The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.

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Inverse problems: A Bayesian perspective

Optimal design methods involving the solution of an adjoint system of equations are an active area of research in computational fluid dynamics, particularly for aeronautical applications. This paper presents an introduction to the subject, emphasising the simplicity of the ideas when viewed in the context of linear algebra. Detailed discussions also include the extension to p.d.e.'s, the construction of the adjoint p.d.e. and its boundary conditions, and the physical significance of the adjoint solution. The paper concludes with examples of the use of adjoint methods for optimising the design of business jets.

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An Introduction to the Adjoint Approach to Design

https://www.southampton.ac.uk/~nwb/lectures/GoodPracticeCFD/Articles/Validation_SAND2002-0529.pdf

Verification and Validation in Computational Fluid Dynamics

Preface Introduction 1. Fluid field equations and modal analysis in rigid containers 2. Linear forced sloshing 3. Viscous damping and sloshing suppression devices 4. Weakly nonlinear lateral sloshing 5. Equivalent mechanical models 6. Parametric sloshing (Faraday's waves) 7. Dynamics of liquid sloshing impact 8. Linear interaction of liquid sloshing with elastic containers 9. Nonlinear interaction under external and parametric excitations 10. Interactions with support structures and tuned sloshing absorbers 11. Dynamics of rotating fluids 12. Microgravity sloshing dynamics Bibliography Index.

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Liquid Sloshing Dynamics

It is well known that when wind turbines are deployed in large arrays, their efficiency decreases due to complex interactions among themselves and with the atmospheric boundary layer (ABL). For wind farms whose length exceeds the height of the ABL by over an order of magnitude, a “fully developed” flow regime can be established. In this asymptotic regime, changes in the streamwise direction can be neglected and the relevant exchanges occur in the vertical direction. Such a fully developed wind-turbine array boundary layer (WTABL) has not been studied systematically before. A suite of large eddy simulations (LES), in which wind turbines are modeled using the classical “drag disk” concept, is performed for various wind-turbine arrangements, turbine loading factors, and surface roughness values. The results are used to quantify the vertical transport of momentum and kinetic energy across the boundary layer. It is shown that the vertical fluxes of kinetic energy are of the same order of magnitude as the power...

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Large eddy simulation study of fully developed wind-turbine array boundary layers

https://pure.rug.nl/ws/files/2941055/2005JCompPhysKleefsman.pdf

A Volume-of-Fluid based simulation method for wave impact problems

https://pure.rug.nl/ws/files/3005906/2003JCompPhysVerstappen.pdf

Symmetry-preserving discretization of turbulent flow

A proper orthogonal decomposition (POD) of the flow in a square lid-driven cavity at Re=22,000 is computed to educe the coherent structures in this flow and to construct a low-dimensional model for driven cavity flows. Among all linear decompositions, the POD is the most efficient in the sense that it captures the largest possible amount of kinetic energy (for any given number of modes). The first 80 POD modes of the driven cavity flow are computed from 700 snapshots that are taken from a direct numerical simulation (DNS). The first 80 spatial POD modes capture (on average) 95% of the fluctuating kinetic energy. From the snapshots a motion picture of the coherent structures is made by projecting the Navier–Stokes equation on a space spanned by the first 80 spatial POD modes. We have evaluated how well the dynamics of this 80-dimensional model mimics the dynamics given by the Navier–Stokes equations. The results can be summarized as follows. A closure model is needed to integrate the 80-dimensional system ...

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Proper orthogonal decomposition and low-dimensional models for driven cavity flows

A quasi-simultan eous method is described to calculate laminar, incompressible boundary layers interacting with an inviscid outer flow. The essential feature of this method is an interactive boundary condition prescribing a linear combination of pressure and displacement thickness which models the behavior of the outer flow. This way the quasi-simultaneous method avoids difficulties incurred when either direct or inverse methods are used, resulting in fast convergence of the iterative procedure involved. The method is consistent with the asymptotic triple-deck theory. Results will be presented for two problems which exhibit strong interaction between the viscous and inviscid regions: a boundary layer with a separation bubble, and the flow near the trailing edge of a flat plate. fj Nomenclature cf = skin friction coefficient, cf-Re h = mesh size in x direction H = thickness of computational boundary-layer domain L = characteristic length M,N = number of grid points in y and x direction, respectively p = pressure RJ = remainder term in discrete interaction law Re = Reynolds number , Re = U0L I v u = component of flow velocity parallel to surface Ue>ue0 = value of u at edge of boundary layer with and

Arthur Veldman

Papers

A Volume-of-Fluid based simulation method for wave impact problems

Symmetry-preserving discretization of turbulent flow

Proper orthogonal decomposition and low-dimensional models for driven cavity flows

New, quasi-simultaneous method to calculate interacting boundary layers

New, quasi-simultaneous method to calculate interacting boundary layers