3. Scramjet inverse design approach

The inverse design approach relies on extracting the configuration of interest from the environment in which it operates. For this design process the centerline

geometry of a given 2-D scramjet configuration is explicitly constructed using the following design inputs: freestream Mach number, M∞, scramjet forebody length, L, shock angle, β, caret angle, α, cruising flight altitude, H∞, and isolator backpressure ratio, Pin/Pexit. All freestream flow-field properties are extracted from the Mach number and altitude [3–6]. This information is used in the construction, analysis and definition of the three fundamental aerodynamic zones, namely; the 'primary shock' zone AB, the 'reflected shock zone', BC, and the 'isolator zone', CD as presented in Figure 4. Also presented in Figure 4 is a 2-dimensional conceptual representation of the flow-field physics associated with supersonic flow interaction over a wedge and in a constant area duct. Details of this flow-field physics and its exploitation in the inverse design approach are explained in the next section. Also addressed is the derivation of the actual 3-dimensional forebody section. This is a

Figure 2.

Figure 3.

53

Dual-mode scramjet concept.

Inversely Designed Scramjet Flow-Path DOI: http://dx.doi.org/10.5772/intechopen.85697

Illustration of the cross section of the scramjet.

Figure 1. Pod-mounted scramjet concept.

Generally speaking, the concepts associated with scramjet engines appear at first glance to be very simple. This however is very misleading as attempts develop a working scramjet engine that has proven to be quite an engineering challenge. Several aspects of scramjet engine development are at various stages of development. These include supersonic fuel-air mixing, aero-thermodynamic heat dissipa-

This chapter focuses on the design concepts for the forebody, inlet, and isolator sections of an innovative scramjet engine geometry and some of its flow physics.

As stated earlier, the scramjet concept represents the latest evolution in the series of air-breathing jet engines. Combustion in these engines occurs under supersonic conditions. Scramjet engines are seen as the propulsion system that is at the heart of hypersonic vehicles/platforms. Every scramjet conceptual engine design and engines flown to-date all have a common set of components or sub-sections. Figure 1 presents these components/sub-sections for a pod-mounted conceptual scramjet design. These components/sections are the forebody section, the inlet section, isolator section, combustor section, and the diffuser-nozzle section. Ideally, the engine concept presented should be able to function over a wide range of Mach numbers. This gives rise to the idea of a morphing ramjet/scramjet or dual mode scramjet configurations as presented in Figure 2 [1]. Figure 2a, presents the dual mode scramjet engine, Figure 2b, the pure scramjet mode and Figure 2c, the pure ramjet mode.

A typical dual mode scramjet configuration as that presented in Figure 3, was inversely carved out of supersonic and hypersonic flow-fields. The design framework used in the design of the forebody, inlet and isolator sections forms the core of

The inverse design approach relies on extracting the configuration of interest from the environment in which it operates. For this design process the centerline

tion from both skin friction and internal combustion, and other thermal management problems associated with operating an engine at exceedingly high temperatures for extended periods of time. Combustion chamber components could experience temperatures on the order of over 3033 K (5000°F). At these temperatures most metals melt and fluids (air and fuel) ionize, making the physics of their

2. Inverse scramjet 2-D centerline design approach

Hypersonic Vehicles - Past, Present and Future Developments

associated behavior unpredictable.

this chapter.

Figure 1.

52

Pod-mounted scramjet concept.

3. Scramjet inverse design approach

Figure 2. Dual-mode scramjet concept.

