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**Chapter 3 ****Formulation and Validation of Engine Analysis Code**

Delving into the simulation efforts of this work, this chapter discusses and catalogs the various analysis methods considered and the reasons for those selected. Additionally, the interaction of each system component is described and the overall scramjet simulation is compared against the data of previous studies.

**Engine Component Breakdown**

The work described here, as well as other studies, approaches the task of engine sim-ulation by breaking up the engine into general components and considering the interactions between them. Engine components considered in this work comprise of the isolator-duct, the combustor, and the expansion exhaust nozzle. Before moving forward with each component, it is important to describe the overall system arrangement and put forth the notation used for key locations in the system. For hypersonic vehicles, these key locations, or stations, are defined by the following diagram which uses a typical wave-rider geometry similar to that of the X-43 and X-51 flight vehicles [43].

Figure 3.1: Diagram of flow stations for an air-frame integrated scramjet [11] of the propulsion system in addition to the internal flows. Due to the inherent 2-D nature of external flows, they will be excluded from this analysis. This reduces the areas of anal-ysis from stations 0-10 down to stations 2-6 spanning the isolator compression system, the combustor, and the nozzle. The remaining portions of this section detail the methods used for each of these three components.

To assist in the development and validation of the flowpath analysis code, the geometry and supporting studies from the HIFiRE program are used. From the flowpath development process described in *Gruber 2008*, the HIFiRE simulation geometry that will be used can be seen in Fig. 3.2 and the coordinates for the listed points are found in Fig.

**Engine Component Breakdown**

This 2D geometry was selected as it has a cavity requiring the simulation of re-circulation regions, as well as multiple cases which to compare the model. The geometry also has secondary fuel injection points downstream of the cavity, however, those will not be considered in this work. The geometry is first discretized into a series of points that define its cross-section, as seen in Fig. 3.4

The points plotted in Fig. 3.4 will be those that flow-path data is recorded and are prescribed at the beginning of the analysis. It should be noted that the more points considered for analysis, the more computational time will be required.

**Inlet/Isolator**

As mentioned above, this work begins its analysis at the entrance to the isolator defined as station 2 in 3.1. These components typically consist of an internal flow compression component upstream of the combustor, and uses a train of shock waves to generate a pressure rise [1]. While these shock trains and pressure generation are critical to the operation of scramjets, it is important that the isolator duct is sized correctly to contain the shock train. In the case that the leading shock extends upstream of station 2, the inlet may unstart and lead to an engine flame-out [1].

The analysis of the isolator component is split into two parts: one that analyzes the flow upstream of the shock train, and one for the flow within the shock train. Upstream of the leading shock, the flow is modeled as a viscous, adiabatic supersonic internal flow, and is assumed to be attached. Additionally, it is assumed to be in chemical equilibrium reducing the computational requirements.

Before analyzing the flow of the isolator, it is necessary to determine the leading edge of the shock-train. To find this, the empirical relation from *Waltrup and Billig, 1973* is employed for circular ducts, and the modified versionnis used for rectangular ducts [44]. In Eqns. 3.1 and 3.2, *D* and *H* represent the duct diameter and duct height, respectively, while the subscripts 2*;* 3 represent the canonical stations represented in Fig. 3.1. While at the beginning of the analysis only *p*_{2} is known, through a procedure described in a following section it is possible to obtain an estimate of *p*_{3} for use with the shock train equations.

Upstream of the shock train, the flow is considered to be a viscous, adiabatic flow, meaning that there is an entropy gain due to viscous forces, but there is no change in the total enthalpy. With these assumptions, the differential conservation equations needed are Eqs. 2.20, 2.21, and 2.22, which can be re-arranged to yield the equations taken from *Hill and Peterson, 1992*. With these two equations as well as the adiabatic and chemical equilibrium assumptions, it is now possible to iterate the temperature and pressure at each axial location of interest and close upon a solution.

