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HYPERSONIC FUEL/AIR MIXING ENHANCEMENT B Y CANTILEVERED RAMP INJECTORS IN THE PRESENCE OF

WAVY WALLS

Derrick Alexander

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Graduate Department of Aerospace Science and Engineering University of Toronto

Copyright @ ZOO 1 by Derrick Alexander

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HYPERSON~C FUEL/AIR M~xING ENHANCEMENT BY CANT~LEVERED ~ M P INJECTORS IN THE PRESENCE OF WAVY WALLS

Derrick Alexander

Master of Applied Science Graduate Department of Aerospace Science and Engineering

University of Toronto

2001

Abstract

In an effort to develop hypersonic air-breathing propulsion systems wavy walls were added to

a cantilevered ramp injector to increase the fuel / air mixing in a shock-induced combustion

ramjet (shcramjet) engine. Numerical snidies of various wavy wall configurations in the vicin-

ity of the cantilevered camp injector were undertaken using three-dimensional, multispecies

Navier-Stokes solvers. Laminar simulations established the amplitude of the wavy walls has

a much greater influence upon the resulting flow field than the wall wavelength. The mixing

initialIy increases with amplitude and then decreases as the shocks formed from the wavy wall

dismpt the main mixing vomces in the flow. The addition of wavy walls allows for an in-

crease in mixing efficiency of approximately IO%, but in the best case incurs the sarne degree

of losses. Subsequent turbulent studies demonstrated similar flow fields to the laminar cases.

with greater the mixing due to increased diffusion, but at the cost of greater Iosses. As such the

mixing efficiency vs. total pressure loss ratios found in the turbulent cases are worse than those

found in the laminac cases. It was also found that the effect of the wavy waU is reduced with

increasing boundary layer height, but the mixing is augmented due to greater dissipation in the

slower tlow. This sîudy suggests the addition of wavy walls to cantilevered ramp injectors rnay

not be desid, since tbey do not provide a sigdicant benefit with minimal detrimental effects.

First of dl , 1 would like to thank tu my thesis supervisor, Prof J.P. Sislian for his excellent guidance and encouragement provided throughout the duration of my Master's program.

1 would also Iike to gratefully acknowledge my frimily and friends, whose help and encouragement have been unfailhg over the course of my studies.

Finaily, 1 would like to thank al1 the hculty, staff, and students at UTIAS for their help and sup- port. Speciai ehanks to my colleagues, namely Jurgen Schumacher, Bernard Parent, Giovanni Fusiaa, Viorel Turcu, Jason Etele, Emothy Hui, and Tom Schwartzentntber for their fiend- ship, helpful suggestions, and IhitfuI discussions during the course of my research program.

Abstract ii

Acknowledgemenb iii

List of Tables vii

List of Figures viii

Nomenclature xi

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . - . . . . 1 1.2 Scope of Current Study . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . 5

1.2.1 Objective md Approach . . . . . . . . . . . . . . . . . . . . - . . . - 5 1.2.2 Overview of the Present Study . . . . . . . . . . . . . . . . . . . . . . 7

2 Numericd Method: Laminar Flow 8 2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . - 8 2.2 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . - - . 13 2 3 Spatial Discretization . . . . . . - . . . . . . . . . . . . . . . . . . . . . - - . 15 2.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . - - . 17 2.5 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . - . . 19

3 Numerical Method: lhrbulent How 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Governing Equations 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Spatial Discretization 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Temporal Discretization 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Boundary Conditions 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Convergence 27

4 Overview of Cases Studied 28 . . . . . . . . . . . . . . . . . . . . . . 4.1 Shear Layer Mixing with Wavy Walk 28

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Background 28 . . . . . . . . . . . . . . . . . . . . 4.1.2 Simulation of Shear Layer Mixing 29 . . . . . . . . . . . . . . . . . . . 4.2 Design Parameters of Cantilevered Injectors 31

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Injector Geometry 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Flow Field Properties 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Grid Generation Algorithms 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Laminar Flow Solver 35 . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Turbulent Flow Sotver 38

5 Flow Field Results 42 5.1 Flow Fields with Changes in Wavy Wall Amplitude and Wavelength . . . . . . 44

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Pressure Fields 45 . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Flow and Fuel Jet Behavior 49

. . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Representative Streamlines 57 . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Boundary Layer Effects 59

5.2 Flow Fields with Turbulence and Changes in Bomdary Layer Height . . . . . . 60 . . . . . . . . . . . . . . . . . . . . 5.2.1 Pressure Fields and Flow Behavior 60

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Fuel let Behavior 64

6 Flow Field Analysis 7 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Circulation 76

. . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Baroclinic Vorticity Generation 79 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Mixing Efficiency 81

. . . . . . . . . . . . . . . . . . . . . . . 6.4 Maximum Hydrogen Mass Fraction 85 . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Fuel Jet Cross-Sectional Area 87

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Fuel Jet Penetration 89 . . . . . . . . . . . . . . . . . . . . . . . 6.7 Mass Averaged Total Pressue Loss 92

7 Conclusions and Recommendations 96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Summary and Conclusions 96

7.2 Suggestions for Further Research . . . . . . . . . . . . . . . . . . . . . . . . . 99

A Flux Jacobian Matrices 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.l Laminar How 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Turbulent Flow 102

Bibliography 105

4.1 Cantilevered Ramp hjector Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 Air md Fuel Freesueam Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.1 Wavy Wall Parameters of Cases Examined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

vi i

7 Shock-Induced Combustion Ramjet (SHCIUMJET) Configuration . . . . . . . . . . . . . . . . . . Ramp injector Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 CantiIevered Ramp injector Vortex Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Caatilevered R m p Injector Shock Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Cantilevered Ramp hjector with Wavy Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Baroclinic Vonicity Generation in the Presence of a Wavy Wall . . . . . . . . . . . . . . . . . . . 6

Velocity Contours in Flat Wall Duct Gilreath et al . Basdine Case . . . . . . . . . . . . . . . . . 31 Velocity Contours with the Addition of a Wavy Wall. Gilteah et al . Prirnary Case . . . . . . . . . 31 Cantilevered Ramp injector Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Laminar Fiow Solver Computational Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Computational Grid in Domain 1 for Laminat Flow Solver . . . . . . . . . . . . . . . . . . . . . 38 S m w i s e Plane of the Computational Grid for the Turbulent Flow Solver at the injector Sym- meuy PIane(x3 = Ox = Om) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lo Spanwise Planes of the Computational Gnd for the Turbulent flow Solver . . . . . . . . . . . . . 40 Streamwise Planes of the ComputationaI Gnd for the TurbuIent Flow Solver . . . . . . . . . . . . 41

Convergence History for Case 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Pressure Contours at hjector Edge . (.r3 = 0.5.t = 0.01m). Case 1 . . . . . . . . . . . . . . . . . . 45 Pressure Contours at injector Edge. (x3 = 0.52 = 0.01m). Case 2 . . . . . . . . . . . . . . . . . . 45 Pressure Contours at hjector Edge . (x3 = 0.5X = 0.0 lm). Case 3 . . . . . . . . . . . . . . . . . . 46 Ressure Contours at injector Edge. (x3 = 0 5 = 0.Olm). Case 4 . . . . . . . . . . . . . . . . . . 16 Pressure Contours at hjector Edge. (.r3 = 0.5.t = 0.0 lm). Case 5 . . . . . . . . . . . . . . . . . . 46 Pressure Contours at injector Edge. (x3 = 02% = 0.01m). Case 6 . . . . . . . . . . . . . . . . . . 46 Pressure Contours at Amy Symrneuy Plane. (x3 = 1.V = 0.03m). Case 4 . . . . . . . . . . . . . 47

