Tim Redford- Effects of Incomplete Fuel-Air Mixing on the Performance Characteristics of Mixed Compression, Shock-Induced Combustion Ramjet (Shcramjet) Engines

8/3/2019 Tim Redford- Effects of Incomplete Fuel-Air Mixing on the Performance Characteristics of Mixed Compression, Shoc

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EFFECTS OF INCOMPLETE FUEL-AIRMIXING O N THE PERFORMANCE

CHARACTERISTICS OF MIXEDCOMPRESSION, SHOCK-INDUCEDCOMBUSTION RAMJET (SHCRAMJET)

ENGINES

Tim Redford

A thesis subrnitted in conformitywith the requirementsfor the Degree of Master of Applied Science

Department of Aerospace Science and EngineeringUniversity of Toronto Institute for Aerospace Studies

@ Copyright by T.Redford 1998.


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Abstract

An investigation of the effects of incornplete fuel-air mixing in a mixed compression.shock-induced combustion, r m j e t (shcramjet) model was carried out. The variableequivalence ratio is applied to a planar, inviscid. chemicdy reacting, shcramjet flow-field. This flowfield is simulated numerically wi th a Lower-Upper Symmetric Gauss-Seidel (LU-SGS) scheme to solve the Euler equations at steady state. The chemistrymodel consists of 33 reaction and 13species.

The results show that incomplete fuel-air mixing leads to longer ignition lengths inthe combustor section for rich mixtures ( 4 > l ) , or both shock-induced and detona-tion wave combustion. In t h e extreme fuel rich regions, combustion is inhibi ted by theamount of fuel in the flow. Lean mixtures (Q< 1) result in little combustion towardsthe centre of the 0ow. Acceptable losses in th e overall performance characteristics aredemonstrated when cornpared to other mixed and external compression shcramjets.with hornogeneous and inhomogeneous fuel-air mixtures. Wi th th e effects of incom-plete fuel-air mixing, the mixed compression shcramjet remains a viable concept forhypersonic propulsion.


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AcknowledgmentsFia t and foremost, I would like to express my sincere t hanks t o my thesis supervisor.Dr. J.P. Sislian. His expertise and guidance have been priceless in t h e completionof the present study. 1would also like to thank my colleagues, past and present:Rudy Dudebout, Jurgen Schumacher, Giovanni Fusina. Samir Fahs. Kevin Linfieldand Rick Oppitz for their suggestions, opinions a n d assistance during t h e course ofthis research.

1 can't express how much 1 appreciate the support of rny family, financially andencouragingly, over t he years. Without them, 1 wouldn't be where I am oday. And,finally, I want to thank my close friend, Neil Buckley, who has b e n there for methrough t hick and t hin.

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Dedicated in memory ofMajor Meredith Hastings, B.Eng.

1908-199'2


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Contents

Abstract

AcknowledgrnentsNomenclature vii1 Introduction 1

1.1 Introduction . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . 1-1.2 Motivation for Preseot Study . . . . . . . . . . . . . . . . . . . . . I1.3 Scope of Present Study . . . . . . . - . . . . . . . . . - . . - - . - . . 91.4 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Shcramjet Design Methodology 112.1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . I l2.2 Inlet Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Fueihjection . . . . . . . . - . . . . . . . . . . - . . . . . . 142.2.2 Compression System . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Combustion System . . . . . . - . . . . . . . . . . . . . . . . . . . . . 202.4 Expansion System . . . . . . . . . . - . . . . . . . . . . . . . . . . . . 242.5 Integmted Shcramjet . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Numericd Method 313.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . 313.2 Numericd Method and Goveming Equations . . . . . . . . . . . . . . 32


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Nomenclature

Roman Symbolsacoustic speedmass fraction for species kstreamline characteristic cu v enegative/right ruming characteristic curveposit ive/left running characteristic curvespecific heat at constant pressurespecific interna1 energyfuellair mass ratioto td energyflux vector in (-directionflux vector in 11-directionheat of reaction of H2 i th airJacobian of species production source terms Wspecific enthaply of species kIdentity Matrixfuel specific impulsemetric Jacobianforward reaction coefficient of reaction rbackward reaction coefficient of reaction rmass flow rate of fuelmass flow rate of air

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Mach numberMolecular weight of species kpressuredynamic pressurecolumn vector of conservative variablesuniversal gas constantgas constant for a specific gasentropy of the flowcolumn vector of source termstimetemperat urespecific t hruslongitudinal velocity componentcont ravariant velocity in t he (-direct iontransverse velocity componentcontravariant velocity in the q-direct ionlongitudinal coordinate of physical reference frametransverse coordinate of physical reference framecolumn vector of species production source terms

Greek SymboIs

Mach anglespecific heat ratiosemiangle of wedgeoblique shock wave anglelongitudinal coordinate of computat onal reference frametransverse coordinate of computational reference frame00w angleeigendues of flux vector Jacobian


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densitydensity of species kelementd density of species kplanar/axisymmetric parameterequivalence ratio

Superscripts and Subscripts

1 index of spatial node dong longitudinal directionj index of spatial node along transverse directionk species nurnbern index of time step or iterationOG freestream condit ons


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Chapter 1Introduction

1.1 IntroductionSince the beginning of the space race in 1957, with the launching of the Russiansatellite Sputnik, rockets have been used exclusively as a means of propulsion forvehicles into orbit and beyond. Even with the reusable shuttle concept, deliveringa payload into orbit is extremely costly (between $2000 - 95000 per pound just toachieve low earth orbit). This high cost can be attributed mainly to the fraction ofthe mass of the payload to the mass of the vehicles, and to the cost of refurbishingthe vehicles. The oxygen to burn the fuel takes up about 213 of the entire mass ofan orbital launch vehicle (Pratt and Heiser 1994). This leaves a very small limit tothe mass of the paylod, the instrumentation, and the factor for redundant systemswhich can be carried with conventional rockets. Unfortunately, at such an expensivecost, this is the od y means currently used to reach orbit safely.

One way of enabling orbital launch vehicles to be more cost effective is to usethe oxygen in the air to achieve as high a velocity (and hence the kinetic energy) aspossible prior to leaving the atmosphere - the point a t which rockets would be requiredfor thrust. Eliminating the need to carry the oxidizer on board launch vehicles woulddecrease the gross mass, allowing for a greater amount of payload to be carried intoorbit, as well as provide significant savings in complexity and cost. This would meancreating an air-breathing engine that would be capable of operating efficientlyup to


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velocities as high as Mach 24, the velocity required to reach low-earth orbit.The design of an air-breathing propulsion system, capable of launching a vehicle

from sea level to orbital velocities, requires reliability and operating efficiency over alarge Bight Mach Xumber range (Mach O to 55). By breaking down this large velocityenvelope into the domains of subsonic, supersonic, and hypersonic flight. one or morepropulsion systems of choice can be shown for each.

To date, turbojets are well understood throughout much of the transonic range andcan operate up to velocities of Mach 3 in particular designs. Above this speed. con-ventional turbojet engines show a significant drop in efficiency. requiring air-breathingpropulsion systems of another form.

Ramjet engines are predominantly used for propulsion at supersonic velocitiesand are th e systems of choice within the Mach nurnber 3-6 regime (Pratt and Heiser1994). As depicted in Fig. 1.1, the freestream air enters the planar diagram of t h eramjet from the left. A system of oblique shocks, generated by the flow against

Vehicle Boundary F uel

Subsonic

f k l njeCu>= name~ld~.Oblique Shock Wave C o w ~ oundary

BumerDiffuser L

Figure 1.1: Schematic cross-section of a subsonic-combustion ramjet engine

-- Exhaust Nozzie

the diffuser, compresses and decelerates the flow. A normal shock transforms theflow from supersonic to subsonic speed and is further decelerated in a divergent duct.In slowing the aidow to subsonic speeds, these shocks increase the pressure andtemperature of the flow so that when the fuel is added in the combustor it willignite. Lastly, the high temperature combustion products are accelerated back up tosupersonic speed through a convergent-divergent nozzle, producing thrust.


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Unfortunately, there is a lirnit to the freestrearn velocity for which a rarnjet isefficient. As the diffusor decelerates the flow to sonic speeds, the kinetic energy loss istransformed into an interna1 energy gain, resulting in an increase in temperature andpressure. This rise in temperature and pressure can become so elevated that moleculardissociation can occur, causing a 106s of chernical energy in t h e flow available forthrust. Excessive temperatures and pressures can also be irnpractical for the materialsand structural design systems. These losses become pronounced at a freest ream M a c hnumber between Mach 5 and Mach 7; the range in which flow deceleration to suhsonicspeed becomes wasteful.

