## Publications

76 results found

Arslan A, Fantuzzi G, Craske J,
et al., 2023, Rigorous scaling laws for internally heated convection at infinite Prandtl number, *Journal of Mathematical Physics*, Vol: 64, Pages: 1-24, ISSN: 0022-2488

We prove rigorous scaling laws for measures of the vertical heat transport enhancement in two models of convection driven by uniform internal heating at infinite Prandtl number. In the first model, a layer of incompressible fluid is bounded by horizontal plates held at the same constant temperature and convection reduces the fraction of the total dimensionless heat input per unit volume and time escaping the layer through the bottom boundary. We prove that this fraction decreases no faster than O(R−2), where R is a “flux” Rayleigh number quantifying the strength of the internal heating relative to diffusion. The second model, instead, has a perfectly insulating bottom boundary, so all heat must escape through the top one. In this case, we prove that the Nusselt number, defined as the ratio of the total-to-conductive vertical heat flux, grows no faster than O(R4). These power-law bounds improve on exponential results available for fluids with finite Prandtl number. The proof combines the background method with a minimum principle for the fluid’s temperature and with Hardy–Rellich inequalities to exploit the link between the vertical velocity and temperature available at infinite Prandtl number.

Gori F, Wynn A, Laizet S, 2023, Sensitivity of wind farm wake steering strategies to analytical wake models, Pages: 669-677

The aerodynamic interactions between wind turbines arranged in farm layout lead to annual energy production losses ranging from 10% to 30%. Wake steering represents a promising strategy in wind farm control for power loss mitigation. The purpose of this work is to assess the sensitivity of optimal wake steering strategies to both analytical wake model choice and optimisation parameters. Using the FLOw Redirection and Induction in Steady State (FLORIS) framework, different wake models are employed to optimise a 4 × 4 farm layout for power maximisation. Model comparison findings indicate significant discrepancies in absolute power predictions for optimal set-points, as well as in optimal decision variables, with different or even opposite optimal yaw angle settings. Initialisation sensitivity results show that solutions corresponding to local extrema lead to potential power losses up to 14% compared to the global maximum for power production. Moreover, wind farm power function is observed to be multi-modal and discontinuous, suggesting that care must be taken when using gradient-based methods in wake steering optimisation.

Diessner M, O'Connor J, Wynn A,
et al., 2022, Investigating Bayesian optimization for expensive-to-evaluate black box functions: Application in fluid dynamics, *FRONTIERS IN APPLIED MATHEMATICS AND STATISTICS*, Vol: 8

Goizueta N, Wynn A, Palacios R, 2022, Adaptive sampling for interpolation of reduced-order aeroelastic systems, *AIAA Journal: devoted to aerospace research and development*, Vol: 60, Pages: 6183-6202, ISSN: 0001-1452

A new strategy for the interpolation of parametric reduced-order models of dynamic aeroelastic systems is introduced. Its aim is to accelerate the numerical exploration of geometrically nonlinear aeroelastic systems over large design spaces or multiple flight conditions. The parametric reduced-order models are obtained from high-dimensional models by Krylov-subspace projection. They are subsequently interpolated to acquire realizations inexpensively anywhere in the parameter space, where having the state space as opposed to interpolating an output metric permits the use of linear analysis tools. The interpolation scheme is heavily conditioned by the available training points; thus, a novel methodology based on adaptive sampling and a combinatorial use of the available true-systems knowledge is presented, whereby we reuse all known data of the true models as different combinations of training and testing data to build statistical surrogates of the interpolation error across the parameter space and refine the sampling in those regions that need it. This minimizes the number of costly to evaluate function calls and ensures that parameter space regions are sampled according to the underlying system dynamics. The initial implementation of this adaptive sampling strategy is demonstrated on a very flexible wing with a complex stability envelope.

