Thermoacoustic Stability of Quasi-One-Dimensional Flows–Part I: Analytical and Numerical Formulation

2004 ◽  
Vol 126 (4) ◽  
pp. 637-644 ◽  
Author(s):  
Dilip Prasad ◽  
Jinzhang Feng
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Problem Formulation ◽  
Mean Flow ◽  
Sonic Point ◽  
Impedance Boundary ◽  
One Dimensional ◽  
Natural Modes ◽  
Local Wave ◽  
Numerical Formulation

A numerical method is developed for transient linear analysis of quasi-one-dimensional thermoacoustic systems, with emphasis on stability properties. This approach incorporates the effects of mean flow variation as well as self-excited sources such as the unsteady heat release across a flame. Working in the frequency domain, the perturbation field is represented as a superposition of local wave modes, which enables the linearized equations to be integrated in space. The problem formulation is completed by specifying appropriate boundary conditions. Here, we consider impedance boundary conditions as well as those relevant to choked and shocked flows. For choked flows, the boundary condition follows from the requirement that perturbations remain regular at the sonic point, while the boundary conditions applicable at a normal shock are obtained from the shock jump conditions. The numerical implementation of the proposed formulation is described for the system eigenvalue problem, where the natural modes are sought. The scheme is validated by comparison with analytical and numerical solutions.

Thermoacoustic Stability of Quasi-One-Dimensional Flows: Part I — Analytical and Numerical Formulation

2004 ◽  
Author(s):  
Dilip Prasad ◽  
Jinzhang Feng
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Problem Formulation ◽  
Mean Flow ◽  
Sonic Point ◽  
Impedance Boundary ◽  
One Dimensional ◽  
Natural Modes ◽  
Local Wave ◽  
Numerical Formulation

A numerical method is developed for transient linear analysis of quasi-one-dimensional thermoacoustic systems, with emphasis on stability properties. This approach incorporates the effects of mean flow variation as well as self-excited sources such as the unsteady heat release across a flame. Working in the frequency domain, the perturbation field is represented as a superposition of local wave modes, which enables the linearized equations to be integrated in space. The problem formulation is completed by specifying appropriate boundary conditions. Here, we consider impedance boundary conditions as well as those relevant to choked and shocked flows. For choked flows, the boundary condition follows from the requirement that perturbations remain regular at the sonic point, while the boundary conditions applicable at a normal shock are obtained from the shock jump conditions. The numerical implementation of the proposed formulation is described for the system eigenvalue problem, where the natural modes are sought. The scheme is validated by comparison with analytical and numerical solutions.

Thermoacoustic Stability of Quasi-One-Dimensional Flows—Part II: Application to Basic Flows

2004 ◽  
Vol 126 (4) ◽  
pp. 645-653 ◽  
Author(s):  
Dilip Prasad ◽  
Jinzhang Feng
Keyword(s):  
Boundary Conditions ◽  
Acoustic Energy ◽  
System Stability ◽  
Combustion Stability ◽  
Mean Flow ◽  
One Dimensional ◽  
Numerical Formulation ◽  
The Mean ◽  
Flow Gradients ◽  
The Stability

In this paper, applications of a previously developed numerical formulation (Prasad, D., and Feng, J., 2004, “Thermoacoustic stability of Quasi-One-Dimensional Flows—Part I: Analytical and Numerical Formulation,” J. Turbomach., 126, pp. 636–643. for the stability analysis of spatially varying one-dimensional flows are investigated. The results are interpreted with the aid of a generalized acoustic energy equation, which shows that the stability of a flow system depends not only on the nature of the unsteady heat, mass and momentum sources but also on the mean flow gradients and on the inlet and exit boundary conditions. Specifically, it is found that subsonic diffusing flows with strongly reflecting boundary conditions are unstable, whereas flows with a favorable pressure gradient are not. Transonic flows are also investigated, including those that feature acceleration through the sonic condition and those in which a normal shock is present. In both cases, it is found that the natural modes are stable. Finally, we study a simplified ducted flame configuration. It is found that the length scale of the mean heat addition affects system stability so that the thin-flame model commonly used in studies of combustion stability may not always be applicable.

