Application of modern tools for the thermo-acoustic study of annular combustion chambers
Franck Nicoud University Montpellier II – I3M CNRS UMR 5149 and CERFACS Introduction
A swirler (one per sector)
November, 2010 VKI Lecture 2 Introduction
One swirler per sector 10-24 sectors
November, 2010 VKI Lecture 3 Introduction
Y. Sommerer & M. Boileau CERFACS
November, 2010 VKI Lecture 4 Introduction
November, 2010 VKI Lecture 5 SOME KEY INGREDIENTS
• Flow physics – turbulence , partial mixing, chemistry, two-phase flow , combustion modeling , heat loss, wall treatment, radiative transfer, …
• Acoustics – complex impedance , mean flow effects, acoustics/flame coupling, non-linearity, limit cycle, non-normality, mode interactions, …
• Numerics – Low dispersive – low dissipative schemes, non linear stability, scalability, non-linear eigen value problems , …
November, 2010 VKI Lecture 6 PARALLEL COMPUTING
• www.top500.org – june 2010
# Site Computer
1 Oak Ridge National Laboratory Cray XT5-HE Opteron Six Core 2.6 GHz USA
2 National Supercomputing Centre in Shenzhen (NSCS) Dawning TC3600 Blade, Intel X5650, NVidia Tesla China C2050 GPU
3 DOE/NNSA/LANL BladeCenter QS22/LS21 Cluster, PowerXCell 8i 3.2 USA Ghz / Opteron DC 1.8 GHz, Voltaire Infiniband
4 National Institute for Computational Sciences Cray XT5-HE Opteron Six Core 2.6 GHz USA
5 Forschungszentrum Juelich (FZJ) Blue Gene/P Solution Germany
6 NASA/Ames Research Center/NAS SGI Altix ICE 8200EX/8400EX, Xeon HT QC 3.0/Xeon USA Westmere 2.93 Ghz, Infiniband
7 National SuperComputer Center in Tianjin/NUDT NUDT TH-1 Cluster, Xeon E5540/E5450, ATI Radeon
China HD 4870 2, Infiniband more or processors DOE/NNSA/LLNL eServer Blue Gene Solution
8 Tflops more or USA
9 Argonne National Laboratory Blue Gene/P Solution USA 000 70 458
10 Sandia National Laboratories Cray XT5-HE Opteron Six Core 2.6 GHz USA
November, 2010 VKI Lecture 7 PARALLEL COMPUTING
• www.top500.org – june 2007 (3 years ago …)
# Site Computer DOE/NNSA/LLNL BlueGene/L - eServer Blue Gene Solution 1 United States IBM (factor (factor 7) Oak Ridge National Laboratory Jaguar - Cray XT4/XT3 8) (factor 2 United States Cray Inc. NNSA/Sandia National Laboratories Red Storm - Sandia/ Cray Red Storm, 3 United States Cray Inc. IBM Thomas J. Watson Research Center BGW - eServer Blue Gene Solution 4 United States IBM Stony Brook/BNL, New York Center for Computional New York Blue - eServer Blue Gene Solution 5 United States IBM ASC Purple - eServer pSeries p5 575 1.9 DOE/NNSA/LLNL 6 GHz United States IBM Rensselaer Polytechnic Institute, Computional Center eServer Blue Gene Solution 7
United States IBM more or processors NCSA Abe - PowerEdge 1955, 2.33 GHz, Infiniband 8 United States Dell Tflops more or Barcelona Supercomputing Center MareNostrum - BladeCenter JS21 Cluster, 9 Spain IBM 000 10 56 Leibniz Rechenzentrum HLRB-II - Altix 4700 1.6 GHz 10 Germany SGI
November, 2010 VKI Lecture 8 PARALLEL COMPUTING
• Large scale unsteady computations require huge computing resources, an efficient codes … Speed up Speed
processors November, 2010 VKI Lecture 9 PARALLEL COMPUTING
November, 2010 VKI Lecture 10 Thermo-acoustic instabilities
Premixed gas
The Berkeley backward facing step experiment.
