15773305 Design of a Supersonic Nozzle

8/13/2019 15773305 Design of a Supersonic Nozzle

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MINISTRY OF AVIATION

AERONAUTICAL RESEARCH COUNCIL

REPORTS AND MEMORANDA

Design of a Supersonic No!e

"# A$ McCABE, P%$D$

LONDON& HER MA'ESTY(S STATIONERY OFFICE

)*+,

PRICE 15s. 6d. NET

R$ & M$ No$ -../


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Design of a Supersonic Nozzle. I

"# A$ McCABE, P%$D$

T%e Mec%anics of F!ui0s Depar12en13 Uni4ersi1# of Manc%es1er

COMMUNICATED "Y PROF$ N$ H$ 'OHANNESEN

Reports and Memoranda No. -../5March,. )*+.

Summary.-T%is paper presen1s a 2e1%o0 of 0esign of a supersonic no!e 6%ic% incorpora1es recen1

0e4e!op2en1s in co2pressi7!e f!o6 1%eor#$ Con1inuous cur4a1ure of 1%e con1our is ensure0 7# 0efining

a con1inuous gra0ien1 of Pran01!8Me#er ang!e a!ong 1%e no!e a9is$ T%e f!o6 in 1%e 1%roa1 6as

ca!cu!a1e0fro2 resu!1s gi4en 7# Ha!! :)*+;ori1# of 1%e ne6 6or? is 1o 7e foun0 in 1%e Appen0ices$ Appen0i9 " presen1s a 2e1%o0 of

0e1er2ining 1%e co8or0ina1es a!ong a par1icu!ar c%arac1eris1ic 7# in4er1ing 1%e resu!1s gi4en 7# Ha!!)

:)*+;


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LIST OF CONTENTS-continued

Appen0i9 "$ T%e Ca!cu!a1ion of a C%arac1eris1ic in 1%e T%roa1 Region

Appen0i9 C$ E=ua1ions for 1%e Me1%o0 of C%arac1eris1ics

Appen0i9 D$ A Me1%o0 1o Re0uce 1%e Error in 1%e Mean S!ope of a C%arac1eris1ic

D$) Es1i2a1ion of Mean S!ope a1 a @enera! Poin1

D$; Es1i2a1ion of Mean S!ope Near a "ranc% Line

Appen0i9 E$ De1er2ina1ion of 1%e No!e Con1our 7# In1egra1ing a!ong a C%arac1eris1ic

Lis1 of S#27o!s

References

Ta7!e I

I!!us1ra1ions 8 Figures )1o

De1ac%a7!e A7s1rac1

Car0s

1. Introduction.

1.1. enera! Introduction.

A!1%oug% 1%e !i1era1ure on 1%e 0esign of supersonic no!es is e91ensi4e 1%ere are 4er# fe6 1a7u!a1e0

or0ina1es a4ai!a7!e for 6or?ing sec1ion Mac% nu27ers grea1er 1%an ;/$ Man# of 1%e or0ina1es 6%ic% are

a4ai!a7!e are 7ase0 on 0esign 2e1%o0s 6%ic% are 1o so2e e91en1 unsa1isfac1or#$ T%e errors in1ro0uce0are in0ee0 s2a!! 7u1 1%e# are cer1ain!# no1 neg!igi7!e in co2parison 6i1% 1%e 2anufac1uring 1o!erances

6%ic% are i2pose0$ In 4ie6 of 1%e a04ance0 s1a1e of ?no6!e0ge in 1%e fie!0 of co2pressi7!e f!o63 i1 s%ou!0

no6 7e possi7!e 1o 0esign 1%e in4isci0 core of a supersonic no!e =uic?!# an0 sufficien1!# accura1e!# 1o

2a?e 1%e 0esign errors s2a!! co2pare0 6i1% 1%e 2anufac1uring errors$ It2a# 7e argue0 1%a1 suc% accurac#

is no1 re=uire0 in 4ie6 of 1%e uncer1ain1# in1ro0uce0 7# a00ing a ca!cu!a1e0 7oun0ar# !a#er a!!o6ance3

7u1 1%is 0oes no1 see2 sufficien1 reason for in1ro0ucing a fur1%er uncer1ain1# 7# using an inaccura1e

