The Study of boundary layer control in a turbopump diffuser with fluid injection

Rochester Institute of Technology

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Theses

Thesis/Dissertation Collections

1-1-1996

The Study of boundary layer control in a

turbopump diffuser with fluid injection

Diego Garcia Pastor

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Recommended Citation

THE STUDY OF BOUNDARY LAYERCONTROL IN A TURBOPUMP

DIFFUSER WITH FLr ) iNJECTION

by

Diego Garcia Pastor

A Thesis Submitted

10

Partial Fulfillment

of the

Requirements of the Degree of

MASTER OF SCIENCE

10

Mechanical Engineering

Approved by :

Dr.

A.

Ogut - Thesis Advisor (R.I.T)

Dr.

A.

Nye (R.I. T)

Dr. P. Venkataraman (R. I. T)

Dr.

C.

Haines (Department Head, R.I. T)

Department of Mechanical Engineering

College of Engineering

Rochester Institute of Technology

Rochester, N.Y 14623

Table

of

Contents.

CHAPTER 1

.

Introduction.

Page 1

1.1

Project Justification. Page 1

1.2ProjectObjectives. Page 5

1.3 ProjectDescription. Page 7

CHAPTER

2.

Principles

of

Turbulent Flow.

Page 9

2.1

Governing

Equations. Page 9

2.2

Methodology

ofAnalysis. Page 10

2.3 Turbulent Flow

Modeling

Procedures. Page 13

CHAPTER 3

.

Principles

of

Diffusion.

Page 23

3.1 DescriptionandSpplications. Page23

3.2 Diffuser Stall. Page 26

CHAPTER 4.

Boundary

Layer

Control

by

Fluid

Injection.

Page 32

4.1 Flow Separation. Page 32

4.2 Controlof

Boundary

Layer Separation

by

Fluid Injection. Page 32

4.2.1 ActiveandPassive Fluid Injection. Page 33

CHAPTER 5. Finite Element Method.

Page 39

5.1 General Concepts. Page 39

5.2 FIDAP. Page42

CHAPTER

6.

Fidap

Diffuser Model.

Page45

6.2 Mesh. Page 45

6.3

Nondimensinalization.

?cge46

6.4

Element

Selection. Page46

6.5 Solution Technique. Page 48

6.6 FIDAP Near WallModel. Page48

6.7Inletand

Boundary

Conditions. Page 49

CHAPTER

7.

Results.

Page54

7. 1 Design Flow Case. Page 54

7.2 60%ofDesign Flow Case. Page65

7.3 20% ofDesign Flow Case Page 76

7.4 Fluid Injection. Page87

7.4. 1 60%- 3% FlowCase. Page 87

7.4.2 60%- 7% Flow Case. Page 98

7.4.3 60%- 1 0% Flow Case. Page 109

7.4.420%- 3% Flow Case. Page 120

7.4.5 20%- 7% Flow Case. Page 131

7.4.620%- 10% Flow Case. Page 1 42

CHAPTER 8.

Summary

of

Results.

Page 153

CHAPTER 9. Conclusions.

Page 1 56

CHAPTER 10. References

Page 159

List

of

Symbols

AR

-Area

Ratio

Bt

-Throat

Blockage

b

Span

c,

-Coefficient

Matrix

c SpeedofSound

c, ,c2

-Empirical

Constants

CM

Empirical Constant

CP

Pressure

Recovery

Coefficient

pideal Ideal Pressure

Recovery

Coefficient

Dh

-Hydraulic Diameter

At Change in Time

E Empirical Constant

F,

- CoefficientMatrix

fi

Body

Force Term

I Turbulent

Intensity

IM

- InjectionCoefficient

k Turbulent Kinetic

Energy

k*

- Dimensionless TurbulentKinetic

Energy

K

- Global StifihessMatrix

L Wall LengthofDiffuser

Lj

- Distance between

M

-Coefficient

Matrix

m

-Mass Flow Rate

Mt

-Throat Mach Number

N -

Coefficient

Matrix

p

-Fluid

Pressure

Ptj

-Jets Stagnation Pressure

qs - Heat Flux

Rs

-Throat ReynoldsNumber

Red

-ReynoldsNumber

R,

- Residual Errors

T Temperature

t Time

U GlobalDisplacement

u LocalDisplacement

V

Velocity

V Friction

Velocity

u+

Dimensionless

Velocity

Vv

inlet Inlet

Velocity

vx,vy

-Blowing Velocity

Components

v, Throat

Velocity

V VolumeofFinite Element

x

-Length

-Dimensionless

Length

y+

-Dimensionless Normal DistanceFrom Diffuser Wall

9

m - Momentum Thickness

20

-DivergenceAngle ofDiffuser

5'

-Displacement

Boundary

Layer Thickness

-TurbulentDissipation

s *

-DimensionlessTurbulent Dissipation

rp

-Column VectorofInterpolation Function

y

-Column VectorofInterpolation Function

&

-Column VectorofInterpolation Function

rj

-Efficiency

7

-Arbitrary

Field Variable

tj

-Timeaverage of ₇₇

n

Fluctuating

Termof ₇₇

K VonKarman Constant

M AbsoluteFluid

Viscosity

Mt

Eddy Viscosity

Mo Shear

Viscosity

V Kinematic Fluid

Viscosity

e Incidence Angle

p Fluid

Density

ac

-Turbulent SchmidtNumber

<rk

-TurbulentPrandtl Number

rw

-Total Shear Stress(Laminar&

Turbulent)

r

List

of

Figures.

Figure 1 - MK49-F

HighPressure

LH2

Turbopump

Page 5

Figure2 - High

Velocity

Ratio Diffuser Crossover (Rocketdyne H20

Tester)

Page 6

Figure3 - Passage Definition Page 7

Figure4- Near Wall Model Page 14

Figure 5 - Stress

Profiles intheNear Wall Region Page 18

Figure 6- Universal

Velocity

Profiles in Near Wall Region Page 20

Figure 7- Universal Kinetic

Energy

Profile in Near Wall Region Page 21

Figure 8 - Universal Viscous Dissipation Profile in Near Wall Region Page 22

Figure 9 - Diffuser

Geometry

Page 25

Figure 10 - Prandtl's

Boundary

LayerConcept Page28

Figure 11 - Diffuser Pressure

Recovery

Chart Page 30

Figure 12 - DiffuserFlow Regime Chart Page 31

Figure 13 - NACA 23012 Page 35

Figure 14 - Effect _ofFluid Injection _on

Boundary

Layer Page 36

Figure 15 - Model's

Geometry

Page44

Figure 16 - Mesh Page 47

Figure 17

-Velocity

Triangles Page 51

Figure 18

-Velocity

on

Symmetry

Plane (Design

Flow)

Page 56

Figure 19

-Velocity

onShroud Side Plane (Design

Flow)

Page 56

Figure 20

-Velocity

onBottomPlane (Design

Flow)

Page 57

Figure 21 - 2-D Viewof

Velocity

on Shroud Side(Design

Flow)

Page57

Figure23

-Close-Up

of

Velocity

atOutleton Shroud Side (Design

Flow)

Page 58

Figure24

-Close-Up

of

Velocity

atOutlet onBottom Plane (Design

Flow)

Page 59

Figure25

-Close-Up

of

Velocity

atInlet on Shroud Side (Design

Flow)

Page 59

Figure26 - Pressure Contour (Design

Flow)

Page 60

Figure 27- Pressure

Along

theCenterline(Design

Flow)

Page 60

Figure 28

-Inlet Pressure (Design

Flow)

Page 61

Figure 29- OutletPressure (Design

Flow)

Page 61

Figure 30 - Kinetic

Energy

Contour (Design

Flow)

Page 62

Figure 3 1 - Dissipation Contour (Design

Flow)