November, 2010 VKI Lecture 11 Thermo-acoustic instabilities
• Self-sustained oscillations arising from the coupling between a source of heat and the acoustic waves of the system
• Known since a very long time (Rijke, 1859; Rayleigh, 1878)
• Not fully understood yet …
• but surely not desirable …
November, 2010 VKI Lecture 12 Better avoid them …
AIR LPP SNECMA
LPP injector (SNECMA) FUEL
November, 2010 VKI Lecture 13 Flame/acoustics coupling
COMBUSTION
Modeling problem Wave equation
ACOUSTICS
Rayleigh criterion: Flame/acoustics coupling promotes instability if pressure and heat release fluctuations are in phase
November, 2010 VKI Lecture 14 A tractable 1D problem
IMPOSED IMPOSED FRESH GAS FLAME BURNT GAS VELOCITY PRESSURE n , τττ = = u ,0(' t) = 0 T1 T2 4T1 p (' L, t ) 0
0 L/2 L Kaufmann, Nicoud & Poinsot, Comb. Flame , 2002
γp − L q (' x,t) = 0 × n×u (' L / 2 ,t −τ )×δ x − γ −1 2 n : interactio n index n −τ model for the unsteady heat release τ : time lag
November, 2010 VKI Lecture 15 Equations
= = = u ,0(' t) 0 T1 T2 4T1 p (' L, t ) 0
0 L
L L 0 < x < : < x < L : 2 2 ∂2 p' ∂2 p' ∂2 ∂2 − 2 = p' 2 p' c1 0 − c = 0 ∂t 2 ∂x2 ∂t 2 2 ∂x2 CLASSICAL ACOUSTICS CLASSICAL ACOUSTICS 2 wave amplitudes 2 wave amplitudes
TWO JUMP RELATIONS 644444444 744444444 8 + + L 2/ L 2/ − [][]− = − = × ()−τ p' L 2/ 0 and u' L 2/ n u' L / ,2 t
November, 2010 VKI Lecture 16 Dispersion relation
• Solve the 4x4 homogeneous linear system to find out the 4 wave amplitudes
• Consider Fourier modes ℑ(ω)< 0 : damped mode p (' x,t) = ℜ(pˆ(x)e− jωt ) ℑ()ω > 0 : amplified mode
• Condition for non-trivial (zero) solutions to exist
ωL ωL 3 1 ne jωτ −1 cos × cos 2 − − = 0 jωτ + 4c1 4c1 4 4 ne 3
Uncoupled modes Coupled modes November, 2010 VKI Lecture 17 Stability of the coupled modes
• Eigen frequencies ωL 3 1 ne jωτ −1 cos 2 − − = 0 jωτ + 4c1 4 4 ne 3
• Steady flame n=0 : ω = 4c1 ± ± 2 + π = m 0, arccos 2m , m 2,1,0 ,... L 3
• Asymptotic development for n<<1 : 4c ω = ω − n 1 [cos (ω τ )+ j sin (ω τ )]+ o(n) m m 0, 9Lsin ()ω L / 2c m 0, m 0, 14444m40, 4441 244444444 3 Complex pulsation shif t
Kaufmann, Nicoud & Poinsot, Comb. Flame , 2002
November, 2010 VKI Lecture 18 Time lag effect
• The imaginary part of the frequency is 4c sin (ω τ ) − n 1 m 0, ()ω 9Lsin m 0, L / 2c1
(ω )< • Steady flame modes such that sin m 0, L / 2c1 0
• The unsteady HR destabilizes the flame if T sin ()ω τ > 0 ⇒ 0 < ω τ < π[2π ] ⇒ 0 <τ < m 0, []T m 0, m 0, 2 m 0, unstable unstable unstable unstable τ 0 Tm 0, / 2 Tm 0, 3Tm 0, / 2 2Tm 0, 5Tm 0, / 2 3Tm 0,
November, 2010 VKI Lecture 19 Time lag effect
• The imaginary part of the frequency is 4c sin (ω τ ) − n 1 m 0, ()ω 9Lsin m 0, L / 2c1
(ω )> • Steady flame modes such that sin m 0, L / 2c1 0
• The unsteady HR destabilizes the flame if T sin ()ω τ < 0 ⇒ π < ω τ < 2π[2π ] ⇒ m 0, <τ < T []T m 0, m 0, 2 m 0, m 0, unstable unstable unstable τ 0 Tm 0, / 2 Tm 0, 3Tm 0, / 2 2Tm 0, 5Tm 0, / 2 3Tm 0,
November, 2010 VKI Lecture 20 Numerical example
Uncoupled modes = T1 300 K L = 0.5 m
Steady flame UNSTABLE UNSTABLE
Unteady flame n=0.01 τττ=0.1 ms STABLE STABLE Imaginary frequency Imaginary frequency (Hz) (Hz)
Real frequency (Hz)
November, 2010 VKI Lecture 21 OUTLINE
1. Computing the whole flow
2. Computing the fluctuations
3. Boundary conditions
4. Analysis of an annular combustor
November, 2010 VKI Lecture 22 OUTLINE
1. Computing the whole flow
2. Computing the fluctuations
3. Boundary conditions
4. Analysis of an annular combustor
November, 2010 VKI Lecture 23 BASIC EQUATIONS reacting, multi-species gaseous mixture
November, 2010 VKI Lecture 24 BASIC EQUATIONS energy / enthalpy forms
Sensible enthalpy of species k
Specific enthalpy of species k
Sensible enthalpy of the mixture
Specific enthalpy of the mixture
Total enthalpy of the mixture
Total non chemical enthalpy of the mixture
E = H – p/ ρ Total non chemical energy of the mixture November, 2010 VKI Lecture 25 BASIC EQUATIONS Diffusion velocity / mass flux
Exact form
Practical model
Satisfies mass conservation November, 2010 VKI Lecture 26 BASIC EQUATIONS Stress and heat flux
µC λ = p Pr
November, 2010 VKI Lecture 27 Turbulence
1. Turbulence is contained in the NS equations 2. The flow regime (laminar vs turbulent) depends on the Reynolds number :
Velocity Length scale Ud Re = ν viscosity laminar turbulent Re November, 2010 VKI Lecture 28 RANS – LES - DNS
DNS LES
RANS Streamwise velocity Streamwise time
November, 2010 VKI Lecture 29 About the RANS approach
• Averages are not always enough (instabilities, growth rate, vortex shedding)
• Averages are even not always meaningful
T T2 Oscillating flame
T1 T2>T1 T1 time
Prob(T=T mean )=0 !!
