3.1 Boundary conditions at shock and freestream

Integration of Eqs. (5) and (6) requires the starting values u and v at the value of θ = θ<sup>2</sup> in front of the shock. A convenient and aerodynamically significant approach is to select the Mach number in front of the shock M<sup>2</sup> and the aerodynamic shock angle θ<sup>23</sup> as the starting variables. The flow deflection through the shock, δ23, is found from the equation relating Mach number, shock angle and flow deflection [28]:

$$\tan \delta\_{23} = \frac{2 \cot \theta\_{23} (M\_2^2 \sin^2 \theta\_{23} - 1)}{2M\_2^2 (\gamma + 1 - 2 \sin^2 \theta\_{23})} \tag{9}$$

The angular location of the shock, which is the starting value for the variable of integration, θ, is then:

$$
\theta\_2 = \theta\_{23} - \delta\_{23} \tag{10}
$$

This ensures that the flow behind the shock is parallel to the axis, which is the most common requirement of flow entering a combustor. The starting values for the radial and circumferential Mach numbers are then:

$$
\mu\_2 = M\_2 \cos \theta\_{23} \tag{11}
$$

<sup>1</sup> Such singularities are discussed in [29, 45, 46]. Their appearance, in any given flow, should be taken as a warning that whatever symmetry assumption(s) have been made may not hold in the physical airflow.

Hypersonic Vehicles - Past, Present and Future Developments

$$v\_2 = -M\_2 \sin \theta\_{23} \tag{12}$$

The radial variable, r, becomes dependent on u and v and the starting value r<sup>2</sup> at the shock. The value of r2, the shock's length, is arbitrary at this stage. It determines the scale size of the streamline and its utility becomes relevant when considering morphing and wavecatching in Sections 1.3 and 7. Note that, prior to integration of Eqs. (5) and (6), and calculation of the intake surface shape, we could calculate the intake's efficiency, using the total pressure ratio as measure,

$$p\_{t3}/p\_{t2} = \left[\frac{(\chi+\mathbf{1})k^2}{(\chi-\mathbf{1})k^2+\mathbf{2}}\right]^{\frac{\gamma}{\gamma-1}} \left[\frac{\chi+\mathbf{1}}{2\chi k^2-\chi+\mathbf{1}}\right]^{\frac{1}{\gamma-1}}\tag{13}$$

and the capability from the exit Mach number,

$$M\_3^2 = \frac{(\chi + \mathbf{1})^2 M\_2^2 k^2 - 4 \left(k^2 - \mathbf{1}\right) \left(\chi k^2 + \mathbf{1}\right)}{\left[2\chi k^2 - (\chi - \mathbf{1})\right] \left[(\chi - \mathbf{1})k^2 + \mathbf{2}\right]} \tag{14}$$

Figure 5. Although the CFD code is not "told" anything about conicality, the conical nature of the flow is well represented by the CFD calculations. Both methods predict a uniform exit flow downstream of the conical shock (courtesy Dr. Ogawa). This is an illustration of the use of Busemann flow as a benchmark for verifying the application of a CFD code to internal flow. Graphical results of an integration of Eqs. 5–7 are shown in Figure 6 for a Busemann intake that reduces the Mach

Busemann intake contour (black curve) with conical shock (red) and cone of inflection points (green). Mach number distribution (blue). Pressure distribution, normalized with respect to exit pressure (red, on the right side ordinate), for an intake that reduces the Mach number from 5.22 to 1.93 with a total pressure recovery of

At the entry, Busemann flow joins to the freestream at a conical Mach wave. The Mach number normal to this wave, v = �1, which makes both Eqs. (5) and (6) have a zero in their denominators. At the conical Mach wave u þ v cot θ is also zero so that Eqs. (5) and (6) have a 0/0-type singularity. This makes it impossible to start the integration at a specific freestream Mach number so as to progress in a clockwise (downstream) direction towards the shock. An infinite number of streamlines are possible and unique boundary conditions cannot be specified at the freestream. The starting value of r<sup>2</sup> is arbitrary; it determines the scale size of the streamline and its utility becomes relevant when considering morphing and wavecatching in

This section describes some features of Busemann-type intake flow that are unique to axisymmetric conically symmetric flow. First, there is the geometric simplicitly that arises from the axial and conical symmetries. These symmetries require that conditions on a circle, which circumscribes the axis, are constant and conditions are constant also on any circular cone surface whose axis is aligned with

the symmetry axis and whose apex is confocal with all other such cones.

number from 5.22 to 1.93 with a total pressure recovery of 0.94.