Page 62

Figure 32

-Vorticity

Contour (Design

Flow)

Page 63

Figure 33 - Inlet

Velocity

Profile (Design

Flow)

Page 63

Figure 34- Outlet

Velocity

Profile (Design

Flow)

Page 64

Figure 35

-Velocity

on

Symmetry

Plane(60% ofDesign

Flow)

Page 67

Figure 36

-Velocity

on Shroud Side Plane (60%ofDesign

Flow)

Page 67

Figure 37

-Velocity

onBottomPlane(60% ofDesign

Flow)

Page 68

Figure 38 - 2-D View_of

Velocity

on Shroud Side (60%ofDesign

Flow)

Page 68

Figure 39 - 2-D View_of

Velocity

onBottom Plane (60%ofDesign

Flow)

Page 69

Figure 40

-Close-Up Velocity

at Outleton Shroud Side(60% ofDesign

Flow)

Page 69

Figure 41

-Close-Up Velocity

atOutlet onBottom Plane (60%ofDesign

Flow)

Page 70

Figure 42

-Close-Up

of

Velocity

atInleton Shroud Side(60% ofDesign

Flow)

Page 70

Figure 43 - Pressure Contour(60% ofDesign

Flow)

Page 71

Figure 44 - Pressure

Along

theCenterline (60% ofDesign

Flow)

Page 71

Figure46- Outlet Pressure (60%

ofDesign

Flow)

Page 72

Figure47 - Kinetic

Energy

Contour(60%ofDesign

Flow)

Page 73

Figure48 - DissipationContour

(60%ofDesign

Flow)

Page 73

Figure49

-Vorticity

Contour (60% ofDesign

Flow)

Page 74

Figure50- Inlet

Velocity

Profile(60%ofDesign

Flow)

Page 74

Figure5 1 - Outlet

Velocity

Profile(60% ofDesign

Flow)

Page 75

Figure 52

-Velocity

on

Symmetry

Plane(20% ofDesign

Flow)

Page 78

Figure 53

-Velocity

on Shroud Side Plane (20%ofDesign

Flow)

Page 78

Figure 54

-Velocity

onBottom Plane(20% ofDesign

Flow)

Page 79

Figure 55

-2-D Viewof

Velocity

on Shroud Side (20%ofDesign

Flow)

Page 79

Figure 56- 2-D View

of

Velocity

onBottom Plane(20%ofDesign

Flow)

Page 80

Figure 57

-Close-Up Velocity

atOutleton Shroud Side (20% ofDesign

Flow)

Page 80

Figure 58

-Close-Up

Velocity

at OutletonBottom Plane(20%ofDesign

Flow)

Page 81

Figure 59

-Close-Up

of

Velocity

atInlet on Shroud Side (20% ofDesign

Flow)

Page 81

Figure 60 - Pressure Contour(20% _ofDesign

Flow)

Page 82

Figure 61 - Pressure

Along

theCenterline (20%ofDesign

Flow)

Page 82

Figure 62- Inlet Pressure(20% _ofDesign

Flow)

Page 83

Figure 63 - Outlet Pressure (20% ofDesign

Flow)

Page 83

Figure 64 - Kinetic

Energy

Contour (20% ofDesign

Flow)

_{Page 84}

Figure 65 - Dissipation Contour (20%ofDesign

Flow)

Page 84

Figure 66

-Vorticity

Contour(20%ofDesign

Flow)

_Page 85

Figure 67 - Inlet

Velocity

Profile (20% ofDesign

Flow)

Page 85

Figure69

-Velocity

^n

Symmetry

Plane(60%

-3%)

Page 89

Figure70

-Velocity

onShroud Side Plane (60%

-3%)

Page 89

Figure71

-Velocity

onBottom Plane (60% -3%) Page 90

Figure72

-2-D Viewof

Velocity

on Shroud Side (60%

-3%)

Page 90

Figure 73

-Close-Up

of

Velocity

atDiffuser's Throat (60%

-3%)

Page 91

Figure74

-Close-Up

of

Velocity

atDiffuser's Outlet (60%

-3%)

Page 91

Figure 75 - 2-D View _of

Velocity

onBottomPlane(60%

-3%)

Page 92

Figure 76

-Close-Up Velocity

atOutletonBottomPlane(60%

-3%)

Page 92

Figure 77

-Close-Up

of

Velocity

atInleton Shroud Side (60%

-3%)

Page 93

Figure 78 - Pressure Contour (60%

-3%)

Page 93

Figure 79 - Pressure

Along

theCenterline (60%

-3%)

Page 94

Figure 80 - Inlet Pressure(60%

-3%)

Page 94

Figure 8 1 - OutletPressure(60%

-3%)

Page 95

Figure 82 - Kinetic

Energy

Contour (60%

-3%)

Page 95

Figure 83 - Dissipation Contour(60%

-3%)

Page 96

Figure 84

-Vorticity

Contour(60%

-3%)

Page 96

Figure 85 - Inlet

Velocity

Profile(60%

-3%)

Page 97

Figure 86- Outlet

Velocity

Profile(60%

-3%)

Page 97

Figure 87

-Velocity

on

Symmetry

Plane(60%

-7%)

Page 100

Figure 88

-Velocity

on Shroud Side Plane(60%

-7%)

Page 100

Figure 89

-Velocity

onBottomPlane (60%_-7%) Page 1 0 1

Figure 90 - 2-D Viewof

Velocity

on Shroud Side (60%

-7%)

Page 101

Figure 91

Close-Up

of

Velocity

atDiffuser'sThroat (60%

Figure92

-Close-Up

of

Velocity

atDiffuser'sOutlet (60%

-7%)

Figure93 - 2-D

Viewof

Velocity

onBottom Plane (60%

-7%)

Figure94

-Close-Up

Velocity

at OutletonBottomPlane (60%

-7%)

Figure95

-Close-Up

of

Velocity

atInleton Shroud Side (60%

-7%)

Figure96

-Pressure Contour (60%

-7%)

Figure97

-Pressure

Along

theCenterline(60%

-7%)

Figure 98 - Inlet Pressure(60%

-7%)

Figure 99 - Outlet Pressure (60%

-7%)

Figure 100 - Kinetic

Energy

Contour (60%

-7%)

Figure 101 - Dissipation Contour (60%

-7%)

Figure 102

-Vorticity

Contour (60%

-7%)

Figure 103 - Inlet

Velocity

Profile (60%

-7%)

Figure 104 - Outlet

Velocity

Profile (60%

-7%)

Figure 105

-Velocity

on

Symmetry

Plane(60%

-10%)

Figure 106

-Velocity

on Shroud Side Plane (60%

-10%)

Figure 107

-Velocity

onBottom Plane(60% -10%)

Figure 108 - 2-D View _of

Velocity

onShroud Side(60%

-10%)

Figure 109

-Close-Up

of

Velocity

atDiffuser's Throat(60%

-10%)

Figure 1 10

-Close-Up

of

Velocity

atDiffuser's Outlet(60%

-10%)

Figure 1 1 1 - 2-D Viewof

Velocity

onBottom Plane (60%

-10%)

Figure 1 12

-Close-Up Velocity

atOutletonBottom Plane (60%

-10%)

Figure 1 13

-Close-Up

of

Velocity

atInlet on Shroud Side (60%

-10%)

Figure 1 14 Pressure Contour(60%

-10%)

Page 102 Page 103 Page 103 Page 104 Page ' 104 Page ' 105

Page 1 05

Page 106

Page 1 06

Page 107

Page ] 07

Page ]08

Page 108

Page 11

Page 1 11

Page 1 12

Page 12

Page ] 13

Page 13

Page 1 14

Page 1 14

Page 1115

Figure 1 15

-Pressure

Along

theCenterline (60%

-10%)

