CHAPTER-1

INTRODUCTION

Nozzle is a main component in Rocket that is used to convert the Chemical-Thermal Energy generated in the Combustion Chamber into Kinetic energy. The Nozzle converts the low velocity, High Pressure, High Temperature gas in the thrust Chamber into High Velocity gas of lower Pressure and temperature. De Laval Nozzle found that the most efficient conversion occurred when the nozzle first narrowed, increasing the speed of the jet to the speed of sound. Computational Fluid Dynamics (CFD) is an engineering tool that assists experimentation .Its scope is not limited to fluid dynamics.CFD could be applied to any process which involves transport phenomena with it. To solve an engineering problem we can make use of various methods like the analytical method Experimental methods using prototypes. The analytical method is very complicated and difficult. The experimental methods are very costly .If any errors in the design were detected during the prototype testing, another prototype is to be made clarifying all the errors and again tested.

A nozzle is a device that increases the velocity of a fluid at the expense of pressure. Nozzle is a part of rocket which is used for the expansion of combustion gases through it and produces thrust. Nozzle is a passage used to transform pressure energy into kinetic energy. During the combustion of fuel, chemical energy is converted into thermal energy and pressure energy. The combustion gases at this stage are at a high pressure and temperature and these gases under such high pressure expand through the nozzle during which the pressure energy is converted into kinetic energy which in turn moves the vehicle in a direction opposite to that of the exhaust gases, according to Newton’s third law of motion. Two primary functions of nozzle are – First, they must control the engine back pressure to provide the correct and optimum engine performance, which is done by jet area variations. Second, they must efficiently convert potential energy of the exhaust gas to kinetic energy by increasing the exit velocity, which is done by efficiently expanding the exhaust gases to the atmospheric pressure.

A new application of the CFD and optimization method is established in the present case for process development for optimization converging-diverging nozzle flow by expecting the highest level of accuracy. This work is divided into two parts. First, a validation and verification of the nozzle model using experimental and computational data is presented. Afterwards the paper deals with the optimization of nozzle contours for having the maximum thrust by using the numeric-cal model to be constructed. The nozzle optimization problem can be defined as the following: given a set of external parameters and geometric con-strains, find a nozzle wall contour such as the resulting thrust produced by the nozzle is maximal. The external parameters are: ambient pressure, temperature and chamber pressure, temperature. The geometric constraints are the nozzle length and throat diameter. Air as ideal gas has been used for the operational fluid.

Nozzle is used to convert the chemical-thermal energy generated in the combustion chamber into kinetic energy. The nozzle converts the low velocity, high pressure, high temperature gas in the combustion chamber into high velocity gas of lower pressure and temperature. Swedish engineer of French descent who, in trying to develop a more efficient steam engine, designed a turbine that was turned by jets of steam. The critical component – the one in which heat energy of the hot high-pressure steam from the boiler was converted into kinetic energy – was the nozzle from which the jet blew onto the wheel. De Laval found that the most efficient conversion occurred when the nozzle first narrowed, increasing the speed of the jet to the speed of sound, and then expanded again. Above the speed of sound (but not below it) this expansion caused a further increase in the speed of the jet and led to a very efficient conversion of heat energy to motion. The theory of air resistance was first proposed by Sir Isaac Newton in 1726. According to him, an aerodynamic force depends on the density and velocity of the fluid, and the shape and the size of the displacing object. Newton?s theory was soon followed by other theoretical solution of fluid motion problems. All these were restricted to flow under idealized conditions, i.e. air was assumed to posses constant density and to move in response to pressure and inertia. Nowadays steam turbines are the preferred power source of electric power stations and large ships, although they usually have a different design-to make best use of the fast steam jet, de Laval?s turbine had to run at an impractically high speed. But for rockets the de Laval nozzle was just what was needed.

Computational Fluid Dynamics (CFD) is an engineering tool that assists experimentation. Its scope is not limited to fluid dynamics; CFD could be applied to any process which involves transport phenomena with it. To solve an engineering problem we can make use of various methods like the analytical method, experimental methods using prototypes. The analytical method is very complicated and difficult. The experimental methods are very costly. If any errors in the design were detected during the prototype testing, another prototype is to be made clarifying all the errors and again tested. This is a time-consuming as well as a cost-consuming process. The introduction of Computational Fluid Dynamics has overcome this difficulty as well as revolutionised the field of engineering. In CFD a problem is simulated in software and the transport equations associated with the problem is mathematically solved with computer assistance. Thus we would be able to predict the results of a problem before experimentation. The current work aims at determining an optimum divergent angle for the nozzle which would give the maximum outlet velocity and meet the thrust requirements. Flow instabilities might be created inside the nozzle due to the formation if shocks which reduce the exit mach number as well as thrust of the engine. This could be eliminated by varying the divergent angle. Here analysis has been conducted on nozzles with divergent angles Experimentation using the prototypes of each divergent angle is a costly as well as a time consuming process. CFD proves to be an efficient tool to overcome these limitations. Here in this work the trend of various flow parameters are also analysed.