4. Aerodynamic features of Busemann intake

3.2 Singularity at entry

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

Figure 6.

0.94.

Sections 7.1 and 7.2.

87

where <sup>k</sup><sup>2</sup> <sup>¼</sup> <sup>M</sup><sup>2</sup> <sup>2</sup> sin<sup>2</sup> θ<sup>23</sup> is the square of the shock-normal Mach number component. In fact, we could prescribe a desired efficiency, pt3=pt2, and calculate <sup>k</sup><sup>2</sup> from Eq. (13); also prescribe the downstream Mach number M<sup>3</sup> and calculate M<sup>2</sup> by inverting Eq. (14). Then θ<sup>23</sup> = sin�<sup>1</sup> (k/M2), u<sup>2</sup> = M<sup>2</sup> cos θ<sup>23</sup> and v<sup>2</sup> = �M<sup>2</sup> sin θ23. After this, θ<sup>2</sup> and δ<sup>23</sup> are found as above and the integration performed, on increasing θ, until (u + v cot θ) ≥ 0. The ability to specify the downstream Mach number and an intake efficiency, before doing the integration, makes this approach particularly suitable for preliminary intake design selection. Note, however, that all is not roses, since the integration yields a freestream Mach number that may not be the desired one. An iteration, on the input conditions, pt3=pt<sup>2</sup> and <sup>M</sup>3, or <sup>k</sup><sup>2</sup> and <sup>M</sup><sup>2</sup> has to be performed to arrive at the desired intake design Mach number. This inconvenience is the direct result of, and the price paid for, the convenience and simplicity achieved by imposing the flow to be conically symmetric and by imposing the outflow conditions. It turns out that, using the T-M equations, the flow curvature and gradients of pressure and Mach number can also be found at the shock wave before the complete integration is done (Sections 4.4–4.6).

Eqs. (5) and (6) are then numerically integrated from θ<sup>2</sup> to θ<sup>1</sup> = π–μ<sup>1</sup> in an upwind direction with an increasing θ. Since θ<sup>1</sup> is not known a priori, the integration is continued until the normal-to-the-axis (cross-stream) Mach number (u sin θ + v cos θ) becomes zero or positive, indicating that the freestream has been reached. The calculated shape and Mach number contours of such an integration are shown in the top half of Figure 5.

Note the conical nature of the contours. The calculated Busemann shape is then used as input to a CFD code to predict the flow as shown in the lower half of

#### Figure 5.

Flow Mach number contours in the axisymmetric Busemann intake for inviscid flow. Top half is obtained by integrating the Taylor-Maccoll equations. Bottom half is a CFD calculation [by Ogawa] of flow in the same intake shape as the top half.

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

#### Figure 6.

v<sup>2</sup> ¼ �M<sup>2</sup> sin θ<sup>23</sup> (12)

γ þ 1 <sup>2</sup>γk<sup>2</sup> � <sup>γ</sup> <sup>þ</sup> <sup>1</sup> � � <sup>1</sup>

<sup>2</sup>γk<sup>2</sup> � ð Þ <sup>γ</sup> � <sup>1</sup> � � ð Þ <sup>γ</sup> � <sup>1</sup> <sup>k</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup> � � (14)

(k/M2), u<sup>2</sup> = M<sup>2</sup> cos θ<sup>23</sup> and v<sup>2</sup> = �M<sup>2</sup> sin θ23.

γ�1

(13)

The radial variable, r, becomes dependent on u and v and the starting value r<sup>2</sup> at the shock. The value of r2, the shock's length, is arbitrary at this stage. It determines the scale size of the streamline and its utility becomes relevant when considering morphing and wavecatching in Sections 1.3 and 7. Note that, prior to integration of Eqs. (5) and (6), and calculation of the intake surface shape, we could calculate the

γ�1

<sup>2</sup>k<sup>2</sup> � <sup>4</sup> <sup>k</sup><sup>2</sup> � <sup>1</sup> � � <sup>γ</sup>k<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �

<sup>2</sup> sin<sup>2</sup> θ<sup>23</sup> is the square of the shock-normal Mach number compo-

intake's efficiency, using the total pressure ratio as measure,

Hypersonic Vehicles - Past, Present and Future Developments

pt3=pt<sup>2</sup> <sup>¼</sup> ð Þ <sup>γ</sup> <sup>þ</sup> <sup>1</sup> <sup>k</sup><sup>2</sup>

and the capability from the exit Mach number,

<sup>3</sup> <sup>¼</sup> ð Þ <sup>γ</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup>

before the complete integration is done (Sections 4.4–4.6).

