Asymptotic Approximations for the Sound Generated by Aerofoils in Unsteady Subsonic Flows

195 Pages·2015·6.16 MB·English

by  Lorna Ayton (auth.)

Springer Theses
Recognizing Outstanding Ph.D. Research
Lorna Ayton

Asymptotic
Approximations for
the Sound Generated
by Aerofoils in
Unsteady Subsonic
Flows
Springer Theses
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Lorna Ayton
Asymptotic Approximations
for the Sound Generated
by Aerofoils in Unsteady
Subsonic Flows
Doctoral Thesis accepted by
the University of Cambridge, UK
123
Author Supervisor
Dr. Lorna Ayton Prof. NigelPeake
Centrefor Mathematical Sciences Centrefor Mathematical Sciences
University of Cambridge University of Cambridge
Cambridge Cambridge
UK UK
ISSN 2190-5053 ISSN 2190-5061 (electronic)
SpringerTheses
ISBN978-3-319-19958-0 ISBN978-3-319-19959-7 (eBook)
DOI 10.1007/978-3-319-19959-7
LibraryofCongressControlNumber:2015941352
SpringerChamHeidelbergNewYorkDordrechtLondon
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For Grandad,
I wish you could see this.

’
Supervisor s Foreword
ItisapleasuretowritethisforewordtoLornaAyton’sthesis.LornastartedherPh.D.
inOctober2011intheDepartmentofAppliedMathematicsandTheoreticalPhysics
(DAMTP) at Cambridge, with me as her supervisor. After making spectacular
progressonanumberoffrontsshesubmittedthethesisinJune2014,andwasduly
awardedherPh.D.inOctober2014.Onthebasisofthesuperbqualityofherwork,
shehasbeen elected toa highlyprestigiousResearch FellowshipatSidney Sussex
College, Cambridge.
Lorna’sresearchliesintheareaofnoisegenerationbyaircraft,andinparticular
isconcernedwith oneofthekeynoisesources,namelytheinteraction ofunsteady
flowwiththebladinginsidetheengine.Thisisobviouslyaproblemofrealconcern
for people living close to the busiest airports, and to the manufacturers designing
engines to meet ever tighter noise certification requirements. There are significant
difficulties inherent in addressing this issue, not least due to the highly complex
nature of the flow and the engine geometry, and despite intense international
research efforts many challenges still remain.
As befits someone working in DAMTP, Lorna’s work is highly mathematical,
and involves the use of a range of advanced techniques, including matched
asymptoticexpansionsandtheWienerHopfmethod,tosolveasuiteofsimplified,
but highly relevant, model problems. One issue concerns the way that turbulence
interacts with a real aerofoil to produce noise in the presence of background shear
flow, and Lorna is the first person to produce a solution to what is a highly
technically demanding problem. Another issue addressed here is the way in which
the turbulence is distorted very close to the aerofoil nose, and Lorna’s predictions
areseen tobeinverygoodagreementwith experiment. Athirdissueisconcerned
withthewayinwhichsoundgeneratedwithintheaeroengineinteractswithblading
as it propagates outwards (so-called rotor blockage). A unifying theme throughout
the thesis is the importance of using the correct blade geometry; many previous
papersinthisareahaveusedsimple‘flatplates’,butLorna’sgreatachievementhas
been to extend her theory sufficiently to allow realistic aerofoil profiles to be
modelled.
vii
viii Supervisor’sForeword
Insummary,ithasbeenagreatpleasuretoworkwithLorna,andIcongratulate
her on producing such an excellent thesis.
Cambridge, UK Nigel Peake
May 2015 Professor of Applied Mathematics
Abstract
This thesis considers the sound generated by unsteady perturbations interacting
with solid aerofoils in background steady flows, in an attempt to further develop
analyticmodelsforthenoisegeneratedbybladeswithinturboengines.Specifically,
high-frequency unsteady gust and sound wave perturbations are considered and
asymptotic results are obtained for, primarily, the far-field noise.
Previous analytic work has examined high-frequency gust–aerofoil interactions
insteadyuniformflowsusing rapiddistortiontheory,andhasfocusedonaerofoils
with simple geometries. We extend this to deal with aerofoils with more realistic
geometries (by including camber, thickness, and angle of attack), as well as con-
sidering the new topic of sound–aerofoil interactions in steady uniform flows for
aerofoilswithrealisticgeometries.Theassumptionofasteadyuniformflowislater
relaxed and we investigate the sound generated by high-frequency gust–aerofoil
interactions in steady shear flows.
Throughoutalloftheaforementionedwork,thekeyprocessinvolvesidentifying
variousasymptoticregionsaroundtheaerofoilwheredifferentsourcesdominatethe
generation of sound. Solutions are obtained in each region and are matched using
the asymptoticmatching rule. The dominant regionsproducing noise are the local,
“inner”,regionsattheleadingandtrailingedgesoftheaerofoil.Approximationsfor
thefar-fieldnoise(inthe“outer”regions)aretheprincipalresults,howeveronecan
also extract approximations for the unsteady pressure generated on the surface
of the aerofoil.
The surface pressure generated by high-frequency gust–aerofoil interaction in
uniformflow isfound tocontainasingularity attheleading-edge stagnationpoint,
thus the final piece of work in this thesis focuses more closely on turbulent inter-
actions with solid body stagnation points in uniform flow, eliminating this
singularity.
ix