5.9 RecKculation Regioii at XI = O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.10 Pressure dong Represenrative Fuel Stredines. Cases 1-4 . . . . . . . . . . . . . . . . . . . . . 49 5.1 1 Velocity Vectors atxi = Om. Case L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.12 Hydrogen Mas Fraction Contours for Case I (a, = QT . w = E) . . . . . . . . . . . . . . . . . . 51 5.13 Hydrogen Mass Fraction Contours for Case 4 (a, = 0.2 187f. w = IO.at) . . . . . . . . . . . . . 51 5.14 Spanwise Veiocity Vectors and Hydrogen Mass Fraction Contours at XI = 17.5.t . . . . . . . . . . 53 5.15 Hydrogen M a s Fraction Contours at hjector Edge. (.r3 = 0.5.f = 0.01m). Case 1 . . . . . . . . . 53 5.16 Hydrogen M a s Fraction Contours at Injector Edge. (x3 = O.% = 0.OIm). Case 2 . . . . . . . . . 53 5.17 Hydrogen M a s Fraction Contours at Injector Edge. (x3 = O.% = 0.0 lm) . Case 3 . . . . . . . . . 54 5.18 Hydrogen Mass Fraction Contours at Injector Edge . (x3 = 0.5.t = 0.0 Im) . Case 4 . . . . . . . . . 54 5.19 Hydrogen Mass Fraction Contours al Injector Edge. (x3 = 0.5 = 0.0 lm). Case 5 . . . . . . . . . 54 5.20 Hydrogen Mass Fraction Contours at [njector Edge . (x3 = 0.52 = 0.0 Lm) . Case 6 . . . . . . . . . 54 5.21 Spanwise Velocity Vectors and Hydragen Mas Fraction Contours at XI = 25.0.f . . . . . . . . . . 55 5.22 Streamiines from Fuel in Smng Vortices: Originating at (2.5.f. tX. O.25.f) . . . . . . . . . . . . . 58 5.23 Velociry Contours at hjector Edge. (x3 = O.5f = 0.0 lm) . Case S . . . . . . . . . . . . . . . . . . 59 5.24 Velocity Contours at the Amy Symmetry Plane. (r3 = 1.S = 0.03m). Case 1 . . . . . . . . . . . 59 5.25 Pressure Contours at injector Edge . (x3 = 0.5.t = 0.0 lm) . Case 7 . . . . . . . . . . . . . . . . . . 40 5.26 Pressure Contours at Injector Edge . (x3 = 0.V = 0.0 lm) . Case 8 . . . . . . . . . . . . . . . . . . 61 5.27 Pressure Contours at Injector Edge . (13 = 0.51 = 0.0 1 m). Case 9 . . . . . . . . . . . . . . . . . . 61 5.28 Pressure Contours at Injector Edge. (r3 = 0.5.f = 0.0 lm) . Case Ib . . . . . . . . . . . . . . . . . 61 5.29 Velocity Vectors atxi = Om . Case 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.30 Velocity Contours at hjector Edge . (x3 = 0 3 = 0.0 lm) . Case 8 . . . . . . . . . . . . . . . . . . 63 5.3 1 Pressure Along Representative FueI Saeanilin es. Cases 7-9 and Zb . . . . . . . . . . . . . . . . . 64 5.32 Hydrogen Mass Fraction Contours for Case 8 (a ,. = 0.0655.F. w = IO.Qt) . . . . . . . . . . . . . 66 5.33 Velocity Contours at injecter Edge . (x3 = 0% = 0.01~). Case 8 . . . . . . . . . . . . . . . . . . 67 5.34 Velocity Contours at injector Edge . (x3 = 0.5X = 0.OIm). Case 9 . . . . . . . . . . . . . . . . . . 67 5.35 .r Location of Strong Vortex Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 536 Hydrogen Mass Fraction Contours at xi = 2.52 = 0.05rn . . . . . . . . . . . . . . . . . . . . . . 69 537 Hydmgen Mass Fraction Contours atxi = 5.E = 0.10m . . . . . . . . . . . . . . . . . . . . . . 69 5.38 Hydrogen Mass Fraction Contours at xi = T5.f = 0 . 15m . . . . . . . . . . . . . . . . . . . . . . 70 5.39 Hydrogen Mass Fraction Contours at XI = 10.Qf = O.2Om . . . . . . . . . . . . . . . . . . . . . . 70 5.40 Hydrogen Mas Fraction Contours at x [ = 12% = 015m . . . . . . . . . . . . . . . . . . . . . . 71

. . . . . . . . . . . . . . . . . . . . . . 5-41 Hydrogen Mass Fraction Contourç a& xi = 17.52 = 0J5m 71 5.42 Hydrogen Mass Fraction Coatours at XI = 22% = 0,45m . . . . . . . . . . . . . . . . . . . . . . 72 5.43 Hydrogen Mass Fraction Contuurs at xi = 27.55 = 035m . . . . . . . . . . . . . . . . . . . . . . 72

. . . . . . . . . . . . . . . . . . . . . . 5.44 Hydrogen Mass Fraction Contours atxi = 32.V = 0.65m 73 5.45 Hydrogen Mass Fraction Contours at xi = 375.t = 0.75rn . . . . . . . . . . . . . . . . . . . . . . 73

. . . . . . . . . . . . . . . . . . . . . . 5.46 Hydrogen Mas Fraction Contours at .K r = 42.% = 0.8Sm 74

. . . . . . . . . . . . . . . . . . . . . . 5.47 Hydragen Mass Fraction Contours at xi = 47.%= 0.95m 74

. . . . . . . . . . . . . . . . . . . . . . 5.48 Hydrogen Mass Fraction Contours at xi = 52.5f = 1.0Sm 75

6.1 Circulation ( Non-diensionaiized by VI-a ) for Cases 1-6 . . . . . . . . . . . . . . . . . . . . . 77 . . . . . . . . . . . . . . . 6.2 Circulation ( Nondimemionaiized by VI.&) for Cases 1.7.9. and 2 b 78

6.3 Baroclinic Source Term ( Non-dimensioaalized by (pP). ) for Cases 1-6 . . . . . . . . . . . . . . 80 6.4 BarocMc Source Tem ( Non-dimensiodized by (pP). 1 for Cases 1.7.9 . and 2 b . . . . . . . . 81 6.5 Mixing EBictency for Cases 1-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.6 MixingEffiçiencyforCases1.7.9. and2b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

. . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 M&ximum Hydrogen Mas Fraction for Cases 1-6 85 6.8 Maximum Hydrogen Mass Fnction for Cases 1.7.9. and 2 b . . . . . . . . . . . . . . . . . . . . 86

. . . 6.9 Fuel Jet Cross-Sectional Area ( Non-dimensionalied by initiai Fuel Jet Area ) for Cases t -6 88 6 . I O Fuei let Cross-Sectiooal Area ( Nondiiensionatized by laitid Fuel let Area ) for Cases 1. 7.9 .

and2b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.1 1 Jet Penetration for Cases 1-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.12 let Penehation for Cases 1.7.9. and 2b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.13 Mass Averaged Total Pressure for Cases 1-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6 . I4 Variation in Mass Averaged Total Ptessure and Mixing E fficicncy with Wavy Wall .4m plitude and

. Wmlength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . 94 . . . . . . . . . . . . . . . . . . . . . . . 6.15 Mass AveragedTamlhssmforCases 1.7.9.andZb 95

7.1 Possible Wavy Wall hplemeantion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Subscripts and Superscripts

dimension index dimension index dimension index dimension index species index species index tirne step index effective tenn including conmbuaon of nubutence turbulence term enay condition

Roman Symbols

tinearization ma& of convective flux, F sound speed amplitude of wavy wall linearïzation mauk of -ion rerm, C linearizatian ma& of source term. S mass fraction ofspecies j specific heat at constant pressure LüSGS operator discretkation operator in the cp dimension molecular diguson coefficient of species j

binary diffusion coetticient berween species j and m distance h m a wall to the k t intenorceii total energy of Eow interna1 energy convective flux vector difiùsion term limiter funcrion specific enthalpy identity matrix ce11 index in dimension ip metric Jacobian conductivity maaix turbulent kinetic energy LUSGS opentor Mach number molar of species j total nwnber of dimensions in the flow problern total number of gas species in the fiow

PR-

mass averaged total pressure turbulent kinetic energy production rem Prandtl number coaservative variables denotes a general variable residual universal gas constant source term

enwPY Schmidt number temperature

Ume LUSGS operator contravariant vetocity in the cp dimension veIocity component in the x, direction wavelength of wavy wall generalized coordinatte direction in the q dimeasion Cartesian cootdinate diction in the g dimension

Greek Symbols

a jump m chatacteristic speed

0 memc function

upwind Limiter function nond-mensioaalized cimiaaon value Kronecker delta entropy correction factor c o m t muring efficiency memc funcrion c h e d conductivity e i g e d u e of the Iinearizatioa matrix A dyaamic viscosity eigenvaiue hction convergence parameter density collision diameter of species j mole fraction of species j specitlc dissipation rate of tucbulent kinetic euergy non-diiensionalized baroclinic vonicity generation value temperature dependent polynomials

Background and Motivation

One of the main concems in the aeiospace industry today is the advent of better and cheaper

access into earth orbit. This demand will ody increase as communications, global positioning systems, and surveying systems increase the demand for satellite launches. Further d o m the

road is the possibility of opening up space to tourism and the dream of manned exploration. in addition there has for years been hope of estabiishing high speed civilian transports. The global outiook in life means interaction and commerce ail over the world demanding vehicles that wiil allow fast transfers to any poht on the globe.

The main obstacle to this bright future is the cost of launching anything into space, or on

a sub-orbital flight path. The reason for this is the continueci reliance upon disposable rocket technology. One of the most viable alternatives in high velocity flight is the use of hypersonic air-breathing engines such as supetsonic ramjets (scramjets) or shock induced combustion mm- jets (shcramjets). These allow for the use of the air in the atmosphere as the oxidizer over the majority of the flight path and castaway the requirement for oxidizer to be carried on board a vehicle. ApproximateIy sixty percent of a rocket's take-off weight at present is the weight of the oxidizer. Not having to c a q thïs weight and the weight of the oxidizer's subsequent sup- port structure means instead larger payioads, more passengws, or more systems can be carried. Additionally sçramjet and shcramjet engines are empioyed in reusable plane like machines

which in itself makes the cost, per kilogram of payioad, drop drarnatically.

While the benefits of using such engines are great, so are the difficulties associated with irnplementing them. The air must be kept at supersonic speeds while traveiing through the engine in order to prevent the huge tosses and heating that would be associated with slowing

the air to subsonic speeds as in a ramjet engine. The longer the length of the engine that is to perform the burning the larger the weight that must be carrieci, the heat load that must be

dissipated, and the drag that must be overcome. Ultimately this means the shortest engine possible is desired. Unfortunately coupüng this requirement with that of supersonic air flow

dictates that the air and fuel wi11 traverse the engine in times on the order of milliseconds."

in a typical scramjet the air entes and is compresseci in the inlet by various shock structures, fuel is injected, mixed, and combusted with the air in the combustor and then exhausted through a divergent nozzle providing the engine thnist. in a scramjet engine the fuel mixes and bums in the combustor, but in a shcramjet engine, as shown in Figure 1.1, it is desired to premix the fuel and air prior to a quick detonation wave induced buming region. This is simply a shock which wiIl increase the temperature of the 0ow past the combustion temperanue of the fuel, typically hydrogen which combusts at approximately 900 EC, and provide a quick and consequently more

efficient buming region? For either engine one of the main difficulties lies in the mixing of the fuel and air that must be accomplished at a molecuiar level in order for combustion to occur.

incompiete mixing results in incomplete combustion and detenoration of thnist performance.

