2. Taylor-Maccoll equation(s) and Busemann flow

Busemann [1] described an axisymmetric, conical flow that starts in the uniform freestream, compresses and contracts isentropically and passes through a conical shockwave to become uniform and parallel to the freestream flow. Courant and Friedrichs [2] make a brief reference to Busemann flow, suggesting its use as an air intake. Molder and Szpiro [3] used the Taylor-Maccoll equations to calculate the inviscid Busemann flow and present a capability/efficiency performance map for the flow as a hypersonic air intake. Experiments, at Mach 8.33, on a full Busemann intake and on modular, wavecatcher surfaces, based on Busemann flow were conducted by Mölder and Romeskie [4] and by Jacobsen et al. [25]. VanWie and Molder [12] suggested applications of the Busemann intake to hypersonic flight vehicles.

The Busemann intake shape is analytically defined by only two numerical parameters [3]. This has made it easily "transportable" and led to its proposed use as a benchmark standard for internal flow CFD verification [38], and a basis for more

general studies of intake flows as well as experiments for such issues as flow starting [18, 25, 27, 31–37], viscous effects [15, 39], truncation [16, 18], drag measurement [11], wavecatcher configurations [13], leading edge blunting [10] and cross section morphing [7]. Viscous effects and truncation and stunting are found in [15, 18, 39]. Experimental results for full and modular Busemann intakes are found in [11, 13, 18, 21, 23, 25, 33, 37, 40]. A four-module Busemann-based intake on a scramjet engine was launched at Mach 5, from a large ballistic gun [41].

single remaining spatial variable—the conical angle, θ. This offers great simplicity in flow analysis where a wide variety of intake surfaces is available for selection of surface shapes that yield both a high compression and a high efficiency for the intake. Furthermore, the presence conical flow means that all shocks, facing conical flow are also conical and therefore of constant strength, at any angular position. The flows are not only uniform but also irrotational—generally a desirable feature for flow that leaves the intake to enter a combustion chamber. These features of conical flow and, in particular, Busemann flow, which is by nature an internal, compressive flow, make the basic Busemann streamline shape an attractive candidate for

Flow which is both axially and conically symmetric is best described in spherical polar coordinates (r, θ, ϕ) where r is distance measured radially out from the origin, θ is the angle measured counterclockwise from the downstream direction and ϕ is the circumferential coordinate around the axis of symmetry (Figure 4). For Busemann flow the origin is at the apex of the conical shock, on the center line of symmetry (xx). The flow velocity components in the radial and angular directions are designated as U and V. Drawing similar triangles along the streamline, in

Busemann flow, and axisymmetric conical flow are governed by the

dU

The coordinates (r, θ); the Mach number (M) and its radial and angular components u and v.

<sup>d</sup><sup>θ</sup> cot <sup>θ</sup> <sup>þ</sup>

" #

d2 U dθ<sup>2</sup>

� dU

<sup>d</sup><sup>θ</sup> <sup>U</sup> dU

dθ þ

dU dθ

" # !

d2 U dθ<sup>2</sup>

¼ 0

(2)

Taylor-Maccoll equation, the same equation that governs the supersonic flow over an axisymmetric cone at zero angle of attack. The original Taylor-Maccoll equation is a non-linear, second order total differential equation with the spherical polar angle, θ, as independent variable and the radial flow velocity, U, as dependent

dr=dθ ¼ rU=V ¼ ru=v (1)

constructing an air intake for a hypersonic flight vehicle's engine.

2.2 Flow symmetry: coordinate axis: flow direction

Figure 4 gives the streamline equation:

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

variable [42, 43].

<sup>1</sup> � <sup>U</sup><sup>2</sup> � dU

dθ � �<sup>2</sup> " # <sup>2</sup><sup>U</sup> <sup>þ</sup>

γ � 1 2

Figure 4.

83

The high performance [24] and analytical simplicity of the Busemann intake has made it a subject for some 60 publications.