Arslan A, Fantuzzi G, Craske J, et al., 2022, Rigorous scaling laws for internally heated convection at infinite Prandtl number, Publisher: arXiv

New bounds are proven on the mean vertical convective heat transport, ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯, for uniform internally heated (IH) convection in the limit of infinite Prandtl number. For fluid in a horizontally-periodic layer between isothermal boundaries, we show that ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯≤12−cR−2, where R is a nondimensional `flux' Rayleigh number quantifying the strength of internal heating and c=216. Then, ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯=0 corresponds to vertical heat transport by conduction alone, while ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯>0 represents the enhancement of vertical heat transport upwards due to convective motion. If, instead, the lower boundary is a thermal insulator, then we obtain ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯≤12−cR−4, with c≈0.0107. This result implies that the Nusselt number Nu, defined as the ratio of the total-to-conductive heat transport, satisfies Nu≲R4. Both bounds are obtained by combining the background method with a minimum principle for the fluid's temperature and with Hardy--Rellich inequalities to exploit the link between the vertical velocity and temperature. In both cases, power-law dependence on R improves the previously best-known bounds, which, although valid at both infinite and finite Prandtl numbers, approach the uniform bound exponentially with R.

Kumar A, Arslan A, Fantuzzi G,
et al., 2022, Analytical bounds on the heat transport in internally heated convection, *Journal of Fluid Mechanics*, ISSN: 0022-1120

We obtain an analytical bound on the mean vertical convective heat flux$\langle w T \rangle$ between two parallel boundaries driven by uniforminternal heating. We consider two configurations, one with both boundaries heldat the same constant temperature, and the other one with a top boundary held atconstant temperature and a perfectly insulating bottom boundary. For the firstconfiguration, Arslan et al. (J. Fluid Mech. 919:A15, 2021) recently providednumerical evidence that Rayleigh-number-dependent corrections to the only knownrigorous bound $\langle w T \rangle \leq 1/2$ may be provable if the classicalbackground method is augmented with a minimum principle stating that thefluid's temperature is no smaller than that of the top boundary. Here, weconfirm this fact rigorously for both configurations by proving bounds on$\langle wT \rangle$ that approach $1/2$ exponentially from below as theRayleigh number is increased. The key to obtaining these bounds are innerboundary layers in the background fields with a particular inverse-powerscaling, which can be controlled in the spectral constraint using Hardy andRellich inequalities. These allow for qualitative improvements in the analysisnot available to standard constructions.

Muñoz-Simón A, Wynn A, Palacios R, 2022, Some modelling improvements for prediction of wind turbine rotor loads in turbulent wind, *Wind Energy*, Vol: 25, Pages: 333-353, ISSN: 1095-4244

This paper investigates the accuracy of three aerodynamic models to compute loads on wind turbine rotors under turbulent inflow: Blade Element Momentum (BEM); Unsteady Vortex LatticeMethod (UVLM) and Large Eddy Simulation with Actuator Line (LES-AL). Turbulent inflow conditions are numerically generated with a new approach that combines control of turbulence andrealistic velocity spectrum by using Mann boxes and LES simulations, respectively. Several deficiencies of the tested models are found and overcome through proposed improvements. First,the BEM assumption of independent radial sections does not hold in turbulent cases with longblades. Thus, a spatial filter to account for the interaction of radial sections in BEM is designedthrough the analysis of these interactions with UVLM. Second, the absence of viscous drag inUVLM is observed to lead to a very high rotor power coefficient, and it is shown that this canbe mitigated by including drag in UVLM with a BEM-like-approach through look-up tables. Third,the free wake model in UVLM, required to accurately capture rotor thrust, significantly increasescomputational cost. For this reason, a new wake discretisation scheme for the wake convectionequation in UVLM is proposed, in which a coarse discretisation is employed far from the solidsurfaces, which significantly reduces the computational time. Finally, these improvements andthe performance of the three fidelities are analysed in a reference 10 MW wind turbine rotordemonstrating, in general, good agreement.

