Table Of ContentSpringer Theses
Recognizing Outstanding Ph.D. Research
Peter Jonathan Baddoo
Analytic
Solutions
for Flows
Through
Cascades
Springer Theses
Recognizing Outstanding Ph.D. Research
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Peter Jonathan Baddoo
Analytic Solutions for Flows
Through Cascades
Doctoral Thesis accepted by
University of Cambridge, Cambridge,
United Kingdom
123
Author Supervisor
Dr. PeterJonathanBaddoo Dr. Lorna J.Ayton
Department ofMathematics Department ofApplied Mathematics
Massachusetts Institute of Technology andTheoretical Physics
Boston, MA, USA University of Cambridge
Cambridge, UK
ISSN 2190-5053 ISSN 2190-5061 (electronic)
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This thesis is dedicated to my grandparents
Paul, Janet, Michael and Barbara.
’
Supervisor s Foreword
ItisapleasuretowritethisforewordforPeterBaddoo’sthesis.HebeganhisPh.D.
in October 2015, in the Department of Applied Mathematics and Theoretical
Physics (DAMTP) at Cambridge University with me as his supervisor, and was
awarded his Ph.D. in October 2019. Peter was my first student, so we very much
embarkedontheadventureofhis Ph.D.together,andIamincrediblyproudofthe
progress he made throughout his four years of study. His unique theoretical work
has led to him being awarded an EPSRC Doctoral Prize Fellowship (held at
Imperial College London). Following this, he will head to MIT as an Applied
Mathematics Instructor.
Peter’s work lies in the area of aerofoil acoustics and aerodynamics, with
applicationstoaircraftnoise,inparticular,theeffectsofcascadesofaerofoilsrather
than isolated blades. Much study has been done previously on isolated blades, but
thecomplexityofacascadehasleftthisareamostlyoutofreachfromatheoretical
standing until now. Peter’s mathematical ability has allowed him to solve key
problems in cascade acoustics and aerodynamics, which are of course inspired by
real concerns within the aviation sector: noise pollution from aircraft must be
reduced in order to meet ever-tightening noise certification requirements.
Peter’s thesis begins with a rigid, impermeable cascade project, which alone
wouldbeworthyofaPh.D.Heisthefirstpersontoaccuratelytheoreticallypredict
the noise scattered by a rectilinear cascade of realistic-geometry aerofoils in tur-
bulence, aswould occur during rotor-stator interaction inan aeroengine. To do so,
PeteremploysbothRiemann–HilbertandWiener–Hopftechniquestosolveforthe
steady and unsteady flow respectively. It cannot be overstated how technical this
workis,andhowmuchofastepforwarditmakestothefield;mostpreviouswork
was unable to include aerofoil geometry thus only modeled the aerofoils as flat
plates.
To further this work, Peter then permits the aerofoils within the cascade to be
compliant, either through the physical adaptations of aerofoil porosity or a locally
reacting flexible surface. These adaptations, inspired by the silent flight of owls,
greatlycomplicatethesystemofequationsthatmustbesolved,however,Peteronce
again achievesfastandaccurate theoreticalresults. Heisthe first person topredict
vii
viii Supervisor’sForeword
thenoise reduction possibleduetoadaptedsurfacesinacascade versus traditional
rigid surfaces which is a truly impressive achievement.
Peter finally focusses on multi-layered cascades, that is, multiple cascades of
blades one after another in the streamwise direction. To do so, he uses the
Schottky–Klein prime function which permits conformal mapping from canonical
circularorannulardomainstoaseriesofaerofoils.Thispowerfultoolallowshimto
calculate the steady potential flow through a range of complicated cascade for-
mations in the canonical domain, and thus be able to predict the effect of the for-
mation on the overall performance.
Insummary,ithasbeenapleasuretoworkwithPeter.Ithankhimforbeingan
exceptional student, and congratulate him on his excellent thesis.
