Adjoint-based aerodynamic design optimisation in hypersonic flow

Abstract

The drive for improved flexibility and reusability in satellite launch systems has brought a resurgence in the popularity of scramjet-powered access-to-space launch vehicle concepts. A challenge in scramjet-powered accelerator vehicles is achieving positive thrust margins at the high-speed end of their flight envelopes. As a consequence, scramjet-powered vehicles typically rely on highly integrated airframe-engine configurations. However, due to the tight integration, the geometries and the flow physics are complex. An automated optimisation method seems an excellent candidate approach to explore the complex design space of hypersonic vehicles. This thesis focuses on the development and application of a particular optimisation method, adjoint-based optimisation, to aid in efficient aerodynamic design in hypersonic flows.Shape optimisation of scramjet-powered vehicles requires many design parameters to capture the geometric detail, and, since the flow physics is complex, the fidelity of Reynolds-Averaged Navier-Stokes (RANS) analyses is desirable. The combination of many design parameters and an expensive objective function evaluation drives the need for gradient evaluation methods that scale well with the number of design variables. An advantage of the adjoint method is that all shape sensitivities for an objective function are evaluated at the cost of only one flow solution and one adjoint solution. As a result of its efficiency, the adjoint method has become widely used for aircraft design, evolving to the design optimisation of full configurations. Despite the wide use of adjoint methods in aircraft optimisation, there has been very little application to hypersonic vehicle design.Several high-speed adjoint solvers have been reported in the literature, however, a majority of these works have only verified the adjoint sensitivities, few have followed on to demonstrate the method for use in design optimisation. The contribution of this work is the description and application of a discrete adjoint solver in high-speed compressible flow optimisation.What is unique in this work is a demonstration that complex-step differentiation works well to linearise a second-order spatially accurate unstructured RANS solver. In particular, the approach presented in this work utilises the k − ω turbulence model in high-speed ducted flow configurations.As part of this work, to provide flow analysis with a rapid turn-around, an unstructured steady-state RANS solver driven by a Jacobian-Free Newton-Krylov method was developed. Turbulence is modelled using the two-equation k − ω turbulence model. The Newton method is globalised by using the pseudo-transient approach. A restarted GMRES method is used to solve the system of linear equations arising when solving for the Newton steps. Evaluation of the matrix-vector products required in the GMRES algorithm is accomplished by Frechet derivatives using imaginary perturbations in the complex plane. This is necessary to achieve robust convergence of the types of turbulent hypersonic flows considered in this work. Equation scaling ensures that the linear solver provides an adequate solution of the linear system, especially for turbulent flows, where the flow and turbulence variables differ by several orders of magnitude. Incomplete lower-upper preconditioning with zero-fill is used to accelerate linear system convergence. To achieve adequate residual convergence for flows with embedded shocks, limiter freezing is required to prevent early stall. The developed flow solver is verified using several methods, including the Method of Manufactured Solutions.Several validation cases from the literature are presented to establish the appropriateness of the implemented physical models for design analysis in high-speed flow.An accompanying discrete adjoint solver was developed to provide efficient computation of the required shape sensitivities. The primary complication of linearising the flow solver routines is handled via a complex-step derivative approach. Targeted differentiation is employed to provide an efficient means of constructing the adjoint operator. The adjoint gradients are verified against a complex variable direct-differentiation method.The flow and adjoint solvers are coupled to the open-source optimisation library, DAKOTA, to perform design optimisation in high-speed flows. Design surfaces are parameterised using Bezier curves and mesh deformation is achieved by the inverse distance weighting method. Two optimisation applications are presented here as a demonstration of the development work: (a) wave drag minimisation of an axisymmetric body; and (b) hypersonic inlet design optimisation. For the case of axisymmetric bodies, the optimal shape determined by this work compares favourably to several minimum-drag power-law bodies published in the literature. The second application is the redesign of the P2 hypersonic inlet. The chosen objective function aimed at removing the reflected cowl shock whilst obtaining the desired compression ratio. The results presented show that the optimiser has removed the reflected shock while achieving the desired compression ratio, at no cost to the inlet performance metrics.The conclusions are that the developed discrete adjoint-based optimisation framework does work well in a hypersonic flow context and that the use of complex-step differentiation is a key enabler in the implementation. In particular, the inlet example presented in this work demonstrates the efficiency, accuracy, and applicability of discrete adjoint-based optimisation to design analysis in turbulent high-speed flow.