Hypersonic Vehicle Design For Mach 5

8/18/2019 Hypersonic Vehicle Design For Mach 5

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Contents

1 Introduction 4

2 Design Objectives 5

3 Literature Survey 6

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Weight and Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.5 Structures and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Aerodynamics 9

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Conical Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.3 Hypersonic Small Disturbance Theory . . . . . . . . . . . . . . . . . . . . . 104.4 Waverider Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.5 Aerodynamic Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.6 3-D CFD simulation of Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.7 Vertical Fin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.8 Aerodynamic Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Propulsion 21

5.1 Combustion Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Nozzle design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Offset design and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.4 CFD Results of inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Weight and Sizing 29

6.1 The HASA technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Structures and Material Selection 32

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.2 Material Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.3 Structural and Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . 33

8 Fuel Selection 36

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.2 Hydrogen Fuel of the Future? . . . . . . . . . . . . . . . . . . . . . . . . . . 368.3 Range in Cruise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.4 Fuel as Thermal Protection System . . . . . . . . . . . . . . . . . . . . . . . 37

9 References 38

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Division of Work

Each team member was allotted a set of topics and sub-systems to cover and some teammembers were responsible for sub system integration.

• Abhishek Singh : Propulsion,Inlet Design and report integration.• Maneesh Kumar : Weight and Sizing Analysis, Sensitivity Analysis, Waverider Body

Construction in CATIA , CAD Modelling

• Kush Sreen : Aerodynamics,Waverider Generation (Conceptual) , Inlet Design (Cod-ing),CFD Analysis, Structural Analysis and report integration.

• Rohit Potla : Fuel Selection and Thermal Protection System Design.

• Sanjeev Yadav : Materials Selection.

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1 Introduction

Since the Wright Brothers took to the sky in 1903, mankind has endeavoured to fly higherand faster. This ambition was nurtured through the two world wars and a major epochin manned flight took place when Chuck Yeager in the Bell X-1 prototype became the

first man to break the sound barrier. The proverbial frontier of flight was pushed evenfurther. The next major challenge in atmospheric flight was of course to fly at Hypersonicspeeds. This has indeed proved a major challenge to aerospace designers. The extremeaero-thermodynamic loads coupled with inefficient air breathing engines at Mach 6 andhigher have made this dream elusive. But since the 1990’s with the advent of new materials,progress in the development of a Scram-jet engine and computational techniques progressin Hypersonic flight has been substantial. A flying prototype the X-43 has already provedits mettle.

This design project envisages to do the same but with a goal of creating a Hyper-sonic Cruise vehicle capable of carrying large payloads.A payload capacity of 10 tons with

a flight range of 10,000 km is set as an objective. A complete preliminary design analysisof the proposed vehicle is carried out. The final vehicle configuration obtained is thoroughaerodynamic, structural and propulsion analysis.

In this project we will look at the major sub-systems present in a hypersonic vehicle.Of-course all systems play crucial role in the flight of any aircraft but certain system havethat criticality. Due to lack of available data and the limited scope of this project we willtry an address those crucial sub systems with the aim of developing a realisable prototype.So you may feel some very critical sub-systems or analysis may be missing but his is doneto have paid more emphasis on the sub systems already being developed.

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2 Design Objectives

The primary objectives set for the hypersonic vehicle are-

• Fly at a cruise velocity of Mach 5.

• Cruising altitude of 30 Km.• A payload of 10,000 Kg.

• Cruising range of 10,000 Km.

These design objectives are set as the final goal which the final vehicle configuration mustsatisfy. During the design process all the above parameters are taken as fixed inputs.

With a high load carrying capacity and long range this vehicle is intended as a mili-tary vehicle capable of long range missions. It can carry listening equipment to get high

resolution data from the enemy territory or carry out quick response and long rangebombing. The high cruise altitude and velocity make it impossible for enemy anti-airdefences to bring down the vehicle.

A major challenge in the design of the vehicle was to ensure a satisfactory L/D ra-tio. Normal airfoils don’t produce lift in the hypersonic regime. The whole body of theaircraft must be used to produce lift. To ensure high efficiency of lift production flow fromthe bottom surface must not leak on to the top surface. This method of lift generation iscalled compression lift and a Waverider geometry is employed.

This method of lift generation leads to one major problem of excessive drag due to

wave drag and skin friction drag owing to the large wetted area of the body. A propulsionsystem capable of delivering enough thrust to keep the aircraft in steady flight has to bemade. A ramjet engine is used to develop the thrust required. The selection of fuel caneither be LH 2 or Synthetic fuel JP-10.

The final consideration was the structural and thermal loads experienced by the air-craft frame. Modern materials like Ti-Al alloys to withstand the loads at temperatures inthe range 800-1000 K are used in the outer skin of the aircraft. Active Thermal ProtectionSystems(TPS) is used to cool the skin using fuel.

Vehicle Specifications are listed in Appendix VIII

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3 Literature Survey

3.1 Introduction

This section will be broken up into literature reviews done for each of the subsystems of

the entire design. Most of the information surveyed is through research papers and doctoralthesis. These have been duly credited in the reference section.The aim of the literature survey was to understand the core concepts related to each sub-system and the integration with the other systems. The design of the NASA X-43 aircraftserved as a guide for our project. John D. Anderson’s Hypersonic and High TemperatureGas Dynamics was a rich source of information on how to deal with hypersonic flows andwas extensively used especially in the aerodynamic design phase.

3.2 Weight and Sizing

HASA-Hypersonic Aerospace Sizing Analysis for the Preliminary Design of Aerospace Ve-

hicles, a NASA report is used in the initial weight and sizing analysis for the vehicle. Thereport contains detailed methodologies to obtain the mass and volume estimates of the var-ious components of the aircraft.The method employed is an iterative one and it requires some unknown parameters to initiatethe process. The report has various aircraft configurations listed to help in this regard. Afterdetermining the final mass/volume of the aircraft the next next is to determine the individualstructural mass components. Empirical relations exist to determine these parameters.

3.3 Aerodynamics

Anderson’s Hypersonic and High Temperature Gas Dynamics was used to fundamentals of

hypersonic flow. The hypersonic small perturbation theory which is essential to derive theempirical relations of lift and drag coefficients of the waverider is presented in great detail.The Mach independence condition at hypersonic speeds is also covered by it.To generate a waverider geometry a basic methodology is proposed using the wedge andconical shock method. Using the conical shock method requires understanding of the Taylor-Maccol numerical solution to conical shocks. The doctoral thesis of Mr Graham L. Felthamis used as a reference while building a numerical solution to the conical flow. The viscousoptimization process for waverider geometry is presented again in Anderson’s book.

Figure 1: Waverider and its Shock Cone.

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The aerodynamic coefficients are evaluated using the methodology developed in Experimen-tal results of a Mach 10 conical-flow derived waverider to 14-X hypersonic aerospace vehicle .This method relies purely on the geometrical parameters of the waverider geometry whichhave already been obtained. This allows us to compare results of the analytical method witha CFD analysis done later.

The design of the vertical tail is done using a plane wedge formulation as devised in Integrated Design of Hypersonic Waveriders Including Inlets and Tailfins . A caret shaped vertical tailis designed using this approach and its aerodynamic coefficients computed.Aerodynamic Heating and shear force are computed using the reference temperature methodas mentioned in Anderson.These methods help to design a waverider geometry along a conical shock wave the designedgeometry is such that the conical shock wave will sit on the leading edge. This is partic-ularly advantageous as no flow leakage from the compression surface will take place thusdramatically increasing the L

D ratio of the entire vehicle.

3.4 PropulsionThe propulsion design was initiated from Sadraey(caption). The working of ramjet engineand the essential upstream parameters required for a ramjet engine to operate were decided.From this survey it was established that a ramjet engine is capable of working at Mach 5under the given conditions.

After the ramjet upstream parameters had been fixed next was the inlet design.Preliminary

Figure 2: Ramjet Engine and Inlet Ramps

Design of a 2D Supersonic Inlet to Maximize Total Pressure Recovery(caption) was used todesign the inlet ramps. The paper discussed the number of inlet ramps required to achievethe required inlet conditions. Also by increasing the number of ramps increased the pressurerecovery ratio but made the fluid flow more complicated so an optimal number of ramps wasselected.