Figure 3. Illustration of the cross section of the scramjet.

geometry of a given 2-D scramjet configuration is explicitly constructed using the following design inputs: freestream Mach number, M∞, scramjet forebody length, L, shock angle, β, caret angle, α, cruising flight altitude, H∞, and isolator backpressure ratio, Pin/Pexit. All freestream flow-field properties are extracted from the Mach number and altitude [3–6]. This information is used in the construction, analysis and definition of the three fundamental aerodynamic zones, namely; the 'primary shock' zone AB, the 'reflected shock zone', BC, and the 'isolator zone', CD as presented in Figure 4. Also presented in Figure 4 is a 2-dimensional conceptual representation of the flow-field physics associated with supersonic flow interaction over a wedge and in a constant area duct. Details of this flow-field physics and its exploitation in the inverse design approach are explained in the next section. Also addressed is the derivation of the actual 3-dimensional forebody section. This is a

3.2.2 Design points at station B

Inversely Designed Scramjet Flow-Path DOI: http://dx.doi.org/10.5772/intechopen.85697

constant γ is set at a value of 1.4.

3.2.3 Design points at station C

using Eqs. (3)–(6).

T T<sup>∞</sup> ¼

Pt, <sup>2</sup> Pt,<sup>∞</sup>

55

Using the input data, the location of design point B, can be computed with the

In addition, using trigonometric relationships design point B1 is evaluated as

The coordinates for design point B2 are evaluated in the following manner: B2x = Bx, B2y = Bxtan(β), and B2z = 0. The wedge angle is represented by theta (θ) and the shock angle is represented by beta (β). Using the Mach number and the shock angle beta (β), the wedge angle theta (θ) can be obtained with the use of the Theta-Beta-Mach (θ-β-M) relationship [2–7] given as seen in Eq. (1). In Eq. (1) the

M<sup>2</sup>

The design points at station C is extracted from the wedge angle, θ, and the flow-field properties behind the primary shock wave, AB2, as seen in Figure 4. Determination of the location of design point C is a little more involved and is

� � <sup>1</sup> <sup>þ</sup> ½ � ð Þ <sup>γ</sup> � <sup>1</sup> <sup>=</sup><sup>2</sup> ð Þ <sup>M</sup><sup>∞</sup> sin ð Þ<sup>β</sup> <sup>2</sup> h i

b.This Mach number, coupled with the free stream parameters are then used with the oblique shock relations derived in [5] for the evaluation of all of flowfield properties behind the primary shock, AB2. The flow-field properties, pressure, P, temperature, T, density, ρ, and total pressure, Pt,2, are evaluated

<sup>¼</sup> <sup>2</sup>γð Þ <sup>M</sup><sup>∞</sup> sin <sup>β</sup> <sup>2</sup> � ð Þ <sup>γ</sup> � <sup>1</sup>

h i ð Þ <sup>γ</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>M</sup><sup>∞</sup> sin <sup>β</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>

<sup>¼</sup> ð Þ <sup>γ</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>M</sup><sup>∞</sup> sin <sup>β</sup> <sup>2</sup>

γ�1

c. B2C1 as seen in Figure 4 represent the reflected shock wave. This reflected shock wave is a the result of a flow-field behind the primary shock wave, AB2,

imaginary wedge, with wedge angle θ at design point B2. This imaginary wedge

with a supersonic Mach number, M, once more being deflected by an

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

<sup>γ</sup>ð Þ <sup>M</sup><sup>∞</sup> sin ð Þ<sup>β</sup> <sup>2</sup> � ð Þ <sup>γ</sup> � <sup>1</sup> <sup>=</sup><sup>2</sup>

h i (2)

ð Þ <sup>γ</sup> <sup>þ</sup> <sup>1</sup> (3)

ð Þ <sup>M</sup><sup>∞</sup> sin <sup>β</sup> <sup>2</sup> (4)

γ�1

(6)

h i

ð Þ <sup>γ</sup> � <sup>1</sup> ð Þ <sup>M</sup><sup>∞</sup> sin <sup>β</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup> (5)

ð Þ γ þ 1 <sup>2</sup>γð Þ <sup>M</sup><sup>∞</sup> sin <sup>β</sup> <sup>2</sup> � ð Þ <sup>γ</sup> � <sup>1</sup>

" # <sup>1</sup>

<sup>∞</sup> sin <sup>2</sup><sup>β</sup> � <sup>1</sup>

� � � � (1)

<sup>∞</sup>ð Þþ γ þ cos 2β 2

use of the following relations: Bx = L, By = 0, and Bz = 0.