November, 2010 VKI Lecture 30 The basic idea of LES
Resolved scales Modeled scales E(k)
kc k
November, 2010 VKI Lecture 31 LES equations
• Assumes small commutation errors
• Filtered version of the flow equations :
November, 2010 VKI Lecture 32 Laminar contributions
• Assumes negligible cross correlation between gradient and diffusion coefficients:
November, 2010 VKI Lecture 33 Sub-grid scale contributions
• Sub-grid scale stress tensor to be modeled τ sgs = ρ ~ ~ − ρ ij uiu j uiu j
• Sub-grid scale mass flux to be modeled sgs = −ρ ~ ~ + ρ J k, j Yku j Yku j
• Sub-grid scale heat flux to be modeled sgs = −ρ ~ ~ + ρ q j E u j E u j
November, 2010 VKI Lecture 34 The Smagorinsky model
• From dimensional consideration , simply assume: ~ 1 ~ 1 ~ ~ 1 ∂u~ ∂u τ sgs − τ sgs δ = 2µ S − S δ , with S = i + j ij kk ij sgs ij kk ij ij ∂ ∂ 3 3 2 x j xi
µ = ρ ( ∆)2 ~ ~ sgs Cs 2Sij Sij
• The Smagorinsky constant is fixed so that the proper
dissipation rate is produced, Cs = 0.18
November, 2010 VKI Lecture 35 The Smagorinsky model
ε =τ sgs ~ ≈ µ ~ ~ • The sgs dissipation is sgs ij S ij 2 sgs S ij S ij always positive
• Very simple to implement , no extra CPU time
• Any mean gradient induces sub-grid scale activity and dissipation, even in 2D !! = = µ ≠ V W 0 but sgs 0 = ≠ because U U (y) and S12 0 No laminar-to-turbulent Solid wall transition possible
• Strong limitation due to its lack of universality. Eg.: in a channel flow, Cs=0.1 should be used
November, 2010 VKI Lecture 36 The Dynamic procedure (constant ρ) • By performing NS , the following sgs contribution appears
sgs ∂ ∂ u u ∂ ∂ (τ +τ ) sgs ρ ui + ρ i j = − P + ij ij τ = ρu u − ρu u ∂ ∂ ∂ ∂ ij i j i j t x j xi x j • Let’s apply another filter to these equations ∂ τ +τ sgs ∂ ∂ u u ∂ ij ij sgs ρ ui + ρ i j = − P + τ = ρu u − ρu u A ∂ ∂ ∂ ∂ ij i j i j t x j xi x j • By performing NS , one obtains the following equations ∂ τ + sgs ∂ ij Tij sgs ∂u ui u j ∂ P = ρ − ρ ρ i + ρ = − + Tij ui u j uiu j B ∂ ∂ ∂ ∂ t x j xi x j
• From A and B one obtains
sgs =τ sgs − ρ + ρ Tij ij ui u j ui u j
November, 2010 VKI Lecture 37 The dynamic Smagorinsky model • Assume the Smagorinsky model is applied twice
1 2 τ sgs − τ sgs δ = 2ρ()C ∆ 2S S S ij 3 kk ij s ij ij ij 1 2 T sgs − T sgs δ = 2ρ(C ∆) 2S S S ij 3 kk ij s ij ij ij • Assume the same constant can be used and write the Germano identity sgs =τ sgs − ρ + ρ Tij ij ui u j ui u j
2 1 2 1 2ρ(C ∆) 2S S S + T sgs δ = 2ρ()C ∆ 2S S S + τ sgs δ − ρu u + ρu u s ij ij ij 3 kk ij s ij ij ij 3 kk ij i j i j
C dynamically obtained 2 = s Cs M ij Lij from the solution itself
November, 2010 VKI Lecture 38 The dynamic procedure
The dynamic procedure can be applied locally :
• the constant depends on both space and time • good for complex geometries, • but requires clipping (no warranty that the constant is positive)
L M C 2 ≡ ij ij M ij M ij
M ij depends on the SGS model
Lij can be computed from the resolved scales
November, 2010 VKI Lecture 39 How often should we accept to clip ?
1. Example of a simple turbulent channel flow
Local
% clipping % Dynamic Smagorinsky Y+ 2. Not very satisfactory , and may degrade the results
Plane-wise Dynamic U+ Smagorinsky DNS
Y+ November, 2010 VKI Lecture 40 The global dynamic procedure
The dynamic procedure can also be applied globally :
• The constant depends only on time • no clipping required, • just as good as the static model it is based on • Requires an improved time scale estimate
L M C 2 ≡ ij ij M ij M ij
M ij depends on the SGS model
Lij can be computed from the resolved scales
November, 2010 VKI Lecture 41 Sub-grid scale model
1. Practically, eddy-viscosity models are often preferred 2. The gold standard today is the Dynamic Smagorinsky model
From the Dynamically grid or filter strain rate : 2S S Computed width ij ij µ = ρ 2 × ∆2 ×( )−1 sgs C time scale
3. Looking for an improved model for the time scale
November, 2010 VKI Lecture 42 Description of the σ - model µ = ρ ( ∆)2 ×( )−1 • Eddy-viscosity based: sgs C time scale • Start to compute the singular values of the velocity gradient tensor (neither difficult nor expensive) •
Null for axi-symmetic flows Null for isotropic
AND σ (σ −σ )(σ −σ ) near-wall ()−1 = 3 1 2 2 3 time scale 2 behavior σ 3 1 is O(y ) !!