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The nozzle design is done using the Method of Characteristics(MOC). Zuckrow(caption) isused to design and implement the MOC code. The nozzle contour and propulsion parameters( thrust, exit Mach) are obtained from the MOC code.

3.5 Structures and Materials

The various material properties were collected from different data sources like wikipedia, engineer-materials toolbox and other related sources. Papers on hypersonic structuralanalysis were used to carry out the loading tests on the vehicle geometry.

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4 Aerodynamics

4.1 Introduction

One of the main challenge of flying at hypersonic speeds is to generate lift efficiently. Con-

ventional airfoils and even supersonic wedge airfoils don’t produce enough lift. The entireaircraft body has to be used as a lifting surface to generate sufficient lift.In such geometryit has to be ensured that the flow from the bottom compression surface does not leak tothe upper surface which has low pressure , this is essential to produce a suitable L

D ratio.

Waverider geometries are used , in which the shock wave remains attached to the leadingedge, to generate lift at hypersonic velocities.Aerodynamic Heating becomes significant as Mach number become higher. Due to the largewetted area of the waverider configuration surface temperatures of the skin can exceed 1,000K. By estimating the heat flux through the aircraft skin an active TPS can be devised tomaintain temperatures at serviceable conditions.

4.2 Conical Shocks

Figure 3: Supersonic flow past a Cone

Supersonic flow past a cone creates a conical shock wave. Numerical solutions like theTaylor-Maccoll analysis can be used to find the flow field.

γ − 12

1 − V

2

r −

dV rdθ

2

2V r + cotθdV rdθ

+ d2V r

dθ2

− dV r

dθ

V r

dV rdθ

+ dV r

dθ

d2V rdθ2

= 0 (1)

V θ = dV r

dθ (2)

The Taylor-Maccoll analysis proceeds as follows. With a known free stream Machnumber M and known cone angle c, assume a value for the oblique shock angle s. The

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oblique shock relations provides the values of change in flow variables across the shock, andthe deflection a of the flow through the shock. Determine the velocity components V r andV θ immediately downstream of the shock. The differential equation can be evaluated untilthe value of V θ is equal to zero. V θ = 0 is the normal velocity condition at the surface of thecone.

In the same manner an inverse solution can also be found where the shock angle is knownand the cone angle is found out.

4.3 Hypersonic Small Disturbance Theory

A frequently employed approach in aerodynamics, instead of sing the flow velocity as adependent variable we deal with the change in velocity with respect to the free stream ,the perturbation velocity. These are a special approximate form of the Euler equationsapplicable to hypersonic slender bodies. The velocity resolved in x and y directions is

u = V ∞ + u (3)

v = v (4)

The ’’ represents the perturbation velocities which are very small when compared tothe freestream velocity. The Hypersonic small disturbance equations are

∂ρ∂x

+ ∂ (ρv)

∂y +

∂ (ρw)∂z

= 0 (5)

ρ∂u

∂x + ρv

∂u

∂y + ρw

∂u

∂z = −∂ρ

∂x (6)

ρ∂v

∂x + ρv

∂v

∂y + ρw

∂v

∂z = −∂ρ

∂y (7)

ρ∂w

∂x + ρv

∂w

∂y + ρw

∂w

∂z = −∂ρ

∂z (8)

∂

∂x

p

ργ

+ v

∂

∂y

p

ργ

+ w

∂

∂x

p

ργ

= 0 (9)

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The hypersonic small disturbance equations are a set of coupled, non-linear differen-tial equations for which no analytical solutions have been found out.

4.4 Waverider Construction

Now as we have established the importance of a waverider geometry this section will dealwith the actual creation of the waverider geometry using the methods discussed above.From literature (caitation of anderson) the shock angle (β ) to construct an optimizedgeometry for M=5 is β =12o.Using the Taylor-Maccoll analysis for β =12o the cone specifications are

Declination of Base(θ) Shock Angle(β ) Cone Angle(δ ) Arc on Cone Base(φ) Length of cone(l) length of vehicle(lw)6.804 12 4.265 54.377 91.4 80

Table 1: Cone Parameters for generating Waverider

Figure 4: Generating Cone and Waverider

Any geometry whose leading edge forms part of the shock cone is a valid waverider. Infinitedesigns are possible. Based on the initial sizing analysis the dimensions of the waverider arefixed. The waverider is constructed using this data. The waverider obtained is constructedusing an inviscid formulation and due to large wetted area has large skin friction drag tominimize skin friction drag viscous optimization of the waverider is done. This involves

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creating an expansion fan on the upper surface of the waverider. A 2o taper is provided fromthe tip to the base of the waverider on the upper surface.

Figure 5: Final Waverider Geometry

4.5 Aerodynamic Coefficients

The lift and drag coefficients are used are obtained from the hypersonic small disturbancetheory is

Planform Area(S plan) Wetted Area(S wet) (σ) Arc on Cone Base(φl) δ θcb1351.831 2892.83 2.8135 54.377 4.265 6.804

Table 2: Geometrical Parameters of generating cone

C L = l2

S plan

δ 2σ3

σ2 − 1 φl0

1 − Rcb

σ

cos φdφ (10)

where

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Rcb = θcb

σ (11)

C L = 0.0388 (12)

The drag computed is both skin friction and wave drag the empirical relations forboth are

C D = C Dp + C f S wetS plan

(13)

C Dp = l2

S plan

δ 4σ2

σ2 − 1 φl0

1 −

Rcb

σ

2− ln

Rcb

σ

2dφ (14)

C Dp = 3.738 × 10−3 (15)

C f = 0.664F 0F 1√

Re(16)

F 0 = 4l2δ

S ∞

lwl

12 φl

0

1 − Rcb

σ

12

dφ (17)

F 1 = S ∞ + (1 + γM 2δ 2)

1

2 S wcS wet

(18)

C f = 6.49071 × 10−4 (19)

C D = 4.7627 × 10−3 (20)

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The LD

ratio obtained is

L

D = 8.146 (21)

The upper bound for the LD

ratio for hypersonic flight is given by the relation

L

D

max

= 6(M ∞ + 2)

M ∞(22)

For a freestream mach number of M ∞=5

LD

max = 8.4 (23)

We can see that the LD

ratio is quite near the maximum value achievable thus theselected geometry is an optimized one.

4.6 3-D CFD simulation of Body

A full body 3-D CFD simulation was carried out to verify the analytical results for thewaverider geometry. The meshing was done in pointwise software and a unstructured mesh

with 6.5 million cells was created.The solver utilized was CFD++ and a 2 equation K- turbulence model was used.Thesolution showed good convergence after 1500 iterations after which the solution was stopped.Post processing was done in CFD-Post software. Apart from the L

D ratio it was of interest

to see if the conical shock actually remain attached to the geometry. The LD

ratio obtainedfrom the CFD is

L

D

CF D

= 7.97 (24)

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Figure 6: Mesh of the Waverider

We can see this is a good match with the analytical results obtained with a deviation of 2.1percent.

In figure 7 you can see the contours of Mach number on the compression surface.The cyan and light green represent the conical shock and it is evident from the figure thatthe conical shock is attached to the leading edge of the waverider. This also validatesthe Taylor-Maccoll Conical flow solution which predicted a Mach number of M=4.771downstream of the conical shock.

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Figure 7: Mach Number Contours of the compression Surface

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4.7 Vertical Fin

The design of the vertical fin is based on the same principle as the waverider , to keep theshock attached to the leading edge , with one difference the generating surface for a fin isa wedge instead of a cone . This results in caret shape of the tail fins. The wedge angle isδ

f in=2o and W

f in=1.5m.

