Top-hat profile

Properties are assumed uniform across the plume at any elevation, z. This is called a top-hat profile as compared to the more empirically correct Gaussian profile given in Equation (10.1). 

Later we shall include combustion and flame radiation effects, but we will still maintain all of assumptions 2 to 5 above. The top-hat profile and Boussinesq assumptions serve only to simplify our mathematics, while retaining the basic physics of the problem. However, since the theory can only be taken so far before experimental data must be relied on for its missing pieces, the degree of these simplifications should not reduce the generality of the results. We shall use the following conservation equations in control volume form for a fixed CV and for steady state conditions ... 

Using the entrainment relationship (Equation (10.3)), in terms of the top-hat profile,... 

The initial conditions for the velocity components are set up so that there is a tubular shear layer aligned along the 2 -direction at time t = 0. The tv-velocity has a top-hat profile with a tan-hyperbolic shear layer. Stream wise and azimuthal perturbations are introduced to expedite roll-up and the development of the Widnall instability. The details can be found in [7]. The initial velocity field is made divergence-free using the Helmholtz decomposition. The size of the computational domain (one periodic cubical box) is 4do on each side. 

More elaborate pulse shapes have been developed over the years which aim to produce a near top-hat profile yet retain uniform phase for all excited resonances within a predefined frequency window. These operate without the need for purging pulses or further modifications, allowing them to be used directly in place of hard pulses. They are typically generated by computerised procedures which result in more exotic pulse envelopes (and acronyms Fig. 9.11) that drive magnetisation vectors along rather more tortuous trajectories than the simpler Gaussian-shap>ed cousins. Trajectories are shown in Fig. 9.15... 

Shown in Fig. 10 are calculated Z-scan curves for both beam profiles with the same value for d>Q. Here d>o is the nonlinear phase shift at r = 0 when the sample is at the focal point z = 0 and is the on-axis intensity at the focal point. Note that the peak-valley transmittance difference, Tp. = 7 p T, obtained with the top-hat profile is approximately 2.5 times greater than that with the Gaussian beam. 

Analyzing the boundary conditions given by Eq. (5.129), it can be concluded that the last two terms in Eq. (5.148) are zero and, therefore, the integral within the first term is independent of x. Considering a top-hat profile for the velocity in X = 0 yields the equation... 

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