Computational Fluid Dynamics (CFD) is an engineering tool that assists experimentation. Its scope is not limited to fluid dynamics; CFD could be applied to any process which involves transport phenomena with it. To solve an engineering problem we can make use of various methods like the analytical method, experimental methods using prototypes. The analytical method is very complicated and difficult. The experimental methods are very costly. If any errors in the design were detected during the prototype testing, another prototype is to be made clarifying all the errors and again tested. This is a time-consuming as well as a cost-consuming process. The introduction of Computational Fluid Dynamics has overcome this difficulty as well as revolutionized the field of engineering. In CFD a problem is simulated in software and the transport equations associated with the problem is mathematically solved with computer assistance. Thus we would be able to predict the results of a problem before experimentation. The current work aims at determining an optimum divergent angle for the nozzle which would give the maximum outlet velocity and meet the thrust requirements. Flow instabilities might be created inside the nozzle due to the formation if shocks which reduce the exit mach number as well as thrust of the engine. This could be eliminated by varying the divergent angle. Here analysis has been conducted on nozzles with divergent angles Experimentation using the prototypes of each divergent angle is a costly as well as a time consuming process. CFD proves to be an efficient tool to overcome these limitations. Here in this work the trend of various flow parameters are also analyzed.

The modern research considering computational fluid dynamics is that it involves software testing and no prototype is need to build during designing stages and hence solution can be obtained faster and at less cost. Computational Fluid Dynamics become a popular tool for solving various problems and the physical aspects are governed by three aspects.

• Mass is conserved.

• Newton’s Second law is observed.

• Energy is conserved.

These factors are expressed in terms of the equation which is either integrals or differential equations. CFD is the art of study of replacing theses integrals or differential equations in terms of discredited algebraic forms which in turn are solved to obtain a number for flow field’s values at the discrete point in time or space. The final product of the study of CFD is a collection of the numbers in contrast to closed form analytic solution which is applicable in the practical solution. Flows and related phenomenon can be described by the partial differential equation, which cannot be solved analytically except in some special cases. To obtain the approximate solution, we have to use a Discretization technique which approximated the differential equation that can be later be solved by computer. The present work goes for deciding an ideal focalized point approached rocket engine nozzle used in a rocket Engine. The acceleration of the combustion gases from exit nozzle at hypersonic velocities is considered by COMSOL software. This CFD specialized beta CAE software system presents in detail all the steps taken to read a CAD file of a geometry simplification, in order to create a good quality shell mesh, Improve the mesh by manually optimizing the shape of the Macro Area and Disparate edge and throat span of the nozzle, which would give the most extreme outlet acoustic flow speed and meet the push prerequisites. This could be disposed of by various the divergent edges. Here investigation has been directed on the nozzle with disparate points with angles.

The nozzle is used to convert the chemical-thermal energy generated in the combustion chamber into kinetic energy. The nozzle converts the low velocity, high pressure, high temperature gas in the combustion chamber into high velocity gas of lower pressure and temperature.

The inlet Mach number is less than one, Convergent section accelerates it to sonic velocity at the throat and further accelerated to supersonic velocities by the diverging section.

Figure 1 – Convergent-divergent nozzle

In this project the designing and analysis of CD nozzle geometry is done in the CFD (Computational Fluid Dynamics software). Firstly the design of nozzle is made in Gambit software and then the nozzle geometry is further analyzed in fluent software in order to analyze the flow inside the CD nozzle and to get the view of the behavior of fluid inside the convergent-divergent section of nozzle.

A multidisciplinary analytic model of a aerospike rocket nozzle has been developed, this model includes predictions of nozzle thrust, nozzle weight, and effective vehicle gross-liftoff weight (GLOW). The linear aerospike rocket engine is the propulsion system proposed for the X-

2 The model has been developed to demonstrate multidisciplinary design optimization (MDO) capabilities for relevant engine concepts, assess performance of various MDO approaches, and provide a guide for future application development1.

Fluid–structure interaction (FSI) is the interaction of some movable or deformable structure with an internal or surrounding fluid flow. Fluid–structure interaction problems and multiphysics problems in general are often too complex to solve analytically and so they have to be analyzed by means of experiments or numerical simulation. Research in the fields of computational fluid dynamics and computational structural dynamics is still ongoing but the maturity of these fields enables numerical simulation of fluid-structure interaction. There are Two main approaches exist for the simulation of fluid– structure interaction problems:

II. Monolithic approach: In monolithic approach consisting the equations, governing the flow and the displacement of the structure are solved simultaneously, with a single solver. This approach becomes complicated but requires less time.

JJ. Partitioned approach: the equations governing the flow and the displacement of the structure are solved separately, with two distinct solvers. We shall used partitioned approach where fluid dynamic equations are solved using separate solver i.e. CFD and structural equations are solved using Finite element method in ANSYS. For industrial designing purpose this methodology is more reliable than the monolithic approach as design criteria for flow design and structural design is satisfied individually but it takes more time.

