Aerospace Science and Technology 44 (2015) 125–134 Contents lists available at ScienceDirect Aerospace Science and Technology www.elsevier.com/locate/aescte Compressible modes in a square lid-driven cavity Leandro F. Bergamo a,∗,1 , Elmer M. Gennaro b , Vassilis Theofilis c , Marcello A.F. Medeiros a,∗ a São Carlos School of Engineering – University of São Paulo, Department of Aeronautical Engineering, Av. Trabalhador São Carlense, 400, São Carlos, Brazil b São Paulo State University – UNESP, Campus de São João da Boa Vista, Brazil c School of Aeronautics, Universidad Politécnica de Madrid, Madrid, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 13 March 2014 Received in revised form 13 March 2015 Accepted 14 March 2015 Available online 25 March 2015 Keywords: Compressible flow Global linear stability analysis Compressible global linear instability Cavity flow This paper studies the effect of compressibility on the linear stability of a two-dimensional lid-driven cavity flow in the subsonic regime. The base flow is generated by high fidelity direct numerical simulation and a biglobal mode instability analysis is carried out by a matrix forming approach. The eigenvalue problem is discretized by high-order finite differences and Arnoldi algorithm is used to reduce the size of the problem. The solution procedure uses sparse matrix techniques. Influence of Mach number on the modes known from incompressible calculations is presented, showing that compressibility has a stabilizing effect. New modes that appear only for compressible flows are presented and their relationship with duct acoustics is investigated.  2015 Elsevier Masson SAS. All rights reserved. 1. Introduction Linear stability plays a role in the transition from a laminar to a turbulent state. The classical linear stability theory deals with par- allel, or almost parallel flows, such as shear layers and boundary layers, yielding a one-dimensional eigenvalue problem. If the flow is non-parallel, but still two-dimensional, one can assume period- icity in the third direction and the stability analysis can be cast into a two-dimensional eigenvalue problem. Often, this approach is referred to as a biglobal, or global, linear stability analysis [1,2], and is a natural extension of the classical linear stability theory. The lid-driven cavity (LDC) flow is amenable to biglobal lin- ear stability analysis and various studies investigated this flow in the incompressible limit [3–5]. In these studies, a critical Reynolds number of ≃ 8000 was obtained for two-dimensional disturbances in a square lid-driven cavity. It was also found that three-dimensional disturbances become unstable much earlier, at Re ≃ 782 [5]. The most unstable three-dimensional mode is sta- tionary, followed by three-dimensional travelling modes that also become unstable. The compressibility effects on the LDC flow has not received much attention. It may be said that those effects are not very important for this problem in comparison with open cavity. Nev- ertheless there are some aspects that are interesting to examine. Flows are generally compressible, and in many of them the com- * Corresponding author. E-mail address: leandro.bergamo@usp.br (L.F. Bergamo). 1 Tel.: +55 16 3373 8285; fax: +55 16 3373 8346. pressibility has stabilizing effects. The boundary layer is a classical example. Nevertheless this is not true for all flows, the attach- ment line boundary layer is a counter example [6]. It is important to verify this issue in the cavity flows. Moreover, acoustic waves propagating in ducts are a common application, the typical case of sound waves under confinement. If a duct wall is moving, it can have an effect on such acoustic waves. Such waves are embedded in our analysis. It may also be that new unstable modes arise due to compressibility, even thought there is no previous indication of that, or that other stable compressible mode can be identified that have physical importance. All these aspects are addressed in the current paper. It would certainly be important to extend such analysis to open cavity flows in the compressible regime and the current paper may constitute a step in this direction. As opposed to the open cavity, the LDC flow has well defined boundary conditions and permits a more rigorous assessment of the quality of the numerical proce- dures presented here. The current work uses the so called matrix forming approach for biglobal instability analysis [1,2]. This approach can lead to eigenvalue problems of very large matrices. Memory becomes a bottleneck, but the use of finite differences for spatial discretiza- tion can increase the sparsity of the associated matrix. Moreover, the Krylov subspace techniques also reduce the memory needed [7,8,6], even though an LU decomposition is required. Here, these operations were performed by using a multi-frontal solver [9]. This is demonstrated by [10] and [6] to be an efficient alternative to the massive parallelization of the corresponding dense solution ap- proach used by [11]. http://dx.doi.org/10.1016/j.ast.2015.03.010 1270-9638/ 2015 Elsevier Masson SAS. All rights reserved.