<sup>θ</sup> <sup>¼</sup> atan 2 cot <sup>β</sup> <sup>M</sup><sup>2</sup>

approached systematically as outlined in the following steps:

sin ð Þ β � θ

P P<sup>∞</sup>

> ρ ρ∞

" # <sup>γ</sup>

<sup>¼</sup> ð Þ <sup>γ</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>M</sup><sup>∞</sup> sin <sup>β</sup> <sup>2</sup> ð Þ <sup>γ</sup> � <sup>1</sup> ð Þ <sup>M</sup><sup>∞</sup> sin <sup>β</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>

<sup>2</sup>γð Þ <sup>M</sup><sup>∞</sup> sin <sup>β</sup> <sup>2</sup> � ð Þ <sup>γ</sup> � <sup>1</sup>

(see Figure 4), is obtained using Eq. (2),

<sup>M</sup> <sup>¼</sup> <sup>1</sup>

a. First, the Mach number, M, behind the primary shock wave, AB2

follows: B1x=Bx, B1y = Bxtan(θ) and B1z = 0.

Figure 4. Conceptual 2-D centerline cross-section of the forebody-inlet-isolator scramjet section with flow physics representation.

two-step process, where in step one, the 2-D construction of the 'forebody', domain A-D, is conducted. Step two is where the 3-D geometry is obtained.

### 3.1 Aerodynamics of the 2-D 'forebody' configuration

Consider the 2-D cross-sectional illustration of the scramjet forebody-inlet-isolator section presented in Figure 4. Now consider a supersonic flow travelling parallel to the x-axis of a 2-D wedge. Supersonic aerodynamics dictates that the flow is deflected first by the oblique shock wave, AB2, originating from the leading edge, A, of the wedge. The flow is deflected a second time by a reflected shock wave, B2C1 emanating from the cowl lip at point B2, of the inlet. The flow enters the isolator duct and travels once more in a direction that is parallel to the x-axis. To ensure that the flow in the isolator duct remain supersonic the freestream Mach number must be greater than 3.0 and the shock wave angle, β, greater than 12 and less than 30 degrees.

The flow-field behavior within the isolator duct is of paramount importance. This flow-field may consist of a system of oblique or normal shocks, as visualized in Figure 4. Driving this behavior is the flow-field vicious interactions with the isolator duct walls. The isolator's non-dimensional length, L/H, and the pressure differential at the duct's entrance and exit also enhance the flow-field's behavior.

### 3.2 Derivation of the 2-D 'forebody-inlet-isolator' configuration

The 'forebody-inlet-isolator' concept presented in Figure 4 relies on determining the geometric design points located at stations A, B, C and D, along the x-axis of the scramjet. This is accomplished by use of the oblique shock relations described in [2–7] and the 'isolator' relations that were experimentally derived in [8–9]. It is assumed that in Figure 4 the flow travels in the x-direction, and that the construction of the 'forebody' configuration starts at design point, A. The following account details the logic used to define the locations of design points A, B, C and D:

#### 3.2.1 Design point at station A

The design point at station A is considered the origin of the scramjet design coordinate system, therefore, design point A coordinates are evaluated as follows, Ax = 0, Ay = 0, and Az = 0.

## 3.2.2 Design points at station B

Using the input data, the location of design point B, can be computed with the use of the following relations: Bx = L, By = 0, and Bz = 0.

In addition, using trigonometric relationships design point B1 is evaluated as follows: B1x=Bx, B1y = Bxtan(θ) and B1z = 0.