Filter-Based Time-Domain Impedance Boundary Conditions for CFD Applications

2008 ◽  
Author(s):  
Andreas Huber ◽  
Philipp Romann ◽  
Wolfgang Polifke
Keyword(s):  
Boundary Conditions ◽  
Time Domain ◽  
Problem Formulation ◽  
Mean Flow ◽  
Acoustic Boundary Conditions ◽  
Impedance Boundary ◽  
Filter Model ◽  
Impedance Boundary Conditions ◽  
Computational Fluid Dynamics Cfd ◽  
New Formulation

For flow simulations, proper boundary conditions are essential for realizing a well-posed, physically meaningful and numerically stable problem formulation. This is particularly difficult for compressible flow, where in general boundary conditions have to be imposed both for mean flow and acoustic quantities. For the acoustic variables, boundary conditions can be formulated in terms of the acoustic impedance or alternatively the reflection coefficient, which are general a complex-valued, frequency dependent quantity. The present work presents a novel, efficient and flexible approach to impose time-domain impedance boundary conditions (TDIBC) for computational fluid dynamics (CFD): The acoustic boundary conditions are represented as a discrete filter model with appropriately optimized filter coefficients. Using the z-transformation the filter model is transferred to a time-domain formulation and applied to the CFD environment in form of advanced filter realizations. Validation studies using various acoustic boundary conditions have been carried out with the new formulation. The results demonstrate that the method works in an accurate and robust manner.

Thermoacoustic Stability of Quasi-One-Dimensional Flows: Part II — Application to Basic Flows

2004 ◽  
Author(s):  
Dilip Prasad ◽  
Jinzhang Feng
Keyword(s):  
Boundary Conditions ◽  
Acoustic Energy ◽  
System Stability ◽  
Combustion Stability ◽  
Mean Flow ◽  
One Dimensional ◽  
Numerical Formulation ◽  
The Mean ◽  
Flow Gradients ◽  
The Stability

In this paper, applications of a previously developed numerical formulation (Prasad and Feng 2004) for the stability analysis of spatially varying one-dimensional flows are investigated. The results are interpreted with the aid of a generalized acoustic energy equation, which shows that the stability of a flow system depends not only on the nature of the unsteady heat, mass and momentum sources but also on the mean flow gradients and on the inlet and exit boundary conditions. Specifically, it is found that subsonic diffusing flows with strongly reflecting boundary conditions are unstable, whereas flows with a favorable pressure gradient are not. Transonic flows are also investigated, including those that feature acceleration through the sonic condition and those in which a normal shock is present. In both cases, it is found that the natural modes are stable. Finally, we study a simplified ducted flame configuration. It is found that the length scale of the mean heat addition affects system stability so that the thin-flame model commonly used in studies of combustion stability may not always be applicable.

Computational Fluid Dynamic Prediction of Noise From a Cold Turbulent Propane Jet

2008 ◽  
Author(s):  
Tanya S. Stanko ◽  
Derek B. Ingham ◽  
Michael Fairweather ◽  
Mohamed Pourkashanian
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Fluid Dynamic ◽  
Broadband Noise ◽  
Navier Stokes ◽  
Mean Flow ◽  
Dynamic Prediction ◽  
Design Changes ◽  
Turbulent Jet Flow ◽  
Velocity Information

Numerical solutions of a turbulent jet flow are used to provide velocity information throughout a simple cold turbulent propane jet at a Reynolds number of 68,000. Predictions provided by the Reynolds-averaged Navier-Stokes simulations, based on a Reynolds stress turbulence model, are compared with experimental data available in the literature. The effect of the modelled inlet boundary conditions on the predicted flow field is described, and the discrepancy between the simulation results and experiment measurements is found to be less than the corresponding variations due to uncertainness in the experimental boundary conditions. In addition, these solutions are used as the basis for noise predictions for the jet based on Lighthill’s theory using the Goldstein broadband noise source formalization that postulates axisymmetric turbulence superposed on the mean flow. The latter model provides an aeroacoustic tool that is reasonable in identifying components or surfaces that generate significant amounts of noise, thereby providing opportunities for early design changes to aircraft and gas turbine components.