Figure 1 16

-Inlet Pressure (60%

-10%)

Figure 1 17- Outlet Pressure

(60%

-10%)

Figure 1 18 - Kinetic

Energy

Contour (60%

-10%)

Figure 1 19

-DissipationContour (60%

-10%)

Figure 120

-Vorticity

Contour (60%

-10%)

Figure 121 - Inlet

Velocity

Profile(60%

-10%)

Figure 122 - Outlet

Velocity

Profile(60%

-10%)

Figure 123

-Velocity

on

Symmetry

Plane (20%

-3%)

Figure 124

-Velocity

on Shroud Side Plane (20%

-3%)

Figure 125

-Velocity

onBottom Plane (20% _-3%)

Figure 126- 2-D View

of

Velocity

on Shroud Side (20%

-3%)

Figure 127

-Close-Up

of

Velocity

at Diffuser's Throat (20%

-3%)

Figure 128

-Close-Up

of

Velocity

atDiffuser's Outlet (20%

-3%)

Figure 129 - 2-D View_of

Velocity

onBottom Plane (20%

-3%)

Figure 130

-Close-Up Velocity

at OutletonBottom Plane(20%

-3%)

Figure 131

-Close-Up

of

Velocity

at Inlet on Shroud Side (20%

-3%)

Figure 132 - Pressure Contour(20%

-3%)

Figure 133 - Pressure

Along

theCenterline (20%

-3%)

Figure 134 - Inlet Pressure (20%

-3%)

Figure 135 - Outlet Pressure (20%

-3%)

Figure 136- Kinetic

Energy

Contour(20%

-3%)

Figure 137 DissipationContour (20%

-3%)

Page ]116

Page 1116

Page ] 17

Page 1 17

Page

'

18

Page 1 18

Page

'

19

Page 19

Page 22

Page ] 22

Page 23

Page 123

Page 1124

Page 1124

Figure 138

-Vorticity

Contour (20%

-3%)

Figure 139- Inlet

Velocity

Profile (20%

-3%)

Figure 140-

Outlet

Velocity

Profile (20%

-3%)

Figure 141

-Velocity

on

Symmetry

Plane (20%

-7%)

Figure 142

-Velocity

on Shroud SidePlane (20%

-7%)

Figure 143

-Velocity

onBottom Plane(20%-7%)

Figure 144- 2-D View

of

Velocity

on Shroud Side (20%

-7%)

Figure 145

-Close-Up

of

Velocity

atDiffuser's Throat(20%

-7%)

Figure 146

-Close-Up

of

Velocity

atDiffuser's Outlet (20%

-7%)

Figure 147- 2-D View_of

Velocity

onBottom Plane (20%

-7%)

Figure 148

-Close-Up Velocity

atOutlet onBottom Plane(20%

-7%)

Figure 149

-Close-Up

of

Velocity

atInleton Shroud Side (20%

-7%)

Figure 150- PressureContour (20%

-7%)

Figure 151 - Pressure

Along

theCenterline (20%

-7%)

Figure 152

-Inlet Pressure (20%

-7%)

Figure 153 - Outlet Pressure (20%

-7%)

Figure 154 - Kinetic

Energy

Contour(20%

-7%)

Figure 155 - Dissipation Contour(20%

-7%)

Figure 156

-Vorticity

Contour (20%

-7%)

Figure 157- Inlet

Velocity

Profile (20%

-7%)

Figure 158 - Outlet

Velocity

Profile(20%

-7%)

Figure 159

-Velocity

on

Symmetry

Plane (20%

-10%)

Figure 160

-Velocity

on Shroud Side Plane(20%

-10%)

Page 129 Page 130 Page 130 Page 133 Page 133 Page ' 134 Page ' 134

Page 1135

Page 135

Page 1136

Page 136

Page 137

Page 37

Page 38

Page 138

Page 139

Page ]139

Page 1 40

Page 1 40

Page ]141

Page 1141

Page ]144

Figure 161

-Velocity

onBottomPlane (20%_-10%) Page 145

Figure 162- 2-D

View of

Velocity

on Shroud Side(20%

-10%)

Page 145

Figure 163

-Close-Up

of

Velocity

atDiffuser's Throat (20%- 1

0%)

Page 146

Figure 164

-Close-Up

of

Velocity

atDiffuser'sOutlet (20%

-10%)

Page 146

Figure 165 - 2-D View

of

Velocity

onBottom Plane (20%

-10%)

Page 147

Figure 166

-Close-Up

Velocity

atOutlet onBottomPlane(20%

-10%)

Page 147

Figure 167

-Close-Up

of

Velocity

atInlet onShroud Side (20%

-10%)

Page 148

Figure 168- PressureContour (20%

-10%)

Page 148

Figure 169

-Pressure

Along

theCenterline (20%

-10%)

Page 149

Figure 170- Inlet Pressure (20%

-10%)

Page 149

Figure 171 - Outlet Pressure (20%

-10%)

Page 150

Figure 1 72 - Kinetic

Energy

Contour (20%- 1

0%)

Page 1 50

Figure 1 73 - DissipationContour(20%- 1

0%)

Page 1 51

Figure 1 74

-Vorticity

Contour(20% - 1

0%)

Page 1 51

Figure 175 - Inlet

Velocity

Profile(20%

-10%)

Page 152

Figure 1 76 - Outlet

Velocity

Profile (20% - 1

ABSTRACT.

Future Space Transfer Vehicles

(STV)

will berequiredtoperform missions (orbital

transfer,

Lunar/Mars

transferand

descents)

forwhich

deep

engine

throttling

isneeded. In ordertodothis the turbopumps thatpropelthe SpaceTransferVehiclesneedtobeableto operate atdifferent flowrates. The current state oftheart cryogenicfuel and oxidizer

turbopump

designs donot operatewellat off-designflowrates_mainlydueto stalland flow separationinthediffusersection.

Thepurpose ofthisThesisisto analyzethebehaviorofthefluidinthe diffuserandthe

vanelessand vaned region oftheMK49-F

turbopump

atdifferent flowratesandtouse fluid

injectionas a_waytoreducetheflowseparation presentinthevaned diffuser. Tomeetthis

objective afinite elementbased _code,

FIDAP,

wasusedtobuildathree-dimensionalmodel

based on previous worksdoneonthevaneddiffuser. Previousworks studiedthebehaviorof thefluidinthevaneddiffuserwithout

taking

into considerationthevanelessdiffuser.

Fromtheresults _obtained,itwas observedthatflowseparationhasoccurred atthe

bottomplaneofthevaned

diffuser,

whentheflowrate was reducedto 60%ofthedesign

flow,

and inthe

top

plane ofthevanelessregion. Theseresults are different from theones obtained inprevious workswheretheflowseparation wasfoundinthe

top

plane ofthevaned

diffuser. This showsthatwiththeadditionofthevaneless_{section, the}flow behaviorchanges

significantly.

Fluid injectionwas applied atthebottomplane ofthevaneddiffuserthrough six

differentslits at20% and 60%ofthedesign flow. Variousrates of_{fluid injection}weretested

fortheireffectivenessin_suppressingor_eliminatingtheflowseparation.Results showedthatat

CHAPTER

1. Introduction

1.1 Project Justification

The

future

space missions planned

by

NASAwill useSpaceTransfer Vehiclesto

perform orbitaltransfermissions andLunar/Marstransferand descents. Inordertodo

this_properly, thevehicle musthaveadeep-engine

throttling

_capabilitythatcanbe

obtained

by

designing

ahighperformanceliquidhydrogen

(LH2)

turbopump

which can

efficientlyoutput atdifferentflowrates.

