Eigenmodes of Ducted Flows With Radially-Dependent Axial and Swirl Velocity Components

This report characterizes the sets of small disturbances possible in cylindrical and annular ducts with mean flow whose axial and tangential components vary arbitrarily with radius. The linearized equations of motion are presented and discussed, and then exponential forms for the axial, circumferential, and time dependencies of any unsteady disturbances are assumed. The resultant equations form a generalized eigenvalue problem, the solution of which yields the axial wavenumbers and radial mode shapes of the unsteady disturbances. Two numerical discretizations are applied to the system of equations: (1) a spectral collocation technique based on Chebyshev polynomial expansions on the Gauss-Lobatto points, and (2) second and fourth order finite differences on uniform grids. The discretized equations are solved using a standard eigensystem package employing the QR algorithm. The eigenvalues fall into two primary categories: a discrete set (analogous to the acoustic modes found in uniform mean flows) and a continuous band (analogous to convected disturbances in uniform mean flows) where the phase velocities of the disturbances correspond to the local mean flow velocities. Sample mode shapes and eigensystem distributions are presented for both sheared axial and swirling flows. The physics of swirling flows is examined with reference to hydrodynamic stability and completeness of the eigensystem expansions. The effect of assuming exponential dependence in the axial direction is discussed

Accuracy Enhancement for Higher Derivatives using Chebyshev Collocation and a Mapping Technique

We study a new method in reducing the roundoff error in computing derivatives using Chebyshev collocation methods. By using a grid mapping derived by Kosloff and Tal-Ezer, and the proper choice of the parameter ff, the roundoff error of the k-th derivative can be reduced from O(N 2k ) to O((N jln fflj) k ), where ffl is the machine precision and N is the number of collocation points. This drastic reduction of roundoff error makes mapped Chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large N . We also study several other aspects of the mapped Chebyshev differentiation matrix. We find that 1) the mapped Chebyshev methods requires much less than ß points to resolve a wave, 2) the eigenvalues are less sensitive to perturbation by roundoff error, and 3) larger time steps can be used for solving PDEs. All these advantages of the mapped Chebyshev methods can be achieved while maintaining spectral accuracy. 1 Introduction In [5], we..

Étude de stabilité et simulation numérique de l'écoulement interne des moteurs à propergol solide simplifiés

Cette thèse vise à modéliser les instabilités hydrodynamiques générant des détachements tourbillonnaires pariétaux (ou VSP) responsables des Oscillations De Pression dans les moteurs à propergol solide longs et segmentés par interaction avec l acoustique du moteur. Ces instabilités sont modélisées en tant que modes de stabilité linéaire globaux de l écoulement d un conduit à parois débitantes. En supposant que les structures pariétales émergent d une perturbation de l écoulement de base, des modes discrets et indépendants du maillage utilisé sont calculés. Dans ce but, une discrétisation par collocation spectrale multi-domaine est implémentée dans un solveur parallèle afin de s affranchir de la croissance polynomiale des fonctions propres et de la présence de couches limites. Les valeurs propres ainsi calculées dépendent explicitement des frontières du domaine, à savoir la position de la perturbation et celle de la sortie, et sont ensuite validées par simulation numérique directe. On montre alors qu elles permettent bien de décrire la réponse à une perturbation initiale de l écoulement modifié par une rupture de débit pariétale. Ensuite, la simulation d une réponse forcée par l acoustique se fait sous forme de structures tourbillonnaires dont les fréquences discrètes sont en accord avec celles des modes de stabilité. Ces structures sont réfléchies en ondes de pression de même fréquences remontant l écoulement. Finalement, la simulation numérique et la théorie de la stabilité permettent de montrer que le VSP, dont la réponse est linéaire vis-à-vis d un forçage compressible comme l acoustique, est le phénomène moteur des Oscillations De Pression.The current work deals with the modeling of the hydrodynamic instabilities that play a major role in the triggering of the Pressure Oscillations occurring in large segmented solid rocket motors. These instabilities are responsible for the emergence of Parietal Vortex Shedding (PVS) and they interact with the boosters acoustics. They are first modeled as eigenmodes of the internal steady flowfield of a cylindrical duct with sidewall injection within the global linear stability theory framework. Assuming that the related parietal structures emerge from a baseflow disturbance, discrete meshindependant eigenmodes are computed. In this purpose, a multi-domain spectral collocation technique is implemented in a parallel solver to tackle numerical issues such as the eigenfunctions polynomial axial amplification and the existence of boundary layers. The resulting eigenvalues explicitly depend on the location of the boundaries, namely those of the baseflow disturbance and the duct exit, and are then validated by performing Direct Numerical Simulations. First, they successfully describe flow response to an initial disturbance with sidewall velocity injection break. Then, the simulated forced response to acoustics consists in vortical structures wihich discrete frequencies that are in good agreement with those of the eigenmodes. These structures are reflected into upstream pressure waves with identical frequencies. Finally, the PVS, which response to a compressible forcing such as the acoustic one is linear, is understood as the driving phenomenon of the Pressure Oscillations thanks to both numerical simulation and stability theory.TOULOUSE-ISAE (315552318) / SudocSudocFranceF