Duran, I

Morgans, A

English

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THE EFFECT OF HYDRODYNAMIC INSTABILITIES ON THEPROPAGATION OF ENTROPY WAVES THROUGH NOZZLESAND ON THE GENERATION OF INDIRECT COMBUSTION NOISEIgnacio Duran and Aimee MorgansImperial College of Londonemail: i.duran@imperial.ac.ukIndirect noise is generated due to the acceleration of entropy waves through the turbine stages.Since these waves are mainly composed of density fluctuations, the acceleration is also unstable:any deformation of the wavefront is amplified by the Rayleigh Taylor instability. The initiallyperturbed entropy wave generates a vorticty field due to the acceleration which deforms the wavefront. In this work we show that the growth rate of this instability is large compared to theresidence time of the entropy wave in a typical aero-engine turbine.1. IntroductionThe generation of noise inside the combustion chamber is an issue for aero-engine manufacturersfor two main reasons. On one hand, the noise generated propagates to the outlet of the engine con-tributing to the global aircraft noise. On the other hand, the noise generated by the flame is reflectedin the turbine stages, generating acoustic modes, which further perturb the flame and generates inturn more noise. This feedback loop can become unstable, leading to a thermoacoustic instability asshown by Crocco and Cheng [1]. In typical aero-engines this undesired feedback usually limits thefunctional domain of the aero-engine to operating points with higher fuel consumption and pollutantemissions. The study of the noise generated by the flame is therefore a priority for the reduction offuel consumption, pollutants and noise. The flame generates noise through two main mechanisms, asfound by Marble and Candel [9]: the direct mechanism, in which the unsteady heat release generatesacoustic waves, and the indirect mechanism, where the entropy waves (hot spots) generate noise whenconvected through the turbine blades. Leyko et al. [7] used a simple 1D model to show that the noisegenerated by the indirect mechanism can be up to an order of magnitude larger than the direct one.For this reason the study of entropy waves and their propagation is of great importance.The study of combustion noise relies strongly on the propagation of both the acoustic and theentropy waves through non-homogeneous flows. This is of great importance for indirect noise in par-ticular, where the acoustic waves are generated due to the acceleration of the entropy waves throughthe turbine stages. The first methods developed relied on the compact assumption (low-frequency) toprovide the transfer functions of 1D nozzles [9] and 2D blade rows [2]. These methods have beenextended recently for the non-compact case [10, 4, 14, 3, 5]. These low-order models assume in gen-eral that the entropy wave is only convected through the blade rows with a simple quasi-1D velocity.ICSV22, Florence, Italy, 12-16 July 2015 1The 22nd International Congress of Sound and VibrationThis is generally not the case, since the entropy wave can be deformed due to the mean flow inhomo-geneities, and is subject to hydrodynamic instabilities. The first mechanism was studied by Morganset al. [11], showing that the wave was strongly deformed by the mean flow shear dispersion, but thatglobal entropy levels remained constant. However, the deformation of the wave is expected to havean impact on the noise generated. The objective of this work is to study the second mechanism forthe deformation of the entropy wave: hydrodynamic instabilities. These hydrodynamic instabilitiesare related to the Rayleigh Taylor mechanism [8, 15]: since the entropy wave is essentially a den-sity gradient, its acceleration due to the mean flow is unstable and any perturbation of the wavefrontwill grow exponentially. Two differences arise with typical Rayleigh Taylor instability: first, in theusual case of a water-air interface under gravity, the density ratio is large O (103) while the acceler-ation term is weak (gravity). For typical entropy waves convected through turbine stages the densityratio is weak (typically 40K hot spots give an Atwood number A = ρ′/ρ ∼ 10−1), while the ac-celeration term is several orders of magnitude larger than gravity (the ratio being O (105) for typicalaero-engines). The second difference lies in the nature of the interface: while the typical RayleighTaylor instability assumes the interface to be infinitely thin, in an entropy wave the interface is of theorder of O(2/λ), where λ is the wavelength of the waves. The thickness of this interface delays thegrowth of the Rayleigh Taylor instabilities developing in the wave front, as will be seen in section2. This interface thickness is also time dependent: as the entropy wave is accelerated the wavelengthincreases as λ = u/f where u is the mean axial velocity and f the frequency of the wave.The rest of this article is organised as follows: in section 2, the effect of a finite-length interface onthe traditional Rayleigh Taylor instability is studied. This is of great importance since entropy wavesare generally low frequency, and therefore the density gradient is finite. In section 3 a simple modelis derived to calculate the growth rate of the instability on the wave front. Numerical simulations arepresented in 4 and conclusions are drawn in 5.2. The effect of finite-length interfaceIn this section the effect of a finite-length interface on the Rayleigh-Taylor instability will beassessed. This will be done in an initially steady flow in which the external force driving the instabilityis gravity. The initial configuration is shown in figure 1(a): a heavy fluid is located on top of a lighterone. The interface between both can be characterised by a gradient with a characteristic length of1/K. In the case where this length tends to zero, the growth rate of any perturbation in the interface(of wavelength k) is given by the well known result of the Rayleigh Taylor instability,(1) αRT =√ρ1 − ρ2ρ1 + ρ2akwhere ρ1 and ρ2 are the heavy and light densities, a is the acceleration term and k is the wavelengthof the initial perturbation.This result is valid when the interface is infinitely thin (i.e. thin compared with the inverse ofthe wavelength of the perturbation k). When a finite-length interface is considered, the growth of thesmall scale perturbations is slowed. LeLevier et al. [6] developed an approach to obtain the growthrate of the Rayleigh Taylor instability for finite-length interfaces. They assumed an initial densityfield in the form,ρ1 −12δ exp (−Ky) y > 0(2)ρ2 +12δ exp (Ky) y < 0(3)where δ = ρ1 − ρ2 and K defines the gradient at the interface (of characteristic length 1/K). Con-sidering an initial perturbation of the interface in the form cos(kx), and a velocity potential in the2 ICSV22, Florence, Italy, 12-16 July 2015The 22nd International Congress of Sound and Vibration(a) t̂ = 0 (b) t̂ = 1.625 (c) t̂ = 3.25 (d) t̂ = 4.875 (e) t̂ = 6.5Figure 1: Density isolines with K̂ = 0.8.formφ = exp (−Ky) cos(kx)f(t) y > 0(4)φ = − exp (Ky) cos(kx)f(t) y < 0(5)where f is the amplitude of the perturbation and is a function exclusively of time. LeLevier et al. [6]integrated the linearised Euler equations in the y direction in both the heavy and light fluids separately,and then applied matching conditions in the interface to obtain an expression for the function f(t),d2fdt2− a kKk +Kδρ1 + ρ2f = 0(6)This leads to a growth rate(7) α =√aδρ1 + ρ2√kKk +Kwhich can be written as(8) α = αRT√K̂1 + K̂where K̂ = K/k is the ratio between characteristic length of the perturbation and of the interface.The equation shows that the actual growth rate of a finite-length interface does not tend to infinity asthe length scale of the perturbation tends to zero (k →∞), but is instead bounded by(9) αmax =√aKδρ1 + ρ2Using numerical simulations, the initial growth rate of perturbations was calculated for differentvalues of K̂. Figure 1 shows the evolution of the density isolines as a function of the dimensionlesstime, t̂ = tαRT , for the case K̂ = 0.8, computed using OpenFOAM [12]. Figure 2 plots the value ofα/αRT as a function of K̂ for both numerical simulations and the analytical expression. (8).ICSV22, Florence, Italy, 