https://ntrs.nasa.gov/api/citations/19860016597/downloads/19860016597.pdf

Preconditioned methods for solving the incompressible low speed compressible equations

: This book has several purposes, including a broad historical summary of combustion instabilities in propulsion systems; a concise compilation of the main mechanisms for instabilities in liquid and solid fueled rockets, ramjets, thrust augmentors and gas turbines; development of a theoretical framework for investigating unsteady motions in combustion systems; and accessible surveys of the basic material required to understand the subject. Emphasis is placed throughout the book on observed behavior. For well-understood reasons, the best, and in many respects most useful quantitative data, have been obtained for solid propellant rockets. Hence that type of device occupies a special position in the subject. The book comprises nine chapters and eight annexes. Material in the annexes is not essential for reading and broadly understanding the main part of the book, but is likely interesting for those choosing to do research on the subject. The nine chapters divide into three parts.

Unsteady Motions in Combustion Chambers for Propulsion Systems

Review of preconditioning methods for fluid dynamics

High-order accurate, low numerical diffusion methods for aerodynamics

Patients referred for treatment of tracheal stenosis typically are asymptomatic until critical narrowing of the airway occurs, which then requires immediate intervention. To understand how tracheal...

https://www.physiology.org/doi/pdf/10.1152/japplphysiol.01063.2006

Tracheal stenosis: a flow dynamics study.

The computation of steady incompressible flows by an Euler implicit algorithm is studied using both the incompressible equations and the low Mach number compressible equations. The incompressible equations are handled by adding an artificial time derivative to the continuity equation. This allows both the pressure and velocity to be obtained implicitly. In one-dimensional problems, both systems converge rapidly, even at low Mach numbers where the eigenvalues are very stiff. In two dimensions where approximate factorization is required, the presence of stiff eigenvalues is highly detrimental. Stiffness can be avoided in the incompressible equations by selecting an appropriate "pseudo"-Mach number. This insures reliable convergence and results in an efficient incompressible flow algorithm. In the case of compressible equations, the Mach number cannot be chosen arbitrarily and the contamination introduced by approximate factorization must be removed. A matrix preconditioning factor that accomplishes this is developed and demonstrated. With this modification, the convergence rate is the same as in the incompressible case and is independent of Mach number. Rapid convergence is observed at Mach numbers as low as 6.05.

Application of time-iterative schemes to incompressible flow

The diagonalized ADI algorithm of Pulliam and Chaussee has been extended to include a timederivative preconditioning that permits efficient computation of flows over a wide range of Mach numbers. The algorithm has also been modified so that time-accurate analyses are possible through the use of dual-time stepping. The dual-time stepping has been incorporated in such a way that no additional factors are needed to handle the dual-time term in the left-hand-side inversion process. The preconditioning can accommodate general equations of state such as constant density (incompressible), perfect gas, or supercritical fluids. Unsteady vortex shedding from a circular cylinder is presented as an example of the capability of the algorithm.

A Preconditioned Dual-Time, Diagonalized ADI Scheme for Unsteady Computations

Two model problems were formulated to investigate the benefits of using higher order upwinded differencing schemes to convect a vortex. In these model problems, one or two ideal vortices were specified at the upstream boundary and convected through a simple fluid domain in the direction of the vorticity vector. Numerical tests were performed using an implicit upwinded finite volume method with orders of accuracy varying from first to fifth order. The effect of grid refinement was examined to determine the relative advantage to higher order accuracy vs additional grid points. The calculations demonstrated that the fifth-order scheme with 14 points across the vortex diameter convected a well-aligned vortex almost perfectly. The fifth-order scheme provided a factor of 10 reduction in the number of points over the third-order scheme in each crossflow plane. The corotating double-vortex model problem was used to study the effect of vortex skewness relative to the grid. A vortex-convection skewness parameter was defined that indicates the number of grid points required in the axial direction for adequate preservation of a vortex skewed relative to the grid.

Investigation of high-order upwinded differencing for vortex convection

A combined experimental and computational characterization of combustion instabilities in a lean, premixed backward-facing step combustor was performed. Specifically, the instabilities of interest were those encountered as the equivalence ratio was reduced to levels approaching the combustor's lean extinction limit. A quasi-one-dimensional unsteady analysis with loss and heat-addition models was developed to simulate combustion instabilities for a premixed step combustor. Experimental results indicated that the magnitude of a longitudinal acoustic disturbance grew with decreasing equivalence ratio, until it eventually triggered a lower-frequency, high-amplitude instability. Numerical results compared with experimental data demonstrated that the analysis can capture the critical frequencies observed in the combustor model. Coupling the heat release with the step flow velocities in the analysis produced an instability at the dominant resonant frequency of the combustor. Experimentally, low-frequency instabilities were visible as a flapping of the flame and increased in severity with decreasing equivalence ratio until they caused combustor blowout.

Investigation of Instabilities in a Lean, Premixed Step Combustor

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Papers

Application of time-iterative schemes to incompressible flow

A Preconditioned Dual-Time, Diagonalized ADI Scheme for Unsteady Computations

Investigation of high-order upwinded differencing for vortex convection

Investigation of Instabilities in a Lean, Premixed Step Combustor

Investigation of high-order upwinded differencing for vortex convection