This Mach number range is the beginning of the hypersonic regime and is char-acterized by the physical flow phenomena of high temperatures, thin curved shockand entropy layers. viscous interaction and rarefied low-density Bow effects (Anderson1989). At Mach 6 and higher, it would be more effective to burn the fuel at super-sonic speeds than to slow the flow down and incur al1 the losses involved. This typeof engine is referred to a s a supersonic combustion ramjet, or scrarnjet. Similar to thebasic design of the rarnjet, the inlet decelerates and compresses the flow via a seriesof shocks. Since the flow is not slowed to subsonic speeds this avoids the occurrenceof a normal shock, leading to lower pressures and temperatures, and overall improvedefficiencies. As the 0ow is supersonic throughout the diffuser, this necessitates anentirely convergent body geometry (see Fig. 1.2).

The compressed flow enters the combustor section where fuel is added and burned,al1 at local supersonic speeds. Ignition occurs by ensuring the diffuser geometryincreases the temperature and pressure to a point such that, as the fuel is added in thecornbustor, it will burn with t he entering oxygen. Here, the significance of achievingrapid and thorough mixing is essential because the time available for burning is shortdue to the speed of th e flow in this section. Because the exiting flow is supersonic,the nozzle is designed to be continuously divergent to provide t h s t by transforrningthe thermal energy of th e products of combustion into kinetic energy.

This design is limitedby th e length of the combustor section. Since the combustorexperiences the highest temperatures and pressures, heavy structural support and


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FreestreamFlow

ObliqueShockWaves

Figure 1.2: Schematic cross-section of a scramjet engine

active cooling would make it by far the most costly section by inass. As the velocitiesentering the combustor increase, a longer section would be needed to allow for theburning of the fuel. If it were too short, the unused fuel and oxygen would passthrough the entrance of the nozzle before it had time to mix and combust.

The count ries involved in scramjet research today include France, Germany, Japan,Russia, and the United States of America. Pursuit into this technology is mainlydriven by the need for a cost effective and dependable way of transporting payloadsinto orbit. Among d l th e programs researching scramjet propulsion systems, theUnited States' National Aero-Space Plane (NASP) program was the most prosper-ous, vigorous, and publicized. The NASP program began in 1986 and was al1 butcancelled in 1994 due to budget considerations; this after approximately $2 billionwas spent on it (Chase and Tang 1995). The main purpose of the NASP programwas to develop a h l l y reusable, single stage to orbit, horizontal takeoff and land-ing aircraft, capable of &ght at hypersonic speeds with an air-breathing propulsionsystem. NASA's intent was to reduce the launching cost by an order of magnitudeand to duplicate the reliability, flexibility, operability, and cost effective potentials ofaircraft to an orbital launch vehicle.

The scramjet used in th e NASP project was a type of diffisive burning, super-sonic combustion ramjet. Another class of scramjet is th e shock-induced combustion


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ramjet, or shcramjet. In this design (Fig. 1.3) the fuel is injected into the air in theinlet region. The air flow is still compressed and decelerated through a series of

ObliqueDetonationWaveFreestream Flow- - - - - _ _- - - - - _ _

Figure 1.3: Schemat ic cross-section of a shock-induced combust ionramjet engine

shocks, but it is limited in that the mixture cannot reach high enough pressures andtemperatures that might prematurely ignite the fuel. In fact, he design of the inlet isto adjust the body geometry and the position of the cowl so that the shock system willcornpress the fuel-air mixture to a point just short of the ignition threshold. Then,as the fuel-air mixture enters the combustor section, it m e t s an advantageously po-sitioned oblique shock generated by the combustor wedge. Passing through this finalshock, the temperature and pressure are increased over the threshold of ignition andcombustion occurs; hence the narne shock-induced combustion ramjet.

The wo processes in which the buniing c m ake place are as follows: first, simpleshock-induced combustion results when the combustion process does not influence thepreceding shock wave as ignition occurs farenough downstream. Second, a detonation

- wave is generated when the ignition delay is short enough that the combustion frontcouples with the preceding shock wave. The shock wave then becomes iduenced bythe heat release of the combustion (Atamanchuk and Sislian 1991).

In cornparhg the shock-induced combustion ramjet with the diffusive burningramjet, a number of advantages and disadvafltages can be found for each design.


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In both cases we have compression created in the inlet from a number of plannedshocks. In the shcrarnjet case, because there are lower temperatures and pressureentering the combustor, there are fewer losses due to the reduced deceleration of theflow. This is mainly due to the compression being shared between the inlet and atthe combustion-inducing shock a t the mouth of the combustor. Also, since t h e fuelis being injected in the inlet, it allows for more time for diffusion of the fuel into thefiow. The scramjet design has the fuel injected in the combustor which would requirea longer combustor section to allow for full mixing. W ith fuel mixed in the air as theflow enters the combustor section, combustion in the shcrarnjet can occur over a shortdistance, thereby reducing the combustor length. High pressures and temperatures inthe combustor section would also require heavy structural support and active coolingsystems. Reduction of the combustor section would translate to considerable savingsin cooling load and structural weight - economically substantial considering presentpayloads at only 2-4% of the total vehicle weight.

The disadvantages of th e shock-induced combustion rarnjet share some of those ofthe diffusive burning scramjet . Since neither design has the capability of static t hrust,another propulsion system would be required to propel them to their lower velocitylimit. Of significant importance to the shcramjet is the prevention of prematureignition of the fuel in the inlet. Unlike the scramjet,burning of the fuel in the air priorto the combustor could have catastrophic consequences if not properly considered.This would require research into areas such as the injection system, to minimizelosses and premature burning, and wall cooling in the inlet to prevent ignition in theboundary layer. As the detonation wave has yet to be used as a burning mechanism inpresent propulsion systems, this would be another area which would require furthertheoretical and experimentd evidence for proof of stability. Th e terms detonationwave engine and shock-induced combustion ramjets (shcrarnjets) represent th e sameconcept design and will be used interchangeably throughout this work.


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1.2 Motivation for Present StudyGoing as far back as 1947, detonation wave engines were conceptualized first by Roy(1947). He showed that normal shock waves could ignite fuel-air mixtures, causingthe burning of the fuel. Following this. is was not until Dunlap et al. (19.58) con-ceptualized using an oblique detonation wave engine in a hypersonic vehicle. Gross( 1959) proved t hrough experimentation that strong? steady detonat ion waves can beachieved when the Mach number of the flow is greater than t hat of the Chapman-Jouguet Mach number. He noted that. for a detonation wave. the pressure loss isequal to that of a corresponding shock wave. He concluded that - t he high speed ofsuch detonation waves with their corresponding small thickness imply a heat releaseper unit volume per unit time far exceeding any ot her previously obtained chernicalcombustion systemsn

Rubins and Rhodes (1963) performed experiments with Hz nd air and concludedthat, not only was a Chapman-Jouguet wave a special case of shock-induced corn-bustion, but that kinetic equations could be used in determining beforehand the wallgeometry downstream of the shock. Following this, they pointed out that for a rangeof %ightMach numbers the dissociation and recombination problems, produced by theexcessive temperatures after detonation, could be controlled by changing the angleon the ignition inducing wedge.

Using a configuration involving multi-shock intake flows, Townend (1970) pre-sented a method for the use of detonative combustion in the optimization and designof hypersonic rarnjets. The paper presents a method by which multi-shock intakes, o ptimized by the Oswatitsch criterion, are matched analytically to strong or Chapman-Jouguet detonation waves followed by exhaust nozzles with a specified, or zero, loss.Th e Oswatitsch criterion is a condition, o ri gin dy observed by Oswat its-:h, where theentropy rise through a series of shocks is minimized when the shock strength (changein pressure) for each shock is equal. Townend found that the detonations should beoblique or conical to impose less structural load on the ramjet. He showed that nor-mal detonation waves axe undesirable for cruising vehicles, t hat Chapman- Jouguet


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detonations are found to be cornpetitive with heat addition at constant pressure, andin some cases problems such as boundary layer separation. premature ignition. anddissociation may be avoided.

Similar studies involving oblique detonation wave rarnjets wi th mult i-shock. inter-nai and external compression int ake systems, were completed by Morrison ( lg8.1980).This investigation included areas such as losses incurred through fuel injection. theeffects of chemistry. and performance estimations of ramjet configurations. By incor-porating more realistic working gas properties to calculate performance parameters.Ostrander et al. (1987)extended the work of Morrison. But both their works. and theresearch of Townend, was restricted to a one-dimensional, cycle-type analysis whichdoes not include non-equilibrium chemistry.