From the hypersonic small disturbance theory

Figure 8: Caret wing generated from a wedge

sin β

sin δ =

γ + 1

4 +

γ + 14

2+

1

M 2∞

sin2 δ f in(25)

β f in

= 12.828o (26)

The fin parameters and aerodynamic coefficients are

lf in = W f in

2sin β f in tan

φfin2

(27)

C Lfin =

4W f inlf in(M 2∞ sin2 β f in

−1)

M 2∞(γ + 1) (28)

C Dfin = C Lfin tan δ f in (29)

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W f in φf in β f in lf in C Lfin C Dfin1.5 60 12.828 5.85 0.1359 0.00474

Table 3: Fin Parameters

4.8 Aerodynamic Heating

To estimate the heat flux through the skin of the aircraft the reference temperature methodis used. The incompressible boundary layer equations are used but the fluid properties areevaluated at a reference temperature T ∗. The wall temperature is taken as T wall=1200K.This value is obtained from the CFD result. The coefficient of heating is

C H = √ 30.332(P r∗)

−2

3

√ Rex∗ (30)

Rex∗ =

ρ∗uex

µ∗(31)

P r∗ = µ∗C p

∗

k∗(32)

The velocity is taken as 1510 m/s the free stream velocity and rest of the values arecomputed at 30 km altitude (see appendix). The reference temperature is calculated as

T ∗

T ∞= 1 + 0.032M 2

∞ + 0.58

T wallT ∞

− 1

(33)

T ∗ = 972.9K (34)

ρ∗ = p∞

RT ∗= 0.00435kg/m3 (35)

µ∗

µref =

T ∗

T ref

32 T ref + 110

T ∗ + 110 (36)

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Figure 9: CFD Results of temperature on compression surface

µ∗ = 4.08×

10−5kg/ms (37)

Rex∗ = 6.435 × 106 (38)

P r∗ = 0.4822 (39)

C H ∗ = 3.686

×10−4 (40)

√ P r∗ =

T aw − T ∞T stag − T ∞

⇒ T aw = 1015.14K (41)

q w = ρ∗u∞C H C p(T w − T aw) (42)

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q w = 449.3451W/m2 (43)

The heat flux through the skin of the aircraft is 449.3451 W/m2.This has to be managedthrough a TPS which is discussed in a later section.

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5 Propulsion

To ram means to force in. In ramjet air is forced into the engine air intake by the sheerdrive of the speed of flight. Ramjet, in principle, can work at subsonic speed but it can bepractical only at higher supersonic speed.

In a ramjet, air undergoes compression in the diffuser, then fuel is added and burnt inthe burner, and then the combustion products expand through the nozzle. It is helpful toconsider first a simplified model of an ideal ramjet. For ideal ramjet it is assumed thatcompression and expansion processes are reversible and adiabatic, that combustion occursat constant pressure, that the air/combustion products properties (specific heat ratio γ andthe gas constant R) are constant throughout the engine, and, although this is not necessary,that the outlet pressure is equal to the ambient pressure, in other words, that the nozzle isin the design regime.

The preliminary design of the inlet is divided into the following subtasks. The first

subtask is the selection of the inlet configuration including selection of the cross sec-tional shape of the supersonic part, selection of compression method, selection of thenumber of ramps or oblique shocks, and selection of the subsonic diffuser. The second sub-task is the determination of the optimization criteria to maximize the total pressure recovery.

For a hypersonic inlet, the free stream is decelerated to the subsonic engine face entryspeed through a suitable shock system (including oblique shocks and a normal shock) anda subsonic diffuser. The shock system will decelerate the flow to a subsonic number, andthe diffuser will further reduce the flow speed to the engine face entry speed. The designcriterion is to maximize total pressure recovery.

Figure 10: sketch of the whole inlet system with on design shock positions

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So have tried out with different possibility such as one ramp followed by normal shock , 2ramp ( i.e two oblique shock followed by normal shock ) and 3 ramp ( i.e 3 oblique shockfollowed by normal shock ). We could have gone with more ramps but it would madedesign more complex and at off design conditions and a lot of spillage may occur. Thereforewe decided to go with 3 ramps (3 oblique shocks) and normal shocks. Next task was to

determine ramp angles which will lead maximum pressure recovery.The inlet is to be designed at the cruise conditions of flight Mach number 5.0 and flightaltitude 40,000 ft. At the on-design point, the oblique shock waves from the two externalramps intersect at the cowl leading edge, and the third oblique shock reflects upward tointersect the junction of the final ramp and the throat section.Given flight altitude H , the free stream air specific heats ratio γ can be determined. Giventhe free stream Mach number , and specifying the normal shock up-stream Mach number ,there are 8 basic equations for the 3 oblique shocks.

tan δ 1 = 2cot θ1(M 20 sin2 θ1 − 1)2 + M 20 (γ + 1 − 2sin2 θ1)

(44)

M 1 = (γ + 1)2M 4

0 sin2 θ1 − 4(M 20 sin2 θ1 − 1)(γM 40 sin2 θ1 + 1)

2γM 20 sin2 θ1 − (γ + 1)((γ − 1)M 20 sin2 θ1 + 2)

(45)

tan δ 2 = 2cot θ2(M

2

1 sin2 θ2 − 1)

2 + M 21 (γ + 1 − 2sin2 θ2) (46)

M 2 = (γ + 1)2M 4


2γM 21 sin2 θ2 − (γ + 1)((γ − 1)M 21 sin2 θ2 + 2)

(47)

tan δ 3 = 2cot θ3(M

2

2 sin2 θ3 − 1)

2 + M 22 (γ + 1 − 2sin2 θ3) (48)

M 3 = (γ + 1)2M 4


2γM 22 sin2 θ3

−(γ + 1)((γ

−1)M 22 sin

2 θ3 + 2) (49)

M 0 sin θ1 = M 1 sin θ2 (50)

M 1 sin θ2 = M 2 sin θ3 (51)

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Figure 11: Sketch of the whole inlet system with on design shock positions

Using oblique and normal shock initially we performed iterations to find out what all rampangles would be optimum which would lead to maximum pressure recovery. Inlet Mach no.is assumed to be 5 (effect of wedge shock and frictions are not taken in to consideration)Once we got basic estimate about the ramp design we have further modified our design byincorporating wedge shock and friction effect in to our consideration [1], code of which isattached in the appendix.

5.1 Combustion Chamber

Since flow entering the combustion camber is subsonic ( M = 0.6), so we designed asubsonic combustion chamber[2], we kept the maximum combustion chamber temperatureto be 2400K .The total pressure loss in a combustor is predominantly due to two sources.The first is due to the frictional loss in the viscous layers where we can lump in the mixingloss of the fuel and air streams prior to chemical interaction. The second source is due toheat release in an exothermic chemical reaction in the burner. The first is usually modeledas the “cold” flow loss and the second is the “hot” flow loss.For a preliminary level designit is safe to assume stagnation pressure loss in nozzel to be around 5 percentage.

F = ( ṁa + ṁb)V e − ṁaV i (52)

P 0aP o

=

1 +

γ − 1γ

M 2a

γ γ −1

(53)

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Figure 12: Table showing ramp angle and corresponding pressure recovery

P 06P 6

=

1 +

γ − 1γ

M 2a

γ γ −1

(54)

( ṁa + ṁb)h04−

ṁah02 = ṁaQf (55)

ṁa((1 + f )h04 − h02) = ṁf Qf (56)

f =

T 04T 0a

− 1

Qf C pT 04

− T 04T 0a

(57)

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Figure 13: Preliminary inlet design value (including wedge shock and friction effect )

F

ṁa= (1 + f )V e − V i (58)

F

ṁa= M

γRT a

1 + f

T 04T 0a

− 1

(59)

F

ṁa= M

γRT a

1 + f

T 04T 0a

(1 + γ − 1

2 M 2a ) − 1

(60)

5.2 Nozzle design

Nozzle design was done using the MOC analysis. The Matlab code is attached in the Ap-

pendix. The nozzle exit Mach number is 4.26 and the nozzle area ratio is 9.86 .

