence of the neutral part.

Capabilities of the high-order parabolic equation to predict propagation in boundary and shear layers.

The parabolic equation (PE) has proved its capability to deal with the long-range sound propagation in the atmosphere. It also represents an attractive alternative to the ray model to handle duct propagation in high frequencies for the noise radiated by the nacelle of aero-engines. It was recently shown that the high-order parabolic equation (HOPE), based on a Padé expansion with an order of 5, increases the aperture angle of propagation compared to the standard and the wide-angle PEs, allowing prediction close to the cutoff frequency of the duct. This paper is concerned with propagation using the HOPE in heterogeneous flows, including boundary layers above a wall and in shear layers. The thickness of the boundary layer is about dozens of centimeters; while outside it, the Mach number reaches 0.5. The boundary layer effects are investigated showing the refraction effects on a range propagation of 30 m, up to 4 kHz. On both sides of the shear layer, significant differences in the directivity patterns occur. Comparisons with the Euler solutions are considered, establishing the domain of application of the HOPE on a set of flow configurations, including those beyond its theoretical limits. [Work supported by Airbus-France.]

Rotational and Quasiviscous Cold Flow Models for Axisymmetric Hybrid Propellant Chambers

In this work, we present two simple mean flow solutions that mimic the bulk gas motion inside a full-length, cylindrical hybrid rocket engine. Two distinct methods are used. The first is based on steady, axisymmetric, rotational, and incompressible flow conditions. It leads to an Eulerian solution that observes the normal sidewall mass injection condition while assuming a sinusoidal injection profile at the head end wall. The second approach constitutes a slight improvement over the first in its inclusion of viscous effects. At the outset, a first order viscous approximation is constructed using regular perturbations in the reciprocal of the wall injection Reynolds number. The asymptotic approximation is derived from a general similarity reduced Navier–Stokes equation for a viscous tube with regressing porous walls. It is then compared and shown to agree remarkably well with two existing solutions. The resulting formulations enable us to model the streamtubes observed in conventional hybrid engines in which the parallel motion of gaseous oxidizer is coupled with the cross-streamwise (i.e., sidewall) addition of solid fuel. Furthermore, estimates for pressure, velocity, and vorticity distributions in the simulated engine are provided in closed form. Our idealized hybrid engine is modeled as a porous circular-port chamber with head end injection. The mathematical treatment is based on a standard similarity approach that is tailored to permit sinusoidal injection at the head end.

Development of an Incompressible Navier-Stokes Solver Involving Symplectic and Nonsymplectic Time Integrators

This article develops a heat/flow solver for properly simulating incompressible viscous flow at high Reynolds numbers and natural-convection flow at high Rayleigh numbers. Our strategy of simulating these problems is to retain some rich geometric properties embedded in the invisid Euler equations. Thanks to the underlying theory of Clebsch velocity decomposition, which divides the velocity vector as the sum of three velocity components due to the potential, rotational, and viscous contributions, the mixed-type Navier-Stokes equations cast in primitive variables can be fractionally split into three corresponding equations which are coupled to each other. In the pure advection step, the symplectic Runge-Kutta integrator is employed to approximate the temporal derivative term so as to numerically retain the Hamiltonians embedded in the lossless Euler differential equations. In addition, an upwinding scheme with minimized phase error is applied to discretize the first-order spatial derivative terms. In the diffusion step, we approximate the time derivative term shown in the time-dependent parabolic equation using a time-stepping scheme which does not need to be symplectic and is thus different from that used in the first step for the calculation of Euler solutions. We then solve the velocity vector from the projection step, subject to the divergence-free constraint condition. The proposed method is validated through benchmark tests. The predicted results of the incompressible Navier-Stokes and natural-convection equations are also justified for the problems investigated at high Reynolds and high Rayleigh numbers, respectively.