Null for 2D or 2C flows Null only if g ij = 0
November, 2010 VKI Lecture 43 Sub-grid scale contributions
• Sub-grid scale stress tensor τ sgs = ρ ~ ~ − ρ → µ ij uiu j uiu j sgs based model
• Sub-grid scale mass flux of species k and heat flux sgs = −ρ ~ ~ + ρ → J k, j Yku j Yku j sgs = −ρ ~ ~ + ρ → q j E u j E u j
• In practice, constant SGS Schmidt and Prandtl numbers
µ µ C Dsgs = sgs ; Sc sgs = 0.7 λ = sgs p ; Pr = 0.5 k ρ sgs k sgs sgs Sc k Pr sgs
November, 2010 VKI Lecture 44 Sub-grid scale heat release
• The chemical source terms are highly non-linear (Arrhenius type of terms)
• The flame thickness is usually very small (0.1 - 1 mm), smaller than the typical grid size
Poinsot & Veynante, 2001
November, 2010 VKI Lecture 45 The G-equation approach
• The flame is identified as a given surface of a G field
Poinsot & Veynante, 2001
• The G-field is smooth and computed from
• “Universal “ turbulent flame speed model not available …
November, 2010 VKI Lecture 46 The thickened flame approach
• From laminar premixed flames theory :
• Multiplying a and dividing A by the same thickening factor F
Poinsot & Veynante, 2001
November, 2010 VKI Lecture 47 The thickened flame approach
• The thickened flame propagates at the proper laminar speed but it is less wrinkled than the original flame:
Total reaction rate R 1 Total reaction rate R 2 R • This leads to a decrease of the total consumption: R → R = 1 < R 1 2 E 1 November, 2010 VKI Lecture 48 The thickened flame approach
• An efficiency function is used to represent the sub-grid scale wrinkling of the thickened flame Diffusivit y : a → Fa → EFa Preexponen tial constant : A → A/ F → EA / F
thickening SGS wrincklin g
• This leads to a thickened flame (resolvable) with increased velocity (SGS wrinkling) with the proper total rate of consumption
• The efficiency function is a function of characteristic velocity and length scales ratios usgs ∆ 0 δ 0 SL F L November, 2010 VKI Lecture 49 The thickened flame approach
• Advantages 1. finite rate chemistry (ignition / extinction) 2. fully resolved flame front avoiding numerical problems 3. easily implemented and validated 4. degenerates towards DNS: does laminar flames
• But the mixing process is not computed accurately outside the reaction zone 1. Extension required for diffusion or partially premixed flames 2. Introduction of a sensor to detect the flame zone and switch the F and E terms off in the non-reacting zones
November, 2010 VKI Lecture 50 About solid walls • In the near wall region, the total shear stress is constant. Thus the proper velocity and length scales are based on the wall shear τ stress w: τ ν w uτ = l = ρ uτ • In the case of attached boundary layers, there is an inertial zone where the following universal velocity law is followed
+ 1 + + u + yu τ y u = ln y + C, u = , y = κ uτ ν κ : Von Karman constant, κ ≈ 0.4 u C : " Universal constant" , C ≈ 5.2
November, 2010 VKI Lecture 51 Wall modeling
• A specific wall treatment is required to avoid huge mesh refinement or large errors,
y Velocity gradient at wall assessed from a coarse grid u
Exact velocity gradient at wall
• Use a coarse grid and the log law to impose the proper fluxes at the wall v τττ model τττ model 32 12 u
November, 2010 VKI Lecture 52 About solid walls
• Close to solid walls, the largest scales are small …
ω Boundary layer along the x-direction: Vorticity x
y
u z x κ - steep velocity profile, L t ~ y
- Resolution requirement: ∆y+ = O(1), ∆z+ and ∆x+ = O(10) !!