The currentdesignsofhigh_pressure, multistageturbopumpswithradialvaned

difiiisers donot perform_efficiently at off-designflowrates. The lowerflowratesleadto

poordiffuser_performance, whichcanbe observed

by

theflowseparation andthediffuser

stall, are dueto the impeller discharge_effects, increased

boundary

layer blockageand lackofturbulence

intensity

inthediffuser. This projectinvestigatesa means of

increasing

the diffuserperformanceat off-designflowrates without_{significantly altering} the

geometryofthediffuser.

The final objective oftheProjectattheRochester Institute of

Technology

(RIT)

isto

develop

methodsto improvetheperformance oftheMK49-F High Pressure

LH2

turbopump

by improving

theperformanceofthediffuseratoff-designflowrates. Inorder

toaccomplish_{this goal,} 3-D models ofthevaned diffuserweredeveloped_usingafinite

elementbased _code,FIDAP (Fluid Dynamics Analysis Package).

This Thesispresentsthenext _stepwhichisto

develop

athree-dimensional

turbulentmodel _consistingofthevaneless and vaneddiffuser sections ofthe turbopump.

Thesemodelswill beusedtogainabetter understandingofthe_{resulting flow}patterns and

1

.2

Project

Objectives

Acentrifugal

turbopump

is

a radial flowturbomachinedriven

by

aturbine. A

typical stage of a

turbopump

includes

the

following

functional elements:

1. Aninducer section,within whichthefluid isturnedfroman axialflowdirection

toa radialflow direction.

2. Animpeller_section,withinwhichthefluidflows_radiallythroughtherotor.

3. A_{diffuser section,} withinwhichthefluid _exitingtherotoriscollected and

directedto the_pump exit.

Thesethreefunctionalelements do not_necessarilycorrespondto structural

elements ofthepump. The functionalinducersectionconsists of_everythingfrom the

pumpinletto some radiuswithinthestructuralrotor_assemblywheretheflowoffluid is

nolonger has anyaxial component. The functional impellersection consists ofthatportion ofthestructural rotor_{assembly from}theexit ofthefunctional inducersectionto theexit

tip

oftherotorblades. The functional diffuser section consists of_{everything from}theexit

tip

oftherotorbladesto theexit ofthepump.

Thislastsection ofthe_pump,whichiswhat this_{Thesis studies,} canbedividedinto

twoseparate regions. The firstone wouldbethevaneless sectionthatgoesfromtheexit

tip

oftherotorbladesuntilthefluidgoes intothevaned diffuserandthe second wouldbe

thevaneddiffusersection_,which isone ofthebasiccomponents ofthe system and

responsibleformost oftheconversion oftheinlet dynamicpressure(kinetic energy)to

static pressurerise. Forsubsonic

flows,

thisis

done

by decelerating

thefluid particles

by

providingagradualand continuous increaseofthecross-sectional area and

try

torecover as much oftheinlet dynamicpressure

during

_{steady flow}conditions. Inaddition, it is

However,

the

diffuser

performsinan adverse pressure gradientfieldwhichlimits

hs

_efficiencyand wherethe

flow

separation and stall occur dueto theincompressible and

viscous nature ofthe

flow field.

Flowseparationoccurswhenthe

fluid

particlesina

boundary

layerareslowed

down

by

wallfriction. Ifthe_{flow is sufficientlyretarded,} forexampledueto thepresence

of an adverse pressure_gradient,themomentum ofthoseparticles willbereduced

by

both

thewall shear andthepressure gradient. Intermsof_{energyprinciples, the}kinetic energy

gained, attheexpense ofthepotential _energy, inthefavorablepressure gradient regionis

depleted

by

viscous effectswithinthe

boundary

layer. Intheadverse pressure gradient

region, theremainingkineticenergyisconvertedto potential_energybut it istoosmallto surmountthepressurehillandthemotion ofthenearwallfluidparticles is_eventually arrested. _{At separation,}thereverseflowregion nextto thewall _{abruptlythickens,} the normal component

increases,

and the

boundary

layerapproximations are no longervalid.

Iftheflow doesnot reattachitselfto the diffuserwall itwill dissipate intoturbulent_mixing

andthediffusionprocessfor_providingpressure_recoverywill cometo an end.

Thegoal ofthiswork isto provideinsight intotheeffectiveness offluid injection

as a

boundary

layercontrol method_{in suppressing}or_eliminatingthediffuserstallthat

causesthepoor performance atlowflowrate conditions and alsotodemonstratethatthis

techniqueallows a pressure_recoverythat_{is substantially}higherthaninadiffuserwithout

it. Theworkdescribedhere looksat athreedimensionalcomputational model_consisting

ofthe vaneless and vaneddiffusersections ofthe

turbopump

underdifferent flow

1.3

Project

Description.

TheMK 49-F High Pressure

LH2

Turbopump

used

by

NASAcanbeseenin

Figure 1. This

turbopump

wasdeveloped

by

theRocketdyne DivisionofRockwell

International,

which also developeda simplified version ofit as a single stageWater Tester

torun performancetests. TheMK 49-F isathree_{stage, centrifugal,} highpressure

turbopump

usingliquid hydrogenasthe_workingfluid. A_{high speed, high efficiency}

multistage centrifugal _pumpofthisnature requires continuous passage diffusercrossovers.

These diffusercrossovers act as channelsforthefluid _goingfromone stageto thenext.

Thepressurerisethat canbeachieved _efficientlyina single stageis

limited,

depending

onthe typeof machine.

However,

stages_{may be} combinedtoproduce multistage _{machines,virtually}without limitonthepressurerise.

Acharacteristic oftheMK 49-Fisthatit has 17continuous crossovers passages

betweeneachimpellerstage. Thesepassages servethepurpose of_conveyingthepumped fluidfromtheexit of oneimpellertotheinlet ofthenextimpellerand are ofparticular interest becausethisiswherethe_recoverytakesplace. Asitwas mentioned

before,

Rocketdyne simulatedtheMK 49-F

turbopump

with a single stageWater Testerthatwas

usedto run performancetests. The Water Testeris shown in Figure

2,

whileFigure 3 showstheindividual diffusercrossover section enlarged anddimensioned (Water Tester

scaled_up

by

afactorof2.85).

Referring

toFigure

3,

thefluidpaththroughanindividualcrossover passage can be described. Thefluid leavestheblade

tip

at ahigh_velocityand entersthevaned diffuser. Thevanes guidethefluid fromtheimpeller intothevanelessdiffuser. Once it has entered

thediffuser crossover_{section, the}fluid

flows into

theupcomer whichisa radial outflow diffuserwhere most ofthe_recoverytakesplace. The fluidthenflowsthrougha

turning

Q. E 3 Cl O Xi

u.

3

H

I

-J H) u

3

t/5 Ol

4)

CO

o

TT

4)

u

3 60 IZ

1

rrc

o

O

c

u >

0

w

CO

O

c

o

Oil

ZQ

Figure 2 - High

view

C-C

FLOW INLET

downcomer

which

is

a radialinflow

diffuser,

actingas a passage_wayto the next_pump

stage.

InthisThesis_work, athree-dimensionalmodel _consisting ofthevaneless and

CHAPTER

2.

Principles

of

Turbulent Flow

2.1

Governing

Equations

The

Navier-Stokes

equations describe fluid flow in either laminar or turbulent

state.

Taking

into

accountthe principles of conservation of_mass, momentum and _energy,

these _equations, when _applied, will show us the behavior ofthe flow field within the

turbopump

vaneless and vaned section

by describing

the fluid flow at _every point in the flowregimeforalltime.

uu

=

dui

ct

+UjUi, =_{-p., +pf, +pg,[\}

-p (T-7]+["

("'>

+M1

dT

p cr dt

+ uj

T.j

= _(kT.

j),j+p+ q,

(Eq.