2e1%o0 1o 0esign 1%e in4isci0 core$

A no!e 1o pro0uce a unifor2 s1rea2 of Mac% nu27er -/ 6as re=uire0 an0 i1 6as 0eci0e0 1o 0esign1%e in4isci0 core sufficien1!# accura1e!# for 1%e or0ina1es 1o 7e correc1 1o ////; inc%es$ T%e %a!f %eig%1of 1%e no!e a1 1%e 1es1 sec1ion 6as 1o 7e a7ou1 ;B inc%es3 so 1%a1 1%e %a!f %eig%1 a1 1%e 1%roa1 6as

appro9i2a1e!# /+ inc%es$In supersonic no!e 0esign 1%e con4en1iona! 16o80i2ensiona! no!e is usua!!# consi0ere0 1o consis1

of se4era! regions as s%o6n in Fig$ )$ T%ese are&8

:i< 1%e con1rac1ion3 in 6%ic% 1%e f!o6 is en1ire!# su7sonic3

:ii< 1%e 1%roa1 region3 in 6%ic% 1%e f!o6 acce!era1es fro2 a %ig% su7sonic 1o a !o6 supersonic spee03

:iii< an ini1ia! e9pansion region3 6%ere 1%e s!ope of 1%e con1our increases up 1o i1s 2a9i2u2 4a!ue3

:i4< 1%e s1raig%1ening or ("use2a2i( region in 6%ic% 1%e cross sec1iona! area increases 7u1 1%e 6a!!

$ s!ope 0ecreases 1o ero3 an0

(v) 1%e 1es1 sec1ion3 6%ere 1%e f!o6 is unifor2 an0 para!!e! 1o 1%e a9is$

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A goo0 appro9i2a1ion 1o 1%e 1%roa1 %eig%1 can 7e o71aine0 7# one80i2ensiona! f!o6 consi0era1ions$

T%e error is s2a!!3 7eing of 1%e or0er of R 8; 6%ere R is 1%e ra1io of 1%e ra0ius of cur4a1ure of 1%eno!e6a!! a1 1%e 1%roa1 1o 1%e 1%roa1 %a!f %eig%1 i1 is on!# /) per cen1 6%enR = B;$

Fro2 prac1ica! consi0era1ions 1%e !eng1% of 1%e no!e s%ou!0 no1 7e 1oo grea1$ T%ere is3 %o6e4er3 a

2ini2u2 !eng1% 7e!o6 6%ic% i1 is no1 possi7!e 1o 0esign a no!e 6%ic% 6i!! acce!era1e 1%e f!o6

s2oo1%!#$T%is 2ini2u2 !eng1% is usua!!# regar0e0 as 1%e !eng1% of 1%e no!e 6%ic% %as a s%arp8

e0ge0 1%roa13

1%e f!o6 7eing 1urne0 fro2 a para!!e! sonic s1rea2 1%roug% a cen1re0 e9pansion 6a4e$ So2e 0ou71s %a4e7een raise0 :Lor0 ; :)*+)ac?8suppor1e0 f!e9i7!e s1ee!

s%ee1s$ Ho6e4er3 an inco2pa1i7i!i1# 7e16een 1%e aero0#na2ic an0 s1ruc1ura! re=uire2en1s is

in1ro0uce0 if a poin1 of 0iscon1inuous cur4a1ure e9is1s a1 1%e inf!ec1ion poin1$ T%is %as !e0 1o 1%e

0e4e!op2en1 of con1inuouscur4a1ure no!es 6%ic% are necessari!# of !onger !eng1%3 e$g$ see Riise)-

:)*B.


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a04ances for 1%e ca!cu!a1ion of 1%e f!o6 in 1%e 1%roa1 region$ A1 1%e sa2e 1i2e 1%e oppor1uni1# %as 7een

1a?en 1o inc!u0e a 2a1c%ing region IVa3 6%ic% 7# 2ain1aining con1inui1# of 1%e a9ia! 4e!oci1# gra0ien1

ensures con1inui1# of 1%e no!e cur4a1ure a1 poin1 3 a 0esira7!e fea1ure 2issing in 2an# no!e 0esign

2e1%o0s$

T%e o1%er 2a1c%ing region I!!a 6as co2pu1e0 7# 1%e 2e1%o0 of c%arac1eris1ics s1ar1ing fro2 ?no6n

con0i1ions a!ong HB, !"P an0 PL To ?eep "P s%or1 an0 1o 2ain1ain con1inui1# of 6a!! cur4a1ure i1 is

0esira7!e 1%a1 7o1% Mac% nu27er Man0 Mac% nu27er gra0ien1 7e con1inuous a1 poin1s "an0 P$ T%e

Mac% nu27er gra0ien1 in 1%e ra0ia! f!o6 region is a func1ion of Mac% nu27er on!#3 6%ereas 1%e Mac%nu27er gra0ien1 a1 " is s1rong!# 0epen0en1 upon R. T%e gra0ien1s in 7o1% cases are s%o6n in Fig$ - for