There are many parameters including rocket, such as thrust chamber assemble, thrust chamber injector, oxidizer zone, Gimbal bearing, thrust chamber body, Thrust chamber nozzle extension, nozzle, hypergol cartridge, etc lots of parameters are included there. But till we will study on aerospike nozzle, because our aim is to getting higher thrust. Therefore we had selected nozzle as the main parameter for this work. The thrust chamber nozzle extension increases the expansion ratio of the thrust chamber from 10:1 to 16:1. It is detachable unit that is bolted to the exit end ring of the thrust chamber. The interior of the nozzle extension is protected from the engine exhaust gas environmental by film cooling, using turbine exhaust gases as the coolant. The gases enter the

1.2. Project Objective

The objectives of this project are:

1. To obtain the optimized throat radius for nine different convergent and divergent angles in a rocket engine nozzle using CFD.

2. To get the optimal Mach number of nozzle by applying the optimized throat radius for two different static pressures using CFD.

3. To compare the optimal Mach number with numerically and experimentally.

CHAPTER-2

LITERATURE SURVEY

A convergent-divergent nozzle is designed for attaining speeds that are greater than speed of sound. The design of this nozzle is obtained from the area-velocity relation ( dA / dV ) = -(A/V)(1-M^2) where M is the Mach number (which means the ratio of local speed of flow to the local speed of sound) A is area and V is velocity.

Fig 2.1: Area Velocity Relation

From the above fig we can observe that

• The decrease in Area results in the increase of pressure and decrease in velicity as seen in the above figure at the entry of the nozzle.

• The increase in area results in increasing the velocity at the exit of the nozzle by decreasing the pressure.

2 M1 results in supersonic speeds.

One important point is that to attain supersonic speeds there is a need to maintain favorable pressure ratios across the nozzle.

Presented in this section are the previous works done in supersonic nozzle design pertinent to the current investigation. Since the nozzles designed in this paper are for irrotational, inviscid, isentropic flows, only previous works dealing with these types of nozzles will be discussed. The first part of this section will deal with annular nozzles and the second part will deal with the previous work done on the nozzles.

Fig 2.2 Rocket Nozzle Profiles

Arjun Kundu, Devyanshu Prasad and Sarfraj Ahmed 1 worked on the topic of “Effect of Exit Diameter on the Performance of Converging-Diverging Annular Nozzle Using CFD” and there findings are – The result obtained after the CFD analysis shows that smaller exit diameter gives greater mach number compared to the larger diameter for the same inlet and boundary conditions. K.M. Pandey, Surendra Yadav and A.P. Singh 2 worked on the topic of “Study on Rocket Nozzles with Combustion Chamber Using Fluent Software at Mach 2.1” and his findings are – The pressure and Temperature parameter depends upon air-fuel ratio. Mohan Kumar G, Dominic Xavier Fernando and R. Muthu Kumar 3 worked on the topic of “Design and Optimization of De Laval Nozzle to Prevent Shock Induced Flow Separation” and there findings are – For maximum thrust and efficiency without flow separation due to induced shock, the direction of flow of stream through nozzle should be axial. Venkatesh V, C Jaya pal Reddy 4 worked on the topic of “Modelling and Simulation of Supersonic nozzle using Computational Fluid Dynamics” and there findings are – Contour nozzle gives a greater mach number at exit compared to conical nozzle because contour nozzle gives maximum expansion ratio.

Aerospyke is a type of flow induced nozzle with capacity of continuous altitude compensation. Aerospike nozzle is considered to have better performance at off- design altitudes compared with that of the conventional bell-shaped nozzle since its plume is open to the atmosphere outside and free to adjust, allowing the engine to operate at its optimum expansion at all altitudes. Various experimental, analytic, and numerical research on plug nozzles have been performed. In contrast to the previously conventional nozzle concepts, plug nozzles provide, at least theoretically, a continuous altitude adaptation up to their geometrical area ratio.

Prosun Roy 01 The paper also addresses static pressure optimization and Mach number optimization. The values on the basis of results along by optimal values of nozzle design parameters are obtained from optimization techniques of Taguchi Design. Convergent angle, Divergent angle and Throat radius are considered. Also response of static pressure and Mach number values of CFD analysis in two types of inlet pressure value applied for optimal parameters of nozzle attained.

P.Parthiban 02 Reported two different design approaches for circular plug nozzles, which differ only in the chamber and primary nozzle layout. The designing of conical central body with more sophisticated contouring methods are also established .

P.BijuKuttan ; M.Sajesh 03 carried out different design approaches include plug nozzles with a toroidal chamber and throat (with and without truncation) and plug nozzles with a cluster of circular bell nozzle modules or with clustered quasi rectangular nozzle modules. The latter approach seems to be advantageous because further losses induced by the gaps between individual modules and the flow field interactions downstream of the module exits can be minimized. It has been shown that transition from a round to a square nozzle results in a very small performance loss.

Varun.R 04 reported the details of experimental analysis carried out at DLR facility, Germany. A truncated toroidal sub-scale plug nozzle is used for sea-level hot-run test.