To study the effect of non-homogeneous fuel/air mixing Carnbier et al. (1988)simulated the results of a fuel injection pattern, examining the structure of the det-onation wave under the poorest mixing conditions. They cornpared the results of aflow with a stiochiometric fuelfair mixture with a flow having an equivalence ratiowith a periodic variation rnging from 0.1 <


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utilized a numerical met hod to solve th e Euler equat ions for hypersonic Bowfields fora reacting, stoichiometric mixture of hydrogen and air. This lead to the researchcompleted by Dudebout (1996) where his work studied the aerothermodynarnic per-formance of an external compression shcramjet (shock-induced combustion ramjet)through the numerical simulations of hit previous work. Following this. Schumacher(1995) performed similar investigations on the external compression design bu t witha non-uniform fuel distribution to study the effects of incomplete fuellair mixing.Using similar numerical simulations, Oppitz (1995) designed a mixed compressionshcramjet model. Through this he determined that it was superior. not only to t h eexternal compression design - both with homogeneous fuel/air miring - but 3s a vi-able concept and offers substantial advantages as an alternative mode of propulsionto both scramjets and rockets in the Mach 12-23 flight regime".

1.3 Scope of Present StudyMuch like the study Schumacher ( 1995) accomplished. invest igating t h e effects of non-ideal fuel/air mixing on external compression configurations (similar to Dudebout'swork (Dudebout 1996)), his study will build upon the work of Oppitz (1995) and thernixed compression configurations. The fundamentaJ addition in the present s udy,as with Schumacher, is the introduction of an incomplete fuel/air mixing model.This study solves the hypersonic inviscid flowfields of the planar shcramjet mod-

eh by using a computational fluid dynamics (CFD) scheme explained in chapter 3.To mathematicdy duplicate the shock-induced, finite rate, hydrogen-air combustionprocess a chernical kinetic model consisting of 33 reaction with 13 species (H2, Z,H,O, OH, HzO, HOz, HaOs, N, NO, HNO, N2, O2) was used.

Modelling the incomplete mixing of fuel and air was accomplished using a Gaus-sian distribution of equivalence ratio ranging frorn 0.02 < 4 < 3.2. Over a range offiight Mach numben 9 < M, < 20 the flow properties were calculated for variousplanar configurations and used to determine performance and aerothermodynarniccharacteristics. Cornparisons with the work of Dudebout, Schumacher, and Oppitz


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are used to establish the difference between external and mixed compression config-urat ions, with ideal and non-ideal fuel/air mixing.

1.4 OverviewAccompanying this Introduction, there will be four chapters to follow. Chapter 2details the process behind how t h e three main sections of t h e shcramjet - inlet. corn-bustor, and nozzle - will be designed. It specifies th e requirements that mus t besatisfied for optimized tlight conditions and areas such as the generation of the fueldistribut ion curve.

The numerical simulation will be covered in Chapter 3. This will explain thegoverning equations employed to represent the physical mode1 and the numericalmethod used to solve the flowfield. Coupled with these will be the equations todescribe the combustion process of the 13 species involved within the 33 reactions.

Results of the completed computational simulation will b e detailed in Chapter4 , but only a single case will be used to guide the reader through the investigationof the incomplete mixing. Analysis of the fl oheld will be accomplished throughda ta extracted along streamlines and cross sections. Comparisons against previousworks with homogeneous fuel/air mixtures ( 4 = 1) will be presented, along withthe performance characteristics of other shcramjet designs studied thus far. Theseinclude the fuel specific impulse, net specific t h s t , net thrust per capture area, andthe thermal, propulsive and overall efficiencies. Comparisons will also be made herewith the performance characteristics of rocket propulsion, the present system used topropel vehicles into orbit.

Findly, Chapter 5 will summarize md discuss conclusions formulated throughthe results obtained herein. The implications of a variable 4 will be addressed andto what extent the degradation of the performance parameters resulted. Capping offthis section will be the recomrnendations for ht u r e work.


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Chapter 2Shcramjet Design Methodology

OverviewThe intent in designing the shcramjet propulsion system is to mode1an air-breathinglaunch vehicle capable of transat mosp heric flight. While operating on an accelerat-h g 7climbing trajectory a constant dynamic pressure for the Bight path is adoptedto calculate the oncoming flow. For hypersonic vehicles, the practical flight enve-lope is 500-2000 psf (Pratt and Heiser 1994). For this study a dynamic pressure of- 1400 psf is chosen for cornparison purposes with the external compressiond, -works of Dudebout (1996) and Schumacher (1995), as well as t h e mixed compressionconfiguration developed by Oppitz (1995). T h e dynarnic pressure is defined as

By choosing a flight Mach number (M,), a dynamic pressure (qd,), and assuminga specific heat ratio (7, = 1.4), the freestream pressure (P,) can be determined.From this pressure, the temperature and altitude can be calculated fiom standardatrnospheric tables (Zucrow and Hoffman 1976).

The planar, mixed compression shcramjet will be studied a t the on-design con-dition for the four fight Mach numbers used in the previous works of Dudebout,Schumacher and Oppitz (M, = 10.6, 14.1, 18.8, 23.5). The dekition of on-design


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will be explained later in this section. Consisting of three main parts, the shcramjetengine performs three basic processes: compression in the inlet. combustion in thecombustor, and expansion using the nozzle. The dimensions of each of th e three corn-ponents are altered for each Mach number to achieve the conditions which provideminimum entropy increase and maximum thrust. This is why each shcramjet design isconsidered optimized. Further explanation of the details on each component's designcan be found in the sections to follow.

The depiction of the shcramjet in Fig. 2.1 shows the air enteriog the inlet of t h eengine from the left side, with the top surface of the body positioned dong the x-a is .The positive y-axis is in the downward direction. The inlet serves two main purposes:to cornpress the oncoming air, and to provide the region where fuel is injected andmixed prior to reaching the cornbustor section. In th is study the fuel being consideredis hydrogen (Hz). ue to the complexity involved with simulating fuel mixing, it isassumed that th e streamtube above the cowl tip will consist of a mixture of hydrogenand air. The outer flow, which is below the cowl tip, will be assumes to containonly air. Further details on th e fuel-air ratio across the opening of the inlet will beexplained in section 2.2.1.

Combustor1 Inlet - Non l e -

Figure 2.1: Shcramjet Configuration.

As the flow passes through(BC and CD) the temperature

the external shock (AB) and the two interna1 shocksand pressure of the fuel-air mixture clearly increases.


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The geometry of the inlet is designed, such that the static temperature after the thirdinlet shock remains below the ignition temperature of the mixture. The main purposein keeping the temperature below the ignition threshold is to avoid uncontrolled andspontaneous combustion in the inlet. Following the compression in t h e inlet th e airflow approaches the final shock (DE): the beginning of the combustor. The tempera-ture of the flow is raised above the ignition threshold for Hz s it passes t hrough thisshock, and burning of the fuel then occurs. This ignition technique is the cornerstoneof the shcramjet mode1 - hence the acronym Shoctinduced Combustion RrZMjet -and is the distinctive characteristic distinguishing itself from diffusive burning scram-jets. The buming of the fuel takes place through shock-induced combustion. whichresults when the combustion process does not influence the preceding shock wave.and through a detonation wave when the combustion is rapid enough that it willcouple with t he shock wave (DE). After burning has occurred, the hot, high pressure0ow exits the combustor section and is expanded through the nozzle section. As theflow is expanded the thermal energy generated through the combustion is convertedto kinetic energy, providing t hrust .

As mentioned above, the studies here are for on-design conditions. This meansthat the geometry of the entire shcramjet will be such that the shocks always intersectat their appropriate points on the body and the cowl. Through each flight Machnumber, the positions of these points (and hence the geometry of the vehicle) willshift suitably to adjust to the shocks. The design of the variable-geometry inlet willbe addressed in section 2.2. In section 2.3 the combustor design rnethodology will beaddressed - this will include a discussion on detonation waves and the selection of thecowl-wedge angle and combustor surface. The final component, the oozzle, will bedescribed in section 2.4. Section 2.5 wiU discuss the incorporation of each componentinto an integrated shcramjet engine, and it will detail how the outer cowl surface isgenerated.


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2.2 Inlet DesignLike most other ram engine inlets, one main task of the shcramjet inlet is to cornpressand decelerate th e oncoming air. The second lunction of this inlet design. differingfrom other ram engines. is to provide a region where fuel is injected and mixed p n o rto reaching the combustor section. No flow information is able to propagate upstrearnsince the entire flow throughout t h e engine is supersonic. This allows the entrance tothe inlet - the inflow boundary - to contain the initial flight conditions for the entireflow through the shcramjet. The initial flight conditions. such as the freestreampressure, temperat ure, specific heat ratio. etc. are discussed in th e preceding section.Design considerations of the geometry of the compression system, and details of thefuel-air mixing, will be studied in the following subsections.

