scholarly journals Stability and Limit Cycles of a Nonlinear Damper Acting on a Linearly Unstable Thermoacoustic Mode

2018 ◽  
Author(s):  
Claire Bourquard ◽  
Nicolas Noiray
Keyword(s):  
Growth Rate ◽  
Combustion Chamber ◽  
Limit Cycles ◽  
Feedback Loop ◽  
Unstable Mode ◽  
Rectangular Cavity ◽  
Van Der Pol Oscillator ◽  
Mean Velocity ◽  
Thermoacoustic Instability ◽  
Numerical Predictions

The resonant coupling between flames and acoustics is a growing issue for gas turbine manufacturers. They can be reduced by adding acoustic dampers on the combustion chamber walls. Nonetheless, if the engine is operated out of the stable window, the damper may be exposed to high-amplitude acoustic levels, which may trigger unwanted nonlinear effects. This work aims at providing an overview of the dynamics associated with those limit cycles using a simple analytical model, where a perfectly tuned damper is coupled to the combustion chamber. The damper, crossed by a purge flow in order to prevent hot gas ingestion, is modeled as a non-linearly damped harmonic oscillator, with vortex shedding as the main dissipation mechanism. The combustion chamber featuring a linearly unstable thermoacoustic mode is modelled as a Van der Pol oscillator. The fixed points of the coupled system and their stability can be determined by analyzing the averaged amplitude equations. This allows the computation of a fixed point topology map as function of the growth rate of the unstable mode and the mean velocity through the damper neck. Simulink simulations are also performed and compared to the analytical predictions. Finally, experiments are performed on a simple rectangular cavity, where the thermoacoustic instability resulting from the interaction between heat release and acoustic pressure is mimicked by an electro-acoustic instability. A feedback loop is built, where the signal from a microphone is filtered, delayed, and amplified before being sent to a loudspeaker placed inside the rectangular cavity. The delay and gain of the feedback loop can be modified to change the growth rate of the instability. One Helmholtz damper can be added to the cavity and tuned to the unstable mode of interest. The growth rate reduction capabilities of the damper and the amplitude of the limit cycle in the unstable cases are in good agreement with the analytical and numerical predictions, underlining the potentially dangerous behavior of the limit cycles which should be taken into account for real engine cases.

scholarly journals Stability and Limit Cycles of a Nonlinear Damper Acting on a Linearly Unstable Thermoacoustic Mode

2018 ◽  
Vol 141 (5) ◽  
Author(s):  
Claire Bourquard ◽  
Nicolas Noiray
Keyword(s):  
Growth Rate ◽  
Combustion Chamber ◽  
Limit Cycle ◽  
Nonlinear Effects ◽  
Coupled System ◽  
High Amplitude ◽  
Van Der Pol Oscillator ◽  
Mean Velocity ◽  
Thermoacoustic Instability ◽  
Numerical Predictions

The resonant coupling between flames and acoustics is a growing issue for gas turbine manufacturers, which can be reduced by adding acoustic dampers on the combustion chamber walls. Nonetheless, if the engine is operated out of the stable window, the damper is exposed to high-amplitude acoustic levels, which trigger unwanted nonlinear effects. This work provides an overview of the dynamics of this coupled system using a simple analytical model, where a perfectly tuned damper is coupled to the combustion chamber. The damper, crossed by a purge flow in order to prevent hot gas ingestion, is modeled as a nonlinearly damped harmonic oscillator. The combustion chamber featuring a linearly unstable thermoacoustic mode is modeled as a Van der Pol oscillator. Analyzing the averaged amplitude equations gives the limit cycle amplitudes as function of the growth rate of the unstable mode and the mean velocity through the damper neck. Experiments are also performed on a simple rectangular cavity, where the thermoacoustic instability is mimicked by an electro-acoustic instability. A feedback loop is built, through which the growth rate of the instability can be controlled. A Helmholtz damper is added to the cavity and tuned to the mode of interest. The stabilization capabilities of the damper and the amplitude of the limit cycle in the unstable cases are in good agreement between the experiments and the analytical and numerical predictions, underlining the potentially dangerous behavior of the system, which should be taken into account for real engine cases.

