## Final Activity Report Summary - SHAPEOPT (Aerodynamic shape optimization for minimum transient growth in compressible flow)

Industrial, economical and environmental interests are at stake in the research efforts concerning the optimisation of the shape of aircraft wings so to obtain low aerodynamic drag. The total drag of an aircraft wing is mainly given by the sum of pressure drag, related to shock waves, and viscous drag, whose magnitude depends on whether the flow is laminar or turbulent. Turbulent flow produces a much larger drag; thus important research efforts have been devoted to keeping the flow laminar over the largest possible portion of the wing surface. Transition from laminar to turbulent flow in the boundary layer on an aircraft wing is usually caused by small perturbations which grow as they propagate downstream. These perturbations undergo exponential growth which can be modelled mathematically using normal modes to capture the local behaviour, or a multiple scales (MSC) approach to include non-parallel effects of the spatially growing boundary layer. When the perturbation amplitude is sufficiently large, non-modal growth can lead to the so called bypass transition, which is not associated with exponential instabilities. It has long been known that in pipe flow transition occurs even before any local modes become unstable.

Two main objectives have characterised this project. The first was to increase the fundamental understanding of non-modal growth as a scenario for laminar/turbulent transition in three-dimensional boundary layer flow. The second objective was to incorporate non-modal growth as a transition prediction method and perform shape optimisation to obtain slender bodies with low drag. Since non-parallel effects of the growing boundary layer are to be considered when modelling non-modal growth, the starting point of the first objective was to use the MSC approach. The investigation was concentrated on the non-parallel effects, whose correction in the MSC approach can be written as a function of the left, and right eigen functions, and the inverse of the distance between the discrete modes, obtained from a normal mode analysis.

One of the main conclusions was that the correction of the eigen functions in the MSC approach cannot be evaluated using only the discrete modes which means that also the continuous spectrum somehow must be incorporated. In order to avoid working with the solution of the modal analysis as a sum of discrete modes, and a continuous spectrum it was decided to investigate if also the latter can be written as a sum of discrete modes. The idea of the second objective was to use gradient based optimisation with the aim of minimising an objective function, based on a measure of the non-modal growth, by changing the shape (geometry). The gradient of the objective function with respect to a design parameter representing the surface of the geometry, is efficiently evaluated using adjoint equations. Due to contacts established with The Swedish Defence Research Agency, FOI, Aeronautics Division, programs solving the state and adjoint equations describing inviscid compressible flow are available and can be coupled with the boundary layer equations, and its corresponding adjoint equations in order to perform shape optimisation. Some first results by FOI of shape optimisation have been obtained for two-dimensional flow at transonic flow conditions with the aim of minimising exponentially growing perturbations.

A new line of research was initiated with the analysis of the representation of the continuous spectrum. This was made with ideas from the theory of optical waveguides, and fibre optics, and it was found that the discrete representation of the continuous spectrum are modes which resembles what in optics are called 'leaky waves', ie. waves that are attenuated in the direction of the wave-guide, while they grow unbounded in a direction perpendicular to it. The first application was to compute the correction in the MSC approach. However, such a representation can have many other interesting applications and could help the understanding of 'higher' modes (discrete modes other than the least stable one). It will, in particular, be the starting point in now commencing work.

Two main objectives have characterised this project. The first was to increase the fundamental understanding of non-modal growth as a scenario for laminar/turbulent transition in three-dimensional boundary layer flow. The second objective was to incorporate non-modal growth as a transition prediction method and perform shape optimisation to obtain slender bodies with low drag. Since non-parallel effects of the growing boundary layer are to be considered when modelling non-modal growth, the starting point of the first objective was to use the MSC approach. The investigation was concentrated on the non-parallel effects, whose correction in the MSC approach can be written as a function of the left, and right eigen functions, and the inverse of the distance between the discrete modes, obtained from a normal mode analysis.

One of the main conclusions was that the correction of the eigen functions in the MSC approach cannot be evaluated using only the discrete modes which means that also the continuous spectrum somehow must be incorporated. In order to avoid working with the solution of the modal analysis as a sum of discrete modes, and a continuous spectrum it was decided to investigate if also the latter can be written as a sum of discrete modes. The idea of the second objective was to use gradient based optimisation with the aim of minimising an objective function, based on a measure of the non-modal growth, by changing the shape (geometry). The gradient of the objective function with respect to a design parameter representing the surface of the geometry, is efficiently evaluated using adjoint equations. Due to contacts established with The Swedish Defence Research Agency, FOI, Aeronautics Division, programs solving the state and adjoint equations describing inviscid compressible flow are available and can be coupled with the boundary layer equations, and its corresponding adjoint equations in order to perform shape optimisation. Some first results by FOI of shape optimisation have been obtained for two-dimensional flow at transonic flow conditions with the aim of minimising exponentially growing perturbations.

A new line of research was initiated with the analysis of the representation of the continuous spectrum. This was made with ideas from the theory of optical waveguides, and fibre optics, and it was found that the discrete representation of the continuous spectrum are modes which resembles what in optics are called 'leaky waves', ie. waves that are attenuated in the direction of the wave-guide, while they grow unbounded in a direction perpendicular to it. The first application was to compute the correction in the MSC approach. However, such a representation can have many other interesting applications and could help the understanding of 'higher' modes (discrete modes other than the least stable one). It will, in particular, be the starting point in now commencing work.