A single iteration solving for the *i ^{th}* axial location begins with guessing temperature and pressure values, typically close to those of the previous axial location, (

*i*1). These, along with the species mole fractions, are then used to determine the equilibrium composition using a nonlinear PDE solver such as Cantera [45]. With Cantera, the axial location is equilibrated at the prescribed temperature and pressure, yielding the density , molecular weight

*MW*, and more. Using the adiabatic assumption,which using equilibrium chemistry to solve for

*h*at the new axial location allows for Eq. 3.5 to be rearranged as Since the flow is adiabatic,

*h*

*is known from the previous axial location yielding the velocity at the*

_{t}*i*

*location. Using the newly calculated velocity and current values of temperature and pressure, it is now possible to calculate the derivatives in Eqns. 3.3 and 3.4 using finite difference methods. For this problem, a backwards difference method was applied in the form of.*

^{th}where

*f*represents any function of

*x*[46], which in this case will be flow parameters such as temperature.

With the derivatives calculated, it is now possible to calculate the values of Eqs. 3.3 and 3.4, forming a system of equations solvable by a nonlinear system of equations solver similar to

*fsolve*in MATLAB or Python’s Scipy library. This process is then iterated until the system is within the set tolerances before moving on to the next axial location. For the isolator, this process occurs until all of the axial locations have been calculated or until the system steps past the leading edge of the shock train whose length is determined with one of Eqs. 3.1 and 3.2. Due to the separation which occurs within the shock train, a different simulation method is employed.

**Shock Train Analysis**

While Eqns. 3.1 and 3.2 provide the shock leading edge, it does not provide the means to simulate the flow within the shock train. As shown by Fig. 3.5 the flow separates.

downstream of the shock resulting in a multi-stream flow providing a complex flow system. The complexities of the system make it difficult to capture the flow-field in a one-dimensional code, thus multiple relations both theoretical and empirical have been created to provide the pressure distribution within the shock train.

One such relation can be found from *Weiss, 2010*

which describes the pressure *p* at for a given *x* distance past the shock wave. In this equation, *c *is an empirical constant of 0.114,* w *is the Crocco number, and* w *is the Crocco number at sonic conditions [44].

With the pressure profile known, it is now possible to determine the rest of the one-dimensional flow within the shock train through manipulation and integration of the differential conservation equations. It is important to note that during shock train formation, the flow within it separates, introducing another variable which will be regarded as the core flow area *A _{c}*. A differential element, or finite element, that the conservation equations are applied to typically looks like that in Fig. 3.6. The conservation equations are then integrated across the control volume and control surfaces.

**Isolator-Burner Interaction**

In order to determine the pressure at the combustor entrance, the interaction be-tween the isolator and burner must be considered. If examined separately from the other engine components, the isolator acts as a back pressured nozzle [1, 47]. This reinforces the necessity of a method to accurately determine the pressure at the burner entrance which will give the last piece of information for the isolator.

necessitating a separation/re-circulation zone model. Due to the 1-D nature of the flow sim-ulation in this work and the higher dimension nature of separation zones, the core flow areas in the cavity are prescribed based on the core flow entering the cavity and the wall geom-etry. This is shown to yield relatively accurate simulations without significantly increasing computational requirements.

**Flame-holding and Fuel Injection**

Scramjet fuel injection is an expanding field for research as it is critical to combustor operation and there are a number of fuel injection methods. Commonly seen are wall-injectors, ramp injectors, and strut mounted injectors [1] each having their own respective benefits. Effective injection and mixing is critical as hydrocarbon-based scramjets see unique difficulties stemming from slow overall kinetic rates and short combustor residence times due to the high speeds [48].

As documented in *Gruber, 2008* the HIFiRE geometry uses wall injectors normal to the wall and burns ethylene fuel *C*_{2}*H*_{4}. This is important to the fuel-injection method as injectors normal to the wall do not add any momentum in the axial direction. With this knowledge, a mass, momentum, and energy balance is used at the point of fuel injection assuming none of the fuel has reacted.

