The effect of small pipe divergence on an inviscid, incompressible, near critical axisymmetric swirling flow is investigated. The singular behavior of a regular expansion solution, in terms of the pipe divergence parameter, around the critical swirl of a flow in a straight pipe is demonstrated. This singularity infers that large-amplitude disturbances may be induced by the small pipe divergence when incoming flows have a swirl level near the critical swirl. In order to gain insight to the behavior of flows in this swirl range, a small-disturbance analysis is developed. It is found that a small but finite pipe divergence breaks the transcritical bifurcation of solutions of a flow in a straight pipe into two equilibrium solution branches. These branches fold at limit swirl levels near the critical swirl resulting in a finite gap of swirl that separates the two branches. This suggests that no near-columnar axisymmetric state can exist within this range of incoming swirl around the critical level; the flow must develop large disturbances in this swirl range. Beyond this range, two steady states may exist under the same inlet/outlet conditions. However, when the pipe divergence is increased, this special behavior uniformly changes into a branch of solutions with no fold. A weakly nonlinear approach to study the effect of slight pipe divergence on standing waves in a long pipe is also derived. The behavior of the asymptotic solutions match the bifurcation diagrams from previous theoretical and numerical studies and extends their results. The relevance of the results to axisymmetric vortex breakdown in a diverging pipe is discussed.
The propagation of a laser beam through Rayleigh-Bénard (RB) turbulence is investigated experimentally and by way of numerical simulation. For the experimental part, a focused laser beam transversed a 5 m×0.5 m×0.5 m water filled tank lengthwise. The tank is heated from the bottom and cooled from the top to produce convective RB turbulence. The effect of the turbulence on the beam is recorded on the exit of the beam from the tank. From the centroid motion of the beam, the index of refraction structure constant Cn2 is determined. For the numerical efforts RB turbulence is simulated for a tank of the same geometry. The simulated temperature fields are converted to the index of refraction distributions, and Cn2 is extracted from the index of refraction structure functions, as well as from the simulated beam wander. To model the effect on beam propagation, the simulated index of refraction fields are converted to discrete index of refraction phase screens. These phase screens are then used in a split-step beam propagation method to investigate the effect of the turbulence on a laser beam. The beam wander as well as the index of refraction structure parameter Cn2 determined from the experiment and simulation are compared and found to be in good agreement.
The effect of temperature on the surface tension of soluble and insoluble surfactants was investigated at an air-water interface. Equilibrium surface tension measurements were performed using the Wilhelmy plate technique in which both temperature and concentration were varied systematically. Insoluble surfactants (oleyl alcohol and hemicyanine) and soluble surfactants (Triton X-100 and sodium dodecyl sulfate (SDS)) were used since they are commonly used in hydrodynamic experiments in which the effects of surfactants on free surface dynamics are studied. The principal result of this investigation is that the surface tension of the above-mentioned surfactants decreases linearly with temperature, independent of concentration, with the exception of oleyl alcohol whose surface tension becomes relatively independent of temperature above 23 °C. The adequacy of standard models for surfactant behavior in describing these data is considered.
A linear stability analysis of a family of steady, noncolumnar and axisymmetric, swirling flows that may develop in a finite-length slightly diverging pipe is presented. These flow states are described by the asymptotic analysis of Rusak et al. (1998). There exists a limit level of the incoming flow swirl ratio ωcσ1 which is the corrected critical swirl as a result of the pipe divergence. When the swirl ratio is in a certain range below ωcσ1, two steady states can exist for the same inlet, outlet, and wall conditions: One which describes a near-columnar vortex state and another which describes a swirling flow with a large-amplitude disturbance. When the swirl level is above ωcσ1, no near-columnar, steady, and axisymmetric state exists. The stability of this family of flows is examined by studying the linearized dynamics of an unsteady and axially symmetric perturbation which also satisfies the boundary conditions. The stability analysis shows that ωcσ1 is a point of exchange of stability for the family of the noncolumnar vortex flows. The near-columnar states have a linearly stable mode of disturbance whereas the states with large disturbances are unstable. Also, the near-columnar states lose their stability characteristics as the swirl level approaches ωcσ1. Therefore, the analysis implies that when the swirl level of the incoming flow is above ωcσ1, the flow in the pipe must develop a transition process that involves large-amplitude perturbations and may lead to vortex breakdown states. The effect of the increase of pipe divergence on the flow dynamics and transition to breakdown states is also discussed.
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