### 2.1 Description of Busemann intake flow

The basic Busemann intake surface is axisymmetric (Figure 3). It is a converging duct with its axis aligned with the freestream. When started, it captures freestream flow (M1) in a circular cross section. Since there is no flow deflection at the leading edge there is a zero-strength conical Mach wave (io) from the leading edge at the freestream Mach angle. The flow then starts turning towards the axis, so that flow area decreases and pressure increases in the flow and along the surface (icfs). A maximum turning angle (inflection point) is reached at (f). Turning from (i) to (f) has made the flow convergent so as to compress by convergence. While still convergent and inclined towards the axis, the flow (fs) starts turning away from the axis, passing through a conical shock (os) where it is deflected to become uniform and parallel to the axis at the exit of the intake. This "turning away" of the pre-shock flow lessens the flow deflection requirement of the terminal shock, leading directly to an increase in efficiency. It is this efficiency increase and the convergence in (icfsoi) that contribute directly to the superior performance of the Busemann intake. Flow in the region (icfsoi) is isentropic and irrotational. In the region (icoi) the compression waves from the surface (ic) converge to the focus (o). The compression waves from the rest of the surface (cfs) are incident on the terminal shock. Aside from axial symmetry, this flow is also conically symmetric so that there is a focal point (o) on the axis, from which rays can be drawn, in any direction, such that the flow conditions on any ray are constant. Except for the leading ray (io), the rays are not Mach waves. Axial symmetry makes the rays to be generators of cones so that the flow conditions on circular cones are constant and the conditions are functions of only the conical angle θ (Figure 3). All the streamlines are geometrically self-similar, with shapes that are scalable with distance from the origin. Thus, only one streamline, r = f(θ), needs to be calculated to define the intake surface. Disappearance of the radial dimension (r) as an independent variable, in conically symmetric flow, permits the depiction of all flow conditions on the

#### Figure 3.

Busemann intake contour is icfs. io is a freestream Mach cone. os is a conical shock. Uniform entry flow at (1). Uniform exit flow at (3). Supersonic, isentropic, axially and conically symmetric flow from (1) to (2). Flow crosses oblique conical shock from (2) to (3), C-characteristics in ico focus at o. C-characteristics from cfs are incident on the shock along os. All streamlines have an inflection point on the cone fo. Spherical-polar coordinate system (r, θ) is centered at o with corresponding radial and angular Mach number components u and v.

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

general studies of intake flows as well as experiments for such issues as flow starting [18, 25, 27, 31–37], viscous effects [15, 39], truncation [16, 18], drag measurement [11], wavecatcher configurations [13], leading edge blunting [10] and cross section morphing [7]. Viscous effects and truncation and stunting are found in [15, 18, 39]. Experimental results for full and modular Busemann intakes are found in [11, 13, 18, 21, 23, 25, 33, 37, 40]. A four-module Busemann-based intake on a scramjet