12-16 July 2015 3The 22nd International Congress of Sound and VibrationK/k100101102α/αRT00.511.5Growth rateSimulationTheoryFigure 2: Growth rate of the hydrodynamic instability as a function of K̂.Figure 3: Linear mean velocity profile nozzle. A(x) is the cross-section area and u(x) the velocity.3. The acceleration of entropy wavesAn entropy wave is a hydrodynamic perturbation composed mainly of temperature (or density)fluctuations. The order of magnitude of the entropy wave is related to the density and temperaturefluctuations by(10) O(s′cp)∼ O(ρ′ρ)∼ O(T ′T)This hydrodynamic wave generates acoustic waves when accelerated with the mean flow, as explainedin Marble and Candel [9]. When this occurs, the density fluctuations are subject to a Rayleigh Taylorinstability in which the acceleration term is due to the mean pressure gradient (a = −∇p/ρ) which,for a steady quasi-one-dimensional nozzle flow, is equal to(11) a = ududx.Lets assume for simplicity a linear mean velocity nozzle as done in the second part of Marble andCandel [9] (see figure 3). In such case, the nozzle extends between x1 and x2, and the velocity profileis given by u(x)/c∗ = x/x∗, where x∗ and c∗ denote the axial coordinate and the speed of sound atthe nozzle throat respectively. Under such assumption, the acceleration term is linear in x,(12) a =c2∗x∗xx∗.Since the flow is accelerating, the wavelength (λ) of the perturbation increases with the x coordinate.This gives,(13) K ∼ 2λ∼ 2fu(x)∼ 2fc∗ (x/x∗).4 ICSV22, Florence, Italy, 12-16 July 2015The 22nd International Congress of Sound and Vibration123Mach0.08823.22(a) Annular simulationx0 0.1 0.2 0.3 0.4 0.5M(x)00.511.522.533.5(b) Mean Mach numberFigure 4: Geometry of the numerical simulations(14) α(x) ∼[c∗fx∗ρ′ρkk +K]1/2As an order of magnitude, lets consider a temperature fluctuation of T ′ ∼ 40K in an aero-enginein which T ∼ 800 and with inlet and outlet Mach numbers of M1 = 0.1 and M2 = 3. The densityfluctuations are then ρ′/ρ ∼ 0.05, the speed of sound at the throat c∗ ∼ 550m/s and the characteristiclength of the throat can be taken to be x∗ ∼ 0.1m. Assuming a frequency of 500Hz, typical ofcombustion noise, the initial growth rate of the instability, given by equation (8) is α ∼ 200s−1. Thismeans that the characteristic time of the instability would be τI ∼ 5 ·10−3s, while the convective timeof the entropy wave (the time that it takes for the entropy wave to accelerate from M1 to M2 in sucha nozzle) is τC ∼ 5 · 10−4s. The ratio of both τI/τC ∼ O (10) shows that the effect of the RayleighTaylor instability in such a configuration can have some effect, though the time it takes to develop is ofthe order of 10 times larger than the residence time. However, when considering higher frequenciesand larger amplitudes, the ratio decreases. As seen from equation (8), this type of hydrodynamicinstability has little or no effect on low frequency combustion noise, but may be important for higherfrequency entropy waves, which are deformed by this mechanism.4. Numerical simulationsNumerical simulations are now performed using the code OpenFOAM [12]. The configuration isshown in figure 4: an annular nozzle with inlet/outlet Mach numbers of 0.1 and 3.2 respectively issimulated. The mesh contains 500 cells in the x-direction, 88 in the θ-direction and one in the radialdirection. This has been found sufficient to reduce numerical dissipation.Using NSCBC [13] boundary conditions, an entropy wave is introduced in the simulation domainthrough the inlet. The forcing term is a plane sinusoidal wave of f = 500Hz and an amplitude ofs′/cp = 0.05, with a small azimuthal perturbation in the form of a time delay, with a maximumvalue of 25% of the period of the wave. Figure 4 show the entropy wave propagating through thedomain. While convected, the deformed wave is subject to the Rayleigh Taylor instability mechanism,generating a vorticity field (shown in figure 4 as contour lines). This vorticity is convected with themean flow and contributes to the deformation of the wavefront.5. ConclusionsThe generation of indirect noise is due to the acceleration of hot spots (entropy