Cambridge, UK Dr. Lorna J. Ayton
September 2020
Abstract
Noise reduction is a principal aim of the aviation industry. Particular attention is
devoted to reducing turbomachinery noise, which is an important component of
overall aeroengine noise during the take-off and landing stages. This thesis, there-
fore,presentsanalyticinvestigationsintoturbomachineryflowswiththeaimthatthe
resultingsolutionsbeusedbyaircraftdesignerstoproducequieteraircraft.Inorder
to facilitate exact solutions, the turbine is unwrapped onto the two-dimensional
plane, resulting in a periodic array of blades commonly referred to as a “cascade”
Previous research has been restricted to the case where the cascade consists of a
single impermeable row of flat plates at zero angle of attack. Consequently, this
thesisconsidersthreeseparatescenarioswherethecascadesconsistof(i)bladeswith
realistic geometry, (ii) blades with porosity gradients, and (iii) multiple blades per
period window. In each case, we begin by solving the steady potential flow and
proceed by investigating the effects of unsteady perturbations. This coupled
approach provides analysis from both aerodynamic and aeroacoustic perspectives
whichisessential forachievingpractical noisereductions.Inorder tofindanalytic
solutions, sophisticated complex analysis is employed in the form of singular inte-
gral equations, Riemann–Hilbert problems, the Wiener–Hopf method, and confor-
malmappingsviathetranscendentalSchottky–Kleinprimefunction.Thesemethods
are applied in the context of rigorous asymptotic expansions where the solution is
expandedintermsofasmallparametersuchastheamplitudeofanunsteadyincident
disturbance or the size of the blades. The aerodynamic analysis generates exact
expressions for the surface velocity, drag, lift, and deflection angle whilst the
aeroacoustic solutions furnish exact expressions for the unsteady surface pressure,
sound power output, and far-field sound. These formulae are rapid to compute
compared to CFD simulations currently used in industry, and moreover, they pro-
videfreshinsightintotherolesplayedbybladespacing,geometry,andporosityfor
turbomachinery noise and aerodynamics. Although the solutions in this thesis are
appliedtoturbomachinery,theywillalsobeusefulinotherapplicationssuchassolid
mechanics,poroelasticity,andbiologicalfliersorswimmersoperatinginformation.
ix
Acknowledgements
Imust firstandforemost expressmysincere gratitude tomy supervisor, Dr. Lorna
Ayton,forherunceasingsupportateverystageofmyPh.D.Hermathematicaland
physical insight has been incredibly valuable, and she has always made time to
answer most of my queries. Thank you, Lorna, for your patience and dedication
whilst supervising my Ph.D.—I feel genuinely privileged to have been supervised
by such a brilliant mathematician and I am excited to see what you accomplish in
the future!
The Department of Applied Mathematics and Theoretical Physical (DAMTP)
has provided a stimulating environment for my Ph.D. research. I have benefited
from many scientific discussions with past and present members of the Waves
group, including Nigel Peake, Ed Brambley, Stephen Cowley, Anastasia Kisil,
David Baker, Owen Petrie, and Benshuai Lyu. Thank you, in particular, to Georg
Maierhofer, Matthew Priddin, and Mungo Aitken for reading and commenting on
draftsofthesischaptersandpapers.Thanksalsotomyofficemate,WillEarley,for
helpingmewrestle with Linux and for sharing late night pizzas in thedepartment!
I am immensely indebted to my family for their support. Mum and Dad: thank
you so much for all the opportunities you have given me through your hard work,
dedication,andlove.Youhavesetanexampleformeineveryaspectoflife,andI
aspiretobelikeyou.KateandHarriet:I’msoprivilegedtohaveyouasmysisters,
anddespitebeingtheeldest,Iamcontinuallylearningfromyou.Yourkindnessand
faith areinspirationaltome. James:welcometothefamily! Inmany ways youare
the brother that I always hoped for, but your Catan record leaves something to be
desired.
There are many other people who have not provided mathematical help (actu-
ally,somehaveprovedmoreofahindrance)butareneverthelessworthyofthanks.
Inparticular,thank you toJonnoSaunders,Ed &Clem Hinton, Matthew Johnson,
LeanderCrocker,andChrisSmoove.ThankyoutothecongregationofSt.Andrew
the Great for providing a warm and encouraging spiritual home for the last few
years. I’m especially grateful for the friendship and fellowship of my housemates
ClarkGlasgow,AndyGales,andJonSavage.Thankyouforsittingpolitelythrough
my incomprehensible ravings about the Schottky–Klein prime function (to be
xi