Combustor

- - - - - - - - _ _ _ _ _

Figure 1.1 : Shock-lnduced Combustion Ramjet (SHCRAMJET) Coafiguration.

The mixing problem is futher complicated by the fact that fuel injection parallel to the main air flow direction is desired. This is due ro the fact t&at the thnist obtained h m the injected fuel can become a significant portion of the overall ttinist at the high velocity end of the engine's

CHAPTER 1. INTRODUCTION 3

operating range, with nearly al1 of a scramjet engine's thrust cornes from the injected fuel once

speeds near Mach 15 are reached." Additionally the use of normal fuel injection or oblique

injection gives rise to shocks and total pressure losses that cm produce severe temperatures and

counteract the advantage gained by the greater mixing produceci. Mixing of parallel streams at the high Mach numbers of the engine is made diilicult by compressibility effects which Iimit shear layer ~nixing." As such, injected fuel wiü only spread very slowly into the sunounding

air. To combat this problem &mg augmentation devices are desired. There are a variety of devices available as outlined by Drummond and Carpenter16 and Pratt and ~eiser," but the ones that incoprate the use of fuel thnist by quasi-parallel injection and produce low losses, with a minimum complexity, are ramp injectors and low angle injection systems.

Rarnp injectors are vortex generation mechanisms in which the air is entrained into the fuel via vortices created by the injector and thus the fuel and air are mixed. While mixing may be increased by the production of vortices, it shouId be kept in mind that the energy needed for

their creation rnust be extracted fiom the air flow. Therefore, it is desired to create vortices right at the fueVair boundary where they will enhance mixing, and not in other regions of the flow where the vortices will only represent an energy Ioss. Numerous ramp injectors configurations have been inve~tigated, '~~ but while they pmmoted good mixing with datively low losses

they did not reach the desired mixing levels. tn order to enhance the behavior of rarnp injectors a new variation was advanced at the University of Toronto institute for Aerospace Studies. The

"cantilwered ramp injector" was studied by Schumacher,' and found to enhance the mixing

characteristics of normal m p injectors.

Air Fiow

Expansion Trough

~ m t i l e v d ~njector>

V Y Fuel injection

(a) Conventionai Ramp injector (b) CantiIevered Ramp injector

Figure 1.2: Ramp hjector Arrays

Vortices in such hjectors are generated in two main ways," by u t i l h g the pressure differences around the injector and through the baroclinic effect. The injectors are mounted in a paralle! array fashion on the wall of the engine, as shown in Figure 1.2. Different configura-

tions exist: the waü can expand away h m the flow direction, the injector can protmde into the Row, or there can be a combination of the two. The air which flows on top of the injectors is in a higher pressure zone than the air which flows in-between the injectors, thus the air will "spili" off into these lower pressure regions creating vortices beside the injected fuel. in the case of the cantilevered injector the fiel injection region is raised off of the wall leaving a cavity under

the injector strut. This allows the cross-stream shear to create an addition stmng vortex pair in the cavity under the injector. Thus the fuel now has air on al1 sides, instead of just three, and a much stronger vortex pair is in intimate contact with the fuel region, as show in Figure 1.3.

Figure 1.3: Cantilevered Ramp Injector Vortex Formation.

Air flow +

Shock

Figure 1.4: Cantilevered Ramp injector Shock Formation.

The second main vortex generation mechanism is the barociinic effect. If the wall is ex- panded away fiom the mean flow direction a shock will be formed when the wall is tumed back parallel to the flow, see Figure 1.4. This shock then interacts with the density gradient between the low density fbel and bigh density auto provide vorticity via the barodinic effect. Vorticity is produced when the pressure and density gradients are misalligneci:

where: o z V x u = vorticity, p = density, P = pressure

While conventionai ramp injectors rely upon baroclinic torque for vortex generation, the

presence of the fuel injectors themselves limit shock formation to the regions between the

injectors. By removing the fuel injection point away fiom the wali in the cantilevered case

the air flow experiences a turning angle dong the whole span of the injector array, and as a consequence a much stronger shock is formed. This will in tum lead to much stronger vortices being fonned via the baroclinic effect. These changes were show to enhance mixing of the

fuel and air, by the increased vortex production in the region of the fuel 1 air interface.'

1.2 Scope of Current Study

1.2.1 Objective and Approach

One major drawback of both the conventional ramp and cantilevered ramp injector designs is the fact that the vortex generation mechanisms, used in turn to mix the fuel and air, are d l at the start of the fuel injection region. They rely on the vortices to continue entraining air downstream, while in fact the vortices diminish in strength as they propagate. The current work examines the effect of addig a sinusoicial wall downstrearn of a cantilevered rarnp injector, as stiown in Figure 1.5, in order to obtain a continuing vortex generation mechanism.

Air flow

Figure 1.5: Cantilevered Ramp injector with Wavy Wall

in supersonic flow over a wavy wall there is a positive pressure gradient on the h n t of the

wave and a negative pressure gradient on the back of the wave. Thus there exist alternathg

compression and expansion regions in the 0ow over such a wall. By using a wavy waii the

continuously aiternating pressure gradients will interact with the density gradient between the

hei and air to produce vorticity via the bmclinic e fkt . This is shom schematicaily in Figure 1.6. It was postulateci that the continuously altemathg vorticity in the region of the species interface would inmase the bulk mixing of the hel and air. The rnolecutar mixing needed for combustion would thus be increaseâ due to increased species gradients and an increased interface area between the gases allowing greater diffiision to occur.

Air

Figure 1.6: Barociinic Vorticiry Generation in the Presence of a Wavy Wall

This work v d e d this postdate and examined the effect of several different configurations of the wavy wall. The prirnary effort was to study the effects of ihe amplitude and wvelength of the wall on the mixiig in the flow. Initially, the study was undertaken nurnericaily using a laminar, three dimensionai, Navier-Stokes flow soiver developed by chum ma cher.' The studies were undertaken as non-reacting, representative of shcrarnjet injection where combustion oc-

curs after the mixing region. The mixing increase and the consequent flow losses were found and compared to the cantilevered injector without a wavy wail. A preliminary attempt was be made to determine the preferred amplitude and wavelength quantities, in terms of the best mixing enhancement. Subsequently, the effect of turbulence was examined by incorporating a different fiow solver, containing a two-equation k- o turbulence mode1, developed by Parent.' The additionai effect of the bomdary Iayer height on d g and the preferred wave pfoperties was examine&

1.2.2 Overview of the Present Study

The study perfomed is presented as follows. Chapter 2 outIines the laminar flow solver developed by Schumacher, including the relevant spatial and temporal discretization processes.

Chapter 3 outlines the turbulent flow solver developed by Parent where it differs in formula-

tion fiom the laminar solver. A Iimited shear-Iayer study is presented in Chapter 4 in order to quanti@ the shear-layer effects on mixing in cantiIevered ramp injectors. Also provided in this chapter is the setup of the cantilwered ramp injector, inciuding the injector geometry and the

flow field properties, and an exphnation of the grid generation algorithms used in the numerical processes. The results of the various simulations perfomed are stiown in Chapter 5. Pictures of different flow field variables are used iliustrate the relevant flow patterns and behavior. The various cases were anaiyzed and performance paramecers were found in order to quantify the differences in flow mixing. These parameters are outlined and compared in Chapter 6. Final conclusions fiom the study and recommendations for future investigations are given in Chapter

7.

2.1 Governing Equations

The Navier-Stokes equations can be expressed in their conservative fom in generalized spatial coordinates as:

The consemative variable, Q, the convective flux in the cp dimension, F,, the diffusion term, G, and the source t a , S, are given as:

ïhe metric Jacobian, J-', used in the transformation fiom normal Cartesian coordinates to generalized coordinates is expressed as:

The following notation is used:

nd is the total number of dimensions in the flow problem. n, is the total nurnber of gas species in the flow. X, is the generalized coordinate direction in she rp dimension. x, is similarly the Cartesian coordinate direction in the cp dimension.

v, represents the velocity comporient in the x, duection. V, denotes the convavariant veIocity in the cp dimension, defined as:

X,., stands for the denvative:

The total density, p, is expressed as the sum of the species densities and E is the totd energy, a sum of the internai energy, e, and the kinetic energy.

The pressure, is determineci fiom the equation of state, using Dalton's law of partial pressures.

where the values of Ip, the universal gas constant, and Mi, the molar mass of species j, are given in McBride et al. z6

in the governing equations, Equation 2.1, the conductivity ma&, &,,, can be expressed as

- --

follows.

where Dj*,I is the molecular diffusion coefiicient of species j, ,u is the dynamic viscosity, hi is the enthalpy of species j, K denotes the thermal conductivity, and 0 and P are me& functions.

(2.9)

(2. IO)

where in this expression 6;; is the Kmuecker delta, giving a vahe of i for y = r and O in al1 otber cases.

