The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

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Numerical Heat Transfer And Fluid Flow

SUMMARY
The open-source CFD library OpenFoam® contains a method for solving free surface Newtonian flows using the Reynolds averaged Navier–Stokes equations coupled with a volume of fluid method. In this paper, it is demonstrated how this has been extended with a generic wave generation and absorption method termed ‘wave relaxation zones’, on which a detailed account is given. The ability to use OpenFoam for the modelling of waves is demonstrated using two benchmark test cases, which show the ability to model wave propagation and wave breaking. Furthermore, the reflection coefficient from outlet relaxation zones is considered for a range of parameters. The toolbox is implemented in C++, and the flexibility in deriving new relaxation methods and implementing new wave theories along with other shapes of the relaxation zone is outlined. Subsequent to the publication of this paper, the toolbox has been made freely available through the OpenFoam-Extend Community. Copyright © 2011 John Wiley & Sons, Ltd.

A wave generation toolbox for the open‐source CFD library: OpenFoam®

SWASH: An operational public domain code for simulating wave fields and rapidly varied flows in coastal waters

http://astarte-project.eu/files/astarte/documents/publications/Fuhrmanetal2014_CE.pdf

Numerical simulation of wave-induced scour and backfilling processes beneath submarine pipelines

Simulation of nonlinear wave run-up with a high-order Boussinesq model

The performance of interFoam (a widely-used solver within the popular open source CFD package OpenFOAM) in simulating the propagation of a nonlinear (stream function solution) regular wave is investigated in this work, with the aim of systematically documenting its accuracy. It is demonstrated that over time there is a tendency for surface elevations to increase, wiggles to appear in the free surface, and crest velocities to become (severely) overestimated. It is shown that increasing the temporal and spatial resolution can mitigate these undesirable effects, but that a relatively small Courant number is required. It is further demonstrated that the choice of discretization schemes and solver settings (often treated as a "black box" by users) can have a major impact on the results. This impact is documented, and it is shown that obtaining a "diffusive balance" is crucial to accurately propagate a surface wave over long distances without requiring exceedingly high temporal and spatial resolutions. Finally, the new code isoAdvector is compared to interFoam, which is demonstrated to produce comparably accurate results, while maintaining a sharper surface. It is hoped that the systematic documentation of the performance of the interFoam solver will enable its more accurate and optimal use, as well as increase awareness of potential shortcomings, by CFD researchers interested in the general CFD simulation of free surface waves.

Performance of interFoam on the simulation of progressive waves

The performance of interFoam (a widely used solver within OpenFOAM®) in simulating the propagation of a nonlinear (stream function solution) regular wave is investigated in this work. It is demonst...

/pdf/performance-of-interfoam-on-the-simulation-of-progressive-djaxzbtmpl.pdf

This paper investigates preconditioned iterative techniques for nite di erence solutions of a high-order Boussinesq method for modelling water waves in two horizontal dimensions. The Boussinesq method solves simultaneously for all three components of velocity at an arbitrary z-level, removing any practical limitations based on the relative water depth. High-order nite di erence approximations are shown to be more e cient than low-order approximations (for a given accuracy), despite the additional overhead. The resultant system of equations requires that a sparse, unsymmetric, and often ill-conditioned matrix be solved at each stage evaluation within a simulation. Various preconditioning strategies are investigated, including full factorizations of the linearized matrix, ILU factorizations, a matrix-free (Fourier space) method, and an approximate Schur complement approach. A detailed comparison of the methods is given for both rotational and irrotational formulations, and the strengths and limitations of each are discussed. Mesh-independent convergence is demonstrated with many of the preconditioners for solutions of the irrotational formulation, and solutions using the Fourier space and approximate Schur complement preconditioners are shown to require an overall computational e ort that scales linearly with problem size (for large problems). Calculations on a variable depth problem are also compared to experimental data, highlighting the accuracy of the model. Through combined physical and mathematical insight e ective preconditioned iterative solutions are achieved for the full physical application range of the model. Copyright ? 2004 John Wiley & Sons, Ltd.

David R. Fuhrman

Papers

Numerical simulation of wave-induced scour and backfilling processes beneath submarine pipelines

Simulation of nonlinear wave run-up with a high-order Boussinesq model

Performance of interFoam on the simulation of progressive waves

Performance of interFoam on the simulation of progressive waves

Numerical solutions of fully non‐linear and highly dispersive Boussinesq equations in two horizontal dimensions