Pandey.K.M 05 studied performance analysis on different types of plug nozzles such are linear plug, circular plug and clustered plug nozzle design. The comparison of experimental results with numerical computations is also carried out. The flow field of the toroidal plug nozzle was calculated with a numerical method and reported the Mach number distribution in the combustion chamber and nozzle. Principal physical processes like expansion waves, shocks, and the recirculation base-flow region are in good agreement with experimental data. Both the experiment and the numerical simulation show that the flow separates from the conical plug body before reaching its truncated end.

Natta Et. al., 06 conducted an experimental evaluation of plug nozzle flow field and reported the parameter distribution and performance parameters with emphasizing the separation of the flow from the central plug body for conical contours. The principal flow field developments predicted by these numerical simulations are again in a good agreement with experimental data. Within the frame of this ESA ARPT Program performance and flow behaviour of clustered plug nozzles at different truncations are being examined by European industries and research institutes (K.M. Pandey) with sub-scale cold-flow plug models.

CHAPTER-3

STUDY

The most popular altitude-compensating rocket nozzle to date is the aerospike nozzle, the origin of which dates back to Rocket dyne in the 1950s. This type of nozzle was designed to allow for better overall performance than conventional nozzle designs.

3.1. Nozzle

A nozzle is a mechanical device of varying cross section which controls the direction and characteristics of the fluid (Air or Water) flowing through it. They are used in rocket engines to expand and accelerate the combustion gases, from burning propellants, so that the exhaust gases exit the nozzle at supersonic or hypersonic velocities.

When the fluid flows through the nozzle it exits at a higher velocity than its inlet velocity. This phenomenon occurs due to conservation of mass which states that the rate of change of mass equals to the product of density, area and velocity. m = *A*V m = mass flow rate A = area of flow V = velocity of flow Solving this equation using differentiation, we get the equation,

3.2 NEED FOR NEW DESIGN

The revolution in aerospace propulsion was increased greatly during World War 2. Faster, bigger and more efficient aerospace vehicles were required which led to the birth of Space research organizations like NASA. Speaking about the future, advanced rocket propulsion systems will require exhaust nozzles that perform efficiently over a wide range of ambient operating conditions. Most nozzles either lack this altitude compensating effect or they are extremely difficult to manufacture. In present day, only bell nozzles are used for launch activities. But, these bell nozzles have a major drawback of decreasing in efficiency as altitude increases. This is due to a phenomenon causing loss of thrust in the nozzle at higher altitudes called as separation of the combustion gases. For conventional bell nozzles, loss mechanisms fall into three categories:

Geometric loss results when a portion of the nozzle exit flow is directed away from the nozzle axis, resulting in a radial component of momentum. In an ideal nozzle, the exit flow is completely parallel to the nozzle axis and possesses uniform pressure and Mach number. By calculating the momentum of the actual nozzle exit flow and comparing it to the parallel, uniform flow condition, the geometric efficiency is determined. By careful shaping of the nozzle wall, relatively high geometric efficiency can be realized. A drag force, produced at the nozzle wall by the effects of a viscous high-speed flow, acts opposite to the direction of thrust, and therefore results in a decrease in nozzle efficiency. The drag force is obtained by calculation of the momentum deficit in the wall boundary layer. This viscous drag efficiency is defined as:

The third nozzle loss mechanism is due to finite-rate chemical kinetics. Ideally, the engine exhaust gas reaches chemical equilibrium at any point in the nozzle flow field, instantaneously adjusting to each new temperature and pressure condition. In real terms, however, the rapidly accelerating nozzle flow does not permit time for the gas to reach full chemical equilibrium.

The overall nozzle efficiency is then given by the combined effects of geometric loss, viscous drag and chemical kinetics: ?kin= 0.99(approximate).

A long nozzle is needed to maximize the geometric efficiency; but simultaneously, nozzle drag is reduced if the nozzle is shortened. If chemical kinetics is an issue, then the acceleration of exhaust gases at the nozzle throat should be slowed by increasing the radius of curvature applied to the design of the throat region. The optimum nozzle contour is a design compromise that results in maximum overall nozzle efficiency. Nozzle contours can also be designed for reasons other than for maximum thrust. Contours can be tailored to yield certain desired pressures or pressure gradients to minimize flow separation at sea level. A nozzle contour designed to produce parallel, uniform exit flow, thereby yielding 100 % geometric nozzle efficiency, is called an ideal nozzle

This ideal nozzle is extremely long and the high viscous drag and nozzle weight that results are unacceptable. Some design approaches consider truncating ideal nozzles keeping in mind the weight considerations. Most companies have a parabolic curve-fit program, generally used to approximate Rao optimum contours, which can also be used to generate desired nozzle wall pressures. For nozzles at higher altitudes, vacuum performance is the overriding factor relating to mission performance and high nozzle area ratio is therefore desirable. However, nozzle over-expansion at sea level does result in a thrust loss because the wall pressure near the nozzle exit is below ambient pressure. If the nozzles exit area could be reduced for launch and then gradually increased during ascent, overall mission performance would be improved. The ideal rocket engine would make use of a variable-geometry nozzle that adjusted contour, area ratio and length to match the varying altitude conditions encountered during ascent. This feature is referred to as Altitude Compensation.