In Pursuit of Grid Convergence for Two-Dimensional Euler Solutions

Grid-convergence trends of two-dimensional Euler solutions are investigated. The airfoil geometry under study is based on the NACA0012 equation. However, unlike the NACA0012 airfoil, which has a blunt base at the trailing edge, the study geometry is extended in chord so that its trailing edge is sharp. The flow solutions use extremely- high-quality grids that are developed with the aid of the Karman-Trefftz conformal transformation. The topology of each grid is that of a standard O-mesh. The grids naturally extend to a far-field boundary approximately 150 chord lengths away from the airfoil. Each quadrilateral cell of the resulting mesh has an aspect ratio of one. The intersecting lines of the grid are essentially orthogonal at each vertex within the mesh. A family of grids is recursively derived starting with the finest mesh. Here, each successively coarser grid in the sequence is constructed by eliminating every other node of the current grid, in both computational directions. In all, a total of eight grids comprise the family, with the coarsest-to-finest meshes having dimensions of 32 x 32-4096 x 4096 cells, respectively. Note that the finest grid in this family is composed of over 16 million cells, and is suitable for 13 levels of multigrid. The geometry and grids are all numerically defined such that they are exactly symmetrical about the horizontal axis to ensure that a nonlifting solution is possible at zero degrees angle-of-attack attitude. Characteristics of three well-known flow solvers (FLO82, OVERFLOW, and CFL3D) are studied using a matrix of four flow conditions: (subcritical and transonic) by (nonlifting and lifting). The matrix allows the ability to investigate grid-convergence trends of 1) drag with and without lifting effects, 2) drag with and without shocks, and 3) lift and moment at constant angles-of-attack. Results presented herein use 64-bit computations and are converged to machine-level-zero residuals. All three of the flow solvers have difficulty meeting this requirement on the finest meshes, especially at the transonic flow conditions. Some unexpected results are also discussed. Take for example the subcritical cases. FLO82 solutions do not reach asymptotic grid convergence of second-order accuracy until the mesh approaches one quarter of a million cells. OVERFLOW exhibits at best a first-order accuracy for a central-difference stencil. CFL3D shows second-order accuracy for drag, but only first-order trends for lift and pitching moment. For the transonic cases, the order of accuracy deteriorates for all of the methods. A comparison of the limiting values of the aerodynamic coefficients is provided. Drag for the subcritical cases nearly approach zero for all of the computational fluid dynamics methods reviewed. These and other results are discussed.

Fully nonlinear periodic internal waves in a two-fluid system of finite depth

Periodic travelling wave solutions for a strongly nonlinear model of long internal wave propagation in a two-fluid system are derived and extensively analysed, with the aim of providing structure to the rich parametric space of existence of such waves for the parent Euler system. The waves propagate at the interface between two homogeneous-density incompressible fluids filling the two-dimensional domain between rigid planar boundaries. The class of waves with a prescribed mean elevation, chosen to coincide with the origin of the vertical (parallel to gravity) axis, and prescribed zero period-average momentum and volume-flux is studied in detail. The constraints are selected because of their physical interpretation in terms of possible processes of wave generation in wave-tanks, and give rise to a quadrature formula which is analysed in parameter space with a combination of numerical and analytical tools. The resulting model solutions are validated against those computed numerically from the parent Euler two-layer system with a boundary element method. The parametric domain of existence of model periodic waves is determined in closed form by curves in the amplitude–speed (A, c) parameter plane corresponding to infinite period limiting cases of fronts (conjugate states) and solitary waves. It is found that the existence domain of Euler solutions is a subset of that of the model. A third closed form relation between c and A indicates where the Euler solutions cease to exist within the model's domain, and this is related to appearance of ‘overhanging’ (multiple valued) wave profiles. The model existence domain is further partitioned in regions where the model is expected to provide accurate approximations to Euler solutions based on analytical estimates from the quadrature. The resulting predictions are found to be in good agreement with the numerical Euler solutions, as exhibited by several wave properties, including kinetic and potential energy, over a broad range of parameter values, extending to the limiting cases of critical depth ratio and extreme density ratios. In particular, when the period is sufficiently long, model solutions show that for a given supercritical speed waves of substantially larger amplitude than the limiting amplitude of solitary waves can exist, and are good approximations of the corresponding Euler solutions. This finding can be relevant for modelling field observations of oceanic internal waves, which often occur in wavetrains with multiple peaks.