The Continuous Adjoint Method in Aerodynamic and Conjugate Heat Transfer Shape Optimization, for Turbulent Flows

Abstract

The goal of this PhD thesis is to further develop continuous adjoint-based shape optimization methods of high accuracy and low computational cost, able to deal with one- (fluid mechanics) or many-discipline problems (two disciplines: Conjugate Heat Transfer, CHT). In addition, adjoint-based methods for tracing the Pareto front in multi-objective optimization are proposed and assessed. Finally, gradient-based methods for the analysis and optimization under uncertainties in CHT problems, based on the intrusive variant of the Polynomial Chaos Expansion method are developed, programmed and applied. To facilitate the above methods, continuous adjoint methods, developed in previous PhD theses for incompress- ible fluid flows [174, 125, 70] and successfully applied in industrial aero- and hydrodynamic shape optimizations, are enriched and extended. This PhD thesis is structured along the following four axes: The first axis is about the proper treatment of grid sensitivities i.e. grid nodes variations w.r.t. the design variables, when developing the continuous adjoint method. Internal grid displacement was initially considered in the computa- tion of sensitivity derivatives (SDs) in which grid sensitivities appear within Field Integrals; this method is abbreviated as the FI adjoint. The computation of the grid sensitivities in the interior of the domain requires as many grid displace- ments as the number of the design variables. To avoid this costly computation, in [70], the first continuous adjoint method involving the formulation and so- lution of adjoint grid displacement equations was presented. The so-computed SDs consisted solely of Surface Integrals and, thus, this method is abbreviated as the E-SI (Enhanced-SI) adjoint. ‘‘Enhanced’’ is used to distinguish this method from frequently used adjoint formulations also leading to SDs with surface in- tegrals, in which the corresponding terms are omitted (Severed SI adjoint). In [70], to derive the E-SI adjoint, grid displacement was assumed to be governed by Laplace PDEs, since this was convenient for developing the adjoint method to the grid displacement model (GDM). In this PhD thesis, this assumption is assessed, by taking into account that, in many cases, GDMs other than Laplace PDEs are used. Comparisons between SDs computed for commonly used GDMs (volumetric B-Splines, Delaunay Graph, Inverse Distance Weighting) and the Laplace PDEs conclude that the choice of the GDM has negligible effect on the SDs. This is a relieving finding since there is no need to reformulate the E-SI adjoint, if different GDMs are in use. The second axis of this thesis concerns the extension of the E-SI and FI con- tinuous adjoint methods, developed in previously completed PhD theses in the PCOpt/NTUA for pure flow problems by considering eddy viscosity variations, to CHT problems. In the latter, except for the fluid flow, the energy equation in the fluid and the heat conduction in the solid, are simulated by also accounting for interactions along their interface. To derive the adjoint method in an exact way, the turbulence model PDEs (in this thesis the Spalart–Allmaras is used) are differen- tiated. Through numerical investigations, it is recorfirmed also in CHT problems that the ‘‘frozen turbulence’’ assumption, i.e. any approach that neglects the eddy viscosity variation, during the formulation of the adjoint equations, mitigates the accuracy of SDs. In addition, the proper treatment of grid sensitivities is extended to CHT problems as well. Laplace PDEs are assumed to govern the internal grid displacement of the fluid and solid grids. Adjoint grid displacement equations are derived for the solid domain and those for the fluid, acquired for pure fluid flows, are extended by terms emanating from the differentiation of the fluid energy equa- tion. A comparison between the E-SI and FI methods reconfirm the computational gain from the use of the E-SI adjoint, this time in CHT problems. Also, it is shown that neglecting the internal grid displacement, by using the Severed SI method, damages the SDs accuracy in CHT problems. Studies similar to those performed for pure fluid flows demonstrate that, the choice of the GDM has negligible effect on the SDs, in CHT problems too. In the third axis of this thesis, adjoint methods for pure fluid flows and CHT problems are used to assist gradient-based methods tracing the front of non- dominated solutions, a.k.a. the Pareto front, in multi-objective optimization prob- lems. In this PhD thesis, a prediction-correction algorithm (in some variants) trac- ing the Pareto front is developed. The prediction-correction variants are initialized by a point on the front, obtained by carrying out a single objective optimization for one of the objectives only. In the prediction and correction steps, different systems of equations are derived by treating the Karush-Kuhn-Tucker optimality conditions in two different ways. The costly computation of the exact Hessian matrix of the objective functions, which appear in the equations solved to update the design variables, is avoided. Instead, two alternative approaches are used: (a) the computation of Hessian-vector products driving a Krylov subspace solver and (b) the approximation of the Hessian via the BFGS method. To compute Hessian- vector products at the lowest cost possible, new systems of equations are derived by applying the Direct Differentiation method on the primal and adjoint PDEs. Between the two approaches, the one approximating the Hessian matrices in the prediction step is shown to be the less expensive. It is also demonstrated that omitting the prediction step increases the cost of tracing the Pareto front. Further comparisons of the less expensive prediction-correction variant and weighted-sum approaches for computing Pareto fronts demonstrate the superiority of the former in terms of computational cost. The last axis of this thesis regards shape optimizations with the presence of environmental and manufacturing uncertainties for CHT problems, based on the intrusive Polynomial Chaos Expansion (iPCE). Primal fields in the correspond- ing equations and boundary conditions for a single operation point (deterministic equations) are expanded by using weighted sums of orthogonal polynomials for chaos order equal to 2. Adjoint equations are then derived by differentiating the iPCE primal PDEs. The new systems of equations are solved to compute statistical moments and their derivatives at low cost. To verify the accuracy of the computed moments, comparisons are made between the iPCE method and more computa- tionally expensive ones, such as the non-intrusive PCE (niPCE) and Monte Carlo. Problems, in which uncertainties are related to the flow boundary conditions along the inlet of the fluid domain and the insulation thickness between fluids and solids, are investigated. These uncertainties are considered by performing expansions of the inlet and Fluid-Solid Interface conditions, leading to new con- ditions for the PCE coefficients. Case studies regarding both pure fluid flow and CHT problems, are presented. Developed methods are first validated/tested in 2D problems with pure fluid flows around isolated airfoils or inside ducts and CHT problems with a duct attached to a solid body. Regarding pure fluid flow problems, multi-objective optimizations for isolated airfoils, with contradictory goals being the exerted lift and drag forces, are performed. Regarding CHT problems, an internally cooled turbine cascade vane is optimized targeting min. mean solid temperature. To do so, its contour shape and the positions of the internal cooling holes are modified. The shape of the 2D cooling channel is also optimized for min. total pressure losses subjected to constraints which prevent the solid body from overheating. In addition, the cooling efficiency of a 3D internal cooling system is optimized. The cooling channel is redesigned for two different objective functions: the heat flux absorbed by the coolant and the part of the solid body volume with the highest temperature values. Moreover, shape optimizations are performed for a car-engine cylinder head. The internal cooling channel immersed in the engine block is redesigned for two different targets: min. max. solid temperature values and min. total pressure losses in the coolant. Finally, a multi-objective optimization is performed in a case in which cooling fins are attached to a heated solid body. In specific, a Pareto front is computed, with the same objective functions as in the optimization of the car engine cylinder-head. The necessary software has been programmed in the open-source CFD toolbox OpenFOAM © , which provides a cell-centered, collocated, finite-volume infrastruc- ture for discretizing PDEs and solving linearized systems of equations. The research work was supported by the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), under the HFRI PhD Fellowship grant (GA.no. 1796).