Stability and receptivity of the swept-wing attachment-line boundary layer: a multigrid numerical approach Gianluca Meneghello To cite this version: Gianluca Meneghello. Stability and receptivity of the swept-wing attachment-line boundary layer: a multigrid numerical approach. Fluid mechanics [physics.class-ph]. Ecole Polytechnique X, 2013. English. ￿NNT: ￿. ￿pastel-00795543￿ HAL Id: pastel-00795543 https://pastel.archives-ouvertes.fr/pastel-00795543 Submitted on 28 Feb 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Gianluca Meneghello [email protected] Stability and Receptivity of the Swept-Wing Attachment-Line Boundary Layer: a Multigrid Numerical Approach (with 51 figures) Under the supervision of Prof. Peter J. Schmid and Prof. Patrick Huerre LadHyX - Ecole Polytechnique Defended on February 15, 2013 On the cover: a multigrid approach to Dante’s “Divina Commedia” A mio padre, per aver sempre trovato il tempo di insegnarmi cose nuove Nel mezzo del cammin di nostra vita mi ritrovai per una selva oscura ché la diritta via era smarrita. Ahi quanto a dir qual era è cosa dura esta selva selvaggia e aspra e forte che nel pensier rinova la paura! Tant’è amara che poco è più morte; ma per trattar del ben ch’i’ vi trovai, dirò de l’altre cose ch’i’ v’ho scorte. Dante Alighieri — Inferno, Canto I Contents 1 Introduction and Motivation 1 2 Problem Definition 11 2.1 The governing equations for the base flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 The governing equations for the perturbations . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 The adjoint governing equations for the perturbations . . . . . . . . . . . . . . . . . . . . 16 2.4 Receptivity and sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Numerical Approach 25 3.1 The pressure form of the Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Computational domain and discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4 Considerations on domain size and grid size . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Multigrid 37 4.1 A generic iterative solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Ingredients of a multigrid solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3 Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Correction Scheme - the linear equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.5 Full Approximation Scheme - the non-linear equation. . . . . . . . . . . . . . . . . . . . . 61 4.6 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5 Global analysis 69 5.1 Base flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2 Global analysis of the direct operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.3 Adjoint field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.4 The wavemaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6 Conclusions and Perspectives 93 A Matlab multigrid code 99 vii viii Contents List of Figures 1.1 The effect of nonalignment of the pressure gradient . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Transition mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Swept attachment-line boundary-layer instabilities . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Receptivity and sensitivity analysis flow chart . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Sketch of the geometry used, with reference systems . . . . . . . . . . . . . . . . . . . . . 12 2.2 Possible routes to obtain the adjoint equations . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Velocity field for the flow around a cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Direct and adjoint fields for a cylinder wake . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Wavemaker for a cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1 Solid boundary control cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Conformal mapping from a rectangle to the Joukowsky profile . . . . . . . . . . . . . . . . 29 3.3 The control volume for the finite-volume formulation of the Laplacian . . . . . . . . . . . 31 3.4 Validation of the discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.5 Conformal mapping from the rectangle to the circle. . . . . . . . . . . . . . . . . . . . . . 33 3.6 Pressure coefficient on the cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.7 Domain for base flow computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.8 Zoom of the grid used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.1 The grid hierarchy used in the multigrid process . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Amplification factor for Lexicographic Gauss-Seidel, Poisson equation . . . . . . . . . . . 47 4.3 Amplification factor for non-Lexicographic Gauss-Seidel, Poisson equation . . . . . . . . . 48 4.4 Amplification factor for downstream Gauss-Seidel, convection-diffusion equation. . . . . . 51 4.5 Amplification factor for upstream Gauss-Seidel, convection-diffusion equation . . . . . . . 52 4.6 Amplification factor for pointwise Gauss-Seidel, anisotropic Poisson equation . . . . . . . 54 4.7 Amplification factor for linewise Gauss-Seidel, anisotropic Poisson equation . . . . . . . . 55 4.8 The V-cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.9 The FMG algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.10 The FV-cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.11 Test 1: Convergence history of the residual, Poisson equation . . . . . . . . . . . . . . . . 65 4.12 Test 2: Convergence history of the residual, anisotropic Poisson equation. . . . . . . . . . 66 4.13 Test 3: Convergence history of the residual, Neumann boundary condition . . . . . . . . . 67 4.14 Test 4: Convergence history of the residual, convection-diffusion equation . . . . . . . . . 68 5.1 Multigrid convergence behavior for the computation of base flow . . . . . . . . . . . . . . 72 5.2 Base flow streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.3 Velocity field within the boundary layer of a swept wing . . . . . . . . . . . . . . . . . . . 75 ix