4.5 Surface Mach number gradient in the flow direction, dM/ds

The streamline equation may be written,

$$\frac{ds}{d\theta} = \frac{r}{v}\sqrt{u^2 + v^2} \tag{23}$$

The flow Mach number, M, in terms of its radial and axial components, u, and v, is,

$$M = \sqrt{\mathfrak{u}^2 + \mathfrak{v}^2} \tag{24}$$

So that,

$$M\frac{dM}{d\theta} = M\frac{dM}{ds}\frac{ds}{d\theta} = u\frac{du}{d\theta} + v\frac{dv}{d\theta} \tag{25}$$

giving,

The Busemann Air Intake for Hypersonic Speeds DOI: http://dx.doi.org/10.5772/intechopen.82736

$$\frac{dM}{ds} = \frac{v}{r(u^2 + v^2)} \left[ u \frac{du}{d\theta} + v \frac{dv}{d\theta} \right] \tag{26}$$

where the derivative terms, in the square brackets, are given by the Taylor-Maccoll Eqs. (5) and (6), when multiplied by u and v, respectively.

$$
u \frac{du}{d\theta} = \imath uv + \frac{\gamma - 1}{2} u^2 v \frac{u + v \cot \theta}{v^2 - 1} \tag{27}$$

$$v\frac{dv}{d\theta} = -\mu v + v\left(1 + \frac{\gamma - 1}{2}v^2\right)\frac{u + v\cot\theta}{v^2 - 1} \tag{28}$$

so that,

tip singularity described above; its existence, in the idealized form, has not seen confirmation by experiment or CFD. The cone of streamline inflections is a significant feature for assessing startability of wavecatcher Busemann intakes

Hypersonic Vehicles - Past, Present and Future Developments

4.There is a point of zero curvature also when (u+v cot θ) = 0. The quantity (u+v cot θ) is the component of Mach number normal to the flow axis. For Busemann flow it is zero only where the Busemann flow joins the freestream. Thus, the leading edge of the Busemann flow has not only zero deflection but also zero curvature. Aerodynamically this means that the leading edge wave is neither compressive nor expansive but is a zero-strength Mach wave. The fact that the entering freestream flow is neither deflected nor curved by the Busemann leading edge means that the leading edge of a hypersonic air intake, based on Busemann flow, is ineffective in contributing to the intake's task of reducing the Mach number. This provides an incentive to foreshorten some length of the leading edge surface so as to decrease viscous losses, possibly without incurring serious inviscid flow losses. For M-flow [32] the potential appearance of the condition (u+v cot θ) = 0 is prevented by the appearance of the (v ! �1)-singularity (described below) so that the postshock flow never becomes parallel to the freestream. This is unfortunate from a practical viewpoint since it presents no possibility of grafting any of the flows that have a uniform upstream, such as cone or Busemann flows, to the downstream of M-flow. From a fundamental viewpoint it also presents an obstacle to the possibility of conical shock reflection at the center line of

5. When the angular component of Mach number v ! �1 then D ! ∞; the curvature becomes infinite and the streamline has a cusp or a corner. This indicates a singularity or a limit line at a corner. Neither cone nor Busemann flow exhibit such a limit line. However, it does occur in both M- and W-flows [32].

It is always a positive quantity for all flows and has no drastic characterizing effect on D except to force streamlines to be less curved, to straighten out, at

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u<sup>2</sup> þ v<sup>2</sup>

> > du <sup>d</sup><sup>θ</sup> <sup>þ</sup> <sup>v</sup>

The flow Mach number, M, in terms of its radial and axial components, u, and v, is,

<sup>M</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

hypersonic speeds. Hypersonic intakes become long and slender.

4.5 Surface Mach number gradient in the flow direction, dM/ds

ds <sup>d</sup><sup>θ</sup> <sup>¼</sup> <sup>r</sup> v

<sup>d</sup><sup>θ</sup> <sup>¼</sup> <sup>M</sup> dM ds ds <sup>d</sup><sup>θ</sup> <sup>¼</sup> <sup>u</sup>

<sup>M</sup> dM

3/2, appearing in the denominator of Eq. (21), is just M<sup>3</sup>

<sup>p</sup> (23)

<sup>u</sup><sup>2</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> <sup>p</sup> (24)

<sup>d</sup><sup>θ</sup> (25)

dv

.

in Section 8.

symmetry [31].

