Supersonic Nozzle Design
In Chapter 10, we demonstrated that a nozzle designed to expand a gas from rest to supersonic speeds must have a convergent-divergent shape. Moreover, the quasi- one-dimensional analysis of Chapter 10 led to the prediction of flow properties as a function of x through a nozzle of specified shape (see, e. g., Figure 10.10). The flow properties at any x station obtained from the quasi-one-dimensional analysis represent an average of the flow over the given nozzle cross section. The beauty of the quasi – one-dimensional approach is its simplicity. On the other hand, its disadvantages are (1) it cannot predict the details of the actual three-dimensional flow in a convergent – divergent nozzle and (2) it gives no information on the proper wall contour of such nozzles.
The purpose of the present section is to describe how the method of characteristics can supply the above information which is missing from a quasi-one-dimensional analysis. For simplicity, we treat a two-dimensional flow, as sketched in Figure 13.7. Here, the flow properties are a function of x and y. Such a two-dimensional flow is applicable to supersonic nozzles of rectangular cross section, such as sketched in the insert at the top of Figure 13.7. Two-dimensional (rectangular) nozzles are used in many supersonic wind tunnels. They are also the heart of gas-dynamic lasers (see Reference 1). In addition, there is current discussion of employing rectangular exhaust nozzles on advanced military jet airplanes envisaged for the future.
Consider the following problem. We wish to design a convergent-divergent nozzle to expand a gas from rest to a given supersonic Mach number at the exit Me. How do we design the proper contour so that we have shock-free, isentropic flow in the nozzle? The answer to this question is discussed in the remainder of this section.
For the convergent, subsonic section, there is no specific contour which is better than any other. There are rules of thumb based on experience and guided by subsonic flow theory; however, we are not concerned with the details here. We simply assume that we have a reasonable contour for the subsonic section.
Due to the two-dimensional nature of the flow in the throat region, the sonic line is generally curved, as sketched in Figure 13.7. A line called the limiting characteristic is sketched just downstream of the sonic line. The limiting characteristic is defined such that any characteristic line originating downstream of the limiting characteristic does not intersect the sonic line; in contrast, a characteristic line originating in the small region between the sonic line and the limiting characteristic can intersect the sonic line (for more details on the limiting characteristic, see Reference 21). To begin a method of characteristics solution, we must use an initial data line which is downstream of the limiting characteristic.
Let us assume that by independent calculation of the subsonic-transonic flow in the throat region, we know the flow properties at all points on the limiting characteristic. That is, we use the limiting characteristic as our initial data line. For example, we know the flow properties at points 1 and 2 on the limiting characteristic in Figure 13.7. Moreover, consider the nozzle contour just downstream of the throat. Letting в denote the angle between a tangent to the wall and the horizontal, the section of the divergent nozzle where в is increasing is called the expansion section, as shown in Figure 13.7. The end of the expansion section occurs where в — втм (point 8 in Figure 13.7). Downstream of this point, в decreases until it equals zero at the nozzle exit. The portion of the contour where в decreases is called the straightening section. The shape of the expansion section is somewhat arbitrary; typically, a circular arc of large radius is used for the expansion section of many wind-tunnel nozzles. Consequently, in addition to knowing the flow properties along the limiting characteristic, we also have an expansion section of specified shape; that is, we know 6, 65, and (9X in Figure 13.7. The purpose of our application of the method of characteristics now becomes the proper design of the contour of the straightening section (from points 8 to 13 in Figure 13.7).
The characteristics mesh sketched in Figure 13.7 is very coarse—this is done intentionally to keep our discussion simple. In an actual calculation, the mesh should be much finer. The characteristics mesh and the flow properties at the associated grid points are calculated as follows:
1. Draw a C_ characteristic from point 2, intersecting the centerline at point 3.
Evaluating Equation (13.17) at point 3, we have
$3 + U3 = (А"_)з
In the above equation, 03 = 0 (the flow is horizontal along the centerline). Also,
(А"_)з is known because (A"_)3 = (K – )3. Hence, the above equation can be
solved for v3.
2. Point 4 is located by the intersection of the C_ characteristic from point 1 and the C+ characteristic from point 3. In turn, the flow properties at the internal point 4 are determined as discussed in the last part of Section 13.2.
3. Point 5 is located by the intersection of the C+ characteristic from point 4 with the wall. Since 05 is known, the flow properties at point 5 are determined as discussed in Section 13.2 for wall points.
4. Points 6 through 11 are located in a manner similar to the above, and the flow properties at these points are determined as discussed before, using the internal point or wall point method as appropriate.
5. Point 12 is a wall point on the straightening section of the contour. The purpose of the straightening section is to cancel the expansion waves generated by the expansion section. Hence, there are no waves which are reflected from the straightening section. In turn, no right-running waves cross the characteristic line between points 9 and 12. Asa result, the characteristic line between points 9 and 12 is a straight line, along which в is constant, that is, 612 = O9. The section of the wall contour between points 8 and 12 is approximated by a straight line with an average slope of ^(в% + 612).
6. Along the centerline, the Mach number continuously increases. Let us assume that at point 11, the design exit Mach number Me is reached. The characteristic line from points 11 to 13 is the last line of the calculation. Again, вц = 9ц, and the contour from point 12 to point 13 is approximated by a straight-line segment with an average slope of 5(^12 + віз).
The above description is intended to give you a “feel” for the application of the method of characteristics. If you wish to carry out an actual nozzle design, and/or if you are interested in more details, read the more complete treatments in References 21 and 34.
Note in Figure 13.7 that the nozzle flow is symmetrical about the centerline. Hence, the points below the centerline (T, 2′, 3′, etc.) are simply mirror images of the corresponding points above the centerline. In making a calculation of the flow through the nozzle, we need to concern ourselves only with those points in the upper half of Figure 13.7, above and on the centerline.