2 - Number of grid points: O(R τ ) for wall resolved LES November, 2010 VKI Lecture 53 Wall modeling in LES
• Not even the most energetic scales are resolved when the first off-wall point is in the log layer • No reliable model available yet
Resolved scales Modeled scales E(k)
kc k
November, 2010 VKI Lecture 54 OUTLINE
1. Computing the whole flow
2. Computing the fluctuations
3. Boundary conditions
4. Analysis of an annular combustor
November, 2010 VKI Lecture 55 Considering only perturbations
November, 2010 VKI Lecture 56 Linearized Euler Equations
° assume homogeneous mixture ° neglect viscosity
° decompose each variable into its mean and fluctuation = + f (x,t) f0 (x) f1(x,t)
° assume small amplitude fluctuations f p 1 ≡ ε << ;1 f = ρ, p,T, s = C ln v ρ γ f0 u γ p 1 ≡ ε << ;1 c = 0 0 ρ c0 0 November, 2010 VKI Lecture 57 Linearized Euler Equations
• the unknown are the small amplitude fluctuations , • the mean flow quantities must be provided
• requires a model for the heat release fluctuation q 1 • contain all what is needed, and more …: acoustics + vorticity + entropy
November, 2010 VKI Lecture 58 Zero Mach number assumption
• No mean flow or “ Zero-Mach number” assumption f p f (x,t) = f (x) + f (x,t); 1 ≡ ε << ;1 f = ρ, p,T, s = C ln 0 1 v ρ γ f0 u γ p u(x,t) = u (x) + u (x,t); 1 ≡ ε << ;1 c = 0 0 1 0 ρ c0 0
La : acoustic waveleng th Lf : flame thicknes s
• Probably well justified below 0.01
November, 2010 VKI Lecture 59 Linear equations
∂ρ ∂ 1 + ρ ()+ ⋅∇ρ = u1 Mass : 0div u1 u1 0 0 Momentum : ρ = −∇p ∂t 0 ∂t 1 ∂T p ρ T Energy : ρ C 1 + u ⋅∇T = − p ∇⋅u + q State : 1 = 1 + 1 0 v ∂ 1 0 0 1 1 ρ t p0 0 T0
- The unknowns are the fluctuating quantities ρ 1 , u 1 , T 1 , p 1
- The mean density, temperature, … fields must be provided
- A model for the unsteady HR q1 is required to close the system
November, 2010 VKI Lecture 60 Flame Transfer Functions
• Relate the global HR = Ω to upstream velocity Q1(t) ∫ q1(x,t)d fluctuations Ω
• General form in the frequency space ˆ ω = ω × ω Q( ) F( ) uˆ(xref , )
• n-τ model (Crocco, 1956) , low-pass filter/saturation (Dowling, 1997) , laminar conic or V-flames (Schuller et al, 2003) , entropy waves (Dowling, 1995; Polifke, 2001) , …
• Justified for acoustically compact flames
November, 2010 VKI Lecture 61 Local FTF model
• Flame not necessarily compact • Local FTF model
qˆ(x,ω) ωτ ω uˆ(x ,ω)⋅n = ω × j l (x, ) × ref ref nl (x, ) e qmean U bulk τ nl , l : Two scalar fields
• The scalar fields must be defined in order to match the actual flame response • LES is the most appropriate tool assessing these fields
November, 2010 VKI Lecture 62 Local FTF model
ACOUSTIC + WAVE
FLUCTUATING HEAT RELEASE
ACOUSTIC VELOCITY AT THE REFERENCE POSITION
ωτ ω ωˆ (x,ω) n (x,ω)×e j l x,( ) ∝ T l ω ⋅ uˆ(xref , ) nref
Giauque et al., AIAA paper 2008-2943 November, 2010 VKI Lecture 63 Back to the linear equations
• Let us suppose that we have a “ reasonable ” model for the flame response
• The set of linear equations still needs to be solved ∂ρ Mass : 1 + ρ div ()u + u ⋅∇ρ = 0 ∂t 0 1 1 0 ρ ∂ p1 = 1 + T1 ρ u1 = −∇ State : Momentum : 0 p1 p ρ T ∂t 0 0 0 ∂ ρ T1 + ⋅∇ = − ∇⋅ + Energy : 0Cv u1 T0 p0 u1 q1 ∂t
November, 2010 VKI Lecture 64 Time domain integration
• Use a finite element mesh of the geometry
• Prescribe boundary conditions
• Initialize with random fields
• Compute its evolution over time …
• This is the usual approach in LES/CFD !
November, 2010 VKI Lecture 65 A simple annular combustor (TUM)
Unsteady flame model with characteristic time delay τττ
Pankiewitz and Sattelmayer, J. Eng. Gas Turbines and Power, 2003
November, 2010 VKI Lecture 66 Example of time domain integration
TIME DELAY LEADS TO STABLE CONDITIONS
ORGANIZED ACOUSTIC TIME DELAY FIELD LEADS TO UNSTABLE CONDITIONS
Pankiewitz and Sattelmayer, J. Eng. Gas Turbines and Power, 2003
November, 2010 VKI Lecture 67 The Helmholtz equation
• Since ‘periodic’ fluctuations are expected, let’s work in the frequency space ( )= ℜ( ( ) − jωt ) ( )= ℜ( ( ) − jωt ) p1 x,t pˆ x e u1 x,t uˆ x e ()()= ℜ()− jωt q1 x,t qˆ x e
ρ • From the set of linear equations for 1, u1, p1, T 1 , the following wave equation can be derived
1 r γ p ∇⋅ ∇pˆ +ω 2 pˆ = jω()γ −1 qˆ 0 ρ 0
November, 2010 VKI Lecture 68 3D acoustic codes • Let us first consider the simple ‘steady flame’ case