The coordinates for design point B2 are evaluated in the following manner: B2x = Bx, B2y = Bxtan(β), and B2z = 0. The wedge angle is represented by theta (θ) and the shock angle is represented by beta (β). Using the Mach number and the shock angle beta (β), the wedge angle theta (θ) can be obtained with the use of the Theta-Beta-Mach (θ-β-M) relationship [2–7] given as seen in Eq. (1). In Eq. (1) the constant γ is set at a value of 1.4.

$$\theta = \operatorname{atan} \left\{ 2 \cot \beta \left[ \frac{M\_{\infty}^2 \sin^2 \beta - 1}{M\_{\infty}^2 (\gamma + \cos 2\beta) + 2} \right] \right\} \tag{1}$$

## 3.2.3 Design points at station C

two-step process, where in step one, the 2-D construction of the 'forebody', domain

Conceptual 2-D centerline cross-section of the forebody-inlet-isolator scramjet section with flow physics representation.

Consider the 2-D cross-sectional illustration of the scramjet forebody-inlet-isolator section presented in Figure 4. Now consider a supersonic flow travelling parallel to the x-axis of a 2-D wedge. Supersonic aerodynamics dictates that the flow is deflected first by the oblique shock wave, AB2, originating from the leading edge, A, of the wedge. The flow is deflected a second time by a reflected shock wave, B2C1 emanating from the cowl lip at point B2, of the inlet. The flow enters the isolator duct and travels once more in a direction that is parallel to the x-axis. To ensure that the flow in the isolator duct remain supersonic the freestream Mach number must be greater than 3.0 and the shock wave angle, β, greater than 12 and less than 30 degrees. The flow-field behavior within the isolator duct is of paramount importance. This flow-field may consist of a system of oblique or normal shocks, as visualized in Figure 4. Driving this behavior is the flow-field vicious interactions with the isolator duct walls. The isolator's non-dimensional length, L/H, and the pressure differ-

ential at the duct's entrance and exit also enhance the flow-field's behavior.

details the logic used to define the locations of design points A, B, C and D:

The design point at station A is considered the origin of the scramjet design coordinate system, therefore, design point A coordinates are evaluated as follows,

3.2.1 Design point at station A

Ax = 0, Ay = 0, and Az = 0.

54

The 'forebody-inlet-isolator' concept presented in Figure 4 relies on determining the geometric design points located at stations A, B, C and D, along the x-axis of the scramjet. This is accomplished by use of the oblique shock relations described in [2–7] and the 'isolator' relations that were experimentally derived in [8–9]. It is assumed that in Figure 4 the flow travels in the x-direction, and that the construction of the 'forebody' configuration starts at design point, A. The following account

3.2 Derivation of the 2-D 'forebody-inlet-isolator' configuration

A-D, is conducted. Step two is where the 3-D geometry is obtained.

3.1 Aerodynamics of the 2-D 'forebody' configuration

Hypersonic Vehicles - Past, Present and Future Developments

Figure 4.

The design points at station C is extracted from the wedge angle, θ, and the flow-field properties behind the primary shock wave, AB2, as seen in Figure 4. Determination of the location of design point C is a little more involved and is approached systematically as outlined in the following steps:

a. First, the Mach number, M, behind the primary shock wave, AB2 (see Figure 4), is obtained using Eq. (2),

$$M = \left\{ \frac{\mathbf{1}}{\sin \left( \beta - \theta \right)} \right\} \frac{\left[ \mathbf{1} + \left[ (\gamma - \mathbf{1})/2 \right] (\mathcal{M}\_{\infty} \sin \left( \beta \right))^2 \right]}{\left[ \gamma (\mathcal{M}\_{\infty} \sin \left( \beta \right))^2 - (\gamma - \mathbf{1})/2 \right]} \tag{2}$$

b.This Mach number, coupled with the free stream parameters are then used with the oblique shock relations derived in [5] for the evaluation of all of flowfield properties behind the primary shock, AB2. The flow-field properties, pressure, P, temperature, T, density, ρ, and total pressure, Pt,2, are evaluated using Eqs. (3)–(6).