The temperatures experienced around the cantilwered rarnp injectors are far below the values needed for ionization and thence such effects are not considered. The static temperature, T, in the flow is implicitIy deteminecl by a Newton-Ebphson iteration procedure. Ushg ther- modynamic relations the function f (T) is minimized over n steps, where n in this case is not the n step in time.

a f v ) where: f = - ar

The specific enthalpy, k, entropy, s, and specific heat at constant pressure, cp = %/aT, for a gas mixture are detemùned as m a s weighted average5 of the individual species values. The species d u e s are detennined h m NASA polynomid w e fits of JANAF thermochemicd

tables, with the coe~cients aij. . .ai., found in McBride et

ns R a3 i 3 04.; a5.j 5 h = xc ,h , with h, = - ( a i J ~ + - ~ 2 i - ~ + - T ' f - T +ab, , ) (2.11) ,= 1 % 2 3 4 5

where c, = p,/p is the mass fraction of species j. The dynamic viscosity of a mixture of species, h k , is determined from Wilke's mixing

mie.24

w here

The species mole tiaction, x,, is detennined by:

The species dynamic viscosity, p,, is derived h m kinetic theory assuming that species j is a pure gas.

where a, is the collision diameter of a gas moleeule, in m, and is a temperature dependent poIynomiai representing the reduced collision integrai of the molecule. Both tenns are

tabulated in ~ardiner."

The thermal conductivity of a mixture of gases, K, is found in a similar fashion to the dynamic viscosity, and is given by the Mason and Saxena re~at ion.~~

The species thermal conductivity, Kj, depends on whether the gas is monatrnoic or polyatomic.

For the rnonatomic case only the translational energy is considered, but in the polyatomic case the conmbution of the vibrational energy is added, with the use of the Eucken correction.

15 R for a rnonatomic gas Kj = O -3 5 4 ~ . , (2.19)

R. for a polyatomic gas

The molecular diffision coefficient for species j h u g h a mixture, DjSmol, can be expressed

as the following.

a,,, is the binary diffision coefficient between species j and m, expressed in m2/s.

where the temperature is in K, molecular weight is in kglmol, the collision diameter, cri, is in nm as in the derivation of the species dynamic viscosity, Equation 2.17, the pressure is in

Pa, and a::, similar to fin, is a temperature dependent polymmial representing a collision integra~.'~

2.2 Temporal Discretization

in order to obtain a solution to the goveming equations they are integrated in tirne. impliçit ktegration is used due to its hck of stringeut stabiIity boundaries. This allows large time steps to be used in order to quickly converge a solution ta its steady-state result. The implicit discretization of the goveming equations is denoted as the following.

ïhe superscripts n and n + 1, denote the tirne step upon which the various terms are evaluated and At is the change in time for the step. The noîation for the consemative variables is given in delta form as:

Thus once found the value of EiQ" is added to the old solution, Q", to give the new solution in time, Q~". This requires the evaluation of tenns at the next time step, n + i. in order to maintain an eficient solver only the convective flux tenns will be evatuated at the new time s tq . This is done by the linearization of the conveceive terms about the previous tirne step.

The flux Jacobian matrices, 4, are given explicitly in Appendix A and are defined as:

By incorpomting Equation 2.24 into Equation 2.22 and ignoring the terms of second order or mater the time stepping aigorithm can be writtzn in delta form.

where R is the midual.

In order tu aiieviate the r e q h e n t of inverting the matrix multiplying SQ", wùich is higidy irnpracticai in a three dimensiond problem, a factorization method is sougtit, Yoan and ame es on'^ introduced an implicit algorithm using Lower-upper factorization, LU, and Gauss- SiedeI relaxation, GS. This process is outiined with symmehic Gauss-Siedel relaxation, SGS, in Yoon and ICwakZ9 and can be written as:

where

The flux Jacobian matrices are split such that hey depend on the characteristic directions. The + matrices are designed to have solely non-negative eigenvalues and conversely the - matrices bave solely non-positive eigenvafues. This is done as follows:

t A; = -(&*< 7 maxi1 A(%) 111) (2.32) -

The 5 term is a constant, greater than 1 , used increase the diagonal dominance and thus the stability of the scheme. The A(&) ternis are the eigenvalues of the matrix &, found hm:

det (A, - h,,,I) = O (2.33)

where h,, is any eigenvalue. The inversion of the implicit ternis is perfonned in three separate steps in the LU-SGS

scheme.

6Q' = -&L-'R

SQ" = DSQ*

SQ = U-'SQ"

A Newton-lie iteration is obtained by letting & + a. It is not Strictly a Newton iteration

due to the approximate factorization, but Lmear convergence can be show^.^^ The discretization operators are chosen to be the following.

where the notation q, denotes the variabIe qq at the point i,.

Wth this discretization Equations 2.29 to 2.3 I reduce to the following form.

where

This allows for the use of scaiar inversions of the matrices in Equations 2.39 to 2.41 and

consequently greatly reduces the computatiod load performing the integration in time.

2.3 Spatial Discretization

The residua! of Equation 2.27 is decomposed into two parts, the convective residuai, K,,, and the non-convective residual, R,,.

where

Rd "d Y Rmo, = 1 DqFi and Ln = - 1 D,+, (K&D,,G~) - Sn (2.44

Al1 the non-convective rems are discretized on ceti-centered 2nd order difference stenciIs.

The discretized of the convective ternis is performed by a t o d variation diminishing, TVD, scheme by ~ee? ' The advantage of Yee's scheme over other hi&-order schemes is its high

resolution of shock and contact discontinuities. which is essential in the simulation of hypersonic flow where many strong shock regions will be fourid. There are two variations of this scheme, a symmetric and an upwind scheme. The upwind scheme, in which the numerical

dissipation depends upon the direction of the characteristic speeds, while having the tendency to be slightly more unstable, can provide sharper resolution of shock discontinuities, The full

upwind discretization is as follows.

in order to exactly maintain freestream properties when highiy distorted meshes are used the metric terms in Equation 2.45 mut be evaluated at the ce11 interfaces, i, + 112, and not at the ce11 centered locations, iv and i, + I ." The meaic Jacobian at the ce11 boundary, G:,,~, is taken as the average of the adjacent ce11 centered Jacobian values.

The Rb+ll2 tenns are matrices whose colurnns are the right eigenvectors of the flux Ja- cobians, d;q,l/z, which is constructed using a Roe average to give the value at the interface,

Qi,+,2, fiom the states Qiq and Q,+I. The scheme is second order accurate except at discontinuities where the first order rnonotonicity of the Roe scheme is maintained through the use

of limiters, denoted as g. ai- is representative of jumps in the characteristic variables and is

expressed in the following fom.

kk.I,,lZ, a fumtion of the eigenvalues. Aiq+l/2, of the flux lacobians, dk,I12, is an entropy s a t i w g condition used to exclude non-physical solutions.

where ci, is a small positive parameter defineci by ~ e e " as a bct ion of velocity and sounci

speed in the case of steady state problems to prevent instabilities in the tlow solver.

where È is a small positive constant,

is similar to Aiv with the addition of a yiv term.

where

There are a number of limiten, g, that can be use4 see ~ e e , ~ ' which vary in their properties as a trade off between difiùsivity and stability. An intermediate limiter was chosen and is of the form:

2.4 Boundary Conditions

The interaction of the numerical scheme with the computational boundaries plays a major

role in modifjing the flow field to its fiaal solution. The boundary conditions are therefore irnplemented in a way consistent with the numerical scheme in order to assure the stabiIity of the scheme and convergence of the solution. It should be noted that the boundary ce11 values

are taken to be at the edges of the interior cell's volume, ie. at i, f 1 /2. There are four boundary types used in the present problem, supersonic inflows, supersonic outflows, wall boundaries,

and symmetry planes. The supetsonic infiow is one in which aii of the flow characteristics are aligned in the same

direction, into the 0ow region. The flow vaIues can thus be defined on the inflow pIane fiom the actuai physicai information about the flow. Once defineci these values are held constant

throughout the solution procedure. Similarly on a supersonic outaow the flow characteristics

are al1 aligned out of the flow region to the boundary. in this case a zero-arder extrapoiatiori is used to determine the boundary values h m the k t set of nodes inside the boundacy.

In a Navier-Stokes flow solver the wail boundary conditions are determined h m the physics of viscous flow. This dictates that the velocity at the wail must have a value of zero.

The species concentrations are extrapolated h m the adjoining celIs to the wall using fùst-arder extrapolation.

The determination of the wail temperature can be done in several ways depending upon the

physical problern being simulated. In this case al1 the walls are specified as having a constant temperature.

TWi1 = constant (2.56)

The final condition needed at the wall is that of pressure. The pressure gradient is assumed to be constant normal to the wail, as proposed in H i r ~ h , ~ ~ by speci@ng the pressure be the sarne

as that of the next ceil away h m the wall.

where n denotes the direction normal to the wall.

Thus the following extrapolation is perfomed.

in order to take advantage of the symmetry of many problerns, such that the computatiooal load may be teduced, symmetry planes are inmduced By definition the flow variables on one side should minor the other side. This implies that the velocity perpendicular to the surtace is

zero, V&=perpeadih) = O. For T) # (p the velocity is extrapolateci, in the form of Equation 2.55, fiom the interior nodes. The Cartesian velocity components are then found h m the following:

Al1 other variables are fomd using a vanishing nomal gradient at the symmetry plane.

where n denotes the direction nomal to the wail, and extrapolation of the form of Equation 2.59 is used.