2.2.1 Fuel InjectionThe essence of this study is to examine the effects of incomplete fuel-air mixing onthe performance of mixed compression shcramjets. It must b e emphasized that fuelinjection sources along the wall of the inlet will add fuel to the air flow, while at thesame time the temperature has to remain well below the ignition d u e f the mixture.Potential locations for injectors would be along the body surface between points A andE, and along the cowl surface between points B and D (see Fig. 3.1). Fuel injectionin th e inlet do w s for more time for the fuel and air to mix before encountering thedetonation wave. Due to the high 00w velocities, it is imperative that the fuel beinjected pardel, or at very s m d angles to the flow, so as to avoid shocks whichcould induce ignition. Also, further studies into fuel injectors rnay find that complete

. mixing may prove a difficult task since the fuel has a short amount of time to mixwith the air - recall that the length from A' to B is only 15 m. Th e case for perfectmixing (as studied by Oppitz (1995)) has both a homogeneous and stoichiometricmixture of fuel and air entering the combustor section. A stoichiometric mixture isone by which complete combustion of all the oxygen in the air with all the reactantsin t he fuel would occur (Pratt and Heiser 1994).


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In practice a homogeneous, stoichiornetric mixture may not always be achieved.When burning occurs and there is oxygen in excess, this means that there was notenough fuel added to the flow; this results from a fuei-air ratio less than that ofthe stoichiometric ratio. A parameter used to indicate a deviation from the ideal,stoichiometric mixture is termed the equivalence ratio, d- This equivalence ratio isdefined as the actual fuel-air ratio ( f ) of the flow over the desired stoichiometricfuel-air ratio ( fstiochianclric ):

4 = fstoichimctric

To mode1 the deficiencies of practical injectors for the shcramjet? a distributionof the equivlence ratio, 4, is dictated at the i d o w boundary of the inlet. For themixed compression shcramjet the distribution of d in this study is based upon aset of normal distributions, with maxima located at t h e edges of t h e fiow. Unlikethe external compression shcramjet (Dudebout 1996: Schumacher 1995). the mixedcompression design uses both the inlet body and cowl as regions of fuel injection. -4worst-case fuel-air mixing configuration is assumed; this results in high concentrationsof fuel near both wd l s and Leads to rich fuel-air mixtures (# > 1). As the distancenormal to either fuel injectors increases the amount of fuel decreases towards thecentre of the airstrearn, and results in a lean mixture (4 < 1). The fuel distributionfrom each of the injectors is modelled by a gaussian curve of the f om:

Since the fuel has two injector sources, a summation of two gaussian curves isused. The maxima of t he c w e s are a t th e bou daries of the flow, therefore p in eachhalf of th e equation c m be replaced with O at the body wd l and a distance L, for thebreadth of the inlet between A and A'.


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This leaves a modified equation with four variables to solve, fuffilling the followingfour requirements. The first requirement is to have equal, maximum fuel equivalenceratios at each of the boundaries ( t h e fuel injected from the walls of the inlet wouldproduce the highest magnitudes in concentration close to the sources).

Due to the geometry of the mixed compression shcrarnjet. the body inlet is about3 times the length of the cowl inlet (see Figure 2.1). By visualizing the Fuel diffusioninto the flow, i t can b e said that th e fuel injected from the body side will stretchfurther into the flow with a longer time amilable for it to diffuse compared to thecowl side. Since the approxirnate ratio between the body wail length and the cowlwall length is 213, an estimated position for the minimum equivalence ratio ( 4 z 0)is placed at 213 of the full breadth of the inlet (L); i.e.

Further, by similar reasoning as above, roughly 213 of the total fuel is estimatedto be present between the body wall and this minimum point:

Finally, he resulting equation is integrated and scaled in an iterative fashion toproduce a curve wit h an average equivalence ratio equal to unity (Eq. 2.8).

The above analysis is similar to that for the extemal compression shcrarnjet case(Schumacher 1995). The distribution produced here results in a curve that rangesfrom 4 x 3.5 at either wall boundary to 4 c 0.02 a t the minimum, a length 2/3Lfrom the body wal l (see Fig. 2.2).


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EquivalenceRatio (4)

(a ) Equivalence ratio graph

equivalence ratio distribution

(b) Equivalence ratio profile schematicFigure 2.2: Initial equivalence ratio profile for Mm = 14.1, planar case.


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2.2.2 Compression SystemAlthough the primary function of the idet is to compress the airstream, this must beaccomplished with a minimum entropy rise. Compression of the airRow is performedto provide the combustor with the preferred cycle temperature ratio. This ratio isdefined as the ratio of the inlet exit temperature to the inlet inflow temperatureand, due to the thermodynarnic eficiencies of heat addition. the higher the ratiothe better the performance. Unfortunatel- there is a limit to the cycle ternperatureratio: extremely high temperatures cari cause dissociation of t h e molecules in the flow.leading to a loss in energy. But even before that plateau is reached, the temperaturelimit in the inlet is restricted to tha t of the ignition temperature, which is taken tobe 900K for the H2 uel.

With the mixed compression shcrarnjet design, as the name suggests, the inietutilizes both an external compression region and an internal compression region. .Asdepicted in Fig. 2.3, the first, external shock (AB) is generated from the leading edgeof the shcramjet body at point A. Following this, two internal shocks are generated:one shock, BC, is produced at t he t ip of the cowl and the second shock, CD, is forrnedat the reflection point of the second shock on the body wali, at C. In order to optimizethe inlet, to reduce the entropy production, the Oswatitch criterion (Oswatitsch 1947)is employed, similar to a technique implemented by Townend (1970). Oswatitschshowed that the minimum entropy rise c m be achieved by ensuring that the shockstrength (the pressure rise across a shock) of al1 the shocks are equal.

The geornetry of the inlet was designed from these two criteria: equal strengthshocks and a maximum of 900 K after the third shock. A third constant was to keepthe length from the inlet to the cowl tip, A'B, to 15 m. This length is used to main-tain design similarities with the external compression shcramjets (Dudebout 1996;Schumacher 1995),and with the homogeneous fuel-air case of t he mixed compressiondesign (Oppitz 1995). Using a the rmd ly perfect, inert gas mixture consisting of O*,N a , and H2 the equivalence ratio distribution is determined through the methoddiscussed in section 2.2.1) the planar inlet geometries is determined. Similar to the


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Figure 2.3: Inlet Configuration

procedure used by Oppitz (1995), the first wedge (AC), represented by the angleproduces the f i s t shock which must intersect the cowl tip at X = 15 rn (see Fig. 2.3).By determining the X and Y ceordinates of the points B, C, nd D for an equalstrengthed shock system, the flow deflection angles (Ji: &, and &) are iterativelysolved for until the temperature at the combustor entrance is just below t h e maxi-mum of 900 K. The locations of the shock-wall intersection points can be determinedthrough trigonometric relations and gas-dynamics t heory. As t his is the on-designcase, it is important that al1 the intersection points are determined precisely; anyerror can result in poor determination of the point where the combustor begins.

The entire flowfield is solved with a simplified, non-reacting numerical method -outlined in chapter 3 - and the location of point D is established by determining thecross section of the flow with the lowest standard deviation in pressure. At t his crosssection the variables of the flow are extracted and used for the inflow variables in thedesign of the combustor section.


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2.3 Combustion SystemThe combustor section is the most distinctive section of the shcramjet compared toother hypersonic vehicles. Mainly used by other systems. t h e diffusive combustionprocess involves high pressures and temperatures in the combustor and results in longcombustor sections due to the required time for mixing and burning of the fuel. Byreplacing the long, diffusive burning process with a method of burning the fuel-airmixture in a relatively quicker fashion would Save vehicle weight and reduce drag.Shock-induced combustion does just that: by utilizing an oblique shock wave tocompress the premixed fuel-air 0ow, raising the temperature above the spontaneousignition threshold, burning occurs an a very short Length of time. The resultingprocess of combustion may be so quick in some cases that the burning occurs withina relatively short distance of the shock wave. In this case, designated as a detonationwave, the combustion front will couple and influence, through its heat release, thepreceding shock wave. When there is a delay, long enough between the shock andthe combustion front that they do not influence each other, this combustion front isterrned shock- induced combustion. The combustion process wiil, in many cases, bea combination of the two mechanisms. This process of combustion will be employedherein.

As the airtlow enters the combustor section, an oblique shock (DE) is generated bya wedge on the upper surface of the cowl (see Fig. 2.1). This shock wave, dependent onthe wedge angle,6*, rovides the fuel-air mixture with its last compression, resultingin ignition. Because the wedge angle, J4, is measured from the x-axis the effectiveangle relative to the flow is & - &. Thus, an increase in the angle of 6.4will cause anincrease in the detonation wave angle.

