Numerical study on diverging probability density function of flat-top solitons in an extended Korteweg-de Vries equation

We consider an extended Korteweg-de Vries (eKdV) equation, the usual Korteweg-de Vries equation with inclusion of an additional cubic nonlinearity. We investigate the statistical behaviour of flat-top solitary waves described by an eKdV equation in the presence of weak dissipative disorder in the linear growth/damping term. With the weak disorder in the system, the amplitude of solitary wave randomly fluctuates during evolution. We demonstrate numerically that the probability density function of a solitary wave parameter $\kappa$ which characterizes the soliton amplitude exhibits loglognormal divergence near the maximum possible $\kappa$ value.Comment: 8 pages, 4 figure

Bleed-induced distortion in axial compressors

In this paper, the influence of nonuniform bleed extraction on the stability of an axial flow compressor is quantified. Nonuniformity can be caused by several geometric factors (for example, plenum chamber size or number of off-take ducts), and a range of configurations is examined experimentally in a single stage compressor. It is shown that nonuniform bleed leads to a circumferential distribution of flow coefficient and swirl angle at inlet to the downstream stage. The resultant distribution of rotor incidence causes stall to occur at a higher flow coefficient than if the same total bleed rate had been extracted uniformly around the circumference. A connection is made between the analysis of nonuniform bleed extraction and the familiar DCθ criterion used to characterize inlet total pressure distortion. The loss of operating range caused by the nonuniform inlet flow correlates with the peak sector-averaged bleed nonuniformity for all the bleed configurations tested.This is a metadata record relating to an article that cannot be shared due to publisher copyright

Solitary waves with recirculation zones in axisymmetric rotating flows

In this paper, we describe a theoretical asymptotic model for large-amplitude travelling solitary waves in an axially symmetric rotating flow of an inviscid incompressible fluid confined in an infinitely long circular tube. By considering the special, but important, case when the upstream flow is close to that of uniform axial flow and uniform rotation, we are able to construct analytical solutions which describe solitary waves with `bubbles', that is, recirculation zones with reversed flow, located on the axis of the tube. Such waves have amplitudes which slightly exceed the critical amplitude, where there is incipient flow reversal. The effect of the recirculation zone is to introduce into the governing amplitude equation an extra nonlinear term, which is proportional to the square of the difference between the wave amplitude and the critical amplitude. We consider in detail a special, but representative, class of upstream inflow conditions. We find that although the structure of the recirculation zone is universal, the presence of such solitary waves is quite sensitive to the actual upstream axial and rotational velocity shear configurations. Our results are compared with previous theories and observations, and related to the well-known phenomenon of vortex breakdown

Large-amplitude solitary waves with vortex cores in stratified and rotating flows

Most theoretical studies of solitary waves are for the weakly nonlinear regime, where models such as the Korteweg-de Vries equation are commonly obtained. However, observations of solitary waves often show that these waves can have large amplitudes, to the extent that they may contain vortex cores, that is, regions of recirculating flow. In this work, we report on theoretical asymptotic models, which describe explicitly the structure of solitary waves with recirculation zones, for certain special but important upstream flow configurations. The key feature which enables this construction is that, both for stratified shear flows and for axisymmetric swirling flows, the steady state vorticity equation is almost linear when the upstream flow is almost uniform. That is, for stratified shear flows the upstream flow and the upstream stratification are almost constant, while for rotating flows the upstream axial flow and angular velocity are almost constant. This feature enables the asymptotic construction of solitary waves described by a steady-state generalised Korteweg-de Vries equation in an outer zone, matched to an inner zone containing a recirculation zone. These recirculation zones exist for wave amplitudes just greater than a certain critical amplitude for which there is incipient flow reversal. The recirculation zones have a universal structure such that their width increases without limit as the wave amplitude increases from the critical amplitude to a certain maximum amplitude, but their existence can be sensitive to the actual upstream flow configuration. Applications are made to observations and numerical simulations of large amplitude internal solitary waves, and to the phenomenon of vortexbreakdown

Cuspons, peakons and regular gap solitons between three dispersion curves

A general wave model with the cubic nonlinearity is introduced to describe a situation when the linear dispersion relation has three branches, which would intersect in the absence of linear couplings between the three waves. Actually, the system contains two waves with a strong linear coupling between them, to which a third wave is then coupled. This model has two gaps in its linear spectrum. Realizations of this model can be made in terms of temporal or spatial evolution of optical fields in, respectively, a planar waveguide or a bulk-layered medium resembling a photonic-crystal fiber. Another physical system described by the same model is a set of three internal wave modes in a density-stratified fluid. A nonlinear analysis is performed for solitons which have zero velocity in the reference frame in which the group velocity of the third wave vanishes. Disregarding the self-phase modulation (SPM) term in the equation for the third wave, we find two coexisting families of solitons: regular ones, which may be regarded as a smooth deformation of the usual gap solitons in a two-wave system, and cuspons with a singularity in the first derivative at their center. Even in the limit when the linear coupling of the third wave to the first two vanishes, the soliton family remains drastically different from that in the linearly uncoupled system; in this limit, regular solitons whose amplitude exceeds a certain critical value are replaced by peakons. While the regular solitons, cuspons, and peakons are found in an exact analytical form, their stability is tested numerically, which shows that they all may be stable. If the SPM terms are retained, we find that there again coexist two different families of generic stable soliton solutions, namely, regular ones and peakons.Comment: a latex file with the text and 10 pdf files with figures. Physical Review E, in pres

Loss in axial compressor bleed systems

Abstract Loss in axial compressor bleed systems is quantified and the loss mechanisms are identified to determine how efficiency can be improved. For a given bleed air pressure requirement, reducing bleed system loss allows air to be bled from further upstream in the compressor, with benefits for the thermodynamic cycle. A definition of isentropic efficiency, which includes bleed flow is used to account for this. Two cases with similar bleed systems are studied: a low-speed, single-stage research compressor, and a large industrial gas turbine high-pressure compressor. A new method for characterizing bleed system loss is introduced, using research compressor test results as a demonstration case. A loss coefficient is defined for a control volume including only flow passing through the bleed system. The coefficient takes a measured value of 95% bleed system inlet dynamic head and is shown to be a weak function of compressor operating point and bleed rate, varying by ±2.2% over all tested conditions. This loss coefficient is the correct nondimensional metric for quantifying and comparing bleed system performance. Computations of the research compressor and industrial gas turbine compressor identify the loss mechanisms in the bleed system flow. In both cases, approximately two-thirds of total loss is due to shearing of a high-velocity jet at the rear face of the bleed slot, one-quarter is due to mixing in the plenum chamber, and the remainder occurs in the off-take duct. Therefore, the main objective of a designer should be to diffuse the flow within the bleed slot. A redesigned bleed slot geometry is presented that achieves this objective and reduces the loss coefficient by 31%.Mitsubishi Heavy Industrie

The Modulation of Multiple Phases Leading to the Modified KdV Equation

This paper seeks to derive the modified KdV (mKdV) equation using a novel approach from systems generated from abstract Lagrangians that possess a two-parameter symmetry group. The method to do uses a modified modulation approach, which results in the mKdV emerging with coefficients related to the conservation laws possessed by the original Lagrangian system. Alongside this, an adaptation of the method of Kuramoto is developed, providing a simpler mechanism to determine the coefficients of the nonlinear term. The theory is illustrated using two examples of physical interest, one in stratified hydrodynamics and another using a coupled Nonlinear Schr\"odinger model, to illustrate how the criterion for the mKdV equation to emerge may be assessed and its coefficients generated.Comment: 35 pages, 5 figure