Adjoint Sensitivity Analysis of Hydrodynamic Stability in a Gas Turbine Fuel Injector

2015 ◽  
Author(s):  
Outi Tammisola ◽  
Matthew P. Juniper
Keyword(s):  
Growth Rate ◽  
Sensitivity Analysis ◽  
Combustion Chamber ◽  
Gas Turbine ◽  
Local Stability ◽  
Recirculation Zone ◽  
Base Flow ◽  
Fuel Injector ◽  
Mean Flow ◽  
Thermoacoustic Instability

Hydrodynamic oscillations in gas turbine fuel injectors help to mix the fuel and air but can also contribute to thermoacoustic instability. Small changes to some parts of a fuel injector greatly affect the frequency and amplitude of these oscillations. These regions can be identified efficiently with adjoint-based sensitivity analysis. This is a linear technique that identifies the region of the flow that causes the oscillation, the regions of the flow that are most sensitive to external forcing, and the regions of the flow that, when altered, have most influence on the oscillation. In this paper, we extend this to the flow from a gas turbine’s single stream radial swirler, which has been extensively studied experimentally (GT2008-50278) [8]. The swirling annular flow enters the combustion chamber and expands to the chamber walls, forming a conical recirculation zone along the centreline and an annular recirculation zone in the upstream corner. In this study, the steady base flow and the stability analysis are calculated at Re 200–3800 based on the mean flow velocity and inlet diameter. The velocity field is similar to that found from experiments and LES, and the local stability results are close to those at higher Re (GT2012-68253) [11]. All the analyses (experiments, LES, uRANS, local stability, and the global stability in this paper) show that a helical motion develops around the central recirculation zone. This develops into a precessing vortex core. The adjoint-based sensitivity analysis reveals that the frequency and growth rate of the oscillation is dictated by conditions just upstream of the central recirculation zone (the wavemaker region). It also reveals that this oscillation is very receptive to forcing at the sharp edges of the injector. In practical situations, this forcing could arise from an impinging acoustic wave, showing that these edges could be influential in the feedback mechanism that causes thermoacoustic instability. The analysis also shows how the growth rate and frequency of the oscillation change with either small shape changes of the nozzle, or additional suction or blowing at the walls of the injector. It reveals that the oscillations originate in a very localized region at the entry to the combustion chamber, which lies near the separation point at the outer inlet, and extends to the outlet of the inner pipe. Any scheme designed to control the frequency and amplitude of the oscillation only needs to change the flow in this localized region.

scholarly journals Kelvin-Helmholtz discontinuity in two superposed viscous conducting fluids in a horizontal magnetic field

2008 ◽  
Vol 12 (3) ◽  
pp. 103-110 ◽  
Author(s):  
Aiyub Khan ◽  
Neha Sharma ◽  
P.K. Bhatia
Keyword(s):  
Magnetic Field ◽  
Surface Tension ◽  
Growth Rate ◽  
Unstable Mode ◽  
Two Dimensional ◽  
Horizontal Magnetic Field ◽  
Streaming Velocity ◽  
The Stability ◽  
Conducting Fluids ◽  
Streaming Motion

The Kelvin-Helmholtz discontinuity in two superposed viscous conducting fluids has been investigated in the taking account of effects of surface tension, when the whole system is immersed in a uniform horizontal magnetic field. The streaming motion is assumed to be two-dimensional. The stability analysis has been carried out for two highly viscous fluid of uniform densities. The dispersion relation has been derived and solved numerically. It is found that the effect of viscosity, porosity and surface tension have stabilizing influence on the growth rate of the unstable mode, while streaming velocity has a destabilizing influence on the system.

scholarly journals Use of estradiol benzoate to limit triplets in a low dose FSH model of twinning

2021 ◽  
Author(s):  
weibin zeng ◽  
Lei An ◽  
Yanping Wang ◽  
Shuai Gao ◽  
Yusheng Qin ◽  
...  
Keyword(s):  
Growth Rate ◽  
Treatment Group ◽  
Feedback Loop ◽  
Production Efficiency ◽  
Low Dose ◽  
Follicle Stimulating Hormone ◽  
Estradiol Benzoate ◽  
Negative Feedback Loop ◽  
Beef Heifers ◽  
Excess Number