Beginning with the extracted data from the inlet (section 2.2.2) and the cowlwedge angle, the computational domain c m be determined. Since the end points ofthe combustor are not h o w prior to the formation of the detonation wave, the domainis constructed so that th e length of the combustor, starting a t DD', extends dong thecowl surface DF for a distance long enough to accommodate the cross section at EF


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Figure 2.4: Combustor section configuration.(see Fig. 3.5). With the left boundary DD' containing the inflow variables, and thelower boundary DF defined as a wall boundary (cowl surface), the upper boundaryis given inflow conditions of the variables at point D' and runs parallel to the lowerboundary. The exit plane at the right boundary remains a transmissive boundary.After developing the computational solution using the numerical method outlinedin chapter 3, the body-wall surface, line D'E, is obtained by following a streamlineemanating from point D' to the region beyond the detonation wave. Point E, theintersection point of the streamline and the detonation wave, is determined as thepoint along the streamline having 99% of the exit plane pressure. Point F becornesthe location on the cowl directly below the determined point E,and the exit vaziablesof the combustor are extracted along this cross section for use at the entrance to thenozzle. By this method the geometry of the combustor, for a given wedge angle, canbe determined.

The ntent of the combustor design in the present study is to maximizethe overall


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Figure 2.5: Combustor temperature contours for M, = 11.1, planar shcramjet.net thrust of the shcramjet. Since varying performance results are produced throughdiffering owl wedge angles, it is necessary to choose the wedge angle that deliversthe greatest thrust. The study performed by Oppitz (1995), following the work ofDudebout (1996), results in a minimum entropy increase by choosing an appropriatecorvl wedge angle. Unfortunately, with th e varying form of the detonation wave due tothe fuel distribution, this method is not applicable in the present study, as is the casewith Schumacher (1995). The criteria used in this study, as with Schumacher, is tochoose the combustor which maximizes the o ve rd thrust; assuming that this impliesa rninimization of entropy. Hence, the results are comparable to that of Oppitz (seeChapter 4) and to the performance characteristics of the external compression designsas well.

By utilizing the method of characteristics (see section 2.4) to generate the nozzleand the outer cowl sections of the shcramjet, the ov erd net thrust is calculated usingthe results of the completed inlet and the combustor sections. Figure 2.6 shows a plotof t h e total thrust vs. the cowl wedge angle,&, for the M, = 14.1 case. A clearlymarked maximum thrust exists over th e range of angles used for the cowl wedge andthis was the value (4= 4') used in the design of the M, = 14.1 case.


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CowlWedgeAngle IdqreeslFigure 2.6: Plot of thmst vs. cowl wedge angle for planar shcramjet, M = 14.1


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2.4 Expansion SystemThe nozzle is the final section of the shcramjet design and its role is to expand andaccelerate the flow exiting the combustor, providing the shcramjet with its thrust.Combustion products, at high temperatures and pressures, znter the nozzle wherethe thermal energy of the air-flow is converted to kinetic energy through expansion.The nozzle is designed with a dual wall technique, introduced by Park (1997) ndimplemented by Oppitz (1995). If the nozzle geornetry is designed correctly- theexiting flow from the nozzle should be parailel to the freestream Bow ( i e . ~ 0 ) andat the sarne pressure as the freestream pressure, P,. One way to ensure that theflow exits the nozzle parallel to the freestream flow. and to keep the nozzle lengthshort, is to have sharp tuming flow angles at the nozzle entrance, points E and F.These sharp turns rvill cause Prandlt-Meyer expansion fans, shown in Fig. 2.7. andideal contour walls of the oozzle (E l and FJ)will cancel the characterist ics stemmingfrom the opposite expansion point ( F and E respectively ).

inflow

Noule Boundary,Noule B o u n d a r y r

exit

Figure 2.7: Diagram of an ideal, tw+dirnensional nozzle.Therefore, the 0ow will exit p a rd el t o th e shcramjet axis and internal reflected

shocks will be avoided inside the nozzle. Under the assumption t hat flow throughoutthe nozzle will be supersonic, as well as both chernically and vibrationally frozen,the method of characteristics is employed to determine the nozzle contours. Thisassurnption is supported through the work of Dudebout (1996).


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The method of characteristics (MOC)s f d y explained in the work of Zucrow andHoffman (1977). In this study, the MOC for a supersonic. steady. tw*dimensional,rotatiooal flow is used in designing the nozzle. The rotational method is used in thiscase, compared to t he irrotational case of Oppitz (1995). because there is an entropygradient normal to t h e streamlines in the flow entering the nozzle. This is due o t h evariable distribution of the fuel in the flow and the curved shock in the combustor.Using the flow variables exiting the combustor section. the MOC ut ilizes a streamlineas a characteristic to integrate values dong it and will solve the four flow variables(V,P,B,p) at each point in the nozzle.

Beginning at the line EF. the nozzle will be broken down into the t hree regions asshown in Fig. 2.8: the Mach triangle (EFG), the kernel (EHFG), and the upper andlower transit ion regions (EH1 and FHJ).

Mach Triangle

Figure 2.8: Dual wall nozzle design.

The Mach triangle is solved first by calcdating characteristic lines for each ofthe points on the i d o w line EF. By taking two points, c d hem point 1 and point2 (see Fig. 2.9), the left and right-mnning Mach Lines (or characteristic lines C+and C- espectively) are determined through the following equations (termed the


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characterist ic equations):

where 8 = tan-'(vfu) is t h e average flow angle and a = sin-l(l/M) is the Machangle at each point. The intersecting point of these two lines is called point 4. andby tracing a strearnline back to line 1-2, this new intersection point becomes point 3.Values at point 3 are determined by linearly interpolating the values known at points

Figure 2.9: Interior point process.

1 and 2. The flow properties at point 4 can then be det ennined bj integratingthe compatibility equations (2.12-2.15) dong the characteristics and simultaneouslysolving t h e h i t e difference equations (2.16-2.23).


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The finite differenceequations can be written as

Ay* = X*4x*

where +, -, O denotes the C+, C- nd Co haracteristic curves respectively.By using this method ( cded the interior point process) in an iterative fashion. a

new line of values is generated as depicted in Fig. O ( a This process is repeateduntil a Mach triangle is formed; at that point, Bow variables along lines EG and FGare known (Fig. %.10(b)).

Figure 2.10: Development of Mach triangle characteristic rnesh.


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With the Mach triangle solved, the kernel region (EffFG) is determined usingthe variables along the lines EG and FG, and by using the interior point processas above. In this case. however, flow tuming angles (O,,, and 8 are appliedat points E and F, causing Prandlt-Meyer expansions. -4s mentioned ahove. a largetuming angle will shorten the nozzle length. The fan is descritized by a specifiednumber of expansion waves, n~.,. within the kernel. At E and Ft the turing anglesare increased by steps of the angle size -& nd th e MOC interna1 unit process isapplied to determine the new lines, EH and FH, as depicted in Figure 2.12. Employingthe Prandlt-Meyer equations and a given expansion angle, the Mach number at pointF can be specified. With this, a new characteristic line (C+) t F can be determined:C+ = ta@ + sin-l(l/M)). The intersecting point of th i s line and the C- line fromthe next point on the Mach triangle line, FG, gives a new solution point (a s shown inFig. . l l ( a ) ) . By repeating this, drawing the C+ characteristic from th e new pointand the C- characteristic from t h e next point along FG, anot her point is determined(Fig 2.11(b)). This is repeated until the C- characteristic is used from point G. ivingG'.

Figure 2.11: Solving points in the kernel with the interior point process.

Using the same method from point E, on the upper side of the Mach triangle, asimilar solution line is generated. Solving for the final point at the tip of each kemel

28


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fan is accomplished by using the characteristic lines from Gfand G" s in Fig. 2.1l(e).The angles are repeatedly expanded by increments of until the values of 8,,,,*fanand BrOwe, are reached. This will create the final lines EH and FH (Fig. 2.12). Theturning angles of B. and Bi0,,, are determined such t h a t t h e flow at point H is atthe freest ream pressure and paraiiel with t h e axis (v = 0).

Figure 2.12: Completed kernel region characteristic mesh, M, = 14.1 (n!,, = 50) .By using the variables dong the new lines EH and FH, the walls of the nozzle

can be found by solving the transition regions, EIH and FJH. The transition regionis solved in a similar fashion a s the kernel region.

Theoretically, it is possible in this planar mode1 to expand the nozzle past theaxis. But, for consistency with the other designs it is not done. Further, if the nozzlewere expanded past the axis, this would create a negative thrust on the top of theshcramjet. Also, the nozzle length can becorne exceedingly long with only a marginalincrease in the thrust. So, in order to reduce the drag caused through friction, thenozzle is cut to 95% of its total thnist . This is done by shortening the body nozzle to98% of its total thmst, and the cowl nozzle the remaining amount to obtain the 95%of the total nozzle thmst . This division was chosen to avoid the cowl becorning longerthan the body and for cornparison with the design of Oppitz (1995). Although thisstudy assumes inviscid flow, the design of the nozzle considers viscous effects because


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th e fr iction caused by th e nozzle t ips could be quite large since they would asymptotefor som e dis tance to produce th e desired t h m s t . This is also considered in th e worksof Dudebout (l996), chumacher (1995), nd Oppitz (1995).

