5. Performance of Busemann flow as an air intake (inviscid flow)

An integration of the TM-Eqs. (5) and (6) from the initial conditions (10–12) is terminated when (u + v cot θ) = 0 at the free-stream where we discover the Mach number M1. The results of many such calculations are shown in Figure 14 where each complete Busemann intake calculation is represented by a dot. For each case, a value of M<sup>2</sup> is selected, in our case between 1 and 8, and k (the shock-normal component of M2) is cycled from 1 to M2. For each M<sup>2</sup> and k the total pressure ratio, Eq. (13), and M3, Eq. (14), are calculated. Integration of the T-M equations then leads to the freestream at M1 and a point is plotted on a graph of M<sup>1</sup> vs. M<sup>3</sup> with pt3/pt<sup>1</sup> as parameter, determining the point's color. Every point in this figure represents a Busemann intake calculation from the downstream shock to the freestream. This graph can be used to select a Busemann intake design based on the desired entry The Busemann Air Intake for Hypersonic Speeds DOI: http://dx.doi.org/10.5772/intechopen.82736

Figure 14. Inviscid performance of Busemann intake.

effects. The analytical expressions for all three gradients have the radial coordinate r in their denominators. This requires the gradients to be the mildest on the intake surface and highest at the origin—a desirable condition for orderly wall boundary

Mach number components (u, v) and gradients behind conical shock at Mach 3.

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As a check on the various algebraic results we have plotted them against the acute (20–90°) and obtuse (90–160°) angles of conical shocks for Mach 3 in Figure 13. The left half of this figure is for acute shocks and the right half is for obtuse shocks, i.e., cone flow and M-flow. u (red) and v (blue) are the Mach number components behind the shock in the ð Þ r; θ -directions. Black curves are for the various gradients from the T-M Eqs. (21), (26) and (27). The green curves are for the same gradients as calculated by Curved Shock Theory (CST) [48]. There is perfect agreement between gradients calculated from the T-M equations and those from CST. This is reassuring since the two methods are based on widely differing

5. Performance of Busemann flow as an air intake (inviscid flow)

An integration of the TM-Eqs. (5) and (6) from the initial conditions (10–12) is terminated when (u + v cot θ) = 0 at the free-stream where we discover the Mach number M1. The results of many such calculations are shown in Figure 14 where each complete Busemann intake calculation is represented by a dot. For each case, a value of M<sup>2</sup> is selected, in our case between 1 and 8, and k (the shock-normal component of M2) is cycled from 1 to M2. For each M<sup>2</sup> and k the total pressure ratio, Eq. (13), and M3, Eq. (14), are calculated. Integration of the T-M equations then leads to the freestream at M1 and a point is plotted on a graph of M<sup>1</sup> vs. M<sup>3</sup> with pt3/pt<sup>1</sup> as parameter, determining the point's color. Every point in this figure represents a Busemann intake calculation from the downstream shock to the freestream. This graph can be used to select a Busemann intake design based on the desired entry

layer development on the intake surface.

4.7 Gradients at conical shockwaves

theoretical approaches.

96

Figure 13.

and exit Mach numbers and the total pressure recovery. Any two of these parameters determine the third. For example, it is apparent from the graph that a Busemann intake that reduces the freestream Mach number from 7 to 3 does so with a total pressure recovery of 0.95. This graph represents both components of Busemann intake performance, the capability by M<sup>1</sup> and M<sup>3</sup> and the efficiency by pt3/pt1. Tradeoffs between these are workable with this diagram. As an example, a Busemann intake that reduces the Mach number by a factor of three does so with a total pressure recovery of about 0.90. A more refined and elaborate version of such a performance map is found in [3].
