J. W. Dold

Flame propagation in a nonuniform mixture: Analysis of a slowly varying Triple Flame

For flames propagating through a nonuniform medium a three-flame structure is described. This consists of a fuel-rich premixed flame, a fuel-lean premixed flame, and, starting where these flames meet, a diffusion flame. Such formations have been observed experimentally and probably occur as laminar flamelets in turbulent diffusion flames. A low-heat-release model for such flame structures is developed and solutions are obtained in the limit of slowly varying premixed flames. Under these conditions, it is shown that the Triple-Flame propagation speed depends on the transverse mixture fraction gradient and is bounded above by the maximum adiabatic laminar flame speed of the system.

Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation

The time evolution of a uniform wave train with a small modulation which grows is computed with a fully nonlinear irrotational flow solver. Many numerical runs have been performed varying the initial steepness of the wave train and the number of waves in the imposed modulation. It is observed that the energy becomes focussed into a short group of steep waves which either contains a wave which becomes too steep and therefore breaks or otherwise having reached a maximum modulation then recedes until an almost regular wave train is recovered. This latter case typically occurs over a few hundred time periods. We have also carried out some much longer computations, over several thousands of time periods in which several steep wave events occur. Several features of these modulations are consistent with analytic solutions for modulations ! ;

Flame Propagation in a Nonuniform Mixture: Analysis of a Propagating Triple-Flame

Abstract A situation in which a dilffusion flame reaches an end at some position in a medium of non-premixed reactants is studied. The mixing of reactants that takes place ahead of the diffusion flame leads to the formation of a “triple-flame”, a structure which consists of a fuel-rich premixed flame, a fuel-lean premixed flame, and a diffusion flame that starts where the two premixed flames meet. An important property of such an end-point is its ability to propagate. The limits of low heat release, unit Lewis number and large Zeldovich number are considered. The structure of the triple-flame and the unique relationship between propagation speed and transverse mixture Traction gradient are computed numerically. For the range of values considered here, the end of the diffusion flame is shown to extend itself at a rate that can be substantially reduced, but that remains positive as the gradient of the mixture fraction is increased.

An efficient surface-integral algorithm applied to unsteady gravity waves

A computationally fast method for calculating the unsteady motion of a surface on a two-dimensional fluid is described. Cauchy’s integral theorem is used iteratively to solve Laplace’s equation for successive time derivatives of the surface motion and time-stepping is performed using truncated Taylor series. This allows fairly large time-steps to be made for a given accuracy while the required number of spatial points is minimised by using high order differencing formulae. This reduces the overall number of required calculations. The numerical implementation of the method is found to be accurate and efficient. A fairly thorough examination of this implementation is carried out, revealing that high accuracies are often achievable using surprisingly few numerical surface points. Extensive calculations are also performed using modest computing resources. Some numerical instabilities are identified, although these would not usually be significant in practical calculations. A model analysis reveals that two of these instabilities can be eliminated by using suitable methods of time-stepping. Should the third “steep-wave instability” become significant, it is shown that it can be completely controlled by using high-order smoothing techniques, at little cost to accuracy. Using a routine to ensure asymptotic conservation of energy, this is confirmed by time-stepping a very steep (but stable) wave over thousands of wave-periods. 0 1992 Academx Press. Inc. Since all of the interior properties of a body of fluid undergoing inviscid, incompressible irrotational motion are fully determined by properties at its boundary, it is possible to reduce the calculation of the motion of such a fluid to the evaluation of the motion of its surface alone. In a numerical scheme, the entire motion can thus be modelled using only a point discretisation of the surface. Such an approach was first suggested by Svendsen in 1971 as a means of calculating the motion of two-dimensional gravity waves on water [ 13. Since then the idea has been implemented using a variety of boundary-integral or conformal mapping techniques, by Longuet-Higgins and Cokelet [2], Vinje and Brevig [3], Baker, Meiron, and Orszag [4], Roberts [S], Fornberg [6] and (based on his original formulation) by Svendsen and co-workers [7]. Although this paper presents yet another numerical method, the approach is aimed specifically at producing a computationally fast and efficient numerical scheme in order to be able to study problems of increased complexity. In numerically representing complicated surface motions, ranging from the breaking of waves to the evolution of instabilities in a train of travelling waves, it typically becomes necessary to use a large number of surface points. If running times tend to increase too rapidly with the number of points, then this can destroy any apparent advantage in being able to treat the whole fluid only in terms of its moving surface. The penalty is particularly severe for algorithms that require matrix inversion or factorisation techniques-yielding running times per time step that increase like N3 (the cube of the number of surface points). By using iterative techniques, the penalty can be reduced to running times per time-step that increase in proportion to N2. While being less severe, this still has the effect that increased numbers of surface points become increasingly expensive to calculate. The use of a conformal mapping technique [6] can lead to running times per timestep that vary like N log N. However, this method is severely limited in its ability to resolve steep wave surfaces (as are found in breaking waves), requiring disproportionately many surface points to achieve a given accuracy. It is therefore vital to ensure computational elliciency if it is to be possible accurately to describe wave processes for which large numbers of surface points may be required. The most direct way of achieving this is to employ the lowest number of points that can adequately describe a fluid surface. In this paper, a further improvement is introduced by showing that higher time-derivatives of the motion can be calculated directly using exactly the same technique that is used to determine the velocity of the surface at each timestep. This yields significantly more information about the motion for relatively little extra computation. In turn, this allows larger time-steps to be performed for a given accuracy so that the overall number of calculations can be reduced. The method presented here combines these two ideas into producing a practical numerical algorithm, the properties of which are examined in the following pages. In Appendix A, the formulae are derived that make it