5.3 Offset design and Conclusion

Performance the engine went down as we have operated it on off-design condition i.e awayfrom Machno. = 5, we choose 2 different off-design condition ,one was for Machno.4andother was for Machno.6. Pressure recovery improved for Machno.4, it went up to 0.652for same inlet and maximum chamber pressure. While in case of Machno.6 total pressure

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Figure 14: Thrust vs Area ratio for nozzle

recovery went down to 0.3487.But since our mission requirement was to cruise at Machno. =5 we went ahead with current engine design.Mass flow rate was found out to be 765kg/s and max thrust value to be 550KN fuel flowrate in case of liquid hydrogen was mound out be around 7kg/s. Specific thrust value= 0.718KNs/kg which is also corresponds to graph below for Ideal Ramjet thrust and fuelconsumption.

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Figure 15: Ideal Ramjet thrust and fuel consumption

Figure 16: Performance comparison at off-set design operation

5.4 CFD Results of inlet

A CFD simulation was carried out to check if the inlet shocks did fall on the cowl and thatthe design was operational. From the images it is clear that the inlet design produces the

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required results.

Figure 17: Inlet 2-D unstructured mesh

Figure 18: Inlet CFD results

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6 Weight and Sizing

6.1 The HASA technique

For initial weight and sizing analysis of the suggested aircraft configuration, a literature

review of existing sizing techniques was undertaken. The technique found to be most suitablefor the proposed vehicle was the HASA (Hypersonic Aerospace Sizing Analysis) techniquefor the preliminary design of Aerospace vehicles. It was developed by NASA and has beenimplemented in a number of hypersonic vehicle designs. For the aforementioned analysis,some initial geometrical parameters had to be sourced from relevant references while themission conditions set other parameters like payload and flight velocity (at 30 km of altitude).The following were the assumed initial parameters: From these values, flight parameters like

Figure 19: Assumed Parameters

weight of fuel carried, fuel ratio, initial estimate of total weight, volume and equivalentbody diameter etc. were calculated. These calculations provided an initial estimate of thevehicle size and weight. For arriving at actual parameters suited to the design requirements,following non-dimensional constants were utilized according to the HASA technique:

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Figure 20: Geometrical Parameters

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Once these constants were determined, the actual values of flight parameters confirming tothe designed mission requirements were determined using the formulas given below:

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7 Structures and Material Selection

7.1 Introduction

With the high aero-thermodynamic loads the vehicle experiences at high Mach numbers the

structural integrity of the vehicle becomes a major issue. Conventional aircraft materialscan’t be used due to the high skin temperatures and high temperature alloys or carboncomposites have to be looked into. The aircraft structure must also be able to withstandthe bending moments and a cross-section for the aircraft is identified which can handle theseloads.

7.2 Material Selection

Different Materials are considered for application in the aircraft design and their propertiesare listed

Material Den sity(kg/m3

) Maximum Service Temperature(K) Ultimate Tensile Strength(MPa) Thermal Conductivity(W/mK) Coefficient of Expansion(K −1

Inconel X-750 8280 923 1250 12 12.6×10−6

Ti-Zr-Mo Alloy 10220 1873 685 126 5.3×10−6

Rene 41 super alloy 8240 1243 1241 80 8.1×10−6

Ti-Al Alloy 44200 1000 950 7.2 9.2×10−6

Hafnium Diboride 11900 2300 500 60 7.6×10−6

Carbon Silicon Carbide 2100 1850 310 15 1.3×10−6

Carbon-Carbon Composites 1700 1600 240 40 4×10−6

Table 4: Properties of Materials considered for application in Aircraft Body

Inconel X-750 was chosen for outer skin of X-15.Throughout the life of X-15 Program, In-conel X-750 has demonstrated its reliability and strength where speeds up to Mach no 6.70.Modern day use of Inconel X-750 has been restricted due to its high weight and cost.Ti-Zr-Mo alloy this form of molybdenum alloys possesses high recrystallisation temperatureup to 1800 K and higher strength compared to many current super alloys. Low thermalexpansion coefficient so it can be used in high heat exposure area without incorporatingof large number of joints to prevent buckling due to thermal stresses.Manufacturing of thissuper alloys consume lots of money as well as time. Coverage of large surface area of vehicle

is not possible to this alloys due to possible weight penalties incurred.Rene 41 super alloy a cooling system involving the use of lithium in lieu of conventionalcoolant encased in a Rene 41 honeycomb matrix was proposed by Boeing to solve thermalmanagement problems in sustained hypersonic flight vehicles. Due to high weight penaltiesof this alloy its use is severely limited.Ti-Al alloy has great potential to achieve weight reduction in scram jet/ramjets by up to25-30 percent. The high strength offered also makes this a good prospective material butdue to lower service temperature a active thermal system must be in place while sing this

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material.Hf Br2 and ZrBr2 are both relatively new ceramic compounds that hold great potential toreplacing traditional TPS in hypersonic atmospheric vehicles.Their low ablation rates at hightemp(first sign of ablation at 3000 K ) allow a reusable hypersonic vehicles to be created.But due to high weight their functionality is reduced.

C/SiC carbide consists of carbon fibre reinforced silicon carbide produced using liquid poly-mer infiltration techniques.Despite the good retention in strength to weight ratio at hightemperatures, C/SiC composite is significantly much lower in load bearing properties com-pared to several materials. This is due to micro cracking during complex component fabrica-tion and microstructural voids (empty spaces) generated during carbon fibre weaving. Dueto this reason C/Sic carbide are not used in hypersonic flight.C-C composites are extensively used in hypersonic and reusable vehicles, it has good re-tention at high temperatures up to 1600K and light weight capability for aero structureintegration.Several applications of overcoat sealers are required to be applied on reinforcedcarbon-carbon (RCC) surface to prevent severe oxidation damage.The coating consisted of SiC substrate, a second chemical vapour deposited Sic layer followed by a last layer of Hafnium carbide.After considering the cost and weight penalties the choice of material for the aircraft skin isTi-Al alloy due to its low weight and high strength but due to its low service temperatureC-C composites are preferred at the nose tip of the vehicle.

7.3 Structural and Thermal Analysis

The cross section bulkhead of the aircraft was subjected to bending stresses which whichappear during flight to check the structural integrity of the vehicle. Simultaneously a thermalload study was conducted so a coupled structural-thermal analysis was preformed to check

the effect temperature may have on the properties of the material.

Figure 21: Vehicle cross section geometry

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The material used is Ti-Al alloy with a thickness of 3mm. Assuming this to be uniform thegross weight of the outer skin of the aircraft is 41,000 Kg.The thermal analysis is preformed by applying constant wall temperatures and consideringeffects of only convection and no radiation. The temperature distribution obtained is shownin figure. This temperature distribution is used to apply the temperature specific properties

at different locations during the structural analysis. The structural analysis is carried out

Figure 22: Temperature Distribution

by applying a bending moment of the entire geometry. This bending moment simulates theaerodynamic loads which appear of the body during flight. The bending moment calculatedis 3000 Nm.

Figure 23: Stress Distribution

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The maximum stresses developed are at the corners of the geometry and are around 790 MPa.This is below the ultimate stress of Ti-Al alloy of 1000 MPa. This gives and overall factorof safety of 1.25 . The maximum displacement observed is 14 mm (attached in Appendix)

Figure 24: Strain Distribution

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8 Fuel Selection

8.1 Introduction

Fuel selection plays a crucial role in determining the final weight as well as performance of the

aircraft. The fuel selection also plays an important role in determining the the parameters of the propulsion system. This section deals with the selection between different type of fuelsmainly hydrogen and other organic based synthetic fuels.

