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

Poisson equation lies in the heart of the numerical solutions of incompressible Navier–Stokes equations (NSEs) based on the popular projection methods, which decouple the pressure and velocities fields. This paper presents enhanced numerical solutions of the 2D incompressible NSEs inspired by a newly-developed Generalized Harmonic Polynomial Cell (GHPC) method for the Poisson equation, which has a fourth-order spatial accuracy. To achieve that, the original GHPC method is adapted to accommodate immersed boundaries, necessary when a staggered grid for velocities and pressure is applied. The finite difference method (FDM) is used in the spatial discretization of the diffusion and advection terms. Numerical analyses of lid-driven cavity flow, a Taylor–Green vortex, and flow around a smooth circular cylinder all show encouraging results, confirming the accuracy and efficiency of the new solver. Through comparison with an NSEs solver which applies second-order FDM for the Poisson equation, we demonstrate that the accuracy of the pressure solution can be greatly improved by applying the more accurate GHPC method for the Poisson equation, even though the accuracy of the velocities solutions are limited by the numerical schemes for advection–diffusion equations. The accuracy in the pressure solution can also be translated into CPU time saving to achieve a predefined accuracy. The present immersed-boundary GHPC Poisson equation solver can easily be ‘plugged’ into other existing NSEs solvers utilizing staggered grids.

Enhanced solution of 2D incompressible Navier–Stokes equations based on an immersed-boundary generalized harmonic polynomial cell method

Generalized time scale for wave-induced backfilling beneath submarine pipelines

/pdf/a-new-s-transform-based-fourier-legendre-galerkin-model-for-5fssziqr02.pdf

A new σ-transform based Fourier-Legendre-Galerkin model for nonlinear water waves

The settling velocities of 66 microplastic particle groups, having both regular (58) and irregular (eight) shapes, are measured experimentally. Regular shapes considered include: spheres, cylinders, disks, square plates, cubes, other cuboids (square and rectangular prisms), tetrahedrons, and fibers. The experiments generally consider Reynolds numbers greater than 102, extending the predominant range covered by previous studies. The present data is combined with an extensive data set from the literature, and the settling velocities are systematically analyzed on a shape-by-shape basis. Novel parameterizations and predictive drag coefficient formulations are developed for both regular and irregular particle shapes, properly accounting for preferential settling orientation. These are shown to be more accurate than the best existing predictive formulation from the literature. The developed method for predicting the settling velocity of irregularly-shaped microplastic particles is demonstrated to be equally well suited for natural sediments in the Appendix. 

Settling velocity of microplastic particles having regular and irregular shapes.

The present study formally proved that Reynolds stress models (RSMs) are unconditionally stable in the potential flow regions. RSMs resolve all components of the Reynolds stress, eliminating e.g. the assumed isotropy of turbulence inherent within two-equation models. The present study implemented and applied Wilcox stress-omega turbulence model for simulating breaking waves. It shows that the Wilcox stress-omega turbulence model can predict accurate results from pre-breaking all the way into the inner surf zone, especially for the undertow velocity profiles in the inner surf zone, which even stabilized two-equation models fails to accurately predict.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/hZqqlGbHpkA

/pdf/simulating-breaking-waves-with-the-reynolds-stress-1wdrbnhf6d.pdf

David R. Fuhrman

Papers

Enhanced solution of 2D incompressible Navier–Stokes equations based on an immersed-boundary generalized harmonic polynomial cell method

Generalized time scale for wave-induced backfilling beneath submarine pipelines

A new σ-transform based Fourier-Legendre-Galerkin model for nonlinear water waves

Settling velocity of microplastic particles having regular and irregular shapes.

Simulating breaking waves with the reynolds stress turbulence model