2.5 Integrated ShcramjetShown in Fig. 2.13 is th e in tegr ated shcrarnjet design for th e M, = 11.1 case.Deterrnining the oute r cowl surface is comp leted by connecting t h e cowl tip and t h ecowl tail with a cubic polynomial. An angle of 5' was d ic ta ted a t th e t ip of t h e cowlfor cornparison with previous designs, and th e tail of th e outer cowl ho an angle ofzero degrees so tha t the flow over it is parallel with the freestream flow.

For each Mach number case (10.6, 14.1, 18.8, 23.5) the de terminat ion of a l l thesections is completed and solved numerically using th e method out l ined in t h e n es tchap ter . The g i d fo r the cornputational domain for the full shcramjet case is 600points in the axial direction and 240 nodes in th e transverse direction. 60 of thoseused in t he outer flow. for a total of 144,000 node points.

Figure 2.13: Schematic of the full shcramjet flowfield.


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Chapter 3Numerical Method

3.1 OverviewThe numerical algorithm used to solve the inviscid shcramjet flowfield is coverecthis chapter. The governing equations employed to represent the physical model andthe numericd method used to calculate the fl od el d are explained. Further. t h echernical model used to describe the combustion process of the 13 species is detailed.

The Lower-Upper Syrnrnetric GaussSeidel (LUSGS) method developed by Yoonand Jarneson (1987) is combined with a symmetric total variation diminishing (TVD)scheme developed by Yee (1987) and a chemicd combustion model proposed by Jachi-mowsky (1988). This combination results in a fully implicit, fully coupled, Newton-iteration TVD scheme which is used to solve the chemically reacting, non-equilibrium,inviscid flows at steady state. Dudebout (1996) developed and implemented theabove numerical method, and his algorithm is used in the numerical simulation ofthe shcramjet flowfield in the present study. This chapter will only surnmarize thenumerical method employed. For further details on the computational model and itsvalidation, the work of Dudebout (1996) is recommended.

The sections to follow outline the governing equations and numerical method usedin the present study.


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3.2 Numerical Method and Governing Equations3.2.1 Physical Mode1In the present st udy, the goveming equat ions employed are the simplified Xavier-Stokes equations in which the viscous terms axe neglected: that is. t he Euler equa-tions. The partial differential form the Euler equations (as seen be low) representsthe conservation of mas. rnomentum and energy equations for an inviscid. compress-ible gas mixture. For a chemically reacting, multi-species gas the governing Eulerequations written in conservational law form and expressed in curvilinear coordinatesbecomes:

where the conservative variables are defined as:


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the inviscid flux terms given by:

with the species production and axisymetric source terms as:

and th e contravariant velocities U and V, and the Jacobian J are expressed as:

With the velocity components in the x and y directions given as u and v, respectively,


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th e total energy, E, c m then be de he d as:

Th e parameter o s set to zero for this planar flow study. The term p k representsthe density of the species, where k ranges from l...n. These species include: H.O. OH1 HzO, HOz, H202, ,NO, HNO, and NO2; with p representing the totaldensity of the flow. These species conservation equations are further expressed interms of the conservation of elemental densities of H, O and N. This form assistsin the computational efficiency by reducing the number of species production termcalculations. The elemental densities of p#, pz and p g are expressed as:

where the total density is determined by th e sumrnation of the elemental densities:

The temperature is implicitly resolved by using an iterative Newton-Raphson proce-dure from the definition of total energy,

where k, ranging from l...n, represents the species: H2, *, H,0, H, HaO, HOa7H 2 0 2 , N , N 0 7HNO, Nz, nd NO2. For each species, k, the enthalpy is represented byHk nd includes the enthalpy of formation at 298 K. The enthalpy is obtained from


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the NASA polynomial curve fit of the JANAF (1971) thermochemical tables. given

The specific heat at constant pressure for each species k is determined from asimilar polynomial:

where the coefficientsa i , k ... ak can be obtained from Gardiner Jr . (1984 , -Mk isthe molecular weight of species k (including Hz, ?, 2) and R s the universal gasconstant. The pressure, P, is determined from the equation of state for a mixture ofthermally perfect gases:

The creation and destruction of species k is represented by the source term wband is determined through finite-rate chemistry. The hydrogenlair combustion mode1employed herein is further explained is section 3.4.

3.3 Numerical Method3.3.1 Shock-Capturing SchemeIn the present study, the hypersonic flowlield of the shcramjet contains a numberof strong discontinuities such as shock and detonation waves. Due to the presenceof these discontinuities, the use of Taylor series expansions to spatially discretizethe PDE becornes inapplicable. In order to solve this problem correctly, a numeri-cal technique which is stable and nonoscillatory near discontinuities must be used.Two echniques which satisfi the above criteria are shock-capturing and shock-fitting


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schemes.The shock-fitting scheme uses the Rankine-Hugonoit equations in solving the

shocks at locations known apriori. The dvantage of shock-fitting over shock-capturingis t hat the discontinuit ies are sharply resolved. But the requirement of the locationof the shock positions before hand leads to serious drawbacks in solving arbitraryflowfields. Shock-capturing methods, on the other hand. predict the location of thediscontinuities as a part of the solution. They do not require prior knowledge of theshoclc locations. Unfortunately? the drawbacks with this scheme is that the shocksare %mearedn over several nodes of the numerical grid. But, with sufficient gridresolut ion in the expected regions of the discont inuit ies, the numerical smearing canbe minirnized. This is the method employed in the present study.

A method known as a flux-difference split scheme is used to distinguish betweenupstream and downstream influences. A t each cell interface. t his scheme int roducesphysical properties of the flow into the discretization process by obtaining either t h eexact or approximate solution to the Euler equations. In order to avoid aphysicalsolutions as a result of oscillations at discontinuities, the concept of monotonicity byGodunov (1959) is instituted. A monotone scheme does not permit the creation ofnew local maxima or minima in the spacial domain. The Total Variation Diminishing(TVD) method is governed by two monotonicity preserving properties, and thesesatisfy the needs of higher order resolution schemes. The first property maintainsthat new local maxima or minima cannot be created. The second property followsthat the value of local a local maximum is non-increasing, and the value of a localminimum is non-decreasing.

Yee (1987) developed a symrnetricTVD schemewhich is used in the computationalmodel herein. Symmetry of this model results from its derivation from a centrdscheme and with added artificial dissipation. Compared to other, higher order upwiudTVD schemes, it has the advantage of requiring less operations: thus the reasoningbehind choosing it for the present study.

The Lower-Upper Symmetric Gauss-Seidel (LU-SGS) ethod, developed by Yoonand Jameson (1987), s the relaxation method used in the present study. This fully


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coupled, implicit scheme uses a Newton-iteration method to solve the Euler equationsat steady state. I t is specially designed for chemical reacting flows as the method isunconditionally stable in any number of spatial dimensions. Although it is an implicitscheme, it is no more costly than explicit schemes because block tri-diagonal inversionsare avoided. As a fully coupled scheme, it solves the gasdynamic and chemical reactionequations simultaneously and ha s increased stabili ty in the presence of strong couplingbet ween gasdynarnic and t hemochernical processes. Clearly. an adian tageous met hodto use in t h e presence of a detonation wave's coupling of gasdynamic and chemicaleffects.

Due to the difference in chemical and gas dynamic time scales. it is importantto solve the source term W implicitly a s above. This is because it is advantageousto avoid satisfying the stability of th e smallest time scale as this slows convergencerates. The criteria for convergence in the present study is determined by the changein the average value over al1 nodes of the conservative variable pu. This is given by:

where n represents the time step, M is the number of nodes in the i-direction, andN is the nurnber of nodes in the j-direction. When the value of L s reduced by fourorders of magnitude, convergence is assumed to have been met.

For a more complete description of the LU-SGS scheme, the work of Dudebout(1996) should be consuited.

3.4 Combustion/Chemical Reaction Mode1The present study utilizes Jachimowski's (1988) chernical react ion model, developedspecifi cdy for the National Aerospace Plane program, to simulate the combustionof hydrogen and air (i.e. nitrogen and oxygen). The finite-rate combustion processconsists of 33 seactions between the 13 species (Hz, 0 2 , H, O, OH, &O, HO2, H20 2,N,NO, HNO, Nz, nd NOz). Although vibrational relaxation processes and onization


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are not considered in the model, the effects of dissociation and chemical react ions are.This is because the temperatures in the shcramjet mode1 ar e not expected to becomehigh enough to promote ionization, but high enough to produce dissociation. Vibra-tional effects are ignored because their influence is considered negligible compared tothe other processes involved.