2.5 Convergence

A converged solution is found after the value of the 12 nom of the node convergence, 5, based on the discretized continuity residuais drops below a defined tolerance.

3.1 Governing Equations

The addition of a turbulence model to a flow solver is an especially difficult task when dealiog with hi@ speed, high enthalpy flows. An incorrect turbulence model can effect the overatl flow structure and produce erogenous solutions. Current turbulence models requue a nurnber of "tuning" parameters which parallel the type of flow expected, however there is very little experirnental data available for high Mach number tiows with which to qualifi these models.

The soIution of a numencd problem via gas dynamic mechanisms should thus be obtained

prïor to the implementation of turbulence rnodeling. For this work the two equation k - o turbulence model of ~ i l c o x ~ ~ was used.

The Fame averaged Navier-Stokes equations can be expresseci in their conservative form in generaiized spatial coordinates in a fom similar to the laminar Navier-Stokes equations, as given in Equation 2.1.

The conservative variables, Q, are taken in a siightly different fotm than in the laminar case and contain the nirbuient k and o terrns. The convective flux in the cp dimension, F,, and the

- --

diffusion tenn, G, are also expressed from the Farve averaged Navier-Stokes equations.

where k is the turbulent kinetic energy and o is the specific dissipation rate of the turbulent

kinetic energy.

The definition of the total density, p, remains the sum of the species densities, as given by Equation 2.5, but the turbulent kinetic energy contniution, k, must be added to the total energy,

E.

The pressure is de&ed as in Equation 2.7. For the turbulent case the conductivity mauix cm be expressed as follows.

where the effective diffiision of species j, q, the effective viscosity, p8, and the effective

thermal conductivity, K' contain a molecdar contribution, as seen in Chapter 2, and a turbulent contribution.

PT y = Dj.rno[ + - with Scr = 1.0 PSCT

'P with PrT = 0.9 K* = K+PT-- P b

Additionally

where p~ the nirbulent viscosity.

~k max (k. 0) = 0.09- z 0.09~ O (o. ~ d i v )

Due to turbulence the source term, S, is no longer trivial. The turbulent source term itself is divided into the normal source terms for the Wilcox k - o modei, Sr,,, and the compressibility source terms, Sr,,,,. The compressible source term is the compressible dissipation correction required to account for the reduction in shear layer growth with increased convective Mach nurnber.

Pk is the twbuIeut kinetic energy production tenn.

where f l is used in pIace of the thearetical such that k acquires a non-zero vaiue at the

in-flow and turbulence can be initiated. À is a parmeter used to prwent division by zen, in the source mm while stiil ailowing

k = O in the freestream flow.

where kdiv is a mal1 constant, and the minimum function prevents clipping in flow regions where turbulence does occur.

f (MT) is defined for the Wilcox mode1 as the following.

where MT is the turbulent Mach nurnber, which makes use of the effective sound speed, a.

3.2 Spatial Diseretization

The spatial discretization is performed similar to the laminar flow solver with the residuaI decomposed into two parts, the convective residual, ho,, and the non-convective residual, R,,, debed as in Equation 2.44. ALI the non-convective terms are discretized on cetl-centered 2nd order difference stencils. The discretized of the convective terms again performed by the TlrD scheme by Yee," however the syrnmetric fonn is used in place of the upwind scheme. The full symmetric discretization is as follows.

The metric term in Equation 3.1 5 mua be evaluated at the ceii interfaces, i, & 1 /2, as prwi- ously noted in order to maintain fkestream properiies in highly distorted rneshes.?' Tiie tenns .&,, Riq+li2, and ai, are as previousiy dehed , see Equations 2.46 and 2.47.

One pmblem wilb using the enûopy correction term, k:v+l12, "e fact that the adficial

dissipation it adds c m greatly effect the accuracy of the solution in turbulent boundary layer

regions. It has been found that such a correction is not actually needed in the cantilevered injector cases. Therefore Aig+,,, is simplined h m the defuiition in Equation 2.48 and is debed as the absolute value of the eigenvaiues of Aiv+I/z.

Similar to the upwind case there are a number of Iimiters, g, that cm be used, which Vary in their properties as a trade off between dif i ivi ty and stability3' An intermediate limiter for the symmetric case is given as follows.

ph = minmod (aig- 1 . ai, , ai,+,+ 1) (3.17)

where the minmod function will provide the minimum for solely positive arguments, the max-

imum for solely negative arguments, and zero for mixed arguments.

3.3 Temporal Discretization

The implicit discretization of the governing equations is denoted as the following, as expressed in Equation 2.22.

This c m be expressed in the delta form by the addition of -&R to both sides, which after rearranging is given as follows.

where the A ternis are the diffaence in the tirne seps, ie.: AF; = FF;+' - F: The residual, R, is in the same fonn as Equation 2.27, and is rewritten here.

A factorization method is sought in order to solve Equation 3.20, as in the laminar fiow solver, but in this case approximate factorization wüi be us& The linearization about the previous, n,

time step is performed on the convective, diffusive, and negative source terms,

The flux Jacobian matrices, A,,,, and the linearization of the viscous terms, B, are given

explicitly in Appendix A and are defined as:

As previously stated oniy the negative componwts of the source terms are linearized to give the Jacobians for the nomat turbulent source, C,n,,,,,, and compressibility source, Cr.comp. Tbis is done to ensure stability of the implicit cher ne.^' The Jacobians are Mtten in fui1 in Appendix A.

Approximate factorization allows the use of one dimensional operators to approximate the

factor to be inverted.3'~~~ Additionally the linearization scheme of Chang and MerkleM was

adopted for the viscous tenns. Equation 3.20 can thus be approximated as the foliowing:

It should be noted tbat the C- tenn is only evaluated on the cp = 1 sweep, as denoted by the Kronecker delta, SE. The tems D& are appmxirnaied as the linearization of a first-order Roe scheme with the Roe Jacobian tocalIy fiozen, which has been show CO reduce the work per iteration without increasing the nurnber of iterations required to reach a s0Iution.3~

Due to the reliance of approxirnate fiictorization upon the invariance of the linearization matrices two simplifications are made to the implicit viscous terms The limiter function, g,

is dropped due to its unstable nature when used implicitly and for similar reasons the cross

diffusion terms, &,, when cp # q, are not inciuded.

The cpth sweep in a time integration step is written as the following.

t/2Biv-1 - y!, SEICC

where ~ 0 . ; ~ = -AfiqRjq and the solution for one time step is then given fmm 6Qt = AQ,,~.,, . The pseudo-time step, Aî, is an average of the maximum CFL condition, generally in the

streamwise direction, and the minimum CFL condition.

where a is generally given a value of 0.5. This method increases the speed of convergence beyond using strictly the minimum CR. condition, while stilI exhibiting stable convergence.

3.4 Boundary Conditions

in the turbulent flow solver the boundary cet1 values are treated as cornplete cells and are not assumed to be the edge of the first interior ce11 as in the laminar flow solver. The boundary conditions are the sarne as outiined in section 2.4, except for the stencils which are different due to the fact the boundaries are considered one full ce11 away, not half a cell, from the intenor nodes.

Thus extrapolation to a boundary celI, similar to Equation 2.55, is given by the following.

The waII conditions are provideci by a extrapoiation of the species concentrations, ci, as given by Equation 3.27, the temperature, T, is specified as a constant, and ail velocities, v,, bave a zero value. The turbulent kinetic energy, k, is aiso specified as having a zero vaiue as dictateci by the physics of viscous flow. The pressure at a wall is found h m the efféctive

pressure, P = P - 2/3pk, which is found by a h t order extrapolated fiom the interior cells.

o at the wall is specified for the Wi1cox mode[ as:

where d, is the distance h m the wall node, i,, to the interior node, i, k 1 . The velocities at the symmetry plane are found as given in section 2.4 using the extrapo-

lation of Equation 3.27. The other variables, ci, T, k, and y are exuapolated using Equation 3.27. The pressure can not be found h m simply extrapolation if momentum is to be exactly conserved. Thus the rnomenturn terms pqendicular to the boundary are added to the effective pressure.

where qqiI is detennined from the velocity, v, at i, k 1 and the metrics X, of i,. The pressure is then found h m the eff'tive pressure at the boundary.

3.5 Convergence

The convergence of a node is obtained when the value of 5, based on the maximum between the discretized continuity and energy residuals, falls below a dehed tolerance.

Once al1 the nodes have reached this IeveI the solution is converged.

As a cornparison with other investigations into the use of wavy walls to enhance mixing, the simulation of shear Iayer mixing in a duct was performed. This provides an idea of the effect of the wavy wall on shear layers without the addition of vortex creation mechanisms. Subsequently, the fiil1 configuration of the cantilwered injector array is outlined including the injector geometry and the initiai Aow field properties. Finally in this chapter a description of the computational grids used in both the larninar and turbulent 0uw solvers is provided.

4.1 Shear Layer Mixing with Wavy Walls

4.1.1 Background

The idea to use wavy walls in hypersonic mixing is not a new one. The enhancement of shear layer rnixing in a duct by the addition of wavy walls has previoudy been examined by severai groups. Tarn and HU^*' pursued the premise that 0ow instability is the primary mechanisrn of fiuid d g in supersoctic shear layers. They showed that there are two possible modes of instabiiity for a shear layer in a duct The ûrst is due to the resonant instabiiity when the acoustic modes of the fiow are driven into resonance. When using wavy wails the strength of the resonant insîabiiity can be incfeased by a chuice of waIl wavelength which allows a pair of acoustic wave modes to provide mutuaüy simultaneous spatial forcing.

