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  • ABB (now Alstom) <br /> <br /> TU M&#xFC;nchen <br /> <br /> Doctoral students in M
  • network models in more detail
  • Noise: flame is an amplifier of turbulent fluctuations <br /> Instability: intense interaction with feedback between flow, acoustics, heat release <br /> <br /> <br /> oint p&apos; ; dot Q&apos; ; dt > 0
  • p-v Diagram - thermodynamic cycle <br /> Heat release, pressure in phase -> clockwise -> output of mechanical energy -> instability <br /> <br /> obvious Question: what determines the phase of Q&#x2019; w.r.t. p&#x2019; ? <br /> <br /> <br /> <br /> oint p&apos; ; dv&apos; = - frac{v}{gamma p} oint p&apos; ; dp&apos; + oint p&apos; ; dv&apos;^{(Q)} = <br /> 0 + oint p&apos; ; d{v&apos;^{(Q)}}{t} ; dt sim oint p&apos; ; dot Q&apos; ; dt
  • Flammendynamik <br /> <br /> density -- equiv ratio -- flame speed -- flame surface area -- stoichiometric heat release <br /> <br /> dot Q = <br /> ho_u , S , Red A Black , Delta h
  • Flammendynamik <br /> <br /> density -- equiv ratio -- flame speed -- flame surface area -- stoichiometric heat release <br /> <br /> dot Q = <br /> ho_u , S , Red A Black , Delta h
  • Zeitverzug f&#xFC;r Brennstofftransport ~ L/U <br /> <br /> again: a time lag appears here - which can bring heat release and pressure fluctuations in phase. <br /> <br /> Note there&#x2019;s not one single time lag <br /> <br /> dot Q = <br /> ho_u Red phi S Black A Black , Delta h <br /> <br /> <br /> egin{eqnarray*} <br /> <br /> flct{dot Q} &=& frac{ <br /> ho&apos;_u}{ <br /> ho_u} + frac{phi&apos;}{phi} + frac{S&apos;}{S} \ <br /> <br /> flct{phi} &=& - frac{p&apos;_I (t- au)}{2 Delta p} - frac{u&apos;_I (t- au)}{u_I}, \ <br /> <br /> flct{dot Q} &=& - (1+a) left( frac{p&apos;_I }{2 Delta p} + frac{u&apos;_I }{u_I} <br /> ight) exp ( -i omega au ) <br /> end{eqnarray*}
  • viele r&#xFC;ckgekoppelte Wechselwirkungen ! <br /> interdisziplin&#xE4;r: <br /> Verbrennung / Akustik / Regelungstechnik <br /> <br /> plenty of interactions - we concentrate on <br /> Front kinematics &#x201C;wrinkling&#x201D; <br /> flow instabilities/ coherent structures <br /> equiv. ratio / mixture inhomogeneities <br /> <br /> Note that there is much more than just the flame - combustion system
  • Linear instability is the conventional wisdom. <br /> <br /> a mode is a pattern of vibration
  • one dominant frequency !
  • Limit cycle with negative velocity amplitudes - explain
  • pattern - the number and location of nodes and anti-nodes <br /> <br /> one mode or degenerate pairs of modes are most unstable
  • Explain what a Rijke tube is <br /> f - wave traveling left to right (&#x201D;downward&#x201D;). <br /> g - wave traveling right to left (&#x201D;upward&#x201D;) <br /> k - wave number <br /> Note: you can change (f,g) to (p&#x2019;, u&#x2019;) and vice versa. <br /> <br /> Time difference between departure of f at &#x201C;i&#x201D; and arrival at &#x201C;c&#x2019; gives phase lag <br /> equal to kl. <br /> <br /> <br /> left( egin{array}{c} f_c \ g_c end{array} <br /> ight) = <br /> left( egin{array}{cc} <br /> e^{-ikl} & 0 \ 0 & e^{ikl} <br /> end{array} <br /> ight) <br /> left( egin{array}{c} f_i \ g_i end{array} <br /> ight), ; ; k = frac {omega }{c}. <br /> <br /> f = inv{2} left( frac{p&apos;}{ <br /> ho c} + u&apos; <br /> ight), quad <br /> g= inv{2} left( frac{p&apos;}{ <br /> ho c} - u&apos; <br /> ight).
  • n - interaction index <br /> tau - time lag !!! that&#x2019;s why we can have instability <br /> <br /> Physics: time-lag required for boundary layer to adjust to change in flow speed. (Linearization of King&#x2019;s Law, not exact, see below ! known from hot wire anemometry) <br /> <br /> NB: here I mix time domain and frequency domain (bad habit) <br /> time lag - phase shift in frequency space <br /> <br /> <br /> <br /> <br /> u&apos;_h (t) = u&apos;_c(t) + n u&apos;_c (t- au). <br /> <br /> egin{eqnarray*} <br /> frac{ <br /> ho_h c_h}{ <br /> ho_c c_c} (f_h + g_h ) &=& (f_c+g_c) , \ <br /> f_h - g_h &=& left(1 + n e^{-i omega au} <br /> ight) (f_c-g_c) , <br /> label{eq:nTau} <br /> end{eqnarray*}
  • p&apos; = 0 ; ; o ; ; f_i + g_i = f_x + g_x =0,
  • putting everything together - homogeneous system of equations ! <br /> <br /> what to do with the system is dicussed in the next section. <br /> now: more complicated system <br /> <br /> <br /> mmm & mbox{Matrix} & \ & mbox{of} & \ & mbox{Coefficients} & \ <br /> emmm <br /> v f_i \ vdots \ g_x <br /> ev = <br /> v 0 \ vdots \ 0 <br /> ev .
