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
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
(Munich) The General Context 3
The Line of Combustion Instabilities 4
A Good Example 5
One Approximation (Early) 6
A Computational Result 7
A Simple Laboratory Demonstration 8
The Grand Picture 9
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
Vortex Shedding From a Bluff Flame Holder 11
Simple Vortex Shedding/Driving 12
Vortex Shedding & Acoustics 13
Acoustic Modes and Vortex Shedding 14
Summary of Mechanisms for CI in Solid Rockets 15
Three Sorts of Vortex Shedding in Solid Rockets 16
A Result of a Sub Scale Test 17
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
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
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
Equations With No Approximations 21
Equations for Mean Flow (1) 22
Some Definitions of Special Symbols 23
Equations for Mean Flow (2) 24
Equations With Terms Collected by Order 25
Equations for Fluctuations to Third Order 26
Equations for Linear Stability 27
Second Order Acoustics 28
Nonlinear Wave Equation (Third Order) 29
Linear Equations, Steady Waves 30
First Order Solution, by Iteration 31
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
Two Results of One Dimensional Representation 33
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
Conservation Equations for One Dimensional Flows 35
Source Terms in One Dimensional Flows 36
Sources Due To Flow Through Lateral Boundaries 37
Sources First Order In Fluctuations and M 38
Nonlinear Wave Equation and Boundary Condition 39
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
Linear Stability of Three Dimensional Motions 41
A Spatially Averaged Solution 42
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
Application of the General Method of Time Averaging 44
Application of the General Method of Time Averaging (cont d) 45
Application of the General Method of Time Averaging (cont d) 46
Application of the General Method of Time Averaging (cont d) These are the principal and very useful results of time averaging. 47
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
Rayleigh s Criterion and Linear Stability 49
Derivation 50
Derivation (cont d) Conclusion Rayleigh s Criterion is equivalent to criterion for linear stability when all gains and losses are accounted for. 51
Caltech Dump Combustor 52
Experimental Confirmation 53
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
Representation of Active Control 55
General Block Diagram 56
Original Proposal of Active Control 57
First Use of Fuel for Active Control 58
Effect of Control on Combustion 59
Early TUM Feedback Control of a Liquid Fuel/Air System 60
Test Result for TUM System 61
Siemens Experience I 62
Siemens Experience II 63
Rolls Royce Annular Combustor 64
GE and P&W Combustors Prior to ECCP (1975) 65
GE and P&W Combustors Developed in ECCP 66
Pratt and Whitney Talon Designs (c. 1980 2007) 67
Schematic of the Flow Field in the Talon X Combustor 68
Schematic of Approximate Analyses 69
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
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
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