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L1.2 Linearity and nonlinear theories. Schrödinger’s equation.

In the order of increasing simplifications, by removing some terms of the full 1D Saint-Venant equations aka Dynamic wave equation , we get the also classical Diffusive wave equation and Kinematic wave equation. For the diffusive wave it is assumed that the inertial terms are less than the gravity, friction, and pressure terms.

The diffusive wave can therefore be more accurately described as a non-inertia wave, and is written as:. The diffusive wave is valid when the inertial acceleration is much smaller than all other forms of acceleration, or in other words when there is primarily subcritical flow, with low Froude values. In the SIC Irstea software this options is also available, since the 2 inertia terms or any of them can be removed in option from the interface. For the kinematic wave it is assumed that the flow is uniform, and that the friction slope is approximately equal to the slope of the channel.

1. Introduction

This simplifies the full Saint-Venant equation to the kinematic wave:. The kinematic wave is valid when the change in wave height over distance and velocity over distance and time is negligible relative to the bed slope, e. The 1-D Saint-Venant momentum equation can be derived from the Navier—Stokes equations that describe fluid motion. The x -component of the Navier—Stokes equations — when expressed in Cartesian coordinates in the x -direction — can be written as:.

The local acceleration a can also be thought of as the "unsteady term" as this describes some change in velocity over time. The convective acceleration b is an acceleration caused by some change in velocity over position, for example the speeding up or slowing down of a fluid entering a constriction or an opening, respectively.

Both these terms make up the inertia terms of the 1-dimensional Saint-Venant equation. The pressure gradient term c describes how pressure changes with position, and since the pressure is assumed hydrostatic, this is the change in head over position.

The friction term d accounts for losses in energy due to friction, while the gravity term e is the acceleration due to bed slope. Shallow water equations can be used to model Rossby and Kelvin waves in the atmosphere, rivers, lakes and oceans as well as gravity waves in a smaller domain e.

In order for shallow water equations to be valid, the wavelength of the phenomenon they are supposed to model has to be much larger than the depth of the basin where the phenomenon takes place. Somewhat smaller wavelengths can be handled by extending the shallow water equations using the Boussinesq approximation to incorporate dispersion effects. For tidal motion, even a very deep ocean may be considered as shallow as its depth will always be much smaller than the tidal wavelength.


Nonlinear Diffusive Waves

Shallow water equations, in its non-linear form, is an obvious candidate for modelling turbulence in the atmosphere and oceans, i. An advantage of this, over Quasi-geostrophic equations , is that it allows solutions like gravity waves , while also conserving energy and potential vorticity. However there are also some disadvantages as far as geophysical applications are concerned - it has a non-quadratic expression for total energy and a tendency for waves to become shock waves [28].

Some alternate models have been proposed which prevent shock formation.

Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations

One alternative is to modify the "pressure term" in the momentum equation, but it results in a complicated expression for kinetic energy [29]. Another option is to modify the non-linear terms in all equations, which gives a quadratic expression for kinetic energy , avoids shock formation, but conserves only linearized potential vorticity [30].

From Wikipedia, the free encyclopedia. A set of partial differential equations that describe the flow below a pressure surface in a fluid. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Play media. This section possibly contains original research.

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Retrieved Mirabito Retrieved 2 December Holly Jr. Pelinovsky , Rogue waves in nonlinear hyperbolic systems shallow-water framework , Nonlinearity 24 3 , pp. R1—R18, doi : Pure and Applied Geophysics. Bibcode : PApGe.

Applied Nonlinear Partial Differential Equations - Summer School

Journal of Ocean Engineering and Marine Energy. Applied Mathematics and Computation. Hydraulic Reference Manual. Version 1. Madsen, J. Computer models of watershed hydrology, — Fewtrell, M. Trigg, and J. Modeling and control of hydrosystems. Secondly, the cooling air that flows around the combustor liner and acts as an acoustic insulation layer is relatively much less.

The requirement for fuel flexibility makes these problems more challenging to solve as the chemical composition of the fuel significantly influences the combustion dynamics. Combustion instabilities are characterised by strong pressure oscillations that can destroy the combustor. These are a result of coupling of the flame heat release with flow perturbations through a feedback mechanism that is usually the combustor acoustics. In the multiphysics environment of a combustor there are various processes that cause flow perturbations and hence, the problem of combustion instability is extremely complex.

The flow perturbations could be the result of other dynamic phenomena such as flame flashback and advecting entropy waves and these are the subject of the current work. The former is a highly transient phenomenon characterised by the sudden upstream propagation of the flame. The entropy waves are hot parcels of fluid that are generated by an oscillatory heat release at the flame and advect with the flow to generate acoustic waves known as entropy noise during their passage through the combustor exit nozzle.

The flashback frequency of a flame periodically moving upstream or the frequency of entropy noise could coincide with an acoustic mode of the combustor, thus resulting in combustion instability. Even if the phenomena do not cause combustion instability they are problems in their own right. Flame flashback can damage upstream components of the combustor that are not designed to operate at high temperatures. On the other hand, entropy noise contributes to engine noise.

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Investigation of flashback requires multiple, simultaneous diagnostics without prior knowledge of the relevant time and length scales of the physical processes involved. Detection, accordingly, deals with post-event characterization.

The current work, attempts to detect subtle dynamics prior to flashback using for the first time nonlinear time series analysis tools to process existing pressure time series from flashback experiments. Time and frequency domain methods are unable to detect precursors of flashback as will be demonstrated. However, these conventional methods of time series analysis operate on the assumption that the source of the time series is linear.

The highly transient nature of flashback clearly indicates that the phenomenon is a consequence of nonlinear dynamics. Following standard nonlinear time series analysis, the trajectory of the system in phase space is constructed from the time series data.

Subsequently, the orbit of the trajectory is analysed using a running window to plot its translation error and recurrence quantification measures of its recurrence pattern as a function of time. The translation error analysis is applied to time series from a flashback experiment in stable combustion. The recurrence analysis is applied to time series from a flashback experiment in a different burner with unstable combustion.

In both cases, it is found that the determinism of the system dynamics gradually increases as flashback is approached. The influence of entropy noise on combustion instabilities is still a subject of contention. Experimental investigations are again difficult because entropy noise cannot be distinguished from noise generated at the flame in acoustic measurements. Hence, experimental investigations rely on measuring the temperature perturbations before their passage through the combustor exit nozzle.

However, this brings about another difficulty, that of measuring high frequency temperature oscillations. Nonetheless, once the temperature measurements are made there is a need for a comprehensive theory to convert them to entropy noise.