  • Example: Rijke tube, see above. <br /> <br /> Stability map (for cold flame, heat source in the middle), blue regions indicate stability Time lag tau controls stability <br /> <br /> Note: some mode seems to be always unstable !? No losses! <br /> <br /> <br /> <br /> <br /> <br /> ewcommand{ <br /> Et}{n e^{-iomega t}} <br /> $ cos k_c l_c cos k_h l_h - xi sin k_c l_c sin k_h l_h left(1 + <br /> Et <br /> ight) = 0,$
  • 2 slides for control theory <br /> <br /> G1 - system, G2 - controller, <br /> G&#x2019;s are described by ordinary diff. eqn., which is Laplace-transformed to frequency space <br /> -> simple polynomial expressions <br /> <br /> system without input x oscillates in its eigenfrequencies, which are determined from the open loop gain <br /> <br /> so again, this is just algebra or root finding !? <br /> No, alternative approaches have been developed based on complex mapping <br /> <br /> <br /> <br /> egin{eqnarray*} <br /> y &=& G_1(omega) x&apos; = \ <br /> &=& G_1(omega) left( x - r <br /> ight) = \ <br /> &=& G_1(omega) left( x - G_2(omega) y <br /> ight) . <br /> end{eqnarray*} <br /> <br /> $y = - G_1(omega) G_2(omega) y;$ or <br /> $; G(omega) = -1, <br /> $
  • 2 slides for control theory <br /> <br /> G1 - system, G2 - controller, <br /> G&#x2019;s are described by ordinary diff. eqn., which is Laplace-transformed to frequency space <br /> -> simple polynomial expressions <br /> <br /> system without input x oscillates in its eigenfrequencies, which are determined from the open loop gain <br /> <br /> so again, this is just algebra or root finding !? <br /> No, alternative approaches have been developed based on complex mapping <br /> <br /> <br /> <br /> egin{eqnarray*} <br /> y &=& G_1(omega) x&apos; = \ <br /> &=& G_1(omega) left( x - r <br /> ight) = \ <br /> &=& G_1(omega) left( x - G_2(omega) y <br /> ight) . <br /> end{eqnarray*} <br /> <br /> $y = - G_1(omega) G_2(omega) y;$ or <br /> $; G(omega) = -1, <br /> $
  • diagnostic dummy inserted in a network, defines a mapping with the desired property <br /> <br /> why: sol&#x2019;n of the modified network are not eigenmodes of homogeneous network, <br /> but if f_u and f_d match up, it&#x2019;s as if the diagnostic dummy was not there - and the solution of the modified network is an eigenmode of the original network <br /> <br /> <br /> G(omega) equiv - frac{f_u( omega)}{f_d},
  • omg -> G(omg) <br /> <br /> so, e.g. real axis (blue line) is mapped to blue curve on the r.h.s. <br /> <br /> Whenever the OLTF passes the critical point, it passes the image of an eigenmode <br /> <br /> But I don&#x2019;t know the omg_m&#x2019;s !? <br /> Don&#x2019;t need to know, every time the image passes the &#x2018;&#x2019;critical point&#x2019;&#x2019; -1, an eigenmode is passed. if -1 to the left, Im(omg_m) > 0, stable <br /> if -1 to the right, Im(omg_m) &lt; 0, unstable <br /> <br /> But what is the open loop gain of an acoustic system???
  • indeed, Sattelmayer and Polifke showed that Barkhousen Criteria gives wrong answers, while Nyquist Criterion is o.k. !
  • pressure loss coefficent !?
  • Note: only dominant mode detected - FE does better in this respect.
  • [P] is the vector of unknowns $p&apos;$, <br /> [A] represents the spatial-derivative operator <br /> [B] represents the boundary terms. <br /> [D] source terms.
  • how is such a network analyzed -> next section of talk <br /> now: more acoustic elements (paper and pencil) <br /> (here we had four elements: duct / open end / closed end / heat source <br /> <br /> <br /> <br /> mmm & mbox{Matrix} & \ & mbox{of} & \ & mbox{Coefficients} & \ <br /> emmm <br /> v f_i \ vdots \ g_x <br /> ev = <br /> v 0 \ vdots \ 0 <br /> ev .
