Elementary non-linear processes in aero-acoustics of internal flows

Transcription

1 Elemenary non-linear roee in aero-aoi of inernal flow Avraham Mio Hirhberg Tehnihe Univeriei Eindhoven Workho N3L, 7- may TU Münhen

2 Oline. Wave diorion -D friionle hok wave. Driven reonane in loed ie b-harmoni reonane, rblene, hermal effe 3. Driven reonane in oen ie vore hedding 4. Self-ained oillaion Rijke be

3 f v v v r r r r r σ v r ' ' ', Ma onervaion Newon Loal hermodynami eqilibrim

4 Friionle, Ienroi and One-dimenional, ;, ;, ;,,, v r -D: Eqaion of moion Ienroi flow

5 ] [ Charaerii form: e Eliminae deniy

6 ] [ Charaerii form ± ± d Adding or braing

7 Sond wave U C - C U - U

8 C : d d Charaerii Along he line in he, diagram Θ d C C d C : d Θ d We follow eiher a C line or a C - line.

9 Iniial vale roblem,,,, d d d d Θ Θ C C We find, and a he inereion of he wo haraerii C and C -. d /,

10 Simle wave: ravelling ino a niform region Θ Θ Θ : : d d on d d d C d C C Uniform region raigh line C

11 Fndamenal derivaive d d d Γ Γ

12 Simle wave in alorially erfe ga: Bondary ondiion,,, ;,, γ γ γ C Uniform region Boh and inreae wih inreaing rere beae > γ Γ γ

13 Wave roagaion non-lineariy Single valed olion fail for >

14 Comreion wave: Bondary ondiion,, For > we have a mlile valed olion! d C C d, d, d

15 Simle wave: Bondary ondiion,, d, d, d d d d d d d d

16 Simle wave: Bondary ondiion,, d Γ d / d γ γ d / d

17 Bray ond Beaham, Hirhberg, Mallam moh iee rere ie rere

18 Li of Mrray Cambell Prere in moh iee

19 Prere in rombone Shok wave

20 Inegral onervaion law aro a hok moving wih he hok. m h h φ hok

21 Rankine Hgonio RH Eliminae he veloiy h h h h m φ

22 Comarion of RH wih ienroe v T h h h e γ γ γ ma

23 Weak hok... / 3 T v v ma Weak hok are almo adiabai

24 Seed of weak hok wave [ ]

25 Weak hok: hange in C C C d d d d d d d d d d C Γ :

26 Weak hok eed C C C Γ Γ Γ d d

27 Shok wave in airraf noie K.L. Gee e al JASA 8

28 Weak hok heory Beyer 997, Piere 98 Negleing he effe of enroy hange on he wave roagaion Prediing hok aenaion de o friion and hea ranfer in he hok wave Sawooh olion N-wave Airraf noie Pereion

29 Limi of wave hae Sawooh L Ma L L we negle enroy hange aro he hok

30 Weak hok: area rle Landa L L L L L L Ma onervaion

31 Weak hok: aenaion Landa L L d dl Γ Γ L

32 Weak hok: aenaion Landa L L L L L Γ L

33 Smmary weak hok wave [ ] Eqal area rle

34 Shok rre and vio daming Lighhill 956 ± ± d δ Brger eqaion!

35 Vio daming Cheer 964 δ v ν y, ;, Linear heory for growh vio bondary layer, allow eimae of vio fore.

36 Vio fore H R R S f H y w d w w w Π π ν τ τ ν π µ µ τ For ewie inreae of veloiy : Inegraion over ie eion!

37 Convolion,,, ξ ξ π ν τ ξ ξ ξ ξ d R R f d w Pr γ Faor o inlde effe of hea ranfer.

38 Cheer 964 Negle hange in average enroy Ue non-linear imle wave heory Eimae friion for hin bondary layer from linear heory Thin bondary layer aroah fail for Sondha be Ro 986

39 Reonane of a loed ie driven by a ion Aoi: Amlide limied a reonane by friion L L, ', ', ' L, Re [ Re[ [ e ik e ik]e i ] ω z e iω] ˆ z e ikl e ikl

40 Sb-harmoni eiaion Keller 975; Alha and Thomann 987 ' L, z L L, /T.5 /T hok De o refleion a loe wall, we have long wave roagaion diane!

41 Sb-harmoni eiaion Keller 975; Alha and Thomann 987

42 Thermal effe Ro , Swif 99, Baillie Change of average enroy de o diiaion and hea ondion. Thermo-aoial devie.

43 Trblene in anding wave Merkli and Thomann 975 Flow de o hok Non-niform flow Unable je flow Uniform deeleraion by hok

44 Oen be reonane Dielhor and van Wijngaarden 98 Refleion a oen ie erminaion rel in an inverion of aoi wave Wave eeening from ion o oen end i omenaed on he way bak

45 Conveive non-lineariy

46 Pie erminaion a a

47 Vore hedding lay No lay Thin walled larine ha narrower one hole Keefe 983, Aig, Dalmon and Gilber 4

48 Vore hedding mode deending on iniial ondiion! Dielhor 978 b a a b

49 Self-ained oillaion Sond ore Reonaor

50 Rijke be ie q' w Hea ranfer inde dilaaion dv d q ' w Heaed grid Pie i aoial ma-ring yem Flow

51 ie Rijke be Pie mode Rayleigh dv d ' a ' a Power: ' a P ' a dv d Heaed grid Oimal oiion: U ' a imedane mahing

52 Self-ained oillaion Sond ore Reonaor

53 < P a > Prediion of linear heory rodion ' a diiaion ' a ' a

54 < P a > Saraion de o non-lineariy rodion Saraion ' a diiaion ' a Seady amlide ' a

55 Saraion de o bakflow ' a > d ω d U U ' a ' a

56 Conlion Comle roblem have imle, eay o nderand wrong anwer N.Ro, 986

57 Some referene L. Landa and E. Lifhiz, Méaniqe de Flide, Ediion MIR, Moo 97 P.A. Thomon, Comreible-flid Dynami, MGrawHill, NY 97 A. Piere, Aoi, MGrawHill 98 D. See edior, Fronier in Phyial Aoi, Enrio Fermi ore XCIII Norh-Holland 986

58 A.Krohaalli and C.A. Smih edior, Reen Advane in Aeroaoi, Sringer-Verlag, NY 986 R.T. Beyer, Nonlinear Aoi, ASA, NY 997 K.Nagolnykh and L. Orovky, Nonlinear wave roee in aoi, Cambridge Univeriy Pre, UK 998 M.F.Hamilon and D.T. Blakok, Nonlinear Aoi, Aademi Pre, NY 998 B.O. Enflo and C. M. Hedberg edior, Theory of Nonlinear Aoi in Flid, Klwer Aademi Pb., Dordreh Y.Arégan, A.Marel, V.Pagne, J.-F. Pinon Edior, Sond-Flow Ineraion, Sringer Verlag, Berlin

59 Self-imilar olion vio bondary layer y d f f f y f ν η η π η ν η η 4 e ' " /