$$\frac{P}{P\_{\infty}} = \frac{2\chi (M\_{\infty}\sin\beta)^2 - (\chi - 1)}{(\chi + 1)}\tag{3}$$

$$\frac{T}{T\_{\infty}} = \frac{\left[2\gamma (M\_{\infty}\sin\theta)^2 - (\gamma - 1)\right] \left[\left(\gamma + 1\right) (M\_{\infty}\sin\theta)^2 + 2\right]}{\left(\gamma + 1\right)^2 (M\_{\infty}\sin\theta)^2} \tag{4}$$

$$\frac{\rho}{\rho\_{\infty}} = \frac{(\chi + 1)(M\_{\infty}\sin\beta)^2}{\left(\chi - 1\right)\left(M\_{\infty}\sin\beta\right)^2 + 2} \tag{5}$$

$$\frac{P\_{t,2}}{P\_{t,\infty}} = \left[\frac{(\boldsymbol{\chi}+\mathbf{1})(\boldsymbol{M}\_{\infty}\sin\mathfrak{h})^2}{\left(\boldsymbol{\chi}-\mathbf{1}\right)\left(\boldsymbol{M}\_{\infty}\sin\mathfrak{h}\right)^2+2}\right]^{\frac{1}{\boldsymbol{\chi}-1}} \left[\frac{(\boldsymbol{\chi}+\mathbf{1})}{2\boldsymbol{\chi}\left(\boldsymbol{M}\_{\infty}\sin\mathfrak{h}\right)^2-\left(\boldsymbol{\chi}-\mathbf{1}\right)}\right]^{\frac{1}{\boldsymbol{\chi}-1}}\tag{6}$$

c. B2C1 as seen in Figure 4 represent the reflected shock wave. This reflected shock wave is a the result of a flow-field behind the primary shock wave, AB2, with a supersonic Mach number, M, once more being deflected by an imaginary wedge, with wedge angle θ at design point B2. This imaginary wedge is oriented in such a manner that it ensures that the deflected flow travels parallel to the x-axis, Figure 4. At this stage updated values for the wedge angle, θ and the Mach number, M, are obtained using Eqs. (1) and (2). A reflection shock angle, ϕ is now be defined as ϕ = β<sup>1</sup> � θ. In this expression, β<sup>1</sup> is the reflected shock angle. This reflected shock angle is generated by the interaction of the flow-field with Mach number M and the imaginary wedge with angle θ Note that β<sup>1</sup> is obtained using Eq. (1) and replacing the value of the freestream Mach number, M∞, with that of the supersonic Mach number, M.

all 'unstart' conditions. In this design process, the ratio, Pout/Pn,in, representing the isolator exit pressure, Pout, to the 'normal total' pressure value, Pn,in, is prescribed. Using this approach, the value for Pin/Pout can be determined by using Eq. (9):

<sup>¼</sup> Pout

in <sup>1</sup> <sup>þ</sup> ð Þ ð Þ <sup>γ</sup> � <sup>1</sup> <sup>=</sup><sup>2</sup> <sup>M</sup><sup>2</sup> in � �

Pn,in � � Pn,in

b.The system of 1-D conservation laws result in the following expression for the

in � Pout=Pin � �<sup>2</sup> � <sup>γ</sup> � <sup>1</sup>

<sup>50</sup>ðð Þþ Pout=Pin <sup>1</sup>Þ þ <sup>170</sup>ð Þ ð Þ� Pout=Pin <sup>1</sup> <sup>2</sup> n o

M<sup>2</sup>

where Re<sup>θ</sup> is the inlet Reynolds number based on the momentum thickness. Also, the symbol, H, represents the isolator height that is determined from the y-

c. The coordinates of point D are computed as follows: Dx = Cx + LIsolator, Dy = 0,

d.The coordinates of point D1 are computed as follows: D1x = Dx, D1y = C1y, and

e. The coordinates of point D2 are computed as follows: D2x = Dx, D2y = C2y, and

Finally, with the coordinates of all the design points at all stations, A, B, B1, B2, C, C1, C2, D, D1, and D2, fully defined, the sketch illustrated in Figure 4 can be