The single step, finiterate chemical reactions are govemed by the law of massaction which provides the rate of production of a species from the forward and back-ward reactions. These react ion rate coefficients are given in Table 3.1. In the react ionequations shown, M represents a third-body collision acting as a catalyst in t h e reac-t ion. For a more detailed explanat ion of the chemical reaction model used, the readeris referred to the work of Dudebout (1996).


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1 Keaction 1(1) H 2 + 0 2 + O H + O H(3 H + 0 2 + O H + 0(3) 0 + H 2 + O H + H(4) OH + Ha + H 2 0 + H( 5 ) OH + OH + H 2 0 + O(6) H + OH + M -t H 2 0 + hd(7 ) H + H + M + H 2 + M(8) H + O + M + O H + M(9 ) H + 0 2 + M + HO2 + M(10) HO2 + H + H2 + 0 2( 11 ) HO2 + H + O H + OH(12) HO2 + H + H 2 0 + O(13) HO2 + O + 0 2 + OH(14) HO2 + OH H20 + 0 2(15) HO2 + HO2 + H202 + 0 2(16) H + Hz02 -+ Hz + HO2(17) O + H 2 O 2 + O H + H O 2(18) O H + Hz02 + H 2 0 + HO2(19) M + H 2 0 2-t O H + O H + M(20 0 + O + M + 0 2 + M(21 N + N + M + N 2 + M(2 2 N + 0 2 + N O + 0(2 3 N + N O + N 2 + 0(24 N + O H + N O + H(25) H + NO + M -t HNO + M(26) H + HNO + NO + H2(27) O + BNO + NO + OH(28) OH + HNO + NO + H 2 0(29) HOz + HNO + NO + H 2 0 2(30) HO2 + NO + NO2 + OH(31) H + NO2 + NO + O H(32) O + NO2 + NO + O2(33) M + NO2+ N O + O + M

Table 3.1: Hydrogen-air reaction mode1 of Jachimowski. Forward reaction constantsgiven by kl = ATn e x p ( - E l R T )


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Chapter 4Results

4.1 OverviewThis chapter will present and anylize the results of the completed computationalsimulation of the shcramjet flowfield. To study the effects of the incomplete fuel-airmixing, a single case (M, = 14.1) will be used to detail the investigation of theflowfield. This Mach number case was chosen to be detailed to remain consistentwith the studies of Oppitz (1995) and Schumacher (1995), s it is the same Machnumber case which they analyzed in their respective works.

The analysis of this shcrarnjet mode1 will be accomplished through da ta extracteddong five equally spaced strearnlines and across a number of cross sections through-out the combustor and nozzle. From these the flow variables will be plotted, includingtemperature, pressure, entropy increase, Mach number, velocity, and the mass frac-tions of several of the involved species. The focus of the present analysis will beon the combustor and nozzle sections. This is because the combustor section is theprimary region of chemical processes in the shcramjet, and will show the effects ofthe incompletemixing on the formation of the detonation wave and of shock-inducedcombustion. The nozzle section is the main t h s t production component of the en-tire shcramjet and also the region where recombination of the chemical species occurs.Recombination has a substantial effect on the o v e r d performance of the shcramjetand therefore will be included in the investigation.


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Following this, the performance characteristics for the flight Mach range 9


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Figure 4.2: Com bustor cross-sect ions (Mx= L-L. Planar. qdy,= 1400 psf).

types of processes: s hock-induced combustion results w hen the combustion processdoes not influence th e preceding shock wave. as ignition occurs far enough d o m -stream: and secondly. t hrough a detonation wave generated when the ignition delayis short enough t hat th e combustion front couples with th e preceding shock rvave.Fig. -4.3 shows a plot o f th e tem pera ture contours within th e cornbustor section forthe Mach 14 case. The oblique shock. produced by th e combustor wedge. and thecombustion regions a re easily located in t his colour diag ram . -4 s an aid. Fig. -1.4details the exact location of th e shock-induced combustion and deto nat ion wave re-gions. In this diagram . t h e line AB represents the oblique shock produced by t h ecombustor wedge: t he line CD represents the combustion fronts. Th e point B. alongthe line CD. shows approx irnately where the shock-induced combustion region couples

Figure 1.3:x-Coordinatelm]

Temperature contours in combustor (Mm=14: Planar, qd,, =l4OO psf).43


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with th e oblique shock and becomes a detonation wave. Th is is easily determined,as the dis tance between AC and DC at this point (B) beco me s negligible. Also. achange in th e angle of th e shock f ront at B is due to the coup ling and influence of thecombust ion f ront , that is the beginning of the detonation wave. A slight curvatureof th e shock is dete ctab le throu gh th e f irst half of t he oblique shock, from point A toB. This curvature is du e t o t he variation of the density in th e flowfield as a result ofth e change in th e ma s fra cti on of fuel across it. Between points A and B, there is anoticeable delay in th e ignition of the fuel - the distance between the oblique shockand th e combust ion f ront - th is is the primary definition of shock-induced combus-t ion. Near the cowl wall, th e length f rom th e s tar t of the obl ique shock (A) to thebeginning of the combustion front is termed the inductance length . A s can b e seen.the inductance length for this case is approximately 20 cm; th i s compared to the3.0 cm inductance length of t h e uniformly mixed case studied by Op p i t z (19%). Dueto th e increased length of t h e induc tance region, th e overall length of th e combustorwil l be longer compared to that of a mode1 with a shor t inductance length . This isdu e to th e fact tha t th e shock f ront in th e shock-induced combu st ion region is no t ass t eep as t h a t in th e detonat ion region. kicreases in the combustor length , due to theincomplete mixingof th e fuel, will lead to overall performance losses du e to increasedweight, drag a nd cooling loads.

x-Coordinate [mlFigure 4.4: Combustor schematic of wave positions (M,=14, Planar, qd,=1400 psf).


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4.2.1 Flow Characteristics AnalysisThe analysis of the combustor section will begin by first investigating the flow pa-rameters, followed by an in-depth discussion of the species concent rat ions.

The temperature and pressure at the inflow to the combustor can b e found inFig. 4.5 and Fig. 1.7, respectively. The temperature is nearly 900 K across the Rotv:this temperature is considered to be the threshold for spontaneous ignition of the Hzfuel. .J. slight drop in the temperature towards the cowl side of t h e inflow (D ) can be

Figure 4.5: Temperature profiles at cross-sections in combustor ( M m = 14 ,Planar, qd, =1400 psf) .

attributed to a numerical smearing of the flow at end of the third compression shock.close to th e cowl wall. By looking at the Temperature along the ~ t r e ~ n i i n e s ,seeFig. 4.6), the combustor temperatures can be confirmed to be nearly 900 K. T he cowlsurface streamline passes quickly through the third compression shock and then thecombustor shock. Due o the smearing of the shocks over a number of grid nodes, itbegins at well below the 900 K target - at approximately 700 K - and then increasesquickly to over 1100 K a s it passes through the combustor shock.

A similar pattern can be seen in the pressure diagrams: the cross-section plots(Fig. 4.7) show the pressure to be approximately 25 kPa across the opening of thecombustor. Once again there is a decrease in the pressure towards point D, at thecowl w d . The streamline plots of pressure in Fig. 4.8 show t h e same behaviour asthe temperature plots: close to the wall the cowl surface streamline passes throughthe end of the third compression shock and is slightly lower than the other streams


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Figure 4.6: Temperature along streamlines t hrough combustor (Mm= .1 .Planar, qd,=1400 psf).

due to smearing. The variation in the composition of th e flow will also explain whythe pressure and temperature across the cross-section are not exactly the same. - 4 sthe fuel concentration changes across the flow, the molecular weight of the mixture ata certain point will change also. This results in a slight curvature of t h e compressionshocks, and hence a variable shock strength dong each shock. depending on thedistance normal to the edge of either side of the flowfield.

