Tesis:
Spectral multi-domain methods for the global instability analisys of complex cavity flows.
- Autor: VICENTE BUENDIA, Francisco Javier de
- Título: Spectral multi-domain methods for the global instability analisys of complex cavity flows.
- Fecha: 2010
- Materia: Sin materia definida
- Escuela: E.T.S. DE INGENIEROS AERONAUTICOS
- Departamentos: MOTOPROPULSION Y TERMOFLUIDODINAMICA
- Acceso electrónico:
- Director/a 1º: THEOFILIS, Vassilios
- Director/a 2º: VALERO SANCHEZ,Eusebio
- Resumen: Our present concern is with devising flexible and accurate numerical methods for the solution of the partial-derivative eigenvalue problem (EVP) which governs BiGlobal linear instability of steady flows over complex geometries. BiGlobal instability analysis1 assumes spatial homogeneity in only one of the three spatial directions; the other two directions being resolved in a coupled manner. Consequently, this approach is appropriate for analysis of the temporal or spatial development of small-amplitude (modal or non-modal [74, 75]) perturbations, superimposed upon steady or time-periodic two-dimensional laminar basic states over complex geometries. Such states may comprise one, two, or all three velocity components, all of which may be two-dimensional functions of the resolved coordinates; see [86] for a reasonably recent review of this fast-growing field. Numerical solution of the BiGlobal EVP has been addressed successfully over the last two decades using a variety of approaches. From the point of view of solution methodology of the eigenvalue problem, one may distinguish between straightforward serial (see [86] for a review) or parallel [72, 45] implementations of a subspace iteration variant, or employment of the widely used [97, 42, 19, 27, 12, 5] implementation of the Implicitly Restarted Arnoldi Method (IRAM) in the ARPACK [50] software package for the solution of large scale eigenvalue problems. On the other hand, from the point of view of matrix formation, existing methodologies for the solution of the BiGlobal eigenvalue problem fall in one of two categories, one in which the matrix is formed, stored and processed using dense (serial or parallel) linear algebra technology, and another known as matrix-free/time-stepper algorithms [80, 6, 5, 2]. The early applications analyzed involved simple two-dimensional domains in which the numerical discretization techniques employed were straightforward extensions of those used in the solution of classic one-dimensional linear stability eigenvalue problems. The pioneering studies of inviscid instability of a vortex by Pierrehumbert [71] and viscous Encountered in the literature under different names, such as "Partial-derivative Eigenvalue Problem Stability analysis" [91], "Direct Stability Analysis" [6], or simply "Global" instability analysis [86, 17]", reflecting the novelty of the approach instability analyses in the wake of the circular cylinder by Zebib [98] and the rectangular duct by Tatsumi and Yoshimura [82] fall in this category; all three works employed spectral methods. Almost simultaneously, finite-element methods were also used for the solution of the BiGlobal EVP by Jackson [41] and Morzynski and Thiele [60]. Although finite-element methods are not restricted to the single-domain two-dimensional grids employed in the early spectral analyses, their low formal order of accuracy limits their resolution capabilities. Should sharp gradients need be resolved, as the case is with the amplitude functions of BiGlobal eigenmodes at increasingly high Reynolds numbers, one resorts to using unstructured meshes of ever-increasing density in order to achieve convergence [33]. In doing so, one effectively trades off the efficiency of a high-order method in favor of the flexibility offered by the unstructured mesh discretization. The case is thus set for high-order accurate, flexible and efficient numerical methods in order to solve the BiGlobal EVP. Such an approach has been introduced in the seminal work of three-dimensional instability in the wake of a circular cylinder by Barkley and Henderson [8, 37] in the form of spectral-element discretization on structured meshes. The first application of a spectral/hp—element method [43] to the study of a global instability problem on unstructured meshes was that of Theofilis, Barkley and Sherwin [88], who recovered instability in the wake of a NACA0012 airfoil as the leading BiGlobal eigenmode of the steady wake flow. While the aforementioned spectral/hp—element analyses utilized time-stepping concepts [95], Gonzalez, Theofilis and Sherwin [34] have recently discussed matrix formation and storage as an alternative technique for the solution of the same problem in the context of spectral/hp—element discretization. However, there exist reasonably complex geometries which are decomposable into rectangular subdomains and as such are are tailor-made for the application of multi-domain techniques for the spatial discretization of the governing equations. In an instability analysis context, accuracy is of primary significance, making spectral multi-domain [20, 21, 56] the method of choice in order to accomplish the task of solving the respective global instability problems. Flexibility is gained by permitting different discretizations (and consequent resolutions) within the different subdomains which, in the present two-dimensional eigenvalue problem context, leads to non-conforming subdomain interfaces; the present contribution discusses in-depth the associated numerical tools. Applications which can be tackled with the proposed methods include two classes of flow problems, firstly those in which resolution is selectively required in a specific area of the computational domain, and secondly flows in which the geometry of the domain considered naturally introduces multiple domains. The first class of problems includes those studied in the limiting case of local linear instability, a representative example in that case being resolution of the critical layer region in a compressible boundary- or mixing-layer flow [53, 55]. Archetypal problems representative of the second class include instability of flow over the backward- [7] and forward-facing steps [57] and the open cavity [89, 12]. Chapter one introduces the linear stability theory in the framework of the BiGlobal analysis and its reformulation as an eigenvalue problem. In the same section, the one-dimensional limit, associated with classic linear stability theory, is derived; its well-known results [64] will be used for validation purposes. In chapter three the two-dimensional incompressible Navier-Stokes equations are presented; they will later be integrated in order to obtain steady base flows, subsequently to be analyzed with respect to their three-dimensional instability. Section three introduces the non-conforming multi-domain spectral collocation methodology utilized to discretize both the basic flow and the instability analysis problems. In order to improve readability of this section, many of the technical details are presented as appendices. Implementational details concerning the sparsity pattern introduced by the spectral multi-domain spatial discretization of the BiGlobal EVP, as well as some details of the same problem arising in the context of the basic flow calculation, are discussed in section four. Section five includes results of the validation and verification work performed and concludes with the three-dimensional instability analysis of the two-dimensional L-shaped cavity [63] flow problem. The leading three-dimensional eigenmodes of the L-shaped cavity flow are recovered and the critical Reynolds number for laminar-turbulent transition of the two-dimensional flow is documented.