Holding flame in a scramjet combustor is no easy task due to the time scales issues mentioned earlier in this section, however, the wall-based cavity of the HIFiRE geometry reduces this challenge greatly. To assist in the combustion modeling and to account for the mixing immediately downstream of fuel injection it is assumed that the fuel does not start reacting until entering the cavity.Following a method described in *Heiser and Pratt, 1994*, the pressure at the burner entrance is determined by a two-step process. The first step is to analyze the isolator as a shock-free duct which provides the solution of no flow separation. These results are then used to simulate the combustor, where the location of the highest pressure is referred to as Station *s*. If at this point the local Mach number is less than 0.762 *M*_{2}, the flow is assumed to separate.

The method of the second step is dependent on this separation: if the flow *does not* separate, the isolator is assumed to be shock free and the engine operates as a pure scramjet. If the flow *does* separate, the maximum pressure of the shock free burner is used as the isolator back pressure *p*_{3} and a shock train is assumed to form in the isolator. Additionally, it is then assumed that the combustor operates at constant pressure throughout [1].

**Combustor**

Comprising of multiple sub-units in itself, the combustor is typically defined as the component between stations 3 and 4. Located at the entrance of the combustor are fuel injectors, of which there are many configurations. Following fuel injection is a mixing length and/or a cavity-based flame-holder, which creates re-circulation regions to improve flame-holding capability. These sub-components are then followed by a diverging duct leading to the expansion nozzle.

**Cavity Shock Wave Formation**

Discussed previously in Sec. 2.1.2, when a supersonic flow encounters a turn, an oblique shock forms causing a variety of changes. As a result of the potential for supersonic flow in the cavity, at the end of which there is often a corner in the engine surface that reduces the cross-section down to a choke point. In the case that the cavity is supersonic, a shock wave will form in this area of the engine whose strength is dependent on the geometry of the cavity. Such shock waves can be seen in Fig. 3.7 which depicts the unsteady nature of flat rear wall cavities and the stabilization benefits of a ramped rear wall [14]

To include these shock waves in the simulation is not a difficult task, however, it must be done with care as the cavity is not always supersonic. It was decided to apply a set of conditions that if all are met, a shock is assumed to form. These conditions are:

The flow is supersonic

The cross-sectional area is decreasing

There has not been a shock wave in the cavity already

Once a shock wave is assumed to form, the turning angle is calculated from the geometry and then a shock wave angle is iterated until the Eq. 2.18 is solved. Once is obtained, the normal and tangential velocity components are calculated and Eqs. 2.14 – 2.17 are solved. The results of this calculation are then used to solve the Constant Pressure Plug Flow Reactor which is defined in Sec. 3.2.

**The Need for Non-equilibrium Methods**

As mentioned in previous sections, this work discusses two different methods of combustion analysis: equilibrium chemistry and finite-rate chemical kinetics. While equilib-rium chemistry is much quicker to calculate, the high velocity flow inside scramjet engines typically have residence times too short to reach equilibrium [1]. Due to the similar time scales of the supersonic flow and chemical reactions, it is often more accurate to model the combustion using finite-rate chemistry.

**Contents**

**List of Figures **

**List of Tables **

**1 Motivation and Background Information**

1.1 Efforts for Development

1.2 Uses of Scramjets

1.3 State of Computational Models, Verification and Validation

1.4 Objectives and Work Summary

**2 Hypersonic Airbreathing Propulsion Primer **

2.1 Compressible Flow Summary

2.2 Combustion Summary

2.3 Operational Concepts

2.4 Existing 1-D Scramjet Simulation Codes

**3 Formulation and Validation of Engine Analysis Code **

3.1 Engine Component Breakdown

3.2 Plug-Flow Reactor(PFR) Revisited

3.3 HIFiRE Validation Cases

3.4 Engine Structural Analysis

**4 Optimization of System Components **

4.1 Objective Function Selection

4.2 Constraint Selection

4.3 Algorithm Selection

4.4 Railgun Launched Scramjet Optimization

**5 Closing remarks**

5.1 Conclusions

5.2 Future Work

**Bibliography**

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Optimization of Hypersonic Airbreathing Propulsion Systems through Mixed Analysis Methods