Spatial and Temporal Averaging in Combustion Chambers

Transcription

1 Spatial and Temporal Averaging in Combustion Chambers F.E.C. Culick California Institute of Technology 18 January, 2010 Combustion Workshop IIT Chennai ReferenceRTO 039 1

2 I Introduction II Chief Mechanisms of CI and Unsteady Motions III A Framework of Analysis Based on Spatial Averaging IV Equations for One Dimensional Unsteady Flow ' V Results for p and M ', Linear Motions VI Time Averaging VII Rayleigh s Criterion and Linear Stability VIII The Case for Active Control in Liquid fueled Systems IX Concluding Remarks 2

3 (Munich) The General Context 3

4 The Line of Combustion Instabilities 4

5 A Good Example 5

6 One Approximation (Early) 6

7 A Computational Result 7

8 A Simple Laboratory Demonstration 8

9 The Grand Picture 9

10 I Introduction II Chief Mechanisms of CI and Unsteady Motions III A Framework of Analysis Based on Spatial Averaging IV Equations for One Dimensional Unsteady Flow ' V Results for p and M ', Linear Motions VI Time Averaging VII Rayleigh s Criterion and Linear Stability VIII The Case for Active Control in Liquid fueled Systems IX Concluding Remarks 10

11 Vortex Shedding From a Bluff Flame Holder 11

12 Simple Vortex Shedding/Driving 12

13 Vortex Shedding & Acoustics 13

14 Acoustic Modes and Vortex Shedding 14

15 Summary of Mechanisms for CI in Solid Rockets 15

16 Three Sorts of Vortex Shedding in Solid Rockets 16

17 A Result of a Sub Scale Test 17

18 I Introduction II Chief Mechanisms of CI and Unsteady Motions III A Framework of Analysis Based on Spatial Averaging IV Equations for One Dimensional Unsteady Flow ' V Results for p ' and M, Linear Motions VI Time Averaging VII Rayleigh s Criterion and Linear Stability VIII The Case for Active Control in Liquid fueled Systems IX Concluding Remarks 18

19 Basis of Approximate Analysis Complete equations of motion Express as mean plus unsteady flows Two small expansion parameters: p ; ū Expand equations to O and O 2 ) or O } Expand dependent variables (p,u ) in normal modes (N.B. boundary conditions are for cases of no flow and no combustion) 19

20 Basis of Approximate Analysis (cont d) Form perturbed wave equation for p Expand p (r,t) in modes with time varying amplitudes m Oscillator equations for m are inhomogeneous, coupled and nonlinear At this point, no physical processes are ignored N.B. the dependent variables p, n, satisfy correct (inhomogeneous) B.C. but the expansion basis functions do not. 20

21 Equations With No Approximations 21

22 Equations for Mean Flow (1) 22

23 Some Definitions of Special Symbols 23

24 Equations for Mean Flow (2) 24

25 Equations With Terms Collected by Order 25

26 Equations for Fluctuations to Third Order 26

27 Equations for Linear Stability 27

28 Second Order Acoustics 28

29 Nonlinear Wave Equation (Third Order) 29

30 Linear Equations, Steady Waves 30

31 First Order Solution, by Iteration 31

32 First Order Results for Linear Harmonic Motions (4.85) shows that due to mean flow interactions and other contributions, the system of equations is always non normal except in very special cases. 32

33 Two Results of One Dimensional Representation 33

34 I Introduction II Chief Mechanisms of CI and Unsteady Motions III A Framework of Analysis Based on Spatial Averaging IV Equations for One Dimensional Unsteady Flow ' V Results for p and M ', Linear Motions VI Time Averaging VII Rayleigh s Criterion and Linear Stability VIII The Case for Active Control in Liquid fueled Systems IX Concluding Remarks 34

35 Conservation Equations for One Dimensional Flows 35

36 Source Terms in One Dimensional Flows 36

37 Sources Due To Flow Through Lateral Boundaries 37

38 Sources First Order In Fluctuations and M 38

39 Nonlinear Wave Equation and Boundary Condition 39

40 I Introduction II Chief Mechanisms of CI and Unsteady Motions III A Framework of Analysis Based on Spatial Averaging IV Equations for One Dimensional Unsteady Flow ' V Results for p and M ', Linear Motions VI Time Averaging VII Rayleigh s Criterion and Linear Stability VIII The Case for Active Control in Liquid fueled Systems IX Concluding Remarks 40