Goizueta N, Wynn A, Palacios R,
et al., 2022, Flutter predictions for very flexible wing wind tunnel test, *Journal of Aircraft: devoted to aeronautical science and technology*, Vol: 59, Pages: 1082-1097, ISSN: 0021-8669

The stability boundaries of a very flexible wing are sought to inform a wind-tunnel flutter test campaign. The objective is twofold: to identify via simulation the relevant physical processes to be explored while ensuring safe and non-destructive experiments, and to provide a benchmark case for which computational models and test data are freely available. Analyses have been independently carried out using two geometrically nonlinear structural models coupled with potential flow aerodynamics. The models are based on a prototype of the wing for which static load and aeroelastic tests are available, and the experimental results have been successfully reproduced numerically. The wing displays strong geometrically nonlinear effects with static deformations as high as 50% of its span. This results in substantial changes to its structural dynamics, which display several mode crossings that cause the flutter mechanisms to change as a function of deformation. Stability characteristics depend on both the free-stream velocity and the angle of attack. A fast drop of the flutter speed is observed as the wing deforms as the angle of attack is increased, while a large stable region is observed for wing displacements over 25%. The corresponding wind tunnel dynamic tests have validated these predictions.

Goizueta N, Wynn A, Palacios R, 2021, Fast flutter evaluation of very flexible wing using interpolation on an optimal training dataset, AIAA SCITECH 2022 Forum, Publisher: American Institute of Aeronautics and Astronautics, Pages: 1-21

Machine learning strategies can be efficiently used to accelerate the exploration of the design space or flight envelope of highly flexible aeroelastic systems. In this paper, we explore the use of interpolation between parametric state-space realizations to, with few true systems sampled in the parameter space, produce with adequate accuracy a state-space model anywhere in the parameter space. The location of the sampling points is shown to be decisive thus the selection of these points takes the focus in this work. Several approaches are explored, putting emphasis on adaptive schemes that locate the optimal points in the parameter space that are needed to capture the changing system dynamics. Since the evaluation of the true system is costly, optimization techniques based on statistical surrogate models are sought, which need to be trained but are effective in locating the best locations to use as sampling data. A novel method inspired by Bayesian optimization is used to make the most out of a limited number of known state-spaces by taking different combinations as training and testing data of the statistical surrogate, leading to not only an accurate interpolation framework but also to a reduction of 50% of the number of costly full system evaluations compared to a standard Bayesian optimization set-up. These methods are demonstrated on the Pazy wing, a very flexible wing with a complex stability envelope, whereby we produce a very accurate representation of the flutter envelope at a reduced computational cost.

Wynn A, Artola M, Palacios R, 2021, Nonlinear optimal control for gust load alleviation with a physics-constrained data-driven internal model, AIAA SCITECH 2022 Forum, Publisher: American Institute of Aeronautics and Astronautics, Pages: 1-22

A data-driven strategy is developed to improve the internal models used for predictive control in nonlinear aeroelastic applications. A nonlinear modal formulation of the structure is retained, while an identified quadratic model for both the gravitational forces and the aerodynamics are obtained from a least-squares fit with LASSO regularisation from simulated flights. This is first seen to improve the accuracy of the resulting reduced-order model for open-loop predictions on both gust response and a payload drop problem. Its superior performance as internal model for nonlinear control and estimation is finally demonstrated numerically.

Artola M, Rodriguez C, Wynn A, et al., 2021, Optimisation of Region of Attraction Estimates for the Exponential Stabilisation of the Intrinsic Geometrically Exact Beam Model, 2021 60th IEEE Conference on Decision and Control (CDC), Publisher: IEEE

Fantuzzi G, Arslan A, Wynn A, 2021, The background method: Theory and computations, *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 380, ISSN: 1364-503X

The background method is a widely used technique to bound mean properties of turbulent flows rigorously. This work reviews recent advances in the theoretical formulation and numerical implementation of the method. First, we describe how the background method can be formulated systematically within a broader "auxiliary function" framework for bounding mean quantities, and explain how symmetries of the flow and constraints such as maximum principles can be exploited. All ideas are presented in a general setting and are illustrated on Rayleigh-Bénard convection between stress-free isothermal plates. Second, we review a semidefinite programming approach and a timestepping approach to optimizing bounds computationally, revealing that they are related to each other through convex duality and low-rank matrix factorization. Open questions and promising directions for further numerical analysis of the background method are also outlined.

