2.3 The first-order Taylor-Maccoll equations

The first-order versions of Eq. (1) are the momentum equations, in spherical polar coordinates, in the r and θ directions [44]:

$$d\text{dV}/d\theta = -U + \frac{a^2(U + V\cot\theta)}{V^2 - a^2} \tag{3}$$

$$d\mathbf{U}/d\theta = \mathbf{V} \tag{4}$$

Having the T-M equations in this form reveals their singular nature at v = �1 where the singularity is caused by the (v<sup>2</sup> � 1)-term in the denominators above.<sup>1</sup> The term u + v cot θ, appearing in both numerators, is the component of Mach number normal to the axis. This component is zero for the freestream flow, so that, at the entrance, the Taylor-Maccoll equations take on a 0/0 type singularity and it turns out that (<sup>u</sup> <sup>+</sup> <sup>v</sup> cot <sup>θ</sup>)/(v<sup>2</sup> � 1) has a finite value at the freestream entrance of

As a result of using the Mach number variables u and v, the absence of any explicit reference to total conditions, as well as the sound speed, leads to a more straightforward application of the boundary conditions. A standard, fourth-order Runge-Kutta scheme [45] has been used to integrate the Mach number components, u and v form of Eqs. (5) and (6) and r ¼ fð Þθ , from Eq. (7). The solutions are identical, to eight decimal places, to similar solutions of (5) and (6) in the velocity variables. Eqs. (5) and (6) govern and describe the flow in a Busemann intake and

Eqs. (5) and (6) are simultaneous, first-order, total differential equations that can be solved by standard methods, such as in Ralston and Wilf [45], for the two Mach numbers u and v in terms of θ. The Mach number M is then found from

<sup>u</sup><sup>2</sup> <sup>þ</sup> <sup>v</sup><sup>2</sup> <sup>p</sup> and other thermodynamic values follow from isentropic relations. The shape of the intake surface can also be integrated within the integration routine to give r in terms of θ, r ¼ fð Þθ so that the Cartesian coordinates of the axisymmetric

Integration of Eqs. (5) and (6) requires the starting values u and v at the value

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

The angular location of the shock, which is the starting value for the variable of

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

<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.

<sup>2</sup> sin <sup>2</sup>θ<sup>23</sup> � <sup>1</sup> � �

θ<sup>2</sup> ¼ θ<sup>23</sup> � δ<sup>23</sup> (10)

u<sup>2</sup> ¼ M<sup>2</sup> cos θ<sup>23</sup> (11)

(9)

<sup>2</sup> γ þ 1 � 2 sin <sup>2</sup> ð Þ θ<sup>23</sup>

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

tan <sup>δ</sup><sup>23</sup> <sup>¼</sup> 2 cot <sup>θ</sup><sup>23</sup> <sup>M</sup><sup>2</sup>

2M<sup>2</sup>

Busemann surface shape are found from x ¼ r cos θ and y ¼ rsin θ.

the Busemann intake.

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

flow deflection [28]:

integration, θ, is then:

85

Eq. (7) gives the streamline/surface shape.

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

3. Solution of the Taylor-Maccoll equations

3.1 Boundary conditions at shock and freestream

the radial and circumferential Mach numbers are then:

where a is the speed of sound that can be written in terms of the velocities and the total conditions through the energy equation. The second of these equations is also the irrotationality condition, implying that conical flows are necessarily irrotational. Explicit reference to the speed of sound and total conditions can be circumvented if the equations are recast so as to have the radial and angular Mach number components (u, v) as dependent variables in place of the corresponding velocity components (U, V). The boundary conditions, when expressed as Mach number components at the up- and down-stream sides of conical shocks, are then applicable directly to the solution of the equations. Also, total conditions, which have no influence on the Mach number solution, do not have to be invoked.

#### 2.4 Mach number components (u, v) as dependent variables

The Taylor-Maccoll (T-M) Eqs. (3) and (4) have been recast in terms of the radial and angular Mach numbers u and v, where u = U/a and v = V/a and a is the local sound speed:

$$\frac{du}{d\theta} = v + \frac{\gamma - 1}{2} uv \frac{u + v \cot \theta}{v^2 - 1} \tag{5}$$

$$\frac{dv}{d\theta} = -u + \left(\mathbf{1} + \frac{\gamma - \mathbf{1}}{2}v^2\right) \frac{u + v \cot \theta}{v^2 - 1} \tag{6}$$

These two equations seem more complicated than their parents (3) and (4). However, it will be shown that the use of Mach number components u and v leads to meaningful and useful physical interpretations from Eqs. (5) and (6). Also, the sound speed has been eliminated as a variable.