1)

(Eq.2)

(Eq.3)

Thetype offlowthat isused in this problem allowsfor some simplification

because it is assumed that within the

LH2

turbopump

diffuser,

the flow can be described

as _{turbulent, steady,}

incompressible,

isothermal,

and Newtonian in nature and therefore

theconservation of_energy equation will notbeused andtheequations forconservation of

mass and momentum are reducedto:

Ui.j =₀

(Eq.4)

p Uj U,.j=

-p.i+pf,+[p(u,.j+Uj.,)]j

(Eq.5)

where u

is

thefluid _velocitywith

i,

j

=

1,

2,

& 3 forathree-dimensional _{problem, p is}the

2.2

Methodology

of

Analysis

Flows in turbopumps are

highly

turbulent. In turbulent flow _situations, the fluid

motionis

highly

_{random, unsteady, andthree-dimensional}and becauseofthis theturbulent

motion and _{mass-transfer phenomena} associated withit are _extremely difficultto describe

and thus predict theoretically. It is

believed

that turbulent flows canbe described with the

usethe of_{the time-dependent three-dimensional}Navier-Stokesequations.

The best _way to describe turbulent motion is

by

_using time averaged quantities

rather than

instantaneous

ones _using the conservation laws for mass and momentum.

Thesebasic conservationlaws are expressed

by

the equations 4 and 5. Osborne Reynolds

was thefirst to suggest_usingastatistical approachwheretheequations are averaged over

a time scale which is

long

compared with that of the turbulent motion to obtain the

equationsthat describethedistribution ofthemean_velocityandpressure.

Thisapproach separates the fieldvariables

(velocity,

_uj and _{pressure, p)} intomean

and

fluctuating

quantities _{allowing for}theuse of mean values ofthefieldvariables

(u[

and

p

)

in modeling the large scale flow characteristics. For an _arbitraryfield variable _T|, we

candefineitsmean value_as,

a r+Ar

tj=-

[ndr

(Eq.

6)

A-T *

At

wherethe_averagingtimeAtis

long

compared withthetime scale oftheturbulentmotion.

Theinstantaneousvariable_r\ isthendecomposed as,

where _r\ ' reflects the small scale

fluctuations

associated with turbulence. This decomposition

is

appliedto the

Navier-Stokes

equationswhich are thenintegrated overthetu. ->. interval

(

t,

t

+

At)

resulting inthe

following

equationswhichgovernthemean-flowquantities(theoverbars

indicating

averagedvaluesthatwill

be

dropped fromthispoint

forward)

.

w..>=o (Eq.

8)

p Uj u,.j~-p.i+pf,+

\fi

(u,.j

+

Uj,,)~pu'

u\

j (Eq.

9)

Dueto thenon

linearity

oftheNavier-Stokes equations,the_averaging process

introduces a correlationbetween

fluctuating

velocities_ujuj

Multiplying

this term

by

pgives

thetransportof momentumdueto the turbulentmotion. Theterm

. l+Al

pv!xu\=-~

\pu\u\dx

_(Eq.

10)

At

describesthe transport ofxj-momentuminthedirectionof x; and actsas a stress onthefluid

(

Reynolds stress

)

and it also summarizesthe effect of small scale _{eddy behavior} onthe large

scale mean flow. To solve the Navier-Stokes equations and Eq. 10 requires a _way of

determining

the turbulence correlation. This determination isthe main roadblock _{in analyzing}

turbulent flows. A turbulence model which approximates this correlation _along with the

Navier-Stokes equations forms a closed set of equations which can be solved for the mean

values of_velocity andpressure.

Thetime_averagingtechniquealso provides abasis forsometurbulentflow definitions.

The

intensity

ofthe_velocityfluctuationsisgiven

by

theirmean square value

\u[

J

Halfof

thisvalue

is

definedastheturbulentkineticenergy.

k=

\<<

(Eq.")

Anothercharacteristic

is

the

intensity

ofturbulence inthe

flow,

which is definedas

theroot mean squaredofthe_{velocity fluctuations}to the timemean velocity.

n -J/2

/ =

-u' u'

2 "' "'

36w,

(Eq.

12)

This dimensionless _quantity is used as an indication ofthe turbulence level ofthe

fluidat_anypoint intheflow based onhowmuchthe_velocityfluctuations deviate fromthe

average flow. There are several other relationships that are important in _{understanding}

turbulent flows. The dimensionless_velocity u+

, dimensionless normal distance from the

wall_{y ,} the shear stress atthewallx _, andthefriction_velocityu aredefinedasfollows:

^=^r (Eq.

13)

u

y'-P~^

(Eq.14)

T =T

to, (Eq.

15)

u=,\~ _(Eq.

16)

Based onEquations 13 through 16 and characteristics ofturbulent

flows,

theflow

of y+=5

is

termedtheviscous sublayer. Nearthe centerline oftheflowat adistance

greater than y+=30 exists the

fully

developed turbulent region. In between these two

regions

it is found

whatis known as the buffer region. Figure 4 shows these regions in a

graphical

form

[4]. The regions are defined

by

the different flow characteristics that are

found

within each_region,whichis helpful in

discussing

thecomplexities ofturbulent flow.

2.3 Turbulent Flow

Modeling

Procedures

The eddy viscosity-diffiisivity model often used to model the Reynolds stress is

based on the assumption developed

by Boussinesq [4]

that the fluxes ofmomentum are

proportional to the gradients of the mean flow field.

Boussinesq

introduced the

proportionality parameter which is termed the _{eddy viscosity} and is dependent upon the

turbulence ofthe

flow,

which is a function ofposition. This relationship is defined as

follows:

-pu[ u\=pt(uX]+U),) (Eq.

17)

This approximation allows Eq. 9 to be rewritten as Eq. 5 provided the total

viscosityis identifiedasthesum oftheshear and _eddyviscosities.

P=

p0+Mt (Eq.

18)

The eddy-viscosity concept transforms theproblem ofturbulence _modeling to the

determinationofthedistributionof fif.

Twoturbulencemodelsare_{commonly used,}

(a)

thezero equation_{mixing length}

fully

How

Figure

4 -

Near

Wall

Model.

equations and use the _{eddy-viscosity concept,} but the former model is not conducive to

complex

flows.

Therefore,

the two equation k-e model was used because it is more

effectiveincases offlowseparation and adverse pressure gradientflows.

Thetwo equation k-emodel describestheturbulentkinetic_energy associated with

the small scale _{eddy behavior} as shown in Equation

11,

which suggests that _velocity

fluctuations

can be characterized

by

the single parameter k^ _, which in turn gives the

approximation,

p, xkm (Eq.

19)

Atransport equation for k can be obtained from the Navier-Stokes equations

by

algebraic manipulation. Thistransportequationinvolves6, whichis defined_as,

1 r'+A/

A?J'

L'-J

"

= v

<j <j

=v

77

P

Kj

u'j

dT (Eq.

20)

which represents the viscous dissipation ofturbulent kinetic energy. A second transport

equation for 8 can also be derived from the Navier-Stokes equations. The transport

equations ofturbulent_{kinetic energy}and viscous dissipation_are;

Pujkj=\

*J

+p<-p

(Eq-21)

P

vj

j

=

\<T,

J tj

15

e

e2

+

whereO

is

theviscous

dissipation term,

the_{eddy viscosity relationship,}

Pt=pcM

(Eq.23)

and the

Navier-Stokes

equations form a set of equations that will approximate the

resultingturbulent flow inaninternal passage.