8.2 Hydrogen Fuel of the Future?

As a preliminary various fuels were analysed they are listed below. It is clearly visible the

Fuel Density(kg/m3) Heating Value(KJ/Kg)Liquid Hydrogen 89.88 121

JP-10 940 42.14CH 4(liquefied) 422.36 55.53

JP-4 751 42.8

Table 5: Various Fuels and their properties

great advantage of LH 2 over other hydrocarbon based fuel is its high calorific value which isthree times the others. Its inherent disadvantage is its low density which causes larger tankvolumes and also the fact that it has to be maintained at cryogenic temperatures.Liquefied CH 4 is fuel which is under development but given its relatively low density

compared to other hydrocarbons and not much improvement in calorific value we reject itas a candidate.The last two fuels JP-4 and JP-10 are both hydrocarbon based synthetic fuels and aresimilar in nature but due to better density characteristics we select JP-10.Hydrogen can be used as the primary fuel in an internal combustion engine or in a fuelcell. A hydrogen internal combustion engine is similar to that of a gasoline engine, wherehydrogen combusts with oxygen in the air and produces expanding hot gases that directlymove the physical parts of an engine. The only emissions are water vapor and insignificantamounts of nitrous oxides. The efficiency is small, around 20 percent. Disadvantagesof hydrogen include fuel High Cost,Low Temperatures needed escape can cause fire orasphyxiation. It will explode at concentrations ranging from 4-75 percent by volume in the

presence of sunlight, a flame, or a spark.On the other hand JP-10 is a gas turbine fuel or synthetic fuel.It contains a mixture of (in decreasing order) endo-tetrahydrodicyclopentadiene, exo tetrahydrodicyclopentadiene,and adamantane. It is produced by catalytic hydrogenation of dicyclopentadiene.Itis high-energy density hydrocarbon fuel. It is less expensive to synthesize than otherfuels.JP-10 is speciality fuel that has been developed for demanding applications.Therequired properties are: maximum volumetric energy content, clean burning, and goodlow-temperature performance. To achieve these properties, the fuels are formulated with

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high-density naphthenes in nearly pure form.

8.3 Range in Cruise

The total mass of the propellant is fixed at 50,000 Kg. The mass flow rate of air which hasto be heated from 1192 K to 2400 K for the propulsion system is 765 Kg/ m3. The cruisevelocity is 5400 km/hr. It is quite clear that LH 2 will clearly give much higher range in

Fuel Fuel Flow Rate(kg/s) Range(Km)Liquid Hydrogen 7.66 9900

JP-10 22.03 3500

Table 6: Range comparison of JP-10 and LH 2

cruise for the aircraft. It also satisfies the mission requirement range of 10,000 Km.

8.4 Fuel as Thermal Protection System

An internal forced convection model (caption) is used to calculated the convective heattransfer coefficient of each fuel.

Nu = 4.82 + 0.0185(RE DP r)0.827 (61)

where Nu is Nusselt number the ratio of convective to conductive heat transfer across(normal to) the boundary.

Nu = hX

k (62)

The convective heat transfer coefficients are

JP-10 : hJP −10=77.69 W/K-m2

Liquid Hydrogen : hLH 2=0.6988 W/K-m2

Clearly Liquid hydrogen posses very poor convective properties and cant be used di-rectly in an TPS system some other refrigerant/coolant has to be used in this case.Based on all the above properties we select Liquid Hydrogen as our fuel of choice as itprovides excellent heating value and is the only fuel which can fulfil our design range.

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9 References

1. HASA-Hypersonic Aerospace Sizing Analysis for the Preliminary Design of AerospaceVehicles, NASA-Contractor Report 182226

2. Randall T. Voland, Lawrence D. Huebner, Charles R. McClinton, X-43A Hypersonicvehicle technology development.

3. Mary Kae Lockwood, Dennis H. Petley, John G. Martin, James L. Hunt, Airbreathinghypersonic vehicle design and analysis methods and interactions.

4. Kevin D. Jones, Helmut Sobieczky, A. Richard Seebass , F. Carroll Dougherty, Wa-verider Design for Generalized Shock Geometries.

5. Kashif H. Javaid and Varnavas C. Serghides, AN INITIAL DESIGN METHODOLOGYFOR OPERATIONAL HYPERSONIC MILITARY AIRCRAFT.

6. Frederick Ferguson, Nastassja Dasque and Mookesh Dhanasar, Waverider Design andAnalysis.

7. John D. Anderson Jr, Hypersonic and High Temperature Gas Dynamics

8. John J. Bertin, Russell M. Cummings, Fifty years of hypersonics: where we’ve been,where we’re going.

9. SHEAM-CHYUN LIN and MING-CHIOU SHEN, FLIGHT SIMULATION OF AWAVERIDER-BASED HYPERSONIC VEHICLE

10. Jeffrey S. Robinson, An Overview of NASA’s Integrated Design and Engineering Anal-

ysis (IDEA) Environment

11. H. Alkamhawi, T. Greiner, G. Fuerst, S. Luich, B. Stonebraker, and T.Wray,Hypersonic Aircraft Design

12. N. C. Bissinger , N. A. Blagoveshchensky, A. A. Gubanov , V. N. Gusev , V. P.Starukhin , N. V. Voevodenko , S. M. Zadonsky, Improvement of forebody/inlet inten-tion for hypersonic vehicle

13. Mohammed H Sadraey, Aircraft Design

14. Hongjun Ran and Dimitri Mavris, Preliminary Design of a 2D Supersonic Inlet to

Maximize Total Pressure Recovery

15. Aircraft propulsion –Saeed Farokhi

16. BASHETTY SRIKANTH, K.DURGA RAO, Dr.S.SRINIVAS PRASAD, StructuralAnalysis on Unconventional Section of Air-Breathing Cruise Vehicle

17. http://www.dtic.mil/dtic/tr/fulltext/u2/607169.pdf

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APPENDIX I

Atmospheric Data taken from US standard atmospheric data.

Figure 25: Atmospheric Properties variation with Altitude (yellow is design altitude)

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APPENDIX II

CFD results for velocity in z direction.The iso surface represents zone of 0 vertical velocity.Notice the iso surface attached to the leading edge indicating no flow movement from bottomto top surface.

Figure 26: Iso surface of 0 z velocity

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APPENDIX III

Inlet-Design Matlab Code

% ramp d e s ig n b as ed on 3 o b l i q u e s h oc ks and 1 no rm al s ho ck% c od e w r i t t e n by Kush S r ee n

c l e a r a l lc l c

m0=4.77 ; %fr ee stream mach numbergamma=1.4;

m4 up = 1 . 7 ; %p r e s s u r e r e c o v e r y v a lm3=10;t h et a 1 =0; %i n t i a l g u es s

w h i l e a b s ( m3−m4 up) >=0.01

ku=(gamma+1)∗(gamma+1)∗(m0ˆ4)∗ ( ( s i nd ( t he t a1 ) ) ˆ2 ) ;

a=−4∗((m0ˆ2)∗ (( sin d ( the ta1 ))ˆ2) −1 ) ;b=(gamma∗(m0ˆ2)∗ ( ( s i nd ( t he t a1 ) ) ˆ2 ) +1 ) ;

c=2∗gamma∗(m0ˆ2)∗ (( sin d ( the ta 1 ))ˆ2) −(gamma−1);d=(gamma−1)∗(m0ˆ2)∗ (( sind ( thet a1 ))ˆ 2)+ 2;

e=(ku+a∗b ) / ( c∗d ) ;%a=(((gamma+1)ˆ2)∗((m0)ˆ4)∗ (( si nd ( thet a1 ))ˆ2)−4∗(m0∗m0∗ (( sind ( the ta1 ))ˆ2)∗ (gamma∗m0∗m0∗ ( ( s i nd ( t he t a1 ) ) ˆ2) +1) ) /( ( 2∗gamma∗m0∗m0∗ (( si nd ( the ta 1 ))ˆ2)((gamma−1)∗m0∗m0∗ ( ( s i nd ( t he t a1 ) ) ˆ2 ) +2 ) ) ;m1=e ˆ0 . 5 ;

thet a2=asi nd (( (m0∗ s i n d ( t h et a 1 ) ) /m1 ) ) ; %o p t i m i za t i o n c o n d i t i o n

ku=(gamma+1)∗(gamma+1)∗(m1ˆ4)∗ ( ( s i nd ( t he t a2 ) ) ˆ2 ) ;



e=(ku+a∗b ) / ( c∗d ) ;