The high performance [24] and analytical simplicity of the Busemann intake has

The basic Busemann intake surface is axisymmetric (Figure 3). It is a converg-

ing duct with its axis aligned with the freestream. When started, it captures freestream flow (M1) in a circular cross section. Since there is no flow deflection at the leading edge there is a zero-strength conical Mach wave (io) from the leading edge at the freestream Mach angle. The flow then starts turning towards the axis, so that flow area decreases and pressure increases in the flow and along the surface (icfs). A maximum turning angle (inflection point) is reached at (f). Turning from (i) to (f) has made the flow convergent so as to compress by convergence. While still convergent and inclined towards the axis, the flow (fs) starts turning away from the axis, passing through a conical shock (os) where it is deflected to become uniform and parallel to the axis at the exit of the intake. This "turning away" of the pre-shock flow lessens the flow deflection requirement of the terminal shock, leading directly to an increase in efficiency. It is this efficiency increase and the convergence in (icfsoi) that contribute directly to the superior performance of the Busemann intake. Flow in the region (icfsoi) is isentropic and irrotational. In the region (icoi) the compression waves from the surface (ic) converge to the focus (o). The compression waves from the rest of the surface (cfs) are incident on the terminal shock. Aside from axial symmetry, this flow is also conically symmetric so that there is a focal point (o) on the axis, from which rays can be drawn, in any direction, such that the flow conditions on any ray are constant. Except for the leading ray (io), the rays are not Mach waves. Axial symmetry makes the rays to be generators of cones so that the flow conditions on circular cones are constant and the conditions are functions of only the conical angle θ (Figure 3). All the streamlines are geometrically self-similar, with shapes that are scalable with distance from the origin. Thus, only one streamline, r = f(θ), needs to be calculated to define the intake surface. Disappearance of the radial dimension (r) as an independent variable, in conically symmetric flow, permits the depiction of all flow conditions on the

Busemann intake contour is icfs. io is a freestream Mach cone. os is a conical shock. Uniform entry flow at (1). Uniform exit flow at (3). Supersonic, isentropic, axially and conically symmetric flow from (1) to (2). Flow crosses oblique conical shock from (2) to (3), C-characteristics in ico focus at o. C-characteristics from cfs are incident on the shock along os. All streamlines have an inflection point on the cone fo. Spherical-polar coordinate system (r, θ) is centered at o with corresponding radial and angular Mach number components

engine was launched at Mach 5, from a large ballistic gun [41].

Hypersonic Vehicles - Past, Present and Future Developments

made it a subject for some 60 publications.

2.1 Description of Busemann intake flow

Figure 3.

u and v.

82

single remaining spatial variable—the conical angle, θ. This offers great simplicity in flow analysis where a wide variety of intake surfaces is available for selection of surface shapes that yield both a high compression and a high efficiency for the intake. Furthermore, the presence conical flow means that all shocks, facing conical flow are also conical and therefore of constant strength, at any angular position. The flows are not only uniform but also irrotational—generally a desirable feature for flow that leaves the intake to enter a combustion chamber. These features of conical flow and, in particular, Busemann flow, which is by nature an internal, compressive flow, make the basic Busemann streamline shape an attractive candidate for constructing an air intake for a hypersonic flight vehicle's engine.

### 2.2 Flow symmetry: coordinate axis: flow direction

Flow which is both axially and conically symmetric is best described in spherical polar coordinates (r, θ, ϕ) where r is distance measured radially out from the origin, θ is the angle measured counterclockwise from the downstream direction and ϕ is the circumferential coordinate around the axis of symmetry (Figure 4). For Busemann flow the origin is at the apex of the conical shock, on the center line of symmetry (xx). The flow velocity components in the radial and angular directions are designated as U and V. Drawing similar triangles along the streamline, in Figure 4 gives the streamline equation:

$$r dr/d\theta = rU/V = ru/v \tag{1}$$

Busemann flow, and axisymmetric conical flow are governed by the Taylor-Maccoll equation, the same equation that governs the supersonic flow over an axisymmetric cone at zero angle of attack. The original Taylor-Maccoll equation is a non-linear, second order total differential equation with the spherical polar angle, θ, as independent variable and the radial flow velocity, U, as dependent variable [42, 43].

$$\frac{\gamma - 1}{2} \left[ 1 - U^2 - \left( \frac{dU}{d\theta} \right)^2 \right] \left[ 2U + \frac{dU}{d\theta} \cot \theta + \frac{d^2 U}{d\theta^2} \right] - \frac{dU}{d\theta} \left[ U \frac{dU}{d\theta} + \frac{dU}{d\theta} \left( \frac{d^2 U}{d\theta^2} \right) \right] = 0 \tag{2}$$

Figure 4. The coordinates (r, θ); the Mach number (M) and its radial and angular components u and v.

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].