However,

the equations are nolonger exact

and the results generated must be interpreted as approximate values. The previous

equations contain several empirical constants which require definition. The empirical

constants, Cm, c\ & C2, are set at

0.09, 1.44,

and 1.92 respectivelyfor isothermal flows

andtheturbulent Prandtl and Schmidt _numbers, o^Sc

oe

, were determined to be 1.0and

1.25 respectivelyforthisflowsituation [4],

The above results arefor "high Reynolds number"

flows and areuseful inthe

fully

turbulent regionwhere the _velocity profileis rather flat [4]. Due to theapplication ofthe

no-slip condition at the diffuser _walls, the flow characteristics are subject to _{very steep}

gradients near the wall. The above results will not be applicable over this low Reynolds

number

boundary

layer. The Law-of-the-Wall model provides the link between the

fully

turbulentregion andthe_no-slip,nearwall region.

The Law-of-the-Wall model requires that the region under investigation be _away

from _{any stagnation,} reattachmentand separation_points, withtheflowparallelto the_wall,

no

body

forces _present, and weakpressure gradients present. These assumptions _may not

hold for

diffiisers,

butthe model willbemodified to relaxthese restrictions. A coordinate

system is set _up such that the first axis is tangent to the wall. The tangential momentum

equationreduces

to,

(r

).2

=

where _rm

is

the sum ofthelarninarandturbulent shearstress,

Tto,=Pui2-Puiui (Eq.

25)

where p.

u^2

is the

laminar

shear stress and _{p u'\ u'2 is} the turbulent shear stress. In the

near wall _region, where y+

<

5,

_{the laminar shear stress is dominant. As the} flow

progresses through the buffer

layer,

where 5 < y+

<

30,

_{turbulence is} generated

by

an

increase in turbulent shear stress and a decrease inlaminar shear stress as can be seen in

Figure 5. Inthe

fully

turbulent _region, the turbulent shearstress is dominant. Thisanalysis

is in agreement with accepted

boundary

layer

theory

that indicates that the conventional

fluid_viscosity need _onlybe accounted forwithin a_very narrow region canbe determined

allowingforthe developmentoftheLaw-of-the-Wall model.

Neglecting

the turbulent shear stressintheviscous sublayer_gives,

Ttc, =

P "1,2 (Eq.

26)

which

by

substitution and rearrangement_{leads to,}

u _ p

u _y

u _p

and

by

definition,

thisbecomesthelinear_velocityprofile

u=y

(Eq.

27)

(Eq.'28)

Beyondy+

Figure

5

-

Stress Profiles in

the

Near Wall Region.

so_that.

t _=-putu2 -p u

which

leads

_{to theclassical}

logarithmic

velocityprofile,

Ep

u'y

(Eq.

29)

u 1

= -ln

U K _{\ P J}

(Eq.

30)

where k'\s thevonKarman constant equal to 0.41 forthis situation and E is an empirical

constant equalto 9.0 [4].

Equations 28 and 30 are plotted in Figure 6 along with a typical _velocity profile

forthenear wall region. From dimensional_{analysis, the}equationsfor kand e are given

by,

* =

<""

(Eq.31)

e=~ _(Eq.

32)

ky

The profiles fork and6 in the nearwall region are shown in Figures 7 and _{8 respectively}

[4]. Aswas statedintheearlier

discussion,

several restrictionswere placeduponthe

Law-of-the-Wall model. Modificationsto account for the flow separation and stall present in

the diffuser will be employed withEquations 28 and 30-32 to approximate the near wall

viscous sublayer

buffer

layer

fully

turbulent

region

20

-15

10

5

-10'

101

typical _velocity

prof!!

empirical scatter

10J 103 10*

Figure

6

-

Universal

Velocity

Profiles in

the

Near Wall

Region.

10 20 30 40 50 60 70

Figure

7

-

Universal Kinetic

Energy

Profile in

the

Near Wall Region.

Figure

8

-

Universal Viscous

Dissipation

Profile

in

the

Near Wall Region.

CHAPTER

3.

Principles

of

Diffusion

3.1 Description

and

Applications

Adiffuser'spurposeisto convertthe inlet dynamicpressure ofthefluid to a static

pressure rise andthis

is

_why

they

are basic componentsin aturbomachine. For subsonic

flow

(M<1),

thisis accomplished

by decelerating

the fluidparticles

by

the application of a

gradualincrease ofthecross sectionalflowarea. In consideringthe effect of area variation

on

fluid

properties _{in isentropic}

flow,

we shall concern ourselves _primarily with _velocity

and pressure.

Using

the_equation,

dA dV

=

-[1-M2]

(Eq33)

derived from the differential momentum equation for isentropic

flow,

it can be seen that

for M<1 an area change causes a_velocitychange of oppositedirection (positive dAmeans

negative dV for M<1). A subsonic diffuser requires an increase ofthe passage area to

cause a decrease invelocity. It is also important that the _exitingflow is _steady and has a

uniform_velocityprofileforthenextimpeller stage.

There are several parameters used to describe a diffuser _geometry [5], These

quantities are useful in _analyzing the performance ofthe diffuser flow field. A simple

diffuseris showninFigure 9 [6]. The_geometryof adiffuserisspecified

by

theaspect ratio

, thedivergenceangle20, thelength-to-widthratio , andthe area ratio Losses

Wx

Wx

Wx

in diffusers depend on a number of geometric and flow variables. _{Diffuser data} most

theratio ofstatic pressureriseto inlet dynamicpressure:

Cp=IT-El

(E^34)

-P2H y2,'

where

P2

is the outlet _{pressure, p\ is} the inlet pressure, and v

t is the mean velocity at

the throatwhichisthe straight channel priorto thevanelessdiffuser. The definitionofC_mayberelated to thehead loss:

V'

1

2 LV (AR)2

K

=T"

^-77^T>-C^

^ 35>

For frictionless flow

h^

-0,

which gives us anideal pressure _recovery coefficient

that isafunction_onlyof_geometry,

(ARY

CP^=1-T7^I

(Eq36)

whereARisthe area_ratio, defined as

AR=(\ +^tan0)2

(Eq.

37)

The ratio ofthe actualpressure _recoverycoefficientto the ideal pressure_recovery

coefficientisthe _{diffuser efficiency}r\.

Figure

9

- Diffuser Geometry.

The

diffuser flow

characteristicsfora subsonic andincompressible floware given

by

the

following

_parameters,the throatReynoldsnumber,

Ret,

and the. throat

blockage,

B,

Their

definitions

are given_respectively as:

R.,

p

v,A

(Eq.

39)

B.=

25'

W

(Eq.

40)

where, W is thewidth ofthe throat and

8*

is the displacement

boundary

layer thickness

calculated fromthe_velocityprofile.

'-!KJ*

(Eq.

41)

3.2

Diffuser

Stall

Diffuser stallisaconceptdescribed

by

Prandtl's

boundary

layer

theory

[7]. In

ordertoencounter diffuserstall adifferentphenomenom called

boundary

layer separation

mustfirst occur.Prandtl _predicted, foraflowregime consideredtobetwo-dimensional

and _steady, thata point ofseparationwill occurinan adverse pressure gradient region

d_p_

d Xj

>0 _, whenthe_velocitygradientatthe wallis_zero,

fd^

\dy)

= 0 This

implies

y =0

that the shear stress atthewall is_zero,t

-p

^

\dy)

= 0

y =0

theeffectonthe_velocityprofile offrictional

drag

leading

to thetransitionto stall and flow

reversal.

The

biggest

difficulty

of

designing

and _employing diffiisers is that the maximum

pressure_recovery and peak _efficiency of most diffiisers occurswhen the adverse pressure

gradient

is

greatestor neartheso called stallline. ThiscanbeseeninFigure 13 that shows

thepressure_recoveryas afunction of area ratioforaconstantaspect ratio. There arefour

major regions of stall defined as the no appreciable stall _area, large

transitory

stall _area,

fully

developed two_{dimensional stall area and thejet flow area. Small}

diverging

angles

and area ratios characterize the area of no appreciable stall with the flow _steady and

symmetric with no visible disturbances.