The 3-D design process has as its origin in the inversely design two-dimensional geometry extracted from a 2-D hypersonic flow-field. This is then coupled with the Nonweiler's waverider approach [10] of inversely carving stream surfaces from inviscid flow-fields. A caret waverider, Figure 5, is chosen as an example because it represents a 3-D geometry that was obtained from a 2-D flow-field. This caret waverider geometry is constructed from a single planer shock wave, AB3B4, as seen in Figure 5. A unique feature of this construction process is that at any cross-section of the waverider geometry there is a wedge that is supported by an oblique shock

coordinates of points C2 and C1, in a manner such that, H = C2y – C1y.

Similarly, with the exit Mach number known, the non-dimensional length of the isolator can be evaluated based on the following experimental relationship devel-

( ) � � �<sup>1</sup>

Pin � � (9)

2

(10)

2

in � <sup>1</sup> (11)

Pout Pin

<sup>1</sup> � <sup>γ</sup>M<sup>2</sup>

isolator exit Mach number, Mout [8, 9];

ffiffiffiffiffiffiffiffiffi θ=H p ð Þ Re<sup>θ</sup> 1 4

4. 3-D computer aided design (CAD) design

wave, with these wedges being parallel to the flow.

4.1 Overview of the 3-D design process

Mout <sup>¼</sup> <sup>γ</sup>2M<sup>2</sup>

Inversely Designed Scramjet Flow-Path DOI: http://dx.doi.org/10.5772/intechopen.85697

oped in [8, 9]:

L H � �

and Dz = 0.

D1z = 0.

D2z = 0.

constructed.

57

Isolator ¼


$$C\_{\mathbf{x}} = \left[ 1 + \frac{\tan\left(\theta\right) - \tan\left(\theta\right)}{\tan\left(\theta\right) - \tan\left(\beta\_1 - \theta\right)} \right] B\_{\mathbf{x}} \tag{7}$$


## 3.2.4 Design points at station D

The evaluation of the coordinates of the design points at station D is also a multistep process.

a. First a non-dimensional expression for the 'normal total' pressure value, Pn,in, is derived, Eq. (8). This expression is a function of isolator entrance conditions, where M1 is treated as the Min, and the static pressure, P1, as Pin. Note here that the values of M1 and P1 are obtained from the flow-field properties behind the reflected shock B2C1.

$$\frac{P\_{n,in}}{P\_{in}} = \left[\frac{2\gamma \mathbf{M}\_1^2 - (\chi - \mathbf{1})}{(\chi + \mathbf{1})}\right] \tag{8}$$

In determining the isolator length for a design process, the ratio of the entrance to exit pressures, Pin/Pout, over the range between Pin and Pn,in has to be evaluated. This value is needed to determine the length of an isolator that can reliably prevent is oriented in such a manner that it ensures that the deflected flow travels parallel to the x-axis, Figure 4. At this stage updated values for the wedge angle, θ and the Mach number, M, are obtained using Eqs. (1) and (2). A reflection shock angle, ϕ is now be defined as ϕ = β<sup>1</sup> � θ. In this expression, β<sup>1</sup> is the reflected shock angle. This reflected shock angle is generated by the interaction of the flow-field with Mach number M and the imaginary wedge with angle θ Note that β<sup>1</sup> is obtained using Eq. (1) and replacing the value of the freestream

Mach number, M∞, with that of the supersonic Mach number, M.

d.The flow-field properties behind the reflected shock wave B2C1 are now

the isolator section of the scramjet. Eqs. (3)–(6) are used to derive the additional flow-field properties of pressure, temperature, density and total temperature, p1, T1, ρ<sup>1</sup> and To, behind the reflected shock. Note that in these equations the value for the freestream Mach number, M∞, is now replaced with the value of the Mach number, M, from the flow-field properties behind

e. Having obtained the parameters, θ, β and β<sup>1</sup> all design points at station C can now be derived. The y-coordinate and z-coordinate are defined as Cy = 0, and