M<sup>2</sup>

where <sup>k</sup><sup>2</sup> <sup>¼</sup> <sup>M</sup><sup>2</sup>

in the top half of Figure 5.

Figure 5.

86

intake shape as the top half.

inverting Eq. (14). Then θ<sup>23</sup> = sin�<sup>1</sup>

ð Þ <sup>γ</sup> � <sup>1</sup> <sup>k</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>

M<sup>2</sup>

nent. In fact, we could prescribe a desired efficiency, pt3=pt2, and calculate <sup>k</sup><sup>2</sup> from Eq. (13); also prescribe the downstream Mach number M<sup>3</sup> and calculate M<sup>2</sup> by

After this, θ<sup>2</sup> and δ<sup>23</sup> are found as above and the integration performed, on increasing θ, until (u + v cot θ) ≥ 0. The ability to specify the downstream Mach number and an intake efficiency, before doing the integration, makes this approach particularly suitable for preliminary intake design selection. Note, however, that all is not roses, since the integration yields a freestream Mach number that may not be the desired one. An iteration, on the input conditions, pt3=pt<sup>2</sup> and <sup>M</sup>3, or <sup>k</sup><sup>2</sup> and <sup>M</sup><sup>2</sup> has to be performed to arrive at the desired intake design Mach number. This inconvenience is the direct result of, and the price paid for, the convenience and simplicity achieved by imposing the flow to be conically symmetric and by imposing the outflow conditions. It turns out that, using the T-M equations, the flow curvature and gradients of pressure and Mach number can also be found at the shock wave

Eqs. (5) and (6) are then numerically integrated from θ<sup>2</sup> to θ<sup>1</sup> = π–μ<sup>1</sup> in an upwind direction with an increasing θ. Since θ<sup>1</sup> is not known a priori, the integration is continued until the normal-to-the-axis (cross-stream) Mach number (u sin θ + v cos θ) becomes zero or positive, indicating that the freestream has been reached. The calculated shape and Mach number contours of such an integration are shown

Note the conical nature of the contours. The calculated Busemann shape is then

used as input to a CFD code to predict the flow as shown in the lower half of

Flow Mach number contours in the axisymmetric Busemann intake for inviscid flow. Top half is obtained by integrating the Taylor-Maccoll equations. Bottom half is a CFD calculation [by Ogawa] of flow in the same

" # <sup>γ</sup>

Busemann intake contour (black curve) with conical shock (red) and cone of inflection points (green). Mach number distribution (blue). Pressure distribution, normalized with respect to exit pressure (red, on the right side ordinate), for an intake that reduces the Mach number from 5.22 to 1.93 with a total pressure recovery of 0.94.

Figure 5. Although the CFD code is not "told" anything about conicality, the conical nature of the flow is well represented by the CFD calculations. Both methods predict a uniform exit flow downstream of the conical shock (courtesy Dr. Ogawa). This is an illustration of the use of Busemann flow as a benchmark for verifying the application of a CFD code to internal flow. Graphical results of an integration of Eqs. 5–7 are shown in Figure 6 for a Busemann intake that reduces the Mach number from 5.22 to 1.93 with a total pressure recovery of 0.94.

#### 3.2 Singularity at entry

At the entry, Busemann flow joins to the freestream at a conical Mach wave. The Mach number normal to this wave, v = �1, which makes both Eqs. (5) and (6) have a zero in their denominators. At the conical Mach wave u þ v cot θ is also zero so that Eqs. (5) and (6) have a 0/0-type singularity. This makes it impossible to start the integration at a specific freestream Mach number so as to progress in a clockwise (downstream) direction towards the shock. An infinite number of streamlines are possible and unique boundary conditions cannot be specified at the freestream. The starting value of r<sup>2</sup> is arbitrary; it determines the scale size of the streamline and its utility becomes relevant when considering morphing and wavecatching in Sections 7.1 and 7.2.