Arslan A, Fantuzzi G, Craske J,
et al., 2021, Bounds on internally heated convection with fixed boundary heat flux, *Journal of Fluid Mechanics*, Vol: 992, Pages: R1-R1, ISSN: 0022-1120

We prove a new rigorous bound for the mean convective heat transport ⟨wT⟩, where w and T are the non-dimensional vertical velocity and temperature, in internally heated convection between an insulating lower boundary and an upper boundary with a fixed heat flux. The quantity ⟨wT⟩ is equal to half the ratio of convective to conductive vertical heat transport, and also to 12 plus the mean temperature difference between the top and bottom boundaries. An analytical application of the background method based on the construction of a quadratic auxiliary function yields ⟨wT⟩≤12(12+13√)−1.6552R−(1/3) uniformly in the Prandtl number, where R is the non-dimensional control parameter measuring the strength of the internal heating. Numerical optimisation of the auxiliary function suggests that the asymptotic value of this bound and the −1/3 exponent are optimal within our bounding framework. This new result halves the best existing (uniform in R) bound (Goluskin, Internally Heated Convection and Rayleigh–Bénard Convection, Springer, 2016, table 1.2), and its dependence on R is consistent with previous conjectures and heuristic scaling arguments. Contrary to physical intuition, however, it does not rule out a mean heat transport larger than 12 at high R, which corresponds to the top boundary being hotter than the bottom one on average.

Arslan A, Fantuzzi G, Craske J,
et al., 2021, Bounds on heat transport for convection driven by internal heating, *Journal of Fluid Mechanics*, Vol: 919, Pages: 1-34, ISSN: 0022-1120

The mean vertical heat transport ⟨wT⟩ in convection between isothermal plates driven by uniform internal heating is investigated by means of rigorous bounds. These are obtained as a function of the Rayleigh number R by constructing feasible solutions to a convex variational problem, derived using a formulation of the classical background method in terms of quadratic auxiliary functions. When the fluid's temperature relative to the boundaries is allowed to be positive or negative, numerical solution of the variational problem shows that best previous bound ⟨wT⟩≤1/2 can only be improved up to finite R. Indeed, we demonstrate analytically that ⟨wT⟩≤2−21/5R1/5 and therefore prove that ⟨wT⟩<1/2 for R<65536. However, if the minimum principle for temperature is invoked, which asserts that internal temperature is at least as large as the temperature of the isothermal boundaries, then numerically optimised bounds are strictly smaller than 1/2 until at least R=3.4×105. While the computational results suggest that the best bound on ⟨wT⟩ approaches 1/2 asymptotically from below as R→∞, we prove that typical analytical constructions cannot be used to prove this conjecture.

Artola M, Goizueta N, Wynn A,
et al., 2021, Aeroelastic control and estimation with a minimal nonlinear modal description, *AIAA Journal: devoted to aerospace research and development*, Vol: 59, Pages: 2697-2713, ISSN: 0001-1452

Modal-based, nonlinear Moving Horizon Estimation (MHE) and Model Predictive Control(MPC) strategies for very flexible aeroelastic systems are presented. They are underpinned by an aeroelastic model built from a 1D intrinsic (based on strains and velocities) description of geometrically-nonlinear beams and an unsteady Vortex Lattice aerodynamic model. Construction of a nonlinear, modal-based, reduced order model of the aeroelastic system, employing a state-space realisation of the linearised aerodynamics around an arbitrary reference point, allows us to capture the main nonlinear geometrical couplings at a very low computational cost. Embedding this model in both MHE and MPC strategies, which solve the system continuous-time adjoints efficiently to compute sensitivities, lays the foundations for real-time estimation and control of highly flexible aeroelastic systems. Finally, the performance and versatility of the framework operating in the nonlinear regime is demonstrated on two very flexible wing models, with notably different dynamics, and on two different control setups: a gust-load alleviation problem on a very high aspect ratio wing with slower dynamics, which involves substantial deflections; and flutter suppression on a flexible wing with significantly faster dynamics, where an unconventional nonlinear stabilisation mechanism is unveiled.