3.3. Nozzle

The aerospike nozzle is a bell nozzle with its nozzle profile turned inside out. Flow of combustion gases is directed radially inward towards the nozzle axis. In the annular aerospike nozzle, flow issues from an annulus at a diameter located some radial distance from the nozzle axis. Flow is directed radially inward toward the nozzle axis. This concept is the opposite of a bell nozzle which expands the flow away from the axis along diverging nozzle walls. In an aerospike, the nozzle expansion process originates at a point on the outer edge of the annulus which is referred to as the “cowl-lip.” In a standard bell nozzle, flow expansion continues regardless of what the ambient pressure is, and the flow can continue to over-expand until it separates from the nozzle walls. The linear aerospike, spike consists of a tapered wedge-shaped plate, with exhaust exiting on either side at the “thick” end.

3.4. Working Principle of nozzle:

The function of a rocket nozzle is to direct all gases, generated in the combustion chamber of the engine and accelerated by the throat, out of the nozzle. The key feature of the aerospike engine is that, as the launch vehicle ascends during its trajectory, the decreasing ambient pressure allows the effective nozzle area ratio of the engine to increase. An aerospike nozzle is often referred to as an altitude-compensating nozzle, because of its specific design capability of maintaining aerodynamic efficiency as altitude increases and thus throughout the entire trajectory. At the outer cowl lip, the gas expands to the atmospheric pressure immediately, and then causes serious expansion waves propagating inward at an angle through the gas stream. At the location where the last expansion wave intercepts the spike, the gas pressure is equal to the atmospheric pressure. For the over expanded case, the spike changes the gas to be directed outward, and thus compression waves form and propagate outward at an angle and reflect off the jet boundary as expansion waves. This process then begins again. The aerospike features a series of small combustion chambers along the ramp that shoot hot gases along the ramp’s outside surface to produce thrust in a spike-shaped plume, hence the name “aerospike.”

Figure 3.1 Model of aerospike with flow field

The ramp serves as the inner wall of the bell nozzle, while atmospheric pressure serves as the “invisible” outer wall. The combustion gases race along the inner wall (the ramp) and the outer wall (atmospheric pressure) to produce the thrust force.

3.5. Flow around Nozzle

The main advantage to the annular aerospike nozzle design (both full length and truncated spike) is its altitude compensation ability below or at its design altitude. More specifically, the aerospike will not suffer from the same overexpansion losses a bell nozzle suffers and can operate near optimally, giving the highest possible performance at every altitude up to its design altitude. Above the design altitude, the aerospike behaves much like a conventional bell nozzle. Figure below shows the exhaust flow along an aerospike at low altitudes, design altitude, and high altitudes for a full spike and a truncated spike. Multiple expansion and compression, or shock, waves are evident in the flow in Figure these waves lead to losses in thrust. The outer flow boundary of the aerospike is the atmosphere itself. Unique to aerospike engines operating at their design altitude, engine geometry at the throat determines the expansion ratio of the aerospike nozzle and thus the corresponding engine performance.

Figure 3.2 Exhaust Flow from a Full and Truncated Spike

At the design altitude of the nozzle, the exhaust flow at the chamber exit lip will follow a parallel path to the centreline to the exit plane. Therefore, the expansion ratio for a full-length spike at design altitude is equivalent to the chamber exit lip area divided by the throat area. As the ambient pressure decreases, the hot gas/ambient air boundary expands outward changing the pressure distribution along the spike; as a result, the expansion ratio increases. As the ambient pressure increases (low altitudes), the higher ambient pressure compresses the hot gas/ambient air boundary closer to the spike resulting in an expansion ratio decrease. The pressure distribution change along the spike and the location of the hot gas/ambient air boundary is automatic thus permitting altitude compensation up to the design altitude of the nozzle. Above the design altitude of the nozzle, the pressure distribution along the nozzle wall is constant. The expansion of the flow exiting the combustion chamber is governed by the Prandtl-Meyer turning angle at the throat.

Figure 3.3 Exhaust Flow along a Truncated Aerospike Nozzle

According to the aerospike nozzle numerical analyses the results of the altitude compensation capabilities of an aerospike up to the design altitude are undeniable. Furthermore, the aerospike performs worse at high altitudes compared to bell nozzles with equal expansion ratios (exit area divided by throat area); therefore, to get the benefit of the aerospike, the design pressure ratio and the expansion ratio should be chosen as high as possible. The design pressure ratio is the ratio of the chamber pressure to the ambient pressure; ambient pressure is based on the chosen design altitude. If the spike is truncated, the aerospike advantage at higher altitudes (orbit transfer missions) includes shortened nozzle length and lower mass as compared to an equivalent performance bell nozzle design for orbit transfer missions.