1 r γ p ∇⋅ ∇pˆ +ω 2 pˆ = 0 0 ρ 0
• With simple boundary conditions = ρ ω ⋅ = ∇ ⋅ = pˆ 0 or 0 uˆ n pˆ n 0
• Use the Finite Element framework to handle complex geometries
November, 2010 VKI Lecture 69 Discrete problem
• If m is the number of nodes in the mesh, the unknown is now = [ ]T P pˆ1, pˆ 2 ,..., pˆ m
• Applying the FE method , one obtains
Linear 2 Eigenvalue AP+ω P=0 A : square matrix Problem of size N 1 discrete version of :γ p ∇ ⋅ ∇ 0 ρ 0
November, 2010 VKI Lecture 70 Solving the eigenvalue problem
• The QR algorithm is the method of choice for small/medium scale problems Shur decomposition: AQ=QT, Q unitary, T upper triangular
• Krylov-based algorithms are more appropriate when only a few modes needs to be computed
Partial Shur decomposition: AQ n=QnHn+E n with n << m
• A possible choice: the Arnoldi method implemented in the P- ARPACK library (Lehoucq et al., 1996)
November, 2010 VKI Lecture 71 Solving the eigenvalue problem
November, 2010 VKI Lecture 72 Computing the TUM annular combustor
SWIRLED INJECTORS • Experimentally, the PLENUM first 2 modes are : – 150 Hz (1L) – 300 Hz (1C plenum)
COMBUSTOR
• FE mesh of the plenum + 12 injectors + swirlers + combustor + 12 nozzles
• Mean temperature field prescribed from experimental observations
November, 2010 VKI Lecture 73 TUM combustor: first seven modes
f (Hz) type 143 1L 286 1C plenum 503 2C plenum 594 2L 713 3C plenum 754 1C combustor 769 3L
November, 2010 VKI Lecture 74 An industrial gas turbine burner
• Industrial burner (Siemens) mounted on a square cross section combustion chamber
• Studied experimentally (Schildmacher et al., 2000) , by LES (Selle et al., 2004) , and acoustically (Selle et al., 2005)
November, 2010 VKI Lecture 75 Instability in the LES
• From the LES, an instability develops at 1198 Hz
November, 2010 VKI Lecture 76 Acoustic modes
f (Hz) type 159 1L 750 2L 1192 1L1T 1192 1L1T’ 1350 3L 1448 2L1T 1448 2L1T’
November, 2010 VKI Lecture 77 Turning mode
• The turning mode observed in LES can be recovered by adding the two 1192 Hz modes with a 90°phase shift
PRESSURE FLUCTUATIONS FOR 8 PHASES
LES SOLVER: 1198 Hz ACOUSTIC SOLVER: 1192 Hz Selle et al., 2005 November, 2010 VKI Lecture 78 Realistic boundary conditions
• Complex, reduced boundary impedance
pˆ e.g.: Z = 0 ⇔ pˆ = 0 Z = ρcuˆ ⋅n Aout n Z = ∞ ⇔ uˆ ⋅n = 0 Ain A Z −1 R = in = • Reflection coefficient + Aout Z 1
• Using the momentum equation, the most general BC is
jω ∇pˆ ⋅n − pˆ = 0 cZ
November, 2010 VKI Lecture 79 Discrete problem with realistic BCs
• In general, the reduced impedance depends on ω and the discrete EV problem becomes non-linear: AP + Boundary Terms +ω 2P = 0 1444 2444 3 jω ∇pˆ⋅n= pˆ cZ (ω)
1 1 C • Assuming = + C ω + 2 ; Z ,C ,C parameters ω 1 ω 0 1 2 Z( ) Z0 Quadratic +ω +ω2 = Eigenvalue AP BP CP 0 Problem A, B,C : square matrices of size N November, 2010 VKI Lecture 80 From quadratic to linear EVP
• Given a quadratic EVP of size N: AP+ωBP+ω2CP=0
• Add the variable: R = ω P
• Rewrite : − IR +ω IP = 0 0 − IP I 0P ⇒ +ω = 0 AP + BR +ω CR = 0 A B R 0 CR
• Obtain an equivalent Linear EVP of size 2N and use your favorite method !!
November, 2010 VKI Lecture 81 Academic validation 5.0 m
Z = 1 / 2 + j Z = ∞ Z = 0 Z = −1 / 2 + j
= T1 T2 4T1
ACOUSTIC SOLVER UNSTABLE UNSTABLE UNSTABLE EXACT STABLE STABLE STABLE
November, 2010 VKI Lecture 82 Multiperforated liners
Multiperforated plate (????) Dilution hole (resolved)
Solid plate (Neumann)
Example of a TURBOMECA burner
November, 2010 VKI Lecture 83 Multiperforated liners
• Designed for cooling purpose …
Burnt gas (combustion chamber)
Cold gas (casing)
• … but has also an acoustic effect.
November, 2010 VKI Lecture 84 Multiperforated liners
400 Hz 250 Hz Absorption coefficient Absorption November, 2010 85 Multiperforated liners
• The Rayleigh conductivity (Howe 1979)
ω a St = U
• Under the M=0 assumption:
(impedance-like condition)
November, 2010 VKI Lecture 86 Academic validation
= − f = 382 −18 i Hz f 534 97 i Hz
November, 2010 VKI Lecture 87 Question
• There are many modes in the low-frequency regime
• They can be predicted in complex geometries
• Boundary conditions and multiperforated liners have first order effect and they can be accounted for properly
• All these modes are potentially dangerous
Which of these modes are made unstable by the flame ?