6.The quantity (v<sup>2</sup> + u<sup>2</sup>

So that,

giving,

94

)

The streamline equation may be written,

$$\frac{dM}{ds} = \frac{v}{r(u^2 + v^2)} \left[ \mathbf{1} + \frac{\gamma - \mathbf{1}}{2} \left( u^2 + v^2 \right) \right] \frac{u + v \cot \theta}{v^2 - \mathbf{1}} \tag{29}$$

This is the Mach number gradient, expressed in terms of the coordinates (r, θ) and the corresponding Mach number components (u, v), where the Mach number component values come directly from the integration of the Taylor-Maccoll Eqs. (5) and (6). dM/ds is plotted in Figure 12 (with s measured in the downstream direction).

#### 4.6 Surface pressure gradient in the flow direction, P = (dp/ds)/(ρV<sup>2</sup> )

In the isentropic flow, from the freestream to the shock, the gradients of Mach number and pressure are related by [47],

$$\frac{dM}{M} = -\frac{\mathbf{1} + \frac{(\mathbf{y} - \mathbf{1})}{2}M^2}{\gamma M^2} \frac{dp}{p}$$
 
$$\text{or} \quad \frac{dM}{ds} = -\left[\mathbf{1} + \frac{(\mathbf{y} - \mathbf{1})}{2}M^2\right]MP$$

where dM ds is given by Eq. 26; so that the non-dimensional pressure gradient is,

$$P \equiv \frac{dp/ds}{\rho V^2} = \frac{-v^2}{r(u^2 + v^2)^2} \frac{(u + v \cot \theta)}{(v^2 - 1)}\tag{30}$$

The pressure gradient is expressed in terms of the radial and azimuthal coordinates r and θ and the radial and angular Mach number components u and v. It is plotted in Figure 12. This permits the calculation of the surface pressure gradient from quantities obtained in the T-M calculation of Busemann flow.3 It also means that the surface pressure gradients are known everywhere on the surface of the highly three-dimensional wavecatcher shapes where all the surface gradients are useful as inputs to boundary layer calculations. Towards this end it is noted that, for the flow just upstream of the corner, where the shock impinges, the u and v Mach numbers are given by u<sup>2</sup> and v<sup>2</sup> from Eqs. (11) and (12), so that the gradients immediately before the shock-boundary-layer interaction at the corner can be evaluated just from the prescribed initial conditions using Eqs. (5)–(7), before embarking on a calculation of Eqs. (5) and (6). This enables a selection of initial conditions that is based on considerations involving the shock losses as well as the shock-boundary-layer interaction

<sup>3</sup> These gradient equations are applicable to all types of Taylor-Maccoll flows [48].

Figure 13.

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

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 layer development on the intake surface.

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

High performance intakes have to have a very weak leading edge shock. Such a weak shock is inclined at near the Mach angle. This leads to the length-to-height ratio of the intake to be approximately M, the freestream Mach number. So that high performance intakes, including the Busemann intake, tend to become long and slender with large surface areas that have high shear near the leading edge, causing disproportionately high viscous losses. Surface length also leads to thick boundary layers at the exit with losses and the potential for major flow disruptions by bound-

A comparison of inviscid and viscous flow in the Busemann intake is shown in Figure 15 by Mach number contours. The blue, low Mach number boundary layer, appears in the viscous flow. The effect of the boundary layer has led directly to the presence of a shock from the leading edge and a noticeable change in the flow at the center line, a change of exit Mach number from 5.3 to 4.8 and a reduction in total pressure recovery from 0.97 to 0.43. The boundary layer has a significant effect on

Flow displacement by the boundary layer causes a conical shock to appear and focus to a point on the center line ahead of the Busemann flow focal point and a reflected, conical shock appears downstream that impinges on the surface ahead of the corner, Figure 16. To restore the inviscid flow topology and pressure distribution of the Busemann flow it is necessary to correct the surface shape of the intake

the inviscid flow even when it appears to stay attached.

map is found in [3].

Figure 14.

ary layer separation.

97

6. Boundary layer effects

Inviscid performance of Busemann intake.

The Busemann Air Intake for Hypersonic Speeds DOI: http://dx.doi.org/10.5772/intechopen.82736

#### 4.7 Gradients at conical shockwaves

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 theoretical approaches.
