Supersonic flow, cone

A nozzle is correctly designed for any outlet pressure between P[ and PE in Figure 4.5. Under these conditions the velocity will not exceed the sonic velocity at any point, and the flowrate will be independent of the exit pressure PE = Pb- It is also correctly designed for supersonic flow in the diverging cone for an exit pressure of PEj. 

Cone in Supersonic Flow. The preceding solutions for a flat plate may be applied to a cone in supersonic flow through the Mangier transformation [39], which in its most general form relates the boundary layer flow over an arbitrary axisymmetric body to an equivalent flow over a two-dimensional body. This transformation is contained in Eq. 6.89, which results in transformed axisymmetric momentum and energy equations equivalent to the two-dimensional equations (Eqs. 6.95 and 6.97). Hence, solutions of these equations are applicable to either a two-dimensional or an axisymmetric flow, the differences being contained solely in the coordinate transformations. 

For the case of an arbitrary pressure distribution, it is just as convenient to solve the axisymmetric problem directly. When the solution for the equivalent two-dimensional flow already exists, however, as for a flat plate, then the results for the corresponding axisymmetric problem can be obtained by direct comparison. This correspondence exists for a cone in supersonic flow when the surface pressure is constant. Solutions of Eqs. 6.95 and 6.97 for a flat plate expressed in terms of C, and T] may then be applied to a cone. Illustrative examples are presented in the following subsections. 

It has been shown above that when the pressure in the diverging section is greater than the throat pressure, subsonic flow occurs. Conversely, if the pressure in the diverging section is less than the throat pressure the flow will be supersonic beyond the throat. Thus at a given point in the diverging cone where the area is equal to At the pressure may have one of two values for isentropic flow. 

We considered it likely that placing the chopper between the nozzle and the skimmer might interfere with the supersonic expansion downstream of the nozzle. We were pleasantly surprised to find that this was not the case because we obtained velocity distributions with the chopper located between the nozzle and the skimmer which were identical to the distributions obtained when the chopper was located in the more conventional position downstream of the skimmer. This is undoubtedly due to the nonideality of the flow through the cone which will be described later. 

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