The second instabiliiy mode is a parametric instability which can ody be induced in a spatialiy periodic flow, such as that obtained in the presence of a wavy wall. The growth of the parametric instabilities varies linearly with the amplitude of Mach waves, Ieading to the conclusion that the instability is indeed tied to the Mach waves. Tarn and Hu concluded that a wavy wail cm be used to enhance parametric instabilities and hence the mixing of the supersonic shear layer.

in order to ven@ that mixing in the presence wavy walls did indeed act in the manner described by Tarn and Hu, numerical and experimental investigations were undertaken by GiIreath et al.." The numerical studies determined internai wave drag Iosses were not overly prohibitive for wavy wail instability enhancement to be used for mixing enhancement. Low waii amplitude to wavelength ratios, such as 2% which Tam and Hu set as an upper Iimit for use of linear stability theory, resulted in a total pressure loss of approxirnately 6% and a 0.6% change in the sueam thnist. Larger amplitude to wavelength ratios gave predictably higher losses, with a 10% ratio giving a total pressure loss of 40% and a 6.5% change in Stream thnist. Linear theory was used to find the most unstable wavelength for the numerical test conditions, but no predictions were made on the instabilities themselves or the enhanced shear layer growth

they produce.

Gilreath et al. went on to experimentally verifi whether the flow instabiiities did produce

enhanced mixing. improved rnixing was assumed to be indicated by larger shear layer growth rates, defined as the change in the shear layer size with downstream distance. Using wave- Iengths detexmined in the manner of Tarn and Hu they were unable to infer any increase in growth rate due to the spatial forcing, but did note an increase in the amount of srnal! scale activity.1° [n subsequent experiments, with a more advanced caiculation of the most unstable

wavelength, they recorded an increase in the shear layer growth rate of up to 30 f 1 O%? Most recently Doty and McLaughlinI2 experimentally verified that the excitation of insta-

bilities could indeed impmve the gowth rate of compressible shear Iayers. Varying from the

analysis of Tarn and Hu, the stability theory they used was based on the interaction of the spatial waves h m the wavy wall, the duct acoustic waves, and the Kelvin-Heirnholtz waves in the shear layer.I3 As predicted by their theory, ciiffereut wavelengths fiom previous experi-

ments were found to offer the greatest tlow instabiiity. They found up to a 50% irnpmvement in growth rate compared to the case without wave waIls was possible.

4.1.2 Simulation of Shear Layer Migiag

To examine of the effect of a wavy waU on shear layer mixing and growth the primary and baseline cases of Gieath et al., 1995: were simulateci withthe turbulent flow solver outlined

in Chapter 3. A shear layer is fonned at the end of a splitter plate between a Mach 2.71 and a Mach 1.47 air Stream in a 2.54 cm square duct. In the primary case there is a 2.82 cm wavy waii section on the duct wall of the slower Stream at a distance of 2-54 cm h m the start of the tnixing region. The waves bave a wavelength of 0.94 cm and there is a amplitude to wavelength ratio of 5%. Güreath's baseline, flat wall, case was repeated to illustrate changes in the shear Iayer behavior due to the presence of the wavy wall.

No attempt was made to simulate the 0ow instabilities, or the shear layer p w t h due to the instabilities. While the same flow conditions as Gilreath et al. were used, including the wall wavelength and amplitude as predicied from instability theory, ody the properties of the flow as determineci by the Farve-averaged Navier-Stokes equations were found.

The shear Iayer growth was examined in both two and three dimensions, however no difference in the shear Iayer growth along the center plane was found from the addition of a third dimension. The p d was reîined until a grid converged solution was found. ïhe velocity contours of the wavy wall case are shown in Figure 4.2 and cm be comparai to the case wikout the wavy walls as shown in Figure 4.1. The size of the shear layer c m be taken as the height of the contour region emanating frorn the splitter plate. The shear layer growth rate was found co be identical, 0.040m/m, in the case with and without the wavy wall. No significant change in growth rate due to the wavy wail is expected due to the lack of instabiiity modeling. instability enhancement is inherentIy a cime dependent phenornenon and as such is not likely to appear in any steady state simulations?.

The gcowth rates obtained are within an acceptable range of the experimental values found by Gilteathet al.. For the wavy wall case g r o d rates of O.O33m/m and 0.022m/m were found for the experimenta1 and numerical tests tespectiveiy. This provided an increase in shear layer growth over the basefine case which was measured as 0.029m/rn experimentally.

The presence of the shock structures from the wavy wall does add a fair amount of non- streamwise motion during the primary interaction with the shear Iayer and shifts it cioser ta the top of the duct than in the baseline case. ïhe perturbation of the boundq Iayers by the wavy wall causes an increase in boundary layer size and interaction with the shear layer at a shorter downstream disiance. The pwth of the shear layer is no longer uniform, but has a wavy nature, as can be seen in Figure 4.2, a straight iine can no longer be drawn through the top contours in the shear iayer. The inmease in cotai pressure losses due CO the wavy waf[ is minimal, a 2.2% i n m e , indicating that the enhancement of flow instabilities may be a good mixing tool. The d t s indicate rough agreement with experiment and illustrate the mixing enhancement found in the cantiievered ramp injector simulations wiH not be attti'butable ro ùnproved shear layer înixing.

aot

am

- E - 0 ri *

.am

.rt (m)

Figure 4.1: Velocity Contours in Flat Wall h c t , Gilreath et al. Baseline Case

.ri w Figure 4.2: Velocity Contours with the Addition of a Wavy Wall, Gilreath et al. Primary Case

4.2 Design Parameters of Cantilevered Injeetors

In order to correlate the resdts with previous studies the design of the cantilevered injector array is based on the baseüne cantüevered injector anay of khumacher.' The complete configuration of the Uijector array is defined with the parameters shown in Figure 4.3. The compression angle determines the angie at which the injector mtrudes into the ffow, the expansion angle determines the angle of the trough between the injectors in relation to the duct, and the sweep angle determines the taper of the injector. The height of the injection plane, the height under the injector, and the extendecl cantiIevered length cm be used in conjunction with the compression and expansion angles to h d the total injector length. For ease of comparison a non-dimensionaiization length scde parameter, f, is deûned as the square root of the hel

CHAPTER 4. OVERVIEW OF CASES STUDIED 32

injection a m , x' = 0.02m.

Injector Length , 9

1

* :y Extended Cantilever Length , - , - ' \ Compression Angle t I

- - Wall Wavelength

Sweep Angle ---__.

A m y Symmetry Plane - - - - - - - - - - - _ _ _ _ _ _ _

r

f lnjector Symmetry Plane " - - T Injection Width

Figure 4.3: Cantilevered Ramp injector Parameters

The advantage in simulating an infùiite anay of injectors is that the symmetry of the injec-

tors can be used to greatly reduce the computationa1 problem. As shown in Figure 4.3 there

exist two symmetry planes which allow the entire array to be simulated by ody solving the problem on the width of half of an injector. The array symmetry plane lies between adjoining injectors, with its location determined by the distance between the injectors and the injector symmetry plane lies on the middle plane of a single injector.

1 Ex~ansion Anale 1 7' 11 Com~ression Anale 1 0" 1

1X ( Height Under Injector 13 Extended CantiIevered Lenath 2.5.t 1 iniector Lenrrth I8.û.t

1 ) II - - 1 Distance Between iniectors 1 I E il iniector ~ t d t h I 12 1

Table 4.1 : Cantilevered Ramp injector Parameters

The values of the injector parameters used are Iisted in TabIe 4.1, The greater the values of the compression and expansion angles the greater the strength of the shocks fomed in the flow

and subsequently the stronger the voaiçes formed. The choice of these angles mut provide a baiance behiveen the production of strong vortiçes needed for mixing and the Iosses incurced in vortex formation. One of the changes to Scbumacher's base design, prompted by the ~ s u l t s of his study, was the value of the compression angle. A protmion into the tkestream flow wilI result in shocks and a subsequent pressure incrase. An injector with a zero compression angle is used in order to avoid such a high pressure region, which tends to force the uijected hel to the bottom of the duct and Iimits the mixing in the flow. The lack of a compression

angle shouId also increase the relative contribution of baroclinic vorticity due to the reduction in strong shear leveis around the Uijector.

While a positive sweep angle has been fouad to increase mixing in conventional ramp injectors there is no sweep angle used in this study." The goal of this study is to examine the effects of the wavy wall, which will be evident in the case with or without a sweep angle. The size of the injector is chosen such that it is srnaII enough to be computationally viable to solve, but large enough to mode1 a realistic shcramjet injector. The sizes of the various injector parameters are given in Table 4.1.

The "extended" cantikvered design has been adopted so that the injected fuel is deposited closer to the shock that is initially responsibie for the production of baroctiniç [orque. An extended section of length 2.52 is chosen to be long enough to deposit the fuel closer to the shock, but not so long as to interfere with the shock structure,

The distance between the injectors can have a great effect upon the fuel-air mixing. If they are too close not enough air will flow between them to generate strong vortices needed for mixing. Conversely if they are too far apart there will be air ffow that is either wasted by not mixing or will require too long of a mixing region to achieve acceptable mixing. The injectors in this study are detbed to have two injector widths, E, between them.