3.6. Advantages and Disadvantages of Using an Nozzle:

The aerospike nozzle has 90% overall better performance than the conventional bell shaped nozzle. The efficiency at low altitudes is much higher because the atmospheric pressure restricts the expansion of the exhaust gas. A vehicle using an aerospike nozzle also saves 25-30% more fuel at low altitudes. At high altitudes, the aerospike nozzle is able to expand the engine exhaust to a larger effective nozzle area ratio. An aerospike nozzle with an expansion ratio of 200:1 to 300:1 can increase the thrust and specific impulse by five to six percent. Specific impulse is the total impulse per unit weight of propellant. As of now, the most widely used nozzle type is the bell-shaped nozzle. It has a high-angled expansion section, usually 20-50°, right behind the nozzle throat, which is then followed by a gradual reversal of nozzle contour slope so that the nozzle exit divergence angle is small, usually less than a ten-degree half angle. The drawback to using this type of nozzle is that its optimum design is for a specific altitude, and thus is better suited for multi-stage to orbit. The aerospike design is suitable for Single Stage to Orbit(SSTO)flight. There are rarely some disadvantages on using an aerospike nozzle.

Figure 3.4 Thrust coefficient between under and over expansion flow conditions

The after body induces heat, and to cool means that the performance reduces along with the pressure against the nozzle. Another issue is weight, which as previously stated can be resolved through truncation. During flight a transonic and supersonic regime, generally between Mach 1 and 3, the slipstream effect reduces the aerospike nozzle’s performance due to the external flow over the vehicle because the plume tends to draw in the air flow and thus alters the aerodynamics at the aft end of the vehicle. Finally, the performance is more difficult to evaluate because of the complex flow field and the turbulence involved.

CHAPTER-4

MATHEMATICAL MODEL

The mathematical model selected for this work is the standard K-? model which is one of the Reynolds Averaged Navier-Stoke(RANS) model available in fluent. The standard K-? model is the most widely used transport model. The standard K-? model is a two-equation model and the two model equations are as follows:

The model equation for the turbulent kinetic energy K is:

(1) = Rate of increase of K+ Convective transport= diffusive transport + Rate of production-Rate of destruction

The model equation for the turbulent dissipation ? is:

(2)

=Rate of increase of ?+ Convective transport= diffusive transport + Rate of production-Rate of destruction

The standard values of all the model constants as fitted with benchmark experiments are (Launder and

Sharma, Letters in Heat and mass transport, 1974, 131-138):

C? = 0.09 ; ?k=1.00 ; ??=1.30 ; C?1=1.44 ; C?2=1.92

Now the Reynolds stresses are found out using:

(3)

And the eddy-viscosity is evaluated as:

(4)

The major advantages of this model are that it is relatively simple to implement, it leads to stable calculations, and it is a widely validated turbulence model. The known limitation of this model is that its performance is very poor for flows with strong separation, large streamline curvature and high swirling components. Despite of all these limitations, the model is widely accepted model for initial level screening of alternate designs in compressible flows, combustion engineering etc.

CHAPTER-5

RESEARCH METHODOLOGY

The methodology of this project starts with the past literature with relevant journals to identify the exact problem identification in the muffler area. Problem identification has been initiated with the past journals study to minimize the undesired effects and maximize the performance. So, based on the literature we have found out the back pressure as a key to improving the performance with their different operating pressures. Next level moves to frame an objective of this project based on the problem identification, the objective has been framed as to improve the performance by varying the angles. shows the various design parameters such as length, height, width, mean and with the help of these different design values the full model has been created by using COMSOL Multiphysics software. Next step is to initiate the trial runs to get the exact results (optimized results) before the modelling of the full model. After the successful trial run, the whole Nozzle model has been created by using the COMSOL Multiphysics software which is called generative design. After the creation of generative design different boundary conditions have been assigned i.e. (inlet, outlet, etc.) and also the different operating parameters are assigned. After the successful application of boundary conditions, next level moves to assign the different Noise characterization such as surface acoustic with respect to static pressure. Based on the numerical results and colour plots, different discussions have been made to analyse the Nozzle performance in different angles. Finally this project ends with the conclusion part based on the numerical results Vs Experimental analysis which are obtained from the COMSOL Multiphysics software with respect to different static pressures. If the possible outcome occurs this project tends to go with future work otherwise again it will be refined with different operating and design parameters as flow process

Input parameters for this work are basically Nozzle geometry (contours), weight, thickness of the contours, various constrains like Area ratio, Length of Nozzle. And output is nozzle profile. Generally after the study related to this topic gives the basic shape of nozzle which is parabolic. And we will do study on parabolic contour nozzle by taking the parabola is the basic shape and finding all the results related to it. After that will change the contour shape in different manner’s to comparing study between it. Those different nozzle contour shapes are cubical contour profile, semi cubical parabolic profile. After that finding best results from these.

For analysis in CFD it requires some input for determining the respected results, so this input getting by following equation. For parabolic profile: y = -0.2088×2 + 0.46865. in this x is the total nozzle length. If x varies automatically y will get varied, therefore getting the different wall points of parabolic profile for CFD analysis. Then by taking this wall point it gets nozzle shape profile and then analyzed it which will give the results in terms of pressure. Same procedure is followed for other two profiles i.e cubical profile and semi cubical profile.