Artola M, Wynn A, Palacios R, 2021, Modal-based model predictive control of multibody very flexible structures, 21st IFAC World Congress on Automatic Control - Meeting Societal Challenges, Publisher: Elsevier, Pages: 7472-7478, ISSN: 2405-8963

A model predictive control strategy for flexible multibody structures undergoing large deformations is presented. The dynamics of such structures are highly nonlinear, with local effects introduced by the joint constraints and distributed effects arising from the structure’s increased flexibility, from which arbitrary large deflections and rotations can be expected. A modal-based nonlinear reduced order model of an intrinsic description (based on velocities and strains) of geometrically-exact beams is used to underpin the internal model. This low-order model, constructed using the linearised eigenfunctions of the constrained structures, is a set of nonlinear ordinary differential equations in time (i.e. no algebraic equations are present) thus facilitating analysis and demonstrating successful control. Numerical examples are presented based on a very flexible hinged two-link manipulator.

Artola M, Wynn A, Palacios R, 2021, Modal-based nonlinear model predictive control for 3D very flexible structures, *IEEE Transactions on Automatic Control*, Vol: 67, ISSN: 0018-9286

In this paper a novel NMPC scheme is derived, which is tailored to the underlying structure of the intrinsic description of geometrically exact nonlinear beams (in which velocities and strains are primary variables). This is an important class of PDE models whose behaviour is fundamental to the performance of flexible structural systems (e.g., wind turbines, High-Altitude Long-Endurance aircraft). Furthermore, this class contains the much-studied Euler-Bernoulli and Timoshenko beam models, but has significant additional complexity (to capture 3D effects and arbitrarily large displacements) and requires explicit computation of rotations in the PDE dynamics to account for orientation-dependent forces such as gravity. A challenge presented by this formulation is that uncontrollable modes necessarily appear in any finite dimensional approximation to the PDE dynamics. We show, however, that an NMPC scheme can be constructed in which the error introduced by the uncontrollable modes can be explicitly controlled. Furthermore, in challenging numerical examples exhibiting considerable deformation and nonlinear effects, it is demonstrated that the asymptotic error can be made insignificant (from a practical perspective) usingour NMPC scheme and excellent performance is obtained evenwhen applied to a highly resolved numerical simulation of thePDEs. We also present a generalisation of Kelvin-Voigt dampingto the intrinsic description of geometrically-exact beams. Finally,special emphasis is placed on constructing a framework suitablefor real-time NMPC control, where the particular structure ofthe underlying PDEs is exploited to obtain both efficient finite-dimensional models and numerical schemes.

Goizueta N, Wynn A, Palacios R, 2021, Parametric Krylov-based order reduction of aircraft aeroelastic models, AIAA Scitech 2021 Forum, Publisher: American Institute of Aeronautics and Astronautics, Pages: 1-25

We present an interpolation-based scheme to obtain in real-time linearised aeroelastic models of very flexible aircraft at any flight condition within the flight envelope. First, in an offline phase, a library of reduced-order linear aeroelastic models is computed by finding the nonlinear equilibrium (trimmed aircraft) using a nonlinear, geometrically-exact beam formulation coupled with an Unsteady Vortex Lattice Method. Subsequent linearisation is achieved analytically and followed by reduction using Krylov subspace methods. The interpolation is performed directly on the reduced-order system matrices, which are projected onto a congruent coordinate system for compatibility. Any interpolation scheme of choice can be used, and the size of the models permits real-time computation.

