3 Introduction to Computational Fluid Dynamics and Ocean Modelling
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3.2 Numerical Models
Since the time of Sir Isaac Newton physicists and mathematicians have expressed
the relations governing mechanical systems, the “Laws of Nature”, in terms of differential and integral equations. In general, the collection of a suitable subset of
these equations can be regarded as a mathematical model when applied to predict
the behaviour of a specific mechanical problem. Continuous model equations can be
applied to model certain aspects of phenomena that appear continuous at a macroscopic level, and the deviation between modelled and measured results are usually
considered to be model errors. Since there are no independent phenomena in nature, and in principle no independent way to measure parameters governing natural
processes, a comparison between models and measurements always includes some
probabilistic element. Hence the model error should not be regarded as an absolute
measure of how well the model describes nature.
The classical mathematical models formulated as differential equations are not
well suited for numerical computations. Although computers are capable of performing a huge number of calculations in a short time, they are limited by the finite
precision in number representation and a finite storage space. While the mathematical equations usually define a function for all real number coordinate values
within the model domain (except at discontinuities) and for continuous time, a computer can only assign values to a finite number of distinct points, surface elements
or volumes at discrete time intervals. The process of transferring continuous model
equations to discrete counterparts is called discretization. The choice of a discretization method is intimately connected with the choice of a numerical method used to
solve the problem. The discretization and computational limitations are the origin
of numerical errors that add to the original model errors.
3.2.1 Methods Used in CFD Modelling
In the classical theory of fluid mechanics there are two main branches providing
different view points on how to specify the flow field. The Lagrangian type of specification tracks the motion of material elements of fluid in time, and the history of
each material element constitutes a single path line. The Eulerian type of specification defines flow quantities in terms of functions for the velocity vector of the
fluid throughout the fluid domain at each time instance, which change according to
flow quantities such as density and pressure. Corresponding to these branches, numerical methods for CFD applications can also be categorized in two main classes;
grid-based methods corresponding to the Eulerian viewpoint, and grid-free methods corresponding to the Lagrangian viewpoint. Grid-free methods, in particular
smoothed-particle hydrodynamics (SPH), have been successfully applied to study
highly dynamic problems such as wave breaking and the Rayleigh-Taylor instability. A large number of fluid elements are usually needed to obtain a satisfactory
resolution, in which case the computational cost is large compared to equivalent
67
3.2 Numerical Models
Since the time of Sir Isaac Newton physicists and mathematicians have expressed
the relations governing mechanical systems, the “Laws of Nature”, in terms of differential and integral equations. In general, the collection of a suitable subset of
these equations can be regarded as a mathematical model when applied to predict
the behaviour of a specific mechanical problem. Continuous model equations can be
applied to model certain aspects of phenomena that appear continuous at a macroscopic level, and the deviation between modelled and measured results are usually
considered to be model errors. Since there are no independent phenomena in nature, and in principle no independent way to measure parameters governing natural
processes, a comparison between models and measurements always includes some
probabilistic element. Hence the model error should not be regarded as an absolute
measure of how well the model describes nature.
The classical mathematical models formulated as differential equations are not
well suited for numerical computations. Although computers are capable of performing a huge number of calculations in a short time, they are limited by the finite
precision in number representation and a finite storage space. While the mathematical equations usually define a function for all real number coordinate values
within the model domain (except at discontinuities) and for continuous time, a computer can only assign values to a finite number of distinct points, surface elements
or volumes at discrete time intervals. The process of transferring continuous model
equations to discrete counterparts is called discretization. The choice of a discretization method is intimately connected with the choice of a numerical method used to
solve the problem. The discretization and computational limitations are the origin
of numerical errors that add to the original model errors.
3.2.1 Methods Used in CFD Modelling
In the classical theory of fluid mechanics there are two main branches providing
different view points on how to specify the flow field. The Lagrangian type of specification tracks the motion of material elements of fluid in time, and the history of
each material element constitutes a single path line. The Eulerian type of specification defines flow quantities in terms of functions for the velocity vector of the
fluid throughout the fluid domain at each time instance, which change according to
flow quantities such as density and pressure. Corresponding to these branches, numerical methods for CFD applications can also be categorized in two main classes;
grid-based methods corresponding to the Eulerian viewpoint, and grid-free methods corresponding to the Lagrangian viewpoint. Grid-free methods, in particular
smoothed-particle hydrodynamics (SPH), have been successfully applied to study
highly dynamic problems such as wave breaking and the Rayleigh-Taylor instability. A large number of fluid elements are usually needed to obtain a satisfactory
resolution, in which case the computational cost is large compared to equivalent