  • validation against exponential horn: one dozen elements is o.k. <br /> <br /> Note: not all elements can be derived with paper & pencil - FLAME (coming soon)! <br /> &#x2022; experiment <br /> &#x2022; CFD (next lecture this afternoon) <br /> <br /> But next: stability analysis (we pretend we have a complete network) <br /> <br /> <br /> M = <br /> mm e^{-ik_{x+}l_1} & 0 \ 0 & e^{-ik_{x-}l_1} emm <br /> mm1 & 0 \ 0 & alpha_1 emm <br /> mm e^{-ik_{x+}l_2} & 0 \ 0 & e^{-ik_{x-}l_2} emm <br /> cdots <br /> mm1 & 0 \ 0 & alpha_N emm
  • application: gas turbine - &#x2018;thin&#x2019; means no radial dependence of acoustic field <br /> pure axial mode in annulus - propagates just like a plane wave in pipe <br /> pure aximuthal mode - does not propagate at all <br /> mixed mode <br /> <br /> NB: other approaches &#x2018;link&#x2019; pipe-elements to make an annular &#x2018;mesh&#x2019; - WRONG! <br /> <br /> <br /> <br /> left( egin{array}{c} f_d \ g_d end{array} <br /> ight) = <br /> left( egin{array}{cc} <br /> e^{-ik_{x+}l} & 0 \ 0 & e^{-ik_{x-}l} <br /> end{array} <br /> ight) <br /> left( egin{array}{c} f_u\ g_u end{array} <br /> ight). <br /> <br /> <br /> k_{x_pm} = frac{omega/c}{1-M^2} <br /> left( -M pm sqrt{1 - left( frac{k_perp}{omega/c} <br /> ight)^2 (1-M^2) } <br /> ight), ; k_perp equiv frac{m}{R}.
  • &#x2018;any&#x2019; element, even a swirl burner <br /> how - linearization of mass and momentum conservation <br /> assumption: compact element, i.e. shorter than wave length (Helmholtz-# kL &lt;&lt; 1) <br /> l_eff - inertia of fluid between &#x2018;u&#x2019; and &#x2018;d&#x2019;, <br /> pressure difference leads to acceleration, but not immediate change in velocity. <br /> depends on shape click <br /> zeta - loss coefficient (vanishes for M -> 0). <br /> l_red - compressibility <br /> alpha - area change <br /> <br /> <br /> <br /> <br /> <br /> ewcommand{lf}{l_{mbox{footnotesize <br /> m eff}}} <br /> <br /> ewcommand{lrd}{l_{mbox{footnotesize <br /> m red}}} <br /> [ vD frac{ p&apos;}{ <br /> ho c} \ u&apos; ev_d = <br /> mm 1 & - i, k, lf - zeta M\ -i, k,lrd & alpha emm <br /> vD frac{ p&apos;}{ <br /> ho c} \ u&apos; ev_u. <br /> ] <br /> <br /> <br /> <br /> ewcommand{lx}{l_{mbox{footnotesize <br /> m eff}}} <br /> [ lx approx int_{x_u}^{x_d} frac{A_u}{A(x)} dx. <br /> ]
  • how detect n-n effects in frequency-domain network models ? <br /> > what is the norm

Iitm10.Key Iitm10.Key Presentation Transcript

  • Technische Universität München Workshop on Advanced Instability Methods Jan. 18 - 21, 2010, IIT Madras, Chennai, India Thermo-Acoustic System Modelling and Stability Analysis: Conventional Approaches Wolfgang Polifke Lehrstuhl für Thermodynamik TU München
  • Technische Universität München thanks to ... Jakob J. Keller, Oliver Paschereit, Bruno Schuermans Stephanie Evesque, Christoph Hirsch, Thomas Sattelmayer Alexander Gentemann, Andreas Huber, Roland Kaess, Jan Kopitz, Robert Leandro, Christian Pankiewitz Matthew Juniper, Raman Sujith W. Polifke - AIM Workshop @ IITM, Jan. 2010 2
  • Technische Universität München Outline of Talk Combustion Instabilities Stability Analysis Unsteady Analysis Eigenmodes and Eigenfrequencies Nyquist Plots Energy Balance System Models CFD Computational Acoustics Galerkin Methods Network Models (“Toy Models”) W. Polifke - AIM Workshop @ IITM, Jan. 2010 3
  • Technische Universität München A history of trouble (from Culick, 2006) W. Polifke - AIM Workshop @ IITM, Jan. 2010 4
  • Technische Universität München Physics of Combustion Instabilities a flame is a source of volume a fluctuating flame is a (monopole) source of sound combustion noise & combustion instability Rayleighʼs Criterion: ˙ p Q dt > 0 Rayleigh Index W. Polifke - AIM Workshop @ IITM, Jan. 2010 5
  • Technische Universität München Thermodynamic interpretation of Rayleigh Fluctuations produce acoustic energy, if Rayleigh Index > 0 p p’, u’ t 2' 3' Q’ 2' 2" 1 3' 4' 1 4' t 3" 4" Q’ 1 4" v 2" 3" t W. Polifke - AIM Workshop @ IITM, Jan. 2010 6