Artola M, Goizueta N, Wynn A, et al., 2021, Proof of concept for a hardware-in-the-loop nonlinear control framework for very flexible aircraft, AIAA Scitech 2021 Forum, Publisher: https://arc.aiaa.org/doi/abs/10.2514/6.2021-1392, Pages: 1-26

Nonlinear Moving Horizon Estimation (MHE) and Model Predictive Control (MPC) strategies for very flexible aircraft are presented. They are underpinned by a nonlinear reduced-order model built upon the structure’s natural modes of vibration. This internal model aims for a minimal realisation of the aircraft which retains sufficient information to enable efficient real-time estimation and control. It is based on a modal intrinsic description of geometrically-nonlinear beams and a linearised unsteady vortex lattice aerodynamic model. Numerical evidence has shown that models of this form are able to capture the main nonlinear geometrical couplings at a very low computational cost. This opens the door to MHE and MPC strategies, which are naturally more computationally demanding than other linear conventional strategies, but are more versatile and able to provide control in the usually neglected nonlinear regime. The proposed control framework is tested on models built in an in-house open-source nonlinear aeroelasticity simulation and analysis package, to emulate the controller performance on a realistic plant model. Very satisfactory results are obtained in a flutter suppression problem involving a very flexible clamped wing, where the nonlinearity of the problem is leveraged by the internal model to achieve stabilisation, and a payload drop control of a very flexible HALE aircraft.

Goizueta N, Wynn A, Palacios R, et al., 2021, Flutter prediction for a very flexible wing wind tunnel test, AIAA Scitech 2021 Forum, Pages: 1-17

Two different nonlinear aeroelastic tool sets, SHARPy and the Modal Rotation Method (MRM), have been employed to predict and design a wind tunnel flutter test campaign of a very flexible wing, the Pazy Wing, as part of the 3rd Aeroelastic Prediction Workshop. The first method, SHARPy, uses geometrically exact beams coupled with an Unsteady Vortex Lattice, which is linearised about a deformed configuration, reduced by means of Krylov subspaces and analysed to compute the stability boundaries of the wing. The MRM is based on structural modal data, from either beam models or finite element models, coupled with a doublet-lattice aerodynamic model from ZAERO of the straight wing configuration. The excellent agreement between numerical and experimental data for structural-only and static aeroelastic analyses paves the way for predicting the stability boundaries of the Pre-Pazy wing with sufficient confidence for the safe design of a flutter wind tunnel test campaign.

Artola M, Wynn A, Palacios R, 2021, Generalized Kelvin–Voigt damping models for geometrically nonlinear beams, *AIAA Journal: devoted to aerospace research and development*, Vol: 59, Pages: 356-365, ISSN: 0001-1452

Strain-rate-based damping is investigated in the strong form of the intrinsic equations of three-dimensional geometrically exact beams. Kelvin–Voigt damping, often limited in the literature to linear or two-dimensional beam models, is generalized to the three-dimensional case, including rigid-body motions. The result is an elegant infinite-dimensional description of geometrically exact beams that facilitates theoretical analysis and sets the baseline for any chosen numerical implementation. In particular, the dissipation rates and equilibrium points of the system are derived for the most general case and for one in which a first-order approximation of the resulting damping terms is taken. Finally, numerical examples are given that validate the resulting model against a nonlinear damped Euler–Bernoulli beam (where detail is given on how an equivalent description using our intrinsic formulation is obtained) and support the analytical results of energy decay rates and equilibrium solutions caused by damping. Throughout the paper, the relevance of damping higher-order terms, arising from the geometrically exact description, to the accurate prediction of its effect on the dynamics of highly flexible structures is highlighted.

Muñoz-Simón A, Palacios R, Wynn A, 2020, Scripts illustrating some modelling improvements for prediction of wind turbine rotor loads in turbulent wind

This repository contains some files to complement the publication "Some modelling improvements for prediction of wind turbine rotor loads in turbulent wind".

Muñoz-Simón A, Palacios R, Wynn A, 2020, Benchmarking different fidelities in wind turbine aerodynamics under yaw, The Science of Making Torque from Wind (TORQUE 2020), Publisher: IOP Publishing, Pages: 1-11, ISSN: 1742-6588