The Cartesian directions used to denote the directions in the flow are referred to with the temhology provided hm. The positive xl direction, the direction in which the air and fuel initially flow, is termed the streamwise direction, with the end of the mixing region being domtream. The - 2 h t i o n originates from below the injector, at the wavy wall, and is directed upwards to the top of the duct, above the injector. The.r3 dimension is in the spanwise direction fiom one injector to another. ïhe injecior symmeny plane will be referred to as the inside and the array symmetry plane as the outside.

The wavy waii is implemented as a sine wave origirlating where the expansion trough turns back paraHel to the duct, as cm be seen in Figure 43. ïhere are an W t e number of ways of implementing a wavy wail, but since this is a k t snidy a simple combination serves to demonstrate the relevant effects. AIthough both suie and cosine wave form implementatioas

were examined, the sine wave is supenor since it preserves the first strong shock that is initiaily

responsible for the production of baroclinic torque. The equation of the wavy al1 is given as

the following.

where a, is the wall's wave amplitude and w its wavelength. The length of the duct h m

the fuel injection plane, the mixing length, is approximately the same length, 1.0m = 50.f, in

al1 cases, but varies to accommodate complete wavelengths of the wavy wall. The various

configurations of the wavy wall can be examined by changing the amplitude and wavelength

parameters. An outline of the these d u e s used for the various cases in this study is given in Chapter 5.

Increasing the amplitude of the wavy wall will provide larger pressure gradients between

the compression and expansion regions in the flow. The larger the pressure gradient the stronger

the vortices formed by baroclinic torque, An increase in vorticity will increase the mixing of

the he l and the air. The energy needed to form strong vortices must be extracted fiom the flow

and the losses will aIso increase with higher wall amplitude. The effect of the wavelength of

the wavy wall is a harder parameter to qualiQ. There must be enough distance between the

alternating pressure gradients to allow the vortices to become established and provide mixing

before their direction is merseci. A wavelength which is too small incurs losses providing

vorticity without having the opportmit. to change the flow patterns and increase the mixing.

Conversely having a wavelength which is too large will result in vortices which are acting for

long periods in one direction. This defeats the addition of the wavy wall which is meant to

provide alternating vorticity.

4.2.2 Flow Field Properties

The intent of this study is to mode1 a shcramjet mixing region and the tlow properties are taken to be representative of such an environment. The high Mach number, high enthalpy flow at the start of the injector corresponds to the mixing region in a shcramjet with a tlight Mach number of 14. in ordw to ümit the effects of undue pressure gradients, which would generate

shoch and prematurely ignite the mixture, the fuel entry velocity and pressure are matched to

the initiai conditions of the air. The initiai vetocities of both the fuel and the air are in a strictly streamwise, XI, dimension. The complete set of flow conditions is outlined in Table 42, in whicb oo denotes an entry condition.

The initial height of the duct is set to provide stuichiometric proportions of fuel and air,

Table 4.2: Air and Fuel Freestream Conditions.

Air Fuel (Hz) Ratio (AirtFuel)

resulting in a height of 2.927.t. The air is modeled as a mixture of nitrogen, Ni, and oxygen, O,, in the mass fraction ratio of 3.27:l. The use of pure hydrogen, Hz, as fuel can greatiy enhance the performance of a scramjet or shcramjet compared to conventionai hydrocarbons,

due to its high specific impulse. The surfaces of the injector and the walls of the duct are given a constant temperature of 800K. This is indicative of the surface temperatures reached in a

hypersonic vehicie containing wail cooling systems, which allow temperatures near the limits

iiuposd by maienai consideratiom. The use of a cryogenic fuel allows for its utilization in the coolhg of the airFrame prior to the fuel injection. The stagnation temperature of the fuel will

thus be high, 850K, stightly greater than the wall surfaces which it is cooling.

in the laminar flow soiver there is no modeling of the velocity profile of the fuel jet at-

tempted, it has the same entry velocity over the whole injection plane. In the turbulent flow

solver a constant velocity plane of hydrogen is specified at the start of the extended cantilevered section inside the injector. This allows boundary layers to develop prior to the mixing region.

If the fuel injection were treated in the same manner as the Iaminar case singularities would

devetop at the fuel / air boundary and provide enoneous results.

4.3 Grid Generation Algorithms

Ma

7 3.48 2.01

4.3.1 Laminar Flow Solver

ln order to maintain a suitable computational problem the flow region was subdivided into diffemt domains, as shown in Figure 4.4. The first domain was solved as a two dimensional problem, needed to produce boundary layers as would be found in an actuaI shcramjet. The output h m this domain was input across the entire span of the three dimensional injector section. The solution of the injector was itself subdivided, with the f h t domain containing the iajector. The solution on the outflow boundary of the injector domain was used to provide the d o w conditions on the next computational domain. The solution of the downstream domain

can not effect the solution of the upstream domain. Even though the majority of the fiow is

p=(Pa)

16500 16500 1.0

LW) 900 240 3.75

v,(:)

4130 4130 1.0

pz($)

0.0636 0.0166 3.83

~ ( p a )

1.07 x 108 1.19~ IO6 89.9

T:(K)

8451 850 9.9

supersouic, the boundary layer regions can transmit information upstream. This means the solution obtained in multiple domains may be different from that obtained on a single large domain. It has been found by simulating the three dimensional problem in two domains and in one domain that no noticeable flow field differences are observed Less than a 1% difference in the total pressure loss is associateci with the subdivision, provided the subdivision occurs at the location of a maximum in height of the wavy wall, where the boundary tayer is the t h e s t and the fiow is moving in a streamwise direction.

I D Dornain Dornain 1 Domain 2

Figure 4.4: Laminar Flow Solver Computational Domains

In order to ensure a numencal solution will give realistic flow field values the issue of grid convergence must be addressed. Ideally this is done by solving the same probiem on a finer and finer grid until no variation is seen in the answers. Practically speaking for a three

dimensional problem this becomes very computational expensive. Therefore a grid density is sought which will show the relevant flow field features to an acceptable tolerance. A grid convergence smdy was perfomied by chum ma cher' for the cantilevered ramp injecton and three dimensional domains containhg approximately 1.2 million cells were found to exhibit the relevant flow features. The error produced fiom not having an infînitely dense grid was

estimated h m increasing grid densities to be near 20% in the performance parameters defined in Chapter 6.

Each domain is subdivided into blocks that have one boundary condition or one adjoining block on each of their six sides. Domain 1, containing the cantilevered injecter, consists of 20

blocks, and domain 2, the downstrearn domain, consists of 6 blocks. This subdivision allows clustering of grid nodes in regions which require greater resolution, such as boundary layers.

The clustering aigorithm used dows the number of nodes, NT the degree of clustering, b and 4, and the maximum and minimum position, .qi,+,) and .r(,=l,, for each dimension of each bIock to be specified. The maximum and minimum positions can depend on the other dimensions, ailowing the creation of non-rectangular shapes. The location of a point inside the Hock is then found with the foiiowing algorithru, modifieci fiom that given by Hohan and

where b is a dustering factor which produces more clustering as it approaches 1 and uniform ciustering as it approaches in f i ty . The + term detemiines the location of the ciustering: more near the minimum boundary as + + O, equally on both sides for @ = 0.5, and more near the maximum hundary as 4 + 1. The a term is dependent upon 4 and nuis h m O to 1 as i, goes fiom the start of the block, i, = I , to the end of the block, i, = NT. The values of b and 9 can be changed linearly with one of the other dimensions, tl # cp, such that the clustering in the cp dimension at one end, i, = 1, may be different than the other, i, = N,.

The grid in the Row direction, cp = 1, on the wavy wali sections is generated in a different mannet such that increased clustering is produceci on the steepest portions of the wavy waIl where greater resohtion is required. A simple exponential term, cap, allows clustering at the maximum boundary, a,, + a, the minimum boundary, a,p -+ O, or a uniforni grid spacing, O,, = 1. The gridding about the sinusoidal section cari be increased with the C,, factor in the following aigorithm, which giws a sinusoidal grid clustering contribution at twice the frPquency of the wall.

In order to provide a consistent grid the clustering factors on adjoining faces of blocks must have the same values. The grid Ïs conmcted such that a change in the wavy wail parameters oniy change the grid in blocks containing tfie wavy wall on one side, resuIting in sirnilar dissipation due to the grid in ail cases. Sections of the computationd grid cm be seea in Figure 4.5.

(a) Streamwise Plane at Edge of hjector (x3 = O.5.t = 0.0 lm) (b) Spanwise Plane at In- jection Plane (xi = 1.5.1 = 0.05m)

Figure 4.5: ComputationaI Grid in Domain 1 for Laminar Flow Solver

4.3.2 Turbulent Flow Solver

The turbulent flow solver contains a number of convergence acceleration schemes. One of

the schernes used is a marching computational window which is a small region of the entire streamwise length, Only the cells inside the window are updated in tirne h m rheir initial conditions. The window consists of the full domain in the non-streamwise dimensions, and propagates in the streamwise direction. The left boundary moves when the nodes in the start of the domain are convergai beIow the convergence limit, while the right boundary moves either in response to an dliptical flow region or to maintain a minimum window size. The use of this window rnarching scheme provides a huge savings in the amount of cornputer memory required to nui the simulation without the possibility of creating errors by subdividing the domain. The computationai savhgs allow the entire simulation to be run as a single domain. A grid convergence snidy by Paren$ found a domain consisting of approximately 2.5 million celis was appropriate. The estimate of e m r produced by the grid was made by Parent to be 15% in the performance parameters.

