661 | Mai 2012 SCHRIFTENREIHE SCHIFFBAU Arthur Stück Adjoint Navier–Stokes Methods for Hydrodynamic Shape Optimisation Adjoint Navier–Stokes Methods for Hydrodynamic Shape Optimisation Vom Promotionsausschuss der Technischen Universität Hamburg-Harburg zur Erlangung des akademischen Grades Doktor-Ingenieur (Dr.-Ing.) genehmigte Dissertation von Arthur Stück aus Schleswig 2011 1. Gutachter: Prof. Dr.-Ing. Thomas Rung 2. Gutachter: Prof. Dr. Rainald Löhner Tag der mündlichen Prüfung: 16. November 2011 Bericht Nr. 661 der Schriftenreihe Schiffbau der Technischen Universität Hamburg-Harburg Schwarzenbergstraße 95c D-21073 Hamburg ISBN 978-3-89220-661-3 Abstract Adjoint Navier–Stokes methods are presented for incompressible flow. The sensitivity derivative of scalar hydrodynamic objective functionals with respect to the shape is ob- tained at the cost of one flow-field computation, so that the numerical effort is practically independent of the number of shape parameters. The adjoint Navier–Stokes problem was derived on the level of partial differential equations (PDE) first. According to the frozen-turbulence assumption, variations of the turbulence field with respect to the shape control were neglected in the adjoint analysis. Second, consistent discretisation schemes were derived for the individual terms of the adjoint PDE on the basis of the primal, un- structured finite-volume discretisation. A unified, discrete formulation for the adjoint wall boundary condition and the sensitivity equation is presented that supports both low- and high-Reynolds number boundary treatments. The segregated pressure-correction scheme used to solve the primal problem was also pursued in the adjoint code. Reusing huge por- tions of the flow code led to a compact adjoint module, reduced coding effort and yielded a consistent implementation. Analytical adjoint solutions were tailored to validate the adjoint approach. Moreover, the adjoint-based sensitivity derivative was verified against the direct-differentiation method for both internal and external flow cases. The adjoint method was used for ship-hydrodynamic design optimisation to compute the sensitivity derivative of a wake objective functional with respect to the hull shape. The sensitiv- ity map yields considerable insight into the design problem from the objective point of view. The adjoint-based sensitivity analysis was carried out at model scale to support a manual redesign of the hull and led to an improved wake field. An explicit, filtering- based preconditioning of the sensitivity derivative is first-order equivalent to the concept of “Sobolev-smoothing”. Particularly the sensitivity derivative obtained in conjunction with the low-Reynolds number treatment of boundary walls needed to be filtered before applied to the design surfaces. Guided by the adjoint-based sensitivity derivative, auto- matic shape optimisation runs were performed for 2D and 3D internal flow problems to reduce the power loss. iv Acknowledgements It has been a pleasure to work at the Institute for Fluid Dynamics and Ship Theory at Hamburg University of Technology under the supervision of Prof. Dr.-Ing. Thomas Rung, who gave me a great deal of latitude for my studies in the field of viscous CFD methods for hydrodynamic shape optimisation. He was continuously interested in my work, spent much of his time on inspiring discussions and enthusiastically came up with many valuable ideas. I thank Prof. Dr. Rainald Löhner for giving me the great opportunity to join his team as a visiting researcher at George Mason University and for being second assessor. Thanks are due to Prof. Dr.-Ing. Otto v. Estorff for chairing the examination board. Prof. Dr.-Ing. Heinrich Söding has probably initialised this study by confronting me with the “adjoint” technique in the first place—without ever using the term. I am very grateful for the valuable discussions we have had. I was lucky to work in the core of a growing group of young researchers developing viscous CFD methods for shape optimisation in research projects with partners from academia and industry. I have greatly enjoyed the company and the help of my colleagues at the institute. In particular, I am very thankful to Jörn Kröger for help in geometry manipulation and mesh generation for the ship-hull optimisation study; and to Jörg Brunswig and Manuel Manzke always being helpful in the range of Fortran and FreSCo+. Finally, I thank Anne for having patience with me- and-the-time-consuming-CFD and, most importantly, for her continuous support beyond the CFD. vi Contents Abstract iii Acknowledgements v Nomenclature xi 1. Introduction 1 1.1. BackgroundandMotivation . 1 1.2. StartingPointandDirection . .... 3 1.3. PresentContributions . 4 1.4. PlanoftheThesis................................ 