For cubical contour profile, equation is y= -(0.208e-4)x3+ 0.46865. and for semi cubical parabolic equation is y3 = – (0.0667) x2= 0.10293.Pressure coming from this software is then validated with numerical solution. Generally pressure measurements are presented in two form:

• As a pressure ratio

• in terms of a pressure coefficient(Cp)

Cp=

5 – Pressure of exhaust gas coming from nozzle P? – Ambient Pressure

Cp – Pressure Co-efficient

? – Density of exhaust gases coming from nozzle Vf – Flow velocity of exhaust gases

Vsound – Velocity of sound

5.1. RESERCH APPROACH:

Rapid progress in aerospace technologies requires a constant interplay between analysis, ground testing, and flight testing activities. All three of these activities are equally important. Testing, both on the ground and in flight, keeps analysis well-grounded and relevant, while analysis provides the foundation and explains flight and ground test results 3. In this project I shall draw the drawings of nozzle shape in CFD and then we will got the different results in terms of fluid analysis, but here we shall concentrate on pressure only for further work.

5.2. METHODOLOGY:

Input parameters for this work are basically Nozzle geometry (contours), weight, thickness of the contours, various constrains like Area ratio, Length of Nozzle. And output is nozzle profile. Generally after the study related to this topic gives the basic shape of nozzle which is parabolic. And we will do study on parabolic contour nozzle by taking the parabola is the basic shape and finding all the results related to it. After that will change the contour shape in different manner’s to comparing study between it. Those different nozzle contour shapes are cubical contour profile, semi cubical parabolic profile. After that finding best results from these.

For analysis in CFD it requires some input for determining the respected results, so this input getting by following equation. For parabolic profile: y = -0.2088×2 + 0.46865. in this x is the total nozzle length. If x varies automatically y will get varied, therefore getting the different wall points of parabolic profile for CFD analysis. Then by taking this wall point it gets nozzle shape profile and then analyzed it which will give the results in terms of pressure. Same procedure is followed for other two profiles i.e cubical profile and semi cubical profile.

For cubical contour profile, equation is y= -(0.208e-4)x3+ 0.46865. and for semi cubical parabolic equation is y3 = – (0.0667) x2= 0.10293.Pressure coming from this software is then validated with numerical solution. Generally pressure measurements are presented in two form:

• As a pressure ratio

• in terms of a pressure coefficient(Cp)

Cp=

Pressure of exhaust gas coming from nozzle P? – Ambient Pressure

Cp – Pressure Co-efficient

? – Density of exhaust gases coming from nozzle Vf – Flow velocity of exhaust gases

Vsound – Velocity of sound

CHAPTER-6

DESIGN METHODOLOGY

After getting familiarized with the concepts of the nozzle, let us now get into detail of the design procedure. Therefore this chapter gives a main focus on the design procedure of the different kinds of nozzles. This chapter relates to the application of the above mentioned thermodynamic relations and the parameters required to design nozzle. It mainly consists of designing of a Conical and Contour nozzle.

6.1 Design of complete Nozzle:

Supersonic nozzles are generally specified in terms of the cross sectional area of final uniform flow A and the final mach number M. The nozzle-throat area is obtained by the 1D flow equation, the shortest nozzles that may be designed by the method of reported are those without a straight-walled section. The straightening part immediately follows with the expanding part. The purpose of method of characteristics is to illustrate the design of a supersonic nozzle by the method of computation with the weak waves.

Consider a supersonic nozzle as shown in figure. The subsonic flow in the convergent portion of the nozzle is accelerated to sonic speed at the throat. Generally, because of the multi-dimensionally of the converging subsonic flow, the sonic line is gently curved. We assume the sonic line to be straight at the throat in most of the applications. In the divergent portion downstream of the throat, let ?w be the angle at any point on the duct wall. The portion of the nozzle with increasing ?w is called the expansion section, where expansion waves are generated and propagate in the downstream direction, reflecting from the opposite wall. At a particular point where the ?w is maximum, there is an inflection of the duct in the wall contour.

6.2 Modeling

The Geometry of the nozzle was created using NX-UG

Model-1

MODEL-2

Drafting-1

DRAFTING-2

CHAPTER-7

COMPUTATIONAL PROCEDURE

The finite volume solver, FLUENT 6.3, is used to obtain the numerical solution of the two-dimensional compressible Reynolds averaged Navier–Stokes (RANS) equations in connection with six turbulence models for closure of the RANS equations. The discretized equations, along with the initial condition and boundary conditions, were solved using the segregated solution method. Using the segregated solver, the conservation of mass and momentum were solved sequentially and a pressure correction equation was used to ensure the conservation of momentum and the conservation of mass (continuity equation). Several turbulence models, i.e. the standard k–e model, the extended k–e model, shear-stress transport k–x model, RSM, v2– f and the realizable v2–f model are tested. The extended k–e model differs from the standard k–e model in its constant and it has an additional source term in the e equation. This model was implemented in the code by adjusting the standard k–e model constants and by defining the additional source term using User Defined Functions (UDF). Moreover, the four equation turbulence v2–f models are implemented using User Defined Scalars (UDS) along with UDF.