  • Technische Universität München Thermodynamic interpretation of Rayleigh Fluctuations produce acoustic energy, if Rayleigh Index > 0 p p’, u’ t 2' 3' Q’ 2' 2" 1 3' 4' 1 4' t 3" 4" Q’ 1 4" v 2" 3" t If production of energy > dissipation, instability occurs ! W. Polifke - AIM Workshop @ IITM, Jan. 2010 6
  • Technische Universität München Thermo-Akustische Instabilität Flame dynamics and system acoustics Eingeschlossene Flamme (p’, u’) Q’ (p’, u’) Rückkopplung zwischen Fluktuationen Premix flames are velocity sensitive: ˙ ˙ Q = Q (u ) der Strömung (p’,u’) und der Wärmefreisetzung Q’ -> Selbsterregte Schwingungen ! p System acoustics controls phase pʼ - uʼ: Z = Stabilitätskriterium nach Rayleigh: ! d Q!p! u" > 0. # " W . Polifke / divide et imp era — Ercoftac TecTag / 2 ˙ p Q dt > 0 W. Polifke - AIM Workshop @ IITM, Jan. 2010 7
  • Technische Universität München Heat release in sync with pressure p', u' Q' p' Q' most likely unstable ! W. Polifke - AIM Workshop @ IITM, Jan. 2010 8
  • Technische Universität München Heat release in sync with velocity p', u' Q' p' Q' stable ! W. Polifke - AIM Workshop @ IITM, Jan. 2010 9
  • Technische Universität München Heat release lags velocity p’, u’ Q’ p’ Q’ possibly unstable ! W. Polifke - AIM Workshop @ IITM, Jan. 2010 10
  • Technische Universität München Flame front kinematics Heat release rate of a premix flame: ˙ Q = ρu φSA ∆h W. Polifke - AIM Workshop @ IITM, Jan. 2010 11
  • Technische Universität München Flame front kinematics Heat release rate of a premix flame: ˙ Q = ρu φSA ∆h W. Polifke - AIM Workshop @ IITM, Jan. 2010 12
  • Technische Universität München Modulation of Equivalence Ratio ˙ Q = ρu φSA ∆h ˙ Q ρu φ S = + + Q˙ ρu φ S φ pI (t − τ ) uI (t − τ ) = − − , φ 2∆p uI ˙ Q pI uI = −(1 + a) + exp(−iωτ ) Q˙ 2∆p uI W. Polifke - AIM Workshop @ IITM, Jan. 2010 13
  • Technische Universität München Modulation of Equivalence Ratio ˙ Q = ρu φSA ∆h ˙ Q ρu φ S = + + Q˙ ρu φ S φ pI (t − τ ) uI (t − τ ) = − − , φ 2∆p uI ˙ Q pI uI = −(1 + a) + exp(−iωτ ) Q˙ 2∆p uI W. Polifke - AIM Workshop @ IITM, Jan. 2010 13
  • Technische Universität München Modulation of Equivalence Ratio ˙ Q = ρu φSA ∆h ˙ Q ρu φ S = + + Q˙ ρu φ S φ pI (t − τ ) uI (t − τ ) = − − , φ 2∆p uI ˙ Q pI uI = −(1 + a) + exp(−iωτ ) Q˙ 2∆p uI W. Polifke - AIM Workshop @ IITM, Jan. 2010 13
  • Technische Universität München Modulation of Equivalence Ratio ˙ Q = ρu φSA ∆h ˙ Q ρu φ S = + + Q˙ ρu φ S φ pI (t − τ ) uI (t − τ ) = − − , φ 2∆p uI ˙ Q pI uI = −(1 + a) + exp(−iωτ ) Q˙ 2∆p uI W. Polifke - AIM Workshop @ IITM, Jan. 2010 13
  • Technische Universität München Flame / Acoustic Interactions Fuel Air Flame Combustor Supply Supply Position and Area of Flame Burning p’, u’ u’ Q’ Velocity Equivalence p’ Ratio W. Polifke - AIM Workshop @ IITM, Jan. 2010 14
  • Technische Universität München Stability Analysis needs a System Model “there are no unstable flames” Rayleigh criterion is necessary, but not sufficient. The system controls Impedance at the flame (→ phase between velocity and pressure) Losses of acoustic energy (dissipation and radiation). Intensity, phase and dispersion of convective waves (equivalence ratio, entropy). W. Polifke - AIM Workshop @ IITM, Jan. 2010 15
  • Technische Universität München Outline of Talk Combustion Instabilities Stability Analysis Unsteady Analysis Eigenfrequencies Nyquist Plots Energy Balance System Models CFD Computational Acoustics Galerkin Methods Network Models W. Polifke - AIM Workshop @ IITM, Jan. 2010 16
  • Q = 385 ± 7 W; vmean = 0.0218 ± 0.0002 m/s Technische Universität München Instability in a Rijke tube Experiment by Lumens, Kopitz 2006 W. Polifke - AIM Workshop @ IITM, Jan. 2010 17
  • Technische Universität München Stability Analysis by Unsteady Simulation 1D CFD Model of Rijke tube ˙ with source term for energy Q(t) = u(t − τ ) 30 gauze [m/s] 20 Velocity at [m/s] 10 c v 0 -10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t [s] Polifke et al, JSV, 2001 Time [s] W. Polifke - AIM Workshop @ IITM, Jan. 2010 18
