Compressible Flow Free Textbook

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Comparison And Limitations between the Two Approaches

The two models produce exactly the same results, but the assumptions for the construction of these models are different. In the geometrical model, the assumption is that the velocity change in the radial direction is zero. In the rigorous model, it was assumed that radial velocity is only a function of $\theta$ . The statement for the construction of the geometrical model can be improved by assuming that the frame of reference is moving radially in a constant velocity.

Regardless of the assumptions that were used in the construction of these models, the fact remains that there is a radial velocity at . At this point ( ) these models fail to satisfy the boundary conditions and something else happens there. On top of the complication of the turning point, the question of boundary layer arises. For example, how did the gas accelerate to above the speed of sound when there is no nozzle (where is the nozzle?)? These questions are of interest in engineering but are beyond the scope of this book (at least at this stage). Normally, the author recommends that this function be used everywhere beyond 2-4 the thickness of the boundary layer based on the upstream length.

In fact, analysis of design commonly used in the industry and even questions posted to students show that many assume that the turning point can be sharp. At a small Mach number, $(1+\epsilon)$ the radial velocity is small $\epsilon$ . However, an increase in the Mach number can result in a very significant radial velocity. The radial velocity is ``fed'' through the reduction of the density. Aside from its close proximity to turning point, mass balance is maintained by the reduction of the density. Thus, some researchers recommend that, in many instances, the sharp point should be replaced by a smoother transition.

Next: The Maximum Turning Angle Up: Geometrical Explanation Previous: Alternative Approach to Governing Index

Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21