5 2. Calculation of Constrained Derivatives 7 2.1. FiniteDifferencing .............................. 8 2.2. Complex-Step Differentiation . 10 2.3. Continuous Differentiation . 10 2.4. DiscreteDifferentiation. 14 3. Review of Adjoint-based Shape Optimisation 19 3.1. Continuous and Discrete Differentiation . ....... 19 3.2. PrimalandAdjointAlgorithms . 21 3.3. GradientEvaluation .............................. 23 3.4. TurbulenceTreatment . 24 3.5. AccuracyandConsistency . 25 3.6. Integration into Shape Optimisation . ...... 27 3.7. HydrodynamicandMarineApplications . 28 4. Hydrodynamic Optimisation Problem 31 4.1. GoverningFieldEquations . 31 4.2. BoundaryConditions . 34 4.3. HydrodynamicObjectives . 35 5. Variation of the Optimisation Problem 39 5.1. Concept of Material Derivative . 39 5.2. Variation of the Objective Functional . ....... 40 5.3. Variation of the Navier–Stokes Equations . ...... 43 5.4. Direct Evaluation of Constrained Derivatives . ........ 45 viii Contents 6. Continuous Adjoint Navier–Stokes Concept 49 6.1. Augmented Objective Functional . 49 6.2. Adjoint Navier–Stokes Equations . 50 6.3. AdjointGradientEquation. 55 6.4. CompleteGradientEquation. 56 6.5. Example: Interior vs. Exterior Evaluation of Forces . ....... 57 7. Finite-Volume Discretisation 61 7.1. ScalarTransportEquation . 61 7.2. MomentumEquations ............................. 68 7.3. Pressure-CorrectionScheme . 70 7.4. BoundaryConditions . 73 7.5. SolutionAlgorithm ............................... 78 7.6. ObjectiveFunctionals. 80 7.7. Navier–Stokes Variation . 80 8. Adjoint Discretisation 87 8.1. ScalarTransportEquation . 88 8.2. MomentumEquations ............................. 92 8.3. Pressure-CorrectionScheme . 97 8.4. BoundaryConditions . 100 8.5. GradientEquation ...............................104 8.6. SolutionAlgorithm ............................... 105 9. Verification and Validation Studies 109 9.1. Analytic Solution for Axis-Symmetric Couette Flow . .......109 9.2. Direct-Differentiation Method . 117 10.CAD-free Geometry Concept for Shape Optimisation 137 10.1.EvaluationofMetricTerms . 137 10.2.HandlingofConstraints . 137 10.3. Filtering of Derivatives . 138 10.4.MeshAdaptation ................................141 11.Applications 143 11.1. Sensitivity Analysis for Manual Wake Optimisation . .......143 11.2.Optimisationofa2DT-Junction . 153 11.3. Optimisation of a 3D Double-Bent Pipe . 162 12.Outlook and Conclusions 171 A. Derivation of Adjoint Navier–Stokes Equations 175 A.1. Variation of the Navier–Stokes Equations . 175 A.2.IntegrationbyParts .............................. 175 Contents ix B. Advection Formulations ADV2/3 177 B.1. CartesianCoordinates . 177 B.2. Physical Cylinder Coordinates . 177 B.3. Comparison of Advection Schemes . 179 C. Differential Geometry 181 D. Two-Step Adjoint Pressure-Correction Scheme 183 x Contents Nomenclature The subsequent glossary of terms is not exhaustive. Einstein’s sum convention applies to small-type Latin subscripts unless declared differently. In symbolic notation, the number of underlines corresponds to the order of a tensor. Lower-case Greek αφ under-relaxation factor (0 < αφ < 1) β, β continuous or discrete (shape) control βφ factor for UDS-CDS blending (0 < βφ < 1) γφ diffusion coefficient for scalar variable φ δ variation operator δC convective variation operator (with respect to the position) δG geometric variation operator denoting partial variation with respect to metrics (surface area, boundary unit vector) δL local variation operator (with respect to the flow) δLC material variation operator, consisting of convective and local varia- tions (δLC = δL + δC ) δn boundary-normal position shift (δn = ni δxi) δx position shift δij Kronecker delta ε rate of dissipation of turbulent kinetic energy λ linear weighting factor for cell-face interpolation (0 <λ< 1) µ dynamic viscosity µeff effective viscosity (µ + µT ) µlog auxiliary definition of effective viscosity in terms of logarithmic law of the wall µT eddy viscosity ν kinematic viscosity π hydrodynamic stress tensor ρ density σε turbulent Prantl number for dissipation σk turbulent Prantl number for turbulent kinetic energy τ viscous portion of hydrodynamic stress tensor τ w wall shear stress magnitude φ′ turbulent fluctuations φ, φ continuous or discrete state variable ω specific dissipation rate or turbulent frequency xii Nomenclature Upper-case Greek Γ boundary of flow domain ΓD design surface(s) subject to the shape control (active boundaries) ΓO part of the boundary carrying the objective functional Ω interior flow domain ΩC part of the interior domain carrying the control ΩO part of the interior domain carrying the objective functional Lower-case Roman d vector connecting CV centres P and N (d = xN xP ) d∗ unit vector indicating the force component defined−