  • Technische Universität München Unsteady Simulation + Simulation of (turbulent, reacting), compressible flow captures all relevant phenomena - Computationally expensive - Only the dominant mode is identified - Numerical vs. physical instability - Results can depend on initial perturbation - Boundary conditions (acoustic impedance !) are a problem. W. Polifke - AIM Workshop @ IITM, Jan. 2010 19
  • Technische Universität München Stability Analysis with Eigenmodes and Eigenfrequencies Mode - a pattern of vibration Eigen - German: own, peculiar, characteristic Eigenmode / Eigenfrequency - a mode / frequency that is easily excited in the system - once established, an eigenmode will persist for some time Typically, a system has many eigenmodes. several eigenmodes may be unstable one mode will be most unstable (“dominant mode”) W. Polifke - AIM Workshop @ IITM, Jan. 2010 20
  • Technische Universität München Eigenmodes / frequencies of a Rijke tube computed with a low-order model . p’=0 Q p’=0 i c h x Acoustic waves travel between “i” and “c”, “h” and “x”: fc e−ikl 0 fi ω = , k= . gc 0 eikl gi c 1 p 1 p f= +u , g= −u . 2 ρc 2 ρc W. Polifke - AIM Workshop @ IITM, Jan. 2010 21
  • Technische Universität München Coupling relations at the heat source . p’=0 Q p’=0 i c h x At the heat source: • no pressure drop, ph = pc • time-lagged heat release, uh (t) = uc (t) + nuc (t − τ ). ρh ch (fh + gh ) = (fc + gc ), ρc cc fh − gh = 1 + ne −iωτ (fc − gc ), W. Polifke - AIM Workshop @ IITM, Jan. 2010 22
  • Technische Universität München Boundary conditions . p’=0 Q p’=0 i c h x open / closed ends: p = 0 → f + g = 0, u = 0 → f − g = 0, W. Polifke - AIM Workshop @ IITM, Jan. 2010 23
  • Technische Universität München Rijke tube system matrix . u p’=00 = Q p’=0 i c h x   fi   0  Matrix of  .  =  . . .   .  . .   Coefficients gx 0 Eigenfrequencies fulfill Det (S(ω)) = 0, which yields: cos kc lc cos kh lh − ξ sin kc lc sin kh lh 1 + n e−iωτ = 0, W. Polifke - AIM Workshop @ IITM, Jan. 2010 24
  • Technische Universität München Eigenfrequency vs. time lag (n = 0.1) Re w Im w 1.03 0.04 1.02 1.01 0.02 1 2 3 4 tau Pi 0.5 1 1.5 2 2.5 3 tau Pi 0.99 0.98 -0.02 0.97 exact solution (--------------) vs. weak coupling approximation (- - - -) W. Polifke - AIM Workshop @ IITM, Jan. 2010 25
  • Technische Universität München Eigenfrequency vs. time lag (n = 0.3) Re w Im w 1.1 0.1 1.05 0.05 1 2 3 4 tau Pi 0.5 1 1.5 2 2.5 3 tau Pi 0.95 -0.05 0.9 -0.1 exact solution (--------------) vs. weak coupling approximation (- - - -) W. Polifke - AIM Workshop @ IITM, Jan. 2010 26
  • Technische Universität München Stability map Rijke tube: . p’=0 Q p’=0 u =0 i c h x cos kc lc cos kh lh − ξ sin kc lc sin kh lh 1 + ne−iωt = 0, m=3 Mode - # m=2 m=1 m=0 0 1π 2π ω0τ W. Polifke - AIM Workshop @ IITM, Jan. 2010 27
  • Technische Universität München Remarks on dynamic stability analysis + Results as presented agree with Rayleigh - because losses are neglected. Could be included easily! + Build system of equations in software → network model - Closed-form expressions for the transfer matrices are known only for the simplest configurations. - Eigenfrequencies give only asymptotic, long-time behaviour → not adequate for non-normal analysis. - Iterative search for eigenfrequencies in complex plane can be tedious and incomplete ! - Matrix coefficients must be known for complex-valued frequencies → Problem for TFMs from experimental data W. Polifke - AIM Workshop @ IITM, Jan. 2010 28
  • N; ! < 4 ?*) 5)9'% ); #$ #* -"' (#+"- ",9;=59,*'B: -"'* -"' %1%-'6 #% %-, " < 47 Technische Universität München N; ! % 4: -"'* -"' 5)#*- !F 9#'% #*%#2' -"' G1H0#%- $)*-)0( ,*2 -"'(' 6 Nyquist Criterion in"=59,*'7 E#+0(' @7A #990%-(,-'% 5)%#-#.' ,*2 *'+,-#.' '*$#( #* -"' (#+"-=",9; Control Theory s s G HG H −1 −1 N =1 N =0 Cauchyʼs argument principle: N = Z -); 5)%#-#.' ,*2 *'+,-#.' '*$#($9'6'*-% !"#$%& '()( OP,659'% P ) N − # of anticlockG1H0#%- $)*-)0(7 clockwise encirclements of critical point (-1,i0) Z − # of zeros of the open loop transfer function G(s) P − # of poles in the right half plane K% , 3*,9 ('6,(J: !' *)-' -",- -"' 5)(-#)* ); -"' G1H0#%- 5,-" ;)( " Stable if N-"' P ! = #$=59,*'7 Q-"'(!#%': -"' #*#-#,9 .,90' -"')('6 +#.'% ;)( -"' ('%5)*%' W. Polifke - AIM Workshop @ IITM, Jan. 2010 29 ' ?& < 4MB < 9#6 "#$ ( ?"B