The outflow plot of the temperature frorn the combustor can be seen in Fig. 4.5.This shows that the region of the flodeld where there is little Hz uel, the middleof the flow, here is very little temperature increase due to the Iack of fuel to burn.Fig. 4.6 can be used to see the temperatures along the five streamlines as they passthrough the combustor section. At the middle of the flow, exiting the combustor,the temperature is a s low as 1700 K compared to that of 2900 K at the peaks ofth e temperature cross-section plots. Also, at the edges of the ffow, the temperaturesalso drops significantly - to about 2300 K at the cowl wall and 1600 K at the bodywall. This drastically lower temperatures are mainly due to the high concentrationsof fuel in the flow. This high concentration of H2 t the edge of the fiow, more than 3times the mass fraction of the homogeneously mixed case (Oppitz 1995), nhibits thecombustion process. In essense, the excessHz erves as a coolant near the walls. Bylooking at the streamline contours, the two streamlines that pass through the shock-induced combustion zone namely the Lines labelled SIC and COWIurface - there is


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a slight jump in the temperature profile to approximately 1200 K, and then a secondjump to over 2000 K. This first raise in temperature is due to the oblique shock. Thetemperature of the flow is increased a s it passes through the shock front, but t here isa short amount of time before the combustion process begins: this inductance lengthwas mention previously. Also, dong these two streamlines. there is a slow increasein temperature after the oblique shock. This is due to a compression of t h e flow.caused by the slight curmture of the oblique shock. This will be mentioned againwhen the pressure streamlines are discussed. The Mid st reamline passes t hrough themiddle of the flow but, as this diagram shows, there is only one area of increase in itstemperature. This would suggest that the streamline passes though the detonationwave area of the combustion process, as the shock and combustion front act as oneto increase the temperature.

At the body wall, the temperature increase is only 700 K, suggesting very littleburning of the fuel. This can be explained through a number of reasons. As ment ionedabove, the high equivalence ratio causes the amount of combustion of the fuel todrop. Compared to the cowl side of the flow, the equivalence ratio drops slowertowards the middle of the f loheld on the body side. This means there is more areaof flow experiencing "too much" fuel, and thus a larger area of lower temperature.Another reason for there to be so little increase in temperature at the body wallis that the detonation wave reaches the wall at the very end of the combustor; thebeginning of the nozzle section. By looking again at the combustor temperaturecontour plot (Fig. 4.3) , at point E, where the detonation wave m e t s the body wall,the hot combustion region begins to fd away from the detonation wave. The highconcentration of Hz n this area slows the rate of combustion. Since the nozzle sectionbegins shortly after this region, near point E on the shock, expansion of the flow causesthe temperature to &op and doesn't give the fuel enough time to completely combust.From the pressure cross-section plot (see Fig. 4.7)' the exiting flow has a significantpressure increase compared to that of the entering values. At th e middle of the flowthere is a slight dip in the pressure and this is due to the lower amount of fuel inthat region prior to combustion. With an equivalence ratio of less than one prior to


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1 - i,a0 t.940c CombustorOutflow58 iy 1.m; l E

1 a80r

IO' 'a 25000 Mo00 75000 1OOOOO 125000 150000Pressure [Pa]

Figure 4.7: Pressure profiles at cross-sections in combustor ( M W =4. Planar.qdvn=1400 psf).

ignition there is a significant drop in combustion. a s the temperature plots showed.This decrease in the amount of combustion leads to a lower pressure, although onlyslightly, at the mid-point exit ng the combustor. Furt her along th e cross-sect ion thereis a large jump in the pressure and this is attributed to the detonation wave. Thehigher pressures across the detonation wave are attributed to the shock wave couplingwith the combustion process, resulting in a stronger discontinuity (detonation wave).Following this, t here is a steep drop in the pressure near point E.This is due, again,to th e smearing of the detonation wave a t the edge of the nozzle entrance and to theIack of time for the combustion process to complete before expansion at the nozzleentrance. The plots of th e pressure along the-strearnlinesare found in Fig. 4.8. They

Figure 4.8: Pressure along streamlines through the combustor (M,=14, Pla-nar, qd,=1400 psf).


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also show a slow continuous increase after the oblique shock, due to the compressioncreated by the curvature along the shock itself. The inductance zone c a n be seenas well in the pressure plots of the SIC and COWL streamlines. The double peakalong the SIC plot is due to an expansion wave ust after the combustion begins inthis area. Effects of this c m also be seen in the Temperature and Mach numberstreamlines between 21.9 rn and 22.0 m.

Fig. 4.9 shows the Mach numbers of the streamlines through t h e combustor. Thefiow enters the combustor section at around Mach 5-7. Differences in the Machnumber across the flow are due to the variation of the molecular mass as a result ofthe fuel distribution. Exiting the combustor, the DW streamline dropped the largestamount in Mach number. This is because it passes though the strongest shock front(the detonation wave) and experiences the highest temperature raise. The Mid stream

L - T - iMid7.0E -

5 4.OB - cowlsurfa3.0EO -,

Figure 4.9: Mach number dong streamline through the combustor (M,=14,Planar, qd, =1400 psf).

has the smallest change in Mach number because it passes through t he weakest shock,the area where the least amount of burning is produced from a lack of fuel. Similartrends are f o u d in the velocity plots, Fig. 4.10. The smallest change is in the areaof the lowest equivalence ratio, and the stronger the combustion front the flow passesthrough, the larger the &op in velocity. The inflow of the velocity has a jump near thepoint D. As mentioned previously, this is due to the smearing of the extracted dataacross the third shock and oblique shock fronts. In the outflow data, the slight increaseat E is due to the smearing of the detonation wave at the end of the combustor.


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Figure 4.10: Velocity profiles at cross-sections in combustor (Mm= 14, Pla-nar, qd,=1400 psf).

The dimensionless change in entropy, defined as A S/RI for the combustor sectionk shown in Fig. 4.11. This increase in entropy is calculated as th e difference betweent h e freestream and combustor entropy. The diagram shows that the entropy increase

Figure 4.11: Dimensionless entropy change dong combustor exit cross-section (Mm= 14, Planar, qdF=1400 psf).

is smdest through the regions of Iowest equidence ratio ( 6 e O) , and also at theedges of the flow where there is high equivalence ratio ( 4 > 1). Near point F, theslight increase in entropy at the edge is due to the drop in the equivalence ratiocaused by numerical smearing at the boundry between imer and outer flows of theinlet. The total heat added to the flow, as a result of the combustion is depictedin Fig. 4.12. It c m be seen that, like the entropy plot, the lowest regions of heat


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addition correspond to the area where there is very litt le fuel (4 0) and where thecombustion is inhibited by high concentrations of fuel (4 > 1)- In fact, the points ofmaximum entropy increase and the points of maximum heat addition are at the sameplace across the flow. The amount of heat added to the flow (and entropy increase) islower on the detonation wave side because there is a lower arnount of H20 roductioncompared to th e shock-induced combustion region. This can be seen in t h e plot ofthe H20mass fraction (Fig. 4-20), which will be examined shortly. The drop in heatadded to the flow near the point E is also due to an increase in th e dissociation of Hzand Oawhich absorbs heat in the flow.

Figure 4.12: Heat added to the flow dong combustor exit cross-section(M,=14, Planar, qdw=1400 psf).

4.2.2 Mass Fraction AnalysisThe plots of entropy and heat addition showed that the shock-induced combustionhas higher entropy and heat added than the detonation wave. To understand whythis is so, the analysis of the m a s fractions of the reactants and products of thecombustion are analyzed. In presenting the profiles of the m a s fractions, only thosespecies with values greater than 10-6 are shown (except for N2). This leaves thespecies of H2, 2, , Hl H20, nd H202o be presented through Figs. 4.13-4.21.

The fuel distribution entering and exiting the combustor section can be found inFig. 4.13 with the streamline values in Fig. 4.14. The i d o w values show 4 x 3.2 at


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Figure 4.13: H mass fraction profiles a t cross-sections in combustor(M,=14, Planar, qdm=1400 psf).

the edges of the flow and c$ ir 0.1 toward t he middle of the flow. The drop in theHz ass fraction neor point F is due to leakage, or numerical smearing. in th e inletbetween the inner and outer flows. Increases in the O2 nd N2 concentrations occurdue to this a s well. The remaining fuel at the end of the combustor shows that thereis an incomplete use of the fuel at the edges due to its high mass fraction. From thediagram, there remains nearly two equivalence ratios of fuel or more at either edge(where 4 = 1 is 0.028 kg Ha/kg air). The lower arnount of fuel remaining on the SICside of the flow, as found in the case of Oppitz (1995), s due to the use of the fuelbeing more pronounced through the SIC area. The strearnline plots of the Hm a sfractions are a better demonstration of where the fuel is, or is not, being used. At

P4.OE-2

IL DW%O-2Q=- UK-2I1O-2 SICMid0-9,.m 21:15- 2 1 b - 2 1 4 5 212 2 1 i - P m -POb Pl0 Pl5

Figure 4.14: H m a s(M,=14, fractions alongPlanar, qdm=1400 streamlines through combustorpsf)


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the edges of the flow, where 4 > 3, the amount of fuel used along the BODY andCOWL streams is less than half of the entering values. The COWL stream drops toa minimum value of 0.041 kg H2/kg air (or # z 1.5) and then increases slightly asit approaches the end of th

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