The governing equation consist of the continuity equation and Reynolds-average governing equations for steady compressible turbulent flow coupled with the equation of state, p=?RT. The system of the governing equations can be described as follows:

The continuity equation:

RANS equations:

Energy equations:

5 Nomenclature:

U – Mean quantities

u´- Fluctuating or turbulence quantities

µ- Viscosity

8 – Pressure of exhaust gas coming from nozzle P? – Ambient Pressure

Cp – Pressure Co-efficient

? – Density of exhaust gases coming from nozzle Vf – Flow velocity of exhaust gases

Vsound – Velocity of sound R – Universal gas constant

M – Molecular weight of exhaust gas

The additional fluctuation quantities are the unknown turbulent or Reynolds-stress tensors, while ui´ represents the velocity fluctuations in i-direction.

7.1 Meshing In ANSYS Workbench

After the modelling is completed the meshing is to be done. The module used to perform meshing is Fluid Flow (Fluent). The meshing method used here is Automatic Method and the mesh type is selected as All Quad. The overning equations used in mesh are as follows: It is assumed that there is a unique, single valued relationship between the generalized co-ordinates and the physical co-ordinates which can be expressed as

,

this also implies that,

The functional coordinates are determined by the mesh generation process.Given these functional relationships, the governing equations are transformed into corresponding equations containing partial derivatives with respect to the parametric space.For example

The inverse transformation can be written as follows:

The Poisson Equation that is solved is of the form as in the following equations :

Where P and Q are predefined functions that are used to control grid clustering. Here in this project Meshing plays a main role, since we are obtaining results by varying the Number of divisions in mesh .The number of divisions are varied at the vertical surfaces (inlet and exit) and the inclined surfaces (walls).

7.2. Meshing Pattern

Mesh generation also called as a „grid generation? is an important part of CFD. Meshing for CFD is pivotal in achieving realistic renderings and physical simulations. Realistic rendering and high precision simulations depend on the quality of meshing for CFD.

The mesh obtained initially will be unstructured mesh and this cannot be used to obtain accurate results. Since the edges are prismatic the mesh can be converted into structured meshing by using Mapped Face Meshing. The analysis is done for five types of meshes which are obtained by varying the number of divisions in mesh. The variation of the number of divisions is done on the inlet, exit and on the walls of the nozzle. The following is the nomenclature that is followed to mention the Number of divisions.

Model-1

Model-2

7.3. Boundary Conditions

Pressure inlet

Outlet

Walls

Specification of the boundary zones has to be done in WORKBENCH only, as there is no possibility to specify the boundary zones in FLUENT. Therefore proper care has to be taken while defining the boundary conditions in WORKBENCH. With all the zones defined properly the mesh is exported to the solver. The solver used in this problem is ANSYS FLUENT. The exported mesh file is read in Fluent for solving the problem.

Design-1(Inlet)

Design-1(Outlet)

Designe-2(Inlet)

Design-2(Outlet)

7.4 Solving

FLUENT analysis is carried out on nozzle at different meshing conditions.

7.5 Analysis Procedure

The same procedure is followed for all the 2 types of mesh and the results are validated.

PROCEDURE DETAILS

Problem Setup Type: Density based

General-Solver Velocity : Absolute

Time: Steady

2D space: Planar

Models Energy :On

Viscous: Laminar

Materials Fluid :Air

Density: Ideal Gas

Viscosity: Sutherland

Boundary conditions Inlet : Pressure Inlet

Gauge Total Pressure (pa): 3e5

Outlet : Pressure Out let

Gauge Pressure(pa): 0

Reference Values Compute from : Inlet

Reference Zone : Solid Surface body

Monitors Create-walls –cd 1

Select Print to console and plot.

Initialization Standard Initialization

Compute from Inlet.

Solution Solution Controls – Courant Number=5.

Run Calculation: Enter the Number of iterations,

Click calculation.

CHAPTER-8

RESULT

MODEL-1

Pressure

Velocity

Temperature

Stream Line-Velocity

MODEL-2

Pressure

Velocity

Temperature

Stream Line- Velocity

TABLE-1

Influence of variation in divergent angle on different flow property is shown in below tables.

NO DIVERGENT ANGLE VELOCITY PRESSURE TEMPREATURE

1 27 1.286e+003 1.314e+005 4.823e+002

2 26 1.154e+003 1.154e+001 4.501e+002

CHAPTER-9

CONCLUSION

Above results were found after conducting this investigation. It was observed that oblique shocks are formed during flow through the nozzle, when the divergent angle was 27°. It is observed that the shock is completely eliminated from the nozzle when divergent angle increase to 26.6° and this could be considered as a good design for the nozzle. At the exit section velocity magnitude is found to be increase with increment in divergent angle. Similarly, at throat section velocity magnitude goes on when divergent angle increased. The static pressure decrease with increasing divergent angle. The efficiency of supersonic rock engine nozzle increase as we increases divergent angle of nozzle up to certain limit.