scholarly journals Stokes’s Fundamental Contributions to Fluid Dynamics

2019 ◽  
pp. 115-128
Author(s):  
Peter Lynch
Keyword(s):  
Fluid Dynamics ◽  
Water Waves ◽  
Stokes Equations ◽  
Weather Prediction ◽  
Finite Amplitude ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Basic Equations ◽  
Primary Focus

George Gabriel Stokes made fundamental mathematical contributions to fluid dynamics that had profound practical consequences. The basic equations formulated by him play a central role in numerical weather prediction, in the simulation of blood flow in the body and in countless other important applications. In this chapter the primary focus is on the two most important areas of Stokes’s work on fluid dynamics, the derivation of the Navier–Stokes equations and the theory of finite amplitude oscillatory water waves.

scholarly journals SHOALING OF FINITE-AMPLITUDE WAVES ON PLANE BEACHES

1970 ◽  
Vol 1 (12) ◽  
pp. 21
Author(s):  
Robert K-C Chan ◽  
Robert L. Street
Keyword(s):  
Water Waves ◽  
Stokes Equations ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Finite Difference Equations ◽  
Rectangular Mesh ◽  
Finite Amplitude Waves ◽  
Large Water ◽  
Marker And Cell Method

This work focuses on the shoaling of large water waves with particular application to storm-generated waves and tsunamis The specific objective is the exact simulation on a digital computer of finite-amplitude waves advancing on a beach of constant slope The study is based on the simulation technique called SUMMAC (the Stanford-University-Modified Marker-And-Cell Method) The flow field is represented by a rectangular mesh of cells and by a line of hypothetical particles which defines the free surface Based on the Navier-Stokes equations, finite-difference equations were derived so that the entire flow configuration could be advanced through a finite increment of time The pressure and velocity components are used directly as the dependent variables Through extensive analyses and numerical experiments, this scheme was found to be computationally stable if the cell size and the time increment are properly selected As a specific example, the dynamics of a solitary wave passing from a zone of constant depth onto a sloping beach were simulated Primary attention was focused on the details of the water particle motions and the changes in the amplitude and shape of the wave as it climbed the slope The computed results are compared with the experiments with good agreement.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

Development of SIMPLE-Like Algorithms for Incompressible Navier-Stokes Equations in Liquid Processing of Materials

2012 ◽  
Vol 184-185 ◽  
pp. 944-948 ◽  
Author(s):  
Hai Jun Gong ◽  
Yang Liu ◽  
Xue Yi Fan ◽  
Da Ming Xu
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Stokes Equations ◽  
Simple Algorithm ◽  
Navier Stokes ◽  
Incompressible Fluid Flow ◽  
Numerical Accuracy ◽  
Simple Scheme ◽  
Navier Stokes Equations ◽  
Computational Fluid Dynamics Cfd

For a clear and comprehensive opinion on segregated SIMPLE algorithm in the area of computational fluid dynamics (CFD) during liquid processing of materials, the most significant developments on the SIMPLE algorithm and its variants are briefly reviewed. Subsequently, some important advances during last 30 years serving as increasing numerical accuracy, enhancing robustness and improving efficiency for Navier–Stokes (N-S) equations of incompressible fluid flow are summarized. And then a so-called Direct-SIMPLE scheme proposed by the authors of present paper introduced, which is different from SIMPLE-like schemes, no iterative computations are needed to achieve the final pressure and velocity corrections. Based on the facts cited in present paper, it conclude that the SIMPLE algorithm and its variants will continue to evolve aimed at convergence and accuracy of solution by improving and combining various methods with different grid techniques, and all the algorithms mentioned above will enjoy widespread use in the future.

