Advanced Mathematical Methods for Scientists and Engineers

Fluid dynamic turbulence is one of the most challenging computational physics problems because of the extremely wide range of time and space scales involved, the strong nonlinearity of the governing equations, and the many practical and important applications. While most linear fluid instabilities are well understood, the nonlinear interactions among them makes even the relatively simple limit of homogeneous isotropic turbulence difficult to treat physically, mathematically, and computationally. Turbulence is modeled computationally by a two-stage bootstrap process. The first stage, direct numerical simulation, attempts to resolve the relevant physical time and space scales but its application is limited to diffusive flows with a relatively small Reynolds number (Re). Using direct numerical simulation to provide a database, in turn, allows calibration of phenomenological turbulence models for engineering applications. Large eddy simulation incorporates a form of turbulence modeling applicable when the large-scale flows of interest are intrinsically time dependent, thus throwing common statistical models into question. A promising approach to large eddy simulation involves the use of high-resolution monotone computational fluid dynamics algorithms such as flux-corrected transport or the piecewise parabolic method which have intrinsic subgrid turbulence models coupled naturally to the resolved scales in the computed flow. The physical considerations underlying and evidence supporting this monotone integrated large eddy simulation approach are discussed.

New insights into large eddy simulation

Combustion dynamics constitutes one of the most challenging areas in combustion research. Many facets of this subject have been investigated over the past few decades for their fundamental and practical implications. Substantial progress has been accomplished in understanding analysis, modeling, and simulation. Detailed laboratory experiments and numerical computations have provided a wealth of information on elementary dynamical processes such as the response of flames to variable strain, vortex rollup, coupling between flames and acoustic modulations, and perturbed flame collisions with boundaries. Much recent work has concerned the mechanisms driving instabilities in premixed combustion and the coupling between pressure waves and combustion with application to the problem of instability in modern low NO x heavyduty gas turbine combustors. Progress in numerical modeling has allowed simulations of dynamical flames interacting with pressure waves. On this basis, it has been possible to devise predictive methods for instabilities. Important efforts have also been directed at the development of the related subject of combustion control. Research has focused on methods, sensors, actuators, control algorithms, and systems integration. In recent years, scaling from laboratory experiments to practical devices has been achieved with some successebut limitations have also been revealed. Active control of combustion has also evolved in various directions. A number of experiments on laboratory-scale combustors have shown that the amplitude of combustion instabilities could be reduced by applying control principles. Full-scale terrestrial application to gas turbine systems have allowed an increase of the stability margin of these machines. Feedback principles are also being explored to control the point of operation of combustors and engines. Operating point control has special importance in the gas turbine field since it can be used to avoid operation in unstable regions near the lean blowoff limits. More generally, closed loop feedback concepts are useful if one wishes to improve the combustion process as demonstrated by applications to automotive engines. Many future developments of combustion will use such concepts for tuning, optimization, and emissions reduction. This article proposes a broad survey of these fast-moving areas of research.

Combustion dynamics and control: Progress and challenges

https://hal.archives-ouvertes.fr/hal-00803230/document

Large Eddy Simulations of gaseous flames in gas turbine combustion chambers

This Part I presents a detailed description of a numerical system built and tested with the final goal of reaching an accuracy of 2-3 dB over a meaningful range of frequencies for airliner engine noise, while having low empiricism and a general-geometry capability. The turbulence is treated by Large-Eddy Simulation with grids of around 1 million points, slightly upwind-biased high-order differencing, and implicit time integration. The code can incorporate boundaries and multi-block grids (thus avoiding the centerline singularity), and capture shocks. The sub-grid scale model is de-activated, because on present grids it strongly interferes with transition in the mixing layer. Without unsteady inflow forcing, the shear-layer roll-up and three-dimensionalization are realistic and reasonably insensitive to the grid. The far-field noise is computed using the permeable Ffowcs-Williams/Hawkings (FWH) formulation without external quadrupoles. The treatment of the disk that closes the FWH surface near the outflow ...

Noise prediction for increasingly complex jets. Part I: Methods and tests

In the past 20 years, periodic vortex shedding as a source of acoustic energy inside solid propellant rocket motors has been continuously studied, in connection with several motors that exhibited oscillatory behaviors although they were predicted stable by means of conventional linear stability methods. On that subject, correlations through Strouhal numbers are commonly used. In this article, the meanings of various Strouhal number definitions are discussed. The basic mechanisms for sound production in ducts or chambers are then discussed through simple tools, such as the linear hydrodynamic stability analysis, Flandro's method and the acoustic balance technique, which provide good insights into the physics of the phenomenon. Necessary conditions for vortex-shedding-driven motors are then arrived at and illustrated by actual firing results. Finally, the full numerical approaches are shown to provide unprecedented insight into the mechanisms behind the vortexshedding phenomenon and are believed to open the way to quantitative predictions of frequencies and oscillatory levels.