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%b=(((gamma+1)ˆ2)∗((m1)ˆ4)∗ (( si nd ( thet a2 ))ˆ2)−4∗(m1∗m1∗ (( sin d ( th eta 2 ))ˆ2∗(gamma∗m1∗m1∗ (( sind ( the ta2 ))ˆ2)+ 1))/( ( 2

∗gamma

∗m1

∗m1

∗(( sin d ( the ta 2 ))ˆ2)

−(gamma+1))

∗((gamma

−1)

∗m1

∗m1

∗(( sin d ( th

m2=e ˆ0 . 5 ;

thet a3=asi nd (( (m1∗ s i nd ( t he t a2 ) ) /m2) ) ;

%c=(((gamma+1)ˆ2)∗((m2)ˆ4)∗ ( ( s i nd ( t he t a3 ) ) ˆ2 )−4∗(m2∗m2∗ (( sin d ( thet a3 ))ˆ2)−1)∗(gamma∗m2∗m2∗( ( s i nd ( t he t a3 ) ) ˆ2) + 1) ) / ( ( 2 ∗gamma∗m2∗m2∗ (( sin d ( thet a3 ))ˆ2)−(gamma+1))∗((g

ku=(gamma+1)∗

(gamma+1)∗

(m2ˆ4)∗

( ( s i nd ( t he t a3 ) ) ˆ2 ) ;



e=(ku+a∗b ) / ( c∗d ) ;

m3=e ˆ0 . 5 ;

d=((gamma−1)∗((m4 up)ˆ2)+2)/(2∗gamma∗ (( m4 up)ˆ2)−(gamma−1) ) ;

m4=dˆ0.5;

i f a bs ( m3−m4 up) >=0.01t he t a1=t he t a1 + 0. 01;en d

en d% a n g l e s

a=(2∗ c ot d ( t he t a1 ) ∗ ((m0)ˆ2∗( si nd ( the ta 1 ))ˆ2−1))/(2+(m0)ˆ2∗(gamma

+1−2∗( s i n d ( t h e t a 1 ) ) ˆ 2 ) ) ;del 1=atand( a ) ;


+1−2∗( s i n d ( t h e t a 2 ) ) ˆ 2 ) ) ;

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del 2=atand( a ) ;


+1

−2

∗( s i n d ( t h e t a 3 ) ) ˆ 2 ) ) ;

del 3=atand( a ) ;

%p r e ss u r e r ec o ve ry r a t i o s

%a c r o s s o b l i q u e s h oc ks

a=(((gamma+1)∗(m0)ˆ2∗( si nd ( the ta 1 )) ˆ 2 ) / ( (gamma−1)∗m0ˆ2∗( sin d ( thet a1 ))ˆ2+b=((gamma+1)/(2∗gamma∗m0ˆ2∗( sin d ( thet a1 ))ˆ2−(gamma−1)))ˆ(1/(gamma−1) ) ;pr1=a∗b ;

a=(((gamma+1)∗(m1)ˆ2∗( si nd ( the ta 2 )) ˆ 2 ) / ( (gamma−1)∗m1ˆ2∗( sin d ( thet a2 ))ˆ2+b=((gamma+1)/(2∗gamma∗m1ˆ2∗( sin d ( thet a2 ))ˆ2−(gamma−1)))ˆ(1/(gamma−1) ) ;

pr2=a∗b ;

a=(((gamma+1)∗(m2)ˆ2∗( si nd ( the ta 3 )) ˆ 2 ) / ( (gamma−1)∗m2ˆ2∗( sin d ( thet a3 ))ˆ2+b=((gamma+1)/(2∗gamma∗m2ˆ2∗( sin d ( thet a3 ))ˆ2−(gamma−1)))ˆ(1/(gamma−1) ) ;

pr3=a∗b ;

%a c r o s s n or ma l s h o ck

a=(((gamma+1)∗(m4 upˆ2))/((gamma−1)∗m4 upˆ2+2))ˆ(gamma/(gamma−1) ) ;b=((gamma+1)/(2∗gamma∗(m4 up)ˆ2−(gamma−1)))ˆ(1/(gamma−1) ) ;

pr4=a∗b ;

pr t ot =pr 1∗pr2∗pr3∗pr4

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APPENDIX IV

Nozzle Design MOC Matlab Code

% B ri tt o n J e f f r e y Olson% Ph .D. Ca nd ida te

% S ta nf o rd U ni ve ri st y% D epa rtm ent o f Aero / A s tr o

%%%%%%%% I n t r o d u c t i o n and Backgro und %%%%%%%%% Th is program g i v e s t he i d e a l n o z zl e geo metry u si n g t he m ethod o f % c h a r a c t e r i s t i c s f o r a Quasi−2D Di v e r gi n g N oz z l e .Assume g a s i s% e x ha u st i ng from a c om bu st io n c hamber t h at h as no m ass f l o w r a t e i n .% Using 2D n oz zl e f lo w r e l a ti o n s , an o pt im al t hr oa t a re a i s found t ha t% w i l l p ro du ce t he max amount o f t h ru s t f o r t he g iv en ambient p r e ss u r e% and c o mb us ti on c hamber p ar am et er s . T hi s Area i s a u t o m a ti c a l l y s e t a% f ed i nt o the method o f c h a r a c t e r t i s t i c s p or ti on o f t h at code .The% method o f c h a r a c t e r i s t i c s a l s o u s es t he e x i t Mach number t ha t% c or re sp on ds t o t he i d e al e x it a re a .

%%%%%%%% D ir ec ti on s fo r runn ing the program %%%%%%%%% Run d e f a u l t o r m od if y t he p ro bl em p ar am et er s i n n o z z l e .m and n o z c f d% 1 ) Run th e n o z zl e .m program− This w i l l d es ig n t he n o zz le% 2 ) Run noz mesh .m− This w i l l s e t up a g r id f o r t he newly d es ig ne d% n o z z l e

% 3) Run no z c f d .m− This w i l l s o lv e t he 2−D E ul er e q u at i o n s on t he% n oz zl e mesh and p lo t the r e s u l t .%% ∗∗∗Note∗∗∗% You must run them i n t h i s o rd er wi thout c l e a r i n g v a r i a bl e s .% Each sc r i p t depends on t he one be fo re i t .Also , s o l v e r ha s t he% f un c t i o n s u sed by n oz c fd .m and needs t o be i n the same d i r ec t or%% F i g ur e ( 1 ) : S t at i c t hr us t a s f un ct io n o f e x it a re a% F ig u re ( 2 ) : N oz zl e d e s ig n and p l o t s o f Mach number & P r e ss u r e v s L eng

% F ig ur e ( 3 ) : CFD s im u l at i o n o f d es i gn ed n o zz l e

c l e a r a l l ;c l c ;

% Problem p a ra me te rsT c = 2 4 0 0; % Temp era tu re i n t he c om bu st io n chamber (K)

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P c = 2 86 7 4 9. 7 5 ; % P r es s ur e i n t he co mbu sti on c hamber ( Pa )P amb = 13 30 .1 9 ; % Ambient p r e s s u r e ( Pa )T amb = 2 3 2 . 9 ; % Ambient t em pe ra tu re (K)gamma = 1 . 4 ; % Ra t i o o f S p e c i f i c H ea t s Cp/Cv ( Gamma)W = 2 5 . 4 ; % M ol ec u l a r wei ght of gas ( kg /kmol )

width = 1 5; % N ozz l e width ( meters )h t h = 0 . 23 ; % Throat h e ig ht ( m eter s )

% Method o f C h a r a c t er i s t i c snum = 1 5; % Number o f C h a r a c t e r i s t i c l i n e st h e t a i = . 0 3 ; % I n i t i a l s t e p i n t h e t ap l o t t e r = 1 ; % Se t t o ’ 1 ’ t o p lo t n o z z l e

dh = h t h / 1 0 0 ;m a x i t e r = 1 0 0 0 0 ;R = 831 4/W;

% Part A

%f i n d w he re P b ec om es uh ( 1 ) = h t h ;A s t ar = h t h∗width ;