The interaction between a solitary wave and a submerged semicircular cylinder

Numerical solutions for fully nonlinear two-dimensional irrotational free-surface flows form the basis of this study. They are complemented and supported by a limited number of experimental measurements. A solitary wave propagates along a channel which has a bed containing a cylindrical bump of semicircular cross-section, placed parallel to the incident wave crest. The interaction between wave and cylinder takes a variety of forms, depending on the wave height and cylinder radius, measured relative to the depth. Almost all the resulting wave motions differ from the behaviour which was anticipated when the study began. In particular, in those cases where the wave breaks, the breaking occurs beyond the top of the cylinder. The same wave may break in two different directions: forwards as usual, and backwards towards the back of the cylinder. In addition small reflected waves come from the region of uniform depth beyond the cylinder. Experimental results are reported which confirm some of the predictions made. The results found for solitary waves are contrasted with the behaviour of a group of periodic waves.

Water-Wave Modulation

More than 150 tests have been analyzed in order to describe the dynamically stable profiles of rock slopes and gravel beaches under wave attack. Relationships between profile parameters and boundary conditions have been established. These relationships have been used to develop a computer program. This program is able to predict the profiles of slopes with an arbitrary shape under varying wave conditions, such as those found in storm surges and during the tidal period.This paper investigates the utility of winds obtainable from a numerical weather prediction model for driving a spectral ocean-wave model in an operational mode. Wind inputs for two operational spectral wave models were analyzed with respect to observed winds at three locations in the Canadian east coast offshore. Also, significant wave heights obtainable from the two spectral models were evaluated against measured wave data at these locations. Based on this analysis, the importance of appropriate wind specification for operational wave analysis and forecasting is demonstrated.

Instability and Breaking of a Solitary Wave

The result of a linear stability calculation of solitary waves which propagate steadily along the free surface of a liquid layer of constant depth is examined numerically by employing a time-stepping scheme based on a boundary-integral method. The initial growth rate that is found for sufficiently small perturbations agrees with the growth rate expected from the linear stability calculation. In calculating the later ‘nonlinear’ stage of the instability, it is found that two distinct types of long-term evolution are possible. These depend only on the sign of the unstable normal-mode perturbation which is superimposed initially on the steady wave. The growth of the perturbation ultimately leads to breaking for one sign. Unexpectedly, for the opposite sign, there is a monotonic decrease in the total height of the wave. In this latter case there is a smooth evolution to a stable solitary wave of lesser amplitude but very nearly the same energy.

Steep Unsteady Water Waves: an Efficient Computational Scheme

This report will update the coastal zone practitioner on the National Flood Insurance Program (NFIP) as it affects the implementation of manmade changes along the coastline. It is our intent to place in proper perspective this fast-changing and often difficult to interpret national program. Readers will achieve an overall understanding of the NFIP on the coast, and will be in a position to apply the programs requirements in their efforts. We will begin with a history of the application of the NFIP to the coastal zone. The history of the problems encountered will lead into current regulations, methodologies, and the changes the Federal Emergency Management Agency plans for the future.The spatial variability of the nearshore wave field is examined in terms of the coherence functions found between five closely spaced wave gages moored off the North Carolina coast in 17 meters depth. Coherence was found to rapidly decrease as the separation distance increased, particularly in the along-crest direction. This effect is expressed as nondimensional coherence contours which can be used to provide an estimate of the wave coherence expected between two spatial positions.Prediction of depositional patterns in estuaries is one of the primary concerns to coastal engineers planning major hydraulic works. For a well-mixed estuary where suspended load is the dominant transport mode, we propose to use the divergence of the distribution of the net suspended load to predict the depositional patterns. The method is applied to Hangzhou Bay, and the results agree well qualitatively with measured results while quantitatively they are also of the right order of magnitude.

An Evolution Equation Modeling Inversion of Tulip Flames

Abstract We attempt to reduce the number of physical ingredients needed to model the phenomenon of tulip-flame inversion to a bare minimum. This is achieved by synthesising the nonlinear, first-order Michelson-Sivashinsky (MS) equation with the second order linear dispersion relation of Landau and Darrieus, which adds only one extra term to the MS equation without changing any of its stationary behaviour and without changing its dynamics in the limit of small density change when the MS equation is asymptotically valid. However, as demonstrated by spectral numerical solutions, the resulting second-order nonlinear evolution equation is found to describe the inversion of tulip flames in good qualitative agreement with classical experiments on the phenomenon. This shows that the combined influences of front curvature, geometric nonlinearity and hydrodynamic instability (including its second-order, or inertial effects, which are an essential result of vorticity production at the flame front) are sufficient to reproduce the inversion process.

Edges of flames that do not exist: Flame-edge dynamics in a non-premixed counterflow

A counterflow diffusion flame model is studied revealing that, at least as a part of the quenching boundary is approached in parameter space at low-enough Lewis numbers, an edge of a diffusion flame, or triple flame, has a propagation speed that still advances the burning solution into regions that are not burning. In crossing the quenching boundary, the advancing flame edge remains a robust part of the solution but the flame behind the edge is found to break up into periodic regions, resembling ‘tubes’ of burning and non-burning, accompanied by the appearance of an oscillatory component in the speed of propagation of the edge. In crossing a second boundary the propagation speed of the flame edge disappears altogether. The only unbounded, non-periodic stationary solution then consists of an isolated flame tube, although stationary periodic flame tubes can also exist under the same conditions. In passing back through parameter space, starting with a single flame tube already present, there is no sign of hysteresis and the oscillatory edge propagation reappears at the same point where it disappears. On the other hand, in continuing forwards across a third, final boundary the flame tube is extinguished leaving no combustion whatever. Boundaries in parameter space where different solutions arise are mapped out.