  • Technische Universität München Open loop transfer function of a network model (Polifke et al., 1997, Kopitz & Polifke, 2008) Fuel Supply Burner 1 Air Supply Combustor & Flame With fu (ω) G(ω) ≡ − , fd eigenfrequencies are mapped to the critical point -1 W. Polifke - AIM Workshop @ IITM, Jan. 2010 30
  • Technische Universität München OLTF G(ω) as conformal mapping (Polifke et al., 1997, Kopitz & Polifke, 2008) Im(ω) ω Im(G(ω)) G(ω) Re(ω) + 2i Re(ω) + i ωm -1 Re(G(ω)) Re(ω) Nyquist Criterion: A mode is stable if the critical point “-1” lies to the left of the image curve of the real axis. W. Polifke - AIM Workshop @ IITM, Jan. 2010 31
  • Technische Universität München Stability analysis with Nyquist plot - what is “passage of the critical point” !? - no rigorous proof ! + this is not the “Barkhausen Criterion” The  Barkhausen Stability Criterion is simple, intuitive, and wrong. (http://web.mit.edu/klund) + it is sufficient to know transfer matrices / transfer functions for real-valued frequencies ω ∈ |R. + no iterative searches for eigenmodes. + growth rate can be estimated from the OLTF W. Polifke - AIM Workshop @ IITM, Jan. 2010 32
  • Technische Universität München Eigenmodes of Rijke tube o - Iterative search for eigenmodes ◆ - Nyquist plot 10 8 6 4 Growth Rate [%] 2 0 -2 -4 -6 -8 -10 0 1000 2000 3000 Frequency [Hz] W. Polifke - AIM Workshop @ IITM, Jan. 2010 33
  • Technische Universität München Outline of Talk Combustion Instabilities Stability Analysis Unsteady Analysis Eigenfrequencies Nyquist Plots Energy Balance System Models CFD Computational Acoustics Galerkin Methods Network Models W. Polifke - AIM Workshop @ IITM, Jan. 2010 34
  • Technische Universität München System Modelling Time Domain Frequency Domain Finite Element Finite Volume, CFD Computational Acoustics nonlinear PDEs - linearized PDEs - Navier-Stokes often extended Helmholtz Mode-Based Galerkin Methods Network Models ODE algebraic equations Nonlinear Linearized Equations W. Polifke - AIM Workshop @ IITM, Jan. 2010 35
  • Technische Universität München Computational Fluid Dynamics Idea: use LES (or URANS) for Unsteady Analysis + conceptually straightforward ! - high computational cost ! - acoustic boundary conditions !? - only dominant mode is detected. - insight does not come easy. Nota bene: CFD can also be used to determine transfer functions / matrices or to compute the OLTF ! W. Polifke - AIM Workshop @ IITM, Jan. 2010 36
  • Technische Universität München Computational acoustics Linearized Euler Equations: ∂ρ ∂ρ ∂ρ ∂ui ∂ui + ui + ui +ρ +ρ = 0, ∂t ∂xi ∂xi ∂xi ∂xi ∂ui ∂ui ∂ui 1 ∂p ρ ∂p + uj + uj + − 2 = 0, ∂t ∂xj ∂xj ρ ∂xi ρ ∂xi ∂p ∂p ∂p ∂ui ∂ui + ui + ui + γp + γp = (γ − 1) q . ˙ ∂t ∂xi ∂xi ∂xi ∂xi + contain acoustic, entropy and vorticity modes (→APEs) - numerically unstable W. Polifke - AIM Workshop @ IITM, Jan. 2010 37
  • Technische Universität München Computational Acoustics Starting point: equation for acoustic perturbations: 1 D2 p ∂ 1 ∂p γ − 1 Dq ˙ 2 Dt2 −ρ = 2 . c ∂xi ρ ∂xi c Dt Add a model for the heat release fluctuations: q (x, t) ˙ ub (t − τ (x)) =n . ˙ q(x) ub Apply FE-Solver (time-marching) for Unsteady Analysis ! (Pankiewitz et al, ʼ02 - ʼ04) W. Polifke - AIM Workshop @ IITM, Jan. 2010 38
  • Technische Universität München Eigenmodes of annular combustor (1,0,0) (1,1,0) (1,0,0) cies and 0.8 agreeme 0.7 0.6 CONCL We 0.5 coustic f especial 0.4 simulati bitrary g 0.3 the prop indicate 0.2 and hav 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 τ FE: triangles, low-order: circles REFER Figure 5. FREQUENCIES AND Unstable: filled symbols. CORRESPONDING MODE TYPES [1] S. H FOR DIFFERENT DELAY TIMES. ( , ) LOW ORDER MODEL, (•,◦) W. Polifke - AIM Workshop @ IITM, Jan. 2010 TIME DOMAIN SIMULATION. ( ,•) UNSTABLE MODE, (◦, ) STABLE 39 mix