Abstract Background: Estrogen could limit the nondominant follicles development after the first deviation by inhibition of the FSH secretion through the negative feedback loop, which ensure that the number of dominant follicles would be in a moderate level. Methods: The objective of the present study was to evaluate the effect of estradiol benzoate (EB) on inhibiting the development of nondominant follicles and inducing twin calves in beef heifers. Beef heifers were synchronized using an estradiol (E2)- plus- progesterone (P4)- based and superovulated using small dose follicle- stimulating hormone (FSH) protocol. From days 6.5 to 7.5 every heifer was treated with variety dose of estradiol benzoate (EB) for 3 times with 12 h intervals to eliminate the excess number of dominant follicles. Results: The diameters of the two largest follicles (F1 and F2) continually increased from day 3.5 to day 10. However, the growth rate was constrained by exogenous EB, and the degree of suppression was greatest in the 0.5 mg EB treatment compared with other treatments. As a result, the number of large follicles (≥ 10 mm) was also reduced along with the dose of EB increased. The double/triple ovulations rate, pregnancy rate and twin were all demonstrate the highest in 0.2 mg EB treatment group than in other treatments. Conclusions：The present study describes an efficient protocol that can be used to stimulate the development of a small number of dominant follicles i.e. 2-3 at the deviation stage through a FSH and 0.2 mg EB combine treatment, which can further result in the production of two calves. The appropriate dose of EB treatment during FSH induced superovulation procedure could limit the number of dominant follicles development and eventually increase the calf production efficiency.

The extension of the Miles-Howard theorem to compressible fluids

1970 ◽  
Vol 43 (4) ◽  
pp. 833-836 ◽  
Author(s):  
G. Chimonas
Keyword(s):  
Growth Rate ◽  
Shear Flow ◽  
Upper Bound ◽  
Unstable Mode ◽  
Vertical Gradient ◽  
Flow Speed ◽  
Incompressible Fluids ◽  
Compressible Fluids ◽  
Sufficient Condition ◽  
Parallel Shear Flow

A statically stable, gravitationally stratified compressible fluid containing a parallel shear flow is examined for stability against infinitesimal adiabatic perturbations. It is found that the Miles–Howard theorem of incompressible fluids may be generalized to this system, so that n2 ≥ ¼U′2 throughout the flow is a sufficient condition for stability. Here n2 is the Brunt–Väissälä frequency and U’ is the vertical gradient of the flow speed. Howard's upper bound on the growth rate of an unstable mode also generalizes to this compressible system.

Laser Velocimeter Measurements in Highly Turbulent Recirculating Flows

1984 ◽  
Vol 106 (2) ◽  
pp. 173-180 ◽  
Author(s):  
W. H. Stevenson ◽  
H. D. Thompson ◽  
R. R. Craig
Keyword(s):  
Turbulent Flows ◽  
Turbulence Model ◽  
Quantitative Description ◽  
Sudden Expansion ◽  
Laser Doppler Velocimeter ◽  
Bias Error ◽  
Mean Velocity ◽  
One Dimensional ◽  
Numerical Predictions ◽  
Velocity Bias

This paper presents the results of an extensive study of subsonic separated flows using a laser Doppler velocimeter. Both a rectangular rearward facing step and cylindrical (axisymmetric) sudden expansion geometry were studied. The basic objectives were to resolve the question of whether a velocity bias error does, in fact, occur in LDV measurements in highly turbulent flows of this type and, if so, how it may be eliminated; map the velocity field (mean velocity, turbulence intensity, Reynolds stress, etc.) including the entire recirculation zone; and compare experimental results with numerical predictions based on the k-ε turbulence model. Measurements were carried out using a one-dimensional LDV operating in forward scatter with signal processing by means of a commercial counter-type processor. Results obtained show that velocity bias does occur in turbulent flows and that it can be overcome by proper data acquisition procedures. The results also indicate that the important mean velocity and turbulence quantities can be obtained with reasonable accuracy using a one-dimensional LDV system. Although the k-ε turbulence model provides a good qualitative picture of the flow field, it does not yield a completely adequate quantitative description. Results obtained here illustrate the discrepancies to be expected and provide a basis for further model development.