In terms of Mach number components, the streamline Eq. (1) is,

$$r dr/d\theta = r u/v \tag{7}$$

and the flow Mach number is,

$$M = \sqrt{u^2 + v^2} \tag{8}$$

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

This is the model equation that governs steady, axisymmetric, conical flow of a perfect gas. No explicit algebraic solution has been found, nor are there any numerical schemes for solution of the second-order Eq. (2) as given above. However, the equation can be converted to two first order Eqs. (3) and (4), at the price of acquiring the additional dependent variable, V. But the two equations are now amenable to standard numerical solution methods. Most of these solutions have been done with boundary conditions applicable to flow over an axisymmetric cone [42, 43].

The first-order versions of Eq. (1) are the momentum equations, in spherical

where a is the speed of sound that can be written in terms of the velocities and the total conditions through the energy equation. The second of these equations is also the irrotationality condition, implying that conical flows are necessarily irrota-

circumvented if the equations are recast so as to have the radial and angular Mach number components (u, v) as dependent variables in place of the corresponding velocity components (U, V). The boundary conditions, when expressed as Mach number components at the up- and down-stream sides of conical shocks, are then applicable directly to the solution of the equations. Also, total conditions, which have no influence on the Mach number solution, do not have to be invoked.

The Taylor-Maccoll (T-M) Eqs. (3) and (4) have been recast in terms of the radial and angular Mach numbers u and v, where u = U/a and v = V/a and a is the

> 2 uv

These two equations seem more complicated than their parents (3) and (4). However, it will be shown that the use of Mach number components u and v leads to meaningful and useful physical interpretations from Eqs. (5) and (6). Also, the

> <sup>M</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u<sup>2</sup> þ v<sup>2</sup>

<sup>2</sup> <sup>v</sup><sup>2</sup> � � <sup>u</sup> <sup>þ</sup> <sup>v</sup> cot <sup>θ</sup>

u þ v cot θ

<sup>a</sup>2ð Þ <sup>U</sup> <sup>þ</sup> <sup>V</sup> cot <sup>θ</sup>

<sup>V</sup><sup>2</sup> � <sup>a</sup><sup>2</sup> (3)

<sup>v</sup><sup>2</sup> � <sup>1</sup> (5)

dr=dθ ¼ ru=v (7)

<sup>p</sup> (8)

<sup>v</sup><sup>2</sup> � <sup>1</sup> (6)

dU=dθ ¼ V (4)

dV=dθ ¼ �U þ

tional. Explicit reference to the speed of sound and total conditions can be

2.4 Mach number components (u, v) as dependent variables

du

dv

sound speed has been eliminated as a variable.

and the flow Mach number is,

<sup>d</sup><sup>θ</sup> <sup>¼</sup> <sup>v</sup> <sup>þ</sup> <sup>γ</sup> � <sup>1</sup>

<sup>d</sup><sup>θ</sup> ¼ �<sup>u</sup> <sup>þ</sup> <sup>1</sup> <sup>þ</sup> <sup>γ</sup> � <sup>1</sup>

In terms of Mach number components, the streamline Eq. (1) is,

local sound speed:

84

2.3 The first-order Taylor-Maccoll equations

Hypersonic Vehicles - Past, Present and Future Developments

polar coordinates, in the r and θ directions [44]:

Having the T-M equations in this form reveals their singular nature at v = �1 where the singularity is caused by the (v<sup>2</sup> � 1)-term in the denominators above.<sup>1</sup> The term u + v cot θ, appearing in both numerators, is the component of Mach number normal to the axis. This component is zero for the freestream flow, so that, at the entrance, the Taylor-Maccoll equations take on a 0/0 type singularity and it turns out that (<sup>u</sup> <sup>+</sup> <sup>v</sup> cot <sup>θ</sup>)/(v<sup>2</sup> � 1) has a finite value at the freestream entrance of the Busemann intake.

As a result of using the Mach number variables u and v, the absence of any explicit reference to total conditions, as well as the sound speed, leads to a more straightforward application of the boundary conditions. A standard, fourth-order Runge-Kutta scheme [45] has been used to integrate the Mach number components, u and v form of Eqs. (5) and (6) and r ¼ fð Þθ , from Eq. (7). The solutions are identical, to eight decimal places, to similar solutions of (5) and (6) in the velocity variables. Eqs. (5) and (6) govern and describe the flow in a Busemann intake and Eq. (7) gives the streamline/surface shape.