This paper analyses the aerodynamics of wind turbines under yaw with different modelling fidelities (BEM, BEM with skewed wake model, UVLM and LES-AL). First of all, models are compared in a zero-yaw case to demonstrate their accuracy in prediction of out-of-plane loads and the discrepancy of UVLM in the in-plane loads due to the lack of viscous drag. Secondly, the yaw aerodynamics are described through the advancing/retreating and skewed wake effects, which are appropriately captured by UVLM and LES-AL and lead to an incorrect prediction of the location of maximum and minimum loading along a revolution by BEM. Further, when a skew-wake model is included in BEM, it predicts the correct locations but exhibits overly large loading variations. These predictions are consistent for all yaw angles studied (γ = 10° − 30°). All solvers predict similar decrease of root-bending moments, rotor power and thrust coefficients up to a yaw angle of 10°. However, at larger yaw angles, BEM overpredicts this decrease of coefficients with the yaw angle due to the unsuccessful performance of yaw corrections as opposed to UVLM that inherently accounts for three-dimensional effects. This study demonstrates the need to use computational models that can account for three-dimensional effects in the computation of aerodynamic loads for yaw angles above 10°.

Zheng Y, Fantuzzi G, Papachristodoulou A,
et al., 2020, Chordal decomposition in operator-splitting methods for sparse semidefinite programs, *Mathematical Programming*, Vol: 180, Pages: 489-532, ISSN: 0025-5610

We employ chordal decomposition to reformulate a large and sparse semidefinite program (SDP), either in primal or dual standard form, into an equivalent SDP with smaller positive semidefinite (PSD) constraints. In contrast to previous approaches, the decomposed SDP is suitable for the application of first-order operator-splitting methods, enabling the development of efficient and scalable algorithms. In particular, we apply the alternating direction method of multipliers (ADMM) to solve decomposed primal- and dual-standard-form SDPs. Each iteration of such ADMM algorithms requires a projection onto an affine subspace, and a set of projections onto small PSD cones that can be computed in parallel. We also formulate the homogeneous self-dual embedding (HSDE) of a primal-dual pair of decomposed SDPs, and extend a recent ADMM-based algorithm to exploit the structure of our HSDE. The resulting HSDE algorithm has the same leading-order computational cost as those for the primal or dual problems only, with the advantage of being able to identify infeasible problems and produce an infeasibility certificate. All algorithms are implemented in the open-source MATLAB solver CDCS. Numerical experiments on a range of large-scale SDPs demonstrate the computational advantages of the proposed methods compared to common state-of-the-art solvers.

Fantuzzi G, Nobili C, Wynn A, 2020, New bounds on the vertical heat transport for Bénard-Marangoni convection at infinite Prandtl number, *Journal of Fluid Mechanics*, Vol: 885, Pages: R4-1-R4-12, ISSN: 0022-1120

We prove a new rigorous upper bound on the vertical heat transport for Bénard–Marangoni convection of a two- or three-dimensional fluid layer with infinite Prandtl number. Precisely, for Marangoni number 𝑀𝑎≫1 the Nusselt number 𝑁𝑢 is bounded asymptotically by 𝑁𝑢⩽const.×𝑀𝑎2/7(ln𝑀𝑎)−1/7 . Key to our proof are a background temperature field with a hyperbolic profile near the fluid’s surface and new estimates for the coupling between temperature and vertical velocity.

Artola M, Goizueta N, Wynn A, et al., 2020, Modal-based nonlinear estimation and control for highly flexible aeroelastic systems, AIAA Scitech 2020 Forum, Publisher: American Institute of Aeronautics and Astronautics, Pages: 1-23

Modal-based, nonlinear Moving Horizon Estimation (MHE) and Model Predictive Control (MPC) strategies for highly flexible aeroelastic systems are presented. The aeroelastic model is built from a 1D intrinsic (based on strains and velocities) description of geometrically-nonlinear beams and an unsteady Vortex Lattice aerodynamic model. Construction of a nonlinear modal-based reduced order model of the aeroelastic system, employing a state-space realisation of the linearised aerodynamics around an arbitrary equilibrium point, allows us to capture the main nonlinear geometrical couplings at a very low computational cost. Embedding this model in both MHE and MPC strategies, which solve the system's continuous-time adjoints efficiently to compute sensitivities, lays the foundations for real-time estimation and control of highly flexible aeroelastic systems.