Effect of biodiesel oxidation on deposit growth rate in the combustion chamber

2020 ◽  
Author(s):  
Bambang Sugiarto ◽  
J. A. Hidayat ◽  
M. T. Suryantoro ◽  
H. Setiapraja ◽  
S. Yubaidah ◽  
...  
Keyword(s):  
Growth Rate ◽  
Combustion Chamber

Compressibility effects in a turbulent annular mixing layer. Part 1. Turbulence and growth rate

2000 ◽  
Vol 421 ◽  
pp. 229-267 ◽  
Author(s):  
JONATHAN B. FREUND ◽  
SANJIVA K. LELE ◽  
PARVIZ MOIN
Keyword(s):  
Growth Rate ◽  
Mach Number ◽  
Mixing Layer ◽  
Mixing Layers ◽  
Velocity Difference ◽  
Mean Velocity ◽  
The Mean ◽  
Mach Numbers ◽  
High Mach ◽  
Compressibility Effects

This work uses direct numerical simulations of time evolving annular mixing layers, which correspond to the early development of round jets, to study compressibility effects on turbulence in free shear flows. Nine cases were considered with convective Mach numbers ranging from Mc = 0.1 to 1.8 and turbulence Mach numbers reaching as high as Mt = 0.8.Growth rates of the simulated mixing layers are suppressed with increasing Mach number as observed experimentally. Also in accord with experiments, the mean velocity difference across the layer is found to be inadequate for scaling most turbulence statistics. An alternative scaling based on the mean velocity difference across a typical large eddy, whose dimension is determined by two-point spatial correlations, is proposed and validated. Analysis of the budget of the streamwise component of Reynolds stress shows how the new scaling is linked to the observed growth rate suppression. Dilatational contributions to the budget of turbulent kinetic energy are found to increase rapidly with Mach number, but remain small even at Mc = 1.8 despite the fact that shocklets are found at high Mach numbers. Flow visualizations show that at low Mach numbers the mixing region is dominated by large azimuthally correlated rollers whereas at high Mach numbers the flow is dominated by small streamwise oriented structures. An acoustic timescale limitation for supersonically deforming eddies is found to be consistent with the observations and scalings and is offered as a possible explanation for the decrease in transverse lengthscale.

On the stability of an inviscid shear layer which is periodic in space and time

1967 ◽  
Vol 27 (4) ◽  
pp. 657-689 ◽  
Author(s):  
R. E. Kelly
Keyword(s):  
Growth Rate ◽  
Shear Layer ◽  
Finite Amplitude ◽  
Periodic Component ◽  
Periodic Flow ◽  
Flow Profile ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean ◽  
The Stability

In experiments concerning the instability of free shear layers, oscillations have been observed in the downstream flow which have a frequency exactly half that of the dominant oscillation closer to the origin of the layer. The present analysis indicates that the phenomenon is due to a secondary instability associated with the nearly periodic flow which arises from the finite-amplitude growth of the fundamental disturbance.At first, however, the stability of inviscid shear flows, consisting of a non-zero mean component, together with a component periodic in the direction of flow and with time, is investigated fairly generally. It is found that the periodic component can serve as a means by which waves with twice the wavelength of the periodic component can be reinforced. The dependence of the growth rate of the subharmonic wave upon the amplitude of the periodic component is found for the case when the mean flow profile is of the hyperbolic-tangent type. In order that the subharmonic growth rate may exceed that of the most unstable disturbance associated with the mean flow, the amplitude of the streamwise component of the periodic flow is required to be about 12 % of the mean velocity difference across the shear layer. This represents order-of-magnitude agreement with experiment.Other possibilities of interaction between disturbances and the periodic flow are discussed, and the concluding section contains a discussion of the interactions on the basis of the energy equation.

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