4arious 4a!ues of r" an0R respec1i4e!#3 6%ere r" represen1s 1%e 0is1ance 7e16een 1%e 4ir1ua! origin of1%era0ia! f!o6 an0 1%e sonic circ!e$ Since 1%e ana!#1ic 1%eor# use0 in 1%e 1%roa1 region is on!# 4a!i0

o4er as2a!! sec1ion i1 is c!ear 1%a1 perfec1 2a1c%ing3 i$e$ 6i1% B an0 P coinci0en13 is no1 possi7!e un!ess

1%e 0esignMac% nu27er is c!ose 1o uni1#3 an0 1%us3 a co2pro2ise is usua!!# re=uire0$ In 1%e presen1

caseR 6as c%osen1o 7e e=ua! 1o B3 ) 1o 7e )+8,/ #r" J -8.;,B< an0 a para7o!ic 0is1ri7u1ion of M6as assu2e0 7e16een Ban0 P$ In genera!3 i1 6ou!0 7e 2ore con4enien1 1o 6or? in 1er2s of $ ra1%er1%an M since $ is use0 in 1%e c%arac1eris1ic ca!cu!a1ions$ For 1%is reason 1%e e=ua1ions in 1%e ne91sec1ion are gi4en in 1er2s of 4$

)$;$ The %uestion o& Minimum 'en(th.

T%e a9ia! 0is1ri7u1ion of $7e16een poin1s B an0 P 2us1 7e c%osen in suc% a 6a# 1%a1 !i2% !ines 0ono1 occur in 1%e f!o6 fie!0 insi0e 1%e no!e$ So2e consi0era1ion of 1%e =ues1ion %as 7een gi4en 7# Lor02

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Ho6e4er3 as in0ica1e0 a7o4e3 a1 A $ J 84/ a


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:

paper 1%e 4e!oci1# an0 f!o6 0irec1ion are e9presse0 in 1er2s of x an0 y as e9pansions in po6ers of )R$T%e resu!1s for 16o$0i2ensiona! f!o6 are su22arise0 in Appen0i9 A$ Ho6e4er3 for no!e 0esign 7# 1%e

2e1%o0 of c%arac1eris1ics i1 is 2ore con4enien1 1o 7e a7!e 1o ca!cu!a1e a par1icu!ar c%arac1eris1ic3 i$e$ 1o

?no6 3 an0 y as func1ions of $ an0 e. T%e necessar# in4ersion %as 7een carrie0 ou1 an0 1%e re!e4an1e=ua1ions are gi4en in Appen0i9 "$

To 0e1er2ine 1%e con0i1ions a!ong 1%e no!e a9is 7e16een " an0 P3 1%e Mac% nu27er gra0ien1s a1

1%ese poin1s 2us$1 7e 2a1c%e0 1o$1%ose in regions II an0 III7 respec1i4e!#$ T%e 2e1%o0 of c%arac1eris1ics

e2p!o#s $ ra1%er 1%an M, an0 1%us3 i1 is 2ore con4enien1 1o 6or? in 1er2s of #a$- ra1%er 1%an'9 yO'M< $ T%e func1ion :a4< is 0efine0 7# e=ua1ion :A$;-< in 1%e 4icini1# of poin1 " an03 inO4 yO O4 yO

Fig$ -3 i1 %as 7een p!o11e0 agains1 $ for 4arious 4a!ues ofR.

..Re(ion III a.

Ini1ia!!#3 1%e 7oun0ar# con0i1ions of 1%is region 2us1 7e specifie0 n0 1%ese canno1 7e c%osen ar7i

1rari!#$ T%e Pran01!8Me#er ang!e $ 2us1 7e con1inuous a1 "an0 'FQ an0 for con1inui1# of 6a!! cur4a1ure

&2us1 7e con1inuous$ A!ong 1%e no!e a&&s a =ua0ra1ic 4aria1ion of $ is 1%e si2p!es1 6%ic% sa1isfies

1%ese con0i1ions$ T%is can 7e 6ri11en in 1%e for2

5:;$;T>?;

6%ere

T? J 6a!! 1e2pera1ure

I

T) = !oca! 1e2pera1ure a1 e0ge of 7oun0ar# !a#er

I1 is 0ifficu!1 1o 0eci0e conc!usi4e!# 6%e1%er or no1 1%is assu2p1ion is sa1isfac1or#$ Si7u!?in co2pare0

1%e resu!1s of %is nu2erica! e9a2p!es 6i1% resu!1s o71aine0 fro2 o1%er 1%eories$ Ho6e4er3 %is co2parisons

6ere on!# 7e16een 1%eories of 4ar#ing 0egrees of appro9i2a1ion conse=uen1!# 1%e a0e=uac# of 1%e