The grid formulation is a single block scheme where the entire computational domain is

speciiied inside a rectangular computationai volume. Cells withh the domain can be specified as boundary cells, inactive cells, or as interior cells to be updated intime. In the construction of the grid a pseudo-multiblock formulation is used. Block locations, in reai and computational

space, are specified dong with the degree of clustering in the starting half, subscript s, and ending half, subscript e, of the block. With these values known the location of each grid point in a single dimension in the block can be found using the followhg equations.

where AGi is the distance between adjoining ce11 centers and E' = AG+ 1 /AL is the mesh spacing growth. The total number of nodes in a block is denoted as IV, which is divided into

the number in the starting half, N,, and the ending half, Ne, of the block. The physical length of the block in the cp dimension, is L, .

The exact position of each node dong the cp dimension can be determined h m the four

variables in Equations 4.6 and 4.7. Three variables must be specified with the fourth being

obtained through the use of a root solver. Equation 4.6 is used when the tength of the region,

the mesh spacing growth in one haIf of the block, and one of the following: first node size, iast node size, or second mesh spacing growth, are specified. Equation 4.7 is used when the length of the region, first node size, and last node size, are specified. This method can be used

to establish gridding on the steepest parts of the wavy wail as was done in the laminar grid. In order to cluster grid cells most effectively an aigorithm to determine the grid node lo-

cation for ail points in a volume, on which the spacing on al1 sides is specified, was used. An

itetative procedure is performed which disûibutes the nodes in each dimension in relation to the position of the same i, nodes on the sides of the volume, weighted by their proximity to

each side. Thus compiicated clustering, as found around the wdl surfaces of the injecter, c m be blended into a uniform grid in regions not requiring the same level of clustering.

Once al1 the block regions are specified the grid on the M l domain is known. The use of

the block regions is purely for generating the grïd and the flow solver does not depend upon the individuai block fonnulation. Sections of the turbulent 0ow solver computationai grid are shown in Figures 4.6 to 4.8,

xi lm)

Figure 4.6: Streamwise Plane of the Computational Grid for the Turbulent Flow Solver at the Injector Synimetry plane(.^ = 02 = Om)

(a) injection Plane (XI = 2.5.f = 0.05m)

(b) End of Domain (XI = 525.t = 1 .O5m)

Figure 4.7: Spanwise Planes of the Computational Grid for the Turbulent Flow Soiver

(a) [njectot Symmeay Plane (x3 = Qf = Om)

Figure 4.8: Streamwise Planes of the Computational Grid for the Turbulent Flow Solver

The results of the numerical simulations are presented in this chapter. The flow fields, illus-

trations of the relevant flow variables, will be shown and the important details discussed. An

analysis of the flow fields will be provided in the following chapter.

The effect of four different parameters were studied to ascertain their effect on the fuel /

air rnixing by cantilevered ramp injectors in the presence of wavy waIIs. As stated in Chapter 4 the two main parameters examined were the amplitude and the wavelength of the wavy wall.

lnitially a wavy wall of constant waveiength and variable amplitude was used, then different wall wavelengths at similar amplitudes were tested. The changes produced in the flow field

with the introduction of turbulence modeling and differing boundary layer heights were also exarnined. The simulations perfomed were used to establish the trends in mixing behavior due

to the different parameters. The variable parameters of the cases studied are given in Table 5.1. Shear Iayer mWng experirnents have suggested that the amplitude to wavetength ratio rnay

be a possible scaling tàctor for the mixing changes attributed to wavy ~ a l l s . ' ~ Amplitude to

waveiength ratios may be re-expresseci as ratios of the wave angles, which are an approximation to the angle through which the flow direction changes due to the wavy wail. The wave angle is defked as the angie between the flat duct wall and a Iine h m the start of the sine wave to its maximum amplitude. In order to veriQ whether the wave angle is useful as a scalhg factor the waii wavelengths in cases 5 and 6 were chosen to generate the same 3.5' wave angle as in case 3, using the wall amplitudes h m cases 2 and 4 respectively. This necessitates slightly

Case Number

1 2 2b 3 4 5

Table 5.1: Wavy Wall Parameters of Cases Examineci

L

6 7 8

different mixing lengths than the other cases in order to accommodate an integer nwnber of wavelengths in the computational domains.

The initial boundary layer duct in cases 1 through 8 was the same Iength, 50X = 1 m. The

Amplitude, OU

turbulence modeling in cases 7 and 8 increases the boundary tayer size, as determined by the flow velocity, by a factor of 3.6 at the start of the injector. in order to establish the contributions made to the flow field by differing boundary layer heights a shorter 9.3 1X = 0.186m initial duct

(M), x 104 O

1.309 1.309 3.058 4.374 1.309 4.374

O 1.309

was used in case 9 to provide the same boundary layer height as in the larninar cases. in order to ensure the cornparison between the turbulent cases and the larninar cases was valid despite

x O

0.0655 0.0655 0.1529 0.2187 0.0655

Wavetength, w

14.303 O t O

9 1 1.309

the hct that different flow solvers were used case 2 was r e m with the flow solver developed by Parent. In this case, denoted case 2b, the grid generation of the turbulent cases was useci, but the turbulence modeling was not used.

The initial region used to develop boundary Iayers shows no interesthg effecrs, other than

Wave An&

( O )

O 1.5 1.5 3 -5 5 .O 3.5

(4 O

0.2 0.2 0.2 0.2

0.0856

t O

0.2187 ( 0.2861

that of a duc& and as such will not be s h o w in the subsequent Bow fields. The flow regions which will be shown are d8icdt to reproduce in a two dimensional representation because

x O 1 O 10 1 O 10

4.281 3.5 O

1.5 O

0.0655 0.0655

of the highly tbree dimensional nanue of the flow structures and discontinuities around the cantilevered injector. in order to give an impression of what is actuaîly happening a number of two dimensional cross-sectional planes are taken through the flow at both constant xt vahes

1.5

O 0 2 0.2

and constant .q values. These planes are used to show the mitical variables of the flow as it progress through the mixing region. Carefiil interpmtation of these planes is needed and che three diensionai natute of the flow should be kept in minci- The hydmgen mass hction

T h l e n t Modeling

no no no no no no

Boundary Layer Height

(m), x 10-~ [ x'

contours, pressure contours, velocity vectors, and streamlines in the 0ow are ai! used to provide an impression of the flow bebavior.

3 .O8 3.08 3.40 3.08 3 .O8 3.08

no yes yes

t

0.154 0.154 O. 170 0.154 0.154 O. 154

yes

3 .O8 11.20 t 1.20 3.30

0.154 0.560 0.560 0.165

The flow field results are subdivided into two sections. The primary section deals with the flow fields present with variations in wavy wall amplitude and wavelength. This section is outlined in detaii to give a complete impression of what is occurring in the mixing region. The results showhg the addition of turbulence modehg and the eEect of boundary layer height will be deait with in the following section. A more limited view is given in this section and

pertains to the ciifferences between the tlows and those that have previously been describeci.

5.1 Flow Fields with Changes in Wavy Wall Amplitude and

Wavelengtb

Al1 of the laminar simulations shown subsequently have residuals converged through four or- ders of magnitude h m the initial values. Observations of tùrther convergence shows no ap-

preciable changes in the tlow field. A sample convergence history, for case 6, is provided in

Figure 5.1 which shows the history for domains 1 and 2. The large jump in the residual for domain 1 indicates a change in the Ë value in the entropy correction factor, defined in Chapter 2. A larger value than that of the final solution must initially be used to ensure convergence in domain 1.

Figure 5.1 : Convergence History for Case 6

CHAPTER 5. FLOW FIELD RESULTS 45

5.1.1 Pressure Fields

The addition of the wavy walI is meant to generate alternathg pressure gradients for the production of periodic baroclinic toque. In the region of the fuel / air interface the larger the pressure differences the stronger the vorticity produced. The pressure fields for the fust six cases, which exhibit an increase in the pressure gradients with an increase in wall amplitude, are shown in Figures 5.2 to 5.7. The pressure contours range From 2 000 Pa to 62 000 Pa in increments of 2 000 Pa. The pressure fields observed in al1 cases are similar, but the location of pressure gradients is dependent upon the walI wavelength and the pressure level upon the wall amplitude. The close clustering of the pressure contours denote a shock in the flow.

Figure 5.2: Pressure Contours at injector Edge, (.q = 0.5.f = 0.01m), Case I

Figure 5.3: Pressure Contours at injector Edge, (x3 = 0.53 = O.Olm), Case 2

Figure 5.4: Pressure Contours at Injecror Edge, (x3 = 0.5.f = 0.0 1 m), Case 3

- , . 1 5 =. -. ' E = - > 1 ; .. 2; II. 1 : =: l ! =

Figure 5.5: Pressure Contours at Injector Edge, (x3 = 0.5.t = 0.0 lm), Case 4

Figure 5.6: Pressure Contours at lnjector Edge, (x3 = O S = 0.01m), Case 5

Figure 5.7: Pressure Contours at Lnjector Edge, (q = 0 3 = 0.0 lm), Case 6

As shown in the pressure contour figures there is initially an expansion fan as the flow proceeds into the troughs between the injectors. The pressure drops from between 17 800 and

19 200 Pa in the boundary layer duct d o m to beIow 6 000 Pa in the expansion trou&, A shock is forme

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