  • Technische Universität München FE / iterative subspace method for eigenmodes (Benoit and Nicoud ʼ05, Sensiau et al. 2008) Discretize eqn. for pressure perturbation (with source term) [A][P ] + ω[B][P ] + ω 2 [P ] = [D(ω)][P ]. 1 a) expand in thermo-acoustic coupling strength ≡ n(x) dV. V V ωm = ωm + ωm + O( 2 ), (0) (1) pm = pm + pm + O( 2 ). (0) (1) b) solve iteratively for sequence ω (k) [A] − [D(ω (k−1) )] [P ] + ω (k) [B][P ] + (ω (k) )2 [P ] = 0. W. Polifke - AIM Workshop @ IITM, Jan. 2010 40
  • Table I. E ect of reference position value and the grid resolution for the ÿrst eigen Technische Universität München = 0:003, frequency; = 10−4 s. Cross ‘X’ indicates unfeasible calculation. Results for “Rijke Tube” 0:24 m x ref = x ref = 0:249 m x ref = 0:25 m Benoit and Nicoud ʻ05 Coarse mesh (561 nodes) 270:4 − 0:087i X X Reÿned mesh (5231 nodes) 271:3 − 0:093i 271:4 − 0:088i X Theoretical 271:5 − 0:098i 271:6 − 0:088i 271:6 − 0:088i using the weak-coupling expansion ε=0.003 ε=0.33 ε=1.6 0.4 40 200 expansion expansion expansion theoretical theoretical 100 theoretical 0.2 20 Growth rate Growth rate Growth rate 0 0 0 -100 -0.2 -20 -200 -0.4 -40 -300 0 500 1000 1500 2000 0 500 1000 1500 2000 0 500 1000 1500 2000 Frequency Frequency Frequency Figure 2. Representation in the complex plane of the theoretical and computed eigen frequencies. available in Figure 2 and displayed in the complex plane. As expected, the computed W. Polifke - AIM Workshop @ IITM, Jan. 2010 41
  • Technische Universität München Computational acoustics + time domain: straightforward + fq domain: identify both stable and unstable eigenmodes + modest computational cost. - acoustic boundary conditions, mean flow effects, losses !? - needs input on flame dynamics. Stay tuned ! W. Polifke - AIM Workshop @ IITM, Jan. 2010 42
  • Technische Universität München System Modelling Time Domain Frequency Domain Finite Element Finite Volume, CFD Computational Acoustics nonlinear PDEs - linearized PDEs - Navier-Stokes often extended Helmholtz Mode-Based Galerkin Methods Network Models ODE algebraic equations Nonlinear Linearized Equations W. Polifke - AIM Workshop @ IITM, Jan. 2010 43
  • Technische Universität München Galerkin method ∂ 2 ψm 1D Helmholtz-equation without sources: 2 + km ψm = 0 2 ∂x Eigenmodes ψm (x) = sin(km x) are orthogonal L ψm ψn dx = δmn . 0 “Project” eigenmodes on PDE with source: 2 d ηm γ−1 L 2 + ωm ηm = 2 2 q (x) ψm (x) dx. ˙ dt Em 0 W. Polifke - AIM Workshop @ IITM, Jan. 2010 44
  • Technische Universität München Galerkin + very efficient + can handle non-linearities + for complicated geometries, expansion functions ψ can be computed with FE + eigenmodes of full problem need not be close to the ψʼs of the homogeneous problem - non-normal modes for non-trival boundary conditions - input on flame dynamics is needed, determination of source term may be non-trivial. W. Polifke - AIM Workshop @ IITM, Jan. 2010 45
  • Technische Universität München Outline of Talk Combustion Instabilities Stability Analysis Unsteady Analysis Eigenfrequencies Nyquist Plots Energy Balance System Models CFD Computational Acoustics Galerkin Methods Network Models W. Polifke - AIM Workshop @ IITM, Jan. 2010 46
  • Technische Universität München Network models Fuel Supply Air Supply Burner Flame Combustor   f1  0   Matrix of  .  =  . . .   .  . .   Coefficients gN 0 W. Polifke - AIM Workshop @ IITM, Jan. 2010 47
  • Technische Universität München Contoured duct A(x) 1 2 3 4 (x) e−ikx+ l1 0 1 0 e−ikx+ l2 0 1 0 M= −ikx− l1 −ikx− l2 ··· 0 e 0 α1 0 e 0 αN W. Polifke - AIM Workshop @ IITM, Jan. 2010 48
  • Technische Universität München Non-plane modes in thin annular duct fd e−ikx+ l 0 fu = −ikx− l . gd 0 e gu   2 ω/c  k⊥ m kx± = 2 −M ± 1− (1 − M 2 ) , k⊥ ≡ . 1−M ω/c R W. Polifke - AIM Workshop @ IITM, Jan. 2010 49
  • Technische Universität München Compact element ( l λ ) l Au Ad xu xd     p p  ρc  = 1 −i k leff − ζM  ρc  . −i k lred α u d u u xd Au leff ≈ dx. xu A(x) W. Polifke - AIM Workshop @ IITM, Jan. 2010 50