scholarly journals One-dimensional model for a laser-ablated slab under acceleration

2004 ◽  
Vol 22 (2) ◽  
pp. 183-188 ◽  
Author(s):  
J. RAMÍREZ ◽  
R. RAMIS ◽  
J. SANZ
Keyword(s):  
Subsonic Flow ◽  
Numerical Solutions ◽  
Critical Surface ◽  
Slab Surface ◽  
Dimensional Model ◽  
Initial Value ◽  
Sonic Point ◽  
One Dimensional ◽  
Ablation Pressure ◽  
Simple Dependence

A one-dimensional model for a laser-ablated slab under acceleration g is developed. A characteristic value gc is found to separate two solutions: Lower g values allow sonic or subsonic flow at the critical surface; for higher g the sonic point approaches closer and closer to the slab surface. A significant reduction in the ablation pressure is found in comparison to the g = 0 case. A simple dependence law between the ablation pressure and the slab acceleration, from the initial value g0 to infinity, is identified. Results compared well with fully hydrodynamic computer simulations with Multi2D code. The model has also been found a key step to produce indefinitely steady numerical solutions to study Rayleigh–Taylor instabilities in heat ablation fronts, and validate other theoretical analysis of the problem.

Determination of Optimum Profile of One-Dimensional Cooling Fins

1983 ◽  
Vol 105 (3) ◽  
pp. 317-320 ◽  
Author(s):  
S. K. Hati ◽  
S. S. Rao
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Minimum Principle ◽  
Problem Formulation ◽  
One Dimensional ◽  
A New Technique ◽  
The One ◽  
Optimum Profile

The optimum design of an one-dimensional cooling fin is considered by including all modes of heat transfer in the problem formulation. The minimum principle of Pontryagin is applied to determine the optimum profile. A new technique is used to solve the reduced differential equations with split boundary conditions. The optimum profile found is compared with the one obtained by considering only conduction and convection.

scholarly journals A one-dimensional model for the interaction between cell-to-cell adhesion and chemotactic signalling

2011 ◽  
Vol 22 (4) ◽  
pp. 291-316 ◽  
Author(s):  
K. ANGUIGE
Keyword(s):  
Cell Adhesion ◽  
Discrete Model ◽  
Continuum Limit ◽  
Numerical Solutions ◽  
Problem Formulation ◽  
Dimensional Model ◽  
One Dimensional ◽  
Spatial Oscillations ◽  
Cell To Cell Adhesion ◽  
The Continuum

We develop and analyse a discrete, one-dimensional model of cell motility which incorporates the effects of volume filling, cell-to-cell adhesion and chemotaxis. The formal continuum limit of the model is a non-linear generalisation of the parabolic-elliptic Keller–Segel equations, with a diffusivity which can become negative if the adhesion coefficient is large. The consequent ill-posedness results in the appearance of spatial oscillations and the development of plateaus in numerical solutions of the underlying discrete model. A global-existence result is obtained for the continuum equations in the case of favourable parameter values and data, and a steady-state analysis, which, amongst other things, accounts for high-adhesion plateaus, is carried out. For ill-posed cases, a singular Stefan-problem formulation of the continuum limit is written down and solved numerically, and the numerical solutions are compared with those of the original discrete model.

NUMERICAL ANALYSIS OF FINITE DEPTH PROBLEMS IN SOIL-WATER HYDROLOGY

1971 ◽  
Vol 2 (1) ◽  
pp. 1-22 ◽  
Author(s):  
K. K. WATSON
Keyword(s):  
Numerical Analysis ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Soil Water ◽  
Water Table ◽  
Numerical Solutions ◽  
Finite Depth ◽  
Flow Equation ◽  
One Dimensional ◽  
The One

The types of initial and boundary conditions which may be involved in the flow of water through an unsaturated profile to a water table are discussed. A numerical solution of the flow equation is then outlined and its use in the one-dimensional drainage of a homogeneous profile and in ponded infiltration into a draining profile detailed. Several characteristics of the drainage of deep profiles are discussed together with numerical solutions for the drainage of a stratified profile and infiltration into a draining profile.

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