1%eor# cou!0 no1 7e sa1isfac1ori!# >u0ge0$

Accor0ing 1o Fig$ . of 1%e paper 7# Rogers an0 Da4is 1%e 2ean ra1e of gro61% of 1%e 7oun0ar#8!a#er

0isp!ace2en1 1%ic?ness !ie


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T%e 6oo0en con1ours 6ere s%ape0 fro2 s1raig%1 graine0 o7ec%e 6%ic% 6as !eng1%6ise !a2ina1e0 1o

2ini2ise s%rin?age$ Co8or0ina1es of a s1ee! 2as1er 1e2p!a1e 6ere specifie0 for a9ia! in1er4a!s of /B

inc%es an0 or0ina1es 1o 1%e neares1 //) inc%es$ Af1er pre!i2inar# car4ing 1%e 7!oc?s 6ere scrape0 1o

fi1 1%e 1e2p!a1e$ T%e finis%e0 surface 6as c!ear 4arnis%e0 an0 ru77e0 0o6n 1o gi4e a co2p!e1e0 surface

6%ic% 6as 7o1% s2oo1% an0 %ar0$ To 2ini2ise 6arping 1%e !iners 6ere 2oun1e0 on iinc% 2i!0s1ee!p!a1es running 1%e 6%o!e !eng1% of 1%e

7!oc?s$T%e ca!i7ra1ion e9peri2en1s on 1%ese !iners s%o6e0 1%a1 1%e appro9i2a1e a!!o6ance 2a0e for 1%e

7oun0ar#8!a#er gro61% 6as sufficien1 :see Sec1ion -< an0 1%e Mac% nu27er 0is1ri7u1ion in 1%e 6or?ing

sec1ion 6as reasona7!# unifor23 iiM %a4ing a 2a9i2u2 of 1%e or0er of //)$ T%e no!e ca!i7ra1ion

is s%o6n in Fig$ +$ S1a1ic pressure 2easure2en1s 6ere 2a0e on 1%e p!ane si0e 6a!! a!ong 1%e no!e

a9is an0 i1 can 7e seen 1%a1 1%ere is goo0 agree2en1 7e16een 1%e e9peri2en1a! an0 1%eore1ica! resu!1s$

Sc%!ieren p%o1ograp%s of 1%e f!o6 in 1%e 1es1 sec1ion 0i0 no1 s%o6 an# 6a4e!e1s crossing 1%e 1unne!$T%e specifica1ion of 1%e 2e1a! !iners is 2u/% 2ore rigorous for 1%e carefu! use of an accura1e 0esign

2e1%o0 is 1o no a4ai! if 1%e 2anufac1uring 1o!erances are grea1!# in e9cess of 1%e aero0#na2ic

1o!erances$Due 1o 1%e significance of 1%e s!ope a1 a poin1 on 1%e con1o4r upon 1%e proper1ies of 1%e

f!o6 a!ong 1%ec%arac1eris1ic origina1ing fro2 1%a1 poin13 e91re2e care in 2anufac1ure is necessar# 1o

pro%i7i1 6a4iness3 or3 in 1%e case of 4er# s2a!! cur4a1ure of 1%e con1our3 1%e for2a1ion of s1eps$ A

si2p!e 2eans of re0ucing1%is 1#pe of error is 1o 1i!1 1%e !iner 7!oc?s 1%roug% a fini1e ang!e so 1%a1 a!!par1s of 1%e con1our are inc!ine0a1 an apprecia7!e ang!e 1o 1%e 2i!!ing cu11er$ T%e 2e1a! !iners are 7eing

fa7rica1e0 fro2 a!u2iniu2 a!!o#1o a 2a9i2u2 1o!erance of ////; inc%es$ Ta7!e I gi4es 1%e co8

or0ina1es of 1%e 7oun0ar#8!a#ercorrec1e0 no!e$ T%e surfaces of 1%e co2p!e1e0 con1ours are 1o 7e

s2oo1% 2i!!e0 an0 %an0 finis%e03an0 as 1%e supersonic 6in0 1unne! uses 0r# air as a 6or?ing f!ui0 i1

%as 7een 0eci0e0 1o !ea4e 1%e !inersfree of pro1ec1i4e 1rea12en1$

cAno?!ed(ements.

T%e au1%or 6ou!0 !i?e 1o 1%an? Dr$ I. M$ Ha!! for 1%e %e!p an0 encourage2en1 offere0 in 1%e prepara1ion of 1%is repor1 an0 1%e Depar12en1 of In0us1ria! $an0 Scien1ific Researc% for 1%e pos1gra0ua1e

s1u0en1s%ip 6%ic% ena7!e0 1%is 6or? 1o 7e carrie0 ou1$

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15773305 Design of a Supersonic Nozzle

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