  • Technische Universität München Transfer matrix of (compact) flame Linearize Rankine-Hugoniot relations (conservation of mass, momentum, energy across discontinuity) p p TH uc ˙ Q ξ = − − 1 u c Mc + , ρc h ρc c TC uc Q˙ TH ˙ Q pc uh = uc + − 1 uc − . TC Q˙ pc closure with flame frequency response, ˙ Q u = F (ω) Q˙ u W. Polifke - AIM Workshop @ IITM, Jan. 2010 51
  • Technische Universität München Example network calculations Acoustics in duct system with low-Mach-# flow x L open end - duct - area change - duct - open end W. Polifke - AIM Workshop @ IITM, Jan. 2010 52
  • Technische Universität München Eigenfrequencies with reflection coefficient r = -1 $"$ $ $ #"* /012345267238029:4-4! #"* /0123450637 #"( #"( #"& #"& #"$ # #"$ !"* !"( # ! !"# !"$ !"% !"& !"' !"( !") !"* !"+ # ! !"# !"$ !"% !"& !"' !"( !") !"* !"+ # ,-. ,-. How is instability possible in a system without energy source? W. Polifke - AIM Workshop @ IITM, Jan. 2010 53
  • Technische Universität München Eigenfrequencies with energy-conserving boundary conditions $"$ # $ !"+' /012345267238029:4-4! #"* /0123450637 #"( !"+ #"& !"*' #"$ # !"* !"* !"( !")' ! !"# !"$ !"% !"& !"' !"( !") !"* !"+ # ! !"# !"$ !"% !"& !"' !"( !") !"* !"+ # ,-. ,-. 1 2 p+ ρ = p∞ , 2 p +M = 0, ρc ƒ (1 + M) + g(1 − M) = 0. W. Polifke - AIM Workshop @ IITM, Jan. 2010 54
  • Technische Universität München Riemann Twist The Riemann Twist !$ !" #% & !% #$ &' #" &" &$ Messung von Transfermatrixen (3) W. Polifke - AIM Workshop @ IITM, Jan. 2010 55 Prof. Wolfgang Polifke
  • Technische Universität München Network Models + Fast & Flexible, low computational effort + Great for qualitative / exploratory studies - Not suitable for many geometries of applied interest - Only in frequency domain ? - Non-linear phenomena ? - Non-normal phenomena ? → n3l Workshop in Munich W. Polifke - AIM Workshop @ IITM, Jan. 2010 56
  • Technische Universität München Summary Time Domain Frequency Domain Finite Element Finite Volume, CFD Computational Acoustics nonlinear PDEs - linearized PDEs - Navier-Stokes often extended Helmholtz Mode-Based Galerkin Methods Network Models ODE algebraic equations Nonlinear Linearized Equations W. Polifke - AIM Workshop @ IITM, Jan. 2010 57
  • Technische Universität München My questions on non-normality ... Real-world configurations: how typical / important are n-n effects? which methods / tools are adequate for the study of n-n effects? How to adopt exisiting modelling approaches: how to formulate appropriate (time-domain) network models? what is the proper norm? what are the physically realistic / permissible initial states ? How to describe / identify the flame dynamics ? W. Polifke - AIM Workshop @ IITM, Jan. 2010 58
  • Technische Universität München Announcement and Call for Papers Summer School and Workshop on Non-Normal and Nonlinear Effects in Aero- and Thermoacoustics In aero-acoustics, nonlinear effects play an important role in generation as well as dissipation of sound. Stability limits and limit cycle amplitudes of self-excited aero- or thermoacoustic instabilities are influenced by nonlinearities. For thermoacoustic interactions, standard linear modal analysis can in general not predict the response of the system to finite amplitude perturbations due to the non-normality of the corresponding evolution operator and nonorthogonality of eigenmodes. At TU München, a Summer School / Workshop on non-normality and nonlinearity in aero- and thermoacoustics will be held in May 2010. During the Summer School (May 17 and 18), a series of invited lectures will give an introduction to the workshop topics and present the state of the art. Expected http://www.td.mw.tum.de/n3l-conf-2010 audience are doctoral students with some background in fluid mechanics, flow instabilities, aero- or thermoacoustics, or combustion. Of course, more experienced researchers interested in the workshop topics are also welcome. Tentative list of speakers: W. Polifke - AIM Workshop (École Jan. 2010 Lyon) C. Bailly @ IITM, Centrale 59