M =1;dM1 = . 1 ;f o r i =1: m a x i t er

h ( i ) = h ( 1 ) + ( i −1)∗dh ;Ae( i ) = h( i )∗width ;A Asq = (Ae( i )/ A sta r )ˆ 2;A r at i o ( i )=s q r t ( A A sq ) ;

%Newton Rhapson on Eq. 5. 20 − Ander son t e x tr e s = 1 ;i f i > 1

M = Ma( i −1);en d

w hi l e r e s > . 0 0 1

M2 = M + dM1;f u n a 1 = −A Asq + (1/Mˆ2)∗ ((2/(gamma+1))∗(1+(gamma−1)∗Mˆ2/2))ˆ((gaf u n a 2 = −A Asq + (1/M2ˆ2) ∗ ((2/(gamma+1))∗(1+(gamma−1)∗M2ˆ2/2))ˆ(dv dm = ( f una2−funa1)/dM1;

M = M − funa1/dv dm;r e s = a bs ( f un a 1 ) ;

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en dMa( i ) = M;

% Fi nd P r e s s u r eP ( i ) = P c

∗(1+(gamma

−1)

∗Ma( i )ˆ2/2 )ˆ(

−gamma/(gamma

−1) ) ;

% Find t h ru s t f o r ea ch p o in tTe( i ) = T c /(1+(gamma−1)∗Ma( i )ˆ 2/ 2) ;Tt( i ) = T c /(1+(gamma−1) /2) ;Ve( i ) = Ma( i )∗ sq rt (Te( i )∗gamma∗R ) ;Vt( i ) = sq rt (Tt( i )∗gamma∗R ) ;rho t ( i ) = P( i )/ (R∗Te( i ) ) ;mdot( i ) = rho t ( i )∗Ve( i )∗Ae( i ) ;TT( i ) = mdot( i )∗Ve( i ) + (P( i ) − P amb)∗Ae( i ) ;

i f P ( i ) < P amb%break%C al cu la te t he p r es s ur e i f sh ock wave e x i s t s a t t he e x i t p la neP e x i t = P ( i )∗(1+(gamma∗2/(gamma+1))∗(Ma( i ) ˆ2 −1) ) ;

i f P e xi t


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M ax t hr ust = TT( b ) ;h ol d on ;pl ot (A max , Max thrust , ’ r ∗ ’ )le ge nd ( ’ Thrust Curve ’ , ’Max Thrust ’ )

% Part C% Method o f C h a r a c t er i s t i c s

M e = Ma( b ) ; %Mach number at i d e a l e x i t

%Fi nd t he ta ma x by u s i n g e q u a ti o n 1 1 . 3 3t he ta max = ( 180/ pi )∗ ( s q r t ( (gamma+1 )/(gamma−1))∗ ata n (( s q rt ( (gamma−1)∗(M

% D theta f o r each char l i n ed e l t h e t a = ( t h e t a m a x

− t he t a i ) /( num

−1);

% Find

f o r i =1:num% I n i t i a l i z e mach numeber

f o r j =1:numi f i ==1

%Theta f o r each l i n e ( f i r s t l i n e s )t h et a ( i , j ) = t h e t a i + d e l t h e t a ∗( j −1);nu( i , j ) = th et a ( i , j ) ;K m( i , j ) = th et a ( i , j ) + nu( i , j ) ;K p ( i , j ) = th et a ( i , j ) − nu( i , j );

e l s e i f i > 1

K p ( i , j ) = −K m(1 , i ) ;

% F i nd T het asi f j >= i

t h e t a ( i , j ) = d e l t h e t a∗

( j−

i ) ;

e l s e%the ta ( i , j ) = the ta ( j , i −1);the ta ( i , j ) = thet a ( j , i ) ;

en dnu( i , j ) = th et a ( i , j ) − K p ( i , j ) ;K m( i , j ) = th et a ( i , j ) + nu( i , j ) ;

en d

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% P r andt l−Mey er f un c t i on ( usi ng Newton R hapson)dM = . 1 ; % Lea ve a t a bo ut . 1i f j == 1

M ex ( i , j ) = 1 . 0 0 ;

e l s eM ex( i , j ) = M ex( i , j −1);

en dM = M ex( i , j ) ;

r e s = 1 ;w hi le r e s > . 0 1

M2 = M + dM;f u n v 1 = (−nu( i , j )∗ ( pi /18 0)+ ( sq r t ( (gamma+1)/(gamma−1))∗ atan (( sq rtf u n v 2 = (−nu( i , j )∗ ( pi /18 0)+ ( sq r t ( (gamma+1)/(gamma−1))∗ atan (( sq rtdv dm = ( f unv 2

−funv1)/dM;

M = M − funv1/dv dm;r e s = a bs ( f un v1 ) ;

en dM ex ( i , j ) = M;

% Fi nd t h e a n g l e mumu( i , j ) = (18 0/ pi )∗ as in (1/ M ex( i , j ) ) ;

en d

% Add l a s t p oi nt to ch ar l i n eth et a ( i , num+1) = th et a ( i , num ) ;nu( i ,num+1) = nu( i ,num);K m( i , num+1) = K m( i , num ) ;K p ( i , num+1) = K p ( i , num ) ;

en d

char = zeros (num,num+1 ,2);f o r i =1:num

f o r j =1:num+1

% Draw p o i nt s o f i n t e r s e c t i o n% Point 1 of a l l c har l i n e si f j == 1

c h ar ( i , j , 1 ) = 0 ;c h a r ( i , j , 2 ) = h t h / 2 ;

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en d

% Where f i r s t l i n e h i t s t he symmetry l i n ei f i == 1 & j ==2

c h a r ( i , j , 1 ) = (

−h th /2)/ tan (( pi /180 )

∗( th et a (1 , j

−1)

−mu( 1 , j

−1

c h ar ( i , j , 2 ) = 0 ;en d

% Where a l l o th er l i n e s h i t th e symmetry l i n ei f j == i +1 & j >2

c h a r ( i , j , 1 ) = −c har ( i −1,j , 2) / tan (( pi /1 80 )∗ ( . 5∗ th et a ( i , j−2c ha r ( i , j , 2 ) = 0 ;t e s t ( i , j ) = ( th et a ( i , j−2)−.5∗(mu( i , j−2)+mu( i , j −1 ) ) ) ;t e s t pt y ( i , j ) = c har ( i −1 ,j , 2 ) ;t e s t pt x ( i , j ) = c har ( i −1 ,j , 1 ) ;

en d

% A l l o t h e r data p oi nt s f o r char 1 c al cu l a t e di f i ==1 & j >2 & j ˜= i +1

C p = t an ( ( pi /1 80) ∗ ( . 5∗ ( the ta ( i , j−2)+t he ta ( i , j −1))+.5∗(mu( i C m = t an ( ( pi /1 80) ∗ ( . 5∗ ( t he t a ( j −1,1)+ th et a ( i , j −1))−.5∗(mu( j −A = [ 1 ,−C m;1 ,−C p ] ;B = [ c har ( 1 , 1 , 2 ) − char (1 ,1 ,1 )∗C m ;

cha r (1 , j −1, 2) − cha r (1 , j −1, 1)∗C p ] ;ite rm (1 ,: )= inv (A)∗B ;c h a r ( i , j , 1 ) = i t e r m ( 1 , 2 ) ;

c h a r ( i , j , 2 ) = i t e r m ( 1 , 1 ) ;en d

% A l l o t h e r p oi nt s f or a l l char l i n e s c al cu l a te di f i > 1 & j ˜ = i + 1 & j >2

C p = t an ( ( pi /1 80) ∗ ( . 5∗ ( the ta ( i , j−2)+t he ta ( i , j −1))+.5∗(mu( i C m = t an ( ( pi /1 80) ∗ ( . 5∗ ( the ta ( i −1, j−1)+t he ta ( i , j −1))−.5∗(muA = [ 1 ,−C m;1 ,−C p ] ;B = [ c h a r ( i −1 , j , 2 ) − c har ( i −1 , j , 1 )∗C m ; c h a r ( i , j −1, 2) − char

i t e r m ( 1 , : ) = i nv (A)∗

B;

c h a r ( i , j , 1 ) = i t e r m ( 1 , 2 ) ;c h a r ( i , j , 2 ) = i t e r m ( 1 , 1 ) ;

en den d

en d

% F i l l i n s i m i l a r p oi nt s ( where char l i n e s s ha re p oi nt s )