Vortex-Shedding Phenomena in Solid Rocket Motors

http://gcasalis.free.fr/Files/ast2003.pdf

Instabilities and pressure oscillations in solid rocket motors

This paper describes a new mechanism leading to vortex shedding instabilities in long (large L/D) solid propellant motors. This mechanism is termed "parietal vortex shedding" and has been discovered thanks to numerical simulations of the unsteady, 2D, compressible, Navier-Stokes equations. It seems to involve hydrodynamic instabilities of the mean flow velocity profiles, corresponding to injection induced internal flow (so called Culick or Taylor profiles), that couple with the acoustic frequencies of the chamber. Although this mechanism is found to be very powerful, it seems to need some background noise to feed it. The presented simulations can explain observed instabilities in configurations, without segmentation or without protruding inhibitor rings, of a simplified subscale setup of the Ariane 5 MPS P230 solid propellant motor. Detailed comparisons are proposed and the influence of the propellant combustion response and of the turbulence of the flow field are analyzed by means of recently developed models. INTRODUCTION-OBJECTIVES not precisely known if inhibitor rings are or not destroyed or completely slack; on the other hand non segmented motors have also given rise to a same type of instability as vortex shedding : this is the case of one of the configuration (LP3D) of the LP3 set up presented in a preceding paper. As it is known ' ' , classical linear acoustic balance computations do not give reliable stability predictions in complex internal geometries (such as in the P230 motor), and then an effort was carried out to perform full numerical simulations of the unsteady, compressible internal flow fields. On the other hand, motor internal flows are mostly non-observable, and without numerical simulations it is not possible to describe the path of the aerodynamic instability development. The object of the present paper is to show hydrodynamic instabilities (which drive pressure oscillations) in configurations without shear flows induced by inhibitor rings, and to explain how these instabilities occur by means of numerical simulations. LP3D AND LP3E TEST CASES This work is part of the overall research effort, supported by CNES, accompanying the development of the Ariane 5 P230 MPS solid propellant motor (program ASSM for Aerodynamics of Segmented Solid Motors) and makes use of experimental results obtained during the combustion stability assessment program carried out for BPD and CNES, by delegation of ESA. In this scope, it is the continuation of earlier works about numerical simulations in solid propellant rocket motors ". The Ariane 5 motor, as other large segmented motors (U.S. Space Shuttle and Titan SRM) has been reported to exhibit pressure and thrust oscillations. Until recently, it was believed that such instability was exclusively due to the segmented design : inhibitor rings induce shear layers and vortex shedding driven oscillations. Nevertheless it is * Research scientist, Energetics Dept. 1 Project manager, Energetics Dept., Member AIAA Copyright © 1996 by the American Institute of Aeronautics and Astronautics, Inc., All rights reserved. The test cases are based on two configurations of the LP3 motor. The LP3 motor, which has been presented in reference , is a simplified 1/15 subscale set-up of the Ariane 5 P230 motor used to test several inter-segment arrangements. The configurations of interest here, called LP3D and LP3E, have no prominent obstacles at the mid chamber point, see figure 1. Both configurations have a cylindrical chamber of length 1632 mm and inner diameter 203 mm. At ignition, the main propellant burning surface is cylindrical, inner diameter 90 mm, with a chamfered surface near the aft end. The supersonic outlet nozzle is submerged, with a throat diameter of 56.5 mm, then the total length of the motor is 1650 mm. LP3D and LP3E have a small forward segment of 225 mm length, cylindrical for LP3D (169.5

Parietal vortex shedding as a cause of instability for long solid propellant motors - Numerial simulations and comparisons with firing tests

The numerical solution of laminar, two-dimensional, compressible, and unsteady Navier-Stokes equations is aimed at a complete description of acoustic boundary layers that develop above a burning pro pell ant. Such acoustic boundary layers can be responsible for the so-called flow turning losses. They can govern the local unsteady flow conditions that are seen by the burning propellant to which it finally responds. In those respects, a complete understanding of such acoustic boundary layers is essential to improve existing solid rocket stability prediction codes. The full numerical solution of the Navier-Stokes equations incorporates into the analysis all the features of two-dimension al rocket chamber mean flowfield in a natural manner. After a standing wave pattern is established through forcing at a given frequency, a special Fourier treatment is used to transform the numerical results in a form directly comparable to available linear acoustic data. The presented results indicate that the acoustic boundary layer is substantially thinner than predicted by simplified models. Moreover, its acoustic admittance is found to vary significantly along the chamber, a result that is of major importance to stability predictions. Finally, the acoustic field is found to be rotational over a significant volume of the chamber.

Francois Vuillot

Papers

Vortex-Shedding Phenomena in Solid Rocket Motors

Instabilities and pressure oscillations in solid rocket motors

Parietal vortex shedding as a cause of instability for long solid propellant motors - Numerial simulations and comparisons with firing tests

Acoustic Boundary Layers in Solid Propellant Rocket Motors Using Navier-Stokes Equations

Investigation of integral surface formulations for acoustic post-processing of unsteady aerodynamic jet simulations