Studies on Natural Convection Induced Flow and Thermal Behavior Inside Electronic Equipment Cabinet Model

2001 ◽  
Author(s):  
Masaru Ishizuka ◽  
Guoyi Peng ◽  
Shinji Hayama
Keyword(s):  
Natural Convection ◽  
Thermal Behavior ◽  
Body Surface ◽  
Stokes Equations ◽  
The Body ◽  
Electronic Equipment ◽  
Navier Stokes ◽  
Surface Boundary ◽  
Navier Stokes Equations ◽  
Induced Flow

Abstract In the present work, an important basic flow phenomena, the natural convection induced flow, is studied numerically. Three-dimensional Navier-Stokes equations along with the temperature equation are solved on the basis of finite difference method. Generalized coordinate system is used so that sufficient grid resolution could be achieved in the body surface boundary layer region. Differential terms with respect to time are approximated by forward differences, diffusions terms are approximated by the implicit Euler form, convection terms in the Navier-Stokes equations are approximated by the third order upwind difference scheme. The heat flux at the body surface of heater is specified. The results of calculation showed a satisfactory agreement with the measured data and led to a good understanding of the overall flow and thermal behavior inside electronic equipment cabinet model which is very difficult, if not impossible, to gather by experiment.

Modeling and simulation of membrane process

2017 ◽  
Vol 2 (6) ◽  
Author(s):  
Maciej Staszak
Keyword(s):  
Fluid Dynamics ◽  
Stokes Equations ◽  
Polymer Membranes ◽  
Lattice Models ◽  
Network Models ◽  
Navier Stokes ◽  
Neural Network Models ◽  
Navier Stokes Equations ◽  
Different Types ◽  
Boltzmann Lattice

AbstractThe article presents the different approaches to polymer membrane mathematical modeling. Traditional models based on experimental physicochemical correlations and balance models are presented in the first part. Quantum and molecular mechanics models are presented as they are more popular for polymer membranes in fuel cells. The initial part is enclosed by neural network models which found their use for different types of processes in polymer membranes. The second part is devoted to models of fluid dynamics. The computational fluid dynamics technique can be divided into solving of Navier-Stokes equations and into Boltzmann lattice models. Both approaches are presented focusing on membrane processes.

CFD Analysis of Vortex Shedding in a Circular Cylinder: Flow Induced Vibration Design

2006 ◽  
Author(s):  
Nadeem Ahmed Sheikh ◽  
M. Afzaal Malik ◽  
Arshad Hussain Qureshi ◽  
M. Anwar Khan ◽  
Shahab Khushnood
Keyword(s):  
Circular Cylinder ◽  
Vortex Shedding ◽  
Stokes Equations ◽  
Boundary Layer Separation ◽  
The Body ◽  
Navier Stokes ◽  
Structural Vibrations ◽  
Flow Induced Vibration ◽  
Navier Stokes Equations ◽  
Implicit Approach

Flow past a blunt body, such as a circular cylinder, usually experiences boundary layer separation and very strong flow oscillations in the wake region behind the body at a discrete frequency that is correlated to the Reynolds number of the flow. The periodic nature of the vortex shedding phenomenon can sometimes lead to unwanted structural vibrations. The effect of vibrating instability of a single cylinder is investigated in a uniform flow using the power of computational methods. Fluid structure coupling procedure predicts the fluid forces responsible for structural vibrations. An implicit approach to the solution of the unsteady two-dimensional Navier-Stokes equations is used for computation of flow parameters. Calculations are performed in parallel using a domain re-meshing/deforming technique with efficient communication requirements. Results for the unsteady shedding flow behind a circular cylinder are presented with experimental comparisons, showing the feasibility of accurate, efficient, time-dependent estimation of shedding frequency and resulting vibrations.

scholarly journals Physical and numerical analysis of the front of a gravity current on a horizontal bottom

1998 ◽  
Vol 26 ◽  
pp. 289-295
Author(s):  
Mohamed Naaim ◽  
Thierry Pellarin
Keyword(s):  
Numerical Model ◽  
Stokes Equations ◽  
Gravity Current ◽  
Processing System ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Experimental Approaches ◽  
Two Parameters ◽  
Two Stages

In this paper, numerical and experimental approaches are applied to analyse the dynamics of the front of a gravity current. This study focused on two parameters: internal density and velocity fields. The salt concentration was determined by a potentiometric process. The internal velocities were determined using an optical device and an image-processing system. The structure of the head of the gravity current was analysed. Its density was measured and two stages of evolution were observed. This analysis allows us to coufirm the existence of two important stages. Forxf<xs, where the dynamics depend on the initial condition, the flow consists of a head and body and the front density is constant. Forxf>xs, we show that the density of the front decreases and evolves towards the Hallworth and others (1993) law. From a comparison between the experiments and the numerical model, we show that the numerical model, which is based on Navier–Stokes equations and on thek−Lturbulence model (whereLis the height of the gravity current), can predict well flow in the slump regime and in the inertia–buoyancy regime with smoothed results in the transition from the head to the body of the gravity current.