waves) throughturbine stages. This acceleration is affected by the Rayleigh Taylor instability since the entropy waveICSV22, Florence, Italy, 12-16 July 2015 5The 22nd International Congress of Sound and Vibration-20.48-20.48-20.48-10.24-10.24000010.2410.2410.2420.4820.48x0.1 0.2 0.3 0.4θ123456-1-0.8-0.6-0.4-0.200.20.40.60.81(a) Ux-0.65-0.65-0.325-0.325-0.3250000000000.3250.3250.3250.650.65x0.1 0.2 0.3 0.4θ123456-1-0.8-0.6-0.4-0.200.20.40.60.81(b) UθFigure 5: Plot of the entropy wave propagating through the nozzle. Contour lines of the inducedvelocity.is composed of a density perturbation. In this work we have shown that the growth rate is, as a firstapproximation, proportional to the square root of the frequency, meaning that for low frequencies theeffect is likely to be negligible since the instability does not have time to develop. However, at higherfrequencies (1000− 2000Hz) the characteristic time for the growth of the instability is comparable tothe residence time of the entropy wave in the nozzle, showing that the wave can be deformed stronglyand therefore generate less indirect noise. Numerical simulations confirm that the deformed entropywave generates a vorticity field that deforms the wavefront.REFERENCES1. L. Crocco and S. I. Cheng. Theory of combustion instability in liquid propellant rocket motors.Number Agardograph No 8. Butterworths Science, 1956.2. N. A. Cumpsty and F. E. Marble. The interaction of entropy fluctuations with turbine blade rows;a mechanism of turbojet engine noise. Proc. R. Soc. Lond. A , 357:323–344, 1977.3. I. Duran and S. Moreau. Solution of the quasi one-dimensional linearized Euler equations usingflow invariants and the Magnus expansion. Journal of Fluid Mechanics, 723:190–231, May 2013.4. A. Giauque, M. Huet, and F. Clero. Analytical Analysis of Indirect Combustion Noise in Sub-critical Nozzles. J. Eng. Gas Turbines Power, 134(11):111202, 2012.5. I. Duran and A. Morgans. On the reflection and transmission of circumferential waves throughnozzles. Journal of Fluid Mechanics, Under Review, 2015.6. R. LeLevier, G. Lasher, and F. Bjorklund. Effect of a density gradient on Taylor instability.(UCRL-4459), 1955.6 ICSV22, Florence, Italy, 12-16 July 2015The 22nd International Congress of Sound and Vibration7. M. Leyko, F. Nicoud, and T. Poinsot. Comparison of direct and indirect combustion noise mech-anisms in a model combustor. AIAA Journal , 47(11):2709–2716, 2009.8. Lord Rayleigh. Investigation of the character of the equilibrium of an incompressible heavy fluidof variable density. Proceedings of the London Mathematical Society, 14: 170-177, 1883.9. F. E. Marble and S. Candel. Acoustic disturbances from gas nonuniformities convected througha nozzle. J. Sound Vib. , 55:225–243, 1977.10. W. H. Moase, M. J. Brear, and C. Manzie. The forced response of choked nozzles and supersonicdiffusers. Journal of Fluid Mechanics, 585:281–304, 2007.11. A. S. Morgans, C. S. Goh, and J. A. Dahan. The dissipation and shear dispersion of entropywaves in combustor thermoacoustics. Journal of Fluid Mechanics, 733 R2:1–11, 2013.12. OpenFOAM. http://www.openfoam.org.13. T. Poinsot, T. Echekki, and M. G. Mungal. A study of the laminar flame tip and implications forpremixed turbulent combustion. Combust. Sci. Tech. , 81(1-3):45–73, 1992.14. S. R. Stow, A. P. Dowling, and T. P. Hynes. Reflection of circumferential modes in a chokednozzle. Journal of Fluid Mechanics, 467:215–239, 2002.15. Taylor, Sir Geoffrey Ingram . The instability of liquid surfaces when accelerated in a directionperpendicular to their planes. Proceedings of the Royal Society of London. Series A, Mathematicaland Physical Sciences, 201 (1065): 192-196, 1950.ICSV22, Florence, Italy, 12-16 July 2015 7

The effect of hydrodynamic instabilities on the propagation of entropy waves through nozzles and on the generation of indirect combustion noise

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The effect of hydrodynamic instabilities on the propagation of entropy waves through nozzles and on the generation of indirect combustion noise

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