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f o r i = 2 : numf o r j =2:num

c h ar ( j , i , 1 ) = c ha r ( i −1, j +1 ,1);c h ar ( j , i , 2 ) = c ha r ( i −1, j +1 ,2);

en d

en d

% ∗∗∗∗∗∗Make t he n o z zl e s ha pe and e xt en d t he c ha r l i n e s t o w a ll ∗∗∗∗∗∗

% I n i t i a l s t a r t p o i n t o f t he n oz z l e ( a t t h r o a t )no z ( 1 , 1 ) = 0 ;n oz ( 1 , 2 ) = h t h / 2 ;

% Find a l l the p o i nt s o f t he n o z zl ef o r i = 2 : num

% Find d i f f e r e n t s l o pe s and p oi nt s t o i n t e r s e c tm1 = tan (( pi /18 0)∗( the ta ( i −1,num)+mu( i −1,num)));i f i ==2

m2 = ( pi /180 )∗ theta max ;e l s e

m2 = (( pi /18 0)∗( the ta ( i −1,num+1)));en dm3 = (( pi /18 0)∗( the ta ( i −1,num)));m4 = tan ( (m2+m3) / 2 ) ;

A = [ 1 ,−m4 ; 1 ,−m1 ] ;B = [ noz ( i −1, 2) − noz ( i −1, 1)∗m4 ; c h a r ( i −1,num+1,2) − c har ( i −1,num+1i t e r m ( 1 , : ) = i nv (A)∗B;n oz ( i , 1 ) = i t e r m ( 1 , 2 ) ;n oz ( i , 2 ) = i t e r m ( 1 , 1 ) ;

% Extend char l i n e s t o w al lc har ( i −1,num+2,1)= noz ( i , 1 ) ;c har ( i −1,num+2,2)= noz ( i , 2 ) ;

en d

%L as t l i n em1 = tan (( pi /18 0)∗( th et a (num, num)+ mu(num, num ) ) ) ;m2 = (( pi /18 0)∗( th et a (num−1,num)));m3 = (( pi /18 0)∗( th et a (num, num+ 1) )) ;m4 = tan ( (m2+m3) / 2 ) ;

A = [ 1 ,−m4 ; 1 ,−m1 ] ;B = [ noz(num, 2 ) − noz(num,1)∗m4; char (num, num+1,2) − cha r (num, num+1 ,1)∗m

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i t e r m ( 1 , : ) = i nv (A)∗B;noz ( num+1, 1) = i t e r m ( 1 , 2 ) ;noz ( num+1, 2) = i t e r m ( 1 , 1 ) ;

% Extend char l i n e s to w al lcha r (num, num+2,1)= noz (num+1 ,1 );cha r (num, num+2,2)= noz (num+1 ,2 );

i f p l o t t e r ==1

% P l o t t he n o z z le shapef i g u r e ( 1 ) ; c l f ;s u b p l o t ( 2 , 1 , 1 ) ;pl ot (noz ( : , 1 ) , noz ( : , 2 ) , ’ k ’ , ’ LineWidth ’ ,3 )h ol d on ;[ a , b ] = max ( noz ) ;pl ot (a (1 ) ,A max/width/ 2 , ’ g ∗ ’ )

% P l o t f o r lo o p f o r cha r l i n e sf o r i = 1 : num

f i g u r e (1 )h ol d on ;p l o t ( c h a r ( i , : , 1 ) , c h a r ( i , : , 2 ) )h ol d on ;

pl o t ( c har ( i , : , 1) ,− c ha r ( i , : , 2 ) )en d

% P lo t th e n o zz l e shape ( bottom s i d e )f i g u r e (1 )subpl ot ( 2 , 1 , 1)h ol d on ;pl o t ( noz ( : , 1) , − noz ( : ,2 ) , ’k ’ , ’ LineWidth ’ , 3 )h ol d on ;p l o t ( a ( 1 ) ,−A max/width /2 , ’ g ∗ ’ )t i t l e ( ’ Max T hrust ( minimum l e ng t h ) N oz z l e De si gn ’ )

x l a b e l ( ’ N o z z le l e n g t h (m) ’ )y l a b e l ( ’ N o z z le h e i g h t (m) ’ )l e g e n d ( ’ N o z z le s ha pe ’ , ’ A r e a e x i t ( p r e d i c t e d ) ’ , ’ Char . L i n es ’ )e l s een d

% Find % e r r or s i n A/A∗ and Mexite r r o r A r e a = 1 0 0∗(width ∗2∗noz(num,2) − A max)/(A max)

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e r r o r M a c h = 1 0 0∗( M e − M ex(num,num))/M e

% P lo t Mach Number and p r e s s u r e t hr ou gh n o z z l e u s in g t he q ua si−1D% a re a r e l a t i o n s . ( I s e n t ro p i c ex pa ns io n thro ugh n o zz le )

Mnoz ( 1 ) = 1 . 0 ; % Choked FlowM = Mnoz ( 1 ) ;f o r i =1 : s i z e ( n oz , 1 )

Ae( i ) = 2∗noz ( i , 2) ∗ width ;A Asq = (Ae( i )/ A sta r )ˆ 2;A r at i o ( i )=s q r t ( A A sq ) ;

%Newton Rhapson on Eq. 5. 20 − Ander son t e x tr e s = 1 ;i f i > 1

M = Mnoz( i−

1);

w hi l e r e s > . 0 0 1M2 = M + dM1;f u n a 1 = −A Asq + (1/Mˆ2)∗ ((2/(gamma+1))∗(1+(gamma−1)∗Mˆ2/2))ˆ((gaf u n a 2 = −A Asq + (1/M2ˆ2) ∗ ((2/(gamma+1))∗(1+(gamma−1)∗M2ˆ2/2))ˆ(dv dm = ( f una2−funa1)/dM1;

M = M − funa1/dv dm;r e s = a bs ( f un a 1 ) ;

en dMnoz( i ) = M;en d% Fi nd P r e s s u r ePnoz( i ) = P c∗(1+(gamma−1)∗Mnoz( i )ˆ2 /2) ˆ(−gamma/(gamma−1) ) ;

en d

f i g u r e ( 1 ) ;subpl ot ( 2 , 1 , 2)pl ot (noz ( : , 1 ) , Mnoz, ’ r

∗’ )

h ol d on ;pl ot ( noz ( : ,1 ) , Pnoz/P amb , ’b ∗ ’ )h ol d on ;pl o t ( noz ( s i z e ( noz , 1 ) , 1 ) , M e , ’ go ’ )h ol d on ;pl o t ( noz ( s i z e ( noz , 1 ) , 1) , 1 , ’ go ’ )x l a b e l ( ’ N o z z le l e n g t h (m) ’ )yl ab el ( ’Mach number and P/P c ’ )

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leg end ( ’Mach Number ’ , ’P/P a m b ’ , ’ M e x i t ( pr ed ic te d ) ’ , ’ P a m b/P a m b

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APPENDIX V

CAD drawing of the Geometry

Figure 27: CAD Drawing

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APPENDIX VI

Final Geometry of the Vehicle

Figure 28: CAD Drawing

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APPENDIX VII

Sensitivity Studies

Figure 29: Variation with Range

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variation with payload weight

Figure 30: Variation with Payload Weight

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variation with geometrical parameters

Figure 31: Variation with Geometry

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APPENDIX VIII

Vehicle Geometrical Specifications are

Length(m) Breadth(m) Width(m) Structural Volume(m3) Wetted Area(m2) Total Volume(m3)80 5.7 6.23 1129 1177 1628

Table 7: Vehicle Geometry Parameters (in SI Units)

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APPENDIX IX

Displacement of Structure under Bending loads

Figure 32: Displacement of Body

Hypersonic Vehicle Design For Mach 5

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