scholarly journals Dynamic Domain Decomposition Strategy Coupling Lattice Boltzmann Methods With Finite Differences Approximations of the Navier-Stokes Equations to Study Bodies in Current

2015 ◽  
Author(s):  
Giuseppina Colicchio ◽  
Claudio Lugni ◽  
Marilena Greco ◽  
Odd M. Faltinsen
Keyword(s):  
Domain Decomposition ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Dynamic Change ◽  
Accurate Solution ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Coupling Strategy ◽  
Hybrid Solver

A Domain-Decomposition (DD) strategy is proposed for problems involving regions with slow variations of the flow (A) and others where the fluid features undergo rapid changes (B), like in the case of steady current past bodies with pronounced local unsteadiness connected with the vortex shedding from the structures. For an efficient and accurate solution of such problems, the DD couples a Finite Difference solver of the Navier-Stokes equations (FD-NS) with a Multiple Relaxation Time Lattice Boltzmann method (MRT-LBM). Regions A are handled by FD-NS, while zones B are solved by MRT-LBM and the two solvers exchange information within a strong coupling strategy. Present DD strategy is able to deal with a dynamic change of the sub-domains topology. This feature is needed when regions with vorticity shed from the body vary in time for a more flexible and reliable solution strategy. Its performances in terms of accuracy and efficiency have been successfully assessed by comparing the hybrid solver against a full FD-NS solution and experimental data for a 2D circular cylinder in an impulsively started flow.

scholarly journals A Massively Parallel Hybrid Finite Volume/Finite Element Scheme for Computational Fluid Dynamics

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2316
Author(s):  
Laura Río-Martín ◽  
Saray Busto ◽  
Michael Dumbser
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Finite Element ◽  
Mach Number ◽  
Finite Volume ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Massively Parallel ◽  
Navier Stokes ◽  
Navier Stokes Equations

In this paper, we propose a novel family of semi-implicit hybrid finite volume/finite element schemes for computational fluid dynamics (CFD), in particular for the approximate solution of the incompressible and compressible Navier-Stokes equations, as well as for the shallow water equations on staggered unstructured meshes in two and three space dimensions. The key features of the method are the use of an edge-based/face-based staggered dual mesh for the discretization of the nonlinear convective terms at the aid of explicit high resolution Godunov-type finite volume schemes, while pressure terms are discretized implicitly using classical continuous Lagrange finite elements on the primal simplex mesh. The resulting pressure system is symmetric positive definite and can thus be very efficiently solved at the aid of classical Krylov subspace methods, such as a matrix-free conjugate gradient method. For the compressible Navier-Stokes equations, the schemes are by construction asymptotic preserving in the low Mach number limit of the equations, hence a consistent hybrid FV/FE method for the incompressible equations is retrieved. All parts of the algorithm can be efficiently parallelized, i.e., the explicit finite volume step as well as the matrix-vector product in the implicit pressure solver. Concerning parallel implementation, we employ the Message-Passing Interface (MPI) standard in combination with spatial domain decomposition based on the free software package METIS. To show the versatility of the proposed schemes, we present a wide range of applications, starting from environmental and geophysical flows, such as dambreak problems and natural convection, over direct numerical simulations of turbulent incompressible flows to high Mach number compressible flows with shock waves. An excellent agreement with exact analytical, numerical or experimental reference solutions is achieved in all cases. Most of the simulations are run with millions of degrees of freedom on thousands of CPU cores. We show strong scaling results for the hybrid FV/FE scheme applied to the 3D incompressible Navier-Stokes equations, using millions of degrees of freedom and up to 4096 CPU cores. The largest simulation shown in this paper is the well-known 3D Taylor-Green vortex benchmark run on 671 million tetrahedral elements on 32,768 CPU cores, showing clearly the suitability of the presented algorithm for the solution of large CFD problems on modern massively parallel distributed memory supercomputers.

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