November, 2010 VKI Lecture 88 Accounting for the unsteady flame
• Need to solve the thermo-acoustic problem
1 r γ p ∇⋅ ∇pˆ +ω 2 pˆ = jω()γ −1 qˆ 0 ρ 0
(γ-1)q jℜ()()ω τ x,ω e.g. : jω()γ−1 qˆ = mean n ()x,ω e l ∇pˆ ⋅ n " local FTF model" ρ U l xref ref 0 ,x ref bulk
• In discrete form Non-Linear AP+ωBP+ω2CP=D(ω)P Eigenvalue Problem A, B,C, D : square matrices of size N
November, 2010 VKI Lecture 89 Iterative method
1. Solve the Quadratic EVP +ω +ω2 = AP 0 BP 0CP 0
2. At iteration k, solve the Quadratic EVP [ − ω ] +ω +ω2 = A D( k−1) P k BP kCP 0 ω −ω < 3. Iterate until convergence k k−1 tol
ω is solution of the thermo-acoustic problem k ,P
November, 2010 VKI Lecture 90 Comparison with analytic results
n=5 , τττ=0.1 ms INLET OUTLET
exact modes
Iteration 0
Iteration 1
Iteration 2
Iteration 3
November, 2010 VKI Lecture 91 About the iterative method
• No general prove of convergence can be given except for academic cases
• If it converges, the procedure gives the exact solution of the discrete thermo-acoustic problem
• The number of iterations must be kept small for efficiency (one Quadratic EVP at each step and for each mode)
• Following our experience , the method does converge in a few iterations … in most cases !!
November, 2010 VKI Lecture 92 OUTLINE
1. Computing the whole flow
2. Computing the fluctuations
3. Boundary conditions
4. Analysis of an annular combustor
November, 2010 VKI Lecture 93 BC essential for thermo-acoustics
p’=0 u’=0
Acoustic analysis of a Turbomeca combustor including the swirler, the casing and the combustion chamber C. Sensiau (CERFACS/UM2) – AVSP code
November, 2010 VKI Lecture 94 Acoustic boundary conditions
Boundary Conditions LES or Helmholtz solver
November, 2010 VKI Lecture 95 Analytical approach
• Assumption of quasi-steady flow (low frequency)
M M 1 2
Nozzle Inlet Nozzle Outlet
• Theories (e.g.: Marble & Candle, 1977; Cumpsty & Marble, 1977; Stow et al., 2002; …) • Conservation of Total temperature, Mass flow, Entropy fluctuations
November, 2010 VKI Lecture 96 Analytical approach
1 2
Mass
Momentum
Energy
November, 2010 VKI Lecture 97 Analytical approach
Acoustic wave Acoustic wave Compact systems Entropy wave
November, 2010 VKI Lecture 98 Example: Compact nozzle
M1 M2
1. Compact choked nozzle: ω− −(γ − ) = 1 1 M1 / 2 ω+ +()γ − 1 1 M1 / 2
2. Compact unchoked nozzle:
ω− ( + )[ ( +(γ − ) 2 )− ( +(γ − ) 2 )] = 1 M1 M2 1 1 M1 / 2 M1 1 1 M2 / 2 ω+ ()()− [ ()+ γ − 2 + ()+()γ − 2 ] 1 M1 M2 1 1 M1 / 2 M1 1 1 M2 / 2 November, 2010 VKI Lecture 99 Non compact elements
M1 M2 The proper equations in the acoustic element are the quasi 1D LEE :
Due to compressor or turbine
November, 2010 VKI Lecture 100 Principle of the method
1. The boundary condition is well known at xout (e.g.: p’=0)
2. Impose a non zero incoming wave at xin 3. Solve the LEE in the frequency space
4. Compute the outgoing wave at xin
5. Deduce the effective reflection coefficient at xin
November, 2010 VKI Lecture 101 Example of a ideal compressor
Imposed boundary condition p’ = 0
November, 2010 VKI Lecture 102 Example of a ideal compressor
Compressor
Imposed boundary condition p’ = 0
November, 2010 VKI Lecture 103 Example of a ideal compressor
π = pt2 = c 4 pt1
Compact theory
November, 2010 VKI Lecture 104 Realistic air intake
Intlet B.C
Outlet Choked B.C. ( taken as an acoustic wall )
Acoustic modes of the combustion chamber Need to be computed
November, 2010 VKI Lecture 105 Realistic air intake
Air-Intake duct
Intake BP Compressor HP Compressor
Intlet B.C
Outlet Choked B.C. ( taken as an acoustic wall )
November, 2010 VKI Lecture 106 Realistic air intake
Intake BP Compressor HP Compressor
November, 2010 VKI Lecture 107 Realistic air intake
Intake BP Compressor HP Compressor
November, 2010 VKI Lecture 108 OUTLINE
1. Computing the whole flow
2. Computing the fluctuations
3. Boundary conditions
4. Analysis of an annular combustor
November, 2010 VKI Lecture 109 Overview of the configuration
• Helicopter engine • 15 burners • From experiment: 1A mode may run unstable
November, 2010 VKI Lecture 110 About the computational domain
• When dealing with actual geometries, defining the computational domain may be an issue • Turbomachinery are present upstream/downstream • The combustor involves many “ details ”: combustion chamber, swirler, casing, primary holes, multi-perforated liner – Dassé et al., AIAA 2008- 3007 , … CC + casing CC + casing + swirler Combustion Chamber Casing + swirler + primary holes
700 Hz 500 Hz 575 Hz 609 Hz Frequency of the first azimuthal mode – 1A November, 2010 VKI Lecture 111 Is this mode stable ?