Muñoz-Simón A, Wynn A, Palacios R, 2020, Unsteady and three-dimensional aerodynamic effects on wind turbine rotor loads, AIAA Scitech 2020 Forum, Publisher: American Institute of Aeronautics and Astronautics, Pages: 1-22

The paper investigates the use of vortex methods in the computation of important flows for wind turbine aerodynamics. These flows are characterised by inflow velocity unsteadiness, spatial variations and non-zero spanwise component. First, computational vortex methods are shown to match analytical solutions on simple geometries. Second, these geometries will be studied to reveal the effects of these flows on airfoil aerodynamics: unsteady damping and load reduction when subjected to side slip or spanwise inflow velocity variations. Finally, vortex methods will be compared with traditional BEM methods in full wind turbine configurations under real operating inflows such as shear, yaw and turbulence which are characterised by unsteadiness and three-dimensional effects. Vortex methods allow quantifying the error of BEM methods for these conditions.

Mahfoze O, Moody A, Wynn A,
et al., 2019, Reducing the skin-friction drag of a turbulent boundary-layer flow with low-amplitude wall-normal blowing within a Bayesian optimisation framework, *Physical Review Fluids*, Vol: 4, Pages: 094601-1-094601-23, ISSN: 2469-990X

A Bayesian optimisation framework is developed to optimise low-amplitude wall-normal blowing control of a turbulent boundary-layer flow. The Bayesian optimisation framework determines the optimum blowing amplitude and blowing coverage to achieve up to a 5% net-power saving solution within 20 optimisation iterations, requiring 20 Direct Numerical Simulations (DNS). The power input required to generate the low-amplitude wall-normal blowing is measured experimentally for two different types of blowing device, and is used in the simulations to assess control performance. Wall-normal blowing with amplitudes of less than 1% of the free-stream velocity generate a skin-friction drag reduction of up to 76% over the control region, with a drag reduction which persists for up to 650δ0 downstream of actuation (where δ0 is the boundary-layer thickness at the start of the simulation domain). It is shown that it is the slow spatial recovery of the turbulent boundary-layer flow downstream of control which generates the net-power savings in this study. The downstream recovery of the skin-friction drag force is decomposed using the Fukagata-Iwamoto-Kasagi (FIK) identity, which shows that the generation of the net-power savings is due to changes in contributions to both the convection and streamwise development terms of the turbulent boundary-layer flow.

Beit-Sadi M, Krol J, Wynn A, 2019, Data driven feature identification and sparse representation of turbulent flows, Eleventh International Symposium on Turbulence and Shear Flow Phenomena (TSFP11)

dentifying coherent structures of fluid flows is of greatimportance for reduced order modelling and flow control.Finding such structures in a turbulent flow, however, canbe challenging. A number of modal decomposition algo-rithms have been proposed in recent years which decom-pose snapshots of data into spatial modes, each associatedwith a single frequency and growth-rate, most prominentlydynamic mode decomposition (DMD). However, the num-ber of modes that DMD-like algorithms construct may beunrelated to the number of significant degrees of freedomof the underlying system. This provides a difficulty if onewants to create a low-order model of a flow. In this work,we present a method of post-processing DMD modes forextracting a small number of dynamically relevant modes.This is achieved by first ranking the DMD modes, then us-ing an iterative approach based on the graph-theoretic no-tion of maximal cliques to identify clusters of modes and,finally, by replacing each cluster with a single (pair of)modes.

Artola M, Wynn A, Palacios R, 2019, A nonlinear modal-based framework for low computational cost optimal control of 3D very flexible structures, 2019 18th European Control Conference (ECC), Publisher: Institute of Electrical and Electronics Engineers

A nonlinear modal-based reduced-order model, equipped with an efficient adjoint-sensitivity analysis, is presented as a low computational cost framework for optimal control of very flexible structures, with particular focus on efficiently computing finite rotations. Multiple shooting is shown to improve convergence of a highly nonlinear problem when compared to the single shooting case, with optimisation further accelerated via parallelisation, which suggests the presented approach may be employed for real-time control of very flexible structures.

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