Book contents
- Frontmatter
- Contents
- Preface
- Contributing Authors
- 1 A Few Tools For Turbulence Models In Navier-Stokes Equations
- 2 On Some Finite Element Methods for the Numerical Simulation of Incompressible Viscous Flow
- 3 CFD - An Industrial Perspective
- 4 Stabilized Finite Element Methods
- 5 Optimal Control and Optimization of Viscous, Incompressible Flows
- 6 A Fully-Coupled Finite Element Algorithm, Using Direct and Iterative Solvers, for the Incompressible Navier-Stokes Equations
- 7 Numerical Solution of the Incompressible Navier-Stokes Equations in Primitive Variables on Unstaggered Grids
- 8 Spectral Element and Lattice Gas Methods for Incompressible Fluid Dynamics
- 9 Design of Incompressible Flow Solvers: Practical Aspects
- 10 The Covolume Approach to Computing Incompressible Flows
- 11 Vortex Methods: An Introduction and Survey of Selected Research Topics
- 12 New Emerging Methods in Numerical Analysis: Applications to Fluid Mechanics
- 13 The Finite Element Method for Three Dimensional Incompressible Flow
- 14 A Posteriori Error Estimators and Adaptive Mesh-Refinement Techniques for the Navier-Stokes Equations
- Index
7 - Numerical Solution of the Incompressible Navier-Stokes Equations in Primitive Variables on Unstaggered Grids
Published online by Cambridge University Press: 12 January 2010
- Frontmatter
- Contents
- Preface
- Contributing Authors
- 1 A Few Tools For Turbulence Models In Navier-Stokes Equations
- 2 On Some Finite Element Methods for the Numerical Simulation of Incompressible Viscous Flow
- 3 CFD - An Industrial Perspective
- 4 Stabilized Finite Element Methods
- 5 Optimal Control and Optimization of Viscous, Incompressible Flows
- 6 A Fully-Coupled Finite Element Algorithm, Using Direct and Iterative Solvers, for the Incompressible Navier-Stokes Equations
- 7 Numerical Solution of the Incompressible Navier-Stokes Equations in Primitive Variables on Unstaggered Grids
- 8 Spectral Element and Lattice Gas Methods for Incompressible Fluid Dynamics
- 9 Design of Incompressible Flow Solvers: Practical Aspects
- 10 The Covolume Approach to Computing Incompressible Flows
- 11 Vortex Methods: An Introduction and Survey of Selected Research Topics
- 12 New Emerging Methods in Numerical Analysis: Applications to Fluid Mechanics
- 13 The Finite Element Method for Three Dimensional Incompressible Flow
- 14 A Posteriori Error Estimators and Adaptive Mesh-Refinement Techniques for the Navier-Stokes Equations
- Index
Summary
Abstract
A numerical algorithm for enforcing the conservation of mass in incompressible flow simulation is discussed and the details of the implementations in terms of standard finite volumes and finite elements are given. It is also demonstrated that both segregated and coupled iterative techniques are applicable and numerical results of test cases for two and three dimensional cavity flows are presented.
Introduction
In many applications, numerical simulations of three dimensional incompressible flows are needed. The problem of satisfying exactly the continuity equation, for these flows, is well known [1] [2].
It is conceivable to update the velocity field using the momentum equations but it is not clear how to update the pressure to conserve mass, since no pressure term appears in the continuity equation. Various methods have been introduced to tackle this problem including the penalty method, the artificial compressibility method, the artificial viscosity methods, the projection and the pressure correction methods. (The discussion, here, is limited to methods based on primitive variables. The vector potential and/or velocity vorticity formulations are not covered, see for example the work of Osswald, Ghia & Ghia [19] which still requires staggered orthogonal grids to enforce mass conservation.)
In the primitive variable methods, either staggered grids are used or the continuity equation is modified. For example in the penalty method of Temam [3], a small term proportional to the pressure is added to the continuity equation, while in Chorin's artificial compressibility method [4], the continuity equation is modified by an artificial time dependent term proportional to the time derivative of the pressure.
- Type
- Chapter
- Information
- Incompressible Computational Fluid DynamicsTrends and Advances, pp. 183 - 202Publisher: Cambridge University PressPrint publication year: 1993
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