• The acoustic mode found at 609 Hz has a strong azimuthal component, like the experimentally observed instability
• Its stability can be assessed by solving the thermo-acoustic problem which includes the flame response
• In this annular combustor, there are 15 turbulent flames …
• Do they share the same response ?
November, 2010 VKI Lecture 112 Using the brute force …
Large Eddy Simulation of the full annular combustion chamber Staffelbach et al., 2008
• The first azimuthal mode is found unstable from LES, at 740 Hz • Same mode found unstable experimentally
November, 2010 VKI Lecture 113 Mode structure • In annular geometries, two modes may share the same frequency • These two modes may be of two types • Consider the pressure as a function of the angular position
A+ A-
From simple acoustics:
November, 2010 VKI Lecture 114 Turning mode. A+ = 1 & A- = 0
Companion mode turns counter clockwise
November, 2010 VKI Lecture 115 Standing mode. A+ = A- = 1
Companion mode has its nodes/antinodes at opposite locations
November, 2010 VKI Lecture 116 LES mode. A+ = 1 A- = 0.3
• A simple analytical model can explain the global mode shape obtained from LES
• Can we predict the stability by using the Helmholtz solver ?
November, 2010 VKI Lecture 117 Responses of the burners
• Because of the self-sustained instability, the pressure in front of each burner oscillates • This causes flow rate oscillations • Because the unstable mode is turning, the flow rate through the different burners are not in phase
November, 2010 VKI Lecture 118 Global Response of the flames
• From the full LES, the global flame transfer of the 15 burners can be computed from the global HR and the flow rate signals • For all the 15 sectors , the amplitude of the response is close to 0.8 and the time lag is close to 0.6 ms • This result support the ISSAC assumption …
November, 2010 VKI Lecture 119 ISAAC Assumption
• As far as the flame response is concerned
⇔ 15×
• Independent Sector Assumption for Annular Combustor • Allows performing a single sector LES with better resolution to obtain more accurate FTF • Of course the1A mode cannot appear is such computation • FTF deduced from single sector LES pulsated at 600 Hz
November, 2010 VKI Lecture 120 Local flame transfer function
Typical field of interaction index November, 2010 VKI Lecture 121 Extended the local n-τ model
• Define one point of reference upstream of each of the 15 burners • Use the ISAAC Assumption ω τ ω • the interaction index nl (x, ) , and the time delay l ( x, ) are the same in all sectors
ωτ ω ω × j l x ),( × 1 ω ⋅ 1 nl (x, ) e uˆ(xref , ) nref if x in sector 1 ωτ ω ω × j l x ),( × 2 ω ⋅ 2 nl (x, ) e uˆ(xref , ) nref if x in sector 2 ω ω ∝ ˆT (x, ) M ωτ ω ω × j l x ),( × ˆ 14 ω ⋅ 14 nl (x, ) e u(xref , ) nref if x in sector 14 ωτ ω ω × j l x ),( × 15 ω ⋅ 15 nl (x, ) e uˆ(xref , ) nref if x in sector 15
November, 2010 VKI Lecture 122 Extended the local n-τ model
• Assume ISAAC and duplicate the flame transfer function over the whole domain • In the same way, duplicate the field of speed of sound
November, 2010 VKI Lecture 123 Stability of the 1A mode
• The 1A mode from the Helmholtz solver looks like the 1A from the full LES
Helmholtz LES
• BUT … 1. Its frequency remains close to 600 Hz instead of 740 Hz 2. It is found stable !
November, 2010 VKI Lecture 124 Shape of the 1A mode
• The 1A modes from the Helmholtz and LES solvers resemble … • But a closer look reveals important differences
angle (rad) angle (rad)
Helmholtz solver LES solver
• What is observed in the Helmholtz solver 1. Pressure amplitude (slightly) larger upstream of the burners 2. No phase shift between upstream and downstream
November, 2010 VKI Lecture 125 What’s wrong ?
• The Helmholtz solver is not as CPU demanding and allows parametric studies
• The 1A mode is found stable for time delays smaller than T/2 • At 600 Hz, this corresponds to τ < 0.83 ms • The time delay of the FTF (0.65ms) is below this critical value
November, 2010 VKI Lecture 126 What’s wrong ?
• Remember the full LES oscillates at 740 Hz • This corresponds to a critical value for the time delay of T/2 = 0.67 ms , very close to the prescribed time delay • From the LES , the flow rate and HR are indeed in phase opposition
November, 2010 VKI Lecture 127 Back to the stability of the 1A mode
• Instead of imposing the time delay in the FTF, let’s impose the phase shift between flow rate and HR • The 1A mode is now found unstable by the Helmholtz solver, consistently with LES and experiment • The mode shape is also in better agreement
Helmholtz solver LES solver
November, 2010 VKI Lecture 128 THANK YOU
More details, slides, papers, … http://www.math.univ-montp2.fr/~nicoud/
November, 2010 VKI Lecture 129