However,

on the microscopic

level,

there is an

appearance of _very small stall bubbles _continually regenerated and destroyed on the

diverging

walls. Theformation oflarge stall regions nearthe diffuser throat _{causing large}

fluctuations in the pressure field characterizes the large

transitory

stall region. A large

stationary stall bubble that grows from the diffuser throat _along the wall characterizes

two-dimensional stall. This creates athick turbulentblockage areaatthediffuserexit. The

formation of stall regions on both diffuser walls, with a _{continuing steady}flow _along the

centerline, characterizesthejet flowregion.

Transitory

stall in diffiisers is a phenomenon ofinternal flowthat is _unsteady and

very difficult to predict. In these _unsteady flows , the maximum pressure recovery at

constant diffuser length-to-width_ratio, ,isachieved as

transitory

stall startsto

develop

w\

[8].

Transitory

stall wasfirstrecognized as a result offlow visualization experiments. The

most useful contributions to this topicwere made

by

Reneau,

et al

[10],

who developed

thepressure_recoverychart(Figure

13),

and

by

Fox andKline

[9],

whoperformed diffuser

flow regime studies (Figure 14). The pressure _recovery chart shows how peak pressure

Figure

10

-

Prandtl's

Boundary

Layer Concept.

The Flow Regime Chart developed

by

Foxand Klineisuseful in_predictingstallfor

different diffuser

geometries. The chart bases its findings on the diffuser's geometry

N

characterized

by

and 29.

Wx

The MK49-F

turbopump

diffuser's_geometry is located itselfon the Flow Regime

Chart in the no appreciable stall region nearthe line a-a ofFigurel2.

However,

this chart

is useful for predicting stall in diffiisers at the design,flow rates with no incidence angle

effects. The

incidence

anglesintroduced

by

the_{flow entering}thediffuser fromtheimpeller

blade

tip

effectstheMK49-Fdiffuserasitthrottles throughvariousoff-designflowrates.

It is important to note that diffusers with distorted inlet _velocity profiles exhibit

stall behavior quite different from that found in diffusers with uniform inlet _velocity

profiles, such as the development of a centerline pocket stall ifthe inlet _{velocity flow} is

severely distorted.

The development ofthe turbulent

boundary

layer has a significant impact on the

diffuser performance. If the turbulent

boundary

layer is thick _creating a large throat

blockage,

separation will occur near the inlet ofthe

diverging

section. The fluid particles

decelerate near the wall region under the effect ofan

increasing

pressure gradient and

reduced transverse momentum transfer. As the fluid progresses through the

diffuser,

excessiveblockage occurs_reducingthediffuserefficiency. Inturbopumps_operatingat

off-design flow rates, this lackofturbulence

intensity

and increased frictional

drag

createsthe

environmentforflowseparation.

No

appreciable

stall

Cr

Large

transitory

stall

NAVi

constant

Two-dimensional

stall

Jet

Flow,

JetFl

ow

AR

Figure 1 1

- Diffuser Pressure

Recovery

Chart.

100

90

60

40

30

29

20

IS

10

4

3

2

f.S

M How

Hytrt(t Zona

fZlrf

Q*l0PiP i

Two Qimtntionol Stall

s

Largo

Tronitory

Stolt

Lint of Apprtcioeio Sfol

i I

No Apprteiooio Stoii

! I

i

I

X

^

i i

L9 2 34 fi O 19 20 30 40 0

Figure

12

-

Diffuser Flow

Regime

Chart.

CHAPTER

4.

Boundary

Layer Control

by

Fluid

Abjection.

4.1 Flow

Separation.

Great advances

have

been made in _establishing a firm analytical foundation for

steady,two-dimensionalseparation. Ontheother

hand,

theoreticalornumerical analysisof

three-dimensional or_unsteady separationis less developed. The breakthrough in _unsteady

separation research was achieved

by

Moore, Rott,

and Sears

[22] during

the 1950's. Prior

to their work it was believed that _steady and _unsteady separations have the same

characteristics,namely, the point of_vanishing wall _shear, the termination ofthe

boundary

layer,

and the

beginning

ofthewake orbubble of separated fluid. Rott in 1956 analyzing

the _unsteady flow in the _vicinity of a stagnation _point, noted that the point of_vanishing

wall sheardoes not coincide withthe point of

boundary

layer detachment. In

1958,

while

investigating

a _steadyflowover a_movingwall, hearrivedtotheconclusionthatforaslow

moving wall, separation occurs _when, at some point in the

boundary

layer,

the profile

velocityandshear _{simultaneously}vanish.

4.2 Control

of

Boundary

Layer Separation

by

Fluid Injection.

The term

boundary

layer control includes _any mechanism or process through

whichthe

boundary

layer iscaused to behave

differently

thanit _{normallywould,} werethe

flow

developing

naturallyalong a smooth straight surface.

Separationcontrolis ofimmense importanceto theperformance of

turbomachines,

diffusers,

air and water vehicles, etc. On the other

hand,

in some

instances

it _may be

4.2.1

Active

and

Passive Fluid

Injection.

Near-wall

momentum addition is the usual approach of choice for control of

residual flow separation _remaining after mitigation ofthe causative pressure field or

off-designconditions.

Common

toall the different control methods isthe _supply ofadditional

energyto thenear-wallfluid particles which are

being

retarded inthe

boundary

layer. The

additional longitudinal momentum is provided either from an external source or through

local redirection into the wall region. Passive techniques do not require _auxiliary power,

but do have an associated

drag

_penalty, and include intentional

tripping

oftransitionfrom

laminarto turbulentflowupstream of what wouldbe alaminar separation_point,

boundary

layer fences to prevent separation at the tips ofswept-back _{wings, placing} an _array of

vortex generators on the

body

to raise the turbulence level and enhance the momentum

and _energyintheneighborhood ofthe_wall, rippled

trailing

edge, streamwise _{corrugations,}

stepped afterbodies to form a system of captive vortices in the base of a blunt

body,

or

usinga screentodiverttheflowand increase the_velocitygradient atthewall,

Active methodsto postpone separation require_energy expenditure.

Obviously,

the

energy gained

by

the effective control of separation must exceed that required

by

the

device. A fluid _maybe injected parallel to the wall to augment the shear-layer momentum

ornormal to the wall to enhance the_mixing rate. Either a blower is used or the pressure

differentialthat exists on theaerodynamic

body

itself isutilized to dischargethe fluid into

the retarded region on the

boundary

layer. The latter method is found in nature in the

thumbpinion ofa_pheasant, the split-tailof a

falcon,

orthelayered wings

feathers

of some

birds.

Inman-made

devices,

passive

blowing

throughleading-edge slits and

trailing

edge

flaps

is

commonly used on aircraft wings. Although inthis case direct energy expenditure

the

body

itself.

Nevertheless,

theeffect ofpassive

blowing

onliftand

drag

couldbe

dramatic. This is shownintheFigure 13 fortheNACA 23012airfoil sectionwithno

flap,

with a singletrailing-edge

flap

and with adouble-slottedflap. Compared to the clean(no

flap)

case,with a singletrailing-edge

flap

is usedthemaximumlift is increased

by

about

175%whilethesection

drag

at

Clmax

is increased

by

morethan 180%. _{The corresponding}

numbers when a

double-slotted

flap

isused are_{respectively,} 230and500%.

Direct tangential

injection,

wall

jets,

was and still is the preferred and

straightfoward flow separation control technique since removal and ejection of low

stagnation pressure fluid _{(fluid suction)} can be difficult in some instances since this

techniquerequires a complex arrangement of severalindependentbleedchambersthatare

not always possibletoinstall.

As it was mentioned

before,

the basic principle of fluid

injection,

consists in

bringing

momentum to the flow in order to increase its _ability to overcome, with

minimum

damage,

anadversepressure gradient.

