Numerical study of nonlinear sustained oscillations in a cylindrical open-ended tube (original) (raw)
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NUMERICAL SIMULATION OF SELF-EXCITED THERMOACOUSTIC INSTABILITIES IN A RIJKE TUBE
Self-excited thermoacoustic instabilities or oscillations occur in con"ned geometries and result from a feedback loop between the heat transferred to the #uid from a heat source and the acoustics of the geometry. If the heat input is at times of high pressure, a self-ampli"cation of acoustic #uctuations may lead to high pressure amplitudes. The e!ect can be observed in a Rijke tube, a straight tube with a heating element made from hot wires or gauze that provides the heat input. In the presence of a gas #ow, pressure oscillations are excited at one of the tube's natural frequencies. Two di!erent kinds of Rijke tubes are modelled by using a control volume based "nite di!erence method to solve iteratively the unsteady conservation equations for mass, momentum and energy. The obtained results are in good agreement with experiments. Besides the general behaviour of the oscillating system, non-linear e!ects are also accounted for by the simulations. The non-linearities in the heat transferred to the #uid from the heat source were investigated. These determine the limit cycle amplitudes of the self-excited oscillations.
Weakly nonlinear analysis of thermoacoustic bifurcations in the Rijke tube
In this study we present a theoretical weakly nonlinear framework for the prediction of thermoacoustic oscillations close to Hopf bifurcations. We demonstrate the method for a thermoacoustic network that describes the dynamics of an electrically heated Rijke tube. We solve the weakly nonlinear equations order by order, discuss their contribution on the overall dynamics and show how solvability conditions at odd orders give rise to Stuart–Landau equations. These equations, combined together, describe the nonlinear dynamical evolution of the oscillations' amplitude and their frequency. Because we retain the contribution of several acoustic modes in the thermoacoustic system, the use of adjoint methods is required to derive the Landau coefficients. The analysis is performed up to fifth order and compared with time domain simulations, showing good agreement. The theoretical framework presented here can be used to reduce the cost of investigating oscillations and subcritical phenomena close to Hopf bifurcations in numerical simulations and experiments and can be readily extended to consider, e.g. the weakly nonlinear interaction of two unstable thermoacoustic modes.
Resonance gas oscillations in closed tubes
Journal of Fluid Mechanics, 1996
The problem of gas motion in a tube closed at one end and driven at the other by an oscillating piston is studied theoretically. When the piston vibrates with a finite amplitude at the first acoustic resonance frequency, periodic shock waves are generated, travelling back and forth in the tube. A perturbation method, based on a small Mach number, M and a global mass conservation condition, is employed to formulate a solution of the problem in the form of two standing waves separated by a jump (shock front). By expanding the equations of motion in a series of a small parameter E = M"', all hydrodynamic properties are predicted with an accuracy to second-order terms, i.e. to 2. It is found that the first-order solution coincides with the previous theories of Betchov (1958) and Chester (1964), while additional terms predict a non-homogeneous time-averaged pressure along the tube. This prediction compares favourably with experimental results from the literature. The importance of the phenomenon is discussed in relation to different transport processes in resonance tubes.
The Journal of the Acoustical Society of America, 1999
The fundamental resonant gas osciuation with periodic shock wave in a closed tube is studied by executing large-scale computations of the 2-D Navier-Stokes equations for compressible flow with a high-resolution upwind fimite-difference TVD scheme. In a quasi-steady state of gas osciuation, acoustic streaming (mean mass flow) with large Rs is excited in the tube, where mathrmRs\mathrm{R}smathrmRs is the streaming Reynolds number based on a charactenistic streaming velocity, the tube length, and the kinematic viscosity.
Non-normal and nonlinear dynamics of thermoacoustic instability in a horizontal Rijke tube
The paper focusses on the non-normal and nonlinear effects of thermoacoustic interaction in a horizontal electrically heated Rijke tube. The analysis starts with the governing equations for the fluid flow. The governing equations become stiff as the Mach number of the steady flow and the thickness of the heat source (compared to the acoustic wavelength) are small. Therefore asymptotic analysis is performed in the limit of small Mach number and compact heat source to eliminate the above stiffness problem. Two systems of governing equations are obtained: one for the acoustic field and the other for the unsteady flow field in the hydrodynamic zone around the heater. A theoretical framework is developed to understand the non-normal nature of the thermoacoustic interaction in the Rijke tube. The role of non-normality in the subcritical transition to instability regime is explored.
Numerical simulations of shock propagation and attenuation in narrow tubes are carried out using a onedimensional approach. The discretization of the convective terms is based on the fifth-order weighted essentially non-oscillatory interpolation. The influence of the dissipative processes such as momentum and heat losses is investigated. Viscosity as well as heat losses are found to play a key role in the attenuation of the shock speed as well as the shock intensity in the long-time evolution, demonstrating the transition from a hyperbolic behavior towards a diffusive regime. Specifically, when only strong heat exchanges are considered, numerical tests, corroborated by a simple asymptotic analysis, showed a transition from a hyperbolic adiabatic regime to an isothermal regime. Furthermore, the influence of the scaling parameter ReD/4L s , through the variation of the tube diameter, D, the viscous length scale, L s , and the Reynolds number on the shock propagation behavior is examined.
Strongly nonlinear waves of nonplanar mode in a circular duct
Propagation of weak shock waves and strongly nonlinear acoustic waves of the lowest order nonplanar mode in a circular duct filled with an ideal gas is numerically investigated by using a high resolution upwind TVD scheme. The results show that, when the nonlinearity is moderately strong and the source frequency is moderately high, the initial sinusoidal wave profile can evolve into shocks, although according to the weakly nonlinear theory a nonlinear Schrödinger equation determines the wave motion in a steady state, where a shock wave does not appear. Furthermore, strongly nonlinear waves can induce a vortex-ring-like streaming jet (mean mass flow), by which the density of the gas in the neighborhood of the vortex core decreases more and more as time goes by. The resulting low-density region is not only of high vorticity but also of high entropy and high temperature.
Nonlinear saturation of the thermoacoustic instability
The Journal of the Acoustical Society of America, 2000
A weakly nonlinear theory of the thermoacoustic instability in gas-filled tubes is developed in the time domain by exploiting the difference between the instability time scale and the period of standing waves. By carrying the expansion to fourth order in the perturbation parameter, explicit results for the initial growth, nonlinear evolution, and final saturation are obtained. The dependence of the saturation amplitude upon the temperature difference in the stack, the tube geometry, stack plate spacing, Prandtl number, and other parameters is illustrated.