> tan ð Þ� β tan ð Þθ tan ð Þ� θ tan ð Þ β<sup>1</sup> � θ

f. The coordinates of point C1 are determined as follows: C1x = Cx, C1y = Cxtan(θ),

g. Similarly, the coordinates of point C2 are determined from: C2x = Cx, C2y = B2y,

The evaluation of the coordinates of the design points at station D is also a multi-

a. First a non-dimensional expression for the 'normal total' pressure value, Pn,in, is derived, Eq. (8). This expression is a function of isolator entrance conditions, where M1 is treated as the Min, and the static pressure, P1, as Pin. Note here that the values of M1 and P1 are obtained from the flow-field properties behind the

In determining the isolator length for a design process, the ratio of the entrance to exit pressures, Pin/Pout, over the range between Pin and Pn,in has to be evaluated. This value is needed to determine the length of an isolator that can reliably prevent

<sup>1</sup> � ð Þ γ � 1 ð Þ γ þ 1 

<sup>¼</sup> <sup>2</sup>γM<sup>2</sup>

Bx (7)

(8)

Cz = 0, respectively. The x-coordinate is obtained with the help of

trigonometric relations, and is defined as:

Hypersonic Vehicles - Past, Present and Future Developments

Cx ¼ 1 þ

Pn,in Pin

the primary shock.

and C1z = 0.

and C2z = 0.

step process.

56

3.2.4 Design points at station D

reflected shock B2C1.

obtained in a similar manner as described in 'b' above. Eq. (2) is used to obtain M1, which is the Mach number behind the reflected shock. In Eq. (2) the freestream Mach number, M∞, is replaced with Mach number M. It is very important to note here that M1 represents the Mach number at the entrance to

all 'unstart' conditions. In this design process, the ratio, Pout/Pn,in, representing the isolator exit pressure, Pout, to the 'normal total' pressure value, Pn,in, is prescribed. Using this approach, the value for Pin/Pout can be determined by using Eq. (9):

$$\frac{P\_{out}}{P\_{in}} = \left(\frac{P\_{out}}{P\_{n,in}}\right) \left(\frac{P\_{n,in}}{P\_{in}}\right) \tag{9}$$

b.The system of 1-D conservation laws result in the following expression for the isolator exit Mach number, Mout [8, 9];

$$M\_{out} = \left\{ \frac{\chi^2 M\_{in}^2 \left[ 1 + ((\chi - 1)/2) M\_{in}^2 \right]}{\left( 1 - \chi M\_{in}^2 - P\_{out}/P\_{in} \right)^2} - \left( \frac{\chi - 1}{2} \right) \right\}^{-\frac{1}{2}} \tag{10}$$

Similarly, with the exit Mach number known, the non-dimensional length of the isolator can be evaluated based on the following experimental relationship developed in [8, 9]:

$$\left(\frac{L}{H}\right)\_{\text{Iolar}} = \frac{\sqrt{\theta/H}}{\left(\text{Re}\_{\theta}\right)^{\frac{1}{4}}} \frac{\left\{ 50(\left(P\_{\text{out}}/P\_{\text{in}}\right) + 1) + 170(\left(P\_{\text{out}}/P\_{\text{in}}\right) - 1)^2 \right\}}{M\_{\text{in}}^2 - 1} \tag{11}$$

where Re<sup>θ</sup> is the inlet Reynolds number based on the momentum thickness. Also, the symbol, H, represents the isolator height that is determined from the ycoordinates of points C2 and C1, in a manner such that, H = C2y – C1y.


Finally, with the coordinates of all the design points at all stations, A, B, B1, B2, C, C1, C2, D, D1, and D2, fully defined, the sketch illustrated in Figure 4 can be constructed.