The _efficiency of fluid injection depends on several _parameters, the most

determining being

themomentum

ij

ofthe injectedfluid andthe distance

Lj

betweenthe

injection slits and the separation point. The maximum allowable distance

Lj

can be

estimated

by

first _computing

5j

, the physical thickness ofthe

boundary

layer,

with a

convenient

boundary

layermethod andthen_applyingaseparation criterion.

In Figure 14 itisshown_that, foragiven value of

ij

,the most appropiatedistance

Lj

resultsfromacompromisebetweentwo tendencies:

(1)

Lj

must be

long

enough to allow the _mixing process

by

which momentum is

transferredfrom thejet to the

boundary

layer in sufficienttime to be _really effective. If

not, the adverse pressure gradient will drive back the

insufficiently

accelerated flow

which creates a pocket of

'separated'

3. Double-slotted

flap

v r

^ 2.Slotted_flap

1. Clean (no

flap)

5 0 5 10 _{15 20}

a

(deg.)

a. Liftcurves.

0.10 0.20 0.30

CD

b.

Lift-drag

polars.

Figure

13- _NACA

23012.

a -

fluid injection is

too close to the shock.

P

(b)

b

-fluid injection is

too

far

from

theshock.

Figure 14

-

Effect

of

Fluid

Injection

on

Boundary

Layer.

goes through a minimum

(Fig

14-a)

Such phenomenon is accompanied

by

a dramatic

increase

in the turbulence level which can be the cause of

instabilities

and loss of

efficiency.

(2)

The new

boundary

_layer which builds _{up between} the wall and the jet has a

thickness

;

increasing

withLj. Thus ifthe distance

Lj

is too

long,

8j

will reach avalue

suchthat the pressure gradient(whoseactionincreases inproportion with

S}

)

will be in a

positionto separate the

boundary

layer (Fig. 14-b).

Moreover,

thejet maximum_velocity

decreaseswhen

Lj

increases,

whichtendstoworsentheprocess.

Theinjectedmomentumismost often characterized

by

thecoefficient:

m(uj

-ueo)

IM

=

2 , (b

=

span, 6 =

momentum

thickness)

(Eq.

42)

Heo eo

Thus / represents the ratio between the momentum excess in the injected fluid

(relative to the local speed u^ in the upstream

flow)

to the momentum deficit in the

upstream

boundary

layer. The subscript _e, designatestheconditions at the

boundary

layer

outer edge andthe subscript _o, designatesconditions attheinteractionorigin.

The experiments used to _verify this information found that the control system

becomes inoperativefor

Mo

=2. Indeedatthis_{Mach number,} therequired injectionmass

flow

is

suchthat (forunchanged orifice _size)thejets stagnation pressure _{ptj is} so highthat

the obstacle effect due to thejets expansion separates the

boundary

layer. One must be

aware ofthis possible negative consequence of

fluid

injection,

as it can be seen _{in Figure}

14. In

fact,

by

improving

theinjection system, interactions couldbe controlled _upto

Mo

=

flow _m, the

injection

ofhotairdid not_appreciablyimprove thesystem's performance

This

finding

is

not expected since with hot injected

fluid,

the

IM

can be

considerably increased

(by

more than 350% for a temperature ratio of 2).

Hence,

the

investigators

concludedthat the coefficient definedwiththeinjected momentum

z';

=m

u]

is the_onlyonethatmakes sense_{physically speaking}

Inthe diffuser _studied, theflow separates _along the lower wall ofthe diffuser due

to the pronounced inlet incidence angles. As a _result, fluid injection was applied at the

bottom ofthe diffuser in an attempt to counteract the incidence angle effects and to

energize the

boundary

layer. Several injection angles were studied in order to determine

their effect on the

boundary

layer. The best results were obtained when the fluid was

injected ata35 degreeanglefromthe diffusercenterline at each ofthe six slits.

The outflow offluid across the six slits was uniform. The slits were positioned

along the wall ofthe diffuser from the shroud side to the hub side (Figure 15). The slits

were positioned near the diffuser inlet in order to more _efficiently add momentum to the

decelerating

particles, as was suggested

by

the previous work of Wissinger

[3]

and

Yoshida [2]. The slits are 2mm. in width and are placed 1.25 cm. apart. Several fluid

injection rates were tested in order to investigate their effect on the

boundary

layer

control. The injection rates studied were

3%, 5%,

10% and 15% ofthe total mass flow

rate.

CHAPTER

5. FINITE ELEMENT METHOD

5.1 General

Concepts

The flow field at _any point in the domain ofinterest can be defined using the

Navier-Stokes andtransportequations ofturbulent kinetic _energyand viscous dissipation.

Thenonlinearities present inthese equations each

having

an infinite number ofdegrees of

freedomare solved_usingtheFiniteElement Method (FEM). This techniquebreakes down

the region ofinterest into small geometric regions called finite elements and replaces the

partial differential equations which govern the entire region with _{ordinary differential}

equations or algebraic equations within these regions. All ofthese regions are linked

togethervia common

boundary

conditions and solved as alargesystem of equations _using

matrix algebra. The basic conversion procedurethat FEA (Finite Element

Analysis)

is as

follows:

(1)

Discretization ofthe

domain,

(2)

Derivation ofthe element equations,

(3)

Assembly

of global _equations,

(4)

Imposition of

boundary

_conditions, and

(5)

Solution of

assembled equation.

An Eulerian approach isused to describe the fluid _motion, elements are assumed

tobefixedin space. Within each_element, the dependent variables

(

_{uj, p,}

T,

k,

and e

)

are

interpolated

by

functions of compatible _order, interms of valuesto be determined at a set

of nodal points. For purposes of

developing

the equations for these nodal points

unknowns, an individual element _may be separated from the assembled system

(discretization). The dependentvariablesare approximated

by,

Uj

(

x,t

)

=cpT

Uj

(

t

)

(Eq.

44)

p(x,t)=

vTP(t)

(Eq.45)

T(x,t)

=

3TT(t)

(Eq.46)

k

(

x,t

)

=cpT

K (Eq.

47)

e

(

x,t

)

=cpT_{E (Eq.}

48)

where

Uj, P, T, K,

andEare column vectors of element nodal point unknowns and _{cp, \y,}

and $ are column vectors oftheinterpolation functions.

Substituing

theseapproximations

intotheNavier-Stokesequations andthe transport equationsforkinetic_energyand

viscous dissipationyields a set of equations:

fj

(

cp, vj/,

S,

Uj, P,

T

)

=

Ri

Momentum (Eq.

49)

f"2

(cp,

U[

)

=

R2

Incompressibility

(Eq.

50)

f3

(

cp,

S,

Uj,

T

)

=

R3

Energy

(Eq. 5

1)

f4

(

(p, vj/,

S,

Uj, T,

k,

e

)

=

R4

Transport - k

(Eq.

52)

f5

(

(p, v|/,

a,

Uj, T,

k,

e

)

=

R5

Transport- _e (Eq.

53)

where

Rj, R2, R3,

R4

and

R5

aretheresiduals or errors_resultingfromthe

The

Galerkin

i_-m oftheMethodofWeightedResiduals seeks toreducethese

errorsto_zero, in a weighted_sense,

by

_makingtheresiduals orthogonalto the interpolation

functions

of each element. These_{orthogonality} conditions are expressed

by,

(fi,cp)

=

(Ri,cp)

=₀

(Eq.54)

(f2,M/)

=

(R2,V)

=₀

(Eq.

55)

(f3,3)

=

(R3,cp)

=₀

(Eq.56)

(f4,cp)

=

(R4,cp)

=₀

(Eq.57)

(f5,(p)

=

(R5,(p)

=₀

(Eq.58)

where

(a,b)

denotestheinner_product, defined

by,

(a,b)

=

jaid