41 Linear Stability of Three Dimensional Motions 41

42 A Spatially Averaged Solution 42

43 I Introduction II Chief Mechanisms of CI and Unsteady Motions III A Framework of Analysis Based on Spatial Averaging IV Equations for One Dimensional Unsteady Flow ' V Results for p and M ', Linear Motions VI Time Averaging VII Rayleigh s Criterion and Linear Stability VIII The Case for Active Control in Liquid fueled Systems IX Concluding Remarks 43

44 Application of the General Method of Time Averaging 44

45 Application of the General Method of Time Averaging (cont d) 45

46 Application of the General Method of Time Averaging (cont d) 46

47 Application of the General Method of Time Averaging (cont d) These are the principal and very useful results of time averaging. 47

48 I Introduction II Chief Mechanisms of CI and Unsteady Motions III A Framework of Analysis Based on Spatial Averaging IV Equations for One Dimensional Unsteady Flow ' V Results for p and M ', Linear Motions VI Time Averaging VII Rayleigh s Criterion and Linear Stability VIII The Case for Active Control in Liquid fueled Systems IX Concluding Remarks 48

49 Rayleigh s Criterion and Linear Stability 49

50 Derivation 50

51 Derivation (cont d) Conclusion Rayleigh s Criterion is equivalent to criterion for linear stability when all gains and losses are accounted for. 51

52 Caltech Dump Combustor 52

53 Experimental Confirmation 53

54 I Introduction II Chief Mechanisms of CI and Unsteady Motions III A Framework of Analysis Based on Spatial Averaging IV Equations for One Dimensional Unsteady Flow ' V Results for p and M ', Linear Motions VI Time Averaging VII Rayleigh s Criterion and Linear Stability VIII The Case for Active Control in Liquid fueled Systems IX Concluding Remarks 54

55 Representation of Active Control 55

56 General Block Diagram 56

57 Original Proposal of Active Control 57

58 First Use of Fuel for Active Control 58

59 Effect of Control on Combustion 59

60 Early TUM Feedback Control of a Liquid Fuel/Air System 60

61 Test Result for TUM System 61

62 Siemens Experience I 62

63 Siemens Experience II 63

64 Rolls Royce Annular Combustor 64

65 GE and P&W Combustors Prior to ECCP (1975) 65

66 GE and P&W Combustors Developed in ECCP 66

67 Pratt and Whitney Talon Designs (c ) 67

68 Schematic of the Flow Field in the Talon X Combustor 68

69 Schematic of Approximate Analyses 69

70 I Introduction II Chief Mechanisms of CI and Unsteady Motions III A Framework of Analysis Based on Spatial Averaging IV Equations for One Dimensional Unsteady Flow ' V Results for p and M ', Linear Motions VI Time Averaging VII Rayleigh s Criterion and Linear Stability VIII The Case for Active Control in Liquid fueled Systems IX Concluding Remarks 70

71 Concluding Remarks (I) 1) P ' p r & M r small implies restrictions on the expansion of the PDE. 2) Problems to be treated are dominated by wave motions PRIMARY 3) Spatial averaging built on expansions in eigenfunctions for unperturbed problems with homogeneous B.C. (e.g. rigid walls) 4) The perturbation/iteration procedure produces results satisfying the actual B.C. to the order of the expansions. SECONDARY 5) The formulation allows treatment of steady waves and general time dependent motions. 6) Time averaging may be used to reduce N second order inhomogeneous equations to 2N first order equations valid for slow changes of amplitudes and phases (VERY USEFUL) 71

72 Concluding Remarks (II) 7) Eigenfunctions calculated for actual problems (i.e. (3) plus perturbations) are nonorthogonal. Hence the solutions computed with the perturbation/iteration procedure are non normal in the current jargon. Simplest realistic case is linear steady waves: 8) Many results of the standard analysis based on spatial averaging have compared well with experimental results (e.g. instability, particle damping, ) 9) The outstanding current deficiency of results based on non normality is the absence of quantitative comparisons with alternative analyses and experimental results. 72